SYNTHESIS, PROCESSING, AND THERMOELECTRIC PROPERTIES OF
GERMANIUM-ANTIMONY-TELLURIUM BASED COMPOUNDS AND ALLOYS
By
Jared Brett Williams

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
SYNTHESIS, PROCESSING, AND THERMOELECTRIC PROPERTIES OF
GERMANIUM-ANTIMONY-TELLURIUM BASED COMPOUNDS AND ALLOYS
By
Jared Brett Williams
Society has become increasingly more aware of the negative impacts which nonrenewable
energy sources have on the environment, and therefore the search for new and more efficient
means of energy production has become an important research endeavor. Thermoelectric
modules possess the unique ability to convert wasted heat into useful electrical energy via
solid state processes, which could vastly improve the efficiency of a number of applications.
The materials which accomplish this are typically comprised of semiconductors which exhibit high electrical conductivity, Seebeck coefficient, and thermal resistivity. Together these
properties give us a gauge for the overall efficiency of the thermal to electrical energy conversion. Phase change materials are a class of materials primarily used for optical data storage
in CDs, DVDs, and Blu-Ray discs. Today’s state of the art phase change materials are
based on alloys of GeTe and Sb2 Te3 . These materials have also been found to exhibit high
thermoelectric efficiencies. These high efficiencies stem from their complex crystal structure
and degenerate semiconducting nature. The purpose of this work was to study and engineer the thermoelectric properties of various alloys and compounds which belong to this
family of materials. Specifically studied were the compounds Ge4 SbTe5 and Ge17 Sb2 Te20 .
In each case various synthesis and processing strategies were implemented to increase the
thermoelectric performance and better understand the fundamental electrical and thermal
properties. Finally various proposals for future work on these materials are presented, all of
which are based on the findings described herein.

I would like to dedicate this work to Teresa McConnell. An amazing woman who poured
her love and soul into everyone she knew. Teresa, I know you would be proud. Rest in
peace.
1963-2014

iii

ACKNOWLEDGMENTS

I would like to acknowledge my advisor, Dr. Donald T. Morelli. Thank you for creating
an environment which was nurturing, challenging, and accepting of all questions and ideas.
I would also like to thank my doctoral committee Dr. Richard Lunt, Dr. Yue Qi, and Dr.
Nelson Sepulveda. This work was supported at Michigan State University as part of the
Center for Revolutionary Materials for Solid State Energy Conversion, an Energy Frontier
Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic
Energy Sciences under Award Number DE-SC0001054. I would like to acknowledge financial
support from the Michigan State Graduate School through the University Distinguished
Fellowship. I would like to acknowledge Karl Dersch from the laboratory of Dr. Timothy
Hogan for assistance with spark plasma sintering and thermal diffusivity measurements.
I would also like to acknowledge the laboratory of Dr. Ctirad Uher at the University of
Michigan for the many collaborations over the years. I would like to acknowledge Dr. Alex
Zevalkink and David Smiadak for assistance with high temperature X-ray diffraction.
I would like to thank my wife, Morgan, and children, Addison, Dakota, and Chandler.
Despite what stresses and hardships I faced they always had a way of helping me keep them
in perspective and remember what the truly important aspects in life are. I would like to
acknowledge all of the researchers I had the pleasure of working with at Michigan State
University, particularly Spencer Waldrop, Winston Carr, and Spencer Mather. Thank you
for the many fruitful discussions over coffee and beer as well the countless hours of assistance
in the lab. Go for the Gusto! The success of my life thus far would not stand without the
amazing support and love of my family. To my parents and grandparents, both biological
and by marriage, thank you for the unwavering support and love throughout these years.
iv

Scott Williams, Lisa Adams, Teresa McConnell Williams, Robert Adams, Gayle Fowler,
Allan Fowler, and Daniel Wesley to name a few (if I missed your name I am truly sorry).
To say this would have been impossible without you is an understatement. I can only hope
to spend the rest of my life paying the selflessness forward. To my brothers, thank you for
being a constant source of joy in my life. I would also like to acknowledge Karl Fowler,
you have an amazing mind and bring a smile to my face. I would like to thank all of the
support throughout these years from my friends, Evan Bibb, Jordan Jones, Josh Lee, and
Dallas Viva, and many more which I could not name here simply due to length constraints.
To my undergraduate advisor, Dr. Terry Tritt, thank you for believing in me and laying the
foundations for my career in materials science and solid state physics. I would also like to
the thank mentorship and guidance from Dr. Dale Hitchcock and Dr. Daniel Thompson,
who spent hours in the lab at Clemson University answering questions from an interested
undergraduate. Lastly, I would like to acknowledge the musical writings of Anton´in Dvo˘
ra
´k,
Andy McKee, Hanz Zimmer, B.B. King, and John Mayer.

v

TABLE OF CONTENTS

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

vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Chapter 1 Introduction: Electronic and Thermal Properties of Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Thermoelectric Power Generation . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Thermoelectric Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Electrical Conductivity in Semiconductors . . . . . . . . . . . . . . .
1.2.1.1 Transport in Intrinsic and Extrinsic Semiconductors . . . .
1.2.1.2 Mobility and Conductivity . . . . . . . . . . . . . . . . . . .
1.2.2 Seebeck Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Thermal Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3.1 Electronic Thermal Conductivity . . . . . . . . . . . . . . .
1.2.3.2 Lattice Thermal Conductivity . . . . . . . . . . . . . . . . .
1.2.4 ZT and Methods of Enhancement . . . . . . . . . . . . . . . . . . . .
1.2.4.1 Optimizing Power Factor . . . . . . . . . . . . . . . . . . .
1.2.4.2 Achieving Low Thermal Conductivity . . . . . . . . . . . .

1
2
5
8
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23
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31
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35

Chapter 2 Background: Phase Change Materials
2.1 Phase Change Memory . . . . . . . . . . . . . .
2.2 Structure and Bonding . . . . . . . . . . . . . .
2.2.1 Resonance Bonding . . . . . . . . . . . .
2.2.2 Crystal Structure . . . . . . . . . . . . .
2.2.3 Insulator-Metal Transition . . . . . . . .
2.3 Thermoelectric Properties . . . . . . . . . . . .

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39
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Chapter 3 Experimental Procedures . . . . . . . . . . . . . .
3.1 Materials Synthesis . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Heating Procedures and Powder Processing . . . .
3.1.2 Spark Plasma Sintering . . . . . . . . . . . . . . . .
3.2 Characterization Techniques . . . . . . . . . . . . . . . . .
3.2.1 X-ray Diffraction . . . . . . . . . . . . . . . . . . .
3.2.1.1 Principle of X-ray Diffraction . . . . . . .
3.2.1.2 Structure Factor and Diffraction Intensity
3.2.2 Differential Scanning Calorimetry . . . . . . . . . .
3.2.3 Scanning Electron Microscopy . . . . . . . . . . . .
3.2.4 Hall Measurements . . . . . . . . . . . . . . . . . .
3.2.5 Low Temperature Transport Properties . . . . . . .
3.2.5.1 Electrical Resistivity . . . . . . . . . . . .

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66

vi

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3.2.6

3.2.5.2 Seebeck Coefficient and Thermal Conductivity .
High Temperature Transport Properties . . . . . . . . .
3.2.6.1 Electrical Resistivity and Seebeck Coefficient .
3.2.6.2 Thermal Diffusivity . . . . . . . . . . . . . . . .

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67
70
70
71

Chapter 4 (GeTe)1−x (Sb2 Te3 )x Compounds
4.1 Experimental . . . . . . . . . . . . . . . .
4.2 Results and Discussion . . . . . . . . . . .
4.3 Conclusion . . . . . . . . . . . . . . . . . .

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73
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74
80

Chapter 5 Ge4 SbTe5 . . . . . . . . . . . . .
5.1 Synthesis and Crystal Structure . . . .
5.2 Atomic Substitutions . . . . . . . . . .
5.2.1 Ge4−x Snx SbTe5 . . . . . . . . .
5.2.1.1 Experimental . . . . .
5.2.1.2 Results and Discussion
5.2.1.3 Conclusion . . . . . .
5.2.2 Ge4 Sb1−x Bix Te5 . . . . . . . .
5.2.2.1 Experimental . . . . .
5.2.2.2 Results and Discussion
5.2.2.3 Conclusion . . . . . .
5.3 Electronic Doping . . . . . . . . . . . .
5.3.1 Ge4−x Nax SbTe5 . . . . . . . .
5.3.1.1 Experimental . . . . .
5.3.1.2 Results and Discussion
5.3.1.3 Conclusion . . . . . .
5.3.2 Ge4−x Gax SbTe5 . . . . . . . .
5.3.2.1 Experimental . . . . .
5.3.2.2 Results and Discussion
5.3.2.3 Conclusion . . . . . .
5.4 Summary and Future Work . . . . . .

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102
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109
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115
115

Chapter 6 Ge17 Sb2 Te20 . . . . . . . .
6.1 Probing the Role of Sb Impurities
6.1.1 Synthesis Procedures . . .
6.1.2 Results and discussion . .
6.1.3 Conclusion . . . . . . . . .
6.2 Probing the Role of Ge Impurities
6.2.1 Synthesis Procedures . . .
6.2.2 Results and Discussion . .
6.2.3 Conclusion . . . . . . . . .
6.3 Enhancing ZT Using Composites
6.4 Ge17+x Sb2−x Te20 . . . . . . . . .
6.4.1 Experimental Procedures .
6.4.2 Results and Discussion . .

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6.4.3

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Chapter 7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.1 Future Work and Interesting Projects . . . . . . . . . . . . . . . . . . . . . . 158
REFERENCES

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

viii

LIST OF TABLES

Table 5.1:

Using SEM/EDS the actual composition of Sn substituted samples of
Ge4 SbTe5 were measured, seen here as atomic percent. . . . . . . . . .

91

Table 6.1:

Electrical resistivity, Seebeck coefficient, carrier concentration, and hole
mobility at 300K for Ge17 Sb2 Te20 +Sbx . . . . . . . . . . . . . . . . . . 128

Table 6.2:

The parameters necessary for calculating the thermoelectric properties
of Ge17 Sb2 Te20 using a single parabolic band model. Temperature, T ,
lattice thermal conductivity, κL , effective mass, m∗ (determined from
Seebeck coefficient), and intrinsic carrier mobility, µo , calculated using
the mobility found from Hall measurements. . . . . . . . . . . . . . . . 155

ix

LIST OF FIGURES

Figure 1.1:

The total energy consumed in the United States since 1976 (black line),
with the percent which is rejected represented by purple bars. . . . . . .

2

Modern day thermoelectric module and unicouple. Governed by the
Seebeck effect, electrons and holes will flow according to a temperature
gradient established by a heat source and generate a voltage difference.

3

(a) Thermoelectric efficiency (η) versus ∆T for various ZT values. (b)
ZT versus temperature for Bi2 Te3 , PbTe, Cu12 Sb4 S13 , and AgPbx SbTe2−x .[4, 5, 6, 7] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

When the junction of two dissimilar metals are held at different temperatures a voltage difference will be generated, which is governed by the
Seebeck coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Figure 1.5:

Fermi-Dirac distribution at 0K, 300K, and 1000K. . . . . . . . . . . . .

9

Figure 1.6:

Carrier concentration versus temperature for n-type Si, (ND = 1015 ).
The freeze-out, extrinsic, and intrinsic regimes can be clearly distinguished.[17] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

Figure 1.2:

Figure 1.3:

Figure 1.4:

Figure 1.7:

Electron mobility versus temperature for GaAs, with the temperature dependencies of ionized impurity and acoustic phonon scattering labeled.[17] 16

Figure 1.8:

Seebeck effect in a single conductor. The Seebeck coefficient is defined
as the ratio of the electric field established, E, in the presence of a temperature gradient, ∇T . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Seebeck coefficient versus η, the reduced Fermi energy (EF /kB T ) using
Fermi-Dirac and classical statistics. Notice that at around η = −2 the
values of Seebeck for the two treatments begin to diverge. As η approaches a value of 2, the classical treatment predicts a Seebeck value of
0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

Figure 1.10: Seebeck coefficient versus temperature for 0.1 and 1.0% Na doped PbTe.
Note that the peak in Seebeck coefficient due to intrinsic excitation of
carrier across the bandgap happens at a higher temperature since the
number of holes is greater in 1%Na-PbTe and therefore a higher concentration of excited electrons is needed to compensate.[28] . . . . . . . . .

22

Figure 1.9:

x

Figure 1.11: (a) Model of phonon dispersion curve for a one-dimensional monoatomic
lattice, with equal masses and atomic spacing a. The result is a single
acoustic branch. (b) Phonon dispersion curve for one-dimensional diatomic lattice (unequal masses). The creation of the higher frequency,
low group velocity optic branch is a result of of the two atomic species
with unequal masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Figure 1.12: Thermal conductivity as a function of temperature for sapphire. The
scattering events which are dominant in their temperature regime are labeled: boundary scattering at low temperature, defect scattering in mid
temperature range, and anharmonic phonon-phonon (Umklapp) scattering at high temperature.[12] . . . . . . . . . . . . . . . . . . . . . . . .

31

Figure 1.13: Seebeck coefficient (S), electrical conductivity (σ), power factor (S 2 σ),
and ZT fo an arbitrary thermoelectric material. Optimization of power
factor and ZT are done at relatively high charge carrier concentrations
of approximately 1019 − 1020 cm−3 , which corresponds to heavily doped,
or degenerate, semiconductors. . . . . . . . . . . . . . . . . . . . . . . .

33

Figure 1.14: Thermal conductivity versus mol fraction of InAs in GaAs.[59] . . . . .

36

Figure 2.1:

Figure 2.2:

Figure 2.3:

Figure 2.4:

Figure 2.5:

Schematic of memory unit using phase change materials where the high
resistance state (reset state) is activated with a short, high intensity
electrical pulse and the low resistance state (set state) is activated with
a long, low intensity electrical pulse. . . . . . . . . . . . . . . . . . . . .

41

Ternary phase diagram of Ge, Sb, Te with the pseudo-binary tie-line of
GeTe and Sb2 Te3 and Ge4 SbTe5 highlighted. . . . . . . . . . . . . . . .

41

Crystal structure of GeTe and Sb2 Te3 , which has repeating vacancy layers between Te layers.These structures were generated using VESTA.[91]

45

Rocksalt and Rhombohedral Crystal Structure of Ge2 Sb2 Te5 . These
structures were generated using VESTA.[91] . . . . . . . . . . . . . . .

46

(a)Electrical conductivity versus temperature around the insulator-metal
transition (represented here by dashed line) for GeSb2 Te4 , (b) and (c)
hole concentration and hole mobility, respectively, around the insulatormetal transition. (d) Schematic of the mobility edge, Eµ in relation to
the Fermi energy, EF , which is the governing principle of an Anderson
localization insulator-metal transition. In this schematic, zero energy
is defined as the top of the valence band. The data in this figure was
digitized and replotted from the work of Siegrist, et al. [93] . . . . . . .

48

xi

(a) Heating procedure shown here as temperature (◦C) versus time (h)
(b) Raw elements sealed in silica ampoule before heat treatment (c) Ingot
obtained after melt and anneal process (d) Final disk obtained after
densifying via spark plasma sintering. . . . . . . . . . . . . . . . . . . .

52

Z-axis displacement of sample during SPS versus temeprature for various
holding temperatures (400◦C, 420◦C, 435◦C, 450◦C, 465◦C). A clear
correlation between % theoretical density yield and SPS temperature
can be observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

A schematic which showcases the principle of X-ray Diffraction with a
periodic lattice of atoms. . . . . . . . . . . . . . . . . . . . . . . . . . .

56

Figure 3.4:

The (110) plane in a face-centered cubic crystal structure. . . . . . . . .

57

Figure 3.5:

A schematic of the setup used for performing DSC. Each crucible is
placed on a heater which is precisely monitored to determine the amount
of heat flux needed to heat each crucible. . . . . . . . . . . . . . . . . .

60

A schematic which depicts the principle of energy dispersive X-ray spectroscopy (EDX). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62

A typical mounting setup for measuring the Hall effect, and thereby the
carrier concentration, in a material. . . . . . . . . . . . . . . . . . . . .

63

Schematic of system used for measuring the electrical resistivity, Seebeck
coefficient, and thermal conductivity across the temperature range of 80350K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

Thermal conductivity versus temperature for Bi2 Te3 with and without
the radiation correction, an effect which can cause an overestimation of
thermal conductivity, especially at temperatures near and above room
temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

70

X-ray diffraction patterns for (GeTe)1−x (Sb2 Te3 )x , where x varies from
0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 . As Sb2 Te3 content increases
the rhombohedral crystal structure slowly changes to the rocksalt structure, as can be seen in the inset. . . . . . . . . . . . . . . . . . . . . . .

75

Electrical resistivity versus temperature for (GeTe)1−x (Sb2 Te3 )x , where
x varies from 0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 . . . . . . . . . .

76

Hole Concentration, (a) and Hole Mobility, (b) versus Temeperature for
(GeTe)1−x (Sb2 Te3 )x . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Figure 3.1:

Figure 3.2:

Figure 3.3:

Figure 3.6:

Figure 3.7:

Figure 3.8:

Figure 3.9:

Figure 4.1:

Figure 4.2:

Figure 4.3:

xii

Figure 4.4:

Figure 4.5:

Figure 4.6:

Figure 4.7:

Figure 5.1:

Figure 5.2:

Figure 5.3:

Figure 5.4:

Seebeck coefficient versus temperature for (GeTe)1−x (Sb2 Te3 )x , where
x varies from 0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 . . . . . . . . . .

78

Power factor, (a) and calculated Lorenz number, (b) versus temperature
for (GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%, 4%, 5.9%, 6%,
and 8.3% Sb2 Te3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Thermal conductivity, (a), and lattice thermal conductivity, (b) versus
temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%, 4%,
5.9%, 6%, and 8.3% Sb2 Te3 . . . . . . . . . . . . . . . . . . . . . . . . .

79

ZT versus temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies from
0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 . . . . . . . . . . . . . . . . . .

80

Rocksalt (F m¯3m) crystal structure of Ge4 SbTe5 , with Ge and Sb randomly occupying the cation site and Te on the anion site. . . . . . . . .

81

X-ray diffraction pattern of Ge4 SbTe5 with peaks labeled by their respective miller indices. . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

(a)Temperature dependent X-ray diffraction patterns for Ge4 SbTe5 . (b)
shows the cubic (111) peak splitting at the phase transition temperature
as the cubic crystal structure transforms to rhombohedral. . . . . . . .

84

Specific heat, as determined by DSC, versus temperature for Ge4 SbTe5 ,
with the Dulong-Petit value represented by a dashed line. . . . . . . . .

85

Figure 5.5:

(a) Electrical resistivity versus temperature, (b) Seebeck coefficient versus temperature, and (d) Power factor versus temperature for Ge4 SbTe5 . 86

Figure 5.6:

(a) Thermal conductivity versus temperature and (b) ZT versus temperature for Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87

ZT versus temperature for a number of today’s state of the art chalcogenide thermoelectric materials, with Ge4 SbTe5 showing competitive
values.[117, 118, 119] . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

X-ray Diffraction pattern of Ge4−x Snx SbTe5 , where x=0, 0.5, 1.0, 1.5,
2.0, 2.5, 3.0, 3.5, and 4.0. A shift in diffraction peaks to lower 2θ can be
seen with increasing Sn substitution, which corresponds to an increase
in lattice parameter. Impurity peaks, which were found to be elemental
Sb, are marked by a diamond. . . . . . . . . . . . . . . . . . . . . . . .

90

Figure 5.7:

Figure 5.8:

xiii

˚) versus Sn concentration for Ge4−x Snx SbTe5 . A
Lattice Parameter (A
least-squares method was used to apply the trendline. The linear behavior agrees with Vegard’s law. . . . . . . . . . . . . . . . . . . . . . . . .

91

Figure 5.10: Electrical resistivity versus temperature for Ge4−x Snx SbTe5 where x=0,
0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4. . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 5.11: Seebeck coefficient versus temperature for Ge4−x Snx SbTe5 where x=0,
0.5, 1, 1.5, 2, 2.5, 3, 3.5, and 4. . . . . . . . . . . . . . . . . . . . . . . .

93

Figure 5.12: (a) Thermal conductivity versus temperature for all Sn substituted samples in the temperature range of 80K to 320K. (b) Lattice thermal conductivity at 300K versus stoichiometric amount of substituted Sn. (c)
Thermal conductivity at 723K for 25%, 50%, 82.5%, and100% Sn substitution. (d) Lattice thermal conductivity at 723K versus stoichiometric
amount of substituted Sn. . . . . . . . . . . . . . . . . . . . . . . . . .

94

Figure 5.13: Lattice parameter versus Bi composition in Ge4 Sb1−x Bix Te5 . . . . . . .

97

Figure 5.9:

Figure 5.14: Specific heat versus temperature for Ge4 Sb1−x Bix Te5 , where x = 0, 0.25, 0.50, 0.75, 1.0.
Phase transition temperature versus percent Bi can be seen in the inset
plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Figure 5.15: (a) Thermal conductivity versus temperature for Ge4 Sb1−x Bix Te5 , where
x = 0, 0.25, 0.50, 0.75, 1.0. (b) Lattice thermal conductivity versus temperature, dashed line marking the 1/T temperature dependence of Umklapp scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Figure 5.16: Seebeck coefficient versus temperature for Ge4 Sb1−x Bix Te5 , where x =
0, 0.25, 0.50, 0.75, 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 5.17: Electrical resistivity versus temperature for Ge4 Sb1−x Bix Te5 , where x =
0, 0.25, 0.50, 0.75, 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Figure 5.18: Room temperature (300K) electrical resistivity versus %Bi substituted
with 8%. The parabolic nature is indicative of alloy scattering, as explained by Nordheim’s rule. The trendline was generated using a least
squares fit without knowledge of the Nordheim coefficient of Ge4 SbTe5 .

102

Figure 5.19: Electrical resistivity versus temperature for 0.5%, 1.0%, and 1.5% Nadoped Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 5.20: Seebeck coefficient versus temperature for 0.5%, 1.0%, and 1.5% Nadoped Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

xiv

Figure 5.21: (a) Hole concentration versus temperature for Ge4 SbTe5 and 1.5% Na
doped. (b) Hole Mobility versus Temperature for Ge4 SbTe5 and 1.5%.

106

Figure 5.22: Power Factor versus Temperature for 0.5%, 1.0%, and 1.5% Na-doped
Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Figure 5.23: Thermal conductivity versus temperature for 0.5%, 1.0%, and 1.5% Nadoped Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Figure 5.24: ZT versus Temperature for 0.5%, 1.0%, and 1.5% Na-doped Ge4 SbTe5 .

109

Figure 5.25: X-ray diffraction patterns for Ge4−x Gax SbTe5 , 1%, 5%, and 10% Ga. . 111
Figure 5.26: Electrical resistivity versus temperature for 1%, 5%, and 10% Ga doped
Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 5.27: Seebeck coefficient versus temperature for 1%, 5%, and 10% Ga doped
Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 5.28: (a) Power factor versus temperature for 1%, 5%, and 10% Ga doped
Ge4 SbTe5 . (b) Thermal conductivity versus temperature for Ga-doped
Ge4 SbTe5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Figure 5.29: ZT versus temperature for 1%, 5%, and 10% Ga doped Ge4 SbTe5 . . . . 115
Figure 6.1:

Specific heat versus temperature, measured by DSC, for Ge17 Sb2 Te20 .
The phase transitions can be seen as endothermic peaks at T1 , the transition from metastable rocksalt to rhombohedral, and T2 , the phase transition from the stable rhombohedral phase to the stable rocksalt phase.

