Doping experiments in heavy fermion superconductors


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Doping experiments in heavy fermion superconductors
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vi, 172 leaves : ill. ; 29 cm.
Kim, Jung Soo, 1956-
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Thesis (Ph. D.)--University of Florida, 1992.
Includes bibliographical references (leaves 162-171).
Statement of Responsibility:
by Jung Soo Kim.
General Note:
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University of Florida
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I have been very fortunate in having Professor G.R.

Stewart as my advisor. I should like to thank him for his

valuable advice. The time and effort he devoted to my

graduate education made my years at the University of

Florida stimulating and rewarding. Our discussions and his

comments about many aspects of my work have greatly helped

me to remove inconsistencies, errors, and obscurities all

throughout my graduate career. I am very grateful to my

supervisory committee, Professors D.B. Tanner, P. Kumar, Y.

Takano, and D.E. Clark for their advice and comments.

It is my pleasure to thank Dr. B. Andraka, S.B. Roy,

and C.S. Jee for their valuable discussions and helpful

suggestions. Much of this work has been performed with

their help. And I thank my fellow graduate students for

their cooperation and friendship.

I am very grateful to the members of the Physics

Department machine shop for their excellent technical

support and assistance in the development of experimental

equipment, and I am also very grateful to the members of

the cryogenic group for their support from which I have

profited for many years. In particular, I am indebted to

Amy who helped our group with x-ray data analysis.

Finally, I wish to thank my family for their support.



ACKNOWLEDGMENTS ........................... ........

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


I INTRODUCTION .............................

II REVIEW OF THEORY ..........................

Kondo Effect ....... .....................
Fermi Liquid Theory ........................
Heavy Fermion Superconductivity ............


Sample Preparation .......................
Experimental Procedures ....................

IV NORMAL STATE ...............................

Single-ion Effects in the Formation of the
Heavy Fermion Ground State in UBe13 ...
Magnetic Behavior of Cei-xThxCu2Sia
for x & 0.75 ..........................
Different Magnetic Behavior in Doped UPt3 ..

V SUPERCONDUCTING STATE ......................

Annealing Effect and Critical Field
of Pure UBe13 ....... .................
Investigation of the Second Transition
in U1-xThxBel3 .......................
Impurity Effects on the Superconductivity
of UBei3-xMx .. ........... .............

VI SUMMARY AND CONCLUSIONS ....................

REFERENCES ................ ... .......................

BIOGRAPHICAL SKETCH .................................




















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



Jung Soo Kim

May 1992

Chairman: G.R. Stewart
Major Department: Physics

Experimental results are presented on the doping

effects both in the normal state and the superconducting

state of the heavy fermion superconductors UBe CeCu Si ,
13 2 2
and UPt3. The objective has been to understand the origin

of heavy fermion nature and the unusual superconductivity

in these materials.

In the normal state, U 5f-electron hybridization with

Be s-electrons controls the specific heat 7 and m* in the

U-dilute limit of x > 0.15 in U MBe For
1-x x 13
ao(U1-xMxBe13) s a (UBe 3) in lattice parameter, 7 per

U-mole is due to single ion effects (independent of U

concentration) with a value of about 40 % of pure UBe
value while the outer electronic configuration of M plays

no role. The large low-temperature magnetic susceptibility


in UBe13 is entirely attributable to single-ion effects,

contrary to previous assumptions.

A huge low-temperature x per Ce-mole and M vs. H

saturation are found in Ce Th Cu Si (x > 0.75). Those
1-x x 2 2
magnetic behaviors for x = 0.85 and 0.9 are mostly

consistent with a spin glass interpretation.

I find that Ti in UPt3 suppresses magnetism in

contrast to the other isoelectronic dopants (Hf, Zr) and

suppresses spin fluctuation faster than Hf and Zr possibly

due to the stronger hybridization for 3d- and 4s-orbitals

of Ti with the 5f-orbital of U.

In the superconducting state, we report specific heat

and T on the highest purity UBe3 to date. The high-purity

annealed samples show larger and sharper transitions than

previously reported.

A large 7 of 2300 mJ mole-1 K-2 for high-purity and

long-term annealed U 97Th Be is derived from entropy

consideration. The critical field slopes at both transition

temperatures of Uo. 9Th03Be13 are parallel within our

error, and these results (H '(T ) ( -45 3 T/K, Hc (T ) =
c2 c c2 c
dH (T)/dT at T) are almost identical with the Hc (Tc)
c2 c c2 c
value of pure UBe The equality of Hc (T ), H '(T ) for
13 c2 C1 c2 c2
U Th03Be13 and Hc (Tc) for pure UBe3 is difficult to
0.97 0.03 13 c2 c 13
explain on the basis of current theories that consider the

lower transition as a magnetic superconducting state.


In UBe M (M = Cu, Ni), the more magnetic dopant

(Ni) suppresses superconductivity faster than the less

magnetic dopant (Cu). AC/7T of Al- or Ga-doped UBe3 (Al

and Ga are isoelectronic atoms of B) is much smaller than

that of B-doped UBe13 and decreased with increasing dopant



Performing alloying experiments is one of the

important ways to study the anomalous properties of heavy

fermion systems, since, among other things, one can study

the single-ion behavior in the dilute limit and,

occasionally, induce unusual superconducting behavior. For

these doping experiments I mainly focus on one of the

uranium-based heavy fermion superconductors, UBe .

The main goals of this work are:

(1) to study the single-ion effect to understand the source

the of high effective mass of electrons in heavy fermion


(2) to investigate the the second transition in the

superconducting state of U Th Be because understanding

its unusual superconducting behavior is important for the

present knowledge of superconductivity, and

(3) to investigate the magnetic behavior of doped heavy

fermion superconductors.

Since CeA3 was first characterized in 1975 as a heavy

fermion system containing conduction electrons with very

large effective masses, m at low temperatures [1], this


exciting new class of metallic materials which exhibit

highly correlated electronic behavior at low temperatures

has been the focus of intense investigation over the

intervening years (for review [2, 3, 4, 5, 6]).

The low-temperature specific heat of an ordinary metal

has two contributions, those due to thermal excitations of

conduction electrons and of lattice vibrations, which can

be expressed as C(T) = yT + OT3, where y = yKn2 N(EF) and 3

S12n4Rn/5e3 ( = 1.38 x 1016 erg K1 is the Boltzmann
constant, N(E ) is the electronic density of states at the

Fermi energy EF per unit energy and unit volume, R = 8314

mJ mole-i K-i is the gas constant, n is the number of atoms

per formula unit, and eD is the Debye temperature). With

the consideration of electron-phonon coupling, N(E ) in y

is normalized by N(EF)(1 + A) as 7 oc N(E )(1 + A) where the

parameter A is the coupling strength containing electron

contribution both due to the lattice phonons and to other


Those compounds with heavy conduction electrons, in

common, show enormous low-temperature electronic specific

heat coefficient 7 (= C/T as T approaches 0 K) values

(Table I-1). And 7 is no longer constant and is strongly

temperature dependent (7(T) = C(T)/T at temperature T).

Viewed as a quasiparticle contribution, anomalously large

values of y are evidence for enormous normalized densities

of electronic states at the Fermi energy in these

Table I-1
Ordering temperature and low temperature properties of
heavy fermion compounds compared with those of ordinary

Compounds Temperature y(0)a (O)b p(0) Ac
(K) (nO cm)


U Zn 9.7 550 18.7 1.6
2 17
UCd 5.0 840 38.7 3

NpBe13 3.4 900 56


UBe3 0.9 1100 14.7 18

CeCu Si 0.65 1100 6.5 4.8 10.7

UPt 0.52 420 8.1 0.4 0.5

No ordering

CeAl3 1620 36 0.77 35

CeCu 1300 17.7 7.2 111

UAuPt4 725 15.2

Pd 10 0.71 4.28x10-4 6.4x10-5

a. in (mJ/f-mole K2) normalized per f-atom-mole.
b. in (memu/f-mole).
c. T2-coefficient of p in (gj cm/K2).
For references, see text.

substances (the normalized density of electronic states at

E per unit energy and per unit volume in turn determines

the magnitude of 7). Enormous 7 in these compounds implies

very large effective masses for the conduction electrons.

This provides the name heavy fermion systems now generally

used to describe these compounds. Alternative names are

heavy electron systems or highly interacting fermion

systems. The conduction electrons of these compounds have

effective masses at least two orders of magnitude larger

than the free-electron mass. The ratio C/T for several

heavy fermion systems rises considerably with decreasing

temperature below 10 K reaching enormous value as T

approaches 0 K [7]. On the other hand, in ordinary metals,

the specific heat varies linearly with temperature as T

approaches 0 K with 7 value of the order of 1 mJ mole-' K-2

(Table I-1). In ordinary metals 7 is independent of

temperature at low temperatures in the free electron


The heavy fermion systems have, in comparison to

ordinary metals, also large values of the low-temperature

magnetic susceptibility x (Table I-1). As can be seen in

Figure I-1, at high temperatures, the magnetic

susceptibility of these systems follows a Curie-Weiss law
(X (T) (T + e ) where e is the paramagnetic Curie-Weiss
P p
temperature) with a negative intercept on the temperature

axis (Table 1-2). In the free electron picture,





100 200

T (K)



Figure I-i
temperature of
shows the

Inverse magnetic susceptibility vs.
UBe13 between 1.5 K and 250 K. The inset
temperature dependence of magnetic
(data are from Ott et al. [7]).


low-temperature X values (X = u(EF) where g =

9.27 x 10-21 erg Oe-1 is the Bohr magneton) are of the

order of 10-4 emu mole-1. The electronic behavior of heavy

fermion systems is drastically different from ordinary

metals. The large values of 7 and x for heavy fermion

systems originate from the strong interaction between

electrons indicating the presence of many body effects.

The temperature dependence of the electrical

resistivity p of heavy fermion compounds can be divided

into two types (Figure 1-2). One of the types of

temperature dependence of p is that of UBe13 which is

similar to that of CeCu Si UBe13 has a fairly broad

shoulder near 20 K and a large low-temperature maximum at

about 2.5 K below which p falls sharply. This type of

behavior has been interpreted in the case of CeCu Si [8]

as evidence for coherent Kondo-like behavior. The other

type of p(T) can be found in UPt3 with a rapid rise in p

with increasing temperature at low temperatures and a weak

temperature dependence above about 150 K. UA2 which is

also known as spin fluctuator like UPt3 shows the same kind

of temperature dependence of resistivity. These heavy

fermion compounds have relatively large low-temperature

resistivity p compared to ordinary metals and p 100 uO cm

at room temperature (Table I-1, I-2). The high p values

indicate very strong scattering of the electrons and the

sharp falling of p at low temperatures proves that it is


: o5 5- ---

a CeCusSia
U UPta
5 ------------

0 100 200 300

T (K)

Figure I-2 Resistivity vs. temperature of UBe13 [9],
CeCu2Si2 [10], and UPt3 [11]. The relative magnitudes of
the three sets of data are arbitrary.

Table I-2
High temperature properties and f-atom separation of
heavy fermion compounds.

Compounds -e8 (K)

.fr,(BB) p(T = 300K)a Structure dr (A)


U Zn 105
2 17
UCd 23
NpBe13 42


UBe 53

CeCu Si 140
2 2
UPt 200

No ordering

CeAl 46

CeCu 59
UAuPt 135



























Pd 86 4.6 10

a. in (gQ cm).
b. orthorhombic structure transforms to monoclinic at low
For references, see text.










not a dirt effect. In ordinary metals, p decreases rapidly

with decreasing temperature below 300 K from values that

are about 1-10 gQ cm such as Pd (Table 1-2).

One of the constituents of heavy fermion compounds

contains partially filled 4f- or 5f-electron shells. All

these compounds have f-atom separations, d- (A), greater

than 4 A (Table 1-2). As Hill pointed out [12], for f-atom

separation less than 3.4 A, f-electron overlap between

neighboring f-atoms would lead to a loss of f-electron

local magnetic moments; for f-atom separation larger than

3.4 A, f-electrons have local moments, and consequent

magnetic ordering occurs at low temperatures due to absence

of direct f-f overlap unless strong f-hybridization with

s-, p-, or d-electrons prevents it.

