UNIVERSITY of CALIFORNIA Los Angeles Prediction of High-Performance Thermoelectric Materials and Optimal Electronic Structures F

UNIVERSITY OF CALIFORNIA Los Angeles

Prediction of High-Performance Thermoelectric Materials and Optimal Electronic Structures for Thermoelectricity

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Materials Science and Engineering

Chun Soo Park

2019 c Copyright by Chun Soo Park 2019 ABSTRACT OF THE DISSERTATION

Prediction of High-Performance Thermoelectric Materials and Optimal Electronic Structures for Thermoelectricity

Chun Soo Park Doctor of Philosophy in Materials Science and Engineering University of California, Los Angeles, 2019 Professor Vidvuds Ozoli¸nˇs,Co-Chair Professor Dwight Christopher Streit, Co-Chair

The thermoelectric figures of merit of bulk materials up to date have not overcome zT = 3, and only in rare occasions have they surpassed zT = 2. Bulk thermoelectrics with zT > 3 have been desired but have not yet been theoretically predicted let alone experimentally realized. In this doctoral work, high-performance thermoelectric materials theoretically capable of zT > 4 are identified and characterized using state-of-the-art first-principles com-

putational methods based on density-functional theory. Ba2BiAu and Sr2BiAu full-Heusler compounds in particular are predicted to deliver ultrahigh thermoelectric performances – the latter across all temperature domain from cryogenic to high: 0.3 ≤ zT ≤ 5 at 100 K ≤ T ≤ 800 K. While unfortunately the compounds look not n-dopable and their predicted zT values inaccessible, they constitute a theoretical proof of concept that zT > 4 is within reach for bulk compounds. With the lessons learned from these compounds and others, the optimal electronic structures for intrinsic thermoelectric performance are generally determined for both semiconductors and metals. Highly dispersive bands at off-symmetry points that minimize electron-phonon scattering is optimal for semiconductors, while a flat-and-linear band crossing that leads to electron filtering and high Seebeck coefficients is optimal for metals. May these generalizations help propel discoveries of more high-performance thermoelectrics.

ii The dissertation of Chun Soo Park is approved.

Louis-Serge Bouchard

Bruce S Dunn

Jaime Marian

Vidvuds Ozoli¸nˇs,Committee Co-Chair

Dwight Christopher Streit, Committee Co-Chair

University of California, Los Angeles

2019

iii To my wife, Seungwon Chung, and my parents, No Young Park and Ye Ok Oh

iv TABLE OF CONTENTS

List of Figures ...... ix

Acknowledgments ...... xvii

Vita ...... xix

1 Introduction ...... 1

1.1 Thermoelectricity ...... 1

1.1.1 The Thermoelectric Eﬀect ...... 1

1.1.2 The Onsager-Callen Theory ...... 3

1.1.3 Challenges of the Figure of Merit zT ...... 6

1.2 Research Eﬀorts on zT Improvement ...... 9

1.2.1 Reducing Lattice Thermal Conductivity ...... 10

1.2.2 Enhancing the Power Factor ...... 12

1.3 Research Statement ...... 16

2 Computational Theories & Methods ...... 17

2.1 Transport of Electrons & Phonons ...... 18

2.1.1 Boltzmann Transport – Electrons ...... 19

2.1.2 Boltzmann Transport – Phonons ...... 22

2.1.3 The Lifetime Problem and Scattering ...... 23

2.2 Scattering of Electrons ...... 24

2.2.1 Electron-Phonon Scattering: Part 1 ...... 24

2.2.2 Electron-Phonon Scattering: Part 2 ...... 30

v 2.2.3 Ionized Impurity Scattering ...... 35

2.3 Scattering of Phonons ...... 37

2.3.1 Lattice Anharmonicity ...... 38

2.3.2 Phonon-phonon Scattering ...... 40

3 Preliminary Investigations ...... 45

3.1 Chalcopyrite CuFeS2 ...... 46

3.1.1 Crystal, Magnetic, and Electronic Structures ...... 47

3.1.2 Electron Transport ...... 49

3.1.3 Phonon Transport and Nanostructuring ...... 53

3.1.4 Lessons ...... 56

3.2 Full-Heusler Fe2TiSi ...... 57

3.2.1 Crystal and Electronic Structures ...... 58

3.2.2 Electron Transport ...... 61

3.2.3 Phonon Transport and Alloying ...... 66

3.2.4 Lessons ...... 69

4 Prediction of High-Performance Thermoelectrics ...... 71

4.1 zT = 5 in Full-Heusler Ba2BiAu ...... 72

4.1.1 Crystal & Electronic Structures ...... 72

4.1.2 Scattering and Mobility ...... 75

4.1.3 Thermoelectric Properties ...... 76

4.1.4 Lessons ...... 80

4.2 High-Performance at Cryogenic-to-High Temperatures in Sr2BiAu ...... 81

4.2.1 Electronic Structure ...... 81

vi 4.2.2 Scattering & Mobility ...... 83

4.2.3 Thermoelectric Properties ...... 85

4.2.4 Lessons ...... 87

4.3 Intermetallic, B20-type CoSi ...... 89

4.3.1 Crystal and Electronic Structures ...... 90

4.3.2 Thermoelectric Properties ...... 91

4.3.3 Electron-phonon Scattering ...... 93

4.3.4 Lessons ...... 95

5 Optimal Electronic Structures for Thermoelectricity ...... 96

5.1 Revisiting the Mahan-Sofo Analysis ...... 97

5.2 Revisiting the Role of Eﬀective Mass ...... 99

5.3 Deduction of Optimal Electronic Structures ...... 104

5.3.1 Semiconductors ...... 104

5.3.2 Metals ...... 105

6 Summary and Outlooks ...... 108

6.1 Conclusions ...... 108

6.2 Future Outlooks ...... 110

A Stability, Defects, and Doping of Ba2BiAu and Sr2BiAu ...... 112

A.1 Computational Methods ...... 113

A.1.1 Defect Formation Energies ...... 113

A.1.2 Defect Concentration ...... 115

A.1.3 DFT Calculations ...... 116

A.2 Results & Discussions ...... 117

vii A.2.1 Chemical Potentials & Stability ...... 117

A.2.2 Defects ...... 117

A.3 Conclusions ...... 120

References ...... 121

viii LIST OF FIGURES

1.1 a) Thermoelectric refrigeration induced by applied external voltage. Heat-carrying electrons ﬂow in the direction enforced by the voltage and create temperature gradient. b) Thermoelectric current generation from applied temperature gradient. Charge carriers are excited at the hot end and ﬂow to the cool end...... 2

1.2 History of zT improvement in some of the highest-performing thermoelectric materials to date (left) [HT17] and zT values of state-of-the-art low-temperature thermoelectrics (right) [MLZ18]...... 7

1.3 Natural abundance of elements occurring in earth’s crust (image taken from Wikipedia)...... 8

1.4 The general behaviors thermoelectric transport properties with respect to doping [ST08]...... 9

1.5 A plot of the distributions that determine the magnitudes of σ, ζ, and κE, with respect to energy. The two dashed vertical lines qualitatively divide the energy- domain into four domains where Σ(E) ought to be high or low in order to maximally beneﬁt the n-type “electronic-part zT ”. Domain (1) is the hole-regime. Domain (2) is where low-energy electrons live which contribute more to σ than ζ. Domain (3) is where the medium-energy electrons live and where the relative beneﬁt to ζ is the largest. Domain (4) is populated by high-energy electrons that

contribute mostly to κE. Domain (3) ought to be weighted with high Σ(E) for the best thermoelectric beneﬁt...... 14

2.1 A schematic of the electron and phonon Wannier functions entering the e-ph matrix elements [GCL07]. The square lattice sketches unit cells, the red curves the electron Wannier functions, and the blue curve the phonon perturbation in the Wannier representation. If any one is spatially separated from the others,

gν0νλ(Re, Rph) vanishes...... 34

ix 2.2 a) The interpolated electronic band structure, b) the phonon dispersion, and c) the interaction matrix elements of boron-doped diamond using Wannier interpolation on EPW [GLC10]...... 35

2.3 a) Normal scattering. The entire scattering processes is contain within the ﬁrst Brillouin zone. No thermal resistivity is experienced. b) Umklapp scattering. The interaction results in a crystal momentum that escapes the ﬁrst Brillouin zone, which is essentially results in backscattering due to the periodic lattice. The reciprocal lattice translation conserves momentum...... 41

3.1 a) The body-centered tetragonal unit cell of antiferromagnetic CuFeS2 belonging to the I42d space group. Fe atoms are brown, Cu atoms are blue, and S atoms are yellow. The red arrows indicate spin directions. b) The orbital-decomposed density of states of one Fe atom. c) The electronic band structure. The Fermi level (horizontal red line) is at the optimum doping level for zT at 400 K. d) The atom-decomposed electronic density of states. Cu appears nearly monovalent (3d104s0): Cu 3d contributes signiﬁcantly to the upper valence bands but very little to the conduction bands...... 48

3.2 (Color online) a) The Seebeck coefficient calculated with fitted τ from τ(E) ∝ N(E)−1 scaling. b) The Seebeck coefficient calculated with constant relaxation time approximation. c) Electron conductivity from τ(E) ∝ N −1 then scaled by the ratio with the experimental conductivity [TMI14]. The true reference lifetime is estimated to be 2 × 10−15 s as opposed to 10−14 s used by BoltzTraP. d) The power factor calculated with the scaled σ. Experimental data points from Tsujii

et al are for Cu.97Zn.03FeS2 [TMI14], and those from Li et al are for Cu.97Fe1.03S2 [LZQ14]. The marker colors are associated with temperature (black for 300 K, red for 400 K). e) Average mean free paths of electrons with τ(E) = 2×10−15N −1(E) s, all of which are shorter than 1 nm. Colors indicate bands...... 52

3.3 a) Phonon dispersion. b) The atom-projected phonon density of states...... 54

x 3.4 a) Calculated bulk lattice thermal conductivity. The expected T −1 trend is observed. Experimental data points are included for comparison. b) Cumulative lattice thermal conductivity over contributions from phonon modes of increasing MFP. c) Phonon lifetimes. d) Phonon mean free paths. The horizontal red line indicates 20 nm, the target length scale of nanostructuring considered in this work. 55

3.5 a) zT with nanostructuring of 20 nm grains b) Predicted improvement of zT via ideal nanostructuring (solid blue, 20 nm grain-size and assuming it does not harm the PF) in comparison to bulk (dotted blue). The power factor (dashed red) is overlaid. At each temperature, the optimal doping concentration for maximum zT is selected...... 56

3.6 a) The unit cell of a full-Heusler compound. The green atoms correspond to Fe, the purple atoms to Ti and the yellow atoms to Si. b) The primitive cell. . . . . 59

3.7 a) The electronic band structure of Fe2TiSi at 0 K. b) The total and atom-

projected electronic density of states of Fe2TiSi. Calculated with PAW-VASP. . 60

3.8 The isoenergy surfaces near the CBM (top two) and the VBM (bottom three). . 61

3.9 a) The p-type and b) the n-type thermoelectric power factor without polar-

optical scattering of Fe2TiSi. c) The p-type and d) the n-type power factor with

polar-optical scattering of Fe2TiSi...... 62

3.10 a) The p-type and b) the n-type absolute Seebeck coeﬃcients. c) The p-type and d) the n-type conductivities. Polar optical scattering is included throughout. 64

3.11 Carrier lifetimes of Fe2TiSi with (black) and without (red) polar-optical scattering. 65

3.12 a) Phonon dispersion of Fe2TiSi b) total and atom-projected pDOS of Fe2TiSi. 67

3.13 a) Lattice thermal conductivity of Fe2TiSi and its alloys calculated with phonon-

phonon and mass-disorder scattering b) mean free path of pure Fe2TiSi c) Phonon-

phonon scattering relaxation time at 300 K of pure Fe2TiSi d) Mode group ve-

locity at of Fe2TiSi ...... 68

xi 4.1 Electronic band structure of Ba2BiAu with (red) and without (black) spin-orbit coupling, aligned at the CBM...... 73

4.2 (Color online) Isoenergy surfaces of the topmost valence band (left) and lowermost

conductions band (right) of Ba2BiAu, 0.1 eV below the VBM and above the CBM, respectively. The levels correspond to electron doping concentration of

20 20 ne = 1.2 × 10 and hole doping concentration of nh = 2.7 × 10 , respectively. . 74

4.3 a) Energy-dependent electron-phonon scattering lifetimes at 300 K of Ba2BiAu with (black) and without (red) polar-optical scattering. b) Electron mobilities limited by electron-phonon scattering (eph, solid lines) and by ionized impurity scattering (ii, dotted lines), and hole mobility at 300 K for comparison (eph, dashed line)...... 76

4.4 a) The n-type power factors and the 500 K p-type power factor. b) The n- type Lorenz numbers and the 800 K p-type Lorenz number, where the black

horizontal line indicates LWF. c) The n-type Seebeck coeﬃcients and the 500 K p-type Seebeck coeﬃcient. d) The n-type conductivities and the 500 K p- type conductivity. All curves are plotted with respect to their respective doping concentrations...... 77

4.5 a) The n-type zT and the 800 K p-type zT . b) Lattice thermal conductivity. . . 79

4.6 (Color online) Electronic band structures of Sr2BiAu calculated on Quantum Espresso [GBB09, GAB17] with norm-conserving pseudopotentials and Perdew- Burke-Ernzerhof (PBE) exchange-correlation functional [PBE96], with (black, solid) and without (red, dotted) spin-orbit coupling, aligned at the conduction band minimum. The atom-decomposed density of states is shown on the right.

Isoenergy surfaces of Sr2BiAu are calculated with PBE+SOC, at 0.1 eV below the VBM (left) and above the CBM (right). The levels correspond to electron

20 −3 doping concentration of ne = 1.5 × 10 cm and hole doping concentration of

19 −3 nh = 7.5 × 10 cm , respectively...... 82

xii 4.7 (Color online) The dependence of transport properties of Sr2BiAu on on doping, temperature, and scattering. a) The carrier lifetimes limited by electron-phonon scattering with (black) and without (red) polar-optical scattering. The temperature is 300 K. b) The electron mobility with polar-optical scattering...... 84

4.8 (Color online) a) The n-type power factor against doping concentrations. b) The

n-type power factor of Sr2BiAu (black squares and line) juxtaposed with power factors of some experimentally and theoretically high-performing thermoelectrics (p-type NbFeSb in blue squares [HKM16], p-type n-type PbTe in red triangles

[SKU08]) and theoretical n-type Ba2BiAu in empty circles [PXO19]). c) The n-type Seebeck coeﬃcient. d) The n-type electrical conductivity...... 86

4.9 (Color online) a) The Lorenz number. b) Lattice thermal conductivity...... 88

4.10 (Color online) a) The predicted zT of n-type Sr2BiAu against doping concen-

tration. b) The predicted zT of n-type Sr2BiAu in comparison to state-of-the- arts thermoelectrics across low temperatures [HLG04, PHM08, HWZ15, KLM15, VSC01]. c) Predicted p-type zT . d) Predicted p-type power factor...... 88

4.11 a) The crystal structure of CoSi. b) The simple cubic Brillouin zone with high symmetry points. c) The electronic structure, where the bands are colored ac-

cording to diﬀerent orbitals shown in the legend, namely, s, p, eg and t2g states. d) The electronic density of states. e) The band structure zoomed into the region near the Fermi level...... 90

xiii 4.12 a) Calculated temperature-dependent Seebeck coeﬃcient of pristine CoSi under the RTA (red solid lines), the CRTA (blue dashed lines) and eDOS−1 approximation (orange dash-dotted lines) in comparison with experimental measurements [SLM13, RLZ05, SZP09, SYA07]. (b) Calculated Seebeck coeﬃcient of

CoSi1−xAlx at various fractions of Al substitution on Si site at 300 K compared with experiments [LKH04, LRZ05]. The black dashed lines are given as guides to the eye. c) Calculated temperature-dependent electrical resistivity of pristine CoSi in comparison with experiments [SLM13, RLZ05, SZP09, PKS10]. d) Calculated temperature-dependent power factor of pristine CoSi compared with experimental measurements [SLM13, RLZ05, SZP09]...... 92

4.13 a) Calculated transport distribution function σ(E) under the RTA (red solid lines), the CRTA (blue dashed lines) and the eDOS−1 approximation (orange dash-dotted lines) at 300 K. The light green shaped area indicates the energy window restricted by the energy derivative of Fermi-Dirac distribution. b) Cal- culated energy-dependent carrier lifetimes at 100 K (blue triangles), 300 K (green squares) and 600 K (red circles) compared with scaled eDOS−1 approximation (orange dash-dotted lines)...... 94

xiv 5.1 (Color online) a) The double parabolic set-up. b) Electron and hole mobilities at various scattering regimes versus respective effective masses. c) The p-type and n-type Seebeck coefficients versus respective effective masses. d) The p-type and n-type power factors versus respective effective masses. e) The power factor versus the Fermi level. f) The Lorenz number versus the Fermi level, where the green horizontal line indicates the Wiedemann-Franz value. The quantities are computed at various scattering regimes – electron-phonon, ionized impurity, and the two processes at competition. All units are arbitrary except for that of the Lorenz number. For b, c, and d, the Fermi levels for holes and electrons are fixed

to their band edges for all me-mh pairs. Also me and mh are varied such that

Eint is kept consistent for all pairing of valence and conduction bands (nh ≈ 180

ne). This way, doping concentration necessary to place the Fermi level at each

band extremum has a consistent reference frame. For e and f, me and mh are

kept ﬁxed as EF is varied...... 101

5.2 (Color online) a) The ideal band structure for semiconductor thermoelectrics. b) The ideal band structure for metallic thermoelectrics. Both schematics are qualitative and from the perspective of promoting n-type performance...... 106

A.1 (Color online) a) An example of defected 2×2×2 supercell with a Ba/Sr vacancy

A.2 (Color online) The ternary phase diagrams of a) Sr2BiAu and b) Ba2BiAu. The phase ﬁelds relevant to the the compounds are bounded by red lines...... 118

xv A.3 (Color online) The formation energies of VAu (solid blue lines) and VBi (dotted

red lines) in a) Sr2BiAu and b) Ba2BiAu. The slopes indicate the charges (−2 ∼ +2). The formation energies under the Bi-and-Au-poor condition is lower, and

therefore shown. EF = 0 corresponds to the VBM, and the black vertical lines correspond to the CBM determined using the band gaps calculated with mBJ+SOC.118

A.4 (Color online) The formation energies of BiAu in a) Sr2BiAu and b) Ba2BiAu. The Bi-and-Au-poor condition (solid red lines) and the Bi-and-Au-rich condition (dotted blue lines) are both shown. The former must be targeted since it leads to

higher defect formation energies. EF = 0 corresponds to the VBM, and the black vertical lines correspond to the CBM determined using the band gaps calculated with mBJ+SOC...... 119

xvi ACKNOWLEDGMENTS

My ﬁrst and foremost gratitude goes to professor Vidvuds Ozoli¸nˇs,my academic advisor at both UCLA and Yale University throughout my Ph.D. years. He has not only provided knowledge, insights and resources, but also been an extremely pleasant and admirable person to work for and work with. He has my highest respect as a scholar and a person. If I had to do a second Ph.D., I would do it with him again. Thank you for believing in me, and for enabling me to realize the dream of contributing to clean energy research eﬀorts.

Professor Dwight Streit, a co-chair of my doctoral committee, has assisted me through multiple administrative procedures as the former department chair and also recommended me for a fellowship. He gave me an opportunity to collaborate with his experimental group, which was a very rewarding experience. Thanks to him, I am able to graduate in one piece. I am also grateful to the rest of my committee, professors Jaime Marian, Bruce Dunn, and Louis Bouchard. I would be honored to be recognized worthy of Ph.D by them.

I owe much to Dr. Yi Xia, now at Northwestern University, who was a senior student of the group and helped a newbie in graduate school establish his foundations. He even took me on a hiking trip to Mt. Whitney, which I summited thereby checking oﬀ one of my “American Dreams.” I am fortunate to be friends with him and even call him a collaborator. The work on CoSi, for which Yi was the leading author, is a part of this dissertation. Younghak Kwon of the UCLA math department has also helped me out with some math along the way.

My gratitude extends to my experimental collaborators on synthesis and characterization of black-AsP alloys (Dr. Vincent Gambin of Northrop Grumann and Dr. Eric Young now at Lam Research) and InAs nanowire growth (Dr. Hyunseok Kim, now at MIT). Working with experimentalists is a big interest of mine, and these projects were both interesting and educational for me. I hope I was as good of a co-worker as they were to me.

It has been a sincere pleasure for me to have spent many years with my long-time group- mate friends: Jiatong Chen, Yusheng Kuo, Daniel Eth, and Brad Magnetta. From California

xvii to Connecticut, from serious research discussions to singing karaoke at my bachelor party (which they threw for me), it has been a fun ride. I will miss them, wish them the best of luck, and hope our paths cross again. Seriously, I couldn’t have asked for a better company.

For where I am now, I cannot forget where I began. My eyes opened to scientific research during my undergraduate years at Washington University in St. Louis. I thank my undergraduate research mentors, professors Lan Yang, Pratim Biswas, and S¸ahin Ozdemir¨ (now at Penn State), for guiding my entry into the world of research and shaping my first ever publication. Pivotally, professor Cynthia Lo (now at Ford) was the first one to introduce me to the joy and wonders of ab initio calculations and the electronic structure, for which I now have so much love and awe. Without them, I would not be filing my doctoral thesis today.