119

Figure 6.2:

X-ray diffraction patterns for Ge17 Sb2 Te20 upon heating from room temperature. The first transition from metastable rocksalt to rhombohedral
is not discernible, but the high temperature rocksalt phase can be seen
at approximately 623K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Figure 6.3:

X-ray diffraction patterns for Ge17 Sb2 Te20 upon cooling. The stable
rhombohedral phase is discernible below 570K. . . . . . . . . . . . . . . 120

Figure 6.4:

(a) X-ray diffraction pattern of Ge17 Sb2 Te20 quenched from 1173K, seen
here with Sb and Ge impurities labeled. Quenching from the melt
yields a rhombohedral crystal structure. (b) X-ray diffraction pattern
of Ge17 Sb2 Te20 quenched from the melt and annealed for an additional
24h at 863K. This method produces an impurity free rocksalt phase. . . 122

xv

Figure 6.5:

ZT versus temperature for Ge17 Sb2 Te20 prepared by quenching from the
melt and subsequently annealing. The SEM/EDS elemental maps show
the presence of Sb precipitates. . . . . . . . . . . . . . . . . . . . . . . . 123

Figure 6.6:

(a)The electrical resistivity, (b) ZT, and (c) Seebeck coefficient for Ge17 Sbx Te20 .
By altering the stoichiometry of Sb, and therefore the concentration of
Sb secondary phase, the properties can be tuned. (d) Shows ZT at 773K
versus x, the stoichiometric amount of Sb. . . . . . . . . . . . . . . . . 124

Figure 6.7:

X-ray diffraction patterns for Ge17 Sb2 Te20 , with 0.25, 0.50, 1.0, and
2.0% added Sb. The star marks the diffraction peaks which indicate the
presence of elemental Sb as a secondary phase(PDF#98-000-0095). The
EDS maps clearly show the presence of Sb microstructures in Ge17 Sb2 Te20 .126

Figure 6.8:

(a)Electrical resistivity versus temperature, seen here with ±8% error
bars and (b)Seebeck coefficient versus temperature for Ge17 Sb2 Te20 with
0.25, 0.50, 1.0, and 2.0% added Sb, in this figure the uncertainty of ±2%
associated with measuring the Seebeck coefficient is smaller than the
markers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 6.9:

(a) Power Factor versus temperature for Ge17 Sb2 Te20 +Sbx , seen here
with ±8% error bars. (b) Seebeck coefficient and hole mobility at 300K
as a function of %Sb as an impurity. A clear compliment can be seen between the two properties, which is indicative of a change in the scattering
relaxation time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Figure 6.10: (a)Thermal conductivity versus temperature for Ge17 Sb2 Te20 with 0.25,
0.50, 1.0, and 2.0% added Sb. (b)The lattice contribution to thermal
conductivity versus temperature. (c)Lorenz number versus temperature,
which was determined from a single parabolic band fit of the Seebeck
coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Figure 6.11: ZT versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50, 1.0, and 2.0%
added Sb, seen here with ±13% on Ge17 Sb2 Te20 and Ge17 Sb2 Te20 +
Sb0.195 . A large enhancement was observed for Ge17 Sb2 Te20 with 0.5%Sb
secondary phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Figure 6.12: EDS elemental maps showcasing the presence of Ge impurities in Ge17 Sb2 Te20 +Gex
samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure 6.13: (a) Electrical resistivity versus temperature for Ge17 Sb2 Te20 +Gex , seen
here with ±8% error bars on Ge17 Sb2 Te20 . (b) Seebeck coefficient versus
temperature for Ge17 Sb2 Te20 +Gex . . . . . . . . . . . . . . . . . . . . . 135

xvi

Figure 6.14: (a) Power factor as a function of temperature for Ge17 Sb2 Te20 +Gex ,
with 8 ± % error bars on Ge17 Sb2 Te20 . (b) Hole mobility and room
temperature power factor as a function of %Ge secondary phase. A clear
complement can be discerned. . . . . . . . . . . . . . . . . . . . . . . . 136
Figure 6.15: Thermal conductivity versus temperature for Ge17 Sb2 Te20 +Gex , seen
here with ±8% error bars on Ge17 Sb2 Te20 , which was calculated from
the measurement techniques used. . . . . . . . . . . . . . . . . . . . . . 137
Figure 6.16: ZT versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50, 1.0, and 2.0%
added Ge, seen here with ±13% error bars on Ge17 Sb2 Te20 . A large
enhancement was observed for Ge17 Sb2 Te20 with 0.5%Ge impurity. . . 138
Figure 6.17: ZT as a function of temperature for a number of microstructured composites which utilize (GeTe)x Sb2 Te3 alloys. Specifically Ge12 Sb2 Te15
with precipitates of CoGe2 from [156]. Ge17 Sb2 Te20 with precipitates of
CoGe2 from [156]. Ge10.5 Sb2 Te13.5 with precipitated CoSb3 from [155].
Ge17 Sb2 Te20 with impurities of Sb and Ge from [144] and [157], respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Figure 6.18: X-ray patterns for Ge17+x Sb2−x Te20 showing x = 0, 0.04, 0.16, 0.20,
0.24, 0.30, and 0.40. The rocksalt structure of Ge17 Sb2 Te20 begins to
transition to rhombohedral as Ge content is increased. GeTe is shown as
a reference for the rhombohedral peak splitting taking place. . . . . . . 147
Figure 6.19: As the amount of Ge doped onto the Sb site of Ge17 Sb2 Te20 is increased
the angle of the rocksalt unit cell (normally 90◦ ) is decreased. This can
be understood as a distortion along the [111 . . . . . . . . . . . . . . . 148
Figure 6.20: Electrical resistivity versus temperature Ge17+x Sb2−x Te20 . As the amount
of Ge dopant is increased the electrical is decreased to values below
1.0mΩ cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Figure 6.21: Seebeck coefficient versus temperature for Ge17+x Sb2−x Te20 . The decrease in Seebeck coefficient as the concentration of Ge dopant is increased is further indication that the hole concentration is increased from
doping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Figure 6.22: Power factor versus temperature for Ge17+x Sb2−x Te20 , seen here with
the calculated error bars of approximately ±10% for Ge17 Sb2 Te20 . An
optimal power factor was reached for 15%Ge; higher concentrations caused
a decrease. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xvii

Figure 6.23: (a) Thermal conductivity versus temperature, with calculated error bars
of on average 9%. The increasing trend in total thermal conductivity is
due to an increased electronic contribution, as was verified from calculating the lattice thermal conductivity (b). . . . . . . . . . . . . . . . . 152
Figure 6.24: ZT versus temperature for Ge17+x Sb2+x Te20 , seen here with the calculated error bars of on average ±13% on Ge17 Sb2 Te20 and Ge17.4 Sb1.6
Te20 (20% Ge). There is a systematic decrease with increased Ge doping. 153
Figure 6.25: Hole concentration versus temperature (300 − 700K) for 0%, 10%, 15%,
and 20% Ge doped Ge17 Sb2 Te20 . . . . . . . . . . . . . . . . . . . . . . 153
Figure 6.26: (a) Calculated ZT values as a function of carrier concentration for Ge17 Sb2 Te20
assuming single parabolic band behavior for 300K, 423K, 523K, 623K,
and 773K. Experimental values are plotted as squares for each temperature, showcasing the optimized ZT values of Ge17 Sb2 Te20 . (b) Seebeck
coefficient versus carrier concentration (Pisarenko plot) for Ge17 Sb2 Te20
at 300K, 423K, 523K, 623K, and 773K assuming single parabolic band
behavior, with experimental points shown as squares for each temperature.154

xviii

Chapter 1
Introduction: Electronic and Thermal
Properties of Semiconductors
The United States consumed approximately 97.5 Quads of energy in 2015 alone.[1] One Quad
is equivalent to 1015 BTU of energy, or 1.055x1018 J. In fact, since 1976 the total energy
consumed by the United States has increased more than 35%, and although we are more
technologically advanced, the percentage of that energy which is rejected has also increased,
as can be seen in Figure 1.1.

1

Figure 1.1: The total energy consumed in the United States since 1976 (black line), with the
percent which is rejected represented by purple bars.

This rejected energy is mainly in the form of waste heat and represents a large untapped
energy source. Additionally, the majority of the energy produced was done using resources
which are nonrenewable such as petroleum, coal, and natural gas. The combustion of these
resources has also been shown to emits high levels of CO2 , which contributes to global
warming by the greenhouse effect.[2, 3] Carbon neutral resources such as solar, nuclear,
wind, geothermal, and biomass had a combined total of less than 20%. Thermoelectric
materials could alleviate this issue by using the wasted heat from a number of processes and
directly converting it back into useful electricity.

1.1

Thermoelectric Power Generation

Thermoelectric materials posses the ability to convert wasted thermal energy directly into
useful electrical energy through a direct, solid-state process. Thermoelectric modules are the
2

leading design for waste-heat recovery and cooling applications. These modules consist of
an array of n-type and p-type thermoelectric legs, which are placed in series electrically and
in parallel thermally, as can be seen in Figure 1.2.

Figure 1.2: Modern day thermoelectric module and unicouple. Governed by the Seebeck
effect, electrons and holes will flow according to a temperature gradient established by a
heat source and generate a voltage difference.

In the presence of a temperature gradient the holes and electrons in the p-type and ntype legs, respectively, will respond by generating an electric field until an equilibrium is
established. The result is a voltage difference ∆V which directly balances the temperature
difference ∆T . The ratio of voltage difference to temperature difference is related to an
intrinsic property of all materials, the Seebeck coefficient S. The efficiency of a number
of applications which produce waste heat could be improved by incorporating thermoelectric devices. However, the current utilization of thermoelectric modules is limited to niche
applications, which can be traced to the high cost of synthesis and low efficiency.
The efficiency of a thermoelectric generator is dependent upon the Carnot efficiency and

3

the unitless parameter ZT , as can be seen in equation (1.1).

η=

TH − TC
TH



1 + ZTavg − 1



T

1 + ZTavg + ( T C )




(1.1)

H

where TH , TC , and Tavg are the hot side temperature, cold side temperature, and average
temperature of the thermoelectric legs, respectively. From equation (1.1) it is clear to see
that a high efficiency thermoelectric module can be obtained by increasing the temperature
difference between TH and TC and/or increasing ZTavg . The dimensionless figure-of-merit
ZT and the materials properties which define it will be discussed in greater detail in subsequent sections. The square-root dependence of efficiency on ZT implies that the greatest
gains in efficiency from improving ZT will happen for values less than approximately 4, as
can be seen in Figure 1.3(a).
The majority of today’s state-of-the-art thermoelectric materials exhibit ZT values approximately equal to unity or slightly higher. Some representative compounds include, but
are not limited to, Bi2 Te3 , PbTe, Cu12 Sb4 S13 , and AgPbx SbTe2−x , which can be seen in
Figure 1.3(b).[4, 5, 6, 7] The primary focus of thermoelectrics research has gone into finding
new materials with increased stability, higher ZT values, and more cost effective and earth
abundant comprising elements.

4

Figure 1.3: (a) Thermoelectric efficiency (η) versus ∆T for various ZT values. (b) ZT versus
temperature for Bi2 Te3 , PbTe, Cu12 Sb4 S13 , and AgPbx SbTe2−x .[4, 5, 6, 7]

1.2

Thermoelectric Phenomena

In 1821 Thomas Johann Seebeck found that when the junctions of two dissimilar metals were
held at different temperatures a compass needle would be deflected.[8] This was originally
thought to be due to the generation of a magnetic field from the temperature difference, but
it was soon realized that an electric current was the source. More specifically, the magnitude
of the voltage difference is directly proportional to the temperature difference and related by
S, the Seebeck coefficient (∆V = −S∆T ). This effect stems from a difference in Fermi levels
between the dissimilar materials and the temperature dependence of the Fermi energy in a
metal, as will be discussed later, and is the basis for modern day thermocouples. Using two
materials with well defined Seebeck coefficients, an absolute temperature can be determined
based on the voltage generated from a temperature difference as can be seen in Figure 1.4.

5

Figure 1.4: When the junction of two dissimilar metals are held at different temperatures a
voltage difference will be generated, which is governed by the Seebeck coefficient.

In 1834 the French watchmaker Jean Charles Athanase Peltier observed that by applying
an electrical current through a system similar to that seen in Figure 1.1 heat would either
be rejected or absorbed at the junctions, depending on the direction of the current.[9] The
quantity of heat rejected or absorbed at a junction is related to the Seebeck coefficient, which
is expressed using the following relation:[10]

Π = ST

(1.2)

where Π is the Peltier coefficient, S is the Seebeck coefficient, and T is the temperature.
A third related property is the Thomson effect (γ), which can also be expressed in terms of
the Seebeck coefficient:[11]

dS
γ
=
T
dT

(1.3)

The Seebeck coefficient is the fundamental phenomenon governing thermoelectric power
generation, whereas the Peltier effect is the basis for thermoelectric refrigeration. Both refrigeration and power generation for thermoelectrics are dependent on ZT, the dimensionless
6

figure-of-merit which is derived from intrinsic physical properties:

ZT =

S 2σ
S2
T =
T
κe + κl
ρ(κe + κl )

(1.4)

where S is the Seebeck coefficient, σ is the electrical conductivity (the inverse of which
is ρ, the electrical resistivity), κe is the electronic component of thermal conductivity, and
κl is the lattice component, which sum to yield the total thermal conductivity, and T is
the operating temperature. To achieve high-ZT it can be seen from equation 1.3 that a low
thermal conductivity and high S 2 σ (or power factor, P F ) is desirable. More specifically,
lowering thermal conductivity is generally achieved through lowering the lattice component,
κl , because, for a given temperature the ratio of κσe is constant due to the constraints of the
Weidemann-Franz Law, which is stated as thus:[12]

κe = LσT

(1.5)

where L is the Lorenz number, generally taken as 2.44 ∗ 10−8 WΩK−2 . Additionally, the
Seebeck coefficient, which scales indirectly with carrier concentration, competes with the
electrical conductivity.
Because of the contraindicated nature of the electrical and thermal properties which
comprise ZT, sophisticated materials engineering is needed to achieve values which would
render thermoelectric materials viable for commercial utilization. In the following sections
each of the aforementioned properties will be discussed in detail, and the methods used to
achieve high ZT values in today’s state-of-the-art materials will be presented.

7

1.2.1

Electrical Conductivity in Semiconductors

For the general case of a semiconductor containing both electrons holes the electrical conductivity (σ) is expressed as:[11]

σ = µe en + µh ep

(1.6)

where µe and µh are the mobilities for electrons and holes, respectively, n and p are
the electron and hole concentrations, respectively, and e is the fundamental unit of charge
(1.602x10−19 C). In the case where n

p, or vice versa, equation (1.6) can be further

reduced to contain only the contributions from the dominant carrier. To determine and
understand the nature of the charge carrier concentration and the charge carrier mobility
which determine the electrical conductivity, one must undergo a more rigorous study of the
fundamental way in which electrons behave in materials.
Electrons obey the Pauli exclusion principle, which states that no two electrons can
have the same quantum state. Because electrons are indistinguishable a statistical analysis
is applied to determine the probability of finding an electron at a specific energy, as is
expressed by the Fermi-Dirac distribution:[13]

f (E) =

1
E−EF
1 + e kB T

(1.7)

In equation (1.7) E is the energy, EF is the Fermi energy, kB is the Boltzmann constant
(8.617 ∗ 10−5 eVK−1 ), and T is the temperature. It is clear to see from equation (1.7) that
at T = 0K all states below Ef are filled and all states at energies above the EF are unfilled.
As T increases electrons from lower energy states become excited and fill states with higher

8

energy. This distribution can be seen in Figure 1.5. At high enough temperatures the
Fermi-Dirac distribution approaches that of a classical Boltzmann distribution.[14]

Figure 1.5: Fermi-Dirac distribution at 0K, 300K, and 1000K.

In metals the Fermi energy resides deep within the conduction band, while in an insulator
the band gap is typically too large for thermal excitation, and the Fermi energy therefore is
defined as residing halfway between the top of the highest energy filled band (valence band)
and the lowest energy unfilled band (conduction band) inside the forbidden energy gap. A
semiconductor is defined as a material with a band gap, but typically has bandgap energies
on the order of a few eV. The density of states available for electronic population, g(E)dE,
is defined as the number of states in the energy interval E to E+dE per unit volume. For
the case of free electrons the density of states is defined as:[13]

1 2m 3/2 1/2
g(E) = 2
E
2
2π

(1.8)

An integration of the Fermi-Dirac distribution weighted by the electronic density-of-states
as a function of energy yields the carrier concentration in a solid, as can be seen in equation
9

(1.9).[13]

∞

n=

fn (E)g(E)dE

(1.9)

0

By combining equations (1.8) and (1.9) the electron concentration in a metal can be
expressed as:



∞

n=
0

1.2.1.1

1 2m 3/2 1/2 
E

2
2π 2

1
E−Ef
1 + e kB T


 dE

(1.10)

Transport in Intrinsic and Extrinsic Semiconductors

Unlike metals, semiconductors contain a band gap in their band structure. The bandgap
changes the transport equations, because in order for electronic conduction to take place
electrons must be thermally excited into the conduction band. The carrier concentration
will now be defined in relation to the band edge (EC for the conduction band and EV for
the valence band):

∞

n=

g(E)f (E)dE

(1.11)

EC

where the density of states, g(E) for semiconductors is defined as:[13]
√

2 (E − EC )1/2 ∗ 3/2
g(E) = gc 2
(m )
3
π

(1.12)

where gc is the band degeneracy, which is defined as the overlap of energy bands, and m∗
is the effective mass of the charge carrier, in this case electrons, and

is the Planck constant

divided by 2π ( = h/2π = 1.054x10−34 Js). The effective mass is a tensor quantity which

10

is best understood as the mass of an electron as a result of their acceleration, dvg /dt from
an externally applied force Fext , such as from an electric field.[14]

Fext = −m∗

dvg
dT

(1.13)

which can be rewritten as:

1
m∗ = 2 ∇2k E

(1.14)

m∗ is dependent on the symmetry of the crystal. For example, in Si which has an ellipsoidal energy surface, the effective mass has two components, the longitudinal and the
1/3 . By comtransverse.[15] The total effective mass is therefore written as m∗ = (m∗l m∗2
t )

bining equations (1.11) and (1.12) the carrier concentration can be expressed as:

2
E − EC
n = NC √ F1/2
kB T
π

(1.15)

where F1/2 is the Fermi integral with one-half index, which is used assuming a parabolic
band.[13]

∞

F1/2 =

0

E 1/2 dE
E−EF
1 + e kB T

(1.16)

and NC is defined as the effective density of states:[13]

NC = 2

2πm∗ kB T
gC
h2

(1.17)

For a non-degenerate semiconductor, where Boltzmann statistics are adequate in describ-

11

ing the population of electronic states F1/2 approaches the value of

√

πeη/2 , where η is the

reduced Fermi energy (EF /kB T ). The carrier concentration can then be further simplified
as:[11]

n=

E −EF
− C
kB T
NC e

(1.18)

These equations are similar for describing the transport in the valence band by holes
(p-type). At temperatures greater than 0K electrons can be excited across the band gap
into the conduction band, leaving a hole in the valence band. The law of mass action states
that the number of holes and electrons will be equal in order to preserve charge neutrality,
n = p = ni .[15] The product of n and p at any temperature will therefore be a constant
known as the intrinsic carrier concentration, ni .

np =

n2i

=

−Egap
NC NV e kB T

(1.19)

By rearranging equation (1.19) the Fermi energy with respect to the valence band edge
can be explicitly calculated.[16]

1
k T
NC
EF − EV = Egap + B ln
2
2
NV

(1.20)

Impurity atoms which are not isovalent to the host atom can contribute additional electrons (or holes) to the material, which can change the charge carrier concentration. For
example, B atoms in Si contain one less electron and introduce an acceptor state (hole),
while P atoms contain one extra electron and introduce a donor state (electron). The process
of adding impurity atoms to change the carrier concentration is called doping. To preserve

12

neutrality the total negative charges (electrons and ionized acceptors) must equal the total
+ + p, where
positive charges (holes and ionized donors). In the case of n-type doping, n = ND

the number of ionized donors is given by:[16]




+ = N 1 −
ND
D

1



ED −EF 

(1.21)

1 + 21 e kB T

At low temperature these impurities are said to be frozen-out, meaning there lacks sufficient energy to ionize all of the donors/acceptors. The carrier concentration in this regime
depends on ND , NA , and the donor ionization energy ED .[15]
−E

D
ND − NA
2k
T
B
n≈
NC e
2NA

(1.22)

As the temperature continues to increase, for a non-degenerate semiconductor, all of
the impurities will become excited, but the energy necessary to excite electrons across the
bandgap is not reached. This is called the extrinsic regime, and is marked by a carrier
concentration independent of temperature. At sufficiently high temperature, the energy
needed to excite carriers across the bandgap is reached; this is called the intrinsic regime
and is marked by an equal number of holes and electrons (n ≈ p

ND ). By combining

the law of mass action for intrinsic semiconductors and the neutrality equation for extrinsic
semiconductors the total carrier concentration in the intrinsic regime can be expressed as:

n = 1/2 (ND − NA ) +

(ND − NA )2 + 4n2i

(1.23)

The carrier concentration versus temperature for n-type Si (ND = 1015 ) can be seen in
Figure 1.6, with the freeze-out, extrinsic, and intrinsic regimes labeled accordingly.[17]
13

Figure 1.6: Carrier concentration versus temperature for n-type Si, (ND = 1015 ). The
freeze-out, extrinsic, and intrinsic regimes can be clearly distinguished.[17]

In the case of a heavily-doped, or degenerate, semiconductor the above equations, which
use Boltzmann statistics, are no longer valid; Fermi-Dirac statistics must be used. In this
case the Fermi energy rests within 2 to 3kB T of the conduction band minimum, or valence
band maximum.[11] In this case σ decreases with temperature, as is observed in metals.
Also the law of mass action, np = n2i is no longer valid. In degenerate semiconductors the
high level of impurities, donors or acceptors, are unable to become ionized, and n typically
reaches a saturation value of approximately 1019 − 1020 cm−3 .[12]

1.2.1.2

Mobility and Conductivity

The mobility of charge carriers in a crystal is given by equation (1.24),

µ=

e¯
τ
m∗

14

(1.24)

where τ¯ is the scattering relaxation time. A number of processes affect the relaxation
and therefore mobility, which has an effect on all the transport properties in a material.
According to Matthiesen’s rule, the total charge carrier mobility is the reciprocated sum of
all the mobilities associated with various processes:[12]

1
1
1
1
1
=
+
+ ...
+
+
µ
µL µi µN
µdef

(1.25)

and therefore the scattering relaxation rates also sum accordingly:

1
1
1
1
1
=
+ +
+
+ ...
τ¯
τL τi τN
τdef

(1.26)

where τL represents scattering by acoustic phonons, τi is the scattering from ionized
impurities, τN is scattering from neutral impurities, and τdef is scattering from defects such
as dislocations, grain boundaries, vacancies, etc. Each scattering mechanism contributes
varying degrees of magnitude and temperature dependencies based on the nature of the
scattering center. Three of the most common, and which have the strongest temperature
dependence are τL , τi , and τdef . The scattering from point defects, for example, has a
square root dependence on temperature.[18] The scattering associated with acoustic phonons
is proportional to temperature as τL ∝ T −3/2 .[19] For the case of ionized impurity scattering
the dependence on temperature is as follows, τi ∝ T 3/2 .[20] In actual materials the isolation
of a single scattering mechanism is rare, but rather the relaxation time contains multiple
contributions. For most cases τ can be generally defined as:[16]

τ (E) = τo (T )( ∗ )λ

15

(1.27)

where ∗ is the reduced energy, E − EC /kB T , and λ is the scattering parameter, which
is defined as −1/2, for acoustic phonon and point defect scattering, 0 for optical phonon
scattering below the Debye temperature, and 3/2 for ionized impurity scattering. By considering only acoustic phonons and ionized impurities, the scattering relaxation time can be
expressed as:[13]

τ=

τoL ( ∗ )3/2
τ
( ∗ )2 + ( τ oi )2

(1.28)

oL

The presence of neutral impurities and point defects affect the overall magnitude of the
mobility, but acoustic scattering and ionized impurities cause the strongest temperature dependence. The charge carrier mobility versus temperature for GaAs can be seen in 1.24. The
scattering from ionized impurities and acoustic phonons contributes to the overall mobility
and cause a peak with increasing temperature.

Figure 1.7: Electron mobility versus temperature for GaAs, with the temperature dependencies of ionized impurity and acoustic phonon scattering labeled.[17]

16

Ohm’s law simply states that under the presence of an electric field, E, a current density,
J will be produced which is proportional to the electrical conductivity of the material.[12]

J = σE

(1.29)

By combining equation (1.29) and the Boltzmann transport equation for electrical conductivity, the current density can be fully expanded as:[16]

∞
∂fo
e2 E
k 3 τ (k)
J= 2 ∗
dk
∂k
3π m 0

(1.30)

More commonly however the electrical conductivity is written very generally as:

σ = neµ

(1.31)

In the case of an intrinsic semiconductor, electrons in the conduction band and holes in
the valence band will both contribute to conduction:

σ = q(nµn + pµp )

(1.32)

where µn and µp are the electron and hole mobilities, respectively. The magnitude
and temperature dependence of electrical conductivity varies by the specific material. For
example, in metals and degenerate semiconductors the electrical conductivity decreases with
temperature, while, generally for extrinsic semiconductors, the electrical conductivity will
follow a trend very close to the carrier concentration as a function of temperature. The
magnitude of the electrical conductivity spans multiple orders of magnitude, with metals such
as copper on the order of 107 Ω−1 m−1 and fused quartz on the order of 10−18 Ω−1 m−1 .[21, 22]
17

1.2.2

Seebeck Coefficient

The Seebeck effect is at the crux of thermoelectric materials. At its most essential, it is
defined as the ratio of the voltage generated by a given temperature difference for a specific
material. The following section will give a more detailed explanation of the Seebeck effect
in various materials.
If a material, in an open circuit, is placed under a temperature gradient, ∇T , the charge
carriers will diffuse from hot to cold. The driving force of the temperature gradient is
compensated by an established electric field, E, which arises from the gradient of electrons
across the material. The ratio of the electric field to the temperature gradient is the material
specific Seebeck coefficient, S = E/∇T .[23] A dramatized schematic of this effect can be seen
in 3.14.

Figure 1.8: Seebeck effect in a single conductor. The Seebeck coefficient is defined as the
ratio of the electric field established, E, in the presence of a temperature gradient, ∇T .