The f-electrons in heavy fermion systems show the

transition from localized to itinerant behavior as

temperature is lowered. At low temperatures the heavy

fermion state evolves from the local moments seen at high

temperatures. Some of the f-electrons of these compounds

become itinerant and f-electron moments become strongly

coupled to the conduction electrons at low temperatures,

whereas at high temperatures the f-electrons become

decoupled from conduction electrons forming a weakly

interacting collection of f-electron moments and show some

kind of Curie-Weiss type behavior of the magnetic

susceptibility x. This Curie-Weiss behavior of the magnetic

susceptibilities above 100 K in heavy fermion systems is

indicative of the interaction of local magnetic moments

arising from the partially filled f-shells.

Those compounds with 7 larger than an arbitrary value

of 400 mJ f-mole-1 K-2 following Stewart [2] are classified

here (Table I-1) as heavy fermion systems (it makes sense

to normalize 7 values per f-atom-mole (f-mole) because

these large 7 are due to f-electrons and per f-mole permits

an easy comparison of compounds). With the definition of 7

> 400 mJ f-mole-I K-2, at least nine heavy fermion systems

can be categorized in three groups depending on what kind

of low temperature ground state ordering they have. At low

temperatures these systems either become superconductors

(CeCu2Si2 [13], UBe3 [9], UPt [11]), antiferromagnets

(NpBe13 [14], U2Zn7 [15], UCd1 [16]), or remain normal

without ordering (CeAl3 [1], CeCu6 [17], UAuPt4 [18]). Some

properties of these heavy fermion systems in low

temperatures (Table I-1) and in high temperatures (Table

1-2) are summarized. Other compounds with large 7 but

smaller than 400 mJ f-mole-' K-2 (see Table I-3) are such

as CeRu Si [19, 20], YbCuAl [21], UCus [22, 23], NpSn

[24], NpIr2 [25], URu2Si2 [26, 27], USn [28], CeA 2 [29],
UAl2 [30], UPt5 [18, 31], TiBe2 [32]. Heavy fermion systems

and ordinary metals may likely be linked by these


Table I-3
Properties of some of the compounds whose 7 is large
but below 400 mJ/f-mole K2).

Compounds Ordering 7(0)a x(0)b ,erf Structure d (A)
temp. (K) (B)

CeRu2Si 380 20 2.38 tetragonal 4.2

YbCuAl 260 25.5 4.35 hexagonal 3.65

UCu T = 15 250 6.7 3.52 cubic 4.97
5 N
NpSn3 T = 9.5 242 15 cubic 4.63

NpIr2 T = 6.6 234 15.3 2.4 cubic 3.25

USn3 169 10 2.5 cubic 4.63

UAl2 spin flue. 142 4.3 3.1 cubic 3.38
CeAl T = 3.9 135 32 2.54 cubic 3.48
2 N
UPt5 85 5.7 4.3 cubic 5.25

URu2Si T = 17.0 65.5 1.5 3.51 tetragonal

T = 1.5

TiBe2 spin fluc. 56 8 1.64 cubic

a. extrapolated to T = 0 K from the data above transition
in unit of (mJ/f-mole/K2).
b. in (memu/f-mole).
For references, see text.

The exciting discovery of superconductivity [9, 11,

13] in high effective mass and highly correlated electrons

at low temperatures has been made in some of heavy fermion

materials despite nearly localized electrons, which are

expected not to be favorable to conventional singlet

superconductivity. In those compounds, superconductivity

and magnetism interact within the same f-electrons [33]

rather than having different sets of electrons which are

trying to superconduct and to become magnetic as can be

seen in some magnetic superconductors. It remains of

considerable interest to investigate the interplay between

magnetism and superconductivity in heavy fermion systems.

Heavy fermion superconductors are attracting

considerable interest because of their unconventional

behaviors which might be consistent with anisotropic

pairing. It has been argued 134, 35, 36, 37] that various

features of the superconductivity of the heavy fermion

superconductors seem to be unconventional with the

possibility of non-s-wave pairing of the superconducting

electrons rather than the usual isotropic s-wave type usual

in the BCS theory [38] of superconductivity. The physical

mechanism and the configuration of the superconducting

state of these heavy fermion superconductors are different

from those of the conventional phonon-mediated type of

superconductors. Instead of an interaction between

electrons resulting from the exchange of phonons that leads

to superconductivity in ordinary metals with an essentially

isotropic gap in the spectrum of electronic excitations, it

has been claimed that a Coulomb interaction between heavy

electrons resulting from the exchange of local moment

fluctuations is responsible for unusual superconductivity

in heavy fermion superconductors [39].

For the fundamental understanding of these compounds,

it is quite important to understand the origin of the large

enhancement of the effective mass, the nature of the heavy

fermion normal state, the occurrence of heavy fermion

superconductivity, and anomalous properties of the

superconducting state. Often the ground state properties of

heavy fermion superconductors are very sensitive to doping

with small quantities of impurity atoms. So it is a

question of great interest to understand how the normal

state as well as the superconducting state of heavy fermion

superconductors is affected by substitution of dopants by

pointing out consistencies and correlations that might

exist in the analysis of data. In this sense, doping

investigation of these heavy fermion superconductors has

its own interest and can provide information on the nature

of the heavy fermion ground state and the source of the

superconductivity of the undoped parent compound.

In chapter II, I briefly discuss some theoretical

parts that are needed most for the understanding of

experimental results. In Chapter III, I describe some

experimental setups and procedures such as cryogenics,

resistivity, ac- and dc-susceptibility, and specific heat

measurements. I present and analyze our experimental

results involving the thermal and magnetic properties of

the normal state and the superconducting state in Chapters

IV and V, respectively, and discuss how those states can be

influenced by changes in the chemical composition of the

heavy fermion superconductors. Alloying experimental

results are summarized with concluding remarks in Chapter



The unusual properties of heavy fermion systems have

attracted great deal of attention among solid state

theorists interested in developing microscopic theoretical

concepts for normal state properties and for electron

pairing and in understanding an interaction mechanism

responsible for superconductivity.

Kondo Effect

Anderson [40] invented a model for an single impurity

that had the capability of developing a magnetic moment in

the sea of conduction electrons of a conventional metal.

The Hamiltonian for the single impurity Anderson model is

written as

H = e(k)ck'ck + EC n + n n ,
Ska k2

+ ( + Vkr Ckcf) (II-1)

The operators ck (Cko) create (annihilate) conduction

electrons with energy dispersion c(k) in momentum and spin



states k, o. The operators for the f electrons capable of

multiple occupancy at the impurity site are f (f ), and

the number operators n = ftf with a single-orbital
the number operators n(T c
energy ef and the Coulomb correlation energy between two f

electrons on the same site is U. For example, for the case

of one f electron at the impurity site, U 4 0. A

hybridization matrix element Vfk couples the orbital level

with the conduction states. With a more simplified

hybridization term, the ground state and the thermodynamic

properties, such as the enhanced specific heat and magnetic

susceptibility of the Hamiltonian (II-1), were successfully

calculated in exact results using the Bethe ansatz method

[41, 42, 43 and for the review 44].

It turned out that such an impurity can lose its

magnetic moment because the conduction band of the metal

could screen out the impurity moment at low temperatures

[45, 46] and form a singlet ground state. This formation of

a singlet ground state occurs by a cancellation of the

impurity spin through the spins of the conduction electrons

as described in the Kondo Hamiltonian [47]. The Kondo

Hamiltonian has the form

H = -2JS-s(0) (II-2)

describing the exchange interaction, with coupling constant

J, of a single magnetic impurity of spin IS) = 1/2 with the


conduction electron having spin density s(0) localized

about the impurity. The magnetic screening mechanism causes

a narrow scattering resonance in the one-particle spectrum

near E A convenient tool for exploiting the consequences

of the screening effect is given by Friedel's sum rule


With the Anderson Hamiltonian (II-1), the

hybridization width F of the f level is given by the golden

rule [3]

r = n N(E )IVfk V 2 (II-3)

where N(EF) is the conduction electron density of states at

the Fermi level for the single spin. The energy gain due to

the formation of a singlet state characterizes the Kondo

effect. This energy gain expressed with the Kondo

temperature, T is usually written as

kT = D e f (II-4)

where D is an effective conduction-electron bandwidth and

v is the degeneracy of the f orbital. Equation (11-4)

reminds one of the energy gain in a superconductor due to

the formation of Cooper pairs. Some of the features of

heavy fermion systems can be understood by the single Kondo

impurity problem in an enhanced form because those systems

have at least one magnetic ion per unit cell. Transport


properties reveal the growing importance of coherence in

the low-temperature state of heavy fermion systems with an

increasing concentration of f-ions. But this single

impurity model fails to develop coherence effects at low

temperature so that the resistivity due to the single

impurity saturates at low temperature.

The original Kondo problem consists in treating the

effect of a simple metal containing a dilute concentration

of magnetic impurities. The physical properties of these

systems are described in the framework of the Kondo model

[47]. It can be shown that the Kondo Hamiltonian can be

derived from the more general Anderson Hamiltonian [39]. At

high temperatures (T >> T) the impurity displays local

magnetic behavior, whereas at low temperatures (T << T )

the spin of the impurity is compensated by a conduction

electron cloud. A consequence of this transition is a

narrow peak of the electronic density of states close to

the Fermi level E Many properties of this model can be

understood as a consequence of the formation of this narrow

resonance in the one-particle density of state near E ,

usually called the Abrikosov-Suhl resonance [49, 50].

The exact analytic solutions for the single-site

impurity case have been obtained [51]. The solution is

characterized by a non-degenerate ground state in which the

entropy associated with the partially filled f-states

vanishes linearly in temperature so that the excess


specific heat (associated with the impurity) is yT. The

specific heat, linear in temperature at low temperatures,

reaches a maximum at temperature T beyond which it falls

off with increasing temperature. The magnetic

susceptibility is independent of temperature and has a

higher value than its free electron value. A logarithmic

increase of resistivity with decreasing temperature varies

as ln(T /T) and then saturates as a consequence of the

scattering of the conduction electrons by the compensated

impurity spins.

The singlet formation can be extended to the two

impurity Kondo problem. The effective interaction between

two magnetic impurities in a metal mediated by the

conduction electrons is the long-range RKKY interaction.

The two impurity Kondo problem has been calculated [52]. If

the RKKY interaction is strong and antiferromagnetic, the

two spins form a singlet and there is no Kondo effect at

all. For a large ferromagnetic interaction, the moments

first align and then are compensated in a two stage process

characterized by two Kondo temperatures.

At first sight one might hope to explain the

properties of heavy fermion systems by regarding them as a

collection of independent compensated spins placed on a

lattice. Some authors [53] have suggested simply applying

what is known about a single-site impurity problem by

considering a set of single-site impurities on the lattice.


Thus, the phrase "Kondo lattice systems" has sometimes been

used for the heavy fermion systems. There are, however,

objections to describing these systems as a simple

collection of single-site Kondo impurities. This picture

cannot be true in detail, since in this picture one would

expect all heavy fermion systems to exhibit maxima in the

resistivity, a prediction in conflict with experiments.

This maximum would come about because scattering from a

magnetic site is partly elastic and partly inelastic [39].

When the sites are in a periodic array, only the inelastic

scattering leads to real scattering processes, whereas the

elastic scattering creates band structure in the electron


In the heavy fermion systems, the f-state atoms sit on

the periodic lattice sites and the problem is no longer

exactly soluble. The lattice problem differs from the

single-site problem in that a coherent state can be formed.

The coherence of the electronic state causes the

resistivity to vanish, thus explaining the (relatively) low

residual resistivity. The energy scale on which this

coherence occurs would be the single-site Kondo energy,

which is determined by the weak hybridization process and

is therefore small. Because all the entropy associated with

the degeneracies of the f-electron configurations is on the

scale of the small Kondo energy, the coefficient 1 is much

larger than usual. The magnetic susceptibility is


correspondingly large because the opposition to the

development of a magnetic moment in the ground state has,

as its characteristic energy, the small Kondo energy.

