THE EFFECTS OF DOPING AND PROCESSING ON THE THERMOELECTRIC
PROPERTIES OF PLATINUM DIANTIMONIDE BASED MATERIALS FOR
CRYOGENIC PELTIER COOLING APPLICATIONS
By
Spencer Laine Waldrop

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Materials Science and Engineering — Doctor of Philosophy
2017

ABSTRACT
THE EFFECTS OF DOPING AND PROCESSING ON THE THERMOELECTRIC
PROPERTIES OF PLATINUM DIANTIMONIDE BASED MATERIALS FOR
CRYOGENIC PELTIER COOLING APPLICATIONS
By
Spencer Laine Waldrop
The study of thermoelectrics is nearly two centuries old. In that time a large number of applications have been discovered for these materials which are capable of transforming thermal
energy into electricity or using electrical work to create a thermal gradient. Current use of
thermoelectric materials is in very niche applications with contemporary focus being upon
their capability to recover waste heat. A relatively undeveloped region for thermoelectric
application is focused upon Peltier cooling at low temperatures. Materials based on bismuth
telluride semiconductors have been the gold standard for close to room temperature applications for over sixty years. For applications below room temperature, semiconductors based
on bismuth antimony reign supreme with few other possible materials.
The cause of this difficulty in developing new, higher performing materials is due to
the interplay of the thermoelectric properties of these materials. The Seebeck coefficient,
which characterizes the phenomenon of the conversion of heat to electricity, the electrical
conductivity, and the thermal conductivity are all interconnected properties of a material
which must be optimized to generate a high performance thermoelectric material. While for
above room temperature applications many advancements have been made in the creation
of highly efficient thermoelectric materials, the below room temperature regime has been
stymied by ill-suited properties, low operating temperatures, and a lack of research.
The focus of this work has been to investigate and optimize the thermoelectric properties
of platinum diantimonide, PtSb2 , a nearly zero gap semiconductor. The electronic properties

of PtSb2 are very favorable for cryogenic Peltier applications, as it exhibits good conductivity
and large Seebeck coefficient below 200 K. It is shown that both n- and p-type doping
may be applied to this compound to further improve its electronic properties. Through
both solid solution formation and processing techniques, the thermal conductivity may be
reduced in order to increase the thermoelectric figure of merit. Further reduction in thermal
conductivity using other novel approaches is identified as an area of promising future research.
Continued development of this material has the potential to generate a suitable replacement
for some low temperature applications, but will certainly further scientific knowledge and
understanding of the optimization of thermoelectric materials in this temperature regime.

I would like to dedicate this work to my family. Without their love, support, and
encouragement this would not have been made.

iv

ACKNOWLEDGMENTS

I am reminded and grateful for the many people who have played a role in helping me to this
point in life. I would first like to thank my advisor Dr. Donald Morelli for allowing me this
opportunity to study thermoelectrics these past four years, his thoughtfulness and guidance,
and his willingness to allow me to explore any topic no matter how reaching it was. I would
also like to thank my committee members Dr. Tim Hogan, Dr. Wei Lai, and Dr. Thomas
Bieler for their assistance and direction.
I would like to thank my past and current lab mates, Dr. Gloria Lehr, Dr. Steve Boona,
Dr. Xu Lu, Dr. Vijay Ponnambolam, Mr. Daniel Weller, and Mr. Spencer Mather for their
help and insight. I would like to thank Dr. Dale Hitchcock for providing me a model of a
graduate student and being my first research mentor. I would also like to thank Dr. Terry
Tritt for first introducing and stirring in me an excitement and enthusiasm for solid state
physics.
I also need to single out and thank two colleagues and friends who have had a very special
role in helping me through these past four years. I’d like to thank Dr. Winston Carr for
his never ending patience, insightful conversations spanning everything from thermoelectrics
to economics, and, of course, the proper way to wear a beard. To Dr. Jared Williams I
would like to thank him for his friendship these past six years. He was always there to offer
insight or critique, to help with a hard problem or study for a test, to debate an issue or
philosophize on end, and to offer a word of encouragement whenever it was needed.
I would also like to acknowledge and thank my family for their love and support through
these four years. First, to my mother, who raised me to persevere through anything, value
the small things in life, and know that we can make it through anything together. To my
v

father, who taught me how to slow down, to value honor, and to never turn back unless it’s
for a good night’s rest. To my sister, who always had my back, but never forgot to teach
me humility every chance she got. And finally, to my wife, who taught me to love life again
and to always keep the child at heart.
Finally, I would like to acknowledge that this research was funded by the Air Force Office
of Scientific Research under the Multi-University Research initiative (MURI), Cryogenic
Peltier Cooling, Contract No. FA9550-10-1-0533.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

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

xi

Chapter 1 Introduction . . . . . . . . . . . . . . . . . .
1.1 Thermoelectric Effects . . . . . . . . . . . . . . . .
1.2 Application of the Thermoelectric Effects . . . . . .
1.2.1 Power Generation . . . . . . . . . . . . . . .
1.2.2 Peltier Cooling . . . . . . . . . . . . . . . .
1.3 Thermoelectric Device Performance . . . . . . . . .
1.4 Electronic Transport . . . . . . . . . . . . . . . . .
1.4.1 Energy-Band Diagrams and Material Types
1.4.2 Electrical Conductivity . . . . . . . . . . . .
1.4.3 The Seebeck Coefficient . . . . . . . . . . .
1.4.4 The Power Factor . . . . . . . . . . . . . . .
1.5 Thermal Transport . . . . . . . . . . . . . . . . . .
1.5.1 Electronic Thermal Conductivity . . . . . .
1.5.2 Lattice Thermal Conductivity . . . . . . . .
1.6 Material Selection and Enhancement of zT . . . . .

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1
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35

Chapter 2 Experimental . . . . . . . . . . . . . . . . . . . . . . .
2.1 Materials Synthesis . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Furnace Reaction Method . . . . . . . . . . . . . . . .
2.1.2 Flux Synthesis . . . . . . . . . . . . . . . . . . . . . .
2.2 Material Characterization Techniques . . . . . . . . . . . . . .
2.2.1 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . .
2.2.2 Low Temperature Transport Property Characterization
2.2.3 Hall Effect Measurement . . . . . . . . . . . . . . . . .

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40
40
40
42
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51

Chapter 3 Background on PtSb2 . . . . . . . . . . . . .
3.1 Low Temperature Thermoelectric Materials . . . . .
3.2 PtSb2 . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Crystal Structure and Phase Diagram . . . . .
3.2.2 Electronic Structure and Transport Properties
3.3 Nominally Undoped PtSb2 . . . . . . . . . . . . . . .
3.3.1 Synthesis . . . . . . . . . . . . . . . . . . . .
3.3.2 Structural Properties . . . . . . . . . . . . . .
3.3.3 Electrical Properties . . . . . . . . . . . . . .
3.3.4 Thermal Properties . . . . . . . . . . . . . . .

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53
53
55
55
57
61
61
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67

vii

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Chapter 4 Antimony Site Doping in PtSb2 . .
4.1 Background and Motivation . . . . . . . . .
4.2 Experimental Methods . . . . . . . . . . . .
4.3 Tellurium Dopant Results and Discussion . .
4.3.1 X-ray Diffraction . . . . . . . . . . .
4.3.2 Electronic Properties . . . . . . . . .
4.3.3 Thermal Properties . . . . . . . . . .
4.3.4 zT . . . . . . . . . . . . . . . . . . .
4.4 Selenium Dopant Results and Discussion . .
4.4.1 X-Ray Diffraction . . . . . . . . . . .
4.4.2 Electronic Properties . . . . . . . . .
4.4.3 Thermal Properties . . . . . . . . . .
4.4.4 zT . . . . . . . . . . . . . . . . . . .
4.5 Tin Dopant Results and Discussion . . . . .
4.5.1 X-Ray Diffraction . . . . . . . . . . .
4.5.2 Electronic Properties . . . . . . . . .
4.5.3 Thermal Properties . . . . . . . . . .
4.5.4 zT . . . . . . . . . . . . . . . . . . .
4.6 Germanium Dopant Results and Discussion
4.6.1 X-Ray Diffraction . . . . . . . . . . .
4.7 Indium Dopant Results and Discussion . . .
4.7.1 X-Ray Diffraction . . . . . . . . . . .
4.7.2 Electronic Properties . . . . . . . . .
4.7.3 Thermal Properties . . . . . . . . . .
4.7.4 zT . . . . . . . . . . . . . . . . . . .
4.8 Conclusions . . . . . . . . . . . . . . . . . .

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70
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92
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95
96

Chapter 5 Platinum Site Doping in PtSb2 . . . . . . .
5.1 Background and Motivation . . . . . . . . . . . . . .
5.2 Experimental Methods . . . . . . . . . . . . . . . . .
5.3 Iron Dopant Results and Discussion . . . . . . . . . .
5.3.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
5.3.2 Electronic Properties . . . . . . . . . . . . . .
5.3.3 Thermal Properties . . . . . . . . . . . . . . .
5.3.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
5.4 Ytterbium Dopant Results and Discussion . . . . . .
5.4.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
5.4.2 Electronic Properties . . . . . . . . . . . . . .
5.4.3 Thermal Properties . . . . . . . . . . . . . . .
5.4.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Silver and Niobium Doping Results and Discussion .
5.5.1 X-Ray Diffraction of Silver Doped Samples . .
5.5.2 Electronic Properties of Silver Dopant . . . .
5.5.3 Thermal Properties and zT of Silver Dopant .
5.5.4 X-Ray Diffraction of Niobium Doped Samples

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100
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102
103
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104
107
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116
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120
120

viii

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5.6

5.5.5 Thermal Properties and zT . . . . . . . . . . . . . . . . . . . . . . . 124
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Chapter 6 Methods to Reduce Thermal Conductivity
6.1 Background and Motivation . . . . . . . . . . . . . .
6.2 Experimental Methods . . . . . . . . . . . . . . . . .
6.2.1 Solid Solutions with Palladium and Bismuth .
6.2.2 Ball Milling and Electronic Doping in PtSb2 .
6.2.3 Non-Stoichiometric PtSb2 . . . . . . . . . . .
6.3 Solid Solution Results and Discussion . . . . . . . . .
6.3.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
6.3.2 Electronic Properties . . . . . . . . . . . . . .
6.3.3 Thermal Properties . . . . . . . . . . . . . . .
6.3.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Ball Milling Results and Discussion . . . . . . . . . .
6.4.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
6.4.2 Thermal Properties . . . . . . . . . . . . . . .
6.4.3 Electronic Properties . . . . . . . . . . . . . .
6.5 Non-Stoichiometric PtSb2 Results and Discussion . .
6.5.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
6.5.2 Electronic Properties . . . . . . . . . . . . . .
6.5.3 Thermal Properties . . . . . . . . . . . . . . .
6.5.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Hole Doping Ball Milled PtSb2 . . . . . . . . . . . .
6.6.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
6.6.2 Electronic Properties . . . . . . . . . . . . . .
6.6.3 Thermal Properties . . . . . . . . . . . . . . .
6.6.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
6.7 Electron Doping Ball Milled PtSb2 . . . . . . . . . .
6.7.1 X-Ray Diffraction . . . . . . . . . . . . . . . .
6.7.2 Electronic Properties . . . . . . . . . . . . . .
6.7.3 Thermal Properties . . . . . . . . . . . . . . .
6.7.4 zT . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . .

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126
126
129
129
130
130
131
131
132
135
136
138
138
140
141
146
146
147
150
151
153
153
154
158
160
164
164
165
167
169
170

Chapter 7 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . 175
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

ix

LIST OF TABLES

Table 7.1:

Comparison of the Thermoelectric Properties of Electronically Doped
PtSb2 at the Temperature of Maximum zT . . . . . . . . . . . . . . . . 180

x

LIST OF FIGURES

Figure 1.1:

Schematic of thermocouple displaying Seebeck effect . . . . . . . . . . .

2

Figure 1.2:

Schematic of Peltier Cooler and Thermoelectric Power Generator . . . .

8

Figure 1.3:

COP vs. Delta T with Hot Side Temperature of 300 K . . . . . . . . .

10

Figure 1.4:

Multi-Stage Device COP vs. Delta T with Hot Side Temperature of 300
K and Z of 10−3 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Schematic Energy Band Diagram for Metals, Semiconductors, and Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15

Figure 1.6:

Fermi-Dirac Distribution for Multiple Temperatures . . . . . . . . . . .

18

Figure 1.7:

Schematic of Open Circuit Seebeck Effect . . . . . . . . . . . . . . . . .

25

Figure 1.8:

Diagram of Power Factor, Seebeck Coefficient, and Electrical Conductivity versus Carrier Concentration . . . . . . . . . . . . . . . . . . . . . .

29

Figure 2.1:

Schematic of Spark Plasma Sinterer . . . . . . . . . . . . . . . . . . . .

42

Figure 2.2:

Schematic of Bragg Diffraction . . . . . . . . . . . . . . . . . . . . . . .

45

Figure 2.3:

Schematic of Cryostat Measurement Setup . . . . . . . . . . . . . . . .

48

Figure 2.4:

Schematic of Hall Effect Measurement Setup . . . . . . . . . . . . . . .

52

Figure 3.1:

Schematic of Pyrite Crystal Structure Showing Pt - Sb Local Bonding .

56

Figure 3.2:

Schematic Phase Diagram of the Pt-Sb System . . . . . . . . . . . . . .

57

Figure 3.3:

Schematic Energy Band Diagram of PtSb2 . . . . . . . . . . . . . . . .

58

Figure 3.4:

X-ray Diffraction Pattern of PtSb2 . . . . . . . . . . . . . . . . . . . . .

63

Figure 3.5:

Seebeck Coefficient of PtSb2 . . . . . . . . . . . . . . . . . . . . . . . .

64

Figure 3.6:

Electrical Resistivity of PtSb2 . . . . . . . . . . . . . . . . . . . . . . .

65

Figure 3.7:

Carrier Concentration and Mobility of PtSb2 . . . . . . . . . . . . . . .

66

Figure 1.5:

xi

Figure 3.8:

Total Thermal Conductivity of PtSb2 . . . . . . . . . . . . . . . . . . .

67

Figure 3.9:

Lattice and Electronic Contributions to Thermal Conductivity of PtSb2

68

Figure 4.1:

X-ray Diffraction of Tellurium Doped PtSb2 . . . . . . . . . . . . . . .

73

Figure 4.2:

Seebeck Coefficient of Tellurium Doped PtSb2 . . . . . . . . . . . . . .

74

Figure 4.3:

Electrical Resistivity of Tellurium Doped PtSb2 . . . . . . . . . . . . .

75

Figure 4.4:

Thermal Conductivity of Tellurium Doped PtSb2

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

77

Figure 4.5:

zT of Tellurium Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . . .

78

Figure 4.6:

X-ray Diffraction of Selenium Doped PtSb2 . . . . . . . . . . . . . . . .

79

Figure 4.7:

Seebeck Coefficient of Selenium Doped PtSb2 . . . . . . . . . . . . . . .

80

Figure 4.8:

Electrical Resistivity of Selenium Doped PtSb2 . . . . . . . . . . . . . .

82

Figure 4.9:

Thermal Conductivity of Selenium Doped PtSb2 . . . . . . . . . . . . .

83

Figure 4.10: zT of Selenium Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . . . .

84

Figure 4.11: X-ray Diffraction of Tin Doped PtSb2 . . . . . . . . . . . . . . . . . . .

85

Figure 4.12: Seebeck Coefficient of Tin Doped PtSb2 . . . . . . . . . . . . . . . . . .

86

Figure 4.13: Electrical Resistivity of Tin Doped PtSb2 . . . . . . . . . . . . . . . . .

87

Figure 4.14: Thermal Conductivity of Tin Doped PtSb2 . . . . . . . . . . . . . . . .

88

Figure 4.15: Thermal Conductivity of Te, Se, In and Sn Dopants at 90 K . . . . . .

89

Figure 4.16: zT of Tin Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

Figure 4.17: X-ray Diffraction of Germanium Doped PtSb2 . . . . . . . . . . . . . .

91

Figure 4.18: X-ray Diffraction of Indium Doped PtSb2 . . . . . . . . . . . . . . . . .

92

Figure 4.19: Seebeck Coefficient of Indium Doped PtSb2 . . . . . . . . . . . . . . . .

93

Figure 4.20: Electrical Resistivity of Indium Doped PtSb2 . . . . . . . . . . . . . . .

94

Figure 4.21: Thermal Conductivity of Indium Doped PtSb2 . . . . . . . . . . . . . .

95

xii

Figure 4.22: zT of Indium Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . . . . .

96

Figure 4.23: Effects of Dopant Concentration in Tellurium and Tin Doped PtSb2 . .

97

Figure 5.1:

X-ray Diffraction of Iron Doped PtSb2

. . . . . . . . . . . . . . . . . . 103

Figure 5.2:

Seebeck Coefficient of Iron Doped PtSb2 . . . . . . . . . . . . . . . . . 104

Figure 5.3:

Electrical Resistivity of Iron Doped PtSb2 . . . . . . . . . . . . . . . . 105

Figure 5.4:

Power Factor of Iron Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . 106

Figure 5.5:

Thermal Conductivity of Iron Doped PtSb2 . . . . . . . . . . . . . . . . 107

Figure 5.6:

zT of Iron Doped PtSb2

Figure 5.7:

X-ray Diffration of Ytterbium Doped PtSb2

Figure 5.8:

Seebeck Coefficient of Ytterbium Doped PtSb2 . . . . . . . . . . . . . . 110

Figure 5.9:

Electrical Resistivity of Ytterbium Doped PtSb2 . . . . . . . . . . . . . 112

. . . . . . . . . . . . . . . . . . . . . . . . . . 108
. . . . . . . . . . . . . . . 109

Figure 5.10: Power Factor of Ytterbium Doped PtSb2 . . . . . . . . . . . . . . . . . 113
Figure 5.11: Thermal Conductivity of Ytterbium Doped PtSb2 . . . . . . . . . . . . 114
Figure 5.12: zT of Ytterbium Doped PtSb2 . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 5.13: X-Ray Diffraction of Silver Doped PtSb2 . . . . . . . . . . . . . . . . . 117
Figure 5.14: Seebeck Coefficient of Silver Doped PtSb2 . . . . . . . . . . . . . . . . . 118
Figure 5.15: Electrical Resistivity of Silver Doped PtSb2 . . . . . . . . . . . . . . . . 119
Figure 5.16: X-Ray Diffraction of Niobium Doped PtSb2 . . . . . . . . . . . . . . . . 121
Figure 5.17: Seebeck Coefficient of Niobium Doped PtSb2 . . . . . . . . . . . . . . . 122
Figure 5.18: Electrical Resistivity of Niobium Doped PtSb2 . . . . . . . . . . . . . . 123
Figure 6.1:

Thermal Conductivity Reduction required for high zT . . . . . . . . . . 127

Figure 6.2:

X-Ray Diffraction Patterns of Palladium Substituted PtSb2 . . . . . . . 131

Figure 6.3:

Seebeck Coefficient of Palladium Substituted PtSb2 . . . . . . . . . . . 132
xiii

Figure 6.4:

Electrical Resistivity of Palladium Substituted PtSb2 . . . . . . . . . . 133

Figure 6.5:

Carrier Concentration of Palladium Substituted PtSb2 . . . . . . . . . . 134

Figure 6.6:

Thermal Conductivity of Palladium Substituted PtSb2 . . . . . . . . . 135

Figure 6.7:

Thermal Conductivity of Palladium Substituted PtSb2 at 90 K . . . . . 136

Figure 6.8:

zT of Palladium Substituted PtSb2 . . . . . . . . . . . . . . . . . . . . 137

Figure 6.9:

X-ray Diffraction of Ball Milled PtSb2 . . . . . . . . . . . . . . . . . . . 138

Figure 6.10: Particle Size of Ball Milled PtSb2 . . . . . . . . . . . . . . . . . . . . . 139
Figure 6.11: Thermal Conductivity of Ball Milled PtSb2 . . . . . . . . . . . . . . . . 140
Figure 6.12: Seebeck Coefficient of Ball Milled PtSb2

. . . . . . . . . . . . . . . . . 142

Figure 6.13: Electrical Resistivity of Ball Milled PtSb2 . . . . . . . . . . . . . . . . . 143
Figure 6.14: Carrier Concentration of Ball Milled PtSb2 . . . . . . . . . . . . . . . . 144
Figure 6.15: Seebeck Coefficient of Zirconium Oxide Ball Milled PtSb2 . . . . . . . . 145
Figure 6.16: X-Ray Diffraction of Non-Stoichiometric PtSb2 . . . . . . . . . . . . . . 147
Figure 6.17: Seebeck Coefficient of Non-Stoichiometric PtSb2 . . . . . . . . . . . . . 148
Figure 6.18: Electrical Resistivity of Non-Stoichiometric PtSb2 . . . . . . . . . . . . 149
Figure 6.19: Thermal Conductivity of Non-Stoichiometric PtSb2 . . . . . . . . . . . 151
Figure 6.20: zT of Non-Stoichiometric PtSb2 . . . . . . . . . . . . . . . . . . . . . . 152
Figure 6.21: X-Ray Diffraction of Sn Doped, Ball Milled PtSb2 . . . . . . . . . . . . 153
Figure 6.22: Seebeck Coefficient of Sn Doped, Ball Milled PtSb2 . . . . . . . . . . . 155
Figure 6.23: Seebeck Coefficient of Sn Doped, Ball Milled PtSb2 at 300 K . . . . . . 156
Figure 6.24: Electrical Resistivity of Sn Doped, Ball Milled PtSb2 . . . . . . . . . . 157
Figure 6.25: Thermal Conductivity of Sn Doped, Ball Milled PtSb2 . . . . . . . . . . 159
Figure 6.26: zT of Sn Doped, Ball Milled PtSb2 . . . . . . . . . . . . . . . . . . . . 160
xiv

Figure 6.27: Power Factor of Sn Doped, Ball Milled and Un-Ball Milled PtSb2

. . . 161

Figure 6.28: Hypothetical zT of the Combined Best Properties of PtSb2 Compared
with the measured zT of Bi0.88 Sb0.12 by Smith and Wolfe and Bi2−x Sbx Te3
by Chung et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Figure 6.29: X-Ray Diffraction of Te Doped, Ball Milled PtSb2 . . . . . . . . . . . . 164
Figure 6.30: Seebeck Coefficient of Te Doped, Ball Milled PtSb2 . . . . . . . . . . . 165
Figure 6.31: Electrical Resistivity of Te Doped, Ball Milled PtSb2 . . . . . . . . . . 166
Figure 6.32: Thermal Conductivity of Te Doped, Ball Milled PtSb2 . . . . . . . . . . 168
Figure 6.33: zT of Te Doped, Ball Milled PtSb2 . . . . . . . . . . . . . . . . . . . . 169
Figure 6.34: Comparison of the Seebeck Coefficients of Sn and Te Doped, Ball Milled
PtSb2 at 170 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xv

Chapter 1
Introduction

1.1

Thermoelectric Effects

For over 150 years three effects of materials, the so-called thermoelectric effects, have been
studied due to their potential for applications in everyday life. These effects allow for the
targeted conversion of heat into electrical energy, or vice versa, electrical energy into heat.
Seminal to these effects is the Seebeck effect discovered in 1821 by Thomas Seebeck [1].
The Seebeck effect describes the generation of a voltage between the junctions of two dissimilar materials held at differing temperatures. This effect is best visualized in its application
with thermocouples, as seen in Figure 1.1.

1

Figure 1.1: Schematic of thermocouple displaying Seebeck Effect

The Seebeck effect is governed by the movement of electrons in a material due to the
occurrence of a temperature gradient across it. The additional thermal energy on one side of
the material causes a net migration of the electrons in response to their increased momentum.
Turning our attention back to the thermocouple in Figure 1.1 we can see how this effect
generates a voltage. Depending upon the properties of material A and B their electrons will
respond to the addition of thermal energy with differing magnitude and create a voltage
potential. This effect is summarized by the equation:

∆V = −αAB ∆T

(1.1)

where ∆V , is the voltage generated, αAB , is the Seebeck coefficient of the additive contribution of the pair of materials A and B, and ∆T, the temperature difference. Each material
will have a Seebeck coefficient dependent upon its own properties which will contribute to
the overall parameter αAB .
In what can be thought of as the inverse of the Seebeck effect is the Peltier effect, which

2

was discovered in 1834 by Jean Peltier [2]. Peltier found that when a current is passed
between two dissimilar conjoined materials a cooling or heating effect is found at their
junction. In 1838 when Lenz was able to freeze water or melt ice depending upon the direction
of the current between two dissimilar materials he discovered the relationship between heat
flow, the current, and a coefficient depending upon the material. This relationship is given
by the equation:

Q˙ = (ΠA − ΠB )I

(1.2)

˙ is the heating or cooling generated, Π, is the Peltier coefficient, and I is the
where Q,
current. Direct measurement of the Peltier coefficient of a material presents difficulties due
to irreversible heating processes that happen during this phenomenon.
Finally in 1854 William Thomson, known as Lord Kelvin, discovered the Thomson effect
[3]. This reversible effect defines the heat absorbed or rejected by a material due to a parallel
temperature gradient and current. This effect is given by the equation [4]:

q˙ = τA I(dT /dx)

(1.3)

where q˙ is the heat absorbed per unit length, τA is the Thomson coefficient, I is the current, and dT/dx is the temperature gradient. Thomson additionally, using thermodynamic
arguments, was able to relate the Seebeck coefficient and the Peltier effect using the Onsager
relationships [5]. One of the Kelvin relations is given by the equation:

τA − τB = T (dαAB /dT )

3

(1.4)

which gives the relation between the Thomson coefficient and the differential of the
Seebeck coefficient at a temperature. The other Kelvin relation is much more applicable
to experimental work and gives the relation between the Seebeck coefficient and the Peltier
coefficient:

ΠAB = αAB T

(1.5)

This relation allows for the conversion between the much easier to measure quantity α
and the Peltier coefficient.
The relations and equations above have all been dependent upon the junction of two
dissimilar materials, A and B, but it is advantageous to express these relations as dependent
upon only one material. To this effect superconducting materials, which have a Seebeck
coefficient of zero, have been utilized as a reference to determine the Seebeck coefficient
for a single material. In this manner standardized values have been determined for many
materials, against which an unknown Seebeck coefficient can be measured.

1.2

Application of the Thermoelectric Effects

The thermoelectric effects can be utilized for numerous applications. The most widely used,
thermocouples, allows for the accurate measurement of temperatures across a wide range
depending upon the materials utilized. In 1909 and 1911 Altenkirch derived the maximimum efficiency of thermoelectric power generators and coolers [6, 7] and in 1954 Julian H.
Goldsmid opened up the possibility of using semiconductors as coolers, paving the way for
further research in the field [5]. While the research and development of thermoelectric power
generators and coolers has had periods where little progress took place, this area is now the
4

most widely researched in the thermoelectric effects.