The magnitude of the Seebeck coefficient can vary from a few µVK−1 for metals to as
high as 40000µVK−1 for insulators such as FeSb2 .[24, 25] The Seebeck coefficient can be
thought of as the amount of entropy per carrier (S ≈ C/e, where C is the specific heat).[26]
For a classical electron gas with average energy 3/2kB T , S ≈ kB /e ≈ 87µVK−1 . For metals

18

the relationship can be approximated as follows:

S≈

kB
e

kB T
EF

(1.33)

These equations give a qualitative understanding of the Seebeck effect, but to understand
it for non-degenerate and degenerate semiconductors, which are crucial for thermoelectric
applications, a more general expression must be derived. For materials with parabolic bands
and charge carrier scattering which can be assigned to a simple relaxation time approximation
(τ ≈ E λ ) the Seebeck coefficient can be expressed as:[16]
∞ ∗(5/2+λ)
0
∞ (∗3/2+λ)
0

k
S=− B
e

od ∗
+ λ ∂f
∂ ∗
od ∗
+ λ ∂f
∂ ∗

−η

(1.34)

where λ defines the energy dependence of the scattering relaxation time, ∗ is the reduced
energy, and η is the reduced Fermi energy. Equation (1.34) can be further reduced to the
form which includes Fermi integrals, similar to the case of carrier concentration:

k
S=∓ B
e

(5/2 + λ)F3/2+λ (η)
(3/2 + λ)F1/2+λ (η)

−η

(1.35)

This equation represents the generalized description of the Seebeck coefficient where
Fermi-Dirac statistics are necessary. In the case of a classical non-degenerate semiconductor,
equation (1.35) can be greatly reduced.[11]

k
5
S=∓ B
+λ−η
e 2

(1.36)

where the negative sign applies to n-type materials and the positive sign applies to p-type
materials. At high temperature λ = 1/2 for optical phonon scattering and 3/2 for ionized
19

impurity scattering.[20] It should be noted that in the case of non-covalent materials such
as ionic crystals, the relaxation time approximation is not applicable.[27] Figure 1.9 shows
Seebeck, S versus the reduced Fermi energy, η for the classical treatment (equation (1.36))
and Fermi-Dirac statistics (equation (1.35)). As η goes beyond a value of −2 the difference
in Seebeck coefficient between the classical and quantum treatments become appreciable;
the classical statistics are therefore no longer valid.

Figure 1.9: Seebeck coefficient versus η, the reduced Fermi energy (EF /kB T ) using FermiDirac and classical statistics. Notice that at around η = −2 the values of Seebeck for the two
treatments begin to diverge. As η approaches a value of 2, the classical treatment predicts
a Seebeck value of 0.

In the case of mixed conduction, such as in intrinsic semiconductors and past the intrinsic
temperature of extrinsic semiconductors, the Seebeck coefficient is affected by the electronic
and hole contributions (equation (1.37)).

S≈

Se σe + Sp σp
σe + σp
20

(1.37)

If n = p, equation(1.37) can be reduced to:

S≈

Se µe + Sp µp
µe + µp

(1.38)

Since the signs of the Seebeck coefficient for n-type and p-type materials are negative
and positive, respectively, mixed conduction where the mobilities for electrons and holes are
similar will result in very low values. The conclusion is that a single carrier type is ideal for
optimizing Seebeck coefficient. Figure 1.10 is the Seebeck coefficient versus temperature for
PbTe doped with monovalent Na at 0.1 and 1.0%. As the amount of dopant is increased
the overall magnitude of the Seebeck coefficient is decreased due to the increased carrier
concentration, but the decrease due to mixed conduction occurs at a higher temperature.
Because the hole concentration is higher in 1% Na doped PbTe, a higher number of electrons
is needed to offset the p-type Seebeck coefficient.

21

Figure 1.10: Seebeck coefficient versus temperature for 0.1 and 1.0% Na doped PbTe. Note
that the peak in Seebeck coefficient due to intrinsic excitation of carrier across the bandgap
happens at a higher temperature since the number of holes is greater in 1%Na-PbTe and
therefore a higher concentration of excited electrons is needed to compensate.[28]

In the case of strong degeneracy in a semiconductor (η ≥ −2), the Seebeck coefficient
can be expressed as:[11]

S=∓

π 2 kB (3/2 + λ)
3 e
η

(1.39)

Equation (1.39) can be further simplified under the assumption that the relaxation time,
τ is energy independent as:[29]

2
2kB
∗ T π 2/3
S=
m
3n
3e 2

(1.40)

Equation (1.40) reveals two important qualities about the Seebeck coefficient: 1. S is
directly proportional to m∗ , which implies that a band structure with wide flat bands will
yield a high S value; 2. S is inversely proportional to n and therefore implies an optimization
22

of S 2 n, which contributes to the power factor (S 2 σ) and the overall ZT. Goldsmid predicted
that the optimization of S 2 n happens around 0 ≤ η ≤ 1, which is a heavily doped or
degenerate semiconductor.[11]

1.2.3

Thermal Transport

The heat flow in an isotropic material obeys the relation originally formulated by Fourier:[30]

Q = −κ∇T

(1.41)

where Q is the heat flux density (Wm−2 ), κ is the thermal conducivity (Wm−1 K−1 ),
and ∇T is the temperature gradient (Km−1 ). The process of heat conduction is a diffusive
process, meaning that thermal energy propagates through a material by enduring frequent
collisions caused by scattering sites, such as defects, lattice atoms, impurities, electrons, etc.
The magnitude of thermal conductivity values fluctuate less for different materials than say
electrical conductivity, with maximums ranging from 1000 to 20000Wm−1 K−1 for metals
and non metals at room temperature.[31]
Heat is predominantly conducted through materials by electrons and quantized lattice
vibrations, called phonons, however other mechanisms of heat conduction have been shown
to exist by diffusion of electron-hole pairs[32], diffusion of excitons[33], and transport of
photons[34]. These mechanisms are not pertinent to this work and will not be discussed.
The total thermal conductivity can therefore be defined as the summation of the electronic
and lattice contributions, κ = κe + κL . The following sections will further outline the theory
and importance behind both contributions.

23

1.2.3.1

Electronic Thermal Conductivity

Electrons, in addition to charge, can act as conductors for heat through a material. Metals
such as copper and silver have relatively high thermal conductivity values, which can be
traced to their high electrical conductivity. Recall that the electrical conductivity can be
expressed as σ = neµ, or more fully as:

ne2 l
σ=
2me v

(1.42)

where, n is the electron concentration, l is the electron mean free path, me is the electron
mass, and v is the average thermal velocity (the scattering relaxation time, which was used
in the previous section can be defined as τ = l/v). In a classical treatment the electrons
behave as a kinetic gas, and the thermal conductivity is therefore expressed as:[31]

1
κ = nce vl
3

(1.43)

where ce is the heat capacity per electron. The electronic contribution to heat capacity is
defined as Ce = nce . The Wiedemann-Franz law, a ratio of thermal conductivity to electrical
conductivity, can be expressed using equations (1.42) and (1.43).

κ
2 m e v 2 ce
=
σ
3 e2

(1.44)

By using a quantum statistical treatment, ce fully expressed is proportional to T , and v
is considered constant, so that equation (1.44) becomes:[31]

κ
π2
=
σ
3

kB 2
T
e
24

(1.45)

Equation (1.45) is defined as the Wiedemann-Franz-Lorenz law and shows the ratio of
thermal to electrical conductivity. At a given absolute temperature κ/σT is a constant. The
constant is known as the ideal Lorenz number and is 2.45x10−8 WΩK−2 . Most often however
there is disagreement between the experimental values of Lorenz number for metals and the
ideal value, most notably in the intermediate temperature range, where assuming a classical
treatment is invalid. In actuality, the Lorenz number Lo is governed by the way the electron
scattering time depends on the electron velocity and can take on a range of values from
(1 − 4) ∗ 10−8 WΩK−2 .[14] In a non-degenerate semiconductor, where Boltzmann statistics
are applicable and assuming a simple energy dependence for the scattering relaxation time
(τ = τo E λ ), the Lorenz number can be expressed as:[31]

Lo =

kB 2
(5/2 + λ)
e

(1.46)

where λ determines the energy dependence of the relaxation time. It should be noted
that for most semiconductors, unless heavily doped, the electronic contribution to thermal
conductivity is negligible compared to the lattice thermal conductivity. For example, in the
case of silicon, the electronic contribution at high temperature contributes approximately 5%
to the overall thermal conductivity. In the case of a heavily doped, or degenerate semiconductor, where Fermi-Dirac statistics are necessary, the Lorenz number is given in terms of the
energy dependence of the scattering relaxation time (λ) and Fermi integrals (Fλ+s (η)):[11]


(λ + 5/2)Fλ+3/2 (η)
(λ + 7/2)Fλ+5/2 (η)
L = (kB /e)2 
−
(λ + 3/2)Fλ+1/2 (η)
(λ + 3/2)Fλ+1/2 (η)

2




(1.47)

For a more complete understanding of thermal transport in materials, especially in semiconductors, an understanding of the lattice contribution to thermal conductivity is essential.
25

1.2.3.2

Lattice Thermal Conductivity

For non-metals and materials with a low electronic thermal conductivity relative to the
total thermal conductivity the lattice contribution to thermal conductivity is the dominant
mechanism of thermal transport. The propagation of heat via lattice atoms is done through
the diffusion of phonons. Phonons are quantized modes of vibration for a given lattice
of atoms. By assuming the atomic bonds are approximated as spring-like forces the energy
states for phonons are quantized, as for a quantum harmonic oscillator (E = ω(n+1/2)).[35]
For a monoatomic, linear chain of atoms with mass M , held together with harmonic forces
that have a spring constant of k, and separated by a distance a, the frequency as a function
of wavenumber, q can be expressed as:[31]

ω(q) = 2

k/M |sin(qa/2)|

(1.48)

This relation is called the dispersion curve, and ranges from ±π/a, which is termed the
Brillouin zone, as can be seen in Figure 1.11(a).[12] The dispersion curve for a monoatomic
chain is called the acoustic branch. Notice as the curve approaches the Brillouin zone edge
the group velocity (vg = dω/dq) approaches zero. For a similar linear chain where there are
two atomic species with different masses the creation of a second higher frequency branch
call the optic branch appears, as seen in Figure 1.11(b). The optic branch has low group
velocity, and therefore generally does not contribute to heat conduction.

26

Figure 1.11: (a) Model of phonon dispersion curve for a one-dimensional monoatomic lattice,
with equal masses and atomic spacing a. The result is a single acoustic branch. (b) Phonon
dispersion curve for one-dimensional diatomic lattice (unequal masses). The creation of the
higher frequency, low group velocity optic branch is a result of of the two atomic species
with unequal masses.

In the case of a real crystal, which is 3-dimensional, the dispersion curves seen in Figure 1.11 are oversimplified. For the general case of N types of atoms there will exist three
acoustic branches, one longitudinal and two transverse, and 3(N − 1) optic branches.
The total heat current in a material contributed by all phonon modes is expressed as:[31]

h = ΣN (q) ωvg (q)

(1.49)

where N (q) is the number of phonons with mode q. This distribution is determined from
Bose-Einstein statistics, since phonons may be treated as bosons. The thermal conductivity
is then found by dividing the heat current by the temperature gradient along a material.
By further adopting some of the simplifying assumptions of the Debye model, i.e. a linear
dispersion relation for all branches, and equal group velocities among acoustic branches, the
Boltzmann transport equation for thermal conductivity can be expressed as:[31]

27

1
κ= v
3

l(x)C(x)dx

(1.50)

where x = ω/kB T , l(x) is the phonon mean free path, and C(x)dx is the differential
contribution to the heat capacity (in the Debye approximation). For the simplistic volume
averaged case, developed by Debye, equation (1.50) can be further simplified and expressed
as:

1
κ = Cvl
3

(1.51)

It is typical to describe the potential energy of the bonds in a crystal as harmonic.
If this assumption was true the phonons propagating through a defect-free crystal would
experience no interactions and the thermal conductivity would tend to infinity. However,
even ideal single crystals have finite thermal conductivity in reality. To understand why
there is inherent phonon resistance in all crystals it is vital to understand that the potential
energy of the atomic bonds is not quadratic. The anharmonic displacement of atoms caused
by a phonon will therefore interact with other phonons. The interaction probability for a
three phonon process, two initial phonons of wave vectors, q1 and q2 , and a final phonon,
q3 can be simply written as:[31]

q1 + q2 = q3 + g

(1.52)

where g is a reciprocal lattice vector. If the interaction of q1 and q2 results in a
mode q3 which lies within the Brillouin zone, g = 0. These interactions are called Normal processes.[36] When q1 and q2 sum to a value which is outside the Brillouin zone,

28

g is equivalent to the reciprocal lattice (±2π/a). Adding or subtracting g then folds the
resulting phonon mode back into the Brillouin zone. This process is called Umklapp scattering and reverses the direction of the phonon flux thereby adding resistance to the heat
flow.[37] Umklapp processes give rise to thermal resistance, while normal processes do not
and can be effectively ignored when determining thermal conductivity. At sufficiently high
temperatures, ω/kB T becomes small for all phonon modes and the probability of Umklapp
processes taking place increases. The lattice thermal conductivity due to the anharmonic
Umklapp scattering can be expressed as:[38]

κL = A

3
Ma aθD
2
N 3 T γ2

(1.53)

where A is a material specific proportionality constant, N is the number of atoms per
unit cell, Ma is the atomic mass, a3 is the volume per atom, θD is the Debye temperature,
and γ is the Gr¨
uneisen parameter. The Gr¨
uneisen parameter is defined as the rate of change
of frequency of mode i with the volume of the crystal (γi = −∂lnωi /∂lnV ). The Gr¨
uneisen
parameter is essentially a measure of the anharmonic nature of the bond energy in a material
and is directly related to the thermal expansion coefficient.[31] It is clear to see from equation
(1.53) that the lattice thermal conductivity at temperatures above the Debye temperature
goes as 1/T with temperature.
At low temperature phonon-phonon scattering processes become less prominent. In this
temperature regime crystal/grain boundaries and defects become the dominant scattering
event for phonons. The lattice thermal conductivity associated with phonon scattering at
grain boundaries or crystal boundaries can be expressed as:[14]

29

κboundaries =

LT 3
C

(1.54)

where L is the linear size of the boundaries and C is a materials specific constant. The
thermal conductivity associated with defect scattering is dependent on the number of defects/impurities (N ) in the crystal:[14]

κdef ects = (BN T p )−1

(1.55)

where B is a materials specific constant, and p < 1. All of the various scattering mechanisms for thermal conductivity sum according to Matthiesen’s rule, similar to that for
electrical conductivity.

1
1
1
1
=
+
+
κL
κanharmonic κboundary κdef ects

(1.56)

The temperature dependence of thermal conductivity is determined by many factors, including the bonding nature in a material, the degree of crystallinity, the specific temperature
regime relative to the Debye temperature, and unit cell size to name a few. The thermal
conductivity as a function of temperature for sapphire (crystalline Al2 O3 ) can be seen in
Figure 1.12.[12, 39]

30

Figure 1.12: Thermal conductivity as a function of temperature for sapphire. The scattering
events which are dominant in their temperature regime are labeled: boundary scattering
at low temperature, defect scattering in mid temperature range, and anharmonic phononphonon (Umklapp) scattering at high temperature.[12]

1.2.4

ZT and Methods of Enhancement

The majority of today’s state of the art thermoelectric materials exhibit peak ZT values
around unity or slightly higher. The low efficiency and often expensive manufacturing of
these compounds are to blame for thier lack of commercial use. Specific examples include
compounds such as PbTe and AgSbTe2 -GeTe alloys, which have ZT approaching 2, but
which are brittle, not cost-effective and contain toxic elements.[40, 28] Oxide-based thermoelectrics such as BiCuSeO and SrTiO3 are stable at high temperature, but lack sufficiently
high ZT values.[41, 42] Materials based on the Skutterudite mineral, CoSb3 , have been extensively researched and are now being integrated into radioisotope thermoelectric generators
for power generation applications in deep space voyages.[43, 44] Cu-based compounds have
garnered much attention recently because of their low cost and low thermal conductivity and
31

have been shown to yield ZT values near unity.[45, 46, 47]
The low efficiency of thermoelectric materials, as aforementioned, can be traced to the
contraindicated properties of the dimensionless figure of merit, ZT. In terms of the Seebeck coefficient, S, the electrical conductivity, σ, thermal conductivity, κ, and the absolute
temperature, T , ZT is expressed as:

ZT =

S 2σ
T
κ

(1.57)

By substituting the one carrier approximation for electrical conductivity (σ = neµ) ZT
can be further expressed as:

ZT = eT (S 2 n)

µ
κ

(1.58)

Equation (1.58) more clearly displays the contraindicated nature of ZT. The Seebeck coefficient is, in general, inversely proportional to the carrier concentration, while the electrical
conductivity is directly proportional. There is an optimization point for S 2 n as a function
of carrier concentration. Also, the ratio of mobility, µ to the thermal conductivity, κ is
directly contradictory. Increasing the number of scattering centers in a crystal will lower
the thermal conductivity, but also lower the mobility. Using a single parabolic band model
the optimization of power factor and ZT as a function of charge carrier concentration can
be seen in Figure 1.13. For insulators (≤ 1018 cm−3 ) the electrical conductivity is too low,
while for metallic materials (≥ 1022 cm−3 ) the Seebeck coefficient is too low. Power factor
is optimized for a heavily doped or degenerate semiconductor.

32

Figure 1.13: Seebeck coefficient (S), electrical conductivity (σ), power factor (S 2 σ), and
ZT fo an arbitrary thermoelectric material. Optimization of power factor and ZT are done
at relatively high charge carrier concentrations of approximately 1019 − 1020 cm−3 , which
corresponds to heavily doped, or degenerate, semiconductors.

Adjusting the carrier concentration in a semiconductor is an effective means of optimizing thermoelectric performance. Because the electronic thermal conductivity is tied to the
electrical conductivity, lowering the lattice contribution has been shown to be an effective
means of increasing overall ZT for a number of materials as well.[48] The following sections
will give an outline of the various theoretical and experimental techniques used in the field to
enhance the power factor and decrease the thermal conductivity of thermoelectric materials.

1.2.4.1

Optimizing Power Factor

One of the most routine methods for enhancing and optimizing the power factor in thermoelectrics is through doping. By adding foreign atoms which contribute one less (p-type) or
one extra (n-type) electron than the host atoms, the charge carrier concentration in a semiconductor can be controlled. Many of the properties and equations associated with doping
33

to tune carrier concentration were explained in previous sections.
In 1993 Hicks and Dresselhaus published work outlining the possibility of enhancing the
Seebeck coefficient of materials by manipulating the density of states.[49] The idea can be
explained by examining the Mott equation of the Seebeck coefficient for metals which is
expressed as:

2T
π 2 kB
1
dσ(E)
S=
3e σ(E)
dE E=Ef

(1.59)

By then substituting the differential energy form of electrical conductivity, equation (1.59)
can be expressed as:

S=

2T
π 2 kB
3e

1 dτ (E)
2 dv(E)
1 dg(E)
+
+
g(E) dE
τ (E) dE
v(E) dE E=Ef

(1.60)

where g(E) is the electronic density of states, τ (E) is the electron relaxation time, and
v(E) is the electron velocity. It is clear to see from equation (1.60) that a sharp increase
in the density of states at the Fermi energy is beneficial for the Seebeck coefficient. Recall
however that the typical density of states for a bulk material has a square root dependence on
the energy. Hicks and Dresselhaus showed that by reducing the dimensionality of a material
the density of states as a function of energy could be fundamentally altered. Specifically,
for 2D quantum wells, 1D nanowires, and 0D quantum dots, there exists discontinuities
at quantized energy levels in the density of states. In theory these nanostructures could
be endotaxially incorporated into bulk materials, meaning an alignment of the lattice parameters. This would allow uninhibited electron transport and altered density of states,
and ultimately, and increased Seebeck coefficient. The proof of concept demonstration was
initially reported using PbTe/Pb1−x Eux Te quantum well superlattices synthesized by molec34

ular beam epitaxy.[50, 51] Unfortunately, due to the complicated and intricate synthesis of
lower dimensional nanostructures, little experimental work beyond this has been reported.
Engineering the band structure is another method towards enhancing power factor.
Specifically engineering degeneracy into the band structure increases the effective density
of states and has been shown to increase electrical conductivity and Seebeck coefficient.[52]
This situation can take place when either multiple band extrema overlap at essentially the
same energy (within a few kB T ) or multiple carrier pockets exist within the Brillouin zone
due to crystal symmetry.[53] Many of today’s state of the art thermoelectric materials have
been shown to exhibit band degeneracy, for example Bi2 Te3 [54], PbTe[53], CoSb3 [55], and
SnSe[56].
Aside from doping, the examples provided above can be complicated and expensive to
carry out. Much of the success in thermoelectrics engineering has been achieved through
lowering the lattice thermal conductivity. Some of these principles will be discussed in the
next section.

1.2.4.2

Achieving Low Thermal Conductivity

In 1956 Ioffe first suggested that optimizing the ratio of mobility to lattice thermal conductivity would be an effective route to optimizing ZT.[57] Point defects, defined as defects
which are much smaller than the phonon wavelength, scatter phonons and can lower lattice
thermal conductivity. The classification of point defects can include atomic substitutions,
vacancies, and interstitial atoms. For substitutional atoms which have the same valence
configuration as the host atom, and differ only by mass, a simple perturbation theory can

35

be used to express the scattering rate as:[58]

1
τ∆M

=

cp a3 ω 4 ∆M 2
M
4πv 3

(1.61)

where cp is the ratio of the number of point defects to the number of lattice sites, a3 is
the average volume per atom, ω is the phonon frequency, and v is the phonon group velocity.
It should be noted that the scattering rate associated with mass fluctuation point defects
is independent of temperature and quadratically dependent on the mass difference between
the host and substituted atom. The thermal conductivity versus mol fraction InAs for a
GaAs-InAs solid solution can be seen in Figure 1.14; a reduction in thermal conductivity of
nearly an order of magnitude was achieved.

Figure 1.14: Thermal conductivity versus mol fraction of InAs in GaAs.[59]

In the early 1990s Slack proposed that good thermoelectric materials would, in general,
have a large unit cell, heavy constituent atoms, and high carrier mobilities.[60] These se-

36

lection rules outline what is called the phonon-glass electron-crystal (PGEC) approach to
materials selection.[61] In this model the ideal thermoelectric material would exhibit electronic mobility akin to a single crystal, but have poor thermal properties, like a glass. One
of the first demonstrations of this concept was in the use of Ge-based clathrate materials.[62]
Clathrates have cage-like structures and have been shown to exhibit minimum thermal conductivity values while maintaining high carrier mobility. The Skutterudite compound, generally written as M X3 , where M represents a metal atom and X a pnictide atom, crystallizes
in a large cubic structure with high carrier mobilities.[63] To minimize the thermal conductivity rare earth atoms such as Ce, Yb, and La are substituted into the interstitial sites of
the Skutterudite unit cell.[43] These rare earth atoms are called filler atoms, or rattlers, and
are so named because of the specific frequency mode at which they rattle. This vibrational
mode then scatters phonons and greatly lowers thermal conductivity.[64] The mineral based
compound, tetrahedrite (Cu12 Sb4 S13 ) has recently garnered attention. The material exhibits
attractive thermoelectric properties, with high levels of anharmonicity and a large unit cell,
all of which lead to minimized thermal conductivity.[65] Recently, it was shown through
experimental and computational techniques that the some of the Cu atoms, which have
lone-pair electrons, resonate like a rattler atom, thus classifying it as a PGEC material.[66]
In 2012, Biswas et al used hierarchical structuring in Na-doped PbTe to achieve record
high ZT values.[67] They achieved multi-scaled phonon scattering through alloying, nanostructured precipitation, and grain size reduction. Since the total lattice thermal conductivity
is a sum of the phonon relaxation rates related to various scattering mechanisms, using a
myriad of defects to scatter phonons across multiple wavelengths can help minimize the lattice thermal conductivity. It should be noted however that the synthesis associated with
these materials is timely and expensive, and so commercial applicability is limited. In the
37

following work alloys and compounds based on Ge-Sb-Te phase change materials were synthesized and characterized. A more in-depth discussion of these materials is presented in
chapter 3.

38

Chapter 2
Background: Phase Change Materials
In 1965 Gordon Moore postulated that the number of transistors per square inch on integrated circuits would double every year, which has been shown to correlate well with the
growth of computers.[68] To maintain this trend the size of the modern transistor is now
within the range of non-negligible quantum effects, such as electron tunneling. An alternative area of improving computer performance is memory. Today the primary memory storage
devices in computers are based on random access memory (RAM). RAM technologies have
inherent downsides such as volatility and the necessity of relatively large magnetic fields.
Phase change memory is the next generation of memory technology now being implemented
in some of the new memory designs of companies such as Intel.[69]
Though Stanford Ovshinsky is generally credited with developing phase change materials
for memory applications, it was Alan Waterman in the early 1900s who originally discovered
the resistive switching character of MoS2 . Waterman discovered that with the application
of electrical heating MoS2 would undergo multiple resistive states, namely α: the high
resistance state and β: the low resistance state. These resistive states were both stable
and reversible upon thermal or photonic switching.[70] In 1968 Ovshinsky reported on the
electrically controlled threshold switching of Si-Ge-As-Te compounds while also outlining the
utilization of other chalcogenide alloys for use in memory devices which exploit this switching
39

effect.[71] Based on the pioneering work of Ovshinsky, phase change materials can now be
found in nonvolatile and rewriteable applications such as CD’s, DVD’s, and Blu-Ray discs.
In the following section an explanation of the function and nature of these memory devices,
crystal structure, and applicability in thermoelectrics will be presented.

