HIGH-TEMPERATURE ELASTICITY AND ANHARMONICITY IN LAYERED

THERMOELECTRIC MATERIALS

By

Wanyue Peng

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Materials Science and Engineering – Doctor of Philosophy

2021

ABSTRACT

HIGH-TEMPERATURE ELASTICITY AND ANHARMONICITY IN LAYERED

THERMOELECTRIC MATERIALS

By

Wanyue Peng

The ability to predict materials with desired thermal conductivity from a large material database
can significantly improve the efficiency of experimental work. Lattice thermal conductivity is
controlled by the velocity and relaxation time of phonons (lattice vibrations). Phonon scattering is
closely related to the anharmonic lattice vibrations of a material, while phonon velocity depends
on density and, bond stiffness. In this research, the relationship between structure, bonding, and
thermal properties is discussed in two classes of layered materials, 

2
2 intermetallic compounds
and GeTe - Sb2Te3 alloys.

First, we study the origin of the anomalously low lattice thermal conductivity of MgMg2Sb2
compare to other isostructural 
Mg2Pn2 compounds (
 = Mg, Ca, Yb, and 

 = Sb and Bi ).
By employing high-temperature X-ray diffraction (XRD) and resonant ultrasound spectroscopy
(RUS) techniques, we have shown that the low lattice thermal conductivity is due to previously-
unrecognized soft shear modes and highly anharmonic acoustic phonons in layered MgMg2Sb2.
Combined with the phonon calculations from our collaborators, we attribute the anomalous thermal
behavior of MgMg2Sb2 to the instability of the vibrational modes that originated from the weak
bonding of the Mg, which is too small for the octahedral site. Second, we investigate the phase
stability of the 
Mg2Pn2 system with mixed occupancy of Mg, Ca, Sr, or Ba on the cation
(
) site. We show that the small ionic radius of Mg2+ leads to limited solubility when alloyed
with larger cations such as Sr or Ba. Third, by performing in-situ high-pressure synchrotron X-ray
diffraction, we showed that a few 

2
2 compounds can exhibit phase transitions at high-pressure,
most of which are previously unrecognized. In addition, we observed that the compressibility of
MgMg2Sb2 and MgMg2Bi2 is near-isotropic, whereas other isostructural 

2
2 compounds
show clear signs of anisotropy between the in-plane and out-of-plane compressibility as is typical

of layered compounds. We have analyzed the compressibility, transition pressure/temperature,
anisotropy, as well as the type of phase transition to develop a deeper understanding of the stability
and bond strength of different 

2
2 compounds.

Lastly, we observed and explained the lattice stiffening and flattened lattice thermal conductivity
curve with increasing temperature in GeTe - Sb2Te3 alloys. Unlike most compounds that soften with
increasing temperature, the elastic moduli of (GeTe)17 - Sb2Te3 stiffen with increasing temperature
before the phase transition. We investigate GeTe, Sb2Te3, and (GeTe)17 - Sb2Te3 from room
temperature up to the phase transitions with high-temperature XRD, high-temperature RUS, and
transport property measurements. We attribute the stiffening behavior to the gradual diffusion
of layered vacancies to random vacancies on the cation site, which profoundly impact the elastic
properties and the transport properties of the material.

Copyright by
WANYUE PENG
2021

ACKNOWLEDGEMENTS

I would like to thank my research advisor Alex Zevalkink for giving me guidance and advice all
the time during my Ph.D. years. She is the most patient and understanding advisor I’ve ever met -
trying to answer all my questions, listening to all the ideas, and showing me every skill that I asked
for. None of this work would have been possible without her help.

I would also like to thank Dr. Donald Morelli for generously allowing me to use his lab. I can’t
believe how much less work I would do without the SPS machine. Special thanks to his group
member Spencer Waldrop , Spencer Mathers, Jared Williams, and Dan Weller for teaching me and
helping me out in the lab.

I have also received huge support from Professor Susannah Dorfman and all her group members.
The later few years of my research have been switched to high-pressure synthesis, which we have
no experience in our own group. She is super patient and tolerant during my learning process. We
also borrowed her diamond anvil cells and much less work can be done without her equipment. I’d
also like to thank Ben Brugman to teach me how to use a diamond anvil cell and helped us with the
measurement during the beamtime. Mingda Lyu taught me the skills in loading samples efficiently.
I’d also like to thank Alex to initiate this collaboration that makes everything possible.

I’d also like to thank all of our collaborators. The DACs from James Walsh, the funding from
Jet Propulsion Laboratory, the calculation works from Jeffery Hautier’s group for the Mg3Sb2
project, the discussions and calculations works from Dr. Oliver Delaire’s group, and advice from
Jeff Snyder’s group and Eric Toberer’s group.

I would also like to thank everyone who helped the experiment during the beamtime to make
the high-pressure study possible. Gill, Mario, Alex, Megan, and Ben has been there almost every
single time. Special thanks to Alex and Gill for all the driving. The beamline scientists Dmitry,
Curits, and Dongzhou have offered tremendous help for setting up the experiment and trouble
shooting whenever things go wrong. Thanks to Sergey who gas-loaded all of our cells in time.

v

TABLE OF CONTENTS

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
LIST OF TABLES .
x
LIST OF FIGURES .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 1 BACKGROUND AND INTRODUCTION . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1
. . . . . . . . . . . . . . . . . . . . . .
1.2.2 Umklapp scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3
. . . . . . . . . . . . . . . .
1.2.4 Literature review of temperature-dependent elastic moduli of thermo-

1.1 Thermoelectric materials .
1.2 Thermal transport .

Softening of elastic moduli with temperature

Speed of sound and elastic moduli

1
1
2
3
5
7

.

.

.

1.3 

2
2 Zintl phases .

1.2.5.1
1.2.5.2

electric materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.5 Grüneisen parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Thermal expansion Grüneisen parameter
. . . . . . . . . . . . . 13
Poisson’s ratio Grüneisen parameter . . . . . . . . . . . . . . . . 14
. 15
1.3.1 The formation rule of Zintl phases . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Chemistry of 

2
2 compounds in CaAl2Si2 structure type . . . . . . . . 16
1.3.3 

2
2 compounds as thermoelectrics . . . . . . . . . . . . . . . . . . . . 17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

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

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1.4 Summary of research .

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CHAPTER 2 METHODS .
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2.1 Overview .
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2.2 Synthesis
. .
2.3 Characterization .
.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . 23
. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
In-situ high-pressure synchrotron diffraction in diamond anvil cells (DACs)
28
2.3.3.1
Loading samples in diamond anvil cells . . . . . . . . . . . . . . 29
2.3.3.2 High-pressure X-ray diffraction experiment details . . . . . . . . 30

2.3.1 Resonant ultrasound spectroscopy (RUS)
2.3.2 X-ray diffraction (XRD)
2.3.3

2.3.4 Calculation details for the phonon density of states and mode Grüneisen

parameters of Mg3Sb2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CHAPTER 3 EFFECT OF CATION SIZE ON THE LATTICE DYNAMICS IN 

2
2

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. .
. .

Introduction .
. .

COMPOUNDS .
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.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
.
3.2 Synthesis
3.3 Experimental temperature dependence of elastic moduli and speed of sound . . .
. 37
3.4 Lattice dynamics and Grüneisen parameters of 
Mg2

2 . . . . . . . . . . . . . . 40
3.5 Breaking of Pauling’s radii rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.6 Conclusion .

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vi

CHAPTER 4 LIMITS OF CATION SOLUBILITY IN 
Mg2Sb2 (
=Mg, Ca, Sr, Ba)

ALLOYS .
. .
. .

.
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4.1
Introduction .
4.2 Experimental
.
4.3 Results and discussion .
.
4.4 Conclusion .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
.

CHAPTER 5 HIGH-PRESSURE COMPRESSIBILITY, ANISOTROPY AND PHASE

.

.

.

.

.

. .

Introduction .

TRANSITIONS OF 

2
2 COMPOUNDS . . . . . . . . . . . . . . . . . 52
5.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Phase transition of Mg3Sb2 and Mg3Bi2 at high-pressure . . . . . . . . . . . . . . 55
5.3 Compressibility and anisotropy of Mg3Sb2 and Mg3Bi2 . . . . . . . . . . . . . . . 55
5.4 Compressibility and anisotropy of ternary 

2
2 compounds . . . . . . . . . . . 61
5.5 High-pressure phase transition of ternary 

2
2 . . . . . . . . . . . . . . . . . . 65
5.6 High-pressure high-temperature phase transition and intermediate phase explo-

ration of EuAl2Si2 and SrAl2Si2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.7 Concluding remarks .

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ALLOYS .

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CHAPTER 6 LATTICE HARDENING DUE TO VACANCY DIFFUSION IN (GeTe)
Sb2Te3
. 72
6.1 GeTe-Sb2Te3 homologous series . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 Structure and phase transitions of GeTe-Sb2Te3 alloys . . . . . . . . . . . . . . . . 73
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3 Experimental
6.3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.2 High-temperature resonant ultrasound spectroscopy . . . . . . . . . . . . . 76
6.3.3 High-temperature X-ray diffraction . . . . . . . . . . . . . . . . . . . . . . 76
. 76
. . . . . . . . . . . . . . . . . . . . . . . . 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.4 Structural evolution with increasing temperature . . . . . . . . . . . . . . . . . .
6.5 Temperature-dependent elastic moduli
6.6 Conclusions .

. .
Synthesis

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CHAPTER 7 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1 Possible explanations for the large anharmonicity of Mg3Sb2 . . . . . . . . . . . . 89
7.2 Lattice hardening of GST alloys
. 91
7.3 Opportunities and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

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

.

APPENDIX .
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BIBLIOGRAPHY .

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
. 97
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

LIST OF TABLES

Table 3.1: Maximum temperature and hold time used during spark plasma sintering of


Mg2

2 (
=Mg, Ca, Yb and 

=Sb, Bi) samples.

. . . . . . . . . . . . . . . 36

Table 4.1: The SPS temperatures of (Ca
Mg1−
)Mg2Sb2, (Sr
Mg1−
)Mg2Sb2, (Ba
Mg1−
)

Mg2Sb2, and (Ba
Ca1−
)Mg2Sb2. . . . . . . . . . . . . . . . . . . . . . . . . . 46

Table 5.1: Transition pressure of 
Mg2

2 compounds and its relationship with the ionic

radius. The transition pressure increases as the r





 : r




 increases. . . . . . 67

Table 6.1: Experimental elastic moduli and speed of sound at 300 K for (GeTe)17Sb2Te3,
Sb2Te3, GeTe, and Bi2Te3 samples measured using resonant ultrasound spec-
troscopy.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

. .

. .

.

.

Table A1: Lattice parameters and R
 
 values from refinements of powder XRD patterns

of (Ba
Ca1−
)Mg2Sb2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Table A2: Lattice parameters and R
 
 values from refinements of powder XRD patterns

of (Ca
Mg1−
)Mg2Sb2.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Table A3: Lattice parameters and R
 
 values from refinements of powder XRD patterns
of (Sr
Mg1−
)Mg2Sb2. With the exception of the 
 = 0.9 sample, all alloyed
samples contain both a Sr-rich and a Mg-rich phase.

. . . . . . . . . . . . . . . 94

Table A4: Lattice parameters and R
 
 values from refinements of powder XRD patterns
of (Ba
Mg1−
)Mg2Sb2. All alloyed samples separated into a Ba-rich and a
Mg-rich phase, suggesting zero solubility.

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

. 95

Table A5: The parameters from the 2nd-order Birch–Murnaghan EOS fit for Mg3Sb2
and Mg3Bi2 corresponding to Figure 5.5. The V0 for the ambient structures
(
¯3
1) were obtained via Rietveld refinement of the corresponding room tem-
perature XRD pattern. For the high-pressure phase (
2/
), V0 is unknown,
so it was treated as an open fitting parameter in the Birch–Murnaghan EOS fit.
Table A6: Lattice parameters and uncertainties of the 
¯3
1 Mg3Sb2 phase from powder
XRD refinements using PDXL2. The lattice parameters, and peak shape
parameters are refined. P1 and P2 are pressure readings before and after the
measurement of each pattern. The deviation of the average value between P1
and P2 is used as the pressure uncertainty for the Birch–Murnaghan equation
of state fit.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

. 95

viii

Table A7: Lattice parameters and uncertainties of the 
2/
 Mg3Sb2 phase from powder
XRD refinements using PDXL2. The lattice parameters, peak shape parame-
ters, z and x are refined. The deviation of the average value between P1 and P2
(pressures measured before and after sample exposure) is used as the pressure
uncertainty for the Birch–Murnaghan equation of state fit.

. . . . . . . . . . . . 96

Table A8: Lattice parameters and uncertainties of the 
¯3
1 Mg3Bi2 phase from powder
XRD refinements using PDXL2. The lattice parameters, and peak shape
parameters are refined. The deviation of the average value between P1 and P2
(pressures measured before and after sample exposure) is used as the pressure
uncertainty for the Birch–Murnaghan equation of state fit.

. . . . . . . . . . .
Table A9: Lattice parameters and uncertainties of the 
2/
 Mg3Bi2 phase from powder
XRD refinements using PDXL2. The lattice parameters, peak shape parame-
ters, z and x are refined. The deviation of the average value between P1 and P2
is used as the pressure uncertainty for the Birch–Murnaghan equation of state fit.

. 96

96

ix

LIST OF FIGURES

Figure 1.1: A schematic demonstration of acoustic and optical phonon vibrations. Optical
phonons form standing waves with negligible net group velocity, whereas
acoustic phonons dominate the heat transport of a material due to their large
net group velocity [1, 2].

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

Figure 1.2: An example of the calculated phonon dispersion of the diamond. TA rep-
resents the transverse acoustic branch and LA represents the longitudinal
acoustic branch. All other modes except the three acoustic modes are optical
modes, which are usually of much higher frequency. The slopes of LA and
TA branches at the Ɖ-point are the longitudinal and transverse (shear) speed
of the sound, respectively.

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

Figure 1.3: A schematic demonstration of normal scattering and Umklapp scattering. The
left and right edge of the grey box represent the range of the first Brillouin
zone. 
1 and 
2 are the initial wave vectors before the scattering, and 
3 is
the resulting wave vector after the scattering.

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

Figure 1.4: A schematic demonstration of the temperature dependence of the phonon re-
laxation time of Umklapp scattering compared to other scattering mechanisms
(i.e carrier-carrier scattering and defect scattering). Umklapp scattering be-
comes increasingly important at high-temperature due to the 1/T temperature
dependence of the phonon relaxation time.

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

Figure 1.5: A schematic demonstration of interatomic potential and its relationship with
thermal expansion and elastic moduli. A larger curvature corresponds to a
narrower potential well and thus a smaller thermal expansion and stiffer bond.

.

Figure 1.6: Three selected materials that range from very soft and anharmonic to very
stiff and harmonic. 

 is related to the Einstein temperature, the slope of the
linear region equals the 
/

 ratio, 
0 is the Young’s modulus at the lowest
temperature (∼0 K), and 









 is the Grüneisen parameter [3, 4, 5].
Figure taken from W. Peng et al. [6]

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

3

4

5

7

8

9

Figure 1.7: Temperature-dependence of the Young’s modulus, 
, of the selected thermo-
electric material [7, 8, 9, 10, 11, 12, 13, 14, 12]. Half-heuslers and Si1−
Ge

come to the top to be the stiffest thermoelectric materials due to their short
covalent bonds. The softest crystalline materials are the layered tetradymites
with covalent octahedral layers and vdW bonding. At the lower left cor-
ner, PEDOT, a conducting polymer, is shown for comparison, which is an
extremely soft material. Figure adapted from W. Peng et al. [6] . . . . . . . .

. 11

x

Figure 1.8: The change of the normalized Young’s modulus with temperature of various

compounds [7, 8, 9, 10, 11, 15]. Figure adapted from W. Peng et al. [6] . . . . . 12

Figure 1.9: a) The CaAl2Si2 structure type is formed from [M2
2]2− polyanion slabs sep-
arated by monolayers of six-fold coordinated cation 
. b) Possible element-
combinations to achieve the CaAl2Si2 structure type according to the 8 - N
rule. Figure taken from W. Peng et al. 1.9.

. . . . . . . . . . . . . . . . . . . . 16

Figure 1.10: The 

 of selected 
-type (dashed curves) and 
-type (solid curves) of 

2
2

compounds. Figure taken from W. Peng et al. [16].

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

Figure 2.1: A demonstration of the synthesis approach in this work. Samples were
synthesized by direct ball-milling (under argon) followed by spark plasma
sintering (SPS). The as-synthesized samples have a cylindrical geometry. For
the annealing process, the samples were wrapped in graphite foils and sealed
in glass ampoules under vacuum, which were then held at 500◦
 for ten days.

. 18

. 22

Figure 2.2: a) In pulse-echo ultrasound measurements, the time required for an acoustic
wave to travel across the sample is measured. b) In resonant ultrasound,
the drive transducer emits acoustic waves over a frequency spectrum. Peaks
correspond to the resonant frequencies of the sample. The black dots under
the spectrum are calculated peak locations with the predicted elastic constants.
The inset is an example of the temperature dependence of the peak positions.
This Figure is reproduced from Ref. [17] . . . . . . . . . . . . . . . . . . . . . 25

Figure 2.3: a) Room temperature RUS with a tripod set-up. b) High-temperature RUS set-up. 26

Figure 2.4: The high-temperature X-ray diffraction was performed using a Rigaku 1400HT
stage. The thermocouple goes into the inner part of the platinum tray to in-
crease the accuracy of the temperature measurement.

. . . . . . . . . . . . . . 28

Figure 2.5: A schematic picture of a symmetric diamond anvil cell. Figure taken from

Dong et al. [18]. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Figure 2.6: The X-ray transmission image of Mg3Sb2 and Mg3Bi2 in this study. The low

transmission regions are the positions of the samples.

. . . . . . . . . . . . . . 31

Figure 2.7: The HT-DAC set up. Photo taken from the set-up manual written by Dr. Bin

Chen and his group members in University of Hawaii.

. . . . . . . . . . . . . 32

Figure 3.1: MgMg2Sb2 crystallizes in the CaAl2Si2 structure (space group 
¯3
1), char-
acterized by anionic 
2
2 slabs separated by 
 cations. In MgMg2Sb2 and
MgMg2Bi2, Mg(1) occupies the highly distorted octahedrally-coordinated 
-
site and Mg(2) the tetrahedrally-coordinated 
-site [19]. The Mg-Sb bond
lengths differ significantly in the two sites. Figure taken from W. Peng et al. [20]. 35

xi

Figure 3.2: The experimental lattice thermal conductivity, 

, of MgMg2Sb2 is signif-
icantly lower than isostructural 

2
2 compounds with similar a) density
and b) predicted speed of sound. 

 data can be found in Ref. [16]. The
average speed of sound, 

, was estimated using the calculated elastic moduli
from MaterialsProject.org and experimental densities [21, 22]. The dashed
line is shown as a guide to the eye representing the cube dependence of 

 and
lattice thermal conductivity, 

, if all else remains constant [23]. Compounds
shown in color are the primary focus of the current study. Figure taken from
W. Peng et al. [20].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Figure 3.3: Experimental (asterisk) and computed (circles) elastic moduli of 
Mg2Pn2
compounds tend to decrease as a function of increasing unit cell volume and
bond length. The anomalously low shear moduli of Mg3Sb2 and Mg3Bi2 are
significant, suggesting soft bonding are unique to these two binary compounds
[22].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

.

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.

.

.

.

.

.

.

.

Figure 3.4: Temperature-dependent a) Young’s modulus, b) shear modulus, and c) longi-
tudinal and d) transverse speed of sound measured using resonant ultrasound
spectroscopy. Quantities were normalized to the room temperature value.
Data for Si0.8Ge0.2, PbTe, and SnTe are from Ref. [9, 10, 24]. Figure taken
from W. Peng et al. [20].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Figure 3.5: Experimental thermal expansion of YbMg2Bi2, CaMg2Sb2, CaMg2Bi2, MgMg2Sb2,

and MgMg2Bi2. As we can see, MgMg2Bi2 and MgMg2Sb2 has the high-
est thermal expansion coefficient compared to other 
Mg2

2 compounds.
Figure taken from W. Peng et al. [20].

. . . . . . . . . . . . . . . . . . . . . . 39

Figure 3.6: The partial phonon density of states of MgMg2Sb2, CaMg2Sb2 and CaMg2Bi2
shows that the low-, mid-, and high-frequency regimes are dominated by dis-
placements of the anions, 

, cations, 
, and metal site, 
=Mg(1), respec-
tively. Figure taken from W. Peng et al. [20]. . . . . . . . . . . . . . . . . . . . 40

Figure 3.7: The black lines are the phonon dispersion of MgMg2Sb2, CaMg2Sb2, and
CaMg2Bi2 at 0 K. The volume dependence of the phonon dispersions are
shown through the thickness of the bands, with red and blue representing
positive and negative values of mode Grüneisen parameters, respectively.
Figure taken from W. Peng et al. [20].

. . . . . . . . . . . . . . . . . . . . . . 41

Figure 3.8: The mode Grüneisen parameters as a function of frequency. Figure taken

from W. Peng et al. [20].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xii

Figure 3.9: For octahedral coordination (CN=6), Pauling’s radius ratio rules predict a
minimum stability limit of 






:





 = 0.414. For Ca, Sr, Eu, and
Yb, this rule is satisfied.
In contrast, the Mg cation is too small, which
leads to a distorted octahedral environment and may be responsible for weak,
anharmonic interlayer bonding. Figure taken from W. Peng et al. [20].

. . . . . 44

Figure 4.1: a-b) For the (Ca
Mg1−
)Mg2Sb2 series, the lattice parameters 
 and 
 under-
went a linear change with calcium alloying ratio. c-d) For the (Sr
Mg1−
)Mg2Sb2
series, the lattice parameters indicate a 10% solubility for Sr on Mg site, and
no solubility for Mg on Sr site. e-f) For (Ba
Mg1−
)Mg2Sb2 series, the
lattice parameters show no solubility between Ba and Mg on the cation site.
Note that the MgMg2Sb2 phase can be observed from X-ray diffraction pat-
tern when 
 = 0.8 in the Sr-Mg series and for 
 = 0.8 and 0.9 in the Ba-Mg
series, but the peak intensities are too low for reliable refinement of lattice
parameters. Figure taken from W. Peng et al. [25]. . . . . . . . . . . . . . . . . 48

Figure 4.2: The linearly varying lattice parameters in the (Ba
Ca1−
)Mg2Sb2 alloy indi-
cate complete solid solubility of this system. Figure taken from W. Peng et
al. [25].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

.

.

.

.

.

.

.

.

Figure 4.3: The cation radii difference of 

2
2 calculated from (
1-
2)/
2 for all possi-
ble combinations of cations where 
1 > 
2. In each combination, the smaller
species is listed first. Square symbols represent experimental observations,
while circles represent predictions. The limited solubility in Sr/Mg (present
study) and Sm/Mg (Ref [26]) alloys provide an approximate upper limit for
cation radii mismatch. The complete solubility of Ca/Eu [27, 28, 29], Ca/Yb
[30, 31, 32, 28], and Yb/Eu [28, 33] Mg/Yb [34] have been confirmed by
prior studies. Figure taken from W. Peng et al. [25].

. . . . . . . . . . . . . . . 50

Figure 5.1: Structure types formed by 

2
2 compounds with 
 

 =16 include a) the
trigonal CaAl2Si2 structure type (
¯3
1), b) the orthorhombic BaCu2S2 struc-
ture type (



), and c) the tetragonal ThCr2Si2 structure type (
4/


).
d) The stability range of each structure can be delineated as a function of the
atomic radii, 
 = 
 
/(

 + 0.2
 
), while the polarity of the 
-
 bond is
clearly not a deciding factor [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47,
48, 49, 50, 51, 52, 53, 19, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66].
Figure taken from W. Peng et al. [16].

. . . . . . . . . . . . . . . . . . . . . . 54

Figure 5.2: Powder diffraction patterns of a) Mg3Sb2 and b) Mg3Bi2 at ambient tem-
perature reveal the emergence of new peaks belonging to a high-pressure
phase above 7.8 and 4.0 GPa respectively. Peak bars in blue correspond to
the trigonal ambient and newly-discovered high-pressure phase. The X-ray
wavelength for Mg3Sb2 and Mg3Bi2 is 0.4133 Å and 0.4340 Å, respectively.
Figure taken from W. Peng et al. [6]

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

. 56

xiii

Figure 5.3: The reversibility of the phase transition was confirmed by decompressing the
powder Mg3Bi2 sample. All of the original 
¯3
1 peaks re-emerged below
4.0 GPa. Figure taken from W. Peng et al. [6].

. . . . . . . . . . . . . . . . . . 57

Figure 5.4: Comparison of the a) ambient (
¯3
1) and b) high-pressure (
2/
) structure
of Mg3Sb2. The high-pressure monoclinic structure of Mg3Sb2 can be viewed
as a distorted variant of the original trigonal structure at low pressure. c) The
four Mg coordination environments in the high-pressure structure. Figure
taken from W. Peng et al. [6].

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

. 57

Figure 5.5: The pressure-dependence of the volume per formula unit for Mg3Sb2 and
Mg3Bi2 from powder and single crystal samples are represented by circles
and asterisks, respectively. The zero-pressure bulk modulus, 
0, of both the
ambient- and high-pressure phases were obtained from a 2nd-order Birch-
Murnaghan equation of state fit, represented by the solid lines. Figure taken
from W. Peng et al. [6].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Figure 5.6: a) - b) A comparison of the unit cell of the 
¯3
1 (ambient) and 
2/

(high-pressure) structures. The blue and grey lines are used to outline the
ambient-pressure cell in both structure types, while the cyan lines represent
the interatomic distance, 
, which is equal to a and b in the 
¯3
1 symmetry.
Here, we define a’=b’ and c’, and a’ and g’ to represent the primitive unit
cell after it has lost its trigonal symmetry. Note that these parameters do not
correspond to the true a, b, and c axes of the monoclinic 
2/
 unit cell.
Normalized lattice constants of powder a) Mg3Sb2 and b) Mg3Bi2 before and
after the phase transition. We have defined 
 as the [110] diagonal of the
trigonal AP unit cell, as shown in the inset of panel. c) - d) A comparison
of the normalized lattice constants and interatomic distance, 
, of powder
Mg3Sb2 and Mg3Bi2. The dashed lines in panel c) show the results of a prior
computational study by Zhang et. al.[67]. Figure taken from W. Peng et al. [6].

59

Figure 5.7: a) The pressure-dependence of the unit cell volume for YbMg2Bi2, CaMg2Sb2,
MgMg2Bi2, MgMg2Sb2, EuZn2Sb2, and EuAl2Si2. The solid lines are re-
produced with the corresponding bulk modulus of each compound with the
2nd-order Birch-Murnaghan equation. b) A comparison of the bulk modulus
obtained with different methods: Resonant ultrasound spectroscopy (RUS),
density functional theory (DFT), and high-pressure X-ray diffraction.

. . . . . 62

Figure 5.8: a) The pressure-dependence of the c/a ratio for YbMg2Bi2, CaMg2Sb2,
MgMg2Bi2, MgMg2Sb2, and EuAl2Si2. The slope of MgMg2Bi2 is sig-
nificantly smaller compared to other compounds, which is an indication of
a more isotropic behavior. b) The slopes of c/a are shown as vertical bars.
The corresponding spatial dependence of Young’s modulus from the Mate-
rialsProject.org is shown on top of each bar. The experimental anisotropy
agrees qualitatively with the DFT results. Figure taken from W. Peng et al. [68]. 63

xiv

Figure 5.9: a) Normalized lattice constants along a- or b-axis and b) c-axis for powder
YbMg2Bi2, CaMg2Sb2, Mg3Bi2, Mg3Sb2, and EuAl2Si2. The discrepancies
of the slope of a/a0 between different compounds is significantly larger than
that of c/c0. Figure taken from W. Peng et al. [68]. . . . . . . . . . . . . . . . . 64

Figure 5.10: a) Powder diffraction patterns of YbMg2Bi2, b) CaMg2Sb2, and c) EuAl2Si2
with increasing pressure. Possible phase transitions were observed in all
phases. The X-ray wavelength is 0.4133 Å for YbMg2Bi2 and CaMg2Sb2,
and 0.4340 Å for EuAl2Si2. Figure taken from W. Peng et al. [68].

. . . . . . . 66

Figure 5.11: a) Powder diffraction patterns of EuAl2Si2 at a fixed temperature around
600 ◦C. An abrupt phase transition from the trigonal (
¯3
1) to tetragonal
(
4/


) phase was observed. Note that the pressure shown on the right is
the gas-membrane pressure. b) Powder diffraction patterns of SrAl2Si2 at a
fixed pressure around 1.4 GPa. A gradual phase transition from the trigonal
(
¯3
1) to tetragonal (
4/


) phase was observed. Two-phase patterns
can be spotted above ∼700 ◦C. The X-ray wavelength is 0.3542 Å for both
measurements. Figure taken from W. Peng et al. [68].

. . . . . . . . . . . . . 68

Figure 6.1: The phase diagram of GeTe - Sb2Te3 [69].

