STRUCTURE-PROCESSING-PROPERTY RELATIONSHIPS AND
THERMOELECTRICITY IN STRONGLY CORRELATED CePd3 AND RELATED
COMPOUNDS
By
Stephen Richard Boona

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Materials Science and Engineering – Doctor of Philosophy
2013

ABSTRACT
STRUCTURE-PROCESSING-PROPERTY RELATIONSHIPS AND
THERMOELECTRICITY IN STRONGLY CORRELATED CePd3 AND
RELATED COMPOUNDS
By
Stephen Richard Boona
The discovery of high performance thermoelectric materials would enable the development of a wide range of new thermal and electrical energy management technologies, including the possibility of solid state cryogenic coolers. Such discoveries have been sharply
inhibited to date, however, by the contraindicated nature of the material properties that
comprise the thermoelectric figure of merit ZT . Enhancement of ZT generally requires
overcoming significant materials science and engineering challenges that become even more
restrictive at low temperatures. One possible approach for overcoming these challenges is to
shift away from conventional semiconductors and toward strongly correlated electron materials, as these systems often display unusual combinations of electronic and thermal properties
that may be exploited to achieve high ZT in the cryogenic temperature regime.
This perspective has motivated the current work, in which the thermoelectric properties
of strongly correlated CePd3 and related compounds are explored. This work presents a
comprehensive and systematic study of the ways in which composition and processing modifications affect the electronic, thermal, magnetic, and structural properties of these materials.
The insights gained from this study have led to the highest ZT value reported to date of
all p-type bulk polycrystalline materials at low temperatures. A detailed review of these
developments is provided along with a discussion of how they may be applied to achieve
further enhancements in ZT .

I dedicate this work to my friends and family who have provided me with invaluable
guidance and inspiration in all areas of my life. I am extremely grateful to those who
have sacrificed their time, energy, and resources to support me intellectually, emotionally,
and financially over the last nine years as I have navigated the winding road of my higher
education.
Especially, to Dad − who told me many years ago that the grand challenge of my generation would be finding new ways to produce and use energy. I took those words to heart.

iii

ACKNOWLEDGMENTS

I would like to thank the current and former members of Professor Morelli’s group who have
directly or indirectly contributed to this work, including Dr. Vijay Ponnabalam, Dr. Kevin
Zhou, Dr. Eric Skoug, Dr. Chang Liu, Hui Sun, Gloria Lehr, Winston Carr, and Minyoung
Jeong. I would especially like to thank Xu Lu for his enthusiastic assistance with SEM/EDS
experiments, as well as Dr. Reza Loloee, Travis Thompson, Karl Dersch, and the many
other students and staff at MSU who have helped in various ways with this project.
I am also indebted to my former colleagues and professors in the Physics Department
at Northern Illinois University, especially my master’s thesis advisor Professor Bogdan
Dabrowski. The strong foundation with which they provided me in the theory and practice
of solid state physics has been essential in the derivation of these results.
I would also like to thank some of the many professors and mentors I have had at MSU
who have contributed to my development as a scientist, especially my Guidance Committee
members Dr. Jeff Sakamoto, Dr. Martin Crimp, and Dr. Viktor Poltavets.
I am of course also grateful to Professor Morelli for his invaluable guidance throughout
my doctoral work, especially his encouragement to pursue my research down whatever path it
may lead; the countless hours spent sharing with me his many stories, insights, and wisdom;
the several opportunities he has provided me with to publish and present my results; and
his expert advice on how to aim for and hit my goals that only a true bocce ball champion
can provide.
Finally, I would like to acknowledge that the funding for this research was provided by
the U.S. Air Force Office of Scientific Research as part of the Multidisciplinary University
Research Initiative for Cryogenic Peltier Cooling, contract number FA9550-10-1-0533.
iv

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Chapter 1 Introduction . . . . . . . . . . . . . . . .
1.1 Thermoelectric effects . . . . . . . . . . . . . . .
1.2 Applications of thermoelectric devices . . . . . . .
1.3 Thermoelectric performance . . . . . . . . . . . .
1.4 ZT optimization strategies . . . . . . . . . . . . .
1.4.1 Semiconductors and the PGEC paradigm .
1.4.1.1 Thermal properties . . . . . . . .
1.4.1.2 Electronic properties . . . . . . .
1.4.2 Need for an alternative approach . . . . .
1.4.2.1 The case for metals . . . . . . . .
1.4.2.2 Increasing the Seebeck coefficient
1.5 CePd3 . . . . . . . . . . . . . . . . . . . . . . . .
1.5.1 Structural properties . . . . . . . . . . . .
1.5.2 Electronic and magnetic properties . . . .
1.5.2.1 Electrical resistivity . . . . . . .
1.5.2.2 Hall coefficient . . . . . . . . . .
1.5.2.3 Magnetic susceptibility . . . . . .
1.6 Thermal properties . . . . . . . . . . . . . . . . .

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1
1
2
4
6
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7
12
16
16
20
21
25
27
29
30
32
33

Chapter 2 Experimental methods .
2.1 Materials synthesis . . . . . . . .
2.2 Materials characterization . . . .
2.3 Transport property measurements
2.4 Magnetic property measurements

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35
35
36
37
40

Ce1−x Scx Pd3
. . . . . . . . .
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43
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44
44
44
45
47
49

Chapter 4 Palladium site substitutions: CePd3−x Ptx . . . . . . . . . . . . .
4.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . . . .

51
51

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Chapter 3 Cerium-site substitutions:
3.1 Background and motivation . . . .
3.2 Experimental methods . . . . . . .
3.3 Results and discussion . . . . . . .
3.3.1 Structural properties . . . .
3.3.2 Electronic properties . . . .
3.3.3 Thermal properties . . . . .
3.3.4 General conclusions and ZT

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v

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4.2
4.3

Experimental methods . . . . . . .
Results and discussion . . . . . . .
4.3.1 Structural properties . . . .
4.3.2 Electronic properties . . . .
4.3.3 Thermal properties . . . . .
4.3.4 Magnetic properties . . . . .
4.3.5 General conclusions and ZT

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54
54
54
55
58
59
60

Chapter 5 Interstitial site filling: CePd3 Mx
5.1 Background and motivation . . . . . . . .
5.2 Experimental methods . . . . . . . . . . .
5.3 Results and Discussion . . . . . . . . . . .
5.3.1 Structural Properties . . . . . . . .
5.3.2 Electronic Properties . . . . . . . .
5.3.3 Thermal Properties . . . . . . . . .
5.3.4 Magnetic Properties . . . . . . . .
5.3.5 General conclusions and ZT . . . .

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64
64
67
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69
74
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79
82

Chapter 6 Nanostructured CePd3 . . . . . . . . .
6.1 Background and motivation . . . . . . . . . . .
6.2 Experimental methods . . . . . . . . . . . . . .
6.3 Results and discussion . . . . . . . . . . . . . .
6.3.1 Structural and morphological properties
6.3.2 Electronic properties . . . . . . . . . . .
6.3.3 Thermal properties . . . . . . . . . . . .
6.3.4 Magnetic properties . . . . . . . . . . . .
6.4 “Size-doping” model . . . . . . . . . . . . . . .
6.4.1 Further study . . . . . . . . . . . . . . .

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. 84
. 84
. 86
. 87
. 87
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. 95
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. 100

Chapter 7 Universal trends and combined approaches
7.1 Nanostructured CePd3 Mx . . . . . . . . . . . . . . . .
7.1.1 Background and motivation . . . . . . . . . . .
7.1.2 Experimental methods . . . . . . . . . . . . . .
7.1.3 Results and discussion . . . . . . . . . . . . . .
7.2 Nanostructured CePd2 Pt . . . . . . . . . . . . . . . . .
7.3 Nanostructured CePd2 PtMx . . . . . . . . . . . . . . .

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101
104
104
105
106
112
114

Chapter 8 Related compounds . . . .
8.1 EuPd3 Mx . . . . . . . . . . . . . .
8.1.1 Background and motivation
8.1.2 Experimental methods . . .
8.1.3 Results and discussion . . .
8.2 Ce3 Pt3 Sb4 . . . . . . . . . . . . . .
8.2.1 Background and motivation
8.2.2 Experimental methods . . .

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115
116
116
116
117
121
121
125

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8.3

8.2.3 Results and discussion . . .
CeNi2 Al3 . . . . . . . . . . . . . .
8.3.1 Background and motivation
8.3.2 Experimental methods . . .
8.3.3 Results and discussion . . .

Chapter 9

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127
128
128
130
130

Results and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 134

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

vii

. . . . . . . . . . . . . . . . . . 139

LIST OF FIGURES

Figure 1.1

Plot of the Seebeck coefficient necessary for ZT of unity at various
temperatures for given values of ρ based on L = L0 and κlattice =
0.1 W/m-K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Seebeck coefficient and thermal conductivity ratio of stoichiometric
CePd3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

Figure 1.3

Filled Cu3 Au structure diagram . . . . . . . . . . . . . . . . . . . .

26

Figure 2.1

Schematic of the experimental setup used for transport property measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Schematic of the experimental setup used for Hall coefficient measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Figure 3.1

Lattice parameter of Ce1−x Scx Pd3 . . . . . . . . . . . . . . . . . . .

45

Figure 3.2

Seebeck coefficient of Ce1−x Scx Pd3 . . . . . . . . . . . . . . . . . .

46

Figure 3.3

Electrical resistvity of the Ce1−x Scx Pd3 solid solution . . . . . . . .

47

Figure 3.4

Lattice thermal conductivity of the Ce1−x Scx Pd3 solid solution . . .

48

Figure 3.5

Thermal conductivity ratio of the Ce1−x Scx Pd3 solid solution

. . .

49

Figure 3.6

Thermoelectric figure of merit of the Ce1−x Scx Pd3 solid solution . .

50

Figure 4.1

Reported lattice parameters for CePd3−x Tx (T = Ni, Ag, Rh, Pt)
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

Figure 4.2

Lattice parameters of CePd3−x Ptx . . . . . . . . . . . . . . . . . . .

55

Figure 4.3

EDS analysis results for CePd2 Pt . . . . . . . . . . . . . . . . . . .

56

Figure 4.4

Seebeck coefficient of the CePd3−x Ptx solid solution . . . . . . . . .

56

Figure 1.2

Figure 2.2

viii

Figure 4.5

Electrical resistivity of the CePd3−x Ptx solid solution . . . . . . . .

57

Figure 4.6

Total thermal conductivity of the CePd3−x Ptx solid solution . . . .

58

Figure 4.7

Lattice thermal conductivity of the CePd3−x Ptx solid solution . . .

59

Figure 4.8

Thermal conductivity ratio of the CePd3−x Ptx solid solution . . . .

60

Figure 4.9

Magnetic susceptibility of the CePd3−x Ptx solid solution . . . . . .

61

Figure 4.10

High temperature susceptibility of the CePd3−x Ptx solid solution

.

61

Figure 4.11

ZT of the CePd3−x Ptx solid solution . . . . . . . . . . . . . . . . .

62

Figure 5.1

XRD plots for a representative series of CePd3 Mx with M from rows
two to four of the periodic table . . . . . . . . . . . . . . . . . . . .

70

Some XRD plots for the CePd3 Mx series for M from row five of the
periodic table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

Figure 5.2

Figure 5.3

Lattice parameters of CePd3 Mx as a function of nominal M ionic radii 73

Figure 5.4

Lattice parameters of CePd3 Mx as a function of effective valence . .

73

Figure 5.5

Normalized electrical resistivity of representative samples from the
CePd3 Mx series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

Figure 5.6

Seebeck coefficient of representative samples from the CePd3 Mx series 76

Figure 5.7

Seebeck coefficient at 295 K versus Vef f for representative samples
from the CePd3 Mx series . . . . . . . . . . . . . . . . . . . . . . . .

77

Lattice thermal conductivity at 100K of representative samples from
the CePd3 Mx series . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Magnetic susceptibility of representative samples from the CePd3 Mx
series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

Curie constant versus Vef f for a representative samples from the
CePd3 Mx series . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

Figure 5.8

Figure 5.9

Figure 5.10

ix

Figure 5.11

Hall coefficient for the CePd3 Gax series . . . . . . . . . . . . . . . .

81

Figure 5.12

295 K Seebeck coefficient and high temperature Curie constant of
CePd3 Mx vs 295 K lattice parameter . . . . . . . . . . . . . . . . .

82

Figure 5.13

ZT of representative samples from the CePd3 Mx series . . . . . . .

83

Figure 6.1

XRD patterns of ball milled CePd3 . . . . . . . . . . . . . . . . . .

88

Figure 6.2

SEM micrograph of an arc melted CePd3 control sample . . . . . . .

88

Figure 6.3

Contrast enhanced SEM micrograph of CePd3 after two minutes of
milling and PECS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

Contrast enhanced SEM micrograph of CePd3 after 12 minutes of
milling and PECS . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Particle size and strain of ball milled CePd3 samples as determined
by Williamson-Hall analysis . . . . . . . . . . . . . . . . . . . . . .

91

Figure 6.6

Electrical resistivity of nanostructured CePd3 . . . . . . . . . . . . .

92

Figure 6.7

Seebeck coefficient of nanostructured CePd3 . . . . . . . . . . . . .

93

Figure 6.8

Total thermal conductivity of nanostructured CePd3 . . . . . . . . .

94

Figure 6.9

Lattice thermal conductivity of nanostructured CePd3 . . . . . . . .

94

Figure 6.10

Thermal conductivity ratio of nanostructured CePd3 . . . . . . . . .

95

Figure 6.11

Hall coefficient of nanostructured CePd3

. . . . . . . . . . . . . . .

96

Figure 6.12

Hall mobility of nanostructured CePd3 . . . . . . . . . . . . . . . .

97

Figure 6.13

Simple geometric model for the fraction of unique unit cells at a cube
surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

98

Figure 6.4

Figure 6.5

Figure 6.14

Estimated fraction of unique unit cells at the surface of a CePd3 grain 99

Figure 7.1

Seebeck coefficient (left ordinate) and Curie constant (right ordinate)
as a function of lattice parameter for various CePd3 series . . . . . . 102
x

Figure 7.2

SEM and EDS results for CePd3 Ga0.05 after ball milling and PECS

Figure 7.3

Location of normalized (311) peak in CePd3 Ga0.05 after ball milling
for various times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 7.4

Location of normalized (311) peak in CePd3 Ga0.05 after ball milling
and PECS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Figure 7.5

Seebeck coefficient of CePd3 Gax before and after ball milling and PECS110

Figure 7.6

Hall coefficient of CePd3 Gax before and after ball milling and PECS 111

Figure 7.7

Comparison of lattice thermal conductivity of CePd3 modified in various ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Figure 7.8

Thermal conductivity ratio of CePd3 modified in various ways . . . 113

Figure 8.1

Lattice parameter of EuPd3 Bx . . . . . . . . . . . . . . . . . . . . . 117

Figure 8.2

Seebeck coefficient of EuPd3 Bx . . . . . . . . . . . . . . . . . . . . . 119

Figure 8.3

Normalized electrical resistivity of EuPd3 Bx . . . . . . . . . . . . . 119

Figure 8.4

Linear fit parameters for the normalized electrical resistivity of EuPd3 Bx 120

Figure 8.5

Crystal structure diagram of Ce3 Pt3 Sb4 . . . . . . . . . . . . . . . . 122

Figure 8.6

Crystal structure diagram of CeNi2 Al3

Figure 8.7

Lattice parameter of CeNi2−x Cux Al3 . . . . . . . . . . . . . . . . . 131

Figure 8.8

Seebeck coefficient of CeNi2−x Cux Al3 . . . . . . . . . . . . . . . . . 131

Figure 8.9

Normalized electrical resistivity of CeNi2−x Cux Al3 . . . . . . . . . . 132

xi

107

. . . . . . . . . . . . . . . . 129

Chapter 1
Introduction

1.1

Thermoelectric effects

Ever since mankind’s earliest use of tools, the fundamental purpose of most engineering
technologies has been to facilitate the efficient, reliable, and targeted transfer of energy
among its various forms. The discovery of three related thermodynamic phenomena by
Thomas Seebeck in 1821 [1], Jean-Charles Peltier in 1834 [2], and William Thomson (Lord
Kelvin) in 1854 [3] established the thermoelectric effect as a solid-state mechanism through
which energy could be reversibly transferred between heat and electricity, thus providing
mankind with yet another tool for harnessing the power of nature.
The simplest of these phenomena is the Thomson effect, wherein the coexistence of an
electric current and a parallel temperature gradient in a single conducting material results in
a reversible flow of heat. Thomson showed that the direction of this heat flow is dependent on
the polarity of the charge carriers, thus distinguishing this effect from resistive Joule heating
[3]. Related to this is the Seebeck effect, in which a voltage arises between the junctions of
two dissimilar metals when those junctions are held at different temperatures [1]. Although
Seebeck’s original experiments were performed on circuits of two distinct materials, it is
possible and convenient to describe the Seebeck coefficient, or thermopower, of a single
material. This quantity is defined as the thermally induced voltage per unit temperature
that arises within a material when subjected to a thermal gradient. Conversely, the Peltier
1

effect describes how the insertion of a current source into a circuit composed of two distinct
metals will result in a net absorption or liberation of thermal energy at the metal junctions.
This occurs due to differences in the amount of heat transported by the charge carriers in the
two materials; in order to maintain a constant flow of current in accordance with Kirchoff’s
Law, a net transfer of heat with the local environment at the junction must occur for the
charges to flow smoothly and continuously from one material to the other [2].
These effects can be utilized for a variety of power generation and temperature control
applications. This requires the combination of at least two types of materials: one where
the dominant charge carriers are electrons (n-type) and one where they are holes (p-type).
Thermoelectric devices can be built from alternating legs of these materials connected electrically in series and thermally in parallel. Applying a temperature gradient across the legs
of such a device results in an open circuit potential that can be utilized as a power source.
In principle, this can be done actively or passively to generate electrical energy from the
temperature gradient created by any primary heat source. Conversely, connecting a similar
thermoelectric module to a power supply allows for precise temperature control of the hot
and/or cold sides of the module by controlling the direction and magnitude of the current
supplied to the device [4].

1.2

Applications of thermoelectric devices

The two most common uses of thermoelectric devices are for power generation via thermoelectric generators and temperature control via Peltier coolers.
The appeal of using thermoelectric devices for power generation lies in their versatility
and reliability, both of which stem from the solid-state nature of the thermoelectric effect.

2

These devices can be tailored for use within a wide range of geometries and length scales,
and their lack of moving parts largely eliminates any design problems related to vibrations,
maintenance, and mechanical failure. These positive aspects are generally negated by the
relatively poor thermal-to-electrical energy conversion efficiency of most thermoelectric materials, thus limiting the use of these devices to niche applications such as power generation
on NASA deep space probes and planetary science missions. These limited applications have
nonetheless served as excellent demonstrations of the strong potential this technology holds
for more widespread use, as indicated by the uninterrupted operation of these thermoelectric
generators over multiple decades in the vacuum of space [5].
Many potential terrestrial applications also exist if materials and devices can be engineered to match the cost, reliability, and performance metrics required for deployment in
commercial, industrial, and military arenas. Among these possible uses are waste heat recovery from automobiles and power plants as well as improved home energy efficiency and
remote power generation for recreational and survival situations. [6, 7, 8, 9]
Thermoelectric devices for temperature control are already a relatively common application of the thermoelectric effect. This can be seen in a variety of products, including localized
heaters and coolers in automobile seats and cup holders, thermoelectric “refrigerators” operating on low voltages and/or offering precise temperature control of wine and other consumer
goods, and coolers designed for delicate electronic equipment such as observational telescope
cameras and other types of sensors. Although these thermoelectric modules are not generally useful for applications requiring high power cooling, their flexible form factors, precision
control, and lack of moving or vibrating components again offer important advantages over
alternative technologies such as vapor compression. [7, 8, 9]
One particularly appealing potential arena for applying the thermoelectric effect is in
3

the cryogenic temperature regime, wherein a number of possible uses exist for cooling various types of electronic and laboratory equipment in conjunction with traditional vapor
compression systems and/or as standalone devices. The reliable and solid-state nature of
thermoelectric technology once more makes these devices attractive candidates for applications such as cooling superconducting transmission cables, precision temperature control of
laboratory characterization equipment, and thermal noise reduction in space-based infrared
and gamma ray detectors.
Before any of these uses can be realized, however, significant improvements are necessary
in the properties of the available materials. This is especially true at low temperatures,
where a dearth of useful compounds has forced cryogenic thermoelectric cooling to remain a
purely theoretical application. This has largely motivated the current work, which focuses on
the exploration of new approaches for developing high performance thermoelectric materials
in this temperature range. To properly consider these new approaches, it is useful to first
establish a suitable context for the origins of favorable thermoelectric performance.

1.3

Thermoelectric performance

The thermoelectric performance of any material can be quantified through the dimensionless
figure of merit ZT , which is defined as

ZT =

σα2 T
α2 T
=
,
ρκ
κ

(1.1)

where α is the Seebeck coefficient, ρ (σ) is the electrical resistivity (conductivity), κ is the
total thermal conductivity, and T is the average absolute temperature [10].