Fermi Liquid Theory

In 1956, Landau [54] presented a phenomenological

theory for the macroscopic low temperature properties of an

interacting normal fermion system. Nozibres [55] has

constructed a Fermi liquid model for the behavior of the

conduction electrons around an impurity, and has shown how

the low temperature behavior of the specific heat,

resistivity, and magnetic susceptibility can be expressed

in terms of a few Fermi liquid parameters.

Landau's approach [54] to Fermi liquids is based on

the assumption that the excited states of an interacting

electron system whose elementary excitations are

quasiparticles are classified as a Fermi gas. From the

above assumption the distribution function of the

quasiparticles is a Fermi function but with renormalized

parameters such as the effective mass. The energy change of

the system due to elementary excitation is given by

6E[n (k)] = x c(k)8n (k)
+ I f,(k,k')n (k)an ,(k') (11-5)
2 kk'oo'

because quasiparticles interact with each other and their

energy depends on the deviation of the quasiparticle

distribution function Sn q(k). The matrix f ,(k,k')

describes the quasiparticle interaction. The Landau theory

was derived originally for the description of an isotropic

quantum Fermi liquid such as 3He. Recently this approach

has been used for the strongly anisotropic heavy fermion

systems. The main thing is to determine the f ,(k,k') by

relating them to measurable quantities. In a isotropic

system, the f ,(k,k') can be replaced by a 2 x 2

interaction matrix because f ,(k,k') depend only on the

angle e between k and k' and ra'.

f(8) = fs"() + .7r'fa(9). (11-6)

The coefficients fs(e) and fa(9) can be expanded in terms

of Legendre polynomials

f'(a ) = I=F 1 P(cose) (11-7)
2N (0) 1-o

where F; and F; are the Landau parameters and N*(0) is the

quasiparticle density of states. These Landau parameters

characterize the interactions between the quasiparticles

and enter into measurable quantities to be determined from

them. Examples are: (1) the spin susceptibility of the


2 N (0)
6 ff1 + Fa
= 1+ F a (11-8)

where eff is their effective magnetic moment. The magnetic

susceptibility, which is enhanced by both the effective

mass and a large Fermi liquid interaction parameter Fa, is

m /m
X = X0 (11-9)
o 1+ F

where X is the magnetic susceptibility of a free electron


(2) the electronic specific heat with a T3lnT term [56]

C(T) = yT + ST3 + ST3 n(T/T*) (II-10)

where the enhanced linear coefficient, v, is

3h 2
3 2 (II-11)


3n2 s
10 T2


1 7
B" [(A2 (1 +A -A
2 12
A=a, s

+ (A ( ( A) 2AAA]k (11-13)

Here w = 1 and a = 3 are the respective degeneracies of
S a
the symmetric and antisymmetric states. T is the spin

fluctuation temperature at which the T3lnT and

T3-contribution to the specific heat from spin fluctuations

are equal and opposite. T the Fermi temperature of the

interacting system, is given by

T F (11-14)
2mK .

The Landau scattering amplitudes, A,, are related to the

Landau parameters, F,, by

A^ k 1 (1 = 0, 1, 2.....) (11-15)
S1 + F /(21+1)

The T3lnT-term in Equation (11-10) accounts for the spin

fluctuations. This approach of a homogeneous quasiparticle

system has been used for a description of heavy fermion

systems, in particular UPt [57]. However heavy fermion

systems are strongly anisotropic systems and some


modifications and changes have to be made in dealing with

inhomogeneous systems of quasiparticles.

Low energy excitations of heavy fermion systems can be

described in terms of Landau quasiparticles by assuming

that the Fermi liquid picture holds for heavy fermion

systems. In heavy fermion systems the sharper scattering

resonances near the Fermi level lead, within the framework

of Landau's Fermi liquid theory, to strongly renormalized

electronic quasiparticles with very heavy masses at low

temperatures. Only at temperatures for T << Toh (Tco is

defined as the temperature below which the electronic

specific heat is linear and the electrical resistivity

falls off sharply with decreasing temperature) does one

expect to observe the Landau temperature dependence. Such

Landau limiting behavior is observed for UPt3 at

temperatures below 1.5 K [58], but UBe13 at zero pressure

becomes a superconductor well before it reaches a

temperature at which Landau theory would apply [59]. In

this Fermi liquid theory [60, 61], 7 is proportional to the

quasiparticle density of states or to the effective mass

(Equation II-11). The fictitious energy EF = hk /2m* where
k is determined by the conduction-electron concentration

and the lattice constant alone is about EF = 500 K for 7 z

1000 mJ mole- K-2 [62]. These numbers implies that the

effective mass m is of the order 300 m where m is the
e e
free-electron mass.


Heavy Fermion Superconductivity

The discovery of superconductivity in the heavy

fermion compound CeCu Si2 in 1979 [13] was the starting

point of the rapid experimental and theoretical development

of heavy fermion physics. Since then questions such as

which interaction is responsible for the occurrence of

superconductivity and what the nature of the paired state

is in heavy fermion systems have been discussed in

considerable detail [3, 36, 39]. For heavy fermion

superconductivity, present theoretical work has mostly been

based on the close similarity with the phenomenology of

superfluid 3He and the Fermi liquid theory. This work has

raised the important question of whether an

electron-electron mechanism could be operative in heavy

fermion superconductivity and also speculation about the

possibility of unusual superconducting states, notably

L 0 and triplet pairing.

The fact that the discontinuity in the specific heat

at T has the same order of magnitude compared with the
enhanced specific heat in the normal state indicates

unambiguously that the superconducting pairs are formed by

heavy quasiparticles. These heavy quasiparticles show

itinerant behavior and no pairbreaking despite the

expectation of magnetic pairbreaking usually associated

with the f-electrons. The origin of an attractive


interaction among the quasiparticles, which is necessary

for superconductivity to occur, is one of the important

questions in this field.

The BCS theory of superconductivity describes

successfully superconductivity in ordinary metals which is

due to the electron-phonon interaction yielding a singlet

s-wave ground state. In general, however, the pair wave

function can be either a singlet (L = even) or triplet (L =

odd) function, if we have rotational symmetry in spin space

the singlet spin wave function is antisymmetric under the

exchange of the spins whereas the triplet wave function is

symmetric. The s-wave state refers to an even parity state

which is an isotropic state observed in ordinary

superconductors whereas a p-wave state refers to a state of

anisotropic pairing. The attractive interaction between

electrons that is mediated by phonons leads to an

essentially isotropic gap in the electron spectrum at the

Fermi surface so that the electronic properties fall off

exponentially with decreasing temperature.

It was suggested [34] that superconductivity in heavy

fermion materials was unlikely to be of the conventional

phonon-mediated s-wave attraction mechanism (the definition

of conventional and unconventional superconductors refers

to the symmetry properties of the order parameter. The

order parameter of a conventional superconductor will have

the same point symmetry as the underlying crystal whereas


an unconventional superconductor will have additional

broken symmetry in its order parameter). In liquid He the

superfluid state with strong spin fluctuations is known to

be an anisotropic p-state. It has been shown theoretically

that the interactions mediated through spin fluctuations

favor p-state superfluidity [53]. But direct application of

this work to the heavy fermion superconductors can be

misleading due to the importance of strong spin orbit

coupling and band structure effects [3].

Heavy fermion superconductors are to be considered

possibly possessed of unconventional pairing from the

experimental findings of the power-law dependence of the

low temperature specific heat, ultrasonic attenuation,

thermal conductivity, and NMR relaxation rate in those

systems. In superconductors with anisotropic gaps varying

in both magnitude and phase with zeroes of the gap at

points or on lines on the Fermi surface, a number of

excitations in the low-temperature state exhibit a

power-law instead of an exponential behavior. For example,

a superconducting state specific heat of the form of C (T)

T3 requires that a gap function vanishes at points of the

Fermi surface and C (T) T2 requires that a gap function

vanishes along lines on the Fermi surface. The existence

and positions of zeroes of the superconducting gap on the

Fermi surface is dependent on its symmetry group. The

observation of such power-law behavior is taken as evidence


for anisotropic superconductivity. This anisotropic

superconductivity is expected theoretically when the

mechanism responsible for the superconductivity is not the

usual electron-phonon interaction. It should be pointed out

that such pure power-law dependence are theoretically

predicted only for T < 0.1 T whereas experimentally they

are observed as high as 0.8 T .


The experimental procedure for low temperature

specific heat measurement of a small sample is described in

detail. Many other aspects of experimental work such as

sample preparation procedure, experimental techniques and

measurement processes are described here only briefly.

Sample Preparation

Sample preparation is a very important part of the

research. In order to prepare a sample, stoichiometric

amounts of elements are weighed and melted together under

highly purified argon in the Bihler arc melting apparatus.

The copper base plate has to be cleaned with acetone or

alcohol before melting to avoid impurities in the sample.

The vacuum chamber is cleaned several times automatically

by pumping and purging with highly purified argon to avoid

impurities from the air. A zirconium button, which is also

used for ignition of the arc, is melted as a further

cleaning of the argon gas before each melting process of


the actual sample to reduce certain residual gas

components, especially oxygen.

To reduce the weight loss of the sample after melting,

the vapor pressures of the elements have to be taken into

account. The element with the lowest melting point is

generally melted first. Through contact of the several

elements being reacted together and via liquid-solid

reaction, the elements with the higher vapor pressures are

kept as low in temperature as possible. Low arc intensity

is used for melting samples with high vapor pressure

constituents. Samples are melted several times, turning

them over after each melting to homogenize them. Some

samples are cast into a particular shape in a water-cooled

crucible by using a pressure casting technique inside an

arc furnace [63].

Usually elements with the highest possible purity are

used to investigate superconductors, in order to measure

intrinsic properties. And some elements are etched, if

needed, before melting. Starting materials, sample

qualities and annealing procedures to reduce strains and

improve the homogeneity for each compound will be explained

in related part. Master alloys are prepared to avoid the

handling of very small amounts of materials if one of the

constituents of a sample has a low concentration. Some

air-sensitive samples are kept in sealed pyrex tubing to

prevent contamination due to oxidization.

X-ray powder diffraction data taken from a Philips

diffractometer are used for a determination of the lattice

parameter from slow scan (1.2" per minute in 2e) and for a

quick inspection of the presence of second phase from fast

scan (6" per minute in 2e). A computer program is used to

index and compare the calculated angles and intensities

with the observed peaks in the x-ray pattern.

Experimental Procedures


Measurements of resistivity, ac susceptibility and

specific heat down to 1.1 K have been performed in a 4He

cryostat by pumping the dewar or only an inner 4He pot.

Even lower temperatures down to 0.3 K can be reached by

using a He cryostat.

Figure III-1 shows a simplified diagram of a He probe

inside the vacuum can. Basically this probe consists of two

inner pots for 3He and 4He, and a sample platform attached

tightly with a thermally conducting grease to the copper

block which is connected to the He pot by a brass thermal

link. The sample platform has to be in good thermal contact

with the copper block to minimize the temperature gradient

to the copper block. After precooling to 4.2 K, cooling

down of the 4He pot is obtained by pumping the liquid

-Pumping Line for
Vacuum Can and
Tube for Wires
i\Needle Valve
Copper Heat Sink

Capillary Connecting
He Pot and4He Bath
Heat Sinking Pins and
Copper Block

Heat Sinking Copper Block

Figure III-1 A schematic drawing of cross-section of
inside the high vacuum can with a He cooling system.

helium in 4He pot filled through the capillary that

connects between 4He pot and helium bath (the other He

cryostat that we have built for the zero-field measurements

does not have a 4He pot, so the dewar has to be pumped to

reach 1.1 K for the use of He cooling system). The

suitable impedance of the capillary with the consideration

of both cooling power and minimum temperature for the 4He

pot can be obtained with a variation of the diameter and

length of the capillary. By cooling down the 4He pot by

pumping, the He (boiling point at 760 mm Hg pressure is
3.2 K) is condensed into the 3He pot. During the

condensation of the He, the adsorption pump (a movable

small container of charcoal inside the He pumping line)

attached to the end of a rod is raised to keep it at a high

temperature (above 40 K). After complete condensation of

He gas, the adsorption pump is lowered slowly to cool down

to initiate the adsorption process of the 3He gas and

reduce the vapor pressure of the liquid in the He pot. For

the specific heat measurement below 1 K, the lowest

temperature can be maintained for several hours. The 3He

gas is kept in a closed circuit to prevent losses and to

collect He gas again in the He container after use. The

temperature of the copper block can be controlled by

pushing current through a manganin wire with a resistance

of about 300 0 (block heater) which is wound around the

brass thermal link.