1.2.1

Power Generation

The vast majority of thermoelectric device research is directed towards the recovery of waste
heat. This is due to the need of increased energy sustainability and environmental consciousness in our future world. As a growing number of global entities industrialize and expand,
the energy usage of prior global power users has remained nearly constant which is supporting this drive for increasing power generated through renewable means [8, 9]. This pressure
for increased energy usage from industrialization is further fueled in relation to the expectation of an increased global population [10, 11]. With this increased energy usage much,
if not a majority, of the energy produced will be wasted as rejected heat. Thermoelectric
power generation modules make it possible for a fraction of this rejected heat to be harvested
and recirculated resulting in a significant potential reduction in the amount of total energy
production required.
Recovery of waste heat from the exhaust of automobiles is a potential application which
has garnered a significant amount of interest. This application would increase the gas mileage
of vehicles by removing the need for an alternator, which generates electricity for the vehicle
by sapping energy produced by the engine [12, 13]. However, this application has several
disadvantages and an analysis of them highlights the many applications where thermoelectric
power generators are better applied. Thermoelectric power generators are solid state devices
without any moving parts, are scalable in size, and are extremely reliable in a steady state
environment free of vibration. The majority of vehicles are however constantly in a state
of flux, either heating up or cooling down due to being used multiple times a day and are
also prone to the occasional sudden stop and ever constant vibrations. Putting aside the
5

diminished efficiency of thermoelectric generators outside of their peak temperature, the
mechanical turbulence of a typical car ride and constant thermal cycling demonstrate how
ill-suited thermoelectric generators are for vehicles.
Examining the drawbacks of generators in automobiles highlights the applications for
which thermoelectric power generators are best suited: an environment where a constant
temperature is maintained and is free of vibrations. Two applications where this environment
may be found are in industrial power plants and in satellites using a radioisotope generator
[14]. Satellites using radioisotope generators are an excellent example of the long term
reliability of thermoelectric devices with the Voyager I and II spacecraft having operated
for almost 50 years [15]. Several other applications of thermoelectric power generators, (i.e.
wrist watches, cardiac pacemakers, recreational gear, etc.), have the potential for use in the
future if the efficiency of these devices can be improved.

1.2.2

Peltier Cooling

The application of thermoelectric devices upon which this work is focused is in Peltier cooling. Peltier cooling utilizes the Peltier effect to generate a heat absorption on one side of
a thermoelectric device that can offer targeted and precise refrigeration with all of the advantages which were highlighted for power generators. Other advantages of Peltier cooling
devices are their low weight and small size when compared to mechanical cryocoolers. Mechanical cryocoolers are generally large, heavy devices which generate significant vibrations.
Thus, for many applications, the use of these mechanical cryocoolers creates unsuitable operating conditions, (i.e. vibrations on an electronic component), or substantially increases
the weight or space requirements of the application.
An exciting application of Peltier cooling is in the refrigeration of infrared focal plane
6

array detectors. Much of the impetus for this research has been for military usage in missile
defense and forward looking infrared detectors [16]. For such applications reliability is of
the utmost importance which lends some credence to using a Peltier cooler due to their
inherent reliability. These detectors use materials such as HgCdTe, InSb, InGaAs, and VOx ,
(Vandaium Oxide), with bandgap energies in the range of electromagnetic radiation such
that an incident infrared photon will cause an electron hole pair to be generated in the
material which creates the signal in these devices [17–19]. These detectors, however, need
to be cooled to the tens of Kelvins such that the ratio of noise to signal is minimized. This
noise is termed, ”dark current”, because an aberrant signal is generated not from an incident
photon, but due to thermal excitation, which is why thermal energy must be kept within an
acceptable range in these detectors [20].
Several other possible applications of Peltier coolers lie in spot cooling of X-ray astronomy
detectors, laser equipment, medical equipment and transport, consumable food, residential
and industrial electronics, and in heat generating mechanical processing machines [21]. However, as with the applications of thermoelectrics for power generation, Peltier cooling is also
dependent upon the improvement of the efficiency or coefficient of performance, a metric of
the cooling power of a device in ratio with the power it requires, of these devices [22].

1.3

Thermoelectric Device Performance

Both Peltier coolers and power generators have a similar device schematic, shown in Figure 1.2.

7

Figure 1.2: Schematic of Peltier Cooler and Thermoelectric Power Generator

As can be seen from the figure both devices consist of p- and n-type legs that are connected
electrically via a metal connector with an insulating layer between heat source and sink. This
allows for the thermoelectric legs to be electrically in series while thermally in parallel [23].
The distinction between p- and n-type materials is due to their dominant charge carrier
which gives rise to a positive Seebeck coefficient in the p-type material and a negative
Seebeck coefficient in the n-type material; the specific arrangement of n- and p-legs in series
allows for a current to flow. A further description of p- and n-type materials is given in the
following section.
In examining the Peltier cooler a standard thermodynamic metric for evaluating the
performance of a cooler is the coefficient of performance, (COP), φ, which is a metric defined
by the ratio of cooling performed to the electrical power input [24]. The cooling power, Qc ,
is a result of the Peltier effect transporting thermal energy through the material, a backward
flow of thermal energy through the material limited by the thermal conductivity of the
8

material, and approximately half of the joule heating due to a current passing through the
leg. The equation that defines this is:

1
Qc = (αp − αn )ITc − K∆T − I 2 R
2

(1.6)

Where I is the current input, Tc is the cold side temperature, K is the thermal conductance
of the legs, ∆T is the temperature differential across the legs, and R is the electrical resistance
of the legs.
The power supplied to perform the cooling, W, is simply the electrical power: P=IV.
The voltage supplied is composed of two parts; the part to overcome the Seebeck voltage
generated by a temperature differential across the material and the part to overcome the
electrical resistance of the leg:

W = IV = I[(αp − αn )∆T − IR]

(1.7)

The COP of a device is then:

φ=

(αp − αn )ITc − K∆T − 21 I 2 R
Qc
=
W
I[(αp − αn )∆T − IR]

(1.8)

Several device optimization parameters can be solved for using equations 1.6 - 1.8, such
as the current to be used for maximum cooling power or the maximum ∆T that can be
generated given a set of material properties [25, 26].

Qmax =

α2 T 2
2R

(1.9)

∆Tmax =

9

ZTc2
2

(1.10)

Using an optimal current input the φmax can be obtained:

Tc
φmax =
∆T

T
1 + ZTAvg − Th
c

1 + ZTAvg + 1

(1.11)

where the figure-of-merit, Z, for the thermoelectric module is used, which is distinct from
the often cited thermoelectric material figure-of-merit z. It is found that as Z increases
the COP also increases, Figure 1.3, thus Z, which is only dependent upon properties of the
materials used, should be attempted to be maximized [4].

Figure 1.3: COP vs. Delta T with Hot Side Temperature of 300 K

In the development and engineering of new materials to be used in Peltier coolers, and
thermoelectric devices in general, it would be quite cumbersome to use the Z for the thermoelectric module to compare and quantify the potential of individual materials. It is therefore

10

advantageous to define a material specific figure-of-merit z [27]:

z=

α2 σ
κ

(1.12)

Where σ is the electrical conductivity, (ρ = 1/σ, the electrical resistivity), and κ is the
thermal conductivity. The thermoelectric module Z is found to be a rather complicated
mean of the two z’s of the legs and further complicating study, contact resistances can be
included resulting in a reduced effective Z. Additionally, the geometry of the legs should be
optimized by the equation [24]:

lp Sn
=
ln Sp

σp κp 1/2
σn κn

(1.13)

where l is the length of the leg and S is the cross-sectional area. When the thermoelectric
properties of both the p- and n-type materials are similar, Z is effectively the mean of z of
the legs and materials can be optimized separately, but when the properties begin to differ
these geometric arguments begin to play a more important role. This effect can become so
large that at extremely low temperatures a superconducting material may be better suited
for one of the legs than the typical semiconductor [23].
In the above description the so called thermoelectric properties α, σ, and κ have been
assumed to be constant with temperature, however this is not the case in real materials and
devices. Thus, it is helpful to examine a material at a mean operating temperature, which
then gives a unitless figure-of-merit, zT:

zT =

α2 σ
T
κ M

11

(1.14)

As a result of the dependence on average operating temperature of maximized Z, and
therefore maximum ∆T of cooling, a Peltier cooler would only be applicable for a small
range of temperatures. To tackle this issue thermoelectric modules may be cascaded such
that multiple layers of modules can be tailored for maximum COP in a specific ∆T range
and an overall increased ∆T can be achieved [28]. The main drawback to this system is
that with each subsequent stage there is an increased cooling load required with not only
the heat from the source, but also the heat generated from the preceding stages. In addition
to generating a greater temperature drop across the module a multi-staged cascade may be
used to increase the COP of a module [29]. If the COP of each stage is assumed to be the
same the overall COP of the module may be given as [24]:

φN =

1
(1 + 1/Φ)N − 1

where Φ = N (φ + 1/2) −

1
2

(1.15)

It is found in practical use that for the majority of applications a two stage module is
sufficient instead of increased spending on larger cascade modules as shown in Figure 1.4 [30].
It is additionally found that for temperature decreases that are well below the maximum
∆T that a single staged module is sufficient, but when the necessary temperature drop
approaches the maximum then a two stage cascade may increase the overall COP greatly.

12

Figure 1.4: Multi-Stage Device COP vs. Delta T with Hot Side Temperature of 300 K and
Z of 10−3 K

1.4

Electronic Transport

While the three thermoelectric effects, Seebeck, Peltier, and Thomson effect allow for the use
of thermoelectric coolers and power generators, they are not the only properties of interest
in the search for increased zT. These other parameters include the electrical conductivity, σ
and the thermal conductivity, K. This section will attempt to establish a base of information
on the properties of a material that determine the zT and ultimately how those properties
may be manipulated such that more effective and efficient devices may be created.

1.4.1

Energy-Band Diagrams and Material Types

The Drude model presented by Paul Drude in 1900 predicts the motion of electrons based
upon a kinetic gas theory [31]. It is a classical model where the electrons are imagined as
13

a sea of charges interacting with only the ionic cores of atoms and acted on by an electric
field. The Drude model had many successes, especially with metals, in predicting things like
the electrical resistivity of a material and the Wiedemann-Franz law. The model, however,
fell short in some ways with a predicted electronic heat capacity much larger than that
experimentally observed in metals.
In 1928 Sommerfeld introduced quantum theory to electronic transport using Fermi-Dirac
statistics instead of the classical Maxwell-Boltzman statistics used by Drude [32]. Additionally in 1928 Felix Bloch mathematically described the periodic potentials that interact with
electrons in a crystalline solid with Bloch functions [33]:

Ψk (r) = eik·r uk (r)

(1.16)

where Ψk (r) is the wavefunction of an electron, k is the crystal wave vector of the
electron, r is distance, and uk (r) is a function with the same periodicity as the crystal.
These wavefunctions are the eigenfunctions which satisfy the Schrodinger equation for the
available states of the electron:

2

−
where

2m

∇2 + U (r) Ψ = EΨ

(1.17)

is the reduced Planck’s constant, m is the mass of the electron, and U (r) is the

periodic potential due to the crystal lattice. Due to the periodic nature of the crystal the
wave vector k can always be represented within the first Brillouin zone and another possibly
further elucidating version of Bloch’s theorem is [34]:

Ψ(r + R) = eik·r Ψ(r)
14

(1.18)

which demonstrates that for any translationally moved wavefunction Ψ(r + R) there
exists a wave vector k which will relate this new wavefunction to a wavefunction within the
first Brillouin zone. This wave vector k may be plotted versus energy to give rise to the
extremely useful representation of the available energy states of a material as the energy
band diagram.
A schematic representation of an energy band diagram is shown in Figure 1.5.

Figure 1.5: Schematic Energy Band Diagram for Metals, Semiconductors, and Insulators

This figure shows the underlying differences in the energy band diagram of metals, semiconductors, and insulators and demonstrates why the electronic properties of these categories
of materials vary so dramatically. The energy band diagram of a metal is shown to have
fully occupied electronic states up to the Fermi energy, EF , which is well within the middle
of a band. As was shown by Sommerfeld the charge carriers that are within a few kB T of
the Fermi energy are those that are able to electronically conduct. In the metal there are
15

a plethora of carriers and available empty energy states around the Fermi energy and thus
its electronic properties reflect this; prominently a very high electrical conductivity. It is
in semiconductors and insulators where a band gap, or a region of non-overlapping energy
states where carriers are unable to populate, exists. In the insulator this band gap is so large
that at terrestrial temperatures carriers will not have enough energy to populate free, empty
states, in the conduction band. This is reflected in the electronic properties of insulators
and most prominently in an extremely low electrical conductivity. In the semiconductor the
band gap is found to be within range of thermal energy supplying greater than the band gap
energy to carriers, such that an electron may be able to populate a free energy state in the
conduction band. This gives rise to a wide range of electronic properties that semiconductors
may adopt.
Stemming from the concept of the energy band diagram is that of the effective mass of
a charge carrier and that of the conduction of charge by holes. The effective mass is the
curvature of a band extrema, either the minima of the conduction band, or the maxima of
the valence band. In one dimension, this may be expressed as [23]:

∂ 2E
m∗ = 2
∂k2

−1

(1.19)

While in reality the inertial effective mass is a tensor, which gives the effective mass of
carriers in the different directions of k-space, it is instructive to examine it as though a scalar
for discussion. When the curvature of these band extrema are negative, as in the conduction
band, the effective mass will be negative and the charge carriers are electrons in this case.
When the curvature is positive, as in the valence band, the charge carriers are holes. Holes
are quasi-particles, because their significance is due to the lack of an electron. Conduction

16

by holes is still performed by the migration of electrons, but it is the absence of an electron,
which entails a positive charge, that is moving through a real material.

1.4.2

Electrical Conductivity

The ability of a material to conduct these charge carriers is defined as the electrical conductivity, σ, one of the properties of a material that determines the figure-of-merit, zT:

σ = neµe + phµh

(1.20)

where n is the electron concentration, e is the electronic charge, µe is the electron mobility,
p is the hole concentration, h is the hole charge, and µh is the hole mobility. For most
temperature ranges and purposes of thermoelectric materials it is advantageous to only have
one type of charge carrier conducting in a material. When both carriers are present this
mode of conduction is called bipolar conduction.
To begin the description of the electrical conductivity it is important to understand that,
for the vast majority of equations concerning carrier concentration, only slight modification
is required to switch between equations for electrons and holes. Therefore, in this description
only the equations for electron concentration will be expressed.
Utilizing Fermi-Dirac statistics the electron concentration in a material is given by:

∞

n=

fn (E)g(E)dE

(1.21)

0

where fn (E), the Fermi-Dirac distribution, and g(E), the density of states for an electron

17

in the energy band, are:
1
fn (E) = E−E
F
e kB T + 1
√
8π 2 ∗3/2
g(E) =
m
E − Ec
h3

(1.22)

(1.23)

where kB is the Boltzmann constant, m∗ is the effective mass of the charge carrier, h is
Planck’s constant, Ec is the conduction band minima, and EF is the Fermi energy. The Fermi
energy is defined as the energy, at absolute zero, where all of the states below this energy
are occupied and none above it are. The Fermi-Dirac distribution gives the probability of
an electron existing in an energy state and is shown in Figure 1.6 for various temperatures.

Figure 1.6: Fermi-Dirac Distribution for Multiple Temperatures

Observing the Fermi-Dirac distribution at 0 K it is found that there is a jump discontinuity at energy equal to the Fermi energy. For all energies below the Fermi energy the
probability of an electron existing in this state is 100 % and above this energy equal to 0 %.
18

At non-zero temperatures the distribution is found to be continuous and varying between the
values of 0 and 1 for energies approaching the Fermi level. This creates a non-zero potential
of a charge carrier to exist in an energy state above the Fermi energy when at non-zero
temperature. Of note is the increase in variance at higher temperatures, which results in a
further increased potential of finding a charge carrier in higher energy states.
The equation for carrier concentration can thus be given as:
√
√
E − Ec
8π 2 ∗3/2 ∞
m
n=
dE
E−EF
h3
0
e kB T + 1

(1.24)

which may be simplified to:

2
E
n = Nc √ F1/2 (η) where η = F
kB T
π

(1.25)

and

Nc = 2

2πm∗ kB T 3/2
h2

(1.26)

where F1/2 (η) is the Fermi-Dirac integral and η is the reduced Fermi energy. For an
intrinsic semiconductor, one in which there has been no doping, the number of electrons
and holes are equal due to the fact that as electrons are populated across the bandgap they
generate an equal number of holes from their absence. This result is stated in the law of
mass action:

n ∗ p = n2i

(1.27)

where ni is the intrinsic carrier concentration. This concentration may be obtained by
19

combining equations 1.27 and a simplified 1.24 to obtain:

ni (T ) =

−EG
2k
Nc Nv e B T

(1.28)

This shows that the intrinsic carrier concentration is dependent upon the effective masses
of the charge carriers, due to the dependence of Nc and Nv upon the effective mass, as well as
upon the energy bandgap. For a given temperature the intrinsic carrier concentration will be
lower in a large bandgap material than for a low bandgap material. The ratio of the effective
masses is often more important than their absolute value. This will affect the electrical
conductivity, as seen in Equation 1.20, as well as the other thermoelectric properties in the
intrinsic regime. The Fermi energy of a material is also dependent upon the ratio of these
effective masses such that:

3
1
Ef,int. = Ev + EG + kB T ln
2
4

mv
mc

(1.29)

where the Fermi energy is dependent upon the energy at the top of the valence band, Ev ,
half of the energy of the bandgap, and then adjusted by the natural log of the ratio of the
effective masses. In the majority of materials this addition will only affect the Fermi level by
a small amount and for most purposes an assumption of the Fermi level lying in the middle
of the band gap will be a sufficient approximation.
As mentioned earlier, the electronic properties of semiconductors can vary over a wide
range. This is in part due to the variation in bandgaps in these materials, but also to the
effect of extrinsic doping. Doping is the process of substituting elements in a compound
with elements containing a differing number of valence electrons. In the most simplistic view
the dopant atom will be adding or subtracting electrons from the material system thereby
20

directly affecting the carrier concentration. There are two categories of dopants, n-type,
where the dopant has additional valence electrons relative to that of the atom for which it is
substituting, and p-type, where the dopant has fewer valence electrons. Often, these dopants
will not have a very high solubility in the parent material, but this is not needed to affect
a dramatic change in the electronic properties. An often used example of extrinsic doping
is with silicon where phosphorous may act as an n-type dopant and boron may act as a
p-type. When these atoms are substituted the result is a change in the energy band diagram
where the Fermi energy may be shifted closer to the valence or conduction band depending
upon the addition of p-type or n-type dopant. The Fermi energy is shifted depending upon
if electrons or holes were added to the system, because this will affect the probability of an
energy state being filled. The equation for this shift is given by:

Ef = Ef,int. + 1/2kB T ln

n
p

(1.30)

where the n and p in this equation may not be the same as that for an intrinsic semiconductor.
The dopants, referred to as donors for n-type materials, and acceptors for p-type materials, require an activation energy before they contribute carriers to the system. These
activation energies are quite low compared to the bandgap and thereby ionize at temperatures much lower than intrinsic carriers. The equation for the electron carrier concentration
in this region of ongoing ionization is:

n≈

ND Nc −E /2k T
e D B
2

when ND

1
Nc e−ED /kB T
2

NA

(1.31)

where ND is donor concentration and ED is the activation energy of a donor. Once
21

the temperature has increased beyond the region where ionization is occurring the carrier
concentration is given as:

n=

1
(ND − NA ) +
2

(ND − NA )2 + 4n2i

(1.32)

where NA is the acceptor concentration. This equation can be simplified to n ≈ ND
given that ND

NA and |ND − NA |

ni .

The extrinsic semiconductor can be doped so heavily that the Fermi level is within 1 - 2
kB T of a band edge. When this occurs the semiconductor is referred to as degenerate and
the carrier concentration is very large, similar to that found in a metal, over a wide range of
temperatures. However, for the extrinsic semiconductor that is non-degenerate the carrier
concentration is small at low temperatures but increases as donors or acceptors are ionized
with increasing temperature. As the temperature increases, the activation energy for all
donors or acceptors is met and a region of saturation occurs where no new carriers are being
populated to charge carrying energy states. At temperatures above the saturation region
intrinsic charge carriers will begin populating across the bandgap and carrier concentration
will again increase.
As can be seen in equation 1.20, the carrier concentration is only part of the electrical
conductivity. The other variable is the mobility of that charge carrier, which is defined from
the drift velocity of carriers in response to an electric field, vd = µE. The mobility may be
given as:

µ=

e¯
τ
m∗

(1.33)

where τ¯ is the average scattering time. This average scattering time is dependent upon
22

the dominant scattering mechanism in a material for a given temperature, but may be given
by Matthiessen’s rule:

1
1
1
1
=
+
+
+ ...
τ
τlattice τimpurities τdef ects

(1.34)

where τlattice is the scattering of charges due to the lattice ions, τimpurities is the scattering due to impurities, and τdef ects is the scattering due to defects in the crystal lattice.
Lattice scattering is due to the interaction between charge carriers and phonons traveling
through the crystal lattice. As temperature increases more phonons are generated in the
lattice, which corresponds to more scattering of carriers, thus, as temperature increases the
dominant scattering mechanisms in materials becomes that from the lattice. It is found that
in this case mobility depends on temperature as: µlattice ∝ T −3/2 .
Impurity scattering however is the low temperature dominant scattering mechanism compared to lattice scattering. Impurity scattering is found to depend upon the average thermal
velocity of charge carriers, meaning that the more energy a carrier has the less likely it is
to be scattered. Thus, at low temperatures, where the thermal velocity is low, more impurity scattering will occur than at high temperatures where thermal velocity is high. The
proportionality for this type of scattering is: µimpurity ∝ T 3/2 .
The scattering due to crystal defects is given as an example of a scattering mechanism
where the scattering rate is constant due to either an ordered or disordered defect causing the
scattering of a carrier. Another scattering mechanism found in degenerate semiconductors,
or for non-degenerate semiconductors that are at temperatures in the intrinsic regime, where
a high carrier concentration is obtained, is electron-electron scattering where the carriers in
a material will begin to scatter off of each other. This only begins to play a role at very

23

high carrier concentrations, but can become the dominant scattering mechanism at high
temperatures, or in heavily doped semiconductors.
There are many more types of scattering mechanisms in materials, but for this work
these are found to often be the dominant scattering mechanisms of importance at observed
temperatures. The result of the switch between ionized impurity scattering, which decreases
as temperature increases, and acoustic phonon scattering, which increases as temperature
increases, is that there typically will be a maximum in the mobility over a temperature range.
How the electrical conductivity of a material ultimately responds to changes in carrier
concentration and mobility will differ in all materials and will be greatly affected by the
material synthesis and processing. However, a few generalizations may be made. Typically,
the electrical conductivity of a non-degenerate, extrinsically doped semiconductor will increase as dopant carriers are populated, but will at higher temperatures begin to decrease
as acoustic phonon scattering begins to dominate. For the degenerate semiconductor the
electrical conductivity follows a trend that is much like that of a metal. Starting at low
temperature with a high conductivity, due to the high carrier concentration, there will be a
downward trend as acoustic phonons become a more dominant scattering mechanism.

1.4.3

The Seebeck Coefficient

The thermoelectric effects, with which this work began, are the foundation for the operation
of thermoelectric materials. The most direct of which to measure is the Seebeck effect and
namesake coefficient. As defined previously, the Seebeck coefficient is a reflection of the
response of electrons to a temperature gradient. The Seebeck coefficient is defined such that
an n-type material in open circuit conditions under a temperature gradient will generate
a voltage potential such that the electric field is pointed in the direction of the cold side,
24

(the side with a negative charge buildup). This creates a negative Seebeck coefficient for an
n-type material and the inverse for a p-type material.

Figure 1.7: Schematic of Open Circuit Seebeck Effect

As was introduced in the previous section the Fermi-Dirac function will also have a
gradient along the material due to the temperature differential. Using Boltzmann transport
equations, an equation can be obtained where the Seebeck coefficient is dependent upon the
integral of the partial derivative of the Fermi-Dirac function, density of states, energy, and
scattering terms. Ioffe was able to simplify the equation to [35]:


5
2

+ λ F3/2+λ (η)



k
− η
α=− B
3 +λ F
e
1/2+λ (η)
2

(1.35)

where λ is a term dependent upon the scattering mechanism, η is again the reduced
E

Fermi energy, k FT , and F3/2+λ and F1/2+λ are Fermi-Dirac integrals. This is assuming a
B
single parabolic band model with a single energy dependent carrier relaxation time. The
scattering dependent λ is equal to −1/2 for acoustic phonon scattering dominant energies
and equal to 3/2 for ionized impurity scattering dominant energies.

25

Examining the Seebeck coefficient for metals the equation for the Seebeck coefficient can
be greatly simplified. The Mott-Jones equation gives the Seebeck coefficient for this class of
materials as [36]:

α=−

π 2 kB
3e

kB T
EF

3
+λ
2

(1.36)

Examining this equation, the dependence of the Seebeck coefficient on kB T /EF is immediately obvious. This is due to the fact that only those electrons that are within a few
kB T of the Fermi energy will contribute to the effect.
To better examine the dependence of the Seebeck coefficient to the carrier concentration
in semiconductors the following equation can be shown [37]:

α=

1 dn(E) 1 dµ(E)
π 2 kB
kB T
+
3 e
n dE
µ dE E=E
F

(1.37)

This equation shows that an increased Seebeck coefficient can be obtained by either having a large energy dependence of the carrier concentration or mobility at the Fermi energy. To
obtain the former there needs to be a large change in the density of states at the Fermi energy
and to obtain the latter a strong energy dependent scattering mechanism is required [38].
This additionally shows the dependence of the Seebeck coefficient on carrier concentration
and carrier mobility. As carrier concentration and mobility increase the Seebeck coefficient
is found to decrease.
Just as with the electrical conductivity, when bipolar effects begin to play a role in
the intrinsic regime of a material, the Seebeck coefficient will be altered due to the mixed

26

conduction effects. The equation for this effect is given as [23]:

α=

αn σn + αp σp
σn + σp

(1.38)

Here the Seebeck coefficient contribution of electrons and holes is given and weighted
by the electrical conductivity of those charge carriers. The first important point is that
for thermoelectric applications it is best to have a material of only one carrier type. Any
contribution of an additional carrier will only serve to decrease the overall Seebeck coefficient,
because αn and αp have opposite signs. The next point to take note of is the dependence on
the mobility of the individual carriers. If a material does have bipolar conduction occurring,
then the contribution of the carriers may not be proportional. For example, if the electrons
in a material have a much higher mobility than the holes, then the material may be n-type
for similar electron and hole carrier concentrations.
If a degenerate, i.e. heavily doped, semiconductor with parabolic bands is assumed, with
only one carrier type, then the following may be obtained [39]:

α=

2
8π 2 kB
∗ T π 2/3
m
3n
3eh2

(1.39)

which shows even more directly the dependence of the Seebeck coefficient on the carrier
concentration and the effective mass. Thus, typically for high Seebeck coefficient in extrinsically doped, degenerate semiconductors a low carrier concentration and high effective mass
are desired. This trend in transport properties is directly contrary to what is required for
high electrical conductivity and is one of the so-called ”contra-indicated” properties that
thermoelectric materials possess.