. . . . . . . . . . . . . . . . . . . . 73

Figure 6.2: a) Ambient-temperature rhombohedral structure of GeTe (
3
), b) Sb2Te3
(
¯3
), and c) (GeTe)
Sb2Te3 (
3
) with 
 = 3 used for illustrative pur-
poses. The cation vacancies in (GeTe)
Sb2Te3 are relaxed into ordered
layers, which resemble van der Waals gaps. Note that the hexagonal unit cell
was employed here, with the c-axis perpendicular to the layers. d) At high
temperature, (GeTe)
Sb2Te3 transitions to cubic symmetry with randomly
distributed vacancies on the cation site. It can be visualized as stoichiometric
occupancy of Ge, Sb, and vacancies on the cation site, while 100% of Te on
the anion site [70]. Figure taken from W. Peng et al. [71].

. . . . . . . . . . . . 74

Figure 6.3: a) The c/a ratio, b) lattice parameter, 
, and c) lattice parameter, 
, of
(GeTe)17Sb2Te3 and GeTe. The GeTe data is taken from ref.
[72]. A
reversible phase transition from a rhombohedral structure (
3
) to cubic
rocksalt (

¯3
) at roughly 623 K is observed in both materials. Figure
taken from W. Peng et al. [71].

. . . . . . . . . . . . . . . . . . . . . . . . . . 77

Figure 6.4: Normalized change in a) unit cell volume and b) lattice parameter along
c-axis of (GeTe)17Sb2Te3, GeTe, and Sb2Te3. Data for GeTe was taken
from ref. [72]. Similar volumetric thermal expansions is observed for the
three compounds, despite drastic differences in the c-axis thermal expansion.
Figure taken from W. Peng et al. [71].

. . . . . . . . . . . . . . . . . . . . . . 79

xv

Figure 6.5: a) A two-step phase transition can be observed during the first heating process.
The cubic rocksalt structure transforms to a rhombohedral structure at 523
K, and then to a cubic rocksalt structure at 623 K. A single phase transition
to rhombohedral structure occurred during the cooling process. b) The 2nd
thermal cycle shows a reversible phase transition between the rhombohedral
and cubic rocksalt structure around 623 K. Figure taken from W. Peng et al. [71]. 80

Figure 6.6: a) Lattice parameter 
, and b) lattice parameter, 
 of Sb2Te3 measured as a

function of temperature. Figure taken from W. Peng et al. [71].

. . . . . . . .

Figure 6.7: a) A comparison of the Young’s modulus, Y, of cubic and distorted rock salt
compounds shows that the distorted structure is softer. Bars show calculated
data from ref. [22] and symbols are experimental data (present work: ਭ, liter-
ature data: (cid:3) [73], (cid:5)[10], (cid:52)[74], ◦[9]). 3D surfaces show the compressibility
of the two structure types [22]. b) The temperature dependence of Y for GeTe
[present work], SnTe[10], and PbTe [74] show the slower rate of softening in
GeTe. Figure taken from W. Peng et al. [71]. . . . . . . . . . . . . . . . . . .

. 81

. 83

Figure 6.8: The temperature dependence of the normalized a) Young’s modulus, 
, and
b) shear modulus, G, for GeTe, (GeTe)17Sb2Te3, Sb2Te3, and Bi2Te3. All
compounds besides (GeTe)17Sb2Te3 soften with increasing temperature. c)
The value of the Young’s and shear moduli of (GeTe)17Sb2Te3 across phase
transition. Figure taken from W. Peng et al. [71]. . . . . . . . . . . . . . . . . . 84

Figure 6.9: a) Longitudinal speed of sound and b) Transverse speed of sound of (GeTe)17Sb2Te3,

Sb2Te3, GeTe, and Bi2Te3 measured from room temperature up to 600 K via
resonant ultrasound spectroscopy. c) Longitudinal and shear speed of sound
of (GeTe)17Sb2Te3 measured from room temperature across phase transition.
Figure taken from W. Peng et al. [71].

. . . . . . . . . . . . . . . . . . . . . . 86

Figure 6.10: The thermoelectric properties of the (GeTe)17Sb2Te3 samples used in this
study: a) Total thermal conductivity, lattice thermal conductivity, and elec-
trical thermal conductivity; b) resistivity; c) Seebeck coefficient; and d)
thermoelectric figure of merit. In contrast to the expectation of a decreasing
trend of lattice thermal conductivity, as it is the case for most materials, the
lattice thermal conductivity of (GeTe)17Sb2Te3 remains roughly unchanged
with increasing temperature before the phase transition. Figure taken from
W. Peng et al. [71].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Figure 7.1: Temperature-dependence of the Young’s modulus, 
, of the selected thermo-
electric material [7, 8, 71, 71, 9, 10, 75]. 

2
2 compounds, the alternating
layers of tetrahedral and octahedral coordination may explain their interme-
diate moduli. Mg3Sb2 and Mg3Bi2, as outliers of the 

2
2 compounds,
have a similar stiffness as the tetradymites structures. Figure taken from W.
Peng et al. [6]

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

. .

.

xvi

CHAPTER 1

BACKGROUND AND INTRODUCTION

1.1 Thermoelectric materials

Thermoelectric materials can be used to either convert a temperature gradient to an electrical po-
tential or create a temperature difference from a voltage difference [76], which has a wide range of
applications from our daily life to space explorations. Portable thermoelectric refrigerators have
replaced the traditional refrigeration units with heavy fragile compressors. On a larger scale, ther-
moelectrics have applications in generating electricity from waste heat in the automotive industry
[77]. In recent years, NASA has been developing the radioisotope thermoelectric generators that
can be used for more than seventeen years [78, 79].
In addition to the environmental benefit,
since thermoelectric devices have no moving parts, they are also less prone to failure compared to
traditional powering system.

As powerful as thermoelectric devices may seem, the application of thermoelectric materials is
still heavily limited by their low conversion efficiency. The thermoelectric conversion efficiency is
correlated to its figure of merit,


,

(1.1)



 =



2



which requires a material to have a high Seebeck coefficient, 
, high electrical conductivity, 
, and
low thermal conductivity, 
. This requirement is not easily achieved as these properties are highly
interdependent. Successful thermoelectric materials to date generally have a combination of light
carrier effective mass, high band degeneracy, and either complex unit cells or strongly anharmonic
bonding [16, 23, 80, 81].

1

1.2 Thermal transport

The figure of merit is related to the Seebeck coefficient, electrical conductivity, and thermal
conductivity. A good thermoelectric material almost always benefits from an inherently low
lattice thermal conductivity [23, 81]. Therefore, thermal conductivity is the focus of this research.
Essentially, all mobile particles that have directional propagation in the presence of a temperature
gradient are contributors to the total thermal conductivity of a material [2]. For semiconductors,
the dominant thermal components are charged carriers and lattice vibrations, which are referred to
as the electrical thermal conductivity and the lattice thermal conductivity, respectively.

As shown in Figure 1.1, lattice vibrations in a material can be classified into two types of motion
mechanisms. These periodic quantum excitations in a structure are called phonons, which can be
described as quasi-particles that interact with other particles within the rules of classical mechanics.
The word “phonon” comes from a Greek word meaning “sound or voice”. Opposing vibrations
of atoms can form standing waves, known as optical modes. Due to small average group velocity,
optical phonons are typically not the subject of study in the field of thermal transport. On the
contrary, the waves that propagate in the same directions, known as acoustic phonons, dominate
the heat transport of a material due to their large net group velocity [1, 2]. We can obtain direct
information of acoustic phonons from elastic moduli, which will be discussed in the next few
sections.

Note that even though optical phonons are typically not considered as a primary contributor
to lattice thermal conductivity, there are some studies on the participations of optical phonon
in thermal conduction in specific materials systems, especially for nanostructures in which the
frequencies of the optical phonons is comparable to the size of the lattice. For example, in a study
by Tian et al. [82], DFT calculations have shown that optical phonon can contribute over 20% of
the nanostructure silicon as compared to less than 5% in the bulk material.

2

Figure 1.1: A schematic demonstration of acoustic and optical phonon vibrations. Optical phonons
form standing waves with negligible net group velocity, whereas acoustic phonons dominate the
heat transport of a material due to their large net group velocity [1, 2].

1
3


2
 provides a guide to the lattice thermal conductivity,
A widely used approximation 

 =


, of a material, which is the product of heat capacity, 

, the speed of sound, 
, and phonon
relaxation time, 
 [23]. Compared with the inherent nature of the heat capacity, 
 and 
 can be
more readily controlled by materials scientists. Both of these properties are closely related and are
therefore the focus of this study, both of which are closely related to the chemical bonding and the
structure of the compound.

1.2.1 Speed of sound and elastic moduli

When we compare lattice thermal conductivity across different compounds or classes of materials,
it is tempting to jump to conclusions about different sources of external scattering. In particular,
scattering from defects, grain boundaries, or nanoparticles. While these are important effects, the
impact of speed of sound should always be considered first.

The speed of sound can be represented by the slope of the acoustic branch in a phonon dispersion
when frequency approaches zero (Ɖ-point), which is easily accessible for a lot of materials via DFT
calculations. The speed of sound 
, is also correlated with the elastic moduli, 
, via density, 
,

3

Figure 1.2: An example of the calculated phonon dispersion of the diamond. TA represents the
transverse acoustic branch and LA represents the longitudinal acoustic branch. All other modes
except the three acoustic modes are optical modes, which are usually of much higher frequency.
The slopes of LA and TA branches at the Ɖ-point are the longitudinal and transverse (shear) speed
of the sound, respectively.

by the general relationship of the form 
 ∝(cid:113) 



 . An example of a calculated phonon dispersion of
diamond is shown in Figure 1.2. TA represents the transverse acoustic branch and LA represents the
longitudinal acoustic branch. All modes except the three acoustic modes are optical modes, which
are usually of much higher frequency. The slope of LA and TA at Ɖ-point are the longitudinal
and shear speed of the sound of this material. The transverse speed of sound only depends on
a material’s shear modulus and density, whereas the longitudinal speed of sound is related to
compressibility, shear modulus, and density.

In isotropic materials (i.e. non-directional polycrystals), the speed of sound can be calculated from
the elastic moduli via Eq 1.2 and Eq 1.3, in which 

 is the longitudinal (transverse) speed of sound,


 is the shear speed of sound, 
 is the bulk modulus, and G is the shear modulus of a material.
In materials with lower symmetry (i.e. single crystals or textured polycrystals), the relationship is
more complex, since the speed of sound is dependent on direction and polarization. The complete
single-crystal elastic tensor 

 
 has up to 21 independent terms in a triclinic single crystal. The

4

details of the tensor notations, the tensor component in various symmetries, and anisotropic tensor
properties can be found in ref [83].

(cid:115)


 + 4/3



 
 =

(cid:115)











 =

(1.2)

(1.3)

From these two equations above, we can see that high density and soft bonds (i.e.
low elastic
moduli) typically lead to low group velocity, and vice versa. Bigger atoms are generally related
to softer bonds [84]. Materials with soft bonds, low speed of sound, and high anharmonicity are
generally favorable for thermoelectric materials. Note that soft bonds do not necessarily mean a
mechanically weak sample. In fact, softer bonds are often associated with materials that are less
prone to brittle failure, which is more favorable for practical applications [85].

1.2.2 Umklapp scattering

Figure 1.3: A schematic demonstration of normal scattering and Umklapp scattering. The left and
right edge of the grey box represent the range of the first Brillouin zone. 
1 and 
2 are the initial
wave vectors before the scattering, and 
3 is the resulting wave vector after the scattering.

In a perfect single crystal with no other scattering sources, the phonon relaxation time, 
, is related
to the Umklapp scattering rate, which is an anharmonic phonon-phonon interaction process that

5

exists even in a flawless crystal. Phonon-phonon interaction falls into two categories - Normal scat-
tering and Umklapp scattering. A schematic demonstration of normal scattering versus Umklapp
scattering can be seen in Figure 1.3. 
1 and 
2 are the initial wave vectors before the scattering, and

3 is the resulting wave vector after the scattering. Normal scattering refers to a harmonic process in
which the resulting wave vector, 
3, is inside the first Brillouin zone. The momentum is conserved,
and this type of phonon-phonon interaction does not directly contribute to lattice thermal resistance
[86]. Note that even though normal scattering does not directly contribute to the lattice thermal
conductivity, the resulting longer wave vector can increase the possibility of Umklapp scattering
during the next stage. Umklapp scattering occurs when the sum of the wavevectors, 
(cid:48)
3, exceeds
the first Brillouin zone. Due to the periodic nature, the point outside the first Brillouin zone can be
expressed as a point inside the zone, 
3, which corresponds to a wave flipping back in the opposite
direction [86].

The general temperature dependence of Umklapp scattering, 

, is described in Eq. 1.4 [87, 88,
89, 90, 91],

¯

3


1/3
2
2




 ∝

,

(1.4)

in which ¯
 is the average mass, 
 is temperature, 
 is the speed of sound, 
 is the unit cell volume
per atom, 
 is the phonon frequency, and 
 is the Grüneisen paramters. Among all the scattering
mechanisms, Umklapp scattering is often the dominant mechanism in thermoelectric materials
and it becomes increasingly important at high-temperature due to the 1/T temperature dependence
of the phonon relaxation time, 
. A schematic demonstration of the temperature dependence of
the phonon relaxation time of Umklapp scattering compared to other scattering mechanisms (i.e
carrier-carrier scattering and defect scattering) is shown in Figure 1.4.

In contrast to the easy accessibility of the speed of sound, the scattering rate, 

 is more difficult
to be directly measured experimentally, or have a unified approach to be computed. However, the
rate of softening, the Poisson’s ratio, and thermal expansion are related to anharmonic bonding

6

Figure 1.4: A schematic demonstration of the temperature dependence of the phonon relaxation
time of Umklapp scattering compared to other scattering mechanisms (i.e carrier-carrier scattering
and defect scattering). Umklapp scattering becomes increasingly important at high-temperature
due to the 1/T temperature dependence of the phonon relaxation time.

potential, and can therefore be used as a rough estimate of the Grüneisen parameter, which is
correlated with 

. These will be explained in detail in the next few sections.

1.2.3 Softening of elastic moduli with temperature

The Grüneisen parameter, named after Eduard Grüneisen, quantifies the anharmonicity of a mate-
rial. The Grüneisen parameter describes the vibrational properties of a crystal lattice as a function
of unit cell volume (i.e.
temperatures) [92]. The vibration of atoms in a solid depends on the
potential energy of its chemical bonds as a function of displacement, r. This is given by


(
) = 
(0) + 
2(
 − 
0)2 + 
3(
 − 
0)3...,

(1.5)

where 
 − 
0 is the relative displacement of an atom from its equilibrium position, 
0, and 
2
and 
3 are the magnitude of the harmonic and anharmonic terms, respectively. The degree of
anharmonicity can be represented by 
3/
2 (or 
4/
2 if there is no 
3 terms in certain symmetries),
in which the 
2(
 −
0)2 is the symmetric term and 
3(
 −
0)3 is the term that breaks the symmetry.

7

The elastic modulus, 
, is co-related to the curvature of the vertex of the potential function,
. A larger curvature corresponds to a narrower potential well and thus stiffer

(cid:17)

(cid:16) 
2




2


 =

1

0


=
0

Figure 1.5: A schematic demonstration of interatomic potential and its relationship with thermal
expansion and elastic moduli. A larger curvature corresponds to a narrower potential well and thus
a smaller thermal expansion and stiffer bond.

bond. The curvature of an anharmonic potential well decreases with increasing potential energy
/ atomic displacement / temperature. In the classic Taylor expansion model, as given by Eq 1.5,
the curvature of potential well is a constant as a function of energy in a harmonic potential well.
The elastic modulus is therefore also a constant as a function of temperature. Note that the thermal
expansion for a perfectly harmonic crystal might not be zero, as predicted in a calculation study by
Matthias et al. [93]. However, in this study, we will use the classic model to explain the observed
anharmonic phenomenon. The more anharmonic the potential well is, the more rapidly a solid
will soften with increasing temperature. Longer chemical bonds tend to be softer, but this does
not necessarily imply that they must also be more anharmonic. However, in general, we associate
highly harmonic materials (e.g., diamond) with bonds that are stiff, short, and strong. In contrast,
most anharmonic solids often have some degree of instability in the structure leading to bonds that
are weak in some respects. Alternatively, we can view this relationship from the perspective of the
melting temperatures. Weaker bonds lead to lower melting temperatures, which require the elastic

8

moduli to soften faster since liquids have elastic moduli close to zero.

Figure 1.6: Three selected materials that range from very soft and anharmonic to very stiff and
harmonic. 

 is related to the Einstein temperature, the slope of the linear region equals the 
/


ratio, 
0 is the Young’s modulus at the lowest temperature (∼0 K), and 









 is the Grüneisen
parameter [3, 4, 5]. Figure taken from W. Peng et al. [6]

The elastic response is characterized by the stiffness tensor 

 
. For bulk, polycrystalline samples,
it is more convenient to refer to Young’s modulus, shear modulus, and bulk modulus, all of which
can be derived from the single-crystal elastic tensor. The general trend of elastic moduli with
temperature is shown in Figure 1.6. The elastic moduli stay mostly constant at low temperatures
because low energy vibrations only sample the bottom of the potential well, the curvature of which
almost always approximates the harmonic ideal. At higher temperatures, they start to soften linearly
assuming no phase transition occurs within the measured temperature range.
In 1970, Varshni
put forward an equation from the Einstein-oscillator model to illustrate the relationship between
the stiffness tensor (or any elastic constant), 

 
, Einstein temperature, 

, and measurement

9

temperature, T.



 
 = 
0


 
 − 
/(


/
 − 1),

(1.6)

in which 
0 denotes the elastic-stiffness coefficient at 0 K and 

 is related to the Einstein
temperature. This equation has shown to be the best overall demonstration of the trend in the low-
and mid-temperature regime. The value of s/2 represents the curvature of the transition region,
which also corresponds to the difference between the value of 

 
 at 0 K and the y-intercept of the
linear fit to the higher temperature linear regime. Further, the linear slope of the high-temperature
region without a nearby phase transition can be used to calculate the Grüneisen parameter and
Einstein temperature of the materials. Note that at temperatures lower than 

/50, the elastic
modulus is theoretically expected to follow a T4 dependence [94], which can not be accurately
represented by the Varshni equation.

One of the equations derived from the Varshni equation by Ledbetter et al. [95]



/

 = −
/

 = −3

(
 + 1)





(1.7)

can directly calculate the Grüneisen parameter of a material from the slope of the bulk modulus


/

. 
 is the bulk modulus, 
 is the Boltzmann constant, and 

 is the volume per atom.
The slope of the linear section, 

/

, equals the 
/

 ratio, from which both variables can be
determined when plugged in back to the Varshni equation for a least-square fit. Here, we showed
the corresponding Varshni fit of three materials with varying stiffness and anharmonicity - ReB2,
Cu, and Mg3Sb2. Since Young’s modulus, 
, is more widely available than the bulk modulus,
therefore, we are fitting the trend of the 

/

 instead of 

/

 here. A narrower transition region
to the linear trend means a lower Einstein (Debye) temperature, a faster softening rate, and smaller
s/2, which agrees well with the data shown in the table inset in Figure 1.6.

10

Figure 1.7: Temperature-dependence of the Young’s modulus, 
, of the selected thermoelectric
material [7, 8, 9, 10, 11, 12, 13, 14, 12]. Half-heuslers and Si1−
Ge
 come to the top to be the stiffest
thermoelectric materials due to their short covalent bonds. The softest crystalline materials are the
layered tetradymites with covalent octahedral layers and vdW bonding. At the lower left corner,
PEDOT, a conducting polymer, is shown for comparison, which is an extremely soft material.
Figure adapted from W. Peng et al. [6]

1.2.4 Literature review of temperature-dependent elastic moduli of thermoelectric materials

Figure 1.7 compiled some selected temperature-dependent elastic moduli data across different
classes of thermoelectric materials from the literature to illustrate the range of behavior. All of
the data were collected on bulk polycrystalline specimens using resonant ultrasound spectroscopy.
Details about the method are described in Chapter 2. We can see the dual roles of structure
and composition in determining elastic moduli. Due to the short and stiff covalent bonds, half
heuslers with small atoms are the stiffest thermoelectric compounds, even stiffer than Si-Ge alloys.
Similarly, the cage structures of clathrates are also dominated by stiff covalent bonds, enabling them
to be the second stiffest thermoelectrics with large cations. Within the same structure type, the
differences in atomic size are the determining factor of the differences in the stiffness of compounds.

11

For example, the Ge-based clathrates are generally softer than Si-based clathrates. However, this
does not typically strongly impact the slope, 

/

, which is used for calculating the Ledbetter
Grüneisen parameter, 









. The clathrates in particular are an example of a large range in
Young’s modulus, but not much variation in slope. The difference in Young’s modulus explains
much of the difference in lattice thermal conductivity across materials. For example, half heuslers’
lattice thermal conductivities are inherently high due to their high bond stiffness. In particular,
ZrNiSn has a lattice thermal conductivity of 17.2 W/mK at room temperature, which is on the high
end of TE materials. PEDOT, on the other hand, is a class of electrical conducting polymers with
ultra-low thermal conductivities (∼ 0.4 - 0.9 W/mK) [96]. The Young’s modulus of PEDOT thin
films at ambient temperature is typically around 1 GPa [12, 13, 14].

Figure 1.8: The change of the normalized Young’s modulus with temperature of various compounds
[7, 8, 9, 10, 11, 15]. Figure adapted from W. Peng et al. [6]

Generally, materials with lower melting temperatures soften faster. Figure 1.8 shows the fractional
change in Young’s modulus with temperature. While SiGe only softens by 3% up to 800 K,
traditional thermoelectrics, such as PbTe and SnTe, are typical low melting temperature materials
which have a much faster rate of 20%.

12

1.2.5 Grüneisen parameters

From a microscopic perspective, the mode Grüneisen parameter, 

, can be expressed as



 = − 











,

(1.8)

in which 

 is the corresponding vibrational frequency of an individual phonon mode 
, and V
is the unit cell volume [92]. Each vibrational mode has its unique response to volume change,
which cannot be measured in an experiment without a neutron scattering facility. This makes the
experimental data of anharmonicity prohibitively difficult to obtain.

Being able to accurately estimate the Grüneisen parameters is important to understand the Umklapp
scattering rate, but unfortunately, there is no single, widely used approach to estimating from
experimental data. The reported experimental Grüneisen parameter value is usually a weighted
average of all modes. In this regard, different approximations are incorporated into the original
equation, making the Grüneisen parameters a more attainable property. There are many different
expressions for the thermodynamic Grüneisen parameter that are equally valid, and make use of
different measurable quantities.

1.2.5.1 Thermal expansion Grüneisen parameter

Traditionally, experimentalists only have access to the thermodynamic, or "average" Grüneisen
parameter. This can be obtained by taking a weighted average of the mode Grüneisen parameters
via the specific heat per particle,



 



,




,



 =

,

(1.9)

in which 

,
 is the heat capacity of an individual vibrational mode. When the Einstein heat capacity
and the quasi-harmonic approximation (
 is the only term that is affected by volume) is applied,
combined with Maxwell relations, the equation


 =






 


13

(1.10)

can be derived. 

 is specific heat, 
 is volumetric thermal expansion, 
 is the bulk modulus, and

 is the density of a material. The major downside is that, in the case of a material with very high
positive and negative mode Grüneisen parameters, the weighted average might be a small positive
value, which lead one to be unaware of the large anharmonicity of that material.

1.2.5.2 Poisson’s ratio Grüneisen parameter

Because phonon frequencies are a function of elastic constants, the Grüneisen parameter can
also be obtained through various thermodynamic expressions from elastic moduli. Recently, the
relationship between the bulk modulus, 
, speed of sound, 
, and Grüneisen parameters got a lot
of attention due to its simplicity for both the experimental and DFT calculations [97, 98, 99, 100].
The equation


 =





2

(1.11)

was initially derived from the Debye model in a cubic system, but it has been shown to work for
other symmetry conditions as well. From an elastic tensors’ perspective, it is well-known that


2

=


11 + 2
44

3


(1.12)

and


11 + 2
12

3


 =

(1.13)
According to Eq. 1.12 and Eq. 1.13, we can see that 

2 only equals 
 when C12 = C44. Therefore,
the value of the Grüneisen parameter in Eq. 1.11 can be interpreted as the difference between C12
and C44 (i.e. (C11+2C12)/(C11+2C44)), or in other words, how uniformly the material reacts to a
given strain. By doing transformations using the relationship between the speed of sound, 
, bulk
modulus, 
, and the Poisson’s ratio, 
, one can obtain a single-variable equation

(cid:19)

(cid:18) 1 + 


2 − 3



 =

3
2

,

(1.14)

which is a monotonically increasing function in the range of the Poisson’s ratio values (0-0.5). A
large Poisson’s ratio is a hallmark of lattice instability, which corresponds with a large Grüneisen
parameter in this expression.

14

So far, we have introduced three equations for calculating the Grüneisen parameters: Eq.1.6,
Eq.1.10, and Eq. 1.14. Although all these equations for Grüneisen parameters are derived from the
same origin, they have different scaling and give different values, which can be misleading when
one is trying to compare the Grüneisen parameter of a material with other materials using values
calculated from a different equation. However, a general trend and conclusion of the Grüneisen
parameters can still be made if a variety of equations are used for the selected compounds.

1.3 

2
2 Zintl phases



2
2 Zintl phases have attracted plenty of interest in thermoelectric applications due to their
chemical diversity and tunability. The highest 

 of 1.6 has been achieved in 
-type MgMg2(Sb,Bi)2
compounds. Much of the research to date has been focused on optimizing the electronic properties,
explaining the band degeneracy, and exploring the dopability of these compounds.
In contrast,
little research has been done on their phonon behaviors and lattice thermal conductivity. However,
even though there is not much tunability for these intrinsic properties, there are some interesting
and unanswered questions that can enhance the understanding of the fundamental questions of the
relationship between structure and phonons.

In 

2
2 compounds with CaAl2Si2 (
¯3
1) structure type, 
 is an alkali or alkaline earth
or rare earth metal, 
 is Mg, Al, or a transition metal, and 
 belongs to a IVA-VIA group
element. CaAl2Si2 can be characterized by covalently bonded [Al2Si2]2− bilayer slabs formed
by alternatively oriented AlSi4 tetrahedra sandwiched by monolayers of Ca2+ (see Figure 1.9 a)
[101, 102, 35].

1.3.1 The formation rule of Zintl phases

The name "Zintl" comes from the German chemist Eduard Zintl, who discovered and defined a
subset of intermetallic compounds that consist of covalently-bonded polyanions surrounded by
cations which provide the overall charge balance. Complete charge transfer from the cation to the

15

anion is assumed for the Zintl phases [103, 104]. Zintl phases were first discovered in 1933 [105, 53]
and they have been widely studied on the stability of the structure [38, 39, 36, 37, 45, 46, 47, 56, 50]
as well as the electronic structures [106, 107].

The Zintl-Klemm concept (8-N rule) is frequently used to rationalize the relationship between the
formula and chemical bonding in Zintl phases, in which N refers to the number of valence electrons
available per unit cell per anion [103]. Take 

2
2 compounds with the CaAl2Si2 structure type
as an example. The valence electrons for Ca, Al, and Si are 2, 3, and 4, leading to a total number
of 16 valence electrons per formula unit. The number of valence electrons available per unit cell
per anion is N = 16 / 4 = 4, if we consider both Al and Si as anions. And therefore, 8 - N = 4.
This means that both Al and Si have four-fold coordination, which is consistent with the observed
structure (see Figure 1.9 a).

1.3.2 Chemistry of 

2
2 compounds in CaAl2Si2 structure type

Figure 1.9: a) The CaAl2Si2 structure type is formed from [M2
2]2− polyanion slabs separated
by monolayers of six-fold coordinated cation 
. b) Possible element-combinations to achieve the
CaAl2Si2 structure type according to the 8 - N rule. Figure taken from W. Peng et al. 1.9.

From the electron count in the previous section, we know that the CaAl2Si2 structure has 16 nominal

16

valence electrons per unit cell. In fact, as long as the 16 valence electron rule can be satisfied,
the ternary 

2
2 compounds will form the CaAl2Si2 structure type. This includes mixed
occupancies and interstitials. All the elements that can form the CaAl2Si2 are shown in Figure 1.9
b). By combining the listed elements on each site, there are more than 100 compounds reported to
crystallize in this structure type. This brings a great deal of flexibility in forming and tuning the
composition, which has attracted plenty of interest for 

2
2 compounds as thermoelectrics.

Note that some of the metallic 

2
2 Zintl phases are exceptions for the 8 - N rule, such as


Al2Si2 and 

Al2Ge2 ((

= lanthanide rare earth metal). These have 17 valence electrons
[59] due to the reduced anti-bonding state filling energy. However, since these compounds do not
have a bandgap, they are not typically within the scope of thermoelectrics.