The electrical connections from the top of the probe

are thermalized to the helium bath through the copper block

which projects into the helium bath and they are heat-sunk

again to another copper block below the He pot to minimize

the heat flow from the top which is at room temperature.

These connections are either #40 gauge copper or #40 gauge

manganin wires. Manganin wires transfer less heat from the

top of the probe but generate more heat for the same

current than copper wires because the resistivity of

manganin wires is about 30 times greater than that of

copper wires. Heat transfer from the top can be reduced by

using manganin wires for low current carrying wires. All

pairs of leads are twisted to reduce electrical noise.


Standard four probe dc technique is used for the

measurement of electrical resistivity from 0.3 K to room

temperature. Four 0.002" diameter platinum wires which are

voltage and current leads are attached to a sample by

silver epoxy. Too much current has to be avoided to prevent

heating of the sample because usually the contact

resistance is higher than the sample resistance. The sample

is mounted on a sapphire disk with General Electric varnish

7031 and then placed in good thermal contact to the copper

block with a thermally conducting grease. The resistance of

the sample is obtained by averaging both absolute values of

the resistance for each polarity of current. The electrical

resistivity has been measured also with a standard four

point ac method at a low frequency (87 Hz in our case). All

measurements are automatically performed with computer

controlled equipment and data acquisition.

Ac Susceptibility

A low frequency inductive method is used for the

superconducting temperature measurements. NbTi

superconducting wire (90/10 CuNi, 0.004" with insulation)

is wound 185 turns on a teflon spool for the primary coil

which is fed by a stable low frequency ac generator of a

lock-in amplifier. And copper wire (#44) is wound 2700

turns on both sides in opposite directions (total 2 x 2700

turns) of an astatic pair of secondary coils. After winding

coils, two components of Stycast 1266 are mixed and cured

on top of the coils to prevent unwinding. The sample is

mounted in one side of the secondary coils. The signal is

measured automatically by standard lock-in techniques and

by the interface with a computer with an analog-digital

converter for digitizing the signal. This assembly of coils

allows ac susceptibility measurements of small samples. The

superconducting temperature is determined by a midpoint of

the inductive signal deviation associated with the

superconducting transition.

Dc Susceptibility

Magnetic property experiments have been performed with

a SQUID from Quantum Design in the temperature range of

1.8 K 400 K in magnetic fields up to 5.5 Tesla. A

temperature dependence of susceptibility and M vs. H curve

can be measured by automatic operation controlled by a HP

personal computer.

Specific Heat

The classical definition of specific heat is the

quantity of heat which is needed to raise a unit mass of

sample by a unit degree of temperature while keeping the

property x constant during the rise of temperature.

Cx= lim -- (III-l)
SdT40 dT x.

This specific heat has usually been measured by

adiabatic methods where a small temperature rise (dT) is

caused by a input of heat (dQ) which is a pulse of power

applied to the sample at a given temperature [64]. These

traditional techniques of specific heat measurement require

a thermal isolation of the sample from its surroundings and

a large amount of sample to minimize the effect of

imperfect thermal insulation and heat leak by making

relaxation time t1 very long (the time to reach thermal

equilibrium increases). These traditional techniques have'

some disadvantages in the necessity of a large size sample

and thermal isolation of the sample from its surroundings.

With the development of signal-averaging technique in

lock-in amplifier to extract the signal from noise,

ac-temperature calorimetry was developed in 1968 [65]. This

ac-temperature calorimetry detects the periodic temperature

change in thermal equilibrium with a sensitive lock-in

amplifier by heating the sample periodically.

As shown in Figure 111-2, the sample is coupled to a

heat bath, thermometer, and heater by thermal conductances

K, Ke, and Kh, respectively. When ac current is applied

sinusoidally in time at a frequency of to the platform

heater to generate heat at a rate P0cos2(wt/2), the

temperature of the thermometer (T ) (with a assumption of

infinite thermal conductivities for a thermometer, heater,

and sample) is expressed by

P P f(w)
T8 = Tb + + cos (wt a) (III-2)
2K 2wC

where T, K, and C are temperatures, thermal conductances,

and specific heat for thermometer (e), heater (h), sample

( c,, Th )


Sample Thermometer
(C,, T, ) ( C,, T)


Heat Bath ( Tb= Const )

Figure 111-2 Diagram of thermal coupling of a
sample in ac-temperature calorimetry. The sample is coupled
to a heat bath, thermometer, and heater by thermal
conductances Kb Ke, and K respectively.

(s), and bath (b), respectively. When the relaxation times

re, h and T (they are defined as Te = Ce/Ke, rh = Ch/K ,

and T = C/K with C = Ce + Ch + C ) satisfy the following

conditions of

W2(rT + T) << 1 and t >> 1, (111-3)

f(w) and a are given by

1 -1/2
Sf( ( + TT) ( -4)

1 2 -1/2
sina = 1 + ( u(T + Th) ) (III-5)

In Equation (III-2), Te is derived with a assumption

of infinite thermal conductivity of the sample. For an

actual sample with a finite thermal conductivity, the ac

component of T. is given by

P 1 2K -1/2
T 1 + --- + C?(.2 2 2 b+
c 20C L 22 9 h int 3K
s s


where rint is the response time of the sample. In Equation

(III-6), T = T is the sample-to-bath relaxation time and
2 2 2 1/2
(T + T2 + )1/2 = T is the relaxation time of the
t h int 2

sample with its heater and thermometer to come to thermal

equilibrium. With a condition in Equation (111-3) which is

corrected to include t in the relaxation times to be
2 << 1/ << 1 and if the sample thermal conductivity K

can be kept much larger the sample-to-bath thermal

conductivity Kb then Equation (III-6) can be simplified as

C -- (III-7)

Then the total specific heat C = (C- + Ch + C ) can be

measured by the ac component of temperature of thermometer.

In this method long relaxation times and thermal

isolation of the sample are no longer necessary and the

sample can be quite small. This method usually provides

only a relative measurement of specific heat to detect very

small changes in specific heat and a continuous readout of

specific heat is possible. Because of the disadvantage in

absolute measurement, ac calorimetry has mostly been used

for detecting an anomaly in the specific heat around the

critical temperature.

As an another technique for small sample calorimetry,

thermal relaxation (time constant) method has been

developed since 1972 [66] and has been improved in

measurement techniques [67, 68]. In this thermal relaxation

method, as can be seen in Figure 111-3, one dimensional

model of heat flow is solved within the system consisting

Ge Chip
S*Silver Epoxy
*1/3 of Au-Cu
C Tol Wires
To +AT C ample+C dd&nda *Thermal Grease

K- Four Au-Cu
AT Wires

Copper Ring
To Heat Reservoir

Figure III-3 Schematic diagram of heat flow in the system
used for thermal relaxation method. The sample and the
addenda (sapphire, Ge chip platform thermometer, H31LV
silver epoxy, 1/3 of linking wires, and thermal grease) are
heat-linked to copper ring heat reservoir with Au-Cu
alloyed wires as can be seen in details in Figure III-4.

of sample, addenda, heat linking wires, and block heat

reservoir. When the temperature of the system is stabilized

with a block temperature T power P flows through the

platform heater to raise the temperature of sample and

platform to T0 + AT. If the power is turned off, the

temperature of sample and platform decays exponentially

with a time constant tI (= C/K) through the wires (K is the

thermal conductance of wires)

In the thermal relaxation method, by solving one

dimensional heat flow equation with assumptions of much

larger conductance of sample and platform than that of

wires and small enough AT (AT/T < 1 %) with a stabilized

T the following relations are obtained.

-t/T C
AT(t) = (T To) e where r = (III-8)

In Equation (III-8), the measurement of thermal conductance

K (from P/AT where AT is the temperature rise due to power

P) and r determines C from which the specific heat of

sample can be obtained by subtracting the addenda

correction. In the specific heat measurement using this

method, absolute accuracy is approximately 5 %.

A small addenda contribution is required for the small

sample calorimetry using thermal relaxation method. Figure

III-4 shows the platform used in our laboratory for the

small sample specific heat measurement technique. It

Solder for Au
(In 44, Sn 42, Cd 1

K Copper Ring
Thermal Epoxy
--Silver Pad

Au-Cu Wires

Ge Chip

Figure III-4 (a) The sample platform used for small
sample specific heat measurement consists of a supporting
copper ring and sample holder linked to the copper ring
thermally by four wires.
(b) Bottom side of the sample holder. A Ge
chip thermometer and evaporated heater are mounted and four
wires are electrically connected with EPO-TEK H31LV epoxy.





consists of a supporting copper ring, which is attached

tightly to the grease-wetted (thermally conducting grease)

post of the copper block, and a sapphire disk which is

thermally linked to the copper ring with four wires. The

sapphire disk (typically 3/8" diameter and 0.003" (3 mil)

thickness) is one of the best sample holders. It has high

thermal conductivity (about 1 Watt cm-1 K-1 at T = 4 K) and

low specific heat ( 2 gJ gram1 K-1 at T = 2 K and eD

1035 K [69]) and it is electrically insulating so it is

easy to mount thermometer and to evaporate heater.

The Ge chip for platform thermometer attached to the

sapphire disk with an silver epoxy is connected by two

Au-Cu wires to electrical contacts with Au solder (In 44,

Sn 42, Cd 14 in at. %) heat-sunk on silver pad attached

with a thermally-conducting (electrically-nonconducting)

epoxy adhesive (such as Thermalbond from Thermalloy Inc.)

to the copper ring. As an alternative platform thermometer

to reduce the addenda, an evaporated Au Ge strip is
0.22 0.78
used for ac calorimetry in our laboratory.

The heater is an evaporated 7 at. % Ti Cr strip with

a resistance of about 300 a. It provides very little

variation of resistance with temperature with negligible

addenda and good thermal link to the sapphire disk. It is

connected by two Au-Cu wires in the same way as the Ge chip

thermometer electrically and thermally.

The platform thermometer and platform heater are

connected by four Au-Cu alloyed wires which serve as

electrical contacts and thermal link to the heat reservoir

(copper ring) and mechanical support for the sample holder.

The thermal conductance of the wire depends on the

diameter, and alloy type of the lead wires. Wires with

smaller diameter than 0.001" have small addenda but they

are fragile. Wires larger diameter than 0.003" are durable

and have larger thermal conductance (thermal conductance is

proportional to the cross-sectional area) but addenda

contribution is large (1/3 of the mass of wires contributes

to the addenda [66]). 7 at. % Cu-Au wire has thermal

conductance approximately 1/100 of that of Pure Au wire

[66] and 1/7 of that of 1 at. % Cu-Au wire. In our case, we

used 7 at. % Cu-Au wires and the thermal conductance of

four 0.003" diameter and 1/4" long 7 at. % Cu-Au lead wires

is typically 1 gWatt/K at T = 0.3 K and 8 UWatt/K at T =

2 K. By using these 7 at. % Cu-Au wires, the thermal

conductance can be corrected without measuring again when

we measure specific heat in magnetic field (linear decrease

in thermal conductance with 3 % correction in 12 Tesla

[70]). The 7 at. % Cu-Au wire has a low eD (about 165 K) so

the addenda contribution increases rapidly with increasing

temperature. The conductance of these wires can be

controlled by choosing the suitable diameter and alloy type

of wire depending on the temperature range of experiment

and what kind of samples are usually measured. Working with

too small thermal conductance requires excessive time to

reach thermal equilibrium between the sample holder and the

copper ring when the temperature is changed.