Neither would I be where I am today, nor maybe even who and what I am today, if not for some of my high school teachers at The Northwest School in Seattle. Glen Sterr, my senior philosophy teacher, introduced me to Zen and the Art of Motorcycle Maintenance, which led me through a transformation of a lifetime. Renee Frederickson made life itself just as inspirational as she made chemistry, by simply being who she is. I feel compelled to mention that her signature phrase, “go look it up!” sounds much like research in retrospect. I haven’t quite gotten around to biking across the U.S. as Glen and Renee did, but I can say I have biked across a few states. In turn, math used to be merely dry symbols and formulae for me until I met Tony Kim and Jim Harmon (now both Ph.D.’s). They have made math so natural, relatable, and real – ergo, beautiful.

To my mom and dad, Ye Ok Oh and No Young Park, without your care and love for me ever since 1990, none of this would have been possible. My coming to the U.S. would not have happened if not for your support and sacriﬁce. Thank you for everything: putting up with me through my maturing days and especially for having always been great friends.

Finally, to my wife and dearest Seungwon Chung, since you came into my life, you’ve become the reason I am, the reason I do any of it all. Nothing in life would be worth as much without you. I love you thi∼∼∼∼∼∼∼∼∼∼s much, and let us set out hand-in-hand into the future. “May the stars shine upon the end of your road!” - The Lord of the Rings.

xviii VITA

2014–2019 Ph.D. Researcher Quantum Prediction of Advanced Materials, UCLA First-principles study of high-performance thermoelectrics Vibrational Raman spectra of black-AsP alloys

2017–2018 Visiting Assistant in Research Energy Sciences Institute, Yale University Department of Applied Physics, Yale University First-principles study of high-performance thermoelectrics

Fall 2017 Teaching Fellow Department of Applied Physics, Yale University ENAS 151 - Multivariable Calculus for Engineers

Fall 2016 Reader Department of Materials Science and Engineering, UCLA MSE 203 - Thermodynamics of Materials

Winter 2016 Teaching Assistant Department of Materials Science and Engineering, UCLA MSE 104 - Science of Engineering Materials

2013–2014 Undergraduate Researcher Micro/Nanophotonics Laboratory, Washington University in St. Louis

Fabrication and optical characterization of on-chip TiO2 whispering gallery microresonators

2014 B.S. in Chemical Engineering, Washington University in St. Louis

xix PUBLICATIONS

J. Park, Y. Xia, and V. Ozoli¸nˇs. High thermoelectric power factor and eﬃciency from a highly dispersive band in Ba2BiAu. Phys. Rev. Appl. 11, 014058 (2019).

J. Park, Y. Xia, and V. Ozoli¸nˇs. First-principles conﬁrmation of the Seebeck coeﬃcient and assessment of aanostructuring in mineral thermoelectric CuFeS2. J. Appl. Phys. (under review).

Y. Xia, J. Park, F. Zhou, and V. Ozoli¸nˇs. High thermoelectric power factor in intermetallic CoSi arising from energy ﬁltering of electrons by phonon scattering. Phys. Rev. Appl. 11, 024017 (2019).

E. Young, J. Park, T. Bai, C. Choi, R. DeBlock, M. Lange, S. Poust, J. Tice, C. Cheung, B. Dunn, M. Goorsky, V. Ozoli¸nˇs,D. Streit, and V. Gambin. Wafer-scale black arsenic-phosphorus thin ﬁlm synthesis validated with density functional perturbation theory predictions. ACS Appl. Nano Mater. 1(9) 4737-4745 (2018).

J. Park, S. K. Ozdemir, F. Moniﬁ, T. Chadha, S. H. Huang, P. Biswas, and L. Yang. Titanium dioxide whispering gallery microcavities. Adv. Opt. Mater. 2(8), 711-717 (2014).

xx CHAPTER 1

Introduction

This introductory chapter outlines the history of thermoelectricity and its fundamental physical principles. Afterwards, the chapter reviews recent research eﬀorts on improving thermoelectric material eﬃciency and the challenges therein, culminating with the statement of research goals of this doctoral work.

1.1 Thermoelectricity

1.1.1 The Thermoelectric Eﬀect

The thermoelectric effect dates back to the 19th century. It was first discovered in 1821, when Thomas Seebeck observed that compass needle is deflected near two distinct materials joined at two junctions, forming a closed loop, when one junction is heated up to be hotter than the other. The deflection was due to electrical current that was driven by the temperature gradient, which since then has been known as Seebeck effect. Not long after Seebeck, Jean Peltier in 1831 discovered that, under a similar experimental setting, electrical current driven through one junction induced heat flow such that temperature gradient is established, with the said junction becoming hotter and the other cooler. This Peltier effect and the predating Seebeck effect, some twenty years later, were shown to be mechanistically one and the same by Lord Kelvin. Kelvin theorized that heat flux and electrical current are coupled as are temperature and electrochemical potential gradients. In early twentieth century, works of Lars Onsager [Ons31a, Ons31b] and Herbert Callen [Cal48] completed an integrative theoretical description of interdependencies between forces and fluxes of charge

1 a b

Figure 1.1: a) Thermoelectric refrigeration induced by applied external voltage. Heat-carrying electrons ﬂow in the direction enforced by the voltage and create temperature gradient. b) Thermoelectric current generation from applied temperature gradient. Charge carriers are excited at the hot end and ﬂow to the cool end. carriers, heat, and entropy within non-equilibrium thermodynamics.

Thermoelectricity nowadays has risen to the status of a promising renewable energy technology that converts heat into electricity (by the Seebeck eﬀect) as well as a cooling technology (by the Peltier eﬀect). Schematics of the two processes as adopted in the industry are shown in Fig. 1.1. While one junction is kept hot and the other cold, electrons (in the n-type materials) and holes (in the p-type material) are thermally excited and conduct to the cool side where there are empty high-energy states. The charge carriers then move through the cool junction and annihilate each other, completing a one-way charge transport throughout the closed circuit. This form of energy generation occurs completely within solid-state, as

2 opposed to bulkier heat engines that require moving ﬂuids, pipes, and turbines. Therefore, it should be realizable in both small and large scales and subject to little environmental or design restriction.

The key to developing thermoelectricity into a commercially viable technology is high overall heat-to-electricity conversion efficiency of a thermoelectric module. This is determined fundamentally by the conversion efficiency in each of the n-type and p-type materials that compose the module. A formal description and understanding of this material efficiency and related material properties calls for a review of the Onsager-Callen theory.

1.1.2 The Onsager-Callen Theory

−→ −→ The total energy flux (JE) through a solid can be written as a sum of heat flux (JQ) and −→ −→ −→ −→ the energy flux related to charge carrier flux (JN ): JE = JQ + µeJN , where µe is the electrochemical potential. The associated driving forces for charge carrier flux and total energy

−→ −→ µe −→ −→ 1 ﬂux are F N = ∇(− T ) and F E = ∇( T ), respectively. Assuming linear coupling of the two ﬂuxes, they can be written down as:       −→ −→ µe JN LNN LNE ∇(− T )   =     (1.1) −→ −→ 1 JE LEN LEE ∇( T )

The 2×2 matrix of L’s is a conductivity matrix, whose oﬀ-diagonal elements represent “coupled” conductivities. In 1931, assuming microscopic reversibility of the irreversible processes, Lars Onsager derived general reciprocal relations which in this speciﬁc case leads to

LNE = LEN [Ons31a, Ons31b], a conjecture that previously had been held by Lord Kelvin. This was the important breakthrough from which all subsequent derivations for thermoelectricity grew.

Decoupling electrochemical and temperature gradients, and introducing heat ﬂux instead of the total energy ﬂux, one may re-write the above linear system of equations as:       −→ 1 −→ JN L11 L12 − T ∇(µe)   =     (1.2) −→ 1 −→ JQ L21 L22 − T 2 ∇(T )

3 Onsager’s reciprocal relations translates to L12 = L21, and the following relations arise:

L11 = LNN

L12 = LNE − µeLNN (1.3)

2 L22 = LEE − µe(LNE + LEN ) + µeLNN

Enabled by the Onsager’s relation, Herbert Callen proceeded to relate the conductivity matrix elements to electrical and thermal conductivities through Ohm’s law and Fourier’s

−→ 1 −→ −→ −→ law. Recognizing the electric ﬁeld as E = − e ∇(µe) and charge ﬂux as JC = eJN , the isothermal electrical conductivity can be written as

−→ 2 JC e σ = −→ = L11, (1.4) E T where e is the fundamental charge. thermal conductivity associated with electrons may be obtained under two conditions, under zero electrochemical potential gradient (κE) and zero charge current (κe): L22 κE = 2 T (1.5) L22L11 − L12L21 κe = 2 L11T −→ −→ −→ −→ Now, the off-diagonal elements control the coupling of JC to ∇(T ) and JQ to E , in other words, they control thermoelectricity. Hence, what I would like to call the “thermoelectric conductivity,” or the conductivity arising from temperature gradients, is defined as, −→ JC e ζ = −→ = − L12, (1.6) −∇(T ) T 2 such that the definition of the Seebeck coefficient becomes minus the ratio of the two gradients: −→ −1 E 1 L12 α = σ ζ = −→ = (1.7) −∇(T ) eT L11 Eq. 1.7 makes it clear that the Seebeck coefficient of a material determines the magnitude of electrical voltage that is generated by a given temperature gradient across the material, and is thus a direct measure of thermoelectric potential. This role is even more clearly seen

4 when Eq. 1.2 is rewritten in terms of the newly defined variables: −→    −→  JC σ ζ E −→ =    −→  . (1.8) JQ T ζ κE −∇(T ) If α = 0, the matrix reduces to its diagonal elements, completely decoupling the two fluxes. Therefore, one may say that α lies at the core of a material’s thermoelectric conversion capability. On another side, Onsager’s microscopic reversibility assumption leads to the following expression for entropy flux: −→ 1 −→ −→ J = J + αJ . (1.9) S T Q C The two contributors are thermal entropy carried by heat and electrochemical entropy carried by charge carriers. Herein, α carries the meaning of entropy (α) per charge carrier. −→ On thermal conductivity side, recall that κe was obtained under JN = 0. This indicates

that κe is of purely conductive origin. In other words, it is thermal conductivity of a material −→ in the open-circuit setting. In turn, recall that κE was obtained under ∇(µe) = 0, which does −→ not forbid carrier ﬂow due to ∇(T ). Therefore, κE is thermal “conductivity” of a material in the short-circuit setting when there is no external voltage but temperature gradient built

across the material induces the Seebeck eﬀect. Thus, it is natural to think κE to be the sum of two heat transport contributions, one of conductive origin and the other of convective origin. Indeed, by Eq. 1.7,

2 κE = κe + α σT, (1.10)

where the second term accounts for the convective contribution.

The transport properties defined so far allow for a formal description of thermoelectric efficiency. A simple approach to thermoelectric efficiency is to consider basic circuits theory. Current generation is maximized when the open-circuit voltage is as large as possible while the short-circuit current is the largest possible for a given open-circuit voltage. Maintenance of a large open-circuit voltage requires minimum dissipation. By analogy, these conditions

translate to minimum possible κe and maximum possible κE for a given κe. Their ratio κ α2σT E = + 1 (1.11) κe κe

5 2 states that maximizing α σ while minimizing κe directly results in maximizing thermoelectric current. The first term therefore serves as a figure of merit. Formulation of the ultimate thermoelectric figure of merit, zT , is complete when thermal conductivity by lattice vibration

(phonons) is taken into account and κl is added to κe, yielding

α2σ zT = T. (1.12) κe + κl

Apart from the sole environmental parameter T , all of the ingredients of zT are inherent material properties. A noteworthy property of zT is that it has no theoretical upper limit. The following relation dictates that as zT grows to inﬁnity, the actual conversion eﬃciency,

η, approaches the Carnot eﬃciency, ηC:

Therefore, central to promoting thermoelectricity as a viable clean energy technology is achieving high zT in both p-type and n-type materials. In colloquial terms, Eq. 1.12 loosely translates to: “as high a voltage as possible should develop from a temperature-gradient (α) and the generated charge must conduct as well as possible (σ) while the gradient is maintained via minimal heat dissipation (κ).” A desirable thermoelectric material would feature high α and σ, while exhibiting low κ. These properties may be individually or collectively tuned to our advantage via doping, alloying, engineering microstructures, and varying temperature or pressure [SCK09, ST08, ZLF17]. Once high zT values in materials are achieved, thermoelectricity would be able to make a meaningful contribution to energy recovery potentially wherever heat source or temperature gradient is present.

1.1.3 Challenges of the Figure of Merit zT

Most of research eﬀorts on thermoelectric materials essentially have one ultimate goal in mind: improve zT . Despite much research in the past several decades, however, many of the issues regarding zT improvement have not yet been resolved to a desirable extent. As illustrated in Fig. 1.2 [MLZ18], zT of 3 is still an unbroken barrier, but several issues surface upon a closer look.

One of the issues aﬀecting current thermoelectric materials is that the highest values of zT

6 Figure 1.2: History of zT improvement in some of the highest-performing thermoelectric materials to date (left) [HT17] and zT values of state-of-the-art low-temperature thermoelectrics (right) [MLZ18]. are typically attained in chalcogenides composed of heavy elements [ZLZ14, ZTH16, CWH18, HLG04, WZZ14, HCY18, VSC01, PHM08, KLM15, HWZ15, LZL11]. This trend reflects the tendency of heavy elements to incur soft phonons and low κl, but many heavy elements are either toxic, rare, or expensive. On the other hand, the efficiencies of materials composed of cheap and naturally abundant elements are usually too low for practical applications. This owes to the fact that abundant elements tend to be light in mass, thus hosting stiff phonons and exhibiting high κl. Recent identification of the sulfosalt mineral tetrahedrite [LMX13, LMX15] as an efficient yet economical thermoelectric (p-type) has been a major breakthrough in this regard, but there can never be too many such discoveries.

Another issue is that many of the highest zT values reported up to date have mostly arisen at high temperatures. Cu2Se recorded 2.6 at 850 K or higher [OMS17, ZZL14]. SnSe [ZLZ14] recorded zT ∼ 2.6 at 923 K, alloys of PbTe [HLG04, WZZ14] zT > 2 between 700 ∼ 900 K, and multiple-ﬁlled skutterudites [SYS11] zT = 1.7 at 850 K. Alloys of GeTe [HCY18] recorded zT > 2 at somewhat lower 600 K and above. However, when it comes to low temperatures, say, T ≤ 300 K, the list of competent materials thins very quickly. At room temperature and below, alloys of Bi2Te3 [VSC01, PHM08, KLM15, HWZ15] have re- mained the only materials with zT > 1 for a very long time. This is unfortunate, since many areas of industry including but not limited to refrigeration, spacecraft propulsion, and scal-

7 Figure 1.3: Natural abundance of elements occurring in earth’s crust (image taken from Wikipedia). able everyday heat recovery are poised to tremendously beneﬁt from eﬃcient thermoelectric conversion at room-to-cryogenic temperatures [Bel08, TS06, YC06, Xia15].

The dearth of efficient thermoelectrics, especially at low temperatures, can easily be understood by revisiting the definition of zT . Eq. 1.12 clearly states that a desirable thermoelectric material would feature high α, high σ, but low κ. This brings us to the core problem in thermoelectrics: such a propitious combination is inherently difficult to achieve [ST08, SCK09, ZZG16, Xia15, ZLF17, MLZ18]. Firstly, at low temperatures, zT is bound to suffer from small T . Secondly, regardless of material, high κl is nearly unavoidable at low temperatures. The only way to overcome these handicaps would be for materials to exhibit very high PF, but high PF itself involves further complications. σ is favored by high symmetry of crystal lattice, but so is κ. Even worse, in semiconductors, the PF suffers from the fact that σ and α behave counteractively with respect to doping, as illustrated by Fig.

8 Figure 1.4: The general behaviors thermoelectric transport properties with respect to doping [ST08].

1.4. In metals, the continuous electronic density of states across the Fermi level allow bipolar transport, leading often to negligible α and PF.

1.2 Research Eﬀorts on zT Improvement

The relatively decoupled nature of phonons and electrons have led to the concept widely known as the “phonon-glass, electron-crystal (PGEC)” concept of thermoelectrics. PGEC concisely illustrates that lattice thermal conductivity can be tuned independently (to a certain extent) of electronic transport properties (electrical conductivity and the Seebeck coef- ﬁcient), and vice versa. As a result, a large portion of success in thermoelectrics research

9 until recently has been on lowering of lattice thermal conductivity. Multiple engineering techniques and intrinsic mechanisms in materials that scatter phonons have been identiﬁed and realized. These principles and/or their combinations have signiﬁcantly lowered lattice thermal conductivity, sometimes down to the amorphous limit, and have pioneered break- throughs beyond zT > 2. However, these achievements have not entailed improvements in the power factor, speaking of which, the attempts to systematically design for high power factor has not been as successful.

It is a general consensus of thermoelectrics research community that further improvements in zT must arise from improvements in the power factor. All things considered, identifying systematic principles for improving the PF (while keeping thermal conductivity low) is critical – and this is one of the focal points of this doctoral work. In particular, the importance of high PF outweighs that of low thermal conductivity when heat source is unlimited and/or free, which is often the case in the real world due to the plethora of waste heat and [LKJ16]. Moreover, for the development of next-generation thermoelectrics, it is essential that high PF and zT originate from mechanisms that are intrinsic to a given material, rooted in its electronic structure. Only when the best possible material for intrinsic reasons are identiﬁed could other interventionist engineering techniques provide further improvements. Intrinsic mechanisms do not only tend to be less material-speciﬁc and easier to generalize, but also more amenable to computational studies

This section reviews recent developments in the reduction of lattice thermal conductivity and in the power factor improvement. Theories and ideas regarding the latter are examined in closer details to provide a smooth transition into the statement of research goals and purpose.

1.2.1 Reducing Lattice Thermal Conductivity

Many techniques that engineer low κl have been developed and implemented with success in recent times. Hierarchical micro/nanostructuring introduces grain-boundary scattering [VSM10, LLZ10, MDR09, BHB09, PHM08]. Insertion of rattling impurities induce strong an-

10 harmonicity and phonon-phonon scattering [SMW96, Nol96, Nol00, MMH98, SYS11, TCS15, SL12, NCS98, CNF99]. Synthesis of superlattices exploit the layered nature of the structures, which hinder phonon transport over long distances [DCT07, VSC01]. Introduction of various defects could also enhance phonon scattering and reduce lattice thermal conductivity [KLM15]. Of course, alloying materials with foreign atoms to create solid-solutions is a common route in which phonon scattering via mass-disorder and/or strained lattices is enhanced. Upon appropriate choice of method depending on the material, these interventionist methods have proven to be highly useful approaches to suppress phonon transport. How- ever, artiﬁcially introduced phonon scattering mechanisms are prone scattering electrons also (though not always), which would harm electron/hole mobility and potentially the power factor.

Because phonon-phonon scattering is the only phonon scattering channel that is in-

trinsic to a given compound, all intrinsic physical mechanisms that lead to low κl in bulk materials involve strong phonon anharmonicity one way or another. Strong anharmonicity is known to arise in the presence of easily polarized lone-pair electrons [NOH13, MJH08, SM11, WW05, WPE11], resonant bonding [LEL14, LHM16], or intrinsic rattling (vibrationally independent) atoms [HAX16]. Also, highly complex crystal structures of low symmetry [LMX13, LMX15, GOS05, KBS07, TCB08, TMS10, SMS17, ZHB14, LLZ12, LZZ16, STL15]

are generally associated with low κl, since it is more diﬃcult for atoms to vibrate harmon- ically. Many of the compounds with strong intrinsic anharmonicity have also been noted of their soft phonon modes that even near dynamic instability or a phase-transition boundary [LMX16, SBP16]. Intrinsic mechanisms are extremely appealing in that they do the job without external intervention and that they are much less likely to harm electronic transport.

The importance of successful reduction of κl cannot be overstated. All of the highest- performing thermoelectrics up to date, such as SnSe and Bi2Te3, possess very low κl in common. The next-generation thermoelectric materials will also indispensably exhibit very

low κl. The principles summarized above will continue to play important role as the eﬀort to improving zT beyond 3 continues. To achieve this goal, however, the other half of zT ,

11 namely the power factor, must follow suit.

1.2.2 Enhancing the Power Factor

Approaches to high PF have been relatively less successful. The main reasons for the compli- cation of achieving high PF are that, it is composed of not one but two transport properties to be tuned, α and σ, and their behaviors are usually at competition with each other. An optimal compromise between the two quantities is necessary but challenging. Also, to a large degree, electronic transport properties are tied to the electronic structure, which is more or less a ﬁxed, quantum mechanical property of a given compound and less amenable to con- trolled manipulation than, for instance, phonons. That said, it is very instructive to review a seminal theoretical work on optimal electronic structures and understand its implications – and more importantly its limits in real contexts, which is a part of this doctoral work.