2.1

Phase Change Memory

Dynamic random access memory (DRAM) and static random access memory (SRAM) have
natural drawbacks when compared with today’s state-of-the-art memory storage alternatives.
DRAM and SRAM can currently be used for approximately 100,000 rewrite cycles before
degrading and are volatile, which means data is stored only while power is supplied. Nonvolatile phase change memory (PCM) operates on the same principle as CDs and DVDs, but
utilizes the swinging electrical properties rather than optical properties. In a typical PCM
device, as is illustrated in Figure 2.1, a large, high-intensity electrical (or laser pulse) is used
to reset a local area of phase change material to the amorphous phase. This amorphous
phase is of a high resistance state, while the crystalline state is of low resistance. To set
the state, a slightly longer duration, low-intensity electrical (or laser pulse) is applied to the
phase change material allowing for crystallization. This crystallization happens on extremely
small time-scales (approximately 100s of nanoseconds). Operational devices using a similar
setup to that seen in Figure 2.1 have been demonstrated by companies such as Samsung,
Intel, and IBM.[69, 72, 73] The unique ability of phase change materials to quickly alternate
between the crystalline and amorphous phases, while also demonstrating high stability and
structural integrity is a large requirement. The majority of today’s state of the art phase
change memory alloys can be traced to the binary tie-line of GeTe and Sb2 Te3 , as can be

40

seen in Figure 2.2.

Figure 2.1: Schematic of memory unit using phase change materials where the high resistance state (reset state) is activated with a short, high intensity electrical pulse and the low
resistance state (set state) is activated with a long, low intensity electrical pulse.

Figure 2.2: Ternary phase diagram of Ge, Sb, Te with the pseudo-binary tie-line of GeTe
and Sb2 Te3 and Ge4 SbTe5 highlighted.

The study of GeTe-Sb2 Te3 (GST) alloys for phase change memory applications, typically
41

involves the synthesis of the compounds in thin film form. Thin film deposition techniques
such as chemical vapor deposition (CVD), atomic layer deposition (ALD), and molecular
beam epitaxy (MBE) are all techniques which allow for rapid cooling rates.[74] Using these
techniques with Ge-Sb-Te phase change materials yield the amorphous phases necessary for
storing data. The following sections will discuss the crystal structure, and related phase
transitions, for GeTe-Sb2 Te3 alloys.

2.2

Structure and Bonding

The ability to switch between the amorphous and crystalline states is not a unique property to phase change materials, but the large optical/resistive contrast found in these compounds sets them apart.[75, 71, 76] The origin of the optical contrast between the amorphous
and crystalline states has been a subject of experimental and theoretical investigations.[77]
For traditional semiconductors such as Si and Ge, the transition from crystalline to amorphous yields only a change in the long-range order; the local tetrahedral bonding structure is maintained, but in Ge-Sb-Te compounds the crystalline phase exhibits octahedral
coordination.[78] Specifically, the Ge atoms, Sb atoms, and vacancies randomly occupy the
cation site of the rocksalt crystal structure with Te occupying the anion sublattice. Upon
amorphization the local structure changes to tetrahedral coordination. This also changes the
number of bonds between Ge and Te. The change in local structure and number of bonds
has been shown experimentally and theoretically to be the origin of the optical contrast.[79]

42

2.2.1

Resonance Bonding

It has recently been shown that resonance bonding is a vital characteristic of phase change
materials.[80] Resonance bonding is a term originally coined by Linus Pauling to describe
the bonding nature in benzene.[81] Recently the theory has been extended to explain the
undersaturated bonding nature in GST phase change materials. Resonance bonding is more
generally defined as the superposition of symmetry-equivalent saturated bond configurations,
which can be expressed using equation (2.1).[82]

ψ=√

1
1 + α2

(ΦI + αΦII )

(2.1)

Most materials, such as Si, Ge, GaAs, and CuInS2 have a mixing coefficient, α, which is
very small or very large resulting in the covalent bonding structure of either ΦI or ΦII . In
the case of materials with high levels of resonance bonding α is very close to unity, which
yields a ground state wave function ψ that involves both bonding structures, ΦI and ΦII .
In other words, for typical materials such as Si the average number of valence electrons per
atom is equal to 4. This results in sp3 hybridization and therefore tetrahedral bonding, but
in (GeTe)m Sb2 Te3 alloys the average number of valence electrons per atom is closer to, or
equal to, 3. This bonding scenario is primarily promoted by the p electrons and yields no sp3
hybridization, but rather delocalized electrons which are shared in the octahedral bonding.
The resonance bonding in phase change materials has a direct effect on the electrical
and thermal properties. It has been shown that the high polarizability and incipient ferroelectricity in Ge-Sb-Te phase change materials is due to the strong electron-phonon interactions which are a direct effect of the resonance bonding.[83, 80, 84] The resonance bonding
and associated distorted structure have also been shown to produce the degenerate p-type

43

semiconducting nature seen in (GeTe)m Sb2 Te3 alloys.[85, 86] Also the highly disordered
structure and soft transverse optical phonon modes generated by the resonance bonding
yield low thermal conductivity and low frequency optical phonon contributions to thermal
conductivity.[87, 88, 89]

2.2.2

Crystal Structure

GeTe exists in the rhombohedral crystal structure (R¯3m) below 673K. This rhombohedral
structure can be understood as the distortion of the symmetric 6-fold coordinated rocksalt
structure (F m¯3m) along the [111] crystal direction. This is consistent with the theory of
Peierls distortion. Rather than 6 energy equivalent bonds, the unit cell of GeTe collapses
to allow for 3 short bonds and 3 long bonds.[85] Group IV elements typically undergo the
sp3 hybridization, but this general rule falls short for elements such as Sn and Pb when
relativistic effects become non-negligible and hybridization does not exist.[90]
On the other extreme of the GeTe-Sb2 Te3 pseudo-binary tie-line is Sb2 Te3 whose crystal
structure can also be understood as a distorted cubic, or rhombohedral structure. Unlike
GeTe, Sb2 Te3 has an unequal number of cations and anions. The specific stacking sequence
goes in units T e − Sb − T e − Sb − T e with Te layers weakly bound by van der Waals forces.
The crystal structure of these two compounds can be seen in Figure 2.3.

44

Figure 2.3: Crystal structure of GeTe and Sb2 Te3 , which has repeating vacancy layers between Te layers.These structures were generated using VESTA.[91]

(GeTe)m Sb2 Te3 phase change materials can be best thought of as an intermediary between the crystal structures of GeTe and Sb2 Te3 . In the metastable rocksalt structure,
which forms after the crystallization of the amorphous phase, Ge and Sb atoms, and vacancies randomly occupy the cation site, and Te atoms occupy the anion sublattice, while
in the rhombohedral structure long repeating chains of Ge, Sb, and Te are interrupted by
vacancy layers which connect the Te layers by van der Waals bonding. For the specific case
of Ge2 Sb2 Te5 the rocksalt and rhombohedral structures can be see in Figure 2.4.

45

Figure 2.4: Rocksalt and Rhombohedral Crystal Structure of Ge2 Sb2 Te5 . These structures
were generated using VESTA.[91]

The phase transition from the metastable rocksalt crystal structure seen in (GeTe)m Sb2 Te3
phase change materials to the stable rhombohedral structure is an ordering transition. The
phase transition is structural in nature, but is also accompanied by a change in the electrical
properties. The transition is categorized as an insulator-metal transition and will be further
explained in the following section.

2.2.3

Insulator-Metal Transition

Upon heating, the amorphous phase of GST materials transitions to the metastable cubic
rocksalt structure, as was aforementioned. Upon further heating, the metastable cubic phase
undergoes an ordering transition to the stable rhombohedral phase. This phase transition is
structural in nature as the randomly assorted Ge and Sb atoms, and vacancies, order to form
the layered trigonal R¯3m phase. However, in addition to the structural phase transition a
change in the electrical resistivity is also seen at elevated temperature in GST materials.

46

Insulators can in general be defined as displaying a temperature coefficient of resistivity
dρ
which is less than zero ( dT
< 0), while metals display a positive temperature coefficient of
dρ
> 0). Some materials display a transition in the electrical resistivity from
resistivity ( dT

insulating to metallic. This is categorized as a insulator-metal transition (IMT). One of
the first theoretical investigations of IMTs was given by Mott in 1968.[92] He found that if
the Coulomb energy (EC ) exceeds the Fermi energy (EF ) in materials, insulating behavior
will be observed. Because both EC and EF are strongly dependent on the charge carrier
concentration, there exists a critical charge carrier concentration for which EC will either
always be less than or greater than EF . For GST materials the critical carrier concentration
exceeds the Mott criterion and can not be explained in terms of a Mott transition.
A recent study of the phase change material GeSb2 Te4 found that charge carrier mobility
is anomalously low, with values less than 10cm2 V−1 s−1 for the rocksalt structure.[93] For
most undoped semiconductors charge carrier mobility exceeds values of 100cm2 V−1 s−1 . The
small range of mobility seen for GeSb2 Te4 implies highly localized charge carriers, as is also
observed in amorphous semiconductors.[94] As the temperature of the material is increased
the carrier mobility is increased nearly two orders of magnitude with little change in the
carrier concentration, as can be seen in Figure 2.5.

47

Figure 2.5: (a)Electrical conductivity versus temperature around the insulator-metal transition (represented here by dashed line) for GeSb2 Te4 , (b) and (c) hole concentration and hole
mobility, respectively, around the insulator-metal transition. (d) Schematic of the mobility
edge, Eµ in relation to the Fermi energy, EF , which is the governing principle of an Anderson
localization insulator-metal transition. In this schematic, zero energy is defined as the top
of the valence band. The data in this figure was digitized and replotted from the work of
Siegrist, et al. [93]

Though one would initially assume the IMT is associated with the structural phase
transition, and therefore band gap, there is little change in the band gap from rocksalt to
rhombohedral. Rather the ordering of the vacancies causes an increase in the hole mobility
with little change in the hole concentration, as can be seen in Figure 2.5. This type of IMT
can be described as an Anderson localization transition. The phenomenon is named after
the physicist P.W. Anderson, who originally proposed the possibility of localized electrons in
a semiconductor with a sufficiently high defect or impurity concentration.[95] As Anderson
48

outlined, the mobility edge, Eµ , is the energy at which electrons are localized. If EF lies
within these localized states, insulating behavior is seen, or |Eµ |> |EF |. As the temperature
increases the mobility edge moves towards the Fermi energy(|Eµ |≈ |EF |). Eventually the
mobility edge crosses the Fermi energy, and electronic states become delocalized (|Eµ |<
|EF |). A schematic of this effect can be seen in Figure 2.5(d).
Anderson localization transitions have been observed in photonic systems such as quantumcorrelated materials and Bose-Einstein condensates.[96, 97] The phenomenon has also been
observed in materials such as Lix Fe7 Se8 and GST phase change materials.[98, 99]

2.3

Thermoelectric Properties

Many of today’s state of the art thermoelectric materials are derived from chalcogenide based
compounds. PbTe for example has been studied for thermoelectric power generation since the
1950s and is still being engineered to unrivaled ZT values.[100, 101] Bi2 Te3 and Sb2 Te3 were
first studied in the 1950s and are still used for room temperature thermoelectric applications
and Peltier cooling.[102] GeTe and AgSbTe2 , have also been studied for thermoelectrics since
the 1960s.[40] Much of the success of this class of materials in thermoelectrics is attributed
to low frequency phonons which lend to low lattice thermal conductivity and high charge
carrier mobility.[103] The specific alloys which pertain to state of the art phase change
materials, namely (GeTe)1−x (Sb2 Te3 )x alloys have only recently begun to catch the attention
of the thermoelectrics research community. The science and understanding behind GST
phase change materials is still emerging, and in the following chapters various aspects of
the thermoelectric properties of Ge-Sb-Te based materials will be explored in our quest to
understand and engineer this intriguing class of materials.

49

Chapter 3
Experimental Procedures
The process of characterizing thermoelectric materials requires a number of techniques. For
instance, understanding the crystal structure, microstructure, and elemental composition
of the materials synthesized requires techniques such as X-ray Diffraction and Scanning
Electron Microscopy/Energy Dispersive Spectroscopy. Additionally, characterization of the
electrical and thermal properties is necessary to quantify the ZT and further understand
the fundamental processes which elucidate high-ZT thermoelectric materials. The following
sections will describe the experimental procedures used for the synthesis, processing, and
characterization of the materials studied in this work.

3.1

Materials Synthesis

The following sections will describe the processes used to synthesize the bulk polycrystalline
materials used in this work. Specifically, the general procedure used to achieve highly dense
single phase GeTe-Sb2 Te3 alloys will be presented.

50

3.1.1

Heating Procedures and Powder Processing

The following description of the process used to synthesize Ge-Sb-Te compounds is general
in nature. In sections where additional chemicals and procedures are used a more specific description of the experimental processes will be presented. Stoichiometric amounts of
Alfa Aesar germanium (Ge ingot, zone-refined, 99.9999%), Alfa Aesar antimony (Sb shot,
99.999%, 6 mm and smaller), and Alfa Aesar tellerium (Te lump, 99.999%) were placed in
SiO2 ampoules with 10mm inner diameter. The ampoules were evacuated to approximately
10−5 Torr and sealed using an oxygen-methane torch.
The samples were then heated to 900◦C at a rate of 2◦Cmin−1 and held for 12h. Samples
were quenched using a room temperature water bath. Additional heat treatment at 590◦C for
24h followed by water quenching was performed. These heating procedures were performed
using a ThermoFisher BF51800 box furnace.
The obtained ingot was then crushed and ball-milled using a SPEX MixerMill 8000D
with a stainless steel jar and 6 stainless steel balls for 5min This process ensures a fine, homogeneous powder, which is needed for further densification. The densification of materials
in this work was achieved by spark plasma sintering, which will be discussed further in the
following section. A graphic of the heating procedures and sample through the various steps
of synthesis can be seen in Figure 3.1.

51

Figure 3.1: (a) Heating procedure shown here as temperature (◦C) versus time (h) (b) Raw
elements sealed in silica ampoule before heat treatment (c) Ingot obtained after melt and
anneal process (d) Final disk obtained after densifying via spark plasma sintering.

3.1.2

Spark Plasma Sintering

Sintering is the act of consolidation of a material through the application of heat and/or
pressure without melting, which usually results in an increase in the density.[104] When
sintering ceramic materials, such as those used for thermoelectric applications, the heat and
pressure used to sinter can be generated in a number of ways. Hot-pressing is a method
by which pressure is applied directly to a material while heat is supplied externally. This
process can takes upwards of hours to achieve density yields greater than 90%. The long
sintering times associated with hot-pressing can cause undesirable grain coarsening, leading
to increased grain size and the destruction of favorable thermoelectric performance.[105] An
alternative application of heat during sintering is through the introduction of large electric

52

fields, which is called spark plasma sintering (SPS), electric-field assisted sintering (EFAS),
or pulsed electric current sintering (PECS). In the SPS process high amperage (100A−600A)
DC current is pulsed on for 12s and off for 2s. The applied current induces joule heating
within the sample and die, which allows for rapid temperature achievements. Additionally,
the on/off behavior of the current is also said to induce high energy spark discharge between
neighboring particles within the material, which in turn results in necking. The process of
plasma creation is widely debated however and has yet to be unequivocally proven.[106]
SPS was used for the densification of all bulk thermoelectric materials studied in this
work, which resulted in density yields greater than 95% theoretical in a matter of minutes.
The specific parameters used such as time, temperature, and pressure were dependent on the
composition and will be outlined in each corresponding section later, aside from Ge17 Sb2 Te20 ,
which will be used as an example here. SPS was performed using a Dr. Sinter-Lab spark
plasma sintering system, model SPS-211Lx. Powder of Ge17 Sb2 Te20 was placed in a 10mm
graphite die and sandwiched between two layers of graphite foil and graphite plungers.
Uniaxial pressure was applied on the graphite plungers while a type-K thermocouple was
placed into a hole drilled into the side of the graphite die. A programmed temperature
controller was used to control the ramp rate and holding temperature/time for a given
sample. For example, Ge17 Sb2 Te20 was sintered via the following temperature profile: ramp
to 300◦C in 2min, ramp to 465◦C in 3min, hold at 465◦C for 5min, with an applied uniaxial
pressure of 45 MPa. This procedure was determined through numerous trials for optimization
for Ge17 Sb2 Te20 . All SPS runs were performed under vacuum levels of approximately 10−2
Torr. Software was used to digitally monitor the voltage, current, temperature, pressure,
z-axis displacement, and vacuum level in real time. The data generated from this monitoring
system was used to aid in developing the sintering profile of a given material.
53

Optimizing the sintering temperature and pressure for a given material is an important
and necessary step in the synthesis process. The above sintering profile for Ge17 Sb2 Te20
was utilized to achieve theoretical densities consistently greater than 99%. By adjusting
the sintering temperature and monitoring the z-axis displacement as the sample sinters
one can collect important data for understanding the sintering process. In Figure 3.2 the
displacement of plungers during sintering versus the sintering temperature can be seen.
Notice that increasing the temperature while keeping the hold time of 5min constant allowed
for an increase in displacement which directly translated to a higher density yield.

Figure 3.2: Z-axis displacement of sample during SPS versus temeprature for various holding temperatures (400◦C, 420◦C, 435◦C, 450◦C, 465◦C). A clear correlation between %
theoretical density yield and SPS temperature can be observed.

3.2

Characterization Techniques

The following sections will be a description of the techniques used to characterize the materials synthesized in this work. Most of the measurements were performed at Michigan State
54

University. However, all microscopy was performed using the equipment in the Center for
Advanced Microscopy overseen by Dr. Stanley Flegler, and laser flash techniques were performed by Karl Dersch using the equipment in Dr. Timothy Hogan’s lab in the Engineering
Research Complex.

3.2.1

X-ray Diffraction

X-ray Diffraction was performed using a Rigaku MiniFlex-II at room temperature with CuKα radiation. A small amount of sample was pulverized and mounted onto a glass slide and
measured at a rate of 3◦ min−1 in the range of 15−90◦ 2θ. High temperature X-ray Diffraction
on some of the samples in this work was performed at Oak Ridge National Laboratory in
collaboration with Dr. Edgar Lara-Curzio and Dr. Thomas Watkins.

3.2.1.1

Principle of X-ray Diffraction

X-ray diffraction can yield a wealth of information about the structure and phase purity of
a material. This information can be extracted from the shape, intensity, position along the
2θ axis, and number of diffraction peaks. To understand the principle of x-ray diffraction
it is best to imagine a periodic lattice of atoms, with spacing d. X-rays interact with the
atoms (in this case perfectly elastically) and are scattered off of the atoms, some with the
same incident angle (θ) to the plane normal, which can be visualized in Figure 5.2.

55

Figure 3.3: A schematic which showcases the principle of X-ray Diffraction with a periodic
lattice of atoms.

When the wavelength of the X-rays (λ) is equal to twice the spacing offset dsinθ the result
is constructive interference of the scattered X-rays. The relation between the interplanar
spacing, d, and the wavelength of incident X-rays, λ, is known as Bragg’s law of diffraction
and is expressed in equation (3.1).[107]

nλ = 2dsinθ

(3.1)

The planes of atoms in a given crystal lattice determine the diffraction of X-rays. These
atomic planes are expressed using Miller indices (hkl) which are an expression of their location in reciprocal space. For example, for a face-centered cubic crystal structure, the (110)
plane intersects 6 atoms, as can be seen in Figure 3.4.

56

Figure 3.4: The (110) plane in a face-centered cubic crystal structure.

Each diffraction peak in an X-ray diffraction pattern therefore corresponds to an (hkl)
plane in a given crystal structure. Once a peak is indexed with its respective Miller index the
location on the 2θ axis can be related back to the unit cell parameter. The general definition
for relating the d-spacing to the plane normal vector Khkl can be seen in equation (3.2).

dhkl =

2π
|Khkl |

(3.2)

The plane normal vector is defined as Khkl = hA + kB + lC, where A, B, and C are
defined as follows:

A=

2π(b × c)
2π(c × a)
2π(a × b)
,B =
,C =
a · (b × c)
a · (b × c)
a · (b × c)

(3.3)

In equation (3.3) a, b, and c are the real space lattice vectors. By applying the necessary
symmetry for a given crystal structure equation (3.2) can be greatly simplified. For example,
in the case of a cubic crystal structure the relationship between the lattice parameter a, (hkl),

57

and the interplanar spacing d can be expressed as:

a
dhkl = √
h2 + k 2 + l2
3.2.1.2

(3.4)

Structure Factor and Diffraction Intensity

In an actual material the scattering process for X-rays depends on the type of atom in the
lattice. The Thomson equation, essentially a low-energy limit of Compton scattering, expresses the intensity of coherently scattered electromagnetic waves as being inversely proportional to the square of the mass of the scattering particle.[108] The atomic nucleus therefore
plays a small role in scattering the X-rays, rather the electrons are the dominant scattering
mechanism. The intensity of diffraction for an X-ray diffraction pattern will therefore be
dependent on the atomic number Z for a given atomic species. The atomic scattering factor,
f describes the efficiency of scattering for a given atom, and along with the (hkl) plane,
plays an important role in determining the intensity and angle of a diffraction peak.
The intensity of the beam diffracted is determined in part by the structure factor, Fhkl ,
which can be written as follows:

N

fn e2πi(hun +kvn +lwn )

Fhkl =

(3.5)

i=1

Equation (3.5) also determines the allowed diffraction peaks for a given crystal structure.
For example, for the diamond crystal structure, all of the diffraction peaks must have (hkl)
which are either all odd, or all even and sum to a multiple of four.
In addition to the structure a number of other variables play a role in determining the
intensity of a diffracted beam. The multiplicity factor, p, considers the relative proportion

58

of (hkl) planes which contribute to the same reflection. The Lorentz-polarization factor
2 2θ
. The
subtracts away from the intensity of a diffracted peak and is expressed as 1+cos
2
sin θcosθ

absorption factor A depends on the diffraction geometry being used and accounts for any absorption taking place within the sample. The temperature contribution is expressed as e−2M
where M is dependent on the amplitude and scattering properties of the lattice vibrations
within the specimen. The overall intensity calculation can be seen in equation (3.6).[108]

I=

3.2.2

1 + cos2 2θ
2
|F | p[
]A(θ)e−2M
2
sin θcosθ

(3.6)

Differential Scanning Calorimetry

Differential Scanning Calorimetry (DSC) is a commonly used technique for the thermal
characterization of solids and liquids. The relatively simple and accurate measurement can
yield information about melting/crystallization behavior, solid-solid phase transformations,
glass transitions, degree of crystallinity, specific heat, and more. For the following work
DSC was performed on a Netzsch DSC 200-F3 Maia. Aluminum crucibles were used in the
temperature range of 50-600◦C. All measurements were performed under a constant flow of
Argon. Sample masses ranged from 50mg - 100mg. An Al2 O3 single crystal (100mg) was
used for the calculation of heat capacity.
For each run a baseline measurement was performed, in which an empty sample crucible
and reference crucible were measured within the temperature range of interest. DSC was
performed using the heat flux method. Two aluminum crucibles, each pierced to allow for
ventilation of sublimated gases, are placed on heaters. The heating rates of each are precisely
controlled and the heat flux to each crucible is monitored throughout the measurement. The
sample will change the thermal resistance as compared to the empty reference crucible and is
59

monitored as a function of the measurement temperature. Using the sapphire standard allows
for a ratio method of calculation whereby the heat capacity can be calculated. Figure 3.5 is
a schematic of the measurement setup used.

Figure 3.5: A schematic of the setup used for performing DSC. Each crucible is placed on
a heater which is precisely monitored to determine the amount of heat flux needed to heat
each crucible.

3.2.3

Scanning Electron Microscopy

Scanning Electron Microscopy(SEM) was performed in the Center for Advanced Microscopy
at Michigan State University. The specific instrument used was a JEOL 6610 LV, which was
also equipped with an Oxford EDX for elemental analysis. The JEOL 6610 LV provides high
resolution imaging within the range of 5X to 50,000X.
The most common imaging mode for SEM is through the collection of secondary electrons
which are produced from interactions of the electron beam with the material. The high
energy electrons of the beam interact inelastically with the atoms near the surface of the
material (typically only a few nanometers). These inelastic collisions then produce electrons

60

from the K-shell of the atoms which are collected. The collected electrons are then accelerated
and a photomultiplier converts the signal to a large flux of photons. These photons are then
digitally processed and used to create an image. The secondary electrons are typically very
low energy (≤ 50eV) and therefore originate from the surface of the material.
Some of the interactions between the electron beam and the material are elastic and result
in high energy electrons which are backscattered out of the material. The elastic collisions
are dependent on the mass of the atoms, where the higher mass atoms (higher Z-number) will
scatter more strongly and the lighter mass atoms (low Z-number) will scatter less strongly.
The result of the collection of these high energy electrons is an image where areas with high
average atomic number are brighter than those of low average atomic number.
Furthermore, these high energy interactions also generate X-rays which are characteristic
of the atomic number of the material at hand. Analyzing these X-rays can allow for an
elemental characterization of the material. This form of characterization is therefore called
energy dispersive X-ray spectroscopy (EDS, EDX, XEDS). The principle of EDX is based
on the fact that each atom has a characteristic atomic structure. Upon interaction with the
high energy electron beam an electron will be ejected from an energy level. An electron at
a higher energy state will then relax into the unfilled state thereby creating an X-ray. The
specific wavelength of the X-ray is dependent on the energy level the electron is transitioning
to and from. All of these transitions will produce X-rays which are characteristic of a specific
atomic species, and can therefore be used to analyze the elemental composition of a material.
A basic schematic of this process and the various energy levels analyzed in EDX can be seen
in Figure 3.6.

61

Figure 3.6: A schematic which depicts the principle of energy dispersive X-ray spectroscopy
(EDX).

The naming convention used for the types of electron transitions which produce X-rays
shows first a letter K, L, or M, followed by a Greek letter, such as α or β. The first letter
describes the states into which the higher energy electron is transitioning. The Greek letter
describes the number of states transitioned through to reach the destination state. For
example, an electron transition from the L state to the K state will produce an X-ray named
K α.

