MAGNETISM AND THE KONDO EFFECT IN CERIUM HEAVYFERMION
COMPOUNDS CERIUMALUMINUM3 AND CERIUMLEAD3
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
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 allpowerful, 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 heavyfermions 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 DT^X 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
gradschool 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, Ivn Guzmn, Clinton Kaiser, Dr. Fernando Gmez, 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.
m
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ii
ABSTRACT vii
CHAPTERS
1 INTRODUCTION 1
2 THEORETICAL BACKGROUND 6
2.1 Landau FermiLiquid Theory 6
2.1.1 Theoretical Basis for a Fermiliquid 7
2.1.2 Thermodynamic and Transport Properties 10
2.2 Localized Magnetic Moments in Metals 11
2.2.1 Electronic States of Magnetic Ions 12
2.2.2 Anderson Model 14
2.3 Singleion Kondo Model 18
2.4 Anisotropic Kondo Model 20
2.5 Kondo Lattice 27
2.6 NonFermiLiquid Effects 30
3 PROPERTIES OF CeAl3 AND CePb3 34
3.1 Properties of CeAl3 34
3.1.1 Crystal Structure 34
3.1.2 Specific Heat 36
3.1.3 Magnetic Susceptibility 37
3.1.4 Transport Measurements 40
3.1.5 Nuclear Magnetic Resonance 42
3.1.6 Muon Spin Rotation 45
3.1.7 Neutron Scattering 48
3.1.8 Chemical Substitution Studies 48
3.2 Properties of CePb3 51
3.2.1 Crystal Structure 51
3.2.2 Specific Heat 51
3.2.3 Sound Velocity Measurements 54
3.2.4 Transport Measurements 58
3.2.5 Magnetic Susceptibility 60
3.2.6 Neutron Scattering 63
3.2.7 Chemical Substitution Studies 65
IV
4 MOTIVATION 68
4.1 Importance of CeAl3 and CePb3 68
4.2 Objectives 70
4.2.1 Magnetism and HeavyFermion Behavior in Ce Kondo Lattices 70
4.2.2 Ground State of CeAl3 71
5 EXPERIMENTAL METHODS 73
5.1 Sample Preparation 73
5.1.1 Synthesis 73
5.1.2 Annealing 77
5.2 Diffraction of XRays 78
5.3 Magnetic Measurements 79
5.4 Specific Heat Measurements 80
5.4.1 Equipment 82
5.4.2 Thermal Relaxation Method 86
5.5 Experimental Probes 89
5.5.1 Experiments on CeAl3 90
5.5.2 Experiments on CePb3 91
6 STRUCTURAL AND THERMODYNAMIC PROPERTIES OF CeAl3 AL
LOYS 93
6.1 Lattice Parameter Study of CeAl3 Alloys 93
6.1.1 Lanthanum Doping: Cei_xLaxAl3 96
6.1.2 Yttrium Doping: Cei_xYxAl3 100
6.1.3 Mixed Doping: Ceo.8(Lai_xYx)0.2Al3 108
6.1.4 Summary 113
6.2 Thermodynamic Measurements of Cei_xLaxAl3 Alloys 116
6.2.1 Magnetic Susceptibility 116
6.2.2 Specific Heat 118
6.2.3 Discussion 127
6.3 Thermodynamic Measurements on Cei_xYxAl3 Alloys 134
6.3.1 Magnetic Susceptibility 134
6.3.2 Specific Heat 137
6.3.3 Discussion 139
6.4 Thermodynamic Measurements on Ce0.8(Lai_xYx)0.2Al3 Alloys . 141
6.4.1 Magnetic Susceptibility 141
6.4.2 Specific Heat 144
6.4.3 Discussion 147
6.5 Heat Capacity of Ce0.8La0.2Al3 and Ce0.3La0.7Al3 in Magnetic Fields 153
6.5.1 Results 154
6.5.2 Discussion 157
v
7 MAGNETIC FIELD STUDY OF CePb3 ALLOYS 165
7.1 Specific Heat of CePb3 in Magnetic Fields 165
7.2 SingleIon Kondo Behavior of Ceo.6Lao.4Pb3 in Magnetic Fields . 175
7.2.1 Results 175
7.2.2 Discussion 179
8 CONCLUSION 184
8.1Summary 184
8.1.1 Ideas for Future Work 187
REFERENCES 189
BIOGRAPHICAL SKETCH 198
vi
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 HEAVYFERMION
COMPOUNDS CERIUMALUMINUM3 AND CERIUMLEAD3
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_xMxAl3 alloys, with M = La (0
of CePb3 and Ce0.6Lao.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 heavyfermion systems.
Data for xray diffraction of Cei_xMxAl3 confirmed an anisotropic lattice
volume expansion for M = La (decreasing c/a ratio) and a contraction for M =
Y. The lowtemperature magnetic susceptibility and specific heat of Cei_xLaxAl3
are consistent with Doniachs 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
vn
of Tm for Y x < 0.2 suggests a dependence of this maximum on the absolute
value change in c/a. Magnetic field measurements on Ladoped CeAl3 alloys
revealed that the field dependence of Tm is inconsistent with the anisotropic Kondo
model, with Tm for Ceo.sLao.2Al3 decreasing only by 0.4 K at 14 T. Experiments
on Ceo.8(Lai_xYa;)o.2Al3 revealed that C/T oc \ oc T~1+x for x = 0.4, with A
comparable to that of heavyfermion 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 X^ to lower values, disappearing for H > 6T. The ratio A/72 is field
dependent below 6T. Studies on Ceo.6Lao.4Pb3 revealed that the electronic specific
heat AC of this alloy can be described by the singleion Kondo model in magnetic
fields, with T/c 2.3K. A previously undetected anomaly in C/T was found below
2K, shifting toward higher temperatures with increasing field. This maximum
appears to be a feature of the Kondo model in magnetic fields.
vm
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 singleelectron 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 FermiDirac statistics is justified
by the theoretical framework set by Landau on his Fermiliquid 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
quantummechanically in terms of a fluid of weaklyinteracting particles (Fermi
liquid, see Chapter 2). The properties of this quantum fluid are similar in form
to those of a gas of noninteracting electrons. Landaus Fermiliquid 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 Fermiliquid theory, the synthesis of new materials
displaying unusual properties presented challenges to this wellestablished descrip
tion of condensed matter systems. A large number of these materials exhibit strong
1
2
electron correlations in their normal (paramagnetic) state, stretching the limits of
applicability of Fermiliquid 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 nonFermiliquid systems [1, 2, 3]. In others,
their normalstate properties remarkably agree with Fermiliquid 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 heavyfermion compounds.
Heavyfermion sytems 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
halfinteger spin (fermions), 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 / electrons and the
conduction band reduce the / magnetic moment and give rise to a Fermiliquid
like state at low temperatures. The large effective mass m* is a consequence of the
large density of states at the Fermi energy N(0).
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 Fermiliquid theory, 7 is proportional to both m* and
iV(0). The specific heat of metals in their normal state at low temperatures is
approximated by the following formula [9, 10]:
C = yT + 0T3, (1.1)
where 7 is the electronic contribution and (3 is the Debye contribution from lattice
vibrations. Values of 7 for heavyfermion compounds typically range from several
hundred to several thousand mJ/K2mol, compared to less than one for normal
metals like Cu and Au. The presence of additional contributions to the specific
3
heat makes the determination of 7 more difficult, and 7 is usually represented as
the extrapolated value of C/T at zero temperature.