27

1.4.4

The Power Factor

The thermoelectric power factor of a material is the numerator in the figure-of-merit equation:

P.F. = α2 σ

(1.40)

where P.F. is the power factor. It is found that the power factor depends on the square
of the Seebeck coefficient and the electrical conductivity. As mentioned earlier these two
properties have contra-indicated components. As the carrier concentration increases the
electrical conductivity is found to increase, but the Seebeck coefficient concurrently decreases.
Additionally, if the mobility of a carrier is decreased the Seebeck coefficient tends to increase,
however, the electrical conductivity will decrease. The contra-indication of these properties
results in a maximization of the power factor at certain carrier concentrations assuming
mobility is unchanged. This relationship is called the Pisarenko relationship and is shown
in Figure 1.8.

28

Figure 1.8: Diagram of Power Factor, Seebeck Coefficient, and Electrical Conductivity
versus Carrier Concentration

This figure graphically demonstrates how the power factor will be maximized at a particular carrier concentration for a given material. For most material types a carrier concentration
in the range of 1019 to 1021 yields a maximized power factor. Current and advanced techniques to increase power factor concentrate on band engineering to increase band degeneracy,
carrier concentration doping with elements or materials which generate a minimum increase
in carrier scattering, and preferential carrier scattering mechanisms leading to optimization [40]. However, as will be seen in the following section, a maximized power factor will
not always relate to a maximized zT.

29

1.5

Thermal Transport

The flow of heat through a material is governed by Fourier’s law of heat conduction which
shows that the power conducted per unit area is proportional, by thermal conductivity, κ,
to the ratio of the temperature difference over the distance of separation:

∆T
∆Q
= −κ
∆tA
∆X

(1.41)

Where ∆Q/∆t is the power conducted, A is the area through which the heat flux is traveling, ∆X is the distance of separation, and κ is the thermal conductivity. There are many
components to thermal conductivity including more exotic forms of heat conduction like
electronic spin waves, however in the study of thermoelectrics the majority are represented
by the electronic and lattice contributions.

1.5.1

Electronic Thermal Conductivity

The electronic thermal conductivity, as suggested by its name, is an effect mediated by the
flow of electrons in a material. This effect is dependent upon the electrical conductivity of a
material and is defined by the Wiedemann-Franz law:

κe = L0 σT

(1.42)

where L0 is the Lorenz number. The Lorenz number is well defined for metals based
2

upon a free-electron model with little variation from the definition: L0 = π3

kB 2
e

=

2.44 ∗ 10−8 W ΩK −2 . However, in many semiconducting materials this value can vary due
to additional scattering mechanisms.

30

1.5.2

Lattice Thermal Conductivity

The other majority component to the thermal conductivity in thermoelectric materials is the
lattice contribution, mediated by the quasi-particle phonon. The phonon is a quantization
of the energy transmitted by thermal vibrations through a crystal lattice and was given its
name by Peierls in 1929 [41]. Earlier work by Debye on a continuum lattice examined heat
conduction by lattice vibrations [42]. This work was foundational and is still very useful to
examine, however, it erroneously resulted in an infinite thermal conductivity. Peierls solved
this issue with the correction that higher order terms in the harmonic solutions would yield
a phonon mean free path before being scattered. The lattice contribution is defined by:

1
κl = cv vlt
3

(1.43)

where κl is the lattice contribution to thermal conductivity, cv is the specific heat per unit
volume, v is the speed of sound in the material, and lt is the mean free path of the phonon.
The specific heat and mean free path shown above are dependent upon both temperature
and the wavelength of phonons.
One of the successes of the Debye model was to describe the specific heat with a high
degree of accuracy. Debye assumed that there were 3N vibrational states for N atoms in
a material and that a maximum vibration frequency would be met, fm , which yielded a
temperature, the Debye temperature, which corresponded to the temperature of the highest
normal mode in a lattice [43]:

hfm
ΘD =
kB

where vm = v

31

3N
4πV

1/3

(1.44)

where ΘD is the Debye temperature and N/V is the number density of atoms. Debye
was able to define the specific heat at constant volume as:

cv = 9N kB

T 3 ΘD /T x4 ex
dx
ΘD
(ex − 1)2
0

(1.45)

where x = ω/kB T . This solution leads to two temperature regions of interest: that
below the Debye temperature and that above the Debye temperature. Below the Debye
temperature the specific heat reduces to a cv ∝ T 3 dependency and above the Debye temperature a saturation effect of all available modes occurs termed the Dulong-Petit limit where
cv ≈ 3R
M , with R being the gas constant and M the molar mass.
The mean free path of a phonon is determined by scattering mechanisms in the solid,
lt = vτ . Mattheisen’s rule of scattering can again be used, only this time for phonons, to
understand how one scattering mechanism will dominate over others:

1
τT ot.

=

1
1
1
+
+
+ ...
τB τI
τU

(1.46)

where τT ot is the combined scattering rates, τB is the boundary scattering rate, τI is
scattering due to impurities, and τU is the scattering due to the Umklapp process.
At low temperatures boundary scattering will dominate due to the long wavelength
phonons at those temperatures. In the perfect crystal the mean free path of phonons will be
approximately the smallest dimension of the crystal, L, [44] and the scattering rate can be
approximated by [45]:

−1 ≈
τB

32

vs
L

(1.47)

This type of scattering can be manipulated by the introduction of additional boundaries
or interfaces. Reduction of particle size by a variety of methods, such as ball milling, meltspinning, and spark plasma sintering can yield greatly reduced lattice contributions due to
additional interfacial scattering of phonons [46].
Scattering from impurities is primarily due to disturbances in the periodic arrangement
of atoms in the lattice and how this affects the propagation of phonons. These disturbances
can occur through mass difference in solid solution formations, lattice vacancies, crystal
dislocations, and even the inclusion of differing isotopes of elements [47–50]. The scattering
rate arising from these mechanisms can be defined by [51]:

τI−1 =

VΓ 4
ω
4πv 3

(1.48)

where V is the volume of the unit cell, ω is the wavelength, and Γ is the mass-fluctuation
scattering parameter. The mass-fluctuation parameter contains a ratio of the change in mass
to the mass of the unsubstituted atom squared, which highlights that large differences in
mass will scatter phonons at a higher rate. This scattering mechanism additionally has a
large dependency upon the frequency, τI−1 ∝ ω 4 , which incidentally is very similar to the
relation for Rayleigh scattering of photons.
Phonon scattering by phonon-phonon interactions takes place by two different processes
each involving the interaction of three phonons. The first is termed a normal ”N-” process which is a non-thermally resistive scattering mechanism whereby the phonon-phonon
interaction momentum is conserved [52]. As this process is non-thermally resistive it does
not contribute to the total thermal conductivity. The second process termed Umklapp, or
U-process, however, does contribute to an increased scattering rate and thereby a reduction

33

in the lattice contribution.
Umklapp scattering is a process whereby the resultant interaction of two phonons results
in a third phonon outside of the first Brillouin zone. A reciprocal lattice vector returns
the phonon to inside the 1st Brillouin zone resulting in a net backwards phonon flux [53].
The temperature dependence of the scattering rate for this mechanism was given by Peierls
as [41]:

τU−1 ∝ T n eΘD /mT

(1.49)

Slack further quantified this relationship with the following equation [54]:

τU−1 ≈

γ2
ω 2 T e−ΘD /3T
M v 2 ΘD

(1.50)

where γ is the Gr¨
uneisen parameter, and M is the average atomic mass. The parameter
of note here is the dependency of the scattering rate on the first order of temperature.
The total scattering rate is then given as:

τT−1
ot. =

vs
+ Aω 4 + Bω 2 T e−ΘD /3T
L

(1.51)

From this it can easily be seen that the boundary scattering effects have a more dominant effect at low temperatures, with impurity scattering becoming larger as temperature increases, and finally Umklapp scattering beginning to have a more dominant role
at T > 0.1ΘD .
With both the specific heat and the scattering rate dependencies defined a few generalizations of the lattice thermal conductivity may be made. At low temperatures, where the

34

specific heat goes as T 3 , and boundary scattering is dominant, the thermal conductivity will
also go as a T 3 dependency which has simply been depressed due to the constant boundary
scattering. As temperature increases impurity scattering also begins to increase which leads
to further suppression of the thermal conductivity. As temperature approaches the Debye
temperature, the thermal conductivity will peak at a maximum value. Once the Debye
temperature has been met the specific heat will stay constant as the Umklapp scattering
process begins to dominate. In this region the thermal conductivity will decrease with a
T −1 dependency, matching the behavior observed by Eucken in 1911 [55].

1.6

Material Selection and Enhancement of zT

In the selection of materials for thermoelectric applications a material must be selected that
can balance all of the contra-indicated properties which comprise the figure-of-merit. To
summarize the requirements of high zT a material needs a high Seebeck coefficient, a high
electrical conductivity, and a low total thermal conductivity. An insulator will typically display very low thermal conductivity and a high Seebeck coefficient because of the large band
gap. However, insulators will, again because of the large band gap, have a very low electrical conductivity. The case for metals is slightly more muddled, however, metals typically
have low Seebeck coefficients, and high electrical and thermal conductivities. Thus, it is the
semiconductor which has been identified and most widely researched for its potential in thermoelectric application [56]. Semiconductors may exhibit a wide array of electrical properties
over a large range of temperatures. They additionally may be modified to greatly reduce
their thermal conductivities. Above all else it is the large degree of tuning and tailoring that
may be performed on semiconductors that makes them attractive for use as thermoelectric

35

materials.
As was stated before, the contra-indicated properties of the Seebeck coefficient and electrical conductivity require a tuning of the carrier concentration for power factor optimization,
however for high zT the additional parameters of κe and κl must also be addressed. The
relationship between κe and σ is direct, and any decrease in κe comes at the expense of a decrease in σ, and vice-versa. Additionally, many methods of lowering κl will negatively affect
the power factor by the introduction or exacerbation of scattering effects. It is beneficial to
rearrange the equation for zT to simplify these connections:

zT =

α2
κ
L0 + σTl

(1.52)

This equation removes the dependency on the electrical contribution to thermal conductivity because of the direct connection between it and the electrical conductivity. It
additionally highlights the importance of the ratio of the lattice thermal conductivity to the
electrical conductivity and the importance of a high Seebeck coefficient. The higher the electrical conductivity the lower the ratio of κl /σT will become. In the case where this quantity
becomes much less than 1, the zT equation simplifies to zT ≈ α2 /L0 . Unfortunately in this
case the material has such a metallic electrical conductivity that its Seebeck coefficient will
typically be very low.
Another way in which equation (1.52) may be interpreted is to attempt to minimize
the lattice contribution. However, in the process of minimizing κl the likely result will be
to increase the intensity and number of scattering events for electrons, as well as phonons,
resulting in a dramatic decrease in mobility and thereby a subsequent reduction in the electrical conductivity and the power factor. The interplay of these properties can be displayed

36

in an expanded form of the Pisarenko plot which was shown before in Figure 1.8, where
it becomes clear that it is not a simple matter of tuning the carrier concentration to the
maximum power factor and then decreasing the lattice contribution to thermal conductivity.
It is thus beneficial to begin by selecting materials with favorable properties and then to
take a refined approach to optimizing the variables in zT.
Slack proposed a paradigm of the optimal properties with which a material may be
selected in his phonon-glass electron-crystal model [57]. In this model Slack states that the
phonon mean free path should be that found in amorphous glasses, and the mobility of
carriers should be that found in single crystals. Under these criteria Slack proposed a term,
∗ 3/2
and R, where R = U/κl . Using these terms
U, the weighted mobility where U = µ m
m
0

Slack came to the conclusion that for high zT the best materials will have a high mobility,
a high effective mass, and a low lattice contribution to thermal conductivity. This R term
could then be a filtering term to distinguish materials with high zT values. In his survey
Slack found no material with R values high enough to result in zT values of unity from 77
to 300 K, but outlined the requirements for zT values as high as 4 at room temperature.
A material parameter, B, given the moniker ”quality factor”, is an attempt to define the
exact properties of a material which will result in a high zT. This was first introduced by
Chasmar and Stratton, named the material factor, β [58], and later reworked by Nolas and
Goldsmid [23]. Taking into account the changes in mobility with doping and carrier band
degeneracy the quality factor was given Snyder et al. as [59]:

2
2kB
Cl Nv
B=
T
3π m∗I Ξ2 κl

(1.53)

where Nv is the valence band degeneracy, Cl is the longitudinal elastic constant, m∗I is

37

the conductivity effective mass, and Ξ is the deformation potential. The band degeneracy
relates back to the concept of energy band diagrams, where the bottom of the conduction
band or the top of the valence band may be comprised of several overlapping bands. It is
seen from the above equation that a high band degeneracy with low conductivity effective
mass is preferred. This comes contrary to what is typically desired for the effective mass
and contrary to what Slack proposed. The parameter Cl is a measure of the resistance of a
material to elastic deformation under an applied force and can be thought of as the stiffness
of the atomic bonding, and for high quality factor, should be maximized. Finally, the term,
Ξ, which is a measure of the deformation potential of energy bands created by the interaction
of phonons with the band structure, should be minimized.
The lattice contribution to thermal conductivity in the Umklapp scattering dominated
region, can be approximated as [60, 61]:

κl ∝

M vs3

1

T V 2/3 γ 2

N 1/3

(1.54)

where M is the average mass, V is the unit cell volume, γ is the Gruneisen parameter,
and N is the number of atoms in the primitive unit cell. This equation is especially useful for
the selection of a material because it highlights the importance of having a large number of
atoms in the primitive unit cell and having a high Gruneisen parameter, which is a measure
of the anharmonicity of bonding [62].
A final aspect of material selection that will be mentioned here is selection based upon
the electronic band gap. Early work by Chasmar and Stratton found a band gap of 6
kB T maximized thermoelectric performance in an indirect gap material with Mahan later
expanding that to 10 kB T with similar values being found for direct gap materials [58,

38

63, 64]. Taking the midpoint of these values at 8 kB T the optimum band gap at 150 K
is approximately 0.1 eV and at 300 K is approximately 0.2 eV. Thus, for low temperature
applications and zT maximization a low bandgap material is desired. This can be intuitively
understood by the requirement for one carrier type to be present in a material for the desired
temperature. If the bandgap of a material is too low for a desired operating temperature,
then intrinsic carriers will be populated and the Seebeck coefficient will be decreased. There
is more room for optimization if the bandgap is larger than this 6-10 kB T guide through
carrier concentration doping. Thus, for the work contained herein a low band gap material
will be preferred for low temperature thermoelectric applications.

39

Chapter 2
Experimental

2.1
2.1.1

Materials Synthesis
Furnace Reaction Method

For this work there were three main steps in the synthesis and processing of materials.
The first was the reaction of elements to form the desired compound. This step typically
consisted of a heating phase to supply the required thermal energy to allow for chemical
reaction between the different elements. The second step consisted of any post reaction
powder processing required by the particular project. This would entail any particle size
reduction attempts by ball milling or any material annealing required. The final step in the
synthesis procedure was a densification step whereby a dense pellet was created capable of
having transport measurements taken upon it.
As the specific experimental synthesis parameters varied for many of the materials examined a general guideline of what was performed will be given here. For the majority of
materials the initial reaction was facilitated by furnace heating. For all materials examined,
stoichiometric amounts of the elements were sealed in an evacuated quartz ampoule. The
elements used were supplied from either Sigma Aldrich or Alfa Aesar with a purity of at

40

least 99.99 %. The elements were placed in a furnace which was heated to 800 ◦ C for 4 days.
The reacted compounds were then quenched to room temperature in a water bath. After
the initial reaction samples were typically very porous with low densities, thus for transport
properties measurements a subsequent densification step was necessary.
Samples were ground via mortar and pestle to prepare for X-ray diffraction and either
material processing or densification depending upon the project. Between the initial reaction, any post-reaction processing, and densification every sample was analyzed by X-ray
diffraction on a Rigaku Miniflex-II using Cu-Kα radiation to ensure phase purity.
For projects which entailed particle size reduction a ball milling step was performed using
a SPEX MixerMill 8000D or a MixerMill 8000M in either a stainless steel ball milling jar
and media or a zirconium oxide, (zirconia), milling jar and media. The duration of milling
time differed between projects and will be given in the discussion of the applicable projects.
Samples which were annealed were again placed back into an evacuated quartz ampoule,
heated to their reaction temperature, and held for a specified time.
The final step in material synthesis was densification by spark plasma sintering, SPS, for
all projects in this work. A Thermal Technology model 10-4 system SPS machine was used
for early samples and a Dr. Sinter spark-plasma-sinter 211LX was used for later samples.
To prepare samples for sintering, a mortar and pestle were used to pulverize samples which
had not been ball milled. The powder was then loaded into a graphite die with graphite foil
spacers between two plungers which acted to give uniaxial pressure on the sample and to
apply a path of electric current through the sample. The die and plungers were stationed
between two water cooled graphite rams. A schematic representation and picture of the SPS
system is shown in Figure 2.1.

41

Figure 2.1: Schematic of Spark Plasma Sinterer

SPS, or pulsed electric current sintering, is differentiated from other types of sintering
by the pulsed direct current which is applied across the sample. This pulsed electric current
results in two effects which create sintering in the sample. One of these is joule heating due to
the passage of a current through an internally resistive sample. The other effect is the arcing
of electric current between neighboring particles [65, 66]. Through these two effects samples
with densities higher than 95% density were obtained. The final densified sample is in the
shape of a cylindrical puck and must be cut to perform transport properties measurement
upon.

2.1.2

Flux Synthesis

In the Conclusions and Future Directions chapter a material system will be presented which
used an entirely different initial chemical reaction than other samples. This material contained the element phosphorous which has a low boiling point that made a furnace reaction
42

method impossible due to the failure of quartz ampoules when phosphorous vaporizes. To
synthesize this material a flux method was utilized. For this method a secondary material was added to the reactants which acted as a solvent for the desired compound. This
stabilized the mixture and allowed for the desired product to react [67].
The synthesis procedures outlined here were based upon the work by Odile et al. [68].
Stoichiometric amounts of elements were sealed in an evacuated quartz tube with 85 wt% tin
added as the flux. The heating profile for this reaction consisted of an initial ramp to 250 ◦ C
at 1.0 ◦ C per minute with a dwell time at this temperature of 12 hours. Another ramp was
performed up to 1000 ◦ C at 0.5 ◦ C per minute with a dwell time of 2 days. A slow cooling
step was performed at a rate of 5 ◦ C per hour to 550 ◦ C to allow the desired compound
to form. Once the furnace had cooled to 550 ◦ C a water quench to room temperature was
performed.
Samples were ground via mortar and pestle and examined with X-ray diffraction for the
desired product. To liberate the product from the tin flux a reaction with 6 molar hydrochloric acid was performed. The reaction with hydrochloric acid will generate the creation of
tin chloride, a salt, and hydrogen gas. Thus, the generation of bubbles due to hydrogen gas
formation was an indication that the reaction was proceeding. When hydrogen production
ceased it was assumed that the reaction was complete. Finally, the mixture was washed
with distilled water to dissolve and remove the tin chloride salt. The remaining material
after the wash should be the desired product. After this step typical powder processing and
densification procedures were resumed.

43

2.2
2.2.1

Material Characterization Techniques
X-ray Diffraction

X-ray diffraction was performed to characterize the phase purity of powderized samples. This
technique allows for the analysis of the crystal structure of a sample due to the diffraction
of X-ray radiation from atoms in the lattice. Due to the periodic nature of atoms in crystal
lattices the spacings between planes of atoms can be examined by the angle with which X-ray
radiation diffracts.
As X-ray radiation is diffracted from the crystal lattice these waves may interact by either
constructive or destructive interference. In destructive interference the waves are completely
”out-of-phase” meaning that the crest of one wave is perfectly aligned with the valley of the
wave it is interacting with. Mathematically this is expressed as when the phase of one wave in
a set is at an odd multiple of π relative to the other wave. X-rays in this arrangement are said
to destructively interfere and have an amplitude of zero which does not generate any signal
in X-ray diffraction. Constructive interference however occurs when waves are perfectly ”inphase”, which may differ by an even multiple of π. In this instance the crests of both waves
perfectly align which results in a doubling of the amplitude of the resultant wave. It is these
constructively interfering waves which generate the signal in X-ray diffraction.
Since it is constructively interfering diffracted beams which generate the output signal
in X-ray diffraction the spacing between planes of atoms and the angle with which they
constructively interfere is given by Bragg’s law:

nλ = 2dhkl sin(θ)

44

(2.1)

where λ is the wavelength of incident X-ray radiation, dhkl is the spacing between planes
of atoms, and θ is the angle of incident X-ray radiation in relation to the atomic plane. A
schematic representation of this is given in Figure 2.2.

Figure 2.2: Schematic of Bragg Diffraction

A desktop Rigaku Miniflex II using Cu-Kα radiation with a wavelength of λ = 1.54 ˚
A was
used. To determine the lattice parameter of samples an internal silicon standard was added
to account for any offset in 2θ due to systematic error or a height offset of the diffracted
powder. For this work the samples examined all had cubic crystal structures, thus the lattice
parameters were all equal, a = b = c, and all internal angles equal to 90◦ . Using reciprocal
space dhkl parameters where h, k, and l are the Miller indices of planes, the lattice parameter
could be obtained by:

1
d2hkl

=

h2 + k 2 + l2
a2
45

(2.2)

To analyze the particle size reduction effects of ball milling X-ray diffraction was again
utilized. Particle size reduction results in a broadening of the observed peaks for a material.
The width of the peak, taken at half the maximum intensity, can be related to the particle
size by the Debye-Scherrer equation [69]:

L=

Kλ
βcos(θ)

(2.3)

where L is the crystallite size, K is the shape factor, β is the broadening at half maximum,
and θ is the 2θ position at which the peak maximum occurred. The shape factor is very
often close to unity but depends upon the shape of the crystallite. When determining the
broadening at half maximum it is important to subtract any instrumental broadening, which
may occur due to several factors including imperfectly monochromatic radiation production
from the machine and a non-infinitesimally small diffraction surface.
However, a reduction in crystallite size is not the only sample effect which can lead
to broadened X-ray peaks. The other factor occurs due to any internal strain effects in
the crystal lattice. These can arise from lattice distortion from dislocations, concentration
gradients, and several other factors which cause a displacement of atoms from their most
thermodynamically stable position. The Debye-Scherrer equation is unable to separate the
peak broadening due to lattice strain from the reduction of crystallite size, however the
Williamson-Hall equation has made these two effects separable and is given by [70]:

βcos(θ) =

Kλ
+ 4 sin(θ)
L

(2.4)

Using this equation it is possible to plot βcos(θ) vs. sin(θ) to obtain the y-axis offset
which can be solved for to obtain the crystallite size. This method gives an average crystallite
46

size and should be confirmed with additional methods to quantify exact crystallite sizes.
The technique is nevertheless a powerful tool for comparison between similar samples given
differing processing conditions.

2.2.2

Low Temperature Transport Property Characterization

To measure the thermoelectric properties of each sample a diamond saw was used to cut
samples into a parallelepiped of typical dimensions: 3 mm x 3 mm x 8 mm. These samples
were then measured on an in-house built Janis liquid nitrogen-flow cryostat from 80 to
350 K. Two key concerns for the accurate measurement of transport properties are good
electrical contacts between the sample and the measuring devices and good thermal contact
and sinking between the sample and the heat source and sink. To ensure good electrical
and thermal contact samples were soldered onto a copper base, which additionally acted
as a heat sink, and to the top surface of the sample an 800 Ω resistor was soldered. A
copper sheet was wrapped around the resistor to allow for good thermal contact between
the resistor and the sample and to also be a conduit for the current input. A Keithley
2400-LV Sourcemeter was used to source a known current through the resistor. A Keithley
2001 multimeter was used to measure the voltage drop across the resistor. With the known
current input into the resistor, and the voltage drop across it, the power output, in the form
of Joule heating, was able to be calculated. A current input, again using a Keithley 2400-LV
Sourcemeter, was soldered onto the copper sheet wrapped around the resistor which allowed
for the propagation of current down the sample and output through the copper base. For all
measurements a steady state heat flux condition was created by holding the powered resistor
on for at least ten minutes prior to any measurement. To measure the conduction of heat
through the sample two copper-constantan thermocouples were soldered onto the surface
47

of the sample perpendicular to the flow of heat taking care to span the entire breadth of
the face of the sample. A Keithley 2182A Nano voltmeter was used to measure the voltage
generated by the thermocouples which could then be interpreted as the temperature of the
sample at that point. To reduce the loss of heat by conduction the cryostat was placed
under a vacuum of approximately 10−5 torr. A schematic representation of this setup used
to measure thermal conductivity, electrical resistivity, and the Seebeck coefficient is given in
Figure 2.3.

Figure 2.3: Schematic of Cryostat Measurement Setup

To determine the thermal conductivity of a sample the thermal conductance was first
determined by:

V I
KM = H H
∆T

(2.5)

where KM is the measured conductance, VH is the voltage drop across the resistor, IH is
48

the current passed through the resistor, and ∆T is the measured temperature drop between
the thermocouples. VH IH is the power in the form of Joule heating generated by the resistor.
It is assumed that all of this heat is conducted into the top surface of the sample, further
highlighting the importance of good thermal contact. According to the Stefan-Boltzmann
law a certain amount of this heat will be lost as radiative energy as it propagates through
the sample. This law is given by: P = σAT 4 , where P is the radiative heat lost,

is the

emissivity of the sample, σ is the Stefan-Boltzmann constant, A is the surface area of the
sample, and T is the temperature of the sample. Thus, the actual thermal conductance of
the sample is the measured component minus the radiative heat loss. To account for this
loss in samples with unknown emissivity a standard glass sample was used to determine the
typical radiative loss:

KRad = 0.00146 ∗

T 3
300

(2.6)

where KRad is the radiative heat loss. Using this term the total thermal conductivity of
the sample could be found from the equation:

κ = (K − Krad ) ∗

L
S

(2.7)

where L is the thermocouple probe separation and S is the area of the sample. The
dependency of the conductivity to the dimensions of the sample highlight the importance of
having accurate dimensional measurement. To reduce the inherent error in the thermocouple
separation measurement the thermocouples should be separated as far as possible while still
allowing space for heat and electrical current flow to be homogeneous. To account for this
thermocouples were soldered onto the sample surface a minimum of 2 mm away from the
49

ends of the sample. Thus, a typical thermocouple probe separation was approximately 5-6
mm. The error in this measurement is still estimated to be up to 10% due to inaccuracy in
dimension measurement, system error, and the radiative heat loss term.
To determine the Seebeck coefficient of samples the voltage drop obtained between the
copper contacts of the thermocouples were measured using a Keithley 2000 multimeter with
the heater energized. With the known voltage and temperature drop between thermocouple
VS
probes the Seebeck coefficient may be directly obtained from: α = ∆T
, where VS is the volt-

age drop between copper leads from the thermocouples and the ∆T is the same temperature
drop between the thermocouples as before. The error in this measurement is estimated to
be 5% due to inherent inaccuracy in measuring the temperature gradient and the relatively
low voltages generated due to the Seebeck effect.
The four probe electrical resistivity measurement method was used for samples where
again the voltage drop along the sample was measured between the copper leads of the
thermocouples. For this measurement the sample current is sourced in one direction and
then reversed to account for any contact resistances or the generation of a Seebeck voltage
from Joule heating. The equation describing this is:

S
V
ρ= S ∗
IS L

(2.8)

where IS is the current passed through the sample. Sample dimensions are found to
play a role in evaluating another measured property which further highlights the need for
accurate sample dimensional measurement. The error in this measurement is also estimated
to be 5% primarily due to dimensional measurement.