4

ZT values for state-of-the-art bulk materials are on the order of unity. Values as high as
two have been reported in PbTe-based compounds at temperatures near 900 K [11]. Below
room temperature, however, the maximum values reported generally remain well below unity
for both n-type and p-type materials. The minimum ZT necessary for a thermoelectric cooler
to operate will vary with the absolute temperature of the device as well as the temperature
gradient across it, but ZT values of at least 0.3 or larger are generally necessary for such a
device to establish a useful temperature gradient [4].
The essential challenge in developing high ZT materials at any temperature stems from
the competing nature of the relevant material properties. Since the Seebeck coefficient is a
measure of the charge separation that occurs when a material is exposed to a temperature
gradient, this charge separation will generally be smaller in materials within which charge
flows easily. Thus as ρ decreases, α tends to decrease as well. Similarly, modifying a material
to lower its thermal conductivity is also likely to affect the electrical conductivity, since the
mechanisms useful for impeding heat flow also tend to affect charge flow as well.
The path to increasing ZT therefore lies in finding ways of decoupling and separately
optimizing these properties. The most direct way to accomplish this is by considering that
the total thermal conductivity of most materials is comprised of two main contributions, the
first from quantized vibrations of the atomic lattice known as phonons (κl ), and the second
from the polar thermal diffusion of charge carriers (κe ). Since the lattice contribution to
the thermal conductivity in most semiconductors tends to be orders of magnitude larger
than the electronic contribution, this has led to a common paradigm in thermoelectricity
over the last several decades wherein the numerator of ZT is considered entirely dependent
on a given material’s electronic properties, while the denominator is a function of only the
lattice thermal conductivity. This interpretation of the ZT equation allows for a relatively
5

straightforward description of the transport properties, and intuitively leads into the conclusion of Slack that an ideal thermoelectric material should behave as a “phonon-glass,
electron-crystal” (PGEC); that is, the heat conduction should be minimal and similar to
that of an amorphous solid, while the electronic properties should mimic those of a highly
ordered crystalline material [12]. It is from this perspective that a basic understanding of
thermoelectric optimization strategies can be developed.

1.4

ZT optimization strategies

Although practical limitations have been asserted for the maximum ZT of a semiconductor
based on realistic values of the relevant properties, there is no theoretical upper limit on
the maximum possible ZT of a material in general [13]. In order to explore the ways
through which increases in ZT can be achieved, particularly at low temperatures, it is
useful to consider the current paradigm of semiconductor-like materials that has largely
governed the field of thermoelectricity since its inception. Critical evaluation of this paradigm
shows that although it is not generally a useful approach for achieving high ZT at cryogenic
temperatures, it can still nonetheless be utilized as a basis for developing a more suitable
perspective.

1.4.1

Semiconductors and the PGEC paradigm

Seebeck’s original experiments on elemental and binary p-block semiconductors laid the
initial framework for the field of thermoelectricity. This has traditionally led to a strong
emphasis on various chalcogenide compounds [1]. Seebeck’s work has been expanded upon
over the last century through notable contributions from many different researchers, in-

6

cluding Eucken’s early studies of thermal conductivity in alloys and compounds [14], Ioffe’s
development of the ZT equation and work on heavily doped semiconductors [15], Goldsmid’s
detailed analysis of desirable electronic band structure properties and invention of functional
thermoelectric coolers [16], Slack’s development of the minimum lattice thermal conductivity and PGEC concepts [12], and several others. These contributions have led to significant
advances in the fields of solid state physics and materials science at large.

1.4.1.1

Thermal properties

Thermal conductivity is a complex transport phenomenon that arises from the interaction
of many different processes inside of a material. The electronic and lattice contributions
to the thermal conductivity of a solid are nominally linearly independent of each other.
The electronic contribution can be estimated through the Wiedemann-Franz (W-F) law [17],
which is defined as
κe =

TL
= σT L,
ρ

(1.2)

where L is a material property known as the Lorenz number. The Lorenz number can be
calculated from first principles for a Fermi gas, and the result of this calculation is commonly
known as the Sommerfeld value L0 , which is given by

L0 =

π
3

kB
e

2

= 2.44x10−8 ΩmK −2 ,

(1.3)

where kB is the Boltzmann constant and e is the elementary charge.
The primary assumption in the derivation of W-F is that the electron mean free path is the
same under an electric field as it is under a temperature gradient, i.e. that inelastic electronphonon scattering events are rare. This assumption is generally reasonable, particularly
7

for conventional metals, and strong empirical support for this relationship can be found in
experiments that show L ≈ L0 for many different types materials over a broad range of
temperatures [18]. The Sommerfeld value can therefore be applied to provide at least an
estimate of the electronic thermal conductivity of any material, in turn enabling a reasonably
accurate and self-consistent determination of the lattice contribution as well.
The magnitude and temperature dependence of a material’s lattice thermal conductivity
are both functions of mostly physical parameters such as temperature, density, stiffness,
structure type, strain, defects, and other properties of the crystal lattice. These parameters
are nominally independent of a material’s electronic properties, although the nature of the
chemical bonds can be important for both, and direct electron-phonon scattering as well as
electron-phonon coupling (so-called “phonon drag”) can occur. In addition to these phenomena, the electronic and thermal properties are often indirectly related through a shared
dependence on various scattering mechanisms that affect both the phonon and electron mean
free paths, such as impurities and interfaces [18].
A simple quantitative model of lattice thermal conductivity can be described by

1
κl = Cv vl,
3

(1.4)

where Cv is the lattice heat capacity, v is the phonon group velocity, and l is the phonon
mean free path [19]. The minimum possible thermal conductivity of a material corresponds
with the so-called “amorphous limit” where the average distance between phonon scattering
events is on the order of the interatomic spacing. This minimum value can be estimated from
Equation 1.4 by measuring Cv , setting the mean free path l equal to the average interatomic
spacing, and solving for the speed of sound v =

8

Λ/δ, where Λ is the elastic modulus and δ

is the density. For typical crystalline materials, this minimum lattice thermal conductivity
is on the order of 0.1 W/m-K [12].
A more complete version of this analysis can be performed by incorporating the dependence of these parameters on the frequency of the atomic vibration modes. This is particularly important below a material’s Debye temperature ΘD , where not all of the possible
phonon modes are active. The Debye-Callaway model provides the necessary framework for
this. Ignoring the small correction that arises for non-resistive phonon-phonon interactions,
this model is described by the integral

κl =

1 ω0
C(ω)v(ω)2 τ (ω)dω,
3 0

(1.5)

where τ = l/v is the frequency-dependent relaxation time between phonon scattering events
and ω0 is the frequency of the highest active phonon mode [18]. Based on this equation, the
basic and obvious strategy that emerges for achieving low lattice thermal conductivity is to
engineer materials such that they have low lattice heat capacity, low sound velocity, and/or
a short phonon scattering relaxation time. The complexity of the factors that contribute to
these properties, particularly their dependence on phonon frequency and temperature, makes
it difficult to predict a priori the exact ways in which changes in composition and processing
will affect the absolute magnitude of a material’s thermal conductivity. Several common
strategies have nonetheless emerged for engineering the thermal properties of materials.
In regards to minimizing the group velocity of the phonons, for example, a common
approach is to seek out dense compounds with relatively soft bonds in order to make

Λ/δ

small. Another common approach is to search for materials with large primitive unit cells,
as this structural complexity suppresses the group velocity of the heat-conducting acoustic

9

modes while it simultaneously introduces a number of optical phonon modes that have
small group velocities and therefore do not typically contribute to thermal conductivity [19].
The effects of large unit cells and complex structures have been credited as the source of
low thermal conductivity in materials such as Y3 Al5 O12 garnet [20], β-Boron, and YB68
[21]. The major drawback to these group velocity reduction approaches, however, is that
they make no guarantee that the electronic properties of the material will simultaneously
be within the useful range for thermoelectric applications, although they do least provide
insight into what types of systems may be of interest.
A more common and more robust approach to achieving low lattice thermal conductivity
is to first select materials with desirable electronic properties and then modify them through
processing and composition adjustments in order to decrease the phonon mean free path l.
This can be thought of qualitatively as maximizing the structural disorder of the material in
order to limit the propagation of coherent lattice vibrations. There are many different ways
that this can be accomplished, and although thermal resistivities are not strictly additive,
the effects of multiple phonon scattering mechanisms on the relaxation time can generally
be added as τ −1 =

τi−1 [18]. This allows for the simultaneous utilization of multiple

mechanisms to achieve a maximum reduction in thermal conductivity.
Effective phonon scattering mechanisms include lattice strain, atomic mass mismatches,
vacancies, interstitial atoms, mobile dislocations, grain boundaries, resonant rattling modes
of loosely bound filler atoms [22], and any other phenomena that increase the structural
disorder of the material [18]. When considering which of these mechanisms to employ for
thermal conductivity reduction in a specific compound, it is useful to consider the temperature range within which the material is being designed to operate.
At relatively high temperatures (T > ΘD ), the thermal conductivity is typically domi10

nated by inelastic phonon-phonon scattering. The heat capacity is approximately constant
and equal for all materials to the Dulong-Petit value of 3R, where R is the ideal gas constant
[18]. The phonon mean free path decreases with temperature due to the increased frequency
of phonon-phonon interactions, and this behavior dominates the overall temperature dependence of the thermal conductivity in this range.
At the opposite extreme of T < ΘD , not all phonon modes are active, and longer wavelength vibrations with relatively long mean free paths tend to dominate the thermal conductivity. This means that lattice vibrations will generally propagate unhindered until scattered
by grain boundaries. The magnitude of ΘD typically scales with ∼ Tm /4, where Tm is the
melting point, so no change in grain size or morphology is expected within the range of
T < ΘD [23, 24]. This implies that the phonon mean free path is constant, and therefore the
overall temperature dependence of the thermal conductivity is dominated in this range by
the temperature variation of the lattice heat capacity, which rises as T 3 [18]. Thus for low
temperature applications, l can be effectively reduced through grain size reduction and/or
the introduction of nano-scale features such as precipitates. Although this approach is still
useful at higher temperature as well, it is particularly well suited for cryogenic applications
due to limited concerns over nucleation and growth of the nano-scale features.
In between these two extremes, where T ≈ ΘD , the two opposing trends in κl vs T
meet and a peak is observed. It is within this range, where the phonon mean free paths
are smaller than the grain size but phonon-phonon interactions are not yet dominant, that
other scattering mechanisms such as point defects and solid solution mixing can be utilized
to substantially suppress the thermal conductivity.
The inherent difficulty of thermal conductivity reduction within the context of thermoelectricity is finding ways of introducing structural disorder without negatively impacting
11

the material’s electronic properties. This can be extremely challenging in some instances, as
electrons are also scattered by many of the same mechanisms that scatter phonons. In addition to this, the presence of vacancies, interstitial atoms, and/or substitutional atoms can be
expected to alter the bonding environment and doping level of the material due to differences
in electronegativity and valence configuration between these atoms and the original matrix.
One way of addressing these challenges is to seek out substitutional elements that have
similar electronic properties to the element they are replacing. For example, the relaxation
M
time for mass mismatch phonon scattering scales as τ ∝ ∆M

2

[18], where M is the mass of

the host matrix element and ∆M is the mass difference between the substitutional element
and the host. It is thus often effective to substitute in an isoelectronic element that has a
much different mass from the original matrix atoms in order to minimize τ . This idea has
been successfully implemented to enhance the ZT of a wide range of materials [4, 19].
In spite of the varying effectiveness of these thermal conductivity reduction approaches
in different temperature ranges, each can still have an important effect on the thermal
conductivity at all temperatures. It is therefore useful to take a broad-based approach
regardless of a material’s desired temperature range of operation.

1.4.1.2

Electronic properties

A consideration of electronic band structure phenomena is necessary in order to determine
the types of materials that may be of interest for thermoelectric applications, as well as how
these materials may be successfully modified to improve their properties. The dependence
of ZT on both the Seebeck coefficient and electrical conductivity requires a careful balance
of these contraindicated properties, and such a balance can often be found in highly doped
semiconductors. If the doping level is sufficiently high, the real space separation of the dopant
12

atoms can become small enough that their electronic states overlap and form an impurity
band in the density of states. This generally leads to a mixture of metallic and semiconductorlike electronic properties, and such materials are known as degenerate semiconductors. It
can be shown from first principles that within the simple direct gap parabolic electronic band
model, a degenerate semiconductor will have a Seebeck coefficient of the form

α ∝ m∗ T (n)−2/3 ,

(1.6)

where m∗ is the carrier effective mass, T is the absolute temperature, and n is the carrier
density [25]. Similarly, the electrical conductivity of such a material is of the form

σ=

nτ
1
∝ nµ ∝ ∗ ,
ρ
m

(1.7)

where µ is the carrier mobility and τ is the relaxation time between charge carrier scattering
events. It is convenient to combine these properties together in the way that they appear
in the ZT equation, and the resulting term is known as the “power factor” α2 σ. For a
degenerate semiconductor, this term is of the form

α2 σ ∝ T 2

m∗2 µ
m∗ τ
= T2 √ .
√
3n
3n

(1.8)

Writing the power factor this way makes it apparent why research into thermoelectricity
has traditionally focused on the relationships among n, µ, and m∗ , as the power factor
can be optimized through carefully tuning these parameters. Furthermore, although n, µ,
and m∗ can each vary with temperature, the explicit T 2 dependence of the power factor
provides some explanation for why high ZT values are more commonly observed at high
13

temperatures. The results of these analyses show that the best thermoelectric materials
tend to be degenerate semiconductors with carrier densities on the order of 1019 cm−3 [4].
Embedded within n, µ, and m∗ are strong dependencies on the crystal chemistry of a
given material. The carrier effective mass as well as the density and strength of charge carrier
scattering mechanisms can be strongly affected by tunable parameters such as composition,
electronegativity, and valence configuration [12, 25]. A rigorous analysis of these parameters
through first-principles considerations allows for additional convenient combinations and
descriptions of the properties of a semiconductor, from which further optimization strategies
can be derived. Among these is the thermoelectric “quality factor” developed by Goldsmid
[16] that emphasizes the need for high carrier mobility and effective mass in conjunction with
low lattice thermal conductivity, as well as Mott’s formalism describing the dependence of
the Seebeck coefficient on the logarithmic derivative of the electrical conductivity [26].
It is possible through some of these more complex derivations to arrive occasionally at conclusions that provide guidance counter to that of Equation 1.8. For example, consideration
of electron-phonon interactions and the corresponding deformation potential they introduce
can be used to show that materials with lower carrier effective mass may in some instances
actually be more likely to display high ZT [27]. Thus it is apparent that the traditional
guiding principles for engineering high performance thermoelectric materials are necessarily
flexible depending on the type of material and temperature range being considered. The
relationship described by Equation 1.8 therefore is most useful as a basic framework for
anticipating which parameters may at least be relevant when attempting to optimize the
electronic properties of any material for thermoelectric applications.
The necessity of this flexibility in design principles becomes most apparent at room temperature and below, where the physics of semiconductors makes high ZT values increasingly
14

difficult to obtain. One of the more thorough treatments of this is provided by Slack, who
has proposed that ZT values as large as four may be possible at room temperature in
appropriately doped narrow-gap binary semiconductors that approximate the ideal PGEC
conditions [12]. The results of his derivation indicate that once a semiconductor-like material has been engineered to reach the amorphous limit of lattice thermal conductivity, ZT
then depends nearly exclusively on the weighted mobility of the carriers, which is defined as
∗
U = µ mo
m

1.5

, where mo is the free electron mass.
2

Slack shows that this parameter must be at least 150 cms in order to reach ZT values
V
of unity at room temperature. The need for such large mobility values emphasizes why the
development of high ZT p-type compounds is especially difficult, as the weighted mobility
of the carriers in such systems is often much lower than this.
Slack also acknowledges that very few binary systems appear to exist that actually possess
this desirable combination of electronic band structure properties, although he does suggest
that some suitable ternary semiconductor systems may be found [12]. In either case, it is
necessary to carefully tune the composition so as to maximize U . Since this parameter is
generally affected by the same scattering mechanisms that are useful for reducing κl , however,
2

the threshold of 150 cms is certainly a conservative estimate of the minimum value needed.
V
This conclusion suggests that the conventional semiconductor paradigm may be prohibitively narrow in its description of the relevant electronic properties at cryogenic temperatures. It is therefore worth considering alternative perspectives to determine if one of these
may be more suitable for developing high ZT materials within this temperature range.

15

1.4.2

Need for an alternative approach

The limited usefulness of semiconductors for low temperature thermoelectric applications
suggests that a broader perspective may be necessary if materials with sufficiently high ZT
values are to be developed. It is therefore useful to reconsider the basic form of the ZT
equation in order to more generally evaluate what material systems may be appropriate.

1.4.2.1

The case for metals

This analysis begins by inserting the W-F law of Equation 1.2 into the original form of ZT
in Equation 1.1. After making the standard assumption that only the electronic and lattice
contributions to the thermal conductivity are significant, it is possible to express ZT as

ZT =

α2
,
L(1 + κ )

(1.9)

where κ = κl /κe is the ratio of the lattice and electronic thermal conductivities [4]. Writing
ZT in this way parameterizes the thermoelectric efficiency of all material at all temperature
into a function of only two variables, α and κ . The relative uniformity of L among a wide
variety of bulk materials establishes the basic criteria that high ZT values of unity or larger
will only be observed in materials with Seebeck coefficients on the order of 150 µV/K or
larger, regardless of the magnitude of κ .
In the limiting case of κ >> 1, which generally applies for semiconductor materials, it is
convenient to use the conventional optimization strategy discussed above. That is, κe can be
ignored and ZT can be expressed as ZT ≈ T (α2 σ)/κl . The optimization of ZT then focuses
on finding materials with low lattice thermal conductivity and large power factors. The basic
problem with semiconductors at low temperatures is that the electrical conductivity (and
16

therefore the power factor) tends to become prohibitively small as T decreases. This trend,
combined with the existence of a minimum possible value for the lattice thermal conductivity,
makes large ZT very difficult to obtain using this approach.
Alternatively, the form of Equation 1.9 suggests that high ZT can also be found in materials with small κ . In the limiting case of κ << 1, ZT becomes a simple function of the
Seebeck coefficient only and has the form ZT ≈ α2 /L. Based on the Sommerfeld value of L,
the ZT of such a material can be estimated as ZT ≈ (α/156)2 . This broadens the focus of
ZT optimization to include the less common idea of maximizing the electronic thermal conductivity in addition to the traditional emphasis on reducing the lattice contribution. Figure
1.1 has been included to demonstrate the importance of metallic resistivity for achieving high
ZT at low temperatures. The plot shows the values of α necessary between 10 K and 1000
K to reach a ZT of unity for various fixed ρ, assuming L = L0 and κlattice = 0.1 W/m-K.
These results are consistent with Slack’s argument regarding the minimum U necessary
2
for ZT of unity at room temperature. A semiconductor with a Hall mobility µ of 150 cms , a
V

typical doping level n of 1019 cm−3 , and an effective mass m∗ ∼ m0 would have a resistivity
on the order of 10−5 Ω-m. Although Slack’s calculation allows for some flexibility in regards
to the range of suitable effective mass and doping level values appropriate for such a material,
his result fits within the range of values plotted in Figure 1.1 [12]. The analysis presented
here simply arrives at this conclusion from a more general set of criteria.
From Figure 1.1 it can be seen that even if a material is engineered to reach thermal
conductivity values in the amorphous range of κl = 0.1 W/m-K, achieving κ ≤ 1 at 100 K
requires the material to also display metallic electrical resistivity on the order of 10−5 Ω-m
or less. Since achieving the minimum lattice thermal conductivity condition is not trivial,
it is more reasonable to assume a value for κl of approximately 1 W/m-K or even larger,
17

Figure 1.1: Plot of α necessary for ZT = 1 at various temperatures for the given values of
ρ based on L = L0 and κlattice = 0.1 W/m-K. Increasing κlattice to a more reasonable 1
W/m-K shifts each line up one place.