The heavyfermion character is also reflected in other properties, like mag
netic susceptibility and electrical resistivity. The magnetic susceptibility at high
temperatures follows the CurieWeiss form [9, 10],
C
x~ r + eow
where C is the Curie constant and cw is the CurieWeiss 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 Fermiliquid theory. The
electrical resistivity of metals at very low temperatures is given by
P Po + AT2. (1.3)
Here, p0 is the temperatureindependent term due to scattering off impurities and
defects, and A is the Fermiliquid term. Values for A in heavy fermions are in
the order of tens of /iQcm/K2, much larger than those corresponding to normal
metals.
An intriguing fact of heavyfermion 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 (UCu5, U2Zni7, CeAl2) to superconduct
ing (UBei3, CeCu2Si2), to both magnetic and superconducting (UPt3, URu2Si2,
UPd2Al3, UNi2Al3). The presence of magnetism and/or superconductivity in these
compounds indicates that the heavy Fermiliquid 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 nonFermiliquid (NFL) alloys have been widely studied
4
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 UCu5_xPdx, CeCu6xAux, Ui_xYxPd3, Ce7Ni3 (pressureinduced NFL),
and CeNi2Ge2 [13], U2Pt2In, and U2Co2Sn [14] (NFL compounds).
Among the many unresolved issues in heavyfermion materials is the coex
istence of magnetic and Fermiliquid 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 / 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 wellstudied canoni
cal heavyfermion compounds: CeAl3 and CePb3. Ceriumbased compounds were
chosen because of their simpler electronic configuration. There is only one 4/ spin
per Ce ionic site, as opposed to two or three 5/ 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 heavyfermion physics is presented in Chapter 2. The chapter
begins with an overview of Landaus Fermiliquid 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
Fermiliquid 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
5
properties of both CeAl3 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 CeAl3 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 heavyfermion systems, such as Fermiliquid theory, ionic
configurations in solids, and the Kondo effect.
2.1 Landau FermiLiquid Theory
Landaus 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 singleparticle description of thermodyamic and transport properties
of Fermi systems like liquid 3 He and normal metals like copper, silver, and gold.
Although Landaus Fermiliquid theory has been successfully applied in a large
number of condensedmatter systems, its validity relies on a series of assumptions
that apply mostly to weak interactions and isotropic scattering between fermions.
Heavyfermion systems, often described as having a Fermiliquid ground state,
exhibit strong manyparticle correlations that lead to magnetic order in many cases.
The relation between magnetism and Fermiliquid 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
6
7
between Fermigas and Fermiliquid models are outlined, followed by a description
of thermodynamic and transport properties of the Fermi liquid.
2.1.1 Theoretical Basis for a Fermiliquid
For a system of noninteracting particles obeying FermiDirac 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],
n(e)
1
X __ e{en)/kBT
(2.1)
where kB is Boltzmanns constant and pi = Â£p, the Fermi energy. The spins are
assumed to be quantized along the 2axis.
In the absence of an external field, the energy of a particle becomes e = ep
p2/2m, and the ground state distribution is given by
1 P< Pf
nPa =
(2.2)
I 0 p > pF
where Pf is the Fermi momentum. The ground state energy of the system E0 is
equal to
Â£o = E"pÂ£p (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
groundstate and excitedstates distribution functions:
8fipa = Tlpcr npai (2*4)
where 8npa > 0 corresponds to a particle excitation and 8npa < 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 Eq + ^ ] Ep 8npa.
pa
(2.5)
8
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 onetoone 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, spin1/2 fermions called quasiparticles, with an effective mass m* different
from that of a free electron. These quasiparticles have a sufficiently long lifetime
t between collisions at low temperatures. The condition for the applicability of
Fermiliquid theory is that the uncertainty in the energy of a particle, of order
Ti/t oc (kBT)2, is much smaller than the width of the excitation spectrum of the
Fermi distribution function, of order kBT [18]:
h/r < kBT. (2.6)
This condition applies to a system with excitation energies much smaller than kBT.
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 selfconsistent field from other quasiparticles. This selfconsistent field affects
9
both potential and kinetic energy terms of each individual quasiparticle. The
energy E then becomes a functional E{npa} of the distribution function. The
excitation (quasiparticle) energy, which itself is a functional of the distribution
function, (e = Â£{np(T}), has an additional term corresponding to the interaction
energy between two quasiparticles /pa.pV', each with momentum and spin p
pV, 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 8npa [19]:
Â£pa Â£pa T fpa,p'a' 8npiai f ..., (27)
p 'a'
where Â£pcr is the groundstate quasiparticle energy. As a result, the total energy of
the system is also an expansion in 8npa:
E Eq + 'y ] Epa Snpcr T y ^ fP(t p'a' 8fipa 5np> a' T ... (28)
P<7 pa p'a'
When considering an ensemble of quasiparticles with spins quantized along
different axes, the distribution function npa 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
higherorder scattering processes, like spinorbit coupling, the interaction energy
can be expressed as the sum of symmetric and antisymmetric (spindependent)
terms
fpp' = /pp' + /pp' tt\ (2.9)
where fpp, and fpp, are the symmetric and antisymmetric terms, respectively,
and r, r' are Pauli matrices. Both /*p, and fpp, are dependent on the angle
between p and p', and can be expressed as an expansion in Legendre polynomials,
with coefficients ff and /, in the case of isotropic scattering (spherical Fermi
surface). In some metals, the presence of crystalfield and spinorbit coupling
effects significantly distorts the Fermi surface, changing the angular dependence of
10
/pp, and /pp/. The Landau parameters Fts and F are defined with respect to the
coefficients ff and / corresponding to isotropic scattering:
Ff = 1V(0) ft,
Ft = N(0) ft,
(2.10)
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 Snp(T, 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 = 'yT + aT3 In T + ... ,
(2.11)
where the Sommerfeld coefficient 7 is
27Fkl
7 =
klm* z
N(0) = TiT
(2.12)
3 ' 3 fit
The first term is linear in temperature, and proportional to the effective quasipar
ticle mass m*. The effective mass is related to the freeelectron mass m by
 =1 + ^
(2.13)
where F* is one of the Fermiliquid 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:
h2 72iV(0)
X = 2/rffiV(0) + ... =
4 1 + Fn
+ ... ,
(2.14)
11
where /eff corresponds to the quasiparticle effective magnetic moment, 7 is the
linear coefficient of the specific heat, and Ffi is a Fermiliquid 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]:
2 = 7r2e2m(76.06) / T_\2
P 16N(0)h3 \T*f )
where e is the electronic charge, m is the mass of a free electron, h is Plancks
constant, and TF 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 HartreeFock form. Orbital states within a single ion
are formed by electrons with energies below Â£p, 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 (Fermiliquid 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
nearlyfree electron models. In order to understand the behavior of 4/ magnetic
12
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 + l)(2/ + l) degenerate states, where
n, /, and s are the principal, orbital, and spin quantum numbers, respectively. The
degeneracy is partially lifted by the electronelectron Coulomb interaction, of order
10 eV. These energy levels, called multiplets, are filled up according to Hunds rules
and the Pauli exclusion principle. Once all 2(2/ + 1) levels are fully occupied, the
sum total of spin and orbital angular momenta 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 spinorbit 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 spinorbit interaction is then given by
nso = A(LS) = gnlz + (LS), (2.16)
where g is the electron pvalue, gB is the Bohr magneton, Zeff is the effective
atomic number, and L and S are the total orbital and spin angular momenta,
respectively. The coefficient A is positive when the shell is less than halffilled, and
negative for more than halffilled. 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 multiplet split into
25+1 levels for L > S or 2L+1 levels for L < 5, each carrying a 2J+1 degeneracy.