The sample is mounted with a thermally conductive

grease (such as thermal compound from Wakefield [66]) on

the smooth side of sapphire disk. It is necessary to press

and rub carefully, with supporting the sapphire disk from

the bottom to prevent breaking the lead wires, to remove

trapped air bubbles and to spread the grease evenly between

sample and sapphire disk. Thin samples with flat surfaces

improve thermal contact and minimize the amount of grease

used. If the thermal contact between sample and sapphire

disk is bad or sample itself has poor conductivity, the

temperature decay is off from exponential shape. This r
effect due to large thermal resistance or poor conductivity

of sample can be corrected as discussed by Bachmann et al

[66]. With a r2 effect the temperature is approximated as

T(t) = TO + Ae / + Be 2 (III-9)

The heat capacity of the sample can be extracted by fitting

the relaxation temperature to the sum of two exponentials.

The sapphire disk is cleaned with trichloroethane to

remove grease for every measurement. After mounting the

sample on the sapphire disk the vacuum can is mounted on

the flange with a brass taper joint seal. The taper joint

surface is cleaned with trichloroethane and sealed with a

high vacuum grease (such as high vacuum grease from Dow

Corning) by a brass can before evacuating the vacuum can.

The joint is 4He superfluid-tight.

The Ge platform thermometer is calibrated with the

measurement of temperature and corresponding thermometer

resistance in an ac Wheatstone bridge [71]. These R vs. T

data are fitted with the polynomial of

1 n
= I Ai(lnR)I (III-10)

with n = 4 in our calibration.

The thermal conductance of four wires is measured from

P/AT where P is the power driven to the platform heater to

change the temperature by AT. We measure conductance as a

function of temperature by measuring the current and

voltage across the platform heater to know P and by

measuring the platform thermometer resistance before and

after the power is on (P/AT is the conductance at T =

To+ AT/2). From these two resistance values, AT is

interpolated using the platform thermometer calibration

previously made. Then this temperature dependent

conductance is fitted to a power series to get conductance

calibration such as

K n
= I AT'. (III-11)

This calibration is used by the computer program for

finding interpolated conductance at temperature T. After

measuring thermal conductivity of the wires of platform,

specific heat of standard samples such as Au [72] and Cu

[73] is measured to check our thermal conductivity fit.

For the specific heat measurement we use this thermal

relaxation method because this method is the ideal

technique of small sample calorimetry for our specific heat

measurements of heavy fermion systems. The sample can be as

small as 1 mg because the low temperature specific heat of

heavy fermion systems is very large (relative ratio of

addenda contribution to the total specific heat is much

smaller than for low 7 samples) and the investigation is

usually focused in the low temperature properties (C of

addenda decreases as temperature is lowered). For example,

the addenda contribution to the total C is less than 0.2 %

at 0.3 K and about 3 % at 2 K in the specific heat

measurement of UBe13 due to large 7 of UBe13 and small C of

addenda at low temperature. Addenda contribution to the

total specific heat of standard gold leads to about 10 %

correction at 2 K. For our sample platform, the sources of

addenda contribution at T = 2 K are 0.12 uJ/K for 0.63 mg

of H31LV silver epoxy, 0.044 UJ/K for 1.5 mg of gold,

0.16 iJ/K for 0.12 mg of grease, 0.012 gJ/K for 14.75 mg of

sapphire disk, and 0.018 UJ/K for 3.8 mg of Ge thermometer.

The experimental setup for the measurement of time

constant in our laboratory using thermal relaxation method

is simplified in Figure III-5. The temperature of the

copper block is controlled by the block heater and measured

by the block thermometer at some stabilized temperature T .

Current is then pushed through the platform heater to

dissipate power P raising the temperature of the

platform-sample assembly to To + AT. In this method AT has

to be sufficiently small so that the time constant does not

change appreciably between T0 and To + AT (AT/T < 1 %).

The platform thermometer must have adequate sensitivity to

detect this small AT. The platform thermometer resistance

is measured with an ac Wheatstone bridge whose output is

used for AT calculation of the platform temperature from

the thermometer calibration made previously. This platform

thermometer is one component of Wheatstone bridge which is

driven by lock-in amplifier as a source of ac excitation

current and detected by same lock-in amplifier as a null

detector. An increase in ac bridge excitation current would

lead to a bigger signal, but this is limited by self

heating of the thermometer on the platform. The other

reason to use lock-in amplifier is to increase the

signal-to-noise ratio by filtering out noise at other than

the measurement frequency ( 2700 Hz). This self heating

Wheatstone Platform Platform Block Block
Bridge Thermometer Heater Heater Thermometer

Figure III-5 Block diagram of the experimental setup for
the specific heat measurements of small samples. HP
computer is interfaced with current sources, voltmeter, and
lock-in amplifier through analog-digital converter. The
rest parts excluding experimental equipment are divided
into three parts, block, platform, and Wheatstone bridge.

leads to an error in T. With a help of analog-digital

converter to digitize the signal from the ac bridge, the

time constant is measured with the output from the lock-in

amplifier and repeatedly signal averaged. The signal

integration time on the lock-in amplifier has to be set

less than t /40 to prevent rounding of thermal relaxation

time and for a better z correction.

HP 9000/300 series computer is used to control

experiment through the interface, interpolate temperature

from the thermometer resistance using calibration made

previously, manipulation of T data by taking logarithm, and

fit the data to calculate the time constant. In doing the

least-squares fit, 4000 points from the signal averaged

output of lock-in amplifier for the exponential decay are

analyzed to determine time constant value. In this

procedure, stability of helium bath and elimination of

electrical noise pick up are required to reduce the error

from the fluctuation in the base line. The addenda

contribution to the total specific heat is calculated using

polynomial fits of each element of addenda to obtain the

specific heat of sample.

In order to investigate the response of magnetic

materials to the external magnetic field and to suppress

the superconducting state to get 7 at lower temperatures in

the superconducting materials, the specific heat

measurements are often performed in a magnetic field. In

our laboratory, the superconducting magnet in the helium

bath (at T = 4.2 K) generates magnetic fields up to

14 Tesla or up to 16 Tesla with the pumping of helium bath

to 2.2 K. For the block temperature measurement, Speer

resistance thermometer is calibrated in zero field with

block Ge thermometer and then used with a field correction

according to Naughton et al. [74]. The Ge chip platform

thermometer shows a large magnetoresistance [p(16

Tesla)/p(0) is about 4.42 at 4.2 K] and noise problem in

magnetic fields.


In heavy fermion systems, the low-temperature

electronic specific heat coefficient 7 is enormous implying

that the effective mass of conduction electron is large. In

Fermi liquid theory as can be seen in Equation (II-11), 7

is proportional to the effective mass m [54]. Within the

Fermi liquid model with an assumption of spherical Fermi

surface, the calculation of an effective mass of the

conduction electrons in UBe leads to m values that
covers a wide range of 192m [9], 260m [75], and 296m

[76], respectively.

The question about the source of this large effective

mass of the conduction electrons in heavy fermion systems

still remains unsolved. One curious feature here has been

the independence of the heavy effective mass of the

magnetic ion concentration, as if the heavy fermion mass

were a single-ion effect. Single-ion effects in the

thermodynamic properties have been investigated in some of

cerium compounds. For example, CePb which is magnetically

ordered at 1.1 K, has been studied down to Ce .La .Pb ,
0.1 0.9 3
with C per Ce-mole and x per Ce-mole above 1.4 K exhibiting


total independence of Ce concentration [77]. Also,

Ce La Cu with varying La concentrations (x = 0, 0.2,
1-x x 6
0.5) has been studied, with the result that the specific

heat data per Ce-mole at low temperatures are independent

of x [78]. Also susceptibility values per Ce-mole show very

little deviations up to x = 0.9 [79, 80]. In the following

I describe further results of the same nature.

I have investigated the source of heavy fermion ground

state in a uranium heavy fermion compound by diluting it

with nonmagnetic dopants. I have studied single-ion effects

in specific heat and magnetic susceptibility in U MBe .

It is of considerable interest whether the single-ion

effects can be observed also in a uranium heavy fermion

system. As the source of large effective mass, the

single-ion effects are reminiscent of Kondo effect

associated with magnetic impurities in metals.

As a second attempt to understand the heavy fermion

ground state, I have investigated the magnetic behaviors of

Ce iTh Cu Si2 and U M Pt (M = Ti, Zr, Hf).
1-x x 2 2 1-x x 3
Antiferromagnetic order in Ce iTh Cu Si from x a 0.08 and
l-x x 2 2
coexistence with superconductivity up to at least x = 0.25

have been discovered recently [81]. With further doping by

thorium to the more cerium dilute end, magnetism increases

consistently up to x = 0.85 with huge low temperature X

values and saturation in M vs. H at 2 Tesla and exists up

to very dilute limits of cerium at least x = 0.975. The

development of the magnetic behavior of dilute Ce in

ThCu Si is of significant interest. Understanding the

magnetic correlations of dilute Ce in a nonmagnetic host

lattice such as ThCu Si is one starting point for theories
2 2
of the formation of the heavy fermion ground state in

CeCu Si It is interesting to note that Ce is magnetic

2 2
Y, LuCu2Si2) [82].

I have found that low-temperature magnetic

susceptibilities per U-mole of U -Ti Pt samples decrease

as the concentration of Ti is increased. This magnetic

behavior is of significant interest due to the fact that

UPt3 shows a remarkable closeness to magnetism. So far in

the doping experiments in UPt3 either on the uranium site

or on the platinum site, the magnetic order is induced by

substitutions of Pd [83, 84], Au [85], and Th [83, 85, 86]

at small concentrations.

Single-ion Effects in the Formation of the

Heavy Fermion Ground State in UBe3

I have doped nine different nonmagnetic atoms on the

uranium site in UBe with various concentrations in order
to investigate the important question about the source of

the high effective mass in a uranium heavy fermion system.

We present x-ray diffraction, magnetic susceptibility, and

specific heat data of U M Be1 (M = Hf, Zr, Sc, Lu, Y,

Pr, Ce, Th, and La; 0 s x s 0.995).


I chose UBe13 as a host system to do this experiment

with following reasons:

(1) UBe13 shows all the interesting properties of heavy

fermion systems as described in chapter I.

(2) UBe13 allows a full range of solubility of different M

atoms to investigate the effect of the M atoms, both due to

their electronic configurations and their sizes.

(3) The other heavy fermion superconductors are not

suitable. Doped UPt3 undergoes the occurrence of magnetism

at low temperatures, e.g., U 9Th O Pt shows
0.95 0.05 3
antiferromagnetic ordering at T = 6.5 K [83, 86]. If a

magnetic transition occurs at low temperatures, the correct

value of 7 can not be identified due to the large entropies

associated with magnetism. CeCu Si has large sample

dependence problems, e.g., the difficulty of reproducing

its properties and the sensitivity of electronic specific

heat coefficient r and magnetic susceptibility x to Cu

stoichiometry in CeCu2Si2 [87, 88]. So UBe13 is an ideal

host system to do this experiment for the understanding the

source of the heavy fermion ground state in a heavy fermion


Many MBe13 are nonmagnetic, as can be seen in Table

IV-1. These MBe13 compounds provide a wide range of sizes,

unlimited intersolubility without forming a second phase,

various electronic configurations, and very weak dependence

of lattice parameter on beryllium stoichiometry.

In order to determine what part single-ion effects

play in the formation of large m* below 10 K in UBe13, I

concentrated on the three interconnected effects on the

properties of U MBe :
1-x x 13
(1) the effect of the change of the effective U-U distance

upon dilution,

(2)the effect of the change of the U-Be distance upon

changing lattice parameter (a ) with varying M, and

(3) the effect of the electronic configuration of M.