50

2.2.3

Hall Effect Measurement

The Hall effect is due to the presence of an orthogonal force, the Lorentz force, acting on
moving charge carriers in the presence of a magnetic field [71]. This force is dependent upon
the electric field acting on the charge carriers and the magnetic field by:

F = q [E + (v × B)]

(2.9)

where F is the Lorentz force, E is the electric field, v is the drift velocity of carriers,
and B is the magnetic field. The cross product of v and B yields a transverse component
which causes a migration of charge carriers to one side of a sample. This buildup of charge
generates a measurable voltage across the sample. This voltage is termed the Hall voltage
and is given as:

VH =

IB
nte

(2.10)

where I is the current, t is the thickness of the sample, and n is the charge carrier
concentration. A Hall coefficient, RH , may then be defined as:

V t
−1
RH = H =
IB
ne

(2.11)

Using Hall testing the carrier concentration of a sample may be obtained and subsequently
the carrier mobility may be determined by the relation: σ = neµ; in the single carrier system.
A schematic representation of this system may be seen in Figure 2.4

51

Figure 2.4: Schematic of Hall Effect Measurement Setup

To perform these measurements samples of dimensions ≈ 0.25 mm x 3 mm x 8 mm with a
probe separation of approximately 5 mm were used. The thickness of samples was minimized
to result in higher signal to noise ratios in obtaining the Hall voltage. Samples were mounted
onto a flat copper base with 2 layers of cigarette paper to act as an electronic barrier, yet
still provide good thermal contact. All sample contacts were soldered on to encourage good
electrical contact with the sample. Magnetic fields ranging between 0 to 2 Tesla were used to
measure the Hall voltage from 80 to 350 K as measured with a Lakeshore 421 Gaussmeter.
The standard four probe electrical resistivity measurement method was used to determine the
resistivity of the sample at all temperatures measured using a Lakeshore 370 AC Resistance
Bridge. Samples were sealed in an evacuated atmosphere of approximately 10−5 torr with
a liquid nitrogen flow to reach 80 K. A Lakeshore temperature control meter was used to
stabilize the system at each temperature.

52

Chapter 3
Background on PtSb2
3.1

Low Temperature Thermoelectric Materials

Thermoelectric materials research for low temperature applications, typically room temperature and lower, is an area of the field with relatively less work compared to the mid
temperature and high temperature regions of application. Since the 1950’s the reigning material for room temperature applications has been Bi2 Te3 . Work by Goldsmid and Douglas
established Bi2 Te3 [5, 72, 73] with zT values approaching unity around room temperature.
Over the past sixty years further work devoted to doping and solid solution alloying with
Sb2 Te3 have been able to increase the zT of this material to values above unity up to temperatures of 500 K [74–76]. Possibly of even more importance, Bi2 Te3 and its alloys have
the capability of assuming p- and n-type properties such that a thermoelectric module may
be made from the same base compound. This is highly beneficial because of the need to
have similar thermal expansion coefficients for the materials in a thermoelectric module to
avoid cracking from thermal cycling stresses. This has made Bi2 Te3 the ”go-to” material for
commercial application and use near room temperature.
These optimized Bi2 Te3 alloys are limited to use around room temperature due to a
drop-off of their properties outside of this region. At temperatures lower than 250 K other
53

materials have been found to have superior properties. Work by Smith and Wolfe established
the solid solution of bismuth and antimony as a material which shows superior properties
to Bi2 Te3 at temperatures between 20 and 220 K [77]. Work with Bi-Sb alloys has been
able to show further enhancement due to the presence of a magnetic field, but even in a zero
field, this n-type material, with a Bi rich composition of 15% Sb, showed a figure-of-merit
maximized at 80 K with a zT ≈ 0.5 [78].
Further work in the low temperature region for thermoelectrics has been sparse due to
the dominance of these two compounds. A material that has shown promise as a potential
low temperature candidate is CsBi4 Te6 . This layered material, which is structurally distinct
from Bi2 Te3 , shows superior zT to Bi2 Te3 at low temperatures until approximately 275 K
with a maximum at 225 K of zT ≈ 0.8 [79].
FeSi first garnered interest due to its unusual magnetic properties [80], but later showed
a very large Seebeck coefficient of approximately 500 µV/K at 50 K [81]. This large Seebeck
coefficient was shown to be from a phonon-drag effect, which correlated to a large thermal
conductivity at the same temperature. Many attempts have been made to dope FeSi to
enhance its thermoelectric properties, with the best dopant being found to be iridium [82].
This dopant showed a maximized zT = 0.12 for planetary milled samples at 100 K.
Other possible materials for low temperature application include those of the intermediate
valence systems such as CePd2 Pt with a zT approaching 0.3 below room temperature and
YbScAl [83, 84]. These systems allow for the chemical pressure tuning of the electronic
properties by simple substitution methods and make use of a complex electronic structure
with very sharp electronic density of states due to f-shell electrons. However, they still show
lower performance than the Bi-Sb alloys examined so long ago.
For further enhancement of the figure-of-merit at low temperature extensive work should
54

be performed on known, as well as, unknown materials.

The serendipitous finding of

CsBi4 Te6 as a superior thermoelectric demonstrates the possibility that there are still material candidates as yet unknown for low temperature applications.

3.2
3.2.1

PtSb2
Crystal Structure and Phase Diagram

PtSb2 is a semiconducting material which crystallizes in the cubic pyrite structure with
space group (Pa¯3). To start, a comparison to the rocksalt, NaCl, structure may be made. Pt
atoms occupy the cation Na sites in a face centered cubic arrangement, with positions: (0, 0,
0), (1/2, 1/2, 0), (1/2, 0, 1/2), and (0, 1/2, 1/2). The anion site is occupied by an Sb dimer
pair with positions: ±(x, x, x), ±(-x, x+1/2, 1/2-x), ±(1/2+x, 1/2-x, -x), and ±(1/2-x,
-x, 1/2+x) with internal parameter x = 0.375 [85]. The Sb dimers are oriented parallel
to the <111> directions with a Sb-Sb separation of 2.78 ˚
A [86]. The lattice parameter
through theoretical calculations and experimental work has been found to be a = 0.644
nm [85, 87]. Platinum atoms are octahedrally coordinated with 6 bonded antimony atoms
while antimony atoms are tetrahedrally bonded to three platinum atoms and one other
antimony atom. Figure 3.1 shows a schematic of the pyrite crystal structure generated from
the software Vesta [88].

55

Figure 3.1: Schematic of Pyrite Crystal Structure Showing Pt - Sb Local Bonding

Visible in this schematic is the distortion of the octahedral coordination of Pt where the
angles between Sb-Pt-Sb groups are no longer 90◦ . Due to this distortion local symmetry
is reduced from cubic to trigonal and it can also be seen that there is a rotation of the
octahedral axis away from the cubic lattice of approximately 24◦ [89].
PtSb2 shares the pyrite crystal structure with several compounds, the namesake compound, FeS2 , CoS2 , MnTe2 , RuSe2 and RuTe2 , AuSb2 , PdAs2 and PdSb2 , and PtPn2 , where
Pn = S, As, Sb, and Bi [90] [91]. The most exciting of these from the thermoelectric point
of view are the Pd and Pt based pyrite compounds due to the possibility of making a variety
of solid solutions which could lead to a reduction in the lattice thermal conductivity.
Using nuclear magnetic resonance Mallick and Emtage determined the likely formal valence state of PtSb2 to be Pt2− Sb+
2 [92]. This valence state points to nearly only covalent
bonding in PtSb2 , which is what would be expected due to the small difference in their Pauling electronegativities, χ, Pt(2.28) and Sb(2.05). The Sb-Sb dimers would be expected to
be entirely covalent and the Pt-Sb bond dipole would be very small with a ∆χ = 0.23 which

56

would lend itself to being almost entirely covalent bonding [93]. Work by Lutz et al. have
shown that PtPn2 compounds going down the column of the periodic table have increasingly
stronger bonding which further supported the claim of primarily covalent bonding in this
system [94]. With the strong bonding found in PtSb2 it is unsurprising that this compound
has a high melting point of 1226 ◦ C [95]. Itkin and Alcock presented the phase diagram
of PtSb2 where it is found that PtSb2 forms by a congruent phase transformation from the
liquid phase at an atomic% Sb of 66.7% to the intermetallic PtSb2 [96]. The phase diagram
for PtSb2 is shown in Figure 3.2.

Figure 3.2: Schematic Phase Diagram of the Pt-Sb System

3.2.2

Electronic Structure and Transport Properties

Much of the early work with PtSb2 was performed on single crystals to determine the electronic properties of the compound. This early work established that PtSb2 is a narrow-gap
semiconductor with a bandgap of approximately 0.1 eV [97–99]. Work by Damon et al.
57

examined the magnetoresistance and piezoresistance of PtSb2 single crystals to find a likely
valence band maxima was located on the <100> directions [98]. Emtage built upon this
work using theoretical modeling and made electronic band structure calculations that would
last for the next 40 years. Emtage calculated that it is non-bonded d-orbitals that take part
in the formation of the conduction and valence bands. He found that there are valence band
maxima in the <100> directions and conduction band minima along the <111> directions,
which give rise to an indirect bandgap of EG ≈ 0.08 eV. A schematic representation of the
energy band diagram of PtSb2 is given in Figure 3.3 modified from that generated by [100].
Damon et al. later supported the valence band maxima being on the <100> directions and
further showed that this was from six ellipsoid bands which produced unequal inertial effective masses, m1 , m2 , and m3 with ratios of 0.61:1:1.64 [101]. They additionally countered
Emtage’s earlier finding of an energy dependent band structure and gave a direct gap of
EG > 0.4 eV.

Figure 3.3: Schematic Energy Band Diagram of PtSb2
58

Much of this early work on PtSb2 on single crystals had an interesting finding that even
when samples were undoped a large variation in carrier concentration and even dominant
charge carrier was found. Several groups provided further validation of the energy gap being
EG ≈ 0.1 eV and showed that acoustic phonon scattering was the dominant mechanism
above 50 K, however found a large variation for many single crystal carrier concentrations
ranging between c.c. ≈ 1 ∗ 1016 to 1.5 ∗ 1018 and also disagreed on the dominant carrier
type [102–106]. These deviations are thought to have occurred because of variations in
stoichiometry due to the loss of antimony when melting the compound in preparation for
single crystal growth. More recent work on single crystal and polycrystalline samples showed
that a non-stoichiometry due to antimony loss can create n-type samples at room temperature
and below [100, 107].
Several doping experiments have been performed on PtSb2 to examine its resultant behavior. Dargys and Kundrotas in 1983 examined the effects of tellurium doping on single
crystals and found with low but, unfortunately, unspecified dopant concentration at T ≈ 25
K a very large Seebeck coefficient can be obtained at almost -800 µV /K [108]. It was evident
though that the PtSb2 structure has a limited ability to incorporate dopants from the work
of Laudise et al. and Bennett et al. In an attempt to make a magnetic semiconducting
material Laudise et al. worked with the solid solution of PtSb2 and the magnetic Heusler
compound PtMnSb. However, with concentrations of x=0.25 and higher in PtSb2−x Mnx ;
secondary phases were found [109]. In the work by Bennett et al. the doping of several rare
earth elements were attempted in PtSb2 . Again, it was found that secondary phases were
produced from these dopants which in the case of Pt1−x Ybx Sb2 created a superconducting
material [110]. This superconducting activity is thought however to have come from the
secondary phase in the system of PtSb, which is superconducting at Tc = 2.1 K [111].
59

Successful p-type doping on the Pt site in PtSb2 was obtained by Nishikubo et al. by
Ir doping in a polycrystalline sample [112]. A large enhancement to the power factor was
obtained largely from a decrease in electrical resistivity. A decrease in the Seebeck coefficient
was found at low temperature, but above 300 K, in their lightest doped sample, the Seebeck
coefficient surpassed the un-doped material. The power factor was found to be approximately
18 µW/cm-K2 at 150 K for the un-doped sample with an increase to approximately 43
µW/cm-K2 at 400 K. Diminished thermal conductivity was additionally found in the doped
samples which ultimately gave an enhanced figure-of-merit.
Theoretical work by Mori et al. suggested that the origin of the large power factor found
by Nishikubo et al. was due to a corrugated flat band structure found in PtSb2 [113]. This
work hearkened back to that of Emtage who found that a low energy barrier separated
the maximum in the valence band between another high lying band which created a wide
dispersive region separated by a 0.011 eV energy barrier. Typically, in a metal where the
Fermi energy is within a band, to have a large Seebeck coefficient the Fermi energy must
be near the band edge where the group velocities of electrons and holes will differ. This in
turn reduces the carrier concentration which negatively affects the power factor. Mori et al.
argued that due to the corrugated flat band structure in PtSb2 hole doping did not cause a
large decrease in the Fermi energy which left the large Seebeck coefficient intact. This can
further be seen in theoretical work on electron and hole doping of PtSb2 and experimental
polycrystalline work on tellurium doping [114, 115].
A final consideration for PtSb2 is due to its narrow bandgap. The small bandgap of
only 0.1 eV creates the maximum in the Seebeck coefficient found at approximately 150 K
due to the population of electrons at this temperature from the material entering into the
intrinsic regime at this point. For the application of PtSb2 at very low temperatures this
60

very narrow bandgap is ideal, but for applications approaching room temperature, a wider
bandgap would be superior. This point was given by Mori et al. who concluded that the
materials PtP2 and PtAs2 would be better candidates at higher temperatures due to their
larger bandgaps. This point was proven by Kudo et al. when they examined rhodium doping
of PtAs2 [116]. The thermoelectric properties were examined from 2 to 600 K and found
the Seebeck coefficient increased over the entire temperature range and a maximized power
factor of 65 µW/cm-K2 at approximately 440 K. Thus, the solid solution of PtSb2 with these
materials may be a route of engineering the bandgap in PtSb2 .

3.3
3.3.1

Nominally Undoped PtSb2
Synthesis

As outlined in chapter 2 undoped PtSb2 was synthesized by the solid-state reaction of elemental chunks of Pt and Sb in an evacuated quartz tube at 800 ◦ C. Samples were allowed
four days to react to completion and were subsequently quenched to room temperature via a
water bath. A solid-state reaction was necessitated due to the high melting point of platinum
and the end compound, 1769 ◦ C and 1226 ◦ C respectively. Neither the furnaces used in synthesis or the quartz tubes would have been able to facilitate such a high temperature to allow
for the liquid state to have been met and subsequently water quenched. Additionally, due
to the work of prior research it was known that many samples resulted in non-stoichiometry
due to the loss of antimony from its high vapor pressure at the melting point of PtSb2 . Since
antimony has a low melting point relative to platinum and PtSb2 at 630 ◦ C the synthesis
was allowed to proceed by way of diffusion. At a furnace temperature of 800 ◦ C antimony
would melt and diffuse into and react with platinum to synthesize the desired end product.
61

Experimentation on reducing and prolonging the reaction time generated secondary phases
with a decreased time and produced no benefit from prolonging the reaction time. To prepare samples for X-ray diffraction and subsequent densification via SPS a mortar and pestle
was used to grind samples into a powder.
For nominal PtSb2 two different SPS parameters were used to densify samples. In the
early experimentation of this work the parameters were a slow ramp to 900 ◦ C over 30 minutes
where the sample was then held at this temperature for 30 minutes and then slowly cooled
to room temperature over 30 minutes at a pressure of 40 MPa. This produced samples with
greater than 95% theoretical density, but it was later found that a much shorter sintering time
could be used while retaining the near theoretical density. The shortened SPS parameters
were a ramp to 900 ◦ C over 7 minutes with a dwell time of 10 minutes at 40 MPa of pressure.
After the 10 minutes of sintering the current was turned off, the pressure was removed, and
the sample was allowed to cool to room temperature limited by the rate at which the water
cooled graphite dies could dissipate heat.
Cylindrical samples densified by SPS were then cut by diamond saw and either prepared
for mounting on the cryostat, Hall measurement, or ground for XRD analysis as were all
future samples.

3.3.2

Structural Properties

The X-ray diffraction pattern of PtSb2 is shown in Figure 3.4 compared against the theoretical pattern generated from Vesta using the parameters obtained by Brese et al. [86, 88].
It was found that the obtained pattern matched well with previous literature and yielded a
lattice parameter of a = 0.643 nm as determined by the inclusion of a silicon standard.

62

Figure 3.4: X-ray Diffraction Pattern of PtSb2

At high 2θ double peaks may be observed in the theoretical pattern due to the inclusion
of Cu-Kα2 peaks. These peaks have been removed from the experimental data for ease of
analysis.

3.3.3

Electrical Properties

The Seebeck coefficient of PtSb2 is shown in Figure 3.5 compared with data on polycrystalline
samples by Johnston et al. and Nishikubo et al. [90, 112].

63

Figure 3.5: Seebeck Coefficient of PtSb2

It is shown that using this synthesis procedure the same p-type Seebeck coefficient is
found, however, its overall magnitude is shown to be diminished. Compared with the work
of Nishikubo et al. it is seen that the peak in Seebeck coefficient found in that work was
at a lower temperature, 130 K, than that found in this work, 150 K. This is indicative of
a narrower bandgap in the work by Nishikubo et al. with the onset of bipolar conduction
occurring at a lower temperature and is also indicative of a lower carrier concentration in
their sample at this temperature due to the increased magnitude. However, above 160 K the
magnitudes of both sets of data match and follow the same trend. Compared to Johnston
et al. it is seen that the peak in Seebeck coefficient occurs at the same temperature, yet,
it is only at high temperatures, > 300 K, that the magnitudes begin to converge, elsewhere
the magnitude of the Seebeck coefficient is greater. The overall increased magnitude of
Johnston’s results is indicative of lower carrier concentration or a difference in mobility across

64

the entire temperature region measured. Though diminished compared to these other results
the Seebeck coefficient found, from as prepared PtSb2 , is very large at low temperatures
with a maximum at 160 K of 220 µV/K. This highlights the interest in this compound for
low temperature Peltier coolers: few materials have such large Seebeck coefficients in the
cryogenic temperature regime.
The electrical resistivity of undoped PtSb2 is shown in Figure 3.6, also compared with
the work of Johnston et al. and Nishikubo et al. It should be noted that the y-axis is in a
logarithmic scale to show the wide ranging data.

Figure 3.6: Electrical Resistivity of PtSb2

Evidence for the lower carrier concentration of the Nishikubo et al. sample at low temperature can be gleaned from this figure. At 80 K the electrical resistivity found by Nishikubo
is approximately an order of magnitude higher than that of this work and decreases rapidly
until it matches the electrical resistivity of this work at approximately 130 K. The data of

65

Johnston et al. shows a relatively flat region until approximately 140 K, where it matches
the values of this work and the work by Nishikubo et al. The flat portion of the work of
Johnston et al. is thought to be in the extrinsic regime of carrier concentration which at 140
K enters the intrinsic regime and reduces the electrical resistivity.
Analysis of the electrical resistivity found from this work can best be done in conjunction
with Hall effect data as shown in Figure 3.7.

Figure 3.7: Carrier Concentration and Mobility of PtSb2

It is found that the carrier concentration from 80 to 140 K is relatively unchanged,
however the mobility in this temperature region is increasing with a T0.5 dependency. This
effect is what resulted in the decrease of the electrical resistivity in this region where the
results of Johnston et al. did not. At approximately 150 K the carrier concentration began
to increase rapidly. Fitting the curve where intrinsic carriers are being populated the band
gap may be found to be EG ≈ 0.16 eV. While the carrier concentration is increasing the

66

mobility is found to decrease with a T−3/2 temperature dependency. This is indicative of
acoustic phonon dominant scattering as was also found by Reynolds et al. in this temperature
regime [103]. As the mobility decreases by a factor of 2 from 140 K to 300 K the carrier
concentration increases by an order of magnitude, which resulted in the observed continued
reduction of the electrical resistivity in this temperature region.

3.3.4

Thermal Properties

The total thermal conductivity of PtSb2 is shown in Figure 3.8 compared with that found
by Nishikubo et al.

Figure 3.8: Total Thermal Conductivity of PtSb2

Good agreement is found between the results of this work and that of Nishikubo et al.
At 80 K the thermal conductivity is found to be very large at approximately 63 W/m-K.
This is found to be the maximum over the temperature range examined due to the low

67

Debye temperature of this material. Above this temperature the specific heat is maximized
and Umklapp scattering is the dominant scattering mechanism. The thermal conductivity
is found to decrease with a T−0.77 temperature dependency, which indicates a mix of scattering mechanisms over this temperature region. Compared with the standard commercial
thermoelectric material Bi2 Te3 this thermal conductivity is much too high and should be
lowered for enhanced figure-of-merit.
Examination of the individual contributions to thermal conductivity in Figure 3.9 shows
that the lattice contribution to thermal conductivity is dominant over the entire temperature range. The electronic contribution is determined using the Wiedemann-Franz law as
described in section 1.5.1.

Figure 3.9: Lattice and Electronic Contributions to Thermal Conductivity of PtSb2

The electronic contribution to thermal conductivity begins with a very low value at 80
K, where the electrical resistivity of PtSb2 is at its maximum, and increases over the entire
temperature range as the electrical resistivity is found to decrease. The lattice contribution
68

is found to be over an order of magnitude larger than the electronic contribution and will be
an opportune parameter to minimize for figure-of-merit enhancement.
The magnitude of the figure-of-merit of nominal PtSb2 is low for all temperatures measured, but reaches a maximum between 220 to 250 K at a value of zT ≈ 0.009. Compared
to commercially maximized Bi2 Te3 this value is almost two orders of magnitude too low.
However, as will be shown in the forthcoming chapters several avenues of enhancement are
possible in PtSb2 that can increase its figure-of-merit substantially.

69

Chapter 4
Antimony Site Doping in PtSb2
4.1

Background and Motivation

Electronic doping provides a ”tuning knob” by which the carrier concentration and ultimately the power factor of a thermoelectric material may be optimized. There are several
considerations when optimizing the power factor of a narrow bandgap material, which will
be shown here, but this method has been proven to increase the figure-of-merit.
The most dramatic of these considerations occurs in the Seebeck coefficient. It is well
known that as the carrier concentration of a material increases the Seebeck coefficient decreases, thus, when the nominally p-type PtSb2 is p-type doped it is expected that this will
increase the carrier concentration and subsequently decrease the Seebeck coefficient. However, an interesting effect occurs when a p-type material is doped with a donor atom. At
high dopant concentrations the nominally p-type material will exhibit a negative Seebeck
coefficient. This denotes that electrons are now the dominant charge carrier in this system as
compared to the intrinsic holes. As the concentration of n-type dopant is decreased the magnitude of the Seebeck coefficient will increase due to fewer competing carriers and the peak
in the Seebeck coefficient will shift to lower temperatures due to fewer intrinsic hole carriers
needing to be populated to overcome the effect of the extrinsic electrons. At some very low
70

concentration, dependent upon the material and intrinsic carrier concentration, the Seebeck
coefficient will reach a maximum in the magnitude of the negative Seebeck coefficient it will
display. After this maximum has been met a subsequent decrease in the dopant concentration will not dope a sufficient concentration of electrons to create an n-type material, and
the typical p-type Seebeck coefficient will re-emerge.
This chapter will focus on electronic doping on the antimony site of PtSb2 . In previous
literature few dopants have been examined on the antimony site with even fewer making
a comprehensive examination over low temperature. The work by Dargys and Kundrotas
examined the effects of low dopant concentrations of tellurium on the antimony site. Their
work found significant enhancement in the Seebeck coefficient at low temperature reaching
approximately -800 µV/K as well as observing a change in dominant carrier type [108]. In
the work by Nishikubo et al. it was stated that the power factor of Sn doped PtSb2 was
increased to approximately 20 µW/cm-K2 [112]. Further, theoretical work by Saeed et al.
showed the enhancement that may be found in electron and hole doped PtSb2 at room
temperature and higher [114].
With this prior research it was shown that significant improvements in the power factor
may be made, thus it was the goal of this work to perform a comprehensive examination
of the potential thermoelectric enhancement of PtSb2 via n- and p-type dopants on the
antimony site.

4.2

Experimental Methods

To examine the effects of n-type dopants selenium and tellurium were selected due to the
additional valence electron that they have compared with antimony. To examine the effects

71

of p-type dopants tin, germanium, and indium were selected since they have fewer valence
electrons than antimony. Samples with compositions PtSb2−x Dpx , where Dp = Se, Te,
Sn, Ge, or In were synthesized with dopant concentrations ranging between 0 < x < 0.15.
While dopant concentrations varied, the n-type donor concentrations were kept low and
p-type acceptor concentrations spanned the entire range due to variations in the electronic
effects of each dopant. Samples were synthesized in the same manner as outlined for nominal
PtSb2 though it should be noted that tellurium doped samples utilized the SPS parameters
which entailed the 30 minute dwell time while all other dopants utilized the shortened SPS
parameters.

4.3
4.3.1

Tellurium Dopant Results and Discussion
X-ray Diffraction

The X-ray Diffraction patterns of tellurium doped PtSb2 are shown in Figure 4.1.

72

Figure 4.1: X-ray Diffraction of Tellurium Doped PtSb2

It was found that there were no observable secondary phases with tellurium dopant
concentrations of up to x = 0.04. It was additionally found that there were no shifts to
lower 2θ values as tellurium dopant concentration increased which would be expected from
an increasing lattice parameter. This is thought to be due to the low overall concentration
of tellurium doped and the fact that the size difference between tellurium and antimony is
small.