13
The second interaction responsible for the splitting of degenerate energy lev
els of a multiplet is due to the surrounding ions. Crystalfield effects represent the
influence of Coulomb interactions from neighboring charges on localized states.
The crystalfield 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,
^cef ~ ^cEF(r) = ~e^2 i i > (217)
i ij lr rV?'l
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
Keep can be expressed in polar coordinates and expanded in terms of the spherical
harmonics Ylm{6, ). The result is an expansion in powers of (r) and of the angular
momentum operators L2 and Lz (or J2, Jz). The crystalfield 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
pointgroup symmetry.
In solids with magnetic ions, the relative strength of spinorbit and crystal
field energies depends on the localized character of the wave function corresponding
to the incompletelyfilled shell. The spinorbit interaction increases as the distance
from the nucleus decreases (Hso oc (1/r3)). The crystalfield contribution HCef, on
the other hand, increases with the radial extent of the wave function. For electrons
in incomplete d orbitals, HcEF > Hso due to their direct interaction with orbitals
from neighboring ions. In contrast, electrons in incomplete / orbitals are very
localized and reside close to the nucleus. Therefore, the spinorbit interaction is
very large (> 0.1 eV), and the crystalfield contribution 7YCef comparatively smaller
(> 0.01 eV). As a consequence, the lowestlying multiplet is first split by the spin
14
orbit interaction, and each of these levels is split further by the crystal field. The
ground state of the system is the crystalfield ground state. For example, in Ce3+,
there is only one 4/ electron (S ), and the lowestlying multiplet corresponds
to L = 3. Hso splits the multiplet into two 6fold degenerate levels: \J = ) and
 J = ). The lowestenergy level (J = ) is then split by 7YCEF into a doublet
and a quartet for cubic crystal symmetry and into three doublets in the case of
hexagonal symmetry. For a crystalfield doublet ground state, the effective total
angular momentum of Ce3+ is J = \.
2.2.2 Anderson Model
The fundamental problem in magnetic alloys (including heavyfermion 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 / orbitals in irongroup and rareearth ions, respectively. The following discus
sion will focus on localized / states. Electrons in a partiallyfilled / 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 / level.
This problem was described by Anderson [22] in the following Hamiltonian for a
single impurity embedded in a freeelectron environment:
n^ = Ho + Hof + Hff + Hcf, (2.18)
The first term is the unperturbed freeelectron Hamiltonian:
Wo = EÂ£k"i (219>
kcr
Here, is the energy of a freeelectron state with wave number k and spin
and rika is the number operator
15
Tlkcr Q'kcr^'kcri (2.20)
with aj^ and aka 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 / level:
tto/ = Â£Â£/n/a, (2.21)
G
where Ef corresponds to the energy of the / level and
nfa = a)aafa (222)
The third term represents the onsite Coulomb repulsion between two / elec
trons of opposite spin:
Tiff = UrifiTifi, (2.23)
with U the Coulomb integral between the two / states, and n/j and n/ the number
operators for / states with up and down spin, respectively. The last term denotes
the mixing between conduction electrons and the / orbital:
Hcf = ^2 Vkf(ak*afv + a/aaka) (2.24)
k G
Here 14/ is the hybridization matrix element between localized and conduction
electronic states.
The effect of the Anderson Hamiltonian on the localized / 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 / levels with respect to each
other. The hybridization term 14/ is responsible for a broadening of the / levels,
which determines the overlap between the lowest / state and the Fermi energy.
These levels are represented by a Lorentzian of width 2T, where
r = 7rvk/2Jv(o),
(2.25)
16
and N(0) is the density of states at the Fermi energy. Figure 2.1 illustrates the
density of states of up and downspin freeelectrons and localized levels for different
relative strenghts of Coulomb repulsion and mixing width. For U \4/, the
localized up and downspin levels (d or /) have a small width 2r and are well
separated by U. The downspin 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 upspin 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
heavyfermion state when the impurity concentration is of the order of Avogadros
number and the magnetic ions achieve the periodicity of the crystal lattice. For
U ~ 114/1, 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 downspin levels, leading to mixed valence and
the formation of a weak local magnetic moment. In the limit I4/ 0, both
levels have the same energy and occupancy and the impurity loses its magnetic
moment.
By studying the limit in which F<^Ef, Schrieffer and Wolff [24] were able to
perform a canonical transformation on the Anderson Hamiltonian that eliminates
the hybridization term 14/ Instead, the transformed Hamiltonian is expressed in
terms of Ho, Hof, H//, and an exchange interaction between /ion and conduction
electron spins
Kx = EJkk'SkS/, (2.26)
kk'
where Sk and S/ 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' ~ k/r, and Jkk> becomes
17
NE) NU nH
NUE)
L2r
ni (a
NU NUE) NU
Figure 2.1: Spinup and spindown electronic density of states distributions for a
localized d orbital embedded in a sea of conduction electrons. Upper left: U =
Vk/ = 0; upper right: U14/; lower left: UVk/; lower right: U
0) (from Mydosh, 1993) [23].
18
JkFkF J 2VkF/ + U) < (2.27)
where J is the Kondo coupling constant. In this manner, the Anderson Hamilto
nian effectively transforms into the Kondo Hamiltonian in the limit T
W(0)J<1).
2.3 Singleion Kondo Model
The Kondo problem is that of a single localized magnetic impurity in a metal
lic host. This scenario corresponds to the abovementioned U oo limit of the
singleimpurity 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 crystalfield ground state of Ce3+. As the temperature decreases, the local
ized / orbital hybridizes with the conduction band, spinflip scattering increases,
and a scattering resonance appears near the Fermi level, known as the Kondo
or AbrikosovSuhl resonance. The Hamiltonian describing these processes is the
Kondo (or sd) Hamiltonian, of the form
^Kondo=jE^)(^S),
(2.28)
where J is the effective coupling constant between / and conduction electrons (as
in Eq. 2.27), S is the localized spin, and s and r represent the zth 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 \ep Ef ),
\VkFf\2
J OC
I Â£f Ef I
(2.29)
where Â£p is the Fermi energy. A perturbation treatment of 7iKondo beyond the
Born approximation leads to an expansion of the thermodynamic and transport
properties in powers of JN(0) In(kBT/D). Here, iV(0) is the density of states at the
19
Fermi energy and D is the bandwidth of scattering states. The electrical resistivity
was calculated by Kondo [25] using thirdorder perturbation theory:
kBT\
P = Pb 1 + 2JN{0) In
D
(2.30)
The constant term
3 77T7T J2 0(0 ^
PB = 2^hte~FS{S + l)'
(2.31)
obtained from the Born approximation, is a residual resistivity term due to the
presence of the magnetic impurity. The thirdorder term diverges at low tempera
tures. The specific heat and magnetic susceptibility due to the impurity are given
by
C = (J7V(0))47t2S(S + l)fcB ( 1 + 4JiV(0) In
kBT
D
+ ...
(2.32)
and
X =
gyBs(g + i)
3/crT
kBr
1 + JN(0) ^1 JN(0) In D
l
(2.33)
respectively. The perturbation treatment for J < 0 breaks down at a temperature
fcBr* = Z)exp(^y (2.34)
The temperature T'k is called the Kondo temperature.
