"High temperature thermoelectric transport in
quaternary copper selenides and ternary
Zintl-antimonides"
Dissertation
zur Erlangung des akademischen Grades
"Doktor der Naturwissenschaften"
am Fachbereich Chemie,
Pharmazie und Geowissenschaften der
Johannes Gutenberg-Universität Mainz
Wolfgang G. Zeier
geboren in Bad Brückenau
Mainz 2013
Dekan:
1. Berichterstatter
2. Berichterstatter
Tag der mündlichen Prüfung: 29/05/2013
ii
Für meine Familie
Die experimentellen Arbeiten, die dieser Arbeit zugrunde liegen, wurden in dem Zeitraum
vom 1. Oktober 2010 bis 1. Februar 2013 unter der Leitung von
an der Johannes Gutenberg-Universität Mainz und
durchgeführt.
The work for this thesis was carried out at the Johannes Gutenberg-Universität Mainz and the
California Institute of Technology in the period between October 1st 2010 and February 1st 2013,
under the guidance of and .
© 2013
Wolfgang G. Zeier
All Rights Reserved
iv
Publications
1. Heinrich C.P., Day T., Zeier W.G., Snyder G.J., Tremel W., "Anion substitution induced
point defect scattering in the thermoelectric solid solution Cu2ZnGeSe4−xSx" in prepara-
tion
2. Zeier W.G., Pei Y., Day T., Schechtel E., Heinrich C.P., Kieslich G., Snyder G.J., Tremel
W., "Influence of the chemical potential on the carrier effective mass in the thermoelectric
solid solution Cu2Zn1−xFexGeSe4" in preparation
3. Zeier W.G., Heinrich C.P., Day T., Panithipongwut C., Kieslich G., Snyder G.J., Tremel
W., "Bond strength dependent electronic phase transformation in the solid solution
Cu2ZnGeSe4−xSx" in preparation
4. Kieslich G., Veremchuck I., Antonyshyn I., Zeier W.G., Birkel C.S., Weldert K., Heinrich
C.P., Visnow E., Panthöfer M., Burkhard U., Grin Y., Tremel W., "Using crystallographic
shear to reduce lattice thermal conductivity: high temperature thermoelectric characteri-
zation of the spark plasma sintered Magneli phases WO2.90 and WO2.722" Energy Environ.
Sci. - submitted
5. Zeier W.G. Pei Y., Pomrehn G., Day T., Heinz N., Heinrich C.P., Snyder G.J., Tremel W.,
"Phonon scattering through a local anisotropic structural disorder in the thermoelectric
solid solution Cu2Zn1−xFexGeSe4" J. Am. Chem. Soc. - Article ASAP doi:10.1021/ja308627v
6. Zevalkink A.*, Zeier W.G.*, Pomrehn G.*, Schechtel E., Tremel W., Snyder G.J., "Ther-
moelectric transport properties of Sr3GaSb3 - A chain based Zintl compound" Energy
Environ. Sci. 2012, 5, 9121-9128 doi:10.1039/C2EE22378C (*equal contribution authors)
7. Birkel C.S., Zeier W.G., Douglas J., Lettiere B., Mills C.; Seward G., Birkel A., Snedaker
M., Zhang Y., Snyder G.J., Pollock T., Seshadri R., Stucky G., "Rapid microwave prepa-
ration of thermoelectric TiNiSn and TiCoSb half-Heusler compounds" Chem. Mater 2012,
24, 2558-2565 doi:10.1021/cm3011343
8. Zeier W.G.*, Zevalkink A.*, Schechtel E., Tremel W., Snyder G.J., "Thermoelectric proper-
ties of Zn-doped Ca3AlSb3" J. Mater. Chem. 2012, 22, 9826-9830 doi:10.1039/C2JM31324C
(*equal contribution authors)
9. Zeier W.G., LaLonde A., Gibbs Z.M., Heinrich C.P., Panthöfer M., Snyder G.J., Tremel
W., "Influence of a nano phase segregation on the thermoelectric properties of the p-type
doped stannite compound Cu2+xZn1−xGeSe4" J. Am. Chem. Soc. 2012, 134, 7147-7154
doi:10.1021/ja301452j
10. Zeier W.G., Panthöfer M., Janek J., Tremel W.,Thermoelektrische Verbindungen. Strom
aus Abwärme" Chem. Unserer Zeit 2011, 45, 188-200
doi:10.1002/ciuz.201100393
v
11. Zevalkink A., Toberer E.S., Zeier W.G., Flage-Larsen E., Snyder G.J., "Ca3AlSb3: an
inexpensive, non-toxic thermoelectric material for waste heat recovery" Energy Environ.
Sci. 2011, 4, 510-518 doi:10.1039/C0EE00517G
12. Zeier W.G., Roof I.P., Smith M.D., zur Loye H.-C., "Crystal growth of Ln3GaO6 (Ln = Nd,
Sm, Eu and Gd): Structural and optical properties" Solid State Sci. 2009, 11, 1965-1970
doi:10.1002/chin.201005015
13. Roof I.P., Jagau T.-C., Zeier W.G., Smith M.D., zur Loye H.-C., "Crystal growth of a
new series of complex niobates, LnKNaNbO5 (Ln = La, Pr, Nd, Sm, Eu, Gd, and
Tb): Structural properties and photoluminescence" Chem. Mater. 2009, 21, 1955-1961
doi:10.1021/cm9003245
vi
Acknowledgments
First of all I have to thank Prof. Dr. for everything he has done for me during my
time as an undergrad and during grad school. I doubt that anything that I have experienced
in the last years would have been possible without him. I could not have asked for a better
boss in terms of academic and mentoring support. Thank you for always sending me abroad
for more research, for letting me conduct the research for my thesis with , and giving me
the space for my research, while being there with support and advice, whenever I needed it.
There are no words to express my gratitude and appreciations. Thank you!
To my research advisor, Dr. . It has been a pleasure to work with you and I
appreciate the way I was adopted as a visiting grad student. Research at Caltech has been a
lot of fun, due to a perfect mixture of motivation, knowledge, and funny discussions about
just anything. I have learned more about thermoelectrics and myself than I could have ever
imagined. Thank you for always trying to push me in any way possible for you. Thank you
for everything!
To the group I want to say thank you for being a lot of fun. Thank you, ,
for making me enjoy synthesis at home. Sharing a glove box and collaborating with you has
been fun and very productive as well. I am glad that we met in our first year, we would
not stand where we are now, if we hadn’t. I have to thank for all the fruit-
ful discussions and all the ranting with me. Thank you Dr. for all the help
and support I got from you. Especially the initial work on the powder consolidation and all
of out discussions about thermoelectrics and soccer. , , and
, thanks for the fun in the office and definitely the fun during all the nights
we spent out partying. Together with Dr. I couldn’t have asked for better
friends in Mainz. Also thanks to Dr. for helping me find a post doc position
and for the fruitful collaborations we had. I would like to thank Dr. for
his initial help with the structural refinements and understanding X-ray refinements. Many
thanks to for the help with some of the sample preparations. I also have to
thank for her help with the differential scanning calorimetry in Mainz.
I also want to thank everyone in the group for putting up with a sometimes ob-
noxious and loud german. I have enjoyed working with everyone of you and I will always
remember the good nights and group meetings with a lot of beer. I will miss working with
you guys. I have to thank for our collaboration within the Zintl project. It
was a lot of fun, I learned a lot from you, and I think we were a good team. I will always
appreciate having been able to work with you. Thank you for the Hall measurements at Jet
Propulsion Laboratory, whenever they were necessary. , you have been a great
friend for all these years and I will miss you. I have to thank you for helping me with the
Scanning Electron Microscopy but your being around was more important. I would like to
thank Dr. and for their help with the hot press and instruments.
, I always had a lot of fun hanging out with you and I am very happy to call you my
vii
friend. I am very grateful for Dr. and help with understanding
and modeling thermal transport. Furthermore I have to thank Dr. for leav-
ing me some of his figures on PbTe and band engineering for this thesis. I will miss spicy
lunch! Thank you, , for the help with the band gap measurements. Special thanks
to for performing the band structure calculations, which have been very
helpful to understand the transport properties of these materials. I also have to thank Dr.
and Prof. Dr. , I have learned a lot from you guys during my first
stay at Caltech in 2010. Thank you for everything, including the help
with the high temperature XRD stage. It has been a pleasure to work with so many intelligent
scientists. I also want to thank Prof. Bill Johnson, who provided the equipment for the ultra-
sonic measurements of the samples.
I want to thank the Carl-Zeiss foundation for the fellowship and with it the financial support
during my time as a Ph.D. student. This fellowship helped me to be more independent in my
research. Furthermore I want to thank the Graduate School of Excellence "MAterials Science
IN MainZ" (MAINZ) and their financial support through the Excellence Initiative (DFG/GSC
266), which has made it possible to attend conferences and summer schools. Especially the
support for the flights to Pasadena has been very helpful for this thesis. I also have to thank
the Jet Propulsion Laboratory, National Aeronautics and Space Administration, for their fi-
nancial support within the Zintl project and for providing measurement time whenever it was
needed. Many thanks to Dr. for discussions and all the help and advice you gave
me. Furthermore I want to thank , Dr. and Dr. , who
always helped me with any issues regarding paperwork and MAINZ.
Finally, my family and friends have to be acknowledged. My parents have always supported
me during my studies and my Ph.D., and the financial and mental support during this time
made this thesis possible. Thank you for always being supportive of all the crazy travels and
times I have gone through. I doubt this will end. I will always be grateful for everything
you two have done for me. My sister , you have always been there for me whenever I
needed an advice or just someone to listen to, also my brother-in-law , and especially
my cute nephew . I love you guys. Thank you for being a great
roommate and awesome friend in Mainz. Meeting you in my first week in Mainz in 2005
made this time of my life a very good one. Thanks for putting up with all my problems
and rants in the last years. I wish you and a happy life and I will always be around
to somehow give back what you guys gave me in the last years! Dr. all
these years would have been less fun without us teaming up, I can’t express how much I
appreciate my time in Mainz with you. and , I think we were
all an awesome team during our studies and I am very happy to have met you guys. Of
course thanks to all the other great people that I have met in Mainz. I will always look
back to the time with you guys and smile. Thank you Dr. , ,
, Dr. , and Dr. for being such great roommates in
Pasadena. Living with you has always been a pleasure, I love you guys, and I will always
viii
miss this. Thank you for your friendship, many things would have not
been possible without you. , , , and
have been mentioned above but their names should be here as well. , thank you
so much for being around in the states, too. A lot of my life would have been less fun without
you being around, knowing that I can always count on you. I am looking forward to being
roommates during my time as a post doc.
ix
Abstract
This manuscript discusses the structural chemistry of quaternary, tetrahedrally bonded cop-
per selenides and Zintl compounds and their thermoelectric properties. Different materials
compositions are investigated, the influence of different dopants considered, and effects of the
structural and micro structural aspects on the thermoelectric transport are discussed. In the
ongoing search for new materials for thermoelectric applications such as waste heat recovery
and power generation, materials composed of inexpensive and earth abundant elements need
to be explored and their thermoelectric efficiencies improved. The thermoelectric efficiency is
determined by the figure of merit zT=α2T/ρκ. Thus, an ideal thermoelectric material requires
a large Seebeck coefficient α, low electrical resistivity ρ, and low thermal conductivity κ.
Engineering nanostructure in bulk thermoelectric materials has recently been established
as an effective approach to scatter phonons, reducing the phonon mean free path, without si-
multaneously decreasing the electron mean free path for an improvement of the performance
of thermoelectric materials. In this work, the synthesis, phase stability, and thermoelectric
properties of the solid solutions Cu2+xZn1−xGeSe4, x = 0 - 0.1 are reported. The substitution
of Zn2+ with Cu+ introduces holes as charge carriers in the system and results in an enhance-
ment of the thermoelectric efficiency. Nano sized impurities, formed via phase segregation at
higher dopant contents, have been identified and are located at the grain boundaries of the
material. The impurities lead to enhanced phonon scattering, a significant reduction in lattice
thermal conductivity, and therefore an increase in the thermoelectric figure of merit in these
materials. This study also reveals the existence of an insulator-to-metal transition at 450 K.
Then, inspired by the promising thermoelectric properties of Cu2+xZn1−xGeSe4, the syn-
thesis and characterization of the solid solution Cu2Zn1−xFexGeSe4 is described. Upon sub-
stitution of Zn with the isoelectronic Fe no charge carriers are introduced in these intrinsic
semiconductors. However, a change in lattice parameters, expressed in an elongation of the
c/a lattice parameter ratio with minimal change in unit cell volume, reveals the existence of a
3-stage cation restructuring process of Cu, Zn, and Fe. The differences in bonding interaction
between the cations and the anions lead to different bond lengths and bond angles, resulting
in the observed trend in the lattice parameters. The resulting local anisotropic structural dis-
order leads to phonon scattering not normally observed resulting in an effective approach to
reduce the lattice thermal conductivity in this class of materials.
In the family of Zintl antimonides, Ca3AlSb3 seems to be an interesting compound with a
peak zT of 0.8 obtained at 1050 K upon doping with Na+ for Ca2+. A previous investigation
of Ca3AlSb3 shows an intrinsic semiconductor with a very low lattice thermal conductivity
(0.6 Wm−1K−1 at 870 K) due to a relatively complex crystal structure. This work explores
the influence of the substitution of Al with Zn on the thermoelectric transport properties of
the Ca3Al1−xZnxSb3 system, and compares the effects of Na- and Zn-doping on hole carrier
concentrations, mobilities of the charge carriers, and the thermoelectric figure of merit. Fur-
thermore a report on the influence of the grain size in these materials on the thermoelectric
transport will be made. Ultimately, it is found that while larger grain size has little effect on
the high temperature properties, the reduction of grain boundary surface area improves the
figure of merit at intermediate temperatures. This work shows the common strategy to use
x
nano structured materials does not always lead to an improved figure of merit. Other com-
peting factors such as a reduction in mobility may overwhelm the improvement of the lattice
thermal conductivity.
Finally, the closely related compound Sr3GaSb3 is investigated. Although the crystal struc-
ture of Sr3GaSb3 contains infinite chains of corner-linked tetrahedra, in common with Ca3AlSb3
and Ca5Al2Sb6, it has twice as many atoms per unit cell (N=56). This contributes to the
exceptionally low lattice thermal conductivity (κL = 0.45 Wm−1K−1 at 1000 K) observed in
Sr3GaSb3. High temperature transport measurements reveal that Sr3GaSb3 is a nondegener-
ate semiconductor (consistent with Zintl charge-counting conventions) with relatively high
p-type electronic mobility (∼ 30 cm2V−1s−1 at 300 K). To rationally optimize the electronic
transport properties of Sr3GaSb3 in accordance with a single band model, doping with Zn2+
on the Ga3+ site was used to increase the p-type carrier concentration. In optimally hole-
doped Sr3Ga1−xZnxSb3 (x= 0.0 - 0.1), a maximum figure of merit of greater than 0.9 at 1000 K
is demonstrated.
xi
Abstract - German
Dieses Manuskript beschreibt den Zusammenhang zwischen der Strukturchemie von quater-
nären, tetraedrisch gebundenen Kupfer-Seleniden und Zintl-Verbindungen und deren ther-
moelektrischen Eigenschaften. Unterschiedliche Zusammensetzungen der Materialien werden
untersucht, der Einfluss verschiedener fester Lösungen betrachtet und die Auswirkungen der
Struktur und des Gefüges auf den thermoelektrischen Transport diskutiert. In der fortwähren-
den Suche nach neuen Materialien für thermoelektrische Anwendungen, wie Wärmerück-
gewinnung und Stromerzeugung, müssen kostengünstige Materialien und häufiger vorkom-
mende Elemente erkundet und ihre thermoelektrische Effizienz verbessert werden. Die ther-
moelektrische Effizienz wird durch die Gütezahl zT=α2T/ρκ bestimmt, bei der ein ideales
Thermoelektrikum einen großen Seebeck-Koeffizienten α, einen niedrigen spezifischen elek-
trischen Widerstand ρ und eine geringe thermische Leitfähigkeit κ besitzt.
Nanostrukturierung von thermoelektrischen Materialien ist ein etabliertes und äußerst wirk-
sames Konzept zur Streuung von Phononen, wodurch die mittlere freie Weglänge der Phononen
verringert wird und somit zu einer Verbesserung der Leistung der thermoelektrischen Mate-
rialien führt. In dieser Arbeit wird die Synthese, die Stabilität und die thermoelektrischen
Eigenschaften der festen Lösungen Cu2+xZn1−xGeSe4, x = 0 - 0,1 betrachtet. Die Substitution
von Zn2+ mit Cu+ erzeugt Löcher als Ladungsträger und führt zu einer Erhöhung des ther-
moelektrischen Wirkungsgrades. Nanometer große Verunreinigungen, durch eine Phasenseg-
regation entstanden, wurden identifiziert und sind an den Korngrenzen des Materials zu
finden. Durch diese Verunreinigungen ergibt sich eine signifikante Reduktion der Wärmeleit-
fähigkeit des Gitters aufgrund einer erhöhten Streuung der Phononen an den Korngrenzen,
welches eine Erhöhung des thermoelektrischen Gütefaktors in diesen Materialien nach sich
zieht. Des Weiteren zeigt diese Arbeit auch die Existenz eines Isolator-Metall-Überganges bei
450 K.
Inspiriert durch die vielversprechenden thermoelektrischen Eigenschaften von Cu2+xZn1−x
GeSe4, wird die Synthese und die Charakterisierung der festen Lösung Cu2Zn1−xFexGeSe4
beschrieben. Die Substitution von Zink mit dem isoelektronischen Eisen erzeugt keine Ladungs-
träger in diesem intrinsischen Halbleiter. Allerdings wird eine Änderung der Gitterparam-
eter mit dem Dotierungsgrad offenbart, der sich in einer Streckung des c/a-Verhältnisses
ausdrückt und bei x = 0,7 ein Maximum aufweist, was durch das Vorliegen eines dreistu-
figen Kationenumstrukturierungsprozesses von Cu, Zn und Fe erklärt werden kann. Die
Unterschiede im Bindungscharakter zwischen den Kationen und Anionen führen zu unter-
schiedlichen Bindungslängen und Bindungswinkeln, welches sich in dem beobachteten Trend
der Gitterparametern ausdrückt. Die daraus resultierende lokale anisotrope strukturelle Un-
ordnung führt zu einer Streuung der Phononen und zu einem bisher nicht bekannten, effek-
tiven Ansatz zur Reduktion der Wärmeleitfähigkeit des Gitters in dieser Materialklasse.
In der Klasse der Zintl Antimonide scheint Ca3AlSb3 eine interessante Verbindung sein, mit
einer Gütezahl zT von 0,8 bei 1050 K. Vorangegangenen Untersuchungen von Ca3AlSb3, in
der ein Substitution von Kalzium mit Natrium vorgenommen wurde, zeigen einen intrinsis-
chen Halbleiter mit sehr niedriger Wärmeleitfähigkeit des Gitters von 0,6 Wm−1K−1 bei 870 K,
was auf eine komplexe Kristallstruktur zurückzuführen ist. In dieser Arbeit wird der Einfluss
xii
der Substitution von Aluminium mit Zink auf die thermoelektrischen Transporteigenschaften
untersucht und ein Vergleich der Auswirkungen einer Substitution von Natrium und Zink in
Hinblick auf die Ladungsträgerkonzentrationen, die Mobilitäten und die thermoelektrische
Gütezahl angestellt. Des Weiteren wird der Einfluss der Korngrößen in diesen Materialien
auf den Transport untersucht. Es zeigt sich, dass die Größe der Körner bei hohen Temper-
aturen wenig Einfluss auf den Transport hat, während bei mittleren Temperaturen die ther-
moelektrische Gütezahl verbessert wird. Diese Arbeit zeigt, dass die allgemein anerkannte
Strategie zur Nanostrukturierung nicht immer zu einer Verbesserung der thermoelektrischen
Eigenschaften führen muss, sondern auch konkurrierende Faktoren wie die Verringerung der
Mobilität der Ladungsträger und eine Änderung der thermischen Leitfähigkeit gegeneinander
abgewogen werden sollten.
Abschließend wird die verwandte Verbindung Sr3GaSb3 untersucht, welche, aufgrund der
größeren Elementarzelle, eine außergewöhnlich niedrige Wärmeleitfähigkeit des Gitters von
0,45 Wm−1K−1 bei 1000 K besitzt. Messungen der Transporteigenschaften bei hohen Tem-
peraturen zeigen einen nicht entarteten Halbleiter mit relativ hohen Mobilitäten der Löcher
(∼ 30 cm2V−1s−1 bei 300 K). Um die thermoelektrischen Eigenschaften gezielt zu optimieren
wurde Gallium mit Zink substituiert, um mehr Lochzustände zu erzeugen, was zu einer ther-
moelektrischen Gütezahl von 0,9 bei 1000 K führt.