18

which then requires a further reduction in resistivity to ρ ≤ 10−6 Ω-m to maintain κ ≤ 1,
and ρ ≤ 10−7 Ω-m to improve this ratio to κ ≤ 0.1. Meanwhile, even if these low electrical
resistivity and lattice thermal conductivity values are achieved, it is still necessary to have
a Seebeck coefficient of at least 150 µV/K or larger to reach ZT values near unity.
This latter requirement is the problem that arises with conventional metals, which all
have Seebeck coefficients on the order of only a few microvolts per kelvin [17]. This is two
orders of magnitude lower than the range of values needed to achieve reasonably high ZT .
For this reason, metals have traditionally been dismissed outright as candidate systems for
thermoelectric applications [25]. Within this same temperature range, however, conventional
semiconductors have resistivity values orders of magnitude too large to make high ZT possible. These trends together summarily explain why essentially no high ZT materials have
yet been developed at cryogenic temperatures.
Based on this analysis, it becomes apparent that one or more of the traditional assumptions in thermoelectrics must give way if thermoelectric cryocoolers are to ever become a
reality. This may be done by finding materials that violate the W-F law, display Lorenz
numbers much smaller than the Sommerfeld value, and/or have lattice thermal conductivity
values much lower than the “amorphous limit” of 0.1 W/m-K. Although each of these approaches may be theoretically possible, the few reported attempts at pursuing them suggest
that they are most likely to be realized only in thin films, heterostructures, and/or through
other extrinsic effects [28, 29]. The most likely path to achieving high ZT values in bulk
materials at cryogenic temperatures therefore appears to be through the development of compounds with metallic ρ ≤ 10−6 Ω-m and semiconductor-like α∼200 µV/K. To explore how
this unusual combination of properties may be possible, it is useful to consider the available
mechanisms for introducing large Seebeck coefficients into otherwise metallic materials.
19

1.4.2.2

Increasing the Seebeck coefficient

The search for large α begins by considering a simple dimensional analysis, which shows that

α∝

(charge)(voltage)
energy/temperature
entropy
voltage
∝
∝
∝
.
temperature
(charge)(temperature)
charge
charge

(1.10)

The Seebeck coefficient can thus be thought of as a measure of the entropy per charge carrier.
This broad qualitative analysis can be extended by considering the fundamental postulate
of statistical mechanics, which relates entropy S and degeneracy g through the relationship
S ∝ ln g [30]. Based on this relationship, it can be reasoned that α should be large when
there are a large number of nearly energetically equivalent atomic states close to the average
energy of the charge carriers.
This general notion is essentially equivalent to the argument developed by Mahan and
Sofo, who more formally and more rigorously showed that α should be large in materials
with sharp peaks in the electronic density of states (DOS) near the Fermi level [31]. This
theory suggests that the closer these sharp features are to a pure delta function, the larger
α should be. Since this theory makes no direct restrictions on the nature of the electrical
resistivity, it may be used as a general guide to engineering materials with simultaneously
large α and metallic ρ.
The validity of this paradigm can be demonstrated by considering the origin of favorable
thermoelectric properties in many different materials. The simplest example of this is in
materials with highly symmetric crystal structures, as a more symmetric real-space arrangement of atomic potentials can be used to create a more symmetric and therefore degenerate
Fermi surface in momentum space. This particular argument has been invoked to explain, for
example, how “valence band convergence” contributes to favorable thermoelectric properties
20

in PbTe, which forms in the cubically symmetric rocksalt structure [32].
Other more complex mechanisms for increasing charge carrier degeneracy have also been
developed. Among these is the manipulation of configurational and spin-orbital degeneracy in
compounds such as transition metal oxides that rely on hopping mechanisms for conduction
[33]. Other viable approaches include the introduction of electronic resonant states near the
Fermi level (e.g. Tl-doped PbTe [34]) and the exploitation of strong electronic correlations
in materials containing rare earth elements [35].
Of these possibilities, the two that appear most likely to result in bulk materials with
metallic ρ and large α are resonant level effects in semi-metals and strongly correlated electron effects in rare earth intermetallic alloys. The work of Mahan and Sofo directly supports
this conclusion, as they describe f electronic levels as “nature’s closest approximation to a
delta function” in the DOS [31]. It is therefore of central importance to the development
of cryogenic Peltier cooling technology to further explore these material systems and better understand how structure-processing-property relationships may be exploited in them to
achieve enhancements in ZT .

1.5

CePd3

Since strongly correlated rare-earth based compounds have been identified as perhaps the
best way to achieve simultaneously metallic ρ and large α [31], it is appropriate to consider
the origin of these effects in order to understand how they may be controlled and optimized.
An array of unusual electronic, structural, and magnetic properties are often observed
in rare-earth compounds as a result of interactions among the f electronic states and the
conduction band [36]. These interactions can sometimes result in a phenomenon known

21

as intermediate valency, wherein the rare-earth atoms display a non-integer time averaged
valence configuration. In simple terms, this is due to the oscillation of electrons between the
nominally localized 4f states and the delocalized conduction band that causes the rare earth
ions to vary in time between two valence configurations.
This effect is similar to mixed valency (MV), which is another commonly observed phenomenon in these materials. These effects are different, however, in that MV materials
contain rare earth ions in distinct valence configurations that do not nominally change with
time, while the rare earth ions in IV compounds are each continuously fluctuating between
different configurations [36].
The fluctuating valence of rare earth ions in IV materials often corresponds with a large
peak in the electronic density of states that often results in unusually high Seebeck coefficients
of tens or even hundreds of microvolts per kelvin [37]. This effect tends to be especially
pronounced in compounds containing cerium and ytterbium, wherein the presence of one
4f electron in cerium and one 4f hole in ytterbium often results in p-type and n-type
behavior in these materials, respectively. Other rare earth elements, in particular europium
and samarium, have also been known to display mixed and/or intermediate valence effects
in certain compounds [38]. The abundance of both p-type and n-type materials displaying
these correlated electron effects is encouraging in relation to their potential development for
thermoelectric applications.
CePd3 is one particularly well known p-type IV compound that has been extensively
studied over the last several decades for various purposes. A clear indication of the IV effect
in this material appears in the form of a significant deviation of the lattice parameter from
the lanthanide contraction trend seen in the other RPd3 compounds, where R is a trivalent
rare-earth element [39]. This lattice parameter anomaly is common for both IV and MV
22

compounds, and it arises from the significant size difference between the effective atomic
radii of the R elements in their different valence states [39]. This property is also often
accompanied by anomalies in the magnetic susceptibility, electronic specific heat, electrical
resistivity, M¨ssbauer absorption spectra, and x-ray absorption near edge spectra, which can
o
be used to distinguish between IV and MV effects [40].
CePd3 is an especially attractive candidate for cryogenic TE cooling due its desirable
combination of ρ ∼ 10−6 Ω-m concurrent with α ∼ 115µV/K near 125 K. Both of these
properties fall exactly within the range of values needed to achieve reasonably high ZT
at low temperatures. In the limiting case of κ << 1, CePd3 has a maximum potential
ZT ≈ (115/156)2 ≈ 0.5.
The main problem with this material is its relatively high lattice thermal conductivity
that results in an unfavorably large κ of approximately 5 near 100 K that drops to below
unity above 200 K. These trends are summarized in Figure 1.2, which shows typical data
for CePd3 available in the literature [41]. The combination of these properties results in
ZT values for unmodified CePd3 of approximately 0.1 at 100 K and 0.2 at 300 K. The
fundamental challenge that emerges is thus finding suitable mechanisms for reducing κ
and/or enhancing α in this material in order to increase ZT .
Several attempts have been made over the last several decades to improve this material’s
thermoelectric properties. With two lone exceptions, however, every reported attempt has
shown a net decrease in ZT . Although the introduction of point defects through solid solution
mixing and interstitial site filling appears to sharply reduce the thermal conductivity [42, 43],
the IV state also tends to rapidly collapse when the physical and/or electronic structure of
the material is perturbed. This reverts it back into a conventional metal, at which point it
becomes a poor thermoelectric material.
23

Figure 1.2: Seebeck coefficient α (dashed line) and thermal conductivity ratio κ (solid line)
of stoichiometric CePd3 [41]. The semi-arbitrary target values of 156 µV/K for α and unity
for κ are indicated by the horizontal dashed lines connected to the left and right ordinates,
respectively.

24

Since the basic goal of any thermal conductivity reduction strategy is to maximize the
structural disorder of the material, the effects of these strategies tend to be inherently at odds
with the sensitivity of the IV state to such perturbations. Finding solutions to this impasse
requires the consideration of a broad range of structure/processing/property relationships
that are not generally well understood in these materials. A comprehensive approach is thus
needed in order to map the connections among the relevant material properties and then
determine what mechanisms may be appropriate for enhancing ZT . Since CePd3 is widely
considered a good prototype IV compound, any insights gained concerning the relationships
among its properties are likely to be useful in other similar materials as well.
In order to probe these relationships, several series of CePd3 -based compounds were
synthesized, characterized, and analyzed for this study. The following is a brief and general
review of the literature on this compound intended to provide an appropriate context for
interpreting these results. Included in this review are several relevant structural, electronic,
and magnetic property motifs that serve as useful reference points for comparing how the
material’s properties change in each of the series discussed subsequently.

1.5.1

Structural properties

CePd3 is a relatively brittle compound that forms in the cubic Cu3 Au (L12 ) structure with
cerium atoms occupying the corners of the cube and the palladium atoms occupying the faces,
as illustrated in Figure 1.3. The simplicity of this compound’s chemical formula and crystal
structure allow for a robust variety of substitutions on both the cerium and palladium sites,
including complete solid solutions with many different isostructural RPd3 and CeT3 -type
compounds, where R is a group III element and T is a transition metal or noble metal.
The effects of such substitutions on the electronic, magnetic, structural, and thermo25

Figure 1.3: Diagram of the filled Cu3 Au crystal structure of CePd3 . The cerium atoms
(dark gray) occupy the corners and the palladium atoms (medium gray) occupy the face
centers. The central 1b interstitial site is identified here by the light gray sphere. This site
is unoccupied in stoichiometric CePd3 , but it can be partially filled with various elements
M to form CePd3 Mx .

26

electric properties of this compound have been studied extensively, with no reports of any
structural ordering in any of the Ce1−x Rx Pd3 or Ce(Pd1−x Tx )3 compounds for which data
are available. This allows for a large phase space of possible substitutions on one or both
of the lattice sites without concern over the secondary effects of structural ordering on the
lattice dynamics and electronic structure of the material.
The central 1b octahedral interstitial site can also be partially filled with various elements,
and much work on compounds such as CePd3 Bx and CePd3 Six with 0 < x < 1 has been
reported [39, 44, 45, 46, 47, 48, 49, 50, 51, 52]. This partial filling has the effect of locally
transforming the filled unit cells into an analog of the perovskite ABO3 structure. Extended
structural ordering effects have been widely reported in some of these compounds [53, 54,
55, 56, 57], along with saturation limits for x that vary with the identity of M [44, 55]. In
spite of this extensive study, the origins of these effects and their explicit impact on the
thermoelectric properties of the material are generally not well understood.
Further details on the crystal structure and morphology of the material are presented
below in the introductions to the relevant chapters.

1.5.2

Electronic and magnetic properties

A consequence of the IV effect in CePd3 is that the cerium atoms fluctuate between the
magnetically active Ce3+ and non-magnetic Ce4+ configurations at a frequency on the order
of 1013 Hz [58]. This fluctuation between magnetic and non-magnetic states results in a
random distribution of magnetic moments throughout the lattice that rapidly appear and
disappear. Since the average valence of the cerium atoms is approximately +3.5 at room
temperature [59, 58], roughly half of the unit cells contain a trivalent cerium atom at any
given time. When a conduction electron encounters one of these atoms that has an electron
27

in its 4f shell, the conduction electron can undergo spin-flip scattering with this 4f electron.
This process is broadly known as the Kondo effect, and it is often apparent in the transport
properties of such materials through a ln T contribution to the electrical resistivity [60]. The
strength and nature of this interaction can vary significantly in different materials depending
on things like the spatial distribution of the magnetic moments and the Fermi energy of the
conduction band. In CePd3 , the high concentration of magnetic ions leads to the description
of this material as a “Kondo lattice”, and the strength and nature of the Kondo interaction
falls within the range referred to as a “lightly doped Kondo insulator” [61].
Every Kondo system has at least one important energy scale described by the Kondo
temperature TK ; below TK , various magnetic ordering effects can occur, while above TK
incoherent spin-flip scattering can have important consequences for the electronic properties [60]. Various theoretical models exist for describing how the Kondo effect can alter a
material’s electronic and magnetic properties [62]. In some cases, it can have an important
impact on the electronic density of states in the form of a so-called “Kondo resonance” or
“Kondo gap”, which can often result in a large Seebeck coefficient in these materials [35]. In
this way, the value of TK can be connected to the magnitude and temperature dependence
of the Seebeck coefficient in Kondo-type systems [63, 64].
The Kondo effect is also explicitly important due to the direct connection it establishes
between a material’s magnetic properties and the temperature dependence of its electrical
resistivity [60]. This connection allows for a broader perspective regarding how the electronic,
magnetic, and structural properties of the material are related, including the ways that
changes in these properties correlate with composition and processing modifications.

28

1.5.2.1

Electrical resistivity

The temperature dependence of the resistivity for a Kondo-type metal such as CePd3 can
be very generally described as a sum of contributions from different terms [65]. Each of
these terms is more or less prominent in different temperature regimes depending on the
relative magnitudes of the Debye temperature ΘD and the Kondo temperature TK . This
relationship is described by an extension of Matthiessen’s rule [66], and is given by

ρ = ρ0 + a0 T + b0 T 2 + c0 T 5 + d0 ln T,

(1.11)

where ρ0 is the residual resistivity at 0 K resulting from defects, a0 is the coefficient of the
linear contribution from lattice vibrations (electron-phonon scattering) for T ∼ ΘD , b0 is the
coefficient of the Fermi liquid (electron-electron scattering) contribution typically relevant
for very low temperatures only, c0 is the coefficient of the higher order contribution from
lattice vibrations (electron-phonon scattering) for T << ΘD , and d0 is the coefficient of the
Kondo term (spin-flip scattering).
The magnitude of d0 in paramagnetic Kondo metals such as CePd3 can be described by
the relationship
d0 ∝ J

m∗
n

J
EF

2

,

(1.12)

where J is the exchange energy between the localized and itinerant electrons and EF is the
Fermi energy [60]. One key aspect of this result is that the sign of J determines whether
this additional contribution to the resistivity increases or decreases with temperature. In
the case of CePd3 , d is negative and large such that the resistivity decreases as ln (1/T ) for
TK < T ∼ ΘD .

29

This type of temperature dependence is one of the defining traits of a metallic Kondo
system, as it is highly unusual for resistivity to decrease with increasing temperature in a
normal metal [17]. One way of experimentally estimating TK is to determine the lower limit
of the temperature range within which the ln (1/T ) description applies [60]. Since the nature
of the interaction between the conduction electrons and the local 4f states changes below
this temperature, it can be anticipated that other electronic and magnetic properties will
show dramatic changes near TK as well.
Generally speaking, the Kondo effect, the IV state, and large α should always be observed
together in CePd3 , since all three share similar origins. It is useful then to study how TK
changes with various modifications schemes, as information can be extrapolated from these
trends regarding the limiting conditions within which large α will be observed. Since the
peak values of ρ and α can both vary with TK [42], this information may also allow the
properties of the material to be tuned in such a way that they are optimized within a
specific temperature range.
Based on Equation 1.12, variations in n and EF introduced through different mechanisms may also have significant impacts on the strength of the Kondo effect. This can affect
the relative magnitude of the coefficients in Equation 1.11, in turn changing the magnitude
and temperature dependence of ρ. Both n and EF play an important role in the magnitude and temperature dependence of the Seebeck coefficient as well, emphasizing the strong
interdependence of all the electronic and magnetic properties of the material.

1.5.2.2

Hall coefficient

The Hall coefficient RH is generally a reliable measure of the free carrier density n in most
materials where electronic transport is dominated by one type of carrier. This relationship
30

is given by
V d
1
RH = − H =
,
IB
ne

(1.13)

where VH is the Hall voltage, d0 is the sample thickness perpendicular to the direction of
the applied magnetic field B, I is the excitation current, and e is the elementary charge [17].
In strongly correlated systems, the combination of unusual electronic and magnetic effects
has been shown to result in anomalous features in the Hall coefficient that can limit the
effectiveness of this metric as a direct probe of the carrier density in these materials. For
example, the sign of RH changes from positive to negative in CePd3 below approximately 50
K [67]. This type of sign change is usually indicative of a crossover in the majority carriers
from holes to electrons [17]. In CePd3 , however, the sign of the Seebeck coefficient is positive
for all T > 0, suggesting that holes remain the dominant carriers at all temperatures. This
inconsistency can be resolved by quantum mechanical considerations that show the sign of
RH must be positive in this material for T >> TK , but it can vary for T ≤ TK [68]. This
implies that RH is not an appropriate measure of the carrier density for T ∼ TK , although
it may be more useful for T >> TK .
Even at higher temperatures, however, some reports exist that indicate the presence of
an anomalous Hall effect in CePd3 that is caused by skew scattering of the free carriers
resulting from an asymmetric splitting of the Kondo resonance in a magnetic field [65]. For
T > 180 K ∼ TK , RH can be described by the relationship

RH = R0 +

A
,
T +θ

(1.14)

where R0 is a constant similar to the values measured in non-magnetic analog materials and
A and θ are temperature independent material parameters [67]. Based on the combination
31

of these observations of the Hall coefficient at both high and low temperatures, RH can only
loosely be utilized as a measure of the carrier density in this material.
Comparisons of n values determined by RH measurements in CePd3 can be made with
other measurements, such as optical reflectivity, that have been successfully used to determine a carrier density in CePd3 on the order of 1021 cm−3 [61, 69, 70] with a carrier effective
mass of approximately 40 times the free electron value [69]. These experiments produce consistent results within the range T ∼ TK , suggesting that the high temperature correction
does not significantly affect the utility of RH for measuring n.

1.5.2.3

Magnetic susceptibility

CePd3 can generally be described as a paramagnetic material, as it displays a positive
magnetic susceptibility χ without evidence of spontaneous ordering [44]. A Curie-Weiss
description is therefore appropriate for fitting the temperature dependence of χ [21]. This
description is given by
χ = χ0 +

C
,
T −θ

(1.15)

where χ0 is a temperature independent term arising from Pauli paramagnetism, C is the
Curie constant, and θ is the Ne´l temperature, below which antiferromagnetic ordering
e
occurs.
The value of C is dependent on various magnetic properties, and for a two-state system
it is proportional to the square of the total magnetic moment of the material [17]. Since
trivalent cerium carries a magnetic moment while tetravalent cerium does not, C will change
with the average valence of the cerium atoms, and can thus be utilized as a probe of the IV
effect as the material is modified in various ways.

32

Direct determination of the effective moment of the cerium atoms in CePd3 is not trivial
using this approach, as it requires at least a few assumptions regarding the nature of the
magnetic properties. Equation 1.15 includes two unknown parameters (χ0 and θ), and thus
a measurement of χ as a function of T is insufficient to completely determine C. The value
of χ0 may nominally be calculated from first principles for a conventional metal, and it
is usually small compared to χ [17]. This often leads to the assumption that this term
can be neglected. However, given the close correlation in CePd3 between the behavior of
the conduction electrons and valence of the cerium atoms, there are subtle relationships
embedded among χ0 , θ, and C that can make it difficult to justify this assumption.
It is nonetheless useful to approximate Equation 1.15 as

χ = χ0 +

C
,
T

(1.16)

which allows for the experimental determination of C from plots of χ versus 1/T . When
this fit function is applied over a relatively small range of temperatures, it often produces
an approximately linear relationship among the variables. This allows for a self-consistent,
semi-quantitative determination of C from which comparisons can be made regarding the
magnetic moment of the cerium atoms in different CePd3 -based compounds.

1.6

Thermal properties

One of the unique aspects of IV compounds in relation to thermal conductivity is the proposed existence of anomalous electron-phonon interactions that occur as the valence state
and size of the R atoms concomitantly oscillate. Experimental evidence for these “breathing”

33

effects have been reported in certain valence fluctuating compounds such as Sm0.75 Y0.25 S
[71] and SmS [72]. For this type of interaction to be possible, the characteristic “fluctuation
temperature” Tf , which is proportional to the width of the 4f peak in the DOS and is associated with the frequency of valence oscillations, should be lower than the material’s Debye
temperature ΘD [73].
These parameters in CePd3 have been found to be 220 K [74] and 292 K [73], respectively,
suggesting that it may be possible to observe the breathing effect in this compound. Some
experimental evidence exists to support this possibility in the form of anomalous force constants in the <100> direction for T ≥ 140 K observed in phonon dispersion data obtained
through neutron diffraction experiments [75]. Optical absorption experiments indicate that
this breathing term may correspond with an optical mode, which suggests that it does not
have a significant impact on heat propagation in the material [76]. Although the impact of
this breathing phenomenon on the thermal conductivity is not trivial to predict, it can be
anticipated that this effect may be relevant when considering which possible solid solutions
may be useful for thermal conductivity reduction.
In addition to this consideration, it has been shown that the Lorenz number of correlated
metals is likely to be larger than the nominal Sommerfeld value L0 derived for a free electron
gas [65]. This deviation is estimated to be as large as a factor of two, which suggests that
using L0 to determine the electronic thermal conductivity of materials like CePd3 provides
only a conservative estimate of the lattice contribution. Since typically κe ∼ κl in these
materials, it is important to keep this in mind when evaluating the effectiveness of any
thermal conductivity reduction strategy, as the lattice contribution may be even smaller
than the calculated values suggest.

34

Chapter 2
Experimental methods
The same general procedures were carried out for all of the series of materials examined
in this study. Certain material systems required some deviation from these procedures,
and these deviations have been specified in the appropriate sections. Characterization and
measurement conditions were held constant in all cases to the extent possible in order to
facilitate uniform comparison of material properties.