1.3.3 

2
2 compounds as thermoelectrics

Although CaAl2Si2 itself is metallic, the majority of the antimonides and some of the bismides in
this structure type are narrow band gap semiconductors. The interest in the thermoelectric research
of 

2
2 compounds started in 2005, when the thermoelectric properties in Ca
Yb1−
Zn2Sb2
solid solution were first reported by Gascoin et al. [31]. Compared with other classes of Zintl
phases, one of the prominent advantages of CaAl2Si2-type compounds is their large variety and
unprecedented tunability, providing many opportunities for controlling the transport properties.
Since then, large amount of efforts on optimizing and understanding the electronic structure,
including band engineering [108, 109, 110, 111], control of defect concentrations [112, 113],
aliovalent doping [114], and point defect phonon scattering [31, 115, 116, 27, 33, 28, 32, 117] have
emerged. Recently, excellent thermoelectric performance was reported by several independent
groups in 
-type Mg3(Sb,Bi)2 alloys, with 

 up to 1.6 [118, 108, 119, 113, 120]. This record-high


 surpasses all previous results for isostructural compounds, which have been exclusively 
-type.

To date, experimental and theoretical investigations of Mg3Sb2 have focused on the electronic

17

Figure 1.10: The 

 of selected 
-type (dashed curves) and 
-type (solid curves) of 

2
2
compounds. Figure taken from W. Peng et al. [16].

properties [118, 121], the multi-valley conduction band [108, 113, 109, 111], and increasing
mobility [120, 122]. In contrast, the anomalously low 

 of both 
- and 
-type Mg3Sb2 has not
been investigated, though it plays an equally important role in leading to the high 

. Here, we will
discuss the origins of low lattice thermal conductivity from the perspective of thermal expansion,
lattice softening, compressibility, and phase transitions, shedding light on the rarely discussed topic
of anharmonicity in 

2
2 compounds.

1.4 Summary of research

In this research, we will discuss the origins of low lattice thermal conductivity in two different
classes of layered thermoelectric materials from the perspective of thermal expansion, lattice
softening, solubility, speed of sound, compressibility, and phase transitions. The main focus of the
thesis is on the 

2
2 compounds described above, shedding light on the rarely discussed topic

18

of anharmonicity in 

2
2 compounds.



2
2 compounds are a particularly attractive class of Zintls for thermoelectrics due to large
variety and unprecedented tunability. By employing high-temperature X-ray diffraction (XRD) and
resonant ultrasound spectroscopy (RUS) techniques, we have shown that the low lattice thermal
conductivity is due to previously-unrecognized soft shear modes and highly anharmonic acoustic
phonons in layered MgMg2Sb2. Even though different Grüneisen parameter equations give different
values, the results repetitively show higher anharmonicity for Mg3Sb2 and Mg3Bi2 compared to
isostructural 

2
2 compounds. Combined with the phonon calculations from our collaborators,
we attribute the anomalous thermal behavior of MgMg2Sb2 to the instability of the vibrational
modes that originated from the weak bonding of the Mg, which is too small for the octahedral site.
The limited solubility of Sr and Ba on the Mg cation site (i.e. 
-site) further confirmed the small
ionic radius of Mg2+.

High-pressure characterizations are not widely used in studying thermoelectric materials, as their
application is usually at high temperature, not pressure. However, 

− 



 high-pressure structural
characterization can provide another dimension of variables to observe the stiffness and stability
of a material, which can help to understand relationship between structure and phonons of the
lattice thermal conductivity. By performing in-situ high-pressure synchrotron X-ray diffraction, we
showed that a few 

2
2 compounds can exhibit phase transitions at high-pressure, most of which
are previously unrecognized. In addition, we observed that the compressibility of MgMg2Sb2 and
MgMg2Bi2 is near-isotropic, whereas other isostructural 

2
2 compounds show clear signs of
anisotropy between the in-plane and out-of-plane compressibility as is typical of layered compounds.
We have analyzed the compressibility, transition pressure/temperature, anisotropy, as well as the
type of phase transition to develop a deeper understanding of the stability and bond strength of
different 

2
2 compounds.

19

Lastly, to extend our understanding of the structure - property relationships in layered materials, we
investigate the GeTe - Sb2Te3 homologous series, which is characterized by van der Waals gaps with
tunable layer thickness. Unlike most compounds that soften with increasing temperature, the elastic
moduli of (GeTe)17 - Sb2Te3 stiffen with increasing temperature before the phase transition. We
attribute the stiffening behavior to the gradual diffusion of layered vacancies to random vacancies
on the cation site, which correspondingly leads to a flattened temperature dependence of the elastic
moduli.

20

CHAPTER 2

METHODS

2.1 Overview

For this work, polycrystalline samples of CaMg2Sb2, YbMg2Sb2, CaMg2Bi2, YbMg2Bi2, MgMg2Sb2
(alternatively written as Mg3Sb2), MgMg2Bi2 (alternatively written as Mg3Bi2), EuAl2Si2, SrAl2Si2,
GeTe, Sb2Te3, Bi2Te3 were synthesized via ball milling followed by spark plasma sintering. Due
to the geometry of the die used for sintering, all the as-synthesized samples are bulk samples with
cylindrical geometry. All sintered samples have densities above 97% of the theoretical density,
and less than 5% of the impurities were observed based on the relative peak intensity of X-ray
diffraction. (GeTe)17Sb2Te3 was made by our collaborator, Jared Williams, via a slightly different
processing method, which will be described separately in the following sections.

The primary characterization techniques used in this work include high-temperature X-ray diffrac-
tion with finely ground powder samples, room-temperature X-ray diffraction with bulk samples,
high-temperature resonant ultrasound spectroscopy with bulk cylindrical samples, and 

 − 




X-ray diffraction in diamond anvil cells with powder or single-crystalline samples. The synthesis,
X-ray diffraction, and resonant ultrasound spectroscopy were performed at Michigan State Univer-
sity, whereas 

-



 X-ray diffractions in diamond anvil cells were performed at Advanced Photon
Source in Argonne National Laboratory. The details are demonstrated below.

2.2 Synthesis

All 

2
2 compounds, GeTe, Sb2Te3, and Bi2Te3 discussed in this paper were synthesized
by direct ball-milling of stoichiometric elements followed by spark plasma sintering (SPS). The
corresponding stoichiometric elements were cut into small pieces in an argon filled glovebox, loaded
into stainless steel vials with two 10 mm diameter stainless balls, and milled for one hour using a

21

SPEX mill. Ball-milling was performed under argon atmosphere for 1-hour using a stainless steel
SPEX mill. The powder was then loaded into graphite dies with 10 mm bores and sintered under a
pressure of 31 MPa using a Dr. Sinter SPS-211LX. The samples were heated to a target temperature
in 5 minutes, and then held at the target temperature for 10 minutes. The sintering temperatures
and times were customized for each compound depending on their phase diagrams. Details for
the specific compounds can be found within each chapter. A heating rate of 50-80◦
/min was
used to reach the target temperature, typically 550-850◦
, followed by a 10-minute hold. The
densities of all the samples were obtained from direct measurements of mass and geometry. The
pressure was removed immediately when cooling started. The phase purity of the samples was
confirmed using a Rigaku Smartlab X-ray diffraction system with Cu K
 radiation. All sintered
samples have densities above 97% of the theoretical density, and less than 5% of the impurities
were observed based on the relative peak intensity of X-ray diffraction. The details of each sample
will be described in the corresponding chapters.

Figure 2.1: A demonstration of the synthesis approach in this work. Samples were synthesized
by direct ball-milling (under argon) followed by spark plasma sintering (SPS). The as-synthesized
samples have a cylindrical geometry. For the annealing process, the samples were wrapped in
graphite foils and sealed in glass ampoules under vacuum, which were then held at 500◦
 for ten
days.

22

In some cases, an annealing process would be done if the equilibrium state of the material is a major
concern. All the samples used for solubility study in section 3.2 were annealed after sintering. The
samples were wrapped in graphite foils and sealed in glass ampoules under vacuum, which then
held at 500◦
 for ten days. Quenching is preferred for high-temperature phase observation.

(GeTe)17Sb2Te3 were made by our collaborator, Jared Williams. Samples were prepared using
stoichiometric amounts of elemental germanium ingot (99.9999%), antimony shot (99.999%), and
tellurium lump (99.999%) from Alfa Aesar. The elements were sealed in quartz ampoules under
vacuum, which were then heated to 1173 K and held for 12 hours and water quenched. The
material was then ball-milled for 5 minutes and densified using Spark Plasma Sintering (SPS) at a
temperature of 673 K and a pressure of 40 MPa for 15 minutes. The structure was identified using
a Rigaku X-ray diffractometer. After consolidating in the SPS, the (GeTe)17Sb2Te3 sample was
initially in a metastable cubic structure due to the rapid cooling rate, as described in ref [123]. A
single heating cycle to 673 K was sufficient to regain the thermodynamically stable rhombohedral
structure, as verified by X-ray diffraction. The densities of the samples yield at least 96% of the
theoretical densities.

2.3 Characterization

2.3.1 Resonant ultrasound spectroscopy (RUS)

Traditionally, the measurements of elastic moduli rely on pulse-echo (Figure 2.2 a), in which a large
piece of sample was required, and multiple measurements need to be done to obtain longitudinal
and transverse velocities [124]. The plane wave approximation, as well as the signal from undesired
sources, lowers the reliability of the result.

Resonant ultrasound spectroscopy (RUS) provides the most accurate, efficient, yet non-destructive
characterization of elastic moduli in solids [125, 24, 126], which has advantage to extract a complete
set of elastic tensor from a single measurement. As shown in Figure 2.2 b), the driving force is a

23

wide range of vibrational frequencies generated from a piezoelectric transducer, and the response
is the resonances from the sample collected by another transducer [125, 24]. Peaks can be observed
when the excitation frequency matches the Eigen-frequency of the sample. The spectrum from a
single measurement contains sufficient Eigen-modes to extract the complete elastic tensor, assuming
size, shape, and density of the sample are known parameters [126]. The mounting in RUS only
requires a very light touch between the sample and the transducers, which avoids the problem of
thermal expansion mismatch, offering a significant advantage in high-temperature measurement
[24, 126].

Elastic moduli are obtained by matching the calculated resonances with observed peaks, which
can be computationally more accessible by using samples of well-defined geometries such as
parallelepipeds, cylinders, and spheres [127]. The calculated resonances are for perfect geometries
with free oscillations. In a real experimental setup, however, deviation from perfect geometry is
inevitable, and the sample is not entirely free from external forces. Take a cylinder sample as
an example, the modes from a perfect geometry should belong to either torsional, extensional or
flexural mode. The first two types are single modes, and the third type of mode occurs in pairs,
which appear to be doublets in a spectrum. When a small defect, such as a crack, appears in the
sample, the resonant frequencies will be perturbed. For doublets, the two components of the pair
modes might not have the same amount of shifts. Therefore, splits of doublets and slight shifts of
singlets and would be observed in an imperfect sample.

All the elastic modulus investigations in this paper were done using polycrystalline samples.
Polycrystalline samples are isotropic, which means it only requires two independent elastic constants
C11 and C44 for the complete description of the elastic behavior regardless of the space group.
The elastic moduli, longitudinal and transverse speed of sound of polycrystalline samples can be
calculated applying the isotropic equations:


12 = 
11 − 2
44

24

(2.1)

Figure 2.2: a) In pulse-echo ultrasound measurements, the time required for an acoustic wave to
travel across the sample is measured. b) In resonant ultrasound, the drive transducer emits acoustic
waves over a frequency spectrum. Peaks correspond to the resonant frequencies of the sample. The
black dots under the spectrum are calculated peak locations with the predicted elastic constants.
The inset is an example of the temperature dependence of the peak positions. This Figure is
reproduced from Ref. [17]


 =


12


11 + 
12


 = 2(
 + 1)
44


 = 
44

25

(2.2)

(2.3)

(2.4)



44

2
44 + 3


(cid:115)


 =


 
 =


 + 4
3




(cid:115)






,



 =

(2.5)

(2.6)

(2.7)

in which 
 is Young’s modulus, 
 is shear modulus, 
 is the Poisson’s ratio, 
 is bulk modulus,


 is the longitudinal speed of sound, and 

 is the transverse speed of sound.

Room temperature RUS Cylindrical samples were mounted on a tripod transducer setup (Figure
2.3 a)). One transducer induced vibrations and the remaining two detected the specimen resonances.
The sinusoidal driving frequency was swept from 0 to 500 kHz. Each scan was typically completed
within a minute.

Figure 2.3: a) Room temperature RUS with a tripod set-up. b) High-temperature RUS set-up.

High-temperature RUS The temperature-dependent elastic moduli were measured by resonant
ultrasound spectroscopy using a custom modification of a Magaflux-RUS quasar 4000 system in a
furnace with a flowing Ar atmosphere (Figure 2.3 b)). Buffer-rods are glued to the transducer that
extended into the furnace for sample mounting. After mounting the sample onto the buffer-rods, the

26

system would stay in vacuum overnight before backfilled with argon. The maximum temperature
of this set-up is 500◦
 due to the melting point of the glue on the buffer-rods.

RUS data analysis The RUS data was analyzed using commercial Quasar2000 CylModel soft-
ware to match the predicted resonant frequencies with observed peaks.

2.3.2 X-ray diffraction (XRD)

Room temperature XRD Room temperature X-ray diffraction is done using a Rigaku Smartlab
X-ray diffraction system with copper K
 radiation to identify the compositions. Phase purity of
the samples was confirmed via peak matching within the ICSD database, and lattice parameters
were obtained using Rietveld refinement using PDXL2 [128, 129]. Rietveld refinement uses the
least square approach to refine line profile with the peak shape, eak shape parameters, preferred
orientation, and the structure factor to match the measured spectrum [130], which has a significant
advantage over other methods, such as peak indexing, especially in cases of peak overlapping.

High-temperature XRD Thermal expansion was measured using a Rigaku Smartlab XRD sys-
tem equipped with a high-temperature stage. The samples were ground into fine powders that were
then placed on a graphite foil on top of a platinum tray. The measurements were performed under
vacuum to prevent oxidation. The thermocouple goes into the inner part of the platinum tray to
increase the accuracy of the temperature measurement (shown in Figure 2.4). The heating rate is 10
K/min with a 1-minute hold, and sample height alignments were performed before each measure-
ment to account for the combined thermal expansion of the holder and sample. Lattice parameters
were refined at each temperature using PDXL2. The volume thermal expansion coefficients 

 can
be calculated from the refined unit cell volume 
 using equation

(2.8)



 =

1










27

Similarly, linear thermal expansions along different crystallographic directions can be obtained
with








(2.9)



 =

1



Figure 2.4: The high-temperature X-ray diffraction was performed using a Rigaku 1400HT stage.
The thermocouple goes into the inner part of the platinum tray to increase the accuracy of the
temperature measurement.

2.3.3

In-situ high-pressure synchrotron diffraction in diamond anvil cells (DACs)

A diamond anvil cell (DAC) is a hand-size device that can generate pressures up to a few megabars
depending on the size of the anvil. DACs are specialized at 

-



 measurements of micron-size
samples with high-focus beams. This gives the advantage of measuring anisotropic compressibility,
observing phase transitions, study the properties of non-quenchable phases, and synthesize novel
materials.

28

2.3.3.1 Loading samples in diamond anvil cells

A schematic picture of a symmetric diamond anvil cell is shown in Figure 2.5. The figure is taken
from Dong et al. [18]. The anvils consist of two opposing diamonds. The diamonds are usually cut
into 16 facets of with a hexadecagonal surface on the top, which refers to as the culet. The culet size
typically ranges from 30 
m to 800 
m. Smaller culets are used for high pressure, whereas bigger
culets are used for finer pressure steps. The maximum pressures for 200, 300 and 800-micron
anvils are approximately 80 GPa, 60 GPa, and 15 GPa respectively.

Figure 2.5: A schematic picture of a symmetric diamond anvil cell. Figure taken from Dong et al.
[18].

A gasket made of a hard material is sandwiched between the two anvils to protect the diamonds
as well as creating a sample chamber. Rhenium or tungsten gaskets are the most common non-
X-ray transparent choices, whereas lighter materials, such as beryllium, are chosen when X-ray
transparency is required.

For experiments that require hydrostatic pressure, a liquid pressure transmitting medium is required

29

to avoid strain variation across the sample. Gas-loading is one of the most common ways to fill the
gasket chamber with pressure medium, typical choices are noble gases and nitrogen. Lighter gases
such as He and Ne have lower peak intensities and higher solidification pressures, but due to their
small atomic sizes, they are hard to be compressed at high pressures. In addition, smaller molecules
diffuse into the diamond surface easier than bigger ones, which means they have higher chances to
damage the anvils. Larger molecules such as Ar and Kr are softer and easier to compress, but they
solidify at a lower pressures and can generate high-intensity peaks. Therefore, these heavy gases
have issues of reduced-transparency and less hydrostatic pressure.

2.3.3.2 High-pressure X-ray diffraction experiment details

Our experiments rely on 

-



 X-ray diffraction to obtain compressibility and detect possible
new phases while increasing pressure and temperature.


-



 high-pressure X-ray diffraction
experiments were conducted at the Advanced Photon Source (APS) in Argonne National Laboratory,
beamlines 13-BM-C (GSECARS) and 16-BM-D (HPCAT). The samples were synthesized via ball-
milling and SPS as described above. The as-synthesized bulk samples were ground into powder
with grain sizes smaller than 1-micron, which was then pressed into flakes before loading into the
DACs. The benefit of using flakes rather than powder is for the convenience of sample-loading,
which is typically done with the static electrical force from the tip of a tungsten needle. Loading a
compressed piece of sample can also prevent the situation in which all the powders move towards
the edge of the gasket during gas-loading. Ruby or gold was placed next to each sample for pressure
readings. Diamond anvils with culet sizes of 300 and 800 
m were used, with rhenium gaskets
from H-Cross pre-indented to thicknesses of ∼45 
m for 300 
m culets and ∼130 
m for 800 
m
culets, respectively. The pre-indented gaskets were drilled in the center to form a hole about half
of the corresponding anvil size using an electrical discharge machine (EDM) at Michigan State
University or the laser micro-machining system at HPCAT [131].

Neon was loaded as the hydrostatic pressure medium using the COMPRES/GSECARS gas-loading

30

Figure 2.6: The X-ray transmission image of Mg3Sb2 and Mg3Bi2 in this study. The low trans-
mission regions are the positions of the samples.

system for all samples. For all the measurements, a gas membrane setup was used to remotely
increase or decrease pressure as needed and the pressure was read via the 

-



 ruby fluorescence
system before and after each data collection [132].

The distance and orientation of the detector was calibrated using a CeO2 standard. The beam size
was 12 
m (horizontal) x 18 
m (vertical) FWHM at GSECARS and 4 
m x 4 
m FWHM at
HPCAT. The detector was an online Pilatus 1M at GSECARS and a Mar345 image plate at HPCAT.
The position of the samples can be identified from the difference of the X-ray transmission between
the sample and diamond (as shown in Figure 2.6). The low transmission regions are the positions
of the samples. The pressure was controlled remotely by an inflatable steel membrane during all
the measurements.

For the high-temperature high-pressure measurements, a heating unit was set up inside the cell. The
heating unit consists of heating wires wrapping around a customized alumina ring with insulation
layers on top and bottom of the heating units to create a heating chamber with minimal heat loss
(see Figure 2.7). The heating was done under vacuum. K-type thermocouples were glued as close

31

Figure 2.7: The HT-DAC set up. Photo taken from the set-up manual written by Dr. Bin Chen and
his group members in University of Hawaii.

to the anvil as possible for accurate temperature reading. The rest of the set up is the same as
room-temperature ones.

Data analysis for high-pressure X-ray diffraction Dioptas was used for raw data processes.
Rietveld refinements for all diffraction patterns were performed using the PDXL2 software [128,
129]. Equation of state fits were performed with the EosFit7 software [133].

2.3.4 Calculation details for the phonon density of states and mode Grüneisen parameters

of Mg3Sb2

The first principles calculations were performed by our collaborators Guido Petretto, Gian-Marco
Rignanese, and Geoffroy Hautier. The ABINIT software package [134, 135, 136] was used
to perform density functional theory (DFT) and density functional perturbation theory (DFPT)
simulations to obtain phonon properties and elastic constants [137, 138, 139, 140]. The exchange-
correlation energy was approximated using the PBEsol [141] functional, that has proven to provide

32

accurate phonon frequencies compared to experimental data [142]. Norm-conserving pseudopo-
tentials [143] extracted from the PseudoDojo pseudopotentials table version 0.3 [144] were used
for all the elements. The Brillouin zone was sampled with 8 × 8 × 5 Monkhorst-Pack grids
[145, 146, 147]. Due to the well-known underestimation of the band gap by standard DFT, we limit
our analysis to Mg3Sb2, CaMg2Sb2 and CaMg2Bi2, which are correctly predicted to be insulating
within the adopted approximations. The temperature-dependent thermodynamical properties were
obtained in the framework of the quasiharmonic approximation from the phonon dispersion curves
calculated at different fixed volumes 
 (while still relaxing the position of the atoms and the shape
of the unit cell). Mode Grüneisen parameters were obtained as the logarithmic derivative of the
phonon frequencies with respect to the volume:



 = − 













,

(2.10)

where 
 is volume and 

 is the mode frequency. The averaged Grüneisen parameter 
 has been
calculated as the square root of the mode-averaged-squared Grüneisen parameter[148]


,


,



 =

,

(2.11)

where the summation is over all the modes and all the 
-points in the Brillouin zone. 

,
 is the
mode contribution to the heat capacity calculated at the Debye temperature 
D

(cid:118)(cid:117)(cid:116)

,
 
2


,
 

,

(cid:118)(cid:117)(cid:116) 5Ò
Þ ∞
Þ ∞

3
2




D = 
−1/3

0 
2
(
)


0 
2
(
)



,

(2.12)

where 
 is the number of atoms per units cell and 
(
) is the phonon density of states.

33

EFFECT OF CATION SIZE ON THE LATTICE DYNAMICS IN 

2
2 COMPOUNDS

CHAPTER 3

(Adapted from W. Peng et al. [20]. An unlikely route to low lattice thermal conductivity: Small
atoms in a simple layered structure. 2018. Joule, 2(9), 1879-1893.)

3.1 Introduction



2
2 compounds with CaAl2Si2 (
¯3
1) structure type have attracted plenty of interest as
thermoelectric materials because of their chemical diversity and tunability. 
 is an alkali or
alkaline earth or rare earth metal, 
 is Mg, Al, or a transition metal, and 
 belongs to a group
14 - 16 element. CaAl2Si2 can be characterized by covalently bonded [Al2Si2]2− bilayer slabs
formed by alternatively oriented AlSi4 tetrahedra sandwiched by monolayers of Ca2+ (see Figure
3.1). Mg3

2 (

 = Sb, Bi) has the same structure type as the ternary 

2
2, thus, there are two
distinct bonding environments for Mg. The Mg on the 
-site, defined as Mg(1) here, is octahedrally
coordinated by six Sb or Bi. In contrast, Mg(2) on the 
-site is tetrahedrally coordinated by four
anions (Sb or Bi).

With only five atoms in a unit cell and strong ionic bonding between the slabs, the lattice thermal
conductivity of 

2
2 compounds is not exceptionally low compared to other Zintl phases. In
the past decades, much research has been done in tuning electronic properties with compositions
[149, 114, 31, 115, 116, 27, 33, 28, 32, 117]. Due to the intrinsic cation vacancies of 

2
2
compounds, as demonstrated in ref. [149], all the compounds have been 
-type until the discovery
of the first 
-type MgMg2Sb2, which has the highest thermoelectric figure of merit 

 = 1.6
[118, 108, 119, 113, 120]. Since then, discussions around the explanations of band degeneracy
[108, 109, 110, 111] and 
-type origin [118, 121, 113] have been the main focus of this compound.
In contrast, little research has been done on MgMg2Sb2’s thermal conductivity.

34

Figure 3.1: MgMg2Sb2 crystallizes in the CaAl2Si2 structure (space group 
¯3
1), characterized
by anionic 
2
2 slabs separated by 
 cations. In MgMg2Sb2 and MgMg2Bi2, Mg(1) occupies the
highly distorted octahedrally-coordinated 
-site and Mg(2) the tetrahedrally-coordinated 
-site
[19]. The Mg-Sb bond lengths differ significantly in the two sites. Figure taken from W. Peng et
al. [20].

There are some interesting and unanswered questions around the low lattice thermal conductivity
of MgMg2Sb2 (alternatively written as Mg3Sb2). As shown in Figure 3.2 a), the experimental
lattice thermal conductivity of MgMg2Sb2 is one of the lowest compared to isostructural 

2
2
compounds despite its low density [16]. A single-crystal study by Song et al. [150] has confirmed
that the low lattice thermal conductivity is an inherent property rather than a result of defect
scattering. However, the origin of the low lattice thermal conductivity of MgMg2Sb2 has not been
investigated, and a detailed study of the phonon density of states is also lacking.

The purpose of this study is to explore the origins of the anomalously low lattice thermal conductivity
in MgMg2Sb2. By employing high-temperature XRD and RUS studies of 
Mg2

2 compounds
(
 = Mg, Ca, Yb, and 

 = As, Sb and Bi), the temperature dependence of elastic moduli and
thermal expansion can be obtained. Combined with the phonon calculation results from our
collaborators, one possible origin of the unrecognized soft shearing modes and highly anharmonic
acoustic phonons in MgMg2Sb2 and MgMg2Bi2 are revealed.

35

Figure 3.2: The experimental lattice thermal conductivity, 

, of MgMg2Sb2 is significantly lower
than isostructural 

2
2 compounds with similar a) density and b) predicted speed of sound. 


data can be found in Ref. [16]. The average speed of sound, 

, was estimated using the calculated
elastic moduli from MaterialsProject.org and experimental densities [21, 22]. The dashed line is
shown as a guide to the eye representing the cube dependence of 

 and lattice thermal conductivity,


, if all else remains constant [23]. Compounds shown in color are the primary focus of the current
study. Figure taken from W. Peng et al. [20].

3.2 Synthesis


Mg2

2 compounds with 
=Mg, Ca, Yb and 

=Sb, Bi were synthesized by direct ball-milling
of the elements followed by spark plasma sintering. The corresponding stoichiometric elements
(99.8% Mg shot, 99.5% Ca shot, 99.9% Yb chunk, 99.99% Sb from Alfa Aesar and 99.99%
Rotometal Bi) were used. The details of balling and sintering can be found in Chapter 2. The
maximum temperature and hold time used during spark plasma sintering are shown in Table 3.1.
All samples were at least 97% of the theoretical density. Phase purity was confirmed using a Rigaku
X-ray Diffraction system, showing that samples contained less than 3% of secondary phases.

Table 3.1: Maximum temperature and hold time used during spark plasma sintering of 
Mg2

2
(
=Mg, Ca, Yb and 

=Sb, Bi) samples.

Temp. (◦
)
Time (min)

Mg3Sb2

850
15

650
10

650
10

CaMg2Sb2 YbMg2Sb2 Mg3Bi2

CaMg2Bi2 YbMg2Bi2

600
10

700
10

700
15

36

3.3 Experimental temperature dependence of elastic moduli and speed of

sound

Figure 3.3: Experimental (asterisk) and computed (circles) elastic moduli of 
Mg2Pn2 compounds
tend to decrease as a function of increasing unit cell volume and bond length. The anomalously
low shear moduli of Mg3Sb2 and Mg3Bi2 are significant, suggesting soft bonding are unique to
these two binary compounds [22].

The elastic moduli of solids tend to become softer with increasing bond length [151]. Within
compounds in the same structural pattern, if the unit cell volume increases, the elastic moduli are
therefore expected to decrease. As shown in Figure 3.3, this trend is observed in ternary 
Mg2

2
compounds in both the experimental (
=Mg, Ca, Yb and 

=Sb, Bi) and computational elastic
moduli (
= Mg, Ca, Sr, Ba and 

=P, As, Sb, Bi) obtained from the MaterialsProject.org [21].
Note that we omitted the computed elastic moduli of rare-earth-containing compounds due to poor
agreement with experiment. Mg3Sb2 and Mg3Bi2, however, are outliers in Figure 3.3. The bulk
and shear modulus of Mg3Sb2 and Mg3Bi2 are much softer compared to compounds with similar
unit cell volume. The weak bonding in Mg3Sb2 was reported in an earlier study of the elastic
moduli of 

2
2 compounds [152], but the impact on 

 was not previously recognized.

37

Figure 3.4: Temperature-dependent a) Young’s modulus, b) shear modulus, and c) longitudinal and
d) transverse speed of sound measured using resonant ultrasound spectroscopy. Quantities were
normalized to the room temperature value. Data for Si0.8Ge0.2, PbTe, and SnTe are from Ref.
[9, 10, 24]. Figure taken from W. Peng et al. [20].

High-temperature resonant ultrasound spectroscopy was used to obtain the elastic moduli of poly-
crystalline 
Mg2

2 samples with 
=Mg, Ca, Yb and 

=Sb, Bi at different temperatures. Poly-
crystalline samples are isotropic, which means it only requires two independent elastic constants
C11 and C44 for the complete description of the elastic behavior. The elastic moduli, longitudinal
and transverse speed of sound of polycrystalline samples can be calculated applying the isotropic
equations in the experimental section.

Figure 3.4 shows the normalized temperature dependence of the Young’s and shear moduli of

Mg2

2 (
=Mg, Ca, Yb and 

 = Sb, Bi) along with a few well-known compounds Si0.8Ge0.2

38

[24], PbTe [9] and SnTe [10] as a comparison. Si0.8Ge0.2 is known to be more harmonic and stiffer,
whereas PbTe and SnTe are typical anharmonic and soft materials.