3.2.4

Hall Measurements

The charge carrier concentration is an important parameter in understanding the thermolectric properties of a material. To directly measure the carrier concentration Hall measurements were performed. These measurements are based upon the Hall effect, originally established by Edwin Hall in 1880.[109] Hall found that when a magnetic field is applied to a
material, orthogonal to the current flow, the electrons are deflected resulting in a measurable
transverse voltage, ∆VH . A schematic of the setup for measuring the Hall effect can be seen
62

in Figure 3.7, where B is the applied magnetic field, IS is the applied current, ∆VH is the
Hall voltage, ∆VS is the resistive sample voltage, which is used to calculate the electrical
resistivity, w is the sample width, t is the thickness, and p is the probe seperation associated
with measuring ∆VS .

Figure 3.7: A typical mounting setup for measuring the Hall effect, and thereby the carrier
concentration, in a material.

With no magnetic filed present, charge carriers move along the length of slab and no
transverse voltage is generated. The deflection of charges due to the presence of a magnetic
field B with drift velocity v is a result of the Lorentz force, defined as:

F = qv × B

(3.7)

As a result of the Lorentz force, negative charges will accumulate on one side of the
sample, which results in the establishment of an electric field, EH , that balances with the
Lorentz force, equation (3.8):
qEH = qvdx Bz
63

(3.8)

The current density in the y direction can be defined as Jy = qnvdy , and therefore
equation (3.8) can be expressed as:

EH = (

1
)Jy Bz
en

(3.9)

Equation (3.9) can be further expressed in terms of the measured quantities: VH , the
Hall voltage, Bz , the applied magnetic field, n, the carrier concentration, e, the elementary
unit of charge, and t, the sample thickness.

VH =

Iy Bz
net

(3.10)

From equation (3.10) we can define a parameter called the Hall coefficient as:

RH =

−1
ne

(3.11)

The resistance across the Hall probes for various values of the magnetic field were measured using a 1T magnet. By plotting the resistance versus magnetic field and measuring the
slope, the carrier concentration was determined. By also using ∆VS along with the known
probe separation p the electrical resistivity was calculated along with the carrier concentration which allowed for a calculation of the carrier mobility (µ = σ/ne, where µ is the
mobility, σ is the electrical conductivity, n is the carrier concentration). The Hall measurements were done under a vacuum of approximately 10−5 Torr, from 80K − 300K using liquid
nitrogen to reach cryogenic temperatures.

64

3.2.5

Low Temperature Transport Properties

The electrical resistivity, Seebeck coefficient, and thermal conductivity were measured on a
custom-built system between the temperatures of 80K and 350K. Liquid nitrogen is used to
reach cryogenic temperatures. The setup seen in Figure 3.8 was built on a Janis cryogenic
measurement system. A sample cut into a parallelepiped is attached to a copper base, which
acts as a thermal sink, using silver epoxy. A resistor of approximately 800Ω was wrapped
in copper and attached to the top of the sample (approximate dimensions: 8mm x 2mm x
2mm), which was also done using silver epoxy. Two small copper strips were attached to
the face of the sample using silver epoxy which act as solder points for the thermocouple.
Phosphor-bronze wiring was used to inject electrical current to the heater, IH , and through
the sample, IS . The material was chosen because of its high electrical conductivity but
relatively low thermal conductivity, which keeps the thermal conduction losses to a minimum.
Two thermocouples made from copper and constantan alloy are soldered to the two copper
probes with known separation, l.

65

Figure 3.8: Schematic of system used for measuring the electrical resistivity, Seebeck coefficient, and thermal conductivity across the temperature range of 80-350K.

3.2.5.1

Electrical Resistivity

The two copper wires in the thermocouples are used to measure the voltage difference generated by the current injected through the sample, ∆VS . IS is applied in both the positive
and negative directions so that the voltage contribution from the Peltier effect can be subtracted out. Using ∆VS and IS the resulting resistance, RS can be determined and used to
calculate the electrical resistivity. Using the sample dimensions of width, (w), thickness, (t),
which together give a surface area, (S) and probe separation, (l) the electrical resistivity is
calculated using equation (3.12).

ρ = RS

S
l

(3.12)

IS and ∆VS were supplied and measured, respectively, using Keithley 2000 multimeters. The dimensions of the sample were measured using calipers. There is intrinsic error
66

associated with the voltage measurement and the measurement of the sample dimensions.
The uncertainty associated with measuring the electrical resistivity using this setup can be
estimated using equation (3.13).

∆ρ = ρ

(

∆R 2
∆l
∆w 2
∆p 2
) + ( )2 + (
) +(
)
R
l
w
p

(3.13)

For a sample with an electrical resistivity of 1mΩ cm, dimensions of 3mmx3mm, and
a probe separation of 4mm the uncertainty is approximately ±8%. The majority of the
uncertainty associated with the electrical resistivity stems from the error in measuring the
dimensions and size of the copper contacts used for the probes.

3.2.5.2

Seebeck Coefficient and Thermal Conductivity

The Seebeck coefficient is a parameter determined directly from the measured voltage difference, ∆V , that is generated by a temperature difference, ∆T , and is therefore independent
of geometry, as can be seen in equation (3.14).

S=

∆V
∆T

(3.14)

The error associated with measuring the Seebeck coefficient is dependent only on the
uncertainty associated with the multimeters used to measure the voltage readings from the
copper wires and thermocouple. The error for Seebeck coefficient is equal to approximately
±2%. The thermal conductivity, on the other hand, is dependent on dimensions and therefore
contains more uncertainty. Current is supplied to the resistive heater, IH , on top of the
2 R. This heat is driven through
sample, which in turns induces joule heating equivalent to IH

the sample creating a temperature gradient. The associated ∆T is measured and used to
67

calculate a thermal conductance K =

2 R)
(IH
∆T .

The thermal conductivity is then calculated

using the dimensions of the sample and the thermal conductance, as can be seen in equation
(3.15).

κ=K

l
S

(3.15)

where l is the probe separation and S is the surface area; calculated by length of sample,
l times the width, w. The uncertainty associated with measuring the thermal conductivity
looks similar to the uncertainty calculation for electrical resistivity. The uncertainty is
dependent, of course on the value of thermal conductivity, but for a setup similar to that
aforementioned for the error calculation in electrical resistivity, and a thermal conductivity
around 2Wm−1 K−1 , the average uncertainty is approximately ±7%. The specific equation
for the calculated uncertainty of thermal conductivity used in this work can be seen in
equation (3.16).

∆κ = κ (

∆IH 2
∆V
∆dT 2
∆l
∆h 2
∆w 2
) + ( H )2 + (
) + ( )2 + (
) +(
)
IH
VH
dT
l
h
w

(3.16)

When measuring the thermal conductance of a material using this steady-state technique
it is assumed that the all of the power generated by the resistive heater flows through
the material, but near room temperature heat loss due to radiation becomes significant.
According to the Stefan-Boltzmann law, the total energy radiated per unit surface area of a
black body per unit time is given by equation (3.17).[110]

W = σAT 4

68

(3.17)

where σ is the Stefan-Boltzmann constant (5.070 x 10−8 Wm−2 K−4 ), A is the surface
area, and T is the temperature.[12] The thermal conductance is therefore expressed as:

dW
= 4σAT 3
dT

(3.18)

Equation (3.18) is an expression for the amount of conductance lost to radiation. It is
therefore important to subtract out this contribution to thermal conductivity in order to
obtain reliable measurements near room temperature (300K). The actual calculation used
to subtract this radiative contribution can be seen in equation (3.19).

κ = [Kmeasured − Kloss (

T 3 l
)]
300 S

(3.19)

In equation (3.19) the value for Kloss is taken to be 0.00146WK−1 . This value has been
determined by independent standardized measurements in our laboratory. The thermal
conductivity, with and without the radiation correction, of a sample of Bi2 Te3 can be see in
Figure 3.9.

69

Figure 3.9: Thermal conductivity versus temperature for Bi2 Te3 with and without the radiation correction, an effect which can cause an overestimation of thermal conductivity,
especially at temperatures near and above room temperature.

The additive effect of radiation can especially be seen at temperatures near and above
300K. However as the temperature continues to increase, the ability to accurately subtract
the radiative contribution becomes more difficult using this steady-state measurement technique. Other methods for measuring thermal conductivity must therefore be employed as
will be discussed in the following sections.

3.2.6

High Temperature Transport Properties

3.2.6.1

Electrical Resistivity and Seebeck Coefficient

The electrical resistivity and Seebeck coefficient at high temperature (300 - 1073K) were
measured using a ZEM-3 from ULVAC-RIKO, Inc. The same geometry used for cryostatic
transport measurements was used for this procedure as well. The sample rests between two
electrodes which apply pressure to ensure Ohmic contacts for inputting electrical and heat
currents. Two spring loaded thermocouples form pressure contacts to the side of the sample
70

for measuring a voltage difference and temperature difference. The mounting chamber is
evacuated using a roughing pump and back-filled with helium. Measurements can be performed within ±5% uncertainty between room temperature and 1073K. At each temperature
multiple temperature differences are established and the Seebeck coefficient and electrical
resistivity are measured for each and averaged.

3.2.6.2

Thermal Diffusivity

At higher temperatures measuring thermal conductivity using a steady-state technique, as
was used for the aforementioned crysostat measurements, becomes too difficult. Radiation
losses become larger making the uncertainty and noise levels too high for accurate measurement. The alternative is to use a non-steady-state technique. There a number of techniques
which can be used, but for bulk polycrystalline samples measuring the thermal diffusivity and
calculating the thermal conductivity is reliable. The relationship used to calculate thermal
conductivity at high temperatures can be seen in equation (3.20).

κ = α∆Cp

(3.20)

where α is the thermal diffusivity, ∆ is the volumetric density, and Cp is the heat capacity.
Laser-flash measurements were used to measure the thermal diffusivity at Michigan State
University on a Netzsch LFA-457 MicroFlash system. The laser-flash technique uses a laser
induced heat pulse on one side of a disk-shaped sample and measures the time associated
with heating the reverse side of the sample. The response is time-dependent and used to
calculate the time to half maximum t1/2 . In the adiabatic case the thermal diffusivity is

71

calculated as follows:[111]
α = 0.1388

d2
t1/2

(3.21)

Samples for this measurement were cut from the sintered pellets with the approximate
dimensions of 10mm diameter and 1mm thickness. Density was measured at room temperature using an ethanol-based Archimedes method, which is independent of geometry. The
specific heat was measured using a Netzsch DSC 200-F3 Maia, as was aforementioned. The
uncertainty associated with calculating thermal conductivity using equation (3.20) can be
quite high and was calculated using equation (3.22). The values for uncertainty depend on
the specific sample, but on average are around ±10%.

∆κ = κ (

∆Cp 2
∆α 2
∆ρ 2
) +(
)
) +(
α
Cp
ρ

(3.22)

The figure-of-merit, ZT, at high temperature is a calculation using ρ, S, and κ. The
uncertainty associated with ZT can be calculated using equation (3.23), which is typically
on the order of ±13%.

∆ZT = ZT

(

2∆S 2
∆ρ 2
∆κ 2
) +(
) +(
)
S
ρ
κ

72

(3.23)

Chapter 4
(GeTe)1−x(Sb2Te3)x Compounds
Some of the first thermoelectric studies of (GeTe)1−x (Sb2 Te3 )x alloys were conducted by
Christakudis, et al in 1991 and Konstantinov, et al in 2001.[112, 113] Each found lattice
thermal conducitity values below 2Wm−1 K−1 and ZT values comparable to PbTe in the
temperature range of 500K − 800K. Rosenthal, et al in 2011 gave a detailed analysis of the
crystal structure and thermoelectric properties of GeTe-rich (GeTe)1−x (Sb2 Te3 )x alloys.[114]
In the following sections a discussion of the crystal structure, phase transition, and thermoelectric properties of (GeTe)1−x (Sb2 Te3 )x will be presented.

4.1

Experimental

Samples with composition (GeTe)1−x (Sb2 Te3 )x , with x = 0.02, 0.04, 0.059, 0.06, and 0.083,
were synthesized by placing stoichiometric amounts of Ge, Sb, and Te in evacuated and
sealed ampoules. The specific samples x = 0.059 and x = 0.083 correspond to Ge12 Sb2 Te15
and Ge17 Sb2 Te20 , both of which have been discussed in the literature.[114] The samples
were melted at 900◦C, held for 12h, and quenched. The resulting ingots were ball-milled for
5min and the powder densified using SPS at 400◦C, for 15min, under 40MPa of pressure.
Samples for X-ray diffraction, thermoelectric characterization, DSC, and Hall measurements

73

were cut from the sintered pellet using a circular diamond saw. Thermal conductivity was
calculated using the measured thermal diffusivity, the specific heat (which was assumed as
the Dulong-Petit value), and the measured volumetric density.

4.2

Results and Discussion

The X-ray diffraction patterns for all samples can be seen in Figure 4.1. Samples up to
x=0.059 showed a single phase rhombohedral crystal structure. For greater concentrations
of Sb2 Te3 impurities of elemental Sb were seen in the patterns (PDF#98-000-0095). The
rhombohedral structure of GeTe has characteristic double peaks around 43◦ 2θ. The peaks
are generated because of the rhombohedral unit cell of GeTe, which can be thought of as
having a slight distortion in the [111] direction of the rocksalt crystal structure. As the
Sb2 Te3 concentration is increased the separation of these peaks decreases, until x = 0.06,
which begins to show a single peak at 43◦ 2θ, as can be seen in the Figure 4.1 inset. For
x = 0.083 the doublets are completely replaced by a single peak, which corresponds to the
full extension of the distorted rocksalt angle back to 90◦ , aside from some peak shoulders
which could correspond to the lingering existence of some off-cubic phase.

74

Figure 4.1: X-ray diffraction patterns for (GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%,
4%, 5.9%, 6%, and 8.3% Sb2 Te3 . As Sb2 Te3 content increases the rhombohedral crystal
structure slowly changes to the rocksalt structure, as can be seen in the inset.

GeTe displays an increasing electrical resistivity with increasing temperature from values
of less than 0.1mΩ cm at 80K to approximately 0.6mΩ cm above 800K. The phase transition
from rhombohedral to cubic at approximately 600K is accompanied with a slight dip in
electrical resistivity as well. As Sb2 Te3 is substituted for GeTe the electrical resistivity is
increased nearly two orders of magnitude, with the 8.3% Sb2 Te3 (Ge12 Sb2 Te15 ) having the
highest values. The electrical resistivity versus temperature for (GeTe)1−x (Sb2 Te3 )x can be
seen in Figure 4.2.

75

Figure 4.2: Electrical resistivity versus temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies
from 0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 .

The increase in electrical resistivity can be traced to an increased concentration of vacancies in the crystal structure which causes a decrease in the hole mobility as the Sb2 Te3
concentration is increased. These vacancies act as additional scattering sites, which scatter
both charge carriers and phonons. It is therefore expected that the thermal conductivity will
be decreased as well. The hole concentration and hole mobility, as measured and determined
by Hall measurements, can be seen in Figure 4.3.

76

Figure 4.3: Hole Concentration, (a) and Hole Mobility, (b) versus Temeperature for
(GeTe)1−x (Sb2 Te3 )x .

The Seebeck coefficient is also increased as a function of Sb2 Te3 concentration, as can be
seen in Figure 4.4. Based on the behavior of the Seebeck coefficient and electrical resistivity
it was believed that Sb2 Te3 was causing a change in the hole concentration. However,
the Hall measurements revealed a decrease hole mobility by approximately two orders of
magnitude, with little change observed in the hole concentration. The small decrease in
the hole concentration is most likely due to an Sb deficiency in the compound, as small Sb
impurities were seen in the X-ray diffraction patterns.
The high hole mobility found in GeTe yields low electrical resistivity and power factor
values in excess of 50µWcm−1 K−2 above 600K. Substituting Sb2 Te3 causes a systematic
decrease in the power factor, with peak values approximately equal to 20µWcm−1 K−2 for
8.3% Sb2 Te3 , which can be seen in Figure 4.5(a).

77

Figure 4.4: Seebeck coefficient versus temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies
from 0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 .

Figure 4.5: Power factor, (a) and calculated Lorenz number, (b) versus temperature for
(GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%, 4%, 5.9%, 6%, and 8.3% Sb2 Te3 .

The linear behavior of the Seebeck coefficient below the phase transition temperature
allows for a more accurate determination of the Lorenz number, which is in turn used to
calculate the electronic thermal conductivity. Using a single parabolic band approximation,
78

the Seebeck coefficient was numerically fit and used to determine the reduced Fermi enE

ergy (η = k fT ) using equation (1.36). After calculating η, the Lorenz number was then
B
calculated using equation (1.48). The calculated Lorenz number for (GeTe)1−x (Sb2 Te3 )x
significantly deviates from the free electron value, as can be seen in Figure 4.5(b). The thermal conductivity was decreased as the Sb2 Te3 concentration was increased. This is due to
the diminishing electronic and lattice contributions to thermal conductivity, as can be seen
in Figure 4.6. It is believed that the vacancies introduced with Sb2 Te3 substitution scatter
phonons and caused the decrease in lattice thermal conductivity.

Figure 4.6: Thermal conductivity, (a), and lattice thermal conductivity, (b) versus temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%, 4%, 5.9%, 6%, and 8.3%
Sb2 Te3 .

By substituting Sb2 Te3 for GeTe a 46% increase in ZT is achieved. The ZT value at
723K shows a turnover with increasing Sb2 Te3 concentration, as can be seen in Figure 4.7,
with the highest ZT being displayed in the 5.9% Sb2 Te3 doped compound, otherwise written
as Ge17 Sb2 Te20 . This large enhancement in ZT is explained primarily by the introduction
of vacancies, which introduced new scattering sites. This resulted in a large increase in the
electrical resistivity and Seebeck coefficient, but also allowed for a significant decrease in
thermal conductivity.
79

Figure 4.7: ZT versus temperature for (GeTe)1−x (Sb2 Te3 )x , where x varies from 0%, 2%,
4%, 5.9%, 6%, and 8.3% Sb2 Te3 .

4.3

Conclusion

From the ternary phase diagram of Ge, Sb, and Te, the psuedo-binary alloys of GeTe and
Sb2 Te3 yield known state of the art phase change materials for nonvolatile memory storage.
From the work seen here, and others since the 1990s, it is clear that (GeTe)1−x (Sb2 Te3 )x
compounds additionally hold promise for thermoelectric power generation applications. It
was found that increasing the Sb2 Te3 content increased the concentration of vacancies within
these compounds. ZT values greater than 1.5 were seen for a number of the GeTe rich alloys.
The following chapters will outline the various chemical and materials techniques used to
optimize the synthesis of Ge4 SbTe5 and Ge17 Sb2 Te20 and enhance the ZT, all while striving
for a more fundamental understanding of the materials.

80

Chapter 5
Ge4SbTe5
Ge4 SbTe5 is a phase change material which is located just off of the GeTe – Sb2 Te3 tieline. This compound is currently used for high-definition DVDs and Blu-ray devices, but
unlike other phase change materials found on the aforementioned tie-line it contains no
vacancies.[115] It exists in a metastable rocksalt (F m¯3m) phase at room temperature with
Ge and Sb randomly assorted on the cation site and Te atoms occupying the anion site, as
can be seen in Figure 5.1, which was created using VESTA.[91]

Figure 5.1: Rocksalt (F m¯3m) crystal structure of Ge4 SbTe5 , with Ge and Sb randomly
occupying the cation site and Te on the anion site.

81

Unlike other state-of-the-art phase change materials Ge4 SbTe5 contains no vacancies in
the rocksalt unit cell, and with a band gap of approximately 0.7 eV, it was believed that
it would exhibit interesting thermoelectric properties in the medium temperature regime
(400-600K).[115] The following sections will outline the synthesis and various doping and
substitutional approaches used to study this compound.

5.1

Synthesis and Crystal Structure

Samples of Ge4 SbTe5 were synthesized by placing stoichiometric amounts of Ge (99.9999
%, Alfa Aesar, zone-refined ingot), Sb (99.999%, Alfa Aesar, shot 6mm and down) and Te
(99.999+%, Alfa Aesar, lump) in silica ampoules. The ampoules were evacuated to approximately 10−5 Torr to prevent oxidation during synthesis. Samples were then melted by
heating to 900◦C at a heating rate of 2◦C/min. The ingot obtained was then ballmilled for
5min in a stainless steel jar using a SPEX-5000 vibratory mill. The sample was densified
using spark plasma sintering (SPS) at 400◦C for 15min under 40MPa of uniaxial pressure.
Volumetric density was measured using the Archimedes method. The sintered pellets, which
had density greater than 95% theoretical density, were then cut for thermoelectric measurements and characterization.
Room temperature X-ray diffraction was used to verify the crystal structure of Ge4 SbTe5 .
The lattice parameter was found by assigning the allowed diffraction planes to each peak in
the pattern and calculating the interplanar spacing, which is related to the angle of diffraction
by Bragg’s law of diffraction: [108]

nλ = 2dsinθ

82

(5.1)

where λ = 1.5418˚
A for CuKα radiation. The interplanar spacing is related to the unit cell
parameter and Miller indices, for cubic crystal structures, via equation (2.4). The diffraction
pattern with (hkl) peaks labeled can be seen in Figure 5.2. The calculated lattice parameter
was 5.95 ˚
A for Ge4 SbTe5 .

Figure 5.2: X-ray diffraction pattern of Ge4 SbTe5 with peaks labeled by their respective
miller indices.

This rocksalt phase, however is metastable and upon heating undergoes a phase transformation to a rhombohedral, or distorted cubic, structure. This high temperature phase
belongs to the space group R¯3m, also known as the tetradymite structure. Similar to the
layered structures of Bi2 Te3 and Sb2 Te3 the high temperature phase of Ge4 SbTe5 exists with
alternating layers of Ge,Sb, and Te with inter-penetrating layers of vacancies. Temperature
dependent X-ray diffraction of Ge4 SbTe5 in the temperature range of 50◦C − 450◦C was
performed at Oak Ridge National Laboratory, the result of which can be seen in Figure 5.3.

83

Figure 5.3: (a)Temperature dependent X-ray diffraction patterns for Ge4 SbTe5 . (b) shows
the cubic (111) peak splitting at the phase transition temperature as the cubic crystal
structure transforms to rhombohedral.

The phase transformation is structural in nature, but also corresponds to the Curie temperature of Ge4 SbTe5 . The ferroelectric nature of Ge4 SbTe5 , and other alloys of GeTe and
Sb2 Te3 , which exists at low temperature, stems from the high levels of resonance bonding,
as aforementioned. DSC measurements of Ge4 SbTe5 as a function of temperature can be
seen in Figure 6.1. An endothermic peak was observed in the heat capacity at 523K, which
corresponds to the structural phase transition.

84

Figure 5.4: Specific heat, as determined by DSC, versus temperature for Ge4 SbTe5 , with
the Dulong-Petit value represented by a dashed line.

The phase transformation is also considered an Anderson localization transition, in which
a material goes from displaying semiconductor like carrier transport to metallic transport.
The electrical resistivity reflects this behavior with a sharp decrease from approximately
8mΩ cm to 2mΩ cm at 523K, as can be seen in Figure 5.5(a). The Seebeck coefficient reflects
similar behavior with a slight dip at the phase transition temperature, but continues to rise
thereafter to values around 250µVK−1 (Figure 5.5(b)). The result of this electronic phase
transition is an increase in power factor to values well above 20µWcm−1 K−2 , which can be
seen in Figure 5.5(c)

85

Figure 5.5: (a) Electrical resistivity versus temperature, (b) Seebeck coefficient versus temperature, and (d) Power factor versus temperature for Ge4 SbTe5 .

The thermal conductivity maintains values approximately equal to 1Wm−1 K−1 , except
for an increase around the phase transition temperature due the spike in specific heat. The
thermal conductivity as a function of temperature can be seen in Figure 5.6(a). Because of
the large power factor values and low thermal conductivity the ZT displays values greater
than unity, reaching 1.5 around 800K. As can be seen in Figure 5.6(b), the best thermoelectric performance is seen at temperatures above the phase transformation.

86

Figure 5.6: (a) Thermal conductivity versus temperature and (b) ZT versus temperature for
Ge4 SbTe5 .

These values of ZT are competitive with state of the art thermoelectric materials, as can
be seen in Figure 6.5.[116] Little research has been conducted on bulk Ge4 SbTe5 . With low
thermal conductivity and high power factor it was believed that Ge4 SbTe5 would be an ideal
candidate for study. The following projects aimed to engineer the thermoelectric properties
through various doping and substitutional techniques.

87

Figure 5.7: ZT versus temperature for a number of today’s state of the art chalcogenide
thermoelectric materials, with Ge4 SbTe5 showing competitive values.[117, 118, 119]

5.2

Atomic Substitutions

Isovalent atomic substitutions are an effective way to improve the ZT of thermoelectric
materials. As aforementioned, substituting atoms which have the same valency but different
mass increases the scattering of heat-carrying phonons. This has been shown to be an
effective way to lower the lattice component of thermal conductivity, and with a matching
valence configuration, no change in the carrier concentration should be observed. It should
be noted, however, that the mass and electronegativity difference of substituting isovalent
atoms can induce additional scattering on the charge carriers, which would also lower charge
carrier mobility. In the following sections the thermoelectric properties of two solid solutions
on Ge4 SbTe5 will be presented, namely Ge4−x Snx SbTe5 and Ge4 Sb1−x Bix Te5 .