2.1

Materials synthesis

Samples were synthesized by combining pure elements together in the appropriate stoichiometric ratios and placing them onto a water-cooled copper hearth inside of a small arc
furnace. The elements were obtained from commercial suppliers Sigma Aldrich and Alfa
Aesar. The furnace chamber was evacuated by a rotary pump then refilled with flowing
argon gas at a pressure of approximately 1 atm. Rare earth elements known to rapidly
oxidize were mechanically polished immediately before use in an argon-filled glove box. A
current of 50 A to 100 A was passed from the arc melter anode through the pieces to melt
them together. The resulting ingots were flipped and remelted several times to promote
homogeneity and ensure all starting pieces had been melted. Mass losses as a result of this
process were generally on the order of 1% to 2% and assumed to be stoichiometric unless
otherwise noted. The typical total mass of the samples was between 1.5 g and 5 g.

35

Additional synthesis steps were required for samples in the CePd3 Mx and the nanostructured series. These additional steps have been specified in the respective chapters below.

2.2

Materials characterization

X-ray diffractometry (XRD) was utilized on the samples after synthesis to examine their
crystal structure, lattice parameters, and phase purity. Small pieces of each sample were
crushed into powder with a mortar and pestle then placed on a glass slide and inserted
into a Rigaku Miniflex II x-ray diffractometer with a Cu K-α x-ray source of λ = 0.154
nm in order to obtain diffraction patterns. The Jade 9.0 software package with the JCPDS
XRD pattern database was used to characterize the phases present based on these patterns.
Rietveld refinement [77] of the XRD patterns was performed using the open-source program
General Structural Analysis System (GSAS) [78, 79] based on the appropriate structure type,
atomic coordinates, and occupation numbers for each of the materials examined.
Scanning electron microscopy (SEM) in conjunction with energy dispersive x-ray spectroscopy (EDS) was also performed on several of the samples in order to examine their
microstructure, homogeneity, and phase purity. This analysis was performed at the Michigan State University Center for Advanced Microscopy using a JEOL 64000V SEM. Powder
samples were adhered to the aluminum sample holders with carbon tape, while larger pieces
were secured to the holder with epoxy. Unless otherwise noted, all analyses were performed
on fracture surfaces.

36

2.3

Transport property measurements

The thermoelectric properties of the samples were measured in a pair of Janis flow cryostats
using a four probe technique. After the appropriate synthesis steps were completed, the
samples were cut with a diamond tipped rotary saw into rectangular prismatic bars approximately 2 mm x 2 mm x 5 mm to 10 mm. One end of these bars was soldered directly to a
rectangular copper bar with a small amount of phosphoric acid used as a flux to facilitate
the wetting of the solder between the sample and the copper surface.
A small heater consisting of an 800 Ω resistor wrapped inside a sheath of copper foil was
soldered onto the opposite end of the sample. The copper bar was attached to the cryostat
and copper/constantan thermocouples were attached to the sample at two different points
separated by distance l between the heater and the base. The entire apparatus was then
evacuated to a pressure of approximately 10−7 torr. The temperature of the sample was
controlled by a flow of liquid nitrogen outside of a primary copper radiation shield balanced
by a block heater inside of the cryostat base. The entire apparatus was allowed to reach
thermal equilibrium at each temperature point by waiting approximately 25 minutes before
any data were collected. This experimental setup is illustrated in Figure 2.1.
The electrical resistivity was measured by applying a current Iρ of approximately 0.1 A
through the length of the samples and measuring the voltage Vρ between the copper leads of
the two thermocouples. The current was reversed and the voltage was measured again, and
this value was averaged with the first measurement in order to minimize the effects of any
contact resistance and Seebeck voltages arising between the sample and the contacts. The

37

Figure 2.1: Schematic of the experimental setup used for transport property measurements.
For interpretation of the references to color in this and all other figures, the reader is referred
to the electronic version of this dissertation.

38

resistance ρ was then determined by the relationship

ρ=

Vρ
Iρ

wd
,
l

(2.1)

where w and d are the sample width and thickness, respectively. The total error in this measurement is estimated to be approximately 3%, resulting mostly from the limited precision
of the sample dimension measurements.
After this measurement, the heater was activated by passing a current IH of approximately 4 mA through the 800 Ω resistor. The system was allowed to reach a dynamic thermal
equilibrium by sitting for five minutes before collecting any data. The Seebeck coefficient
S = (∆Vα )/(∆T ) was determined by measuring the voltage difference ∆Vα between the copper leads of the thermocouples then dividing this by the temperature difference ∆T between
the leads. The error in this measurement is estimated to be approximately 3% based on
the sensitivity of the thermocouple probes for determining the temperature gradient and the
relative magnitude of the Seebeck voltages generated by the sample and the measurement
apparatus.
The thermal conductance was then calculated based on the relationship

I V
Kmeasured = H H ,
∆T

(2.2)

where VH is the voltage drop measured across the resistive heater. A radiation loss correction
of
Ksample = Kmeasured − Kradiation

(2.3)

was applied, where Ksample is the true thermal conductance of the material and Kradiation
39

is determined by the relationship

Kradiation = (0.00146)

T 3 W
,
300
K

(2.4)

where T is the average absolute temperature of the sample. The value of 0.00146 W/K was
determined by measuring a similarly sized sample of fused quartz at 300 K while the sample
was disconnected from the copper base.
The total thermal conductivity κ of the sample was then determined by the relationship

κ = Ksample

l
.
wd

(2.5)

Typical errors in the thermal conductivity data are estimated to be 10%, resulting from a
combination of the limited precision of the sample geometry measurements, the radiation
loss correction, and the accuracy of the thermal gradient measurements.

2.4

Magnetic property measurements

The magnetic susceptibility of some of the samples was measured in a Quantum Design
superconducting quantum interference device (SQUID) magnetometer in the Physics and
Astronomy Department at Michigan State University from 2 K to 300 K in a constant
magnetic field of 5.5 T. The samples were prepared by placing them into a thin plastic
pouch that was then rolled up and placed into a plastic straw. This straw was inserted into
the SQUID, then the chamber was evacuated and the sample was cooled to 2 K. The samples
were allowed to sit at each temperature point for approximately 15 minutes in order to reach
thermal equilibrium before the susceptibility was measured.
40

The Hall coefficient RH of several samples was measured from 80 K to 300 K using a
standard five probe technique. A rectangular prismatic bar of each sample was prepared by
cutting larger ingots with a diamond saw. These bars were then sanded to create sufficiently
thin samples with dimensions of approximately 2 mm x 0.3 mm x 5 mm. The samples
were placed on top of the copper base of a cryostat with two layers of insulating paper in
between, with a non-conducting varnish used to adhere each layer to the next. Silver-based
adhesive was used to connect silver wire to the samples in the appropriate arrangement,
as indicated in Figure 2.2. A magnetic field was applied perpendicular to the slab and the
resulting Hall voltage was measured. The field strength was varied from approximately -1.5
T to +1.5 T. A LakeShore 370 AC resistance bridge accurate to 10 nΩ was used to measure
the Hall resistance while pulsing an excitation current of approximately 30 mA. The Hall
coefficient was then extracted from plots of the measured Hall voltages versus applied field.
The magnitude of the carrier density n was estimated by the relationship given by Equation
1.13.

41

Figure 2.2: Schematic of the experimental setup used for Hall coefficient measurements.

42

Chapter 3
Cerium-site substitutions:
Ce1−xScxPd3
3.1

Background and motivation

Solid solutions of the type Ce1−x Rx Pd3 have been reported to form for a wide variety of R
elements, including gadolinium [80], yttrium [80, 81, 82, 83], neodymium [83, 84], lanthanum
[83, 85], thorium [84], scandium [86], uranium [87], and manganese [88]. Available reports
on the thermoelectric properties of these alloys typically show a substantial decrease in the
Seebeck coefficient as x is increased for all R elements, although relatively large substitution
levels near 20% can sometimes be obtained before any significant change in α occurs [83].
This relatively large substitution window suggests that it may be possible to reduce κl
through solid solution mixing without substantially affectng α. It is not clear from these previous studies how effective cerium site substitutions are in reducing the thermal conductivity,
as very few of them report such data.
Solid solution mixing has a maximum impact on κl when the substitutional element has
a maximum mass and/or size mismatch with the host matrix [18]. Because of this, scandium
appears to be an optimum choice for exploring the potency of cerium site substitutions as
a means of thermal conductivity reduction due to its significantly smaller atomic size and
mass relative to cerium.
43

Previous work by Gambke, et al. has demonstrated that the Ce1−x Scx Pd3 solid solution
forms for all x [86]. The lattice parameter generally decreases linearly with x, although this
dependence appears to have two different slopes for x < 0.3 and x > 0.3. The decrease is
relatively steep for 0 < x <0.3 then becomes more gradual for x > 0.3. Gambke attributes
this to a combination of Vegard’s law superimposed on a change in the size of the cerium
atoms resulting from a shift in the average valence toward the smaller 4+ state that saturates
for x ≈ 0.3. Since large α is not expected outside the range of x where the IV effect is
observed, the range of compositions examined in this study was limited to 0 < x < 0.25.

3.2

Experimental methods

Samples of 0 < x < 0.25 were synthesized and measured in accordance with the procedures
outlined in Chapter 2. No deviations from these procedures were necessary for any of the
compositions in this series, and no heat treatments were applied to any of the samples after
arc melting.

3.3
3.3.1

Results and discussion
Structural properties

XRD patterns obtained for each of the samples indicated single phase samples of the expected
Cu3 Au structure type. No obvious signs of impurity phases or structural ordering were
apparent in the diffraction patterns.
Included in Figure 3.1 are the lattice parameter values determined for each sample
through Rietveld refinement of the corresponding XRD patterns. An approximately lin-

44

Figure 3.1: Lattice parameters of the Ce1−x Scx Pd3 solid solution from Gambke (triangles)
[86] and from this study (circles) as determined by Rietveld refinement of XRD patterns.
The solid line is a linear fit to the data.
ear decrease is observed as x is increased, consistent with previous reports [86].

3.3.2

Electronic properties

The Seebeck coefficient data have been included in Figure 3.2. Consistent with the general
trends widely reported for other Ce1−x Rx Pd3 materials, a relatively large peak of approximately 100 µV/K is observed near 125 K for samples with 0 < x < 0.1. Above this
substitution level, the magnitude drops steadily and the peak flattens out.
This change in the magnitude and temperature dependence of the Seebeck coefficient
correlates with the increased substitution of scandium atoms into the cerium sites. Although
the nearest-neighbor environment of the cerium atoms is not directly affected by this type
of substitution, the smaller size of the scandium atoms induces a contraction of the lattice
45

Figure 3.2: Seebeck coefficient of Ce1−x Scx Pd3 .
parameter that appears to create a chemical pressure on the cerium atoms, pushing them
toward the smaller Ce4+ configuration. This alters the IV state and thus indirectly affects
the Seebeck coefficient.
The electrical resistivity results have been included in Figure 3.3 and show no significant
variation in magnitude for any of the compositions studied. The temperature variation
appears to go through a maximum for x = 0.10 at 80 K before decreasing again. The
magnitude of the Kondo temperature TK appears to be shifting to temperatures below 80
K as x is increased. The temperature dependence of the x = 0.25 sample is distinctly flat
compared to the others, indicating the beginning of a transition away from the ρ ∼ ln (1/T )
Kondo regime to conventional metallic behavior of ρ ∼ T . The onset of this transition is
consistent with the lattice parameter trends in Reference [86].

46

Figure 3.3: Electrical resistivity of the Ce1−x Scx Pd3 solid solution.

3.3.3

Thermal properties

Figure 3.4 has been included to show the temperature dependence of the lattice thermal
conductivity for each sample in this series. It is apparent from these data that scandium
substitution has only a modest affect on the lattice thermal conductivity, as all compositions
show values approximately 1 W/m-K to 1.5 W/m-K lower than those of the unmodified
compound. These values correspond with reductions in magnitude of approximately 25%.
No trend is apparent between the scandium content and the observed reductions, as all
compositions with 0.05 < x < 0.25 show nearly identical values within the error of the
measurement throughout the entire temperature range examined.
The ineffectiveness of cerium site substitutions in reducing the lattice thermal conductivity may be explained by the so-called “breathing” of the cerium atoms discussed previously
that occurs as these atoms rapidly change size at a time scale on the order of the atomic
47

Figure 3.4: Lattice thermal conductivity of the Ce1−x Scx Pd3 solid solution.
vibrations. Since scandium gives the maximum possible size and mass mismatch with the
cerium atoms, any potential reduction in the lattice thermal conductivity resulting from
substitutions on this site should be most easily observed in this system. Since no significant
reduction has been observed, it appears that these lattice sites do not play a significant role
in the net transport of thermal energy through the lattice, consistent with the idea that the
“breathing” term corresponds with an optical phonon mode(s).
Although a modest reduction in the lattice thermal conductivity was observed in these
samples, the corresponding increase in electrical resistivity produced a simultaneous decrease
in the electronic thermal conductivity. As a result, the value of κ is not improved for any
of the compositions studied. The results of this analysis are included in Figure 3.5.

48

Figure 3.5: Thermal conductivity ratio of the Ce1−x Scx Pd3 solid solution.

3.3.4

General conclusions and ZT

The combination of a decrease in the Seebeck coefficient and an increase in κ resulted in no
net enhancement in ZT for any of the samples in this series, as seen in Figure 3.6.
Based on the data presented here in conjunction with the available reports on similar
Ce1−x Rx Pd3 studies, it appears that cerium-site substitutions are an ineffectual approach
for improving the thermoelectric properties of the material. Although the nearest-neighbor
environment of the cerium atoms remains unchanged, the chemical pressure introduced by
the substitution of the smaller scandium atoms pushes the cerium atoms toward the smaller
tetravalent configuration. The thermal conductivity reduction reported here is relatively
large compared to the results of other studies for cerium-site substitutions, but this improvement is insufficient to produce any enhancement in ZT . The proposed “breathing”
effect of the cerium atoms as they fluctuate in size may be related to the limited efficacy of
49

Figure 3.6: Thermoelectric figure of merit of the Ce1−x Scx Pd3 solid solution.
cerium-site substitutions for reducing κl . The implication of these results is that substitution
on the palladium site should have a substantial effect on the lattice thermal conductivity.
Beyond CePd3 , these results suggest that substitutions on the R site in valence fluctuating compounds with Tf < ΘD may not generally be effective for decreasing the thermal
conductivity, depending on the type of phonon mode that correponds with the “breathing”.

50

Chapter 4
Palladium site substitutions:
CePd3−xPtx
4.1

Background and motivation

The results of the Ce1−x Scx Pd3 study in Chapter 3 suggest that the cerium site plays only
a minor role in phonon propagation. This in turn implies that palladium site substitutions
should be an especially effective approach for reducing the thermal conductivity. Reports
on CePd3−x Rhx and CePd3−x Agx solid solutions by Lackner [43] have shown that this is
generally true, as that work has demonstrated a relatively substantial reduction in lattice
thermal conductivity of approximately 50% for those systems as x is increased.
Those reports also indicated that these substitutions have a significant impact on the
other properties of the material, including the Seebeck coefficient, electrical resistivity, and
lattice parameter. Although these changes tend to be detrimental overall, Lackner does
report a slight increase in ZT for one sample of CePd2.7 Rh0.3 . He shows that ZT values
up to approximately 0.15 at 100 K and 0.25 at 300 K can be reached in this compound as
a result of balancing the reductions in both κl and α. These ZT values correspond with
increases of approximately 50% and 10%, respectively, over the base compound. A similar
study by Sales [37] briefly mentions an increase in ZT up to approximately 0.15 at 100
K for CePd2.655 Pt0.3 , although no other transport data or compositions in the series were
51

Figure 4.1: Reported lattice parameters for CePd3−x Tx (T = Ni [89], Ag [89], Rh [59], Pt
[90]) series. Solid lines are a guide for the eye.
included in that report. It is worth noting that these are the only two reports that have ever
indicated an increase in ZT relative to unmodified CePd3 , and both result from palladium
site substitutions.
Based on these limited studies, it appears that there may be additional opportunities
for improved thermoelectric performance within the CePd3−x Tx phase space. A simple way
to filter through the possibilities for T is to consider the strong correlation between the IV
state and the lattice parameter. This connection implies that any large change in the size
of the unit cell is likely to correspond with substantial changes in the IV state and therefore
the electronic properties. The available lattice parameter data for various solid solutions of
T = Ni [89], Ag [89], Rh [59], and Pt [90] have been compiled in Figure 4.1.
The trends in Figure 4.1 for silver and rhodium substitution appear to be relatively
straightforward, as these atoms are larger and smaller, respectively, than palladium, resulting
52

in a corresponding increase and decrease in the unit cell size. An unusual result can be seen,
however, for the nickel-substituted samples. In this series, the lattice parameter first increases
with x up to approximately 0.05, then decreases for all x above 0.05. This initial increase
coincides with a change in the cerium valence towards the larger trivalent state [89], which
saturates at x = 0.05. The cerium atoms thus have a fixed size for all x above 0.05, and a
simple Vegard’s law behavior is observed in this range as the lattice parameter begings to
decrease with increasing x [89]. A similar but more subtle trend can be seen in the samples
containing rhodium, where a saturation of the cerium atoms in the smaller 4+ configuration
occurs for x ≈ 0.6 [59], corresponding with a kink in the trend.
In contrast to these trends, the lattice parameter of the platinum-substituted series does
not appear to change much as x is increased. Platinum has a nearly identical valence
configuration, size, and electronegativity relative to palladium [91], which suggests that
the substitution of one for the other does not significantly change the nearest neighbor
environment of the cerium atoms. The relative uniformity of the lattice parameter data
supports this idea, as do data for χ and ρ in this series [90].
Although no Seebeck coefficient data for this system have been previously reported, it
can be anticipated from these trends that α will also be close to the original values seen in
CePd3 . In conjunction with this expected preservation of α, the substantial mass difference
between platinum (195.08 amu) and palladium (106.42 amu) should sharply decrease the
lattice thermal conductivity. This hypothesis, encouraged by the limited results of Sales on
similar compounds [37], has provided strong motivation for further study.

53

4.2

Experimental methods

Samples of CePd3−x Ptx with 0 < x < 1 were synthesized and measured in accordance
with the procedures outlined in Chapter 1. A maximum of x = 1 was chosen based on the
work of Rambabu, et al. who reported this substitution level to be the solubility limit of
platinum into the palladium sites of the Cu3 Au structure [92]. Due to the proximity of the
compositions studied herein to the reported solubility limit, SEM and EDS experiments were
performed to search for evidence of any secondary phases.

4.3
4.3.1

Results and discussion
Structural properties

Analysis of XRD patterns for each of the CePd3−x Ptx samples indicated the presence of the
expected Cu3 Au-type phase with no apparent secondary phases. A slight decrease in the
lattice parameter was observed as x was increased, as seen in Figure 4.2 and consistent with
previous reports for this solid solution.
Due to the limited solubility of platinum into CePd3 [92], it is possible and perhaps likely
that some ordering of the platinum occurs as the composition approaches the solubility limit.
Although such ordering may have an impact on the transport properties of the material, no
obvious signs of ordering were present in the XRD patterns, consistent with previous reports
[90]. SEM analysis of the fracture surface of the CePd2 Pt samples showed grains on the
order of 10 microns. EDS results have been included in Figure 4.3 and indicate that cerium,
palladium, and platinum were present a ratio of 1:2:1 to within 2% of the expected ratio as
determined by averaging over several area and point spectra. The apparent homogeneity of

54

Figure 4.2: Lattice parameters of the CePd3−x Ptx solid solution determined by Rietveld
refinement. Dashed line is a guide for the eye.
the microstructure and composition support the XRD results that show no signs of impurity
phases in the material.

4.3.2

Electronic properties

Measurements of the Seebeck coefficient have been included in Figure 4.4. Consistent with
the initial hypothesis, large α is observed for all x. This result sharply contrasts with the
trends of nearly every other study of chemically modified CePd3 series for which Seebeck
coefficient data are available, which all show dramatic changes in α as the composition is
changed.
A slight enhancement in α occurs as x is increased that coincides with a more gradual
slope of the α versus T dependence above T ≈ 150 K. Since the substitution of platinum

55

Figure 4.3: EDS analysis results for CePd2 Pt indicating the presence of cerium, palladium,
and platinum in a ratio of 1:2:1 within 2%

Figure 4.4: Seebeck coefficient of the CePd3−x Ptx solid solution. Solid lines are a guide for
the eye.
56

Figure 4.5: Electrical resistivity of the CePd3−x Ptx solid solution.
is not expected to significantly alter the electronic structure, this modest increase in α can
best be explained as a result of the chemical pressure introduced by the slightly smaller
platinum atoms constricting the size of the unit cell. This appears to affect α in a way
similar to mechanical pressure, as the latter has been shown to reduce the lattice parameter
[93] and produce relatively substantial increases in α on the order of 30% when the material
is subjected to several GPa of external pressure [94, 95].
The electrical resistivity data are included in Figure 4.5 and show that a slight increase
in magnitude as well as a change in temperature dependence occur as the platinum content
is increased. These trends are consistent with the available reports in the literature [90]
and indicate a slight decrease in TK . The lack of any large change in ρ implies that the
magnitude of κe is nearly constant as a function of x.

57

Figure 4.6: Total thermal conductivity of the CePd3−x Ptx solid solution.