In 1996, Mahan and Sofo made an attempt to generally determine the optimal electronic structures for thermoelectricity [MS96]. In doing so, they treated the problem as a mathematical optimization problem, and looked to maximize the expression for zT with respect to the functional form of the energy-dependent conductivity,

Σ(E) = N(E)v2(E)τ(E). (1.13) where N(E) is the electronic density of states (eDOS), v2(E) is squared group velocity, and τ(E) is electron lifetimes, all as functions of energy (E). Then, the electrical conductivity, the thermoelectric conductivity, and electronic thermal conductivity are obtained by taking the integral of Σ(E), modulated by the energy-derivative of the Fermi-Dirac distribution f(E), over all energies: Z ∂f σ = e2 Σ(E) − dE, (1.14) ∂E e Z ∂f ζ = Σ(E)(E − E) − dE, (1.15) T F ∂E 1 Z ∂f κ = Σ(E)(E − E)2 − dE. (1.16) E T F ∂E

12 Substituting these expression into Eq. 1.12 and after some mathematical transformations, Mahan and Sofo have determined that a Dirac-delta function is the ideal functional form of Σ(E),

Σ(E) = Σ0δE,E† , (1.17)

where Σ0 is an arbitrary prefactor.

To visualize the beneficial effect of such a narrow Σ(E), first consider a different form of the definition of zT : ζ2/σ zT = T. (1.18) κe + κl Interestingly from this point of view, σ should be low rather than high, and instead ζ should be high. As opposed to Eq. 1.12, Eq. 1.18 loosely states “electric current should arise as much as possible from a temperature-gradient (ζ) and as little as possible from some external

ﬁeld (σ), while the gradient is maintained via minimal heat dissipation (κ).” Ignoring κl for ∂f now, consider the fact that σ integrates Σ(E) scaled by − ∂E , while ζ and κE integrate ∂f 2 ∂f Σ(E) respectively scaled by (EF −E) − ∂E and (EF −E) − ∂E . Then for the “electronic- part zT ” to be high, the functional form of Σ(E) must be such that it weighs the red-shaded region in Fig. 1.5 as much as possible and the blue-and-green-shaded regions as little as possible. Because the regions overlap considerably in energy, for this to be realized, the narrow medium-energy domain (3) must couple to high Σ(E), whereas the other domains to its sides couple to low Σ(E). In particular, it is essential to keep low Σ(E) in domain (1) and prevent minority carriers from negating ζ and hence α, not to mention their further

contributions to σ and κE. For these conditions to be met, it clearly seems that Σ(E) with a narrow distribution peaking in domain (3) would be beneﬁcial.

For Eq. 1.17 to hold, at least one of the three components of Σ(E) must be a delta function of energy. N(E) and v2(E) are directly determined by the electronic structure, whereas the behavior of τ(E) is less trivial to conceive because it depends on the types of scattering processes and only indirectly derives from the electronic structure. Because N(E) is the most accessible quantity among the three to manipulate, the relation in Eq. 1.17 has led to much experimental eﬀorts to utilize materials with and/or engineer a sharply increasing

13 (1) (2) (3) (4) Low Low High Low EF

Figure 1.5: A plot of the distributions that determine the magnitudes of σ, ζ, and κE, with respect to energy. The two dashed vertical lines qualitatively divide the energy-domain into four domains where Σ(E) ought to be high or low in order to maximally benefit the n-type “electronic-part zT ”. Domain (1) is the hole-regime. Domain (2) is where low-energy electrons live which contribute more to σ than ζ. Domain (3) is where the medium-energy electrons live and where the relative benefit to ζ is the largest. Domain (4) is populated by high-energy electrons that contribute mostly to κE. Domain (3) ought to be weighted with high Σ(E) for the best thermoelectric benefit.

14 eDOS near the Fermi level in the general vicinity of domain (3). Thermoelectric materials whose merits owe large eDOS at the band edge include CuFeS2 BLAHBLAHBLAH? Among the band engineering efforts to introduce a sharp increase in the eDOS is the so-called “band alignment” technique. It seeks to tune doping mechanisms and temperature to align pocket energies at the band edge in order to simultaneously benefit σ and α [PXA11, PWS12]. If the band dispersions can be made to remain the same during the energy-alignment process, carrier concentration can be enhanced without damaging mobility. Also when the bands are perfectly aligned in energy, the Seebeck coefficient suffers minimally from the multi-pocket effect, Np 1 X α = α σ , (1.19) N p p p p=1 since all pockets (p) deliver similar σp as weights, where Np is the total number of pockets.

Just like eDOS, the lifetime could influence Σ(E) enough to trigger a large PF. Long overall τ(E) clearly increases σ via Eq. 1.14, but microscopic variations τ(E) can be such that it benefits α and the overall PF – if low-energy electrons in domain (2) in Fig. 1.5 are carefully scattered away. Then, whereas ζ and σ may individually decrease, their ratio α and the PF can significantly increase. Such preferential scattering of low-energy electrons (or holes) has a commonly used name “electron (hole) filtering,” and could be a useful scheme for enhancing the PF. An instructive formula to refer to when understanding α specifically is the Mott formula. When the Fermi level crosses a band (in metals and degenerate semiconductors) and when the Sommerfeld expansion is valid, it is derived as π2k2 T d ln Σ(E) α = − B 3e dE E=EF (1.20) π2k2 T N 0(E) τ 0(E) hv2(E)i0 = − B + + . 3e N(E) τ(E) hv2(E)i E=EF

A large gradient across EF of N(E) or τ(E) may all contribute to high α. Strong electron- ﬁltering can result in a large τ 0(E).

A way of harnessing the filtering effect is by engineering resonant levels near the Fermi level, which offer additional states to participate in scattering of low-energy carriers [HWC12,

HJT08]. The prime example is the Kondo resonance scattering in metallic YbAl3 that has

15 led to the highest ever PF in any bulk materials [RKK02]. For materials with Dirac bands, scattering away electrons with long mean free path (MFP) assures ﬁltering out low-energy electrons due to the monotonically decreasing MFP with energy [LZL18], similar to the

nanostructuring concept for phonons and κl. While occasionally useful, these engineering methods are very sensitive to the compound under consideration, and for many materials may not even be possible to realize. Therefore, a more general approach towards electron ﬁltering and high PF is desirable.

1.3 Research Statement

Considerations of both recent developments in thermoelectrics and future directions of the ﬁeld strongly suggest that principles (preferably intrinsic) for a systematic improvement in the power factor is the next leap to take. Since the power factor is intimately tied to the electronic structure, it is of critical importance to determine optimal electronic structures for high power factor and high thermoelectric performance. In light of these, this doctoral study is comprised of two main research goals, one on the material side and the other on the more general, physical side.

1) Identify and study bulk materials that combine low lattice thermal conductivity and high power factor en route to high zT .

2) Generalize optimal electronic structures for high power factor such that further discoveries of high-performance thermoelectrics may be accelerated.

Accurate characterizations of intrinsic electron and phonon scattering processes are necessary steps to successful characterizations of thermoelectric performances. To achieve this, the state-of-the-art computational methods on solid quantum mechanical theories are employed.

16 CHAPTER 2

Computational Theories & Methods

Density-functional theory (DFT) [KS65] is the workhorse for all electronic structure calculations that appear in this work. DFT routines have been implemented in many software packages, such as Vienna Ab initio Simulations Package (VASP) [KH93, KH94, KF96a, KF96b], Quantum Espresso (QE) [GBB09, GAB17], and Abinit [GAA09, GRV05, GBC02]. The- oretical study of any property of any system requires knowledge of its ground-state electronic (and crystal) structure, which these software packages provide with speed and to a good accuracy. Since DFT is well-documented and very widely used, detailed explanations are not given here. Yet, it is worth going over two main issues regarding plain DFT using local-density approximation (LDA) or generalized-gradient approximation (GGA): 1) its underestimation of the band gap due to the self-interaction error [Per85, PZ81]; 2) its misrepresentation of strongly correlated electrons due to the single-particle picture that theory is based on. These issues are amenable to improvement via hybrid-exchange functionals [Bec88, LYP88, HSE03, HSE04], the so-called DFT+U method [DBS98], the GW method [Hed65, SK06, SK07, SMK07, JT10], and a multitude of the more recently developed exchange-correlation functionals [BJ06, TB09, SRP15]. This work employs appropriate methods of improvement for each material under study.

Similar to DFT, density-functional perturbation theory (DFPT) [Gon95b, Gon95a, BGC01, GL97] has also become a standard method for characterizing a system’s linear responses – harmonic phonons, dielectric tensors, Born eﬀective charges – to external perturbations – electric ﬁeld, lattice vibrations. Therefore, a detailed review of DFPT will also be omit- ted, though this work relies heavily on DFPT and its implementations in the said software packages.

17 Instead, this chapter focuses on theories that are more directly related to thermoelectrics and material transport properties. Boltzmann transport formalism is the basis of electron and phonon transport formulations, leading to computationally tractable expressions for conductivities and the Seebeck coeﬃcient. A large portion of the chapter is dedicated to the methods for calculating electron and phonon lifetimes (inverse scattering rates). Methods for quantum-mechanically treating electron-phonon scattering, ionized impurity scattering, phonon-phonon scattering, and alloy mass-diﬀerence scattering processes are explained in detail. Electron-phonon scattering and phonon-phonon scattering are of particular importance, as they are intrinsic and usually the dominant scattering mechanisms in thermoelectric materials for electrons and phonons, respectively.

For the record, throughout the work the rigid band assumption is made – that the elec- tronicw states as calculated by DFT at 0 K are fixed and the same for all temperatures. This is not a correct picture in the strict sense, since both can change with temperature, sometimes significantly. First of all, at finite temperature materials experience thermal expansion due to anharmonic phonons, which would slightly shift the band structure. Then, electron-phonon interaction that intensifies with temperature also changes the electronic eigenenergies as well [GLC10, APC14]. For instance, phonon-perturbed electronic states have been revealed to be critical in the reproduction the experimental band gap of diamond [GLC10]. However in this work, since the lifetimes of the said phonon-perturbed electronic states are of interest, and the materials considered here have much heavier atomic composition than the likes of diamond (carbon), the rigid band assumption should suffice.

2.1 Transport of Electrons & Phonons

Introduced in the first chapter, the exact physical definitions of the conductivities associated with electron transport within the Onsager-Callen theory reflect their physical meanings and origins. However, they are not the most computationally tractable expressions. Com- putationally useful forms of expressions can be derived within the Boltzmann transport formalism. The resulting relations constitute arguably the most widely adopted method for

18 calculating the said transport properties ab initio. Phonon transport and lattice thermal conductivity are also formulated within Boltzmann transport formalism.

2.1.1 Boltzmann Transport – Electrons

The general form of Boltzmann transport equation is ∂f −→ ∂f −→ ∂f ∂f = v −→ + F −→ + (2.1) ∂t ∂ r diff ∂ p ext ∂t scat where f is Fermi-Dirac distribution of electron population. At steady-state the equation would be zero. The right-hand side identifies three mechanisms by which f changes. The first term on the right-hand side is change in f due to diffusion, and the second term is change in f due to external force. These two terms are well described by the variables position −→ (−→r ), velocity (−→v ), momentum (−→p ), and the force exerted ( F ). However, scattering is a quantum mechanical process by nature, and hence the third term depends on the scattering mechanisms being considered. Thus, it cannot be formulated in a simple manner by classical mechanical considerations alone. A common practice that avoids these difficulties is to assume that the momentum relaxation time due to scattering (τ) is the same for electrons of all eigenenergies (E), crystal momenta (k), and the bands they belong to. This is known as the relaxation time approximation (RTA). It leads to a simple form of the scattering term [JM85]: ∂f f − f df = ∆ = , (2.2) ∂t scat τ τ which essentially states that the deviation of the electron occupation distribution due to scattering (df) from the equilibrium distribution (f) lasts for a lifetime τ. −→ −→ If temperature gradient ∇(T ) is present as well as ∇(µe), df must be expanded to the first order in both [JM85]. It is convenient work with Eq. 1.2 and the gradients of electrochemical

19 potential and temperature. −→ df = τ−→v · ∇(f) −→ −→ ∂f −→ ∂f = τ v · ∇(EF ) + ∇(T ) ∂EF ∂T (2.3) −→ ! −→ ∇(T ) ∂f = τ−→v · e E + (E − E ) − . T F ∂E

Now, assuming no temperature gradient and by identifying the external force as electric ﬁeld, the diﬀusion term can be deemed negligible and isothermal σ can be calculated. At steady-state, both sides of Eq. 2.1 equal to 0. The electron distribution is then [JM85]

−→ ∂f df = eτ−→v · E − . (2.4) ∂E

∂f The term − ∂E peaks at the Fermi level and sharply decreases away from it, limiting that transport to electrons near the Fermi level. This aligns with the intuitive sense as there is both enough electrons and enough empty states for them to move into near the Fermi level.

∂f In fact, − ∂E = kBT f(1 − f), which is maximized when the occupancy is 0.5 - at the Fermi level.

−→ −→ 1 ∂Eν Each carrier contributes e v νk to the whole current, where v νk = ¯h ∂k is the group velocity of electrons with crystal momentum k and band index ν, and whose Cartesian

elements are denoted as vνki. Lifetimes also depend on k and ν, and becomes τνk. Then, the total charge current requires summation over all electrons in all bands and of all k in the ﬁrst Brillouin zone that participate in transport [JM85]:

−→ X Z −→ ∂f J = ev eτ E v − dk. (2.5) C νki νk νkj ∂E ν BZ The indices i and j refer to the Cartesian directions of the two velocities, respectively, due to which the conductivity naturally takes the form of a 3 × 3 tensor. Collecting all of the terms with ν and k indices into Σνk and rearranging Eq. 2.5 one obtains what is essentially Ohm’s law, −→ −→ X Z ∂f J = E e2 Σ − dk, (2.6) C νk,ij ∂E ν BZ

20 and the electrical conductivity is identiﬁed as

X Z ∂f σ = e2 Σ − dk. (2.7) ij νk,ij ∂E ν BZ Converting the integral over BZ into a sum, the conductivity becomes

e2 X ∂f σij = (τvivj)νk − , (2.8) ΩNk ∂E νk E=Eνk

where Ω is the unit cell volume and Nk is the number of k-points included in the sum. Note that the energy-dependent conductivity in Eq. 1.13 can be recovered by projecting Σνk,ij onto the electronic density of states (eDOS). This procedure is equivalent to counting all ν and k that exist within inﬁnitesimally thin constant energy surfaces [JM85, MS06]:

1 X Σ(E) = Σνkδ(E − Eνk). (2.9) Nk νk

The procedure for obtaining σij can be analogously applied to the other transport properties. If only temperature gradient is present and no electric ﬁeld, using the second term on the right-hand side of Eq. 2.3, the coupling term ζ in Eq. 1.8 is constructed:

e X Z ∂f ζij = − Σνk,ij(E − EF ) − dk T BZ ∂E ν (2.10) e X ∂f = − (τvivj)νk(E − EF) − , ΩTNk ∂E νk E=Eνk

This coupling term depends on T and EF , just as σij(T,EF ) is. It is positive for holes

(E < EF ) and negative for electrons (E > EF ). Then the Seebeck coeﬃcient is also a tensor

dependent on T and EF : −1 αij = ζij · σij (2.11)

which adopts the sign of ζij. Note that the eﬀects of group velocity and lifetime are more or less cancelled for α. In fact, under the so-called constant relaxation time approximation

(CRTA), τ = τ0 and is constant, and so τ exactly cancels for α. Of course, this wouldn’t be the case if τ were energy-dependent and hence must be placed inside the integral for inte-

gration. Moving onto thermal energy carried by electrons, κE,ij can similarly be formulated

21 from energy ﬂux relation:

1 X Z ∂f κ = Σ (E − E )2 − dk E,ij T νk,ij F ∂E ν BZ (2.12) 1 X 2 ∂f = (τvivj)νk(Eνk − EF) − . ΩTNk ∂E νk E=Eνk

−14 Eq. 2.8, 2.10 and 2.12 are implemented in BoltzTraP but with constant τ0 = 10 s [MS06].

These equations allow calculation of σ, α, κE and κe at any T and EF . It is a trivial matter of

ﬁnding κe,ij through Eq. 1.11. The EF modulation simulates the extent of doping. Finally, accurate calculations of σ, ζ, α, and κe requires a very dense sampling of the Brillouinz zone. Therefore, a self-consistent DFT calculation at a very dense k-point mesh is essential.

2.1.2 Boltzmann Transport – Phonons

With σ, α, and κe obtained, what remains to be calculated for the full description of zT is κl. This can also be treated within the Boltzmann transport formalism in close analogy. In the presence temperature gradient but no electric field, Eq. 2.1 reduces to the diffusion term and the scattering term. The diffusion term can be expressed with phonon distribution expanded to the first order in T :

∂b −→ ∂b = −∇(T ) · −→v (2.13) ∂t diﬀ ∂T

−→ ∂ω where b is the Bose-Einstein distribution of phonon population and v = ∂q is phonon mode group velocity (not to be confused with carrier group velocity, and ω stands for phonon frequency while q is phonon wavevector). Eq. 2.13 merely states that phonons are carried diffusively down the temperature gradient. Similarly to the charge current flux for electrons, the heat flux due to phonons is expressible as the following sum over phonon modes (indicated by λ), −→ X Z −→ ∂b J = − v τ ∇(T )v dq, (2.14) C λqi λq λqj ∂T λ BZ

22 which is essentially Fourier’s law. Then similarly to σ for electrons, κl for phonons is expressed as X Z ∂b κl = (τvivj)λq dq BZ ∂T λ (2.15) 1 X ∂b = (τvivj)λq . ΩNq ∂T λq ω=ωλq

2.1.3 The Lifetime Problem and Scattering

All ingredients of transport properties except for τ are either directly obtainable from the band structure (or phonon dispersion), which are for all intents and purposes static. How- ever, τ is the result of dynamic scattering processes, and for an accurate theoretical study of transport properties of materials, the quantum mechanical nature of scattering processes must be taken into account. The bottleneck is that quantum mechanical calculation of scattering events involving many bands (and many phonon modes if they are involved) at a dense enough k-point mesh required for accurately describing transport phenomena is very computationally cumbersome. Before introducing the state-of-the-art technique for execut- ing these calculations in the following section, it is worthwhile to go over a few approximate approaches to τ.

A rudimentary way to avoid such a heavy lifting is by resorting to the CRTA. It is still occasionally used in research for quick and qualitative characterization of electron scattering processes, but the problem is that it often times fails to capture even qualitative trends in σ and α. Even less plausible is the CRTA for calculating reasonable values. After all,

−14 τ0 = 10 s used by the default-mode BoltzTraP is groundless for most real materials. When experimental data are available, τ is often scaled by the ratio of experimental conductivities and CRTA conductivities. While the fitting procedure is a definite improvement over the CRTA, it is still data-dependent and ultimately not first-principles. In addition, because the fitted τ is still constant and has no dependence on band, k, or energy, this procedure fails to correctly capture the Seebeck coefficient behavior.

A more reasonable approach is to scale lifetimes by the inverse of the eDOS (τ ∝ eDOS−1),

23 by which the energy-dependence of τ is approximately taken into account. The validity of this approximation hinges on the fact that phase spaces for electron scattering are often proportional to the eDOS, and it is especially valid when acoustic phonon scattering process is dominant [WWL17]. For example, this inverse-eDOS scaling has correctly reproduced the anomalous positive Seebeck coeﬃcients measured in Li [XV14]. When valid, it is a vast improvement over the CRTA that requires essentially equal computation time. As a matter of fact, this method is employed successfully, in combination of experimental ﬁtting, in the study of CuFeS2 – to be discussed in the next chapter.

2.2 Scattering of Electrons

As the last section punctuates, eﬃcient and accurate characterization of electron scattering events is the major concern in theoretical studies of not just thermoelectrics but electron transport behaviors in general. In thermoelectric materials, arguably the most important and almost always the dominant scattering process is electron-phonon (e-ph) scattering, including those by the deformation-potential mechanism and by long-ranged polar-optical interactions. In semiconductor thermoelectric materials, ionized impurity scattering could play some role due to the typically high doping requirements. This section introduces the state-of-the-art method for explicitly calculating electron-phonon scattering matrix elements based on perturbation theory and Wannier interpolation: the primary method of choice for studies of thermoelectric compounds in this thesis. The Brooks-Herring theory for ionized impurity scattering is discussed afterwards.