88

5.2.1

Ge4−x Snx SbTe5

Tin represented a logical choice as an atomic substitution for the Ge site in Ge4 SbTe5 . Sn is
inexpensive, relatively abundant, and nontoxic to humans and the environment. Buller, et
al recently studied the effects of Sn for Ge substitution and Se for Te substitution in thinfilms of Ge8 Sb2 Te11 . They found that these substitutions affected the electronic bandgap
and altered the overall electrical properties.[120] Welzmiller, et al studied Ge2 Sb2 Te5 and
the effects which Sn substitution had on the thermoelectric properties.[121] More recently
Welzmiller studied Cd doping in (GeTe)m (Sb2 Te3 ) alloys and (SnTe)m (Sb2 Te3 ) alloys.[122]
Additionally, the atomic mass of Sn is greater than that of Ge, so it was believed that this
isovalent substitution would provide additional phonon scattering without a detrimental
decrease to the impressive electrical properties of Ge4 SbTe5 .

5.2.1.1

Experimental

Samples of Ge4−x Snx SbTe5 were synthesized using the method described in earlier sections,
where stoichiometric amounts of Ge, Sn, Sb, and Te were massed out and placed in 10mm
silica ampoules. The ampoules were then evacuated to approximately 10−5 T orr and sealed.
The samples were heated to 900◦C at a rate of 2◦C/min, held for 12 hours, and water
quenched. Sn was substituted for Ge in 12.5mol % intervals, which for Ge4−x Snx SbTe5
corresponds to x=0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0. X-ray diffraction was measured
at room temperature on all samples, the patterns of which can be seen in Figure 5.8. As
Sn was substituted some of the diffraction peaks either decreased or increased in intensity,
which was explained by the difference in structure factor between Ge and Sn. The diffraction
peaks showed a uniform shift to lower 2θ with increasing Sn substitution. This is consistent
with an increase in the cubic lattice parameter of the unit cell. To calculate the lattice
89

parameter a Si standard was used during X-ray diffraction to correct for 2θ offset during the
measurements. The calculated lattice parameter can be seen in Figure 5.9. There is a linear
increase in the lattice parameter of the rocksalt unit cell with increasing Sn substitution
concentration, which is in agreement with Vegard’s law; essentially a rule of mixtures for the
lattice parameter of a solid solution.[123]

Figure 5.8: X-ray Diffraction pattern of Ge4−x Snx SbTe5 , where x=0, 0.5, 1.0, 1.5, 2.0, 2.5,
3.0, 3.5, and 4.0. A shift in diffraction peaks to lower 2θ can be seen with increasing Sn
substitution, which corresponds to an increase in lattice parameter. Impurity peaks, which
were found to be elemental Sb, are marked by a diamond.

90

Figure 5.9: Lattice Parameter (˚
A) versus Sn concentration for Ge4−x Snx SbTe5 . A leastsquares method was used to apply the trendline. The linear behavior agrees with Vegard’s
law.

With the increase of Sn substitution, secondary phases of elemental Sb were observed in
the X-ray diffraction patterns and SEM/EDS (PDF#98-000-0095). Despite the presence of
these Sb impurities the rocksalt crystal structure was maintained. The stoichiometry was
therefore verified using SEM/EDS, which can be seen in Table 5.1.
Table 5.1: Using SEM/EDS the actual composition of Sn substituted samples of Ge4 SbTe5
were measured, seen here as atomic percent.

All samples were ball-milled in a stainless steel ball mill jar for 5min and densified using
SPS at 350◦C for 15min under 40MPa of pressure. The sintering temperature was decreased
91

from previous samples of Ge4 SbTe5 to account for substantial cracking observed in high Sn
concentration substituted samples. The pellets were then polished and cut for thermoelectric
measurements.

5.2.1.2

Results and Discussion

With successful substitution of Sn for Ge it was expected that the electrical resistivity would
increase due to increased charge carrier scattering, however the electrical resistivity was
decreased as a function of Sn concentration. This decrease in the electrical resistivity is
indicative of a doping effect, as was reflected in the Seebeck coefficient. However, because Sn
should substitute for Ge without a change in the valence configuration, it was hypothesized
that the substitution caused a decrease in the band gap of Ge4 SbTe5 . The variation of
bandgap with alloying in chalcogenide based compounds is a well known phenomenon. If,
for instance, one was to alloy two binary compounds, AB and BC, the optical band gap
should vary quadratically with composition, as can be seen in equation (5.2).[124]

Eg (x) = [x AC + (1 − x) BC ] − bx(1 − x)]

(5.2)

where b is defined as the bowing parameter.[124, 125] The electrical resistivity versus
temperature (80K-823K ) and Seebeck coefficient versus temperature for all Sn substituted
samples can be seen in Figure 5.10 and Figure 5.11, respectively. A decrease in electrical
resistivity and Seebeck coefficient was observed, which in turn resulted in a decreased power
factor. However, the original motivation for introducing the atomic substitutions was to
induce additional phonon scattering and thereby decrease lattice thermal conductivity.

92

Figure 5.10: Electrical resistivity versus temperature for Ge4−x Snx SbTe5 where x=0, 0.5, 1,
1.5, 2, 2.5, 3, 3.5, and 4.

Figure 5.11: Seebeck coefficient versus temperature for Ge4−x Snx SbTe5 where x=0, 0.5, 1,
1.5, 2, 2.5, 3, 3.5, and 4.

A decrease in the lattice thermal conductivity at room temperature (300K) was observed,

93

while at high temperature (723K) little to no decrease in the lattice contribution to thermal
conductivity was observed. A plot of the thermal conductivity at 300K and 723K versus
% Sn substitution, thermal conductivity versus temperature, and a closer look at the high
temperature thermal conductivity can be seen in Figure 5.12.

Figure 5.12: (a) Thermal conductivity versus temperature for all Sn substituted samples
in the temperature range of 80K to 320K. (b) Lattice thermal conductivity at 300K
versus stoichiometric amount of substituted Sn. (c) Thermal conductivity at 723K for
25%, 50%, 82.5%, and100% Sn substitution. (d) Lattice thermal conductivity at 723K versus stoichiometric amount of substituted Sn.

For temperatures well above the Debye temperature for a given material the dominant
scattering mechanism becomes phonon-phonon scattering, or Umklapp scattering, as was
explained in Chapter 1. The Debye temperature of Ge4 SbTe5 can be approximated using
the relation developed by Slack, which states that for a given crystal structure the product
94

¯ , the average volume per atom, δ, and the Debye temperature,
of the average atomic mass, M
θ (equation (5.3)) is constant.[126]

¯ δθ3 = constant
M

(5.3)

For the rocksalt crystal structure of Ge4 SbTe5 the Debye temperature was found to
be 126K. Therefore the lack of substantial reduction in lattice thermal conductivity at high
temperature is consistent with the qualitative understanding that Umklapp scattering should
be dominant. The lack of decrease in the lattice thermal conductivity at high temperature
ultimately resulted in an increase in the total thermal conductivity because of the increasing
electronic contribution, as can be seen in Figure 5.12.

5.2.1.3

Conclusion

The original motivation for this work was based on the premise that Sn substituted for Ge in
Ge4 SbTe5 would decrease thermal conductivity and increase the overall ZT. However it was
observed that the Sn substitution led to a decrease in the electrical resistivity and Seebeck
coefficient, which was believed to be due to the contracted bandgap. This change in bandgap
also led to a substantial increase in the electronic contribution to thermal conductivity. The
ZT was decreased for all samples with Sn substitution greater than 25%. This is mainly
attributed to the increased thermal conductivity.
Though this study was not effective in enhancing the thermoelectric performance of
Ge4 SbTe5 , a new understanding of the relation between group IV elemental substitutions and
the electrical properties was achieved. In the following section a study on the thermoelectric
properties of Ge4 Sb1−x Bix Te5 , a solid solution on the Sb site, will be presented.

95

5.2.2

Ge4 Sb1−x Bix Te5

In 1999 Kuznetsov, et al studied GeBi4 Te7 and found that stoichiometry was an important
factor in altering the Seebeck coefficient and electrical resistivity of GeTe-Bi2 Te3 compounds.[127]
This was one of the first examples of studying the thermoelectric effects on Bi in GeTe phase
change materials. Matsunaga, et al studied the stability and formation of GeTe-Bi2 Te3
alloys.[128] Shelimova, et al studied these alloys and found that the charge carrier type
varied based on the specific composition synthesized.[129] More recently Wu, et al found
that small amounts of Bi2 Te3 substitution in GeTe enhanced the ZT by more than 100%.
The substitution of Bi for Sb, especially for Ge4 SbTe5 , has yet to be studied for thermoelectric applications although these compounds have been researched for various memory
applications.[130] The following section is a description of the synthesis and thermoelectric
characterization of bulk Ge4 Sb1−x Bix Te5 .

5.2.2.1

Experimental

Stoichiometric amounts of Ge, Sb, Te, and Bi (99.999%, Alfa Aesar) were sealed in evacuated
silica ampoules. The samples were melted at 900◦C for 12h and water quenched. The
resulting ingots were ball milled and sintered using SPS at 400◦C for 15min under 40MPa
of pressure. Samples were cut from the sintered pellets wich were then used for structural
and thermoelectric characterization, the specific details of which were described in previous
sections.

5.2.2.2

Results and Discussion

The X-ray diffraction patterns revealed a shift in the 2θ position of the (hkl) peaks. This
shift in 2θ is indicative of a change in the lattice parameter. Substituting Sb with the larger
96

Bi atom resulted in an increase in the lattice parameter, as can be seen in Figure 5.13.

Figure 5.13: Lattice parameter versus Bi composition in Ge4 Sb1−x Bix Te5 .

Differential scanning calorimetry (DSC) was performed to analyze the specific heat. With
increasing Bi concentration the melting point was found to decrease to approximately 783K.
The phase transition temperature also decreased from 620K for Ge4 SbTe5 to 601K for
Ge4 BiTe5 . The Cp versus temperature and phase transition temperature, TC , versus Bi
content can be seen in Figure 5.14.

97

Figure 5.14: Specific heat versus temperature for Ge4 Sb1−x Bix Te5 , where x =
0, 0.25, 0.50, 0.75, 1.0. Phase transition temperature versus percent Bi can be seen in the
inset plot.

The primary goal of substituting Sb with Bi was to scatter heat-carrying phonons and
lower the lattice contribution to thermal conductivity. The thermal conductivity, which
was calculated using the volumetric density, specific heat, and thermal diffusivity shows a
decreasing trend with increasing Bi concentration. The lattice thermal conductivity was
estimated using the Weidemann-Franz law to calculate the electronic contribution and subtracting from the total thermal conductivity. The result of the Bi substitution was a general
decrease in the lattice thermal conductivity and the total thermal conductivity, as can be
seen in Figure 5.15.

98

Figure 5.15: (a) Thermal conductivity versus temperature for Ge4 Sb1−x Bix Te5 , where
x = 0, 0.25, 0.50, 0.75, 1.0. (b) Lattice thermal conductivity versus temperature, dashed
line marking the 1/T temperature dependence of Umklapp scattering.

As was noticed with Ge4−x Snx SbTe5 , the Seebeck coefficient decreased with increased
Bi substitution, as can be seen in Figure 5.16. The decrease in Seebeck coefficient is most
likely due to a narrowing of the electronic bandgap, which would shift the Fermi energy and
yield higher charge carrier concentrations.

99

Figure 5.16: Seebeck coefficient versus temperature for Ge4 Sb1−x Bix Te5 , where x =
0, 0.25, 0.50, 0.75, 1.0.

Unlike in the case of Ge4−x Snx SbTe5 , the electrical resistivity was not decreased with
the substitution of Bi. At temperatures below the phase transition the electrical resistivity
is increased to 17mΩ cm for 50% Bi substitution and decreased with further substitution
to values approximately equal to 13mΩ cm for Ge4 BiTe5 . The electrical resistivity versus
temperature can be seen in Figure 5.17.

100

Figure 5.17: Electrical resistivity versus temperature for Ge4 Sb1−x Bix Te5 , where x =
0, 0.25, 0.50, 0.75, 1.0.

The electrical resistivity follows a trend that is consistent with traditional alloys. As a
solid solution is formed the solute atoms scatter charge carriers. Recall that according to
Matthiesen’s rule the scattering relaxation rates for the various processes which contribute
to the electrical resistivity sum as:

τ −1 = τT−1 + τI−1 + ...

(5.4)

τT is the relaxation time associated with scattering from thermal vibrations, and τI is
the relaxation time associated with scattering from positively charged ions in the crystal
lattice. The scattering relaxation time which is shortest contributes the most. As more
disorder is introduced into the system by means of randomly distributed solute atoms the
scattering relaxation time associated with ions will decrease, thereby becoming more prominent in affecting electronic transport. Nordheim’s rule relates the resistivity associated with
101

impurity atoms to the atomic fraction X of the solute atoms and the Nordheim coefficient,
C. When Matthiesen’s rule and Nordheim’s rule are combined the resistivity of an alloy can
be estimated with knowledge of the matrix resistivity ρmatrix and the Nordheim coefficient
according to equation (5.5).[12]

ρ = ρmatrix + CX(1 − X)

(5.5)

The electrical resistivity of Ge4 Sb1−x Bix Te5 displays this quadratic behavior, as can be
seen in Figure 5.18.

Figure 5.18: Room temperature (300K) electrical resistivity versus %Bi substituted with
8%. The parabolic nature is indicative of alloy scattering, as explained by Nordheim’s rule.
The trendline was generated using a least squares fit without knowledge of the Nordheim
coefficient of Ge4 SbTe5 .

5.2.2.3

Conclusion

Because of the increase in electrical resistivity and decrease in Seebeck coefficient the power
factor was decreased to values below 5µWcm−1 K−2 above the phase transition temperature.
Despite a decrease in thermal conductivity, the overall ZT was decreased across the entire
temperature range of measurement. Both Sn and Bi were shown to worsen the thermoelec102

tric properties of Ge4 SbTe5 . These decreases in thermoelectric performance were attributed
primarily to changes in the band gap, which arose from the substitution of group IV and V
elements. Future work on lowering the thermal conductivity using other isovalent substitutions such as Si, Se, or S should be investigated. In the sections to follow a description of
the work on impurity doping in Ge4 SbTe5 will be presented.

5.3

Electronic Doping

Doping foreign atoms which either introduce a donor or acceptor into a crystal lattice is one of
the most effective methods of tuning carrier concentration, and can therefore lead to increases
in the ZT of thermoelectric materials. Welzmiller, et al studied Mn doping in Ge4 Sb2 Te7 ,
which resulted in a ZT enhancement as well as the introduction of a ferromagnetic phase
from the Mn ions.[131] Welzmiller, et al also observed a nearly three-fold increase in ZT by
doping Cr on the Ge site in GeSb2 Te4 .[132] In 2015 Welzmiller, et al studied the effects of Cd
doping on (GeTe)n Sb2 Te3 and (SnTe)n Sb2 Te3 compounds and found that the Cd increased
the density of states effective mass and led to an increase in the Seebeck coefficient, which
allowed for slight increases in the power factor, and ZT overall.[122] In the following sections
the work done on doping Ge4 SbTe5 with Na and Ga will be presented.

5.3.1

Ge4−x Nax SbTe5

Alkali metals are monovalent with one valence electron filling the s-shell. Ge is nominally
divalent, and if substituted by an alkali metal such as Na or Li, could in principle be doped
p-type. Similar chalcogenide materials, such as PbTe and AgSbSe2 have exhibited increased
ZT by Na doping.[133, 134] The Na doping in these compounds allowed for an optimization

103

of the carrier concentration, but also, because of the difference in mass and atom size between
Na and the host atom, displayed a decrease in thermal conductivity. Schroder, et al studied
the effects which Li doping has on the thermoelectric properties of GeTe-Sb2 Te3 alloys, but
the effects of Na doping on the Ge site in GeTe-Sb2 Te3 compounds has yet to be studied for
thermoelectric applications.[135]

5.3.1.1

Experimental

Samples of 0.5%, 1.0%, and 1.5% Na doped Ge4 SbTe5 , which corresponds to Ge3.98 Na0.02 SbTe5 ,
Ge3.96 Na0.04 SbTe5 , and Ge3.94 Na0.06 SbTe5 , respectively, were synthesized by placing stoichiometric amounts of Ge, Sb, Te, and Na2 Te (99.9%, 60 mesh) in evacuated silica ampoules.
Because of the volatility of Na, the ampoules were first coated with a layer of carbon. The
samples were melted for 12h and quenched in water. The purity of the synthesized compounds was verified using room temperature X-ray diffraction. All diffraction patterns were
single phase and showed no changes in d-spacing or intensity, implying the successful incorporation of Na atoms on the Ge site.

5.3.1.2

Results and Discussion

Increasing the concentration of Na doped for Ge caused a decrease in electrical resistivity
and Seebeck coefficient, which is consistent with an increase in the hole concentration. The
electrical resistivity can be seen in Figure 5.19 and the Seebeck coefficient can be seen in
Figure 5.20.

104

Figure 5.19: Electrical resistivity versus temperature for 0.5%, 1.0%, and 1.5% Na-doped
Ge4 SbTe5 .

Figure 5.20: Seebeck coefficient versus temperature for 0.5%, 1.0%, and 1.5% Na-doped
Ge4 SbTe5 .

High temperature Hall measurements were performed at the University of Michigan by

105

Alexander Page under the supervision of Dr. Ctirad Uher on Ge4 SbTe5 and 1.50% Na doped
Ge4 SbTe5 . No change in the hole concentration was observed below the phase transition,
but at temperatures above the phase transition the hole concentration was increased approximately 80%. The hole concentration and hole mobility (calculated from the one carrier
approximation for electrical conductivity σ = peµh ) can be seen in Figure 5.21(a) and (b),
respectively.

Figure 5.21: (a) Hole concentration versus temperature for Ge4 SbTe5 and 1.5% Na doped.
(b) Hole Mobility versus Temperature for Ge4 SbTe5 and 1.5%.

The power factor experienced a slight increase with 0.5% Na, however the increase is well
within the measurement uncertainty of ±12%. The higher doping amounts of 1.0% Na and
1.5% Na yielded decreases in the power factor to values significantly below 20µWcm−1 K−2 .
The power factor versus temperature for Ge4 SbTe5 , 0.5% Na, 1.0% Na, and 1.5% Na doped
Ge4 SbTe5 can be seen in Figure 5.22.

106

Figure 5.22:
Ge4 SbTe5 .

Power Factor versus Temperature for 0.5%, 1.0%, and 1.5% Na-doped

In compounds such as PbTe, as was aforementioned, Na doping is a means to tune the
charge carrier concentration, but also has also been shown to decrease the thermal conductivity. As can be seen in Figure 5.23, the thermal conductivity near room temperature
experienced no change. For 1.50% Na doped the thermal conductivity was decreased from
values around 1.5Wm−1 K−1 to 1.0Wm−1 K−1 at temperatures above the phase transition
temperature. This decrease could be due to the scattering of phonons from the mass difference between Ge and Na.

107

Figure 5.23: Thermal conductivity versus temperature for 0.5%, 1.0%, and 1.5% Na-doped
Ge4 SbTe5 .

5.3.1.3

Conclusion

The small decrease in thermal conductivity and the decrease in power factor resulted in a nonneglible decrease in the ZT of Na-doped Ge4 SbTe5 . ZT versus temperature for Ge4 SbTe5
and the Na-doped samples can be seen in Figure 5.24. This decrease in ZT stems from
essentially over-doping. As was mentioned in the introduction, the thermoelectric power
factor and ZT of a given material is optimized as a function of carrier concentration. Based
on the results for Na doping it would seem that the maximum in power factor and ZT should
be at, or below, the carrier concentration of undoped Ge4 SbTe5 .

108

Figure 5.24: ZT versus Temperature for 0.5%, 1.0%, and 1.5% Na-doped Ge4 SbTe5 .

5.3.2

Ge4−x Gax SbTe5

The element Ga is found in the group III column of the periodic table and has the the
electronic configuration of [Ar]3d10 4s2 4p1 , which has one less valence electron in the pshell than Ge. Ga as a dopant for Ge would therefore introduce one acceptor per dopant
atom. In this study Ga was used as a dopant for Ge in an attempt to increase the hole
concentration, which would lower the electrical resistivity and Seebeck coefficient and allow
for an optimization of ZT.

5.3.2.1

Experimental

Stoichiometric amounts of Ge, Sb, Te, and Ga (Alfa Aesar, 6mm diameter pellets, 99.9999%)
were placed in silica ampoules and evacuated. The tubes were then sealed and thermally
treated as was mentioned for Ge4 SbTe5 in previous sections. Phase purity was confirmed

109

using room temperature X-ray diffraction. All samples were ball milled for 5 minutes and
densified using spark plasma sintering for 15min at 400◦C under 40MPa of pressure. Volumetric density was measured using an ethanol based Archimedes method. The resulting
pellet was then cut for thermoelectric characterization measurements at both cryogenic temperatures and high temperatures. The specific samples synthesized were Ge3.96 Ga0.04 SbTe5 ,
Ge3.8 Ga0.2 SbTe5 , and Ge3.6 Ga0.4 SbTe5 , which correspond to Ga concentrations of 1%, 5%,
and 10%, respectively.

5.3.2.2

Results and Discussion

Because of the small atomic size difference between Ga and Ge there was a negligible shift in
the d-spacing of the X-ray diffraction patterns between samples, as can be seen in Figure 5.25.
For 5% and 10% Ga doped, the secondary phase Ga2 Te3 was formed and observed in the
X-ray diffraction pattern (PDF#00-035-1490). It is conjectured that the Ga2 Te3 secondary
phase was not observed in the X-ray pattern of the 1% Ga doped sample due to the low
concentration of Ga2 Te3 present.

110

Figure 5.25: X-ray diffraction patterns for Ge4−x Gax SbTe5 , 1%, 5%, and 10% Ga.

Ga2 Te3 forms a defect zinc-blende crystal structure, space group F ¯43m. An example
of the crystal structure, which was generated using Vesta software from JP-Minerals, can
be seen in Figure 5.25.[91] Ga2 Te3 contains a high concentration of vacancies with 31 of
all anion sites being an empty atomic site. The compound therefore exhibits an interesting
number of physical properties, such as a naturally occurring superstructure from the ordered
vacancy planes which lend to unusually low thermal conductivity.[136] Ga2 Te3 has a bandgap
of approximately 1.1eV at room temperature, and, with the high vacancy concentration,
lends to a large electrical resistivity and p-type Seebeck coefficient.[137] Additionally, the
compound has been investigated for use in phase change memory applications.[138]
Most often, secondary phases are considered undesirable for studies in thermoelectrics
research, in part because of the work by Bergman and Levy, which found that under idealistic assumptions the ZT of a composite could never exceed that of the highest ZT of the
comprising materials.[139] The compounds of Ga doped Ge4 SbTe5 were nonetheless fully
111

characterized. The electrical resistivity, Figure 5.26, began to exhibit insulating behavior
at temperatures below the phase transition temperature, with increases from approximately
8mΩ cm to greater than 50mΩ cm for 10% Ga doped Ge4 SbTe5 . This is in-line with the
observance of an increasing amount of Ga2 Te3 secondary phase. The Seebeck coefficient
showed a marginal increase, despite the increased electrical resistivity, as can be seen in
Figure 5.27. The small increase in Seebeck coefficient could be explained by the thermal
excitation of carriers from Ga2 Te3 , which would produce competing carriers and thereby
offset any contribution to Seebeck coefficient.

Figure 5.26: Electrical resistivity versus temperature for 1%, 5%, and 10% Ga doped
Ge4 SbTe5 .

112

Figure 5.27: Seebeck coefficient versus temperature for 1%, 5%, and 10% Ga doped
Ge4 SbTe5 .

Despite the high electrical resistivity, the small enhancements in Seebeck coefficient resulted in an increased power factor, to values as high as 23.1µWcm−1 K−2 . The presence of
these secondary phases also decreased the thermal conductivity across the entire temperature range. The power factor and thermal conductivity versus temperature can be seen in
Figure 5.28(a) and (b), respectively.

113

Figure 5.28: (a) Power factor versus temperature for 1%, 5%, and 10% Ga doped Ge4 SbTe5 .
(b) Thermal conductivity versus temperature for Ga-doped Ge4 SbTe5 .

The ZT versus temperature for Ge4−x Gax SbTe5 can be seen Figure 5.29. For the higher
Ga concentrations, namely 5% and 10%, the ZT was decreased, but, surprisingly, for 1%
Ga-doped Ge4 SbTe5 there was a significant increase in ZT, with a peak at 823K greater
than 1.6. Though the ZT appears to continue rising, the temperature was kept in a range
of stability for Ge4 SbTe5 (< 900K).

114

Figure 5.29: ZT versus temperature for 1%, 5%, and 10% Ga doped Ge4 SbTe5 .

5.3.2.3

Conclusion

The increase in ZT despite the existence of secondary phases contradicts the theoretical work
of Bergman and Levy, but is not uncommon in thermoelectrics research. Using secondary
phases to tune the power factor while simultaneously maintaining a low thermal conductivity
has been used in materials such as CoSb3 and PbTe to achieve some of the highest ZT values
to date.[140, 67] Using secondary phases to enhance the thermoelectric performance becomes
an important concept in the remainder of this work.