4.3.3

Thermal properties

The total thermal conductivity of the CePd3−x Ptx series is included in Figure 4.6. The
effects of platinum substitution on the thermal conductivity are dramatic, especially in comparison to the relatively small changes observed in the Seebeck coefficient and resistivity
described above.
As seen in Figure 4.7, the reduction in the total thermal conductivity can be attributed
almost entirely to a large reduction in the lattice contribution. Since platinum substitution
introduces only a slight change in the lattice parameter, this reduction in κl can be explained
as a result of the large mass mismatch between the platinum and palladium atoms and not
from strain effects. This result also emphasizes the potency of palladium-site substitutions
in scattering phonons, in contrast to the minimal effects seen for Ce1−x Scx Pd3 .
In order to consider the net effects of the changes in the lattice thermal conductivity and
58

Figure 4.7: Lattice thermal conductivity of the CePd3−x Ptx solid solution.
electrical resistivity, Figure 4.8 has been included to show how κ changes with platinum
content.
Overall, the substitution of platinum into the palladium sites results in a favorable decrease in κ at all temperatures. When combined with the improvement in Seebeck coefficient
seen in Figure 4.4, these samples show an overall improvement in thermoelectric performance
for all x and T .

4.3.4

Magnetic properties

It is useful to examine the magnetic properties of this material in order to better demonstrate
the subtle ways that the platinum substitution affects the cerium valence. It has been
previously shown that the substitution of platinum into CePd3 results in a slight change in
the cerium valence toward the 4+ state [90, 92]. In order to confirm this in the compounds

59

Figure 4.8: Thermal conductivity ratio of the CePd3−x Ptx solid solution.
studied here, magnetic susceptibility measurements were performed on samples of CePd3 ,
CePd2.5 Pt0.5 , and CePd2 Pt. The results of these experiments are included in Figure 4.9.
The changes in the susceptibility are relatively minor and consistent with those reported
in the literature. An examination of the high temperature (200 K to 300 K) dependence
of χ versus 1/T was used to determine the Curie constant. This analysis is included in
Figure 4.10 with the resulting values of C labeled on the plot. The decreasing values of C
with increasing platinum content support the hypothesis that the cerium valence is shifting
towards the tetravalent state as x is increased.

4.3.5

General conclusions and ZT

The combination of large α and reduced κ in these materials results in increases in ZT for
all x at all T . The ZT of both CePd2 Pt samples studied here match each other exceptionally
60

Figure 4.9: Magnetic susceptibility of the CePd3−x Ptx solid solution.

Figure 4.10: High temperature susceptibility of the CePd3−x Ptx solid solution plotted
against 1000/T. Solid lines are linear fits of the form χ = χ0 + C/T . The values of C
were extracted from these fits and are indicated on the figure in units of emu-K/mole-Oe.
61

Figure 4.11: ZT of the CePd3−x Ptx solid solution.
well. A maximum value of approximately 0.3 occurs above 250 K that corresponds with an
improvement of approximately 35% over the control sample. This value is the highest ever
reported for any CePd3 -based material.
Three main conclusions emerge from this study. The first is support for the idea that
palladium site substitutions are generally more effective for reducing the lattice thermal conductivity relative to cerium-site substitutions. The second conclusion is that careful selection
of substitution elements based on factors such as their size, electronegativity, and valence
configuration can allow for a minimal perturbation of the local cerium bonding environment,
which can enable substitutions on the order of 30% without having any significant impact
on the IV state. The third important conclusion is that chemical pressure introduced by the
substitution of smaller elements into the lattice can mimic the effects of applied mechanical
pressure. This idea is not new, but the important distinction demonstrated here is that

62

the substitution element should be carefully selected to match the chemical properties of
the host matrix if the effect is to be properly isolated. The importance of this is readily
demonstrated by considering the studies involving silver, nickel, and rhodium substitution
mentioned above, in which dramatic changes in the lattice parameter and electronic properties were observed as the palladium content was decreased. Although these elements do
introduce chemical pressure, they also cause changes in the local cerium environment and
the electronic structure, making it essentially impossible to differentiate the impact of each
mechanism separately.
Since platinum appears to be the optimal T element available for substitution onto the
palladium site, this approach may already be exhausted as a way of improving the thermoelectric properties of the material. Additional study of this approach may be possible by
doubly substituting onto the palladium site with platinum and a small amount of another
element such as rhodium. Given the minimal improvement in the thermal conductivity that
was gained in Figure 4.7 by increasing the platinum content from x = 0.5 to x = 1, however,
it does not seem likely that any further reduction can be achieved through this route.

63

Chapter 5
Interstitial site filling: CePd3Mx
5.1

Background and motivation

Work by Lackner, et al. [42] has shown that partially filling the central interstitial site of the
cubic Cu3 Au structure with boron is an effective way of lowering the thermal conductivity
of the material through the introduction of point defect phonon scattering. The problem
with this approach, however, is that the electrical resistivity and Seebeck coefficient are also
sharply affected by the presence of the boron atoms. The combination of these factors led to
no net enhancement in ZT for any of the CePd3 Bx compounds reported in Lackner’s study.
It is therefore important to better understand the specific ways in which these filler atoms
affect the electronic properties of the material in order to fully assess the potential of this
approach for increasing ZT .
These changes in the electronic properties can be attributed to the rapid change in cerium
valence toward the Ce3+ state that occurs in materials like CePd3 Bx and CePd3 Six as x
is increased. This valence change has been observed through lattice parameter, magnetic
susceptibility, neutron diffraction, and specific heat data for these compounds [39, 44, 45, 46,
47, 48, 49, 50, 51, 52]. Additional reports by other groups on the electrical resistivity and
magnetic susceptibility of samples filled with varying amounts of beryllium [96], aluminum
[55, 97], gallium [55], and germanium [55] show similar shifts in the cerium valence toward
the trivalent state as the filling fraction is increased, suggesting that a general trend may
64

exist for all M .
In addition to these studies of partially filled samples, work by multiple research groups
has shown that an ordered Ce8 Pd24 M phase forms for certain M elements such as gallium
[53, 54], indium [53, 54], tin [53, 54], antimony [53], lead [53, 54], bismuth [53], germanium
[54, 55, 56], aluminum [54, 55, 97], and manganese [57], as well as possibly zinc and tellurium
[54]. The formation of this superstructure is driven by a slight displacement of the palladium
atoms from the face centers of the filled unit cells in response to the presence of the M atoms
in the central interstitial sites. When these displacements are large enough, it is no longer
possible to have M atoms in adjacent unit cells, and the ordered phase emerges. This
corresponds with an effective unit cell that can be described by shifting the origin to (0.5,
0.5, 0.5) and placing each M atom at the shared corner of eight simple CePd3 unit cells.
It is interesting to note that certain elements such as aluminum, gallium, and germanium
have been reported to have both continuous solubility for 0 < x < 0.125 and form the ordered
phase for x = 0.125. This can most likely be explained by differences in the heat treatment
conditions applied to the samples in the different studies. Those that reported the formation
of the ordered phase were working with samples that were annealed for approximately three
weeks at 900◦ C [53, 54, 56], while those that reported continuous solubility did not provide
details regarding any heat treatments applied to the samples after arc melting, which suggests
that no annealing was performed [55, 97].
It is also worth noting that although some M elements may be present in both the
disordered and ordered phases, certain other M elements such as boron and silicon do not
appear to order regardless of any heat treatment applied [54]. Thus a range of structural
ordering behavior can be observed as a function of the identity of the M element as well
as the processing conditions used to prepare the material. Given the important effects that
65

such structural ordering can have on transport properties, variations in composition and
processing of CePd3 Mx are both important to consider.
The lattice thermal conductivity may be particularly sensitive to the formation of these
ordered phases. If the interstitial atoms are no longer randomly dispersed, their presence in
the structure may simply introduce additional coherent vibration modes. This suggests that
the ordered phases are not likely to display the same thermal conductivity reduction that
would be expected in disordered materials.
This ordering is also likely to have an important impact on the electronic structure of the
material, and limited studies of the thermoelectric properties of some ordered Ce8 Pd24 M
samples reveal trends consistent with this idea. Although decreases in the Seebeck coefficient
and increases in the electrical resistivity are observed in these materials that are similar to
those reported by Lackner for disordered boron-filled samples [42], these effects appear to
have at least some dependence on the nominal size of the M elements. This suggests that
within the ordered phases, the extent to which the palladium atoms are displaced from the
unit cell faces can be correlated with changes in the electronic properties [54].
Based on these reports, there are clearly many complex relationships among the size
of the unit cell, the cerium valence, and the properties of these filler atoms. There does
not appear to be a consensus, however, regarding what mechanisms are ultimately driving
the changes observed in the electronic, magnetic, and structural properties. For example,
some reports exist suggesting that these changes are driven entirely by chemical pressure
and therefore depend primarily on the size of the filler elements, while others suggest that
factors such as the electron count of M may be equally or more important than the atomic
radii [44, 54, 96].
The current study has thus been motivated by the effectiveness of interstitial site filling for
66

lowering the thermal conductivity of CePd3 in conjunction with the ambiguity surrounding
how these filler atoms affect the electronic transport properties. The following is an attempt
to untangle some of these complex relationships.

5.2

Experimental methods

Samples of CePd3 Mx (M = Na, Mg, Ca, B, Al, Si, Ga, Ge, In, Sn, Te; 0 < x < 0.25)
were prepared through the standard arc melting procedure described above. Some additional
processing steps were necessary for M elements with high vapor pressures and/or low boiling
points, such as sodium, magnesium, calcium, and tellurium.
For samples containing these volatile elements, initial attempts were made to synthesize
binary M -Pd precursors in order to facilitate stoichiometry control during arc melting. The
basic requirement of any solid state precursor is that the volatile element be present in a
ratio greater than or equal to its concentration in the final compound. It is also desirable for
the precursor to be a line compound that melts congruently in order to ensure stoichiometry
control during intermediate processing steps. Although precursor compounds with higher
melting points tend to be more stable when arc melted, they are also generally more difficult
to form through solid state reactions at low temperatures. The optimal reaction temperature
is thus determined by the properties of the precursor phase as well as the volatile element,
which can vary with the absolute pressure inside of the reaction chamber.
No stable binary phases exist between palladium and the Group I elements [98]. Synthesis
of the binary compounds of palladium and Group II elements was generally difficult to
control, as there are a relatively large number of possible binary phases close in stoichiometry,
and the alkali earth elements have a strong tendency to react with quartz and other types

67

of reaction vessels [99, 100].
Instead of using this precursor approach, samples containing sodium, magnesium, and
calcium were successfully synthesized by simply re-melting pieces of pre-made CePd3 on top
of pieces of the filler element inside the arc melter. Although mass losses as a result of this
process were relatively small, and although the transport properties of these samples deviate
substantially from the control sample in ways consistent with the successful substitution of
the M elements into the interstitial site, significant vaporization during melting may have
occurred in some or all of the samples produced this way.
The M -Pd binary precursor approach was successful in the preparation of samples containing tellurium. Te7 Pd20 samples were prepared by combining pure tellurium and palladium in the appropriate ratio in an evacuated quartz ampoule, which was then heated
above the melting point of the desired phase. This was combined with additional palladium
and cerium pieces and arc melted in the standard way to produce the desired stoichiometry. Overall mass losses in the arc melter were less than 1%, suggesting that this precursor
approach is effective for the implementation of tellurium into the interstitial site.
An attempt was also made to synthesize samples containing sulfur in hopes of introducing an electron acceptor into the interstitial site. To do so, PdS powder was purchased
commercially (Sigma Aldrich), sealed in an evacuated quartz ampoule, then melted and recrystallized in order to produce larger pieces suitable for arc melting. Although mass losses
from this process were less than 1%, the resulting XRD patterns showed what appear to be a
combination of Ce2 S3 , CePd5 , and CePd3+x . The presence of these impurity phases is only
subtly apparent in the XRD patterns and becomes more prominent as the sulfur content is
increased.
Additional compositions were also attempted with M = Sr, Ba, and Ag for x = 0.05,
68

but the resulting ingots appeared to have obvious impurity phases coating their surfaces.
In addition to this apparent phase separation, XRD and transport property analysis of
the interior portions of these ingots closely resembled the behavior expected of unmodified
CePd3 . Thus it appears that these M elements are not easily incorporated into the interstitial
site through this synthesis approach, if they can be at all. One interesting consequence
of this is that although CePd3 can reportedly tolerate non-stoichiometry variations from
approximately 74.6 at.% palladium to 76.2 at.% palladium (corresponding with compositions
of CePd2.94 to CePd3.27 ) [101], and although silver has been repeatedly shown to substitute
successfully into the palladium site in CePd3−x Agx [43, 89], stoichiometric CePd3 appears
to completely reject excess silver. This suggests that the defect mechanism(s) that enable
the structure to tolerate such a wide range of Ce:Pd ratios do not arise from an extension of
the palladium sublattice and introduction of cerium vacancies, but instead likely correspond
with PdCe anti-site defects that are not energetically favorable when silver is involved.

5.3
5.3.1

Results and Discussion
Structural Properties

Two distinct trends emerged in the XRD patterns obtained for the CePd3 Mx samples. The
first of these trends is in relation to the onset of the ordered phase. A gradual shift in the
lattice parameter was observed for all M elements from rows two through four of the periodic
table, as seen in the XRD patterns obtained for a representative series of samples included
in Figure 5.1. This behavior suggests that these elements are all continuously soluble into
the interstitial site, consistent with the trends reported for samples containing aluminum,
gallium, and germanium that were not annealed [55, 97]. The patterns for sulfur filled
69

Figure 5.1: XRD plots for a representative series of CePd3 Mx with M from rows two to four
of the periodic table. The gradual shift in lattice parameter suggests a continuous solubility
of these M into the structure.
samples have been included as well, with the subtle impurity peak at ∼38◦ visible in the
CePd3 S0.10 sample.
In contrast to this trend, Figure 5.2 shows XRD patterns collected for samples containing
M elements from row five of the periodic table. No reports exist of gradual solubility for these
M elements. The results in Figure 5.2 are consistent with this idea, as they appear to show a
mixture of two Cu3 Au-type phases with slightly different lattice parameters. The locations of
the first sets of peaks are consistent with the spacings expected for the respective Ce8 Pd24 M
superstructures, while the second set matches the original locations of the CePd3 pattern.
The lack of any substantial shift in the CePd3 peaks suggests that there is essentially no
solubility of these M elements into the central interstitial site except through the formation
of the ordered phase, which appears to form immediately from the melt with no additional

70

Figure 5.2: Some XRD plots for the CePd3 Mx series for M from row five of the periodic
table. The split peaks indicate a mixture of CePd3 and Ce8 Pd24 M phases.
heat treatment required.
Careful examination of the XRD patterns in Figure 5.1 reveals what appear to be small
superlattice reflections in the samples containing gallium and germanium between 30 < 2θ <
50. These minor peaks appear to grow in relative intensity from the x = 0.05 to x = 0.10
samples for both gallium and germanium. The presence of these peaks, combined with the
reported effects of annealing on similar samples, suggests that the palladium atoms are only
slightly displaced from their nominal position on the face centers of the unit cell in these
compounds. This displacement appears to be sufficiently small that it is still possible to place
the filler atoms randomly throughout the structure, and ordering is not strictly necessary
to accommodate their presence in the lattice. Although it appears that the solubility of
these atoms can thus be described as continuous, the subtle perturbation they introduce to
the structure may have a non-negligible effect on the electronic and/or thermal properties
71

beyond what occurs as a result of purely randomly distributed defects such as boron and
silicon, especially as the composition approaches x = 0.125.
The second trend that emerges in the structural properties is unique to the M elements
from rows two through four of the periodic table. The large variety of M elements included
in this study allows for a direct comparison of the CePd3 Mx lattice parameters with the
properties of the M elements, which in turn allows for a clarification regarding the origin of
the unit cell expansion in these compounds
To determine if the expansion of the unit cell is dependent at all on the negative chemical
pressure of the M atoms in the interstitial site, Figure 5.3 has been included to show the
refined lattice parameters for these samples as a function of the nominal atomic radii of the
M elements. No apparent trend is visible in the data; if anything, the lattice parameter
appears to decrease as the nominal size of the M atoms becomes larger.
This suggests that the unit cell expansion is electronic in origin. Based on this idea, it
is useful then to define the “effective valence” Vef f of the interstitial site, which describes
the average number of valence electrons per unit cell introduced by the M atoms. This
parameter is defined as
Vef f = (x)(NV ),

(5.1)

where x is the filling fraction and NV is the number of valence electrons of M . Figure 5.4
shows the lattice parameter data plotted as a function Vef f .
A linear trend can be seen in the data, as indicated by the fit line included in the
plot. This establishes a clear correlation between the size of the unit cell and the electronic
properties of the filler atoms.

72

Figure 5.3: Lattice parameters of CePd3 Mx as a function of nominal M ionic radii. No clear
correlation can be seen between these two variables.

Figure 5.4: Lattice parameters of CePd3 Mx as a function of Vef f . These variables appear
to be linearly dependent to first order.
73

5.3.2

Electronic Properties

Figure 5.5 shows the electrical resistivity of a representative group of samples normalized to
their values at 350 K. A small amount of variation was observed in the absolute magnitude
of ρ at 350 K among these samples, although they all fell within a relatively narrow range
from approximately 0.9 µΩ-m to 1.3 µΩ-m. These changes in the absolute magnitude are
necessarily related to variations in the coefficients of Equation 1.11. It is difficult to explicitly
determine these parameters, however, as there is no clearly dominant mechanism contributing
to the overall behavior. For example, the residual resistivity of CePd3 has been shown to
vary by over an order of magnitude from 0.14 µΩ-m up to 2.9 µΩ-m among samples with
only slight variations in their nominal stoichiometry [102]. Thus the changes seen here may
simply be attributable to variations in the Ce:Pd ratio. Regardless of this, the variations in
magnitude are relatively small, and as such there does not appear to be any useful way of
controlling the transport properties through these considerations.
It is possible, however, to determine systematic trends in the temperature dependence of ρ
by comparing the trend in Figure 5.5 to the behavior reported for samples partially filled with
boron [42, 46], silicon [45], aluminum [55, 97], gallium [55], and germanium [55] where only
x was varied. In those reports, the peak in the resistivity is seen to simultaneously increase
in magnitude and shift to lower temperatures, indicating a drop in TK . For high enough
filling levels, the Kondo term vanishes and ρ(T ) is dominated by the linear temperature
dependence common for metals. This latter behavior can be seen for CePd3 B0.25 in Figure
5.5.
It is difficult to determine the location of the resistivity peaks for these samples due to the
limited temperature range over which these data were collected. A qualitative description

74

Figure 5.5: Normalized electrical resistivity of representative samples from the CePd3 Mx
series.
of the temperature dependence can still be used to show that these samples follow the
same overall trends as those in the literature, in that the resistivity peak shifts to lower
temperatures before it eventually saturates, decreases in magnitude, and approaches linear
behavior. The novelty of the current work lies in the demonstration that these changes in
ρ(T ) are can be generally described as functions of Vef f , regardless of the chemical identity of
the specific filler atom. This trend is in agreement with the changes in the lattice parameter
seen in Figure 5.4.
Figure 5.6 shows α(T ) for a representative series of samples filled with various p-block
elements. The identities of the specific filler elements have been removed to emphasize the
primary dependence of the results on Vef f only. The temperature variation is approximately
linear for all of the samples with a slope that decreases gradually as Vef f is increased. The
magnitude of the Seebeck coefficient systematically decreases until eventually becoming n75

Figure 5.6: Seebeck coefficient of representative samples from the CePd3 Mx series. Solid
lines are a guide for the eye.
type for Vef f = 0.4 and T > 200 K.
A substantial change in α(T ) is seen in the sample with Vef f = 0.75. In this sample,
the Seebeck coefficient is small, positive, and has a linear temperature dependence over the
entire range examined. This behavior indicates a complete collapse of the IV state and is
similar to that previously reported for samples with Vef f ≥ 0.6 [42, 45, 46]. For smaller
values of x, however, a nearly linear variation in α at 295 K occurs as a function of Vef f .
Figure 5.7 has been included to demonstrate this trend.

5.3.3

Thermal Properties

Consistent with the trends reported for CePd3 Bx [42], a substantial decrease on the order of
50% was observed in the lattice thermal conductivity of these partially filled samples. Figure

76

Figure 5.7: Seebeck coefficient at 295 K versus Vef f for representative samples from the
CePd3 Mx series. Solid line is a linear fit to the data.
5.8 has been included to show a cross section of κl values for various samples at 100 K. The
values have been plotted as a function of the atomic mass of the corresponding filler element
and grouped by the filling level x.
Two trends are apparent in the data. The first is that for fixed x, κl decreases as the
mass of M is increased up to the saturation point of approximately 30 amu, above which no
further reduction in κl is observed. The second trend is that for fixed M mass, κl decreases
as x is increased up to 0.05, above which a similar saturation effect is observed.
The two dominant parameters that emerge for reducing the thermal conductivity are
thus the filling fraction and the mass of M . Additional parameters such as lattice strain and
the onset of ordering may also have an effect on κl , but the data in Figure 5.8 suggest that
these effects have a relatively insignificant impact on the overall lattice dynamics of these
materials.
77

Figure 5.8: Lattice thermal conductivity at 100K of representative samples from the
CePd3 Mx series. A saturation in the thermal conductivity reduction is observed as a function
of M atomic weight as well as the filling fraction x.