Samples were prepared by arc melting together uranium

(99.9 %), M (typically 99.99 %), and Brush Wellman Be

(99.8 %) in a purified Ar atmosphere. The beryllium used

was always premelted separately to try to remove gaseous

impurities. U MBe compounds present difficulties in
1-x x 13
preparation. Remelting twice to insure homogeneity requires

extreme care in bringing the arc very slowly and at low

power up to the already formed bead to avoid exploding due

to thermal stresses. Since the melting point of UBe3 is so

high (2000 C), beryllium vapor loss is unavoidable upon

Lattice parameter and

Table IV-1
NMel temperature of MBe13 compounds.

M (A) T(K)


Data are from Bucher et al. [89].

< 0.45

T = 0.97


melting. With experience and uniformity in preparation,

additional beryllium can be added in the weighing procedure

(a 6 % excess allows for three meetings) to compensate for

this loss.

All lattice parameters, a of U _M Be13 are obtained

from x-ray diffraction (Table IV-2). No second phases are

observed. In U M Be with a change in a both d, and
X x 13 O H-M
due change. This change in dus will be important in

discussing the results because of the strong hybridization

effects between U 5f-electrons and Be s-electrons that must

be present to prevent the U 5f-electrons from localizing,

since d u(5.13 A) is well beyond the Hill limit.

Results and Discussion

1. Magnetic susceptibility

The dc susceptibilities of U MBe samples at
1-x x 13
T = 1.8 K, which is the lowest temperature of our

magnetometer, are shown in Table IV-2. The value of x at

T = 1.8 K for a given U M Be13 is normalized per U-mole

via [((U_ MBe ) x x x(MBe13)]/(1 x). The values of

dc susceptibility are relatively insensitive to minor

amounts of either excess uranium or excess beryllium.

Within a 5 8 % range of intentional stoichiometry

variation, samples prepared show susceptibilities equal to

the value of stoichiometric UBe The Curie-Weiss law

Table IV-2
Magnetic parameters and lattice parameters for U1-xMxBe13.

Sample x(T=1.8 K) erf(A /U-mole) -e (K) a (A)

UBe 15.1 0.2 3.08 53 10.256

U Y Be 16.1 3.31 76
0.95 0.05 13
U Y Be 14.7 3.62 108 10.245
0.8 0.2 13
U Y Be 16.7 3.88 105 10.240
0.6 0.4 13
U Y Be 13.6 3.3 98 10.235
0.4 0.6 13
U Y Be 14.3 3.15 81 10.229
0.2 0.8 13
U Y Be 17.0
0.1 0.9 13
U Y Be 17.9 10.228
0.05 0.95 13
YBe13 0.32 (per mole YBe 3)

U Sc Be 13.6 3.35 99
0.97 0.03 13
Uo Sc 1Be3 13.2 3.39 89 10.227
0.85 0.15 13
U Sc Be 12.0 3.42 116 10.179
0.5 0.5 13
U Sc Be 12.6 3.2 101 10.102
0.1 0.9 13
ScBe13 0.027 (per mole ScBe 3)

U La Be 15.9 3.25 66
0.97 0.03 1
U La Be 17.9 3.36 82 10.282
0.85 0.15 13
U La. Be 15.8 10.357
0.5 0.5 13
U 0La Be 16.2 3.29 10.435
0.1 0.9 13
LaBe3 -0.26 (per mole LaBe )
13 -13'

Table IV-2

Sample x(T=1.8 K) .ef(fB/U-mole) -9 (K) a (A)

U Th Be 16.27 3.39 74 10.292
0.85 0.15 13
U Th Be 15.2 3.36 84 10.303
0.69 0.31 13
U Th 47Be13 16.1 3.35 67 10.319
0.53 0.47 13
U Tho Be 16.0 3.13 42 10.381
0.1 0.9 13
ThBe3 0.090 (per mole ThBe 3)

U Ce Be 15.6 3.54 100 10.264
0.9 0.1 13
U Ce Be 15.8 10.270
0.85 0.15 13
U Ce 9Be13 12.3 3.0 10.364

CeBe13 1.87 (per mole CeBe13)

U 8Lu Be 15.3 3.89 109 10.229
0.85 0.15 13
U Lu Be 14.5 3.31 77 10.173
0.1 0.9 13
LuBe3 0.46 (per mole LuBe )

U. Pro. Be3 20.4 10.357
0.1 0.9 13
PrBe3 45.3 (per mole PrBe )

U .1Zr. 9Be13 14.5 3.42 101 10.062

ZrBe3 0.095 (per mole ZrBe )

Uo. Hf. Be3 12.4 3.53 31 10.026

HfBe3 -0.014 (per mole HfBe )
13 13

effective moments were calculated from the straight line

behavior of x-1 vs. T between 100 K and 400 K for the

U 1M Be13 samples. In order to calculate uf per U-mole,

the appropriate x x x (MBe13) needs to be subtracted. For

low doping, this is a negligible correction. For large

doping, i.e., for low uranium content, this involves taking

the difference of two similarly sized numbers, the result

of which is multiplied by the large factor 1/(1 x). Our

magnetometer's accuracy is not sufficient to warrant this

procedure for x 2 0.95, due to the increasingly large error


In Table IV-2, the large magnetic susceptibility at

low temperatures in nonmagnetically ordered heavy fermion

systems is thought to be linked with the process by which

local moments, present at room temperature (as seen in the

Curie-Weiss behavior of X-1), are compensated and prevented

from ordering at low temperatures. As seen in Table IV-2,

this large low-temperature magnetic susceptibility in UBe3

appears to be entirely due to single-ion effects, i.e.,

independent of uranium concentration. For U1 Y Be3, a
i-x x 13
total of seven differing x values gives an average

X(T = 1.8 K) per U-mole of 15.8 memu U-mole-1, i.e.,

essentially unchanged from that of pure UBe13. As can be

seen in Figure IV-1, magnetic susceptibilities per U-mole

for wide range of x in U1 YBe13 lie in 10 % range. For

M isoelectronic with Y, i.e., Sc and La, either a slight



A x-0.2
N x-0.4
U x=0.6
A x-O.8

I j ** A

0 I I
0 50 100 150

T (K)

Figure IV-1 Magnetic susceptibilities per U-mole of
various Ui-xYxBel3. All curves for wide range x lie within
10 % range compared to the curve of pure UBe13. This
supports that x per U-mole is independent of

decrease or increase respectively in x(T = 1.8 K) per

U-mole is observed. This effect may be linked to the change

in U-Be distance in the case of Sc and La, with little

change for Y. The X(T = 1.8 K) results for the other

U M Be samples similarly show only rather small
i-x x 13
variations versus the value for pure UBe13, with the

exception of U 0Pr .Be3 which, based on the value for

pure PrBe may be approaching magnetism. This result,

that the large magnetic susceptibility observed at low

temperature in UBe13 is essentially independent of uranium

concentration, is an important finding of the present work.

2. Resistivity

In the resistivity measurement, most of the errors

( 10 %) are due to uncertainties in the cross-sectional

area of the bar and in the voltage contact separation.

Possible additional errors would be due to undetected

cracks giving too high a value for p. And small differences

in beryllium stoichiometry between samples may affect the

absolute values of resistivity obtained to some degree. A

preliminary study on our part on samples of UBe13 and
13. 85
UBe94 indicates a lowering of p for both concentrations

as compared to that reported for pure UBe13.

At the lowest temperatures, UBe13 has a peak in p

versus T theorized to be due to coherence between the

uranium 5f scattering sites below 2.2 K with incoherent

scattering at high temperatures leading to the high

absolute values observed. As small amounts of yttrium are

added, the coherence peak moves to lower temperatures as

the doping introduces additional incoherent scattering

(which also causes the increase in p(T 4 0)). This same

qualitative behavior for low thorium doping has been

previously observed [90]. The resistivity peak at 2.2 K in

the data of pure UBe13 broadens and shifts slightly lower

in temperature for x = 0.01, and becomes a flat plateau up

to 1.7 K for x = 0.02. At higher temperatures we observe a

decrease in the resistivity as x increases in U Y Be in
1-x x 13
full range of x. However, still at x = 0.9, the resistivity

remains quite high at low temperatures (Figure IV-2) in

comparison with the resistivity of pure YBe13 with a Kondo

upturn in p below 125 K. The comparison between the

resistivities of U M Be13 for M = Y, Sc, La, and Th is
O.1 0.9 13
shown in Figure IV-2. Clearly, Sc and Y have a different

low-temperature behavior in their resistivity than La and

Th. The atoms with equal or smaller radii than that of

uranium (Sc, Y) show a Kondo upturn in the resistivity at

low temperatures, while the larger La (isoelectronic to Y

and Sc) and Th show a more metallic behavior.

3. Specific heat

In pure UBe13, C/T rises fast at low temperatures. But

C/T per U-mole is suppressed at low temperatures with a



C ""*-..... MYSc ......--.

40 .*M = Th

0 100 200 300

T (K)

Figure IV-2 Resistivity vs. temperature for Uo. Mo. Bes
for M = Y, Sc, La, and Th. An upturn in resistivity below

125 K for Y doping, and 80 K for Sc doping, is observed,
with p (T ) much lower for the Sc doped sample. For............ La

and Th doping, a shoulder at around 50 K is observed, as
0 100 200 300

T (K)

Figure IV-2 Resistivity vs. temperature for Uo.iMo. 9Be13
for M = Y, Sc, La, and Th. An upturn in resistivity below
125 K for Y doping, and 80 K for Sc doping, is observed,
with p (T -* 0) much lower for the Sc doped sample. For La
and Th doping, a shoulder at around 50 K is observed, as
can be seen in pure UBe13 (p for pure UBe13, see Figure

small amount of Y (Figure IV-3). Upon doping [91, 92], with

about 3 -5 at. % M, superconductivity is suppressed. Upon

further doping, C/T per U-mole becomes even less

temperature dependent (Figure IV-3), thus allowing me to

present C/T data in the dilute limit (or even up to

90 at. % U) at T = 1 K as being representative of y. In

order to check whether y can be represented as the C/T

value at T = 1 K I measured specific heat of U Sc Be
0.1 0.9 13
down to 0.3 K. C/T at T = 0.45 K of this sample increased

less than 5 % compared to C/T at T = 1 K. A complication in

determining C/T in the concentrated limit in UBe13 is the

peak in C at around 2.5 K which shifts upon doping (Figure

IV-3 (inset)), but remains present up to 5 10 % doping -

thus affecting the value of C/T near the peak, and

necessitating lower-temperature measurements to correctly

determine 7 for x < 0.15. As can be seen in Figure IV-4,

this peak, which broadens and shifts to higher temperature

in the specific heat of U0 Y 3Be and U Sc Be ,
0.97 0.03 13 0.97 0.03 13
almost disappears in U Th Be The specific heat of
0.97 0.03 13
U Sc Be (Sc is isoelectronic to Y and has even
0.97 0.03 13
smaller radius than that of Y) shows similar behavior as

that of U Y03 Be with a C/T value at T = 1 K of 650

mJ U-mole-1 K-2 within 5 % difference compared with that

of Uo Y .0Be Whereas, the specific heat of

U La Be (La is also isoelectronic to Y and Sc and
has larger radius than that of Th) shows similar behavior
has larger radius than that of Th) shows similar behavior

900 _20
900.-.-300, ,*

7 1800 -
r-I 0AAJ
S--I -* *r.

E 1300

5 500 vo

L k ST (K)


100 a. .- .- .
0 50 100

T2 (K2)

Figure IV-3 Specific heat per U-mole divided by
temperature vs. temperature squared for U1-xYxBela (x = 0,
triangles; x = 0.02, squares; x = 0.05, inverted triangles;
x = 0.2, diamonds). The rapid decrease in low temperature
C/T with small amounts of doping is evident, with small
changes above 3 K. The C/T per U-mole vs. T2 data for
U-dilute regime, 0.2 I x s 0.995, Ui-xYxBe3i all lie within
10 % range of the curve for x = 0.2.