2.2.1 Electron-Phonon Scattering: Part 1

Band-and-momentum-dependent carrier scattering rates and lifetimes associated with e-ph interaction (τeph,νk) are essential for accurate description of transport phenomena. These quantities will be obtained directly by calculating their scattering matrix elements at various momenta and energy near the Fermi level. Because direct computation of e-ph scattering

24 matrix elements at a given k-point requires rather heavy computational load, the scheme is to compute them directly only on a coarse mesh in the BZ ﬁrst. Then to sample a denser mesh, interpolation using Wannier functions [GCL07, NGM10, PMV16, Giu17] will be used. This computational approach for electron-phonon scattering rates and lifetimes adopted in this work derives from the general theory of electron-phonon interaction and phonon-renormalized electronic states developed by Allen, Heine, and Cardona – thereafter known as the Allen- Heine-Cardona (AHC) theory. It is a second-order perturbation theory that treats harmonic phonon perturbation and is hence compatible with DFPT. While anharmonic phonons in theory in theory perturbs electrons as well as harmonic phonons, only the perturbation by harmonic phonons within the AHC theory is considered since it is the predominant characteristic of phonons.

a In terms of atomic displacements ui of atom a in the direction i, the total energy of a lattice in equilibrium is expanded as

1 X 1 X E = E + Φabuaub + Φabcuaubuc + ··· . (2.16) 0 2! ij i j 3! ijk i j k ab,ij ijk,abc

The first-order term does not appear because at the equilibrium atomic configuration, net force between all atoms vanish. Cutting off Eq. 2.16 at the second order describes the energy

1 P ab a b as 2! ij,ab Φij ui uj, where 2 ab ∂ E Φij = a b (2.17) ∂ui ∂uj are the second-order interatomic force constants (IFCs), or harmonic IFCs, while all higher- order terms reﬂect anharmonicity. Interatomic forces on all atoms can be explicitly calculated by DFT using the Hellmann-Feynman theorem: * + −→ ∂E ∂Hˆ

F = − = − ψ ψ . (2.18) ∂u ∂u Diﬀerentiating Eq. 2.18 once more and keeping notations for bands and momentum, the second-order derivative and the corresponding shift in electronic eigenenergy for band ν and

25 momentum k due to the perturbation of phonon mode λ and momentum q are: 2 ∂ Eνk ab ab a b = Dνk,ij + Fνk,ij ∂ui ∂uj * + " * + ! # (2.19) ∂2H 1 ∂ψ ∂H = ψ ψ + νk ψ + (ai ↔ bj) + c.c. νk a b νk a b νk ∂uqi∂uqj 2 ∂uqi ∂uqj DW SE ∂Eνk ∂Eνk ∂Eνk 1 X 1 a† b ab ab = + = √ eλqieλqj Dνk,ij + Fνk,ij (2.20) ∂nλq ∂nλq ∂nλq 2ωλq M M ab,ij a b The double-sided arrow indicates the previous term with the indices switched, and c.c. stands for complex conjugation of the previous term. Clearly, the second derivative of electronic energy depends on not only both the ﬁrst and second derivative of the Hamiltonian, but also the ﬁrst derivative of unperturbed electronic states, obtained by DFT or more accurately by

ab DFT+GW. The first term on the right-hand side, Dνk,ij, is called the Debye-Waller (DW) ab term. The second term involving the first-order modifications only, Fνk,ij, is called the self-energy (SE) term, or sometimes Fan term.

For the scope of this work, let us focus on the SE contribution only. Notice the bra-ket of the SE term in Eq. 2.19 that includes a ﬁrst-order derivative of the ground-state electronic wavefunction ψ with respect to atomic displacement. Then by the ﬁrst-order perturbation theory, this term can be formulated in terms of only ψ as follows [PAG14, PGJ15, GLC10]:

D ∂H ED ∂H E * + ψ ψ 0 ψ 0 ψ ∂ψ ∂H νk ∂ua ν k+q ν k+q ∂ub νk νk X −qi qj a b ψνk = ∂u ∂u Eνk − Eν0k+q qi qj ν0 (2.21) 2 X |gν0νλ(k, q)| = Eνk − Eν0k+q ν0 * s + h¯ ∂V X iq·Rl al 0 0 gν νλ(k, q) = ψν k+q e al eλqj ψνk (2.22) 2Maωλq ∂u ajl qj where gν0νλ(k, q) are the electron-phonon interaction matrix elements at the level of harmonic phonons. Index l runs over unit cells within the supercell on which the calculation is done. The operator ∂λqV is change in the self-consistent potential with respect to atomic displacements and takes screening into account and consists of contributions from all atoms. The ﬁrst-order approximation of this is a direct product of a DFPT linear response calculation. Hence, gν0νλ(k, q) can also be calculated in approximation within the DFPT scheme.

26 In order to collect contributions from all e-ph scattering events to SE, one needs to add terms from phonon absorption processes in which electrons are excited: Eνk +hω ¯ λq − Eν0k+q; and phonon emission processes in which electrons are grounded: Eνk −hω¯ λq −Eν0k+q. Each term must be scaled by appropriate carrier and phonon state occupations. Finally, a summation

ab 0 of Fν,ij over q, λ, and ν results in the ﬁnal electron self-energy correction Ξ dependent on ν and k [NGM10]:

Ξνk = 1 X 2 b(ωλq,T ) + f(Eν0k+q,EF,T ) b(ωλq,T ) + 1 − f(Eν0k+q,EF,T ) |gν0νλ(k, q)| + Nq Eνk +hω ¯ λq − Eν0k+q − iη Eνk − hω¯ λq − Eν0k+q − iη ν0λq (2.23) Note that the factor −iη has been added to the denominator in order to broaden the energy range allowed to scattering events. This is a reasonable step because in a real-life material, electronic eigenenergies and phonon frequencies are incessantly ﬂuctuating within a given range, while in Eq. 2.23 only the unperturbed ground-state E and ω are used to enforce energy conservation. Also, −iη creates a Lorentzian lineshape function, which collapses to a widthless line upon taking the η → 0 limit. Therefore the denominator may be replaced with a representative delta function on the numerator that conserves energy. This self- energy correction due to phonon perturbation can then be decomposed into a real part and an imaginary part, Ξ = ΞRe + iΞIm [GCL07, NGM10, GLC10, SK15], which are respectively

ΞRe = π X 2 (b(ωλq,T ) + f(Eν0k+q,EF,T )) δ(Eνk +hω ¯ λq − Eν0k+q) |gν0νλ(k, q)| dE Nq Eνk +hω ¯ λq − Eν0k+q dE (2.24) ν0λq (b(ω ,T ) + 1 − f(E 0 ,E ,T )) δ(E − hω¯ − E 0 ) + λq ν k+q F νk λq ν k+q dE Eνk − hω¯ λq − Eν0k+q dE and

ΞIm = π X 2 δ(Eνk +hω ¯ λq − Eν0k+q) |gν0νλ(k, q)| (b(ωλq,T ) + f(Eν0k+q,EF,T )) dE Nq dE (2.25) ν0λq δ(Eνk − hω¯ λq − Eν0k+q) +(b(ω ,T ) + 1 − f(E 0 ,E ,T )) dE . λq ν k+q F dE

27 Let us ﬁrst examine what the two terms have in common. The factor π arises from Lorentzian approximation. Note that, if phonon energy is much smaller than smaller than electronic

transition energy (¯hωλq << Eνk − Eν0k+q), the delta function essentially serves as the eDOS projection. This is the adiabatic approximation. Approximating the delta function with DFT-calculated the eDOS could be a valid and eﬃcient procedure for semiconducting materials with dispersive pockets in their band structures near the Fermi level. Electrons occupying

these pockets interact predominantly with phonons of small ωλq and q, e.g., acoustic modes around the Γ-point, due to energy and momentum conservation. Therefore, the adiabatic approximation is reasonable. However, the ﬂatter the band, the worse this approximation,

because ﬂat bands are allowed interactions with phonons with wider ranges of ωλq and q

(have larger phase space) in which case ωλq in the delta functions cannot be ignored.

Now, what each of these mean physically must be clariﬁed. ΞRe is the actual modiﬁcation of electronic energy due to phonon perturbation, but only the SE contribution to it. Going back to Eq. 2.19, one sees that the complete description of the second derivative of the total energy involves the Debye-Waller contribution as well. However, these terms constitute the energy shift of electronic states due to phonon-perturbation which, as mentioned in the opening of the section, is neglected for the scope of this study. Instead, the major interest lies with the lifetimes of those phonon-perturbed states. This is where the imaginary part comes into play. ΞIm is the linewidth (the full-width) of the self-energy, and determines the time-decay of phonon-perturbed electronic state. This fact becomes more obvious by considering the following time-evolution operator:

− i (Ξ +iΞ )t − i Ξ t 1 Ξ t e h¯ Re Im = e h¯ Re e h¯ Im . (2.26)

Eq. 2.26 demands that phonon-perturbed electronic states have characteristic inverse life-

times (decay rates) proportional to ΞIm/h¯:

2 τ −1 = Ξ . (2.27) eph,νk h¯ Im

The factor of 2 arises from the fact that lifetime is related only to the half-width of SE, on the positive side of the time-axis.

28 Long-ranged polar-optical scattering rate is treated and added separately. The matrix element for this is derived from solving the Poisson’s equation with a Coulombic potential: s L X 1 X (q + G) · Z · eaλq i(q+G)·r 0 gν0νλ(k, q) = i ∞ hψν k+q| e |ψνki , (2.28) 2Maωλq (q + G) · · (q + G) a G6=−q

∞ where is the high-frequency dielectric constant, G is the reciprocal lattice vector and Ma is the mass of atom a involved in the polar-optical interaction.

It is worth emphasizing that, Eqs. and 2.25 and 2.27 calculate the lifetime of a perturbed state – does not calculate the electron relaxation time during transport strictly speaking. In order to obtain the electron transport relaxation time, one must re-scale Eq. 2.25 by projecting the crystal momenta of perturbed electron states, i.e., phonon-scattered electrons, onto the intended transport direction, namely

˜ ΞIm = π X 2 δ(Eνk +hω ¯ λq − Eν0k+q) |gν0νλ(k, q)| (b(ωλq,T ) + f(Eν0k+q,EF,T )) dE Nq dE ν0λq −→ −→ ! δ(Eνk − hω¯ λq − Eν0k+q) v νk · v ν0k+q +(b(ωλq,T ) + 1 − f(Eν0k+q,EF,T )) dE 1 − −→ −→ . dE v νk v ν0k+q (2.29) −→ If the perturbed-state velocity ( v ν0k+q) lies in the same direction as the unperturbed-state −→ velocity ( v ν0k, assumed to be the direction of transport), then the dot product is 1 and there is no momentum relaxation to be spoken of. This makes intuitive sense, since if all electrons are scattered towards the direction of intended transport, this is not really scattering from the transport point of view, and conductivity would not suﬀer at all. In reality, electrons are scattered in all directions due to perturbation. In fact, in neglecting the velocity treatment, Eq. 2.27 inadvertently assumes isotropic random scattering. The −→ −→ more v ν0k+q aligns in direction with v νk, the longer the transport relaxation time. To be clear, this velocity-projection treatment is ignored in the studies here for two reasons. First, it can only increase predicted τeph. Ergo, it could lead to overestimation of σ and the PF, which is to be avoided. Second, it is not known to make a large diﬀerence in τeph anyway for thermoelectrics purposes. Therefore, the lifetimes that enter the Boltzmann integrals

29 appearing in Eq. 2.8∼2.12 are all “state” lifetimes deﬁned with Eq. 2.25, not Eq. 2.29.

Of course, e-ph scattering aﬀects phonon transport as well as it does electron transport. In a manner similar to that for electrons, the imaginary part of self-energy due to e-ph interaction can be calculated as follows,

ΠIm = (2.30) 2π X 2 δ(Eνk +hω ¯ λq − Eν0k+q) |gν0νλ(k, q)| (f(Eνk, µ, T )) + f(Eν0k+q, µ, T )) . Nk dE ν0λk and the corresponding inverse lifetime of phonons

2 τ −1 = Π . (2.31) phe,λq h¯ Im

However, for thermoelectric materials, lattice thermal conductivity is typically limited by phonon-phonon scattering if not by scattering processes associated with alloys. Since the eﬀect on κl should be minimal, e-ph scattering aﬀects phonon transport is not taken into account in this work.

2.2.2 Electron-Phonon Scattering: Part 2

For all of the theory development, one problem still remains from the computational stand- point. The central piece to all theories and approaches leading up to τeph,νk is the e-ph matrix elements gν0νλ(k, q). The major issue regarding the implementation of these theories is computational load. Computing gν0νλ(k, q) for a high densities of q-mesh and k-mesh imposes a heavy computational load. DFPT calculation for one q-point is comparable in load to self-consistent electronic energy minimizations [NGM10]. How is then τeph,νk computable over many points in the Brillouin zone given the huge computational cost to calculate each, and when an accurate description of (τeph,νk) requires high-density sampling of gν0νλ(k, q) in the Brillouin zone as gν0νλ(k, q) may vary rapidly with q and k? A reasonable approach is to calculate gν0νλ(k, q) at select irreducible q-points and interpolate them over the rest of the Brillouin zone. For this purpose, Wannier interpolation using the so-called maximally localized Wannier functions (MLWF) is employed.

30 Before Wannier interpolation is discussed, a brief introduction to MLWF is in order. A Wannier function is an object in the real space that lives in a unit cell within a supercell.

The number of unit cells in the supercell is the same as the number of k-points (Nk) used to calculate Bloch functions from which Wannier functions are constructed. Wannier functions are deﬁned as the Fourier transforms of Bloch functions, vice versa:

X −ik·Re wν(r − Re) = wνRe = e ψν(k) k (2.32) 1 X ik·Re ψν(k) = e wνRe Nk Re

where Re is the position vector for electronic Wannier function pointing at a given unit cell

from the reference unit cell 0e in the supercell, and r is the Wannier function coordinate. When an electronic band is isolated from all other bands, such a Wannier function is uniquely obtained up to an arbitrary phase factor of the parent Bloch function. However, for a composite set of bands, a set of generalized Wannier functions is more appropriate to deﬁne. These objects are created by taking into account the freedom of Bloch bands to mix, which is the composite version of the freedom of arbitrary phase factors of individual Bloch functions [MV97]:

X −ik·Re 0 0 Wν(r − Re) = WνRe = e Uν νkψν (k) 0 ν k (2.33) 1 X ik·Re † 0 ψν(k) = e Uνν0kWν Re Nk 0 ν Re 0 where Uν0b is a unitary transformation matrix that mixes bands ν and b, and as a result,

Wν0 no longer corresponds to a single Bloch state as Wν does. This unitary transformation

preserves the center (rν = hrib) of Wν in the unit cell in which it resides, but it does not preserve the spread of Wannier function in the unit cell. The deﬁnition of the total spread of all Wannier functions is [MV97]

X 2 2 χ = ( r ν − rν) (2.34) ν MLWF are obtained by minimizing the total spread, i.e., maximizing localization, with

respect to choices of unitary transformers Uν0bk. The globally maximally localized Wν are

real, whereas locally maximally localized Wν are complex. However, even complex Wν serve

31 our intent to use them for interpolation well as long as they are localized enough. Entangled bands can also be treated after disentanglement pre-processing steps [SMV01]. Much deeper details on the Marzari-Vanderbilt algorithm and its signiﬁcance can be found elsewhere [MV97, SMV01]. Calculation of MLWF will be done using the wannier90 package [MYL08].

Next up is Wannier interpolation, which is implemented in the EPW package [GCL07, NGM10, PMV16], which is compatible with DFT and DFPT outputs of QE. Wannier interpolation starts out by transforming electronic eigenstates, phonon eigenvectors, and

gν0νλ(k, q) in the reciprocal space (the Bloch representation) to the real space (the Wannier representation). The electronic Hamiltonian in the Wannier representation is

D ˆ E H 0 = WνR Hk W 0 Re,Re e νRe 0 (2.35) X −ik·(Re−Re) † = e UkHkUk k

0 It may be assumed that Re = 0e, residing in the reference unit cell. Then Eq. 2.35 can be back-transformed to the Bloch representation to be interpolated at as many arbitrary k0 as one desires: ! 0 1 X ik ·Re † 0 0 Hk = Uk e H0eRe Uk0 (2.36) Nk Re In essence, this procedure is a Fourier interpolation followed by diagonalization. How to

determine Uk0 may not be immediately obvious. Certainly, these cannot be determined during the process of obtaining MLWF because electronic eigenstates and eigenvalues are not known at an arbitrary k0 for which self-consistent DFT calculation has not been done.

However, it must be that Uk0 must diagonalize the quantity inside the bracket in Eq. 2.36.

In other words, it is not that a known Uk0 diagonalizes the Fourier transformed Hamiltonian,

but Uk0 is the product of its diagonalization. Thus obtained Uk0 can then be used to ﬁnd

ψν(k) by Eq. 2.32.

A similar procedure applies to harmonic phonons as well. For them, the central eigenvalue problem in the reciprocal space is

2 Dqeq = ωqeq (2.37)

32 where D(q) is the dynamical matrix at q deﬁned as the Fourier transform of the 3N × 3N

ab matrix of the second-order IFCs Φij . The dynamical matrix elements are thus

Φab,Rph ab X ij iq·Rph Dq,ij = √ e (2.38) MaMb Rph

Rph is the position vector for phonon Wannier function pointing at a given unit cell from the

reference unit cell 0ph in the supercell, and N is the total number of atoms in the supercell. In a similar manner as Eq. 2.35, transforming Eq. 2.38 to the Wannier representation,

0 X −iq(Rph−Rph) † D 0 = e eqDqeq (2.39) Rph,Rph q

0 Since the vector diﬀerence Rph − Rph is what matters, it can again be assumed that 0 Rph = 0ph, the reference unit cell. Then Eq. 2.39 can be back-transformed to the Bloch representation to be interpolated at as many arbitrary q0 as one desires:   1 0 † X iq Rph 0 0 Dq = eq0  e D0ph,Rph  eq (2.40) Nq Rph

Again, eq0 is not known from DFPT, but since it must diagonalize the quantity inside the bracket in Eq. 2.40, it can be obtained by diagonalizing it.

0 At this point, both Uk0 for electrons and eq0 for phonons are known at desired k -mesh q0-mesh much denser than the original k and q-mesh for which self-consistent calculations were run. These allow Wannier interpolation of the e-ph matrix elements at k0 and q0:

1 X −i(k·Re+q·Rph) † † gν0νλ(Re, Rph) = e Uk+qgν0νλ(k, q)Ukeq (2.41) Nq k,q   1 0 0 0 0 X i(k ·Re+q ·Rph) † gν0νλ(k , q ) = Uk0+q0  e gν0νλ(Re, Rph) Uk0 eq0 (2.42) Nk Re,Rph “Minimized” χ does not directly reveal whether Wannier functions are good enough for accurate interpolation, as one may not know whether it has converged to the global minimum or even a good enough local minimum. The MLWF minimization process is very sensitive to the initial guess of orbitals, and often times would not converge to a good enough local

33 Figure 2.1: A schematic of the electron and phonon Wannier functions entering the e-ph matrix elements [GCL07]. The square lattice sketches unit cells, the red curves the electron Wannier functions, and the blue curve the phonon perturbation in the Wannier representation. If any one is spatially separated from the others, gν0νλ(Re, Rph) vanishes. minimum. Then the quality of Wannier interpolation can be then checked in a few ways. First, Wannier-interpolated band structure must exactly reproduce the DFT band structures.

This should be a sign that Eq. 2.35 and 2.36 were performed accurately, meaning that Uk is good enough. Second is spatial decay of gν0νλ(Re, Rph). For well localized Wanniers, the matrix elements must decay quickly. As illustrated by Fig. 2.1, gν0νλ(Re, Rph) vanishes if either Re or Rph is suﬃciently far away from the reference unit cell. The quality of Wannier interpolation can be checked by plotting Wannier-interpolated band structure and comparing it to DFT band structures. Excellent agreements appear when all of the Wannier functions are strongly localized.

With gν0νλ(k, q) calculated at a denser mesh of k and q through Wannier interpolation,

τeph,νk can be calculated at many more k-points. This leads to a more accurate treatment of band-and-k-dependent lifetimes of carrier transport. Also, this allows calculation of ΣRe at

34 Figure 2.2: a) The interpolated electronic band structure, b) the phonon dispersion, and c) the interaction matrix elements of boron-doped diamond using Wannier interpolation on EPW [GLC10].

many more k-points. which serves to renormalize electronic eigenenergies as will be discussed in the next subsection. Eqs. 2.22, 2.25, 2.27, 2.30, 2.31, and Eq. 2.35∼2.42 are all implemented in the EPW package. The current version of EPW only supports norm-conserving pseudopotentials. Though accurate and powerful, implementation of EPW computation is a costly process. When the number of Wannier functions required to guarantee strong localization exceeds 30, or for system sizes beyond ten atoms per unit cell, Wannier interpolation may not become very practical. As previously hinted at, scaling τeph,νk with the inverse of the eDOS (τ(E) ∝ N −1(E)) can yield transport properties quickly and relatively accurately especially when acoustic phonon scattering dominates. Such a method will be used at discretion.

2.2.3 Ionized Impurity Scattering

Metallic thermoelectrics need little to no doping because enhancement of carrier concentration is not required. Semiconductor thermoelectrics on the other hand are typically heavily doped, and scattering by ionized dopants could inﬂuence carrier lifetimes and mobilities.

35 A standard method for treating ionized impurity scattering is the Brooks-Herring theory [Bro55, CQ81]. Just as phonon-perturbation of interatomic potentials is the scattering potential for electron-phonon scattering, a Coulombic potential is the scattering potential for ionized impurity scattering. Particularly, a screened Coulombic potential represents a more accurate scattering potential, Ze V (r) = e−βsr, (2.43) 4πr which is also sometimes known as the Yukawa potential. Z is the eﬀective charge of the impurity (assumed to be 1 in this work), is the static dielectric constant of the host material, and r is the distance (of electrons) from the impurity center. The inverse Thomas-Fermi screening distance is given by

v ∞ −0.5 u 2 R y ! u ne Γ(1.5) y+η dy β = 0 1−e , (2.44) s t ∞ y0.5 kBT R Γ(0.5) 0 1+ey−η dy

where n is the carrier concentration around the impurity, η = EF is the reduced Fermi kBT energy, and Γ(x) is the Gamma function. Calculation of the matrix elements is simpliﬁed under the ﬁrst-order Born approximation, which essentially assumes that scattering is elastic: the outgoing scattered wavefunction and the incident wavefunction are identically plane

0 waves only with diﬀerent momenta such that ψin = exp(ik · r) and ψout = exp(ik · r). This approximation is also known as weak-potential approximation since the wavefunction is not altered due to collision. Then the Fermi’s golden rule for an isotropic parabolic band leads to the following approximate form of carrier lifetime limited by ionized impurity scattering, √ 2 1 3 −1 16 2π m 2 E 2 β τ = log(1 + β) − , (2.45) ii NZ2e4 1 + β

where m is the eﬀective mass, N is the concentration of ionized impurities, and

4k2 8mE β = = , (2.46) 2 2 2 βs h¯ βs where the second equality holds for a parabolic band. Clearly, the higher the N the heavier

the ionized impurity scattering and the shorter the τii. The screening distance increases with

growing , to the improvement in τii and µii.