Over the measured temperature range, the elastic moduli of MgMg2Sb2 and MgMg2Bi2 soften
by an impressive amount of ∼25% in contrast to a ∼5% decrease for CaMg2

2 and YbMg2

2
samples. The softening rate of the elastic moduli with temperature, as explained in Chapter 1,
can be directly correlated with anharmonicity. The results are an indication that MgMg2Sb2 and
MgMg2Bi2 are significantly more anharmonic compared to other 

2
2 compounds. Additional
evidence can be seen from thermal expansion (see Figure 3.5). MgMg2Bi2 and MgMg2Sb2 has a
higher thermal expansion coefficient compared to other 
Mg2

2 compounds, which again shows
their anharmonic bonding nature.

Figure 3.5: Experimental thermal expansion of YbMg2Bi2, CaMg2Sb2, CaMg2Bi2, MgMg2Sb2,
and MgMg2Bi2. As we can see, MgMg2Bi2 and MgMg2Sb2 has the highest thermal expansion
coefficient compared to other 
Mg2

2 compounds. Figure taken from W. Peng et al. [20].

39

3.4 Lattice dynamics and Grüneisen parameters of 
Mg2

2

To better understand the origin of anharmonic behavior in MgMg2Sb2 as well the discrepancies of
thermal expansion and elastic moduli, a computational investigation of lattice dynamics is necessary.
The calculation results shown below are done in collaborations with Guido Petretto, Gian-Marco
Rignanese, Geoffroy Hautier at Université catholique de Louvain. The phonon properties and elastic
constants were obtained from density functional theory (DFT) and density functional perturbation
theory (DFPT) simulations using the ABINIT software package [134, 135, 136].

As shown in the stacked partial phonon density of states (PDOS) of MgMg2Sb2, CaMg2Sb2, and
CaMg2Bi2 in Figure 3.6, the anion is responsible for the low frequency modes whereas the cation
(
) and metal site (
=Mg(1)) vibrations are distributed in the mid- and high-frequency regime,
respectively. Given that low-frequency acoustic vibrations dominate the lattice thermal transport, it
is natural for one to expect that the cation species does not have much impact on the anharmonicity
of these compounds, which is clearly not the case for 
Mg2

2 compounds.

Figure 3.6: The partial phonon density of states of MgMg2Sb2, CaMg2Sb2 and CaMg2Bi2 shows
that the low-, mid-, and high-frequency regimes are dominated by displacements of the anions, 

,
cations, 
, and metal site, 
=Mg(1), respectively. Figure taken from W. Peng et al. [20].

40

The interactions between the cation and anion site are somehow having a major impact on the acous-
tic modes in MgMg2Sb2. A comparison of the phonon dispersions of MgMg2Sb2, CaMg2Sb2, and
CaMg2Bi2 can demonstrate this point. The slopes of the transverse and the longitudinal acoustic
branches represent the transverse and longitudinal phonon velocities, respectively. As shown from
the black lines in Figure 3.7, a smaller slope as well as the "dips" around the 
-, 
-, and 
-point
on the transverse acoustic branch of MgMg2Sb2 set this compound as an outlier.

Figure 3.7: The black lines are the phonon dispersion of MgMg2Sb2, CaMg2Sb2, and CaMg2Bi2
at 0 K. The volume dependence of the phonon dispersions are shown through the thickness of the
bands, with red and blue representing positive and negative values of mode Grüneisen parameters,
respectively. Figure taken from W. Peng et al. [20].

As mentioned in Chapter 1, the frequency shift with the unit cell volume is defined as the mode
Grüneisen parameter, representing local anharmonicity of a certain vibration mode.



 = − 













,

(3.1)

where 
 is volume and 

 is the mode frequency. To quantify the degree of anharmonicity of
each acoustic vibrational mode, the volume dependence of the frequency are also shown Figure
3.7 via the colored bands. The thickness of the bands represent the normalized mode Grüneisen
parameters, from which 
-, 
-, and the 
-point of the transverse phonons in MgMg2Sb2 are
identified as modes with large Grüneisen parameters and therefore most unstable.

The red and blue color bands in Figure 3.7 represent the positive and negative values of mode
Grüneisen parameters, respectively. A locally negative thermal expansion in a material with a

41

positive average thermal expansion coefficient is usually a result of a local metastable mode. This
can be seen in Figure 3.8, the absolute values of the mode Grüneisen parameters of MgMg2Sb2 can
reach as high as ∼5 at the low-frequency regime, but the similar amount of positive and negative
distribution of the value lead to a low average mode Grüneisen parameter. As a comparison,
the acoustic mode Grüneisen parameters of CaMg2Sb2 only contain densely distributed positive
values. A negative thermal expansion corresponds to a negative Grüneisen parameter value from
definition, but it is contributing to the overall anharmonic phonon-phonon scattering. The large
mode Grüneisen parameters of MgMg2Sb2 lead to high anharmonic phonon-phonon scattering
rate, leading to its anomalously low lattice thermal conductivity.

Figure 3.8: The mode Grüneisen parameters as a function of frequency. Figure taken from W. Peng
et al. [20].

One might notice that the thermal expansions of Mg3Sb2 and Mg3Bi2, in contrast to the drastic
difference in elastic moduli, are only slightly higher compared to other 

2
2 compounds (Figure
3.5. While elastic moduli and speed of sound is related solely to the slope of acoustic branches
upon Ɖ-point, the thermal expansion is an effect from the average of all the vibrational modes in
both the acoustic and optics regimes. The negative mode Grüneisen parameters in the acoustic
branches, for example, can lead to a lower average Grüneisen parameter and therefore reduced
thermal expansion.

42

3.5 Breaking of Pauling’s radii rule

Given the results above, it comes to the question of the fundamental reasons behind the unusual
phonon properties of MgMg2Sb2. The cation size plays an important role in the stability of
CaAl2Si2 structure type relative to other 

2
2 structures (i.e. BaCu2S2 and ThCr2Si2 structure
types) [16, 153, 45]. In the sphere packing model proposed by Pauling for ionic solids, the smallest
stable cation to anion radius for octahedral coordination is given by 






:





 = 0.414 [154].

Figure 3.9 shows the estimated 






:





 for 
Mg2

2 compounds, in which compounds
with 
=Mg have 






:





 below the stability limit, while compounds with larger cations
are predicted to be stable in a six-fold coordinated environment.
If the radii ratio is less than
the minimum, the polyhedron is more likely to get distorted in order to depart the anions from
touching. This distortion is featured in both MgMg2Sb2 and MgMg2Bi2, in which large octahedral
bond angle variances of 26.53◦ and 36.01◦ can be observed. As a comparison, the octahedral bond
angle variance in CaMg2Sb2 and CaMg2Bi2 is only 1.77◦ and 5.04◦. The cation radii were from
ref [155, 156] using the values for 2+ valence and 6-fold coordination. The anionic radii were
estimated by taking the average 
-

 bond length and subtracting the corresponding cation radii,
yielding 

 = 1.93 Å, 
 

 =2.07 Å, 


 = 2.23 Å, and 


 = 2.29 Å.

3.6 Conclusion

Inherently low lattice thermal conductivity is typically associated with dense materials or com-
pounds with lattice complexity. The unusually low lattice thermal conductivity of Mg3Sb2, in
contrast, shows that it is possible for a simple structure with low density to have high phonon
scattering rates and low lattice thermal conductivity when structural instability is presented.

This was shown experimentally by the rapid decrease of the speed of sound and elastic moduli in
Mg3Sb2 and Mg3Bi2 with increasing temperature, which is a direct consequence of the softening
of the acoustic modes. By combining ab initio phonon calculations and high-temperature elasticity

43

Figure 3.9: For octahedral coordination (CN=6), Pauling’s radius ratio rules predict a minimum
stability limit of 






:





 = 0.414. For Ca, Sr, Eu, and Yb, this rule is satisfied. In contrast, the
Mg cation is too small, which leads to a distorted octahedral environment and may be responsible
for weak, anharmonic interlayer bonding. Figure taken from W. Peng et al. [20].

measurements, we showed that Mg3Sb2 and Mg3Bi2 are highly anharmonic, in contrast to the Ca-
and Yb-containing 
Mg2

2 compounds investigated here. Large mode Grüneisen parameters,
both negative and positive, were predicted in the acoustic branches of Mg3Sb2 and Mg3Bi2, which
are expected to have a large contribution to thermal transport.

We attribute this unique behavior to the small radii of Mg, which is undersized for the octahedrally-
coordinated cation site. The poor fit of the Mg cation is suspected to lead to weak interlayer
bonding, which was observed as the soft shear modes both experimentally and computationally.
This behavior ultimately results in the highly anharmonic behavior of the acoustic branches and
thus low thermal conductivity. These results suggest more broadly that soft shear modes resulting
from undersized cations provide a potential path to low lattice thermal conductivity in ionic layered
structures.

44

CHAPTER 4

LIMITS OF CATION SOLUBILITY IN 
Mg2Sb2 (
=Mg, Ca, Sr, Ba) ALLOYS

(Adapted from W. Peng et al. [25]. 2019. Limits of cation solubility in 
Mg2Sb2 (
= Mg, Ca, Sr,
Ba) alloys. Materials (Basel, Switzerland), 12(4).)

4.1 Introduction

Given that the small cation size is the origin of anharmonicity in MgMg2Sb2 and the smallest
cation that can occupy the octahedral site (
) of 
Mg2

2 is Mg, the next question is what
is the impact of the cation size on solubility. Complete solubility has been reported in most


2
2 isotructural alloys with the CaAl2Si2 structure type regardless of the substitution site
[30, 31, 32, 28, 27, 28, 29, 28, 33, 34].

Thus far, much of the 

 optimization of 
-type MgMg2Sb2 has focused on alloying or doping on
the anion site.(e.g. alloying Bi and doping Te on the Sb site of MgMg2Sb2). In contrast, for most
of the previous work on the isostructural 
-type 

2
2 compounds, alloying on the cation site (

= Mg, Ca, Sr, Ba, Sm, Eu, Yb) was a frequent approach to improve and optimize thermoelectric
performance. For example, the mixed occupancy of 
 = Ca, Yb, and Eu in the 
Zn2Sb2 [27, 28, 31]
and 
Cd2Sb2 systems [32, 28, 33, 29], and more recently in the 
Mg2

2 system, have been
investigated of their effect on enhancing the thermoelectric performance [34, 111, 30].

Alloying on the cation site is appealing in two major perspectives. Recall 

 = 
2


 
. Previous
evidence has shown that the site disorder often increases the ratio of electronic mobility to lattice
thermal conductivity [30, 31, 32], which is beneficial to a higher 

. Equally importantly, alloying
subtly affect the defect formation energy for cation vacancies, which is the dominant intrinsic defect
in this structure type [157, 149]. This allows for optimization of the carrier concentration without
introducing additional dopants.

45

Although isovalent alloying on the cation site in 
-type MgMg2

2 has not yet been reported, one
would expect that similar 

 enhancements could be achieved through this strategy (assuming that

-type doping can still be achieved for 
 ࣔ Mg.)

In the present study, we reported the solubility limits of larger cations (e.g., Ca, Sr, Ba) alloyed with
Mg on the 
 cation site in 
Mg2Sb2 compounds. Even though alloying on the 
 site seems to be
an applicable approach to further optimize the thermoelectric properties of 
-type MgMg2Sb2, the
present study shows that the small ionic radius of Mg2+ leads to limited solubility of any cation
larger than Ca.

4.2 Experimental

Table 4.1: The SPS temperatures of (Ca
Mg1−
)Mg2Sb2, (Sr
Mg1−
)Mg2Sb2, (Ba
Mg1−
)
Mg2Sb2, and (Ba
Ca1−
)Mg2Sb2.
x=0.1
810
x=0.1
800
x=0.3
750
x=0.1
700

(Ca
Mg1−
)Mg2Sb2
(Sr
Mg1−
)Mg2Sb2
(Ba
Mg1−
)Mg2Sb2
(Ba
Ca1−
)Mg2Sb2

Temp (◦
)
Temp (◦
)
Temp (◦
)
Temp (◦
)

x=0.8
670
x=1
700
-
-
-
-

x=1
650
-
-
-
-
-
-

x=0.6
710
x=0.8
750
-
-
x=1
700

x=0.7
690
x=0.9
750
-
-
-
-

x=0.5
730
x=0.7
750
x=1
700
x=0.9
700

x=0
850
x=0
850
x=0
850
x=0
650

x=0.2
790
x=0.2
750
x=0.5
700
x=0.3
700

x=0.3
770
x=0.4
750
x=0.8
700
x=0.5
700

x=0.4
750
x=0.6
750
x=0.9
700
x=0.7
700

Samples of (Ca
Mg1−
)Mg2Sb2 (
=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1), (Sr
Mg1−
)Mg2Sb2
(
=0, 0.1, 0.2, 0.4, 0.6, 0.7, 0.8, 0.9, 1), (Ba
Mg1−
)Mg2Sb2 (
=0, 0.3, 0.5, 0.8, 0.9, 1), and
(Ba
Ca1−
)Mg2Sb2 (
=0, 0.1, 0.3, 0.5, 0.7, 0.9, 1) were synthesized via direct ball-milling of
elements followed by spark plasma sintering (SPS). The corresponding stoichiometric elements
(99.8% Mg shot, 99.5% Ca shot, 99% Sr chunk and 99%+ Ba rod, and 99.99% Sb from Alfa
Aesar) were used. The details of balling and sintering can be found in Chapter 2. The maximum
temperature and hold time used during spark plasma sintering are shown in Table 4.1. The densities
of all the samples were obtained by measurement of mass and geometry, yielding at least 97% of the
theoretical density. All of the alloyed samples were annealed after sintering to ensure homogeneity.

46

The samples were wrapped in graphite foils and sealed in glass ampules under vacuum, which were
then held at 500◦
 for ten days. The samples were quenched to room temperature in air. X-ray
diffraction was performed before and after annealing. The lattice parameters and R
 
 values from
refinements of powder XRD patterns can be seen in the Appendix.

4.3 Results and discussion

Among 
Mg2Sb2 and 
Mg2Bi2 compounds, Mg is the smallest cation that can occupy the
octahedral site (
). The ionic radius of Mg2+ in an octahedral environment is 0.72 Å, which is
significantly smaller than that of Ca2+ (1.00 Å), Sr2+ (1.18 Å) or Ba2+ (1.35 Å) [155, 156]. The
divalent rare-earth metals Sm, Eu, and Yb can also occupy the cation site, having ionic radii in
between that of Ca and Sr. In the present study, alloyed samples in the series (Ca
Mg1−
)Mg2Sb2
(
=0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 1), (Sr
Mg1−
)Mg2Sb2 (
=0, 0.1, 0.2, 0.4, 0.6, 0.7, 0.8,
0.9, 1), (Ba
Mg1−
)Mg2Sb2 (
=0, 0.3, 0.5, 0.8, 0.9, 1) were synthesized to investigate the phase
stability when cations of increasingly divergent ionic radii occupy the 
 site in 
Mg2Sb2. Note
that Ca, Sr, and Ba are too large to occupy the tetrahedrally-coordinated 
 site of the 

2
2
compounds. Therefore, with increasing amount of Ca, Sr, or Ba, the observed mixed occupancy is
solely presented on the 
 = Mg site, not on the 
 = Mg site.

For the (Ca
Mg1−
)Mg2Sb2 series, we find that the lattice parameters undergo a linear change
with calcium alloying for 
=0-1 (Figure 4.1a) and b)), showing that (Ca
Mg1−
)Mg2Sb2 forms
a complete solid solution according to Vegard’s rule [158]. In contrast, alloying Mg with larger
cations (Sr or Ba) leads to phase separation into a Mg-rich phase and Mg-poor phase, indicating a
eutectic-like phase diagram.

The lattice parameters of (Sr
Mg1−
)Mg2Sb2 and (Ba
Mg1−
)Mg2Sb2 are shown in Figure 4.1c)-
f). For the Sr-Mg alloy, a slight decrease in the lattice parameters 
 and 
 indicates a small (roughly
10%) solubility for Mg on the Sr site, but no solubility of Sr on the Mg site. In the Ba-Mg alloy,

47

Figure 4.1: a-b) For the (Ca
Mg1−
)Mg2Sb2 series, the lattice parameters 
 and 
 underwent
a linear change with calcium alloying ratio. c-d) For the (Sr
Mg1−
)Mg2Sb2 series, the lattice
parameters indicate a 10% solubility for Sr on Mg site, and no solubility for Mg on Sr site. e-f) For
(Ba
Mg1−
)Mg2Sb2 series, the lattice parameters show no solubility between Ba and Mg on the
cation site. Note that the MgMg2Sb2 phase can be observed from X-ray diffraction pattern when 

= 0.8 in the Sr-Mg series and for 
 = 0.8 and 0.9 in the Ba-Mg series, but the peak intensities are
too low for reliable refinement of lattice parameters. Figure taken from W. Peng et al. [25].

no solubility of Ba on the Mg site, or of Mg on the Ba site was observed. Note that the lattice
parameters shown in Figure 4.1 were measured after annealing at 500◦
 for ten days.

48

Figure 4.2: The linearly varying lattice parameters in the (Ba
Ca1−
)Mg2Sb2 alloy indicate com-
plete solid solubility of this system. Figure taken from W. Peng et al. [25].

In the ionic metal model proposed by Hume-Rothery for substitutional solid solutions, differences
in ionic radius, polarizability, structure, valence, and electronegativity are the key factors affecting
the solubility [159, 160, 161]. Here, the atomic size difference is expected to play a dominant role
in the stability [161, 162]. To estimate an upper limit for size mismatch on the cation site in the

Mg2Sb2 compounds discussed here, we use the limit established by the partial solubility of the
Sr-Mg system. For substitutions of a small cation by a larger one, the upper limit size mismatch is
estimated by (


-


)/


 = 64%. For substitutions of a larger cation by a smaller one (e.g., Mg
on the Sr site) the limit is given by (


-


)/


=39%. To test these limits, the Ba
Ca1−
Mg2Sb2
series was synthesized. The radii difference of Ba to Ca is 35% and Ba to Ca is 25.9%, both
of which are smaller than the critical size difference. As shown by the linearly increasing lattice
parameters in Figure 4.2, the Ba
Ca1−
Mg2Sb2 alloy is found to be a complete solid solution, as
predicted.

A survey of prior alloying studies of isotructural 

2

2 compounds suggests that the limit
proposed here is likely to be generic to the cation-site alloying. As shown in Figure 4.3, the ionic
radius ratio, (
1-
2)/
2, for all possible combinations of cations where 
1 > 
2 are summarized
here. The ionic radii were obtained from ref [155, 156] using the values for 2+ valence and

49

6-fold coordination. Square symbols indicate cation combinations that have been experimentally
attempted, while circles represent our predictions.
Indeed, this figure illustrates that it is only
possible to exceed the predicted size mismatch limit by alloying with Mg on the cation site.
All other combinations have sufficiently similar ionic radii (e.g., Ca-Yb [30, 31, 32, 28], Ca-Eu
[27, 28, 29], Yb-Eu [28, 33], Mg-Yb [34].) to form complete solid solutions.

Figure 4.3: The cation radii difference of 

2
2 calculated from (
1-
2)/
2 for all possible
combinations of cations where 
1 > 
2. In each combination, the smaller species is listed first.
Square symbols represent experimental observations, while circles represent predictions. The
limited solubility in Sr/Mg (present study) and Sm/Mg (Ref [26]) alloys provide an approximate
upper limit for cation radii mismatch. The complete solubility of Ca/Eu [27, 28, 29], Ca/Yb
[30, 31, 32, 28], and Yb/Eu [28, 33] Mg/Yb [34] have been confirmed by prior studies. Figure
taken from W. Peng et al. [25].

One notable exception in the literature is the (Sm,Mg)Mg2Sb2 system, investigated in 2006 by
Gupta et al. [26]. Depending on synthesis conditions, alloying with Sm on the Mg site was shown
to lead to either phase separation or to the formation of a superstructure in which Mg and Sm occupy
alternating cation monolayers. The ionic radii of Sm2+ is similar to that of Sr2+, which appears to
be around the upper limit for ionic radius on the 
 = Mg site. However, no superstructure formation
was observed after quenching from high temperature for any of our alloyed samples. The upper size
limit established here may provide guidance for doping on the Mg site in MgMg2Sb2. Recently,

50

La3+ and Y3+ on the Mg site were successfully used as an 
-type dopants [163, 164]. The ionic
radii of La3+ are similar to that of Yb2+, as are the radii of the majority of trivalent lanthanides, and
Y3+ has a even smaller radii between Yb2+ and Mg2+. This suggests that the radii of these 
-type
dopants will not be a primary factor limiting their solubility. We note, however, that complete solid
solubility would never be expected for alio-valent dopants in Zintl phases. Further, in alio-valent
doping, the size of the dopant is only a minor factor. Other factors controlling solubility such as the
valence of the dopant and the impact that the dopant has on the chemical potential of other types
of defects.

4.4 Conclusion

In 

2
2 alloys, the existence of complete solid solubility is found to depend strongly on the
difference between the ionic radii of the alloyed species. For mixed occupancy on the cation
site, the partial solubility of Mg on the Sr site in the (Mg
Sr1−
)Mg2Sb2 series indicates that the
size mismatch of Sr and Mg can be used as an approximate upper limit to guide future alloying
and doping studies.
Indeed, among all cations that are known to occupy the 
 site in 

2
2
compounds, we find that only Mg is sufficiently small to lead to phase separation, and only when
alloyed with cations with radii equal to or larger than Sr (e.g., Ba, Eu, or Sm).

51

CHAPTER 5

HIGH-PRESSURE COMPRESSIBILITY, ANISOTROPY AND PHASE TRANSITIONS

OF 

2
2 COMPOUNDS

(Adapted from M. Calderon and W. Peng et al. [165] and W. Peng et al. [6].
[165] Anisotropic structural collapse of Mg3Sb2 and Mg3Bi2 at high pressure. 2020. Chemistry
of Materials (submitted).
[6] High-pressure behavior of layered 

2
2 compounds. 2020. (in preparation))

5.1 Introduction

High-pressure is an effective tool for the discovery of novel phases of known materials, synthesis
of new compounds, as well as studying structure-property relationships. The most fascinating
aspect of pressure is that it allows the variation of bonding environment without varying chemical
compositions, which can avoid introducing new variables to the system and provides an opportunity
for 

-



 observations without disturbance [166].

In the past few years, the alloys between Mg3Sb2 and Mg3Bi2 have become increasingly interesting
as 
-type thermoelectric materials, with a thermoelectric figure of merit higher than any other
isostructural Zintl phases as well as the traditional thermoelectric materials such as Bi2Te3 [167,
168, 120, 118]. One of the major reasons for such a high figure of merit of 
-type Mg3Sb2 is
because of their exceptionally low lattice thermal conductivity [169, 170, 171, 118, 67, 172]. The
results described in Chapter 3 [20] have explained the anomalously low lattice thermal conductivity
of Mg3Sb2 using a combination of resonant ultrasound spectroscopy and DFT phonon calculations,
giving insights into the rapid softening of the speed of sound, soft shear acoustic modes, and large
Grüneisen parameters originated from the loosely-bonded Mg-Sb octahedron [20].

As described in the previous sections, Mg3Sb2 and Mg3Bi2 are binary members of the CaAl2Si2

52

structure type (
¯3
1), making them part of a broader family of 

2
2 Zintl compounds. Tradi-
tionally, 

2
2 compounds are described as layered structures [106, 102, 35], which consists of
the covalent [
2
2]2− slabs and the ionically-bonded interlayer 
2+ cations to provide the overall
charge balance [115]. However, as suggested by the first-principles study by Zhang et al., this
layered structure description is not suitable in the case of Mg3Sb2 and Mg3Bi2 [67] even though
they share the same structural description, since the Mg(1)-Sb and Mg(2)-Sb bonds in Mg3Sb2 are
quite similar with respect to the degree of charge transfer from Mg to Sb. The study further showed
the calculated near-isotropic compressibility of Mg3Sb2 under pressure to prove this quasi-isotropic
bonding prediction [67].

Thus far, much of the experimental studies related to the elastic behavior of the 

2
2 com-
pounds were focused on the temperature-dependence. However, equally importantly, the pressure-
dependence of the same set of material can provide a different aspect of insights - but the data on this
relationship is generally lacking. In fact, there have been only a handful of high-pressure studies of
compounds in the CaAl2Si2 structure type, despite their long history and technological importance
[173, 174, 175]. The understanding of lattice stability and bonding environment can benefit the
understanding of fundamental thermoelectric properties such as lattice thermal conductivity, which
can be studied with high-pressure experiments.

In addition to the ability to obtain anisotropic compressibility, for a given composition, pressure
can simulate different bonding and atomic size conditions, which can be a leading factor for phase
transitions. Specifically, the structure type of 

2
2 compounds is controlled partially by the
cation to anion size ratio [16]. This generates an even higher tendency for the ambient phase to
transform to other competing structures types when pressure is applied. In a study by Zevalkink et
al. [174], SrAl2Si2 transformed from the CaAl2Si2-type to the ThCr2Si2-type structure at 3 GPa
/ 700 K. In a more recent study, pressure-induced polymorphism between the trigonal structure
(
¯3
1) and monoclinic structure was reported by Gui et.al [176], in which single-crystal CaMn2Bi2

53

Figure 5.1: Structure types formed by 

2
2 compounds with 
 

 =16 include a) the trigonal
CaAl2Si2 structure type (
¯3
1), b) the orthorhombic BaCu2S2 structure type (



), and c)
the tetragonal ThCr2Si2 structure type (
4/


). d) The stability range of each structure can be
delineated as a function of the atomic radii, 
 = 
 
/(

 + 0.2
 
), while the polarity of the 
-

bond is clearly not a deciding factor [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 19, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. Figure taken from W. Peng et al.
[16].

exhibits a displacive structural transition to a monoclinic structure (
21/
) between 2-3 GPa at
room temperature. Therefore, it is likely that the 

2
2 phases being selected for this elastic
moduli study can also transform into polymorphic forms at high-pressure/temperature.

In the present study, we investigate the high-pressure behaviors experimentally using 

-



 syn-
chrotron X-ray diffraction of Mg3Sb2, Mg3Bi2, YbMg2Bi2, CaMg2Sb2, and EuAl2Si2 under
different pressure and temperature conditions at pressures up to ∼50 GPa, and temperatures up to
900 K, revealing disparities in the anisotropic compressibility of the ambient and high-pressure
structures in Mg3Sb2, and the relative elastic anisotropy of the selected 

2
2 compounds. Phase
transitions have also been observed for the first time in several compounds at different pressures and
temperatures. This work provides direct experimental evidence of the bond strength and relative
phase stability of 

2
2 compounds,

54

5.2 Phase transition of Mg3Sb2 and Mg3Bi2 at high-pressure

High-pressure powder diffraction experiments were performed to investigate the compressibility
and possible high-pressure phase transitions of the Mg3Sb2 and Mg3Bi2. As shown in Figure
5.2, the ambient structure pattern shifted consistently with pressure up to ∼7.8 GPa for Mg3Sb2
and ∼4.0 GPa for Mg3Bi2. Above these two pressures, new peaks as well as abrupt changes of
compressibility can be observed, indicating a pressure-induced phase transition.

To determine whether or not the phase transition is reversible, we performed a decompression
experiment on Mg3Bi2. The ambient-pressure 
¯3
1 structure was completely recovered when the
pressure was reduced (see Figure 5.3), indicating the non-quenchability of the high-pressure phase
and confirming that the new peaks at high-pressure are not a result of decomposition.

The high-pressure structure was solved on a Mg3Sb2 single crystal. The synthesis of the single
crystal and structural solution were done by our group member, Mario Calderon. The details of the
single crystal diffraction and crystallographic data can be found in ref [165]. A structure solution
for Mg3Sb2 at 7.8 GPa was reached in the monoclinic 
2/
 space group. As shown in Figure
5.4, the high-pressure monoclinic structure of Mg3Sb2 can be viewed as a distorted variant of the
original trigonal structure at low pressure. The tetrahedrally coordinated Mg(2)2Sb2 slab in the
ambient phase transforms into a layer with alternating tetrahedral and square pyramidal coordination
environments at high pressure. The same 
2/
 structure solution provides a satisfactory fit for
Mg3Sb2 and Mg3Bi2 powder data at high pressure, suggesting that both phases undergo the same
high-pressure phase transition.

5.3 Compressibility and anisotropy of Mg3Sb2 and Mg3Bi2

To investigate the impact of cation size on the pressure response of 

2
2 structures, we first
study the the binary members of the 

2
2 family:
the ambient- and high-pressure structures
of Mg3Sb2 and Mg3Bi2. The pressure-dependence of the volume per formula unit for Mg3Bi2

55

Figure 5.2: Powder diffraction patterns of a) Mg3Sb2 and b) Mg3Bi2 at ambient temperature reveal
the emergence of new peaks belonging to a high-pressure phase above 7.8 and 4.0 GPa respectively.
Peak bars in blue correspond to the trigonal ambient and newly-discovered high-pressure phase.
The X-ray wavelength for Mg3Sb2 and Mg3Bi2 is 0.4133 Å and 0.4340 Å, respectively. Figure
taken from W. Peng et al. [6]

and Mg3Sb2 from powder and single crystal samples is shown in Figure 5.5, represented by
circles and asterisks, respectively. The pressure-dependence of the powder and single crystal data

56

Figure 5.3: The reversibility of the phase transition was confirmed by decompressing the powder
Mg3Bi2 sample. All of the original 
¯3
1 peaks re-emerged below 4.0 GPa. Figure taken from W.
Peng et al. [6].