xiii

Contents
Publications v
Acknowledgments vii
Abstract x
Abstract - German xii
Table of Contents xiv
List of Figures xviii
List of Tables xxiii
List of Symbols, Notations, and Abbreviations xxiv
1 Introduction 1
1.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Transport Phenomena and Thermoelectric Energy Conversion . . . . . . . . . . . 2
1.3 Thermoelectric Figure of Merit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Enhancement of Thermopower . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.2 Band Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Reduction of Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Pertinent Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 Cu I I2M GeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 Zintl-Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.3 Ca3AlSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.4 Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Summary of Research and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Experimental Methods 19
2.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Quaternary Copper Selenides . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Zintl-Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3 Pressure-Assisted Sintering . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Chemical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
xv
Contents
2.3.2 Characterization of Transport Properties . . . . . . . . . . . . . . . . . . . 24
2.4 Electronic Structure Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Cu MI I2 GeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Chemical and Structural Results 31
3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Cu2Zn1−xFexGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Cu2+xZn1−x−yFeyGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Ca3AlSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4 Electrical Transport 45
4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Electronic Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.1 Electronic Transport in a Single Parabolic Band . . . . . . . . . . . . . . . 46
4.2.2 Electronic Transport Equations within the Single Parabolic Band Ap-
proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Cu2Zn1−xFexGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Cu2+xZn1−x−yFeyGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.6 Ca3AlSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7 Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Thermal Transport 63
5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Thermal Transport Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.1 Lattice Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.2 Minimum Lattice Thermal Conductivity . . . . . . . . . . . . . . . . . . . 70
5.2.3 Electronic Contribution to the Thermal Conductivity . . . . . . . . . . . . 71
5.3 Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Cu2Zn1−xFexGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.5 Cu2+xZn1−x−yFeyGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.6 Ca3AlSb3 and Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 Thermoelectric Efficiency 85
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Thermoelectric Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.3 Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4 Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4 . . . . . . . . . . . . . . . . . . . 87
6.5 Ca3AlSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.6 Sr3GaSb3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
xvi
Contents
7 Conclusion 91
8 Supporting Information 93
8.1 X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.2 Scanning Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.3 Speed of Sound Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.4 Additional Transport Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.5 Additional Pictures and Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Bibliography 121
xvii

List of Figures
1.1 Schematic of the Seebeck effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Overview of zT-values for different p-type and n-type materials. . . . . . . . . . 4
1.3 Influence of the carrier density on the thermoelectric transport properties. . . . . 5
1.4 Influence of multiple bands and resonant states on the DOS. . . . . . . . . . . . . 6
1.5 Light and heavy valence bands in PbTe and their influence on the figure of merit
with carrier concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 First Brillouin zone of PbTe and temperature dependent valence band energies. 8
1.7 Influence of the band gap on the figure of merit. . . . . . . . . . . . . . . . . . . . 9
1.8 Crystal structures of stannite and kesterite-type Cu MI I2 GeSe4. . . . . . . . . . . 11
1.9 Crystal structure of Ca3AlSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.10 Crystal structure of Sr3GaSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Schematic of the induction hot press. . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Schematic measurement setup for the Seebeck coefficient. . . . . . . . . . . . . . 25
2.3 Schematic measurement setup for the Hall coefficient and resistivity. . . . . . . . 26
2.4 Schematic speed of sound measurement and laser flash diffusivity. . . . . . . . . 27
3.1 Powder diffraction data and absorption spectrum of Cu2ZnGeSe4. . . . . . . . . 32
3.2 TGA/DSC data for Cu2ZnGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Lattice parameter of Cu2+xZn1−xGeSe4 with composition and temperature. . . . 34
3.4 Occupations and residual electron densities of Cu and Zn in Cu2ZnGeSe4. . . . 35
3.5 Scanning electron micrographs of Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . 36
3.6 Shift of Miller indices (040) and (008) in Cu2Zn1−xFexGeSe4. . . . . . . . . . . . . 37
3.7 Non-Vegard like change of lattice parameters in Cu2Zn1−xFexGeSe4. . . . . . . . 38
3.8 3-stage cation restructuring process from stannite to kesterite structure type. . . 38
3.9 Band structure and Fermi surface of stannite - Cu2ZnGeSe4 and kesterite -
Cu2FeGeSe4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.10 Lattice parameter ratio c/a of Cu2.025Zn1−yFeyGeSe4 . . . . . . . . . . . . . . . . 41
3.11 X-ray diffraction data and SEM micrographs of Ca3Al1−xZnxSb3 . . . . . . . . . 42
3.12 X-ray diffraction data and SEM micrographs of Sr3Ga1−xZnxSb3 . . . . . . . . . 43
3.13 Density of states and band structure of Sr3GaSb3. . . . . . . . . . . . . . . . . . . 44
4.1 Resistivity and Hall-carrier concentrations of Cu2+xZn1−xGeSe4. . . . . . . . . . 51
4.2 Doping effectiveness and Hall-mobilities of Cu2+xZn1−xGeSe4. . . . . . . . . . . 52
4.3 Seebeck coefficients and Pisarenko relation of Cu2+xZn1−xGeSe4. . . . . . . . . 52
4.4 Temperature dependence of the electrical resistivity of Cu2Zn1−xFexGeSe4. . . . 53
xix
List of Figures
4.5 Temperature dependence of the Seebeck coefficients of Cu2Zn1−xFexGeSe4. . . . 54
4.6 Carrier effective mass of Cu2Zn1−xFexGeSe4 vs. measured Hall carrier concen-
tration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.7 Hall carrier concentration and Hall mobility of Ca3Al1−xZnxSb3. . . . . . . . . . 56
4.8 Resistivity and Seebeck coefficients of Ca3Al1−xZnxSb3. . . . . . . . . . . . . . . . 57
4.9 Pisarenko plot of Ca3Al1−xZnxSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.10 Hall carrier concentration and Hall mobility of Sr3Ga1−xZnxSb3. . . . . . . . . . 59
4.11 Resistivity and Seebeck coefficients of Sr3Ga1−xZnxSb3. . . . . . . . . . . . . . . . 59
4.12 Pisarenko plot of Sr3Ga1−xZnxSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1 A periodic one-dimensional chain of identical masses, connected by springs. . . 65
5.2 Schematic phonon dispersion curves for a monoatomic and diatomic lattice . . . 65
5.3 Total thermal conductivity and thermal diffusivity of Cu2+xZn1−xGeSe4 as a
function of temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.4 Calculated temperature dependent Lorenz number and electronic thermal con-
ductivity of Cu2+xZn1−xGeSe4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5 Temperature dependence of the lattice thermal conductivity of Cu2+xZn1−xGeSe4. 73
5.6 Longitudinal and transverse sound response data of Cu2ZnGeSe4. . . . . . . . . 74
5.7 Influence of nano scaled impurities on the grain boundaries on the thermal
transport of Cu2ZnGeSe4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.8 Total thermal conductivity of Cu2Zn1−xFexGeSe4 as a function of temperature. . 75
5.9 Temperature dependent Lorenz numbers and electronic thermal conductivity
of Cu2Zn1−xFexGeSe4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.10 Temperature dependent lattice thermal conductivity of Cu2Zn1−xFexGeSe4. . . . 77
5.11 Lattice thermal conductivity of Cu2Zn1−xFexGeSe4 at room temperature show-
ing a 15% reduction due to disorder scattering . . . . . . . . . . . . . . . . . . . . 78
5.12 Lattice thermal conductivity of Cu2.025Zn0.975−yFeyGeSe4 and Cu2Zn1−xFexGeSe4
at room temperature showing a 15% reduction due to disorder scattering. . . . . 79
5.13 Total thermal conductivity and lattice thermal conductivity of Ca3Al1−xZnxSb3
with temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.14 Total thermal conductivity and lattice thermal conductivity of Sr3Ga1−xZnxSb3. 81
5.15 Lattice thermal conductivities of Ca3AlSb3 and Sr3GaSb3 in respect to their crys-
tal structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.1 Power factor and figure of merit of Cu2+xZn1−xGeSe4. . . . . . . . . . . . . . . . 87
6.2 Temperature dependence of the power factor PF of Cu2Zn1−xFexGeSe4. . . . . . 87
6.3 Temperature dependence of the figure of merit zT of Cu2Zn1−xFexGeSe4. . . . . 88
6.4 Figure of merit with temperature and carrier concentration of Ca3Al1−xZnxSb3. 89
6.5 Figure of merit with carrier concentration and temperature of Sr3Ga1−xZnxSb3 . 90
8.1 X-ray diffraction data of Cu2+xZn1−xGeSe4, x = 0.00. . . . . . . . . . . . . . . . . 93
8.2 X-ray diffraction data of Cu2+xZn1−xGeSe4, x = 0.025. . . . . . . . . . . . . . . . . 94
8.3 X-ray diffraction data of Cu2+xZn1−xGeSe4, x = 0.05. . . . . . . . . . . . . . . . . 94
8.4 X-ray diffraction data of Cu2+xZn1−xGeSe4, x = 0.075 . . . . . . . . . . . . . . . . 95
xx
List of Figures
8.5 X-ray diffraction data of Cu2+xZn1−xGeSe4, x = 0.1. . . . . . . . . . . . . . . . . . 95
8.6 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.1. . . . . . . . . . . . . . . . . 96
8.7 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.2. . . . . . . . . . . . . . . . . 96
8.8 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.3. . . . . . . . . . . . . . . . . 97
8.9 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.4. . . . . . . . . . . . . . . . . 97
8.10 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.5. . . . . . . . . . . . . . . . . 98
8.11 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.6. . . . . . . . . . . . . . . . . 98
8.12 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.7. . . . . . . . . . . . . . . . . 99
8.13 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.8. . . . . . . . . . . . . . . . . 99
8.14 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 0.9. . . . . . . . . . . . . . . . . 100
8.15 X-ray diffraction data of Cu2Zn1−xFexGeSe4, x = 1.0. . . . . . . . . . . . . . . . . 100
8.16 X-ray diffraction data of Cu2+xZn1−x−yFeyGeSe4, x = 0.025, y = 0.975. . . . . . . 101
8.17 X-ray diffraction data of Cu2+xZn1−x−yFeyGeSe4, x = 0.025, y = 0.8. . . . . . . . . 102
8.18 X-ray diffraction data of Cu2+xZn1−x−yFeyGeSe4, x = 0.025, y = 0.7. . . . . . . . . 102
8.19 X-ray diffraction data of Cu2+xZn1−x−yFeyGeSe4, x = 0.025, y = 0.6. . . . . . . . . 103
8.20 X-ray diffraction data for Ca3Al1−xZnxSb3, x = 0.00. . . . . . . . . . . . . . . . . . 103
8.21 X-ray diffraction data for Ca3Al1−xZnxSb3, x = 0.01. . . . . . . . . . . . . . . . . . 104
8.22 X-ray diffraction data for Ca3Al1−xZnxSb3, x = 0.02. . . . . . . . . . . . . . . . . . 104
8.23 X-ray diffraction data for Ca3Al1−xZnxSb3, x = 0.05. . . . . . . . . . . . . . . . . . 105
8.24 X-ray diffraction data for Sr3Ga1−xZnxSb3, x = 0.00. . . . . . . . . . . . . . . . . . 105
8.25 X-ray diffraction data for Sr3Ga1−xZnxSb3, x = 0.02. . . . . . . . . . . . . . . . . . 106
8.26 X-ray diffraction data for Sr3Ga1−xZnxSb3, x = 0.05. . . . . . . . . . . . . . . . . . 106
8.27 X-ray diffraction data for Sr3Ga1−xZnxSb3, x = 0.07. . . . . . . . . . . . . . . . . . 107
8.28 X-ray diffraction data for Sr3Ga1−xZnxSb3, x = 0.1. . . . . . . . . . . . . . . . . . . 107
8.29 X-ray diffraction data for GaSb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.30 SEM micrographs of Cu2Zn1−xFexGeSe4, x = 0.1 and x = 0.2. . . . . . . . . . . . 109
8.31 SEM micrographs of Cu2Zn1−xFexGeSe4, x = 0.3 and x = 0.4. . . . . . . . . . . . 109
8.32 SEM micrographs of Cu2Zn1−xFexGeSe4, x = 0.5 and x = 0.6. . . . . . . . . . . . 110
8.33 SEM micrographs of Cu2Zn1−xFexGeSe4, x = 0.7 and x = 0.8. . . . . . . . . . . . 110
8.34 SEM micrographs of Cu2Zn1−xFexGeSe4, x = 0.9 and x = 1.0. . . . . . . . . . . . 110
8.35 SEM micrograph of a fractured surface of Ca3Al1−xZnxSb3, x = 0.00. . . . . . . . 111
8.36 SEM micrograph of a fractured surface of Ca3Al1−xZnxSb3, x = 0.01. . . . . . . . 111
8.37 SEM micrograph of a fractured surface of Ca3Al1−xZnxSb3, x = 0.02 and x = 0.05.111
8.38 SEM micrograph of a polished surface of Ca3Al1−xZnxSb3, x = 0.00 and x =0.01. 112
8.39 SEM micrograph of a polished surface of Ca3Al1−xZnxSb3, x = 0.02 and x =0.05. 112
8.40 SEM micrographs of fractured surfaces of Sr3Ga1−xZnxSb3, x = 0.00 and x = 0.02. 112
8.41 SEM micrographs of fractured surfaces of Sr3Ga1−xZnxSb3, x = 0.05 and x = 0.07. 113
8.42 SEM micrographs of polished surfaces of Sr3Ga1−xZnxSb3, x = 0.00 and x = 0.02 113
8.43 SEM micrographs of polished surfaces of Sr3Ga1−xZnxSb3, x = 0.05. . . . . . . . 113
8.44 Longitudinal and transverse sound response data of Cu2Zn0.5Fe0.5GeSe4. . . . . 114
8.45 Longitudinal and transverse sound response data of Cu2FeGeSe4. . . . . . . . . 114
8.46 Longitudinal and transverse sound response data of Ca3AlSb3. . . . . . . . . . . 115
xxi
List of Figures
8.47 Longitudinal and transverse sound response data of Sr3GaSb3. . . . . . . . . . . 115
8.48 Temperature dependent Hall carrier concentrations of Cu2Zn1−xFexGeSe4. . . . 116
8.49 Seebeck coefficients and resistivities of Cu2.025Zn0.975−yFeyGeSe4. . . . . . . . . . 116
8.50 Temperature dependence of Hall carrier concentrations and Hall mobilities of
Cu2.025Zn0.975−yFeyGeSe4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.51 Resistivities and Hall mobilities of Sr3Ga0.93Zn0.07Sb3 measured on disks cut in
different directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.52 Hall carrier concentration of Sr3Ga0.93Zn0.07Sb3 measured on disks cut in differ-
ent directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.53 Temperature dependence of κ and L of Cu2.025Zn0.975−yFeyGeSe4. . . . . . . . . . 118
8.54 Temperature dependence of κel and κL of Cu2.025Zn0.975−yFeyGeSe4. . . . . . . . . 118
8.55 Temperature dependence of L and κel of Ca3Al1−xZnxSb3. . . . . . . . . . . . . . 118
8.56 Temperature dependence of L and κel of Sr3Ga1−xZnxSb3. . . . . . . . . . . . . . 119
8.57 Temperature dependent power factor and zT of Cu2.025Zn0.975−yFeyGeSe4. . . . . 119
8.58 Thermal stability of consolidated disks of Cu2ZnGeSe4. . . . . . . . . . . . . . . . 120
xxii
List of Tables
1.1 Crystallographic data of stannite-type Cu MI I2 GeSe4. . . . . . . . . . . . . . . . . 11
1.2 Crystallographic data of kesterite-type Cu MI I2 GeSe4 . . . . . . . . . . . . . . . . 12
1.3 Crystallographic data of Ca3AlSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Crystallographic data of Sr3GaSb3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Lattice parameters of Cu2+xZn1−xGeSe4 . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Temperature dependent lattice parameters of Cu2ZnGeSe4. . . . . . . . . . . . . 34
3.3 Lattice parameters of Cu2Zn1−xFexGeSe4. . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Lattice parameters of Cu2.025Zn0.975−yFeyGeSe4. . . . . . . . . . . . . . . . . . . . 40
xxiii

List of Symbols, Notations, and Abbreviations
a - lattice parameter
a0 - equilibrium lattice parameter
a(x) - composition dependent lattice parameter
ap - lattice parameter of pure end member
α - Seebeck coefficient / Thermopower
αA - absorption coefficient
αT - thermal expansion coefficient
B - quality factor
BSE - back scattered electron
BTE - Boltzmann transport equations
BT - bulk modulus
b - lattice parameter
b(XX) - number of bonds
C - heat capacity / specific heat
c - lattice parameter
D - sample density
DM - mass diffusion coefficient
∆T - temperature difference
DFT - density functional theory
DOS - density of states
DSC - differential scanning calorimetry
d - thermal diffusivity
dqT - Thomson heat
dqP - Peltier heat
E - electric field
Eg - band gap
EDS - energy dispersive X-ray spectroscopy
E(~k) - band dispersion
EA - activation energy
e(X) - number of valence electrons per anion
e(M) - number of valence electrons per metal cation
e - electron charge
e - reduced carrier energy
eF - Fermi energy
e - anharmonicity parameter
xxv
List of Tables
FP-LAPW - full-potential linearized augmented plane-wave
f - Fermi-Dirac distribution function
f0 - Fermi-Dirac distribution function at equilibrium
fi - fraction of atoms i
GGA - generalized gradient approximation
g(ω) - vibrational density of states
Γexp - experimental disorder scattering parameter
Γm - mass disorder scattering parameter
ΓS - strain disorder scattering parameter
γ - Grüneisen parameter
h - Planck constant
η - reduced electrochemical potential
ηC - Carnot efficiency
ηe - electrical efficiency
i+ - electrical current
Je - electrical current density
KS - spring constant
kB - Boltzmann constant
~k - electron wave vector
κ - thermal conductivity
κL - lattice thermal conductivity
κel - electronic thermal conductivity
p
κL - lattice thermal conductivity of a crystal without disorder
κmin - minimum lattice thermal conductivity
κb - bipolar contribution to the thermal conductivity
L - Lorenz number
l - phonon mean free path
lD - diffusion length
λ - scattering parameter
M - atomic mass
M̄ - average mass per atom
m∗b - band effective mass
m∗d - density of states effective mass
µT - Thomson coefficient
µ - charge carrier mobility
µH - Hall mobility
µ0 - intrinsic carrier mobility
N - number of atoms
NV - band degeneracy
N(E) - density of states
n - charge carrier concentration/density
nH - Hall carrier concentration
xxvi
List of Tables
ni - occupation number of vibrational states
ν - wave number
νg - phonon group velocity
νp - phonon phase velocity
∇T - temperature gradient
P - power
PF - power factor
PBE - Perdew, Burke, and Enzerhof
p - momentum
πa,b - Peltier coefficient
QBSD - quadrant back scattering detector
qJ - Joule heat
q - heat flow
R - electrical resistance
R - ideal gas constant
Ra - atomic distance
Ra,0 - equilibrium atomic distance
RH - Hall coefficent
Rwp - profile residual
rH - Hall factor
ρ - electrical resistivity
SEM - scanning electron microscopy
SE2 - secondary electron detector
σ - electrical conductivity
T - temperature
TC - temperature on the cold side
TH - temperature on the hot side
TB-mBJ - modified Becke-Johnson semi-local exchange potential
TGA - thermogravimetric analysis
t - thickness
τD - diffusion time
τ - scattering relaxation time
τN - phonon scattering relaxation time for normal scattering
τU - phonon scattering relaxation time for Umklapp scattering
τB - phonon scattering relaxation time for boundary scattering
τPD - phonon scattering relaxation time for point defect scattering
τEP - phonon scattering relaxation time for electron-phonon scattering
ΘD - Debye temperature
u - disorder scaling parameter
V - voltage
V - volume
VEC - valence electron concentration
xxvii
List of Tables
V0 - minimum band energy
vd - electron drift velocity
vm - average velocity of sound
vl - longitudinal speed of sound
vt - transverse speed of sound
XRD - X-ray diffraction
ZT - device thermoelectric figure of merit
zT - materials thermoelectric figure of merit
Z - formula units per unit cell
ω - phonon frequency
ωD - Debye frequency
Ω - average volume per atom
xxviii
1 Introduction
Todays world is facing threats like climate change due to the combustion of fossil fuels, while
the worlds demand for more and more energy is ever more increasing. The slowly receding
supply of fossil fuels and with it the increasing price has led to a renewed interest in renewable
energy technologies. For example Germany, which decided in 2011 to shut down all nuclear
power plants, is having a closer look into new technologies like e.g. photovoltaics, wind
energy, batteries, supercapacitors, and thermoelectricity.
Especially thermoelectric materials can be a potential route to increase the efficiency of ex-
isting technologies through the recovery of waste heat and its conversion into useful electrical
energy.[1, 2, 3, 4]
It is therefore necessary to explore new compounds for a possible use as thermoelectric
materials with a focus on understanding the structure and property relationships in order to
be able to optimize thermoelectric efficiencies.
1.1 Summary
This chapter starts with an overview on the fundamentals of thermoelectric energy conversion.
The basic transport phenomena, thermoelectric effects, and the required materials properties
are discussed in detail. Combining these effects and the materials properties leads to the
dimensionless thermoelectric figure of merit, zT. In modern thermoelectric materials the
figure of merit approaches unity, however higher figure of merits are needed for a more
efficient energy conversion.
An overview over the state of the art materials is given along with a discussion on some
approaches for higher figure of merits in thermoelectric materials, including the optimization
of the charge carrier density, resonant scattering, band engineering, and approaches to a lower
thermal conductivity. This chapter is mainly designed to give an overview on the transport
phenomena and provide a rough literature review. A more thorough description of transport
will be made at the beginning of each transport chapter.
The remainder of this chapter focuses on the literature overview and background of the
materials considered in this thesis: Cu2MI IGeSe4, Ca3AlSb3, and Sr3GaSb3. In particular the
section on Zintl-phases will touch some of the reasons why this class of materials is suited
for the investigation of structure-property relationships in thermoelectric materials. Finally,
the discussion will close with a summary and a motivation of the research conducted for this
thesis.
1
1 Introduction
1.2 Transport Phenomena and Thermoelectric Energy Conversion
Basic solid state transport phenomena of an electrical conductor or semiconductor can usu-
ally be investigated through two very basic methods. Either an electric field, a temperature
gradient, or both can be applied to observe different effects.[5]
Applying an electric field E across a material will lead to an electrical current in the material
which is defined by the the electrical current density Je and the electrical conductivity σ:[6, 7]
Je = σE . (1.1)
On the other hand, applying a temperature gradient∇T will lead to a heat flow q (when no
electric current is allowed to flow) due to the thermal conductivity κ of a material, with:[6, 7]
q = −κ∇T . (1.2)
Figure 1.1: Schematic of the Seebeck effect. Charge carriers diffuse in response to a tem-
perature gradient from the hot side to the cold side, until the resulting voltage
counterbalances the heat flux.
Both transport phenomena, electrical conductivity and thermal conductivity, are intrinsic
properties of a bulk material. However, there is a phenomenon which causes an electrical flow
in response to a temperature gradient, without an external electrical field. This, so called ther-
moelectric current, is caused by the Seebeck effect. In 1821, Thomas Johann Seebeck, realized
that an electric current flows around a closed circuit made up of different electrical conductors
when the junctions are subject to different temperatures.[8, 9, 10, 11, 12] The Seebeck effect
originates in the appearance of an electromagnetic force, which is caused by the diffusion of
charge carriers, electrons or holes, along the temperature gradient from the hot side to the
cold side. This is best visualized by an analogy to a gas in a tube. The gas molecules/atoms
at the hot side have a higher thermal energy and diffuse to the cold side. In other words,
the hot gas expands and the cold gas contracts.[13] This effect is also observed for electrons
and holes (see Figure 1.1). The diffusion from the hot to the cold side causes a voltage V
2
1.2 Transport Phenomena and Thermoelectric Energy Conversion
and the carriers diffuse until the resulting voltage counterbalances the heat flux. For small
temperature differences ∆T, the resulting thermoelectric voltage is found to be proportional
to the temperature difference, with the Seebeck coefficient α:[8]
V
α = . (1.3)
∆T
The Seebeck coefficient (or thermopower) is negative when the conduction is dominated
by electrons and positive when holes are the dominant charge carrier type. The Seebeck
coefficient is an intrinsic property of a solid material under open circuit conditions, however,
it can only be measured when in contact with another material (i.e. closed circuit conditions),
which leads to the common misconception that a junction between two materials is necessary
for a thermoelectric voltage to occur.
However, if we consider the situation where an electric current flows from one substance
(a) to another (b), then heat may either be absorbed or given out at the junction, depending
on the direction of the current.[5] This effect has been reported by Jean Peltier in 1834, when
he passed a small external current across a bismuth-antimony thermocouple. This Peltier heat
dqP was found to be proportional to the magnitude i+ and the duration of the applied external
current:[8]
dqP = π i+a,b . (1.4)
The coefficient of proportionality is the so called Peltier coefficient πa,b. The Peltier effect, as
well as the Seebeck effect, is a reversible effect and should not be confused with the irreversible
Joule heating qJ , which occurs when an electrical current flows through a material with the
resistivity ρ (qJ = Jeρ).[5, 8]
In 1854 William Thomson (later Lord Kelvin) applied the first and second law of thermo-
dynamics on thermoelectricity and decided there should be a thermodynamical connection
between the Seebeck and the Peltier effect. After analyzing the situation and realizing that
experimental evidence was not in agreement with the theoretical connection, he postulated
a third thermoelectric effect, the so called Thomson effect.[5, 8] An electrical current that
passes through a conductor, when there is also a temperature gradient ∇T present, produces
a Thomson heat dqT:[5]
dqT = µT Je∇T . (1.5)
The Thomson coefficient µT depends on the temperature of the conductor and has a pos-
itive sign if heat is absorbed, i.e. when a current flows to the hot region. The Kelvin (or
Thomson) relations relate Seebeck coefficient α, Peltier coefficient πa,b, and Thomson coeffi-
cient µT:[5]
Tdα
µT = and πa,b = Tα . (1.6)dT
Considering all transport phenomena and thermoelectric effects, good thermoelectric ma-
terials should have a large Seebeck coefficient α. Furthermore, the heat flow q from the hot to
the cold side should be as small as possible to keep a stable temperature gradient ∇T along
3
1 Introduction
the material. Therefore, small thermal conductivities are desired, as can be seen in Equation
1.2. Good thermoelectric materials also need to have good electrical conductivities σ (or a
small electrical resistivity ρ = 1/σ), to keep the Joule heating low and the power output P
high (with P = V2/R, for a total resistance R).[7] However, the combination of these materials
properties is usually difficult to obtain, since they are highly interdependent. For example,
high α usually corresponds to a low σ and vice versa. Therefore different strategies need to
be employed to obtain a good thermoelectric performance.
1.3 Thermoelectric Figure of Merit
A thermoelectric generator converts heat q into electrical power P with a certain electrical
efficiency ηe. This efficiency strongly depends on the temperature difference ∆T and is limited
by the Carnot efficiency ηC (ηC = ∆T/TH) and is typically defined as:[9, 14, 15, 16]
√
· √ 1 + ZT − 1ηe = ηC , (1.7)
1 + ZT + TCTH
for an optimized current. With the device figure of merit ZT and the temperatures of the
hot side and the cold side, TH and TC, respectively. Since the device figure of merit is not
easily calculated, the average materials figure of merit zT can be used to assess the efficiency
of thermoelectric material:[15, 17]
α2σ
zT = T . (1.8)
κ
Figure 1.2 shows published figure of merits for some common state of the art thermoelectric
materials.
Figure 1.2: Overview of zT-values for different p-type (left) and n-type (right) materials.[18]
TAGS stands for (GeTe)1−x(AgSbTe2)x. Note that different materials have their
maximum figure of merit in different temperature regimes.
As mentioned above, high thermoelectric efficiencies are achieved for high Seebeck coeffi-
cients, high electrical conductivities (or low electrical resistivities), and low thermal conduc-
tivities. However, these materials properties are strongly coupled and the desired combination
is difficult to obtain.
4
1.3 Thermoelectric Figure of Merit
The thermal conductivity, for instance, has two primary contributions, a lattice thermal
conductivity κL and an electronic contribution κel :[3, 19]
α2σ
zT = T . (1.9)
κL + κel
While the lattice contribution to the thermal conductivity can be independently altered (see
Section 1.3.3 and Chapter 5), the electronic part directly depends on the electrical conductivity,
expressed in the Wiedemann-Franz law:[3, 9, 14, 20]
κel = LσT = LneµT , (1.10)
with the Lorenz number L (L = 2.44x10−8 WΩK−2 for free electrons), the charge carrier
concentration n, the mobility of the charge carrier µ, and the electron charge e. As mentioned,
a high electrical conductivity is important for a good thermoelectric material, however, as can
be seen in Equation 1.10, this also leads to a higher electronic contribution to the thermal
conductivity.
Figure 1.3: Influence of the charge carrier concentration on the Seebeck coefficient α, the elec-
trical conductivity σ, the power factor α2σ, and the figure of merit zT. While the
electrical conductivity increases with increasing carrier concentration, the Seebeck
coefficient decreases. It can be seen, that an optimum carrier concentration exists,
which will lead to a maximum figure of merit in a material.
In addition to a low thermal conductivity a high power factor PF (PF = α2σ) is required
for a high figure of merit. Equation 1.10 shows the dependence of the electrical conductivity
on the charge carrier concentration n and the mobility of the charge carriers µ. However,
the Seebeck coefficient does also strongly depend on the charge carrier concentration of the
material. For a degenerate system (high carrier concentrations) the Seebeck coefficient is given
by:[3, 19]
8π2k2 ( ) 2
α = B m∗
π 3
T , (1.11)
3eh2 d 3n
5
1 Introduction
while at lower carrier concentrations (non-degenerate semiconductor) α is proportional to
ln(1/n).[9, 10] This means the Seebeck coefficient decreases with increasing carrier concentra-
tion. Here m∗d stands for the density of states effective mass, kB is the Boltzmann constant,
and h the Planck constant. Furthermore, even the Lorenz number is not constant and does to
some extent depend on the location of the Fermi level and therefore the carrier concentration
(see Equation 5.30).[21] The influence of the charge carrier concentration on the Seebeck coef-
ficient, the electrical conductivity, and the electronic contribution to the thermal conductivity
reveals the need to precisely control the charge carrier density in a material, so that reasonable
values of α and σ are obtained at an optimum carrier concentration (see Figure 1.3).
Achieving an optimum carrier concentration in a material is crucial for a maximum figure
of merit of a given thermoelectric material. However, different approaches to higher figure of
merits have been chosen as well. For instance, the thermopower of a material can be increased
by increasing the density of states of a material at the Fermi level, either by resonant impurities
or multiple bands via band engineering. Furthermore, reducing the thermal conductivity via
alloying, nano structuring, and employing glass-like materials has also been shown to increase
the figure of merits in some materials.
1.3.1 Enhancement of Thermopower
As seen in Section 1.3 a high Seebeck coefficient is desired, while an optimum carrier concen-
tration for a decent electrical conductivity might result in too low carrier concentrations for an
actual high Seebeck coefficient. However, considering Equation 1.11, the Seebeck coefficient
does not depend on charge carrier density only. The density of states effective mass m∗d at
the Fermi level has an influence on α as well, and materials with a high effective mass are
desired.[19, 22, 23, 24, 25]
Figure 1.4: Schematic of the influence of multiple bands and resonant states on the density
of states (DOS). Both mechanisms increase the Seebeck coefficient, either through
a larger number of conducting bands (NV) or a higher density of states effective
mass m∗d at the Fermi level EF.[25]
6
1.3 Thermoelectric Figure of Merit
A high density of states and therefore a high density of states effective mass can be achieved
due to either a large number of conducting bands NV (also called band degeneracy),[26] or
via flat bands giving a high band effective mass m∗b :[9, 24, 25]
m∗ = N2/3m∗d V b . (1.12)
One approach to a high density of states (or flat bands) at the Fermi level can be achieved
through the introduction of resonant states via doping. This concept has first been proposed
by Heremans et al. in 2008.[27] Doping bulk PbTe with Tl introduces new energy levels (or
resonant states) leading to a higher density of states and an increased Seebeck coefficient in
this material. This has also successfully been shown for doping with Al and Cr in PbTe or
PbSe.[28, 29, 30] Figure 1.4 shows the influence of resonant states and multiple bands on the
density of states at the Fermi level.[25]
Figure 1.5: Left: Light and heavy valence band in PbTe for different symmetry points in the
Brillouin zone. PbTe exhibits a valence band with a larger (heavier) effective mass
at lower energies. Right: Figure of merit zT for different carrier concentrations n
and the influence of the different valence bands. After shifting the Fermi level to
lower energies due to a larger doping level, the heavier valence band contributes
to the transport and increases the zT.[18, 25]
However, a high band effective mass (i.e. flat bands) will unfortunately result in a low
electron (or hole) mobility, because the mobility is inversely proportional to the effective mass
(µ ∝ 1/m∗b).[3, 19, 24] And a low charge carrier mobility will inevitably lead to a low electrical
conductivity (see Equation 1.10). For a material with an optimized carrier concentration, the
figure of merit zT depends on the quality factor B, which is determined by the lattice thermal
conductivity κL, the mobility of the charge carriers, the band effective mass, and the band
degeneracy (number of bands contributing to the conduction):[18]
µN m∗3/2
B ∝ V b . (1.13)
κL
Besides increasing the effective mass for a high quality factor, there are approaches to
increase the band degeneracy, and lowering the lattice thermal conductivity for a high ther-
7
1 Introduction
moelectric figure of merit. These approaches will be discussed in the following sections.
1.3.2 Band Engineering
Following the expression for a high quality factor B (Equation 1.13), thermoelectric materials
with a large number of band degeneracy NV are beneficial for a high figure of merit. In other
words, having multiple bands with comparable energies (within a few kBT) contributing to
the electronic transport from either degenerate band extrema (orbital degeneracy) or multiple
carrier pockets in the Brillouin zone (see Figure 1.6) due to high crystal symmetry. While it
is not possible to change the NV of a material (since this would mean a change in symmetry
and crystal structure), it is possible to converge several bands to achieve a larger, combined
NV .[18, 25, 31, 32]
Lead chalcogenides, for example, have a highly symmetric crystal structure (NaCl-type)
and therefore a very high band degeneracy. The band structure of PbTe shows a light valence
band at the L-point, with a direct band gap Eg to the conduction band, and a heavier valence
band at the Σ-point, which is lower in energy (see Figure 1.5 left). A higher doping level would
inevitably move the Fermi level into the direction of the heavier valence band and therefore
increase the thermopower, leading to a higher figure of merit for both bands contributing to
the transport at an optimum carrier concentration (see Figure 1.5 right).[18]
Furthermore, both valence bands exhibit different temperature dependencies (see Figure
1.6). At a certain temperature both bands converge, leading to an effective band degeneracy
of 12-16 and therefore increasing the figure of merit to zT ∼ 1.8.[31]
Figure 1.6: Left: Number of carrier pockets in the first Brillouin zone of PbTe, showing a
band degeneracy of NV equals 4 for the L-point and 12 for the Σ-point, due to
the exceptionally high crystal symmetry. Right: Temperature dependence of the
two valence bands. At a specific temperature both bands converge, leading to a
combined band degeneracy of 12-16 and therefore an increased figure of merit in
this temperature regime.[18, 31]
Increasing the band gap of a material via alloying also leads to an increase in the figure of
merit of a material.[25] At higher temperatures, minority carriers get excited across the band
gap and the thermopower starts to decrease due to the contribution of minority carriers. If
8
1.3 Thermoelectric Figure of Merit
there are electrons and holes present, the resulting Seebeck coefficient will be the sum of their
contributions, with:[8]
αnσn + αpσp
α = , (1.14)
σn + σp
and therefore the thermoelectric voltage will be lowered. Hence, a decrease of the figure
of merit will occur at higher temperatures due to the decreasing thermopower. Alloying a
compound to widen the band gap will therefore not only shift the maximum of the zT to
higher temperatures, it will also lead to higher figure of merits, since larger values of the
Seebeck coefficient can be obtained at higher temperatures (see Equation 1.11 and Figure 1.7).
Figure 1.7: Influence of the band gap on the temperature dependence of the figure of merit.
For larger band gaps the carrier activation across the band gap occurs at higher
temperatures, leading to a higher peak and integrated zT, before the influence of
the minority carriers lead to a decrease in the thermoelectric voltage.[25]
1.3.3 Reduction of Thermal Conductivity
In order to obtain high thermoelectric efficiencies a low lattice thermal conductivity κL is
desired for a large quality factor. The lattice thermal conductivity of a material is caused by
lattice vibration waves or phonons, propagating through the lattice and therefore carrying
heat. These phonons have group velocities νg, mean free path lengths l between scattering
events, and the amount of heat transported is proportional to the heat capacity C. This leads
to a lattice thermal conductivity in analogy to kinetic gas theory of:[19]
1
κL = Cνgl . (1.15)3
For materials with low lattice thermal conductivities, materials with low heat capacities are
required, which can usually be found in materials with weak bond strengths, heavy atoms, or
large unit cell volumes.[19]
Lowering the phonon mean free path in a material is one possible approach to a lower
9
1 Introduction
lattice thermal conductivity. This can easily be achieved via alloying with other elements to
create point defect scattering,[33] successfully shown for thermoelectric materials in Heusler
compounds,[34] skutterudites,[35, 36] lead telluride,[37] and SiGe.[38, 39] Point defect scat-
tering (or alloy scattering) is an effective way to scatter high frequency phonons, which are
usually not heavily scattered by phonon-phonon scattering.[33, 40, 41] Furthermore the use of
nano structuring or interfaces to reduce the phonon mean free path has shown to be effective
for scattering low frequency (large wavelength) phonons, achieving low κL.[42, 43, 41, 44, 45,
46]. However, oftentimes these point defects, interfaces, and nano structures not only scat-
ter phonons but charge carriers as well, leading to a reduction in the mobility and therefore
decreasing the quality factor (Equation 1.13).[41]
This problem has lead to the search for materials within the "phonon-glass electron-crystal"
concept.[47] Utilization of the materials’ structure, e.g. use of filling atoms in skutterudites,[48,
49, 50, 51] or clathrates,[52, 53, 54] exhibit glass-like thermal properties, with still high mobil-
ities due to a crystalline behavior.