36 The standard Brooks-Herring theory as represented by Eqs. 2.44∼2.46 has several lim- itations for applications to real materials and bands. One is the isotropic band assumption characterized by single scalar m, whereas real bands are very often anisotropic to varying degrees. A form of non-parabolic correction was made by Rode [RK71], under which the screening term is modiﬁed as follows, √ 2 1 3 16 2π m 2 E 2 τ = (B log(1 + β) − B )−1, (2.47) ii NZ2e4 1 2

where the newly introduced coeﬃcients are 2β2c2 3β4c4 sB = 1 + s + s , 1 k2 4k4 4 2 2 4 4 (2.48) 2 2 2 2 (3βs +6βs k −8k )c 4k + 8(βs + 2k )c + k2 B2 = 2 2 . βs + 4k Eq. 2.45 for an isotropic parabolic band is a special case of Eqs. 2.47 and 2.48 where c = 0

4k2 β such that B1 = 1 and B2 = 2 2 = . Eqs. 2.47 and 2.48 with the non-parabolic βs +4k 1+β correction is implemented in the aMoBT package [FIL15].

2.3 Scattering of Phonons

There are many potential sources of phonon scattering, such as defect/impurity scattering, grain-boundary scattering, alloy scattering via mass-disorder or straining of lattice, electron- phonon scattering, and phonon phonon scattering. Among these, only the last two processes are intrinsic to a given single-crystal compound. Between the two, phonon-phonon scattering arising from anharmonicity is almost always the dominant scattering mechanism. Hence, correct characterizations of lattice anharmonicity and phonon-phonon scattering rates are integral to study of lattice thermal conductivity. The lack of eﬃcient method for calculating lattice anharmonicity had been a major bottleneck for theoretical studies of lattice thermal conductivity. In this section, after a short prelude, the compressive sensing lattice dynamics (CSLD) technique developed in 2014 [ZNX14] is introduced as the state-of-the-art method for rapidly and accurately calculating anharmonic interatomic force constants (IFCs). Then the perturbative method for formulating phonon-phonon scattering matrix elements and

37 Boltzmann transport formalism for lattice thermal conductivity are discussed. A method for treating mass-disorder scattering is also brieﬂy discussed in alloy contexts.

2.3.1 Lattice Anharmonicity

abc Calculation of the third-order IFCs (Φijk ) in Eq. 2.16 for a periodic crystal has been a daunting computational task. Traditionally, the brute force method to calculate IFCs has been the finite difference method. This method require displacement of all atoms in a supercell in equilibrium in various directions. In the frozen-phonon method, atoms are displaced as stipulated by each specific phonon mode being simulated, effectively “freezing” the mode in time. This must be done for all modes at signified wavelengths. In the finite displacement method, a set of supercells corresponding to a set of irreducible displacements is systematically generated. DFT provides interatomic forces for each supercell via Eq. 2.18. From the displacement and force data, IFCs can be extracted by solving the following linear equation curtailed at the second or the third order:

F = −AΦ     ua,1 1 ua,1ub,1 ... Φab,1 ... Φab,L i 2 i j ij ij (2.49)  . . .    = −  . . .   abc,1 abc,L  . . .  Φijk ... Φijk      a,L 1 a,L b,L . . . ui 2 ui uj ...... where A has a total of (3N)ML rows for L supercells, M unit cells per supercell, and N atoms

ab per unit cell. The finite displacement method is effective for extracting Φij , and can calculate abc Φijk as well. Its problem lies with the hundreds or even thousands of displaced supercells that abc 3 2 abc abc are needed for calculating Φijk . There are (3M) N Φijk . Each Φijk , to be approximated by two second-order derivatives each approximated by two first-order derivatives, requires force calculation on four supercell of different displacement configurations. Since one supercell calculation outputs (3M)N forces, as many as 4(3M)2N supercells must be calculated. Of course this number can be reduced by taking advantage of crystal symmetry and applying cutoff radius beyond which atomic clusters are deemed non-interacting. Even then, the number of calculations needed is astounding.

38 Recently, a quick and robust method of calculating IFCs up to as many orders as desired was developed. This new, revolutionary method, called compressive sensing lattice dynamics (CSLD) [ZNX14], uses the technique it is named after and solves Eq. 2.49 to extract IFCs of many orders in one shot. Compressive sensing has recently been applied to cluster expansion [NHZ13, NOR13] and sped up the calculation by a large extent. It also conceived new theoretical objects such as compressed Wannier modes [OLC13], which are compactly supported version of the traditional Wannier functions, whose derivation does not require pre-calculated Bloch functions. Theoretically, there is no limit to the number of columns that the displacement matrix A in Eq. 2.49 may have, whereas it must have ﬁnite rows. The equation in its true form is thus an underdetermined problem. CSLD reformulates the equation as a convex optimization problem by adding an L1 norm: µ Φ = argmin kΦk + cs kF + AΦk2 (2.50) cs Φ 1 2 2

The first term on the right-hand side is the L1 norm and the second term is the familiar least-squares term, or the L2 norm. µcs is the lone parameter that controls the tradeoff between the L1 and L2 norms, with higher µcs The L1 norm drives the solution to that having the least number of non-zero solutions, i.e., the simplest solution. Implementation of this minimization requires force calculations on only a handful of supercells, in each of which atoms are randomly displaced up to typically 0.06 A.˚ Cutoff radii are applied for pairs, triplets, quadruplets, and on. For the computational techniques employed in this thesis, fitting up to the third order is enough, meaning only triplets are required. Each of these clusters are symmetrized. Afterwards, the split-Bregman algorithm [GO09] executes minimization and IFCs can be extracted. For its accuracy and superior computational efficiency, CSLD will be the major tool by which IFCs are calculated.

Another method to be utilized alongside CSLD is DFPT, which is a purely analytical method, so it does not involve force calculations on multiple supercells. It calculates second- order IFCs by adopting atomic displacements as perturbations to Kohn-Sham self-consistent Hamiltonian from DFT. In a similar manner, by taking homogeneous electric ﬁeld as the perturbation, dielectric permittivity tensors and Born eﬀective charge tensors on atoms can

39 be computed. The former are related to the second derivate of total energy with respect to electric field. The latter are related to the second derivative of the total energy once with respect to atomic displacement and once with respect to electric field. Dielectric tensors and Born effective charges become necessary constituents in resolving splitting of longitudinal and transverse optical modes at the long-wavelength limit.

2.3.2 Phonon-phonon Scattering

The discussion of phonon-phonon scattering is limited to the three-phonon (third-order) process because it occurs the most often and is sufficient for accurately describing κl in most materials. While four-phonon (fourth-order) scattering may occasionally play a non- negligible physical role, it does not necessarily lead to lower κl, as in PbTe [Xia18]. In a three-phonon process, two phonons may collide and create a third phonon, a.k.a., phonon absorption, or one phonon may split into two phonons, a.k.a. phonon emission. During so called “normal scattering,” momenta of two phonons add up to the third phonon momentum, −→ −→ −→ −→ −→ −→ which remains within the first Brillouin zone: q1 + q2 = q3 and q3 < |G| where G is the reciprocal lattice vector. Hence, as Fig. 2.3a. shows diagrammatically, there is only forward propagation of phonons, and κl is not limited. During Umklapp scattering however, involved −→ −→ −→ phonons lose an overall momentum equivalent to G because q3 > |G| and this requires −→ −→ −→ −→ a translation of q3 back into the first Brillouin zone: q1 + q2 = q3 + G. As seen in Fig.

2.3b., this means that the third phonon with |q3| is reverse propagating, and therefore resists phonon transport. This results in suppression of κl. It is important to note, however, that by creating phonons of progressively higher momenta, which may then participate in Umklapp scattering, normal scattering indirectly does contribute to phonon resistance.

By Fermi’s golden rule, phonon-phonon scattering rates are given by + ∂bλq hπ¯ + 2 bλ0q0 − bλ00q00 = V δ(ωλq − ωλ0q0 − ωλ00q00 ) ∂t 4 ωλqωλ0q0 ωλ00q00 ph (2.51) − ∂bλq hπ¯ − 2 bλ0q + bλ00q00 + 1 = V δ(ωλq − ωλ0q − ωλ00q00 ) ∂t ph 4 ωλqωλ0q0 ωλ00q00 The delta function enforces conservation of energy during scattering, which for computational

40 a b

q1 q2

q 2 q 3 q3’ q3

Figure 2.3: a) Normal scattering. The entire scattering processes is contain within the ﬁrst Brillouin zone. No thermal resistivity is experienced. b) Umklapp scattering. The interaction results in a crystal momentum that escapes the ﬁrst Brillouin zone, which is essentially results in backscattering due to the periodic lattice. The reciprocal lattice translation conserves momentum.

41 purposes is broadened using a Gaussian function. Phonon frequencies (ω) are associated with three phonon modes denoted by λ that participate in a given scattering event. The + sign applies for phonon absorption and the − sign for phonon emission. The centerpiece of this equation is the interaction potential V . In the package ShengBTE [LCK14], the third-order, anharmonic potential is used for V . Then eﬀectively, the third-order potential acts as ﬁrst- order perturbation to the harmonic approximation, which is of the second order. Then one can write the interaction matrix elements as

a b c ± X X abc eλqieλ0q0jeλ00q00k V 0 00 0 00 = Φ √ (2.52) λλ λ qq q ijk M M M abc ijk a b c e are eigendisplacements of three atoms a,b and c that are involved in the process. The i, j and k indices denote the Cartesian directions of motions of each atom, and λ, λ0 and λ00 indicate the phonon modes associated with each atomic motion. The third-order interatomic force constants (IFC) are the electronic energy diﬀerentiated thrice with respect to three atomic displacements: 3 abc ∂ E Φijk = a b c (2.53) ∂ui ∂uj∂uk abc Obtaining Φijk for all atomic triplets and all phonon modes is a formidable computational task. CSLD has dramatically eased this process [ZNX14].

One more scattering mechanism that is implemented in ShengBTE is scattering due to mass disorder of ions that occupy a same site. Though not applicable to pure compounds, mass disorder scattering has an appreciable eﬀect on alloys where diﬀerent atoms occupy a same site. 2 2! ∂bλq πω X X Ma,s = f 1 − ∂t 2 s M¯ mass a s a (2.54) a∗ a 2 eλq · eλ0q0 δ(ωλq − ωλ0q0 ) where the index a runs over atoms in the unit cell and s runs over its substituting species, whose relatives frequencies of occurrence are fs. Substituting Eq. 2.13, Eq. 2.51 and Eq. 2.54 into Eq. 2.1, one can solve for perturbation-driven deviation in phonon distribution dn due to thermal gradient, limited by phonon-phonon scattering and mass disorder scattering. The scattering term renormalize phonon velocity and relaxation time. For each phonon mode

42 λ, the population deviation is

−→ −→ −→ ∂bλ dbλ = τλ( v λ + ∆ λ) · ∇(T ) ∂T (2.55) −→ −→ −→ hω¯ λ = τλ( v λ + ∆ λ) · ∇(T )bλ(bλ + 1) 2 kBT −→ −→ ∆ λ adds to v λ the eﬀect of non-equilibrium distribution of phonon population due to phonon-phonon scattering and mass disorder scattering:

0 −→ 1 X ∂bλq 0 ω −→0 −→0 ∆ λq = τph ( v + ∆ ) Nq ∂t ω λ0q0 mass + + 00 0 1 X ∂bλq 00 ω −→00 −→00 0 ω −→0 −→0 + τph ( v + ∆ ) − τph ( v + ∆ ) (2.56) Nq ∂t ω ω λ0q0λ00q00 ph − − 00 0 1 X ∂bλq 00 ω −→00 −→00 0 ω −→0 −→0 + τph ( v + ∆ ) + τph ( v + ∆ ) 2Nq ∂t ω ω λ0q0λ00q00 ph

+ + − − ! −1 1 X ∂bλq 1 X ∂bλq X ∂bλq τph,λq = + + (2.57) Nq ∂t 2 ∂t ∂t λ0q0λ00q00 ph λ0q0λ00q00 ph λ0q0 mass where Nq is the number of q-points sampled. Similar to the case of carrier transport, each −→ −→ phonon mode contributeshω ¯ λ( v λ + ∆ λ) to the total heat transport. Then, the total heat ﬂux requires summation over all phonons modes and all q-points:

−→ 1 X ∂b −→ JQ = − τphvi(vj + ∆j) ∇(T ) ΩNq ∂T λq λq (2.58) −→ = −κl,ij ∇(T ) where vλi are phonon group velocity vector elements in direction i. The second equality is a restatement of Fourier’s law. Transforming the sum over q into an integral and substituting

∂b for ∂T , the lattice thermal conductivity tensor elements are

2 h¯ X 2 κl,ij = 2 b(b + 1)ω τphvi(vj + ∆j) λq (2.59) kBT ΩNq λq

Eq. 2.13, 2.51, 2.52, 2.54, 2.56, 2.57 and 2.59 are implemented in ShengBTE. The software −→ calculates κl by iteratively solving for ∆ λ and τλ. The purpose of this iteration is to in- corporate the long-term eﬀect of normal scattering processes, which, as mentioned above,

43 serve to create high-q phonons which eventually participate in Umklapp processes and are −→ backscattered. This eﬀect is well captured by the ∆ λ and τλ renormalization process. Iter-

ation ﬁnishes when thus calculated κl,ij reaches convergence by a given threshold. At this

point, one may say the response dbλ to perturbation has converged to its steady-state value. The scattering rates are calculated only at irreducible wedge of q-points within the Brillouin zone.

κl can also be calculated at hypothetically limited mean free path. This would simulate lattice thermal conductivity when small grain sizes (via nanostructuring) limit propagation of phonons with mean free paths larger than the grain sizes via grain boundary scattering. Then, the following lattice thermal conductivity receives contribution only from phonons with mean free paths smaller than the grain sizes:

2 " # h¯ X 2 vivj κ˜l,ij = 2 b(b + 1)ω −→ (2.60) kBT ΩNq v λq λq It must be noted that, though typically very accurate, at very low temperatures where quantum mechanical eﬀects manifest or in the presence of strongly anharmonic phonons, this ﬁrst-order perturbation approximation may fail to accurately capture phonon transport behaviors, leading to incorrect κl.

The key ingredients for calculating κl by the ﬁrst-order perturbation theory are the second and third-order IFCs. The second-order IFCs are necessary for establishing phonon −→ dispersion relation, from which v λ, ωλ and eλ are obtained. The third-order IFCs clearly enter in the interaction potential, from which all scattering rates are calculated.

44 CHAPTER 3

Preliminary Investigations

These preliminary studies on chalcopyrite mineral CuFeS2 and full-Heusler Fe2TiSi were performed with two goals in mind: 1) investigate useful thermoelectrics with environmentally benign and cost-eﬀective elements; 2) clarify what types of band structures lead to high (or low) Seebeck coeﬃcient and power factor. It turns out that neither material would be a

great thermoelectric, though CuFeS2 is the more promising compound amenable to a large

reduction of lattice thermal conductivity via nanostructuring. Fe2TiSi cannot overcome the very high lattice thermal conductivity even with isovalent alloying, not to mention that it as never been synthesized in bulk. While falling short of the first goal, the two compounds offer important insights for a key ingredient for high-performance semiconductor thermoelectrics, namely high mobility. High mobility relieves conductivity’s dependence on carrier concentration, allowing high conductivity to form and cooperate with the high Seebeck coefficient at low doping. Also, the compounds provide clarifications regarding the origin of high Seebeck coefficient in semiconductors.

In CuFeS2, the high n-type Seebeck coeﬃcient arises from a sharp energy-gradient of

group velocities, not the electronic density of states. Fe2TiSi shows that dispersive-only bands are as capable as flat-and-dispersive bands of generating high Seebeck coefficient. In fact, the well-known flat-and-dispersive bands do not offer an intrinsic advantage for the See- beck coefficient relative to the purely dispersive bands, and simply invites more undesirable scattering of the majority carriers by phonons. In all, a band edge characterized by a sharp density of states is not a very important feature for semiconductor thermoelectrics. These are important lessons that lead to the identification of truly high-performing thermoelectrics, composed of purely dispersive pockets at off-symmetry points, discussed in the next chapter.

45 3.1 Chalcopyrite CuFeS2

The contents of this section is under review in Journal of Applied Physics

Chalcopyrite CuFeS2 is a naturally abundant and environmentally benign mineral and a common copper ore. Multiple experimental studies have been performed on this compound as an economical and potentially useful n-type thermoelectric [LZQ14, TMI14, TM13, LTL13, TMT15, AKT17], and maximum zT of 0.33 at T = 700 K has been measured in bulk alloys. Attempts to reduce the grain size down to as low as 20 nm (so-called “nanostructuring”) have resulted in a considerable decrease in lattice thermal conductivity [LTL13, TMT15], but left room for optimizing synthesis as the power factor was also damaged. It is then of interest to know how eﬃcient the mineral could be if nanostructured ideally. Also, recent

computational studies of CuFeS2 have not been able to achieve a desired level of agreement with experimentally measured high Seebeck coeﬃcients [TKS17a, TKS17b], leaving room for clariﬁcation of their origin.

In this study, using the inverse density of states scaling of electron lifetimes, a close agreement with experimental n-type Seebeck coeﬃcients is obtained. The high values of the Seebeck coeﬃcient are attributed to the strong energy-gradient of group velocities, not density of states. Moreover, the failure of the high α to translate to high power factor is

due to low mobility. CuFeS2 serves as an example highlighting the critical role played by mobility in optimizing conductivity and the Seebeck coeﬃcient, and that heavy-mass bands are to be avoided when designing for high power factor. Further, it is determined that ideally

doped and nanostructured n-type CuFeS2 could theoretically achieve 0.25 ≤ zT ≤ 0.8 at 300 K ≤ T ≤ 700 K. The analysis of phonon mode lifetimes and mean free paths indicates that nanostructuring of 20 nm grains is capable of signiﬁcantly reducing bulk lattice thermal conductivity, resulting in more than tripling of zT across a broad temperature range. If realized, CuFeS2 could be a useful thermoelectric for medium temperatures, particularly due to its aﬀordable nature as a mineral. However, if not ideally synthesized, its zT may not be high enough for it to be useful.

46 3.1.1 Crystal, Magnetic, and Electronic Structures

DFT calculations are performed with the Hubbard U correction (DFT+U) using PAW pseudopotentials and the PBE exchange-correlation functional in VASP. As CuFeS2 is known to be an antiferromagnet with a Neel Temperature of 823 K, all calculations were performed un-

der a spin-polarized setting with magnetic moments on the two Fe atoms. Ueﬀ = U − J = 3 eV correction was applied to the strongly correlated Fe 3d orbitals via the formulation by Dudarev et al [DBS98]. The value of 3 eV was chosen because it results in lattice parameters very close to the experimentally determined lattice parameters [ZGY14]. Our choice of DFT+U over more rigorous hybrid-exchange methods stems from the need to densely sample the Brillouin zone when calculating for electronic transport properties, for which hybrid-exchange is prohibitively expensive.

The primitive cell was relaxed with both antiferromagnetic (AFM) and ferromagnetic (FM) initial conditions. The ground state energy of the AFM state is lower than the FM state by 0.553 eV per primitive cell, indicating that AFM is the true ground state conﬁguration. Also, the large energy diﬀerence indicates that thermal disorder of spins is negligible in the temperature range of interest in this work (300∼700 K). Fully relaxed ground state AFM unit cell lattice parameters are a = b = 5.327 A˚ and c = 10.528 A,˚ with c/a = 1.976. These are in close agreement previous theoretical studies as well [KGV14, LNP04].

The AFM conﬁguration, shown in Fig. 3.1a, converged to magnetic moments of ±3.62 µB for the two Fe atoms, while the FM assumption converged to 3.65 µB for both. The Fe magnetic moment in AFM has been measured by a neutron diﬀraction study to be 3.85 µB [DCD58], and has been calculated using the B3LYP hybrid functional to be 3.8-3.89 µB [CAL15, MCM16], both in a close vicinity of our value. The lower and upper Hubbard bands are centered at -5.8 eV and 1.2 eV respectively, as seen Fig. 3.1b, and their widths are respectively 0.52 eV and 0.96 eV.

The calculated electronic band structure is shown in Fig. 3.1c. There are three nearly degenerate pockets near the conduction band minimum (CBM). The pockets at the N-point and the Z-point are identical in energy, while the pocket at the Σ-point is only 0.05 eV

47 a b

c d

Figure 3.1: a) The body-centered tetragonal unit cell of antiferromagnetic CuFeS2 belonging to the I42d space group. Fe atoms are brown, Cu atoms are blue, and S atoms are yellow. The red arrows indicate spin directions. b) The orbital-decomposed density of states of one Fe atom. c) The electronic band structure. The Fermi level (horizontal red line) is at the optimum doping level for zT at 400 K. d) The atom-decomposed electronic density of states. Cu appears nearly monovalent (3d104s0): Cu 3d contributes signiﬁcantly to the upper valence bands but very little to the conduction bands.

48 higher; the dispersion curves are rather ﬂat around these points, indicating heavy eﬀective masses. The calculated band gap is 0.67 eV, in a much closer proximity to the 0.53 ∼ 0.6 eV range measured by optical absorption [AGP56a, AGP56b, TSK74] than is 1.82 eV calculated with B3LYP [CAL15, MCM16].