Figure 5.4: Comparison of the a) ambient (
¯3
1) and b) high-pressure (
2/
) structure of
Mg3Sb2. The high-pressure monoclinic structure of Mg3Sb2 can be viewed as a distorted variant
of the original trigonal structure at low pressure. c) The four Mg coordination environments in the
high-pressure structure. Figure taken from W. Peng et al. [6].

Mg3Sb2 agree well within the measured temperature range. The unit cell volume obtained from
the powder data decreases abruptly above approximately 7.8 GPa and 4.0 GPa for Mg3Sb2 and
Mg3Bi2, respectively. Note that even though the the pattern at 7.8 GPa is shown in Figure 5.2, is
not included here. The on-going phase transition made the pattern hard to be accurately refined.

57

Figure 5.5: The pressure-dependence of the volume per formula unit for Mg3Sb2 and Mg3Bi2
from powder and single crystal samples are represented by circles and asterisks, respectively. The
zero-pressure bulk modulus, 
0, of both the ambient- and high-pressure phases were obtained from
a 2nd-order Birch-Murnaghan equation of state fit, represented by the solid lines. Figure taken
from W. Peng et al. [6].

The bulk modulus at zero-pressure, 
0, of the ambient- and high-pressure phase of each compound
was fitted using the powder diffraction data with the 2nd-order Birch-Murnaghan equation of state,
which are displayed as the solid curves in Figure 5.5. The 
0 for the ambient structures (
¯3
1)
were obtained via Rietveld refinement of the corresponding room temperature XRD pattern. For the
high-pressure phases (
2/
), since 
0 is unknown, they were treated as an open fitting parameter
in the Birch–Murnaghan EOS fit. The zero-pressure bulk modulus of AP- and HP-Mg3Sb2 is 38
GPa and 46 GPa, respectively, while the zero-pressure bulk modulus of AP- and HP-Mg3Bi2 is
37 GPa and 49 GPa respectively. The zero-pressure bulk moduli of AP-Mg3Sb2 and AP-Mg3Bi2
obtained in this study is comparable to the results of resonant ultrasound spectroscopy (36 GPa and
38 GPa, respectively) [20] and DFT (42 GPa and 37 GPa, respectively) [22]. The HP structures
of both compounds are slightly stiffer than the AP structures, similar to the behavior reported for

58

CaMn2Bi2 [176]. The exact fitted parameters from the 2nd-order Birch–Murnaghan EOS fit for all
four phases are shown in Table A5 in the Appendix. The uncertainties of the pressure and lattice
parameters is shown in Tables A6-A9 in the Appendix and the parameters of the 2nd-order and 3rd
order Birch-Murnaghan fit can be found in Table A5 in the in the Appendix.

Figure 5.6: a) - b) A comparison of the unit cell of the 
¯3
1 (ambient) and 
2/
 (high-pressure)
structures. The blue and grey lines are used to outline the ambient-pressure cell in both structure
types, while the cyan lines represent the interatomic distance, 
, which is equal to a and b in the

¯3
1 symmetry. Here, we define a’=b’ and c’, and a’ and g’ to represent the primitive unit cell
after it has lost its trigonal symmetry. Note that these parameters do not correspond to the true a,
b, and c axes of the monoclinic 
2/
 unit cell. Normalized lattice constants of powder a) Mg3Sb2
and b) Mg3Bi2 before and after the phase transition. We have defined 
 as the [110] diagonal of
the trigonal AP unit cell, as shown in the inset of panel. c) - d) A comparison of the normalized
lattice constants and interatomic distance, 
, of powder Mg3Sb2 and Mg3Bi2. The dashed lines in
panel c) show the results of a prior computational study by Zhang et. al.[67]. Figure taken from
W. Peng et al. [6].

59

The question of whether or not Mg3

2 (

 = Sb, Bi) are layered structures has been under debate
[80]. For a typical layered compound, the out-of-plane axis is significantly more compressible
than the in-plane axis [177, 178, 179].
In fact, anisotropic compressibility is a key feature of
layered structures, in particular those characterized by weak interlayer van der Waals bonding.
However, even though Mg3

2 share the same structural features as the ternary layered 

2
2
compound family, the in-plane (Mg(2)-

) and out-of-plane (Mg(1)-

) bonding in Mg3

2 has
been shown to be chemically similar despite their difference in the bonding environment. This was
reported in a recent computational study of the pressure-dependence of AP-Mg3Sb2 by Zhang et
al. [67] predicted nearly isotropic compressibility of the 
- and 
-axes. As shown in Figure 5.6 c),
our experimental powder diffraction data (circle symbols) is consistent with Zhang’s predictions
(dashed lines) up to 8 GPa. Further, powder data for Mg3Bi2 up to 4 GPa (Figure 5.6 d)) reveals
that the 
-axis and 
-axis of AP-Mg3Bi2 compress at near-identical rates, suggesting a even more
isotropic behavior than AP-Mg3Sb2. The exact values and uncertainties of each powder data point
in Figure 5.6 can be found in Tables A6-A9 in the Appendix.

When the ambient Mg3

2 structure (
¯3
1) transforms to the high-pressure Mg3

2 structure
(
2/
), the volume collapse is highly anisotropic with respect to the relative compression of the
unit cell axes, as can be seen in Figure 5.6. At pressures above the phase transition, we continued
to employ the unit cell axes from the trigonal 
¯3
1 phase to analyze the 
2/
 structure since the
monoclinic cell is four-times the size of the trigonal cell and with axes re-defined in a different angle.
The 
 lattice parameter is the out-of-plane direction, and the 
 lattice parameter is the in-plane
directions. Here, we have newly defined 
 as the [110] diagonal of the trigonal unit cell, which is
also an in-plane direction. In the 
¯3
1 structure, due to the fixed 120 degree angle, 
, between the

 and 
 axes, the relationship 
 = 
 is fixed regardless of the pressure. From Figure 5.6, we can see
that the phase transition in both compounds involves a collapse in the 
- and 
-direction, while 

remains unchanged. The reduced symmetry of the 
2/
 structure allows the angle to deviate from
120 degrees, such that 
 < 
 after the phase transition. The drastic collapse along the 
-direction

60

is a result of the transition of the distorted [Mg(1)-Sb6] octahedra to a square planar coordination
environment. The experimental evidence of this statement can be found in the single crystal data
in ref [165]. Furthermore, at pressures exceeding the phase transition, we find that the 
-axis of
the HP structure is more compressible than the 
-axis, which is quite different from the isotropic
compressibility of the AP structure.

Note that the nearly-isotropic in-plane and out-of-plane compressibility in ambient-pressure 
¯3
1
Mg3

2 (

=Sb, Bi) does not mean that the octahedral Mg(1)-

 bonds are equal in strength to
the tetrahedral Mg(2)-

 bonds. The force constant calculations reported by J. Ding et al. have also
shown that the octahedral Mg(1)-

 bonds are significantly weaker than the tetrahedral Mg(2)-


bonds [180]. Experimentally, the single crystal data collected in the present study has also given
similar result, in which the octahedral volume decreases more rapidly than the tetrahedral volume
(The data can be found in the original paper.).

The relatively weak octahedral Mg(1)-

 bonds help to explain the anomalously weak shear
modulus and soft transverse phonon modes reported in Mg3

2 compounds. These instabilities
are in turn responsible for the low thermal conductivity and excellent thermoelectric performance
of Mg3

2 compounds [20, 180].

5.4 Compressibility and anisotropy of ternary 

2
2 compounds

Here, we show the pressure response of the unit cell volume and the lattice parameters of YbMg2Bi2,
CaMg2Sb2, and EuAl2Si2, in comparison with the Mg3Bi2, Mg3Sb2 data with the same measure-
ment and data processing approach in the prior section. The zero-pressure bulk modulus, 
0
of each compound were fitted with the 2nd-order Birch-Murnaghan equation of state, which are
reproduced as solid curves shown in Figure 5.7 a). The 
0 obtained in this study for all compounds
are comparable to those from high-temperature resonant ultrasound spectroscopy [20] and the DFT
results (see table in Figure 5.7 b) [22].

61

Figure 5.7: a) The pressure-dependence of the unit cell volume for YbMg2Bi2, CaMg2Sb2,
MgMg2Bi2, MgMg2Sb2, EuZn2Sb2, and EuAl2Si2. The solid lines are reproduced with the
corresponding bulk modulus of each compound with the 2nd-order Birch-Murnaghan equation.
b) A comparison of the bulk modulus obtained with different methods: Resonant ultrasound
spectroscopy (RUS), density functional theory (DFT), and high-pressure X-ray diffraction.

From the RUS measurements, only the average bulk modulus in all directions is reflected.
In
contrast, the in-situ high-pressure measurements provide a convenient approach to track the progress
of the relative compressibility along the a- and c-axis, which give information on which direction is

62

being affected the most. Here, we explored different types of 

2
2 compounds with the trigonal
CaAl2Si2 structure type to study the relationship between anisotropy and the bulk modulus within
this structure type.

Figure 5.8: a) The pressure-dependence of the c/a ratio for YbMg2Bi2, CaMg2Sb2, MgMg2Bi2,
MgMg2Sb2, and EuAl2Si2. The slope of MgMg2Bi2 is significantly smaller compared to other
compounds, which is an indication of a more isotropic behavior. b) The slopes of c/a are shown
as vertical bars. The corresponding spatial dependence of Young’s modulus from the Material-
sProject.org is shown on top of each bar. The experimental anisotropy agrees qualitatively with the
DFT results. Figure taken from W. Peng et al. [68].

63

Figure 5.9: a) Normalized lattice constants along a- or b-axis and b) c-axis for powder YbMg2Bi2,
CaMg2Sb2, Mg3Bi2, Mg3Sb2, and EuAl2Si2. The discrepancies of the slope of a/a0 between
different compounds is significantly larger than that of c/c0. Figure taken from W. Peng et al. [68].

Figure 5.8 a) shows that the c/a ratio of all 

2
2 compounds are decreasing near-linearly with
pressure but at different rates. Since all the compounds are in the same structure type, one might not

64

expect to see a major difference in the c/a trend with increasing pressure. However, as can be seen
from Figure 5.8 a), the slope of the MgMg2Bi2 is almost zero, indicating a much higher isotropic
behavior compared to other 

2
2 compounds. Our prior experimental study of the pressure-
dependence of Mg3Sb2 and Mg3Bi2 [165] observed near-isotropic compressibility of the 
- and

-axes. The EuAl2Si2, in contrast, show a faster decreasing rate of c/a, which also corresponds
with more anisotropic behavior. A faster rate suggests a bigger difference of the compressibility in
the a-b plane compared to that along the c-axis, whereas a slower rate suggests a smaller difference,
which in turn indicates a more isotropic behavior. To further illustrate this, Figure 5.8 b) shows the
calculated spatial dependence of the Young’s modulus from MaterialsProject.org database.

When we compare the c/a result with the bulk modulus values, a clear relationship can be found:
A lower bulk modulus corresponds with more isotropic compressibility, whereas a higher bulk
modulus corresponds with more anisotropic compressibility. This can be a result of less compress-
ibility along the c-axis with decreasing anisotropy or more compressibility along the ab-plane with
decreasing anisotropy. The normalized lattice constants, a/a0 and c/c0 (Figure 5.8), showed that
the discrepancies of the slope of a/a0 are significantly larger than that of c/c0. This indicates that
the bond strength of 
-
 in the covalent slabs is a key factor in the anisotropy of the elastic moduli
of 

2
2 compounds. The more ionic the 
-
 bond is, the weaker the covalent slab is, and
therefore, the more isotropic the elastic modulus is.

5.5 High-pressure phase transition of ternary 

2
2

To investigate the relative stability of the 

2
2 compounds mentioned above, we kept adding pres-
sure until the phase transition occurred. As shown in Figure 5.10, phase transitions of YbMg2Bi2,
CaMg2Sb2, and EuAl2Si2 occured between 10.2 - 11.3 GPa, 14.1 - 19.1 GPa, and 13.7 - 16.0 GPa
were observed, all of which were not previously reported. The peak bar for high-pressure phase of
YbMg2Bi2 was made with the same structure as CaMn2Bi2 (
21/
) [176]. However, since we
did not do single-crystal diffractions to confirm any of these three high-pressure compounds, we do

65

Figure 5.10: a) Powder diffraction patterns of YbMg2Bi2, b) CaMg2Sb2, and c) EuAl2Si2 with
increasing pressure. Possible phase transitions were observed in all phases. The X-ray wavelength
is 0.4133 Å for YbMg2Bi2 and CaMg2Sb2, and 0.4340 Å for EuAl2Si2. Figure taken from W.
Peng et al. [68].

66

not know the exact high-pressure structures. As shown in Table 5.1, within selected the 
Mg2

2
(

=Sb,Bi) compounds, the transition pressure increases as the r





 : r




 increases, which is
an indication of increasing phase stability.

Table 5.1: Transition pressure of 
Mg2

2 compounds and its relationship with the ionic radius.
The transition pressure increases as the r





 : r




 increases.

Transition pressure (GPa)
r





 : r






Mg3Bi2 Mg3Sb2 YbMg2Bi2 CaMg2Sb2
4.0 - 4.5
14.1 - 19.1

10.2 - 11.3

0.44

0.45

7.8 - 8.5

0.32

0.31

5.6 High-pressure high-temperature phase transition and intermediate phase

exploration of EuAl2Si2 and SrAl2Si2

The CaAl2Si2 (
¯3
1) structure type for 

2
2 compounds is one of the several structures that
can be formed, which is also the structure type being discussed in the previous sections. Another
structure type of 

2
2 compounds is the tetragonal ThCr2Si2 (
4/


). Studies have shown
that smaller cations tend to form CaAl2Si2 structure type, whereas ThCr2Si2 structure is preferred
when larger cations is presented [181, 45, 16]. The increase of pressure can result in an increase of
coordination number (i.e. SiO4 to SiO6) [182]. The cation coordination is six in the CaAl2Si2 and
eight in the ThCr2Si2 structure type, respectively. Pressure-induced polymorphism between these
two structure types was first observed in ref [183], in which SrAl2Si2 transformed from CaAl2Si2
structure to ThCr2Si2 structure type above 3 GPa / 700 K with a multi-anvil press.

It has been pointed out by Huster et al.
that a possible intermediate structure BaCu2S2 existed
from a topological point of view [184], since there is no group-subgroup relation between the
low-pressure trigonal phase and high-pressure tetragonal phase. However, no experimental studies
have been done. In this study, our goal was to 1) collect in-situ X-ray diffraction data of SrAl2Si2
during heating to observe the transition process from CaAl2Si2 to ThCr2Si2 structure type. 2)
Observe the phase transition of another CaAl2Si2 structure with big cations. For this purpose, we

67

Figure 5.11: a) Powder diffraction patterns of EuAl2Si2 at a fixed temperature around 600 ◦C.
An abrupt phase transition from the trigonal (
¯3
1) to tetragonal (
4/


) phase was observed.
Note that the pressure shown on the right is the gas-membrane pressure. b) Powder diffraction
patterns of SrAl2Si2 at a fixed pressure around 1.4 GPa. A gradual phase transition from the
trigonal (
¯3
1) to tetragonal (
4/


) phase was observed. Two-phase patterns can be spotted
above ∼700 ◦C. The X-ray wavelength is 0.3542 Å for both measurements. Figure taken from W.
Peng et al. [68].

68

chose EuAl2Si2, as Eu has a similar ionic radius with Sr.

To explore the possible intermediate phase, we slowed down the reaction rate via two approaches.
The first was to keep the system at a constant temperature, while gradually increasing pressure
(Figure 5.11 a). Note that a high-enough temperature is kinetically necessary to activate the
reaction due to the reconstructive nature of this phase transition. We have shown in Figure 5.10
c) that a different type of phase transition would take place when the system is not exposed to a
high enough temperature. In this regard, we chose the temperature to be 600 - 700 ◦C for our
first approach, which is high enough to drive the kinetics but low enough to allow a few pressure
steps to observe the transition process. Two thermocouples were attached for temperature reading
at different locations. The thermocouple that was attached to the anvil read a temperature of 610
±10 ◦C. The thermocouple that was attached to the heater read a temperature 685 ±10 ◦C. The
ruby fluorescence spectrum is a function of temperature, which shifts and broadens as temperature
increases. Using ruby as a pressure calibrate is not ideal at high-temperature. Therefore, the
pressure of the gas-membrane instead of ruby pressure was used in Figure 5.11 a), which is
positively correlated with the pressure of the DAC. Each data collection took approximately 5
minutes to complete.

The second approach is to keep the pressure fixed at around 1-2 GPa, while gradually increasing
temperature, as shown in Figure 5.11 b). The room pressure reading was 1.4 GPa according to the
ruby fluorescence. The pressure is expected to creep up a small amount with increasing temperature
due to the thermal expansion of the pressure medium. One thermocouple was attached to the heater
for temperature reading. Each data collection took approximately 5 minutes to complete.

As shown in Figure 5.11 a) the phase transition from trigonal (space group: 
¯3
1) to tetragonal
EuAl2Si2 (space group: 
4/


) was reported for the first time. The trigonal phase is a semi-
conductor, whereas the tetragonal phase is typically a superconductor. The successful observation

69

of the high-pressure phase also represents the discovery of a new superconductor.

No intermediate phase was observed with both approaches. Figure 5.11 a) shows an abrupt transition
between the two structures. In Figure 5.11 b), two-phase patterns can be spotted above ∼700 ◦C,
which can serve as further evidence that the reaction mechanism could be a fundamentally different
pathway. However, we cannot exclude the possibility that an intermediate phase existed for a few
seconds and we missed to capture it. Thus, we are planning to perform the same experiment at a
higher energy beamline in the future, which can collect diffraction patterns in a matter of seconds.
We also have an on-going collaboration with Dr. Stevanovic Vladan’s group at Colorado School of
Mines to computationally explore the transition pathway for a deeper understanding of this phase
transition.

5.7 Concluding remarks

The present work observed the high-pressure phase transition of 

2
2 compounds in powder
Mg3Sb2, Mg3Bi2, CaMg2Sb2, YbMg2Bi2, and EuAl2Si2 samples above 7.8 GPa, 4 GPa, 19.1 GPa,
11.3 GPa and 16.0 GPa, respectively. The reversibility of the phase transition was confirmed with
powder Mg3Bi2. Single-crystal diffraction at high pressure confirmed the monoclinic high-pressure
structure (
2/
) of Mg3Sb2 and Mg3Bi2, which is a distorted variant of the ambient-pressure
structure. The transition to the high-pressure structure was shown to involve a highly anisotropic
collapse of the lattice parameters along different crystallographic directions.

The anisotropy of the elastic moduli of the powder Mg3Sb2, Mg3Bi2, CaMg2Sb2, YbMg2Bi2,
and EuAl2Si2 samples was compared within the ambient structures. Mg3Sb2 and Mg3Bi2 exhibit
near-isotropic compressibility, whereas the ternary compounds are more anisotropic. When we
compare the c/a decrease per GPa in powder Mg3Sb2, Mg3Bi2, CaMg2Sb2, YbMg2Bi2, and
EuAl2Si2 samples, we found that the bond strength of 
-
 in the covalent slabs is a key factor
in the anisotropy of the elastic moduli of 

2
2 compounds. The more ionic the 
-
 bond

70

is, the weaker the covalent slab is, and therefore, more isotropic the elastic modulus is. The
normalized lattice constants, a/a0 and c/c0 also confirmed that the discrepancies of the in-plane
compressibility is significantly larger than that of the out-of-plane direction. Note that even though
the ambient-pressure structures of Mg3Sb2 and Mg3Bi2 exhibit isotropic compressibility, analysis
of the single crystal data shows that the octahedral Mg-

 bonds does not have the same strength as
the tetrahedral Mg-

 bonds. The the octahedral ones are more compressible than the tetrahedral
ones due to longer bond lengths.

For the high-temperature high-pressure phase transition of EuAl2Si2 and SrAl2Si2, no intermediate
phase was observed, indicating that the reaction mechanism could be a fundamentally different
pathway as Huster et al. predicted. However, we cannot exclude the possibility that an intermediate
phase might existed for a few seconds and we missed to capture it. The phase transition from
trigonal to tetragonal of EuAl2Si2 was reported for the first time. The successful observation of
the high-pressure phase also represents the discovery of a new superconductor.

These high-pressure results provide a unique perspective and deeper understanding of the stability
and bonding environment of 

2
2 compounds. We hope that this work will inspire the next
generation of thermoelectric research to explore in the high-pressure dimension.

71

LATTICE HARDENING DUE TO VACANCY DIFFUSION IN (GeTe)
Sb2Te3 ALLOYS

CHAPTER 6

(Adapted from W. Peng et al. [71] "Lattice hardening due to vacancy diffusion in (GeTe)
Sb2Te3
alloys." 2019. Journal of Applied Physics, 126(5), 055106.)

6.1 GeTe-Sb2Te3 homologous series

To extend our understanding of the structure - property relationships in layered materials, we
investigate the GeTe - Sb2Te3 homologous series.
(GeTe)
Sb2Te3 alloys, also known as GST
alloys, have been widely studied for nonvolatile memory storage applications due to their rapid and
reversible transition between amorphous and crystalline phases with the accompanying changes
in electrical and optical properties [185, 186, 187, 188, 189, 190, 191, 192, 193, 194]. The
amorphous phase is achieved by rapid-quenching from the liquid state to the solid phase, whereas
the cubic crystalline phase is obtained with a slower cooling rate. More recently, GST alloys have
also attracted a great deal of excitement as 
-type thermoelectric materials in the intermediate
temperature range due to their low thermal conductivity, power factor up to 40 
Wcm−1K−2, and
impressive thermoelectric figure of merit (

) in samples with optimized carrier concentrations
[123, 195, 196, 197, 198, 199]. The highest 

 values have been reported in Ge-rich alloys; J.
Williams 

 

. reported a peak 

 of 2.2 at 750 K in (GeTe)17Sb2Te3 doped with an excess of 5%
Sb [197], and Xu 

 

. achieved a 

 of 2.4 at 773 K in undoped (GeTe)17Sb2Te3 after extended
annealing [198].

Compared to other other rock salt (e.g. GeTe, PbTe, and SnTe) and van der Waals (vdW) compounds
(e.g. Bi2Te3 and Sb2Te3), one of fascinating features of GeTe-Sb2Te3 is their tunable layers. As
shown in the phase diagram in Figure 6.1, up to about 7% of Sb2Te3 can be inserted into the
GeTe structure within the solid-solution region to form the (GeTe)
 - Sb2Te3 homologous series.

72

Different thicknesses of layers in the building block can subtly affect the transport properties

Figure 6.1: The phase diagram of GeTe - Sb2Te3 [69].

within the different temperature range due to the complicated effect of phonon scattering, carrier
concentration, interlayer bonding, defect layers, and phase transition temperatures. For example, in
a study by Rosenthal et al. [200], they found that (GeTe)
 - Sb2Te3 compounds with high values
of GeTe contents (i.e. high 
 values) typically have rather low 

 values at low temperatures but
reach the highest 

 values in the high-temperature range. In contrast with the relatively simple
scattering mechanism in most telluride thermoelectrics, GeTe-Sb2Te3 opens more opportunities to
have the highest 

 in the application range.

6.2 Structure and phase transitions of GeTe-Sb2Te3 alloys

In contrast to the amorphous - crystalline transition for memory applications, the phase transition for
in this study and for thermoelectric applications in general is the rhombohedral to cubic transition
within the solid-state.

73

Figure 6.2: a) Ambient-temperature rhombohedral structure of GeTe (
3
), b) Sb2Te3 (
¯3
),
and c) (GeTe)
Sb2Te3 (
3
) with 
 = 3 used for illustrative purposes. The cation vacancies in
(GeTe)
Sb2Te3 are relaxed into ordered layers, which resemble van der Waals gaps. Note that
the hexagonal unit cell was employed here, with the c-axis perpendicular to the layers. d) At high
temperature, (GeTe)
Sb2Te3 transitions to cubic symmetry with randomly distributed vacancies
on the cation site. It can be visualized as stoichiometric occupancy of Ge, Sb, and vacancies on the
cation site, while 100% of Te on the anion site [70]. Figure taken from W. Peng et al. [71].

As shown in Figure 6.2 a) and b), GeTe crystallizes in a distorted rocksalt structure with rhombohe-
dral symmetry (
3
) at ambient temperature. At 643 K, GeTe transforms into the cubic rocksalt
structure (

¯3
) [72]. Pure Sb2Te3, in contrast, does not exhibit a temperature-dependent phase
transition as a solid. It forms the layered tetradymite structure (
¯3
) [201, 202], which is charac-
terized by five-atom thick, Te-terminated covalent slabs separated by van der Waals (vdW) gaps, as
shown in Figure 6.2 b).

(GeTe)
Sb2Te3 alloys form a homologous series, which can be visualized as 
 layers of GeTe
inserted into the covalent Sb2Te3 slabs, expanding them into 2
+5 atomic-layer thick 3-D blocks
[200, 203, 204, 205, 206], as illustrated in Figure 6.2 c). Like GeTe, GST alloys transform into the
cubic rocksalt structure at high temperature. However, unlike the rapid displacive phase transition

74

found in pure GeTe, the phase transition in (GeTe)
Sb2Te3 alloys is reconstructive in nature,
necessitating significant cation diffusion [207, 208, 209, 200, 198]. As illustrated in an 

 − 




study by Xu 

 

. [198], the vdW gaps in (GeTe)17Sb2Te3 at low temperature can be thought
of as ordered layers of cation vacancies, which spontaneously diffuse into the surrounding matrix
with increasing temperature. This behavior is reversible; upon cooling, the vacancies re-order into
layers, and the expanded tetradymite-like structure is recovered. The exceptional mobility of the
cation vacancies in GST alloys, as well as the flexibility in spacing between vacancy layers has
been demonstrated by several recent studies [205, 206].

6.3 Experimental

6.3.1 Synthesis

Samples of (GeTe)17Sb2Te3, GeTe, Sb2Te3, and Bi2Te3 were synthesized starting with stoichio-
metric quantities of Ge (zone-refined ingot, 99.9999%), Sb (6mm shot, 99.999%), Bi (granules,
1-2 mm, 99.997% ) and Te (chunk, 99.999%) from Alfa Aesar. GeTe, Sb2Te3, and Bi2Te3 were
synthesized via direct ball-milling of the elements followed by spark plasma sintering (SPS). The
powder was heated to 738 K in 5 minutes and held at the target temperature for 5 minutes for
GeTe, to 738 K in 10 minutes and held for 15 minutes for Sb2Te3, and to 723 K in 10 minutes and
held for 5 min for Bi2Te3 under a uni-axial pressure of 31 MPa using a Dr. Sinter SPS-211LX
spark plasma sintering (SPS) press. (GeTe)17Sb2Te3 was synthesized by our collaborator, Jared
Williams, with a slightly different approach (see Chapter 2 for details). The densities of all samples
were obtained via measurements of mass and geometry, yielding at least 96% of the theoretical
densities. Phase purity and compositions of the samples was confirmed via peak matching within
the ICSD database, in which no impurity phase could be observed. More experimental details can
be found in Chapter 2.

75

6.3.2 High-temperature resonant ultrasound spectroscopy

The temperature-dependent elastic moduli and speed of sound for the selected compositions were
obtained by resonant ultrasound spectroscopy (RUS) [24]. The elastic moduli were measured in
∼10 K intervals from room temperature up to 723 K for (GeTe)17Sb2Te3, 530 K for Bi2Te3, and
600 K for Sb2Te3 and GeTe. The signal was lost for the GeTe sample above 600 K, so the elastic
moduli above phase transition were not captured. More details of the high-temperatur RUS setup
can be found in Chapter 2.

6.3.3 High-temperature X-ray diffraction

Thermal expansion of (GeTe)17Sb2Te3 and Sb2Te3 were measured from 303 K to 673 K and 303
K to 573K, respectively, using a Rigaku Smartlab X-ray diffractometer equipped with the Rigaku
HT1500 high-temperature stage. The samples showed no hysteresis upon further thermal cycling
after the first cycle, in which a reversible phase transition between the rhombohedral and cubic
rocksalt structure could be observed. More details can be found in Chapter 2.

6.4 Structural evolution with increasing temperature

High-temperature powder X-ray diffraction was used to investigate the evolution of the lattice
parameters with increasing temperature in (GeTe)17Sb2Te3 and Sb2Te3. These results are compared
with the corresponding data for GeTe from the previous literature [72]. The (GeTe)17Sb2Te3
patterns share nearly the same peak positions both in the rhombohedral phase and in the cubic
phase as the corresponding structure of GeTe without any superstructure peaks. The absence of
a well-defined superstructure in the alloyed composition has been observed in several previous
reports [198, 200]. Even though the vacancy layers (
.
., van der Waals gaps) in (GeTe)
Sb2Te3
alloys are nominally spaced at 2
+5 atom intervals, the layer thickness exhibits a large degree of
variability, which is dependent on 
 and the processing route. Samples with larger 
 and faster
quenching lead to more variability. However, even afer two weeks of annealing, some variability of

 can still be seen. Therefore, the GeTe unit cell was used for the refinement of the (GeTe)17Sb2Te3

76

Figure 6.3: a) The c/a ratio, b) lattice parameter, 
, and c) lattice parameter, 
, of (GeTe)17Sb2Te3
and GeTe. The GeTe data is taken from ref. [72]. A reversible phase transition from a rhombohedral
structure (
3
) to cubic rocksalt (

¯3
) at roughly 623 K is observed in both materials. Figure
taken from W. Peng et al. [71].