1.4 Pertinent Materials
1.4.1 Cu MI I2 GeSe4
This section is in parts an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154
and J. Am. Chem. Soc 2013 - accepted manuscript.[55, 56] Reproduced with permission of the
American Chemical Society Copyright 2012 and 2013.
In an attempt to broaden the quest for abundant materials with low thermal conductivities
Chen et al. have reported on the transport properties of quaternary tetrahedrally bonded
Cu2(Zn/Cd)SnSe4.[57, 58, 59] Despite being wide band gap compounds with low electron
mobility, these materials exhibit fairly good thermoelectric performance upon "doping" due
to the very low intrinsic thermal conductivities, leading to a maximum zT of 0.95 at 850K for
Cu2ZnSn1−xInxSe4 .[57, 58, 59, 60]
There exists a whole family of structurally related quaternary compounds of the formula
Cu2-II-IV-S4(Se4), with II = Mn, Fe, Co, Ni, Zn, Cd, Hg and IV = Si, Ge, Sn, crystallizing
in either tetragonal or orthorhombic crystal structures.[61] Many of these compounds have
been synthesized and characterized, for instance for their electronic properties,[62, 63, 64]
optical properties,[65] thermal properties,[66], and magnetic properties.[67, 68] The structural
robustness of this class of compounds makes systematic substitution and hence the study of
their properties possible.[67, 69, 70, 71, 72] Recently Cu2ZnSnSe4−xSx (CZTSSe), and related
compounds, have attracted attention for the use in photovoltaics, due to possible band gap
tuning,[73, 74, 75, 76] leading to efficiencies around 10% of laboratory fabricated photovoltaic
devices.[73, 77, 78]
Structurally derived from the diamond structure, the structures of these so called "adaman-
tine" compounds, have been synthesized and structurally characterized.[61, 63, 80] In these
diamond like structures, each atom is coordinated by four nearest neighbors, forming a tetra-
hedrally bonded structure.[79] In the tetragonal unit cell system there exist two structurally
similar but distinct structure types, namely stannite and kesterite. The crystal structures of
10
1.4 Pertinent Materials
Figure 1.8: Crystal structures of stannite-type (left) and kesterite-type (right) Cu MI I2 GeSe4.
Cu atoms are yellow, Ge atoms are gray, MI I atoms are blue (stannite) or green
(kesterite), and Se atoms are red, with the tetrahedral coordination of the elements
indicated.[55] Note the difference in the ordering of the metal cations. While Cu
resides on the 4d-Wyckoff position (0, 12 ,
1
4 ) in the stannite type, it shares this lattice
site with MI I in the kesterite structure.[79]
the stannite-type and the kesterite-type Cu MI I2 GeSe4 are shown in Figure 1.8.
These multinary chalcogenides, form binary, ternary, and quaternary compounds where
the two substructures are populated with cations and anions, respectively. The binary com-
pounds AI IXVI (e.g. ZnSe) crystallize in the cubic-sphalerite structure type (space group
F4̄3m). Lowering the symmetry by ordered cation substitution with two metals leads to the
chalcopyrite structure type AIBI I IXVI2 (I4̄2d), with the mineral chalcopyrit CuFeS2 as the proto-
type. This ordered substitution doubles the translational period along the z-direction leading
to a tetragonal structure crystal system. Due to different interactions between the metals and
the anions, resulting in different bond length and bond angles, a tetragonal distortion takes
place resulting in a c/2a-ratio < 1.[79]
A further decrease in symmetry leads to the stannite type structure (I4̄2m) or (with even
further reduced symmetry) the kesterite type structure (I4̄). This decrease in symmetry is
achieved by substitution of elements but also through metal ordering, e.g. metal ordering
leads from stannite (prototype Cu2FeSnS4) to kesterite (prototype Cu2ZnSnS4).[79] In the stan-
nite type Cu is located on the Wyckoff position 4d (0, 12 ,
1
4 ), M
I I occupies 2a (0,0,0), and Ge is
located on site 2b (0, 0, 12 ), while selenium coordinates all cations on Wyckoff position 8i-site
(0.2449, 0.2449, 0.1298). All site occupancies are 100% and each unit cell contains two chemical
Table 1.1: Crystallographic data of stannite-type Cu2MI IGeSe4 (space group I4̄2m).[61]
Atoms Wyckoff Position Point symmetry Occupation
MI I 2a (0, 0, 0) 4̄2m 1.0
Cu 4d (0, 1 12 , 4 ) 4̄.. 1.0
Ge 2b (0, 0, 12 ) 4̄2m 1.0
Se 8i (0.2449, 0.2449, 0.1298) ..m 1.0
11
1 Introduction
Table 1.2: Crystallographic data of kesterite-type Cu MI I2 GeSe4 (space group I4̄).[80]
Atoms Wyckoff Position Point symmetry Occupation
MI I 2d ( 12 , 0,
1
4 ) 4̄.. 1.0
Cu1 2a (0, 0, 0) 4̄.. 1.0
Cu2 2c (0, 12 ,
1
4 ) 4̄.. 1.0
Ge 2b ( 12 ,
1
2 , 0) 4̄.. 1.0
Se 8g (0.756, 0.756, 0.8722) 1 1.0
formulas (Z=2).[61, 81] The crystallographic positions for both structures are given in Tables
1.1 and 1.2.
In the kesterite type structure (see Figure 1.8), however, Cu is located at the Wyckoff po-
sition 2a-site and the 2c-site with equal parts. The MI I cation occupies the 2d-site sharing it
with Cu.[79] In other words, while MI I and Ge share the z = 0 and z = 12 metal layers, Cu
occupies the z = 14 and z =
3
4 layer only, whereas in kesterite all layers are occupied with Cu
sharing with either the MI I-cation or the Ge.[80]
Both structure types are closely related and there is a high degree of possibility for anti site
disorder, especially since the kesterite structure evolves from metal ordering, occurring con-
currently with a symmetry decay from the stannite type. There is further indication that the
synthetic procedure might have an influence on the obtained crystal structures.[79, 81, 82]
Given the high degree of similarity of the prototype structures of kesterite and stannite,
Cu2ZnSnS4 and Cu2FeSnS4, respectively, many investigations have been undertaken to an-
alyze the solid solutions of this system. Kesterite and stannite are found coexisting adjacent
with no indication of miscibility,[80] while there seems to be a discontinuity of the lattice
parameters, showing an inversion of the trend of the unit-cell parameters, observed in the
range 60-70 mole % in Cu2ZnSnS4.[80, 83] Neutron diffraction studies by Susan Schorr on
these multinary chalcogenides suggest a 3-stage cation restructuring process originating from
a crossover from the stannite to the kesterite structure type.[81, 82]
In this thesis, the solid solutions Cu2+xZn1−xGeSe4, Cu2Zn1−xFexGeSe4, and
Cu2+xZn1−x−yFeyGeSe4 have been synthesized and characterized for a possible application as
thermoelectric materials. The solid solution of Cu2Zn1−xFexGeSe4 has been synthesized by
Caldera et al.,[84] and the lattice parameters have been reported. These materials have been
mainly characterized for their magnetic properties, and Cu2FeGeSe4, for example, exhibits
antiferromagnetic behavior with a Neel-temperature of 20 K.[68]
1.4.2 Zintl-Phases
This section is an adapted reproduction of the Diploma thesis from Wolfgang Zeier, 2010.[85]
Zintl phases are intermetallic solids, which include an electropositive metal, such as alkali
and alkaline earth metals, and a metallic but more electronegative partner (mostly semimetals,
e.g. Si, P, As, Sb). These compounds contain chemical bonds that demonstrate both ionic and
covalent character.[86, 87]
The Zintl-Klemm formulation provides an explanation for the properties of these com-
12
1.4 Pertinent Materials
pounds. The more electropositive metal transfers all valence electrons to its more electroneg-
ative partner, and the resulting anion forms a covalently bonded superlattice, structured ac-
cording to its valence electron concentration (VEC).[86, 87] This model is referred to as the
Zintl-Klemm-Busmann concept.[86]
For a binary compound MmXx, with the cation M and the anion X, the VEC is given by:[88]
m · e(M) + x · e(X)
VEC(X) = , (1.16)
x
where e(M) and e(X) are the number of valence electrons for the metal and anion, respectively.
The number of bonds b(XX), formed by the anions, is given by:[88]
b(XX) = 8−VEC(X) . (1.17)
This formulation is simply a restatement of the octet rule for the formation of elemental
structures. In the Zintl-Klemm-Busmann concepa these compounds form the structures of the
elements of the 4th to 7th main group, according to their valence electron configuration.[88]
Consider the Zintl compound NaTl. Sodium is the more electropositive partner, so it trans-
fers one electron to thallium to form Tl−. This results in a VEC of four for Tl−, which is
the same valence electron configuration as of carbon. According to the octet rule carbon has
to form four bonds, which results in the diamond structure. Thus in NaTl, the Tl− forms a
diamond-like, tetrahedral coordinated superlattice with covalent Tl-Tl bonds and Na+ in the
tetrahedral holes.[87]
Other examples are the compounds FeS2 (pyrite) and CaSi. In pyrite the iron transfers
two electrons to two sulfur atoms, which leads to a formal S− with a VEC of 7 and S2−2 -pairs
form in analogy to Cl2.[88] As expected from the Si2−-species in CaSi, with a VEC of 6, chains
similar to S0 can be found.
Clearly the term Zintl phase can apply to more than just binary compounds, it comprises
a vast number of compounds (ternaries, clusters with molecular motifs, etc.), and a detailed
discussion can be found elsewhere in the literature, e.g. [86].
Due to the covalent character of the anionic bonding in the superlattice and the ionic cation-
anion interaction, the properties of the Zintl phases differ from the properties of intermetallic
compounds.[87] Similar to ionic materials a band gap (< 1 eV) forms between the filled anionic
valence band and the empty cation conduction band.[87] Zintl phases are also referred to as
polar intermetallics and they exhibit properties of both ionic and covalent materials, such as
brittleness but metallic or semiconducting behavior.[87, 88]
Materials with good thermoelectric properties can be found in a wide range of compounds.
Some thermoelectrics have high mobilities and low effective masses of the charge carriers,
while more ionic materials exhibit low mobilities and high effective masses.[3] The interdepen-
dence of the transport properties on the figure of merit is the reason why good thermoelectric
compounds might be completely different materials. This leads to an important question: Are
there interesting compounds located in the middle of this wide range between intermetallics
and ionics with good thermoelectric properties?
One possible class of compounds are these Zintl phases. As mentioned they exhibit both
13
1 Introduction
ionic and metallic behavior, which led to a recent research focus on Zintl phases for thermo-
electric materials, e.g. [89, 19, 90].
Probably the best example for a focus on Zintl phases is the compound Yb14AlSb11. Doping
on the Al-site with Mn2+ leads to a zT > 1.3 for Yb14Mn1−xAlxSb11 at the optimized doping
level of x = 0.6 and 0.8.[91] The high zT of this material is mainly due to the glass-like thermal
conductivity.[19, 89, 92] The complexity of the structure with a large number of atoms per unit
cell (104 atoms) leads to an intrinsically low lattice thermal conductivity.[19, 89] Furthermore,
the heavy atoms of this compound result in a low group velocity of the phonons, which
leads to an additional decrease of the lattice thermal conductivity.[19] Magnetic investigations
suggest a contribution of the d5-spin-state of the Mn2+-cations to the high zT-value due to
coupling with the holes.[90, 91, 93] The compound Yb14AlSb11 seems to be a very promising
candidate and a very good example for the research on Zintl phases, due to good electrical
properties and a low thermal conductivity - which leads to a successful combination of the
electron-crystal phonon-glass approach - and the ability of precise doping on the Yb or Al-
site.[19, 89].
As seen in Section 1.3 the thermoelectric properties strongly depend on the carrier con-
centration of a material, and a control over the carrier concentration is necessary for a good
thermoelectric material.[19] Zintl phases are valence compounds with semiconducting behav-
ior. The anions form the valence band, with only small cation contributions, and the cations
the conduction band. Due to the small impact of the cations on the electronic structure of the
valence band, acceptor doping on the cation site allows a precise adjustment of the Fermi level
and carrier concentration, usually without a huge disruption of the band structure.[19, 94]
Zintl phases offer the desired characteristics for thermoelectric materials. They often ex-
hibit complex crystal structures resulting in a low thermal conductivity, they form small band
gap semiconductors, and their rich crystal chemistry leads to the possibility of precise tuning
of the transport properties.[94] On the fundamental level, there is a need to explore Zintl com-
pounds and their thermoelectric properties in respect to their crystal structure. A comparison
of the electrical and thermal transport properties might lead to an insight in the structure-
property relationships in Zintl compounds and maybe opening new approaches to enhanced
thermoelectric efficiencies.
1.4.3 Ca3AlSb3
This section is in parts an adapted reproduction, from Energy Environ. Sci. 2011, 4, 510-518
and J. Mater. Chem 2012 22, 9826-9830.[95, 96] Copyright 2011 and 2012 The Royal Society of
Chemistry.
First reported by Cordier et al.,[97] Ca3AlSb3 crystallizes in the orthorhombic crystal system
(space group Pnma) with 28 atoms in the unit cell. Four Sb atoms coordinate the Al atoms,
forming slightly distorted tetrahedra. These tetrahedra connect to one-dimensional corner-
sharing infinite chains in the b-direction, in analogy to the structure of (SiO3)2−n in the chain-
silicates.[98] A perspective along the AlSb4 tetrahedral chains can be viewed in Figure 1.9.
The Al-Sb distances of 270.9-273.9 pm are shorter than the sum of the radii for the metals
or the ions and are in the range of the covalent radii. Therefore a mainly covalent Al-Sb bond
14
1.4 Pertinent Materials
Figure 1.9: Crystal structure of the orthorhombic Zintl-phase Ca3AlSb3 (space group Pnma)
viewed along the b-direction (left). Ca atoms are green, Al atoms blue, and Sb
atoms are orange. Al is tetrahedrally coordinated by Sb and forms linear chains
(right) of corner sharing tetrahedra in the direction of b.[97]
seems to exist,[97] probably due to the similar electronegativities of antimony and aluminium.
The chains are arranged in a pseudo hexagonal pattern with the Ca2+ cations between the
chains, with a charge of -1 for the double bonded pnicogen atoms and a charge of -2 for the
single bonded antimony.[97] This is in accordance with the Zintl-Klemm-Busmann concep (see
Section 1.4.2). A charge of -1 for Sb leads to a valence electron concentration VEC of 6, which
has to form two bonds according to the octet rule, while a charge of -2 leads to a VEC of 7
resulting in one bond. The aluminium has to have a valence state of -1, leading to a VEC of 4,
which results in the observed tetrahedral coordination.
The Zintl-phase Ca3AlSb3 has recently been investigated for its thermoelectric transport
properties and the electronic and thermal transport properties have been reported.[95] Through
substitution of sodium for calcium, control and optimization of the hole carrier concentration
Table 1.3: Crystallographic data of Ca3AlSb3.[97]
Space group a / Å b / Å c / Å Z
Pnma 12.835(5) 4.489(2) 14.282(5) 4
Atoms Wyckoff Position Point symmetry Occupation
Sb1 4c (0.0393, 14 , 0.3509) .m. 1.0
Sb2 4c (0.1129, 14 , 0.8890) .m. 1.0
Sb3 4c (0.2555, 14 , 0.6199) .m. 1.0
Ca1 4c (0.2711, 14 , 0.2806) .m. 1.0
Ca2 4c (0.8487, 14 , 0.5050) .m. 1.0
Ca3 4c (0.0598, 14 , 0.1104) .m. 1.0
Al 4c (0.0659, 14 , 0.7032) .m. 1.0
15
1 Introduction
Table 1.4: Crystallographic data of Sr3GaSb3.[99]
Space group a / Å b / Å c / Å β / ◦ Z
P21/n 11.762(4) 14.509(5) 11.749(4) 109.96 8
Atoms Wyckoff Position Point symmetry Occupation
Sr6 4e (0.0772, 0.0665, 0.1638) 1 1.0
Sb6 4e (0.1099, 0.7587, 0.3988) 1 1.0
Sb1 4e (0.1148, 0.0908, 0.4263) 1 1.0
Ga2 4e (0.1292, 0.2450, 0.3219) 1 1.0
Sr2 4e (0.1370, 0.7502, 0.1579) 1 1.0
Sr3 4e (0.1395, 0.4291, 0.1817) 1 1.0
Sb5 4e (0.1556, 0.4001, 0.4498) 1 1.0
Sr1 4e (0.3504, 0.5758, 0.0230) 1 1.0
Sb4 4e (0.3658, 0.2372, 0.3199) 1 1.0
Ga1 4e (0.3719, 0.0911, 0.1888) 1 1.0
Sr5 4e (0.4041, 0.2836, 0.0915) 1 1.0
Sb3 4e (0.6089, 0.0936, 0.2046) 1 1.0
Sr4 4e (0.6187, 0.4044, 0.4622) 1 1.0
Sb2 4e (0.6496, 0.4329, 0.2261) 1 1.0
has been obtained, since sodium has one less valence electron to donate compared to calcium
which leads to the introduction of holes as charge carriers. Ca3AlSb3 was found to have an
exceptionally low lattice thermal conductivity (0.6 Wm−1K−1 at 1050 K), due in large part to its
complex unit cell. The combination of a reasonable bandgap, good carrier concentration con-
trol, and low thermal conductivity yields a thermoelectric figure of merit of 0.8 at 1050 K.[95]
1.4.4 Sr3GaSb3
This section is in parts an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-
9128.[100] Reproduced with permission of The Royal Society of Chemistry Copyright 2012.
Sr3GaSb3 is the only compound known to crystallize in a unique, chain-based structure
that, according to Schäfer et al., can be considered a connecting link between the structures
based on either linear chains or isolated tetrahedra-pairs formed by all other known A3MPn3
compounds (A = Ca, Sr, Ba, M = Al, Ga, In, and Pn = P, As, Sb).[99] The monoclinic structure,
with 56 atoms per unit cell, of Sr3GaSb3 (space group P21/n) contains infinite, close packed,
non-linear chains of corner sharing tetrahedra, characterized by four-tetrahedra repeat units,
as shown in Figure 1.10 a). This four-tetrahedra periodicity in Sr3GaSb3 leads to a larger
unit cell than that of Ca3AlSb3, which is only one tetrahedron wide along the chain direction.
The chains in Sr3GaSb3 are aligned along the [101] direction, shown pointing into the page
in Figure 1.10 b). A single chain is highlighted in red. Note that while the chains appear to
overlap with nearby chains to form zigzagging planes, staggering along the [101] direction
prevents strong interactions between neighboring chains.
In accordance with the Zintl-Klemm-Busmann concept for Ca3AlSb3, Sr3GaSb3 forms these
tetrahedra resulting in a valence-balanced compound. While the basic bonding environment
16
1.5 Summary of Research and Motivation
a) b) 
b b
c a c a
Figure 1.10: a) Sr3GaSb3 contains unique chains of corner-sharing GaSb4 tetrahedra, formed
from four-tetrahedra repeat units. Ga atoms are green, Sb are orange, and Sr
atoms are not shown. b) The structure of Sr3GaSb3 viewed down the [101] axis,
with chains going into the page (Sr atoms shown in blue). In both orientations, a
single chain is highlighted in red.[100]
is the same as in the Ca3AlSb3 compound, the bigger size of the Sr2+ cation inevitably leads
to a different structure, compared to Ca3AlSb3.
1.5 Summary of Research and Motivation
This thesis addresses the structural and thermoelectric properties of materials and tries to link
the structure to the properties in Zintl compounds and quaternary copper selenides. Different
materials compositions are investigated, the influence of different dopants considered, within
transport models from common solution to the Boltzmann transport equation, and effects of
structural and micro structural aspects on the thermoelectric transport are discussed.
For most of this work, the compound Cu2ZnGeSe4 and its solid solutions are considered.
This class of compounds has a high structural robustness regarding the substitution with dif-
ferent elements,[86] which should inevitably lead to the possibility of precise doping in these
materials to obtain an optimum charge carrier concentration. Combined with the low intrinsic
thermal conductivities of these compounds an attempt is made to investigate and to institute
a whole, new range of possible thermoelectric materials. Different materials compositions are
investigated in order to combine structural and doping effects for a thorough characteriza-
tion of their impact on the materials properties. First the solid solution Cu2+xZn1−xGeSe4
is investigated for its transport properties, upon the substitution of Zn2+ with Cu+ and the
introduction of charge carriers. It is shown that the substitution not only introduces holes,
but also creates a charge imbalance in this material, leading to a phase segregation, which in
turns lowers the thermal conductivity in this material.
Then the system Cu2Zn1−xFexGeSe4 is considered. The isoelectronic substitution of Zn
with Fe and the inherent structural complexity resulting from alloying influences the thermal
transport properties greatly. To finalize the analysis of this class of compounds, the solid so-
17
1 Introduction
lution Cu2+xZn1−x−yFeyGeSe4 is characterized in an attempt to combine the previous results,
showing a certain degree of band engineering, leading to a dependence of the effective mass
on the composition.
In addition to the work on the Cu MI I2 GeSe4 compounds, this thesis discusses the trans-
port in the Zintl compounds Ca3AlSb3 and Sr3GaSb3. The solid solutions of Ca3Al1−xZnxSb3
and Sr3Ga1−xZnxSb3 were synthesized and the thermoelectric transport properties measured.
The influence of the grain size and the grain boundaries on the transport is discussed and
an attempt will be made to relate the difference of the thermal transport properties of both
compounds to the difference in their crystal structures.
Zintl compounds, characterized by covalently bonded “substructures,” surrounded by
highly electropositive cations, have recently attracted attention due to their often complex
crystal structures.[19, 89] In general, the lattice thermal conductivity of Zintl compounds
trends with unit cell complexity;[101, 102] e.g. Yb14AlSb11, containing 104 atoms per unit
cell, exhibits one of the lowest known lattice thermal conductivities of any bulk material.[91]
In contrast, the electronic mobility in Zintl compounds is more difficult to predict, though
the nature of the covalently-bonded substructure is thought to play a major role. For exam-
ple, a relatively high electronic mobility has been demonstrated in AM2Sb2 (A = alkaline or
rare earth, M = transition metal) compounds which form 2-dimensional covalent planes (µ ∼
100 cm2V−1s−1),[103, 104, 105], while Yb14AlSb11, characterized by isolated AlSb4 tetrahedra
and Sb3 moieties, exhibits extremely low carrier mobility (∼ 5 cm2V−1s−1).[91] This is not true
in all cases however; despite highly covalent, 3-dimensional anionic structures, the clathrate
compounds generally do not exhibit high carrier mobility, due to disorder on the anionic
sites.[54, 106]
Bridging these well-characterized 3-, 2-, and 0-dimensional structures are the relatively
little-known 1-dimensional, chain-forming Zintl compounds. The Zintl compounds Ca3AlSb3
and Sr3GaSb3 form distinct structures, characterized by infinite chains of corner-sharing AlSb4
or GaSb4 tetrahedra, respectively. While these compounds exhibit similar crystal structures,
Sr3GaSb3 has twice as many number of atoms per unit cell and a significantly higher density
compared to Ca3AlSb3. Their similar but distinct crystal structures, the possibility of dop-
ing and therefore optimized carrier concentrations provide a good model system to relate
structure to thermoelectric properties in this class of compounds.
18
2 Experimental Methods
2.1 Summary
This chapter discusses the synthesis and characterization of the pertinent materials of this
thesis and the synthetic procedures for the different materials are given in detail in the first
part of the chapter. The quaternary copper selenides were synthesized via a classical "shake
and bake" route and mechanical alloying has been employed for the synthesis of the Zintl
phases.
The last section focuses on the different characterization techniques and the instruments
that were used for the chemical characterization as well as the characterization of the trans-
port properties. While some techniques will only be roughly described, a more thorough
description of the transport measurements will be provided.
2.2 Synthesis
The synthesis of inorganic, crystalline solids is generally limited by mass diffusion. The pri-
mary goal during synthesis is thus to reduce the time for diffusion to occur and therefore for
the atoms to react with each other. In general, the diffusion time (τD) is given by:[7]
τD = l2D/DM , (2.1)
where lD is the diffusion length and DM is the mass diffusion coefficient. Since the diffusion
coefficient has an energy activated behavior in a solid, synthesis is often employed at high-
temperatures, which results in relatively large diffusion coefficients. This is sometimes called
the "shake and bake" route, which employes thorough mixing of the elements followed by
high temperature annealing steps to obtain phase pure compounds. One alternative to this
route is the reduction of lD, which can be achieved via mechanical alloying (or high energy
ball-milling).
The final synthetic step is the sintering of powders to produce dense samples that can be
characterized by a variety of techniques. The samples discussed in this work were sintered
via pressure-assisted sintering, or hot pressing (for details see Section 2.2.3).
2.2.1 Quaternary Copper Selenides
Cu2+xZn1−xGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154.[55]
Reproduced with permission of the American Chemical Society Copyright 2012.
19
2 Experimental Methods
Bulk samples of polycrystalline Cu2+xZn1−xGeSe4 with compositions of x = 0, 0.025, 0.05,
0.075, and 0.1 were prepared via solid state reactions using elemental powders of Cu (Alfa
Aesar, 99.999 %), Zn (Sigma Aldrich, 99.995 %), Ge (Chempur, 99.99 %), and Se (Alfa Aesar,
99.999 %). Phase purity of the starting materials was verified by X-ray diffraction and all
synthetic procedures were carried out in a N2 dry box. Annealing was performed in evacuated
quartz ampoules, which were preheated at 1073 K under dynamic vacuum for 5 hours to
ensure dry conditions.
The starting elements were thoroughly ground, sealed in quartz ampoules, heated to 923 K
at 5 K/min, and annealed for 48 hours. The ampoules were cooled at 5 K/min and the ob-
tained materials were ground, sealed in quartz ampoules again, and then reannealed at 1073 K
for 96 hours with the above mentioned heating and cooling rates. It was found that the second
annealing step was necessary to prevent the formation of the binary and ternary compositions.
The quartz ampoules were 10-12 cm in length and 11 mm inner diameter with a maximum
amount of 1.5 g of starting materials within the ampoule. This ampoule geometry was found
to prevent significant loss of selenium at higher temperatures, indicated by red selenium
precipitation present on longer ampoules. The obtained powders were hand ground and con-
solidated into 1-1.5 mm thick, 12 mm diameter disks at 873 K for 5 hours under a pressure of
40 MPa by induction hot pressing in high density graphite dies.[107] The resulting samples
have more than 97 % theoretical density, determined from the geometric densities.
Cu2Zn1−xFexGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2013 - accepted manuscript.[56]
Reproduced with permission of the American Chemical Society Copyright 2013.
Bulk samples of polycrystalline Cu2Zn1−xFexGeSe4 with compositions of x = 0, 0.1, ...,
0.9, 1.0 were prepared via solid state reactions using elemental powders of Cu (Alfa Aesar,
99.999 %), Zn (Sigma Aldrich, 99.995 %), Fe (Alfa Aesar, 99.998 %), Ge (Chempur, 99.99 %),
and Se (Alfa Aesar, 99.999 %). Phase purity of the starting materials was verified by X-ray
diffraction and all synthetic procedures were carried out in a N2 dry box. Annealing was
performed in evacuated quartz ampoules, which were preheated at 1073 K under dynamic
vacuum for 5 hours to ensure dry conditions.
The starting elements were thoroughly ground, sealed in quartz ampoules, and annealed
in a first step for 48 hours at 923 K and 673 K, for x < 0.4 and x ≥ 0.4, respectively. In a
following second step, the harvested powders were reannealed for 96 hours at 1073 K and
873 K, respectively. Heating and cooling rates for all procedures in the horizontal tube fur-
naces were 5 K/min. It was found that the second annealing step was necessary to prevent
the formation of the binary and ternary compositions. The quartz ampoules were 10-12 cm
in length and 11 mm inner diameter with a maximum amount of 1.5 g of starting materials
within the ampoule. This ampoule geometry was found to prevent significant loss of selenium
at higher temperatures, indicated by red selenium precipitation present in longer ampoules.
The obtained powders were hand ground and consolidated into 1-1.5 mm thick, 12 mm diam-
eter disks at 873 K for 5 hours under a pressure of 40 MPa by induction hot pressing in high
density graphite dies.[107] The resulting samples have more than 95 % theoretical density,
20
2.2 Synthesis
determined from the geometric densities.
Cu2+xZn1−x−yFeyGeSe4
Bulk samples of polycrystalline Cu2+xZn1−x−yFeyGeSe4 with compositions of x = 0.025 and y
= 0.975, 0.8, 0.7, and 0.6 were prepared via solid state reactions using elemental powders of Cu
(Alfa Aesar, 99.999 %), Zn (Sigma Aldrich, 99.995 %), Fe (Alfa Aesar, 99.998 %), Ge (Chempur,
99.99 %), and Se (Alfa Aesar, 99.999 %). Phase purity of the starting materials was verified by
X-ray diffraction and all synthetic procedures were carried out in a N2 dry box. Annealing
was performed in evacuated quartz ampoules, which were preheated at 1073 K under dynamic
vacuum for 5 hours to ensure dry conditions.