3.1.2 Electron Transport

Electron transport properties of CuFeS2 are analyzed assuming conventional band transport mechanism. Polaron formation in CuFeS2 is a theoretical possibility, but has little support experimentally. Numerous studies have reported increasing electrical resistivity with temperature [TMI14, TM13, LZQ14, AKT17], indicative of band transport in a degenerate semiconductor as opposed to activated polaron hopping. Regarding scattering processes, it has previously been determined that electron-magnon scattering in Cu1−xFe1+xS2 samples manifests at 200 K and below [AKT15], well below the temperature range of interest in this study. At 300 K and above, scattering by (acoustic) phonons is likely to be the limiting mechanism for electron lifetimes, as indicated by the previously measured µ ∝ T −3/2 dependence of the mobility[LZQ14, AKT17].

Electronic transport properties at temperatures up to 800 K and n-type doping concentrations up to 1022 cm−3 were calculated using the BoltzTraP software [MS06]. BoltzTraP uses the constant relaxation time approximation (CRTA) by default, which fails in materials with strongly energy-dependent electron lifetimes τ(E) and also ignores their temperature dependence. To alleviate the CRTA, the electron lifetimes (τ) are scaled by the inverse of the electronic density of states (eDOS−1). This method of approximating the energy-dependent τ(E) reﬂects the fact that the higher the eDOS, the larger the phase space for scattering of electrons by acoustic phonons, and hence the shorter the lifetimes. While it ignores the scattering matrix elements, the eDOS−1 approximation is expected to work well for materials where the carrier scattering is dominated by acoustic phonons [WWL17, XV14, XPZ19]. In reporting transport properties (σ, α, etc.) it is taken into account that CuFeS2 is tetragonal and hence anisotropic. Because nanostructures are eventually considered where grains are

49 oriented in random directions, all transport properties are space-averaged. Speciﬁcally, the harmonic means of the xyz-components for each transport property are reported, which is the lower Wiener bound for composite materials. The upper Wiener bound given by the arithmetic mean is larger by at most 5% for σ.

−1 CuFeS2 stands out for its high n-type Seebeck coefficient. The τ(E) ∝ eDOS scaling method reproduces experimental α with good accuracy, as seen in Fig. 3.2a. This result is a further testament to the validity of the method and that the effects of the neglected scattering matrix elements would be small. Nevertheless, our calculations slightly underes- timate the experimental Seebeck coefficients. To explain the origin of high α and explain the remaining discrepancies, it is instructive to refer to the well-known Mott formula. It formulates the Seebeck coefficient as a log-derivative of the energy-dependent conductivity (Σ(E) = N(E)τ(E)v2(E)), and is a valid expression for metals and degenerate semiconductors: π2k2 T d ln Σ(E) α = − B 3e dE E=EF (3.1) π2k2 T N 0(E) τ 0(E) hv2(E)i0 = − B + + . 3e N(E) τ(E) hv2(E)i E=EF Here, N(E) is the eDOS and hv2(E)i is squared group velocity. Under the τ(E) ∝ N(E)−1 approximation, the eDOS term and the lifetime term cancel each other, reducing the Mott formula to π2k2 T hv2(E)i0 α = − B (3.2) 3e hv2(E)i E=EF indicating that the Seebeck coefficients under τ(E) ∝ eDOS−1 scaling arises from the energy dependence of the electron group velocities. Under the exact treatment of τ(E), the precise cancellation of the two terms would of course not occur due to the further energy-dependence contributed by the scattering matrix elements. The other extreme (no cancellation) is in fact represented by the CRTA, under which the Mott formula becomes π2k2 T N 0(E) hv2(E)i0 α = − B + . (3.3) 3e N(E) hv2(E)i E=EF Comparing Fig. 3.2a and Eq. 3.2 against Fig. 3.2b and Eq. 3.3 clearly reveals that the contribution of N 0(E) to the magnitude of α is small and actually leads to an overestimation.

50 The results indicate that the origin of the high Seebeck coeﬃcient is primarily the strong energy-dependence of group velocities, not eDOS as commonly thought. Nevertheless, it can be illustrated that the energy-dependence of eDOS is still very much correlated with high α. For a single parabolic band, it turns out

hv2(E)i0 = E−1, (3.4) hv2(E)i

while

N 0(E) 1 = E−1, (3.5) N(E) 2

rendering the two terms identical in nature diﬀering only by a constant factor. In fact, the larger prefactor in Eq. 3.4 indicates that group velocities are capable of imposing much

larger inﬂuence on the Seebeck coeﬃcient than eDOS, as is the case in CuFeS2.

Because the lifetimes scaled by eDOS−1 and hence computed σ could still diﬀer from experimental values by the reference lifetime, the calculated σ is rescaled by its ratio with the experimentally measured value near 1020 cm−3 [see Fig. 3.2c]. The resulting PF, as demonstrated by Fig. 3.2d, agrees with experimental measurements well. The results indicate that that in spite of the high α, the PF is limited because σ is small for all but very high carrier concentrations. The culprit is the low µ of the ﬂat pockets at the CBM, which forces σ to depend on heavy doping, which in turn reduces accessible α. Previous measurements reported higher σ in Cu1−xFe1+xS2 than in Cu1−xZnxFeS2 given x = 0.03 even though the µ was higher in the latter. Comparisons indicate that the higher carrier concentration in the former (3.2 × 1020 cm−3) than in the latter (1.5 × 1020cm−3) [TMI14, TM13] is responsible for this (an Fe atom contributes more conduction electrons than a Zn atom). In spite of the

higher σ, however, Cu1−xFe1+xS2 attained lower PF, indicating that high µ is the priority for optimizing the PF. This is a good example showing that low mobility is highly undesirable for thermoelectrics because it further separates the inherently distant regions of high Seebeck coeﬃcient from those of high electronic conductivity with respect to the doping level. The corollary is that high mobility permits lower doping levels, leading to a better compromise between σ and α. These observations are consistent with the bulk of the literature that

51 a b

c d

Figure 3.2: (Color online) a) The Seebeck coefficient calculated with fitted τ from τ(E) ∝ N(E)−1 scaling. b) The Seebeck coefficient calculated with constant relaxation time approximation. c) Electron conductivity from τ(E) ∝ N −1 then scaled by the ratio with the experimental conductivity [TMI14]. The true reference lifetime is estimated to be 2 × 10−15 s as opposed to 10−14 s used by BoltzTraP. d) The power factor calculated with the scaled σ. Experimental data points from Tsujii et al are for Cu.97Zn.03FeS2 [TMI14], and those from Li et al are for Cu.97Fe1.03S2 [LZQ14]. The marker colors are associated with temperature (black for 300 K, red for 400 K). e) Average mean free paths of electrons with

τ(E) = 2 × 10−15N −1(E) s, all of which are shorter52 than 1 nm. Colors indicate bands. highlights the importance of small eﬀective mass and high mobility for high thermoelectric performance [PLW12, GST17, GRL17, ?, PXO19].

3.1.3 Phonon Transport and Nanostructuring

Because of the limited PF, one must look to low κl for improvement in zT . For reliable

calculation of κl via perturbation theory, an accurate characterization of the phonon anharmonicity is essential. Harmonic and anharmonic interatomic force constants (IFC) are calculated directly from DFT using the highly eﬃcient CSLD technique. A training set of 18 conﬁgurations of a 512-atom supercell with randomly displaced atoms are employed to

ﬁt harmonic and third-order anharmonic IFCs using the standard CSLD L1 regularization procedure. The quality of the ﬁtted IFCs was tested by cross validation, which resulted in a

force fitting error below 2.7%. Finally, κl was calculated at temperatures up to 800 K by iteratively solving the linearized Boltzmann transport equation using the ShengBTE package. The third-order IFCs were used to construct phonon-phonon scattering matrices, using also Born effective charge and electronic polarizability tensors calculated using density-functional perturbation theory [BGC01]. To partially account for alloy effects, mass-disorder scattering matrices were also was constructed from atomic mass ratios and phonon eigenvectors.

The calculated phonon dispersion and phonon DOS are shown in Fig. 3.3. The acoustic modes arise largely from vibrations of Cu. Low energy optical modes receive the largest contributions from Fe atoms. High-frequency optical modes are dominated by the lightest S atoms. Overall, the Fe-S bond appears stiﬀer than Cu-S bond, in agreement withLa˙zewski et al [LNP04].

Calculated bulk κl values as seen in Fig. 3.4a are within experimentally determined range. The calculation suggests that mass-diﬀerence scattering of phonons does not result in a signif-

icant reduction in either Cu1−xFe1+xS2 or Cu1−xZnxFeS2 (dotted line in Fig. 3.4a). However, a much larger reduction in κl has been measured in alloyed CuFeS2 [LZQ14, AKT17]. There- fore, that strained lattice and not mass-disorder is the main route by which alloying reduces

κl, an observation that is consistent with the conclusions drawn in the two studies.

53 a b

Figure 3.3: a) Phonon dispersion. b) The atom-projected phonon density of states.

The predicted bulk zT values are shown in Fig. 3.5a. The results are in close agreement with the measured zT , where Cu0.95Fe1.05S2 achieved 0.05 at 300 K, 0.16 at 500 K, and 0.33 at

700 K [LZQ14]. It is clear bulk κl values are too high to yield useful zT for a bulk compound and must be lowered. Figs. 3.4b∼d indicate that nanostructuring [LLZ10, MDR09] can be a promising technique to boost zT by reducing κl. Low-energy acoustic modes around 10 meV and below exhibit the longest lifetimes and mean free paths (MFP), and are thus responsible for the largest contribution to thermal conductivity. Fig. 3.4b shows that by introducing additional phonon scattering mechanism capable of limiting MFP would lower

κl down to a fraction of its original value, thereby improving zT . If all modes with MFPs

−1 −1 longer than 20 nm were to be scattered, κl would reduce to 1 W m K or less at all temperatures. A recent attempt at nanostructuring to 20 nm grains successfully reduced

κl by a factor of three, but at the cost of reduced σ [TMT15]. Ideally, since the electron MFPs are much shorter (see Fig. 3.2e), nanostructuring of this extent should do no damage to the PF. Thus, the reduction in experimental σ must have risen from other inadvertently introduced side-eﬀects in the nanostructuring process, such as increased defects or deviation from ideal carrier concentrations. The maximum beneﬁt attainable by nanostructuring 20 nm grains is studied by assuming the PF is unharmed by nanostructuring.

54 a b

c d

Figure 3.4: a) Calculated bulk lattice thermal conductivity. The expected T −1 trend is observed. Experimental data points are included for comparison. b) Cumulative lattice thermal conductivity over contributions from phonon modes of increasing MFP. c) Phonon lifetimes. d) Phonon mean free paths. The horizontal red line indicates 20 nm, the target length scale of nanostructuring considered in this work.

55 a b

Figure 3.5: a) zT with nanostructuring of 20 nm grains b) Predicted improvement of zT via ideal nanostructuring (solid blue, 20 nm grain-size and assuming it does not harm the PF) in comparison to bulk (dotted blue). The power factor (dashed red) is overlaid. At each temperature, the optimal doping concentration for maximum zT is selected.

The resulting zT improves by many factors, as shown in Fig. 3.5, and especially so at low temperatures. Successful realization of even smaller grains could further lower κl

and improve zT . Fig. 3.5c shows that nanostructuring CuFeS2 can be an especially useful technique for its use near 400 K where zT improvement is very large. Moreover, operation at a this temperature range as opposed to high temperatures would help the maintenance of the nanostructure itself, since the grain-coarsening process is subdued at low temperatures. That way, the improved eﬃciencies may be preserved for longer.

3.1.4 Lessons

Accounting for the strong energy-dependence of electron lifetimes near the conduction band edge gives an accurate account of the experimentally observed Seebeck coeﬃcients in CuFeS2. In contrast, CRTA noticeably overestimates the negative value of the Seebeck coeﬃcient. The improvement is due to the fact that in the Mott formula the lifetime has exactly the same energy dependence as the density of states term, but with an opposite sign. Under CRTA, the density of states and the group velocity terms both add in the overall Seebeck, while in the τ ∝ eDOS−1 treatment the former is canceled and only the term arising from

56 hv2(E)i remains. This oﬀers a fresh perspective on the relative contributions of the density of states, lifetime, and group velocities in the Mott formula. The energy-dependence of the eDOS at the band edge is correlated with the energy-dependence of the squared group velocity [both give the same E−1 dependence of α(E) in the parabolic limit, see Eqs. 3.4 and 3.5], the former is exactly canceled by the lifetime term when τ ∝ eDOS−1. The fact that α only depends on the energy dependence of the group velocity carrier over to the non-parabolic case. Quite generally, scaling of the electron lifetimes by the inverse density of states is a quick and reliable method of estimating the Seebeck coeﬃcient and the power factor. In particular, it could be adopted as a standard approach 0-th order approach in high-throughput computations, especially when reliable experimental resistivity data are available for the compound in question or for related compounds.

The key weakness of CuFeS2 is that the low mobility of the heavy-mass pockets distances the peaks of the Seebeck coefficient and conductivity, limiting the PF. CuFeS2 leaves us an important reminder that high mobility is the key to bridging high Seebeck coefficient and conductivity, and hence must be prioritized over features that benefit them individually.

Also predicted here is that doping and nanostructuring (of 20 nm grains) could reduce

κl could improve zT to at most 0.25 at 300 K and 0.8 at 700 K. Being a mineral, CuFeS2 is composed of cheap and environmentally benign elements. If ideally doped and nanostructured, it could oﬀer an economical alternative to the state-of-the-art n-type thermoelectrics. If not, the mineral itself may not be too useful after all.

3.2 Full-Heusler Fe2TiSi

The contents of this section have been published in Physical Review Applied [PXO19].

The lesson from CuFeS2 is that heavy-mass bands alone do not lead to high power factor due to low mobility. What if heavy-mass bands and light-mass bands coexisted? Indeed, one of the recent concepts for high power factor focuses on the role of electronic bands that exhibit anisotropic features common in low-dimensional materials [HD93, DCT07, PCS13], such as

57 coexistence of flat and dispersive directions at the band edge. The idea behind this approach is that the flat portions provide high entropy due to high eDOS and induce high Seebeck coefficient, while the dispersive portions facilitate mobile transport due to light effective mass. To that end, let us next consider Fe2TiSi, one of Fe-based full-Heusler compound analogous in structure to Fe2VAl that were advertised as a potentially great n-type material due to the flat-and-dispersive conduction bands [BHW15, YON13].

As Fig. 3.7 shows, its conduction bands represent a quintessential example of a ﬂat- and-dispersive band structure. Previous theoretical studies of Fe2TiSi have resorted to various approximations for computing carrier scattering rates. Either a constant lifetime

−14 was borrowed from experimentally determined τ0 = 3.4 × 10 s of similar (though actually distinctly different) n-type Fe2VAl [BHW15, YON13]. In this study, explicit treatment of electron-phonon scattering using the Wannier-interpolation approach shows that n-type power factor is in fact rather modest and comparable to the p-type power factor, which arises from three purely dispersive valence band pockets. The electron lifetimes are much shorter than the hole lifetimes due to strong acoustic phonon scattering accommodated by the flat conduction band, which offers a very large phase space. Also, the flat-and-dispersive conduction bands do not appear to offer an intrinsic advantage in the Seebeck coefficient relative to the purely dispersive valence bands. Overall, it is determined that purely dispersive bands are more favorable for high power factor than flat-and-dispersive bands.

3.2.1 Crystal and Electronic Structures

Fe2TiSi is a full-Heusler compound, essentially a rock-salt structure (of Ti and Si) with all eight of its diamond sublattice positions occupied (by Fe). Its primitive cell is that of face-centered cubic with additional atoms in the interior. For the reasons elaborated in the methods section, calculation of ground-state lattice parameter and electronic structures of the primitive cell was performed using two diﬀerent schemes: using norm-conserving (NC) pseudopotentials with the PBE exchange-correlation functional on QE, hereafter NC-QE, and using projector-augmented wave (PAW) pseudopotentials with PBE on VASP, hereafter

58 a b

Figure 3.6: a) The unit cell of a full-Heusler compound. The green atoms correspond to Fe, the purple atoms to Ti and the yellow atoms to Si. b) The primitive cell.

PAW-VASP.

The ground state lattice parameter as calculated by NC-QE is 5.796 A,˚ and that by PAW-VASP is 5.685 ˚(A). These values bracket experimental value of 5.72 A˚ determined

from X-ray diﬀraction on 110 nm thin ﬁlm [BHW13]. To our knowledge, Fe2TiSi has not been synthesized at a bulk scale, limiting opportunities for comparisons. The 0 K band gap

(∆Eg) is 0.340 eV when calculated with plain NC-QE and 0.456 eV when calculated from

plain PAW-VASP. The higher ∆Eg of the latter rises from the smaller lattice parameter it predicts. Fourier transform infrared (FTIR) spectroscopy on a 110 nm thin ﬁlm has

determined ∆Eg to be 0.4 eV [BHW13]. Our values again bracket this experimental gap in a consistent manner. Band structures calculated in the two ways were qualitatively the same with minor discrepancies along some k-point paths.

The conduction band edge features two bands that are very different in character. The lowest conduction band is flat in Γ − X direction and dispersive in other directions such as Γ-L. The second band is degenerate at the Γ-point with the lowest band but dispersive in all directions. Flat bands are responsible for rapidly increasing the eDOS at the band edge. These traits are also observed in isoenergy surfaces of the two bands near the CBM as presented at the top of Fig. 3.8. The dispersive band exhibits spheroidal isosurface, representative of low effective mass (m) in all directions. The anisotropic band exhibits

59 a b

Figure 3.7: a) The electronic band structure of Fe2TiSi at 0 K. b) The total and atom-projected electronic density of states of Fe2TiSi. Calculated with PAW-VASP. isosurface that resembles three orthogonal pipes that intersect at the Γ-point, as it has been previously calculated and studied [BHW15]. m is low along the pipes, corresponding to the ﬂat direction, and high around the pipes.

Coexistence of ﬂat and dispersive bands at the CBM can be attributed to the octahedral coordination of Fe atoms and that Fe eg orbitals dominate the CBM, as seen in the eDOS . From group theory it is understood that octahedral crystal ﬁeld splits d orbitals into three degenerate t2g orbitals and two degenerate eg orbitals. The former lower in energy because they are oriented away from other ions at octahedral vertices, while the latter rise in energy because they point straight at them hence experiencing more repulsion. Also as a direct consequence of their octahedral coordination, Fe atoms in Fe2TiSi form a parallel set of sheets in all three directions. As a result, Fe atoms are interconnected by Fe eg orbitals in all three orthogonal direction. Each orbital then must facilitate carrier movement in the well-connected direction(s), e.g., Γ-L, and suppress movement but boost the eDOS in poorly connected direction(s), e.g., Γ − X.

Unlike the conduction band, the top three valence band pockets at the Γ-point are all very

60 Figure 3.8: The isoenergy surfaces near the CBM (top two) and the VBM (bottom three).

dispersive, and more or less parabolic reminiscent of free electrons. The isoenergy surfaces of all three topmost valence bands are spherical as shown at the bottom of Fig. 3.8, ensuring ﬂuent transport in all directions.

3.2.2 Electron Transport

First, electron transport is considered without considering polar-optical scattering. The maximum n-type PF at each T ranges from 55 mW m−1 K−2 at 100 K to 7.6 mW m−1 K−2 at 600 K. As shown in Fig. 3.9a, σ reaches over 106 Ω−1m−1 between 100 ∼ 300 K at high ne. As shown in Fig. 3.10b, α is indeed high, reaching easily over −500µV/K at 300 K and

below at low ne. However, in spite of the seemingly favorable band structure for high both α (ﬂat) and σ (dispersive), and the seemingly high individual values of the two, the n-type

PF does not fully beneﬁt from them. This is because the regions of ne where these two quantities individually peak are too far apart, indicative of low mobility. In the high doping regime beyond 1020 cm−3, where σ approaches its peak, α is already in a steep decline. The ﬁgures show that the peak power factor regions is roughly between 1020 ∼ 1021 cm−3, a range in between the high spots of σ and α. At each peak PF doping concentration, α is merely

61 a b

c d

Figure 3.9: a) The p-type and b) the n-type thermoelectric power factor without polar-optical scattering of Fe2TiSi. c) The p-type and d) the n-type power factor with polar-optical scattering of Fe2TiSi.

−175 µV/K at 100 K and −181 µV/K at 600 K. Hence, even though the anisotropy in band structure manages to induce high σ and high α individually, the synergy is incomplete due to heavy electron-phonon scattering.

The opposite is the case for p-type properties, as will be seen next. The eDOS at the valence band maximum (VBM) is minimal at best despite triple degeneracy of the bands due to the dispersive nature of the bands. This may not seem like a favorable band structure for high Seebeck coeﬃcient. Yet, analysis of e-ph scattering (again, without polar-optical scattering, for the time being) reveals that Fe2TiSi has an even higher p-type PF. The maximum PF at each T ranges from 88 mW m−1 K−2 at 100 K to 8.6 mW m−1 K−2 at 600

62 K. The higher p-type PF is primarily rooted in the diﬀerential scattering phenomena in the conduction bands and the valence bands. First, one sees in Fig. 3.10a that σ is extremely high regardless of doping or T . It is well higher than n-type σ overall. This reﬂects much

higher hole mobility made possible by the extremely high τeph near the VBM, as the red dots

in Fig. 3.11 illustrate. At 100 K, τeph at the VBM is nearly 100 times longer than that at the CBM. Even at 800 K, the trend persists.