77

patterns.

Note that the hexagonal instead of rhombohederal notation are employed for all three compounds
for the convenience of symmetry. As shown in Figure 6.2, with the hexagonal notation, the c-axis
is perpendicular to the layers, while the a- and b-axes are parallel to the layers.

First compare the unit cell volume (per formula) of GeTe and the (GeTe)17Sb2Te3. As shown in
Figure 6.3 b) - c), the unit cell of (GeTe)17Sb2Te3 is larger along the a-axis and smaller along the
c-axis compared to GeTe in the entire temperature range below the phase transition. The expansion
in the a-b plane for the (GeTe)17Sb2Te3 is due to the substitution of the larger Sb cations on the Ge
site, while the contraction along the c-axis is caused by the cation vacancy layer perpendicular to c.
A reversible phase transition from the rhombohedral structure (
3
) to the cubic rocksalt structure
(

¯3
) occurs in both GeTe and (GeTe)17Sb2Te3 at ∼623 K. The data in Figure 6.3 were measured
during heating of the sample, the cooling data in Figurexx shows that the phase transition is fully
reversible.

As shown in Figure 6.3 a), the c/a ratio, provides a convenient approach to track the progress of

the phase transition, since cubic symmetry is achieved when c/a = 2(cid:112)3/2. As the temperature

approaches the transition to the cubic structure, GeTe has been shown previously to expand in
the a-direction, while contracting in the c-direction, corresponding to the gradual opening of
the rhombohedral angle [72]. Since the same rhombohedral-to-cubic phase transition occurs in
(GeTe)17Sb2Te3, one might expect to see a similar contraction along the c-axis approaching the
transition temperature. However, as shown in Figure 6.4 b), the thermal expansion of the c-axis in
(GeTe)17Sb2Te3 is almost zero. This suggests that vacancy re-ordering is playing a role. In the
high-temperature cubic phase, symmetry dictates that the cation vacancies be randomly dispersed.
However, the transition from ordered vacancies to random vacancies cannot occur instantaneously at
the transition temperature. Instead, the vacancy diffusion occurs gradually as temperature increases,

78

Figure 6.4: Normalized change in a) unit cell volume and b) lattice parameter along c-axis of
(GeTe)17Sb2Te3, GeTe, and Sb2Te3. Data for GeTe was taken from ref. [72]. Similar volumetric
thermal expansions is observed for the three compounds, despite drastic differences in the c-axis
thermal expansion. Figure taken from W. Peng et al. [71].

leading to a gradual expansion of the structure along the c-axis.

To estimate the extent of vacancy re-ordering that occurs between 300 K and 620 K, we look to the
c/a ratio. In the case of randomly dispersed vacancies, the c/a ratio in (GeTe)17Sb2Te3 should be
identical to that of GeTe, assuming that the rhombohedral distortion lessens at the same rate in both
compounds and the size effect of the random vacancies dispersed into the structure can be ignored

79

due to their small amount. And indeed, with increasing temperature, we see that the c/a ratio
of (GeTe)17Sb2Te3 and GeTe slowly converge as we approach the phase transition temperature,
indicating that the vacancy layers have mostly disappeared. Above the phase transition, the c/a ratio
is necessarily identical in both compounds due to the cubic symmetry. The magnitude of the 
 and

 in both compounds in the cubic symmetry are also similar, since the size-effect of the randomly
distributed cation vacancies is mostly cancelled out by the partial substitution of the larger Sb on
the Ge site.

For comparison, Figure 6.4 a) - b) shows the temperature-dependent unit cell volume and lattice
parameter, 
, of GeTe, Sb2Te3, and (GeTe)17Sb2Te3, normalized to the room temperature values.
Although the c-axis of Sb2Te3 expands more rapidly with increasing temperature, the volumetric
thermal expansion of all three compounds is quite similar. The lattice parameters 
 and 
 for
Sb2Te3 are shown separately in Figure 6.6.

Figure 6.5: a) A two-step phase transition can be observed during the first heating process. The
cubic rocksalt structure transforms to a rhombohedral structure at 523 K, and then to a cubic rocksalt
structure at 623 K. A single phase transition to rhombohedral structure occurred during the cooling
process. b) The 2nd thermal cycle shows a reversible phase transition between the rhombohedral
and cubic rocksalt structure around 623 K. Figure taken from W. Peng et al. [71].

Note that the as-synthesized sample need to be annealed in order to show the reversibility. Upon

80

cooling during the spark plasma sintering (SPS) process, the (GeTe)17Sb2Te3 sample was es-
sentially "quenched" in a meta-stable cubic phase (

¯3
) due to relatively slow kinetics of the
diffusion process, as described in ref.
[123]. During the first post-SPS heating cycle, a two-
step phase transition can be observed: first, a phase transition from the metastable cubic rocksalt
to the stable rhombohedral (
3
) structure occurs beginning at around 523 K (given sufficient
time, annealing at a lower temperature would presumably have the same effect). The rhombohe-
dral structure undergoes another phase transition to the cubic rocksalt structure at 623 K. During
cooling, only one phase transition to the rhombohedral structure at 623 K occurred (Figure 6.5
a). The sample showed no hysteresis upon further thermal cycling after the first cycle, in which
a phase transition between the rhombohedral and cubic rocksalt structure around 623 K can be
observed both during heating and cooling, indicating stable structures and reversible transitions of
(GeTe)17Sb2Te3 (Figure 6.5 b).

Figure 6.6: a) Lattice parameter 
, and b) lattice parameter, 
 of Sb2Te3 measured as a function of
temperature. Figure taken from W. Peng et al. [71].

6.5 Temperature-dependent elastic moduli

Figure 6.7 a) - b) compares the calculated and experimental Young’s modulus of polycrystalline
rock salt compounds at room temperature. Density functional theory (materialsproject.org [22]),
shown as the vertical bars, predicts i) that the distorted rock salt structure is softer than the cubic

81

rock salt structure, and ii) that the cubic structure becomes softer as we proceed to larger cations
with longer bonds (consistent with room temperature experimental data for SnTe and PbTe from ref.
[74, 73, 9, 10]). The average Ge-Te bond length in the distorted structure is longer than that in the
cubic phase, leading to an overall weaker Young’s modulus in comparison with the cubic structure.
The distorted rock salt structure is also anisotropic (illustrated by the calculated 3D compressibility
surfaces [22]) as a result of the distortion of the Ge position, which has three longer and three
shorter Ge-Te bonds. This in turn leads to weaker bonds in the 
-direction, and stronger bonding
in the 
 − 
 plane. Note that RUS measurements using polycrystalline samples do not capture the
anisotropy of rhombohedral GeTe, but they do confirm the prediction of softer Young’s modulus
compared with cubic compounds SnTe and PbTe.

A comparison of cubic PbTe and SnTe with rhombohedral GeTe reveals a slower rate of softening
in GeTe (Figure 6.7 b)). In the absence of phase transitions, the rate of softening of a material is
typically correlated with the degree of anharmonicity of the acoustic phonons [20, 17], which is
usually correlated with thermal expansion. Structural changes, however, can have a greater impact
on the elastic moduli than thermal expansion. In the case of GeTe, there is a phase transition from a
distorted rocksalt structure to a cubic rocksalt structure. We attribute this anomalous slow softening
to the gradual decrease in the rhombohedral distortion in GeTe with increasing temperature. In the
current study, we did not obtain data for GeTe above the phase transition due to limitations of our
RUS equipment, however, we would predict a step-like stiffening of the Young’s modulus upon the
transition to cubic symmetry. This step-like stiffening was observed in a previous investigation of
single crystalline Ge0.2Sn0.8Te at the phase transition at 220 K [210].

We next turn to a comparison of the temperature-dependent shear and Young’s moduli (
 and 
,
respectively) of GeTe, (GeTe)17Sb2Te3, and Sb2Te3 (see Figure 6.8 a) - b). Although all three
compounds have similar rates of volumetric thermal expansion, as established in Figure 6.4 a), they
exhibit drastic differences in the temperature dependence of the elastic moduli. For comparison,

82

Figure 6.7: a) A comparison of the Young’s modulus, Y, of cubic and distorted rock salt compounds
shows that the distorted structure is softer. Bars show calculated data from ref. [22] and symbols
are experimental data (present work: ਭ, literature data: (cid:3) [73], (cid:5)[10], (cid:52)[74], ◦[9]). 3D surfaces
show the compressibility of the two structure types [22]. b) The temperature dependence of Y for
GeTe [present work], SnTe[10], and PbTe [74] show the slower rate of softening in GeTe. Figure
taken from W. Peng et al. [71].

we also measured the high-temperature elastic moduli of Bi2Te3, reported here for the first time.
The rate of softening in Sb2Te3 and Bi2Te3, which does not exhibit a phase transition at high
temperature, is comparable to that of other thermoelectric materials such as PbTe [9] and SnTe
[10].

The (GeTe)17Sb2Te3 alloy is unique in that the elastic moduli actually harden gradually with in-
creasing temperature. This gradual hardening is followed by an abrupt increase of the elastic moduli

83

Figure 6.8: The temperature dependence of the normalized a) Young’s modulus, 
, and b)
shear modulus, G, for GeTe, (GeTe)17Sb2Te3, Sb2Te3, and Bi2Te3. All compounds besides
(GeTe)17Sb2Te3 soften with increasing temperature. c) The value of the Young’s and shear moduli
of (GeTe)17Sb2Te3 across phase transition. Figure taken from W. Peng et al. [71].

at the transition from rhombohedral to cubic symmetry (Figure 6.8 c). Finally, at temperatures
above the phase transition, softening of the elastic moduli can be observed, which is expected for

84

most compounds due to thermal expansion. Even though the abrupt increase in Y and G at the
phase transition was expected, and is consistent with previously reported behavior for Ge0.2Sn0.8Te
as well as the 







 behavior of GeTe. The gradual hardening of (GeTe)17Sb2Te3 leading up
to the phase transition is unusual.

We attribute the gradual hardening of (GeTe)17Sb2Te3 to two independent factors. The first is
the gradually decreasing rhombohedral distortion with increasing temperature. As mentioned in
the previous paragraphs, this effect also contribute to the relatively slow softening rate of GeTe
compared to compounds with similar thermal expansion. The second and more significant effect is
the dynamic reconfiguration of the cation vacancies with increasing temperature, which uniquely
present in the alloy. When the vacancies are ordered into layers, the neighboring slabs are connected
by weak Te-Te van der Waals interactions. Whenever a vacancy diffuse onto a cation site, a cation
simultaneously diffuse onto the vacancy site. These local bondings formed between the cation and
Te are stronger than the initial weak Te-Te van der Waals bonds. As temperature increases, more
and more cations diffuse onto the vacancy sites, leading to stronger and stronger bonding between
the slabs. This gradual diffusive re-ordering of the vacancies lead to the gradual stiffening of the
lattice. The randomly distributed vacancies represent only 5% of the cation sites, which is not
enough to have significant impact on the overall stiffness. Upon the phase transition, the elastic
moduli of (GeTe)17Sb2Te3 has an abrupt increase as the vacancy layers diminish. The behavior
of the elastic moduli in (GeTe)17Sb2Te3 is reversible, showing the same temperature-dependence
during heating and cooling.

The lattice hardening due to the phase transition in (GeTe)17Sb2Te3 leads to increasing speed
of sound as a function of temperature (shown in Figure 6.9), in stark contrast to most solids.
While the impact of lattice hardening is not significant enough to lead to increasing lattice thermal
conductivity with temperature, since the the inverse temperature dependence of Umklapp scattering
is simply too strong, it likely leads to a flatter lattice thermal conductivity. The relatively flat lattice

85

Figure 6.9: a) Longitudinal speed of sound and b) Transverse speed of sound of (GeTe)17Sb2Te3,
Sb2Te3, GeTe, and Bi2Te3 measured from room temperature up to 600 K via resonant ultrasound
spectroscopy. c) Longitudinal and shear speed of sound of (GeTe)17Sb2Te3 measured from room
temperature across phase transition. Figure taken from W. Peng et al. [71].

thermal conductivity of the (GeTe)17Sb2Te3 sample used for this study is shown in Figure 6.10 c).

The stiffening of the elastic moduli with increasing temperature is expected to be a general feature
of all GeTe-Sb2Te3 as well as GeTe-Bi2Te3 alloys [211], as long as they exhibit a rhombohedral-
to-cubic transition involving vacancy diffusion. Further, we expect the stiffening rate to be more
pronounced with increasing vacancy ratio in the structure (i.e., with decreasing value of 
.)

Table 6.1: Experimental elastic moduli and speed of sound at 300 K for (GeTe)17Sb2Te3, Sb2Te3,
GeTe, and Bi2Te3 samples measured using resonant ultrasound spectroscopy.

(GeTe)17Sb2Te3 Sb2Te3 GeTe Bi2Te3

Young’s modulus (GPa)
Shear modulus (GPa)
Longitudinal speed of sound (m/s)
Transvere speed of sound (m/s)
Theoretical density (g/cm3)

6.6 Conclusions

55
22
3300
1900
6.09

45
18
2950
1750
6.12

43
17
2980
1740
5.91

47
18
2800
1550
7.47

In the current study, we have combined high-temperature X-ray diffraction and elasticity mea-
surements to reveal anomalous lattice hardening in (GeTe)17Sb2Te3, despite its positive thermal
expansion coefficients. A positive thermal expansion coefficient is usually associated with lattice

86

Figure 6.10: The thermoelectric properties of the (GeTe)17Sb2Te3 samples used in this study:
a) Total thermal conductivity, lattice thermal conductivity, and electrical thermal conductivity;
b) resistivity; c) Seebeck coefficient; and d) thermoelectric figure of merit.
In contrast to the
expectation of a decreasing trend of lattice thermal conductivity, as it is the case for most materials,
the lattice thermal conductivity of (GeTe)17Sb2Te3 remains roughly unchanged with increasing
temperature before the phase transition. Figure taken from W. Peng et al. [71].

softening, since bonds weaken as they become longer. The elastic moduli of GeTe, Sb2Te3 and
Bi2Te3, in contrast, showed the expected softening behavior with increasing temperature. The un-
usual behavior of (GeTe)17Sb2Te3 is assumed to originate from the gradual diffusion of vacancies
that accompanies the phase transition from a layered structure to the cubic rocksalt structure. As the
the ordered layers of vacancies switched sites with the cations, the attractions between neighboring
layers presumably increases, which in turn stiffens the lattice. We expect this behavior to be a

87

generic feature of all GeTe-Sb2Te3 as well as GeTe-Bi2Te3 alloys, playing a fundamental role in
the thermal conductivity in this important class of thermoelectric materials.

88

CHAPTER 7

CONCLUDING REMARKS

This work was focused largely on understanding the elastic moduli of two classes of layered
thermoelectric materials: 

2
2 Zintl phases and GeTe - Sb2Te3 alloys. We learned that the
stability of the bond controls the elastic moduli and anharmonicity, which is partially determined
by the size ratio between the cation and anion, bond length, and bond type in the structure. In
addition, approaching phase transitions have an impact on the elastic modulus of materials.
In
Figure 7.1, the data measured in this study is added to the literature temperature-dependent elastic
moduli data of typical thermoelectric materials to illustrate the relationship between structure and
elastic modulus.

In general, compounds with octahedral coordination, such as tetradymites and rock salts, are softer
than those with tetrahedral coordination. This is because the bonds of a close-packed octahedron
are typically longer than a close-packed tetrahedron according to Pauling’s close-packed rule. In the
case of 

2
2 compounds, the alternating layers of tetrahedral and octahedral coordination may
explain their intermediate to low moduli. Within the same structure type, the differences in atomic
sizes are the determining factor of the differences in the bond length and therefore have an impact
on the stiffness of compounds. Among 

2
2 Zintls, for example, the Bi-based compounds are
generally softer than Sb-analogues due to the larger size of Bi.

7.1 Possible explanations for the large anharmonicity of Mg3Sb2

Mg3Sb2 and Mg3Bi2, as outliers of the 

2
2 compounds, have a similar stiffness as the
tetradymites structures due to their anomalously low shear moduli due to the small size of Mg.
One of the main conclusions from our study of the elastic properties of 

2
2 compounds is that
Mg3Sb2 and Mg3Bi2 have large anharmonicity and anomalously low lattice thermal conductivity.
This finding is further confirmed by the comparison of Young’s modulus, softening rate, thermal

89

Figure 7.1: Temperature-dependence of the Young’s modulus, 
, of the selected thermoelectric
material [7, 8, 71, 71, 9, 10, 75]. 

2
2 compounds, the alternating layers of tetrahedral and
octahedral coordination may explain their intermediate moduli. Mg3Sb2 and Mg3Bi2, as outliers
of the 

2
2 compounds, have a similar stiffness as the tetradymites structures. Figure taken
from W. Peng et al. [6]

expansion, and mode Grüneisen parameters in the previous chapters. One possible explanation is
the poor fit of the cation in the octahedral environment. In the sphere packing model proposed by
Pauling for ionic solids, the smallest stable cation to anion radius for octahedral coordination is
given by 






:





=0.414 [154]. Figure 3.9 shows that for all the 

2
2 compounds, 
 = Mg
ones are the only compounds that have 






:





 below the stability limit, while compounds
with larger cations are predicted to be stable in a six-fold coordinated environment. This results
in large octahedral bond angle variances in Mg3Sb2 and Mg3Bi2 in order to depart the anions
from touching and may explain why only compounds with 
 = Mg (not those with A=Ca, Sr, Yb,
Eu) exhibit anomalous thermal properties. The unstable octahedra lead to a low shear modulus in
Mg3Sb2 and Mg3Bi2.

90

Another possible explanation is that the Mg(2)-Sb ionic bond is weaker than a typical 
 − 

covalent bond in 

2
2 compounds, which leads to an overall less stable "covalent" slabs and
lower in-plane thermal conductivity in Mg3Sb2. This idea was first reported by a DFT study by
Zhang et. al [67]. Due to the high electronegativity of Mg compared to other possible element
choices for the 
-site, the Mg(2)-Sb bond is showing a stronger ionic character compared to most
M-X bonds in a 

2
2 compound (e.g. CaZn2Sb2). Our results provide experimental evidence
for this prediction. The 

-



 high-pressure synchrotron X-ray diffraction showed that the bond
strength of 
-
 in the covalent slabs is a key factor in the anisotropy of the elastic moduli of 

2
2
compounds. The more ionic the 
-
 bond is, the weaker the covalent slab is, and therefore, the
more isotropic the elastic modulus is. A more isotropic elastic modulus results in a lower average
elastic modulus and lower average lattice thermal conductivity.

7.2 Lattice hardening of GST alloys

Even though the phase transition of GST alloys has been previously investigated with 

-




TEM, the elastic moduli during the phase transition were reported for the first time in this study.
Unlike most compounds that soften with increasing temperature, the elastic moduli of (GeTe)17
- Sb2Te3 stiffen with increasing temperature before the phase transition. This is the only type of
thermoelectric material that has observed to have this behavior due to a non-defect reason. Further,
we observed the flattening of lattice thermal conductivity of (GeTe)17 - Sb2Te3 as a result of the
lattice stiffening. We concluded that the unusual behavior of (GeTe)17Sb2Te3 is originated from the
gradual diffusion of vacancies that accompanies the phase transition from a layered structure to the
cubic rocksalt structure. This behavior is expected to be a generic feature of all (GeTe)
Sb2Te3 and
(GeTe)
Bi2Te3 alloys. Future works can be directed at investigating the temperature-dependent
elastic moduli of other types of GST compounds to exploit the relationship between the rate of
lattice stiffening with the thickness of the layer, as well as the impact of lattice stiffening on lattice
thermal conductivity.

91

7.3 Opportunities and challenges

Temperature-dependent resonant ultrasound spectroscopy and 

-



 high-pressure X-ray diffrac-
tion are powerful tools for the characterization of elastic properties and investigation of anharmonic-
ity of materials. Few works have been directed at measuring and utilizing the elastic properties in
the thermoelectric community so far. In addition, high-pressure characterization opens a new di-
mension to discover the unknown phase transitions of thermoelectrics. Combined with the property
measurements through the phase transition, direct observation of the structure-property relation-
ship can be achieved. Even though elastic moduli alone cannot be independently used predict the
thermoelectric performance of materials, they can provide guidance for systematically searching
high 

 materials with low thermal conductivity wit DFT calculations. The main challenge of the
ultrasound measurement is the limitations on the geometry of the sample and difficulty in solving
the spectrum for low symmetry structures. Besides, the speed of sound and elastic moduli only
contain information of the lowest frequency phonons. A further area of opportunity is to combine
elasticity measurements with other in-situ techniques. For example, the frequencies of selected
optical modes can be investigated via Raman spectroscopy and inelastic scattering can be used to
target at any desired phonon mode.

92

APPENDIX

93

Table A1: Lattice parameters and R
 
 values from refinements of powder XRD patterns of
(Ba
Ca1−
)Mg2Sb2.

After SPS
a
4.64913(18)
4.65967(10)
4.6821(2)
4.7077(10)
4.7328(2)
4.76027(14)
4.76767(15)

x
0
0.1
0.3
0.5
0.7
0.9
1

c
7.56089(2)
7.61872(15)
7.7356(4)
7.8544(16)
7.9790(4)
8.0907(3)
8.1293(3)

R
 

3.66
2.99
4.33
5.97
4.64
2.81
2.82

After annealing 10 days
a
-
4.6563(3)
4.68539(15)
4.7046(2)
4.7328(2)
4.75453(16)
-

c
-
7.6121(4)
7.740300(3)
7.8483(4)
7.9790(5)
8.0838(3)
-

R
 

-
2.22
3.56
3.46
1.83
3.73
-

Table A2: Lattice parameters and R
 
 values from refinements of powder XRD patterns of
(Ca
Mg1−
)Mg2Sb2.

After SPS
a
4.56436(14)
4.5760(2)
4.5856(2)
4.58928(15)
4.59639(11)
4.60429(12)
4.61042(9)
4.62359(10)
4.62980(7)
4.64026(8)
4.64913(18)

x
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

c
7.2310(2)
7.2631(2)
7.3033(4)
7.3319(2)
7.36216(17)
7.3853(2)
7.41467(14)
7.45927(17)
7.49189(14)
7.52549(14)
7.56089(2)

R
 

3.66
3.54
3.86
2.89
3.38
2.62
2.72
2.12
2.74
1.98
2.42

After annealing 10 days
a
-
4.56854(18)
4.57815(16)
4.58691(11)
4.59622(11)
4.60254(11)
4.60919(11)
4.61850(7)
4.63103(16)
-
-

c
-
7.25428(17)
7.2932(2)
7.3304(2)
7.36684(17)
7.3905(18)
7.41438(18)
7.44851(13)
7.4971(2)
-
-

R
 

-
3.85
3.7
3.16
3.33
3.1
2.71
2.21
2.34
-
-

Table A3: Lattice parameters and R
 
 values from refinements of powder XRD patterns of
(Sr
Mg1−
)Mg2Sb2. With the exception of the 
 = 0.9 sample, all alloyed samples contain both a
Sr-rich and a Mg-rich phase.

After SPS
Sr-rich phase
a
-

x
0
0.2 4.6841 (2)
0.4 4.6850 (3)
0.6 4.68527 (19)
0.7 4.68489 (17)
0.8 4.69262 (8)
0.9 4.68927 (12)
1
4.70068 (7)

c
-
7.7482 (7)
7.7570 (5)
7.7568 (3)
7.7597 (3)
7.78273 (17)
7.7681 (2)
7.82196(13)

Mg-rich phase
a
4.56436 (14)
4.57706 (10)
4.5736 (3)
4.5711 (4)
4.5694 (6)
-
-
-

R
 

c
3.66
7.2310 (2)
7.2805 (2)
2.9
7.2721 (15) 2.19
7.2599 (8)
1.9
7.2643 (10) 1.83
2.18
-
2.47
-
-
1.59

After annealing 10 days
Sr-rich phase
a
-
4.6827(4)
4.6852(2)
4.68857(11)
4.6891(9)
4.68644(19)
4.68727(9)
-

c
-
7.7533(8)
7.7594(4)
7.7671(19)
7.765(15)
7.7651(3)
7.76557(16)
-

Mg-rich phase
a
-
4.56333(17)
4.5639(3)
4.5637(5)
4.5664(2)
-
-
-

c
-
7.2389(3)
7.2406(4)
7.2407(8)
7.2377(7)
-
-
-

R
 

-
2.28
2.92
1.81
1.83
2.03
-
-

94

Table A4: Lattice parameters and R
 
 values from refinements of powder XRD patterns of
(Ba
Mg1−
)Mg2Sb2. All alloyed samples separated into a Ba-rich and a Mg-rich phase, suggesting
zero solubility.

After SPS
Ba-rich phase
a
-

x
0
0.1 4.7615 (3)
0.3 4.7645 (3)
0.5 4.7716 (3)
0.8 4.76800 (15)
0.9 4.7624 (18)
1
4.76767 (15)

c
-
8.1174 (8)
8.1252 (5)
8.1319 (2)
8.1250 (3)
8.122 (3)
8.1293 (3)

Mg-rich phase
a
4.56436 (14)
4.5649 (15)
4.5653 (4)
4.5682 (4)
4.564 (4)
4.5746 (17)
-

R
 

c
7.2310 (2)
3.66
7.2327 (18) 3.06
2.71
7.2335 (5)
7.2302 (9)
2.83
3.64
7.1924 (8)
3.19
7.2295 (5)
-
2.82

Mg-rich phase
a
-

After annealing 10 days
Ba-rich phase
a
-
4.7647 (3)
4.7642 (4)
4.76237 (14)
4.7665 (3)
4.76172 (18)
-

c
-
8.1269 (10) 4.56246 (15)
4.56622 (18)
8.1249 (8)
8.1222 (3)
4.56307 (17)
-
8.1236 (4)
-
8.1196 (3)
-
-

R
 

-

c
-
7.22828 (19) 2.85
2.73
7.2372 (3)
7.2346 (4)
2.47
3.31
-
3.8
-
-
-

Table A5: The parameters from the 2nd-order Birch–Murnaghan EOS fit for Mg3Sb2 and Mg3Bi2
corresponding to Figure 5.5. The V0 for the ambient structures (
¯3
1) were obtained via Rietveld
refinement of the corresponding room temperature XRD pattern. For the high-pressure phase
(
2/
), V0 is unknown, so it was treated as an open fitting parameter in the Birch–Murnaghan
EOS fit.

AP - Mg3Sb2 
2/
 Mg3Sb2 AP - Mg3Bi2 
2/
 Mg3Bi2

V0 (Å3)
K0 (GPa)
K
 (GPa)

130.44
38.4
4

122.5
45.7
4

138.79
36.9
4

128.7
48.9
4

Table A6: Lattice parameters and uncertainties of the 
¯3
1 Mg3Sb2 phase from powder XRD
refinements using PDXL2. The lattice parameters, and peak shape parameters are refined. P1 and
P2 are pressure readings before and after the measurement of each pattern. The deviation of the
average value between P1 and P2 is used as the pressure uncertainty for the Birch–Murnaghan
equation of state fit.

P1 (GPa) P2 (GPa)

0.58
0.64
0.73
0.91
1.26
1.94
3.06
3.38
3.90
4.36
4.81
5.48
6.70

0.58
0.64
0.75
0.88
1.26
1.94
3.06
3.38
3.90
4.36
4.90
5.57
6.70

a (Å)

4.5376(17)
4.5337(17)
4.5304(16)
4.5269(18)
4.513(2)
4.4919(18)
4.464(2)
4.452(2)
4.4403(16)
4.4274(15)
4.4122(18)
4.3986(19)
4.3792(17)

c (Å)
7.195(3)
7.185(3)
7.180(3)
7.170(3)
7.146(3)
7.107(3)
7.055(3)
7.038(3)
7.016(4)
6.997(3)
6.971(3)
6.954(16)
6.913(4)

V (Å3)
128.29(8)
127.90(8)
127.62(8)
127.25(9)
126.02(9)
124.19(9)
121.73(10)
120.78(9)
119.80(9)
117.53(9)
117.53(9)
116.51(13)
114.81(9)

95

Table A7: Lattice parameters and uncertainties of the 
2/
 Mg3Sb2 phase from powder XRD
refinements using PDXL2. The lattice parameters, peak shape parameters, z and x are refined.
The deviation of the average value between P1 and P2 (pressures measured before and after sample
exposure) is used as the pressure uncertainty for the Birch–Murnaghan equation of state fit.