At low temperatures (T TV), the impurity spin strongly couples with
the conduction electron spin polarization, forming a manybody singlet that com
pletely suppresses the localized magnetic moment at T = 0. In this range, the
thermodynamic and transport properties can be described by Fermiliquid theory
due to the absence of an impurity spin. The zerotemperature susceptibility of the
impurity is inversely proportional to the Kondo temperature [26],
(2.35)
1 \2 1.29
Xo = I z9Pb
nkBTK'
and the linear coefficient of the specific heat 7 is given by
20
7=129^. (2.36)
The ratio of the magnetic susceptibility to the electronic specific heat coefficient
7, called the Wilson ratio, is given by
Xo = 3 fgÂ¡iB\2
7 2\TrkB)
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 zerofield 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 at large fields gfiBH kBTx,
where g is the gfactor of the magnetic impurity. The zerofield magnetic suscep
tibility shows a Curielike increase for T > Tk, and then saturates until it reaches
a temperatureindependent value well below Tk. A maximum associated with the
Schottky anomaly of the specific heat appears around Tk for g[iBHjkBTK = 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, Jj., where The Hamiltonian is given by
susceptibility
21
10 10* 10* 10 10 10* 10
Figure 2.2: Specific heat of a S =  Kondo impurity as a function of T/TK for
different magnetic fields (H > gg,BH/kBTK) [30].
Figure 2.3: Magnetic susceptibility of a S =  Kondo impurity as a function of
T/Tk for different magnetic fields (H > gÂ¡iBH/kBTK) [30].
22
Hakm = Y, 4,A* + ^ (4TCfc'i5 + 4Cfc'T5+) +
k,a Z fcfc'
(cfccfc'T cfcicfcU)5z + gfiBhSz, (2.38)
z jfcfc'
where 4a and are the conduction electron creation and anihilation operators,
5+ and S' are the impurity spin raising and lowering operator eigenvalues, and Sz
is the impurity spin value in the z direction. The first term in 7YAKM represents the
conductionelectron energies, the second and third terms represent the inplane
(J) and easyaxis (Jy) 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 (Jy < 0) is given in
terms of Jy and J_ as [21, 33]
kBT^ = Dex p
1
MO) yjf Jl
x tanh 1
(2.39)
where N(0) is the density of states and D is the bandwidth. The exponential
dependence of the Kondo temperature in the parameter J\\ 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 spinboson Hamiltonian [34, 35, 36, 37],
which describes the dissipation in the dynamics of a twolevel system by an Ohmic
bosonic bath. A mapping of the spinboson model [38] onto the AKM has been
exploited to calculate the thermodynamic properties of the former model. Further
more, the parameters of the spinboson model have recently been used to describe
the properties of the AKM applied to the heavyfermion system Cei_a;LaxAl3 [15].
The spinboson Hamiltonian has the form
23
7"SB c\ ^ c\ ^ ^ ^Ol
z z a
\qo ^ E ,?* t ; (a + 4) (240)
2 a yj TTIqUJch
Here ox and az 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
twolevel system to the bath, with coupling constants Ca. In the case of Ohmic
dissipation, the spectral function of the system is J{u) = 2n a uj for u> coc, where
a is a measure of the strength of the dissipation and uc is a cutoff frequency. For
q^O, the tunneling energy A (h = 1) is renormalized into
aaaa + ) +
1
Ar = A
(2.41)
with Ar/kB equivalent to the Kondo temperature Tk in the AKM.
The low temperature behavior of both spinboson and AKM systems is that
of a Fermi liquid. The linear coefficient of the specific heat per total mole is given
by [35, 36]
7x2kl
7
a
3At
Na = a
7r2 R
3TÂ¡
K
(2.42)
where Na is Avogadros number and R = kBNA is the gas constant, and the
magnetic susceptibility of the spinboson model per total mole at T = 0 is
_ g2glNA = g2v2BNA
XsB 2Ar 2kBTK
where g is the ^factor of the impurity spin. The susceptibility of the AKM at
T = 0 differs from Xsb by a factor of a: Xakm =c*Xsb The Wilson ratios for both
models are related as follows:
d 4 7T2kl Xakm 0
/tAKM 0 7
3 (j/b)2 7
24
R
SB
4 7r2kl Xsb
3(^b)2 7
2
a
(2.44)
where RAKU=aRSB.
The thermodynamic properties of the AKM are given in terms of the exchange
interactions (J) and (J), and therefore can also be expressed in terms of the
parameters a and Ar of the spinboson model [35, 36]:
Ar p Jj_,
2
(2.
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 J\\. 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/Ar, 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 kBTxsB has a finite value at T = 0, as in the isotropic
Kondo model, and reaches the freespin 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 freespin behavior.
The behavior for a finite bias e>0 is described in Fig. 2.5 for a = 0.2. The
quantity e is equivalent to a magnetic energy gpBh acting on the impurity spin in
the AKM. The temperature Ar is renormalized by e, and becomes [37]
Ar = ^A? + e2. (2.46)
a =
11 tan"
7T
npJ\\
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 lowtemperature susceptibility strongly decreases as a function of the
25
parameter e. It also shows a maximum for fields of order Ar and above, with a
temperature position that increases with Ar.
2.5 Kondo Lattice
Certain types of metallic compounds, including heavyfermion 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 AbrikosovSuhl resonance of width Tk appears
in the density of states near the Fermi level for T T^. In the case of spin
 Kondo scatterers, the resonance lies right at the Fermi energy. This feature
indicates the crossover to a strongcoupling regime in the scattering between /
and conduction electrons, growing in size as the number of impurities approaches
Avogadros number NA. Consequently, there is a substantial increase in the density
of states at Â£p. Figure 2.6 illustrates the evolution of the AbrikosovSuhl resonance
for different temperatures. In heavyfermion compounds, the 4/ 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 Fermiliquid theory. At high temperatures (T T/e), the
AbrikosovSuhl 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 Tcoh,
the electronic properties change from those described by scattering off independent
Kondo impurities to those reflecting the periodicity of the lattice via Blochs theo
rem. This crossover is usually described in terms of a maximum in the temperature
dependence of the specific heat as C/T and the electrical resistivity around T ~Tcoh.
A consequence of coherence is an increase of indirect exchange interactions between
impurity spins. At distances larger than the 4/ radius (r4f <0.5) but less than
MxJT)
26
Figure 2.4: Thermodynamic properties of the anisotropic Kondo model for e = 0
and different values of a. a) Specific heat expressed as ArC/kBT vs T/Ar. b)
Universal static susceptibility curves expressed as kBTxSb vs T/Ar [37].
27
o 1/5
ct= 1/5
Figure 2.5: Thermodynamic properties of the AKM for a 0.2 and different values
of e (in units of Ar). a) Specific heat as ArC/kBT vs kBT/Ar. b) Susceptibility
curves expressed as Arxsb vs kBT/Ar [37].
28
Figure 2.6: Density of states of a nonmagnetic Kondo lattice at different temper
atures, showing the evolution of the giant AbrikosovSuhl resonance [39].
29
the size of the Kondo compensation cloud for a single impurity, the presence of
closelyspaced uncompensated spins leads to the RudermanKittelKasuyaYosida
(RKKY) interaction between localized / orbitals
Wrkky = J(r)SiSj, (2.47)
where
/7V\ J cos(2kFr)
J{r) ~ { ]
is the RKKY coupling at large distances, J is the Kondo coupling, and kF is
the Fermi wavevector. In most heavyfermions 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 TF 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 4/ 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 (7k 3>TRKKY), magnetic order
disappears altogether and the material is a nonmagnetic Kondo lattice. Heavy
fermion compounds exist in the region around the magneticnonmagetic phase
boundary, and those with a magnetic ground state exhibit mostly antiferromag
netic order.
30
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 singleimpurity 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 singleimpurity 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 Fermiliquidlike. A large number of
intermetallics falling in this category are commonly referred to as nonFermiliquid
(NFL) systems. Their thermodynamic and transport properties can in some cases
be described by either logarithmic divergences or powerlaw behavior according to
different theoretical models [3, 12].