4.3.2

Electronic Properties

The results of low n-type dopant concentrations on a narrow bandgap semiconductor are
shown very well by the results of tellurium doping. Examination of the Seebeck coefficient
data in Figure 4.2 showed that a change in dominant charge carrier occurred with very low
concentrations of tellurium dopant.

73

Figure 4.2: Seebeck Coefficient of Tellurium Doped PtSb2

In the x = 0.02 and 0.04 samples an n-type sample was found with a low magnitude
across all temperatures measured due to the high concentration of extrinsic carriers. With
a reduction in the concentration of tellurium dopant it was found that the n-type character
remained and further that the magnitude of the Seebeck coefficient was found to be increased
at lower temperatures. These samples also showed a maximum, or peak in their trend, which
decreased in temperature as dopant concentration was reduced. At low temperatures, in
these low dopant concentration samples, the Seebeck coefficient increased with temperature
until the thermal energy met that of the bandgap. At this temperature intrinsic hole carriers
were then populated in the system which led to the competition of holes and electrons and
caused a decrease in the magnitude of the Seebeck coefficient. At high enough temperatures,
when the weighted contribution to the Seebeck coefficient of holes had surpassed that of
electrons, the overall measured Seebeck coefficient was found to return to a positive value.

74

This effect was best shown in the x = 0.0005 sample with a maximum Seebeck coefficient
of approximately -210 µV/K between 120 to 140 K. As temperature increased, and intrinsic
carriers began to be populated, the sample became p-type between 250 and 260 K.
Comparing the results of the Seebeck coefficient with those of the electrical resistivity,
shown in Figure 4.3, further evidence is shown for the bipolar conduction effects in this
narrow bandgap material.

Figure 4.3: Electrical Resistivity of Tellurium Doped PtSb2

Starting with the lightest dopant concentration of x = 0.0005 there is a marked reduction in the electrical resistivity at low temperatures compared with that of nominal PtSb2 .
The effect of extrinsic doping is seen by the relatively flat trend in resistivity until such
temperatures are met where intrinsic carriers begin to populate in the material. It was seen
that at approximately 170 K the resistivity of this sample converged with that of undoped
PtSb2 at which point the intrinsic carriers were driving the electrical resistivity to lower

75

values. The electrical resistivity was found to be further reduced in the x = 0.001 and 0.002
samples illustrating the effect of increasing the dopant concentration. These samples again
showed the onset of intrinsic carrier population at approximately 170 K with a change in the
slope of their data that eventually converged with that of undoped PtSb2 . At the highest
concentrations examined in this work, the x = 0.02 and 0.04 samples, a metallic electrical
resistivity was found which was characterized by a flat trend. In these samples it was only
at approximately 350 K where this data converged with that of the undoped PtSb2 .
The power factor of all doped samples showed enhancement at low temperatures. At 80
K undoped PtSb2 has a power factor of 3.5 µW/cm-K2 while that of the x = 0.04 sample is
17.5 µW/cm-K2 . As temperature increased the power factors of the lightly doped samples
were maximized at varying temperatures, but the x = 0.02 and 0.04 samples, which showed
the lowest electrical resistivity, were found to have a minimum between 120 to 130 K and
then to progressively increase. The x = 0.04 sample showed the highest power factor at 350
K of approximately 24 µW/cm-K2 .

4.3.3

Thermal Properties

Examining the response of the thermal conductivity to n-type doping, shown in Figure 4.4,
it was found that there was a general decrease as dopant concentration was increased.

76

Figure 4.4: Thermal Conductivity of Tellurium Doped PtSb2

At low temperatures the reduction was further pronounced due to the increased effect
that ionized impurities have at low temperature compared to the effects of phonon-phonon
scattering at higher temperatures. However, it was found that even with the reduction shown
in these results further minimization techniques should be performed in future work.
Analysis of the electronic and lattice contributions to thermal conductivity showed that
the lattice contribution was still the dominant contribution in this material. At low temperature the electronic contribution was over an order of magnitude less than the lattice
contribution and as temperature increased the lattice contribution was still approximately
five times as large as the electronic contribution at 350 K.

4.3.4

zT

The figure-of-merit of tellurium doped PtSb2 is shown in Figure 4.5.

77

Figure 4.5: zT of Tellurium Doped PtSb2

At low temperatures the figure-of-merit of all doped samples was found to be increased
over that of the undoped PtSb2 . The overall magnitude of this enhancement was small
at low temperature where this work was focused. At varying temperatures in all but the
highest doped samples the figure-of-merit of undoped PtSb2 eventually surpassed those of
the doped samples. The most prominent values in this figure were those of the highest
dopant concentrations, x = 0.02 and 0.04. These samples showed a large enhancement over
that of undoped PtSb2 , most pronounced in the region around room temperature, achieving
a zT of 0.059 at 350 K. Comparing this value to that of commercial Bi2 Te3 puts this into
perspective and shows that an enhancement of approximately 20 times is necessary for room
temperature application. Comparison to that of Bi-Sb alloys at 80 K shows the need for
several orders of magnitude improvement.

78

4.4
4.4.1

Selenium Dopant Results and Discussion
X-Ray Diffraction

The X-ray diffraction patterns obtained for selenium doping are shown in Figure 4.6.

Figure 4.6: X-ray Diffraction of Selenium Doped PtSb2

It was shown that for all selenium doping concentrations no secondary phases were
present. Examination for a change in lattice parameter showed no shift, which was likely
due to the low concentrations of dopant added in this work.

4.4.2

Electronic Properties

An examination of the Seebeck coefficient of selenium doped PtSb2 may be made from
Figure 4.7.

79

Figure 4.7: Seebeck Coefficient of Selenium Doped PtSb2

It was found that as with tellurium doping the n-type selenium dopant generated a
negative Seebeck coefficient for many doping concentrations. Examining the highest dopant
concentrations first the x = 0.01 and 0.02 samples showed a fairly flat trend in Seebeck
coefficient. It is of interest to note the cross over in the sample with the highest magnitude
between the x = 0.01 and 0.02 samples between the temperatures of 190 to 210 K. This was
due to the population of intrinsic carriers creating an overall decrease in the magnitude of
the x = 0.01 sample more than the x = 0.02 sample due to the fewer number of extrinsic
carriers in the x = 0.01 sample. Moving now to the x = 0.001 and 0.002 samples the effect of
a low extrinsic carrier concentration being acted upon by the population of intrinsic carriers
was seen. At low temperatures the Seebeck coefficient of these samples had a relatively large
magnitude that decreased as intrinsic hole carriers were populated which competed with the
extrinsically doped electrons. Finally, examining the x = 0.0005 sample it was found that

80

the maximum in the Seebeck coefficient was likely to occur at lower temperatures than were
examined here.
Comparing these results with those of the tellurium doped samples an interesting effect
may be observed. The x = 0.0005 composition for selenium doping versus the same composition for tellurium doping demonstrated that selenium was likely not doping as many extrinsic
carriers into the PtSb2 electronic structure as tellurium did. Evidence of this came from the
result that the Seebeck coefficient did not have a maximum and was still decreasing at 80 K,
as well as, the lower temperature at which the sample transitioned to p-type when compared
to tellurium at this composition. Since the peak in the Seebeck coefficient for the selenium
sample was at a temperature lower than 80 K its carrier concentration was likely lower than
the tellurium sample. Additionally, the transition to p-type at a lower temperature indicated
that fewer intrinsic carriers were required to dominate over the extrinsic carriers. Thus, it is
likely that fewer extrinsic carriers were doped from selenium versus that of tellurium at the
same composition.
Comparing the element tellurium to selenium it is found that beyond the size difference
of the atoms the selenium atom has a Pauling electronegativity of χ = 2.55, which is much
larger than that of tellurium or antimony. The degree of electronegativity of an element is
a characterization of the degree by which the element attracts electrons to itself. Thus, if
the more electronegative selenium is doped into PtSb2 it is less likely to fully donate the one
additional valence electron to the material system than tellurium.
The electrical resistivity of selenium doped PtSb2 is shown in Figure 4.8.

81

Figure 4.8: Electrical Resistivity of Selenium Doped PtSb2

It was found that the electrical resistivity of the x = 0.0005 sample supported the claim
that an effective decrease in the carrier concentration compared to Te at the same composition was observed. At low temperatures it was seen that the x = 0.0005 sample had a
higher electrical resistivity than undoped PtSb2 , which is different from the tellurium doped
sample with the same composition. This effect may have arisen from a decrease in the mobility or a decrease in the carrier concentration and Hall measurements would be a future
route to probe this result. Examining those samples with a higher dopant concentration the
more expected low temperature decrease in the electrical resistivity was observed. This data
matched well with that found from tellurium doping and showed the same convergence of
electrical resistivity towards that of undoped PtSb2 as temperature increased.

82

4.4.3

Thermal Properties

The thermal conductivity of selenium doped samples is shown in Figure 4.9.

Figure 4.9: Thermal Conductivity of Selenium Doped PtSb2

Relatively little change in the thermal conductivity was found for the low dopant concentrations examined. The outlier in this data set was found for the x = 0.02 sample. Comparison of the thermal conductivity of this sample with that of a similar tellurium dopant
concentration showed very good agreement. Thus, it is not expected that this particular sample showed any larger decrease in the thermal conductivity than would be expected. These
results along with those of tellurium highlight the importance of minimizing the thermal
conductivity in future experimentation.

4.4.4

zT

The figure-of-merit of selenium doped samples is shown in Figure 4.10
83

Figure 4.10: zT of Selenium Doped PtSb2

The lackluster individual transport properties for selenium doping culminated in a reduced figure-of-merit for all dopant concentrations other than over a small temperature range
at low temperature. From an examination of the power factor it is expected that further
doping beyond the x = 0.02 concentration may lead to some enhancement, but the overall
magnitude for these samples was low enough that it is unlikely a significant improvement in
zT would be found before a solubility limit was met for selenium on the antimony site.

4.5
4.5.1

Tin Dopant Results and Discussion
X-Ray Diffraction

The X-ray diffraction patterns obtained for tin doping are shown in Figure 4.11.

84

Figure 4.11: X-ray Diffraction of Tin Doped PtSb2

It was found from this X-ray pattern that a likely solubility limit exists for tin doping on
the antimony site in PtSb2 . Low concentrations of tin dopant showed no secondary phases
present, however, at concentrations at and above x = 0.1, secondary phases of tin were found
to exist demarcated by a star in the figure. There are no Pt-Sn phases which crystallize in
the pyrite structure, thus it was not unexpected that there would be a solubility limit for
tin dopant on the antimony site even with the similarity between the two elements. This
resembled the limited solubility found by the work of Laudise et al. [109] and Bennett et
al. [110] where secondary phases with low concentrations of dopant were found.

4.5.2

Electronic Properties

The Seebeck coefficient of tin doped PtSb2 is shown in Figure 4.12.

85

Figure 4.12: Seebeck Coefficient of Tin Doped PtSb2

These results showed the effect of p-type doping on a nominally p-type material. It
was found for all dopant concentrations examined that a decrease in the Seebeck coefficient
was the result of an increase in the carrier concentration due to p-type doping. As the
concentration of dopant was increased it was found that the Seebeck coefficient decreased in
response. The maximum in the Seebeck coefficient for these samples was shifted to higher
temperatures compared with undoped PtSb2 and is only observable in the lowest doped
sample of x = 0.01. This sample showed a maximum at approximately 250 K. Doping in
the x = 0.01, 0.02, and 0.04 samples was found to decrease the magnitude of the Seebeck
coefficient at low temperatures, but increase it over that of the undoped PtSb2 at higher
temperatures. This was due to the high concentration of extrinsic holes that had been
doped into the system. Thus, when the Seebeck coefficient of the undoped PtSb2 was
decreasing due to the population of intrinsic carriers, the heavily doped x = 0.01 sample was

86

relatively unchanged with intrinsic carrier population and its Seebeck coefficient increased
with temperature.
This conclusion was supported by the electrical resistivity data on these samples shown
in Figure 4.13.

Figure 4.13: Electrical Resistivity of Tin Doped PtSb2

It should be noted that the y-axis for this figure is given in a logarithmic scale to show
the large change in the electrical resistivity demonstrated by these samples. At 80 K the
electrical resistivity was reduced by a factor of 24 in the x = 0.01 sample. All doped samples
showed a large reduction in electrical resistivity and displayed a metallic trend characterized
by a linearly increasing electrical resistivity with increasing temperature. This positive trend
was due to increasing scattering of carriers as temperature increased from phonons. In the
x = 0.01 and 0.02 samples a convergence with undoped PtSb2 was found at temperatures
above room temperature, however for higher dopant concentrations the large extrinsic carrier

87

concentration reduced the electrical resistivity beyond that of undoped PtSb2 . The solubility
limit of Sn in PtSb2 was further exemplified by the similar electrical resistivities found in
the x = 0.1 and 0.15 samples.

4.5.3

Thermal Properties

The thermal conductivity of Sn doped samples is shown in Figure 4.14.

Figure 4.14: Thermal Conductivity of Tin Doped PtSb2

Of note are the low temperature results in the highest dopant concentration samples, x
= 0.1 and 0.15. These samples showed a low and flat trend, while all other samples showed a
decreasing thermal conductivity with temperature. It is thought that the secondary phases
in the highest Sn dopant samples caused additional boundary scattering and is likely the
cause for the decreased thermal conductivity at low temperature.
Examination of these results would at first appear to show that Sn was able to reduce the

88

thermal conductivity superior to that found from tellurium or selenium doping. However,
this was not the case, due to the increased dopant concentration examined in the work with
Sn. In Figure 4.15 the reduction in thermal conductivity found for all antimony site dopants
at 90 K is shown with the abscissa in a logarithmic scale.

Figure 4.15: Thermal Conductivity of Te, Se, In and Sn Dopants at 90 K

This figure shows that, as a general trend, as dopant concentration increased the total
thermal conductivity decreased. Thus, when comparing the overall reduction in thermal
conductivity found for tellurium and selenium doping, for low dopant concentrations, with
those of tin, for higher dopant concentrations, the larger reduction found for tin is more
easily understood. The greater reduction in thermal conductivity found in higher dopant
concentration samples is likely due to increased phonon scattering on impurities.

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4.5.4

zT

The figure-of-merit of Sn doped PtSb2 is shown in Figure 4.16.

Figure 4.16: zT of Tin Doped PtSb2

It was found that for all Sn doped samples that at room temperature and higher the
figure-of-merit was enhanced to values similar to those obtained from tellurium doping. In
the tellurium doped system a dopant concentration of x = 0.02 resulted in the highest zT
at 350 K of approximately 0.059. In the tin doped system the highest zT was found in x =
0.04 again with a value of 0.059 at 350 K. At low temperatures the lightly doped samples
of composition x = 0.01, 0.02, and 0.04 were found to have zT values larger than that
of undoped PtSb2 . It should be noted again though that these values are very low when
compared to the Bi-Sb alloys or commercialized Bi2 Te3 .

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4.6
4.6.1

Germanium Dopant Results and Discussion
X-Ray Diffraction

The X-ray diffraction patterns of germanium doped PtSb2 are shown in Figure 4.17.

Figure 4.17: X-ray Diffraction of Germanium Doped PtSb2

These X-ray diffraction patterns showed the very limited solubility of germanium in
PtSb2 . In the lowest dopant concentration of x = 0.005 a secondary phase was found in the
system and in the x = 0.01 sample a large amount of secondary phase was found. Comparison
to the results of Sn doping in PtSb2 would lead one to expect a solubility limit closer to a
composition of x = 0.1, however, likely due to the greater size difference between germanium
and antimony, a single phased sample was unable to be synthesized.
Due to the lack of single phased samples from germanium doping the transport properties
of this dopant were not examined in this work.

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4.7
4.7.1

Indium Dopant Results and Discussion
X-Ray Diffraction

The X-ray diffraction patterns of indium doped PtSb2 are shown in Figure 4.18.

Figure 4.18: X-ray Diffraction of Indium Doped PtSb2

The diffraction patterns for indium doping were found to coincide with that of the PtSb2
structure. Extending to higher concentrations of dopant suggested a solubility limit for
indium at the composition of x = 0.05, due to secondary phases of PtIn2 at higher concentrations. For compositions of x = 0.05 or less no shift in the lattice parameter was found. It
is likely that indium did not generate secondary phases at the very low concentrations that
germanium did due to the greater size difference between antimony and germanium than
antimony and indium.

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4.7.2

Electronic Properties

The Seebeck coefficient for indium doped samples, shown in Figure 4.19, suggests that indium
dopes onto the antimony site as a p-type dopant.

Figure 4.19: Seebeck Coefficient of Indium Doped PtSb2

It was expected that indium would p-type dope in PtSb2 due to indium containing two
fewer valence electrons than antimony. The Seebeck coefficient behaved as expected from a
p-type dopant with a general decrease in the magnitude and a slight shift in the temperature
in peak Seebeck coefficient values towards higher temperatures.
Comparing indium with tin it was expected that indium would effectively dope two
acceptor states for every atom of indium compared to the single acceptor state which was
expected from tin doping. It can be concluded that this did not occur due to the larger
magnitude in the Seebeck coefficient and the higher electrical resistivity for comparable
doping compositions.

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The electrical resistivity of indium doped PtSb2 is shown in Figure 4.20.

Figure 4.20: Electrical Resistivity of Indium Doped PtSb2

The typical p-type doping behavior was again found in this data with a reduction in
the electrical resistivity of these samples at low temperature. As temperature increased
the electrical resistivity was found to converge to that of undoped PtSb2 due to the onset
of intrinsic carriers. To directly compare compositions of indium with those of tin the x
= 0.01 composition is taken for discussion. At 80 K the indium x = 0.01 sample showed
an electrical resistivity of 2.5 mOhm-cm while the tin x = 0.01 sample showed a value of
0.3 mOhm-cm, nearly an order of magnitude difference. Some of this difference may be
due to increased scattering from the extra mass variance between indium and antimony as
compared with tin and antimony. However, another consideration is that indium only has
three valence electrons and it is being incorporated into a site for a tetrahedrally bonded
element. It is likely that indium, with three valence electrons, was thermodynamically less

94

favorable to dope onto this tetrahedrally bonded site than tin, with four valence electrons,
which ultimately caused a decrease in the effective carriers it doped per atom.

4.7.3

Thermal Properties

The thermal conductivity of indium doped PtSb2 is shown in Figure 4.21.

Figure 4.21: Thermal Conductivity of Indium Doped PtSb2

The expected minor reduction in thermal conductivity was found for these lightly doped
samples. With only a small decrease in the electrical resistivity the electronic contribution
was not found to increase substantially and the impurity scattering introduced by doping
suppressed the thermal conductivity across all temperatures measured.

4.7.4

zT

The resultant figure-of-merit values obtained for indium doped PtSb2 is shown in Figure 4.22.

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Figure 4.22: zT of Indium Doped PtSb2

Due to the lackluster improvement in power factor at high temperature, from a decreased
Seebeck coefficient and slightly reduced electrical resistivity, as well as a minor decrease in the
thermal conductivity, the zT for indium doped samples showed only minimal enhancement
above 200 K.

4.8

Conclusions

A comparison of the results found for tin and tellurium doping on the antimony site is
enlightening of the effects of p- and n-type doping in a narrow bandgap system. Unlike
materials with larger bandgaps the narrow bandgap system is in a state of intrinsic carrier
population at much lower temperatures and thus is often characterized by the effects of bipolar conduction. The electrical conductivity is characterized by the conduction of both holes
and electrons though both terms are weighted by their respective mobilities. The Seebeck
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coefficient, on the other hand, is reduced due to bipolar conduction. When an extrinsic
dopant of opposite type to the nominal material is introduced those extrinsic carriers will
compete with the nominal carriers in establishing the Seebeck coefficient. Examining a system with a large dopant concentration, of opposite type to the nominal system and extrinsic
carrier concentration much greater than the nominal carrier concentration, creates a change
in the sign of the Seebeck coefficient and yields a low magnitude due to the large number of
carriers. As the dopant concentration is decreased, the magnitude of the Seebeck coefficient
is increased and a maximum is found due to the onset of intrinsic carrier population. As
the dopant concentration is further decreased the Seebeck coefficient is expected to increase
significantly, and the maximum will shift to lower temperatures, until such a concentration
is met where there will be a sign change. Subsequent reduction in the dopant concentration
will return the Seebeck coefficient to its nominal state. Refer to Figure 4.23 to examine this
effect.

Figure 4.23: Effects of Dopant Concentration in Tellurium and Tin Doped PtSb2
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This figure shows the Seebeck coefficients at 140 K of varying concentrations of tin
and tellurium doped PtSb2 , with an undoped sample for reference. The compositions of tin
dopant are given as a negative on this figure to connote the loss of electrons with this dopant,
while tellurium composition is positive to connote the gain of electrons. Examination of this
figure displays the above mentioned effects for tellurium dopants. As the concentration of
tellurium was reduced, the magnitude of the Seebeck coefficient increased and at some very
low concentration would result in a maximized Seebeck coefficient for the electron dominant
system. At any concentration less than this, the result would be a return to positive values.
The effects of tin dopant displayed the typical decrease with increasing carrier concentration
that is expected.
From the study of n- and p-type doping on the antimony site it was found that PtSb2
could be made to assume a positive or negative Seebeck coefficient dependent upon the
dopant. It was additionally found that a maximization in the power factor was observed
for both donor and acceptor dopants in the lightly doped samples above room temperature.
In both tellurium and tin doped samples the highest power factors occurred at 350 K with
respective values of 24 µW/cm-K2 and 26 µW/cm-K2 , both with the composition x = 0.04.
At low temperatures, between 80 and 200 K it was found that n-type dopants resulted
in enhancement while p-type dopants generated a decrease. Neither n- or p-type dopants
were found to significantly decrease the thermal conductivity beyond that expected for an
ionized impurity scattering site. It was additionally noted that the magnitude of the thermal
conductivity was prohibitive to the generation of large figure-of-merits.
Ultimately, the figure-of-merits of n- and p-type doping on the antimony site led to enhancement in the tellurium and tin dopants primarily due to their power factor enhancement
above room temperature. Thus, an endeavor to be presented in subsequent chapters is an
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attempt to further increase the figure-of-merit by reducing the thermal conductivity while
additionally tuning the carrier concentration to enhance the power factor.

99

Chapter 5
Platinum Site Doping in PtSb2
5.1

Background and Motivation

From the results of electronic doping on the antimony site it was shown that the thermoelectric properties of PtSb2 could be greatly affected with very low concentrations of dopant.
Doping on the antimony site with elements of varying valence shell electron arrangements
will largely create an alteration in the Sb p-shell. Doping on the Pt site, however, will largely
alter the electronic configuration and the interaction of valence electrons in the Pt d-shell.
From the work of Emtage et al. it was shown that the interaction of the 5d and 6s bands
of platinum are what contribute to the minima of the conduction band being in the <111>
direction as opposed to the minima being in the same k-space as the valence band maxima, the <100> direction [99]. This leads one to presume that alterations in the electronic
structure of these 5d or 6s bands by electronic doping on the platinum site may affect the
minima of the conduction band and may lead to even more dramatic effects than antimony
site doping.
Prior work by Nishikubo et al. showed that the Pt site could successfully be doped with
iridium which acted as a p-type dopant [112]. This generated a large decrease in the electrical
resistivity and a decrease in the Seebeck coefficient. Similar to what was found for antimony
100

site doping, the electronic doping of iridium resulted in an increase in the power factor and
a decrease in the thermal conductivity resulting in an increase in the figure-of-merit.
Beyond testing for possible routes to increase the figure-of-merit by p- or n-type doping
on the platinum site there were two other routes examined here. The first examined was
the effect of iron doping. It is well known that FeSb2 is a narrow bandgap material which
exhibits an extremely large Seebeck coefficient between -45 mV/K and -30 mV/K [117,118].
This Seebeck coefficient is 2 to 3 orders of magnitude higher than the typical thermoelectric
material and has generated a large amount of research and attention. FeSb2 has been
identified as a strongly correlated electron system which is characterized by a hybridization
gap around the Fermi energy [119, 120]. This hybridization gap is due to the interaction of
the conduction band with d- or f-bands and leads to a very large anisotropy of the density of
states on either side of the gap [121, 122]. It is this large anisotropy in the density of states
that leads to the large Seebeck coefficients in these materials [123].
Another route examined in this chapter was that of rare-earth doping with ytterbium.
Several studies have shown the promising electronic properties of ytterbium compounds i.e.
YbAl3 and YbAgCu4 [124–127]. Both of these compounds show large power factors at low
temperature, behavior that is likely due to the intermediate valence state of Yb between the
2+ and the 3+ states. The chemical pressure tuning of the valence state of these compounds
is another route which may be possible with Yb doping, though it is likely that only low
concentrations of Yb would be soluble in the PtSb2 structure which may limit the effects of
this technique. A final possible route with rare-earth doping would be with cerium due to
the similarity of the 4f configuration of Ce with that of Yb. Yb is found on the far end of
the lanthanides with one less electron than a full f-shell and Ce is on the near end with one
electron occupying the f-shell. Additionally, the Ce containing rare-earth compounds show
101

good electronic properties at low temperatures and are likely to have interesting effects as a
dopant [128–130].
Thus, the effects of n- and p-type dopants, as well as iron and ytterbium doping on the
platinum site will be examined in this chapter.

5.2

Experimental Methods

To examine the effects of transition metal doping on the platinum site the elements iron,
silver, and niobium were selected. To examine the effects of rare-earth doping the element ytterbium was selected. Compounds with compositions Pt1−x Dpx Sb2 were synthesized where
Dp = Fe, Ag, Nb, and Yb, with concentrations ranging between 0 < x < 0.1. Samples were
synthesized in the manner outlined in Chapter 2 without any post reaction processing. It
should be noted that iron and ytterbium doped samples utilized the 30 minute dwell time
SPS parameters while silver and niobium dopants utilized the shortened SPS parameters.
One final consideration was made for the ytterbium doped samples. Ytterbium is known
to react with oxygen, thus care was taken with these samples to limit their exposure. This
was especially important during the initial heat treatment during which sample reaction
was taking place. During the heat treatment it was discovered that unless the silica, SiO2 ,
ampoules were coated with a layer of carbon the ytterbium would react with the oxygen in
the sides of the ampoule. Thus, for all ytterbium containing samples a layer of carbon was
coated on the inside of the silica ampoules by deposition from boiled ethanol.

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5.3

Iron Dopant Results and Discussion

Iron has two fewer valence electrons than platinum, and would theoretically be a p-type
dopant. However, that is only part of the reason to explore iron doping, the other being that
FeSb2 is a strongly correlated system that generates a large Seebeck coefficient. Therefore,
to explore the possible generation of this effect in PtSb2 iron doping was performed.