P1 (GPa) P2 (GPa)

8.43
9.08
10.25
13.33
15.02
17.48
18.56

a (Å)

15.062(11)
15.012(15)
14.910(16)
14.693(19)
14.638(16)
14.495(16)
14.450(18)

b (Å)
4.201(3)
4.187(4)
4.169(4)
4.144(5)
4.126(4)
4.094(4)
4.085(5)

c (Å)
7.599(3)
7.595(4)
7.549(3)
7.479(4)
7.420(3)
7.348(4)
7.322(4)

V (Å3)
425.8(4)
423.0(6)
415.7(6)
403.4(7)
396.4(6)
385.8(6)
382.2(7)

beta (deg)
117.67(2)
117.62(3)
117.63(4)
117.65(5)
117.79(4)
117.78(4)
117.84(4)

8.60
9.11
10.44
13.53
15.29
17.65
18.70

Table A8: Lattice parameters and uncertainties of the 
¯3
1 Mg3Bi2 phase from powder XRD
refinements using PDXL2. The lattice parameters, and peak shape parameters are refined. The
deviation of the average value between P1 and P2 (pressures measured before and after sample
exposure) is used as the pressure uncertainty for the Birch–Murnaghan equation of state fit.

P1 (GPa) P2 (GPa)

1.49
2.48
2.92
3.52

1.49
2.55
3.00
3.59

a (Å)

4.6005(12)
4.5672(11)
4.5559(11)
4.5365(11)

c (Å)
7.293(8)
7.236(7)
7.217(7)
7.185(7)

V (Å3)

133.67(16)
130.73(13)
129.74(13)
128.06(13)

Table A9: Lattice parameters and uncertainties of the 
2/
 Mg3Bi2 phase from powder XRD
refinements using PDXL2. The lattice parameters, peak shape parameters, z and x are refined.
The deviation of the average value between P1 and P2 is used as the pressure uncertainty for the
Birch–Murnaghan equation of state fit.

P1 (GPa) P2 (GPa)

4.51
5.21
5.80
6.56
7.10
8.00
8.60
9.30
10.02

4.6
5.25
5.91
6.68
7.26
8.00
8.60
9.3
10.02

a (Å)

15.618(19)
15.548(15)
15.481(13)
15.426(14)
15.388(15)
15.320(14)
15.284(14)
15.233(13)
15.181(14)

b (Å)
4.329(3)
4.327(2)
4.315(2)
4.310(2)
4.292(3)
4.281(2)
4.276(2)
4.269(2)
4.260(2)

c (Å)
7.907(3)
7.8773(19)
7.850(2)
7.831(2)
7.801(2)
7.7722(19)
7.7573(19)
7.7368(18)
7.7143(18)

V (Å3)
475.4(7)
471.0(5)
466.2(5)
462.9(5)
457.7(5)
452.7(5)
450.1(5)
446.6(5)
442.8(5)

beta (deg)
117.20(6)
117.27(4)
117.24 (4)
117.26(4)
117.34(4)
117.36(4)
117.39(4)
117.40(4)
117.42(4)

96

BIBLIOGRAPHY

97

BIBLIOGRAPHY

[1]

[2]

[3]

Robert Berman and Paul G Klemens. Thermal conduction in solids. Physics Today, 31:56,
1978.
Terry M Tritt. Thermal conductivity: theory, properties, and applications. Springer Science
& Business Media, 2005.

Yoko Suzuki, Jonathan B Levine, Albert Migliori, Jim D Garrett, Richard B Kaner, Victor R
Fanelli, and Jonathan B Betts. Rhenium diboride’s monocrystal elastic constants, 308 to 5
K. The Journal of the Acoustical Society of America, 127(5):2797–2801, 2010.

[4] W.C. Overton Jr. and John Gaffney. Temperature variation of the elastic constants of cubic

elements. I. Copper. Physical Review, 98(4):969, 1955.
Lattice dynamics and local structure in 
Mg2Sb2 thermoelectric Zintl phases. 2020 (In
preparation).

[5]

[6] Wanyue Peng, Ashiwini Balodhi, and Alexandra Zevalkink. High-temperature elastic mod-
uli: a tool for understanding chemical bonding in thermoelectric materials. 2020 (in prepa-
ration).

[7]

[8]

[9]

Norihiko L Okamoto, Takahiro Nakano, Katsushi Tanaka, and Haruyuki Inui. Mechanical
and thermal properties of single crystals of the type-I clathrate compounds Ba8Ga16Ge30
and Sr8Ga16Ge30. Journal of Applied Physics, 104(1):013529, 2008.
G Rogl, A Grytsiv, M Gürth, A Tavassoli, C Ebner, A Wünschek, S Puchegger, V Soprunyuk,
W Schranz, E Bauer, et al. Mechanical properties of half-heusler alloys. Acta Materialia,
107:178–195, 2016.

Fei Ren, Eldon D Case, Joseph R Sootsman, Mercouri G Kanatzidis, Huijun Kong, Ctirad
Uher, Edgar Lara-Curzio, and Rosa M Trejo. The high-temperature elastic moduli of
polycrystalline Pb Te measured by resonant ultrasound spectroscopy. Acta Materialia,
56(20):5954–5963, 2008.

[10] Robert D Schmidt, Eldon D Case, Jennifer E Ni, Rosa M Trejo, Edgar Lara-Curzio, Rachel J
Korkosz, and Mercouri G Kanatzidis. High-temperature elastic moduli of thermoelectric
SnTe1±
-y SiC nanoparticulate composites. Journal of Materials Science, 48(23):8244–
8258, 2013.

[11] R.B. Schwarz and J.F. Vuorinen. Resonant ultrasound spectroscopy: applications, current

status and limitations. Journal of Alloys and Compounds, 310(1-2):243–250, 2000.

[12] Udo Lang, Nicola Naujoks, and Jurg Dual. Mechanical characterization of PEDOT: PSS

thin films. Synthetic Metals, 159(5-6):473–479, 2009.

98

[13] Baoyang Lu, Hyunwoo Yuk, Shaoting Lin, Nannan Jian, Kai Qu, Jingkun Xu, and Xuanhe

Zhao. Pure PEDOT: PSS hydrogels. Nature communications, 10(1):1–10, 2019.

[14] Francesco Greco, Alessandra Zucca, Silvia Taccola, Arianna Menciassi, Toshinori Fujie,
Hiroki Haniuda, Shinji Takeoka, Paolo Dario, and Virgilio Mattoli. Ultra-thin conductive
free-standing pedot/pss nanofilms. Soft Matter, 7(22):10642–10650, 2011.

[15] Dean Cheikh, Kathleen Lee, Wanyue Peng, Alexandra Zevalkink, Jean-Pierre Fleurial, and
Sabah K Bux. Thermoelectric properties of scandium sesquitelluride. Materials, 12(5):734,
2019.

[16] Wanyue Peng, Sevan Chanakian, and Alexandra Zevalkink. Crystal chemistry and thermo-
electric transport of layered 

2
2 compounds. Inorganic Chemistry Frontiers, 5(8):1744–
1759, 2018.

[17] Alex Zevalkink, David M Smiadak, Jeff L Blackburn, Andrew J Ferguson, Michael L
Chabinyc, Olivier Delaire, Jian Wang, Kirill Kovnir, Joshua Martin, Laura T Schelhas, et al.
A practical field guide to thermoelectrics: Fundamentals, synthesis, and characterization.
Applied Physics Reviews, 5(2):021303, 2018.

[18] Zhaohui Dong and Yang Song. Novel pressure-induced structural transformations of inor-

ganic nanowires. In Nanowires-Fundamental Research. InTech, 2011.

[19] M Martinez-Ripoll, A Haase, and G Brauer. The crystal structure of 
-Mg3Sb2. Acta Crystal-
lographica Section B: Structural Crystallography and Crystal Chemistry, 30(8):2006–2009,
1974.

[20] Wanyue Peng, Guido Petretto, Gian-Marco Rignanese, Geoffroy Hautier, and Alexandra
Zevalkink. An unlikely route to low lattice thermal conductivity: small atoms in a simple
layered structure. Joule, 2(12):1879–1893, 2018.

[21] G Ceder and K Persson. The materials project: A materials genome approach. DOE Data

Explorer, osti. gov/dataexplorer/biblio/1077798, 2010.

[22] Maarten De Jong, Wei Chen, Thomas Angsten, Anubhav Jain, Randy Notestine, Anthony
Gamst, Marcel Sluiter, Chaitanya Krishna Ande, Sybrand Van Der Zwaag, Jose J Plata, et al.
Charting the complete elastic properties of inorganic crystalline compounds. Scientific Data,
2:150009, 2015.

[23] Eric S Toberer, Alex Zevalkink, and G Jeffrey Snyder. Phonon engineering through crystal

chemistry. Journal of Materials Chemistry, 21(40):15843–15852, 2011.

[24] G Li and JR Gladden. High temperature resonant ultrasound spectroscopy: a review.

International Journal of Spectroscopy, 2010.

[25] Wanyue Peng and Alexandra Zevalkink. Limits of Cation Solubility in 
Mg2Sb2 (
 = Mg,

Ca, Sr, Ba) Alloys. Materials (Basel, Switzerland), 12(4), 2019.

99

[26] Shalabh Gupta, Ashok K Ganguli, and John D Corbett. Mg5.23Sm0.77Sb4: an ordered
superstructure derived from the Mg3Sb2 structure type. Inorganic Chemistry, 45(20):8175–
8178, 2006.

[27] Tessera Alemneh Wubieneh, Pai-Chun Wei, Chien-Chih Yeh, Szu-yuan Chen, and Yang-
Yuan Chen. Thermoelectric Properties of Zintl Phase Compounds of Ca1−
Eu
Zn2Sb2.
Journal of Electronic Materials, 45(3):1942–1946, 2016.
Jing Shuai, Yumei Wang, Zihang Liu, Hee Seok Kim, Jun Mao, Jiehe Sui, and
Zhifeng Ren. Enhancement of thermoelectric performance of phase pure zintl compounds
Ca1−
Yb
Zn2Sb2, Ca1−
Eu
Zn2Sb2, and Eu1−
Yb
Zn2Sb2 by mechanical alloying and
hot pressing. Nano Energy, 25:136–144, 2016.

[28]

[29] Alexander Ovchinnikov and Svilen Bobev. Zintl phases with group 15 elements and the
transition metals: A brief overview of pnictides with diverse and complex structures. Journal
of Solid State Chemistry, 270:346–359, 2019.
Jing Shuai, Zihang Liu, Hee Seok Kim, Yumei Wang, Jun Mao, Ran He, Jiehe Sui, and
Zhifeng Ren. Thermoelectric properties of Bi-based Zintl compounds Ca1−
Yb
Mg2Bi2.
Journal of Materials Chemistry A, 4(11):4312–4320, 2016.

[30]

[31] Franck Gascoin, Sandra Ottensmann, Daniel Stark, Sossina M Haïle, and G Jeffrey Snyder.
tuned transport properties of the compounds

Zintl phases as thermoelectric materials:
Ca
Yb1−
Zn2Sb2. Advanced Functional Materials, 15(11):1860–1864, 2005.

[32] Qigao Cao, Hui Zhang, Meibo Tang, Haohong Chen, Xinxin Yang, Yuri Grin, and Jingtai
Zhao. Zintl phase Yb1−
Ca
Cd2Sb2 with tunable thermoelectric properties induced by
cation substitution. Journal of Applied Physics, 107(5):053714, 2010.

[33] Hui Zhang, Michael Baitinger, Mei-Bo Tang, Zhen-Yong Man, Hao-Hong Chen, Xin-Xin
Yang, Yi Liu, Ling Chen, Yuri Grin, and Jing-Tai Zhao. Thermoelectric properties of
Eu(Zn1−
Cd
)2Sb2. Dalton Transactions, 39(4):1101–1104, 2010.

[34] Ting Zhou, Jun Mao, Jing Jiang, Shaowei Song, Hangtian Zhu, Qing Zhu, Qinyong Zhang,
Wuyang Ren, Zhiming Wang, Chao Wang, et al. Large reduction of thermal conductivity
leading to enhanced thermoelectric performance in p-type Mg3Bi2-YbMg2Bi2 solid solu-
tions. Journal of Materials Chemistry C, 7, 2019.

[35] Gerhard Cordier and Herbert Schäfer. Neue intermetallische Verbindungen im anti-Ce2O2S-
Strukturtyp. / New Intermetallic Compounds in the anti-Ce2O2S-Structure Type. Zeitschrift
für Naturforschung B, 31(11):1459–1461, 1976.

[36] Peter Klüfers and Albrecht Mewis. AB2X2-Verbindungen im CaAl2Si2-Typ, III Zur Struktur
der Verbindungen CaZn2P2, CaCd2P2, CaZn2As2 und CaCd2As2/AB2X2 Compounds with
the CaAl2Si2 Structure, III / The Crystal Structure of CaZn2P2, CaCd2P2, CaZn2As2, and
CaCd2As2. Zeitschrift für Naturforschung B, 32(7):753–756, 1977.

100

[37] Peter Klüfers and Albrecht Mewis. Zur Struktur der Verbindungen BaZn2P2 und BaZn2As2
/ The Crystal Structure of BaZn2P2 and BaZn2As2. Zeitschrift für Naturforschung B,
33(2):151–155, 1978.

[38] Albrecht Mewis. 2-Verbindungen im CaAl2Si2-Typ, IV Zur Struktur der Verbindungen
CaZn2Sb2, CaCd2Sb2, SrZn2Sb2 und SrCd2Sb2 / AB2X2 Compounds with the CaAl2Si2
Structure, IV The Crystal Structure of CaZn2Sb2, CaCd2Sb2, SrZn2Sb2, and SrCd2Sb2.
Zeitschrift für Naturforschung B, 33(4):382–384, 1978.

[39] Albrecht Mewis. AB2X2-Verbindungen im CaAl2Si2-Typ, V Zur Struktur der Verbindungen
CaMn2P2, CaMn2As2, SrMn2P2 und SrMn2As2 / AB2X2-Compounds with the CaAl2Si2
Structure, V The Crystal Structure of CaMn2P2, CaMn2As2, SrMn2P2, and SrMn2As2.
Zeitschrift für Naturforschung B, 33(6):606–609, 1978.

[40] Erwin Brechtel, Gerhard Cordier, and Herbert Schäfer. Neue ternäre Erdalkaliverbindun-
gen des Arsens mit Mangan / New Ternary Alkaline Earth Compounds of Arsenic with
Manganese. Zeitschrift für Naturforschung B, 33(7):820–822, 1978.

[41] Erwin Brechtel, Gerhard Cordier, and Herbert Schäfer. Darstellung und Kristallstruktur von
BaMn2Sb2, BaZn2Sb2 und BaCd2Sb2 / Preparation and Crystal Structure of BaMn2Sb2,
BaZn2Sb2 and BaCd2Sb2. Z. Naturforsch. B, 34(7):921–925, 1979.

[42] R Rühl and W Jeitschko. New pnictides with Ce2O2S-type structure. Materials Research

Bulletin, 14(4):513–517, 1979.
Inga Schellenberg, Matthias Eul, Wilfried Hermes, and Rainer Pöttgen. A 121Sb and
151Eu Mössbauer spectroscopic investigation of EuMn2Sb2, EuZn2Sb2, YbMn2Sb2, and
YbZn2Sb2. Zeitschrift für anorganische und allgemeine Chemie, 636(1):85–93, 2010.

[43]

[44] AV Morozkin, O Isnard, P Henry, S Granovsky, R Nirmala, and P Manfrinetti. Synthesis
and magnetic structure of the YbMn2Sb2 compound. Journal of Alloys and Compounds,
420(1):34–36, 2006.

[45] Peter Klüfers and Albrecht Mewis. AB2X2-Verbindungen mit CaAl2Si2-Struktur. Zeitschrift

für Kristallographie-Crystalline Materials, 169(1-4):135–148, 1984.

[46] Peter Klüfers, Hermann Neumann, Albrecht Mewis, and Hans-Uwe Schuster. AB2X2-
Verbindungen im CaAl2Si2-Typ, VIII [1] / AB2X2 Compounds with the CaAl2Si2 Structure,
VIII [1]. Zeitschrift für Naturforschung B, 35(10):1317–1318, 1980.

[47] Peter Klüfers, Albrecht Mewis, and Hans-Uwe Schuster. AB2X2-Verbindungen im
CaAl2Si2-Typ, VI Zur Struktur von ThCu2P2, YbZn2P2 und YbZnCuP2 sowie der ver-
wandten Verbindungen YbCu3P2 und YbCu2P2. Zeitschrift für Kristallographie-Crystalline
Materials, 149(3-4):211–225, 1979.

[48] Gerhard Zwiener, Hermann Neumann, and Hans-Uwe Schuster. Magnetische Eigenschaften
von AB2X2-Verbindungen im CaAl2Si2-Typ/Magnetic Properties of AB2X2 Compounds
with the CaAl2Si2 Structure. Zeitschrift für Naturforschung B, 36(9):1195–1197, 1981.

101

[49] A. Nateprov, J. Cisowski, J. Heimann, and I. Mirebeau. A new ternary compound YbZn2As2

with mixed valency of Yb. Journal of Alloys and Compounds, 290(1):6–9, 1999.

[50] Zelinska, O. Ya and Tkachuk, A.V. and Grosvenor, A.P. and Mar, A. Structure and physical
properties of YbZn2Sb2 and YbCd2Sb2. Chemistry of metals and alloys, (1):204–209, 2008.
[51] A. Artmann, A. Mewis, M. Roepke, and G. Michels. AM2X2-Verbindungen mit CaAl2Si2-
Struktur. XI. Struktur und Eigenschaften der Verbindungen ACd2X2 (A: Eu, Yb, X: P, As,
Sb). Zeitschrift für anorganische und allgemeine Chemie, 622(4):679–682, 1996.

[52] Robert Juza and Reinhard Kroebel. Über eine Hochtemperaturmodifikation des Magnesiu-
marsenids und eine ternäre Phase Mg2MnAs2 gleicher Struktur. Zeitschrift für anorganische
und allgemeine Chemie, 331(3-4):187–199, 1964.

[53] E. Zintl and E. Husemann. Bindungsart und Gitterbau binärer Magnesiumverbindungen.

Zeitschrift für physikalische Chemie, 21(1):138–155, 1933.

[54] Klaus Deller and Brigitte Eisenmann.

Ternäre Erdalkali-Element (V)-Verbindungen
AMg2B2 mit A= Ca, Sr, Ba und B= As, Sb, Bi/Ternary Alkaline Earth-Element (V)-
Compounds AMg2B2 with A= Ca, Sr, Ba and B= As, Sb, Bi. Zeitschrift für Naturforschung
B, 32(6):612–616, 1977.

[55] Andrew F. May, Michael A. McGuire, David J. Singh, Radu Custelcean, and Gerald E. Jelli-
son Jr. Structure and properties of single crystalline CaMg2Bi2, EuMg2Bi2, and YbMg2Bi2.
Inorganic Chemistry, 50(21):11127–11133, 2011.

[56] Franziska Wartenberg, Christian Kranenberg, Regina Pocha, Dirk Johrendt, Albrecht Mewis,
Rolf-Dieter Hoffmann, Bernd D Mosel, and Rainer Poettgen. Neue pnictide im CaAl2Si2-
Typ und dessen existenzgebiet / New pnictides with the CaAl2Si2 type structure and the
stability range of this type. Zeitschrift für Naturforschung B, 57(11):1270–1276, 2002.

[57] A.A. Murav’eva, O.S. Zarechnyuk, and E.I. Gladyshevskii. Systems Y–Al–Si (Ge, Sb) in

the range 0 to 33.3 at percent Y. Technical report, Lvov State Univ., 1971.

[58] Reinhard Xesper and Hans Georg von Schnering. GdAl2Si2, eine unerwartete Verbindung
im CaAl2Si2-Typ / GdAl2Si2, an Unexpected Compound of the CaAl2Si2-Type. Zeitschrift
für Naturforschung B, 37(12):1514–1517, 1982.

[59] C. Kranenberg, D. Johrendt, and A. Mewis. Untersuchungen zum Existenzgebiet des
CaAl2Si2-Strukturtyps bei ternären Siliciden. Zeitschrift für anorganische und allgemeine
Chemie, 625(11):1787–1793, 1999.

[60] Christian Kranenberg, Dirk Johrendt, Albrecht Mewis, Rainer Pöttgen, Gunter Kotzyba,
Carsten Rosenhahn, and Bernd D Mosel. Structure and properties of the compounds LnAl2X2
(Ln= Eu, Yb, X= Si, Ge). Solid State Sciences, 2(2):215–222, 2000.

[61] Nataliya Muts, Roman Gladyshevskii, and Evgen Gladyshevskii. Crystal structures of the
compounds PrAl2Si2, Pr3Al4Si6 and PrAlSi2. Journal of Alloys and Compounds, 402(1):66–
69, 2005.

102

[62] SJ Andersen, CD Marioara, A Frøseth, R Vissers, and HW Zandbergen. Crystal structure of
the orthorhombic U2-Al4Mg4Si4 precipitate in the Al–Mg–Si alloy system and its relation
to the 
’ and 
” phases. Materials Science and Engineering: A, 390(1):127–138, 2005.

[63] P. Schobinger-Papamantellos and F. Hulliger. The magnetic structure of EuAl2Si2. Journal

of the Less Common Metals, 146:327–335, 1989.

[64] H. Flandorfer, D. Kaczorowski, J. Gröbner, P. Rogl, R. Wouters, C. Godart, and A. Kostikas.
The systems Ce–Al–(Si, Ge): phase equilibria and physical properties. Journal of Solid
State Chemistry, 137(2):191–205, 1998.

[65] Christian Kranenberg, Dirk Johrendt, and Albrecht Mewis. The stability range of the
CaAl2Si2-type structure in case of LnAl2Ge2 compounds. Solid State Sciences, 4(2):261–
265, 2002.

[66] Olaf Reckeweg and Francis J DiSalvo. About binary and ternary alkaline earth metal nitrides.

Zeitschrift für anorganische und allgemeine Chemie, 627(3):371–377, 2001.
Jiawei Zhang, Lirong Song, Mattia Sist, Kasper Tolborg, and Bo Brummerstedt Iversen.
Chemical bonding origin of the unexpected isotropic physical properties in thermoelectric
Mg3Sb2 and related materials. Nature Communications, 9(1):1–10, 2018.

[67]

[68] Wanyue Peng, Mario Calderón-Cueva, Mingda Lyu, Benjamin Brugman, Gill Levental,
Ashiwini Balodhi, Megan Rylko, Susannah Dorfman, and Alexandra Zevalkink. High-
pressure behavior of layered 

2
2 compounds. 2020 (in preparation).

[69] Yougui Liao. Practical electron microscopy and database, 2006.

[70]

Jamo Momand, Felix R.L. Lange, Ruining Wang, Jos E. Boschker, Marcel A. Verheijen,
Raffaella Calarco, Matthias Wuttig, and Bart J. Kooi. Atomic stacking and van-der-Waals
bonding in GeTe–Sb2Te2 superlattices. Journal of Materials Research, 31(20):3115–3124,
2016.

[71] Wanyue Peng, David M. Smiadak, Michael G. Boehlert, Spencer Mather, Jared B. Williams,
Donald T. Morelli, and Alexandra Zevalkink. Lattice hardening due to vacancy diffusion in
(GeTe)
Sb2Te3 alloys. Journal of Applied Physics, 126(5):055106, 2019.

[72] H. Wiedemeier and P.A. Siemers. The thermal expansion of GeS and GeTe. Zeitschrift für

anorganische und allgemeine Chemie, 431(1):299–304, 1977.

[73] Y. Gelbstein, G. Gotesman, Y. Lishzinker, Z. Dashevsky, and MP Dariel. Mechanical
properties of PbTe-based thermoelectric semiconductors. Scripta Materialia, 58(4):251–
254, 2008.

[74]

Jennifer E. Ni, Eldon D. Case, Kristen N. Khabir, Ryan C. Stewart, Chun-I. Wu, Timothy P.
Hogan, Edward J. Timm, Steven N. Girard, and Mercouri G. Kanatzidis. Room temperature
Young’s modulus, shear modulus, Poisson’s ratio and hardness of PbTe-PbS thermoelectric
materials. Materials Science and Engineering: B, 170(1-3):58–66, 2010.

103

[75] Mathias Woydt and Hardy Mohrbacher. The tribological and mechanical properties of

niobium carbides (NbC) bonded with cobalt or Fe3Al. Wear, 321:1–7, 2014.

[76] Francis J. DiSalvo. Thermoelectric cooling and power generation. Science, 285(5428):703–

706, 1999.

[77] Bradley Orr and Aliakbar Akbarzadeh. Prospects of waste heat recovery and power genera-

tion using thermoelectric generators. Energy Procedia, 110:250–255, 2017.

[78] George Schmidt, Leonard Dudzinski, and Thomas Sutliff. Radioisotope power: A key
technology for deep space exploration. In 6th International Energy Conversion Engineering
Conference (IECEC), page 5640, 2011.

[79] Ajay Misra. Overview of nasa program on development of radioisotope power systems with
high specific power. In 4th International Energy Conversion Engineering Conference and
Exhibit (IECEC), page 4187, 2006.
Jiawei Zhang, Lirong Song, and Bo Brummerstedt Iversen. Insights into the design of ther-
moelectric Mg3Sb2 and its analogs by combining theory and experiment. Npj Computational
Materials, 5(1):76, 2019.

[80]

[81] G. Jeffrey Snyder and Eric S. Toberer. Complex thermoelectric materials. In Materials for
sustainable energy: a collection of peer-reviewed research and review articles from Nature
Publishing Group, pages 101–110. World Scientific, 2011.

[82] Zhiting Tian, Jivtesh Garg, Keivan Esfarjani, Takuma Shiga, Junichiro Shiomi, and Gang
Chen. Phonon conduction in PbSe, PbTe, and PbTe1−
Se
 from first-principles calculations.
Physical Review B, 85(18):184303, 2012.

[83] Robert E. Newnham. Properties of materials: anisotropy, symmetry, structure. Oxford

University Press on Demand, 2005.

[84] Wolfgang G. Zeier, Alex Zevalkink, Zachary M. Gibbs, Geoffroy Hautier, Mercouri G.
intuition in thermoelectric

Kanatzidis, and G. Jeffrey Snyder. Thinking like a chemist:
materials. Angewandte Chemie International Edition, 55(24):6826–6841, 2016.

[85] Yaniv Gelbstein. Pb1−
Sn
 Te alloys: Application considerations. Journal of electronic

materials, 40(5):533–536, 2011.

[86] David G. Cahill and Robert O. Pohl. Lattice vibrations and heat transport in crystals and

glasses. Annual review of physical chemistry, 39(1):93–121, 1988.

[87] P.G. Klemens. Theory of the thermal conductivity of solids. Thermal conductivity, 1:1–68,

1969.

[88] Micheline Roufosse and P.G. Klemens. Thermal conductivity of complex dielectric crystals.

Physical Review B, 7(12):5379, 1973.

[89] G. Leibfried and E. Schl. Omann thermal conductivity of dielectric solids by a variational

technique nachr. Akad. Wiss. G€ ottingen II a (4), 71, 1954.

104

[90] Carl L. Julian.

Theory of heat conduction in rare-gas crystals.

137(1A):A128, 1965.

Physical Review,

[91] Glen A. Slack and S. Galginaitis. Thermal conductivity and phonon scattering by magnetic

impurities in CdTe. Physical Review, 133(1A):A253, 1964.

[92] E. Grüneisen. Theory of solid state of monatomic elements. Annalen der Physik, 39:257–306,

1912.

[93] Matthias Agne, Shashwat Anand, and G. Jeffrey Snyder. Inherent anharmonicity of harmonic

solids. submitted, 2020.

[94] Günther Leibfried and Wolfgang Ludwig. Theory of anharmonic effects in crystals. In Solid

state physics, volume 12, pages 275–444. Elsevier, 1961.

[95] Hassel Ledbetter, Ming Lei, Allen Hermann, and Zhengzhi Sheng. Low-temperature elastic

constants of YBa2Cu3O7. Physica C: Superconductivity, 225(3-4):397–403, 1994.

[96] Teahoon Park, Chihyun Park, Byeonggwan Kim, Haejin Shin, and Eunkyoung Kim. Flexible
pedot electrodes with large thermoelectric power factors to generate electricity by the touch
of fingertips. Energy & Environmental Science, 6(3):788–792, 2013.

[97] D.S. Sanditov and M.V. Darmaev. Effective Modulus of Elasticity and Grüneisen Parameter
of Chalcogenide Glasses in the As–Tl–S System. Inorganic Materials, 55(6):617–622, 2019.
[98] D.S. Sanditov and V.N. Belomestnykh. Relation between the parameters of the elasticity
theory and averaged bulk modulus of solids. Technical Physics, 56(11):1619–1623, 2011.
Interrelation between
anharmonicity and lateral strain in quasi-isotropic polycrystalline solids. Technical physics,
49(8), 2004.

[99] Vladimir Nikolaevich Belomestnykh and Elena Pavlovna Tesleva.

[100] Vladimir Nikolaevich Belomestnykh. The acoustical grüneisen constants of solids. Technical

Physics Letters, 30(2):91–93, 2004.

[101] G.J. Miller and S.M. Kauzlarich. Chemistry, Structure, and Bonding of Zintl Phases and

Ions. Kauzlarich, S., Eds, 1996.

[102] Chong Zheng, Roald Hoffmann, Reinhard Nesper, and Hans Georg Von Schnering. Site
preferences and bond length differences in CaAl2Si2-type Zintl compounds. Journal of the
American Chemical Society, 108(8):1876–1884, 1986.