78

Figure 5.9: Magnetic susceptibility of representative samples from the CePd3 Mx series.

5.3.4

Magnetic Properties

The temperature dependence of the magnetic susceptibility for several samples in this series
has been plotted in Figure 5.9. The changes observed in these data are again qualitatively
similar to the changes seen for samples filled with boron [42, 46], silicon [45], aluminum
[55, 97], gallium [55], and germanium [55] where only x was varied. The increasing magnitude
at low temperatures that occurs as Vef f is increased indicates a shift toward trivalent cerium
in these materials.
The magnitude of C extracted from plots of χ versus 1/T (200 K ≤ T ≤ 300 K) have
been included in Figure 5.10. An approximately linear relationship is observed between these
parameters, again linking the electronic properties of the M atoms to the overall properties
of the material.
The Hall coefficient of the CePd3 Gax series was also measured. The results of these
79

Figure 5.10: Curie constant versus Vef f for representative samples from the CePd3 Mx series.
Solid line is a linear fit to the data.
experiments are included in Figure 5.11 and show a general decrease in RH as x is increased.
Although the Hall coefficient in CePd3 is not necessarily a reliable or direct measure of
the free carrier density, it is still possible to utilize RH as a tool for probing changes in
transport properties. Based on conventional interpretations of RH , there are two simple
explanations for why a decrease would be observed as the material is doped. If the material
has only one type of carrier, then a decrease in RH = 1/ne is an indication that the carrier
density has increased.
A decrease in RH is also expected, however, for a system into which a second type of
mobile carrier has been introduced. RH for such a material can be expressed as

RH =

pµ2 − nµ2
e
h
,
e(pµh + nµe )2

80

(5.2)

Figure 5.11: Hall coefficient for the CePd3 Gax series.
where n (p) is the electron (hole) density and µe (µp ) is the electron (hole) mobility [17].
This interpretation offers a more suitable explanation for the change in RH apparent
in Figure 5.11, as the addition of gallium into the structure increases the number of free
electrons that act to compensate the intrinsic p-type behavior of the material. It is also
useful then to consider the form of the electrical resistivity for a two carrier system, which
can be expressed as [17]
ρ=

1
1
=
.
σ
e(pµp + nµn )

(5.3)

With these considerations in mind, the behavior of the CePd3 Ge0.10 sample further
supports the idea of charge carrier compensation in these materials. This sample displays a
broad minimum in ρ in Figure 5.5 at approximately 200 K that coincides with a change in
the sign of α visible in Figure 5.6, which together indicate that the majority carriers switch
from holes to electrons at this temperature.
81

Figure 5.12: 295 K Seebeck coefficient (left ordinate) and high temperature Curie constant
(right ordinate) of CePd3 Mx vs 295 K lattice parameter. Solid lines are linear fits to the
data.

5.3.5

General conclusions and ZT

The results of this study have demonstrated that the Seebeck coefficient, lattice parameter,
and Curie constant of CePd3 Mx can all be simultaneously tuned through adjusting a single
parameter, Vef f . To emphasize this shared dependence, the room temperature Seebeck
coefficient and high temperature (200 K < T < 300 K) Curie constant have been plotted
against the corresponding room temperature lattice parameter for these samples in Figure
5.12.
Furthermore, it has been demonstrated here that although both Vef f and κl are dependent on the filling fraction x, their separate dependence on the valence configuration of M
and the mass of M , respectively, allow for these transport parameters to be decoupled to a
limited extent. As a result, it is possible to achieve modest enhancements in ZT through
82

Figure 5.13: ZT of representative samples from the CePd3 Mx series.
interstitial filling, as illustrated in Figure 5.13. The insertion of elements such as magnesium,
which has a low number of valence electrons and a relatively large atomic mass, can be used
to produce modest increases in ZT at 80 K from 0.07 in CePd3 to 0.11 in CePd3 Mg0.05 .
Based on these overall results, it does not appear that interstitial site filling by itself
will lead to any substantial improvements in the thermoelectric performance of the material.
If a suitable mechanism can be found for increasing the hole concentration in the material,
however, it may be possible to combine this mechanism with interstitial site filling in order to
cancel out the competing doping effects, potentially leading to a material with a low lattice
thermal conductivity but favorable electronic properties..

83

Chapter 6
Nanostructured CePd3
6.1

Background and motivation

In the region T < ΘD , thermal conductivity tends to be dominated by longer wavelength
phonons with relatively long mean free paths that are scattered primarily by grain boundaries
[18]. Nanostructuring is therefore often an especially effective thermal conductivity reduction
strategy at cryogenic temperatures. Since ΘD = 292 K for CePd3 [73], reducing the grain
size of the material should be an effective way to decrease κl below room temperature.
Two previous reports exist concerning nanostructured CePd3 that suggest nanostructuring may also alter the electronic properties of the material. The first is a series of spectroscopic and diffraction experiments performed by Lin, et al. on 5 nm – 10 nm particles
created through laser ablation [103]. Two primary results emerged from the data included
in that report. The first is from the magnetic susceptibility of the nanoparticles, in which a
systematic increase in the low temperature susceptibility is observed as the particle size is
reduced. This increase is one of the signatures of a valence shift in the cerium atoms towards
the 3+ state, and Lin, et al. estimate this change to correlate with approximately 25% of
the cerium atoms in the trivalent configuration. In contrast to this, they also report x-ray
absorption data that show a shift in the average valence of the cerium atoms toward the
4+ configuration. These results together strongly suggest that size effects have important
consequences for the cerium valence state in nano-scale CePd3 particles.
84

The second report is a study of magnetization, nuclear magnetic resonance (NMR), and
computational analysis of 30 nm, 60 nm, and 150 nm CePd3 particles created through high
speed ball milling in toluene. These samples also show systematic increases in the magnetic
susceptibility, in addition to a change in the NMR Knight shift that indicates an increase
in the average 4f occupancy as the particle size is reduced. The authors of this study
caution that defects and/or surface oxidation introduced by mechanical grain size reduction
techniques may be at least partially responsible for the observed changes, but overall their
results appear consistent with the idea that particle size reduction can have a direct impact
on the IV state [104].
From a broader perspective, several reports exist describing the impact of surface effects on the valence state of rare earth elements and compounds that further support this
idea. One example is that of samarium metal, for which angle resolved x-ray photoelectron spectroscopy (ARXPS) experiments have shown that a relatively large concentration
of Sm2+ ions are found at the sample surface, while the bulk consists of predominantly
Sm3+ [105, 106]. These ARXPS results are supported by compelling calculations of surface
energy differences between Sm2+ and Sm3+ ions that reveal the former to exist in a lower
energy state when exposed at a surface [105]. The authors of these studies also acknowledge
that the ARXPS observations may simply be a result of oxide or hydride formation, but
either way, these studies support the idea that exposure at an interface may have important
consequences for the valence state of rare earth elements in certain alloys and compounds.
The behavior of several other rare-earth compounds has been studied as a function of
grain size. Size-induced changes in electronic and magnetic properties have been widely
reported through various experimental techniques in CeAl2 [107, 108, 109], Ce3 Al [107],
Ce3 Al11 [107], CePt2 [108, 110], Ce2 Pt5 [108], TbAl2 [111], TbCu2 [111], GdAl2 [111, 112],
85

YbAl3 [113], Ce(Ru0.4 Rh0.6 )2 Si2 [114, 115], CeRu2 Si2 [115], CeAu2 Si2 [109], and CePd2 Si2
[109]. The samples studied in these other works were produced through a variety of techniques including ball milling, laser ablation, and flash evaporation, demonstrating the apparent universality of the size-dependent trends regardless of the specific processing technique
used to create the nanoparticles.
These considerations suggest that nanostructuring CePd3 may introduce a new mechanism for tuning the cerium valence, in addition to substantially decreasing the lattice thermal
conductivity. This combination has provided significant motivation for the work described in
this chapter, which explores the transport properties of several series of bulk nanostructured
CePd3 samples. This study appears to be the first time that the bulk transport properties
of a nanostructured correlated metal have been explored.

6.2

Experimental methods

Samples of CePd3 were prepared through arc melting in the standard way described above.
The resulting ingots were transferred into an argon-filled glove box then placed inside of a
65 mL stainless steel milling jar. Two stainless steel balls 0.5” in diameter and four stainless
steel balls 0.25” in diameter were added to the jar, which was then sealed with paraffin tape
and removed from the glove box. The samples were milled for various lengths of time in a
SPEX 8000M vibratory mill, and the resulting powders were removed in air. No obvious
signs of spontaneous or gradual oxidation were present during any of these steps.
These powders were sintered into pellets through Pulsed Electric Current Sintering
(PECS) (also known as Spark Plasma Sintering, or SPS) by pressing them at 56 MPa of
uniaxial pressure at 820◦ C for one hour. A one hour hold was initially selected in an at-

86

tempt to regrow the grains and search for signs of contamination from the ball milling and
PECS processes. Even after a one hour hold, however, examination of the samples through
XRD and SEM indicated that the samples were composed of nano-scale grains. As such, the
same processing conditions were selected for all subsequent experiments in order to maintain
consistency.

6.3
6.3.1

Results and discussion
Structural and morphological properties

No shift was observed in the lattice parameter for any of the ball milled samples, as seen
in the XRD patterns included in Figure 6.1. The patterns are consistent with the expected
Cu3 Au-type structure. Significant broadening of the peaks is observed that increases with
milling time, indicative of the presence of nano-scale features. The lack of a peak shift is
consistent with the results of Ref. [103] but disagree with those of Ref. [104], in which a
shift to lower angles was observed as the grain size was reduced. Although no obvious second
phases are apparent, the substantial broadening of the peaks severely restricts the overall
effectiveness of XRD for detecting any impurity phases.
The results of SEM and EDS analysis are consistent with the results of the XRD data.
For reference, Figure 6.2 has been included to show a representative SEM micrograph of
arc melted CePd3 collected at a magnification of 1,300x. The scale bar represents 10 µm,
and relatively large grains on the order of several microns are clearly visible in the image.
Analysis of this sample at higher magnifications (not included) did not indicate any fine
structure to these grains.
In contrast to this, Figure 6.3 and Figure 6.4 have been included to show representative
87

Figure 6.1: XRD patterns of CePd3 samples ball milled for various times t, as labeled.

Figure 6.2: SEM micrograph of an arc melted CePd3 control sample recorded at a magnification of 1,300x.
88

Figure 6.3: Contrast enhanced SEM micrograph magnified 100,000x of a CePd3 sample ball
milled for two minutes and sintered through PECS.
micrographs of the fracture surfaces of ball milled samples after sintering through PECS.
The magnification of both images is 100,000x and the included scale bars represent 100 nm.
Although relatively large agglomerates on the order of 100 nm and larger can be seen in both
images, these agglomerates appear to be composed of smaller individual grains on the order of
10 nm to 20 nm. EDS analysis showed the presence of a small peak corresponding to oxygen
in the samples, but a comparison of these spectra with those obtained for microstructured
CePd3 did not reveal any significant differences between the relative intensity of this peak
as a function of processing conditions.
The size of these nano-scale grains was estimated by applying the Williamson Hall (W-H)
method to the XRD patterns of the powders after ball milling. This analysis is based on the

89

Figure 6.4: Contrast enhanced SEM micrograph magnified 100,000x of a CePd3 sample ball
milled for 12 minutes and sintered through PECS.
relationship
β2

=

2
β0

0.9λ
+
D cos θB

2

+ (4 tan θB )2 ,

(6.1)

where β is the full width at half maximum (FWHM) of the XRD peak located at Bragg angle
θB , β0 is the contribution to the FWHM from instrument broadening, 0.9 is a geometric
form factor, λ is the wavelength of radiation used in the experiment, D is the particle size,
and

is the strain [116].

This equation was used to analyze the patterns in Figure 6.1. The results of this analysis
are included in Figure 6.5, where D and

have been plotted as functions of the milling

time. The estimated uncertainty in the calculations of D and

are ±2 nm and ±0.05%,

respectively. A saturation in both the grain size reduction and strain increase is observed
as the milling time is increased. This effect is common for ball milled samples and results

90

Figure 6.5: Particle size and strain of ball milled CePd3 samples as determined by
Williamson-Hall analysis (Equation 6.1). Dashed lines are a guide for the eye.
from a combination of the finite frequency and maximum kinetic energy of the collisions in
the jar, combined with the mechanical properties of the samples and milling media. A slight
decrease in the calculated strain is observed as the milling time is increased from 30 minutes
to 60 minutes. This variation is within the estimated error of the calculation, but it may also
be a real result of the rearrangement and annihilation of dislocations with increased milling
time [117].
Although SEM analysis on the sintered pellets indicates the presence of nano-scale features that appear to be consistent with the results of Figure 6.5, this W-H analysis was
performed on the powders directly after ball milling and not after sintering. This was done
due to the presence of agglomerates in the sintered samples, which seem to limit the accuracy of the W-H method for determining the crystallite size. Thus the actual grain size

91

Figure 6.6: Electrical resistivity of nanostructured CePd3 .
and strain present in the sintered pellets used for transport property measurements may
be slightly different from the values indicated in Figure 6.5, but these values can at least
be used as a self consistent way of interpreting any changes observed in the thermoelectric
properties.

6.3.2

Electronic properties

A general increase in the magnitude of the electrical resistivity of the ball milled samples can
be seen in Figure 6.6. The changes in ρ appear to saturate in the samples milled for time
t ≥ 12 minutes, mimicking the saturation in the grain size reduction seen in Figure 6.5. A
systematic change in the temperature dependence is also observed for T < 150 K, indicative
of a change in the Kondo temperature TK .
A monotonic decrease in the magnitude of the Seebeck coefficient as a function of milling

92

Figure 6.7: Seebeck coefficient of nanostructured CePd3 .
time is apparent in Figure 6.7. This reduction in the magnitude of α also appears to saturate
in a fashion similar to the W-H results in Figure 6.5.

6.3.3

Thermal properties

The total and lattice thermal conductivity of the samples are included in Figure 6.8 and
Figure 6.9, respectively. A reduction of ∼50% is achieved for the samples milled for t ≥ 12
minutes, emphasizing the effectiveness of this approach in scattering phonons. In both of
these plots, the same saturation effect is observed as a function of milling time, further supporting the notion that these changes are closely tied with the grain size reduction achieved
as a result of ball milling.
Although this approach is particularly effective for reducing the absolute magnitude of
the lattice thermal conductivity, it is important to also consider changes in the thermal

93

Figure 6.8: Total thermal conductivity of nanostructured CePd3 .

Figure 6.9: Lattice thermal conductivity of nanostructured CePd3 .

94

Figure 6.10: Thermal conductivity ratio of nanostructured CePd3 .
conductivity ratio κ . The relatively large increase in ρ that occurs as the grain size is
reduced results in a decrease in κe that closely mirrors the reduction in κl , resulting in no
significant improvement in κ for any of the nanostructured materials. These results are
summarized in Figure 6.10.

6.3.4

Magnetic properties

Measurements of the Hall coefficient were performed on these samples in order to search
for evidence of changes in the free carrier density. The results of these measurements are
included in Figure 6.11. A monotonic reduction in RH occurs as the milling time is increased,
which suggests that the density of p-type carriers is systematically increasing as the grain
size is reduced. These results are qualitatively different from those of the CePd3 Gax samples
described by Figure 5.11, which show a much larger decrease in RH at low temperatures.
95

Figure 6.11: Hall coefficient of nanostructured CePd3 .
Thus although the magnitude of α and TK appear to change in both of these series in similar
ways, the mechanisms that drive these changes appear to be fundamentally different.
Since the ball milled samples do not appear to be chemically modified in any way, it
is reasonable to assume that a one carrier model can be used to interpret the changes in
RH . From RH and ρ it is possible to determine the Hall mobility µ through the relationship
µ = 1/(ρen) = RH /ρ. The results of this analysis are included in Figure 6.12 and show
that a significant decrease in the Hall mobility of the carriers occurs as the grain size of the
samples is reduced, likely a direct result of increased interfacial scattering.
This apparent change in the free carrier density is somewhat surprising, as no compositional modifications have been made, and contamination effects would not be expected to
saturate with milling time. In order to help explain these changes within the context of grain
size reduction effects, the following “size-doping” model is proposed.

96

Figure 6.12: Hall mobility of nanostructured CePd3 .

6.4

“Size-doping” model

It is convenient to consider a ”size-doping” model to directly connect the observed grain
size reduction with the corresponding changes in the transport properties. Inspired by the
observed surface effects in other materials as described above [107], this model begins with
the assumption that incomplete atomic coordination at the grain boundary collapses the IV
effect and converts these 4f electron levels into highly localized states.
Filling these 4f levels at the surface requires some electrons to be pulled from the bulk.
Since the average cerium valence in CePd3 is approximately +3.45 at room temperature [58,
59], each unique unit cell exposed at the grain surface thus requires on average approximately
0.45 additional electrons to fill the cerium 4f shell. The effective hole concentration of the
bulk should be expected to increase as a result of these electrons being pulled from the
conduction band and localized at the surface.
97

Figure 6.13: Simple geometric model for the fraction of unique unit cells at a cube surface.
A basic geometric argument can be invoked to estimate the proportion of unit cells at
the surface of a nano-scale grain. Since CePd3 forms in a cubic structure, it is appropriate to
consider a perfect cubic crystal composed of l cubic unit cells. The total number of unique
unit cells N exposed at the surface of such a cube is equal to

N = l3 − (l − 2)3 .

(6.2)

The fraction of total unit cells in the material exposed at the surface is then given by
l3 − (l − 2)3
N
2 3
=
=1− 1−
.
l
l3
l3

(6.3)

A plot of N/l3 versus l has been included in Figure 6.13 to demonstrate the significant change
in surface area-to-volume ratio that occurs for a cubic crystal as the cube size is reduced.
98

Figure 6.14: Estimated fraction of unique unit cells at the surface of a CePd3 grain.
For a grain approximately 1,000 unit cells across, less than 1% of the cells are located
at the surface. As the grain shrinks in size, however, more cells become exposed; about 6%
of the cells are at the surface in a 1003 cell cube, and nearly 50% are exposed for a cube 10
cells across. Thus surface effects may be negligible for grains composed of 1,0003 cells, but
they may become significant in grains that are only tens or hundreds of unit cells in length.
This geometric model can be extended to a physical system by multiplying l by the
lattice parameter of the material. The results of this analysis are included in Figure 6.14
for CePd3 . Based on the results of this simple analysis, 20 nm grains of CePd3 will contain
approximately 12% trivalent cerium, while this ratio nearly doubles to 23% for 10 nm grains.
This model agrees with Lin’s study on laser ablated samples, which showed that approximately 25% of the cerium atoms in a 9 nm particle were in the 3+ configuration [103]. This
falls precisely within the range of values expected based on the results of Figure 6.14.

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In spite of this success, this model should be considered only a basic approximation of
the expected behavior of the material, as real materials are made of grains with spanning a
range of sizes that are often faceted and full of defects. This variation can make the actual
surface area-to-volume ratio difficult to accurately predict and measure. The presence of
point and line defects introduced through the milling process may also have a significant
impact on the IV state [110]. Full consideration of these other possibilities would improve
the accuracy and utility of this model for interpreting and predicting the behavior of the
material.

6.4.1

Further study

Further grain size reduction may be possible through the use of different milling media, such
as tungsten carbide instead of stainless steel. Other techniques for producing nanoparticles
in sufficient quantities for powder processing, such as wet chemistry, may also be possible.
It would be worthwhile to compare the results of such studies to the current work in order
to examine the universality of these results, although the sensitivity of the CePd3 system to
even small amounts of chemical contamination may make isolation of size effects difficult.
More generally, the results of this study appear to support the idea that grain size
reduction can alter the electronic properties of strongly correlated materials. Since correlated
electron effects are central to the behavior of many interesting and useful materials, it may
be possible to modulate these effects simply by reducing the grain size to the nano-scale,
thus tailoring the materials’ properties without modifying their composition or structure.
Further study of this idea in other rare earth and transition metal compounds may yield
interesting results in this regard.

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Chapter 7
Universal trends and combined
approaches
A careful comparison of the properties observed in the different series discussed above leads
to additional insights regarding the apparently universal nature of some of these trends.
The first universal trend that emerges is in regards to the relationship among Seebeck
coefficient, lattice parameter, and Curie constant in the CePd3 Mx and CePd3−x Ptx series.
In both cases it was observed that an approximately linear correlation exists among all
three of these properties that results from their shared dependence on the average valence
configuration of the cerium atoms. Extending this analysis further, it is also possible to
compare the temperature dependence of the lattice parameter in stoichiometric CePd3 with
the Seebeck coefficient at the same temperatures [118]. The results of this combined analysis
are included in Figure 7.1, and show that a very similar relationship appears to exist among
all three properties in all three series.
This universality is surprising when considering the different mechanisms that dominate
the behavior of each series. The properties of the CePd3 Mx system have a strong dependence
on the “effective valence” of the interstitial site, indicating that the changes in this system
are entirely electronic in origin. In CePd3−x Ptx , however, the observed changes appear to
be predominantly a result of chemical pressure introduced by the slightly smaller platinum

101

Figure 7.1: Seebeck coefficient (left ordinate) and Curie constant (right ordinate) as a function of lattice parameter for various CePd3 series. Temperature dependent lattice parameter
data are from [81].