The source of these high hole lifetimes is traced back to the band structure. The VBM

features three very similar bands, corresponding to the three t2g orbitals. These bands are degenerate at the Γ-point and, more importantly, stay nearly identical within a considerably large reciprocal space around it. The three of them together do not cover a very big energy- momentum space, which is necessary for accommodating a large number of scattering events.

When an electron from band ν at k with energy νk is scattered by a phonon, it needs a

final state with ν0 k0 to be scattered into while conserving energy and momentum. The more 0 variant of ν0 k0 , k combination available, the more scattering events. However, despite the presence of three bands, the number does not translate to many extra available final states because they are too much alike. Plus, all three are dispersive, so none of them contributes significantly to the eDOS, as shown by the small magnitude near the VBM in Fig. 3.7b. These characteristics are in stark contrast to those of the CBM, where the flat band offers plethora of states for electrons to be scattered into. With limited amount of scattering in the

valence bands, τeph could only be high there. The diﬀerential extent of the phase spaces for the conduction and valence bands can be conceived from comparing their isoenergy surfaces in Fig. 3.8.

Due to such weak acoustic phonon scattering in the valence bands, it turns out that polar-optical scattering is the dominant scattering mechanism there. Polar-optical scattering

−1 1 is less sensitive to the eﬀective mass (generally goes as τ ∝ m 2 ) [NSG01], and therefore discriminates band-shapes less than acoustic phonon scattering does. When it is fully treated, the n-type PF grows somewhat higher than the p-type PF due to relatively subdued hole lifetimes, as reﬂected by the black dots in Fig. 3.11. However, polar-optical scattering is

63 a b

c d

Figure 3.10: a) The p-type and b) the n-type absolute Seebeck coeﬃcients. c) The p-type and d) the n-type conductivities. Polar optical scattering is included throughout.

64 300 K

Figure 3.11: Carrier lifetimes of Fe2TiSi with (black) and without (red) polar-optical scattering.

amenable to suppression due to screening provided by free carriers, which would somewhat improve polar-optical-limited hole lifetimes and the p-type PF. Considering this, overall, the p-type performance delivered by the dispersive-only valence bands is likely to be at least comparable to the n-type performance.

A noteworthy character of the p-type PF is that it peaks in the low hole-doping regime,

19 −3 20 −3 10 cm ≤ nh ≤ 10 cm , while typical thermoelectric materials favor higher levels of doping. This follows from very high µ, which allows σ to grow high enough to complement α in the low doping regime. Take 300 K for instance. The peak power factor forms at

19 −3 5 −1 −1 nh ≈ 2 × 10 cm . In these conditions, the p-type σ already hovers above 10 Ω m ,

4 −1 −1 whereas at the same T and ne, n-type σ is not even 10 Ω m . Also at these conditions, p-type α is nearly 300 µV/K which is much larger than n-type α in the maximum n-type power factor region. Taken individually, the overall magnitude of p-type α is smaller than that of n-type α, perhaps to some dismay at the ﬁrst glance. Nevertheless, the fact that α and σ collaborate better for holes boosts the p-type PF even beyond the n-type PF. These observations emphasizes that bridging σ and α is much more important than generating individually high σ and α. Accordingly, these observations again highlight the importance of µ in realizing high PF.

A few notable observations are made at this stage. When polar-optical scattering is fully incorporated, the p-type and n-type Seebeck coeﬃcients are nearly equivalent at the

20 −3 21 −3 respective optimal doping concentrations of nh ∼ 10 cm and ne ∼ 10 cm [see Fig.

65 3.10a and Fig. 3.10b]. At 500 K, for instance, the optimal n-type α is higher merely by 3 µV K−1, which only contributes a 1% increase in the n-type PF over the p-type PF, a negligible amount. Meanwhile, the optimal n-type σ is higher than than the optimal p-type σ by a factor of nearly 1.18, leading to an 18% relative increase in the n-type PF. Since the valence bands have longer lifetimes and group velocities, this phenomenon can only be explained by the presence of extra fourth and ﬁfth dispersive branches at the CBM over three at the VBM. The fact that the valence bands produce comparable PFs in spite of being only triply degenerate is a promising result.

These observations lead to the following conclusion: relative to the p-type performance, the n-type performance draws benefits not from any enhancement in the Seebeck coefficients that the flat band is thought to provide (and does not), but rather from the number- advantage in dispersive pockets. In other words, the p-type PF is on par with the n-type PF because of the disadvantage in dispersive pocket multiplicity. If additional pocket(s) were present, purely dispersive bands could easily deliver higher conductivity and higher PF than flat-and-dispersive ones. A higher dispersion likewise would lead to even better thermoelectric properties via longer lifetimes and higher group velocities with little difference in the Seebeck coefficient.

3.2.3 Phonon Transport and Alloying

In order to test whether Fe2TiSi could be a useful thermoelectric, its κl should be calculated.

While the light atomic composition and high crystal symmetry bodes ill for κl, possible substitutional alloy compositions are considered that may reduce κl to become low enough.

Phonons are here solved up to the third order using CSLD. Comparisons of harmonic dispersion with those calculated by DFPT via NC-QE and PAW-VASP indicate that force constant ﬁtting is accurate and plausible. CSLD correctly captures all attributes of harmonic phonons other than the LO-TO splitting at the Γ-point, its inherent shortcoming. As phonon dispersion and phonon density of states (pDOS) diagrams show in Fig. 3.12a,b, acoustic modes and low-energy optical modes arise largely from vibrations of Fe atoms and to a

66 a. b.

Figure 3.12: a) Phonon dispersion of Fe2TiSi b) total and atom-projected pDOS of Fe2TiSi. lesser degree from Ti atoms. Contribution of Si to acoustic modes is minimal, but that to optical modes is significant. This correctly reflects the lightest mass of Si atoms and strongest covalency that they develop. Slightly larger phonon frequencies predicted from PAW-VASP are attributed to the slightly larger (1.9 %) lattice parameter that NC-QE predicts. An equivalent but smaller lattice stiffens phonons.

Intrinsic κl is limited by ph-ph scattering. Analysis of mode group velocity (vg) and ph- ph relaxation time (τph) and mean free path through perturbation theory indicates that all modes contribute signiﬁcantly to heat transport. Acoustic modes exhibit by far the highest vg, as depicted in Fig. 3.13d, and are most populated, especially at low-T . Optical modes, however, exhibit much longer τph, as depicted in Figs. 3.13b and 3.13c. As these long τph are paired with not insigniﬁcant vg, contribution to κl of these optical modes is also sizable at high temperatures when they are populated enough. Overall, the intrinsic κl of Fe2TiSi is much too high to make the compound by itself a useful thermoelectric in any temperature domain. At room temperature it is lofty 60 W m−1 K−1, and even at 800 K, it marks just below 30 W m−1 K−1. Alloying heavier elements is therefore a necessary treatment. It would

67 a. b.

c. d.

Figure 3.13: a) Lattice thermal conductivity of Fe2TiSi and its alloys calculated with phonon-phonon and mass-disorder scattering b) mean free path of pure Fe2TiSi c)

Phonon-phonon scattering relaxation time at 300 K of pure Fe2TiSi d) Mode group velocity at of Fe2TiSi not only introduce mass-disorder scattering that reduces κl beyond the ph-ph limit, but also soften phonons and suppress vg.

Speciﬁcally, four isovalent substitutions of heavier elements are considered: Ge for Si, Sn for Si, Zr for Ti and Hf for Ti. Isovalent subsitutions are considered since they would not qualitatively alter the electronic structure or electron transport properties. The convex hull formed by the elements appearing in a given alloy is calculated using the formation energies of binary and ternary compounds found on Materials Project [JOH13]. Our analysis indicates that that energies of Fe2TiSi1−xGex alloys lie more or less right on the convex hull formed by the commensurate ratios of Fe2TiSi and Fe2TiGe. This suggests energetic stability of these alloys. However, their κl are not satisfactorily low. At 10% substitution, Fe2TiSi0.9Ge0.1

68 −1 −1 has κl of about 13 W m K at 300 K. Though the other substitutions involve larger

mass-diﬀerence, their alloys are unfortunately energetically unstable. Analysis of SnSi and

ZrTi resulted in prohibitively large formation energies of 0.49 eV and 0.43 eV per impurity,

respectively. HfTi was found to have still large but smaller formation energy of 0.3 eV

per impurity. Of all alloys considered, Fe2Ti1−xHfxSi alloys beneﬁt the most from heat

transport suppression. After the treatment of HfTi mass disorder, even at 4% substitution,

−1 −1 Fe2Ti0.96Hf0.04Si marks 5 W m K at 300 K. In an eﬀort to amend stability, the vibrational

entropy of formation (∆Sv) is calculated, in the hope that positive ∆Sv will stabilize the

alloy. Unfortunately, ∆Sv for Fe2Ti1−xHfxSi turned out to be negative. Also, even if 4%

HfTi substitution were possible, zT would only be 0.5 at most.

3.2.4 Lessons

Both flat-and-dispersive band structures [HD93, DCT07, PCS13, BHW15] and bands with small effective mass have been associated with high thermoelectric performance [PLW12]. As reflected by the most recent review article on thermoelectric materials, the controversy regarding the ideal band shapes and effective mass for thermoelectricity continues [MLZ18].

Though Fe2TiSi has received attention as a promising n-type material, its dispersive-only valence bands are found to be more promising band structures for high PF. The fact that

Fe2TiSi simultaneously possesses the two sets of bands that are drastically diﬀerent in character provides a direct comparison of the performances of the two types of band structures. Highly dispersive bands exhibit longer carrier lifetimes due to the more limited phase space for acoustic phonon scattering. The combination of long lifetimes and high group velocities results in high mobility. The importance of high mobility as the bridge between the gap of the Seebeck coeﬃcient and conductivity with respect to carrier concentration is reiterated.

Fe2TiSi shows yet again that a cooperation of the two quantities through mobility (the p- type case) is more important than them becoming individually high (the n-type case). This is an important clariﬁcation going forward in power-factor research. In fact, it ought to be

mentioned that the prototype compound, Fe2VAl, has high n-type PF arising precisely from

69 its dispersive conduction band pocket at the X-point, not some ﬂat-and-dispersive bands.

Another important point is that the ﬂat conduction band does not provide an inherent advantage in the Seebeck coeﬃcient, against popular belief. The n-type PF in Fe2TiSi in fact relies on high electrical conductivity promoted by the high number of dispersive pockets at the CBM (three at X and additional doubly degenerate band at Γ). In contrast, the VBM is only triply degenerate at Γ but still yields comparable p-type PF, which would easily outperform the n-type PF the degeneracy factor were higher. This observation is precisely what led to the study of Ba2BiAu and Sr2BiAu, to be discussed in the next chapter.

Lastly, Fe2TiSi by itself would not be a useful thermoelectric due to insurmountably high lattice thermal conductivity. Even with generous 4% HfTi substitutional alloying, which reduces it to 5 W m−1 K−1 via mass-disorder scattering, the maximum achievable zT would be 0.5.

70 CHAPTER 4

Prediction of High-Performance Thermoelectrics

For high power factors in semiconductor thermoelectrics, the lessons learned from CuFeS2 and Fe2TiSi point in one direction. One must look for band structures with numerous highly dispersive bands that can support high mobility by minimizing phonon scattering. The ensuing question is, could such a band structure exist for a compound with low lattice thermal conductivity? While there have been plenty of compounds with low lattice thermal conductivity or with high power factor, no material yet predicted or synthesized has harbored them both as to generate zT ≥ 3.

Reported in this chapter is the identification and characterization of Ba2BiAu and Sr2BiAu, also full-Heusler compounds, as high-performance thermoelectrics. They have recently predicted to possess ultralow lattice thermal conductivities arising from intrinsically strong anharmonicity due to rattling of Au atoms [HAX16]. Predicted here is that they also possess very high power factors that arise from multitudes of highly dispersive bands at off- symmetry points (six and ten, respectively) that minimize electron-phonon scattering due to small phase spaces. As a result, Ba2BiAu is predicted to be theoretically capable of achieving zT = 5 at 800 K and zT = 1.4 at 300 K. Better yet, Sr2BiAu is predicted to be theoretically capable of achieving even higher zT across most temperatures, notably zT = 0.3 at cryogenic 100 K and zT = 1 at 200 K and zT = 2 at 300 K, in addition to zT = 5.1 at 800 K. Unfortunately, these performances are unlikely to be realized experimentally due to unfavorable defect formations [see Appendix] and calls for further search for similar but more realizable high-performance materials. A study of CoSi (led by Dr. Yi Xia) is also presented as a case of high Seebeck coefficient and power factor in metals – also originating from intrinsic behaviors of electron-phonon scattering that lead to electron filtering.

71 4.1 zT = 5 in Full-Heusler Ba2BiAu

The contents of this section have been published in Physical Review Applied [PXO19].

Reported herein is the prediction of unprecedented zT = 5 at 800 K in n-type full-

Heusler Ba2BiAu. This material is a member of the family of full-Heusler compounds recently discovered to be stable and to exhibit ultralow intrinsic lattice thermal conductivity arising from anharmonic rattling of heavy atoms [HAX16]. Very similar to Fe2TiSi, this compound also exhibits a coexistence of flat and dispersive directions at the valence band maximum, which has drawn interest for a possibility for high p-type power factor. However, explicit treatment of electron-phonon scattering leads to the prediction that Ba2BiAu features high n-type PF (as high as 7 mW m−1 K−2) originating from a highly dispersive conduction band with multiple degenerate pockets. By comparison, the p-type PF arising from the flat-and- dispersive valence band is significantly lower due to much increased phase space for (acoustic) phonon scattering of holes, with no advantage in the Seebeck coefficient provided by the flat band – to the further confirmation of the observations made in Fe2TiSi.

4.1.1 Crystal & Electronic Structures

Refer to Fig. 3.6 for the crystal structure, which is identical to that of Fe2BiAu. Re- laxation of the crystal structure and self-consistent calculation of the electronic structure are performed using two diﬀerent schemes: NC-QE and PAW-VASP. The PBE exchange- correlation functional is used for both cases. A plane-wave energy cut-oﬀ of 120 Ry and an energy convergence threshold of 10−8 Ry are used for NC-QE calculations. 600 eV and 10−7 eV are respectively used for PAW-VASP calculations. Convergence with respect to k-point mesh is safely ensured.

The ground-state lattice parameter of Ba2BiAu as calculated by NC-QE is 8.30 A,˚ and that calculated by PAW-VASP is 8.42 A.˚ Ba2BiAu has not been synthesized at all to date, so experimental data are unavailable for comparison. NC-QE and PAW-VASP yield more or less the same band structures for both compounds near their energy gaps. The 0 K band gap

72 Figure 4.1: Electronic band structure of Ba2BiAu with (red) and without (black) spin-orbit coupling, aligned at the CBM.

of Ba2BaAu is 0.44 eV when calculated with plain NC-QE and 0.45 eV when calculated with PAW-VASP. A more accurate band gap is likely predicted by the Tran-Blaha-modiﬁed Becke- Johnson pseudopotential functional (mBJ) [BJ06, TB09] with spin-orbit coupling (SOC),

which yields ∆Eg of 0.56 eV.

Fig. 4.1 shows the electronic band structure of Ba2BiAu. The notable features are a highly dispersive conduction band pocket along Γ − X and a flat-and-dispersive valence band pocket at the L-point. These features translate to 1) a pipe-like isoenergy surface at the VBM, where effective mass (m) is large along the pipes, corresponding to the flat direction, and small around the pipes, corresponding to the more dispersive directions; 2) a set of oblates in the middle of the six Γ − X directions for the isoenergy surface at the CBM [see Fig. 4.2]. These oblates reflect that m is very small longitudinally and larger but still small transversely, characteristic of dispersive pockets at off-symmetry points in the Brillouin zone. Both types of band structures have been associated with high PF. It is also noted that while SOC induces a slight curving of the otherwise flat VBM, it leaves the dispersive CBM pocket

73 Figure 4.2: (Color online) Isoenergy surfaces of the topmost valence band (left) and lower-

most conductions band (right) of Ba2BiAu, 0.1 eV below the VBM and above the CBM,

20 respectively. The levels correspond to electron doping concentration of ne = 1.2 × 10 and

20 hole doping concentration of nh = 2.7 × 10 , respectively.

practically unchanged. Eﬀective masses at the CBM are mxx = 0.069, myy = mzz = 0.51 without SOC and mxx = 0.067, myy = mzz = 0.49 with SOC. Since our main interest here is the n-type performance, and SOC has negligible inﬂuence on both the conduction band and phonon dispersion [HAX16], SOC is neglected when calculating the electron-phonon scattering rates.

An advantage of the conduction band pocket is that it comes with sixfold pocket multiplicity, while the VBM pocket at the L-point is fourfold degenerate. Pocket multiplicity, or band degeneracy, directly beneﬁts σ by enhancing carrier concentration for a given Fermi level and removes the detrimental eﬀect on α caused by band misalignment [PXA11]. No other portion of any band is within 0.3 eV of the CBM, making the pocket the sole contrib- utor to electron transport up to 800 K. Also, the particularly small m of the CBM pockets is expected to constrict the phase space for acoustic phonon scattering – the typical limiting carrier scattering process in thermoelectrics.

74 4.1.2 Scattering and Mobility

The two major scattering mechanisms at work in thermoelectric materials are treated in this study: electron-phonon scattering and ionized impurity scattering. For the former, electron-phonon interaction matrix elements are explicitly calculated over a coarse 8 × 8 × 8 k-point mesh (for electrons) and a 4 × 4 × 4 q-point mesh (for phonons) using DFPT [PAG14, PGJ15]. Then maximally localized Wannier functions [MV97, SMV01, MYL08] are used for an eﬃcient interpolation of the matrix elements onto dense 40 × 40 × 40 k-and-q- point meshes using the EPW software [GCL07, NGM10, PMV16]. This treatment accounts for both intravalley and intervalley scattering. Long-ranged polar-optical scattering matrix elements are calculated separately and added to the total matrix elements on the dense kmesh [VG15]. The carrier lifetimes are obtained from the imaginary part of electron self- energy, which is calculated by summing the matrix elements over the dense phonon mesh. With lifetimes at hand, Eqs. 2.8–2.12 are solved for transport properties.

The mobility limited by ionized impurity scattering (µii) is estimated using the aMoBT software [FIL15], which implements an improved form of the Brooks-Herring theory that takes into account screening and band non-parabolicity [RK71, CQ81]. The overall mobility

(µ) is obtained by combining µii with µeph according to the Matthiesen’s rule,

−1 −1 −1 µ = µeph + µii , (4.1)

and is used to rescale σ.

The calculated τ due to electron-phonon scattering at 300 K are shown in Fig. 4.3a. If polar-optical scattering is excluded, electron lifetimes near the CBM are roughly two orders of magnitude longer than those of the hole states near the VBM. This shows that (acoustic) phonon scattering is indeed weak for the CBM states, as expected due to the limited scattering phase space that a dispersive band provides. Inclusion of polar-optical scattering reduces τ by an order of magnitude at the CBM, and it is the dominant scattering mechanism for electrons as reﬂected by the mobility calculations [see Fig. 4.3b]. This mirrors the characteristics of other dispersive-band, high-µ materials such as InSb and GaAs.

75 a b 300 K

Figure 4.3: a) Energy-dependent electron-phonon scattering lifetimes at 300 K of Ba2BiAu with (black) and without (red) polar-optical scattering. b) Electron mobilities limited by electron-phonon scattering (eph, solid lines) and by ionized impurity scattering (ii, dotted lines), and hole mobility at 300 K for comparison (eph, dashed line).

Screening effects, which are not accounted for but could play a role in suppressing polar- optical scattering at high carrier concentrations, would only increase the predicted τ and µ. However, its effect would be counteracted by electron-electron scattering that also intensifies with carrier concentration. The contribution of ionized impurity scattering is for the most part small unless the compound is very heavily doped at 300 K. Hole mobility is lower by nearly an order of magnitude due to shorter lifetimes let alone lower group velocities.

4.1.3 Thermoelectric Properties

The calculated n-type PF tops out at 7 mW m−1 K−2 at 500 K [see Fig. 4.4a]. This is among the highest n-type PFs for bulk semiconductors. At the same temperature, p- type half-Heusler NbFeSb, the highest-PF semiconductor to date, exhibits 8 mW m−1 K−2

[HKM16]. The high PF of Ba2BiAu is attributed in part to the high σ arising from the sixfold pocket multiplicity and weak (acoustic) phonon scattering that elevates τ near the CBM. Also, the band generates high n-type α which exceeds −300 µVK−1 at 800 K, as shown in Fig. 4.4c and 4.4d. The maximum PF requires degenerate doping through the CBM, which is typical of phonon-limited electron transport. Ionized impurity scattering is

76 a b

c d

Figure 4.4: a) The n-type power factors and the 500 K p-type power factor. b) The n-type Lorenz numbers and the 800 K p-type Lorenz number, where the black horizontal line indicates LWF. c) The n-type Seebeck coeﬃcients and the 500 K p-type Seebeck coeﬃcient. d) The n-type conductivities and the 500 K p-type conductivity. All curves are plotted with respect to their respective doping concentrations. found to lower the peak n-type PF by 25% at 300 K and 15% at 500 K and 800 K.