2.6 NonFermiLiquid Effects
Current models of nonFermiliquid phenomena can be divided into two
groups: theories describing a possible singleion origin to these effects and those
attributing them to intersite interactions. A member of the first group is the
twochannel 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 heavyfermion
systems like Ui_xThxBei3 [48]. In this model, NFL behavior is associated with
fluctuations of the quadrupolar degrees of freedom, rather than spin fluctuations.
Another possible singleion mechanism towards nonFermiliquid behavior
is Kondo disorder [49, 50, 51]. The material exhibits a random distribution of
31
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)/dTx acquires a lognormal form for
strong disorder:
P(TK) = (4*,)* exp ln2[poJe~" ln(Â£F/r*)]} <249>
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 T/<
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
nonFermiliquid 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 uj* 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 hcu* C kBTc [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, nonFermi
liquid effects in heavyfermion systems arise as a consequence of a nearzero anti
ferromagnetic transition temperature, so that a quantummechanical treatment is
necessary. The thermodynamic properties are dominated by the collective modes
due to critical fluctuations rather than by Fermiliquidlike elementary excitations,
and are described by various scaling laws [53, 54] depending on the effective dimen
32
sionality and the nature of the magnetic transition. As a result, the system is said
to have a generalized (nonLandau) Fermiliquid ground state, with an enhanced
quasiparticle mass m* due to the presence of longrange 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]. NonFermiliquid 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,
spinglass, or ferromagnetic transition [58]. The temperature dependences 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.
33
Figure 2.7: Phase diagram of the Kondo lattice [39], illustrating the different
dependences of Tk and TRKKY 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 CeAl3
3.1.1 Crystal Structure
The compound CeAl3 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 and c 4.608 [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 3. A study of the structure of rare
earth trialuminides[62] attributed the formation of a particular structure and its
c/a ratio to the rareearth/aluminum ratio Rre/RAi 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 Rre/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
35
Figure 3.1: Hexagonal Ni3Sn structure of CeAl3.
Figure 3.2: Hexagonal Ni3Sn structure of CeAl3 (extended scheme).
36
Table 3.1: Cell Content of Ni3Sn structure of CeAl3 [64].
Atom
Multiplicity
(Wyckoff notation)
X
Coordinates
y
z
Ce
2c
1/3
2/3
1/4
2/3
1/3
3/4
Al
6 h
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 A1 nearest neighbors, at a distance dCe.M = 3.27 ,
and 6 Ce nearest neighbors at a distance dCe.Ce 4.428 [63]. The central Ce
atom is surrounded by six nearest neighbors (3 A1 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 caxis direction, but rather at an angle. These offaxis neighboring
distances might have some implications regarding the hybridization between Ce
and A1 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 7 extrapolated from C/T vs T2 is 7 = 1250 mJ/K2 mol. This max
imum in C/T has been the subject of intense controversy about the ground state of
CeAl3. It was initially proposed that its origin is due to the formation of a Kondo
37
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 4 T [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 23T[72]
indicated a decrease in C/T values below 45K (more than 15% at IK) and an
increase in values above the same temperature (around 20% near 10 K). 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
P1/6. There is no sign of the specific heat anomaly at a pressure of 0.4kbar. The
coefficient 7 is reduced from 1250 mJ/K2 mol at atmospheric pressure to about
550mJ/K2 mol at 8.2kbar. Values of C/T are essentially constant below IK for
pressures around and above 2 kbar.
An attempt was also made to measure specific heat on very small single crys
tals of CeAl3 [74]. The results proved to be sampledependent. 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 (y(T = 0) 29.5 memu/mol), consistent
38
T {K)
Figure 3.3: Magnetic field and pressure dependence of the specific heat of CeAl3.
Upper part: C/T vs T of CeAl3 in magnetic fields up to 4 T (OT: circles, 2T:
diamonds, and 4T: triangles) [68]. Lower part: Pressure dependence of C/T vs T
for CeAl3 up to 8.2kbar [73].
X O 03 emu/mol)
39
T (K)
Figure 3.4: Magnetic susceptibility of CeAl3 below 10 K [75]. The inset shows the
inverse susceptibility.
40
with Fermiliquid behavior. The inverse susceptibility follows CurieWeiss law
above 150 K, with an effective magnetic moment close to that of a free Ce3+ ion,
/ieff = 2.54b, and cw = 30 6K. The susceptibility of single crystals above
4K was also measured with the field parallel (x) and perpendicular (x) to the
caxis [76]. The susceptibility along the caxis X is at least three times as large as
X_l 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 Kondolike increase down to 50 K, a maximum around 35 K,
possibly signaling the crossover from singleimpurity to Kondolattice behavior,
and a sharp decrease below 10 K. At temperatures below 100 mK, the resistivity
has the form of a Fermiliquid, with a coefficient A 35 /xQcm/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 0.40.5 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 2T [77]. In singlecrystal measurements, the electrical resistivity
in zero field along the basal plane is more than twice that along the caxis [76, 81].
The field dependence of the A coefficient parallel to the caxis shows a peak around
41
p
2*0
220
200
180
160
1*0
120
IOO
80
60
*o
20
OAtj
ekclncol resistivity
O
too
200
T[X]
500
Figure 3.5: Transport measurements on CeAl3. Upper part: Electrical resistivity
below 300 K [78]. Lower part: p vs T2 below 100 mK [20].
42
2 T. The authors found this result to be in qualitative agreement with theoretical
models describing weaklyantiferromagnetic metals.
3.1.5 Nuclear Magnetic Resonance
Measurements on 27A1 nuclear magnetic resonance (NMR) on CeAl3 down
to 0.3 K by Nakamura et al. [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 spinlattice 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 Korringalike (Ti 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 fim. The nonuniform strains cre
ated by preparing the powder can have a dramatic effect on the physical properties
of CeAl3 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 Ce3Aln, as
proposed by Wong and Clark [83]. Gavilano et al. also measured the NMR spec
tra of partially oriented powder (caxis 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
43
P(B)
Figure 3.6: Magnetoresistance of CeAl3 down to 100 mK [79].
Echo intensity
44
Figure 3.7: NMR spectra of partially oriented powder at 6.968 MHz for different
temperatures [70].
45
less than 0.05B. The presence of magnetic correlations in CeAl3 argues against a
simple interpretation of its ground state in terms of a nonmagnetic Fermiliquid.
3.1.6 Muon Spin Rotation
The only muon spin rotation (SR) experiments on pure CeAl3 available to
date are those of Barth et al. [69, 84]. The authors measured the timedependent
muon polarization on two polycrystalline samples, as seen in Fig. 3.8. The muon
polarization signal was described as the sum of several timedependent 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 spindensitywave behavior [85]. The
presence of the local field at the muon sites was interpreted as the development
of shortrange, quasistatic magnetic correlations in CeAl3 below 0.7 K. As the
temperature decreases, these correlated moments, estimated to be around 0.5B,
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 GdAl3 (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 ab planes.
 polarization (o.u.)
Figure 3.8: Muon polarization as a function of time in zero external field
T = 0.05, 0.5, and IK [69].
47
Energy transfer [meV]
Figure 3.9: Magnetic contribution to the inelastic scattering function of CeAl3 at
T 20 and 40 K [87]. The solid line is a fit to a threeLorentzian model. The
dotted lines represent the individual fit components.
48
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+
\J = ) multiplet splits into three doublets under the influence of a crystalline
electric field (CEF): Ty :  ), T8 :  ), and r9 :  ). In cerium heavy
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 0, and an inelastic peak at an energy that coincides with the
characteristic energy of crystalfield excitations.