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atoms occupying the palladium sites. The extension of this trend to the temperature dependence of stoichiometric CePd3 shows that similar behavior can be observed as a function of
thermal expansion as well.
This type of broad perspective may be useful for determining additional ways of increasing
ZT . Although each of the mechanisms discussed in the preceding chapters has on its own
at best a limited effectiveness for improving the thermoelectric performance of the material,
it may be possible to combine them together in such a way that the properties are improved
beyond what each approach can achieve separately. This basic idea has been attempted
before in studies inspired by the general correlation between the Seebeck coefficient and the
lattice parameter of CePd3 often observed as the material is modified in various ways, as
seen in Figure 7.1. These efforts have typically focused on combining mechanisms that shift
the lattice parameter in opposing directions, such that κl can be reduced while the lattice
parameter (and thus α) may remain unchanged.
The first such attempt was reported by Ijiri and DiSalvo [83], who explored the effects of
dual substitutions on the cerium site. In their study, they examined a series of samples where
cerium was partially replaced in equal parts by large lanthanum and small yttrium atoms, as
well as regular solid solutions of Ce1−x Yx Pd3 and Ce1−x Ndx Pd3 . Their results very clearly
indicated that even if the average interatomic spacing was held constant, the thermoelectric
properties of the material were substantially changed in ways not expected based on the
simple combination of Mattheissen and Nordheim-Gorter rules for solid solutions.
The neodymium substituted samples do follow the expected trends reasonably well up
to approximately 40% doping. The incoherence of the cerium sublattice above this doping
level appears to dramatically affect the properties of the material, which the authors suggest
may be a result of the differences in the d and f levels of the two elements. This may also
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be correlated with a change in the spatial distribution of magnetic and non-magnetic atoms
that alters the nature of the Kondo interaction.
The Ce1−x (La0.5 Y0.5 )x Pd3 samples showed even larger changes in the transport properties relative to those seen in Ce1−x Ndx Pd3 , even though these samples also had a very
similar average lattice parameter to CePd3 . This suggests that in addition to doping effects,
local strains and distortions in the lattice can have significant effects on the IV state. Based
on these results, it is not valid to assume that simply maintaining the average lattice parameter at a constant value is sufficient to maintain the desired thermoelectric properties as
well [83].
The second attempt at combining approaches was reported by Lackner, et al. [43], who
tried to balance the lattice parameter reduction of rhodium substitution with the corresponding expansion observed with boron filling and/or silver substitution. Although this
approach did result in a Seebeck coefficient increase in some Ce(Pd1−x Rhx )3 B0.05 samples
relative to their singly modified parent compounds, none of these combinations resulted in
ZT values above 0.2. The results of this study nonetheless serve as proof of principle that it
may be possible to combine thermal conductivity reduction mechanisms together in such a
way that the Seebeck coefficient can be at least partially restored to its original value, which
may result in ZT values not otherwise attainable through each approach alone.

7.1
7.1.1

Nanostructured CePd3Mx
Background and motivation

The combined results of the CePd3 Mx and nanostructured CePd3 series described above
indicate that these approaches may alter the material in opposite ways. In particular, it
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appears that partially filling the interstitial site adds electrons to the conduction band,
while nanostructuring pulls electrons from the conduction band to fill the 4f shells of cerium
atoms at the grain surfaces. This suggests that combining these approaches may result in
a material with low lattice thermal conductivity and no overall change in the charge carrier
density within the bulk of the grains.
If successful, the cancellation of these effects should be readily apparent in the Seebeck
and Hall coefficients. Furthermore, the strong linear dependence of the lattice parameter
on Vef f implies that a nanostructured sample partially filled with M atoms should exhibit
a smaller lattice parameter shift compared to a sample of the same composition without
nano-scale features if the doping effects do in fact cancel out.
This combination of approaches may not completely restore the electronic properties of
the material to those of the unmodified compound, however, as a substantial fraction of
unit cells containing metallic trivalent cerium should still be present at the grain boundaries.
This may result in a mixture of behaviors expected for a composite of metallic RPd3 and
IV CePd3 . In order to explore these possibilities directly, several samples of nanostructured
CePd3 Gax were prepared and examined for this study.

7.1.2

Experimental methods

Samples of CePd3 Gax (0 < x < 0.05) were synthesized by arc melting the constituent
elements in the typical manner. Gallium was chosen as the filler element for these studies
primarily due to its large mass and high boiling point, both of which enable easier stoichiometry control during arc melting. The arc melted samples were ball milled and subsequently
sintered at 820◦ C under 56 MPa of uniaxial pressure. The milling and sintering times were
varied in an attempt to study the effects of processing variations on the transport properties
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and phase purity of the material.

7.1.3

Results and discussion

Initial experiments were performed on a sample of CePd3 Ga0.05 ball milled for 30 minutes
and sintered at 820◦ C for one hour under 56 MPa of uniaxial pressure in a manner identical
to the procedures followed for the original CePd3 nanostructured series. The results of XRD
experiments on this sample before ball milling showed a characteristic shift of the peaks to
lower angles, indicating an expansion of the unit cell consistent with the general trend for
all CePd3 Mx samples. After ball milling, XRD peak broadening was observed that closely
resembled that of the other ball milled CePd3 samples. A slight shift in the location of the
diffraction peaks back toward the positions expected for unmodified CePd3 was also observed
after milling. This shift became even more prominent after sintering, wherein the peaks
appeared to almost completely return to the nominal positions. These initial results were
consistent with the lattice parameter shift expected for a sample in which the two competing
doping effects were balanced. Analysis of the transport properties of this sample revealed that
the magnitude and temperature dependence of the Seebeck coefficient, electrical resistivity,
and thermal conductivity were consistent with a mixture of behaviors seen separately in the
ball milled and CePd3 Mx series, further suggesting that dopant compensation had occurred.
SEM and EDS analysis was performed on this sample in order to more thoroughly examine its phase purity, and the results of this analysis are included in Figure 7.2. These images
show that although the gallium appears to be well distributed throughout the entire sample,
a large cluster can be seen in the circled area that corresponds with a decrease in cerium
content within the same region. The shape of this gallium-rich region appears to match the
general outline of the grains in that area. This apparent segregation of the gallium into a
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Figure 7.2: SEM and EDS results for CePd3 Ga0.05 after ball milling and one hour of sintering
through PECS. Dashed lines have been added to highlight the primary area of interest where
segregation of gallium into an impurity phase is apparent in the elemental mapping results.
secondary phase suggests that only a fraction of the gallium initially added to the compound
remained dissolved in the bulk after the mechanical nanostructuring process. The composition of this secondary phase was identified through EDS point and area spectra collected
from the circled region in Figure 7.2 to be within the range GaPd1.8 (64 at.% palladium) to
GaPd2.1 (67.7 at.% palladium). Analysis of the Ga-Pd binary phase diagram indicates that
this phase is likely GaPd2 [119].
The presence of this secondary phase indicated that the initial processing conditions were
suboptimal. In order to explore whether this segregation occurred during ball milling, sintering, or both, another sample of CePd3 Ga0.05 was synthesized and ball milled for 12 minutes
instead of 30. As seen in Figure 7.3, XRD analysis of this sample shows an approximately
identical shift in the lattice parameter between the two samples after milling, as indicated by

107

Figure 7.3: Location of normalized (311) peak in CePd3 Ga0.05 after ball milling for various
times. The peak shifts to lower angles after gallium is added to the interstitial site, but it
begins to shift back toward its original position after ball milling.
the approximate overlap of the (311) diffraction peaks included in the figure. The positions
of the original CePd3 and CePd3 Ga0.05 peaks have also been included for comparison, and
the relative peak intensities have been scaled to emphasize the broadening that occurs as a
result of ball milling.
The similarities in these patterns suggest that this shift arises primarily from dopant
compensation and not from dopant segregation, as it does not appear to have any significant
dependence on the milling time. The sample milled for 30 minutes has a slightly broader
peak width and is shifted to a slightly higher angle, which can both be explained as likely
arising from minor variations in the average grain size between the two samples.
A comparison was next made between samples sintered for one minute and one hour.
These results can be seen in Figure 7.4, which shows the diffraction pattern of the (311)

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Figure 7.4: Location of normalized (311) peak in CePd3 Ga0.05 after ball milling and PECS.
While the peak shifts back almost to its original position in unmodified CePd3 after one
hour of sintering, only a small peak shift is observed in the sample sintered for one minute.
peak for these CePd3 Ga0.05 samples before and after ball milling and PECS. From this
figure it is apparent that the sample sintered for one minute has a substantially smaller peak
shift than the sample sintered for one hour. This result suggests that the dopant segregation
occurs primarily during the sintering stage, and as such, a one minute hold was utilized for
all further samples studied in this series.
An additional sample of CePd3 Ga0.02 was arc melted, ball milled for 12 minutes, and
sintered for one minute to compare with the properties of the CePd3 Ga0.05 sample. The
results of transport property measurements for these two materials are consistent with the
expected trends for dopant compensation. In particular, the plot of the Seebeck coefficient
in Figure 7.5 shows that both of these samples display α values approximately 10 µV/K to
25 µV/K larger than those of the nanostructured samples without gallium. The data for the

109

Figure 7.5: Seebeck coefficient of CePd3 Gax before and after ball milling and PECS.
corresponding gallium-filled microstructured samples have also been included for comparison.
From these data, two trends are apparent. The first is that the temperature dependence
of α near 100 K in the gallium filled nanostructured samples more closely resembles that of
the unfilled nanostructured samples than it does the gallium-filled microstructured samples,
although a shift in the peak temperature can still be seen relative to the CePd3 control
sample. In addition to this, the magnitude of α is larger in the nanostructured sample
containing 2 at.% gallium than it is in the 5 at.% sample. These results together suggest that
the gallium content is the dominant doping mechanism controlling the Seebeck coefficient in
these materials, although comparison with the microstructured gallium-filled samples shows
that the grain size reduction also has an important effect on the overall behavior.
The Hall coefficient of the nanostructured CePd3 Ga0.02 sample was measured in order to
further explore the effects of combining these approaches. The results of this measurement

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Figure 7.6: Hall coefficient of CePd3 Gax before and after ball milling and PECS. The included error bars represent an estimated uncertainty of 10%.
are included in Figure 7.6 and match reasonably well with the results of the CePd3 and
CePd3 Ga0.02 samples, especially for T ≥ 200 K, where RH is a more accurate probe of
the carrier density. The included error bars represent an estimated uncertainty of 10%
arising from the limited precision of the sample thickness measurement and the typical
errors associated with the linear fit of the resistance versus field plots. All three of these
samples display Hall coefficients slightly larger than the unfilled nanostructured sample,
further suggesting that the gallium partial filling dominates the overall behavior of the free
carriers in these materials.
The lattice thermal conductivity was also significantly reduced and matched well with the
expected behavior for nanostructured CePd3 to within experimental error. The magnitude
of the resistivity for the nanostructured samples decreased slightly as the gallium content
was increased. The combination of these trends resulted in a marginal but favorable decrease
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in κ . Even when combined with the relative improvement in α, however, these results failed
to produce any increase in ZT relative to the microstructured CePd3 control sample.
In spite of this failure, this study demonstrates that combining these approaches appears
to result in a partial cancellation of their negative effects on the thermoelectric performance
of the material. Given the large number of possible choices for composition and processing
conditions, there may still be a specific combination that will result in an increase in ZT ,
and the results included here thus establish a useful basis for further study.

7.2

Nanostructured CePd2Pt

Although thermal resistivities are not strictly additive [18], it is possible to introduce multiple
phonon scattering mechanisms into the same material in order to maximize the thermal
conductivity reduction. In order to explore the possibility of combining point defect and
grain boundary scattering, a sample of CePd2 Pt was arc melted in the manner previously
described. The sample was milled for 12 minutes and sintered through PECS for one minute
at 820◦ C under 56 MPa of uniaxial pressure.
Measurements of the transport properties of this sample indicate a mixture of effects
observed in the separate series. Most notably, the results of the lattice thermal conductivity
reduction are included in Figure 7.7, which shows that κl is lower in this material than it is
in either the nanostructured and platinum substituted samples are on their own.
In addition to these data, the results of the thermal conductivity ratio are included in
Figure 7.8. From this figure it can be seen that the combined effects of these approaches on
the lattice thermal conductivity are sufficiently large to enable a reduction in κ that neither
approach can achieve on its own.

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Figure 7.7: Comparison of lattice thermal conductivity of CePd3 modified in various ways.

Figure 7.8: Thermal conductivity ratio of CePd3 modified in various ways. Solid lines are a
guide for the eye.
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7.3

Nanostructured CePd2PtMx

In the final series of CePd3 -based compounds examined in the course of this study, three
different modification strategies were combined in an attempt to further enhance ZT . Samples of CePd2 PtGax were arc melted, ball milled, and sintered through PECS in the manner
described above. The behavior of the Seebeck coefficient closely resembled that of the nanostructured CePd3 Ga0.02 samples in Figure 7.5, with the only difference being a slight change
in the temperature dependence of the linear portion (T > 150 K) that mimicks the behavior
of CePd2 Pt seen in Figure 4.4. The change in thermal conductivity was also very similar to
that of the nanostructured CePd2 Pt samples seen in Figure 7.7 and Figure 7.8.
Although a favorable reduction in κ can be achieved by combining these mechanisms,
the value of α remained too low for any improvement in ZT to be observed. Nonetheless, this
study has demonstrated that combining chemical substitutions and/or processing methods
in novel ways may enable net enhancements in the thermoelectric properties of materials
such as CePd3 that may not be otherwise achieved through each approach separately. The
limited number of combinations attempted in this work leaves a substantial range of joint
composition and processing conditions available for further study.

114

Chapter 8
Related compounds
Given the sensitivity of the IV state in CePd3 to even small changes in composition and
structure, it is generally very difficult to find ways of decreasing the thermal conductivity
without also affecting the electronic properties. For this reason, it is desirable to identify
other base compounds that are not already optimally doped so that the thermoelectric
properties may be simultaneously optimized.
Several other strongly correlated materials are known to display intermediate valence,
mixed valence, and/or Kondo-type behavior that may give rise to favorable thermoelectric
properties. These compounds exist within a wide range of stoichiometry ratios that can be
described by the general chemical formula Rx Ty Mz , where R is a lanthanide or actinide
element with 1 < x < 3, T is a transition metal or noble metal element with 1 < y < 3, and
M is a p-block element with 0 < z < 4.
These compounds form in various crystal structures that also span a wide range of complexity, including the relatively simple perovskite-like RT3 Mz (Cu3 Au-type; e.g. CePd3 Bx
[39]), the hexagonal RT2 M3 (PrNi2 Al3 -type; e.g. CeNi2 Al3 [120]), and the complex cubic
R3 T3 M4 (Y3 Au3 Sb4 -type; e.g. Ce3 Pt3 Sb4 [121]). The available data on thermoelectric
properties of these compounds suggest that some of them may be close to the ranges necessary for high ZT . In order to explore these possibilities, a few of these materials were
synthesized and analyzed as part of this study.

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8.1
8.1.1

EuPd3Mx
Background and motivation

All of the lanthanide elements crystallize into the Cu3 Au structure when combined with
palladium in a 1:3 ratio, and all of these compounds have been reported to combined with
boron in a 1:3:1 ratio to form a perovskite analog phase [39]. EuPd3 B displays a lattice
parameter significantly larger than the value that would be expected based on the trends for
RPd3 B, and this anomaly is a strong indication of IV/MV behavior in this material.
Examination of lattice parameter, magnetic susceptibility, and XANES data for the
EuPd3 Bx series combined with density functional theory (DFT)-based calculations have confirmed that the europium atoms exist in both the 2+ and 3+ configuration for 0.2 ≤ x ≤ 0.5.
Only Eu3+ seems to be present for x ≤ 0.2, while the solubility of boron into the interstitial
site saturates for x > 0.5 [122]. Although the temperature dependence of these spectroscopic
analyses suggest that the europium atoms exist in the MV and not IV state, it appears from
these studies that boron doping nonetheless allows for precise control of the 4f peak relative
to the Fermi level. For this reason, it is worth examining this series to search for signs of
enhanced Seebeck coefficient.

8.1.2

Experimental methods

Samples of EuPd3 Bx were prepared by arc melting pieces of europium, palladium, and boron
in the appropriate stoichiometric ratios. Mass losses as a result of arc melting were on the
order of 1 %. Europium has a lower vapor pressure and a higher oxygen affinity than cerium,
which introduces additional concerns regarding phase purity and sample stoichiometry. In
order to address these concerns, SEM and EDS experiments were performed on these samples
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Figure 8.1: Lattice parameter of EuPd3 Bx . Solid lines are a guide for the eye.
after arc melting.

8.1.3

Results and discussion

XRD analysis of the samples after arc melting indicated single phase materials in the Cu3 Autype structure with no obvious impurity phases present. Results of lattice parameter refinements are included in Figure 8.1 and show a systematic increase with increasing boron
content. Consistent with previous reports [122], a kink is observed for x ≈ 0.25.
Although the XRD results suggest these materials are single phase, SEM and EDS analysis (not included) show what appear to be small amounts of an unidentified oxide impurity
phase not detected through XRD. These samples tended to be porous and brittle, which
may be at least partially a result of the presence of this secondary phase. For this reason,
it is difficult to attribute the absolute magnitudes of the electrical resistivity and thermal

117

conductivity to the intrinsic properties of the materials. The Seebeck coefficient and temperature dependence of the resistivity may, however, still be reliably determined, as they
depend predominantly on the electronic structure and are mostly insensitive to porosity and
small volume fractions of impurity phases.
The measured Seebeck coefficient as a function of temperature for each sample has been
included in Figure 8.2. A general trend emerges in these data that is consistent with the
lattice parameter results as well as the DOS calculations in the literature [122]. For samples
with 0 < x < 0.25, the Seebeck coefficient is small, negative, and increases in magnitude
linearly with increasing temperature, consistent with typical metallic behavior. For x = 0.25,
the temperature dependence of the Seebeck coefficient is linear but with a positive slope,
and it crosses over from n-type to p-type at ∼275 K. The sample with x = 0.275 also shows
linear temperature dependence with a positive slope and larger positive values. A change in
behavior is observed for x ≥ 0.3, wherein the temperature dependence is no longer linear, and
a maximum as a function of composition occurs at approximately α = 25 µV/K for x = 0.3.
This behavior is overall consistent with a crossover from a simple metallic system to one
in which magnetic and/or other types of scattering processes dominant to the electronic
properties.
The temperature dependence of the electrical resistivity also shows variations consistent
with the transition in the lattice parameter and Seebeck coefficient mentioned above. The
results of these measurements are included in Figure 8.3. A systematic change in behavior
is observed as the temperature dependence gradually shifts from the ρ ≈ ρ0 + a0 T behavior
expected for a conventional metal to the ρ ≈ ρ0 + d0 ln T behavior expected for a system in
which spin-flip scattering dominates.
It is interesting to note that the sign of d0 is positive in this system, while it is negative
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Figure 8.2: Seebeck coefficient of EuPd3 Bx . Solid lines are linear fits to the data, while
dashed lines are a guide for the eye.

Figure 8.3: Normalized electrical resistivity of EuPd3 Bx . Solid lines are linear fits to the
data, while the dashed line is fit to ρ/ρ350 = ρ0 /ρ350 + (d0 /ρ350 ) ln T .
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Figure 8.4: Linear fit parameters for the normalized electrical resistivity of EuPd3 Bx . Solid
and dashed lines are a guide for the eye.
in CePd3 Bx . This indicates that the exchange energy between the local and conduction
electrons is positive in this material.
These changes in the coefficients of ρ(T ) can be quantified by analyzing the slope and
intercept of the linear fit lines in Figure 8.3. Figure 8.4 has been included to show the
trends in these parameters as a function of x. No significant change is observed until the
composition reaches x ≥ 0.25, at which point a linear increase in a0 /ρ350 and decrease in
ρ0 /ρ350 are seen as x is further increased. Since the behavior of the system appears to evolve
gradually from ρ(T ) ≈ ρ0 + a0 T to ρ(T ) ≈ ρ0 + d0 ln T , it is possible that these changes in
the fit parameters are simply indicative of an increasing contribution from the d0 ln T term
that does not become significant until x > 0.4.
Although reliable thermal conductivity data could not be obtained from the samples
examined in this study, the measured values were on the order of 1 W/m-K to 10 W/m-K,
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and a decrease in magnitude was generally observed as x was increased. These results are
consistent with the trends for CePd3 Mx . The Seebeck coefficient of the doped samples did
not reach the range of 100 µV/K necessary for high ZT , which suggests that this system is
not suitable for thermoelectric cooling. A systematic change in the transport properties was
observed that is consistent with a change in the dominant scattering mechanism between
the metallic and mixed valence regions. Additional studies with other M filler elements may
yield further insights into the general behavior of the material.