1500- vrvV 1
7I M 'A-A- A A
E A *

S 1000 -
U o** A pure UBes
a U M-Y
V M-Sc
S* M-La
0 5 10

T (K)

Figure IV-4 Specific heat vs. temperature for pure UBe13
and Uo.97Mo.o3Be13 (M = Y, Sc, La, Th). The peak observed
at 2.5 K in pure UBe13 broadens and shifts to higher
temperature for M = Y, Sc whereas the peak shifts to lower
temperature and almost disappears for M = La, Th.

as that of U Th Be with a C/T value at T = 1 K of
0.97 0.03 13
800 mJ U-mole-1 K-2. These similar behaviors in specific

heat for smaller dopant atoms (Sc, Y) and larger dopant

atoms (La, Th) are consistent with the resistivity

behaviors as can be seen in Figure IV-2. In the present

work, we focus on the nonconcentrated regime, where the

peak in specific heat is no longer present.

Upon substitution for U in UBe three different

effects (effects due to the differing electronic nature of

M, due to change in de, and due to change in d )
U-U U-Be
occur. To check the effect of electronic nature of M, we

present C/T at T = 1 K (C/T at T =1 K can be safely

considered as 7 for x 2 0.15) versus composition for

isoelectronic Sc, Y, and La in Figure IV-5. For Y, where

a (YBe ) ao(UBe ) (Table IV-1), C/T at T = 1 K falls

sharply with increasing doping in the regime x < 0.15. (The

exact details of this fall are determined with the shift of

the peak in C with doping). However, z per U-mole of

U Y Be for 0.20 s x s 0.995, with almost no scatter in
1-x x 13
the data, remains constant (independent of U concentration)

at a value of 420 40 mJ U-mole-1 K-2 (Figure IV-5). 7 is

normalized per U-mole by ( Cmeu/T x x

C(MBe 3)/T)/(1 x). For Sc, where a (ScBe3) < ao(UBe )

(Table IV-1), 7 is similar to that of Y with the constant y

reached in the dilute limit more nearly 350 mJ U-mole-1 K2.

The case of La is different. C/T at T = 1 K does not fall


U Sc
i T La
E %

-" 500

% A


0 0.5 1.0


Figure IV-5 Specific heat value per U-mole at T = 1 K
for Ui-xMxBel3, M is isoelectronic (Y, Sc, and La), vs.
doping level x. Sc and Y behave similarly, with a rapid
decrease in C/T at T = 1 K with only a few percent doping,
followed by an approximately constant value for x a 0.15.
La, however, causes a smooth decrease of C/T at T = 1 K
with increasing concentration with no plateau region

as quickly with increasing x in the concentrated regime.

More importantly, v for U La Be continues to fall
1-x x 13
monotonically even in the dilute limit.

In order to investigate this difference in

7-dependence on x for Sc and Y versus La, the specific heat

7 values of other U M 1Be samples in the nonconcentrated
1-x x 13
(x ; 0.15) limit have been measured. All the data are

presented in Figure IV-6 in log 7 vs. lattice parameter as

a first conclusion that may be drawn from these data. For x

= 0.03 samples, 7 values are determined by measuring

specific heat down to 0.3 K as can be seen in Figure V-ll

(b) and Figure V-14. The electronic nature of M is

evidently not significantly affecting 7 of the remaining U

ions in the lattice. For example, as may be seen in the

right bottom corner of Figure IV-6, Ce, Pr, Th, and La,

which have widely varying outer shell electronic

configurations, give 7 per U-mole of U M 9Be for these
0.1 0.9 13
atoms is approximately the same. On the other hand, as

pointed out in Figure IV-5, La-doped UBe13 samples show

different 7 value behavior compared to the isoelectronic

Sc- and Y-doped UBe3 with same concentrations.
def in this nonconcentrated regime does not appear to

be an important variable for determining 7 and m As we

saw for Sc and Y, and M atoms with a (MBe13 ) < a (UBe ),

including Lu, Zr, and Hf, have a 7 per U-mole almost

independent of concentration as shown in Figure IV-6. For


3.5 *
Th, A x-0.9
C x-0.5
Sa xLa
Pr v x-0.31
la 3.0 x-0.15 -
) 3. pure UBes A Co
SLa x-0.03
E i La
I Lu Sc Y Th
A ,Th

E 2.5 -- Lu
C Hf Zr Sc Sc Th E La

2.0 A A
W Pr A La
cD Th

1.5 -* --
10 10.1 10.2 10.3 10.4 10.5

ao (A)

Figure IV-6 Specific heat v per U-mole of U1-xMxBel3 vs.
lattice parameter ao in logloy vs. ao in the
nonconcentrated, x & 0.15 limit and for x = 0.03. For
smaller dopant atoms than uranium (e.g., M = Y, Sc), an
essentially constant region of y per U-mole vs. x is found.
For larger dopant atoms (e.g., M = La, Th), 7 per U-mole
falls monotonically with increasing ao, independent of
which metal M is used.

these cases, z is independent of def for x ; 0.15. For the
case where a (MBe ) >a (UBe13) (Ce, La, Th, and Pr), it

could be argued that v plotted versus de" would give a
monotonic plot. However, atoms of essentially identical

de0f from the smaller a group on the same plot would have
U-U 0
much different I values compared to atoms with larger a .

Thus with almost same de for U Th Be and
U-U 0.1 0.9 13
U Sc Be the y's are a factor of 3.6 different.
0.1 0.9 13
However, when the data are plotted versus a as shown in

Figure IV-6, which stresses due as the important

parameter, 7 varies smoothly with a For the case where

a0(UM Be13) > ao(UBe13), log 10 values decrease linearly

as lattice parameter a increases with slower decreasing

rate for La than other atoms (Ce, Pr, Th).

One way to check this hypothesis that d is the

factor that determines 7 in the nonconcentrated regime in

U -M Be3 would be to mix two M atoms in a given sample to
1-x x 13
achieve a given a and see if y for this U (M M ) Be
0 1-x I 2'x 13
material has the 7 expected from the plot in Figure IV-6.

Using the lattice parameter data shown in Table IV-1, we

have prepared and characterized U 0La 39Sc .Be1 ,
0.1 0.39 0.51 13
expected to have a lattice parameter equal to that for

Uo Yo Be13. X-ray diffraction on this sample reveals

broadened high angle lines as expected due to a strain from

the three varying size atoms on the U-site. The lattice

parameter is, however, as predicted. x(T = 1.8 K) 18

memu U-mole-" compared to the value of 15.8 memu U-mole-"

for U Y Be which has essentially the same lattice
0.1 0.9 13
parameter. The specific heat y per U-mole is 435

mJ U-mole-1 K2, compared with the expected value of 420

mJ U-mole-1 K-2. This result nicely supports the hypothesis

that d determines 7 per U-mole in U M Be in the
U-Be 1-x x 13
nonconcentrated regime.

As can be seen in Figure IV-6, high 7 values per

U-mole above 500 mJ U-mole-1 K-2 are observed in highly

correlated regime for only a narrow region of a (10.24 A -

10.30 A). In dilute regime, from the dependence of y on a ,

it appears that it is the hybridization between the uranium

5f-electrons and the beryllium s-electrons, which depends

critically upon separation, that dominates the behavior of

7 per U-mole upon dilution. If the U-Be separation remains

the same as found in UBe13 (2.15 A) or smaller, then the

hybridization remains effective in the single-ion,

U-concentration independent mechanism that creates a z per

U-mole of about 375 mJ U-mole-" K-2 or about 40 % of that

in pure UBe13. For increases in the U-Be separation above

that found in pure UBe13, even in the 1 2 % range,

clearly the creation of this still large dilute limit

constant 7 and accompanying m does not occur.

What is the mechanism that creates the dilute limit y

of about 450 mJ U-mole-1 K-2 in U Y Be Figure
0.IV-7, or, in general, the of 420.995 13 4 U-mole
IV-7, or, in general, the 3 of 420 40mJ U-mole-' K-2



Iy Uo0.05Yo.85BeI3

E 20
I-- U.'00aY'0.mBe13
o.ooYBe3- --es-

0 I
0 50 100

T2 (K2)

Figure IV-7 Specific heat divided by temperature vs.
temperature squared for two dilute limit Ui-xYxBel3 samples
and pure YBe13. The solid lines are fit to the single-ion
Kondo model of Schotte and Schotte [93]. The parameters
obtained from the fits are the following: for x = 0.995, TK
= 22 K, 7 = 9.5 mJ mole- K-2 and eD = 1000 K; for x =
0.95, TK = 23 K, 7 = 22 mJ mole-I K-2, and eD = 750 K.

observed independent of x for x a 0.15 for those M atoms

with a (MBe 3) < a (UBe13)? The one model thought to be

valid for Ce systems in the dilute limit is of course the

Kondo model. The specific heat of dilute Kondo impurities

can be calculated [93]. Figure IV-7 shows the unnormalized

C/T data below 10 K for YBe, U Y 995Be13, and
13 0.005 0.995 13
U0 Y Be The slope 3 of a C/T vs. T2 plot for a

normal metal is inversely proportional to the Debye

temperature, a measure of the lattice stiffness, via =-

[1940 x n/3(mJ/mole K4)]1/3 x 10.

For YBe13, the low-temperature data in Figure IV-7

give eD = 1060 K. This value is expected due to the large

amount of Be (whose eD = 1480 K). eD = 820 K for LaBe3 ,

930 20 K for LuBe13 [89]. As may be seen in Figure IV-7,

as U is added to YBe13, the heavy fermion upturn in C/T

occurs at low temperatures, obscuring the simple

determination of eD. However, a fit may be made to the C/T

data for U Y Be3 and U Y Be using the
0.005 0.995 13 0.05 0.95 13
model of Schotte and Schotte [93]. The result of this

fitting procedure (C/T = C odo/T + 7 + T2) is shown in

Figure IV-7.

The Kondo temperature T K, and p terms obtained and

listed in the caption to Figure IV-7 are not too

unrealistic. In particular, the value of eD obtained from p

(1000 K) for x = 0.995 is quite close to the value of 970 K

obtained from the pure YBe1 data. No credence should be

attached to either the value itself for T or its lack of

change with composition, as the scatter in the data limit

the accuracy of T .

Magnetic Behavior of Ce Th Cu Si for x t 0.75
1-x x 2 2

As pointed out earlier, the investigation of magnetic

behavior of Ce Th Cu Si is important for the
1-x x 2 2
understanding of magnetic correlation of dilute Ce in a

nonmagnetic host lattice. Low temperature magnetic

susceptibility per Ce-mole increases with increasing Ce

content showing a maximum with a huge value around x =

0.85. It is important to study the magnetic behavior of

this system for the understanding the heavy fermion ground



Samples (x = 0.75, 0.8, 0.85, 0.875, 0.9, 0.925, 0.95,

0.975) have been prepared using Ce from Ames, Th crystal

bar, and 99.9999 % pure Cu and Si from Johnson Matthey. The

samples were arc-melted three times in a zirconium-gettered

high pure argon atmosphere with turning after each melting

to insure homogeneity. A portion of each sample was wrapped

in tantalum foil to avoid possible contamination from the

contact with quartz during annealing. These samples were

sealed under vacuum in quartz tubes and then annealed at

9000 C for eight days. Samples were made with excess Cu

(Ce 1Th Cu Si instead of Ce ThxCu Si ) to minimize
-x x 2.2 2 1-x x 2 2
the presence of uncompensated Ce3* spins. All the samples

were made using the same procedure to minimize the sample

dependence problem. X-ray diffraction of all of the samples

indicate single phase, as expected from the same tetragonal

structure with very similar lattice parameters in both

CeCu Si and ThCu Si .
22 2 2

Results and Discussion

The dc magnetic susceptibility data of

CexTh Cu2.2Si below 10 K for 0.8 s x s 0.9 are plotted

in Figure IV-8. Samples were cooled in H = 100 gauss by

turning on the field at T = 30 K and cooling down to 1.8 K

to measure susceptibility with increasing temperature.

Magnetic susceptibility data are normalized per Ce-mole by

subtracting the contribution from ThCu 2Si2 (X(T = 1.8 K)

= -0.02 memu mole-I) and then multiplying by 1/(1 x). The

low temperature x data per Ce-mole show a huge value

(22.8 emu mole-I for x = 0.85), which primarily is seen to

build (see Figure IV-8) below 6 K.