In addition to the quasielastic peak, the most recent measurements [87] dis
played a single inelastic peak at an energy e~6.4meV for T = 20 K (Fig. 3.9). With
the help of previous singlecrystal magnetic susceptibility data [76], the authors cal
culated the crystalfield parameters for CeAl3 and determined the ground state to
be r9 :  ), followed by T8 :  ) at 6.1 meV (T = 71 K), and Ty : 
at 6.4 meV (T = 74 K). By comparing the parameters to those of other rareearth
trialuminides with Ni3Sn structure, they concluded that the hybridization of Ce
4/ 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 CeAl3 to date are those of La
impurities on the Ce sites. Recent specific heat studies of Cei_xLaxAl3, 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
CeAl3. 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
49
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 Ladoped 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 / 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 basalplane triangular lattice of Ce ions [86,
89]. As the Ce ions are substituted by nonmagnetic 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
inplane interactions are much stronger than the interactions between two adjacent
planes.
More recent neutron scattering and SR studies on Cei_xLaxAl3 [15, 90] have
shown that the temperature at which the maximum in the specific heat for a; = 0.2
develops coincides with both the appearance of an inelastic peak in the neutron
scattering function and the divergence of the SR relaxation rate. The divergence
of the muon relaxation rate was interpreted as evidence for either shortrange mag
netic correlations, as found for pure CeAl3 [69, 84], or longrange 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 temperaturedependent, 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
50
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 SR results. Instead, the AKM proved
to be useful in providing an explanation for the anomalies in terms of a singleion
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 Ladoped CeAl3 alloys in Chapter 6.
Only one study reports doping of CeAl3based alloys on the A1 ligand sites [61].
Corspius et al. found that the alloys were singlephased 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(Al0.9Sn0.i)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 transiton. All samples except those with Ga impurities exhibit discrepan
cies between zerofieldcooled and fieldcooled susceptibilities, and only those above
x 0.1 show a timedependent maximum (spinglasslike). 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 absolutevalue change
(increase or decrease) in the c/a ratio.
51
Table 3.2: Cell Content of Cu3Au structure of CePb3 [64].
Atom
Multiplicity
(Wyckoff notation)
X
Coordinates
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 facecentered cubic Cu3Au 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 facecentered
positions. The structure forms directly from the melt at 1170C on the CePb
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 = 4.8760.002 [92], corresponding to a lattice volume
V = 115.93 3.
Figure 3.11 shows the Cu3Au 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 fee 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 ~ci/y/2 3.448 .
3.2.2 Specific Heat
The lowtemperature 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/K2mol, and the extrapolated electronic coefficient 7
reaches a value around 1000 mJ/K2mol. The effect of high magnetic fields was
(mJVK Ce mol)
52
4000
3000
2000
o
1000 L
0
Figure 3.10:
A A
m
Ce La R 1
1 x x 3
x0.05
K0. 1
x0.2
*...! I 4 ... .1 I 1 . .
1 2 3 4 5 6 ?
T CK)
Specific heat of Cei^La^A^ alloys (x 0.05, 0.1, and 0.2) [71].
Figure 3.11: Cubic CU3AU structure of CePb3.
53
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].
54
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 is suppressed down to 0.6 K; above 0.7 GPa, the temper
ature of this pressureinduced typeII antiferromagnetic phase increases from 0.6 K
to 1 K at 1.3 GPa. Figure 3.13 illustrates the temperaturepressure phase diagram,
with T/v decreasing up to 0.7 GPa and increasing at higher pressures. This behav
ior is rather unusual since a continuous decrease of T/v 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 CePb3 [96]. In addition, contrary to
other Ce Kondo lattices like CeC^zAu^, (x > 0.1) [97], and CeRu2Ge2 [45], no
pressureinduced suppression of 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 highfield boundary corresponds to an unknown
phase, possibly a spinflop state [98]. The exact nature of this fieldinduced phase
remains to be determined by neutron diffraction experiments. Nevertheless, the
discovery of this fieldinduced transition in the (110) direction motivated further
investigation of the properties of CePb3 single crystals in magnetic fields.
55
Figure 3.13: Transition temperaturepressure 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).
56
Ll l 1 1
0 5 10 15 20 B (T)
Figure 3.14: Magnetic field dependence of the relative change in sound velocity for
the (cn Ci2)/2 elastic mode at 10 MHz [98].
57
Figure 3.15: Magnetic contribution to the electrical resistivity of CePb3 at H = 1T
below room temperature. The inset shows the resistivity between 0.2 and 4 K at
H = 0.93 T [93].
58
3.2.4 Transport Measurements
In order to measure the electrical resistivity of CePb3, it is important to
measure in magnetic fields of order 1T 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, Kondolike 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 crystalfield 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 11.5kbar. This result
is consistent with an increase of the Kondo temperature T'k The magnetore
sistance was recently measured along the (110) crystallographic direction [100].
Two fieldinduced anomalies were found for the magnetoresistance curves below
400mK at 5 and 9.5T, respectively (see Fig. 3.16). The resistivity increases up
to 5T, decreasing sharply above the first transition, and becoming almost field
independent after the second. A magnetic fieldtemperature 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 fieldinduced 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 (10 0) direction, at which point the sharp
59
Figure 3.16: Magnetoresistance curves between 1 and 16 T for temperatures in the
range 20mK to 8K. The magnetic field is along the (110) direction [100].
60
drop at 5 T could not be detected. The lowtemperature resistivity was found to be
proportional to T2 with a fielddependent A coefficient. At 5T, A reaches a max
imum, the range of T2 dependence becomes smaller, and the resistivity acquires
a linear term, all coinciding with the fieldinduced transition. This enhancement
of A with field points to a corresponding enhancement of the specific heat coeffi
cient 7, as the ratio A/72 is expected to remain constant for heavy fermions [101].
At 10 T, there is a small bump in the A coefficient, indicating a transition to a
ferromagneticallypolarized 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 2.6 kG. 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 CurieWeiss behavior, and gives a high temperature effective moment
= 2.5 B, and a CurieWeiss temperature 0CW = 25 K. An investigation of
the pressure dependence of the inverse susceptibility [99] found an increase of 0CW
from 0 to 15kbar, 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 spinflop) phase transition [103]. Their phase diagrams along
the (1 00) and (11 0) directions indicated that the range of the fieldinduced phase
depends on the crystallographic direction. Between 20 and 600 mK, with H
(10 0), the range is about IT, while for H  (110), it is close to 5T. The phase
diagram determined from ac susceptibility data along (11 0) is in agreement with
previous studies, as shown in Fig. 3.18.
61
Figure 3.17: Magnetic susceptibility of a CePb3 polycrystal below 4K at H
2.6kG [102],
62
e
X
i 1 r
Paramagnetic State
H//<110>
~~Ol
A
tV
'<5
SpinFlop phase
a
\
\
\
Xt
o 
Antiferromagnetic State
\
Qa \
X
X
X
M
' %
w
J i_
0.0 0.2 0.4 0.6 0.8 1.0
12
Temperature (K)
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]).
63
3.2.6 Neutron Scattering
Neutron scattering studies are essential in the determination of the ordered
moment at low temperatures and the crystalfield parameters of heavyfermion
systems. The CU3AU cubic structure of CePb3 provides a high degree of crystal
symmetry. In the cubic environment of Ce3+ ions in CePb3, the crystalfield (CEF)
potential splits the  J = ) multiplet into a T7 doublet and a T8 quartet [104]:
r7> = a !> 6 T !>
_ *>l Â§> + I T 1>
where a = ()1 / 2 and 6 = ()1/f2.