5.3.1

X-Ray Diffraction

X-ray diffraction data on iron doped samples is shown in Figure 5.1

Figure 5.1: X-ray Diffraction of Iron Doped PtSb2

It was found that for all dopant concentrations examined the single phase diffraction pattern for nominal PtSb2 was observed. Due to the size difference between iron and platinum,
when compared to germanium doping on the antimony site, it was surprising that while in
the case of germanium no solubility was found, up to 5 atomic% of Fe can be incorporated
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into the structure without the appearance of a secondary phase. However, FeSb2 crystallizes
in the orthorhombic marcasite structure, which is related to the pyrite structure of PtSb2 ,
and may have allowed a higher solubility limit for this dopant.

5.3.2

Electronic Properties

In Figure 5.2 the data obtained for the Seebeck coefficient of this dopant is shown.

Figure 5.2: Seebeck Coefficient of Iron Doped PtSb2

Iron doping was found to have the conventional effect of p-type doping in this material.
With a slight dopant concentration, as in the x = 0.002 sample, a decrease in the nominal
positive Seebeck coefficient was found. The maximum in the Seebeck coefficient was also
found to be shifted to a higher temperature of approximately 225 K. At this temperature the
Seebeck coefficient of this composition surpassed that of nominal PtSb2 . With increasing
concentration of iron dopant a further decrease in the magnitude in the Seebeck coefficient

104

was found. In the x = 0.005 sample a further shifted maximum was found at approximately
300 K, and in concentrations higher than this the Seebeck coefficient was steadily increasing
even up to 350 K.
Examining the electrical resistivity in Figure 5.3, further evidence of iron acting as a
p-type dopant is shown. It should be noted that due to the large reduction in electrical
resistivity a logarithmic scale on the ordinate was utilized.

Figure 5.3: Electrical Resistivity of Iron Doped PtSb2

Starting again with the lightest iron dopant, the x = 0.002 sample, almost an order of
magnitude reduction in the electrical resistivity was found. A slight increase in resistivity with temperature was found until the onset of intrinsic carriers brought about a slight
decrease starting at approximately 210 K. Further iron doping reduced the resistivity and
generated a metallic trend.
A large enhancement was found in the power factors of these samples, shown in Figure 5.4.

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Figure 5.4: Power Factor of Iron Doped PtSb2

These Fe doped samples all showed enhancement above 210 K, where the Seebeck coefficient of nominally undoped PtSb2 was reduced due to the population of intrinsic carriers,
and for the x = 0.01, 0.05, and 0.1 samples was found to still be increasing at 350 K. At
low temperatures it was only the low dopant composition samples where the power factor
was not suppressed relative to undoped PtSb2 , yet even for these samples the values were
not greatly enhanced until above 210 K. The largest power factor obtained for the doped
samples was approximately 25 µW/cm-K2 from 280 to 320 K.
It is curious comparing this power factor with that of tin and tellurium dopants on the
antimony site that all three samples showed maximum power factors of approximately 25
µW/cm-K2 above room temperature. This indicates that there is relatively little change in
the electronic band structure, other than p-type doping effects, from the dopant of iron in this
system, where it was theorized that the 5d and 6s bands largely constitute the conduction

106

band minima. Since an element with a different d-shell valence electron configuration from
that of platinum was doped here it was expected that more than the typical p-type doping
effects would be found. It is possible that since iron was acting as a p-type dopant in this
system this may have shifted the Fermi energy down to lower energies, away from the altered
conduction band, which may have minimized any effects which otherwise would have been
noted.

5.3.3

Thermal Properties

The thermal conductivity of iron doped PtSb2 is shown in Figure 5.5.

Figure 5.5: Thermal Conductivity of Iron Doped PtSb2

As with other p-type dopants the effects on the thermal conductivity were minimal and
resulted from ionized impurity scattering. The sample with the lowest total thermal conductivity was found to be the x = 0.01 sample due to the increased electronic contribution

107

from the further doped x = 0.05 and 0.1 samples.

5.3.4

zT

The resultant zT of iron doped samples is shown in Figure 5.6.

Figure 5.6: zT of Iron Doped PtSb2

Very similar to the figure-of-merit found for tin and tellurium doped samples the iron
doped samples showed large enhancement above room temperature. The maximum zT was
obtained in the x = 0.005 sample at 350 K with a value of approximately 0.057, though
the x = 0.01 and 0.05 samples were found to be within systemic error of this value. At
low temperatures a slight enhancement of the figure-of-merit was found for doped samples.
These results demonstrated that, as with doping on the antimony site, a large reduction in
the thermal conductivity was needed for substantial zT values to be obtained.

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5.4
5.4.1

Ytterbium Dopant Results and Discussion
X-Ray Diffraction

The X-ray diffraction patterns for ytterbium doped PtSb2 are shown in Figure 5.7.

Figure 5.7: X-ray Diffration of Ytterbium Doped PtSb2

Examination of these patterns shows no extraneous peaks corresponding to a secondary
phase and no shift in the lattice parameter. It should be noted that the composition of x
= 0.1 may be a solubility limit for ytterbium in the PtSb2 structure due to the occurrence
of secondary phases for any larger composition. In trials with higher compositions an x
= 0.15 sample yielded secondary phases of antimony and a x = 0.2 sample yielded phases
of platinum ytterbium, Pt3 Yb, and platinum antimony, PtSb. The finding of secondary
phases in samples with high concentrations of ytterbium dopant matched what was found
by Bennett et al. [110].

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5.4.2

Electronic Properties

A comparison of the effect of varying concentrations of ytterbium dopant on the Seebeck
coefficient may be found in Figure 5.8.

Figure 5.8: Seebeck Coefficient of Ytterbium Doped PtSb2

First turning our attention to the lightest doped sample, the x = 0.01 composition, a
significant decrease in the Seebeck coefficient and a slight shift in the maximum was found
when compared to undoped PtSb2 . Moving to the next highest concentration, the x = 0.02
sample, an increase in the magnitude was found at low temperatures compared to the x
= 0.01 composition, which at approximately 170 K met that of the x = 0.01 sample and
followed the same trend as temperature increased. The x = 0.05 sample further increased
the magnitude of the Seebeck coefficient across all temperatures measured, and showed the
same range of temperatures where it was maximized. The final composition measured, the
x = 0.1 sample, showed the largest Seebeck coefficient of the doped samples with a slight

110

shift in peak to lower temperatures of approximately 10 to 20 K, compared to the other
compositions. As dopant concentration increased, the temperature at which the Seebeck
coefficient of these samples surpassed undoped PtSb2 decreased. At high temperatures all
samples were found to converge to approximately 80 to 90 µV/K at 350 K.
From these results it was clear that for this dopant there was not simply an n- or p-type
doping process occurring here. As was found from the results of Fe, Sn, and Te doping a
p-type dopant will decrease the magnitude of the Seebeck coefficient, and shift the peak
to higher temperatures as dopant concentration is increased. An n-type dopant will at
high enough concentration change the sign of the material to negative, and with decreasing
concentration lead to an increase in the magnitude, and a shift in the maximum to lower
temperatures. However, with ytterbium doping the magnitude of the Seebeck coefficient
initially dropped to a low value for low concentrations and subsequently increasing that
concentration led to an increase in the magnitude.
Observing now the effects of ytterbium doping on the electrical resistivity in Figure 5.9
further evidence of this unusual transport behavior is shown.

111

Figure 5.9: Electrical Resistivity of Ytterbium Doped PtSb2

It was found that in the lowest dopant concentration sample, x = 0.01, between 80
and 110 K the electrical resistivity was higher than undoped PtSb2 but quickly dipped
below due to its steeply decreasing slope. At temperatures where intrinsic carriers began to
be populated, the slope decreased and converged with the electrical resistivity of undoped
PtSb2 at approximately room temperature. All other dopant concentrations were found to
have significantly decreased electrical resistivity at low temperature. All of these dopant
compositions were found to display very similar electrical resistivities of approximately 1
mOhm-cm, which decreased slightly as intrinsic carriers were populated and converged with
undoped PtSb2 , again at approximately room temperature.
If an element is doped into a material and it is not isoelectronic to the atom it is replacing, then a change in the electrical resistivity between differing dopant concentrations
is expected. Thus, it was unusual that increasing concentrations of ytterbium caused no

112

significant change in the electrical resistivity between the x = 0.02, 0.05, and 0.1 samples.
For this to happen there was either no change in the mobility of carriers and no change in the
carrier concentration, or the changes in the mobility and the carrier concentration perfectly
compensated each other.
Comparing the results of the Seebeck coefficient with the results from electrical resistivity,
we see clear changes in the Seebeck coefficient in spite of the constant resistivity. The
effects observed here may have their origin in the chemical-pressure driven intermediate
valence behavior of Yb, which is known to produce large Seebeck coefficient in the cryogenic
temperature range [84]. Further work to determine the cause of this effect would entail the
use of Hall experimentation and magnetic measurements.
Turning to the power factor for these doped samples a large increase was found at low
temperature. This is shown in Figure 5.10.

Figure 5.10: Power Factor of Ytterbium Doped PtSb2

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The maximized power factor was found in the x = 0.1 sample to have a value of approximately 42 µW/cm-K2 between 180 to 200 K. This is the highest power factor for PtSb2 found
in this work and double the values found for other dopants in this temperature region. This
power factor was even comparable to that of commercial Bi2 Te3 in this temperature region.
With the large enhancement found over undoped PtSb2 , and the increasing power factor
with increasing Yb concentration, higher dopant concentrations were attempted. However,
as outlined in the X-ray Diffraction section, higher concentration samples were unable to be
synthesized without secondary phases.

5.4.3

Thermal Properties

The thermal conductivity of ytterbium doped PtSb2 is shown in Figure 5.11.

Figure 5.11: Thermal Conductivity of Ytterbium Doped PtSb2

For all samples a small increase was found in the thermal conductivity and was most

114

prominent at low temperatures in the x = 0.01 and 0.1 samples. The electronic contribution
to thermal conductivity was found to be increased relative to undoped PtSb2 , but was still
more than an order of magnitude lower than the lattice contribution at room temperature
and two orders of magnitude lower at 80 K. Ultimately, the lattice contribution to thermal
conductivity was the major contributor to the increase in the total thermal conductivity in
these samples.

5.4.4

zT

Turning our attention to the figure-of-merit of ytterbium doped samples in Figure 5.12.

Figure 5.12: zT of Ytterbium Doped PtSb2

The increased power factor of these samples ultimately resulted in an enhancement in the
figure-of-merit most prominent below room temperature. The highest dopant concentration,
x = 0.1, resulted in a zT = 0.03 at 250 K. Comparison of this result with those found previ-

115

ously for n- or p-type doping showed that this value was not uncommon at this temperature.
For instance, tellurium doped samples showed zT = 0.027 and tin doped samples showed
zT = 0.038 at 250 K. However, this enhancement was centered around the temperature region targeted by this research, and if a decrease could be made in the thermal conductivity,
without affecting the electrical properties, the zT would be further enhanced.

5.5

Silver and Niobium Doping Results and Discussion

The transition metals, silver and niobium, were selected to examine the effects of doping on
the platinum site with a large difference in number of valence electrons, nascent valencies,
and electronegativities relative to platinum.

5.5.1

X-Ray Diffraction of Silver Doped Samples

Examining first the X-ray diffraction pattern for silver as a dopant it was found that with
even the lowest concentrations of silver dopant there existed secondary phases.

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Figure 5.13: X-Ray Diffraction of Silver Doped PtSb2

It was found that in the x = 0.005 sample a platinum antimony phase existed, demarcated
with a cross, and in the higher dopant concentration samples an antimony phase was found,
demarcated with an asterisk. It was unexpected that a low concentration of silver dopant
would have this effect due to its proximity in the periodic table to platinum, however, a
similar effect was seen with germanium doping on the Sb site.

5.5.2

Electronic Properties of Silver Dopant

The Seebeck coefficient measured for silver doping displayed the additional secondary phase
in this material in Figure 5.14.

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Figure 5.14: Seebeck Coefficient of Silver Doped PtSb2

The lowest concentration, which displayed the secondary phase of platinum antimony,
was found to have a greatly decreased Seebeck coefficient reflecting the incorporation of the
metallic platinum antimony compound. Samples with larger dopant compositions showed
a secondary phase of antimony which is reflected in the Seebeck coefficient by a shift in
the temperature of the maximum to lower temperatures, as well as a slight increase in the
magnitude. From the x = 0.01 sample to the x = 0.04 sample a decrease was found in the
magnitude of the Seebeck coefficient, but no shift was found for the temperature at which
the peak occurred.
Due to the secondary phases and impurities in this system, the expected n-type dopant
behavior was not observed. A similar change in sign of the Seebeck coefficient as that found
from tellurium doping on the antimony site was expected in these silver doped samples.
However, a separate effect, ionized impurity scattering, was likely observed. It has been

118

shown that due to the large dependence of the Seebeck coefficient on the mobility of carriers
that manipulation of the mobility, by increased scattering from ionized impurities, may result
in an enhancement in the thermoelectric properties [131, 132]. This effect requires a balance
between the enhancement in the Seebeck coefficient and the reduction in the carrier mobility,
which Ioffe first theorized as a possible application [133]. It is theorized that those carriers
which are lower in energy are preferentially scattered by the ionization of these impurity
states and that this leads to an increase in the Seebeck coefficient [134].
An examination of the electrical resistivity displays the increased scattering from impurities in Figure 5.15.

Figure 5.15: Electrical Resistivity of Silver Doped PtSb2

It was found for all silver doped samples that the electrical resistivity was increased at
low temperatures until the onset of intrinsic carriers brought about a reduction. This data
demonstrated the lack of electronic doping from small concentrations of silver dopant. If

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silver had been n-type doping on the platinum site, a similar electrical resistivity as that
found from tellurium doping on the antimony site would be expected. In combination with
the results for Seebeck coefficient, these results pointed toward a change in the mobility of
carriers with the lack of a large change in carrier concentration, however, Hall experiments
to determine carrier concentration and mobility would be required for a full examination.
An examination of the power factor displayed an overall reduction from the incorporation
of secondary phases in this structure. The slight enhancement in the Seebeck coefficient at
low temperatures did not offset the increase in the electrical resistivity.

5.5.3

Thermal Properties and zT of Silver Dopant

The thermal conductivity of these samples was found to be largely unaffected by the incorporation of impurities. All samples showed thermal conductivity values that were within the
error of the system and thus the additional scattering from secondary phases was offset by
their contribution to the thermal conductivity.
Ultimately, the figure-of-merit for these silver doped samples was found to be reduced
for all compositions measured. Without a reduction in the thermal conductivity the lack of
enhancement in power factor only produced a diminished zT for these samples.

5.5.4

X-Ray Diffraction of Niobium Doped Samples

Turning now to the doping of niobium on the platinum site the X-ray diffraction pattern of
this dopant is given in Figure 5.16.

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Figure 5.16: X-Ray Diffraction of Niobium Doped PtSb2

Unlike the silver doped samples, the niobium dopant only showed an additional secondary
phase of antimony in the x = 0.05 sample. However, as will be shown in the Seebeck
coefficient and electrical resistivity data, it was very likely that a secondary phase was present
in all samples which was simply too low of a concentration to generate a signal in X-ray
diffraction. It was further found that for even greater concentrations of niobium, x > 0.05,
that additional secondary phases were found on X-ray diffraction of niobium diantimonide,
NbSb2 .
The Seebeck coefficient data for niobium doped samples is shown in Figure 5.17.

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Figure 5.17: Seebeck Coefficient of Niobium Doped PtSb2

A similar increase and shift of the peak in Seebeck coefficient to lower temperatures
found from silver doping was also found here in the lowest concentrations. This pointed
again towards the effects of ionized impurity scattering as a likely cause of this increase in
the Seebeck coefficient at these low concentrations. As the concentration was increased the
Seebeck coefficient was found to be diminished over all temperatures in the x = 0.02 sample.
In the x = 0.05 sample where a secondary phase was readily observed in X-ray diffraction
the Seebeck coefficient was found to be diminished due to the presence of antimony.
This scattering was again reflected in the electrical resistivity for these samples, shown
in Figure 5.18.

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Figure 5.18: Electrical Resistivity of Niobium Doped PtSb2

The large degree of scattering that was found for silver doping was again found for this
data in the x = 0.005 sample. This gave further evidence of a secondary phase in this sample
that was too low to be found in X-ray diffraction. As niobium concentration increased it
was found that the electrical resistivity decreased due to the increasing concentration of
secondary phases of antimony and niobium antimonide.
As with the silver dopant, the power factor of niobium doped samples was found to
be reduced over all temperatures, except in the x = 0.01 sample, yet this was observed
only at low temperatures. The increase for this sample was due to the enhanced Seebeck
coefficient magnitude at low temperatures in combination with the relatively unchanged
electrical resistivity of the sample compared to undoped PtSb2 .

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5.5.5

Thermal Properties and zT

Similar to the silver doped samples these samples showed no significant change in thermal
conductivity due to doping. Again, it is likely that the increased scattering due to secondary
phase boundary scattering was offset by the thermal conductivity of those secondary phases.
The slight increase in power factor coupled with an unchanged thermal conductivity led
to a slight enhancement of the figure-of-merit at low temperature in the x = 0.01 sample.
All other samples for all temperatures measured showed a diminished zT due to a decreased
power factor and an unchanged thermal conductivity.

5.6

Conclusions

In effect, there were three different studies performed on doping the platinum site of PtSb2 .
The first examined here concerned iron doping in an attempt to incorporate the strongly
correlated electron effects which are present in FeSb2 to increase the Seebeck coefficient and
thereby the figure-of-merit. It was found that iron acted as a p-type dopant on the platinum
site and did indeed yield increased figure-of-merit, yet not by any incorporation of the effects
found in FeSb2 . This however, gave further evidence of the enhancement potential of n- or
p-type doping of transition metals.
The successful p-type doping of iridium and iron on the platinum site instigated an investigation for an n-type dopant, thus, silver, with an additional valence electron, was examined.
Due to the generation of secondary phases the effect of this dopant was unable to be examined, however, the effects of ionized impurity scattering were observed for these samples. It
was further found that niobium dopant produced a similar ionized impurity scattering effect,
which resulted in a slight enhancement in the power factor at low temperatures.
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The incorporation of rare earth dopants in PtSb2 is an exciting avenue due to the curious
results that may be obtained. While the figure-of-merit was not particularly enhanced at
low temperature relative to simple electronic n- or p-type doping, the trend of the maximum
in zT, which was a shift to lower temperature and an increase in magnitude, was exactly
what was desired for this project. The thermal conductivity of this sample was, as with all of
these samples, the largest hindrance to large figure-of-merit, but the power factor obtained
by ytterbium doping was significantly enhanced at low temperature.
Further work on doping the platinum site of PtSb2 could examine the effects of gold
doping in an attempt to n-type dope on the platinum site. Another avenue would be a study
into the effects of cerium doping on the platinum site and a comparison of those results with
those found from ytterbium doping. Ultimately, future work on PtSb2 should incorporate
methods of reducing the thermal conductivity while also tuning the carrier concentration
through doping on the platinum site, as well as, the antimony site.

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Chapter 6
Methods to Reduce Thermal
Conductivity

6.1

Background and Motivation

From the prior work on electronic doping it was determined that a large reduction in the
thermal conductivity of PtSb2 was required for significant enhancement of the figure-of-merit.
Figure 6.1 displays the large reduction in thermal conductivity that would be required to
generate significant zT values in PtSb2 . This figure shows the result of several hypothetical
reductions in the total thermal conductivity on the zT assuming no reduction in the power
factor from this effect.

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Figure 6.1: Thermal Conductivity Reduction required for high zT

This figure demonstrates that for un-doped PtSb2 the total thermal conductivity would
need to be reduced by two orders of magnitude to approach values of commercial Bi2 Te3 .
In electronically doped samples, with optimized carrier concentration, a reduction in the
thermal conductivity by a factor of 20 to 30 would still be required to result in near unity
zT, again, independent of any modifications to the electronic properties.
Several methods of reducing thermal conductivity have been examined in prior research.
One method is the generation of a solid solution of two elements or compounds, e.g. Si-Ge or
(PbTe)x (PbSe)1−x . In this method elements with a difference in mass are alloyed together
to cause point defect scattering. A large body of theoretical and experimental work has
been performed to display the effectiveness of this procedure [135–138]. In the generation
of a complete solid solution it is important for the two end compounds to have similar
crystal structures otherwise it is likely that the materials will merely mix and generate a two

127

phased material. The more differences there are between the two end compounds the more
scattering there will be, however, these large differences will make the full solid solution less
thermodynamically favorable. For example, if a trigonal system with large lattice parameter
is alloyed with a cubic system with small lattice parameter, then a large amount of strain
will be present in the lattice even with low concentrations which may stop the formation of
a full solid solution.
For PtSb2 in the pyrite structure this means that other materials also in the pyrite
structure should be targeted for their potential use because these materials in theory will
more dependably form a solid solution. It has been shown that the materials: PdAs2 and
PdSb2 , and PtPn2 , where Pn = S, As, Sb, and Bi also crystallize in the pyrite structure
[90, 91]. Thus, all of these materials would be possible candidate materials for use in the
generation of a solid solution. The material PdSb2 was selected due to its similar crystal
structure and lattice parameter to PtSb2 , as well as to avoid using the toxic element arsenic
and phosphorous with a low boiling point.
Another often used means of reducing the thermal conductivity is the reduction of particle
size by ball milling or melt spinning [139–142]. Boundary scattering acts as a method to
reduce the lattice contribution to thermal conductivity which is most prominent at low
temperatures but will cause a reduction over all temperatures. The reason why particle size
reduction affects low temperature thermal conduction to such a large degree is due to the
length scale of the wavelength of phonons at low temperature. With low thermal energy,
phonons will have a low frequency, and thereby a long wavelength. This long wavelength is
scattered preferentially by those effects which are on the same length scale. Of these effects,
grain boundaries will scatter these large length-scale, long wavelength phonons greater than
some of the smaller length-scale effects like point defect scattering.
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Thus, the effects of ball milling on the low temperature properties of PtSb2 will be
examined here. Further, electronic doping will be performed on ball milled PtSb2 to enhance
the electronic properties of these thermally tailored materials. In the course of examining
ball milled PtSb2 a series of experiments were performed to examine the effects of nonstoichiometry on the low temperature thermoelectric properties. It had previously been
shown that at room temperature the dominant carrier type of PtSb2 could be changed by the
effects of non-stoichiometry, thus this was examined as a possible route of enhancement [100].

6.2
6.2.1

Experimental Methods
Solid Solutions with Palladium and Bismuth

To examine the reduction in thermal conductivity due to the generation of a solid solution
in PtSb2 the compound PdSb2 was selected. Samples with the composition Pt1−x Pdx Sb2
were synthesized where x = 0.01, 0.05, 0.1, 0.25, 0.5, and 1.0. All samples were synthesized
as outlined in the experimental section except for a variation in the SPS parameters which
had to be altered due to the lower melting point of PdSb2 of 678 ◦ C [143]. With increasing
concentrations of palladium substitution the SPS temperature was reduced incrementally to
account for the reduction in melting point. The x = 0.01 and 0.05 samples were sintered
at 850 ◦ C, x = 0.1 sample at 800 ◦ C, x = 0.25 sample at 700 ◦ C and the x = 0.5 and 1.0
samples at 450 ◦ C. All samples were sintered for 5 minutes with a pressure of 40 MPa.

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6.2.2

Ball Milling and Electronic Doping in PtSb2

To examine the reduction in particle size of PtSb2 from ball milling, samples were synthesized
using the typical furnace heat treatment, but before densification were ball milled in a
stainless steel ball mill jar back filled with argon in increments of 10 minutes. The sample
was examined with X-ray diffraction after each 10 minute increment of milling. A zirconium
oxide jar and media was used for comparison with the results of the stainless steel jar and
media for one examination. After 100 total minutes of milling, samples were densified by SPS
at 500◦ C for three minutes at a pressure of 40 MPa. A reduction in sintering temperature
was necessitated by the lowered melting point of ball milled samples due to the smaller
particle size of the powder PtSb2 .
To examine the effects of electronic doping on ball milled PtSb2 the two elements Sn
and Te were again utilized for comparison with un-ball milled samples. The compositions
examined were PtSb2−x Dpx where Dp is either Sn or Te with compositions ranging between
0<x

0.15 for tin dopant and 0 < x

0.04 for tellurium dopant. All samples required

modified SPS parameters due to the reduction in the melting point of ball milled samples.
These parameters entailed SPS at 500 ◦ C for three minutes at a pressure of 40 MPa.

6.2.3

Non-Stoichiometric PtSb2

To determine the effects of non-stoichiometry in PtSb2 , samples were synthesized with compositions of PtSb2+x and -0.05

x

0.1. These samples were synthesized using the furnace

heat treatment and SPS parameters outlined in the experimental section.

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6.3
6.3.1

Solid Solution Results and Discussion
X-Ray Diffraction

It is known that PdSb2 crystallizes in the same pyrite structure as PtSb2 with a very similar
lattice parameter of 0.645 nm [85]. It was thus expected that a full solid solution between
these two end compounds would form. Contrary to this expectation it was found that
at concentrations of Pd of x = 0.1 and higher, the two end compounds failed to form a
single phase and that the two phases of PtSb2 and PdSb2 were present in all of these high
concentration samples. At concentrations lower than x = 0.1 no secondary phases were
found, thus it is only the high concentration samples which are shown in Figure 6.2.

Figure 6.2: X-Ray Diffraction Patterns of Palladium Substituted PtSb2

Observed in Figure 6.2 are the high 2θ peaks for the x = 0.1, 0.25, and 0.5 compositions
where the PdSb2 phase is demarcated with an asterisk.

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6.3.2

Electronic Properties

When examining the electronic properties of palladium substituted PtSb2 it is important
to remember that in the x = 0.1 and higher concentrations there is evidence of two phases
being present in these samples. It is expected that a full solid solution between these two
compounds would generate differing electronic properties than those found from the alloying
of these two compounds. Shown in Figure 6.3 are the Seebeck coefficients of these samples.

Figure 6.3: Seebeck Coefficient of Palladium Substituted PtSb2

As the concentration of palladium substitution was increased a general decrease in the
Seebeck coefficient was found. The temperature at which the peak in Seebeck coefficient
occurred was generally found to slightly increase as concentration increased. In the x = 0.5
and 1.0 samples the Seebeck coefficient was found to be greatly decreased and exemplary
of what would be expected of a metal. The Seebeck coefficient of PdSb2 is found to range
between one to three µV/K and is the expected very low value often found for metals. This is

132

consistent with band structure calculations, which predict metallic behavior for PdSb2 [144].
Unlike p-type doped samples the Seebeck coefficient was not found to have any composition
with a positive trend above room temperature. These samples appear to show an incremental
decrease in the Seebeck coefficient with increased Pd substitution which would be expected
from an increasing concentration of a metallic phase.
The electrical resistivity of palladium substituted samples is shown in Figure 6.4, with a
logarithmic ordinate axis due to the large reduction displayed by the samples.