The starting elements were thoroughly ground, sealed in quartz ampoules, and annealed
in a first step for 48 hours at 673 K. In a following second step, the harvested powders were
reannealed for 96 hours at 873 K. Heating and cooling rates for all procedures in the horizon-
tal tube furnaces were 5 K/min. It was found that the second annealing step was necessary
to prevent the formation of the binary and ternary compositions. The quartz ampoules were
10-12 cm in length and 11 mm inner diameter with a maximum amount of 1.5 g of starting
materials within the ampoule. This ampoule geometry was found to prevent significant loss
of selenium at higher temperatures, indicated by red selenium precipitation present in longer
ampoules. The obtained powders were hand ground and consolidated into 1-1.5 mm thick,
12 mm diameter disks at 873 K for 5 hours under a pressure of 40 MPa by induction hot press-
ing in high density graphite dies.[107] The resulting samples have more than 95 % theoretical
density, determined from the geometric densities.
2.2.2 Zintl-Phases
Ca3Al1−xZnxSb3
This section is an adapted reproduction, from J. Mater. Chem 2012 22, 9826-9830.[96] Repro-
duced with permission of The Royal Society of Chemistry Copyright 2012.
Bulk, polycrystalline Ca3Al1−xZnxSb3 samples with compositions x = 0.00, 0.01, 0.02, and
0.05 were prepared by ball milling followed by hot pressing from elemental Ca (Sigma-
Aldrich, 99.9%, dendritic pieces), Al (Alfa Aesar, 99.9%, shot), Zn (Alfa Aesar, 99.99%, shot),
and Sb (Alfa Aesar, 99.9999%, shot). The elements were slivered into small pieces and loaded
into stainless-steel vials with stainless-steel balls in an Ar dry box. The precursors were milled
for 90min using a SPEX Sample Prep 8000 Series Mixer/Mill. The resulting powder was con-
solidated by induction hot pressing in high density graphite dies.[107] The die was slowly
heated up to 973 K in two hours then held at this temperature at a pressure of 40 MPa for four
hours, and five hours for x = 0.01, respectively. After a stress free two hour cool down, the
resulting ingots were sliced into 1-1.5 mm thick disks, 12 mm in diameter, and characterized.
Sr3Ga1−xZnxSb3
This section is an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-9128.[100]
Reproduced with permission of The Royal Society of Chemistry Copyright 2012.
21
2 Experimental Methods
Bulk, polycrystalline Sr3Ga1−xZnxSb3 samples with compositions x = 0.00, 0.02, 0.05, 0.07,
and 0.1 were prepared by ball milling followed by hot pressing, starting with elemental Sr, Zn,
Sb, and GaSb. A GaSb precursor was synthesized from Ga (99.99 % shot, Sigma-Aldrich), and
Sb (99.9999 %, shot, Alfa Aesar) in an evacuated quartz ampoule which was held at 1000 K
for 12 hours followed by water-quenching. The refined X-ray diffraction data of the GaSb
precursor is shown in the Supporting Information in Chapter 8. The elements (99 % pieces Sr,
99.99 % Zn shot, and 99.9999 % Sb shot from Alfa Aesar) were cut into small pieces and loaded
with the GaSb precursor in stoichiometric amounts into stainless-steel vials, with stainless-
steel balls in an Ar dry box. The mixtures were milled for 90 min using a SPEX Sample Prep
8000 Series Mixer/Mill. The resulting powder was consolidated by induction hot pressing
in high density graphite dies.[107] The die was heated to 873 K in 2 hours and held for 1.5
hours, followed by a 30 min ramp to 973 K, at which point a pressure of 120 MPa was applied.
These conditions were maintained for 2 hours, followed by a 2 hour pressure-free cool down.
The resulting ingots, which were sensitive to air and moisture, were sliced into disks, using a
low-speed diamond saw and non-aqueous lubricant, for characterization. The characterized
specimens were roughly 1-2mm in thickness.
2.2.3 Pressure-Assisted Sintering
A rapid hot press has been employed for the consolidation of the powdered materials. The
custom-built hot press system at Caltech has been developed to rapidly consolidate thermo-
electric materials over a large temperature range (400-2300 K) within an inert atmosphere, this
is referred to as rapid hot pressing. The heat in this system is provided by an induction coil
operated in the RF range applied directly to a graphite die acting as a susceptor. Utilizing the
hydraulic control system during processing, the consolidation of samples can be monitored to
determine the minimum required pressing time.[107] The synthesized powder or ball-milled
elemental powders are loaded into a high density graphite die in an argon glove box.
Before the power is loaded into a die, a grafoil sheet is used to cover the inside of the die,
furthermore a circular piece of grafoil and a graphite anvil (spacer) are inserted into the base
of the die. The settled powder is then compressed by hand using a long graphite plunger, after
which the final grafoil, spacers, and (shorter) plunger are be inserted. In case of the copper
selenides, more spacers have been used in order to obtain three consolidated disks in one
press run. The basic setup of the pressing procedure and a picture taking at Caltech can be
seen in Figure 2.1. After loading the sample into the die and placing the die in the induction
coil, the chamber door is closed, a small load of about 5 MPa, and a vacuum is applied to the
chamber. When an acceptable vacuum has been reached, the desired load is applied to the
die, the chamber is backfilled with Ar and flowed through the chamber, and the induction
heating power supply is turned on and heating begins.
22
2.3 Characterization
Figure 2.1: Schematic of induction hot press setup (left) and picture of hot press taken at
Caltech (right).[107] The powdered sample is loaded into the die and covered with
grafoil on all sides. Pressure is attained with a graphite plunger and the induction
coil is utilized to heat up the die.
2.3 Characterization
2.3.1 Chemical Characterization
X-ray diffraction measurements for the copper selenides were performed on a Siemens D5000
powder diffractometer with a Braun M50 position sensitive detector and CuKα1 radiation [Ge
(220) monochromator] with a step size of 0.0078°. Data were collected over a time of 14 hours.
X-ray diffraction measurements on the Zintl-phases and high temperature X-ray diffraction
were performed on a Philips PANalytical X’Pert Pro with CuKα radiation and a step size of
0.0084◦. The usual time for data collection was two hours. For the high temperature diffrac-
tion, the samples were kept at the target temperatures for 30 min before the measurement
started, in order to ensure equilibrium conditions. Rietveld and Pawley refinements were per-
formed with TOPAS Academic V4.1,[108] applying the fundamental parameter approach, as
implemented in TOPAS, using the crystallographic data from Section 1.4. The instrumental
measurement uncertainty for the determination of the lattice parameters is approximated to
be 0.003 Å.
Scanning electron microscopy (SEM) images and energy-dispersive X-ray spectroscopy
(EDS) of the consolidated materials were taken using a Zeiss 1550 VP SEM in the Quadrant
Back Scattering Detector (QBSD or BSE) and the Secondary Electron Detector (SE2) mode. For
scanning electron microscopy, the samples were embedded in epoxy including graphite as a
conductive filler, and then polished with different grid size to achieve smooth surfaces for the
measurement. The sequential arrangement was: sanding paper with 240 grid, then 400, and
600 followed by 800. Afterwards pastes with polishing particles from 3 µm to 0.3 µm are used.
For the study of grain sizes via fractured surfaces of the samples, the disks were broken prior
to a measurement and directly mounted for SEM.
23
2 Experimental Methods
Optical absorption spectra of the roughened consolidated disks were measured at room
temperature by a Cary 5000 UV-visible-near-IR spectrometer equipped with an integrating
sphere. Absorption coefficients were calculated from the diffuse reflectance using the Kubelka-
Munk function.
Combined thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC)
were performed using a Netzsch STA 449.
2.3.2 Characterization of Transport Properties
Thermal diffusivity was measured using a Netzsch laser flash diffusivity instrument (LFA
457), samples were coated with a thin layer of graphite to maximize the samples’ emissivity
and absorptivity to obtain strong radiation signals in the measurement. The data were an-
alyzed using a Cowan model with pulse correction. Heat capacity (C) was estimated using
the method of Dulong-Petit (C = 3R, kB per atom) and theoretical densities were calculated
from the molar mass and the lattice parameters for each composition obtained from X-ray
diffraction. The Seebeck coefficient was calculated from the slope of the voltage vs. tempera-
ture gradient measurements from Chromel-Nb thermocouples.[109] Electrical resistivity, Hall
coefficient, and carrier concentration were measured using the Van der Pauw technique under
a reversible magnetic field of 2 T for the copper selenides and 1 T for the Zintl phases using
pressure-assisted contacts. A current of 20 mA has been used to obtain Hall data of the copper
selenides and 100 mA for the Zintl phases. All measurements were performed under dynamic
vacuum and on multiple samples for each composition. Shown measurement data represent
both heating and cooling data. The combined uncertainty for all measurements involved in zT
determination is ~20 %. Normal and shear ultrasonic measurements were performed at room
temperature to extract the longitudinal and transverse sound velocities, respectively. Honey
was used as the couplant on a Panametrics NDT 5800 pulser/receiver, which was employed
with a Tektronix TDS 1012 digital oscilloscope. The measurements were performed in Prof.
Bill Johnson’s laboratory at Caltech.
The measurement techniques for the transport properties will be described in the following.
Seebeck coefficient
For a detailed description of the Seebeck coefficient metrology, for different measurement
setups, and methods the reader is referred to the articles by Martin et al.[110] and Iwanaga
et al.[109]. In principle, the measurement of the Seebeck coefficient of a bulk material is
relatively simple. Knowing the temperature difference and the voltage between two points of a
material gives the Seebeck coefficient, see Equation 1.3. There are a few primary requirements
for "good" Seebeck coefficient measurements. First the voltage and temperature must be
measured at the same location and at the same time, second the probes must be in very good
thermal and electrical contact with the specimen, and third the occurring voltage should be
low.[109, 110]
With the measurement setup of Figure 2.2, these requirements are believed to be met.[109]
The disk shaped samples are loaded into the Seebeck system (Figure 2.2 c)) and lateral spring
loading ensures a good thermal contact between the thermocouples and the sample. A piece
24
2.3 Characterization
Figure 2.2: a) Schematic build-up of a thermocouple with a crossed-wire geometry. b)
Schematic of the uniaxial 4-point method for the measurement of the temper-
ature and voltage across the sample. The upper and lower green blocks rep-
resent the heaters and heat sinks, the center yellow block the sample, and the
blue narrow rods represent the thermocouples. c) Schematic setup of the Seebeck
instrument.[109] The heaters are used to create a temperature gradient along the
sample and the thermocouples are simultaneously used to measure the voltage
and temperature.
of grafoil is placed between the sample surface and the thermocouple to prevent chemical
reactions at higher temperatures. The thermocouples (see Figure 2.2 a)), made of Chromel-
Nb wires, are designed to measure the voltages and temperatures at the same location, to
minimize errors.[109]
In this setup, the temperature of the top heater and the bottom heater (Figure 2.2 b)) oscil-
late around one value of temperature, which is used as the average value of the temperature.
The corresponding ∆V are recorded and plotted vs. the corresponding ∆T. The slope of the
voltage response to the temperature gradient yields the Seebeck coefficient. Typical devia-
tions between data sets are within ±5 µVK−1 and an accuracy of 5 %-10 % is estimated for
reproducible Seebeck data.[109]
Hall coefficient and Resistivity
The electrical conductivity and the Hall coefficient are important materials properties in the
field of thermoelectrics, especially since transport properties are strongly dependent on the
charge carrier concentration of a material. In a Hall measurement, the charge carrier con-
centrations can be directly calculated from the Hall coefficient. Through simultaneous mea-
surement of the resistivity and the Hall coefficient, the Hall carrier mobility can be inferred.
Especially the temperature dependence of the Hall mobility can be used to extract information
about the scattering mechanism of the charge carriers.[111]
In the used instrument, the Van der Pauw technique is utilized for the measurement of
the specific resistivity and Hall coefficient. Pressure assisted contacts are used to eliminate
25
2 Experimental Methods
the need for contact glue and paste, for a lower contact area (i.e. lower contact resistance)
and to prevent chemical reactions with the sample. These contacts are placed on the edge
of the sample disk and are all turned by 90 degrees in respect to each other (see Figure 2.3
left). To measure the resistivity, a current is passed between to adjacent contacts, while the
voltage is measured between the other two. The Hall coefficient is obtained by passing a
current diagonal across the sample in a magnetic field perpendicular to the sample plane. For
a detailed description of the Hall measurement, the reader is referred to references [111] and
[112].
While the electrical resistivity can be calculated from the resistance, the Hall coefficient RH
is calculated from the average change in resistance ∆R when changing the field ∆B:[111]
RH = t∆R/∆B , (2.2)
with the sample thickness t. Within the free electron or single parabolic band model, the Hall
carrier concentration nH can be calculated from:
1
nH = , (2.3)eRH
with the elementary charge e. Especially at high temperatures where there is carrier activation
across the band gap, the measured carrier concentrations tend to be overestimated. The Hall
mobility µH is calculated from the Hall coefficient and electrical resistivity ρ:
RH
µH = . (2.4)
ρ
The typical measurement uncertainty is estimated to be 5 %. However, badly placed con-
tacts and too thick samples tend to increase this error significantly. Therefore thin disks are
mandatory for less noise in the Hall data.
Figure 2.3: Schematic of the 4-point setup in Van der Pauw geometry, with pressure assisted
contacts (left) and a picture of the Hall measurement system taken at Caltech
(right).[111]
26
2.3 Characterization
Thermal diffusivity
Thermal diffusivity was measured using the laser flash thermal diffusivity method (see Figure
2.4 a)). In this method the sample is irradiated by a short laser pulse. The temperature rise
of the opposite side of the sample is then monitored using an IR detector and the thermal
diffusivity d is determined from the sample thickness and the time constant of the temperature
rise. The thermal diffusivity is related to the thermal conductivity κ as:[112]
κ = dDC , (2.5)
with the heat capacity C and the density of the sample D. The sample disks and the standard
are spray-coated with graphite on both sides, to minimize the influence of the reflectivity of
the sample surface on the measurement.
Figure 2.4: a) Schematic of the laser flash method. A LASER irradiates the sample via a short
pulse, while the temperature of the opposite side of the sample is monitored by an
IR detector. b) Schematic of the speed of sound measurement. The transducer in-
troduces sound waves into the bulk sample, which travel to one end of the sample
before they are reflected back. Honey is used as a couplant.
The measurement uncertainty of the laser flash diffusivity measurement is 3 %. However,
errors in density and thickness of the samples will contribute to the overall measurement
uncertainty of 5 %.
Speed of sound
Figure 2.4 ba) shows a schematic of the speed of sound measurement setup. A pulser/receiver
introduces sound waves (frequency 1 kHz), longitudinal and transverse, respectively, into the
bulk sample. These propagate to the end of the sample where they are reflected. The response
is recorded via a digital oscilloscope. The speed of sound of the sample can be determined
from the time between the pulse and the response and the thickness of the sample. Errors
in the measurement mainly arise from the error of the thickness and if the two surfaces of
27
2 Experimental Methods
the sample disk are not parallel. These measurements are independent of the grain sizes of a
material since sound waves are long wavelength phonons. However, the quality of the data
generally depends on the quality of the samples and both surface defects and internal defects
reduce the quality of the data. Therefore, the data was collected multiple times to minimize
errors in the analysis.
2.4 Electronic Structure Calculations
Electronic structure calculations have been performed under collaboration by Gregory Pom-
rehn, within the Snyder group at Caltech. These results are shown in this thesis to address
and further understand some of the transport properties presented in this work.
2.4.1 Cu2MI IGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2013 - accepted manuscript.[56]
Copyright 2012 American Chemical Society.
Density Functional calculations were performed with the WIEN2K code [113] based on
the full-potential linearized augmented plane-wave (FP-LAPW) method under the gener-
alized gradient approximation (GGA) as parameterized by Perdew, Burke, and Ernzerhof
(PBE).[114] A plane wave basis cutoff was RmtKmax = 7, in terms of the smallest muffin tin
radius and maximum plane wave vector respectively. Muffin tin radii for Cu, Zn, Fe, Ge and
Se were 2.4, 2,5, 2.4, 2.3 and 2.1 a.u. respectively. Calculations were performed at the theo-
retical ground state lattice parameters as determined by a structural minimization of the unit
cell. Atomic positions were relaxed to converged forces below 0.025 eV/ on each atom. An
anti-ferromagnetic state was used for the initial magnetization of the density. Total energy
convergence was achieved in a self consistent calculation using with a shifted 8x8x4 k-point
mesh, which reduced to 32 symmetrically unique k-points. The modified Becke-Johnson (TB-
mBJ)[115] semi-local exchange potential was employed, which has been shown to improve the
band gap across a wide variety of semiconductors,[116] but may still fall short in these ma-
terials. This method was utilized to produce the density of states and electronic dispersions
shown in this work.
2.4.2 Sr3GaSb3
This section is an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-9128.[100]
Copyright 2012 The Royal Society of Chemistry.
Density Functional calculations were performed with the WIEN2K code [113] based on the
full-potential linearized augmented plane-wave (FP-LAPW) method under the generalized
gradient approximation (GGA) as parameterized by Perdew, Burke, and Ernzerhof (PBE).[114]
A plane wave basis cutoff of RmtKmax = 9 was used, in terms of the smallest muffin tin radius
and maximum plane wave vector respectively. Muffin tin radii were 2.4 a.u. for Sr and 2.5
a.u. for both Ga and Sb. Calculations were performed at the theoretical ground state lattice
parameters as determined by structural minimization of the unit cell. Atomic positions were
28
2.4 Electronic Structure Calculations
relaxed to converged forces below 0.025 eV/Å on each atom. Total energy convergence was
achieved in the self consistent calculations with a shifted 6x4x6 k-point mesh, which reduced
to 36 symmetrically unique k-points. The modified Becke-Johnson (TB-mBJ) [115] semi-local
exchange potential was employed, which has been shown to improve the band gap in sp
bonded semiconductors.[116] This method was used to produce the density of states and
electronic dispersions shown in this work.
29

3 Chemical and Structural Results
3.1 Summary
The purity and composition of the synthesized compounds is discussed in this chapter. Specif-
ically powder X-ray diffraction is utilized to address phase purity. The Cu MI I2 GeSe4 materials
are all found to be phase pure, except for the composition Cu2.1Zn0.9GeSe4, where the impu-
rity phase Cu2−xSe is present with approximately 1-2 wt%, due to a charge imbalance. The
Zintl phases Ca3AlSb3 and Sr3GaSb3 both show impurities of the binary and other ternary
antimonides.
Rietveld and Pawley refinements have been performed on room temperature and high
temperature X-ray diffraction data. While the Zintl phases and Cu2+xZn1−xGeSe4 do not ex-
hibit a dependence of the lattice parameters on the composition, both Cu2Zn1−xFexGeSe4 and
Cu2+xZn1−x−yFeyGeSe4 show a non-Vegard like behavior of the lattice parameter indicating a
restructuring process, occurring upon substitution. Rietveld and occupational analyses of the
compound Cu2+xZn1−xGeSe4 show a cation deficiency on their specific lattice sites, indicative
of an interstitial occupancy of Cu and Zn.
Scanning electron microscopy has been utilized to verify phase purity, investigate grain
boundaries, and microstructures of the materials. For instance micrographs of the solid solu-
tion Cu2+xZn1−xGeSe4 show the impurity phase Cu2−xSe, formed through a phase segrega-
tion, as nano sized particles located at the grain boundaries between the interfaces of the bulk
material.
The chemical and structural information on the compounds, presented in this chapter, are
crucial for the understanding of the electronic and thermal transport properties discussed in
the following chapters, since the structures and microstructures strongly affect the transport
in these materials.
3.2 Cu2+xZn1−xGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154.[55]
Reproduced with permission of the American Chemical Society Copyright 2012.
All samples were checked for phase purity prior to any transport measurements and se-
lected powder diffraction data is shown in Figure 3.1 for each composition. In all compositions
with x < 0.075 all reflections can be indexed to Cu2ZnGeSe4 and no secondary phase is ob-
served. However, the samples with the nominal composition of Cu2.1Zn0.9GeSe4 show an
impurity phase of Cu2−δSe with a weight percentage of 1-2 wt% indicated by Rietveld re-
finement. Complete diffraction data can be seen in the Supporting Information (Chapter 8).
Lattice parameters were refined and no significant change was observed upon doping Zn2+
31
3 Chemical and Structural Results
Figure 3.1: Powder diffraction data a) and b) for Cu2+xZn1−xGeSe4 including profile fit, profile
difference and profile residuals from the corresponding Rietveld refinement. The
intensities are plotted as the square root to show low intensity reflections as well.
The inset in b) shows extra reflections indexed to Cu2−δSe. Absorption spectrum
c) of Cu2ZnGeSe4 with the band gap extrapolation indicated (Eg ∼ 1.4 eV).
with Cu+ due to the relatively similar ionic radii of the species (see Figure 3.3 left). The refined
parameters are given in Table 3.1.
An occupational analysis of the specific lattice sites shows occupancies of Cu+ and Zn2+
below 100 % (see Figure 3.4 left). Back calculating the residual electron densities via Rietveld
refinement shows a significant amount of the missing electron density on interstitial sites in
this material (Figure 3.4 right). However, a detailed analysis of the interstitial occupancy
cannot be performed via standard laboratory X-ray diffraction. Due to the same number of
valence electrons of Cu+ and Zn2+, these cations exhibit the same scattering form factors
for X-rays and can therefore not be distinguished via X-ray diffraction. In addition to the
interstitial occupancy, anti site disorder of Cu and Zn cannot be ruled out, and it is therefore
necessary to perform neutron diffraction and possible a pair distribution function analysis to
further elucidate the underlying nature of the structure in this class of materials.
Combined TGA/DSC analysis of the material from ambient temperature to 1100 K revealed
a slightly exothermic effect around 450 K corresponding to a phase transition (Figure 3.2),
which has not been reported previously.[66, 65] To further investigate this phase transforma-
tion, X-ray diffraction has been performed on the consolidated disk at different temperatures
(see Figure 3.3 right). The lattice parameters of the sample with the composition Cu2ZnGeSe4
(Table 3.2) increase linearly with temperature, which can be explained through thermal ex-
32
3.2 Cu2+xZn1−xGeSe4
Table 3.1: Lattice parameters a, c, and occupation of Zn and Cu, and profile residuals Rwp for
Cu2+xZn1−xGeSe4, from the corresponding Rietveld refinement.
x a / Å c / Å occ (Zn) occ (Cu) Rwp
0.0 5.6128(2) 11.0480(2) 0.83 0.89 6.6
0.025 5.6084(3) 11.0397(6) 0.78 0.86 6.7
0.05 5.6139(7) 11.0470(1) 0.84 0.89 6.6
0.075 5.6130(1) 11.0489(5) 0.86 0.90 6.8
0.1 5.6096(4) 11.0542(2) 0.84 0.89 6.8
Figure 3.2: TGA/DSC data for Cu2ZnGeSe4, with the slightly exothermic effect (black circle)
indicating a phase transition.
pansion of the unit cell. Therefore no structural phase transformation can be detected via
standard laboratory X-ray diffraction. At present, the nature of this phase transformation
remains unclear and a detailed investigation and analysis is necessary.
At room temperature the optical data in Figure 3.1 c) plotted as (αAhν)2 against photon
energy, where αA is the absorption coefficient, h the Planck constant, and ν the wave number,
gives an estimated optical band gap Eg of 1.4 eV, which is consistent with the reported data of
this class of wide band gap material.[66]
Scanning electron microscopy of polished and fractured surfaces showed very high densi-
ties of the samples, in agreement with the high density values measured after hot pressing.
Selected SEM micrographs of polished surfaces for different doping contents are shown in
Figure 3.5. SEM micrographs are shown in backscattered (QBSD) and secondary electron
mode (SE2) to distinguish between impurities, voids and the residue of polishing particles.
Energy-dispersive X-ray spectroscopy was utilized to verify the stoichiometry.
Contrasts in the micrographs are due to different orientations of the grains of the com-
pound Cu2+xZn1−xGeSe4 and grain sizes are between 10-20 µm. Scanning electron microscopy
at higher magnifications reveal impurity phases present in the material that were undetected
by X-ray diffraction. The impurity phase has been attributed to a segregation triggered by
higher contents of Cu. Samples with the nominal composition of Cu2.05Zn0.95GeSe4 show the
33
3 Chemical and Structural Results
Figure 3.3: Left: Lattice parameter of Cu2+xZn1−xGeSe4 at room temperature with composi-
tion. No significant change is observed upon doping Zn2+ with Cu+ due to the rel-
atively similar ionic radii of the species. Right: Lattice parameters of Cu2ZnGeSe4
with temperature. A linear increase with temperature is apparent, indicative of
thermal expansion of the unit cell. No structural phase transformation is visible
with laboratory X-ray diffraction.
two impurity phases CuSe and Cu2−δSe as white and black spots, respectively. However,
higher Cu contents mainly lead to the formation of Cu2−δSe for x = 0.075 and x = 0.1, which
is in accordance with the X-ray diffraction data for x = 0.1 (Figure 3.1). This phase segregation
is possibly a result of a charge imbalance in the material. Substitution of Zn2+ with Cu+ leads
to the formation of holes, or Se− (for localized charges). For higher doping levels the excess
of Cu leads to a phase segregation to form the energetically more stable species Se2− in the
secondary phase. It is possible that above a certain amount of excess Cu, all the extra Cu
segregates in the form of Cu2−δSe. The amount of the impurity phase should then be propor-
tional to the amount of extra Cu. These impurities have grain sizes of 250 nm and smaller and
are mainly located at the grain boundaries.
The Rietveld refinements of powder samples prior to the hot pressing procedure and data
taken from consolidated samples do not show any significant texture in this material. We
therefore believe that disks of this polycrystalline material are suitable for a thermoelectric
characterization.
Table 3.2: Lattice parameters a, c, and profile residuals Rwp for Cu2ZnGeSe4 with temperature,
from the corresponding Pawley refinement.
T / ◦C a / Å c / Å Rwp
25 5.6128(2) 11.0480(2) 6.6
100 5.6176(2) 11.0523(5) 5.9
150 5.6213(2) 11.0565(5) 5.6
180 5.6235(1) 11.0600(4) 5.4
200 5.6251(2) 11.0635(7) 12.3
250 5.6281(2) 11.0737(6) 11.7
34
3.3 Cu2Zn1−xFexGeSe4
Figure 3.4: Left: Occupations of Cu+ and Zn2+ on their specific lattice sites in Cu2ZnGeSe4.
Both elements show significantly lower occupations for the expected occupation of
100 %. Right: Residual electron density from the corresponding Rietveld refine-
ment showing the missing Cu and Zn electron density on interstitial sites.
Significant weight loss does not occur until 973 K, while purged with Argon, however,
measurements in dynamic vacuum lead to a significant evaporation of selenium above 723 K.
The influence of different maximum temperatures on the consolidated disks can be seen in
the Supporting Information in Chapter 8. Therefore, all measurements were performed up to
673 K to ensure thermal stability of the samples.
Table 3.3: Lattice parameters a, c, V, c/a-ratio, and profile residuals Rwp for
Cu2Zn1−xFexGeSe4, from the corresponding Pawley refinement.
x a / Å c / Å V / Å3 c / a Rwp
0.0 5.6121(2) 11.0480(2) 347.96 1.969 6.6
0.1 5.6080(3) 11.0536(2) 347.64 1.971 4.5
0.2 5.6036(2) 11.0661(1) 347.48 1.975 4.4
0.3 5.6079(9) 11.0935(3) 348.88 1.978 4.2
0.4 5.6022(6) 11.1046(3) 348.52 1.982 4.3
0.5 5.5879(6) 11.1025(1) 346.68 1.987 3.8
0.6 5.5901(5) 11.1122(1) 347.25 1.988 3.9
0.7 5.5827(2) 11.1176(2) 346.50 1.991 3.8
0.8 5.5918(2) 11.1076(1) 347.32 1.986 3.6
0.9 5.5956(0) 11.0924(8) 346.84 1.980 4.0
1.0 5.6014(6) 11.0703(8) 347.35 1.976 5.7
3.3 Cu2Zn1−xFexGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2013 - accepted manuscript.[56]
Reproduced with permission of the American Chemical Society Copyright 2013.
All samples were checked for phase purity prior to any transport measurements and the
refined powder diffraction data is shown in the Supporting Information (Chapter 8) for each
35
3 Chemical and Structural Results
Figure 3.5: Scanning electron micrographs of polished surfaces for different doping contents x
in Cu2+xZn1−xGeSe4 in backscattered (QBSD) and secondary electron mode (SE2).
White spots (blue circle) and black spots (red circle) in samples with x = 0.05 reveal
minor impurities of CuSe and Cu2−δSe, respectively. Doping contents of x = 0.075
and x = 0.1 mainly show impurities of Cu2−δSe with sizes of 250 nm and smaller,
mainly located at the grain boundaries.
composition. In all compositions all reflections can be indexed to Cu2Zn1−xFexGeSe4 and no
secondary phases are observed. Refined lattice parameters are shown in Table 3.3. Powder
X-ray diffraction data of samples prior to the hot pressing procedure and data taken from
consolidated samples do not show any significant texture in this material. Therefore, prop-
erties measured on disks of this polycrystalline material are expected to be isotropic within
expected experimental uncertainty and represent a scalar average of the tensor properties.