(3.1)
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 T7 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 0.55//b at 30 mK. A similar incommensurate structure has also
been detected for CeAl2 [107], another cubic heavyfermion 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 CeCu2S2. This statement has important implications regarding a
possible role of crystalline anisotropy in regulating the competition between the
Kondo and RKKY energy scales.
64
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 dashdotted line corresponds to the inelastic com
ponent.
65
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 singleimpurity effects in a concentrated heavyfermion
system. The specific heat, magnetic susceptibility, and electrical resistivity all
scale with Ce concentration. Electrical resistivity measurements revealed that the
crystalfield 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 = The Kondo temperature is constant throughout
the series, implying a constant value of J. The transition temperature T/v 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 CeCe distance,
rather than to an overall change in J. Indeed, Cei_xLaxPb3 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 Cei_xLaxPb3 seem to be
unaffected by the electronic environment surrounding the Ce3+ ions, experiments
on Cei_xMxPb3 (M = Y, Th) [109] confirmed that the singleimpurity 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 magneticlike
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 Cei_xMxPb3.
AC(^moleCeK)
66
Figure 3.20: Electronic specific heat vs T/Tk for Cei_xLaxPb3 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 TÂ¡< = 3.3 K.
67
Chemical substitution studies were also performed on both /ion and ligand
sites of the CePb3 structure. In Ce(Pbi_a;Mx)3 studies with M = Tl, In, and
Sn [110, 111], the antiferromagnetic transition temperature decreased toward zero
for a Sn concentration x = 0.4, and increased for both Tl and In. For the latter
two dopants, there is a maximum towards the center of the x phase diagram.
Substitution of Sn for Pb on the ligand sites suppresses Tyv and greatly increases
the Kondo temperature [112, 113].
CHAPTER 4
MOTIVATION
This chapter begins with a discussion on the importance of the study of CeAl3
and CePb3, followed by a presentation of the objectives of the current study.
4.1 Importance of CeAl3 and CePb3
Both CeAl3 and CePb3 are canonical, welldocumented heavyfermion sys
tems, with values of the 7 coefficient surpassing 1 J/K2mol, crystalfield 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 heavyfermion systems, based on the Kondo effect
and Fermiliquid 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 nonFermiliquid effects. These are all topics of
current interest, yet they are among the least understood aspects of heavyfermion
physics. Any information obtained from the study of the above two compounds
might be utilized in the development of new interpretations for the heavyfermion
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 heavyfermion 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 CeAl3 as the first heavy
68
69
fermion compound [20]. Despite its significance in the field of stronglycorrelated
electron systems, CeAl3 is probably one of the least understood among these com
pounds. Ever since its discovery, it has been considered a canonical, nonmagnetic
heavyfermion system. Yet later experimental results (see Chapter 3) challenged
its nonmagnetic status, and pointed to a possible magneticallyordered ground
state for CeAl3. Whether the ground state in this compound is magnetic or not
has been a longstanding debate, and remains an important topic in the study of
heavyfermion 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, 7 1000 mJ/K2 mol. The electrical re
sistivity has a large T2 coefficient, A 45 /iQcm/K2, and the ratio A/72 is around
4 x 10~5 OcmK2 mol2/J2. When taking into account the relatively large value of
7 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 heavyfermion 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 singleion
scaling of thermodynamic and transport properties in a concentrated 4/ system.
The study of CeixLa^Pbs by Lin et al. [96] revealed that the normal state of
alloys over the range (0
(see Chapter 3). It is the only Ce heavyfermion system to date exhibiting such
behavior. The reason why such a concentrated system can exist with apparently
noninteracting 4/ sites remains unclear.
70
4.2 Objectives
4.2.1 Magnetism and HeavyFermion Behavior in Ce Kondo Lattices
The studies on CeAl3 and CePb3 alloys presented in this dissertation are
motivated by a fundamentally important topic in heavyfermion research: the
need for a full understanding of the interdependence between magnetic correlations
and/or magnetic order and the heavyfermion state. The ground state of rareearth
intermetallics is generally described in terms of the competition between two energy
scales, Tk and TRKKY, discussed in Chapter 2. The former represents a singleion
effect due to the local Kondo interaction between conduction electrons and the /
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/K2mol. This is the least understood area
of the Doniach phase diagram. The applicability of this model to heavyfermion
Kondo lattices, in particular to CeAl3 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 heavyfermion ground states: the Wilson ratio R and
the KadowakiWoods ratio. The experimental Wilson ratio R [5] is defined as
n2klxo/l2efl, where y0 is the zerotemperature susceptibility and /eff is the effective
moment at room temperature. Values of R are usually much larger for magnetically
71
ordered than for nonmagnetic Kondo lattices [5]. Nevertheless, the experimental
ratios for CeAl3 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 heavyfermion compounds, the empirical relation A/72 lies somewhat
close to the KadowakiWoods ratio A/72 = 1 x 105 QcmK2 mol2/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.i [114], the only published study of the field dependence
of this ratio. In order to verify whether A/72 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 magneticallyordered heavyfermion system.
Previous thermodynamic and transport measurements on Ce0.6La0.4Pb3 [96]
suggested a singleion mechanism for the heavyfermion behavior in this system. A
study of the specific heat in magnetic field of Ce0.6Lao.4Pb3, a nonmagnetic coun
terpart of CePb3, was conducted in this dissertation to search for further evidence
of a singleion Kondo origin for the heavyfermion state in Cebased systems.
4.2.2 Ground State of CeAl3
The experiments on CeAl3 alloys presented in this dissertation are motivated
by the existing controversy about the ground state of CeAl3. 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 stronglycorrelated 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
72
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 /rSR studies, casting serious doubt on the
socalled coherence interpretation [68]. However, it is not clear at the present time
whether the magnetic correlations are shortranged, 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 singleion 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 singleion
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 EdmundBiihler arc furnace under a highpurity argon atmo
sphere. The arcmelting apparatus consisted of a stainlesssteel vacuum chamber
with a watercooled 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 1cm.
The Cu hearth on the arcmelter 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
highpurity Ar. After this procedure was repeated three to four times, the cham
73
Temperature
74
Figure 5.1: Phase diagram of CeAl [91].
75
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 highpurity Ar goes
through a purifier before entering the arcfurnace. 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 Cei_xMa;Al3 (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 A1 atoms. The synthesis of CeAl3
alloys is always accompanied by the formation of a large amount of the secondary
phases CeAl2 and Ce3Aln. The presence of these unwanted phases is substantially
reduced by proper annealing conditions.
The ceriumaluminum phase diagram has been studied by several groups [91],
its latest addition being CeAl3 [62]. It contains four other compounds: Ce3Aln,
CeAl2, CeAl, and Ce3Al (see Fig. 5.1). Both CeAl2 and Ce3Al form directly
from the liquid solution, CeAl and Ce3Aln form peritectically, and CeAl3 forms
peritectoidally at 1135C. A peritectic reaction is one in which the compound
76
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 CeAl2 and
/?Ce3Aln, which in turn melts into CeAl2 crystallites embedded in a liquid solution
matrix.
The transformation of a mixture of CeAl neighboring phases into the CeAl3
phase upon cooling has a marked effect on the way samples crystallize. The pres
ence of secondary phases is the cause of many sample dependences of thermo
dynamic and transport measurements. Polycrystals synthesized by arc melting
consist of a mixture of CeAl3 with large amounts of CeAl2 and Ce3Aln. Anneal
ing has been found to reduce the proportion of secondary phases to the point of
becoming undetectable by conventional xray diffraction methods. Magnetic sus
ceptibility measurements on annealed samples are an efficient way of detecting the
above second phases, since CeAl2 is antiferromagnetic below 3.8 K, and Ce3Aln 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
Lanthanumdoped 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 Ce0.6La0.4Pb3 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
77
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.