Figure 6.4: Electrical Resistivity of Palladium Substituted PtSb2

It was found that PdSb2 displayed a very low electrical resistivity, nearly three orders
of magnitude less than PtSb2 , with a positive trend for all temperatures measured. It was
found that as Pd composition was increased the electrical resistivity decreased due to the
inclusion of the metallic secondary phase. Examining the trend of the electrical resistivity
it was found that for low Pd concentration samples a decrease was found as temperature

133

increased due to the population of intrinsic carriers in the PtSb2 phase. In the x = 0.25 and
0.5 samples this was replaced with an increasing with temperature electrical resistivity due
to carrier scattering from the lattice.
Examining the carrier concentration and carrier mobility results from Hall experiments
in Figure 6.5 it was found that a large change in the carrier concentration was what created
the reduction in electrical resistivity for these samples.

Figure 6.5: Carrier Concentration of Palladium Substituted PtSb2

A large difference in carrier concentration between PtSb2 and PdSb2 from Hall experiments of nearly four orders of magnitude was found. As the concentration of the secondary
phase, PdSb2 , was increased the carrier concentration was also found to increase. The onset
of intrinsic carriers in the PtSb2 phase was found for all Pd substituted samples further
demonstrating that a solid solution was not generated. The mobility of these samples was
shown to incrementally decrease with increasing palladium substitution which is what would

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be expected from the addition of a secondary phase.

6.3.3

Thermal Properties

The thermal conductivity of the Pt1−x Pdx Sb2 samples studied here is shown in Figure 6.6.

Figure 6.6: Thermal Conductivity of Palladium Substituted PtSb2

It was found that as palladium composition was increased, a general decrease in the
total thermal conductivity was found for all temperatures. The sample with the lowest
thermal conductivity was the x = 0.5 composition, which would be the expected result from
alloying. The total thermal conductivity of PdSb2 was shown to be relatively flat across all
temperatures measured with a small decrease as temperature increased. It was found for the
PdSb2 sample that the electronic contribution was on the order of the lattice contribution
due to the extremely low electrical resistivity of the sample. The effects of Pd substitution
were most prominent at low temperature, shown in Figure 6.7.

135

Figure 6.7: Thermal Conductivity of Palladium Substituted PtSb2 at 90 K

A large reduction in total thermal conductivity with low concentrations of Pd substitution was observed, which extended to further reduction in the higher Pd substitution
compositions. The expected lowest thermal conductivity was found for the x = 0.5 sample
due to the large amount of scattering from the two phases present.

6.3.4

zT

The figure-of-merit of palladium substituted samples is shown in Figure 6.8.

136

Figure 6.8: zT of Palladium Substituted PtSb2

It was found that all samples with composition x

0.25 showed enhancement in the

figure-of-merit. It was only in the x = 0.01 and 0.05 samples that the power factor was
enhanced beyond nominal PtSb2 , but due to the large reduction in the thermal conductivity
of the x = 0.1 and 0.25 samples an enhanced zT was still generated. Samples with concentrations x > 0.25 had such a reduced power factor that the lowered thermal conductivity did
not offset this value and led to very low zT values.
While the enhancement to the figure-of-merit was only slight for these samples, these
results demonstrated the interplay of thermal conductivity reduction and power factor manipulation found from alloying PtSb2 with a metal. It is expected that if a full solid solution
had been generated in this experimentation the resultant electronic properties, as well as
thermal properties would be significantly different. Most notably the scattering mechanisms
would be expected to change from a boundary scattering dominated system to point defect

137

dominated.

6.4
6.4.1

Ball Milling Results and Discussion
X-Ray Diffraction

To examine the effects of ball milling on the thermal conductivity of PtSb2 a study was
performed to examine the optimal time of milling to result in a minimized particle size. A
sample of PtSb2 was milled in increments of 10 minutes and shown in Figure 6.9 are the
X-ray patterns for 0, 50, and 100 minutes of ball milling time.

Figure 6.9: X-ray Diffraction of Ball Milled PtSb2

A significant amount of peak broadening was observed in ball milled samples. As can be
seen from the above figure, additional broadening was found when comparing 100 minutes of
ball milling to only 50 minutes. As was discussed in the experimental section, the broadening

138

of a peak may be related to both particle size reduction, as well as increased strain in the
material. To separate these effects and to determine how long samples should be milled to
reach a minimum particle size the Williamson-Hall equation was employed. The results of
this analysis are given in Figure 6.10 where particle size is given on the primary axis and
full width at half maximum, FWHM, values are given on the secondary axis.

Figure 6.10: Particle Size of Ball Milled PtSb2

Williamson-Hall analysis demonstrated that a large decrease in the particle size was
found from only 10 minutes of ball milling. After 10 minutes subsequent milling generated
further reduction until approximately 80 minutes where additional milling did not result in
further reduction in particle size. Also in this figure is the FWHM of the (311) plane. It was
found that the FWHM of this plane showed a slower broadening of its peak, however the
broadening was again found to be maximized at approximately 80 minutes. The FWHM,
again, contains information not only on particle size but also internal strain and is useful

139

as a secondary proof that milling should be stopped at 100 minutes. It should additionally
be noted that to substantiate any quantitative particle size further experimentation should
be performed, such as scanning electron microscopy analysis. However, for this study, the
most important aspect was not the exact size of particles but the required time of ball
milling to reach a minimized particle size. Thus, to account for error, it was assumed that to
generate a minimized particle size in this material a ball milling time of 100 minutes should
be performed.

6.4.2

Thermal Properties

As the objective of this study was to examine the reduction in thermal conductivity obtained
from ball milling the thermal properties will be presented first. Shown in Figure 6.11 is the
thermal conductivity of ball milled PtSb2 .

Figure 6.11: Thermal Conductivity of Ball Milled PtSb2

140

A very large reduction in thermal conductivity was found from ball milling PtSb2 , most
pronounced at low temperatures, which matched what was expected from theory. The
reduction found at 80 K was slightly larger than an order of magnitude changing from approximately 63 W/m-K to 5.7 W/m-K. Unlike that found for nominal PtSb2 a reduction
from acoustic phonon scattering was not found as temperature increased, which is exemplary of Matthieson’s rule in this sample. Matthieson’s rule states that the largest scattering
mechanism will dominate over other present mechanisms, which is what was found in these
results. A large increase in boundary scattering from particle size reduction led to boundary
scattering being the dominant mechanism for these temperatures. Thus, the typical reduction in thermal conductivity as temperature increased due to acoustic phonon scattering was
not observed.
Comparing the lattice contribution to thermal conductivity versus the electronic contribution showed that the lattice contribution was the majority component to the total thermal
conductivity for all temperatures examined, as well as, the largest contributor to the reduction observed here.

6.4.3

Electronic Properties

Focusing now on the electronic properties measured from ball milled samples an interesting
result was observed in the Seebeck coefficient, shown in Figure 6.12.

141

Figure 6.12: Seebeck Coefficient of Ball Milled PtSb2

The most prominent effect of ball milling was a change in the sign of the Seebeck coefficient from p-type to n-type. This meant that the dominant carrier in the system had changed
from holes in the nominally undoped system to electrons in the ball milled system. A similar
change in type had been observed from n-type doping on the antimony site of PtSb2 , however
such a change resulting from ball milling was very unexpected. It was additionally noted
that the magnitude of the Seebeck coefficient had decreased, but the temperature where
the peak was found remained the same. The existence of a peak in the Seebeck coefficient
indicated the onset of intrinsic carriers in the system and that while a shift in the dominant
carrier had been generated, the material remained a narrow bandgap semiconductor.
There were a few possible causes for the change in dominant carrier in this system. One
possibility is that the stainless steel ball mill jar may have incidentally doped Fe into the
sample from the high energy milling. Another possibility was that boundary scattering on

142

that length scale was preferential to holes and thereby dropped their contribution to the Seebeck coefficient. The last possibility was that antimony sublimated during SPS densification
which had altered the electronic properties. This sublimation may have occurred due to the
reduced melting point of ball milled PtSb2 .
Looking beyond the Seebeck coefficient, the electrical resistivity is shown in Figure 6.13.

Figure 6.13: Electrical Resistivity of Ball Milled PtSb2

A comparison of the electrical resistivity of ball milled PtSb2 to nominal demonstrated
an unexpected decrease in the electrical resistivity at low temperature. A reduced particle
size will reduce the mean free path of charge carriers, as well as phonons, and increase the
electrical resistivity by decreasing the charge carrier mobility. Thus, the decrease in electrical
resistivity which was found at low temperatures was very odd. This result gave an indication
of a change in the band structure of this sample. The trend of the electrical resistivity was
found to decrease at a slower rate than un-ball milled PtSb2 as temperature increased and

143

resulted in the expected increase in electrical resistivity.
From the results on the electronic properties it was expected that there was a significant
change in the electronic band structure of the material. To examine how this affected the
carrier concentration and mobility Hall experiments were performed on these samples. Shown
in Figure 6.14 is the carrier concentration data obtained.

Figure 6.14: Carrier Concentration of Ball Milled PtSb2

It was found that due to ball milling a large change in carrier concentration of approximately an order of magnitude had taken place. Both samples displayed the onset of intrinsic
carriers at approximately 150 K and increased as temperature increased. The mobility was
found to have been reduced dramatically from ball milling to approximately 30 cm2 /V-s at
low temperature and remained relatively flat as temperature increased. Given these results
the most likely cause of the change in dominant carrier type in the Seebeck coefficient and
decreased electrical resistivity at low temperature was either iron doping from the milling or

144

the loss of antimony during densification.
From prior investigations in doping iron on the platinum site it was unlikely that the
cause of the irregularities in the electronic properties were due to iron doping. In that study
it was found that iron acted as a p-type dopant and would not produce a negative Seebeck
coefficient when doped on the platinum site. However, to rule out the possibility of iron
acting as a dopant a similar milling study was performed with a zirconium oxide ball mill
jar and media, since it did not contain any iron, to determine if a similar effect to that found
for the stainless steel jar and media would be observed.
The Seebeck coefficient of zirconium oxide milled PtSb2 is shown in Figure 6.15.

Figure 6.15: Seebeck Coefficient of Zirconium Oxide Ball Milled PtSb2

It was found that the Seebeck coefficient of ball milled PtSb2 using a zirconium oxide jar
and media still produced a sign change. A slight shift in the temperature of the peak was
observed but these results, coupled with the previous work on iron doping, motivated the

145

conclusion that the sign change was not due to iron deposition during ball milling.
Thus the most likely reason for the sign change was a change in the stoichiometry of
PtSb2 due to the loss of antimony during densification. Thus, a study of the effects of
non-stoichiometry on PtSb2 was performed.

6.5

Non-Stoichiometric PtSb2 Results and Discussion

Prior experimental work demonstrated that antimony sublimation would occur during the
synthesis of PtSb2 while elemental precursors were undergoing a heat treatment to create a
melt [102–106]. Samples synthesized in this manner showed large variations in their transport
properties from this sublimation and unless specific methods were used to reduce antimony
sublimation it would occur to some extent. Thus, when it was found that the melting point
of PtSb2 powder had been decreased from particle size reduction, the sublimation of antimony was identified as a possibility. To examine the degree to which antimony sublimation
may have occurred in ball milled samples, and further to probe the possible thermoelectric
enhancement of PtSb2 by non-stoichiometry this study was performed.

6.5.1

X-Ray Diffraction

The X-ray diffraction pattern of non-stoichiometric PtSb2 is shown in Figure 6.16.

146

Figure 6.16: X-Ray Diffraction of Non-Stoichiometric PtSb2

It was found that with antimony excess samples a secondary phase of antimony was
observed demarcated with an asterisk and in antimony deficient samples a secondary phase
of platinum antimony, PtSb, was observed demarcated with a cross. These results match
well with what would be expected from their respective non-stoichiometry. Upon addition
of a silicon standard no shift in the lattice parameter was observed which is in agreement
with the formation of a secondary phase.

6.5.2

Electronic Properties

The Seebeck coefficient of non-stoichiometric PtSb2 is shown in Figure 6.17.

147

Figure 6.17: Seebeck Coefficient of Non-Stoichiometric PtSb2

Examining the electronic properties of these non-stoichiometric samples good agreement
was found with prior literature on non-stoichiometric PtSb2 at room temperature [100]. In
this prior work it was found that antimony deficiency led to the generation of a negative
Seebeck coefficient at room temperature, which matched well with what was found here in
the x = -0.05 sample. With a light deficiency of antimony a negative Seebeck coefficient was
found at low temperatures as displayed in the x = -0.02 sample. As temperature increased,
and intrinsic carriers were populated, this sample returned to positive values due to the large
hole concentration generated by intrinsic carrier population. The x = -0.05 sample showed
the results that were being probed for in this study. It was found that across all temperatures
measured a negative Seebeck coefficient was generated from this composition of antimony
deficiency in PtSb2 . This result matched well with the negative Seebeck coefficient generated
from ball milled samples and was taken as strong evidence that ball milling caused a loss of

148

antimony during SPS from sublimation. Examining the antimony excess samples, a decrease
in the magnitude of the Seebeck coefficient was found. The decrease in magnitude was
observed to intensify as the concentration of antimony excess increased. A slight reduction
in the temperature of the peak with a similar decreasing value as temperature increases was
observed in these antimony excess samples.
Examining the electrical resistivity of non-stoichiometric PtSb2 in Figure 6.18 a large
amount of variability is found at low temperature.

Figure 6.18: Electrical Resistivity of Non-Stoichiometric PtSb2

At 80 K the electrical resistivity spans an order of magnitude between the most deficient
sample and that with the largest antimony excess. Starting with the x = 0.1 sample a
large increase in the electrical resistivity was found. As intrinsic carriers were populated
in the PtSb2 phase the electrical resistivity was found to decrease rapidly and match the
value of stoichiometric PtSb2 at approximately 200 K. As antimony content was decreased

149

the electrical resistivity at 80 K was also found to decrease relative to the x = 0.1 sample.
In samples with antimony deficiency a reduction in the electrical resistivity was found at
80 K relative to stoichiometric PtSb2 . As antimony content was reduced to the x = -0.05
sample the trend of the electrical resistivity became flat across all temperatures measured.
All samples were found to approach the electrical resistivity of stoichiometric PtSb2 at
temperatures above room temperature.
The results from studying the electrical resistivity suggest that, as with silver and niobium doping on the platinum site, ionized impurity scattering occurred in the antimony
excess samples. The shift to lower temperatures of the peak in the Seebeck coefficient,
the reduction in magnitude with increased antimony excess, and the large increase in the
electrical resistivity at low temperatures all point towards ionized impurity scattering in
these samples. In the antimony deficient samples a more complex mechanism is at play,
but appears to be similar to n-type doping. Comparing the Seebeck coefficient of the x =
-0.02 samples with that found for low dopant concentrations of selenium a similar result
was found. The negative values at low temperatures which crossed the x-axis and became
p-type as temperature increased were very similar to that found for selenium doping. The
electrical resistivity of this sample also displayed similarity to n-type doping. Finally, the
x = -0.05 sample displayed a Seebeck coefficient and electrical resistivity similar to lightly
n-type doped samples.

6.5.3

Thermal Properties

The thermal conductivity of non-stoichiometric PtSb2 is shown in Figure 6.19.

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Figure 6.19: Thermal Conductivity of Non-Stoichiometric PtSb2

From this data it was found that antimony deficient samples showed the same thermal
conductivity as nominal PtSb2 within the error of the system, however antimony excess
samples showed a reduction across all temperatures measured. Examining just the lattice
contribution to thermal conductivity it was found that these samples were dominated by
the lattice contribution with the electronic contribution at least an order of magnitude less
for all temperatures. Thus, it is likely that the antimony impurities in the excess antimony
samples lead to greater phonon scattering than did the PtSb phase in antimony deficient
samples.

6.5.4

zT

Figure 6.20 shows the resultant figure-of-merit for these non-stoichiometric samples.

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Figure 6.20: zT of Non-Stoichiometric PtSb2

Drawing comparison to prior experimentation again, the antimony excess samples showed
no improvement in the figure-of-merit just as the silver and niobium dopant samples showed
no improvement. The antimony deficient samples when compared to n-type dopants showed
the effects of light doping, which only had modest improvements in the zT. These antimony
deficient samples however showed no improvement due to their low power factor.
Overall, the results of non-stoichiometric PtSb2 demonstrated that manipulation of stoichiometry in this material is not a viable means of increasing the figure-of-merit. However,
these findings served to solidify the interpretation of results on ball milled PtSb2 and furthered the understanding of this narrow bandgap material.

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6.6

Hole Doping Ball Milled PtSb2

To capitalize upon the large reduction in thermal conductivity produced by ball milling
PtSb2 , a study on tuning the carrier concentration was performed.

6.6.1

X-Ray Diffraction

The X-ray diffraction patterns obtained for Sn doped PtSb2 are shown in Figure 6.21. A
large number of samples were examined in this work and to better visually demonstrate the
results of X-ray diffraction only a subset of the patterns obtained for these samples is shown.

Figure 6.21: X-Ray Diffraction of Sn Doped, Ball Milled PtSb2

In this figure the large amount of broadening due to ball milling is shown to be preserved
for all Sn dopant concentrations. Examination with an internal silicon standard showed
no shift in the lattice from the incorporation of Sn dopant. In Sn doped samples without
any ball milling a secondary phase was found in compositions of x
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0.1, however, in these

ball milled samples only the single phase of PtSb2 is observed for all concentrations examined here. It is possible that due to the broadening of peaks the amplitude of secondary
phases were either suppressed to such a low value that they were insignificant, or that they
were indistinguishable from the broadened peaks of PtSb2 . As will be shown, the electrical
properties of these doped samples followed the expected trend from doping, thus it is likely
the strained lattice of ball milled samples was able to incorporate a higher solubility of Sn
dopant.

6.6.2

Electronic Properties

The Seebeck coefficient of Sn doped, ball milled PtSb2 is shown in Figure 6.22 for a subset of
samples measured. A subset is shown because if all of the data points for all of the samples
were shown it would occlude the effects which are occurring in these samples. Thus, a subset
is shown to better visualize the data. The samples included in the subset were selected to
best demonstrate the trends from Sn doping.

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Figure 6.22: Seebeck Coefficient of Sn Doped, Ball Milled PtSb2

The Seebeck coefficient of nominal PtSb2 was shown in this figure as a reference point
for the values obtained. The lowest concentration of tin doping in these samples, x = 0.001,
showed a decrease in the magnitude of the Seebeck coefficient when compared with undoped,
ball milled PtSb2 . The Seebeck coefficient of this sample approached zero and crossed over
to become p-type at approximately 320 K. This sample demonstrated the effect of bipolar
conduction with the doped holes competing with the electrons in this ball milled sample.
As the tin dopant concentration was increased the sample became p-type due to the large
number of doped holes in the system. This was seen from the transition in Seebeck coefficient
between the x = 0.001 and 0.005 samples. A maximized Seebeck coefficient for all doped
samples was found for the x = 0.04 dopant concentration. From the x = 0.01 to 0.04 sample
an increase in Seebeck coefficient was found, but decreased from x = 0.04 to 0.1 and further
still in 0.15. This effect was exemplary of the typically expected behavior of the Seebeck

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coefficient in a bipolar conductor as the doped holes met a maximum value while competing
with intrinsic electrons which upon further doping led to a reduction.
Comparing the Seebeck coefficients of all samples at 300 K, in Figure 6.23, this maximized
Seebeck coefficient in the x = 0.04 sample is better observed.

Figure 6.23: Seebeck Coefficient of Sn Doped, Ball Milled PtSb2 at 300 K

In this figure the Seebeck coefficient of un-ball milled PtSb2 was not included, but the
undoped, ball milled sample is shown on the y-axis at a composition of x = 0. This figure
clearly shows the effects of Sn doping on the Seebeck coefficient of ball milled PtSb2 . For
the x = 0.001 sample the Seebeck coefficient was still negative because the mobility weighted
Seebeck coefficient contribution from holes was not yet enough to surpass that of intrinsic
electrons. As the dopant concentration was increased the Seebeck coefficient became positive
and rose as the doped holes began to dominate in the system over electrons. At the x = 0.04
composition a maximum in the Seebeck coefficient for these samples was met and further

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doping resulted in a reduction from an increased hole doped carrier concentration.
Examining the electrical resistivity in Figure 6.24 a similar effect was found.

Figure 6.24: Electrical Resistivity of Sn Doped, Ball Milled PtSb2

In this figure the electrical resistivity of nominal PtSb2 is not shown to better visually
demonstrate the data, however, reference should be made to Figure 6.13 for any comparison.
An interesting result was found for low dopant concentration samples. The electrical resistivity was found to be increased in the x = 0.001, 0.005, and 0.01 samples. It was known
from the Seebeck coefficient data that a bipolar conduction effect was occurring in all doped
samples, but this result on its own would not increase the electrical resistivity. It is likely
that these samples demonstrated an increased carrier scattering rate due to dopant impurities. As the Sn dopant concentration was increased it is likely that the mobility was not
affected as greatly by an incremental increase in impurity as was the increase in the carrier
concentration. However, to be certain of this interpretation, future work with these samples

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would entail Hall experimentation to probe the mobility and carrier concentration.
Beginning with the x = 0.04 sample the electrical resistivity was found to be decreased
from doping and in the x = 0.1 and 0.15 samples a similar electrical resistivity was found
compared with previously doped samples. It was likely not a coincidence that the maximum
in the Seebeck coefficient was found in the same sample that first demonstrated a reduced
electrical resistivity. The reduced electrical resistivity was due to the large number of doped
carriers in this system from these higher Sn dopant concentrations. Comparing the electrical
resistivity values found for these dopant concentrations, the carrier scattering effects of ball
milling were prominent. For the x = 0.04 composition the electrical resistivity of the un-ball
milled sample was ρ = 0.19 mΩ-cm, while the ball milled sample was ρ = 2.54 mΩ-cm, an
order of magnitude increase. This result was reflected in the other doped samples as well.

6.6.3

Thermal Properties

The thermal conductivity of Sn doped, ball milled PtSb2 is shown in Figure 6.25.

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Figure 6.25: Thermal Conductivity of Sn Doped, Ball Milled PtSb2

The large reduction in thermal conductivity from ball milling was found to be further
reduced from point defect phonon scattering in these doped samples. To demonstrate the
large reduction found from ball milling and doping in these samples the thermal conductivity
of nominally undoped PtSb2 is also shown. It was found for these Sn doped ball milled
samples that the electronic contribution to the thermal conductivity began to comprise
a significant portion of the total thermal conductivity. Thus, the reduction in the total
thermal conductivity found from point defect scattering in these samples does not follow an
incremental trend of further reduction from increasing Sn dopant concentration. In the x
= 0.1 and 0.15 samples the electronic contributions are 12% and 15% of the total thermal
conductivity at 80 K and reach 38% and 40% at 350 K, respectively. Comparison of the
thermal conductivity of nominally undoped PtSb2 with that of undoped ball milled PtSb2
and the x = 0.04 sample at 80 K a reduction of approximately a factor of 30 was found for

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nominal PtSb2 and a factor of approximately 3 from ball milled PtSb2 . Recall from Figure 6.1
that this large reduction in thermal conductivity was what would be required to result in
a competitive figure-of-merit for PtSb2 , dependent upon the effects of this minimization on
the electronic properties.

6.6.4

zT

The figure-of-merit of Sn doped, ball milled PtSb2 is shown in Figure 6.26.

Figure 6.26: zT of Sn Doped, Ball Milled PtSb2

The figure-of-merit of Sn doped ball milled samples showed a large enhancement over
both nominally undoped PtSb2 and undoped ball milled PtSb2 . The largest zT obtained
in these samples was for the composition of x = 0.04 with a value of approximately 0.07 at
350 K. The x = 0.01 sample was elevated as well when compared with nominally undoped
PtSb2 , as were samples not shown here of x = 0.06 and 0.08. At low temperatures, shown

160

in the inset, the x = 0.1 sample demonstrated the highest figure-of-merit, but not by a large
magnitude compared with ball milled or nominally undoped PtSb2 .
As is the case with all thermoelectric materials, a reduction in the thermal conductivity
by the introduction or exacerbation of a scattering mechanism will impact both the thermal,
as well as, the electronic transport properties. Yet in the typical thermoelectric material
this effect will often still result in an enhancement of the figure-of-merit. This effect was
indeed found in this work as well, however the dramatic change in the sign of the Seebeck
coefficient was atypical. While it is unrealistic to expect the power factor of a material to
be unaffected from significant particle size reduction, if that had been found for these Sn
doped, ball milled samples a very significant increase in the figure-of-merit would have been
found. A comparison of the values found for the power factor of ball milled and un-ball
milled PtSb2 is shown in Figure 6.27.

Figure 6.27: Power Factor of Sn Doped, Ball Milled and Un-Ball Milled PtSb2

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The dramatic decrease in the power factor due to ball milling was found to be the major
limiting factor to large figure-of-merit. Upon analysis it was found that the large decrease in
the power factor did not solely come from the change in the sign of the Seebeck coefficient
in ball milled samples. In fact, since the Seebeck coefficient of ball milled samples returned
to the p-type values found in Sn doped un-ball milled samples, the Seebeck coefficient of the
x = 0.04 samples for ball milled and un-ball milled samples varied by less than 10% over
all temperatures measured. The ball milled sample was even found to have an enhanced
Seebeck coefficient compared to the un-ball milled sample for several temperature ranges.
Thus, it was largely the reduction in the electrical resistivity found from additional boundary
scattering which led to the large decrease in the power factor.
If the power factor was not reduced due to ball milling and could instead be further
enhanced from doping, then substantial zT values could be obtained at low temperature. A
purely hypothetical situation is presented where the best properties obtained from separate
PtSb2 projects are combined to show a projected zT possible. This zT with the combined
best properties from measured samples is compared with Bi0.88 Sb0.12 [77] and Bi2−x Sbx Te3
[79] in Figure 6.28.

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Figure 6.28: Hypothetical zT of the Combined Best Properties of PtSb2 Compared with
the measured zT of Bi0.88 Sb0.12 by Smith and Wolfe and Bi2−x Sbx Te3 by Chung et al.

In this figure the thermal conductivity of Sn doped, ball milled PtSb2 is combined with
the power factor of Yb doped PtSb2 to yield a large zT which surpasses both Bi0.88 Sb0.12 and
Bi2−x Sbx Te3 in the temperature range from 180 to 200 K. It is unlikely that the zT of PtSb2
could be increased to these values due to the interplay between ball milling and the power
factor, however, this demonstrates the substantial enhancements, albeit separately, that were
generated in the course of this work. This also motivates further work into minimizing the
antimony loss during densification of ball milled samples, as well as an examination of Yb
doping on ball milled PtSb2 .

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6.7

Electron Doping Ball Milled PtSb2

To examine both p- and n-type doping on ball milled PtSb2 a series of samples were synthesized using Te as a dopant due to the success of tellurium doping on un-ball milled samples.