5.4

Summary and Future Work

Ge4 SbTe5 is a phase change memory material which does not lie on the GeTe-Sb2 Te3 pseudobinary tie-line. The compound exists in a metastable rocksalt crystal structure upon rapid
quenching from the melt. In this rocksalt crystal structure the Ge and Sb atoms are ran115

domly distributed on the cation site with the Te atoms occupying the anion sublattice.
Upon heating this metastable rocksalt phase undergoes an ordering transition to the stable
rhombohedral phase. This phase transition is accompanied by a drop in electrical resistivity. Despite the decrease in electrical resistivity the Seebeck coefficient continues to increase
while the thermal conductivity stays relatively temperature independent. The ZT reaches
values well above unity, which are comparable to state-of-art PbTe.
It was shown that both Ge4−x Snx SbTe5 and Ge4 Sb1−x Bix Te5 , which were isovalent
atomic substitutions intended to lower the lattice contribution to thermal conductivity, actually decreased the thermoelectric performance. From the decreased trend of the Seebeck
coefficient, it is believed that the substitutions caused a change in the band gap of Ge4 SbTe5 .
Future work could include the study of other substitutions such as Si on the Ge atomic site
or Se on the Te atomic site.
Na, which is predominately monovalent, was substituted for Ge. This was done to increase the hole concentration, as Na would introduce an acceptor for each atom substituted.
Samples with 0.5, 1.0, and 1.5% Na in Ge4 SbTe5 were synthesized and shown to decrease
the electrical resistivity and Seebeck coefficient. Unfortunately the power factor and ZT
were decreased, implying the carrier concentration was too high for the optimization of ZT.
Ga was doped for Ge, which also should have acted as a p-type dopant. The Ga was not
successfully substituted for Ge, but rather formed the impurity phase Ga2 Te3 . Despite
the existence of this secondary phase the power factor was increased, thermal conductivity
was decreased, and the ZT was increased at temperatures above 600K for 1% Ga doped
Ge4 SbTe5 . A more controlled study of the thermoelectric properties of Ge4 SbTe5 -Ga2 Te3
composites should be conducted so as to properly understand the mechanism of power factor
and ZT enhancements. Future dopants could include In, Zn, transition metals, or rare-earth
116

elements, which could introduce a magetic moment and enhance the Seebeck coefficient.

117

Chapter 6
Ge17Sb2Te20
Ge17 Sb2 Te20 rests on the GeTe-rich side of the GeTe-Sb2 Te3 binary tie-line. The formula
can be alternatively written as (GeTe)17 Sb2 Te3 , where Sb2 Te3 represents a roughly 5.9%
substitution in GeTe. Few studies, other than for bulk thermoelectrics have been conducted
on this specific composition. It displays degenerate p-type semiconducting nature and has a
melting point of approximately 670◦C.[141]
The stable crystal structure for Ge17 Sb2 Te20 is the rhombohedral phase, R¯3m. This
phase as aforementioned, has alternating layers of Ge, Sb, and Te atoms, with inter-penetrating
vacancy layers between van der Waals bonded Te layers. A metastable rocksalt phase exists
when synthesized under non-equilibrium conditions, such as chemical vapor deposition or
rapid quenching. This rocksalt phase, upon heating, transitions to the stable rhombohedral
crystal structure at approximately 500K. A second phase transition occurs at temperatures
just below 600K from the rhombohedral phase to a rocksalt phase. The high temperature
rocksalt phase, similar to GeTe, exists only at elevated temperature and corresponds to the
temperature at which the ferroelectric state transitions to a paraelectric state. These phase
transitions can be observed in the specific heat versus temperature for Ge17 Sb2 Te20 , which
can be seen in Figure 6.1.

118

Figure 6.1: Specific heat versus temperature, measured by DSC, for Ge17 Sb2 Te20 . The
phase transitions can be seen as endothermic peaks at T1 , the transition from metastable
rocksalt to rhombohedral, and T2 , the phase transition from the stable rhombohedral phase
to the stable rocksalt phase.

The temperature dependent X-ray diffraction patterns can be seen in Figure 6.2 and
Figure 6.3 for the warming and cooling trends, respectively. Upon warming the first phase
transition from metastable rocksalt to rhombohedral is not clearly discernible. It is believed
that because the transition is an order-disorder transition and is therefore diffusion-limited
there has not been substantial time for the full transition to take place. Upon further
warming the high temperature rocksalt phase can be seen above 623K. Upon cooling, the
stable rhombohedral phase can be discerned, and begins to form around 571K.

119

Figure 6.2: X-ray diffraction patterns for Ge17 Sb2 Te20 upon heating from room temperature.
The first transition from metastable rocksalt to rhombohedral is not discernible, but the high
temperature rocksalt phase can be seen at approximately 623K.

Figure 6.3: X-ray diffraction patterns for Ge17 Sb2 Te20 upon cooling. The stable rhombohedral phase is discernible below 570K.

120

In the following work stoichiometric amounts of Ge, Sb, and Te were placed in evacuated
silica ampoules and melted at 1173K for 12 hours. The samples were then water quenched
to room temperature. Additional annealing was performed at 863K, above the second phase
transition temperature, for 24h and again water quenched. X-ray diffraction was performed
on all ingots obtained. The samples were ball-milled for 5min in a stainless steel jar with
stainless steel balls. The ball-milled powder was then sintered using SPS at 673K, for 5min,
under 40MPa of uniaxial pressure. The sintered pellets were accordingly cut using a diamond
saw for thermoelectric characterization, thermal characterization, and X-ray diffraction.
The X-ray diffraction patterns of Ge17 Sb2 Te20 quenched from the melt and annealed are
displayed in Figure 6.4. When quenched from the melt Ge17 Sb2 Te20 forms in the rhombohedral crystal structure (R¯3m). When further annealed at 590◦C, Ge17 Sb2 Te20 forms in the
rocksalt crystal structure(F m¯3m). For the annealed sample the peaks in the pattern are
quite broad which could be due to large amounts of strain in the system from the quenching
process. By assigning the allowed Miller indices for the diffraction peaks of the rocksalt
phase, the lattice parameter was calculated to be 5.92˚
A, which yields a theoretical density
of 6.256gcm−3 . The X-ray diffraction pattern of melt-quenched Ge17 Sb2 Te2 reveals impurities of elemental Ge and Sb, as can be seen in Figure 6.4. The rhombohedral phase has a
calculated lattice parameter of 4.24˚
A with a rhombohedral angle of 58.24◦ .

121

Figure 6.4: (a) X-ray diffraction pattern of Ge17 Sb2 Te20 quenched from 1173K, seen here
with Sb and Ge impurities labeled. Quenching from the melt yields a rhombohedral crystal
structure. (b) X-ray diffraction pattern of Ge17 Sb2 Te20 quenched from the melt and annealed
for an additional 24h at 863K. This method produces an impurity free rocksalt phase.

Despite having secondary phases in the quenched sample, both the quenched and annealed Ge17 Sb2 Te20 samples were characterized. The thermoelectric properties of both compounds were found to be nearly the same, aside from a slightly higher ZT for the quenched
compound, as can be seen in Figure 6.5.

122

Figure 6.5: ZT versus temperature for Ge17 Sb2 Te20 prepared by quenching from the melt
and subsequently annealing. The SEM/EDS elemental maps show the presence of Sb precipitates.

The performance of the quenched compound with the secondary phases contradicts the
theoretical work of Bergman and Levy from 1991.[139] Their work aimed to determine the
validity of making composites which could overrcome the contraindicated properties of ZT.
Unfortunately it was found that under ideal conditions, the ZT of a composite could not
exceed the highest ZT of any one of the constituents which comprise the composite. As
a preliminary means of gauging the role and validity of enhancing the ZT of Ge17 Sb2 Te20
with secondary phases, the stoichiometric amount of Sb was varied. Specifically, the following compounds were quenched from 1173K and synthesized using the aforementioned
synthesis parameters: Ge17 Sb1.90 Te20 , Ge17 Sb1.95 Te20 , Ge17 Sb2 Te20 , Ge17 Sb2.10 Te20 , and
Ge17 Sb2.20 Te20 .
The electrical resistivity and Seebeck coefficient increased with increasing Sb content,
as can be seen in Figure 6.6(a) and (b), respectively. This behavior would indicate a dependence of the charge carrier concentration on the amount of Sb impurity phase. It was
therefore believed that the Sb, which is defined as a semimetal was decreasing the hole con-

123

centration. The ZT showed a dependence on the Sb content, with a peak value above 2.0
for Ge17 Sb2.1 Te20 , as can be seen in Figure 6.6(c) and (d).

Figure 6.6: (a)The electrical resistivity, (b) ZT, and (c) Seebeck coefficient for Ge17 Sbx Te20 .
By altering the stoichiometry of Sb, and therefore the concentration of Sb secondary phase,
the properties can be tuned. (d) Shows ZT at 773K versus x, the stoichiometric amount of
Sb.

These results are exciting, but lack the proper scientific quality to be deemed valid. Both
Sb and Ge impurities exist in the compound, but also, the existence of these secondary
phases upon quenching implies that Ge17 Sb2 Te20 is not stoichiometric since additional time
is necessary to allow all of the Sb and Ge to diffuse onto their respective lattice sites. Based
on the findings above the following hypothesis was formed:
Seconday phases of elemental Sb and/or Ge provide a route for tuning the electrical and
124

thermal properties of Ge17 Sb2 Te20 , which in turn could be used to optimize the
thermoelectric performance.
To experimentally investigate this hypothesis two separate studies on the effects of elemental Sb and Ge as a secondary phase in Ge17 Sb2 Te20 were conducted. The following
sections will describe in greater detail the results of these projects.

6.1

Probing the Role of Sb Impurities

The goal of this specific project was to understand the mechanisms, if any, Sb impurities play
in affecting the thermoelectric performance of Ge17 Sb2 Te20 . To probe this experimentally,
pristine Ge17 Sb2 Te20 was synthesized and characterized. Additional samples were then
made with controlled amounts of elemental Sb as an impurity. Below are the experimental
procedures, results and discussion, and conclusions from this work.

6.1.1

Synthesis Procedures

To synthesize single phase Ge17 Sb2 Te20 stoichiometric amounts of Ge (Alfa Aesar, zonerefined ingot, 99.9999%), Sb (Alfa Aesar, 6mm shot, 99.999%), and Te (Alfa Aesar, chunck,
99.999%) were sealed in silica ampoules which were evacuated to approximately 10−5 T orr.
The samples were then melted at 1173K for 12h and water quenched. Additional annealing
was done at 863K for 24h and again water quenched. Additional Sb (Alfa Aesar, 6mm shot,
99.999%) was added to the materials before powder processing. The resulting ingots and
added Sb were then ball-milled in a SPEX vibratory mill for 5min using a stainless steel jar
and stainless steel media. The powder was densified using SPS at 673K for 10min under

125

40MPa of uniaxial pressure. The phase purity and crystal structure was verified using room
temperature X-ray diffraction and SEM/EDS in the Center for Advanced Microscopy.

6.1.2

Results and discussion

For Ge17 Sb2 Te20 +Sbx , x was set equal to 0.0975, 0.195, 0.39, 0.78 which correspond to 0.25,
0.50, 1.0, and 2.0 mol% added Sb, respectively. The X-ray diffraction patterns revealed no
change in the crystal structure of Ge17 Sb2 Te20 , as seen from Figure 6.7. As the amount of
additional Sb reached 0.50% and higher the diffraction peaks associated with elemental Sb
could be distinguished.

Figure 6.7: X-ray diffraction patterns for Ge17 Sb2 Te20 , with 0.25, 0.50, 1.0, and 2.0% added
Sb. The star marks the diffraction peaks which indicate the presence of elemental Sb as
a secondary phase(PDF#98-000-0095). The EDS maps clearly show the presence of Sb
microstructures in Ge17 Sb2 Te20 .
126

To further verify the existence of Sb as a secondary phase and probe the general size of
the Sb agglomerates SEM/EDS was performed, which can also be seen in Figure 6.7. Sb
can be clearly distinguished as a secondary phase in Ge17 Sb2 Te20 . The general size of the
Sb clumps range from approximately 1µm to 50µm. The size of the impurities puts the
possibility of nanostructured quantum effects out of the question. It should also be noted
that the impurities show no regular geometry or pattern, they are essentially randomly
oriented and irregularly shaped.
The electrical resistivity for the samples can be seen in Figure 6.8(a). As the Sb content
is increased the electrical resistivity across all temperatures was increased. The Seebeck
coefficient is also increased with increased Sb content (Figure 6.8(b)).

Figure 6.8: (a)Electrical resistivity versus temperature, seen here with ±8% error bars and
(b)Seebeck coefficient versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50, 1.0, and 2.0%
added Sb, in this figure the uncertainty of ±2% associated with measuring the Seebeck
coefficient is smaller than the markers.

Because of the increased Seebeck coefficient and electrical resistivity it was initially believed that the Sb impurities were decreasing the hole concentration. To verify this hypothesis Hall measurements were performed at room temperature. The measurements showed no
significant change in the hole concentration with increased Sb content. Rather, there was a
127

sharp decrease in the hole mobility with the addition of Sb secondary phase. The electrical
resistivity, Seebeck coefficient, hole concentration, and hole mobility at 300K as a function
of Sb content can be seen in Table 6.1.
Table 6.1: Electrical resistivity, Seebeck coefficient, carrier concentration, and hole mobility
at 300K for Ge17 Sb2 Te20 +Sbx .

Because of the increased Seebeck coefficient, the power factor was increased to values well
above 30µWcm−1 K−2 for 0.25% and 0.50% added Sb. For higher concentrations of added
Sb the hole mobility was decreased too far and resulted in a decrease in power factor as can
be seen in Figure 6.9(a). The complement between decreased hole mobility and increased
Seebeck coefficient can be seen in Figure 6.9(b).

128

Figure 6.9: (a) Power Factor versus temperature for Ge17 Sb2 Te20 +Sbx , seen here with ±8%
error bars. (b) Seebeck coefficient and hole mobility at 300K as a function of %Sb as an
impurity. A clear compliment can be seen between the two properties, which is indicative of
a change in the scattering relaxation time.

Similar results in Pb-precipitated PbTe were observed by Heremans, et al.[142] They
found that the secondary phases of Pb acted as additional scattering sites in the PbTe
matrix. The Pb precipitates did not change the charge carrier concentration but lowered
the charge carrier mobility, which in turn resulted in an increased Seebeck coefficient by
changing the energy dependence of the scattering relaxation time.
Recall from chapter 1 that for metals and degenerately doped semiconductors the Mott
relation expresses the Seebeck coefficient as a function of the energy dependent electrical
conductivity, equation (1.60). The electrical conductivity is typically equated to the product
of e, the electron charge, n(E) the energy dependent charge carrier concentration, and µ(E),
the energy dependent charge carrier mobility. The mobility can be further expressed as
the ratio of τ (E), the energy dependent relaxation time, and m∗, the effective mass of the
charge carriers, equation (1.24). The scattering relaxation time can be approximated with
a power-law dependence for semiconductors with parabolic bands, equation (1.28):

129

Because the host matrix of Ge17 Sb2 Te20 was not changed chemically, it is believed that
the effective mass is not altered, but rather the energy dependence of τ (E) was increased,
which resulted in a decrease in mobility and an increase in the Seebeck coefficient. To
further verify and quantify the changing energy landscape of τ (E) the Nernst effect should
be measured.[143]
The thermal conductivity was also decreased with the increasing presence of Sb impurities, which can be seen in Figure 6.10(a). To estimate the electronic contribution to
thermal conductivity the Lorenz number was calculated using a single parabolic band approximation to fit the Seebeck coefficient below the phase transition temperature, determining the reduced Fermi energy, and calculating the Lorenz number accordingly. The Lorenz
number versus temperature can be seen in Figure 6.10(c), with the free electron value of
2.44x10−8 WΩ−1 K−2 marked by the dotted line. By subtracting the calculated electronic
contribution from the total thermal conductivity the lattice contribution was determined.
As can be seen in Figure 6.10(b) the lattice contribution to thermal conductivity was significantly decreased with an increasing presence of Sb secondary phase. This decrease in
thermal conductivity is further justification that the Sb impurities act as scattering sites for
both electrons and phonons in Ge17 Sb2 Te20 .

130

Figure 6.10: (a)Thermal conductivity versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50,
1.0, and 2.0% added Sb. (b)The lattice contribution to thermal conductivity versus temperature. (c)Lorenz number versus temperature, which was determined from a single parabolic
band fit of the Seebeck coefficient.

With a decreased thermal conductivity and optimized power factor the calculated ZT
was increased to values well above 2; seen in Figure 6.11. The largest increase in ZT was
seen for 0.5% added Sb, which represents an average increase of approximately 20% at peak
values. The ZTavg was also increased to approximately 1.4.[144]

131

Figure 6.11: ZT versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50, 1.0, and 2.0% added
Sb, seen here with ±13% on Ge17 Sb2 Te20 and Ge17 Sb2 Te20 + Sb0.195 . A large enhancement
was observed for Ge17 Sb2 Te20 with 0.5%Sb secondary phase.

6.1.3

Conclusion

This study aimed to understand the reason for why impurities of Sb in Ge17 Sb2 Te20 lead to
increased thermoelectric performance. It was found that power factor was enhanced because
of an increase in electrical resistivity and Seebeck coefficient. Hall measurements revealed
that the secondary phase of Sb did not change the hole concentration, but rather decreased
hole mobility because of the increased scattering on the charge carriers in Ge17 Sb2 Te20 . The
thermal conductivity was also decreased because of the ability of the Sb impurities to scatter
heat carrying phonons. The result of these changes in the electrical and thermal properties
was a significant enhancement in the ZT of Ge17 Sb2 Te20 to values well above 2.0. In the
future, studies on the geometry and particle size of the Sb precipitates should be conducted.
Also, the effects of the precipitates on the mechanical properties would be of interest.

132

6.2

Probing the Role of Ge Impurities

Composites of Ge17 Sb2 Te20 and Sb were shown to enhance the thermoelectric performance
of pristine Ge17 Sb2 Te20 . The initial motivation for the project was to investigate the fundamental mechanisms behind the ZT enhancements caused by the secondary phases present
in quenched Ge17 Sb2 Te20 , where both Ge and Sb impurities were present. The following
study is an attempt to understand, if at all different from the case of Sb, what role the Ge
secondary phases play in affecting the thermoelectric properties of Ge17 Sb2 Te20 .

6.2.1

Synthesis Procedures

Single phase rocksalt Ge17 Sb2 Te20 was synthesized using the parameters aforementioned.
0.25%, 0.50%, 1.0%, and 2.0% added Ge was ball milled with single phase Ge17 Sb2 Te20
for 5min in a stainless steel jar and under argon atmosphere. The resulting powder was
densified using the same SPS parameters aforementioned for Ge17 Sb2 Te20 +Sbx . Density of
the sintered pellets was measured using the Archimedes method. All densities were ≥ 95%
theoretical density.

6.2.2

Results and Discussion

The X-ray diffraction patterns for Ge17 Sb2 Te20 +Gex revealed secondary phases of elemental
Ge with concentrations greater than 1.0% Ge (PDF#01-073-7012). Overall the rocksalt
crystal structure was maintained aside from some peak splitting around 70◦ 2θ. This double
peak could be a slight distortion along the [111] direction of the rocksalt crystal structure of
Ge17 Sb2 Te20 . The presence of Ge secondary phases for 0.25% and 0.50%Ge added samples
could not be verified using X-ray diffraction due to the low concentrations of the Ge. To

133

verify the existence of the Ge impurities SEM/EDS was conducted. The results of the EDS
maps can be seen in Figure 6.12.

Figure 6.12: EDS elemental maps showcasing the presence of Ge impurities in
Ge17 Sb2 Te20 +Gex samples.

Finding and identifying the Ge impurities proved to be difficult for two reasons: (1) The
high atomic concentration of Ge in Ge17 Sb2 Te20 produces a dense Ge background during
the EDS elemental analysis, and (2) the size of the Ge agglomerates were typically ≤ 10µm.
However the Ge secondary phases were positively identified and can be seen in Figure 6.12,
with dashed circles outlining the Ge impurities.
The electrical resistivity was increased as the amount of Ge secondary phase was increased. At 300K the electrical resistivity was increased from approximately 1.5mΩ cm to
1.8mΩ cm for Ge17 Sb2 Te20 +2.0%Ge, which is well beyond the ±8% uncertainty associated
134

with measuring electrical resistivity. The Seebeck coefficient was also increased as the concentration of Ge was increased. The highest increase in Seebeck coefficient occurred for
Ge17 Sb2 Te20 +0.50%Ge. Unlike in the case of using Sb impurities the Seebeck coefficient
decreased for the higher concentrations of 1.0%Ge and 2.0%Ge. The electrical resistivity
and Seebeck coefficient can be seen in Figure 6.13(a) and (b), respectively.

Figure 6.13: (a) Electrical resistivity versus temperature for Ge17 Sb2 Te20 +Gex , seen here
with ±8% error bars on Ge17 Sb2 Te20 . (b) Seebeck coefficient versus temperature for
Ge17 Sb2 Te20 +Gex .

By increasing the Ge impurity concentration an optimization in the power factor was
achieved. Specifically the power factor was increased across all temperatures, with peak
values around 35µWcm−1 K−2 . The power factor versus temperature can be seen in Figure 6.14(a). Hall measurements revealed that the Ge impurities act as scattering sites for
the holes and lowered the hole mobility to less than 20cm2 V−1 s−1 . As can be seen in
Figure 6.14(b) the decrease in hole mobility coincides with an optimization of the power
factor.

135

Figure 6.14: (a) Power factor as a function of temperature for Ge17 Sb2 Te20 +Gex , with
8 ± % error bars on Ge17 Sb2 Te20 . (b) Hole mobility and room temperature power factor as
a function of %Ge secondary phase. A clear complement can be discerned.

The increase of the Seebeck coefficient, electrical resistivity, and ultimately power factor,
can be explained by a change in the energy landscape of the scattering relaxation time, τ (E).
This phenomenon was explained in greater mathematical detail in the previous section on
Ge17 Sb2 Te20 +Sbx . In addition to scattering the charge carriers to increase the resistive
nature of the electrical transport in Ge17 Sb2 Te20 the Ge impurities scatter phonons to
decrease the lattice thermal conductivity. The thermal conductivity versus temperature for
Ge17 Sb2 Te20 +Gex can be seen in Figure 6.15.

136

Figure 6.15: Thermal conductivity versus temperature for Ge17 Sb2 Te20 +Gex , seen here with
±8% error bars on Ge17 Sb2 Te20 , which was calculated from the measurement techniques
used.

The Ge impurities led to a decreased thermal conductivity and increased power factor,
which led to an overall increase in the ZT for 0.25% and 0.50% added Ge. The ZT versus
temperature and ZT versus %Ge impurity can be seen in Figure 6.16.

137

Figure 6.16: ZT versus temperature for Ge17 Sb2 Te20 with 0.25, 0.50, 1.0, and 2.0% added
Ge, seen here with ±13% error bars on Ge17 Sb2 Te20 . A large enhancement was observed
for Ge17 Sb2 Te20 with 0.5%Ge impurity.

6.2.3

Conclusion

In this study Ge was added to the pristine phase of Ge17 Sb2 Te20 in controlled amounts
of 0.25, 0.50, 1.0, and 2.0 atomic percent. Similar to the results of Ge17 Sb2 Te20 +Sbx , it
was found that the Ge impurities were approximately a few microns in size and caused an
increase in the electrical resistivity and Seebeck coefficient. The power factor was enhanced
significantly and the thermal conductivity was decreased. Hall measurements showed no
significant change in the hole concentration, rather the hole mobility was decreased with
the introduction of Ge impurities. This was explained by a changing of the scattering
relaxation time as the secondary phase of Ge acted as a scattering site for both charge carriers
and heat carrying phonons. The ZT of Ge17 Sb2 Te20 with the presence of Ge impurities
increased for 0.25 and 0.50% added Ge. The results of the two studies, Ge17 Sb2 Te20 +Sbx

138

and Ge17 Sb2 Te20 +Gex , revealed a promising avenue for engineering new high-ZT materials.
However a more comprehensive understanding of why the enhancements from secondary
phases is possible is necessary and will be presented in the next section.

6.3

Enhancing ZT Using Composites

In 1991 Bergman and Levy published work to examine the thermoelectric properties of
composites in hopes of finding a new way to enhance the ZT of materials. To begin, the
basic macroscopic equation for a thermoelectric material is defined concisely as:

J = Q∇ψ

(6.1)

where




 −JE /e 
J≡

−JS /k


(6.2)



 ∇(eφ) 
∇ψ ≡ 

∇(kT )






σ/e2

(6.3)



σS/ek 
 Q11 Q12  
Q≡
≡

2
σS/ek γ/k T
Q21 Q22

(6.4)

JE is the electrical current density, defined as JE = σE, where E is the electric field
and σ is the electrical conductivity. JS is the heat flux given as T JS = −γ∇T , where γ is
the coefficient of thermal conductivity. The usual thermal conductivity under zero electric
current, κ, can be defined in terms of gamma as κ = γ − T σS 2 . The quantity S is the
139

thermoelectric coefficient, or the absolute Seebeck coefficient, and φ is the voltage. The
thermoelectric figure of merit can therefore be defined in terms of the determinant of Q as
follows:

Q212

ZT =

detQ

=

Q11 Q22
−1
Q212

−1

(6.5)

For composite media a volumed averaged value for the effective transport is defined in
(1)

(2)

(2)

(2)

terms of the macroscopic field E (1) = (E1 , E2 ) and E (2) = (E1 , E2 ). The components
of the volume-averaged effective transport matrix, Qe then become defined as seen below.