Scanning electron micrographs for all compositions are shown in the Supporting Information
(Chapter 8). Scanning electron microscopy confirms phase purity and similar microstructures
with grain sizes between 10-20 µm throughout the whole composition range.
Pawley refinements of the powder X-ray diffraction data reveal a change in lattice parame-
ters upon substitution of Zn with Fe as seen in Figure 3.6. Experimental c/a-ratios (see Figure
3.7) exhibit a non linear trend, showing an elongation of the c-axis with increasing Fe content
until a maximum is reached at 70 % Fe content, where this trend becomes reversed. This non-
Vegard like behavior has been observed by Caldera et al., however they have assumed it to
be not significant giving experimental uncertainty.[84] This discontinuity of the lattice param-
eters has also previously been observed in the solid solution Cu2Fe1−xZnxSnS4.[80, 83, 117]
A phase segregation due to the solubility limit being reached has been ruled out, since no
splitting of reflections at higher diffraction angles is observed (see Figure 3.6). Furthermore
36
3.3 Cu2Zn1−xFexGeSe4
Figure 3.6: Reflections of selected Miller indices (040) and (008), showing the change in lattice
parameters upon substitution. The reflections of the Miller indices do not cross
over as expected for Vegard’s law.
57Fe Mössbauer spectroscopy on this class of compounds shows the existence of Fe2+ only,
therefore ruling out the influence of Fe3+ on the crystal structure.[118]
Neutron diffraction studies by Schorr et al.[79, 81] on the chemically and structurally
very similar solid solution Cu2Fe1−xZnxSnS4 resolved this behavior of the lattice parameters
and the c/a-ratio upon substitution, which can be described as a 3-stage cation restructur-
ing process in the crossover from the stannite (Cu2ZnGeSe4, space group I4̄2m) to kesterite
(Cu2FeGeSe4, space group I4̄) structure type (see Figure 3.8).[79, 81] Upon substitution of Zn,
Fe starts to replace Zn on the 2a-site (0,0,0) as expected (Figure 3.8 b)). In the range of 0.35
< x < 0.7, which is the main restructuring region, Cu depletes the 4d-site (0, 1 , 12 4 ), occupying
the 2a-site and forcing Zn on the 4d-site, while Fe holds its position (2a).[79, 81] At a substi-
tution level of x = 0.7, the cations Fe2+, Zn2+, and Cu+ are equally distributed on the 2a-site
(Figure 3.8 c) ) and Cu and Zn on position 4d, exhibiting the highest c/a-ratio and highest
local anisotropic structural disorder of this solid solution. With further decreasing Zn content,
Cu starts to dominate position 2a and Fe and Cu occupy position 4d in Cu2FeGeSe4 (Fig-
ure 3.8 d)).[79, 81] The differences in bonding interaction between the cations and the anions
lead to different bond lengths and bond angles, resulting in the observed trend in the lattice
parameters.[79]
The electronic dispersion along lines of high symmetry from the DFT calculations are
shown in Figure 3.9 for Cu2ZnGeSe4 in the stannite structure and Cu2FeGeSe4 in the kesterite
structure. Both materials exhibit direct band gaps at the Brillouin Zone center. The stannite
structured Cu2ZnGeSe4 exhibits a degeneracy of two at the valence band maximum. The two
bands split apart away from the Gamma point and each exhibit parabolic behavior with aver-
aged effective mass of 0.81 me and 0.23 me, respectively. The top valence band of Cu2FeGeSe4
37
3 Chemical and Structural Results
Figure 3.7: Lattice parameter ratio c/a with composition x of Cu2Zn1−xFexGeSe4, from a Paw-
ley refinement. The broken line indicates the expected behavior following Vegard’s
law. A 3-stage cation restructuring process of Cu+, Fe2+, and Zn2+ results in the
trend of the lattice parameters.
a) b) c) d) 
Figure 3.8: 3-stage cation restructuring process from the stannite (Cu2ZnGeSe4, a) ) to kesterite
(Cu2FeGeSe4, d) ) structure type. Cu atoms are yellow, Ge atoms gray, Zn atoms
are blue, Fe atoms green, and Se atoms are red. The sizes of the atoms in this
ball-and-stick model are arbitrary and have been chosen for a better structural
representation. With increasing Fe content, Fe starts to replace Zn (a). Then Cu
depletes it’s original site and occupies the Zn site, while forcing Zn on the Cu site,
which results in the kesterite structure (d) type with further decreasing Zn content.
38
3.3 Cu2Zn1−xFexGeSe4
0.5
Cu ZnGeSe
2 4
0
−0.5
0.5
Cu FeGeSe
2 4
0
−0.5
G X M G Z R A Z G
Figure 3.9: Left: Calculated band structures of stannite-type Cu2ZnGeSe4 (top) and kesterite-
type Cu2FeGeSe4 (bottom). Note the heavy band, consisting mainly of Fe d-states,
in the valence band contributing to the transport at higher carrier concentrations.
Right: Fermi surface of kesterite - Cu2FeGeSe4 in the heavy band along kx-ky. The
Gamma centred pocket shows non-parabolic character deeper in the valence band,
while the four degenerate pockets represent the heavy valence band.
39
E − E  (eV)
V E − E  (eV)V
3 Chemical and Structural Results
in the kesterite structure is a single band with parabolic band mass of 0.67 me. Around 82 meV
below the valence band maximum there is another band with predominantly Fe d-state char-
acter. This band has a much heavier effective mass of 1.91 me and a symmetric degeneracy,
NV , of four. It is expected that this band would contribute to electronic transport due to its
proximity to the valence band maximum.
3.4 Cu2+xZn1−x−yFeyGeSe4
For all compositions x was set to 0.025 and only the value of y has been altered through-
out the series of solid solutions to obtain the material Cu2.025Zn0.975−yFeyGeSe4. All samples
were checked for phase purity prior to any transport measurements and the refined powder
diffraction data is shown in the Supporting Information (Chapter 8) for each composition. In
all compositions all reflections can be indexed to Cu2.025Zn0.975−yFeyGeSe4 and no secondary
phases are observed. Refined lattice parameters are shown in Table 3.4.
Pawley refinements of the powder X-ray diffraction data reveal a change in lattice pa-
rameters upon substitution of Zn with Fe as seen in Section 3.3. Experimental c/a-ratios
(see Figure 3.10) exhibit a non linear trend, consistent with the trend seen and explained for
Cu2Zn1−xFexGeSe4, which has been included for comparison.
3.5 Ca3AlSb3
This section is an adapted reproduction, from J. Mater. Chem 2012 22, 9826-9830.[96] Repro-
duced with permission of The Royal Society of Chemistry Copyright 2012.
Following ball milling and hot pressing, Ca3Al1−xZnxSb3 samples with x = 0.00, 0.01, 0.02,
and 0.05 have geometric densities of at least 97% of the theoretical densities. As reported in
Ref. [95], X-ray diffraction (Figure 3.11) and scanning electron microscopy reveal secondary
phases of Ca5Al2Sb6 and Ca14AlSb11 with 5-10 wt% and 2-5 wt%, respectively. SEM anal-
yses of fracture surfaces reveals small grains (< 1 µm in diameter) and porosity (0.1 µm in
diameter) at the grain boundaries. However, the sample with the nominal composition of
Ca3Al0.99Zn0.01Sb3 contains grains that are at least an order of magnitude larger, as shown
with selected SEM micrographs in Figure 3.11. The larger grain sizes is possibly due to the
longer pressing time for this sample. Complete XRD and SEM data can be found in the Sup-
porting Information in Chapter 8. The relatively small grains found in all samples in this
Table 3.4: Lattice parameters a, c, c/a-ratio, and profile residuals Rwp for
Cu2.025Zn0.975−yFeyGeSe4, from the corresponding Pawley refinement.
y a / Å c / Å c / a Rwp
0.975 5.5940(2) 11.0755(1) 1.979 3.8
0.8 5.5937(2) 11.1089(1) 1.986 4.0
0.7 5.5846(1 11.1186(1) 1.990 4.1
0.6 5.5875(1) 11.1084(1) 1.988 4.5
40
3.6 Sr3GaSb3
Figure 3.10: Lattice parameter ratio c/a with composition y of Cu2.025Zn1−yFeyGeSe4 (red data
points), from a Pawley refinement. The broken line indicates the expected behav-
ior following Vegard’s law. For comparison the data of Cu2Zn1−xFexGeSe4 has
been included (black circles) and the x-axis has been labeled in respect to this
solid solution. A 3-stage cation restructuring process of Cu+, Fe2+, and Zn2+
results in the trend of the lattice parameters.
study are in accordance with the broad profiles of the reflections in Figure 3.11 and therefore
no significant change of the lattice parameters upon substitution with Zn can be observed.
3.6 Sr3GaSb3
This section is an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-9128.[100]
Reproduced with permission of The Royal Society of Chemistry Copyright 2012.
Following ball milling and hot pressing, the polycrystalline Sr3Ga1−xZnxSb3 (x = 0, 0.02,
0.05, and 0.07) samples have geometric densities of ∼ 98% of the theoretical density. SEM
analysis using backscattered electrons reveals small, light-colored precipitates (see upper left
panel of Figure 3.12), identified by EDS as the Zintl phase Sr2Sb3. This phase appears to
comprise approximately 1-3 vol% of each sample. X-ray diffraction confirms that Sr3GaSb3
and Sr2Sb3 were the only phases present in samples with x = 0.00 - 0.07 (shown in Figure
3.12). The inset shows a magnified view in which the most prominent Sr2Sb3 reflections are
marked with asterisks. The fraction of this phase was comparable in each of the samples with
x = 0.00 - 0.07. In the sample with nominal composition Sr3Ga0.9Zn0.1Sb3, additional phases
were present, including SrZn2Sb2, indicating that the solubility limit of Zn in Sr3GaSb3 has
been exceeded. The transport properties of this composition has therefore not been included
in this thesis. In all samples, the grain size after hot pressing was extremely small, estimated
from fracture surfaces to be in the sub-micrometer range (upper right panel in Figure 3.12),
in accordance with the broad profiles of the XRD reflections. No obvious texturing was seen
in the diffraction patterns. Complete XRD and SEM data can be found in the Supporting
41
3 Chemical and Structural Results
Figure 3.11: Left: X-ray diffraction data for Ca3Al1−xZnxSb3 including profile fit, profile differ-
ence, and profile residuals from the corresponding Pawley refinement including
the secondary phases of Ca5Al2Sb6 and Ca14AlSb11. The inset shows the reflec-
tions indexed to the impurity phases, observed in all samples. Right: Selected
SEM micrographs illustrating the different grain sizes of the samples. Smaller
grains (left) for the compositions with x = 0.00, 0.02, and 0.05 and larger grain
sizes for x = 0.01 (right). The scale bar is representative for both micrographs.
Information in Chapter 8.
Sr3GaSb3 forms a valence precise crystal structure that can be described within the Zintl
formalism as follows: The highly electropositive Sr (3 Sr2+) atoms donate their valence elec-
trons to the anionic chains. Two of the Sb atoms making up the chains have only one covalent
bond, leading to a valence state of -2, while one Sb atom is shared between two tetrahedra,
leading to two covalent bonds and a valence state of -1. Assigning a valence state of -1 to
the four-bonded Ga leads to overall charge balance and the formation of the covalent anionic
chains.
The basic features of this simple bonding description are reflected in the calculated den-
sity of states, shown in Figure 3.13 a). The prominent Ga and Sb electronic states deep in
the valence band (-4 to -6 eV) and at the conduction band minimum likely correspond to
Ga-Sb bonding and anti-bonding interactions, respectively. The valence band maximum is
dominated by Sb states, likely arising from non-bonding Sb lone-pairs. In contrast, the con-
duction band is formed primarily by Ga-Sb anti-bonding states and by Sr states at higher
energies, consistent with the assumption that Sr donates its valence electrons to form the an-
ionic substructure. A band gap of ∼ 0.75 eV is predicted, qualitatively consistent with the
experimentally observed behavior described in Chapter 4.
The electronic band structure of Sr3GaSb3 is shown in Figure 3.13 b) in the k-space direc-
tion parallel to the GaSb4 chains (Γ-A[101]), and in the two perpendicular directions (Γ-Z and
Γ-A[1̄01]). The k-space points Γ, A and Z are labeled in the Brillouin zone shown in the inset.
The band structure is characterized by an indirect band gap (∼0.75 eV) with the valence band
42
3.6 Sr3GaSb3
Figure 3.12: Left: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile differ-
ence, and profile residuals from the corresponding Pawley refinement including
the secondary phase Sr2Sb3. The inset shows the reflections indexed to the impu-
rity phase marked by asterisks. Right: back-scattered electron image of a polished
Sr3GaSb3 sample reveals the secondary phase Sr2Sb3 as white specks, while sec-
ondary electron imaging of a fracture surface shows the very small grain size
(< 1 µm).
maximum at Γ and the conduction band minimum between Γ and Z. The upper and lower
blue lines correspond to carrier concentrations of 1019 and 1020 holes/cm3, respectively. At
energies very close to the valence band maximum (0 to -0.1 eV), the dispersion is anisotropic,
with the effective mass along the chains (m∗[101] = 0.18 me) smaller than the mass in the per-
pendicular directions (m∗1̄01 = 0.57 me and m
∗
[010] = 0.66 me). In contrast, the two additional[ ]
bands with maxima at ∼ -0.1 eV are nearly isotropic. When electrons are scattered primarily
by acoustic phonons, as is the case in most known thermoelectric materials, the improvement
in carrier mobility conferred by a light band mass outweighs the detrimental effect that low
m∗ has on α. In this case, the thermoelectric quality factor is given by B ∝ Nvm∗κ , where m
∗
i isi L
the mass of a single hole pocket along the conduction direction.[18, 25] This suggests that the
light band mass in Sr3GaSb3, particularly in the direction of the chain substructures (m∗[101]),
may be advantageous.
The number of bands contributing to transport (Nv) at a given carrier concentration influ-
ences a material’s figure of merit, as illustrated by the thermoelectric quality factor. If Nv is
high, it is possible to simultaneously have light bands (small m∗i ) and large density of states ef-
fective mass. In Sr3GaSb3, only one band contributes to transport at energies very close to the
valence band maximum. In heavily doped Sr3GaSb3 (n > 1020 holes/cm3), the contribution
from the two bands with maxima at about -0.1 eV could potentially lead to Nv = 3.
43
3 Chemical and Structural Results
a) 
cumulative densities 
Sr 
Ga 
Sb 
b) 
Figure 3.13: a) Density of states and b) band structure of Sr3GaSb3 reveal an indirect band gap
of ∼ 0.75 eV. The upper and lower blue lines correspond to carrier concentrations
of 1019 and 1020 holes/cm3, respectively. Inset: Brillouin zone of Sr3GaSb3 with
selected k-space directions labeled.
44
4 Electrical Transport
4.1 Summary
This chapter starts with a basic introduction to electrical transport theory. The single parabolic
band approximation will be introduced and with it the chemical potential, the density of
states and effective mass of charge carriers within the parabolic band approximation. The
different scattering types for carriers and the influence on the transport will be reviewed. Then
the different solutions to the Boltzmann transport equations will be given for the electronic
transport properties, which will be the foundation for the modeling of the transport in this
thesis.
The substitution of Zn2+ with Cu+ introduces holes as charge carriers in the system leading
to a lower thermopower and higher electrical conductivities. The resulting carrier concentra-
tions deviate significantly from the charge carrier concentrations expected from simple charge
counting, which can be attributed to the influence of the secondary phases in this system.
Furthermore, a previously unknown insulator-to-metal like phase transformation is observed
around 450 K in this solid solution.
In the solid solutions of Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4 a partially un-
filled, heavy Fe d-band contributes to the transport, supported with band structure calcu-
lations, which leads to higher carrier concentrations in a combination of intrinsic defects.
Seebeck and Hall data suggest a significant change of the band effective masses, depending
on the Fe content, which results in higher Seebeck coefficients.
Both, Ca3AlSb3 and Sr3GaSb3, exhibit non-degenerate intrinsic p-type semiconducting be-
havior showing an influence of the Zn content on the transport properties, due to higher
carrier concentrations in the solid solutions. While both materials exhibit an exponential be-
havior of the temperature dependent Hall mobility due to barriers at the grain boundaries,
e.g. oxide layers or secondary phases, an influence of the grain size on the mobility and the
Hall carrier concentration can be found in the solid solution Ca3Al1−xZnxSb3, which proper-
ties have been compared to recently published data of the sodium doped analogue.[95] Larger
grain sizes lead to higher carrier mobilities due to a reduced scattering of the charge carriers
on the grain boundaries.
4.2 Electronic Transport Theory
The following section is thought to give a rough overview over the basic concepts of the
electronic transport in semiconductors. The concept of the single parabolic band will be
highlighted and a short overview on the different scattering mechanisms will be provided. At
last the common solutions of the Boltzmann transport equations, which have been employed
45
4 Electrical Transport
for the modeling of the important materials properties discussed in this chapter, will be given
here. The following section is in parts an adapted reproduction of the Diploma thesis from
Wolfgang Zeier, 2010.[85]
4.2.1 Electronic Transport in a Single Parabolic Band
The perhaps simplest model for the electrical conductivity of solids is the Drude model, where
the electrons are described as a gas of electrons moving through the lattice of a solid. In this
model, the electrical conductivity is given by a concentration of charge carriers n, which move
into the direction of an applied electric field E with a drift velocity vd. The drift velocity can
be written as:[7]
eτE
vd = ∗ , (4.1)m
with the carried charge e, the effective carrier mass m∗, and the scattering relaxation time
τ. The electrical current density Je (charge per area per time) can be expressed through:
Je = nevd , (4.2)
using Ohm’s law for the electrical current density Je (Equation 1.1):[7]
Je = σE ,
leads to the electrical conductivity σ:[7]
ne2τ
σ = ∗ , (4.3)m
showing the dependence of the electrical conductivity on the number of charged carriers,
their effective mass, and the importance of the scattering relaxation time. The term eτ/m∗ is
defined as the mobility of the charge carriers µ, leading to a familiar expression for σ with
σ = enµ. However, in general, only the electrons near the electrochemical potential (within a
few kBT) and not all carriers in a band contribute significantly to the transport.
Solutions to the Boltzmann transport equation (BTE) are usually employed for the de-
scription and analysis of electronic transport. The electron distribution is described by the
Fermi-Dirac distribution function f , which is an approximation for the non-equilibrium elec-
tron distribution, with:
f = (1 + exp[e− η])−1 . (4.4)
Here e stands for the reduced energy (e = E/kBT) and η for the reduced electrochemical
potential. The Fermi energy eF represents the highest occupied energy level at 0 K and the
electrochemical potential is the energy at which f = 12 for all temperatures above 0 K.[13]
Single Parabolic Band Model
One major assumption for the solutions of BTE is that the energies of the carriers can be
described by an isotropic, parabolic dispersion relationship which is commonly referred to as
46
4.2 Electronic Transport Theory
a parabolic band.
Band theory describes the motion of free, delocalized electrons as waves in a periodic
lattice potential. Superposition of the Bloch functions lead to the band structure of a solid.
In the chemical picture, the band structures and band gaps arise from the energy difference
between atomic orbitals, or between bonding and antibonding interactions.[119] The energy
of a band is a function of the wave vector~k of the electrons and the wave vector is related to
the momentum p of an electron via:[119]
~k = h̄p , (4.5)
with h̄ = h2π . Plots of the energy vs. wave vector E(~k) are therefore useful for the discussion
of the dynamics of electrons in solids.
Electrical conductivity arises when a band is partially filled and therefore the motion of
electrons from an occupied to an unoccupied state becomes possible. This is how band theory
differentiates between electrical insulators and conductors (at 0 K).[119]
The energy of an electron in the conduction band is given via the dispersion relationship,
with the minimum energy of an electron in the band V0:[119]
h̄2~k2
E(~k) = V0 + ∗ . (4.6)2m
This equation shows the origin of the term "parabolic band" due to the square of the wave
vector. The density of states N(E) for a parabolic dispersion relation under the assumption of
V0 = 0 is given by:
(2m∗)3/2√
N(E) = E , (4.7)
2π2h̄3
and increases as the square root of the energy. This model is commonly referred to as the
single parabolic band model (SPB), if only one band contributes to the transport.
Here the electrons are treated within the free electron model, while the concept of the
effective mass m∗ is used to account for the electrons interaction with the lattice.[119] Electrons
with a small effective mass behave as mobile particles. While "heavy" electrons have lower
mobilities and are therefore less itinerant and more localized on the atoms.[119] The effective
mass of a single parabolic band is defined by the relation:[119]
1 1 ∂2E(~k)
=
m∗ h̄2
. (4.8)
∂~k2
This shows that the effective mass for electrons depends on the curvature of E(~k) at any
point in a band.[9, 119] In other words the effective mass of an electron increases with de-
creasing dispersion of the band, for the same number of states. In a wide band, the electron
behaves as a free, nearly completely delocalized particle. In a narrow band however, a more
localized description of the electrons is necessary. The concept of an effective mass can be
explained with the motion of the electrons in the band. In a wide band the energy states of
the electrons are distributed over a wide range of energies and fully delocalized. In a narrow
47
4 Electrical Transport
band, the electrons interact with the lattice and each other. They polarize their surroundings
and a polarization cloud moves through the lattice along with the electrons. The electrons
seem to have a bigger mass compared to a free electron.[119]
The bandwidth of a band depends on the bonding interactions in the solid. If the overlap
of the neighboring orbitals is strong, the atomic orbitals tend to lose their identity and a wide
band is formed.[120, 121] This is the case in materials with small electronegativity differences
(covalent bonds), while ionic compounds have narrow bands due to a poor overlap of the
atomic orbitals.[119, 121] Therefore ionic compounds usually exhibit a very low electrical
conductivity, compared to metals, due to the high effective mass of the charge carriers.
Relaxation Times and Carrier Scattering
The description of transport processes using statistical mechanics, expresses the movement of
all particles by the change rate of their overall distribution function which, in the steady state,
can be expressed as:[122] ( ) ( )
∂ f ∂ f
+ = 0 , (4.9)
∂t f ield ∂t collision
where the first term represents the disturbance of the distribution function by external fields
and the second term the effect of collisions or scattering. The third major assumption in BTE
is the assumption, that the extent of deviation of the distribution function from equilibrium is
linearly to the rate of scattering:[122(] )
∂ f f − f0
= . (4.10)
∂t collision τ
In other words, the relaxation of the distribution function back to equilibrium can be described
by the scattering relaxation time τ.[13]
The electrical transport in a material is associated with various scattering mechanisms such
as scattering on defects, impurities, electrons, holes, and the lattice.[13] The relaxation time
associated with the various scattering mechanisms (s) is given by:[122]
1 s 1
=
τ ∑ , (4.11)s=1 τs
with the relaxation time of the scattering mechanisms:[13, 122]
τs = τ0,se
λ−1/2 , (4.12)
where e is the reduced carrier energy (e = E/kBT), kB the Boltzmann constant, and λ a
variable, which relates to the energy dependence of the carrier relaxation time.[13] The energy
dependence of τ is important for the calculations of the Seebeck coefficient as discussed below
and therefore a rough description of the different scattering mechanisms is important.
In the case of acoustic phonon scattering, lattice vibrations give a relaxation time with λ =
0 and τacoustic ∝ e−0.5. Acoustic phonon scattering mainly occurs at high temperatures where
48
4.2 Electronic Transport Theory
high energy electrons are present and therefore a greater number of states being available for
the electrons to be scattered into. For highly degenerate electrons the mobility µ decays as T−1,
which is the classic metallic behavior at higher T, while µ decays as T−1.5 in a nondegenerate
material.[13, 122]
Ionized impurity scattering due to the Coulombic field of the impurities can be described
via λ = 2. This scattering mechanism is limited to the lower temperature regime and the
mobility increases with temperature, because higher energy electrons have higher velocities
and will be less influenced by local disruptions of the Coulombic field.[122]
Energy independent scattering (λ = 0.5) occurs in the intermediate temperature regimes
and can be explained as a balance between ionized impurity scattering and acoustic phonon
scattering.[13, 122]
4.2.2 Electronic Transport Equations within the Single Parabolic Band
Approximation
This single parabolic band approximation is the basis of the transport equations, which will
be discussed in the following. This section is thought to give an overview over the transport
equations, which have been used for modeling the thermoelectric transport of the pertinent
materials. More information and the derivations can be found in the literature, e.g. Fistful’s
"Heavily Doped Semiconductors"[122] and the Ph.D Thesis by Andrew F. May, 2010[13].
Carrier Concentration
The carrier concentration of a band is obtained through the integration over all occupied states
in the band, in other words an integral over the density of states N(E):
∫ ∞
n = N(E) f (E)dE , (4.13)
0
with the assumption of a minimum band energy of zero (V0 = 0 ). Substituting for N(E)
(Equation 4.7) and f (E) (Equation 4.4) leads to:
( ∗ )2m kBT 3/2 ∫ ∞ e1/2den = 4π 2 . (4.14)h 0 1 + exp[e− η]
Defining the Fermi integrals as:
∫ ∞ ejde
Fj(η) = − , (4.15)0 1 + exp(e η)
the carrier concentration can be written (as: )
2m∗k T 3/2
n = 4 Bπ F (η) . (4.16)
h2 j
The assumption of acoustic phonon scattering, with the scattering parameter λ = 0 relating to
1
the energy dependence of the carrier relaxation time τ, with τ=τ λ−0e 2 ,[10] leads to F1/2(η).
49
4 Electrical Transport
Now, carrier concentrations can be calculated using Equations 4.15 and 4.16 and can be
compared to measured Hall carrier concentrations. The reduced chemical potential η can be
calculated from the measured Seebeck coefficients using Equation 4.15 and 4.20. However,
scattering effects can cause a deviation between n and nH, which has to be accounted for
(with n = rHnH). This can be achieved with the Hall factor rH which is given by:
3 (1/2 + 2λ)F
r = F (η) 2λ−1/2
(η)
H 1/2 , (4.17)2 (1 + λ)2F2λ(η)
which leads to the expression for nH: ( )
2m∗k T 3/2 F
n = 4 Bπ 1/2
(η)
2 . (4.18)h rH
Seebeck coefficient
The Seebeck coefficient within the sing(le∫parabolic band de)scription is given by:[13, 21, 122]∞ 3
kB 20 τ∫e (e− η) ∂ fα = ∂e dee ∞ 3 ∂ f , (4.19)2
0 τe ∂e de
which leads to: ( )
kB (2 + λ)F1+λ(η)α = − η , (4.20)
e (1 + λ)Fλ(η)
when τ=τ eλ−
1
0 2 . For a single parabolic band, the Seebeck coefficient only depends on η,
regardless of the band mass. The value of η can be estimated using Equation 4.20 and the
Fermi integrals (Equation 4.15) and all other transport properties can be calculated. The band
effective mass can be calculated from the chemical potential and the measured Hall carrier
concentration (Equation 4.18) of a given thermopower and with it Pisarenko relations, while
temperature dependent Lorenz number can be calculated with η (Equation 5.30).
4.3 Cu2+xZn1−xGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154.[55]
Reproduced with permission of the American Chemical Society Copyright 2012.
The temperature dependence of the resistivity for different doping concentrations in
Cu2+xZn1−xGeSe4 is shown in Figure 4.1. As expected for a wide band gap semiconductor,[64]
unsubstituted Cu2ZnGeSe4 exhibits a high electrical resistivity at room temperature and de-
creases with increasing temperature. The above (see Section 3.2) mentioned phase transforma-
tion at around 450 K has a significant impact on the electrical resistivity leading to an increase
in resistivity with temperature for all doping levels, similar to metallic transport behavior and
the temperature of the transition seems to be composition dependent. Thus, this transition
can be considered an insulator-to-metal transition.
While substitution introduces charge carriers and results in a decrease of the resistivity over
orders of magnitude, the change in the resistivity does not entirely follow the trend expected
50
4.3 Cu2+xZn1−xGeSe4
Figure 4.1: Resistivity ρ (left) and Hall-carrier concentrations nH (right) of Cu2+xZn1−xGeSe4
as a function of temperature. An insulator-to-metal transition can be seen in the
transport at 450K. Note that the composition with x = 0.05 exhibits the lowest
electrical resistivity which can be attributed to its higher Hall carrier concentration.
for the substitution with Cu2+ for Zn+. The exception to the expected behavior is seen at the
substitution level of x = 0.05 which shows a lower resistivity than x = 0.075 and x = 0.1. This
can be explained by the charge carrier concentrations (Figure 4.1) that result for each doping
level, where a higher than expected carrier concentration was observed for the x = 0.05 sam-
ple. Expected carrier concentrations from simple charge counting and measured Hall-carrier
concentrations are shown in Figure 4.2. At a doping level of x = 0.025 the substitution has the
expected effect on the carrier concentration, however, higher substitution levels deviate sig-
nificantly from the charge carrier concentrations expected from simple charge counting. This
behavior is attributed to the secondary phases in the material seen in the scanning electron
micrographs (Figure 3.5). The formation of Cu2− 2− +δSe reduces the amount of Se and Cu in
the matrix. Considering the case where only Se is removed from the matrix, there would be
electrons left behind in the matrix which would lead to fewer holes in the material, as com-
pared to that expected from charge counting for each level of substitution x. In the case where
only Cu is removed from the matrix, the result would be more holes left behind in the matrix.