[103] Susan Mary Kauzlarich. Chemistry, structure, and bonding of Zintl phases and ions. VCH

Publishers, 1996.

[104] Jürgen Evers. Zintl Phases: Principles and Recent Developments, volume 139. Springer

Science & Business Media, 2011.

[105] E. I. Gladyshevskii, P. I. Kipyakevich, and O.I. Ukr. Fiz. Zh. Bodak. Crystal structure of

CaAl2Si2 and its analogs.

105

[106] Jeremy K. Burdett and Gordon J. Miller. Fragment formalism in main-group solids: applica-
tions to aluminum boride (AlB2), calcium aluminum silicide (CaAl2Si2), barium-aluminum
(BaAl4), and related materials. Chemistry of Materials, 2(1):12–26, 1990.

[107] Chong Zheng and Roald Hoffmann. Complementary local and extended views of bonding
in the ThCr2Si2 and CaAl2Si2 structures. Journal of Solid State Chemistry, 72(1):58–71,
1988.

[108] Jiawei Zhang, Lirong Song, Aref Mamakhel, Mads Ry Vogel Jørgensen, and Bo Brummer-
stedt Iversen. High-performance low-cost n-type Se-doped Mg3Sb2-based Zintl compounds
for thermoelectric application. Chemistry of Materials, 29(12):5371–5383, 2017.

[109] Zhang Jiawei, Song Lirong, Madsen Georg K.H., Fischer Karl F.F., Zhang Wenqing, Shi Xun,
and Iversen Bo B. Designing high-performance layered thermoelectric materials through
orbital engineering. Nature Communications, 7:10892, 2016.

[110] Jing Shuai, Jun Mao, Shaowei Song, Qinyong Zhang, Gang Chen, and Zhifeng Ren. Recent
progress and future challenges on thermoelectric zintl materials. Materials Today Phys.,
1:74–95, 2017.

[111] Jing Shuai, Huiyuan Geng, Yucheng Lan, Zhuan Zhu, Chao Wang, Zihang Liu, Jiming Bao,
Ching-Wu Chu, Jiehe Sui, and Zhifeng Ren. Higher thermoelectric performance of Zintl
phases (Eu0.5Yb0.5)1−
Ca
Mg2Bi2 by band engineering and strain fluctuation. Proceedings
of the National Academy of Sciences, page 201608794, 2016.

[112] Alex Zevalkink, Wolfgang G. Zeier, Ethan Cheng, Jeffrey Snyder, Jean-Pierre Fleurial, and
Sabah Bux. Nonstoichiometry in the Zintl phase Yb1−
Zn2Sb2 as a route to thermoelectric
optimization. Chemistry of Materials, 26(19):5710–5717, 2014.

[113] Kazuki Imasato, Stephen Dongmin Kang, Saneyuki Ohno, and G. Jeffrey Snyder. Band
engineering in Mg3Sb2 by alloying with Mg3Bi2 for enhanced thermoelectric performance.
Materials Horizons, 5(1):59–64, 2018.

[114] Effects of Lanthanum Substitution on Thermoelectric Properties of YbZn2Sb2, au-
thor=Zhang, Xiong and Peng, Kunling and Guo, Lijie and Yan, Yanchi and Zhan, Hen
and Lu, Xu and Gu, Haoshuang and Zhou, Xiaoyuan. Journal of Electronic Materials,
46(5):2611–2615, 2017.

[115] Xiao-Jun Wang, Mei-Bo Tang, Hao-Hong Chen, Xin-Xin Yang, Jing-Tai Zhao, Ulrich
Burkhardt, and Yuri Grin. Synthesis and high thermoelectric efficiency of zintl phase
YbCd2−
Zn
Sb2. Applied Physics Letters, 94(9):092106, 2009.

[116] Vijay Ponnambalam and Donald T Morelli. Thermoelectric Properties of Light-Element-
Containing Zintl Compounds CaZn2−
Cu
P2 and CaMnZn1−
Cu
P2 (
=0.0–0.2). Journal
of Electronic Materials, 43(6):1875–1880, 2014.

[117] Kai Guo, Qigao Cao, and Jingtai Zhao. Zintl phase compounds 

2

2 (
=Ca, Sr, Ba, Eu,
Yb; 
=Zn, Cd) and their substitution variants: a class of potential thermoelectric materials.
Journal of Rare Earths, 31(11):1029–1038, 2013.

106

[118] Hiromasa Tamaki, Hiroki K. Sato, and Tsutomu Kanno.

Isotropic conduction network
and defect chemistry in Mg3+
Sb2-based layered zintl compounds with high thermoelectric
performance. Advanced Materials, 28(46):10182–10187, 2016.

[119] Jiawei Zhang, Lirong Song, Steffen Hindborg Pedersen, Hao Yin, Le Thanh Hung, and
Bo Brummerstedt Iversen. Discovery of high-performance low-cost n-type Mg3Sb2-based
thermoelectric materials with multi-valley conduction bands. Nature Communications,
8:13901, 2017.

[120] Jing Shuai, Jun Mao, Shaowei Song, Qing Zhu, Jifeng Sun, Yumei Wang, Ran He, Jiawei
Zhou, Gang Chen, David J Singh, and Zhifeng Ren. Tuning the carrier scattering mechanism
to effectively improve the thermoelectric properties. Energy & Environmental Science,
10(3):799–807, 2017.

[121] Saneyuki Ohno, Kazuki Imasato, Shashwat Anand, Hiromasa Tamaki, Stephen Dongmin
Kang, Prashun Gorai, Hiroki K Sato, Eric S Toberer, Tsutomu Kanno, and G Jeffrey Snyder.
Phase Boundary Mapping to Obtain n-type Mg3Sb2-Based Thermoelectrics. Joule, 2, 2017.
[122] Jimmy Jiahong Kuo, Stephen Dongmin Kang, Kazuki Imasato, Hiromasa Tamaki, Saneyuki
Ohno, Tsutomu Kanno, and G. Jeffrey Snyder. Grain boundary dominated charge transport
in mg3sb2-based compounds. Energy & Environmental Science, 11(2):429–434, 2018.

[123] Jared B Williams and Donald T Morelli. Understanding the superior thermoelectric perfor-
mance of Sb precipitated Ge17Sb2Te20. Journal of Materials Chemistry C, 4(42):10011–
10017, 2016.

[124] HJ McSkimin. Ultrasonic methods for measuring the mechanical properties of liquids and

solids. Physical Acoustics, 1(part A):271–334, 1964.

[125] Robert G Leisure and FA Willis. Resonant ultrasound spectroscopy. Journal of Physics:

Condensed Matter, 9(28):6001, 1997.

[126] Brian J. Zadler, Jérôme H.L. Le Rousseau, John A. Scales, and Martin L. Smith. Reso-
nant ultrasound spectroscopy: theory and application. Geophysical Journal International,
156(1):154–169, 2004.

[127] Harold H Demarest Jr. Cube-resonance method to determine the elastic constants of solids.

The Journal of the Acoustical Society of America, 49(3B):768–775, 1971.

[128] Integrated X-ray powder diffraction software. The Rigaku Journal, 26(1), 2010.
[129] PDXL Version. 2.0. 3.0 Integrated X-ray Powder Diffraction Software. Rigaku Corporation,

Tokyo, Japan, pages 196–8666, 2011.

[130] HaM Rietveld. A profile refinement method for nuclear and magnetic structures. Journal of

applied Crystallography, 2(2):65–71, 1969.

107

[131] Rostislav Hrubiak, Stanislav Sinogeikin, Eric Rod, and Guoyin Shen. The laser micro-
machining system for diamond anvil cell experiments and general precision machining ap-
plications at the High Pressure Collaborative Access Team. Review of Scientific Instruments,
86(7):072202, 2015.

[132] Guoyin Shen, Yanbin Wang, Agnes Dewaele, Christine Wu, Dayne E. Fratanduono, Jon
Eggert, Stefan Klotz, Kamil F. Dziubek, Paul Loubeyre, Oleg V. Fat’yanov, et al. Toward an
international practical pressure scale: A proposal for an IPPS ruby gauge (IPPS-Ruby2020).
High Pressure Research, pages 1–16, 2020.

[133] Ross J. Angel, Matteo Alvaro, and Javier Gonzalez-Platas. Eosfit7c and a fortran module
(library) for equation of state calculations. Zeitschrift für Kristallographie, 229(5):405–419,
2014.

[134] Xavier Gonze, J.M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.M. Rignanese, Luc Sindic,
Matthieu Verstraete, G. Zerah, F. Jollet, et al. First-principles computation of material
properties:
the ABINIT software project. Computational Materials Science, 25(3):478–
492, 2002.

[135] Xavier Gonze, Bernard Amadon, P-M Anglade, J.M. Beuken, François Bottin, Paul
Boulanger, Fabien Bruneval, Damien Caliste, Razvan Caracas, Michel Côté, et al. ABINIT:
First-principles approach to material and nanosystem properties. Computational Materials
Science, 180(12):2582–2615, 2009.

[136] Xavier Gonze, François Jollet, F. Abreu Araujo, Donat Adams, Bernard Amadon, Thomas
Applencourt, Christophe Audouze, J.M. Beuken, Jordan Bieder, A. Bokhanchuk, and
E. Bousquet. Recent developments in the ABINIT software package. Computer Physics
Communications, 205:106–131, 2016.

[137] Stefano Baroni, Stefano de Gironcoli, Andrea Dal Corso, and Paolo Giannozzi. Phonons and
related crystal properties from density-functional perturbation theory. Reviews of Modern
Physics, 73:515–562, 2001.

[138] Xavier Gonze. First-principles responses of solids to atomic displacements and homogeneous
electric fields: Implementation of a conjugate-gradient algorithm. Phys. Rev. B, 55:10337–
10354, 1997.

[139] Xavier Gonze and Changyol Lee. Dynamical matrices, born effective charges, dielectric
permittivity tensors, and interatomic force constants from density-functional perturbation
theory. Physical Review B, 55:10355–10368, 1997.

[140] D. R. Hamann, Xifan Wu, Karin M. Rabe, and David Vanderbilt. Metric tensor formulation

of strain in density-functional perturbation theory. Physical Review B, (3).

[141] John P. Perdew, Adrienn Ruzsinszky, Gábor I. Csonka, Oleg A. Vydrov, Gustavo E. Scuseria,
Lucian A. Constantin, Xiaolan Zhou, and Kieron Burke. Restoring the density-gradient
expansion for exchange in solids and surfaces. Physical Review Letters, 100:136406, 2008.

108

[142] Lianhua He, Fang Liu, Geoffroy Hautier, Micael J. T. Oliveira, Miguel A. L. Marques,
Fernando D. Vila, J. J. Rehr, G.-M. Rignanese, and Aihui Zhou. Accuracy of generalized
gradient approximation functionals for density-functional perturbation theory calculations.
Physical Review B, 89:064305, 2014.

[143] D. R. Hamann. Optimized norm-conserving vanderbilt pseudopotentials. Physical Review

B, 88:085117, 2013.

[144] Michiel Van Setten, Matteo Giantomassi, Eric Bousquet, Matthieu J. Verstraete, Donald R
Hamann, Xavier Gonze, and Gian-Marco Rignanese. The PseudoDojo: Training and grad-
ing a 85 element optimized norm-conserving pseudopotential table. Computer Physics
Communications, page 10.1016/j.cpc.2018.01.012, 2018.

[145] Hendrik J. Monkhorst and James D. Pack. Special points for Brillouin-zone integrations.

Physical Review B, 13(12):5188–5192, 1976.

[146] Guido Petretto, Xavier Gonze, Geoffroy Hautier, and Gian-Marco Rignanese. Convergence
and pitfalls of density functional perturbation theory phonons calculations from a high-
throughput perspective. Computational Materials Science, 144:331–337, 2018.

[147] Guido Petretto, Shyam Dwaraknath, Henrique P. C. Miranda, Donald Winston, Matteo
Giantomassi, Michiel J. Van Setten, Xavier Gonze, Kristin A. Persson, Geoffroy Hautier, and
Gian-Marco Rignanese. High-throughput density functional perturbation theory phonons
for inorganic materials. Scientific data, 5:180065, 2018.

[148] Georg K. H. Madsen, Ankita Katre, and Chandan Bera. Calculating the thermal conductivity
of the silicon clathrates using the quasi-harmonic approximation. Physica status solidi (A),
213(3):802–807, 2015.

[149] Alex Zevalkink, Wolfgang G. Zeier, Ethan Cheng, Jeffrey Snyder, Jean-Pierre Fleurial, and
Sabah Bux. Nonstoichiometry in the Zintl phase Yb1−
Zn2Sb2 as a route to thermoelectric
optimization. Chemistry of Materials, 26(19):5710–5717, 2014.

[150] Shaowei Song, Jun Mao, Jing Shuai, Hangtian Zhu, Zhensong Ren, Udara Saparamadu,
Zhongjia Tang, Bo Wang, and Zhifeng Ren. Study on anisotropy of n-type Mg3Sb2-based
thermoelectric materials. Applied Physics Letters, 112(9):092103, 2018.

[151] Wolfgang G. Zeier. New tricks for optimizing thermoelectric materials. Current Opinion in

Green and Sustainable Chemistry, 4:23–28, 2017.

[152] Guodong Li, Umut Aydemir, Max Wood, Qi An, William A. Goddard III, Pengcheng
Zhai, Qingjie Zhang, and G. Jeffrey Snyder. Deformation mechanisms in high-efficiency
thermoelectric layered zintl compounds. Journal of Materials Chemistry A, 5:9050–9059,
2017.

[153] Mansura Khatun, Stanislav S. Stoyko, and Arthur Mar.

Quaternary Arsenides
AM1.5Tt0.5As2 (A= Na, K, Rb; M= Zn, Cd; Tt= Si, Ge, Sn): Size Effects in 2-and
ThCr2Si2-Type Structures. Inorganic chemistry, 52(6):3148–3158, 2013.

109

[154] Linus Pauling. The Nature of the Chemical Bond, volume 260. Cornell university press

Ithaca, NY, 1960.

[155] C. Giacovazzo. Editor. Fundamentals of Crystallography. IUCr, 1992.
[156] R.D. T. Shannon and C. Tfc Prewitt. Effective ionic radii in oxides and fluorides. (IUCr)

Acta Crystallographica Section B, 25(5):925–946, 1969.

[157] G. S. Pomrehn, A. Zevalkink, W. G. Zeier, A. van de Walle, and G. J. Snyder. Defect-
controlled electronic properties in 
Zn2Sb2 Zintl phases. Angewandte Chemie International
Edition, 53(13):3422–6, 2014.

[158] Alan R. Denton and Neil W. Ashcroft. Vegard’s law. Physical Review A, 43(6):3161, 1991.
[159] D.A. Goodman, L.H. Bennett, and R.E. Watson. Valency effects and relative solubilities in

transition metal alloys. Scripta metallurgica, 17(1):91–96, 1983.

[160] Geoffrey Vincent Raynor. XII. Atomic and ionic radii.—III. Polarization effects in al-
loys. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science,
26(173):152–165, 1938.

[161] William Hume-Rothery and Geoffrey Vincent Raynor. XI. Atomic and ionic radii.—II.
Application to the theory of solid solubility in alloys. The London, Edinburgh, and Dublin
Philosophical Magazine and Journal of Science, 26(173):143–152, 1938.

[162] Zhijun Wang, Yunhao Huang, Yong Yang, Jincheng Wang, and CT Liu. Atomic-size effect

and solid solubility of multicomponent alloys. Scripta Materialia, 94:28–31, 2015.

[163] Prashun Gorai, Brenden R. Ortiz, Eric S. Toberer, and Vladan Stevanović. Investigation of n-
type doping strategies for Mg3Sb2. Journal of Materials Chemistry A, 6(28):13806–13815,
2018.

[164] S.W. Song, J. Mao, M. Bordelon, R. He, Y.M. Wang, J. Shuai, JY Sun, X.B. Lei, S. Ren,
Z.S .and Chen, et al. Joint effect of magnesium and yttrium on enhancing thermoelectric
properties of n-type Zintl Mg3+
Y0.02Sb1.5Bi0.5. Materials Today Physics, 8:25–33, 2019.
[165] Mario Calderón-Cueva*, Wanyue Peng*, Samantha M. Clarke, Jingxuan Ding, Benjamin
Brugman, Gill Levental, Ashiwini Balodhi, Megan Rylko, Olivier Delaire, James P. S. Walsh,
Susannah Dorfman, and Alexandra Zevalkink. Anisotropic structural collapse of Mg3Sb2
and Mg3Bi2 at high pressure. Chemistry of Materials (submitted), 2020.

[166] John V. Badding. High-pressure synthesis, characterization, and tuning of solid state mate-

rials. Annual Review of Materials Research, 28(1):631–658, 1998.

[167] Maxwell Wood, Jimmy Jiahong Kuo, Kazuki Imasato, and Gerald Jeffrey Snyder. Improve-
ment of Low-Temperature zT in a Mg3Sb2–Mg3Bi2 Solid Solution via Mg-Vapor Annealing.
Advanced Materials, 31(35):1902337, 2019.

110

[168] Jun Mao, Yixuan Wu, Shaowei Song, Qing Zhu, Jing Shuai, Zihang Liu, Yanzhong Pei, and
Zhifeng Ren. Defect Engineering for Realizing High Thermoelectric Performance in n-Type
Mg3Sb2-Based Materials. ACS Energy Letters, 2(10):2245–2250, 2017.

[169] Cathie L. Condron, Susan M. Kauzlarich, Franck Gascoin, and G. Jeffrey Snyder. Ther-
moelectric properties and microstructure of Mg3Sb2. Journal of Solid State Chemistry,
179(8):2252–2257, 2006.

[170] Vijay Ponnambalam and Donald T Morelli. On the thermoelectric properties of Zintl
compounds Mg3Bi2−
Pn
 (Pn= P and Sb). Journal of electronic materials, 42(7):1307–
1312, 2013.

[171] Saneyuki Ohno, Kazuki Imasato, Shashwat Anand, Hiromasa Tamaki, Stephen Dongmin
Kang, Prashun Gorai, Hiroki K Sato, Eric S Toberer, Tsutomu Kanno, and G Jeffrey Snyder.
Phase boundary mapping to obtain n-type Mg3Sb2-based thermoelectrics. Joule, 2(1):141–
154, 2018.

[172] Jun Mao, Hangtian Zhu, Zhiwei Ding, Zihang Liu, Geethal Amila Gamage, Gang Chen, and
Zhifeng Ren. High thermoelectric cooling performance of n-type Mg3Bi2-based materials.
Science, 365(6452):495–498, 2019.

[173] Weiwei Xie, Michał J. Winiarski, Tomasz Klimczuk, and Robert Joseph Cava. A tetragonal
polymorph of SrMn2P2 made under high pressure–theory and experiment in harmony.
Dalton Transactions, 46(21):6835–6838, 2017.

[174] Alex Zevalkink, Matej Bobnar, Ulrich Schwarz, and Yuri Grin. Making and breaking bonds

in superconducting sral4−
si
 (0 ≤ x ≤ 2). Chem. Mater., 29(3):1236–1244, 2017.

[175] Masashi Tanaka, Shuai Zhang, Kei Inumaru, and Shoji Yamanaka. High-Pressure synthesis
and superconductivity of the laves phase compound Ca(Al,Si)2 composed of truncated
tetrahedral cages Ca@(Al, Si)12. Inorganic Chemistry, 52(10):6039–6045, 2013.

[176] Xin Gui, Gregory J. Finkelstein, Keyu Chen, Tommy Yong, Przemyslaw Dera, Jinguang
Cheng, and Weiwei Xie. Pressure-Induced Large Volume Collapse, Plane-to-Chain, Insulator
to Metal Transition in CaMn2Bi2. Inorganic Chemistry, 58(14):8933–8937, 2019.

[177] R.J. Riedner, S.R. Srinivasa, L. Cartz, T.G. Worlton, R. Klinger, and R. Beyerlein.
Anisotropic thermal expansion and compressibility of black phosphorus. In AIP Confer-
ence Proceedings, volume 17, pages 8–20. American Institute of Physics, 1974.

[178] You Xiang Zhao and Ian L Spain. X-ray diffraction data for graphite to 20 GPa. Physical

Review B, 40(2):993, 1989.

[179] S. Stølen, A. Grzechnik, T. Grande, and M. Mezouar. Anisotropic compressibility and

expansivity in layered GeSe2. Solid State Communications, 115(5):249–252, 2000.

[180] Jingxuan Ding, Tyson Lanigan-Atkins, Mario Calderon-Cueva, Arnab Banerjee, Douglas L.
Abernathy, Ayman Said, Alexandra Zevalkink, and Olivier Delaire. Anharmonic lattice
dynamics in Zintl 
Mg2X2 (
 =Mg, Ca, Yb; X =Sb, Bi. In preparation, 2020.

111

[181] Mansura Khatun, Stanislav S Stoyko, and Arthur Mar. Quaternary Arsenides AM1.5Tt 0.5As2
(A= Na, K, Rb; M= Zn, Cd; Tt= Si, Ge, Sn): Size Effects in CaAl2Si2-and ThCr2Si2-Type
Structures. Inorganic chemistry, 52(6):3148–3158, 2013.

[182] Charles T. Prewitt and Robert T. Downs. High-pressure crystal chemistry. Reviews in

Mineralogy and Geochemistry, 37:284–318, 1998.

[183] Alex Zevalkink, Matej Bobnar, Ulrich Schwarz, and Yuri Grin. Making and Breaking Bonds
in Superconducting SrAl4−
Si
 (0 ≤ x ≤ 2). Chemistry of Materials, 29(3):1236–1244,
2017.

[184] J. Huster and W. Bronger. 
- und 
-BaCu2X2 (X = S, Se)–Darstellung von Einkristallen in
Kaliumchalkogenocyanat-Schmelzen. Zeitschrift für anorganische und allgemeine Chemie,
625(12):2033–2040, 1999.

[185] G.I. Meijer. Who wins the nonvolatile memory race? Science, 319(5870):1625–1626, 2008.
[186] Matthias Wuttig and Noboru Yamada. Phase-change materials for rewriteable data storage.

Nature Materials, 6(11):824, 2007.

[187] Geoffrey W. Burr, Matthew J. Breitwisch, Michele Franceschini, Davide Garetto, Kailash
Gopalakrishnan, Bryan Jackson, Bülent Kurdi, Chung Lam, Luis A. Lastras, Alvaro Padilla,
et al. Phase change memory technology. Journal of Vacuum Science & Technology B,
28(2):223–262, 2010.

[188] Simone Raoux, Geoffrey W. Burr, Matthew J. Breitwisch, Charles T. Rettner, Y.C. Chen,
Robert M. Shelby, Martin Salinga, Daniel Krebs, S.H. Chen, H.L. Lung, et al. Phase-change
random access memory: A scalable technology. IBM Journal of Research & Development,
52(4.5):465–479, 2008.

[189] G. Bruns, P. Merkelbach, C. Schlockermann, M. Salinga, M. Wuttig, T.D. Happ, J.B. Philipp,
and M. Kund. Nanosecond switching in GeTe phase change memory cells. Applied Physics
Letters, 95(4):043108, 2009.

[190] J Hegedüs and SR Elliott. Microscopic origin of the fast crystallization ability of Ge–Sb–Te

phase-change memory materials. Nature Materials, 7(5):399, 2008.

[191] Bong-Sub Lee, John R. Abelson, Stephen G. Bishop, Dae-Hwan Kang, Byung-ki Cheong,
and Ki-Bum Kim. Investigation of the optical and electronic properties of Ge2Sb2Te5 phase
change material in its amorphous, cubic, and hexagonal phases. Journal of Applied Physics,
97(9):093509, 2005.

[192] Stanford R Ovshinsky. Reversible electrical switching phenomena in disordered structures.

Physical Review Letters, 21(20):1450, 1968.

[193] Zhimei Sun, Jian Zhou, Andreas Blomqvist, Börje Johansson, and Rajeev Ahuja. Formation
of large voids in the amorphous phase-change memory Ge2Sb2Te5 alloy. Physics Review
Letters, 102(7):075504, 2009.

112

[194] Wei Zhu, Ruisheng Yang, Yuancheng Fan, Quanhong Fu, Hongjing Wu, Peng Zhang, Nian-
Hai Shen, and Fuli Zhang. Controlling optical polarization conversion with Ge2Sb2Te5-
based phase-change dielectric metamaterials. Nanoscale, 10(25):12054–12061, 2018.

[195] Ho-Ki Lyeo, David G. Cahill, Bong-Sub Lee, John R. Abelson, Min-Ho Kwon, Ki-Bum
Kim, Stephen G. Bishop, and Byung-ki Cheong. Thermal conductivity of phase-change
material Ge2Sb2Te5. Applied Physics Letters, 89(15):151904, 2006.

[196] Juan Li, Zhiwei Chen, Xinyue Zhang, Yongxing Sun, Jiong Yang, and Yanzhong Pei.
Electronic origin of the high thermoelectric performance of gete among the p-type group iv
monotellurides. NPG Asia Materials, 9(3):e353, 2017.

[197] Jared B. Williams, Spencer P. Mather, Alexander Page, Ctirad Uher, and Donald T. Morelli.
Increasing the thermoelectric power factor of Ge17Sb2Te20 by adjusting the Ge/Sb ratio.
Journal of Applied Physics, 122(4):045105, 2017.

[198] Xiao Xu, Lin Xie, Qing Lou, Di Wu, and Jiaqing He. Boosting the Thermoelectric Per-
formance of Pseudo-Layered Sb2Te3(GeTe)
 via Vacancy Engineering. Advanced Science,
5(12):1801514, 2018.

[199] Tobias Rosenthal, Philipp Urban, Kathleen Nimmrich, Ludwig Schenk,

Johannes
de Boor, Christian Stiewe, and Oliver Oeckler. Enhancing the Thermoelectric Proper-
ties of Germanium Antimony Tellurides by Substitution with Selenium in Compounds
Ge
Sb2(Te1−
Se
)
+3 (0≤x≤0.5; n≤7). Chemistry of Materials, 26(8):2567–2578, 2014.
[200] Tobias Rosenthal, Matthias N. Schneider, Christian Stiewe, Markus Doblinger, and Oliver
Oeckler. Real structure and thermoelectric properties of GeTe-rich germanium antimony
tellurides. Chemistry of Materials, 23(19):4349–4356, 2011.

[201] THOMAS L Anderson and H BRIGITTE Krause. Refinement of the Sb2Te3 and Sb2Te2Se
structures and their relationship to nonstoichiometric Sb2Te3−
Se
 compounds. (IUCr) Acta
Crystallographica Section B, 30(5):1307–1310, 1974.

[202] Peter Bayliss. Crystal chemistry and crystallography of some minerals in the tetradymite

group. American Mineralogist, 76(1-2):257–265, 1991.

[203] Matthias N. Schneider, Markus Seibald, Patrick Lagally, and Oliver Oeckler. Ambiguities in
the structure determination of antimony tellurides arising from almost homometric structure
models and stacking disorder. Journal of Applied Crystallography, 43(5):1012–1020, 2010.
[204] O.G. Karpinsky, L.E. Shelimova, M.A. Kretova, and J.P. Fleurial. An X-ray study of the
mixed-layered compounds of (GeTe)
 (Sb2Te3)
 homologous series. Journal of alloys and
compounds, 268(1):112–117, 1998.

[205] Andriy Lotnyk, Torben Dankwort, Isom Hilmi, Lorenz Kienle, and Bernd Rauschenbach.


-



 observations of the reversible vacancy ordering process in van der Waals-bonded
Ge-Sb-Te thin films and GeTe-Sb2Te3 superlattices. Nanoscale, 11(22):10838–10845, 2019.

113

[206] Jamo Momand, Ruining Wang, Jos E. Boschker, Marcel A. Verheijen, Raffaella Calarco,
and Bart J. Kooi. Dynamic reconfiguration of van der Waals gaps within GeTe-Sb2Te3 based
superlattices. Nanoscale, 9(25):8774–8780, 2017.

[207] Peter Zalden, Karl Simon Siegert, Stephane Rols, Henry E. Fischer, Franziska Schlich, Te Hu,
and Matthias Wuttig. Specific Heat of (GeTe)
(Sb2Te3)1−
 Phase-Change Materials: The
Impact of Disorder and Anharmonicity. Chemistry of Materials, 26(7):2307–2312, 2014.

[208] Juarez L.F. Da Silva, Aron Walsh, and Hosun Lee. Insights into the structure of the stable
and metastable (GeTe)
(Sb2Te3)
 compounds. Physical Review B, 78(22):224111, 2008.
[209] Andriy Lotnyk, Isom Hilmi, Ulrich Ross, and Bernd Rauschenbach. Van der Waals interfacial
bonding and intermixing in GeTe-Sb2Te3-based superlattices. Nano Research, 11(3):1676–
1686, 2018.

[210] T. Seddon, J.M. Farley, and G.A. Saunders. An acoustic anomaly at the phase transition in

GeTe - SnTe alloy single crystals. Solid State Communications, 17(1):55–57, 1975.

[211] Naor Madar, Tom Givon, Dmitry Mogilyansky, and Yaniv Gelbstein. High thermoelectric
potential of Bi2Te3 alloyed GeTe-rich phases. Journal of Applied Physics, 120(3):035102,
2016.

114