(1)2

Qe11 =

1
V

dV (Q11 E1

Qe22 =

1
V

dV (Q22 E2

Qe12 =

1
V

(2)2

(1)

(1)2

− Q22 E2

(2)2

− Q11 E1

(2)

(2)

)

(6.6)

)

(6.7)

(1)

dV (Q12 (E1 · E2 ) − E1 · E2 )

(6.8)

Qe11 , Qe22 , and Qe12 each contain off-diagonal terms which make the integrals too complex to analytically solve. The following assumptions are therefore invoked in order to simplify the equations: (1) Q12

Q11 and Q12

Q22 ; (2) each component of the composite is

homogeneous and isotropic. The first assumption allows for the coupled terms in equations
(6.6), (6.7), and (6.8) to be neglected, and the second assumption allows the transport coefficients to be pulled out of the integrand since they would be isotropic and homogeneous and
therefore independent of volume. After further simplifying it was found that the effective
figure of merit for the composite could never exceed the figure of merit of any one of the
140

components alone. This negative result was the first theoretical study of composites and
showed that working with composites would not show promising results.
In 1993 Hicks and Dresselhaus established that if composites could be made which were
nanostructured the figure of merit of a material could be substantially enhanced.[49] The
original work by Bergman and Levy assumed that the composite media were large in nature
(µm and larger). Hicks and Dresselhaus found that nanostructured composites did not fall
under the original assumptions of the composite medium theory and could therefore be used
to alter the electronic density of states, which in turn would enhance the Seebeck coefficient.
Energy filtering, which was essentially an extension of the work by Hicks and Dresselhaus,
theoretically showed that using nanostructured secondary phases with specific band gap
offsets relative to the host matrix could result in a “filtering” of the hot electrons, which in
turn would lead to enhanced power factor.[145] Bulk nanostructured thermoelectric materials
have repeatedly been shown to yield enhancements in ZT for a number of materials.[146,
147, 140, 101, 148]
In 1999 Bergman and Fel found that the power factor of a composite mixture of two
materials could be enhanced.[149] When such an enhancement is possible based on the materials it is realized for a microstructure with parallel slabs or coated spheres. In the study,
it was theoretically proposed that composites of Ni-CoSb3 , PbTe-Bi2 Te3 , Ni-YbAl3 , Al(Bi2 Te3 )0.2 (Sb2 Te3 )0.8 , and constantan-(Bi2 Te3 )0.2 (Sb2 Te3 )0.8 would all exhibit increased
power factor. Recently a number of studies have shown that using secondary phases can
lead to enhancements in ZT. Zhang, et al synthesized PbTe with MnTe secondary phases.
The result was an increase in the thermoelectric performance as well as the hardness of the
material.[150] Wu, et al synthesized PbTe-with precipitated PbS secondary phases, which
resulted in a broad plateau in ZT and a ZTavg value of approximately 1.5.[151]. It was
141

recently shown that the incorporation of ZnO as a secondary phase in the chalcopyrite compound CuInTe2 can lead to enhancements in power factor and the ZT overall.[152] A number
of studies on the skutterudite compound CoSb3 have shown that the incorporation of secondary phases such as AgSbTe2 and InSb can result in the decrease of thermal conductivity
as well as an enhancement of power factor and therefore ZT.[153, 140] In 2014 Zou, et al
used the secondary phase of β-Zn4 Sb3 in Cu3 SbSe4 to enhance the ZT value up to 1.3 by
increasing the energy dependence of the scattering relaxation time.[154]
Perhaps more interesting are the recent studies on Ge-Sb-Te compounds and using secondary phases to enhance the thermoelectric performance. In 2015 Fahrnbauer, et al used
microstructured CoSb3 in (GeTe)m Sb2 Te3 alloys to enhance the ZT.[155] In the same year
Fahrnbauer et al found that secondary phases of CoGe2 were effective in enhancing the ZT of
Ge12 Sb2 Te15 , Ge17 Sb2 Te20 , and Ge19 Sb2 Te22 .[156] The results of these studies were similar
in nature to the work described in the previous sections on using Sb and Ge secondary phases
to enhance the ZT of Ge17 Sb2 Te20 . Figure 6.17 shows the ZT versus temperature results
for a number of Ge-Sb-Te compounds from this work and others, which use microstructured
secondary phases to enhance ZT.

142

Figure 6.17: ZT as a function of temperature for a number of microstructured composites
which utilize (GeTe)x Sb2 Te3 alloys. Specifically Ge12 Sb2 Te15 with precipitates of CoGe2
from [156]. Ge17 Sb2 Te20 with precipitates of CoGe2 from [156]. Ge10.5 Sb2 Te13.5 with precipitated CoSb3 from [155]. Ge17 Sb2 Te20 with impurities of Sb and Ge from [144] and [157],
respectively.

The results seen in Figure 6.17 directly contradict the conclusions reached by Bergman
and Levy. The caveat to the potential for using composites in thermoelectric applications can
be pinpointed to the assumptions made by Bergman and Levy in their original paper. The
assumptions that Q12

Q11 and Q12

Q22 have important implications for the overall

conclusions one would reach when developing an effective medium theory for thermoelectrics.
Recall the earlier equation for the figure of merit in terms of Q:

ZT =

Q212
detQ

=

Q11 Q22
−1
Q212

−1

(6.9)

The important part of the equation lies in the components of Q. One can therefore

143

condense the argument down to ∆e , which is defined in equation (6.10).

∆e =

Q2e12
Qe11 Qe22

(6.10)

The simplicity of this equation stems from the assumptions made about the relative
magnitudes of Qe11 , Qe22 , and Qe12 . If one does not assume/neglect the off-diagonal terms
from (6.6), (6.7), and (6.8), ∆e becomes

[Q12

∆e =
(Q11

(1)2

dV E1

(1)

(2)

dV (E1 · E2 ) −

− Q22

(1)2

dV E2

)(Q22

(2)

(1)

dV (E1 · E2 )]2
(2)2

dV E2

− Q11

(2)2

dV E1

(6.11)
)

Equation (6.11) contains terms which are nonsymmetric with respect to the matrix Q,
and essentially imply that the intrinsic electrical and thermal properties of each component
of the composite are coupled and not independent of one another. The original assumptions
on Q11 , Q22 , and Q12 effectively lead to the conclusion that ZT is much less than unity for all
components of the composite. The apparent disagreement between this theoretical work and
the recent experimental findings on composite thermoelectrics then becomes resolved in that,
for materials with ZT ≥ 1 the constraints may not apply. This leaves room for developing
new thermoelectric composite materials with greater thermoelectric efficiency and improved
mechanical stability.

6.4

Ge17+xSb2−xTe20

The charge carrier concentration in semiconductors is a highly sensitive variable which is
important for the optimization of thermoelectric performance. An effective means of tun144

ing the carrier concentration for thermoelectric materials is through the incorporation of
foreign atoms which either introduce an acceptor state (p-type) or a donor state (n-type).
The compound of interest, Ge17 Sb2 Te20 , has yet to be studied for optimization of the carrier concentration. However, other studies on doping various (GeTe)x Sb2 Te3 alloys show
the potential to enhance the ZT of Ge17 Sb2 Te20 . Welzmiller, et al studied Mn doping in
Ge4 Sb2 Te7 , which resulted in a ZT enhancement as well as the introduction of a ferromagnetic phase from the Mn ions.[131] Welzmiller, et al also observed a nearly 300% increase
in ZT by doping Cr on the Ge site in GeSb2 Te4 .[132] In 2015 Welzmiller, et al studied the
effects of Cd doping on (GeTe)n Sb2 Te3 and (SnTe)n Sb2 Te3 compounds and found that Cd
increased the density of states effective mass and led to an increase in the Seebeck coefficient,
which allowed for slight increases in the power factor, and ZT overall.[122] Perumal, et al
used Sb and Bi to optimize the carrier concentration and power factor of GeTe, resulting in
ZT values well in excess of unity.[158, 159]
The following work studied the thermoelectric properties of Ge17+x Sb2−x Te20 . Doping
on the nominally trivalent Sb site with divalent Ge should introduce one acceptor (or hole) per
atomic substitution, increase the carrier concentration, and therefore decrease the electrical
resistivity. Also, by adjusting the Ge and Sb stoichiometry we avoid using additional elements
for doping and therefore keep the cost of synthesis down.

6.4.1

Experimental Procedures

Samples of Ge17+x Sb2−x Te20 (x = 0, 0.04, 0.16, 0.20, 0.24, 0.30, 0.40, which correspond to
0, 2, 8, 10, 12, 15, and 20% Ge, respectively) were synthesized by placing stoichiometric amounts
of elemental Ge, Sb, and Te in evacuated and sealed silica ampoules. The samples were then
melted at 1173K, held for 12h, and water quenched. After quenching, the samples were
145

annealed at 863K, for 24h and water quenched. The obtained ingots were ball milled for
5min. The powder was densified using SPS at 738K for 5min under 45MPa of pressure. All
samples had density ≥ 99% theoretical density. High temperature Hall measurements were
measured by Alexander Page at the University of Michigan, Department of Physics, under
the advice of Dr. Ctirad Uher. The carrier concentration was determined using the Hall
effect, under the single carrier approximation using the relation Rh = 1/pe. The Hall coefficient was measured using an AC 4-probe method in a custom built apparatus from 293K to
773K under a maximum field of ±1T. The samples were in a 1atm Argon environment to
prevent oxidation.

6.4.2

Results and Discussion

All samples were single phase and showed no secondary phases, which was verified using Xray diffraction and SEM/EDS. As Ge was doped onto the Sb site of Ge17 Sb2 Te20 the rocksalt
symmetry began to degrade to a rhombohedral structure. Specifically, at x = 0.16 significant
peak splitting began to take place above 40◦ two-theta and below 30◦ two-theta. The X-ray
patterns for all samples as well as a reference GeTe sample can be seen in Figure 6.18.

146

Figure 6.18: X-ray patterns for Ge17+x Sb2−x Te20 showing x = 0, 0.04, 0.16, 0.20, 0.24, 0.30,
and 0.40. The rocksalt structure of Ge17 Sb2 Te20 begins to transition to rhombohedral as
Ge content is increased. GeTe is shown as a reference for the rhombohedral peak splitting
taking place.

Quenching Ge17 Sb2 Te20 from the annealing temperature of 863K yields a rocksalt crystal
structure with a lattice parameter of 5.92˚
A. The rhombohedral crystal is typically described
in mathematical terms as a hexagonal cell, but can also be visualized as a distortion along
the [111] direction of the rocksalt unit cell. The angle between the a,b, and c crystal
directions can also be used to characterize the rhombohedral crystal structure, i.e. for GeTe
the distorted cubic angle is 88.26◦ .[160] As the amount of Ge doped for Sb is increased the
c-axis lattice parameter of the hexagonal unit cell is increased and the cubic angle of the
rocksalt structure is decreased, as can be seen in Figure 6.19.

147

rystal direction of the rocksalt unit cell. The transition is fundamentally driven by the
expansion of the c-axis lattice parameter of the rhombohedral-hexagonal unit cell.]As the
amount of Ge doped onto the Sb site of Ge17 Sb2 Te20 is increased the angle of the rocksalt
unit cell (normally 90◦ ) is decreased. This can be understood as a distortion along the
[111] crystal direction of the rocksalt unit cell. The transition is fundamentally driven by
the expansion of the c-axis lattice parameter of the rhombohedral-hexagonal unit cell.
Figure 6.19: c

The electrical resistivity as a function of temperature can be seen in Figure 6.20. The
electrical resistivity is decreased with increased Ge content, from a value of 1.6mΩ cm for
Ge17 Sb2 Te20 at 300K to 0.5mΩ cm for Ge17.4 Sb1.6 Te20 (20% Ge) at room temperature.
This trend is suggestive that Ge is replacing Sb and increasing the hole concentration. Most
interesting to note is the changing nature of the phase transition. For Ge17 Sb2 Te20 there is
a sharp decrease in the electrical resistivity, which is due to the phase transition from cubic
to rhombohedral. However, the resistivity of the 20% doped sample displays a true metallic
nature and continues to increase with temperature. It is hypothesized that this occurs
because the increased number of holes makes the electronic transition, which is originally
148

due to an increase in hole mobility, less pronounced, as the hole mobility should decrease
with increased hole concentration.[12]

Figure 6.20: Electrical resistivity versus temperature Ge17+x Sb2−x Te20 . As the amount of
Ge dopant is increased the electrical is decreased to values below 1.0mΩ cm.

The Seebeck coefficient is also decreased with increased Ge dopant, as can be seen in
Figure 6.21. The Seebeck coefficient, in general, is inversely proportional to the Seebeck
coefficient. The decreasing Seebeck coefficient seen in Figure 6.21 is further indication that
the carrier concentration is increased due to Ge doping.

149

Figure 6.21: Seebeck coefficient versus temperature for Ge17+x Sb2−x Te20 . The decrease in
Seebeck coefficient as the concentration of Ge dopant is increased is further indication that
the hole concentration is increased from doping.

Because of the decreased Seebeck coefficient and electrical resistivity the power factor
was increased approximately 30% from a peak value of 35µWcm−1 K−2 for Ge17 Sb2 Te20
to a peak value of 43µWcm−1 K−2 for 15% Ge doped Ge17 Sb2 Te20 . For higher dopant
concentrations the power factor is decreased. The power factor as a function of temperature
can be seen in Figure 6.22.

150

Figure 6.22: Power factor versus temperature for Ge17+x Sb2−x Te20 , seen here with the
calculated error bars of approximately ±10% for Ge17 Sb2 Te20 . An optimal power factor
was reached for 15%Ge; higher concentrations caused a decrease.

The thermal conductivity was increased with increased Ge content, with 20%Ge showing
values in excess of 2 across the entire temperature range. The thermal conductivity as a
function of temperature can be seen in Figure 6.23(a). The increase in thermal conductivity
is believed to come primarily from the electronic contribution to thermal conductivity. To
estimate the lattice thermal conductivity, the electronic contribution to thermal conductivity
was subtracted from the total thermal conductivity. The Lorenz number was calculated
by fitting the Seebeck coefficient to a single parabolic band model with acoustic phonon
scattering assumed to be the dominant scattering mechanism. The result of the calculated
lattice thermal conductivity as a function of temperature can be seen in Figure 6.23(b). All
values display temperature independence with values around 1Wm−1 K−1 . There is a slight
spread in the values, but all agree within the uncertainty of ±10%.

151

Figure 6.23: (a) Thermal conductivity versus temperature, with calculated error bars of
on average 9%. The increasing trend in total thermal conductivity is due to an increased
electronic contribution, as was verified from calculating the lattice thermal conductivity (b).

Despite an increase in the power factor, the ZT for Ge17+x Sb2−x Te20 is decreased as the
concentration of Ge is increased, as can be seen in Figure 6.24. As aforementioned in the
introductory chapters, the thermoelectric properties which comprise ZT are contraindicated
and there is an optimum charge carrier concentration to maximize ZT for a given material
and temperature. The decreased ZT is therefore indicative of overdoping, which would mean
the carrier concentration is either already optimized for Ge17 Sb2 Te20 or should be decreased.
To confirm this hypothesis the hole concentration from 300 − 723K was measured using the
Hall effect on Ge17 Sb2 Te20 and 10%, 15%, and 20% Ge doped Ge17 Sb2 Te20 . The carrier
concentration for the samples as a function of temperature can be seen in Figure 6.25

152

Figure 6.24: ZT versus temperature for Ge17+x Sb2+x Te20 , seen here with the calculated
error bars of on average ±13% on Ge17 Sb2 Te20 and Ge17.4 Sb1.6 Te20 (20% Ge). There is a
systematic decrease with increased Ge doping.

Figure 6.25: Hole concentration versus temperature (300 − 700K) for 0%, 10%, 15%, and
20% Ge doped Ge17 Sb2 Te20 .

To better understand the nature of doping and electrical transport a single parabolic band
153

model was used to determine the maximum ZT as a function of carrier concentration for
Ge17 Sb2 Te20 . The specific parameters used can be seen in Table 6.2. The Seebeck coefficient
versus carrier concentration for 300K, 423K, 523K, 623K, and 723K can be seen along with
experimental values in Figure 6.26(b). There is good agreement between experiment and
model aside from experimental values which are lower than expected. These deviations could
be due to additional contributions to transport from a degenerate light band, which has been
observed in other IV-VI compounds.[161]
The expected ZT values as a function of carrier concentration can be seen in Figure 6.26(a) along with experimental values. The undoped Ge17 Sb2 Te20 was shown to yield
the highest ZT values at each temperature, which was further verified by experiment.

Figure 6.26: (a) Calculated ZT values as a function of carrier concentration for Ge17 Sb2 Te20
assuming single parabolic band behavior for 300K, 423K, 523K, 623K, and 773K. Experimental values are plotted as squares for each temperature, showcasing the optimized ZT
values of Ge17 Sb2 Te20 . (b) Seebeck coefficient versus carrier concentration (Pisarenko plot)
for Ge17 Sb2 Te20 at 300K, 423K, 523K, 623K, and 773K assuming single parabolic band
behavior, with experimental points shown as squares for each temperature.

154

Table 6.2: The parameters necessary for calculating the thermoelectric properties of
Ge17 Sb2 Te20 using a single parabolic band model. Temperature, T , lattice thermal conductivity, κL , effective mass, m∗ (determined from Seebeck coefficient), and intrinsic carrier
mobility, µo , calculated using the mobility found from Hall measurements.

6.4.3

Conclusion

In this study Ge was substituted onto the Sb site in Ge17 Sb2 Te20 in an attempt to increase
the hole concentration and optimize the thermoelectric properties. It was found that the
rocksalt unit cell of Ge17 Sb2 Te20 was distorted along the [111] crystal direction as the Ge
content increased. The electrical resistivity and Seebeck coefficient were consistently and
significantly reduced which resulted in a maximization of power factor, with values around
40µWcm−1 K−2 for Ge17.3 Sb1.7 Te20 . The lattice thermal conductivity was unchanged, but
because of the decreased electrical resistivity the electronic contribution to thermal conductivity increased which resulted in an increased total thermal conductivity. The ZT was
therefore decreased for the higher doping percentages of Ge. Using the results from temperature dependent Hall measurements the thermoelectric properties were modeled using a
single parabolic band model, where it was shown that the ZT is maximized between 300K
and 700K for carrier concentrations coinciding with Ge17 Sb2 Te20 . To increase ZT for this
compound the focus of future work should therefore rest on either reducing lattice thermal
conductivity or increasing hole mobility and/or Seebeck coefficient.
155

Chapter 7
Concluding Remarks
Thermoelectric materials represent a unique research endeavor for alleviating the energy
demands of society. By means of the Seebeck effect, thermoelectric materials can directly
convert wasted heat into electrical energy. Unfortunately the efficiency of power generation
is constrained by a unitless parameter called the figure of merit, ZT, which is dependent on
materials properties which are by their nature contraindicated. Novel materials engineering
such as reduced dimensionality, doping, particle size control, and intuitive materials selection
are needed to enhance ZT and achieve commercial viability.
This work studied the synthesis, processing, and thermoelectric properties of Ge-SbTe based compounds and alloys. GeTe rich alloys of GeTe and Sb2 Te3 were synthesized
using a conventional solid state process and spark plasma sintering. It was found that the
thermoelectric properties were highly sensitive to the concentration of vacancies in theses
alloys. The thermal conductivity was decreased with the introduction of Sb2 Te3 , but the
power factor was also decreased. This resulted in an optimized peak ZT of approximately
1.9 for Ge17 Sb2 Te20 .
The compound Ge4 SbTe5 was studied for thermoelectric applications. This compound
does not lie on the GeTe-Sb2 Te3 tie-line and contains no vacancies in the unit cell. The
compound exhibited inherently high ZT values due to the low thermal conductivity and high
156

temperature metallic state, which resulted in power factor values in excess of 20µWcm−1 K−2 .
Studies of substituting Sn on the Ge atomic site and Bi on the Sb atomic site were conducted.
It was found that the introduction of these point defects successfully lowered the thermal
conductivity. The band gap of Ge4 SbTe5 was sensitive to atomic substitutions and decreased
upon introduction of Bi and Sb, which resulted in an overall decrease of the power factor, and
therefore ZT. Doping was also performed to tune the concentration. Na, which is monovalent,
was substituted for divalent Ge. This resulted in an increase in the hole concentration, as
was hypothesized. The Seebeck coefficient also decreased with Na doping, and resulted in a
decrease in overall power factor and ZT. Ga, which has one less electron than Ge, was also
studied as a dopant. Ga was shown to not be stable as a dopant, even for small amounts
such as 1%. This resulted in the creation of the secondary phase Ga2 Te3 .
The alloy Ge17 Sb2 Te20 was also studied. This stoichiometry was found to exhibit the
highest thermoelectric properties among the GeTe-Sb2 Te3 alloys from Chapter 3. After a
typical solid state synthesis of melting, quenching, and annealing the material was found
to exist in a metastable rocksalt phase. Upon heating a phase transition to the stable
rhombohedral phase takes place. This phase transition occurs because of the ordering of
vacancies, which are randomly assorted on the cation site of the metastable rocksalt unit
cell. The phase transition from metastable rocksalt to rhombohedral has also been classified
as a insulator-metal transition, because of the change in electrical resistivity. It was found
that the method of synthesis has a direct effect on the crystal structure and microstructure
of Ge17 Sb2 Te20 . By quenching directly from the melt, microstructured secondary phases of
Ge and Sb exist and were found to enhance the thermoelectric performance. The effects of
both Sb as a secondary phase and Ge as secondary phase in Ge17 Sb2 Te20 were isolated and
studied. It was found that these secondary phases enhance the charge carrier scattering and
157

provide an enhancement in the Seebeck coefficient. Additionally, the thermal conductivity
was reduced with the introduction of these secondary phases. These enhancements led to
consistent ZT values in excess of 2 for both Sb and Ge secondary phases. Finally, a study
on altering the carrier concentration through doping was conducted, Ge17+x Sb2−x Te20 . It
was found that altering the Ge/Sb ratio was a method towards successfully tuning the hole
concentration. The power factor and thermal conductivity were enhanced, which ultimately
led to decreases in ZT. Fortunately this work also allowed for modeling of the thermoelectric
properties as a function of carrier concentration. It was found that Ge17 Sb2 Te20 has an ideal
carrier concentration. Therefore, to enhance the ZT of this specific compound reductions
in the lattice thermal conductivity through ball-milling and atomic substitutions should be
conducted.

7.1

Future Work and Interesting Projects

There is much work needed on GeTe-Sb2 Te3 alloys and compounds to understand their potential for thermoelectric power generation. This thesis showed the impressive thermoelectric
performance of these materials, specifically Ge17 Sb2 Te20 . There lacks sufficient studies in
the literature on the crystallographic nature of Ge17 Sb2 Te20 . It would therefore be wise to
conduct a more thorough study of the fundamental elastic and crystallographic properties
of this compound. Here we give a brief presentation of ideas for future projects which would
be fruitful for understanding and engineering this interesting class of materials.
Recently, Levin et al showed that rare earth dopants such Yb, Dy, and Er enhanced that
Seebeck coefficient in GeTe and TAGS-85 by introduction of a magnetic moment.[162, 163]
This enhancement in Seebeck resulted in an increase in power factor as well. Based on

158

the similarity between these compounds and Ge17 Sb2 Te20 , it is believed that these dopants
could yield positive results. The study of adjusting the Ge/Sb ratio in Ge17 Sb2 Te20 showed
that the charge carrier concentration of this compound was ideal for optimized ZT. However,
it would be interesting to study the effects of increased vacancy concentration on the thermoelectric properties of Ge17 Sb2 Te20 . Introduction of vacancies could potentially lower the
lattice thermal conductivity and enhance the Seebeck coefficient be introducing additional
point defect scattering for both phonons and holes.
The studies in this work along with others in the field on using secondary phases to
enhance thermoelectric performance warrants substantial attention. Future work on this
subject should include studies on the size of the secondary phases, the importance of geometry, and types of materials used as a secondary phase. It is believed that magnetic materials
and ultra-low thermal conductivity materials could be of particular interest as a secondary
phase.
Lowering the lattice thermal conductivity is an avenue that has not been explored yet.
To achieve this, future studies on reducing particle/grain size using high energy ball milling
would be fruitful. It has been shown in this work and others that substituting isovalent atoms
such as Sn on the Ge site and Se on the Te site can reduce lattice thermal conductivity, but
can also change the electronic bandgap, which reduces the power factor. Co-substituting
Sn and Se, written as Ge17−x Snx Sb2 Te20−y Sey , could be used to control the bandgap and
reduce thermal conductivity. This method would introduce phonon scattering point defects,
but also the bandgap could be specifically engineered as Sn substitution would decrease the
bandgap while Se would increase the bandgap. It should be noted however that the rate of
change for Sn and Se could be unequal and should be studied independently first. Also, the
solubility of Sn and Se has been shown to not be 100% in other Ge-Sb-Te compounds.[164,
159

157] These represent just some of the interesting avenues to further understand the complex
nature of thermoelectric phenomena in Ge-Sb-Te compounds.
Ge-Sb-Te phase change materials have been extensively researched for nonvolatile memory storage and are now the front runner for replacing less attractive RAM based technologies. It is now being realized that these materials, which are Pb-free and easy to synthesize,
are competitive with today’s state of the art thermoelectric compounds. If the thermoelectric
properties could be optimized, one might envision a future application which incorporates
these materials in a hybrid setup. By exploiting the memory storage capabilities and high
thermoelectric performance, the memory technologies of tomorrow could also utilize the
wasted heat in computers and electronic devices.

160

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