In the case of phase segregation where stoichiometric Cu2Se is formed, the individual effects
of less Se and Cu in the matrix would cancel each other out and would not impact the carrier
density. However, the formation of nonstoichiometric Cu2−δSe means that more Se than Cu is
removed from the matrix and the overall result is less holes in the matrix as compared to the
expected number from simple charge counting, the effect of which is shown by the measured
Hall carrier concentrations shown in Figure 4.2. Additionally, it is likely that CuSe forms for x
= 0.05 due to a slight excess of Se and reduces the amount of Cu and therefore leads to more
holes in the matrix.
It should be noted that measured Hall-mobilities (Figure 4.2) at room temperature of the
doped materials are around 2-3 cm2/V·s and not effected by this unexpected trend, lead-
ing to the assumption that the charge carrier concentration mainly dominates the electronic
transport. The Hall voltage is positive, which is expected for p-type charge carriers (holes),
51
4 Electrical Transport
Figure 4.2: Left: Expected charge carrier concentrations ncalc and measured Hall-carrier con-
centrations nH for Cu2+xZn1−xGeSe4. The deviations from the expected charge
carrier concentrations can be attributed to the secondary phases, which alter the
actual stoichiometry leading to defects affecting the transport. Right: Temperature
dependence of the measured Hall-mobilities. The mobilities are low for a semicon-
ductor showing some degree of localization.
and the mobilities are low for a semiconductor but high for localized charges in a very ionic
environment.
Figure 4.3: Seebeck coefficients α vs. temperature (left) of Cu2+xZn1−xGeSe4, showing the
phase transition in the undoped sample. Experimental Seebeck-coefficients as a
function of Hall carrier concentration nH (right) at 360 K. The dotted line, which
was generated using a single parabolic band approximation and an effective mass
of 1.2 me, shows single parabolic band behavior in this materials system.
Figure 4.3 illustrates the effect of the charge carrier concentrations on the thermopower.
The Seebeck coefficients are large and positive, indicating holes as the majority carrier type,
consistent with the carrier concentration and Hall-mobilities acquired via Hall-measurements.
The increase of the Seebeck coefficient with increasing temperature reveals that the maximum
of the thermopower is not yet reached at 670 K, which is consistent with the measured wide
band gap of 1.4 eV in this material (Figure 3.1 c)). Within this measurement range, thermally
activated electrons do not cause a reduction in the thermovoltage. Equations 4.15, 4.16, and
52
4.4 Cu2Zn1−xFexGeSe4
Figure 4.4: Temperature dependence of the electrical resistivity ρ of Cu2Zn1−xFexGeSe4 for x =
0.0 - 0.4 (left) and x = 0.5 - 1.0 (right) showing semiconducting behavior with high
electrical resistivity. A previously reported phase transformation[55] is apparent
for x = 0.0 and 0.1
4.20 have been used to model the relationship between carrier concentration and the Seebeck
coefficient. It is assumed that only one type of carrier is present in a single parabolic band,
with the assumption of acoustic phonon scattering (λ = 0). The effective mass m∗ of 1.2 me,
which was used to calculate the Pisarenko relation (Figure 4.3) at 360 K, was calculated from
the experimental Seebeck coefficient and Hall carrier concentration for x = 0.025.
Because the experimental data fall on or near the generated curve it may be assumed that
a heavily doped, single parabolic band is a good starting model for electronic transport of
this system below the transition temperature. Substitution with Cu does not significantly
change the effective mass or mobility of the holes and therefore confirms that the electronic
transport is mainly governed by the charge carrier densities. This furthermore shows that the
carrier mobility is not affected by nano particles of this length scale, which is expected, since
the mean free paths of electrons and holes in a semiconductor are much shorter than that of
phonons.[41]
In conclusion, a phase transition at 450 K has been identified, and it has an influence on the
transport properties in this material, leading to an insulator-to-metal transition. Substitution
of Zn2+ with Cu+ introduces holes as charge carriers, but also leads to the formation of
impurities by phase segregation which greatly influence the charge carrier concentration.
4.4 Cu2Zn1−xFexGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2013 - accepted manuscript.[56]
Reproduced with permission of the American Chemical Society Copyright 2013.
The temperature dependence of the electrical resistivity ρ for different compositions x
of Cu2Zn1−xFexGeSe4 is shown in Figure 4.4. As expected for these wide band gap semi-
conductors,[55] these compositions exhibit a high electrical resistivity at room temperature,
decreasing with temperature. A previously reported phase transformation[55] (see Section
53
4 Electrical Transport
4.3) is apparent for x = 0.0 and 0.1. No charge carriers are introduced directly through the
substitution with isoelectronic Fe2+ for Zn2+, however measured Hall carrier concentrations
(see Supporting Information in Chapter 8) change across compositions due to the changes in
character of the density of states of the valence band (see Figure 3.9) as well as the presence
of intrinsic defects. However, it has to be mentioned that due to the multi-band behavior (see
Figure 3.9), measured Hall carrier concentrations will not represent chemical carrier concen-
trations since Equation 2.3 does only apply for transport within a single band.
Figure 4.5: Temperature dependence of the Seebeck coefficients α of Cu2Zn1−xFexGeSe4 for
x = 0.0 - 0.4 (left) and x = 0.5 - 1.0 (right) showing intrinsic p-type semicon-
ducting behavior controlled by intrinsic defects. A previously reported phase
transformation[55] is apparent for x = 0.0 and 0.1.
The resistivity of a composition is therefore mainly controlled by the change in mobility,
due to alloy scattering, and charge carrier concentrations. The unchanged intrinsic nature of
these semiconductors, upon substitution of Zn with Fe, can also be seen in the flat slope of the
temperature dependent positive Seebeck coefficients (Figure 4.5), which show intrinsic p-type
conduction.[55, 123] The differences in the Seebeck coefficients arise from the different carrier
concentrations of the samples and an increase in the carrier effective mass with measured Hall
carrier concentrations, as seen Figure 4.6. The effective masses, which have been calculated
within the single parabolic band model at 360 K using Equations 4.18 and 4.20, show a linear
increase with increasing carrier concentrations. Such a trend is usually attributed to a non-
parabolicity of the band, which results in higher effective masses for a shift of the chemical
potential deeper into the band. However, as mentioned above, this trend could also be due to
the multi-band character of the valence band with the Fe-d band coming into play.
It has been shown, that substitution of Zn2+ with Fe2+ does not change charge carrier con-
centration as it is expected for aliovalent doping and the intrinsic nature of the solid solutions
is not changed. However, the multi-band behavior in the valence band seems to play a role on
the measured Hall carrier concentrations and the effective masses of the charge carriers.
54
4.5 Cu2+xZn1−x−yFeyGeSe4
Figure 4.6: Carrier effective mass of Cu2Zn1−xFexGeSe4 vs. measured Hall carrier concentra-
tion showing a linear dependence usually associated with non-parabolic behavior,
but possible a result of the multi-band behavior in this class of material.
4.5 Cu2+xZn1−x−yFeyGeSe4
For all compositions x was set to 0.025 and only the value of y has been altered throughout
the series of solid solutions to obtain the material Cu2.025Zn0.975−yFeyGeSe4 with the carrier
concentrations obtained for Cu2+xZn1−xGeSe4 as seen in Section 4.3. The electronic transport
data of the temperature dependent Seebeck coefficient, resistivity, Hall carrier concentration,
and Hall mobility can be found in the Supporting Information in Chapter 8.
Due to the multi-band character upon Fe-substitution and the difficulty in controlling
charge carrier concentrations in this class of materials all carrier concentrations are higher
than the 7.7x1019 holes/cm−3 of x = 0.025 in Cu2+xZn1−xGeSe4 and therefore a change in the
intrinsic nature of the electronic properties cannot be detected, which reproduces the results
from Section 4.4 very well.
4.6 Ca3AlSb3
This section is an adapted reproduction, from J. Mater. Chem 2012 22, 9826-9830.[96] Repro-
duced with permission of The Royal Society of Chemistry Copyright 2012.
The electronic properties of sodium doped Ca3AlSb3 have previously been investigated
and are largely similar to those reported here with zinc as a dopant. In the following results
the transport properties of the Na-doped sample that resulted in the highest figure of merit
(nominal composition Ca2.94Na0.06AlSb3) has been included as a basis for comparison.[95]
Figure 4.7 shows the measured Hall carrier concentrations as a function of temperature.
Though Ca3AlSb3 is a classic, valence-precise Zintl compound, the undoped material has a p-
type carrier concentration of 1018 holes/cm3 at room temperature due to intrinsic defects. As
55
4 Electrical Transport
expected, the carrier concentration increases upon substitution of Al3+ with Zn2+ due to the
introduction of holes. Doped samples behave extrinsically, with temperature independent car-
rier concentrations up to 700 K, after which the carrier concentration increases abruptly, likely
due to carrier excitation across the band gap (Eg ∼ 0.6 eV[95]). Zinc doping introduces signif-
icantly fewer holes than predicted, in accordance with previous attempts at doping Ca3AlSb3
and the closely related phase, Ca5Al2Sb6.[95, 124, 125] Ultimately, the maximum carrier con-
centration achieved by doping with zinc is lower than that achieved upon sodium doping,
suggesting a lower solubility limit for zinc, assuming substitution of Al with Zn.
Figure 4.7: Left: Hall carrier concentration nH with temperature of Ca3Al1−xZnxSb3. The car-
rier concentration increases upon substitution due to the introduction of holes. The
sharp increase of nH is likely due to carrier excitation across the band gap. Right:
Hall mobility µH with temperature. The increase with temperature at lower T rep-
resents an activation energy process due to grain boundaries. At higher temper-
atures acoustic phonon scattering dominates the carrier mobilities. Experimental
data is compared to Ca2.94Na0.06AlSb3 data (black line) from Reference [95].
Impurities at grain boundaries, such as oxidation products or possibly phases that are too
small to observe by SEM, are already known to impede charge transport in Ca3AlSb3, leading
to reduced mobility at low temperatures.[95] Grain boundaries are also a likely place for Zn
to accumulate, potentially causing the reduced doping effectiveness observed in this study.
This supposition is supported by the anomalous carrier concentration of the x = 0.01 sample,
which is higher than that of the samples with x = 0.02 and 0.05, and may be attributable to its
large grain size (see Section 3.5) and thus lower interfacial surface area.
Grain size also correlates with the carrier mobility of the Ca3Al1−xZnxSb3 samples in this
study. The Hall mobility, calculated from the measured Hall coefficient (RH) and resistivity
(ρ), is shown in Figure 4.7. For samples with sub-micron grains (x = 0.00, x = 0.02, and x =
0.05), the Hall mobility at low temperatures exhibits an exponential temperature dependence
(µH = µ e−EA/kBT0 ) indicative of a barrier at the grain boundaries, such as an oxide layer,
requiring an activation energy to overcome. At higher temperatures (> 600 K) the mobility is
limited by acoustic phonon scattering, for which the temperature dependence is given by µH ∝
T−ν, with ν between 1 and 1.5 for degenerate and non-degenerate behavior, respectively.[126]
However, the sample with the nominal composition of Ca3Al0.99Zn0.01Sb3 exhibits larger Hall
56
4.6 Ca3AlSb3
mobilities than all other compositions, with acoustic phonon scattering dominating transport
above 400 K. This suggests that since larger grains (see Figure 3.11) result in a lower interfacial
surface area, a less pronounced activation process in the temperature dependence of the Hall
mobility in this sample leads to higher mobility at lower temperatures. Additionally, larger
grain sizes may cause a larger carrier mobility at lower temperatures in the sample with x =
0.01 through reduced grain boundary scattering.
Figure 4.8: Left: Resistivity ρ of Ca3Al1−xZnxSb3 exhibiting semiconducting behavior consis-
tent with the Hall mobility. Right: Temperature dependent Seebeck coefficients of
Ca3Al1−xZnxSb3 showing p-type behavior consistent with the p-type carrier con-
centrations. Experimental data is compared to Ca2.94Na0.06AlSb3 data (black line)
from Reference [95].
The temperature dependence of the electrical resistivity (Figure 4.8), shows a decreasing
trend at lower temperatures, consistent with the activated mobility behavior (ρ = 1/neµ). At
higher temperatures the resistivity behaves as expected for a heavily doped semiconductor.
The Seebeck coefficients are positive, consistent with the p-type carrier concentrations, and
the temperature dependence is shown in Figure 4.8. The Seebeck coefficient of undoped
Ca3AlSb3 increases up to a temperature of 700 K, at which point thermally activated electrons
reduce the thermoelectric voltage resulting in a decay of the Seebeck coefficient. As expected
for a heavily doped semiconductor, the Seebeck coefficients decrease with increasing carrier
concentrations. In order to compare the influence of Zn and Na as dopants on the electronic
structure of this material, the Pisarenko relation of this material is shown in Figure 4.9. The
experimental Seebeck and carrier concentrations were measured at 700 K, while the dashed
line was generated using the single parabolic band model described in Section 4.2, with a
valence band effective mass of m∗ = 0.8 me. Both Zn and Na doped materials are well described
by the same model, suggesting that using Zn as a dopant does not affect the band structure
of the compound.
In conclusion, the electronic properties of zinc-doped Ca3AlSb3 were compared to a previ-
ously reported study on sodium as a dopant.[95] A similar band mass is observed regardless
of whether Zn or Na is the dopant. However, zinc exhibits a lower solubility limit than
sodium resulting in lower carrier concentrations, because the optimum carrier concentration
for this material cannot be reached via doping with zinc. This study also shows the effect of
57
4 Electrical Transport
Figure 4.9: Experimental Seebeck coefficients as a function of carrier concentration of
Ca3Al1−xZnxSb3 exhibiting single parabolic band behavior. Both Zn and Na[95]
doped materials are well described by the same model, suggesting that using Zn
as a dopant does not affect the band structure of the compound.
different grain sizes on the thermoelectric transport properties, with larger grains leading to
higher mobilities at lower temperatures, due to reduced scattering of electrons at the grain
boundaries.
4.7 Sr3GaSb3
This section is an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-9128.[100]
Reproduced with permission of The Royal Society of Chemistry Copyright 2012.
Figure 4.10 shows the measured Hall carrier concentrations nH of Sr3Ga1−xZnxSb3 samples
as a function of temperature. Despite Sr3GaSb3 being a valence-precise Zintl compound, the
undoped material has an extrinsic p-type carrier concentration of 4×1018 holes/cm3 at room
temperature, most likely due to intrinsic defects in the crystal structure. The resistivity of
undoped Sr3GaSb3 is very high (see Figure 4.11), decreasing due to thermal activation of
carriers into the conduction band only at temperatures above 800 K.
With each substitution of Zn2+ on a Ga3+ site we expect to introduce one additional free
hole (h+), due to the difference in valence states.[19] In many Zintl compounds, this sim-
ple assumption works well for predicting nH of doped samples (i.e. Yb14Al1−xMnxSb11[91],
Ba8Ga16−xGe30+x[54]), indicating that the dopant primarily substitutes on the intended crys-
tallographic sites. However, in Sr3GaSb3, in common with Ca3AlSb3,[96] doping with Zn
results in only a fraction of the predicted hole concentration. For example, when x = 0.07
(equivalent to a synthetic concentration of 1 at.% Zn) we would predict n = 2.1x1020 h+/cm3.
However, the measured nH is only 5x1019 h+/cm3. If each measured hole is assumed to result
from a Zn atom on a Ga site, than only one fourth of the synthetic Zn content resides on the
58
4.7 Sr3GaSb3
Figure 4.10: Left: Hall carrier concentration nH with temperature of Sr3Ga1−xZnxSb3. The
carrier concentration increases upon substitution due to the introduction of holes.
Right: Hall mobility µH with temperature. The increase with temperature at
lower T represents an activation energy process due to grain boundaries. At
higher temperatures acoustic phonon scattering dominates the carrier mobilities.
intended site, while the remainder may form secondary phases too small to identify or per-
haps becomes trapped at grain boundaries. However, estimating the matrix Zn content from
the measured Hall coefficient can be misleading. Sources of error might include Zn defects
with an effective charge other than +1, formation of compensating n-type defects,[127] or a
Hall factor (rH=n/nH) that deviates significantly from unity.[54]
As a function of temperature, nH in Sr3Ga1−xZnxSb3 is constant at low temperatures, in-
dicative of extrinsic transport, and increases at high temperature as intrinsic carriers are ac-
tivated across the band gap. The increase in nH leads to the decrease in electrical resistivity
with Zn-doping shown in Figure 4.10.
Figure 4.11: Left: Resistivity ρ of Sr3Ga1−xZnxSb3 exhibiting semiconducting behavior consis-
tent with the Hall mobility. Right: Temperature dependent Seebeck coefficients
of Sr3Ga1−xZnxSb3 showing p-type behavior consistent with the p-type carrier
concentrations.
The Hall mobility, µH, is shown in Figure 4.11, calculated from ρ = 1/nHeµH. At low tem-
peratures (T < 500 K), µH exhibits a positive temperature dependence that can be fit using
59
4 Electrical Transport
µH ∝ e−EA/kBT, where EA is the activation energy associated with a potential barrier.[128]
Such behavior may arise from barriers at the grain boundaries, such as oxide layers or sec-
ondary phases, requiring an activation energy to overcome. Our previous study (Section 4.6)
of Ca3AlSb3 indicates that with the correct processing it is possible to avoid this activated
behavior, drastically increasing electrical conductivity.[96] At higher temperatures (> 600 K)
the mobility in Sr3Ga1−xZnxSb3 is limited by acoustic phonon scattering, for which the tem-
perature dependence is given by µ ∝ T−νH , with ν between 1 and 1.5 for degenerate and
non-degenerate behavior, respectively.[126]
The Seebeck coefficients, α, of Sr3Ga1−xZnxSb3 samples, shown in Figure 4.11, are posi-
tive, consistent with the p-type carrier concentrations. The Seebeck coefficient of undoped
Sr3GaSb3 increases up to a temperature of 700 K, at which point thermally activated electrons
result in a decay of α. From the resulting maximum (α −1max ∼ 340 µVK , Tmax ∼ 700 K), a
rough estimate of the band gap using Eg = 2eαmaxTmax [129] yields Eg ∼ 0.5 eV, which is in
general agreement with the results of the DFT calculations (see Figure 3.13). As expected, the
Seebeck coefficients decrease with increasing carrier concentrations. This is best illustrated by
the Pisarenko relation shown in Figure 4.12, with experimental α and nH (cross symbols) ob-
tained at 700 K. The broken curve was generated using the single parabolic band (SPB) model
described in Section 4.2 with a valence band effective mass of m∗ = 0.9 me. All of the doped
samples in this study appear to be well described by m∗ = 0.9 m∗ at 700 K, suggesting that,
within experimental uncertainty, it is not possible to see any evidence of multi band behavior
at these doping levels. While not shown here, the same effective mass also provides a good fit
for the experimental Seebeck coefficients at 300 K and 500 K.
Figure 4.12: Experimental Seebeck coefficients of Sr3Ga1−xZnxSb3 vary with carrier concentra-
tion approximately according to an SPB model at 700 K. The blue curve represents
the predicted SPB behavior of α, assuming m∗ = 0.9 me, µ0 = 17 cm2V−1s−1 and κL
= 0.55 Wm−1K−1.
60
4.7 Sr3GaSb3
To investigate the possibility of anisotropic transport properties, the Hall coefficients and
electrical resistivities of Sr3Ga0.93Zn0.07Sb3 were measured on disks cut in different directions
(perpendicular and parallel to the hot pressing direction) and can be found in the Supporting
Information in Chapter 8. A ∼ 5% disparity in resistivity measured to 1000 K appeared
to stem from a higher hole concentration in the perpendicular slice, while the mobility was
identical in both directions. The difference in the Seebeck coefficients at 300 K was within
the measurement uncertainty of 5%. Together, these results suggest that possible anisotropy
does not significantly influence transport properties in our polycrystalline samples. This is in
agreement with XRD measurements, which show no signs of preferred grain orientation (see
Figure 3.12).
In conclusion, the electronic properties of the Zintl compound Sr3GaSb3 and the influence
of Ga substitution with Zn has been investigated. While zinc exhibits a lower solubility limit
than expected for complete substitution, the properties are representative of a heavily doped,
degenerate semiconductor with simple parabolic band behavior, which is in accordance to
other Zintl compounds.
61

5 Thermal Transport
5.1 Summary
This chapter begins with a general discussion of the thermal conductivity of solids. The dif-
ferent contributions to the thermal conductivity and the heat conduction of the lattice are
highlighted and general concepts are introduced. An attempt will be made to relate basic
concepts such as phonon group velocity and heat capacity to a general chemistry understand-
ing. Then the influence of different scattering mechanisms on the lattice thermal conductivity
will be discussed, since it is necessary for the understanding of the data in the remainder of
this chapter. Furthermore transport equations which are used in this chapter for the modeling
and calculations of the minimum lattice thermal conductivity, the electronic contribution to κ,
and the Lorenz numbers will be given.
In Cu2Zn1−xGeSe4, a statistically significant reduction in the lattice thermal conductivity
can be seen for x = 0.075 and 0.1, which can be attributed to the secondary phases at these sub-
stitution levels. The nanometer scaled impurities are mainly located at the grain boundaries,
leading to an enhanced phonon scattering and therefore a significant reduction in phonon
mean free path. The effect of the reduction leads to a lowering in lattice thermal conduc-
tivity of 0.3-0.5 Wm−1K−1at room temperature, with κL approaching the minimum value of
0.6 Wm−1K−1 at higher temperatures.
The solid solution of Cu2Zn1−xFexGeSe4 shows a dramatic reduction in thermal conduc-
tivity at 70 % Fe, correlating to the 3-stage cation restructuring process described in Chapter 3
where the highest local anisotropic structural disorder occurs at at 70 % Fe content. This dis-
order apparently reduces the lattice thermal conductivity by 15 % resulting in a minimum for
the composition Cu2Zn0.3Fe0.7GeSe4 due to scattering of phonons by local anisotropic strain.
The lattice thermal conductivities of the Zintl phases Ca3AlSb3 and Sr3GaSb3 are inves-
tigated. The effect of different grain sizes on the thermal transport properties of Ca3AlSb3
is discussed, where larger grain sizes lead to higher thermal conductivities at lower tem-
peratures, due to reduced scattering of phonons at the grain boundaries. In the compound
Sr3GaSb3 the lattice thermal conductivity is found to be 0.4 Wm−1K−1 at 1000 K and therefore
over the whole temperature range approximately 0.2 Wm−1K−1 lower than in the Zintl phase
Ca3AlSb3, which can be attributed to the larger number of atoms per unit cell compared to
Ca3AlSb3.
5.2 Thermal Transport Theory
Chapter 1 showed that the application of a temperature gradient ∇T will lead to a heat flow
q due to the thermal conductivity κ of a material (q = −κ∇T, Equation 1.2. The heat flow
63
5 Thermal Transport
from the hot to the cold side of a thermoelectric module should be as small as possible to
keep a stable temperature gradient along the material. Hence, small thermal conductivities
are desired for good thermoelectric materials.
The following section is in parts an adapted reproduction of the Diploma thesis from Wolf-
gang Zeier, 2010.[85] It is thought to give a rough overview over the basic concepts on thermal
transport properties.
5.2.1 Lattice Thermal Conductivity
The lattice thermal conductivity κL arises from quanta of lattice vibrations of a solid, the so
called phonons, which propagate through the lattice with a certain specific heat C, a phonon
group velocity vg, and a mean free path l, given by Equation 1.15 for a treatment via kinetic
gas theory:
1
κL = Cνgl .3
The mean free path characterizes the distance a phonon travels between random scattering
events and is related to the scattering relaxation time τ via τ = l/νg. Considering the frequency
ω dependence of the lattice gives:
1 ∫ ω ∫max 1 ωmax
κL = C(w)νg(ω)l(ω)dω = C(ω)ν2g(ω)τ(ω)dω . (5.1)3 0 3 0
Equation 5.1 demonstrates the importance of the heat capacity, the phonon group velocity,
and the phonon mean free path on the lattice thermal conductivity. These materials properties
will be discussed in the following, giving an overview on the concepts including chemical as
well as structural approaches for lowering C, νg, and l.
Lattice Vibrations
The atoms in a solid oscillate around their equilibrium positions. The vibrations are strongly
coupled to neighboring atoms and are therefore not independent of each other. These vibra-
tions can be described as standing waves in a classical treatment or in analogy to electrons
as quanta of the crystal vibrational field, commonly referred to as phonons.[20] The simplest
way to treat the vibrational behavior is to consider a periodic one-dimensional chain of atoms,
which interact via springs (spring constant KS) as shown in Figure 5.1.
At equilibrium, a chain of N atoms with mass M has an interatomic spacing equal to the
unit cell length a0. Within periodic boundary conditions and neglecting anharmonic forces,
the solution of Newton’s laws leads to the phonon frequencies ω~q for a one-dimensional
chain:[20] √
KS nπ
ω~q = 2 |sin(~qa0/2)| ~q = , (5.2)M a0
with the phonon wave vector ~q. The maximum value of the vibrational frequency of the
1D periodic chain ωmax~q is given by:[20]
64
5.2 Thermal Transport Theory
Figure 5.1: A periodic one-dimensional chain of identical masses M, connected by springs
with a spring constant KS, showing the origin of phonons from atomic vibrations.
The top schematically shows no lattice vibrations while the bottom shows the max-
imum oscillation leading to the maximum vibrational frequency ωmax~q .
√
KS
ωmax~q = 2 . (5.3)M
Thus the vibrational frequency has a periodic behavior with the periodicity (2π/a0) of the
reciprocal lattice. This is usually illustrated in phonon dispersion curves, with the phonon
frequency as a function of the phonon wave vector ~q.[20, 130]
Figure 5.2: Schematic phonon dispersion curves for a given direction of ~q of (a) monoatomic
lattice and (b) diatomic lattice.[130] An increasing number of atoms introduces
more optical phonon modes, which have low phonon group velocities and there-
fore carry less heat than acoustic phonons.
Figure 5.2 shows phonon dispersion curves for a monoatomic lattice (a) and a diatomic
lattice (b). The low-frequency acoustic branches correspond to atoms in the unit cell moving in
the same phase (harmonic vibrations), whereas the high-frequency optical branches represent
the anharmonic lattice vibrations. In a three-dimensional lattice with N atoms per unit cell,
the harmonic and anharmonic vibrations lead to three acoustic and 3N-3 optical branches,
65
5 Thermal Transport
respectively.[9, 20, 130]
As seen in Equations 5.2 and 5.3 the frequencies of the phonons of a one-dimensional
periodic chain strongly depend on the spring constant and the atomic mass. The magnitude
of the spring constant strongly depends on(the bo)nd strength of the atoms in the solid:[20]
∂2E
KS = , (5.4)
∂R2a Ra,0
with the distance Ra of the atoms and the equilibrium distance Ra,0, where a high bond
strength leads to a high spring constant.
Hence, the phonon dispersion decreases with increasing mass of the atoms and decreasing
bond strength between the atoms. This can also be expressed with the group velocity νg of
the phonons:[20] √
∂ω~q K
νg = = a
S
0 cos(~qa0/2) , (5.5)
∂~q M
which resembles the slope of the phonon dispersion curves.[130] Therefore a low bond
strength between the atoms and high atomic masses lead to low phonon group velocities and
a low value of the maximum value of the vibrational frequency.
As seen in Equation 5.1, the lattice thermal conductivity depends on the group velocity
of the phonons, which is difficult to access because of a broad spectrum of phonons present.
However, the high energetic optical phonons have low group velocities and are not effective in
transporting heat energy through the lattice, even though they might affect the heat conduc-
tion due to interactions with the acoustic phonons.[19, 130, 131] This approximation becomes
more valid in materials that contain atoms with a high mass contrast, as well as a variety of
chemical bonds, because these features promote a large energy gap between the acoustic and
optical modes, often leading to lower group velocities.[132, 133] The acoustic phonons are the
main heat conductors and for a material with low thermal conductivity the group velocity of
the acoustic branches should be small.[19] Hence materials with heavy atoms and low bond
strength usually exhibit low thermal conductivities.
Heat Capacity
When the temperature of a solid is raised, the supplied heat increases the occupation of the
vibrational energy states of the atoms. The energy difference of the vibrational states is given
by:
∆E = h̄ω (5.6)
with the vibrational density of states:[134, 135]
4πω2V
g(ω)dω = 3 dω , (5.7)vm
where V is the volume, ω the vibrational frequency, and vm the average velocity of sound.