Our lowest temperature (T = 1.8 K) magnetic

susceptibility per Ce-mole for each sample is plotted vs.

concentration (Figure IV-9). This plot shows a peak at x =

0.85. Thus, clearly the strength of the magnetic behavior

is a maximum around 15 % of Ce in ThCu 2Si and then
2.2 2


A x-O.8
20 x-0.875

E 15 v T

E to A


10 o

0 5 10

T (K)

Figure IV-8 Magnetic susceptibility per Ce-mole vs.
temperature in low field (H = 100 gauss) of various
Cei-xThxCu2.2Si2 for T < 10 K. Samples were cooled in 100

A HO100 gauss A
H-300 gauss

E 15 A

E 10 A-

5 -

0 i .* .
0.7 0.8 0.9 1.0


Figure VI-9 Magnetic susceptibility per Ce-mole vs.
concentration at 1.8 K of Cei-xThxCu2.zSi2 in H = 100 gauss
and H = 300 gauss. Samples were cooled in 100 gauss and 300
gauss, respectively.

falls off on either side of this concentration. These low

temperature magnetic susceptibility data per Ce-mole have a

large dependence on the applied field with more than a

factor of two difference between H = 100 gauss and 300

gauss. This is indicative of moment saturation beginning at

quite low field.

In contrast to the field cooled susceptibility, zero

field cooled susceptibility shows a sharp peak for the two

samples we checked, at T = 3.5 K for x = 0.85 and T = 2.3 K

for x = 0.9, respectively (Figure IV-10). This behavior

with the fact of huge X value at low temperature and cusp

for the zero field cooled x are reminiscent of spin glass

(a collection of magnetic moments whose low temperature

state is a frozen random disordered one in contrast to

conventional magnets with uniform or periodic spins)

behavior [94]. Figure IV-10 indicates the onset of

remanence effects below Tr (freezing temperature). In a

spin glass, the interactions between the moments compete

with each other so that no single configuration of the

spins is favored by all the interactions which must be at

least partially random (frustration). The behavior of dc

susceptibility below the peak in small field depends

strongly on how the experiment is performed. Figure IV-9

shows the onset of irreversible behavior in x below the

freezing transition when the sample is cooled below T in

zero field and the field is applied below Tf (zero field

A %
A %

A x-0.85
* x=0.9



/ A



U.. .

T (K)

Figure IV-10 Magnetic susceptibility per Ce-mole vs.
temperature in H = 100 gauss of Cei-xThxCu2.2Si2. After
cooling in zero field down to 1.8 K, susceptibilities were
measured by going up in temperature in an applied field of
H = 100 gauss (a) up to 10 K, and for going down in
temperature down to 1.8 K (b), and again for increasing
temperature (c) without changing applied field. For (b) and
(c) procedure, same susceptibilities are measured by going
up and down in temperature.


A 0


1 1

--- -

cooling). Whereas the susceptibility is independent of

history (one can measure the same x by going up and down in

temperature) when the field is applied above T and the

sample is cooled in this field to a temperature below T

(field cooling). The observation of this irreversible

behavior below Tf is a remarkable phenomenon in spin


As a further indication of just how strong this

magnetic behavior is, consider Figure IV-11 where the low

temperature inverse magnetic susceptibility is plotted for

Ce Th Cu Si with the extrapolated high temperature
0.15 0.85 2.2 2
Curie-Weiss law behavior. 1/1 versus T plots for our

Ce xTh Cu .Si sample do have a Curie-Weiss behavior
1-x x 2.2 2
(X(T) = C/(T + e )) for T > 100 K with gef = 2.4 0.2
SB/Ce-ion and a negative intercept (i.e. the zero intercept

of 1/x is on the negative T axis) with e = 25 5K for
0.8 s x s 0.925.

I show in Figure IV-12 low temperature (T = 2 K)

magnetization in units of gB per Ce-ion versus applied

magnetic field for x = 0.8, 0.85, 0.9, and 0.95. This

saturation behavior is present only at low temperatures -

significant curvature is still present at T = 10 K but is

mostly absent by T = 20 K (Figure IV-13).

As a nonuniversal property of spin glass, the

hysteresis effects of magnetization are observed due to an

anisotropy which originates from an indirect coupling




I 50 -


VS. temperature of Ceo. isTho. 85Cu2. Si2. The line

temperature (T > 100 K) 11X vs. T data. In order for 11X
data to fall faster (or for X data to rise faster) than the


0 10 20 30 40 50

T (K)

Figure IV-11 Inverse susceptibility (in H = 100 gauss)
vs. temperature of Ceo.isTho.s85Cu2.2Si2. The line
represents Curie-Weiss law obtained by fitting the higher
temperature (T > 100 K) 1/h vs. T data. In order for 1/x
data to fall faster (or for x data to rise faster) than the
Curie-Weiss law behavior, ferromagnetic behavior must be

yT V V



i '



*U .*fE ..EEE U U U
A x-0.8
S x-0.85
V xO0.9

* x-0.95

H (T)

Figure IV-12 Magnetic moment in gB per Ce-ion vs. applied
field of various Cei-xThxCu2.2Si2 at T = 2 K.




A T-4 K
* T-B K
* T-10 K
* T-20 K


4 A




U Vy

* U v
r, 41 0 ''O



H (T)

Figure IV-13 Magnetic moment in sB per Ce-ion vs. applied
field of Ceo.ITho.9Cu2.2Si2 at various temperatures. The
saturation behavior is mostly absent at 20 K.



m n
.x *

0460** 0**

between the spins and might be different in various spin

glass systems. By sweeping the applied field between

positive and negative values at a temperature below T

only, the remanence is observed in the form of hysteresis

loops. Figure VI-14 (a) and (b) show hysteresis cycles for

Ce Th sCu Si and Ce Th Cu Si respectively.
0.15 0.85 2.2 2 0.1 0.9 2.2 2
The samples were cooled down to 2 K (below Tf) in a very

large field of 2 Tesla and after switching off the field a

smaller field (100 gauss) was applied in the same and

opposite directions.

The specific heat data of Ce Th Cu Si in zero
0.1 0.9 2.2 2
field and applied fields are plotted in Figure IV-15. These

data are consistent with those of a spin glass, e.g., CuMn

[95] or [96] showing a broad maximum above Tf (versus the

sharp cusp found at a ferromagnetic ordering temperature)

with no indication of an anomaly right at T (T = 2.3 K

from the x but specific heat peak at T = 2.9 K).

Magnetic field broadens the transition and moves it to

higher temperature as seen for CuMn [95] (and, of course,

also for ferromagnets). The broadness of the observed

specific heat anomaly indicates that the change of the

internal energy might be small due to the comparatively few

degrees of freedom (the specific heat measurement is used

for the information about the excitation of spin glass and

the considerable fraction of the spin degrees of freedom

which are already frozen out around Tf).





*- -* :. = *


rs /
's r

. .4

-40001 ..




H (oe)
40001 1 ..--

2000 -



-40001 .


H (Oe)

Figure IV-14 (a) Hysteresis of Ceo.IsTho.8sCu2.2Si2
measured at T = 2 K. The sample was cooled in 2 Tesla and
after switching off the field, magnetization was measured
in the sequence shown by letter.
(b) Same hysteresis measurement of
Ceo. iTho. 9CU2.2Si2.




E H-2 T
SH-5 T
3000 -
%/ a H-10 T

o A
0 V A 0

E 2000 a

0 5 10

T (K)

Figure IV-15 Low temperature specific heat (normalized
per Ce-mole by subtracting 0.9 x (specific heat of
ThCu2.2Si2) and then divided by 0.1 to give specific heat
u 1000 v A-

0 5 10

T (K)

Figure IV-15 Low temperature specific heat (normalized
per Ce-mole by subtracting 0.9 x (specific heat of
ThCu2.2Si2) and then divided by 0.1 to give specific heat
per Ce-mole) vs. temperature of Ceo. Tho.9Cu2.2Si2 in
fields from 0 Tesla to 10 Tesla.

Because of the random placement of impurity moments

which produce in the vicinity of them a positive or

negative magnetic polarization of the host metal conduction

electrons, we have competing positive or negative

interactions in spin glass systems. With these features of

disorder and competing interactions, some simple models

such as RKKY glasses (an indirect exchange interaction

between impurity moments with oscillatory behavior; RKKY

long range interaction), Edward-Anderson model [97] (random

interaction which depends only on the lattice vector

separation), and Sherrington-Kirkpatrick [98] (Ising

version (retains only a single component of vector spins)

of Edward-Anderson model) were constructed.

The main characteristic property of a spin glass is

that moments are frozen at temperatures below Tf. This spin

freezing results experimentally in a cusp at T in the

magnetic susceptibility. Remanence and also hysteresis are

observed due to this spin freezing. Although we have been

unable to find time dependence (on the scale of 2 min) of

the magnetization taken from H = 5.5 Tesla to 50 gauss, we

have found irreversibility of xdc which illustrates the

onset of remanence effect below T Also X. data (measured

at 87 Hz) shows a peak at 2.7 K. Allowing for sample

variation, this is an acceptable agreement with Xdc (a

similar slight shift of Xd to lower T from ac is seen
max max
in La Gd Al [99]). In specific heat for
1-x x 2

Ce Th Cu Si the peak is at T = 2.9 K, which is (as
0.1 0.9 2.2 2
required for a spin glass) at a higher temperature than the

peak temperature in x at T = 2.3 K. Also hysteresis

cycles are observed.

These data, which are not perfectly in agreement with

a spin glass or spin clustering (there would be

correlations in the impurity positions even if the sample

could be infinitely rapid quenched from the liquid state)

sort of picture, however, best agree with such a

description. The peak in specific heat for

Ce Th Cu Si is at T = 2.1 K; no peak down to 1 K
0.05 0.95 2.2 2
is observed in Ce Th Cu Si With the peak of
0.025 0.975 2.2 2
specific heat for Ce Th Cu Si at T = 2.9 K (Figure

IV-14), it is clear that Ce-Ce overlap is modifying the

expected dilute-limit simple scaling of T with


To check the cause of this strong magnetic behavior in

this Ce dilute system, I intentionally made

Ce La Y .Cu Si with an unit-cell size matching
0.1 0.69 0.21 2.2 2
that of Ce Th Cu Si with the fact that Ce in
0.1 0.9 2.2 2
LaCu 2Si2 and YCu2.2Si shows no signs of magnetism. The
magnetic susceptibility of Ce La Y Cu Si at
0.1 0.69 0.21 2.2 2
T = 1.8 K is 11.8 memu Ce-mole-" (about three orders of

magnitude small). This indicates that the formation of

magnetism is not sensitive to the Ce-nearest-neighbor

distance in this system. The 4-valent Th electrons may be

crucial to the magnetic behavior of Ce Th Cu Si via
1i-x x 2 2
moving of the Fermi energy.

Ce Th Cu 2Si is a system with a rich variety of
i-x x 2.2 2
magnetic behavior for large x. The evident existence of

some ferromagnetic interactions between Ce ions in

Ce _Th Cu 2Si raises the question if they may also be
1-x x 2.2 2
present in pure CeCu2Si2 (screened by Kondo effect

(quenching the impurity moments by the formation of a

collective singlet state and coupling the impurity spin

with the spins of the nearby conduction electrons)) and

perhaps play a role in the formation of the high effective

mass ground state.

Different Magnetic Behavior in Doped UPt

UPt3 is a heavy fermion system of considerable

interest because of its unusual low temperature properties.

This hexagonal compound is likely to be interpreted in

terms of spin fluctuation phenomena because similar low

temperature anomalies in specific heat and temperature

dependence of resistivity in normal state of UPt are found
in the spin fluctuation compounds UA2 [100] and TiBe2

[101] with the exception of the larger 7 value for UPt A
T31n(T/T*) contribution (see Equation II-10) to the

specific heat [102] and a T2-term in the resistivity [103,

104] are suggestive for spin fluctuation phenomena [11,