6.7.1

X-Ray Diffraction

The X-ray diffraction patterns of Te doped, ball milled PtSb2 are shown in Figure 6.29.

Figure 6.29: X-Ray Diffraction of Te Doped, Ball Milled PtSb2

As was found from Sn doping in ball milled PtSb2 , no solubility limit was found for any
tellurium composition examined here. The addition of a silicon standard revealed that the
apparent shift in peaks in Figure 6.29 was due to a 2θ offset from differing powder heights
during X-ray diffraction and did not represent a change in the lattice parameter of these
doped samples.

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6.7.2

Electronic Properties

Examining the effect of tellurium dopant on the Seebeck coefficient of these samples, the
results are shown in Figure 6.30.

Figure 6.30: Seebeck Coefficient of Te Doped, Ball Milled PtSb2

In this figure the results for nominally undoped PtSb2 have been removed since the comparison being made here is between the ball milled samples only and no scale comparison is
necessary to the nominal p-type PtSb2 . It was observed that all doped samples displayed the
expected trend for same type doping, n-type doping in an n-type material. The expectation
was that n-type doping would further increase the concentration of electrons which would
lead to a reduction in the magnitude of the Seebeck coefficient. This effect was indeed found
for all samples which showed that as the concentration of Te dopant increased the magnitude
incrementally decreased. All samples remained n-type over the entire temperature range examined and a peak was observed in the x = 0.005 and 0.01 samples, which had been shifted

165

to higher temperatures. The existence of a peak in these samples, as with the un-doped
sample, indicated the onset of intrinsic carrier population. The x = 0.02, 0.04, and 0.06
samples were found to have continuously increasing values with increasing temperature for
the examined region. It is unknown whether these samples would cross back over the x-axis
to become p-type at some higher temperature due to the competition of holes with these
doped electrons.
The effects of tellurium doping on the electrical resistivity of ball milled PtSb2 may be
seen in Figure 6.31.

Figure 6.31: Electrical Resistivity of Te Doped, Ball Milled PtSb2

The expected results of n-type doping in an n-type material were found again in these
samples. As the concentration of tellurium was increased the electrical resistivity was noted
to decrease. In the x = 0.005 and 0.01 samples the electrical resistivity was found to be
slightly higher than the undoped sample at approximately room temperature due to the

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population of a large concentration of intrinsic carriers in the undoped sample at this temperature. The x = 0.02, 0.04, and 0.06 samples showed very low and flat trends in the
electrical resistivity and were not higher than the undoped values for any temperature.
A comparison of the electrical resistivity of these samples with those found for Sn doped,
ball milled PtSb2 highlighted a curiosity in the Sn doped samples. Comparing the composition of x = 0.005 for both Sn and Te doping it was found that the electrical resistivity
for the Sn sample at 80 K was ρ = 1.06 Ω-m and for the Te sample was ρ = 0.36 Ω-m.
Both samples had the same composition of dopant and would theoretically have very similar
degrees of scattering. Additionally, both dopants should be doping a similar concentration
of charge carriers into the system for the same composition. The only difference is in the
doping type and if reference is made to the electrical resistivities of Te and Sn doped un-ball
milled samples a similar disparity in the degree of reduction is found. In the case of un-ball
milled samples the Sn dopant generated the largest reduction in the electrical resistivity,
while the tellurium dopant, (of opposite type to the parent material), generated a relatively
mild reduction. In the ball milled samples, where the parent material became n-type, the
Sn dopant was found to generate the mild reduction but only in more heavily doped samples, while the same type dopant, Te, generated the relatively much larger reduction. To
more fully probe this effect Hall experimentation should be performed to compare the carrier
concentrations and mobilities of these samples.

6.7.3

Thermal Properties

The large reduction in thermal conductivity found from ball milling was again further reduced
by doping and is shown in Figure 6.32.

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Figure 6.32: Thermal Conductivity of Te Doped, Ball Milled PtSb2

As was found with Sn doping, a large reduction in the thermal conductivity was found
in Te doped, ball milled PtSb2 . For all doped samples further reduction in the thermal conductivity was obtained beyond that of ball milling by doping. Similar values for all dopant
compositions were obtained and it was found that as with the Sn dopant the electronic contribution to the thermal conductivity was a significant portion. From the electrical resistivity
data it was known that the electronic contribution increased with increasing concentration
of dopant and in the x = 0.06 sample the electronic contribution was approximately 45% at
350 K.
Comparing the total thermal conductivity of both the Sn and Te ball milled samples very
similar values were obtained, which displayed the expected similarity in amount of phonon
scattering from dopant inclusion. Upon first inspection of this figure, and the figure for the
thermal conductivity for Sn doped, ball milled samples, it would appear as though doping

168

contributed a very large reduction in the thermal conductivity, which was not found in the
un-ball milled, doped samples. This conclusion is faulty though due to the logarithmic y-axis
giving the illusion of significantly more reduction from doping in these ball milled samples. A
comparison of the total thermal conductivity of doped, ball milled samples versus those that
were only doped shows that the ratio of the reduction was much larger in those which had
been ball milled, but the actual value of that reduction was very similar, which correlates
with what would be expected.

6.7.4

zT

The figure-of-merit displayed by these samples is shown in Figure 6.33.

Figure 6.33: zT of Te Doped, Ball Milled PtSb2

It was found that the figure-of-merit of tellurium doped, ball milled PtSb2 was enhanced
over nominal PtSb2 over all temperatures measured. The low temperature results shown in

169

the insert continued to display low values, but demonstrated that all dopant compositions
enhanced the figure-of-merit. As temperature increased the x = 0.02 sample showed the
highest figure-of-merit of approximately 0.074 at 350 K. Of similar values were the x = 0.04
and 0.06 samples both with values of approximately 0.06 at 350 K. As with all samples
discussed, the absolute magnitude of the figure-of-merit in PtSb2 , doped or otherwise manipulated, remains low but significant enhancement was made from tellurium doping, ball
milled samples.

6.8

Conclusions

From previous research on PtSb2 it was known that significant reduction in the thermal
conductivity would be necessary to significantly enhance the thermoelectric performance of
this material. Thus, the work in this chapter strove to explore methods of reducing the large
thermal conductivity of PtSb2 , to explain some of the interesting results occurring from
those minimization attempts, and to reap the benefits of a reduced thermal conductivity by
tuning the power factor from p- and n-type doping.
The first method of thermal conductivity reduction examined in this chapter concerned
the generation of a solid solution between PtSb2 and PdSb2 . This method of thermal conductivity reduction is well known and between two materials with the same crystal structure
would normally be fully soluble in each other over the entire composition range. It was
found though that with only a 10% substitution of Pd that secondary phases were created.
The generation of secondary phases created a different scattering mechanism which was a
more long range boundary scattering mechanism than the point defect scattering expected
from a solid solution. The results from this investigation demonstrated this reduction in

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the thermal conductivity most pronounced at low temperatures, yet still substantial around
room temperature. The Seebeck coefficient of these samples was found to be diminished due
to the inclusion of a secondary phase which was characterized as a metal. The power factor
was enhanced in two samples which coupled with a reduced thermal conductivity yielded an
enhanced figure-of-merit, but these results were nevertheless lackluster in their magnitude.
Future work with this method would examine the creation of a single phased solid solution
with PdSb2 and α-PtBi2 .
The other well known method of thermal conductivity reduction examined in this work
concerned ball milling as a means of reducing particle size. It was found for PtSb2 that
the particle size was minimized after a total milling time of 100 minutes in a stainless steel
ball mill jar and media. Concurrent with these results it was additionally found that this
milling treatment resulted in the generation of a negative Seebeck coefficient. This result
was unexpected from ball milling and was the impetus for a series of follow up investigations,
which culminated in an examination of the effects of non-stoichiometry in PtSb2 .
In the narrow bandgap material, PtSb2 , deviations from stoichiometry were thought to
have generated a wide dispersion in the transport properties measured in prior literature.
Thus, to explore the cause of the change in the Seebeck coefficient of ball milled samples antimony excess and deficient samples were explored. It was found that antimony excess acted
as an ionized impurity scattering site which decreased the overall thermoelectric properties
of the material. From antimony deficiency it was found that the negative Seebeck coefficient
found in ball milled samples was also found for these samples. This led to the conclusion
that antimony sublimation during densification of ball milled samples is what created the
negative Seebeck coefficient found in these samples. Overall, the thermoelectric potential of
PtSb2 was decreased when non-stoichiometry was introduced in the material, and would not
171

be a future route of enhancement.
To profit from the large reduction in the thermal conductivity found from ball milling
two series of doping investigations were performed using Te and Sn as n- and p-type dopants
respectively. Focusing first upon the thermal properties both dopant types produced further
reduction in the thermal conductivity beyond that which was generated from ball milling.
Upon analysis this further reduction was found to be similar to that found from doping in
un-ball milled samples, and was of similar magnitude between the two dopants. A few useful
comparisons may be made of the two types of dopants concerning the electronic properties
found. A comparison of the effects of the dopants on the Seebeck coefficient at 170 K is
explanatory of the general results from these dopants. This can be seen in Figure 6.34.

Figure 6.34: Comparison of the Seebeck Coefficients of Sn and Te Doped, Ball Milled
PtSb2 at 170 K

This figure hearkens back to that shown in Figure 4.23 and highlights the effect that the
same two dopants, Sn and Te, have on a now n-type parent material. As was done before,
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the compositions of tin dopant are given as a negative to connote the loss of electrons, while
tellurium composition is positive to connote the gain of electrons. The expected trend is
found for tellurium dopant where as the concentration is increased of n-type dopant in an ntype material the Seebeck coefficient decreased due to the increasing carrier concentration.
In the Sn dopant it would be expected that for compositions less than the the x = 0.04
sample the Seebeck coefficient would continue to increase until a sudden change to negative
values was observed for some very low composition. It was instead found that the magnitude
decreased in these low dopant concentration samples until a gradual change to negative values
were observed. This is thought to have occurred due to the competition between the now
intrinsic electron carriers and the extrinsically doped holes.
Also of interest from these results is the increase in the electrical resistivity found for
Sn doped, ball milled samples. This increase in the electrical resistivity relative to undoped
ball milled samples was found in the same low dopant concentration samples which did not
display the expected Seebeck coefficient. In systems with bipolar conduction the addition
of extrinsic carriers of opposite type to the dominant carrier in the parent material should
not create an increase in the electrical resistivity because they are a non-competing, additive
effect. Potential causes for this increase in the electrical resistivity are a decreased mobility
from additional scattering, which was not found for tellurium dopants of similar composition,
or a change in the electronic band structure which generated a decreased carrier concentration. The only way to determine the answer to this question is through the use of Hall
experimentation, which would be a direction of future research in this project.
Ultimately, the figure-of-merit of these doped, ball milled samples showed enhancement
over the parent material and similarly doped PtSb2 . This further enhancement was largely
due to the dramatic reduction in the thermal conductivity of these samples. As was found
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for un-ball milled samples, the highest zT obtained in the Sn and Te samples were very
similar in magnitude and composition. A zT of 0.07 was obtained in the x = 0.04 sample
with Sn doping and a zT of 0.074 was obtained in the x = 0.02 sample with Te doping.

174

Chapter 7
Conclusions and Future Directions

7.1

Conclusions

This work has concerned the thermoelectric enhancement of the narrow bandgap material
PtSb2 for use at low temperatures. The electronic properties of PtSb2 made it a very
promising candidate for the low temperature range due to the large Seebeck coefficient it
displays, as well as the low electrical resistivity found up to room temperature. The bandgap
of PtSb2 is approximately 0.1 eV which offered several avenues of manipulation to probe
enhancement mechanisms which are not typically available in middle and high temperature
thermoelectric materials. The greatest impediment to the expeditious use of PtSb2 at low
temperature is the large thermal conductivity the material exhibits. Thus, in addition to
some of the other work offered here on electronic doping are two routes of reducing this large
thermal conductivity.
An often used technique to enhance the figure-of-merit of a thermoelectric material is
tuning the carrier concentration to maximize the power factor. It is known from the Pisarenko relationship that the power factor of a material is maximized at a specific carrier
concentration where the trade off between the Seebeck coefficient and the electrical resistivity are balanced for maximization. In the typical thermoelectric material to meet this carrier
175

concentration extrinsic carriers will need to be doped into the system. This is typically done
by p-type doping in a p-type material, or n-type doping in an n-type material, to avoid
any decrease in the Seebeck coefficient which would occur from doping a material with the
opposite type of the nominal compound. This is not the case with PtSb2 though due to the
narrow bandgap of the material. When PtSb2 is opposite type doped, the Seebeck coefficient
will change to the opposite type given sufficient dopant concentrations. This effect was seen
in the work performed on antimony site doping in both the ball milled and un-ball milled
studies.
Doping on the antimony site in un-ball milled PtSb2 was performed with n- and p-type
dopants to great effect for Sn and Te dopants, but was lackluster or incapable for In, Se,
and Ge dopants. A large enhancement was found for Sn and Te doping with a factor of
six increase in the above room temperature figure-of-merit as well as significant increases
found in the power factor for these dopants. However, due to the low figure-of-merit for the
nominal material this large enhancement was stymied by its overall low magnitude.
Examining indium as a dopant in comparison to Sn demonstrated a large variability in
the electronic properties that were observed for these two p-type dopants. It was expected
that for every In atom doped an extra hole would be doped in comparison with Sn doping
and that this effect would be indicated by a reduced electrical resistivity for a lower In
composition when compared to a similar Sn composition. This was not found though and
for a given composition the electrical resistivity was higher. A possible explanation for this
effect is the inability of accommodating In in the tetrahedral coordination of the Sb site in
PtSb2 . Thus, in future work this effect should be taken into account when selecting doping
elements.
In the work with Se and Ge doping on the antimony site it was found that Se acted as a
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typical n-type dopant, however, compared to Te, Se did not display as large a reduction in the
electrical resistivity and also yielded a slightly different behavior in the Seebeck coefficient
for similar compositions. This was likely due to a difference in mobility generated from
these two dopants since selenium has a larger mass and size difference from antimony than
tellurium does. For germanium it was found that even for the lowest dopant concentrations
no solubility was observed for this dopant. This was an unexpected result due to the relatively
high solubility of tin on the antimony site, as well as the availability of tetrahedral bonding
in germanium as compared to indium. It is likely that the size difference of germanium
compared to tin created this very limited solubility.
Examining the effects of platinum site doping a few interesting effects were found. Iron
doping on the platinum site resulted in a p-type doping effect and generated a large enhancement in the figure-of-merit that was very similar to that created for the Sn and Te
doped samples. Unlike the results found for germanium, no solubility limit was found for
the iron doped samples examined here. The large mass difference and change in valence shell
electron configuration did not impede the incorporation of iron onto the platinum site for
these compositions.
Turning to the dopants of niobium and silver the low solubility of these dopants was
observed in the x-ray patterns for silver at low compositions and for niobium in the x = 0.05
sample only. The results in the Seebeck coefficient however demonstrated an ionized impurity
scattering effect which ultimately led to these two dopants decreasing the figure-of-merit.
Ytterbium dopant was explored due to the possible f-shell electron interactions it may
have with the PtSb2 electronic structure. It was found that the power factor of ytterbium
doped samples peaked between approximately 180 to 200 K with a very large value close to 42
µW/cm-K2 . This power factor was nearly twice as large as the power factors generated from
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Sn or Te doping and was also centered around the temperature region of focus in this work.
The figure-of-merit found for this dopant displayed superior properties most pronounced at
250 K of zT = 0.03. In comparison to the Sn and Te antimony site doping this figure-ofmerit was not found to be any larger than these dopants at this temperature. It is likely
that additional ytterbium dopant concentration would result in further enhancement in this
temperature regime and begs for additional work.
It was known that the thermal conductivity of PtSb2 was a major impediment to a
large figure-of-merit at low temperature in this material, thus two experimental studies were
performed on lowering this thermal conductivity. The first to be examined was the effect
of a solid solution of PdSb2 with PtSb2 . PdSb2 was expected to have full solid solubility
in the parent material PtSb2 because they both crystallize in the same structure and have
very similar lattice parameters. It was found though that at approximately 10% substituted
palladium a secondary phase of PdSb2 was observed. This secondary phase would cause a
different scattering mechanism than the intended point defect scattering from a solid solution.
Nevertheless, a decrease in the thermal conductivity was observed for these samples and a
slight increase in the power factor for the 5% and 1% substituted samples. This created an
overall increase in the figure-of-merit for all samples up to a 25% substitution.
Exploring the effects of ball milling demonstrated the large reduction in thermal conductivity available from reducing the particle size of PtSb2 . It was found from using incremental ball milling times that a minimized particle size was met after 100 total minutes of
ball milling and that the melting point of the material was drastically reduced due to the
reduction in particle size. Unexpectedly, the Seebeck coefficient of these ball milled samples
changed sign over the temperatures examined here. After investigations into the effects of
non-stoichiometry in PtSb2 it was determined that the change to n-type from ball milling
178

was likely due to the loss of antimony during densification. Non-stoichiometry in PtSb2 was
additionally shown to reduce the figure-of-merit for all temperatures measured and would
not likely be an avenue of future examination for enhancement due to the secondary phases
found for low deficiency and excess.
The final examinations performed in this work entailed a return to electronic doping with
Sn and Te on the antimony site of ball milled samples. This study would make use of the
reduced thermal conductivity from ball milling and attempt to tune the carrier concentration
of the now n-type material. It was found that using both Sn and Te the power factor was
reduced in ball milled samples except for the highest performing concentrations, but due to
the large reduction in the thermal conductivity the figure-of-merit was enhanced for several
compositions. The highest zT found for Sn doped samples was in the x = 0.04 composition
with zT = 0.07 and the highest for Te was in the x = 0.02 composition with zT = 0.074.
It is a remarkable feature of PtSb2 that this material may be made p-type or n-type just
by the element with which it is doped, and further, that very similar zTs are found for both
types of dopants when maximized. When optimizing the carrier concentration for a typical
medium or high temperature thermoelectric material the dopant type chosen will typically
be of the same type as the parent material so as to not negatively impact the Seebeck
coefficient from the addition of bipolar conductors. While this does indeed happen in PtSb2
the impact is negated due to the low temperature at which intrinsic carriers populate in
the system because of its narrow bandgap. As was mentioned previously it is a significant
benefit for thermoelectric materials to be capable of being doped n- or p-type for their use
in a thermoelectric module. These materials should have a very similar thermal expansion
coefficient which will help to reduce the stresses experienced by the materials and decrease
the likelihood of fractures and failures.
179

Table 7.1 gives a comparison of all of the electronic dopants examined in this work and
the thermoelectric properties they exhibit at the temperature of maximum figure-of-merit.
Table 7.1: Comparison of the Thermoelectric Properties of Electronically Doped PtSb2 at
the Temperature of Maximum zT

It is beneficial to make a brief overview of dopants used in this work and their resultant
thermoelectric properties to highlight a few general implications. From Table 7.1 it may
be seen that for the purposes of n-type doping, tellurium provides superior enhancement to
selenium and for p-type doping, tin is the better dopant when compared to indium. This
implies that for doping on the antimony site in PtSb2 , and possibly further to all pnictogen
sites in narrow-gap semiconductors, proximity in the periodic table of the dopant element
to the substituted atom may be key.
When doping on the platinum site, it is likely that proximity is not the only guiding
principle for selecting a suitable dopant. It may be observed from the results found in
niobium, silver, and iron doping that iron was fully incorporated into the structure and
produced a p-type doping effect, while neither niobium nor silver were found to electronically
dope. It is likely that size difference, electronegativity, and valency of the dopant are of
importance for doping on this site. When examining ytterbium, the cause of the enhancement
displayed by this dopant was likely due to a chemical pressure effect on its valency and not
180

simply the result of a change in the carrier concentration. Of all of the dopants examined in
this work, ytterbium may have been the only dopant whose enhancement was not generated
through a change in the carrier concentration and may present an independent route of
enhancement in the figure-of-merit. It is possible that further optimization of the figure-ofmerit may be obtained by combining the effects of ytterbium doping and electronic doping
to result in further enhanced power factors and zT values.
Finally, the electronically doped ball milled samples showed the largest improvement
in figure-of-merit of all samples examined. A slightly higher figure-of-merit was found for
tellurium dopant as compared to tin doping in these ball milled samples. To conclude
this overview, it could be stated that for maximal enhancement in the figure-of-merit at
temperatures approaching room temperature a ball milled, tellurium doped sample should
be used, but for temperatures approaching 250 K ytterbium doped samples display enhanced
performance.

7.2

Future Directions

With the large number of materials crystallizing in the pyrite structure it is likely that a
solid solution may be able to be synthesized with some compounds, which will result in
further enhancement than what was found with PdSb2 . Of further interest beyond that of
making a solid solution is the substitution of bismuth or arsenic for antimony in PtSb2 . It
is known that PtBi2 is a metal with fully overlapping conduction and valence bands, similar
to PdSb2 [144]. While a solid solution of PdSb2 was unable to be generated PtBi2 may not
have the same issue. α-PtBi2 crystallizes in the pyrite structure with a lattice parameter, a
= 0.6691 nm [145]. It undergoes a phase change at temperatures greater than 270 ◦ C to β-

181

PtBi2 in the hexagonal crystal structure [97,146,147]. Hulliger found α-PtBi2 to be metallic
and that the compound PtSbBi could be synthesized with a lattice parameter, a = 0.6570
nm, the expected midpoint between the lattice parameters of the two end compounds [97].
While it is expected that large concentrations of a solid solution of PtSb2 with a metal will
not result in enhancement, low concentrations of α-PtBi2 may increase the figure-of-merit
through a reduction in the thermal conductivity. It is expected that this solid solution may
further narrow the bandgap of PtSb2 with very low substitution compositions since PtSbBi
was found to be a metal. This bandgap engineering may be a way to lower the temperature
where the figure-of-merit is maximized. In the opposite manner PtAs2 has been shown
to have a larger bandgap than PtSb2 , and while a solid solution with this compound is
expected to lower the thermal conductivity, this may also lead to bandgap engineering by
increasing the bandgap of PtSb2 [144]. This is likely to lead to an increased temperature for
the maximum figure-of-merit, but may lead to enhancement around room temperature.
As mentioned earlier in chapter five further rare earth doping on PtSb2 may yield promising results. Ytterbium doping was the only dopant that generated an increase in the power
factor in the low temperature region between 150 to 250 K. Examining greater concentrations of ytterbium may yield increased power factors beyond what was found here, but a
new synthesis method would need to be determined. Oxidation of ytterbium and secondary
phase formation became a challenge in concentrations beyond x = 0.1 and therefore these
concentrations were unable to be examined. Additionally, it is possible that the effect of
ytterbium doping may be independent to a reduction in particle size by ball milling, thus a
possible avenue of future work would examine those effects. With the enhancement found
from ytterbium it is expected that doping with the rare earth cerium may yield similar results. Cerium has one valence electron in the f-shell, and as with ytterbium, may lead to
182

interesting effects stemming from density of states manipulation near the bandgap.
It has been found that a semiconductor which has been engineered to have a more complex unit cell, for instance by taking a group 4 element and splitting it up into group 3
and 5 elements, often leads to a lower thermal conductivity. This is a common strategy
with thermoelectric materials and can lead to enhancement in the figure-of-merit [53]. This
engineering may be done with PtSb2 by substitution of the elements Sn and Te for antimony. Tin and tellurium are found on either column of the periodic table of antimony and
when crystallized PtSnTe forms the orthorhombic Pca21 structure with lattice parameters
a = 0.6476 nm, b = 0.6461 nm, and c = 0.6445 nm [148]. This structure is only slightly
distorted from the pyrite, but the antimony dumbbells found in PtSb2 are replaced by Sn
- Te bonded pairs [149]. This is expected to reduce the thermal conductivity dramatically.
Early experimental work has been promising with this compound which resulted in nearly an
order of magnitude reduction in the thermal conductivity at low temperature. However, in
this early experimental work a two phased sample was generated. Further experimentation
would require optimization of the synthesis parameters to generate a single phased sample, subsequent doping to tune the carrier concentration, and further thermal conductivity
reduction attempts.
Another pyrite structured compound of interest is MnTe2 , which from the work of Hulliger
et al. was shown to have a wide range of electronic properties depending only upon the
synthesis parameters used [90]. The highest Seebeck coefficient for these samples was found
to be approximately 500 µV/K between 80 to 120 K, while all other samples showed a range
of values greater than 200 µV/K above 150 K. The electrical resistivity of these samples
is relatively large for low temperature use but would be expected to decrease with carrier
concentration manipulation. It is expected that MnTe2 would have a similar high thermal
183

conductivity as compared to PtSb2 , but may be amenable to reduction attempts. The large
variability in the electronic properties is thought to arise due to non-stoichiometry stemming
from the tellurium in the system and as such, extensive ball milling may exacerbate this effect.
This effect however may be an avenue to explore carrier concentration in the material as it
was for PtSb2 .
Finally, a compound worthy of investigation is NiP2 , which has been shown to have a
narrow bandgap between 0.5 to 0.7 eV [68,150]. Examination of the electronic band structure
additionally shows a similarity with that found for PtSb2 with an indirect bandgap [144].
NiP2 crystallizes in the monoclinic structure with the space group C2/c [151]. Relatively
little experimental work has been performed on this material, but the room temperature
electrical resistivity has been found to range between 0.42 to 0.09 Ω-cm with a Seebeck coefficient of -392 µV/K [68]. The major benefit of NiP2 over that of PtSb2 is the lower cost to
use nickel and phosphorous compared to that of platinum and antimony. While thermoelectric applications are a niche market where reliability and efficiency are of higher importance
than cost, a lowered cost may help by allowing for additional testing and experimentation
without as much concern for monetary waste. And it is always a possibility that if costs
could be reduced substantially, then extensive adoption may be practical.
As was introduced in the experimental section, some early investigations into this material
have been attempted in this work. After separation of NiP2 and tin flux by reaction with
hydrochloric acid and a water wash the recovered powder was found to be single phased from
X-ray diffraction in the expected monoclinic structure. This powder required densification
by SPS in order to have transport properties measured and after this step the compound was
found to have separated into a secondary phase containing nickel phosphide. Thus, for future
work, experimentation will be necessary on a densification procedure which does not result
184

in the sublimation of phosphorous. This material is a promising candidate due to its similar
electronic band structure to PtSb2 and its earth abundant constituents. If the synthesis of a
single phased sample can be accomplished, further doping and thermal conductivity studies
could be performed to create a cost effective thermoelectric material.
As can be seen there are still several open areas and directions in which the work on PtSb2
can be taken to probe some of the interesting results displayed here, as well as to further
enhance some of the effects found. Additionally, there are several materials in the pyrite or
otherwise structure which show promise as potentially good materials for low temperature
thermoelectric use. The cryogenic region of thermoelectric use is a challenging regime to
increase the figure-of-merit, but one with many possible applications and benefits for society
as well as the world.

185

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