Quantum statistics leads to the occupation number of the different states (ni):[135]
66
5.2 Thermal Transport Theory
1
ni = ( ) (5.8)
exp hωk T − 1B
However the vibrations of a solid are not an elastic continuum, because they are comprised
of atoms. Each atom has three degrees of dynamical freedom and a solid containing N atoms
has 3N degrees of vibrational freedom, which imposes a limit on the maximum frequency
ωmax = ωD (Debye theory). With the Debye frequency ωD the total number of all distinguish-
able phonon modes is:[134]
∫ ωD
3N = g(ω)dω , (5.9)
0
with
( ) 1
3N 3
ωD = vm . (5.10)4πV
Considering the total energy U of the phonon modes thermodynamically leads to the heat
capacity of a system of:[134]
∫ ΘD/T x4ex ( )T 3
C ∝ x − 2 dx . (5.11)0 (e 1) ΘD
The quantity ΘD is known as the Debye temperature:[130, 134, 135]
hω
Θ = DD . (5.12)kB
At temperatures above the Debye temperature with ΘD  T all vibrational states are suf-
ficiently occupied and the value of C reaches the classical value of 3R (Dulong-Petit law).[134]
However, for the materials considered in this thesis, this value is likely to result in an under-
estimation (∼ 10%) of the thermal conductivity at high temperatures.[136]
The Debye temperature ΘD of a material can be easily estimated from the mean speed of
sound vm and the average volume per atom Ω via:[137]( )
v 1/3mh̄ 6π2ΘD = , (5.13)kB Ω
where the average speed of sound is calculated from the longitudinal vl and transverse vt
sound velocities:
3
vm =
v−3 + 2v−3
. (5.14)
l t
Obviously large volumes of the unit cell and small number of atoms (see Equation 5.10),
as well as a high mass of the atoms and a weak bond strength lead to a low Debye cutoff
frequency and a low Debye temperature, which in turn leads to a low heat capacity of a solid.
67
5 Thermal Transport
Scattering Mechanisms
The use of Equation 5.1 not only shows the need for a low heat capacity and phonon group
velocity, but also small phonon mean free paths or small phonon scattering relaxation times.
There are different mechanisms for phonons to get scattered along their propagation through
the lattice and the scattering relaxation time can be written as the sum of the different contri-
butions using Matthiessen’s rule:[39]
τ−1 = τ−1 + τ−1 + τ−1 + τ−1N U B PD + τ
−1
EP , (5.15)
with the different scattering relaxation times for normal phonon scattering (N), Umklapp
scattering (U), boundary scattering (B), point defect scattering (PD), and electron-phonon
scattering (EP), respectively.
For phonon-phonon scattering (normal and Umklapp-scattering), effects by normal pro-
cesses (or N-processes) are ignored since they conserve the phonon wave vector. Furthermore
electron-phonon scattering will not be discussed here since its effect is small compared to the
other scattering mechanisms.[39]
Umklapp processes are 3-phonon processes in which the scattering processes translates
the wave vector into another Brillouin zone, in other words, the crystal momentum changes
due to the scattering event. The inverse phonon scattering relaxation time for Umklapp scat-
tering is proportional to the phonon frequency ω (τ−1 2U ∝ ω ) showing that low frequency
(or long wavelength) phonons are more heavily scattered by these phonon-phonon scattering
processes.[39]
The temperature dependence of κL for phonon-phonon Umklapp-scattering is well de-
scribed by a 1/T dependence which, in materials with low mass contrast and simple crystal
structures, is given by:[102]
(6π2)2/3 M̄v3
κL =
m
2 2/3 2 , (5.16)4π TΩ γ
where M̄ is the average mass, vm the average speed of sound, Ω the volume per atom and
γ the Grüneisen parameter. This can also be expressed in terms of the Debye temperature
shown by Klemens:[138]
( )3
≈ π kB Θ
3
κL 2 M̄a
D , (5.17)
2γ h T
with the lattice parameter a. The G(rüneisen)parameter is defined as:[139]
− dlnωD 3α= = TBTVγ , (5.18)
dlnV T C
with the linear coefficient of thermal expansion αT and the isothermal bulk modulus BT.
The bulk modulus can be experimentally obtained from the density D and the longitudinal
and transverse speeds of sound via: ( )
2 − 4BT = D vl v2t . (5.19)3
68
5.2 Thermal Transport Theory
The Grüneisen parameter relates to the anharmonicity of the lattice vibrations. Low γ is
achieved in materials with very low anharmonicity for example in the tetrahedrally bonded
ZnSe (γ = 0.8), while very anharmonic materials exhibit Grüneisen parameters greater than
1.[139] For example, materials with free electron pairs on one atom in the unit cell exhibit a
high anharmonicity usually resulting in low lattice thermal conductivities.
In the limit of acoustic phonons contributing to the thermal transport only, the temperature
dependence of the lattice thermal conductivity can be expressed via:[19]
(6π2)2/3 M̄ν3 1
κ = mL 4π2 TV2/3 2 1/3
, (5.20)
γ N
showing the effect of the number of atoms N in a unit cell on κL. It is therefore practical to
search for materials with a large number of atoms per unit cell.
Boundary scattering is a very effective mechanism particularly in nanostructured materials
such as nanowires, thin films, and nanocomposites. The use of nano structuring or interfaces
reduces the phonon mean free path due to scattering of the low frequency (large wavelength)
phonons, which carry the largest portion of heat.[42, 43, 41] For example in PbTe 10 % of
the heat at room temperature is carried by phonons with a mean free path of greater than
860 nm[140] and Pei et al.[43] have shown that a 20 % reduction in phonon thermal conduc-
tivity can be expected in PbTe by scattering the phonons with mean free paths greater than
200 nm.
Another very effective way to reduce the phonon mean free path of a material is the uti-
lization of point imperfections or point defects. Scattering by point defects can be achieved
through alloying with other elements,[33] successfully shown for thermoelectric materials in
Heusler compounds,[34] skutterudites,[35, 36] lead telluride,[37] and SiGe.[38, 39] While al-
loying with isoelectronic elements does not introduce charge carriers into a material, it creates
point defect scattering for phonons due to mass differences[141] (or mass field fluctuations)
and strain field fluctuations. This strain arises from different bonding interactions and size
differences between the host and impurity atoms,[34] and expresses itself in the lattice param-
eters.
The scattering relaxation time for point defect scattering is given by:[142]
(
4π νg(ω)ν2 ( ) ( ) )−1p(ω) m 2 r 2
τPD =
i i
V 4 ∑ fi 1− + ∑ fm i 1− , (5.21)ω r
with the phonon group and phase velocities, νg and νp, respectively. Here, fi is the fraction of
atoms with mass mi and radius ri that reside on a site with average mass m and radius r.[142]
The inverse phonon scattering relaxation time for point defects is proportional to the phonon
frequency ω (τ−1PD ∝ ω
4) showing that high frequency (or short wavelength) phonons are more
heavily scattered by these static point imperfections, leading to an effective way of scattering
phonons that are not influenced by phonon-phonon processes.
Following Callaway’s expression to model the effect of point defect scattering on the ther-
mal transport at higher temperatures (ΘD  T)[33] and combining it with the approaches of
Alekseeva et al.[37] and Yang et al.[34] leads to a model in which the fractional occupancies
on the lattice sites can be neglected even if the lattice parameters do not follow Vegard’s law.
69
5 Thermal Transport
If only Umklapp and point defect scattering are the only scattering sources, the ratio of κL of
the crystal with disorder to the crystal without disorder, pκL, is:[34]
κ −1L tan (u)
p = , (5.22)
κL u
with the disorder scaling parameter u, which can be expressed as:[34]
2
u2
π Θ
= D
Ω p
2 κLΓexp , (5.23)hνm
where Ω is the average volume per atom, h Planck’s constant, νm the average lattice sound
velocity, and Γexp the experimental disorder scattering parameter. The disorder scattering
parameter can be expressed as the sum of the contributions of the mass and strain field
fluctuations, Γm and ΓS, respectively. F[or the simple cubic case this c]an be expressed as:[37]( ) ( )
− ∆M
2 a(x)− a 2
Γ pexp = x(1 x) fi + e . (5.24)M̄ xap
Here M̄ stands for the average atomic mass of the compound, ∆M for the mass difference
between the host and impurity atom, e is an adjustable anharmonicity parameter, a(x) the
lattice parameter with composition, and ap the lattice parameter of the pure end member.
It is important to note that the strength of strain field fluctuation effects relative to mass
fluctuation effects depends on more than just the size differences relative to the mass differ-
ences. Important considerations include the nature of the atomic bonding specific to each
atom.[34]
5.2.2 Minimum Lattice Thermal Conductivity
The assumption of a minimum scattering length for phonons as a function of the phonon
frequencies leads to a model of the glassy limit of the lattice thermal conductivity, initially
described by Cahill et al.[143] and recently employed to calculate the lower bound for lattice
thermal conductivities κmin in thermoelectric materials by Toberer et al.[124]. The formulation
is developed for amorphous materials as an extension of Einstein’s model of a random walk
of energy. It assumes that the minimum phonon mean free path is one-half of its wavelength,
leading to:[143]
( ) ( )π 1/3 T 2 ∫ Θ−2/3 i/T x3exκmin = k6 BΩ ∑ vi x dx , (5.25)i Θi 0 (e − 1)2
where the summation is over the two transverse modes vt and the one longitudinal mode
vl , determined by speed of sound measurements, and:[54]( )( )
h̄ 6 2 1/3π
Θi = vi . (5.26)kB Ω
In the high temperature limit, an estimation of the minimum lattice thermal conductivity
70
5.3 Cu2+xZn1−xGeSe4
κmin with an average volume Ω per atom is given by:[21]
1 (π)1/3
κ −2/3min = k Ω (2v + v2 6 B t l
) . (5.27)
5.2.3 Electronic Contribution to the Thermal Conductivity
Measurements of the thermal conductivity of a material always leads to the total thermal
conductivity κ, which is a sum of the three different contributions of the lattice thermal con-
ductivity κL, the electronic thermal conductivity κel , and the bipolar contribution κb:
κ = κL + κel + κb . (5.28)
The electronic contribution to the thermal conductivity can be expressed through the
Wiedemann-Franz law as:[10]
κel = LσT . (5.29)
The value of the Lorenz number L for the free electron model is 2.44 WΩK−2. However, most
materials do not exhibit values of the Lorenz number close to the free electron value and
their temperature dependent Lorenz numbers should be calculated. In the limit of a single
parabolic band, under the assumption of acoustic phonon scattering (λ = 0) the temperature
dependent Lorenz number is found to be:[21]
k2B (1 + λ)(3 + λ)Fλ(η)Fλ+2(η)− (2 + λ)
2F2
L = λ+1
(η)
2 . (5.30)e (1 + λ)2F2λ(η)
The reduced chemical potential η, can be calculated from the experimental Seebeck coeffi-
cients via Equation 4.20 using the Fermi integrals Fj(η).
The term of the bipolar contribution to the thermal conductivity, occurs when a sample
has both holes and electrons significantly contributing to σ. This energy transport in addition
to that carried by the electrons and the holes is due to the creation of electron-hole pairs that
extract an amount of energy. This may be a noticeable contribution to the electronic ther-
mal conductivity when carrier concentrations and the mobilities are equal for electrons and
holes.[130] Since only σ and therefore κel can be readily accessed through the measurements
and the Wiedemann-Franz law, the bipolar term is usually neglected. If a significant signifi-
cant bipolar contribution to the total thermal conductivity occurs, it is visible as an upturn in
the lattice thermal conductivity data at higher temperatures.
5.3 Cu2+xZn1−xGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154.[55]
Reproduced with permission of the American Chemical Society Copyright 2012.
Thermal diffusivity of the material was measured up to 670 K. The total thermal conduc-
tivity was calculated using κ = dDC where d = thermal diffusivity, D = geometric density, C
= specific heat capacity. Here, use of the Dulong-Petit approximation for the heat capacity (C
71
5 Thermal Transport
= 0.34 Jg−1K−1) is likely to result in an underestimation (∼ 10%) of the thermal conductivity
at high temperatures.[136] The temperature dependence of the total thermal conductivity is
shown in Figure 5.3 and the effect of the previously mentioned phase transformation can be
seen at around 450 K. The thermal diffusivity is additionally shown in Figure 5.3.
Figure 5.3: Total thermal conductivity κ (left) and thermal diffusivity d (right) of
Cu2+xZn1−xGeSe4 as a function of temperature. The kink in the data can be at-
tributed to the change in the electronic contribution κel .
The total thermal conductivity of a semiconducting material is a combination o(f lattice, )
electronic, and bipolar contributions (κ = κ LTL + κel + κb). The Wiedeman-Franz relation κel = ρ
was employed to estimate the κel contribution to the thermal conductivity. Temperature de-
pendent Lorenz numbers (L) were calculated within the single parabolic band approximation
using Equation 5.30 under the assumption of acoustic phonon scattering (λ = 0).
The electronic contribution to the total thermal conductivity (Figure 5.4) of the undoped
sample is very low due to the high resistivity and increases with increasing carrier concen-
tration. The phase transformation can be seen as well due to the change in the resistivity at
the transition temperature in this material. The calculated Lorenz numbers are considerably
lower than the value for the free electron model of 2.44 WΩK−2.
Subtraction of κel from κ leaves the lattice and bipolar contribution of the thermal conduc-
tivity Figure (5.5). As expected for this wide band gap semiconductor there is no evidence of
a significant bipolar contribution to the total thermal conductivity in this temperature range,
which would be visible as an upturn in the data at higher temperatures, and the resulting
thermal conductivity is hereafter referred to as lattice thermal conductivity.
The lattice thermal conductivities of Cu2+xZn1−xGeSe4 with x = 0, 0.025 and 0.05 are the
same within the range of the measurement uncertainty for the laser flash diffusivity mea-
surements of 3 %. This can be expected since the mass contrast between Cu and Zn is low
and therefore point defect scattering, due to the mass disorder, can be assumed to be negligi-
ble. The temperature dependence of κL is well described by a 1/T dependence attributed to
phonon-phonon Umklapp-scattering which, in materials with low mass contrast and simple
crystal structures can be described by Equation 5.16. The temperature dependence of κmin is
given in Figure 5.5 calculated via Equation 5.25.
72
5.3 Cu2+xZn1−xGeSe4
Figure 5.4: Calculated temperature dependent Lorenz number L (left) and electronic thermal
conductivity κel (right) of Cu2+xZn1−xGeSe4. Lorenz numbers are smaller than
the value for the free electron model of 2.44WΩK−2. The phase transformation is
visible in the electronic thermal conductivity due to its dependence in the electrical
conductivity.
Figure 5.5: Temperature dependence of the lattice thermal conductivity κL of
Cu2+xZn1−xGeSe4. κL exhibits a 1/T temperature dependence (purple line)
attributed with phonon-phonon Umklapp scattering, slowly approaching the
glassy limit κmin (dashed line) at higher temperatures.
73
5 Thermal Transport
Figure 5.6: Longitudinal (left) and transverse (right) sound response data of an ultrasonic
sound/response measurement of Cu2ZnGeSe4. As expected the longitudinal
sound waves propagate faster than the shear modes in a solid.
Room temperature ultrasonic measurements of undoped Cu2ZnGeSe4 yield the longitudi-
nal vl and transverse vt speeds of sound of 4101 m/s and 2154 m/s, respectively. The average
speed of sound and the Debye temperature ΘD of the material of 2409 m/s and 257 K, re-
spectively, can be calculated via Equations 5.14 and 5.13. Respective speed of sound data for
Cu2ZnGeSe4 can be seen in Figure 5.6. As expected the amplitudes decrease with 1/distance2
and the longitudinal speed of sound is faster as the transverse speed of sound in a solid.
Differences in the data to other compositions of the solid solution Cu2+xZn1−xGeSe4 can be
assumed to be negligible, again due to the very small mass contrast between Cu and Zn.
Furthermore, the differences in grain size and the impurity phases on the grain boundaries
are not expected to change the speed of sound significantly since the sound waves of the
measurement are very long wavelength phonons.
Figure 5.7: SEM micrograph of Cu2.1Zn0.9GeSe4 (left) showing the Cu2−δSe as nano sized im-
purities on the grain boundaries. These impurities lead to enhanced phonon scat-
tering and with it a significant reduction in phonon mean free path. The effect of
the reduction leads to a lowering in the lattice thermal conductivity by 15%. (right)
74
5.4 Cu2Zn1−xFexGeSe4
Using Equation 5.16 and the experimental data of the temperature dependent lattice ther-
mal conductivity, a Grüneisen parameter of γ = 0.8 has been determined for Cu2ZnGeSe4.
This value is reasonable as compared with the room temperature Grüneisen parameter of
ZnSe of 0.8,[139] due to similar crystal structures and bonding interactions.
In this material, a statistically significant reduction (see Figures 5.5 and 5.7) in the lattice
thermal conductivity can be seen for x = 0.075 and 0.1, which can be attributed to the sec-
ondary phases at these substitution levels. The nanometer scale impurities are mainly located
at the grain boundaries leading to an enhanced phonon scattering and therefore a significant
reduction in phonon mean free path. The effect of the reduction leads to a lowering in lat-
tice thermal conductivity of 0.3-0.5 Wm−1K−1 at room temperature, with κL approaching the
minimum value of 0.6 Wm−1K−1 at higher temperatures.
5.4 Cu2Zn1−xFexGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2013 - accepted manuscript.[56]
Reproduced with permission of the American Chemical Society Copyright 2013.
The total thermal conductivity was calculated from the measured thermal diffusivity using
κ = dDC where d = thermal diffusivity, D = geometric density, C = specific heat capacity.
Here, use of the Dulong-Petit approximation for the heat capacity is likely to result in an
underestimation (∼ 10%) of the thermal conductivity at high temperatures.[136] The temper-
ature dependence of the total thermal conductivity is shown in Figure 5.8 and the effect of
the previously mentioned phase transformation can be seen at around 450 K. It has to be
emphasized that the atomic structure for the thermal conductivity calculations described be-
low are guided by the structure of the room temperature phase and not the unknown high
temperature phase.
Figure 5.8: Total thermal conductivity κ of Cu2Zn1−xFexGeSe4 for x = 0.0 - 0.4 (left) and x =
0.5 - 1.0 (right), as a function of temperature.
Temperature dependent Lorenz numbers L (see Figure 5.9 top) were calculated within the
single parabolic band approximation using Equation 5.30 under the assumption of acoustic
phonon scattering (λ = 0). And Lorenz numbers are smaller than the value for the free
75
5 Thermal Transport
Figure 5.9: Temperature dependent Lorenz numbers (top) and electronic thermal conductivity
(bottom) of Cu2Zn1−xFexGeSe4. The degenerate limit of the Lorenz number is
indicated and the calculated Lorenz numbers are close to the non-degenerate limit.
electron model of 2.44 WΩK−2. The temperature dependent electronic contribution to the
thermal conductivity (see Figure 5.9 bottom) has been calculated via the Wiedemann-Franz
relation with the measured electrical resistivities and the calculated temperature dependent
Lorenz numbers.
Subtraction of κel from κ leaves the lattice and bipolar contribution of the thermal conduc-
tivity Figure (5.10). As expected for this wide band gap semiconductor there is no evidence of
a significant bipolar contribution to the total thermal conductivity in this temperature range,
which would be visible as an upturn in the data at higher temperatures, and the resulting
thermal conductivity is hereafter referred to as lattice thermal conductivity. The tempera-
ture dependence of κL is well described by a 1/T dependence attributed to phonon-phonon
Umklapp-scattering (see Equation 5.16). The temperature dependence of κmin is given in Fig-
ure 5.10 calculated via Equation 5.25.
Room temperature ultrasonic measurements of Cu2Zn0.5Fe0.5GeSe4 and
Cu2FeGeSe4 yield the longitudinal vl and transverse vt speeds of sound of 4230 m/s and
2225 m/s for Cu2Zn0.5Fe0.5GeSe4, and 4371 m/s and 2166 m/s, respectively. The average
speeds of sound and the Debye temperatures ΘD of the materials are 2488 m/s and 265 K
76
5.4 Cu2Zn1−xFexGeSe4
(for 50 % Fe content) and 2430 m/s and 260 K (for 100 % Fe content), respectively. All values
are well within the measurement uncertainty of the speed of sound measurements. The simi-
lar speeds of sounds and Debye temperatures throughout this solid solution leads to the very
similar κmin as seen in Figure 5.10. Respective sound response data for Cu2Zn0.5Fe0.5GeSe4
and Cu2FeGeSe4 can be found in the Supporting Information (Chapter 8).
Figure 5.10: Lattice thermal conductivity κL of Cu2Zn1−xFexGeSe4 for x = 0.0 - 0.4 (left) and
x = 0.5 - 1.0 (right), decreasing with temperature due to Umklapp-scattering pro-
cesses. The calculated minimum lattice thermal conductivities are indicated. κL
does not fully approach κmin at the low temperatures yet. The differences in κmin
for different samples of the solid solution is negligible.
The lattice thermal conductivity κL at room temperature (see 5.11) was evaluated by sub-
tracting the electronic contribution (κel = LTρ−1, where L = 1.5 x 10−8WΩK−2 for non de-
generate materials,[10]) i.e. κL = κ − LTρ−1. While the measurement uncertainty for the laser
flash diffusivity measurement is 3 % we have assumed a combined uncertainty for κL of 5 %
at room temperature, taking into account a 5 % error for the resistivity measurement. Upon
substitution with Fe the lattice thermal conductivity decreases with increasing Fe content until
a minimum value is reached at 70 % Fe content, where this trend reverses. Since the mass dif-
ference between Fe and Zn is small, this statistically significant reduction of the lattice thermal
conductivity cannot be explained by the mass contrast only, as indicated in Figure 5.11, where
the expected κL has been calculated within the Callaway model.[33, 34, 37] Furthermore, ef-
fects of the microstructure can be ruled out since similar microstructures and no impurity
phases are apparent in the SEM micrographs.
The structural change that correlates with this dramatic reduction in thermal conductivity
at 70 % Fe is the 3-stage cation restructuring process described in Figure 3.8 where the highest
local anisotropic structural disorder occurs at at 70 % Fe content. This disorder apparently
reduces the lattice thermal conductivity by 15 % resulting in a minimum for the composition
Cu2Zn0.3Fe0.7GeSe4 due to scattering of phonons by local anisotropic strain. As the disorder
scattering of the phonons, due to mass fluctuation and isotropic strain is expected to be very
small from these similar mass and similar volume ions, this shows the potential of signifi-
cant point defect scattering due to local anisotropic strain. As seen in Figures 5.10 and 5.11
77
5 Thermal Transport
Figure 5.11: Lattice thermal conductivity κL of Cu2Zn1−xFexGeSe4 at room temperature show-
ing a 15% reduction due to disorder scattering. An error of 5% is assumed for
κL at room temperature. The broken line indicates the calculated influence of the
mass contrast between Fe and Zn on the lattice thermal conductivity which shows
that the strain related to the lattice constant change has a significant effect on the
lattice thermal conductivity.
the reduction of the lattice thermal conductivity by point defect scattering is more effective
at lower temperatures. With increasing temperatures the high frequency phonons get scat-
tered mainly by phonon-phonon processes reducing their phonon mean free path. In other
words, the dominant scattering mechanism at higher temperatures is phonon-phonon Umk-
lapp scattering and point defect scattering does not contribute hugely to the lattice thermal
conductivity.[103]
5.5 Cu2+xZn1−x−yFeyGeSe4
The total thermal conductivity was calculated from the measured thermal diffusivity using
κ = dDC where d = thermal diffusivity, D = geometric density, C = specific heat capacity.
Here, use of the Dulong-Petit approximation for the heat capacity is likely to result in an un-
derestimation (∼ 10%) of the thermal conductivity at high temperatures.[136] The temperature
dependence of the total thermal conductivity can be found in the Supporting Information in
Chapter 8. It has to be emphasized that the atomic structure for the thermal conductivity
calculations described below are guided by the structure of the room temperature phase and
not the unknown high temperature phase.
Temperature dependent Lorenz numbers L, the electronic contribution to the thermal con-
ductivity and lattice thermal conductivities can be found in the Supporting Information in
78
5.5 Cu2+xZn1−x−yFeyGeSe4
Chapter 8. As seen in Section 5.4 the calculated Lorenz numbers are smaller than the value
for the free electron model of 2.44 WΩK−2. As expected for this wide band gap semiconduc-
tor there is no evidence of a significant bipolar contribution to the total thermal conductivity
in this temperature range and the temperature dependence of κL is well described by a 1/T
dependence attributed to phonon-phonon Umklapp-scattering (see Equation 5.16).
Figure 5.12: Left: Lattice thermal conductivity κL of Cu2.025Zn0.975−yFeyGeSe4 at room temper-
ature showing a 15% reduction due to disorder scattering. For comparison the
data of Cu2Zn1−xFexGeSe4 has been included (black circles) and the x-axis has
been labeled in respect to this solid solution. The broken line indicates the cal-
culated influence of the mass contrast between Fe and Zn on the lattice thermal
conductivity which shows that the strain related to the lattice constant change of
c/a (right) has a significant effect on the lattice thermal conductivity. Right: Lat-
tice parameter ratio c/a with composition y of Cu2.025Zn1−yFeyGeSe4 (red data
points), from a Pawley refinement. The broken line indicates the expected behav-
ior following Vegard’s law. For comparison the data of Cu2Zn1−xFexGeSe4 has
been included (black circles) and the x-axis has been labeled in respect to this
solid solution. A 3-stage cation restructuring process of Cu+, Fe2+, and Zn2+
results in the trend of the lattice parameters.
The lattice thermal conductivity κL at room temperature (see 5.12 (left)) was evaluated by
subtracting the electronic contribution (κ −1el = LTρ , where L = 1.5 x 10−8WΩK−2 for non
degenerate materials,[10]) i.e. κ = κ − LTρ−1L . While the measurement uncertainty for the
laser flash diffusivity measurement is 3 % we have assumed a combined uncertainty for κL
of 5 % at room temperature, taking into account a 5 % error for the resistivity measurement.
As seen in Section 5.4, upon substitution with Fe the lattice thermal conductivity decreases
with increasing Fe content until a minimum value is reached at 70 % Fe content, where this
trend reverses. The mass difference between Fe and Zn is small, this statistically significant
reduction of the lattice thermal conductivity cannot be explained by the mass contrast only,
as indicated in Figure 5.11, where the expected κL has been calculated within the Callaway
model.[33, 34, 37]
This reduction in the lattice thermal conductivity has been attributed to the structural
change in this material, which is due to a 3-stage cation restructuring process described in
Figure 3.8 where the highest local anisotropic structural disorder occurs at at 70 % Fe con-
79
5 Thermal Transport
tent. The solid solution Cu2.025Zn0.975−yFeyGeSe4 supports this explanation, since the lattice
parameter ratio c/a (see Figure 5.12 (right)) and the lattice thermal conductivity reproduce
the results of Section 5.4 fairly well, with a high c/a-ratio leading to a lowering of κL due to
to scattering of phonons by local anisotropic strain.
5.6 Ca3AlSb3 and Sr3GaSb3
This section is an adapted reproduction, from J. Mater. Chem 2012 22, 9826-9830 and Energy
Environ. Sci. 2012, 5, 9121-9128.[96, 100] Reproduced with permission of The Royal Society of
Chemistry Copyright 2012.
Figure 5.13: Total thermal conductivity (left) and lattice thermal conductivity (right) of
Ca3Al1−xZnxSb3 with temperature, with the theoretical minimum lattice thermal
conductivity at higher temperatures. The inset illustrates the different grain sizes
of the samples. Smaller grains (left) for the compositions with x = 0.00, 0.02, and
0.05 and larger grain sizes for x = 0.01 (right). The scale bar is representative
for both micrographs. While the lattice thermal conductivity approaches κmin at
higher temperatures, the grain size influences the thermal transport at lower tem-
peratures. Larger grains lead to higher lattice thermal conductivities due to less
boundary scattering of the phonons.
The total thermal conductivities of Ca3AlSb3 (see Figure 5.13) and Sr3GaSb3 (see Figure
5.14), were calculated using κ = dDC where d = thermal diffusivity, D = geometric density,
C = specific heat capacity. The specific heat capacity was estimated using the Dulong-Petit
law. While generally accurate at room temperature, the Dulong Petit heat capacity often
underestimates Cp at high temperature by as much as 10 %, leading to an overestimation
of zT.[136] The electronic contribution κel to the total thermal conductivity can be estimated
from the Wiedemann-Franz relation (κel = LT/ρ). In this study, the Lorenz numbers L are
calculated from the experimental Seebeck coefficients using a single parabolic band model as
described in Section 5.2.3 and Equation 5.30. The electronic thermal conductivities and Lorenz
numbers for Ca3Al1−xZnxSb3 and Sr3Ga1−xZnxSb3 can be seen in the Supporting Information
in Chapter 8.
Subtracting the electronic term from the total thermal conductivity leaves the lattice ther-
80
5.6 Ca3AlSb3 and Sr3GaSb3
Figure 5.14: Total thermal conductivity (left) and lattice thermal conductivity (right) of
Sr3Ga1−xZnxSb3. The calculated minimum lattice thermal conductivity κmin is
shown as a broken line, with the theoretical minimum lattice thermal conduc-
tivity at higher temperatures. The undoped composition with x = 0 exhibits a
significant bipolar contribution, evidenced by the increase in κL + κb at high tem-
peratures relative to the doped samples.
mal conductivity and the bipolar contribution. No significant bipolar contribution can be
observed for Ca3AlSb3 (see Figure 5.13) in this temperature range, which would be visible
as an upturn in the data at higher temperatures. Of the samples in these studies, only the
undoped Sr3GaSb3 exhibits a significant bipolar contribution, evidenced by the increase in
κL + κb at high temperatures relative to the doped samples, as seen in Figure 5.14.[10] In all
other remaining compositions for Ca3AlSb3 and Sr3GaSb3 the lattice thermal conductivity is
the dominant term, decreasing with a T−1 temperature dependence expected when scattering
is dominated by Umklapp processes (Equation 5.16).[130] In all samples, the lattice thermal
conductivities approach the minimum thermal conductivity κmin (Figure 5.13), which was
calculated using speed of sound measurements and Cahill’s formula for disordered crystals
(Equation 5.25).