CePb3based 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 lOOmtorr. The quartz tubes were
then placed inside a Lindberg furnace and annealed according to a previously
tested prescription. Alloys of Cei_xLazAl3 were annealed at 830C for two weeks,
while those of Cei_xYxAl3 were annealed at 800C for two weeks, then 850C for
five days. Both CePb3 and Ceo.6La0.4Pb3 were annealed at 800C for one week. In
all cases, annealing started with the furnace already at annealing temperature. At
78
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 XRays
Measurements of xray diffraction were used as a means to verify whether
the arc melting and annealing processes led to the formation of the desired crystal
structure. Prom 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 xrays in crystals is based on Braggs Law:
A = 2d sin 9, (5.1)
which for a first order (n = 1) spectrum relates the known Cu Ka wavelength to the
diffraction angle 6 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 (hkl).
The experimental setup consisted of a Phillips APD 3720 diffractometer, an
xray source with a watercooled power supply, and a computer for data acquisition.
The APD 3720 consists primarily of xray 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 9. The xray beam is of known wavelength: a Cu Ka line with
A = 1.540562 .
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
79
to 6/min, and its range to 5 < 29 < 120. 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 structuregenerating soft
ware. This procedure allows for identification of secondaryphase intensity lines
larger than the background intensity (~ 5% of maximum intensity line). For a
cubic system (i. e. CePb3 alloys), the indices for primaryphase lines are obtained
from the following equation [117]:
sin2 9 = Y~x{h2 + k2 + l2).
4 a2
Similarly, for a hexagonal system (CeAl3 alloys),
(5.2)
sin 9 =
4 (.h2 + k2 + l2) Z2
3 a2 + c2
(5.3)
The indices (h k l) and the angles 29 for the highest and narrowest intensity lines
were entered as data points into a leastsquares fitting program, along with the
wavelength and structure type. The roomtemperature lattice parameters and
their uncertainties were then obtained from a leastsquares 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
80
approximately 4.2 K, the liquidhelium 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 0.5cmlong 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 pickup coils
connected to the SQUID. The SQUID voltage is read at different position intervals
accross 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 5T. 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 heavyfermion systems (e.g., Refs. [5]
and [6]).
81
block
thermometer
Cu block
brass can
Figure 5.2: View of the cryostat used for zerofield specific heat measurements
betwen 1 and 10 K.
82
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 liquidHe 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 lockin 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 ATTNT
Plug and Play GPIB board from National Instruments. A 12bit resolution Keith
ley Metrabyte DAS1402 A/D converter board interfaced the PC to the lockin
amplifier. The data acquisition was monitored using two PCbased programs for
thermal conductance and specific heat measurements, respectively. The software
was designed by the author using Lab VIEW version 5.1 for Windows 95/98.
Cryogenic
The cryostats used for zerofield measurements are illustrated in Figs. 5.2
and 5.3. Figure 5.2 shows the probe used in the temperature range 110 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
83
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 0.42 K were conducted using the cryostat described
in Fig. 5.3. After reaching a temperature of 4.2 K following the procedure above,
the 4He 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 4He pot was pumped out to reach a temperature between 1 and 2 K. Al
though this temperature can be sustained for many hours, the 4 He 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 3Hegas 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
niobiumtin wire and two outer coil sections of niobiumtitanium wire. The cryo
stat used below 1 K is the same as in Fig. 5.3, and the one used between 110 K
is illustrated in Fig. 5.4. The main difference between them is the lack of a 3He
enclosure for the highertemperature probe.
84
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.
85
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.
86
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 110 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 0.42 K for its
known magnetoresistance [122]. All thermometers are linked to the block using
thermallyconductive 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 thermallyconductive 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 110 K, the platform thermometer used was
an elongated piece of doped Ge, and the platform wires were made of a Au7%Cu
alloy. A thin piece of Speer carbon resistor and Pt10%Rh platform wires were
used for measurements between 0.4 and 2 K.
87
Figure 5.5: Top view of the sampleplatform/Curing assembly at the bottom of
the cryostat.
88
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 platformsample system by means of a small current (in /xA), the
temperature increases from a base value T0 by an amount AT. When the current
is turned off, the system temperature T(t) decays exponentially to T0:
T(t) = T0 + ATe~t/Tl.
(5.4)
The time constant T\ is proportional to the total heat capacity (sample plus plat
form) Ctotal:
^total
Tl = ,
K,
(5.5)
where k, is the thermal conductance of the wires linking both platform and sample
at T = T0 + AT, and the Cu ring at T = T0. The time constant was obtained
by measuring the time decay of the offnull voltage signal from a Wheatstone
bridge using a lockin 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 0.410 K is 13%. The thermal conductance is given by
k =
P
AT'
(5.6)
Here, P IV is the power applied to the platform heater. The above equations
are valid under the assumption of an ideal thermal contact (Avsample ~ oo) between
sample and platform. In the event of a poor thermal contact between the sample
89
and the sapphire (/isamPie~^) the temperature decay can generally be described as
the sum of two exponentials
T(t) = T0 + Ae~t/n + Be~t,T\ (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 Ti, r2, and k. The thermal conductance is measured separately by applying a
current to the platform heater, calculating the power P IV, and calculating AT
as a result of the power applied to the heater. The accuracy of this measurement
between 0.410 K 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 heavyfermion
systems at low temperatures and test their thermodynamic properties against the
oretical predictions.
The two main types of doping on heavyfermion compounds are Kondohole
and ligandsite 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
90
due to an atomic size difference between the / 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. Ligandsite 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 singlesite 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 heavyfermion compounds ranges from shortrange correlations to
nonFermiliquid behavior to longrange 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 singleimpurity 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 CeAl3
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.
91
5.5.1 Experiments on CeAl3
A doping study of the lattice parameters, specific heat, and magnetic sus
ceptibility of Cei_xMxAl3 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 x/l of Cei_xLaxAl3 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 Doniachs
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 Ce0.8La0.2Al3 and Ce0.3La0.7Al3 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 Ydoped samples will be compared to data as a function of pressure for
CeAl3 to distinguish between the effects of chemical and hydrostatic pressure on
the anomaly in C/T.
Additional Ceo.8(Lai_xYx)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 Ce0.8La0.2Al3 was conducted to create a similar hybridization environ
ment to that of CeAl3 by reducing the lattice volume to that of the undoped
compound. Thermodynamic measurements will allow to test the magnetic inter
92
pretation of the anomaly in C/T by assuming a constant coupling J, yet reducing
TRKKY by increasing the CeCe 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 heavyfermion 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 Fermiliquid 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 fieldinduced transition detected by previous sound velocity and magnetoresis
tance measurements, and for possible nonFermiliquid effects. The data should
be helpful in understanding the effects of a magnetic transition on the nature of
the heavyfermion state.
Results from measurements of the heat capacity of Ceo.6La0.4Pb3 in magnetic
fields up to 14 T will also be discussed in order to investigate further the single
impurity nature of the paramagnetic heavyfermion state of CePb3. The electronic
contribution to the specific heat below 10 K will be compared to predictions for the
S \ singleimpurity Kondo model in magnetic fields. The above measurements
on CePb3 and Ce0.6Lao.4Pb3 allow for an analysis of the electronic coefficient 7 and
the Kondo state in both nonmagnetic and magnetic heavyfermion systems.