8.2
8.2.1

Ce3Pt3Sb4
Background and motivation

Ce3 Pt3 Sb4 belongs to a larger family of R3 T3 M4 compounds that crystallize in the Y3 Au3 Sb4 type structure [121], as pictured in Figure 8.5. These materials are interesting for thermoelectric applications for a number of reasons. The most immediate of these is that the size
and complexity of the unit cell are favorable for intrinsically low lattice thermal conductivity.
Typical κl values for these materials are relatively low and on the order of a few W/m-K,
including as low as 1.5 W/m-K in Sm3 Au3 Sb4 [123]. This matches the lowest value observed
in the CePd3 compounds studied here, which was obtained only after carefully combining
multiple thermal conductivity reduction mechanisms.
The other main reason these materials are interesting for thermoelectric applications is
that they can be loosely classified as narrow-gap semiconductors, and as such they tend to
display relatively large Seebeck coefficient and low resistivity values typical for such compounds. In addition to this, Ce3 T3 M4 materials often display Kondo-type behavior. The
mixture of the Kondo effect and semiconductor-like properties in some of these compounds
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Figure 8.5: Crystal structure diagram of Ce3 Pt3 Sb4 . Cerium atoms (dark gray) are in a distorted dodecahedral coordination with the antimony atoms (light gray), while the platinum
atoms (medium gray) are in a distorted tetragonal configuration. Edge and face sharing
cerium polyhedra are both present.

122

leads to their description as “Kondo insulators” [124], while other compounds in this family
behave like Kondo metals, similar to CePd3 . It is thus possible to observe a wide variety of
behavior in these systems depending on composition.
A demonstration of this variety can be seen in the resistivity of RCu3 Sb4−x Snx compounds with R = La, Ce, Pr, Nd, Sm, and x = 0.05 and 0.10 [125]. While all of the
compounds without cerium show what appears to be a nearly linear temperature dependence in the resistivity with magnitudes on the order of 10−5 Ω-m, the samples containing
cerium show a dramatically different dependence that appears to correspond with the ln (T )
behavior typical for Kondo metals.
Given the large number of isostructural compounds with similar lattice parameters in
this family, an enormous number of possible variations exist that can be formed through
single or double substitutions on the R, T , and/or M sites. Such solid solution mixing can
be expected to result in a robust range of electronic properties while simultaneously inducing
a reduction in the lattice thermal conductivity.
Two of the few complete thermoelectric studies available on this compound explore the
properties of Ce3 Pt3−x Cux Sb4 [124] and Ce3−x Ndx Pt3 Sb4 [126]. In these studies, the authors show that the material can be systematically doped through substitutions on both
the cerium and platinum sites. Copper substitutions on the platinum site, for example, can
reduce the electrical resistivity at 100 K by three orders of magnitude, from 10−2 Ω-m in
Ce3 Pt3 Sb4 to 10−5 Ω-m in Ce3 Cu3 Sb4 . The Seebeck coefficient also changes dramatically
in this system, from a peak value at 100 K of 300 µV/K in Ce3 Pt3 Sb4 , to -175 µV/K in
Ce3 Pt2.5 Cu0.5 Sb4 , to approximately zero in Ce3 Cu3 Sb4 .
This strong sensitivity to composition changes exposes one of the biggest problems related
to this family of compounds, which is that stoichiometry can be very difficult to control
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during arc melting due to vaporization of M elements such as antimony and bismuth. The
sensitivity of these systems to compositional variation is apparent even in the nominally
stoichiometric Ce3 Pt3 Sb4 control samples in Ref. [126]. As the authors show, the room
temperature resistivity varies by an order of magnitude between two nominally identical
samples, and the peak Seebeck coefficient ranges from approximately 230 µV/K to 350
µV/K. The only difference between the samples is that one was measured directly from
arc melting and the other after powder processing, respectively. Since these changes in the
transport properties are comparable to those resulting from compositional modifications, it is
imperative to improve stoichiometry control in these materials in order to better distinguish
between the effects of off-stoichiometry and intentional doping.
Two significant challenges arise that make phase purity difficult to obtain. The first
is that these compounds tend to have relatively high melting points, which means that
arc melting is generally necessary to form them. However, antimony and bismuth tend to
vaporize readily when heated to these temperatures. Excess amounts of approximately 5
at.% of the volatile elements are commonly added to account for this vaporization [126], and
although the resulting XRD patterns are reported to be single phase, the limited resolution
of this technique for detecting secondary phases does not completely eliminate the concerns
over stoichiometry control.
This problem is amplified by the second synthesis challenge common in these materials,
which is that the RT M -type phase is very close in stoichiometry to R3 T3 M4 . Thus even if
problems regarding M vaporization are resolved, any local deviations in stoichiometry may
lead to the spontaneous formation of RT M .
There is an obvious need to develop more consistent synthesis techniques for this system
if it is to be properly studied and explored as a candidate thermoelectric material. This has
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motivated the current brief study, which considers a precursor synthesis approach as a way
of improving stoichiometry control, homogeneity, and overall sample quality.
Inspired by the success of the this approach in the synthesis of CePd3 Mx compounds
containing volatile elements such as tellurium, an analysis of the Ce-Sb [127] and Pt-Sb
[128] binary phase diagrams revealed that only two congruently melting compounds exist
between these systems. The first is CeSb, which melts at 1820◦ C (2093 K), and the other
is PtSb2 , which melts at 1225◦ C (1498 K). The latter compound is coincidentally a narrowgap semiconductor identified by Slack as a possible low temperature thermoelectric material
[12]. Since CeSb has an insufficiently high antimony content to enable the appropriate atomic
ratios in the final compound, PtSb2 is the only suitable precursor phase in either of these
systems.

8.2.2

Experimental methods

Platinum melts at 1769◦ C (2042 K), and antimony at 631◦ C (904 K). A eutectic point exists
in the Pt-Sb system at 31.5 at.% antimony and 630◦ C (903 K), below which a mixture of
Pt3 Sb and Pt3 Sb2 forms [128]. In order to expedite the solid state reaction of the antimony
with the platinum into the PtSb2 phase, the Pt-Sb eutectic phase was first created by mixing
68.5 at.% platinum with 31.5 at.% antimony in an evacuated and sealed quartz ampoule.
This ampoule was placed into a furnace pre-heated to 800◦ C (1073 K) for 90 seconds then
removed and allowed to slowly cool to room temperature. A small metallic bead formed in
the bottom of the ampoule that consisted of 98.6% of the starting mass, suggesting that
this temperature was high enough to cause approximately 13% of the antimony to vaporize
before reacting.
This bead was crushed with a mortar and pestle to increase its surface area, then placed
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into another quartz ampoule with an appropriate amount of antimony to produce a 1:2
atomic ratio of platinum and antimony. The mass loss in the eutectic sample was assumed
to be entirely antimony, and the composition was thus normalized to the amount of platinum
initially present. A slight excess of approximately 1 at.% antimony was further added to
help drive the solid state reaction and compensate for any small amounts of additional
vaporization.
The ampoule was evacuated, sealed, and placed back into a furnace at 900◦ C (1173 K)
for 48 hours, then removed and allowed to slowly cool to room temperature. The resulting
sample was brittle, highly porous, and displayed a dull gray color. This process resulted
in a mass loss of 2.6%, again suggesting that some antimony vaporized before reacting.
Examination under an optical microscope reveled the presence of many small and faceted
crystallites approximately 1 mm in size. XRD analysis indicated the presence of only the
expected PtSb2 pyrite phase.
The stability of this PtSb2 phase was examined by repeatedly melting it directly in the arc
melter. Considerable vaporization was observed as a result of this process that corresponded
with a mass loss of over 50%. XRD analysis after arc melting indicated the presence of
PtSb, suggesting that a relatively large portion of antimony had vaporized. However, since
PtSb2 is only 23.7% antimony by mass, the magnitude of mass loss suggests that at least
some of it was also PtSb2 . Either way, these results suggest that PtSb2 is not an effective
precursor for arc melting, as it does not appear to be any more stable upon melting than
elemental antimony. This may be related to the relatively low melting point of the compound
compared to the temperatures reached inside the arc melter.
A sample of Ce3 Pt3 Sb4 was arc melted directly from the pure elements to compare the
level of mass loss typical for this process, and the results of this experiment produced an
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overall loss of only 1.1% after melting. Assuming the mass lost to be entirely antimony, this
result corresponded with a loss of approximately 13 at.% Sb. XRD patterns of this sample
indicated the clear presence of both CePtSb and Ce3 Pt3 Sb4 . To encourage the formation of
the Ce3 Pt3 Sb4 phase, the sample was placed back in the arc melter with additional antimony
and remelted until a sample with a final mass within 1% of the starting mass was obtained.
XRD results of this final sample still indicated a mixture of CePtSb and Ce3 Pt3 Sb4 .
Based on the procedures described in Ref. [126], this ingot was annealed for one week at
900◦ C (1173 K). XRD analysis of the sample after annealing showed only the presence of
Ce3 Pt3 Sb4 , suggesting that this approach is effective for producing single phase materials.
Upon cutting this sample to prepare it for transport property measurements, however, it
was found to be porous and relatively brittle. These poor mechanical properties are likely a
result of the substantial phase change that appears to have occurred during annealing.
In an attempt to produce a dense pellet suitable for measurements through powder
processing, the material was ball milled in a way similar to the nanostructured CePd3 samples
mentioned above. The CePtSb phase reappeared in the XRD patterns after ball milling and
was still present after the powder was sintered through PECS. Further annealing of the
sample after PECS did not eliminate this phase. Because of these difficulties with producing
dense, phase-pure samples, transport properties could not be reliably measured for this
material.

8.2.3

Results and discussion

Based on the results of this study, it appears that precursor synthesis is not a viable option
for improving the stoichiometry control of Ce3 Pt3 Sb4 -based compounds. PtSb2 is the only
suitable precursor for this system, but it does not appear to be stable during arc melting.
127

Melting at lower temperatures in an induction furnace may allow for improved stability,
and this precursor approach may still be appropriate for other compositions in the R3 T3 M4
family for which suitable R-M or T -M binary compounds exist.
Although single phase materials could be produced through the conventional approach
of arc melting with excess antimony and then annealing, the poor mechanical properties of
these samples made reliable measurements difficult to obtain. Powder processing and further
annealing did not help to produce a dense, single phase material suitable for measurement.
In spite of these difficulties, this conventional approach still appears to be the most
reliable way of producing these materials. This suggests that future studies focusing on
optimization of the milling, sintering, and annealing conditions necessary for creating homogeneous, dense, single phase, powder processed samples may yield useful insights into
improving stoichiometry control.
The results of this study have also shown that single phase PtSb2 powder can be produced through solid state reactions. Although this system is unrelated to the cerium-based
compounds on which the current work is focused, this insight may be useful for the synthesis
of bulk samples of PtSb2 -based compounds suitable for further study as candidate materials
for cryogenic thermoelectric cooling.

8.3
8.3.1

CeNi2Al3
Background and motivation

Based on results of Sun [120, 129], CeNi2 Al3 has been identified as another candidate system
for low temperature thermoelectric applications. The compound crystallizes in the hexagonal
PrNi2 Al3 -type structure, as seen in the diagram in Figure 8.6. This system is an excellent
128

Figure 8.6: Crystal structure diagram of CeNi2 Al3 . Cerium atoms (dark gray) are at the
corners of the rhomboid, the nickel atoms (medium gray) are on the top and bottom faces,
and the aluminum atoms (light gray) are on the side faces and center.
example of one in which the base compound is not optimized, as the undoped material has
a maximum Seebeck coefficient of only ∼50 µV/K at 300 K that decreases with decreasing
temperature. Sun reports that the substitution of copper onto the nickel site results in
the simultaneous optimization all of three thermoelectric properties, increasing α up to
approximately 90 µV/K, decreasing ρ to 10−6 Ω-m, and decreasing κl to under 5 W/m-K
at 100 K for x = 0.8.
In an attempt to reproduce these results and better determine the range of optimal copper
content, samples of CeNi2−x Cux Al3 were synthesized and measured as part of this work.

129

8.3.2

Experimental methods

Guided by the available reports in the literatures [120, 129], samples of CeNi2−x Cux Al3 (0.6
< x <1) were prepared in the standard arc melting technique described in Chapter 2. Mass
losses for these samples were less than 1%. Some of the samples were also crushed through
vibratory milling in the same way as the nanostructured CePd3 samples were prepared above,
then sintered through PECS at 800◦ C for 1 minute under 60 MPa of uniaxial pressure in
an attempt to produce bulk nanostructured samples. Although substantial peak broadening
was observed in the XRD patterns of the ball milled powders, analysis of the XRD patterns
and transport properties after PECS showed no apparent changes relative to the original arc
melted samples, indicating that these PECS conditions are not appropriate for preservation
of the nanoscale features.

8.3.3

Results and discussion

A substantial shift in the lattice parameter was observed based on the XRD patterns obtained
after arc melting, indicating an increase in the lattice parameter as x increases. The results
of lattice parameter calculations are included in Figure 8.7 and are consistent with the values
previously reported for similar compounds [120, 129].
Results of Seebeck coefficient measurements are included in Figure 8.8 and are consistent
with the results of Ref. [120]. Although Ref. [129] reported values as high as 90 µV/K
in samples with x = 0.8 at 80 K, the largest value observed in this study was 65 µV/K at
the same temperature for x = 0.7, essentially identical to the results of Ref. [120]. This
value is substantially higher than would be expected for a conventional metal, but it is still
approximately two to three times smaller than what is needed for ZT > 0.1.

130

Figure 8.7: Lattice parameter of CeNi2−x Cux Al3 . Solid lines are linear fits to the data.

Figure 8.8: Seebeck coefficient of CeNi2−x Cux Al3 .

131

Figure 8.9: Normalized electrical resistivity of CeNi2−x Cux Al3 . A progression in behavior
is observed as x is increased. Dashed lines are fits to ρ/ρ350 = ρ0 /ρ350 + (d0 /ρ350 ) ln T ,
solid lines are fits to ρ/ρ350 = ρ0 /ρ350 + (a0 /ρ350 )T , and the dotted line is a fit to ρ/ρ350 =
ρ0 /ρ350 − (d0 /ρ350 ) ln T + (a0 /ρ350 )T . This evolution indicates a gradual transition in the
nature of the electronic scattering mechanisms with composition.
The electrical resistivity data are included in Figure 8.9. Due to problems with sample
porosity, data for each set have been divided by the corresponding value at 350 K in order
to emphasize the temperature dependence. The values of ρ350 were on the order of approximately 10−6 Ω-m and identical to within a factor of two for all samples, indicating that no
significant change in magnitude occurs as a function of composition.
A gradual change in the behavior of ρ(T ) can be seen as x is increased. For relatively
small x ≤ 0.7, ρ(T ) ≈ ρ0 + d0 ln T , indicating the dominant contribution from magnetic
scattering with a positive exchange energy. For 0.8 ≤ x ≤ 0.9, the data can be fit sufficiently
well with only the linear contribution from phonon scattering of the form ρ(T ) ≈ ρ0 + a0 T .
For x = 1, both the linear and logarithmic terms must be considered in order to achieve an

132

accurate fit to the data, and ρ is of the form ρ(T ) ≈ ρ0 + a0 T + d0 ln T , now with a negative
sign of d0 that indicates a change in the sign of the exchange energy. This change in sign of d0
is consistent with the results of Ref. [129], in which it was established that antiferromagnetic
ordering begins for x ≈ 1.0. The increased importance of the linear term with increasing x
may be a result of changes in the Debye temperature with composition. Overall, this change
in behavior is similar to that seen in EuPd3 Bx , in which the magnitude of the magnetic
scattering contribution to the resistivity can also be controlled through composition.
Based on the results of this work, this specific solid solution does not appear to be a
viable candidate for high ZT at low temperatures. Although the resistivity and lattice
thermal conductivity are both within the range necessary for achieving reasonably strong
thermoelectric performance, the maximum Seebeck coefficient of 65 µV/K corresponds with
a maximum theoretical ZT of only ∼0.17 at 80 K. Further study of the effects of substitutions
on the cerium and aluminum sites may yield additional insights into how the electronic and
thermal properties of the material can be controlled. Since it does not appear that this
type of compositional modification causes any appreciable change in the magnitude of the
electrical resistivity, emphasis should be placed on exploring the appropriate combination of
chemical pressure and doping for maximizing the Seebeck coefficient. If this can be done to
achieve α ∼ 100 µV/K, this material has potential as a more cost effective alternative to the
palladium and platinum based compositions discussed above.

133

Chapter 9
Results and Conclusions
This work has presented a systematic study of structure-processing-property relationships
and thermoelectricity in several series of rare earth intermetallic compounds. The primary
emphasis has been placed on CePd3 , as this material has one of the largest known power
factors of all p-type bulk materials at low temperatures. A thorough combination of composition and processing modifications has been used to explore the available mechanisms
for enhancing the thermoelectric properties of CePd3 , and some of the insights gained from
these studies have been extended to brief examinations of the related compounds EuPd3 Bx ,
CeNi2−x Cux Al3 , and Ce3 Pt3 Sb4 .
The broad goal of this study was to determine what, if any, mechanisms are available for
decreasing the thermal conductivity ratio while increasing or at least maintaining the large
Seebeck coefficient. Three series of chemically modified CePd3 have been used to explore this
possibility based on the available lattice sites on which atoms can be added or substituted.
The elements chosen for these modifications were carefully selected so as to optimize their
impact and/or exhaustively explore their effects, thus establishing upper and/or lower limits
for the maximum possible improvements that can be made to the material through each
approach.
The study of scandium substitutions on the cerium site has definitively demonstrated
that no significant thermal conductivity reduction can be achieved through this route, as
scandium has the maximum possible size and mass mismatch with cerium and would thus
134

be expected to have a maximal effect on the thermal conductivity.
Platinum was chosen for substitutions on the palladium site due to the large mass mismatch and chemical similarities of the two elements, and the results of this study confirm
that this approach is effective for reducing κ while slightly increasing α. This combination
has led to the highest ZT values reported to date for any bulk polycrystalline p-type material
at low temperatures.
The wide variety of elements chosen for partially filling the central interstitial site, including multiple elements that had never been used in this approach before, has demonstrated
that the electronic and thermal properties of the material can be separately controlled to a
limited extent. As long as no structural ordering occurs, the impact of this partial filling
approach on the electronic and magnetic properties is determined primarily by the filling
fraction and valence electron count of M , while the effects on the lattice thermal conductivity depend primarily on the filling fraction and mass of M . Although no appreciable gains
in ZT could be made through this approach, it nonetheless demonstrates a generally useful
way of separately tuning the thermoelectric properties of a material.
In addition to these chemical modifications, this study reports for the first time the
transport properties of a nanostructured correlated metal. The reduction in size of the
CePd3 grains to the nanoscale appears to significantly reduce the lattice thermal conductivity
while simultaneously resulting in what may be described as morphologically-driven doping.
The combination of this approach with interstitial site filling has been demonstrated as a
possible way of double-doping the material such that no significant change in the electronic
properties occurs. Examination of nanostructured CePd2 Pt has also demonstrated that
grain size reduction may be combined with other thermal conductivity reduction strategies
to further reduce κ below values that either approach can achieve on its own.
135

Studies of EuPd3 Bx have shown for the first time that the nature of the electronic
properties in this material can be carefully tuned through adjusting the boron content,
although the Seebeck coefficient does not appear to be large enough to facilitate high ZT .
Based on the results of the CePd3 Mx series studied here, however, it may be interesting
to examine the the effect of introducing other M elements into the interstitial site to see if
larger α can be found.
Attempts at improving stoichiometry control in Ce3 Pt3 Cu4 via solid precursor synthesis
failed to produce a more stable compound suitable for arc melting. The importance of
annealing in obtaining single phase materials synthesized through the conventional approach
was demonstrated, and an attempt at producing this compound through powder processing
approaches was also discussed. The challenge of improving stoichiometry control in this series
remains. The insights gained from CePd3−x Ptx regarding the importance of substituting
chemically similar elements to preserve the electronic properties may be utilized in this
system to further reduce its thermal conductivity after its electronic properties have been
properly tuned.
Finally, CeNi2−x Cux Al3 was also examined as a possible high ZT intermetallic compound. Although copper substitution on the nickel site does appear to simultaneously improve all three thermoelectric properties, the maximum α observed is again insufficiently
large to enable high ZT . Nano-scale powders of this material were successfully produced by
ball milling, but the nanoscale features were not preserved after PECS. This study has thus
established an upper limit on the processing conditions appropriate for producing nanostructured samples of CeNi2−x Cux Al3 . Additional substitutions on the nickel and/or aluminum
sites, guided by the basic trends observed in the several CePd3 series studied here, may lead
to further improvements in the thermoelectric properties of this material.
136

From a broad perspective, this work has thoroughly explored the best available ways of
modifying some of the best known p-type materials for cryogenic thermoelectric applications.
Although this study has produced record high ZT values for bulk polycrystalline p-type materials at low temperatures, these values are likely still too low to make solid state cryocooler
device construction practical. The results of this work may nonetheless be generally useful for providing empirical feedback to theoretical models of the structural, magnetic, and
transport properties of correlated metals, as well as guide future studies of other correlated
systems identified as potential high ZT materials.
If such materials can be found and engineered to achieve sufficiently high ZT , the development of efficient solid state cooling technologies may finally be realized. This development
would open up potentially enormous markets in commercial, industrial, military, and research cooling technologies, and extend even further mankind’s ability to harness the power
of nature.

137

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