Room temperature ultrasonic measurements of undoped Ca3AlSb3 (see Supporting In-
formation Chapter 8) yield longitudinal and transverse speeds of sound of 4170 m/s and
2440 m/s respectively. From these, a mean speed of sound of 2710 m/s and an effective Debye
temperature of 261 K are calculated using Equations 5.13 and 5.14.
The influence of Zn on the lattice thermal conductivity seems to be negligible in these
materials, as reported in the sodium doped Ca3AlSb3.[95] However, the sample with nominal
composition Ca3Al0.99Zn0.01Sb3 exhibits a higher thermal conductivity at lower temperatures.
This can be attributed to the larger grains in this sample (see Figure 5.13). The larger grains
lead to less grain boundary scattering and hence longer phonon mean free paths compared to
the sub-micron grains in the other samples and thus higher lattice thermal conductivities at
lower temperatures. At higher temperatures the effect of the grain boundaries becomes less
pronounced. Phonon-phonon processes reduce the phonon mean free path with increasing
temperature. At higher temperature the phonon mean free path is smaller than the average
grain size and scattering events take place before phonons could become scattered at the
boundaries, leading to a nearly grain size independent lattice thermal conductivity at higher
81
5 Thermal Transport
temperatues. In conclusion the influence of grain boundary scattering is seen on the thermal
transport with larger grain sizes exhibiting less scattering, which in turn leads to higher lattice
thermal conductivities.
Figure 5.15: Left: Lattice thermal conductivities of Ca3AlSb3 and Sr3GaSb3 with tempera-
ture. Right: The distinct chain-like poly-anions formed by the Zintl-antimonides
Ca3AlSb3 and Sr3GaSb3. Sb (orange) resides at the tetrahedral corners, while ei-
ther Ga (green) or Al (blue) resides in the centers. The difference in the crystal
structures and densities lead to the difference of 0.2 Wm−1K−1.
The large number of atoms per unit cell in Sr3GaSb3 (N=56), lead, by definition, to a
phonon dispersion with 3 acoustic modes and 55×3 optical modes. Such complexity in the
dispersion leads to both enhanced Umklapp scattering and flattened, low velocity optical
modes that contribute very little to heat transport.[101, 102] These features, combined with
the small grain size in our Sr3GaSb3 samples (see Supporting Information in Chapter 8), lead
to the extremely low lattice thermal conductivity observed in this material.
To estimate the minimum lattice thermal conductivity κmin for Sr3GaSb3, and thus to de-
termine whether further nano structuring would be beneficial, we have employed Cahill’s
formula for disordered crystals (see Equation 5.25).[143] κmin is dependent on the longitudi-
nal and transverse sound velocities, which were determined from ultrasonic measurements to
be 3750 m/s and 2130 m/s, respectively. These yield an estimated Debye temperature of θD
= 243 K and κ = 0.43 Wm−1K−1min . At high temperature, the lattice thermal conductivity of
Sr3GaSb3 approaches the estimated κmin, in common with Ca3AlSb3, adding to the growing
body of evidence suggesting that large unit cell size leads to the glass-like high temperature
κL observed in a number of thermoelectric materials.
In the left panel of Figure 5.15 are the lattice thermal conductivities of Ca3AlSb3 and
Sr3GaSb3, estimated from the measured diffusivity of Zn-doped samples. Though both com-
pounds exhibit an identical temperature dependence indicative of Umklapp-scattering, κL in
Sr3GaSb3 is about 25% lower across the entire temperature range. Accounting for traditional
82
5.6 Ca3AlSb3 and Sr3GaSb3
factors such as M̄, V, and sound velocity explains only ∼10% of the disparity. This assumes
that the Grüneisen parameter does not vary greatly between these three compounds (see
Equation 5.16), which is reasonable given the same tetrahedral symmetry and the bonding
interaction. The remaining difference in κL can be explained by the exceptionally large unit
cell of Sr3GaSb3 (N= 56), which is twice the size of Ca3AlSb3 (N= 28) and thus significantly
reduces the acoustic lattice thermal conductivity (Equation 5.20). The resulting large unit cell
and comparatively high density in Sr3GaSb3 combine to yield the exceptionally low lattice
thermal conductivity observed in this material (with κL = 0.45 Wm−1K−1 at 1000 K).
83

6 Thermoelectric Efficiency
6.1 Summary
This chapter focuses on the thermoelectric performances of the materials discussed in this the-
sis: Cu2+xZn1−xGeSe4, Cu2Zn1−xFexGeSe4, Cu2+xZn1−x−yFeyGeSe4, Ca3AlSb3 and Sr3GaSb3.
The chapter begins with a short review of the thermoelectric figure of merit, which has
been extensively introduced in the Introduction in Chapter 1. Then the important equations
which have been used for the transport modeling will be introduced, with the goal to assess
the optimum carrier concentration in the materials discussed.
As seen in Chapter 4 the substitution of Zn2+ with Cu+ introduces holes as charge carriers
in Cu2Zn1−xGeSe4 leading to a lower thermopower and higher electrical conductivities. The
resulting carrier concentrations result in an increased power factor compared to the "undoped"
material. This, in combination with the lower lattice thermal conductivity, due to a nano sized
phase segregation of Cu2−δSe, leads to an increased figure of merit of 0.45 at 670 K for x =
0.075.
In the solid solutions of Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4 the increased band
effective mass (see Chapter 4) and higher valley degeneracy, and therefore increased Seebeck
coefficients lead to an increased intrinsic figure of merit zT of 0.3 at 670 K for x = 0.6, 0.7 and
0.8 compared to the end members Cu2ZnGeSe4 (0.1 at 670 K) and Cu2FeGeSe4 (0.1 at 670 K).
In Ca3AlSb3 and Sr3GaSb3, doping with zinc increases the figure of merit due to increased
carrier concentrations, leading to a zT of 0.5 at 900 K for Ca3AlSb3. However the peak zT
values of the sodium doped Ca3AlSb3[95] material cannot be achieved with zinc due to the
lower carrier concentrations, which are well below the optimum carrier concentration. The
doping of Sr3Ga1−xZnxSb3 with zinc leads to an increase in the maximum zT from 0.4 for x
= 0 to greater than 0.9 when x = 0.07. The improved peak zT of Sr3GaSb3 compared with
Ca3AlSb3 can be attributed almost entirely to its lower lattice thermal conductivity, resulting
from the larger unit cell and comparatively high mass density in Sr3GaSb3.
6.2 Thermoelectric Efficiency
The efficiency of a thermoelectric material is assessed through the materials dimensionless
figure of merit zT (see Chapter 1 and Equations 1.8 and 1.9):
α2σ
zT = T
κ
and
85
6 Thermoelectric Efficiency
α2σ
zT = T ,
κL + κel
with the power factor PF = α2σ, determined by the Seebeck coefficient α, the electrical
conductivity σ, the Lorenz number L, and the lattice κL and electronic contribution κel to the
total thermal conductivity κ.
The figure of merit has a maximum at a given optimum carrier concentration as seen in
Figure 1.3. It is therefore necessary to determine the optimum carrier concentration in a given
material to be able to tune the zT to a maximum value at a certain temperature. The figure
of merit can be written as a function of the reduced electrochemical potential η (see Section
4.2):[21]
α2
zT =
L + (ψβ)−1
, (6.1)
where
µ (m∗/m )3/2T5/20 e
β = , (6.2)
κL
(1/2 + 2λ)F
µ = µ 2λ−1/2
(η)
H 0 , (6.3)
(1 + λ)Fλ(η)
and
( )
8πe 2mek
3/2
= Bψ 2 (1 + λ)Fλ(η) . (6.4)3 h
The values of α and L can be calculated via Equations 4.20 and 5.30, respectively. The
electron mass me and µ0 can extracted from µH at the temperature of interest.[21]
This method is based on the assumption of a single parabolic band and is used for the Zintl
phases in this Chapter under the assumption of acoustic phonon scattering (λ = 0).
6.3 Cu2+xZn1−xGeSe4
This section is an adapted reproduction, from J. Am. Chem. Soc 2012, 134, 7147-7154.[55]
Reproduced with permission of the American Chemical Society Copyright 2012.
The power factor and figure of merit are shown in Figure 6.1. The carrier concentration
of 7.7x1019 cm−3 of x = 0.025 leads to the highest power factor in the material, while higher
doping levels introduce too many charge carriers (see Section 4.3). This doping level seems to
achieve the optimum carrier concentration for this material. However, the figure of merit of
the samples with x = 0.075 are higher than those with x = 0.025 due to a statistically significant
reduction of the lattice thermal conductivity induced by the nano sized impurities located at
the grain boundaries (see Section 5.3). The material exhibits the highest figure of merit of
0.45 at 670 K for x = 0.075. The maximum zT is not yet reached in this temperature range,
because the wide band gap of these compounds effectively suppresses the thermal activation
of minority charge carriers. Additionally, the incorporation of a protective coating to prevent
86
6.4 Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4
the evaporation of selenium will stabilize the material for use and measurement at higher
temperatures that should result in higher figure of merit values as seen for Cu2ZnSnSe4 (zT =
0.45 at 700K and 0.9 at 860K).[57, 59]
Figure 6.1: Temperature dependence of the power factor PF (left) and figure of merit zT (right)
for the solid solutions Cu2+xZn1−xGeSe4. The highest power factor is achieved for
the carrier concentration at the doping level of x = 0.025, but the highest figure of
merit is for x = 0.075 due to the reduction of lattice thermal conductivity by nano
sized impurities.
6.4 Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4
Power factors and figure of merits with temperature of the solid solution Cu2Zn1−xFexGeSe4
are shown in Figure 6.2 and 6.3, respectively. The data for the solid solution Cu2+xZn1−x−yFeyGeSe4
can be found in the Supporting Information in Chapter 8.
Figure 6.2: Temperature dependence of the power factor PF of Cu2Zn1−xFexGeSe4 for x = 0.0
- 0.4 (left) and x = 0.5 - 1.0 (right). The highest power factors are achieved for the
materials compositions which exhibit the highest effective masses and Hall carrier
concentrations.
87
6 Thermoelectric Efficiency
The increased Hall carrier concentration due to the defects and the multi-band behavior
from Fe substitution result in an increase of the power factor from 0.1 mWK−2m−1 at 670 K
for Cu2ZnGeSe4 to a maximum of ∼ 0.4 mWK−2m−1 for x = 0.2, 0.5, 0.6, and 0.7, respectively.
As expected, the highest power factors are achieved for the compositions with the highest
Hall carrier concentrations and highest effective masses showing the high influence on the
transport.
Figure 6.3: Temperature dependence of the figure of merit zT of Cu2Zn1−xFexGeSe4 for x =
0.0 - 0.4 (left) and x = 0.5 - 1.0 (right). The high power factors resulting from
high effective masses at higher Hall carrier concentrations in combination with the
reduction of the lattice thermal conductivities due to anisotropic strain lead to a
figure of merit of nearly 0.3 at 670 K for these intrinsic solid solutions.
The combination of the reduced lattice thermal conductivities due to anisotropic structural
disorder resulting from the 3-stage cation restructuring process (as seen in Section 5.4), show-
ing a minimum at x = 0.7, with the high power factors results in the reasonable high figure of
merits of ∼ 0.3 for an intrinsic semiconductor.
In conclusion we have seen an enhancement of the figure of merit in the intrinsic solid
solutions Cu2Zn1−xFexGeSe4 and Cu2+xZn1−x−yFeyGeSe4 due to a reduction of the lattice
thermal conductivity combined with higher effective masses from Fe substitution.
6.5 Ca3AlSb3
This section is an adapted reproduction, from J. Mater. Chem 2012 22, 9826-9830.[96] Repro-
duced with permission of The Royal Society of Chemistry Copyright 2012.
The figure of merit of Ca3Al1−xZnxSb3 is shown in Figure 6.4 (right). Doping with zinc
increases the figure of merit, leading to a zT of 0.5 at 900 K for x = 0.01. Compared with
the Na-doped samples of the previous investigation,[95] all of the Zn-doped samples exhibit
enhanced zT at intermediate temperatures, which can be attributed to their higher carrier
mobilities. The x = 0.01 sample, which has the largest grain size and thus the highest mobility,
shows the largest degree of zT enhancement. However, the peak zT values of the sodium
doped Ca3AlSb3 samples cannot be achieved with zinc due to its limited dopant effectiveness
when compared to Na (see Chapter 4), as seen in Figure 6.4 (left).
88
6.6 Sr3GaSb3
Figure 6.4: Left: Calculated zT vs carrier concentration (broken line) with the experimental
zT values at 700 K. The dashed line was generated using a single parabolic band
model, with an effective mass of 0.8 me, an intrinsic mobility of 15 cm2V−1s−1 and
lattice thermal conductivity of 0.78 Wm−1K−1. Both Zn and Na[95] doped mate-
rials are well described by the same model, suggesting that using Zn as a dopant
does not affect the band structure of the compound. It can be seen, that the peak
zT values of the sodium doped Ca3AlSb3 samples can not be achieved with zinc
due to its limited dopant effectiveness when compared to Na. The deviations of the
samples with x = 0.01 are due to the higher intrinsic mobility due to the larger grain
size. Right: Experimental figure of merits zT of Ca3Al1−xZnxSb3. The reduction of
grain boundary surface area improves the figure of merit at intermediate temper-
atures, seen for x = 0.01. Experimental data is compared to Ca2.94Na0.06AlSb3 data
(black line).[95]
This study shows the effect of different grain sizes on the thermoelectric transport proper-
ties, with larger grains leading to higher thermal conductivities and higher mobilities at lower
temperatures, due to reduced scattering of phonons and electrons at the grain boundaries.
Ultimately, we find that while larger grain size has little effect on the high temperature proper-
ties, the reduction of grain boundary surface area improves the figure of merit at intermediate
temperatures. This work shows the common strategy to use nano structured materials does
not always lead to an improved figure of merit. Other competing factors such as a reduction
in mobility may overwhelm the improvement of the lattice thermal conductivity.
6.6 Sr3GaSb3
This section is an adapted reproduction, from Energy Environ. Sci. 2012, 5, 9121-9128.[100]
Reproduced with permission of The Royal Society of Chemistry Copyright 2012.
The figure of merit of Sr3Ga1−xZnxSb3 is shown in Figure 6.5 as a function of temperature.
The transition from nondegenerate behavior to degenerate semiconducting behavior upon
doping with zinc leads to an increase in the maximum zT from 0.4 for x = 0.0 to greater than
0.9 when x = 0.07. To provide a rough estimate of the optimum carrier concentration in this
material we have employed an SPB model at 700 K (see Figure 6.5). At 700 K, it is reason-
able to assume that extrinsic carriers still dominate transport and acoustic phonons are the
89
6 Thermoelectric Efficiency
Figure 6.5: Left: zT with carrier concentration approximately according to an SPB model at
700 K. The broken curve represents the predicted SPB behavior of zT, respectively,
assuming m∗ = 0.9 m , µ = 17 cm2V−1s−1e 0 and κL = 0.55 Wm−1K−1. The optimum
carrier concentration for a maximum zT has been obtained in these doping levels.
Right: The measured figure of merit of Sr3Ga1−xZnxSb3 exceeds 0.9 when heavily
doped.
primary scattering source. Using input parameters m∗ = 0.9 m , µ = 17 cm2V−1e 0 s−1 and κL =
0.55 Wm−1K−1, the predicted optimum carrier concentration is approximately 5x1019 h+/cm3.
The doped samples in this study have carrier concentrations clustered around this optimum,
explaining their similar zT values.
Relative to the Zintl compound Ca3AlSb3, Sr3GaSb3 exhibits a higher figure of merit across
the entire measured temperature range. The improved peak zT of Sr3GaSb3 (zT ∼ 0.9 at
1000 K) compared with Ca3AlSb3 (zT ∼ 0.75) can be attributed almost entirely to its lower
κL. Shown in Figure 5.15 are the lattice thermal conductivities of each compound, estimated
from the measured diffusivity of Zn-doped samples. Though all three compounds exhibit an
identical temperature dependence indicative of Umklapp scattering, κL in Sr3GaSb3 is about
25% lower across the entire temperature range. Accounting for traditional factors such as
M̄, V, and sound velocity explains only ∼10% of the disparity, assuming that the Grüneisen
parameter does not vary greatly between these three compounds (see Equation 5.16). The
remaining difference in κL can be explained by the exceptionally large unit cell of Sr3GaSb3
(N= 56), which is twice the size of Ca3AlSb3 and thus significantly reduces the acoustic lattice
thermal conductivity.
In conclusion, doping with Zn2+ on the Ga3+ site leads to degenerate semiconducting be-
havior, allowing us to obtain p-type carrier concentrations near the optimum value predicted
by a single parabolic band model. The combination of low lattice thermal conductivity, rea-
sonable electronic mobility, and a sufficiently large band gap to maintain degenerate behavior
at high temperature leads to a zT of 0.9 at 1000 K. Compared with previously studied chain-
based Zintl compounds, both the peak zT and integrated zT of Zn-doped Sr3GaSb3 samples
are significantly higher.
90
7 Conclusion
This thesis focuses on the structural and thermoelectric properties of quaternary copper se-
lenides and Zintl compounds and tries to link the materials structures to their transport prop-
erties.
The thermoelectric and structural properties of the solid solution Cu2+xZn1−xGeSe4 and
the effect of substitution of Zn2+ with Cu+ have been investigated. A phase transition at
450 K has been identified, and it has an influence on the transport properties in this material,
leading to an insulator-to-metal transition. Temperature dependent X-ray diffraction does
not reveal a structural phase transformation suggesting that this phase transformation is an
electronic phase transition only. The substitution introduces holes as charge carriers, but also
leads to the formation of impurities by phase segregation, due to a charge imbalance in this
material, which greatly influences the charge carrier concentration. No striking influence
on the mobilities and effective masses of the charge carriers can be detected suggesting that
the impurities do not affect the percolation of the holes. However, these nanometer sized
impurities are primarily located at the grain boundaries causing a reduction of the lattice
thermal conductivity by ∼15 % due to enhanced phonon scattering at the grain boundaries,
ultimately leading to an enhanced figure of merit in this material.
Furthermore, the systematic substitution of Zn2+ with Fe2+ has been considered in the
solid solution Cu2Zn1−xFexGeSe4. While the intrinsic electronic nature of the material is not
changed upon substitution with Fe2+, a distinct change in the lattice parameter c/a-ratio
has been observed, which can be explained by a 3-stage cation restructuring and disordering
process of Fe, Zn, and Cu. The differences in bonding interaction between the cations and the
anions lead to different bond lengths and bond angles, resulting in the observed trend in the
lattice parameters. The restructuring leads to a reduction of the lattice thermal conductivity by
15 % at 70 % Fe content. As the disorder scattering of the phonons, due to mass fluctuation and
isotropic strain is expected to be very small from these similar mass and similar volume ions,
this work shows the potential of significant point defect scattering due to local anisotropic
strain that also results in a change in c/a-ratio.
The thermoelectric properties of zinc-doped Ca3AlSb3 were compared to a previously re-
ported study on sodium as a dopant. A similar band mass and lattice thermal conductivity
is observed regardless of whether Zn or Na is the dopant. However, zinc exhibits a lower
solubility limit than sodium resulting in lower carrier concentrations and a lower peak figure
of merit, because the optimum carrier concentration for this material cannot be reached via
doping with zinc. This study also shows the effect of different grain sizes on the thermo-
electric transport properties, with larger grains leading to higher thermal conductivities and
higher mobilities at lower temperatures, due to reduced scattering of phonons and electrons
at the grain boundaries. Ultimately, we find that while larger grain size has little effect on the
91
7 Conclusion
high temperature properties, the reduction of grain boundary surface area improves the fig-
ure of merit at intermediate temperatures. This work shows the common strategy to use nano
structured materials does not always lead to an improved figure of merit. Other competing
factors such as a reduction in mobility may overwhelm the improvement of the lattice thermal
conductivity.
At last, we have described a new thermoelectric material, Sr3GaSb3, which has a crystal
structure characterized by chains of corner-linked tetrahedra, similar to the polyanions found
in the previously studied Zintl compounds, Ca5Al2Sb6 and Ca3AlSb3. The large unit cell
and comparatively high density in Sr3GaSb3 combine to yield the exceptionally low lattice
thermal conductivity observed in this study (κ = 0.45 Wm−1K−1L at 1000 K). High temperature
transport measurements reveal that Sr3GaSb3 is a nondegenerate p-type semiconductor, with
a relatively large band gap and high electronic mobility. Substitution of Ga with Zn leads
to degenerate semiconducting behavior, allowing us to obtain p-type carrier concentrations
near the optimum value predicted by a single parabolic band model. The combination of low
lattice thermal conductivity, reasonable electronic mobility, and a sufficiently large band gap
to maintain degenerate behavior at high temperature leads to a zT of 0.9 at 1000 K. Compared
with previously studied chain-based Zintl compounds, both the peak zT and integrated zT of
Zn-doped Sr3GaSb3 samples are significantly higher.
Ultimately, this thesis shows the potential and need of understanding the structure-to-
property relationships in materials for new thermoelectric materials and their improvement.
Especially the different bonding environments, the elemental chemistry, and the crystal struc-
tures have to be considered to understand these properties and to further design new thermo-
electrics.
92
8 Supporting Information
8.1 X-ray Diffraction
Figure 8.1: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Rietveld fit of phase pure Cu2ZnGeSe4.
93
8 Supporting Information
Figure 8.2: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Rietveld fit of phase pure Cu2+xZn1−xGeSe4.
Figure 8.3: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Rietveld fit of phase pure Cu2+xZn1−xGeSe4.
94
8.1 X-ray Diffraction
Figure 8.4: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Rietveld fit of phase pure Cu2+xZn1−xGeSe4.
Figure 8.5: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Rietveld fit of Cu2.1Zn0.9GeSe4. The inset shows the extra
reflections indexed to Cu2−δSe.
95
8 Supporting Information
Figure 8.6: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
Figure 8.7: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
96
8.1 X-ray Diffraction
Figure 8.8: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
Figure 8.9: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
97
8 Supporting Information
Figure 8.10: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
Figure 8.11: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
98
8.1 X-ray Diffraction
Figure 8.12: X-ray diffraction data including profile fit, profile difference, and profile residuals
of the corresponding Pawley fit of phase pure Cu2Zn1−xFexGeSe4.
Figure 8.13: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2Zn1−xFexGeSe4.
99
8 Supporting Information
Figure 8.14: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2Zn1−xFexGeSe4.
Figure 8.15: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2Zn1−xFexGeSe4.
100
8.1 X-ray Diffraction
Figure 8.16: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2+xZn1−x−yFeyGeSe4.
101
8 Supporting Information
Figure 8.17: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2+xZn1−x−yFeyGeSe4.
Figure 8.18: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2+xZn1−x−yFeyGeSe4.
102
8.1 X-ray Diffraction
Figure 8.19: X-ray diffraction data including profile fit, profile difference, and profile residuals
of phase pure Cu2+xZn1−x−yFeyGeSe4.
Figure 8.20: X-ray diffraction data for Ca3Al1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phases of Ca5Al2Sb6 and Ca14AlSb11. The inset shows the reflections
indexed to the impurity phases, observed in all samples.
103
8 Supporting Information
Figure 8.21: X-ray diffraction data for Ca3Al1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phases of Ca5Al2Sb6 and Ca14AlSb11.
Figure 8.22: X-ray diffraction data for Ca3Al1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phases of Ca5Al2Sb6 and Ca14AlSb11.
104
8.1 X-ray Diffraction
Figure 8.23: X-ray diffraction data for Ca3Al1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phases of Ca5Al2Sb6 and Ca14AlSb11.
Figure 8.24: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phase Sr2Sb3. The inset shows the reflections indexed to the impurity
phase marked by asterisks.
105
8 Supporting Information
Figure 8.25: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phase Sr2Sb3.
Figure 8.26: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phase Sr2Sb3.
106
8.1 X-ray Diffraction
Figure 8.27: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement including the
secondary phase Sr2Sb3.
Figure 8.28: X-ray diffraction data for Sr3Ga1−xZnxSb3 including profile fit, profile difference,
and profile residuals from the corresponding Pawley refinement. The inset shows
the impurities at lower Bragg angles.
107
8 Supporting Information
Figure 8.29: X-ray diffraction data for GaSb including profile fit and profile difference, show-
ing a phase pure precursor GaSb.
108
8.2 Scanning Electron Microscopy
8.2 Scanning Electron Microscopy
Figure 8.30: SEM micrographs of Cu2Zn1−xFexGeSe4 in backscattered (QBSD) and secondary
electron mode (SE2).
Figure 8.31: SEM micrographs of Cu2Zn1−xFexGeSe4 in backscattered (QBSD) and secondary
electron mode (SE2).
109
8 Supporting Information
Figure 8.32: SEM micrographs of Cu2Zn1−xFexGeSe4 in backscattered (QBSD) and secondary
electron mode (SE2).
Figure 8.33: SEM micrographs of Cu2Zn1−xFexGeSe4 in backscattered (QBSD) and secondary
electron mode (SE2).
Figure 8.34: SEM micrographs of Cu2Zn1−xFexGeSe4 in backscattered (QBSD) and secondary
electron mode (SE2).
110
8.2 Scanning Electron Microscopy
Figure 8.35: SEM micrograph of a fractured surface of Ca3Al1−xZnxSb3 in secondary electron
mode.
Figure 8.36: SEM micrographs of fractured surfaces of Ca3Al1−xZnxSb3 in secondary electron
mode.
Figure 8.37: SEM micrographs of fractured surfaces of Ca3Al1−xZnxSb3 in secondary electron
mode.
111
8 Supporting Information
Figure 8.38: SEM micrographs of polished surfaces of Ca3Al1−xZnxSb3 in secondary electron
mode.
Figure 8.39: SEM micrographs of polished surfaces of Ca3Al1−xZnxSb3 in secondary electron
mode.
Figure 8.40: SEM micrographs of fractured surfaces of Sr3Ga1−xZnxSb3 in secondary electron
mode.
112
8.2 Scanning Electron Microscopy
Figure 8.41: SEM micrographs of fractured surfaces of Sr3Ga1−xZnxSb3 in secondary electron
mode.
Figure 8.42: SEM micrographs of polished surfaces of Sr3Ga1−xZnxSb3 in secondary electron
mode.
Figure 8.43: SEM micrograph of polished surfaces of Sr3Ga1−xZnxSb3 in secondary electron
mode.
113
8 Supporting Information
8.3 Speed of Sound Data
Figure 8.44: Respective longitudinal (left) and transverse (right) sound response data of an
ultrasonic sound/response measurement of Cu2Zn0.5Fe0.5GeSe4.
Figure 8.45: Respective longitudinal (left) and transverse (right) sound response data of an
ultrasonic sound/response measurement of Cu2FeGeSe4.
114
8.3 Speed of Sound Data
Figure 8.46: Respective longitudinal (left) and transverse (right) sound response data of an
ultrasonic sound/response measurement of Ca3AlSb3.
Figure 8.47: Respective longitudinal (left) and transverse (right) sound response data of an
ultrasonic sound/response measurement of Sr3GaSb3.
115
8 Supporting Information
8.4 Additional Transport Data
Figure 8.48: Temperature dependent Hall carrier concentrations nH of Cu2Zn1−xFexGeSe4 for
x = 0.0 - 0.4 (left) and x = 0.5 - 1.0 (right).
Figure 8.49: Temperature dependence of the Seebeck coefficients α (left) and resistivities ρ
(right) of Cu2.025Zn0.975−yFeyGeSe4.
116
8.4 Additional Transport Data
Figure 8.50: Temperature dependence of Hall carrier concentrations nH (left) and Hall mobili-
ties µH (right) of Cu2.025Zn0.975−yFeyGeSe4.
Figure 8.51: Resistivities ρ and Hall mobilities µH of Sr3Ga0.93Zn0.07Sb3 measured on disks cut
in different directions (perpendicular and parallel to the hot pressing direction).
Figure 8.52: Hall carrier concentrations nH of Sr3Ga0.93Zn0.07Sb3 measured on disks cut in dif-
ferent directions (perpendicular and parallel to the hot pressing direction)
117
8 Supporting Information
Figure 8.53: Temperature dependence of the total thermal conductivity κ (left) and Lorenz
numbers L (right) of Cu2.025Zn0.975−yFeyGeSe4.
Figure 8.54: Temperature dependence of electronic thermal conductivity κel (left) and lattice
thermal conductivity κL (right) of Cu2.025Zn0.975−yFeyGeSe4.
Figure 8.55: Temperature dependence of calculated Lorenz numbers (left) and electronic ther-
mal conductivity κel (right) of Ca3Al1−xZnxSb3.
118
8.4 Additional Transport Data
Figure 8.56: Temperature dependence of calculated Lorenz numbers L (left) and electronic
thermal conductivity κel (right) of Sr3Ga1−xZnxSb3.
Figure 8.57: Temperature dependence of power factor (left) and figure of merit (right) of
Cu2.025Zn0.975−yFeyGeSe4.
119
8 Supporting Information
8.5 Additional Pictures and Images
Figure 8.58: Surface of consolidated disks of Cu2ZnGeSe4 after a laser flash diffusivity mea-
surement up to 723 K (left) and 673 K (right). Above 673 K significant evaporation
of selenium takes place, resulting in the yellowish color of the ternary phases.
120
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