Ultimately, high zT requires high PF coupled with low thermal conductivity. Ba2BiAu

−1 −1 has already been predicted to possess ultralow κl, 0.55 W m K at 300 K and 0.2 W

−1 −1 m K at 800 K, close to the amorphous limit [HAX16]. The κl prediction is shown in Fig. 4.5b, for which the compressive sensing lattice dynamics technique [ZNX14] and the iterative Boltzmann transport scheme [LCK14] are used.

−1 −1 Because of such low κl, heat is mainly carried by electrons (κe ≥ 1 W m K ). Since

κe > κl, small Lorenz number (L) is indispensable for the enhancement of zT . This statement

77 can be generalized beyond Ba2BiAu to any wishful high zT material since it must have low

κl and high PF. As Fig. 4.4b shows, our results indicate that throughout relevant doping

2 2 π kB ranges, L < LWF in the favor of zT , where LWF = 3 e2 is the Wiedemann-Franz value. Such a negative deviation from the Wiedemann-Franz law is attributed to energy-dependent electron lifetime due to electron-phonon scattering. A simple illustration is provided by the non-degenerate approximation to the single parabolic band model, according to which [NSG01] k2 5 L = B + s (4.2) e2 2 where the energy-dependence of scattering controls the constant s. For polar-optical scat- 2 1 3kB 1 tering, s = 2 , which leads to L = e2 < LWF. Under acoustic phonon scattering, s = − 2 2 2kB and L would be even lower at e2 . As the Fermi level moves towards the center of the gap,

L increases above LWF as each electron carries more energy.

The combination of high PF and low thermal conductivity in Ba2BiAu leads to an unprecedentedly high n-type zT = 5 at 800 K. This value is higher by far than any other 3D bulk thermoelectric to date. Predicted at 300 is zT = 1.4, which would also be the highest for an n-type material at room temperature. Fig. 4.5 shows that the peaks of zT form at

noticeably lower doping concentrations compared to those of the PF. Optimal ne for zT in

fact corresponds to the non-degenerate doping regime. The gap between the optimal ne for

the PF and zT is reﬂective of the fact that κe contributes heavily to the total thermal con-

ductivity. Because κe > κl and L is relatively constant with ne, zT naturally favors lower ne where the Seebeck coeﬃcient is maximally utilized. Such a low doping requirement is favorable in terms of weaker ionized impurity scattering as well. Calculations result in zT = 0.8 at 200 K and zT = 0.2 at 100 K. Though not as remarkably high as the values highlighted at higher temperatures, they are still respectable considering the very low temperatures.

Ba2BiAu has previously been projected to have higher p-type PFs than n-type PFs due to the ﬂat-and-dispersive valence band [HAX16]. According to our calculations, however, the highest achievable p-type PF is merely 2.5 mW m−1 K−2 at 500 K, and the highest achievable p-type zT is at best 2 at 800 K. This is ascribed to heavy phonon-scattering of

78 Figure 4.5: a) The n-type zT and the 800 K p-type zT . b) Lattice thermal conductivity.

holes invited by the large phase space associated with the ﬂat VBM, which alone reduces hole mobility below the overall electron mobility [see Fig. 4.3b]. Though SOC adds a small curvature to the VBM and would somewhat improve hole τ and the p-type performance, it is unlikely to make a meaningful diﬀerence. Additionally, the valence band only has fourfold pocket multiplicity at the L-point. Shorter lifetimes and fewer pockets combined

20 −3 with lower group velocities force one to rely on very high hole doping (nh > 10 cm ) to achieve sufficiently high values of σ in p-type case [see Fig. 4.4d]. However, this reduces α to approximately 80 µVK−1 at 500 K [see Fig. 4.4c], resulting in poor overall PF. In all, these observations indicate that highly dispersive bands that minimize acoustic phonon scattering can easily outperform flat-and-dispersive bands. Also, note that the n-type α is even higher in magnitude than the p-type α. This indicates that purely dispersive bands are capable of generating competitive Seebeck coefficients and that flat bands do not provide an intrinsic advantage.

79 4.1.4 Lessons

Ba2BiAu has the potential to achieve unprecedentedly high thermoelectric performance (n- type) marked by zT = 5 at 800 K and 1.4 at 300 K, due to a rare complementation of very high power factor and ultralow lattice thermal conductivity in addition to small Lorenz number. The predictions of here are oﬀered for experimental synthesis and characterization, which would constitute a proof of principle that the long-desired zT above 4 is actually attainable. The high n-type power factor of Ba2BiAu is traced to a highly dispersive conduction band pocket along the Γ−X direction, which constricts the phase space for electron-phonon scattering, promoting long lifetimes and high mobility for electrons. On the other hand, the

ﬂat-and dispersive valence band of Ba2BiAu yields much smaller p-type power factors due to strong (acoustic) phonon scattering accommodated by a large phase space. These trends mirror those observed in Fe2TiSi, in which the conduction bands are ﬂat-and-dispersive.

The commonalities found in Ba2BiAu and Fe2TiSi confirm that highly dispersive bands with multiple pockets ought to be the main focus in the quest for next-generation semiconductor thermoelectrics. Flat-and-dispersive band structures tend to invite too much scattering while not providing notable benefits for the Seebeck coefficient. From this perspective, it is only fitting that the highest known semiconductor power factors occur in half-Heusler NbFeSb, which is known to have a very weak deformation potential [ZZL18] but also happens to possess eight relatively dispersive valence band pockets to help the cause [FZP14]. In turn, InSb, in spite of having single conduction band pocket at its Γ-point, exhibits a fairly high PF (2 ∼ 6 mW m−1 K−2 at 300 ∼ 730 K) due to its extremely high mobility (> 103 cm2 V−1 s−1) arising from the pocket’s extremely small m (0.014) [KHK09, YYK05, BUB59]. If that pocket were even threefold degenerate, the PF would also be roughly triple and make InSb easily one of the highest-PF semiconductors.

The hope is that these examples will stimulate further eﬀort in search of highly dispersive pockets at oﬀ-symmetry points occurring in materials with low lattice thermal conductivity.

In fact, the next material to be introduced features every beneﬁcial aspect of Ba2BiAu and then some more, and is predicted to deliver an even higher thermoelectric performance.

80 4.2 High-Performance at Cryogenic-to-High Temperatures in Sr2BiAu

Trading Ba away for Sr in the same full-Heusler structure, one obtains Sr2BiAu. This swap preserves the sixfold degenerate Γ − X pocket and adds a dispersive pocket at the L-point, which has a fourfold multiplicity. The new L-pocket happens to align in energy with the Γ−X pocket at the conduction band minimum, for a total of ten highly dispersive pockets at the band edge. This leads to ultrahigh n-type power factor that tops out at 12 W m−1 K−1 – the highest ever for an n-type thermoelectric and on par with the p-type power factors of doped NbFeSb. The strong lattice anharmonicity due to the rattling behavior of the Au atoms is also preserved in Sr2BiAu, by which lattice thermal conductivity is kept minimal. The combination of ultrahigh power factors and ultralow lattice thermal conductivities translate to ultrahigh zT across all temperature domains. At cryogenic 100 K, zT = 0.3. At room temperature 300 K, zT = 2. At higher 800 K, zT = 5.1. Sr2BiAu is a further proof that multitude of highly dispersive pockets at oﬀ-symmetry points is the ideal structure for high semiconductor thermoelectric performance. Successful n-doped samples of this compound would likely be a new major breakthrough in thermoelectrics and leave an enduring impact in the ﬁeld.

4.2.1 Electronic Structure

Spin-orbit coupling (SOC) does not affect the conduction band pocket along Γ − X, just as in the case of Ba2BiAu. The L-point pocket is rendered slightly anisotropic by SOC. It becomes slightly heavier in the L-Γ direction and slightly lighter in all other directions. The corresponding energy surfaces as seen in Fig. 4.6 is visibly complex, revealing all ten pockets. Anisotropicity and Fermi surface complexity have been known to benefit the PF [GRL17], barring extreme anisotropicity the likes of a flat-and-dispersive band, which invites too much acoustic phonon scattering and is likely harmful [PXO19]. The PBE+SOC band gap is 0.19 eV, a severe underestimation. The more realistic gap calculated with Tran-Blaha modified Becke-Johnson (mBJ) potential [TB09] and SOC is 0.53 eV.

81 Figure 4.6: (Color online) Electronic band structures of Sr2BiAu calculated on Quantum Espresso [GBB09, GAB17] with norm-conserving pseudopotentials and Perdew-Burke-Ernz- erhof (PBE) exchange-correlation functional [PBE96], with (black, solid) and without (red, dotted) spin-orbit coupling, aligned at the conduction band minimum. The atom-decomposed density of states is shown on the right. Isoenergy surfaces of Sr2BiAu are calculated with PBE+SOC, at 0.1 eV below the VBM (left) and above the CBM (right). The lev-

20 −3 els correspond to electron doping concentration of ne = 1.5 × 10 cm and hole doping

19 −3 concentration of nh = 7.5 × 10 cm , respectively.

82 4.2.2 Scattering & Mobility

In treating electron-phonon scattering, electronic states and electron-phonon interaction matrix elements are computed at an 8 × 8 × 8 coarse k-point mesh, using phonon perturbations computed at a coarse 4 × 4 × 4 q-point mesh using density functional perturbation theory [PAG14, PGJ15]. Then with the EPW package [GCL07, NGM10, PMV16, Giu17], the electronic states, the phonon states, and the interaction matrix elements are interpolated onto dense 40 × 40 × 40 k-point and q-point meshes through maximally localized Wannier functions (MLWF) [MV97, SMV01, MYL08]. Long-ranged polar-optical scattering matrix elements are added on the dense k-mesh [VG15]. The imaginary part of the resulting electron self-energy leads directly to band-and-k-dependent electron lifetimes (τνk) limited by electron-phonon scattering. With τνk as ingredients, the Boltzmann transport equations

(implemented with BoltzTraP modiﬁed to intake τνk) are solved for electron transport properties. In performing Eqs. 2.8–2.12, the band structure calculated with SOC and the band gap value from mBJ+SOC are utilized. Since the eﬀect of SOC on the electronic structure or phonons is not severe, SOC is neglected for the more computationally cumbersome analysis of electron-phonon scattering. Ionized-impurity scattering is also neglected because the static dielectric constant is very high (96 as calculated with PBE+SOC), which would largely screen ionized-impurity scattering.

Fig. 4.7a shows that electron and hole lifetimes in Sr2BiAu are similar in magnitude at the VBM and the CBM, with or without polar-optical scattering. This poses a contrast to the lifetime behaviors in the previously studied Ba2BiAu, where holes had signiﬁcantly shorter lifetimes than electrons due to the ﬂat-and-dispersive valence band and lone dispersive conduction band. In comparison, Sr2BiAu has a more dispersive valence band and higher eDOS in the conduction band due to the extra L-pocket. Overall, Sr2BiAu exhibits somewhat shorter electron lifetimes than Ba2BiAu. This is expected since the presence of the additional L-pocket would invite more intervalley scattering. Interestingly, electron mobilities in Sr2BiAu are higher than those of Ba2BiAu at 300 K, but lower at 800 K [see Fig. 4.7b]. This indicates that as temperature rises the added phase space for intervalley

83 a b 300 K

800 K Sr2BiAu Ba2BiAu

100 K Sr2BiAu Ba2BiAu

Figure 4.7: (Color online) The dependence of transport properties of Sr2BiAu on on doping, temperature, and scattering. a) The carrier lifetimes limited by electron-phonon scattering with (black) and without (red) polar-optical scattering. The temperature is 300 K. b) The electron mobility with polar-optical scattering.

84 scattering in the former increasingly manifests. The question that arises then is whether the reduction in electron lifetimes due to the extra pocket is significant enough that its harmful effect on the PF outweighs the beneficial effect of increased electron concentration, also due to the extra pocket. Which effect is to prevail?

4.2.3 Thermoelectric Properties

The n-type α values, shown in Fig. 4.8c, are nearly identical to if slightly lower than those

of Ba2BiAu. Because the two bands are energy-aligned, the two pockets deliver the same α for a given Fermi level, maintaining high overall α. Also, high pocket multiplicity (Np = 10) increases carrier concentration per Fermi level thereby enhancing σ, leading to very high

PFs. The n-type PFs of Sr2BiAu, as shown in Figs. 4.8a and 4.8b, behaves very similarly to the measured PF of the p-type NbFeSb, only slightly lower, across all temperature domains.

In essence, Sr2BiAu is a natural realization of the concept of band-convergence engineering that is known to beneﬁt the PF in general [PXA11].

While pocket multiplicity is generally deemed beneficial for thermoelectrics, it remains possible that the added phase space for intervalley scattering could outweigh the benefit by lowering carrier lifetimes and mobilities [XTL18, NV16]. Comparison of the PF with that of Ba2BiAu offer insights to the role of the L-pocket and intervalley scattering accordingly

−1 −2 introduced. Sr2BiAu surpasses Ba2BiAu in the PF by up to several mW m K at 500 K and below, but this is not so at T > 500 K [see Fig. 4.8b]. At higher temperatures,

Ba2BiAu attains higher PFs. This crossover indicates that the additional dispersive pockets at the L-points ultimately beneﬁt the PF at low temperatures, where the dominant scattering mechanism (intravalley scattering) remains the same with or without the extra pockets and only carrier concentration per Fermi level is increased. As intervalley scattering intensiﬁes with temperature, the associated reduction in lifetimes and mobility harms the PF more than

it is beneﬁtted. Sr2BiAu therefore makes a strong case that the presence of extra pockets aligned the Fermi level will beneﬁt thermoelectric performance at a low temperature regime below where the additional intervalley scattering starts to critically interfere.

85 a b

c d

Figure 4.8: (Color online) a) The n-type power factor against doping concentrations. b)

The n-type power factor of Sr2BiAu (black squares and line) juxtaposed with power factors of some experimentally and theoretically high-performing thermoelectrics (p-type NbFeSb in blue squares [HKM16], p-type n-type PbTe in red triangles [SKU08]) and theoretical n-type

Ba2BiAu in empty circles [PXO19]). c) The n-type Seebeck coeﬃcient. d) The n-type electrical conductivity.

86 Several eﬀects at high doping concentrations that are not explicitly taken into account would counterbalance one another’s inﬂuence on the PF. First, the screening of optical phonons by high-density itinerant electrons would subdue polar-optical scattering, thereby

elevating τeph,νk. However, this would be counteracted by increased electron-electron scattering and ionized-impurity scattering. The overall eﬀect of the unaccounted phenomena is therefore deemed small.

Fig. 4.9a demonstrates that the Lorenz number (L) is consistently below the Wiedemann-

Franz value (LWF), a feature characteristic of transport dominated by either acoustic phonon

scattering or polar-optical scattering [NSG01]. The small magnitude of κe relative to σ is

important because κl is so low [see Fig. 4.9b], as determined by a previous study [HAX16].

The combination of very high PF and very low κl in Sr2BiAu results in ultrahigh intrinsic zT from cryogenic to high temperatures. Notable in particular is the high performance zT = 0.3 ∼ 2 at 100 ∼ 300 K, which is a boon to the rarity of eﬃcient thermoelectrics at such low temperatures. Comparisons made in Fig. 4.10b clearly shows that the theoretical performance of Sr2BiAu is record-high at nearly all temperatures for bulk materials. Because

κe is the dominant thermal conductivity and L is rather constant, zT prefers lower ne than the PF where the high Seebeck coeﬃcient develops. The p-type zT is not as high in comparison, marking 1.5 at best around 700 K. This owes to the fact p-type PF is signiﬁcantly lower due to both lower hole mobility and the fewer valence band pocket multiplicity of four.

4.2.4 Lessons

Two very dispersive conduction band pockets at oﬀ-symmetry points (for a total of ten pockets) generate very high power factors across all temperatures, very close to the highest-ever semiconductor power factors of the p-type NbFeSb. The power factors are particularly high at low temperatures where weak intervalley scattering spares high mobility and high pocket multiplicity improves conductivity. Comparisons with Ba2BiAu indicate that at higher temperatures, the harm done to mobility by the added intervalley scattering outweighs the beneﬁts from high carrier concentration per Fermi level. Generalizable from this analysis is

87 a b

Figure 4.9: (Color online) a) The Lorenz number. b) Lattice thermal conductivity. a b

c d

Figure 4.10: (Color online) a) The predicted zT of n-type Sr2BiAu against doping concentration. b) The predicted zT of n-type Sr2BiAu in comparison to state-of-the-arts thermoelectrics across low temperatures [HLG04, PHM08, HWZ15, KLM15, VSC01]. c) Predicted p-type zT . d) Predicted p-type power factor.

88 that thermoelectric performance would not monotonically increases with pocket multiplicity, and there would be an optimal pocket multiplicity where high mobility and carrier concentration per Fermi level are ideally balanced for high power factor. The predicted zT values are particularly high at cryogenic-to-room temperatures – the temperature domain that general lacks eﬃcient thermoelectric materials. If experimentally realized, Sr2BiAu would be a next-generation thermoelectric with high-performance at all temperatures.

4.3 Intermetallic, B20-type CoSi

The contents of this section have been published in Physical Review Applied [XPZ19].

While several high-performing semiconductor thermoelectrics have been identified and studied this work, metallic thermoelectrics have been overlooked. As stated in the introductory chapter, metals are typically not great thermoelectric materials due to the continuous electronic density of states across the Fermi level. Without a natural filter (band gap) to separate majority carriers from minority carriers, metals often have negligibly low Seebeck coefficients. Eq. 3.1 however makes it clear that the density of states is not the sole con- tributor to the Seebeck coefficient: high Seebeck coefficient could theoretically arise from steep energy-gradients of electron lifetimes. Indeed, several metallic compounds are associated with unusually high Seebeck coefficients. A prominent example is YbAl3 [RKK02], whose ultrahigh Seebeck coefficient and power factor at very low temperatures originate from Kondo scattering enabled by magnetic impurities. The question is then whether such a beneficial scattering phenomenon could occur intrinsically via electron-phonon scattering.

The B20-type CoSi is another intermetallic compound that turned in high Seebeck coefficient measurements [SLM13, RLZ05, SZP09, SYA07], but the underlying mechanism has not been clear. The theoretical study in this work clarifies that CoSi exhibits strongly energy- dependent electron-phonon scattering near the Fermi level that scatters holes and low-energy electrons. This efficient form of energy filtering leads to high n-type Seebeck coefficient and power factor. The presence of a heavy band crossing the linear, massless (Dirac-like) bands

89 (e)

Figure 4.11: a) The crystal structure of CoSi. b) The simple cubic Brillouin zone with high symmetry points. c) The electronic structure, where the bands are colored according

to diﬀerent orbitals shown in the legend, namely, s, p, eg and t2g states. d) The electronic density of states. e) The band structure zoomed into the region near the Fermi level. just above the Fermi level is essential as it oﬀers a scattering pathway for low-energy electrons. The band also bends below the Fermi level rather than above, providing a large phase space for hole scattering by phonons across a wide energy range but allowing electrons to maintain generally high lifetimes. Combined with large Fermi velocities of the massless bands, a remarkably high power factor higher of 8 mW m−1 K−2 is predicted at temperatures higher than 300 K. Our study constitutes a proof of concept that coexistence of massless and heavy bands can lead to very high power factors in intermetallics.

4.3.1 Crystal and Electronic Structures

The B20-type CoSi belongs to space group P 213, bearing a simple cubic structure with four Co atoms and four Si atoms in the primitive cell. The bands near the Fermi level are mainly composed of Co 3d states. Away from the Γ-point, the band character shifts from eg to t2g states. The point of interest is that massless bands and a heavy band appear at and around

90 the Γ-point near the Fermi level, giving rise to a quadratic increase in the eDOS followed by a sharp increase [see Fig. 4.11c and Fig. 4.11d]. The heavy band bends below the Fermi level, providing comparatively larger eDOS for holes than for electrons.

4.3.2 Thermoelectric Properties

The calculated α under the CRTA, the RTA and the eDOS−1 approximation compared with experimental measurements are shown in Fig. 4.12a. For the RTA calculation, momentum- and-energy-resolved e-ph scattering matrix elements and lifetimes are computed by DFPT and Wannier interpolation. First, the CRTA yields small positive α, which is calculated due to the larger eDOS below the Fermi level than above and because the CRTA neglects energy- dependent lifetimes. All experimental measurements display negative α from modest −20 µV/K at 100 K to much larger −80 µV/K at 300 K and above. Indeed, when e-ph scattering is fully treated, the RTA calculation agrees excellently with experiments and reproduces the observed temperature dependence of negative α. Even the simpler eDOS−1 approximation is a signiﬁcant improvement upon the CRTA and leads to negative α comparable with that obtained by the RTA, save for the overestimation of α at low temperatures (≈ 100 K). Experimentally measured values of the PF vary quite a bit, as shown in Fig. 4.12d, but the RTA calculations agree reasonably well with one set of measurements [SZP09], showing that a maximum PF of about 8 mW m−1 K−2 for pristine CoSi is achievable at 600 K.

A previous theoretical study on the B20-type CoGe with similar electronic structure has concluded that the high negative α originates from the asymmetric band structure about the Fermi level [KOS12]. However, without considering energy-dependent carrier lifetimes, temperature and doping dependencies deviate considerably from experimental results. For α to be large, asymmetric Σ(E) is required (high in domain 3 of Fig. 1.5) while for σ to be large, an overall high Σ(E) near the Fermi level is all that is needed. Fig. 4.13a displays the calculated Σ(E) under the CRTA, RTA and eDOS−1 approximation at 300 K. It is evident that Σ(E) under the CRTA deviates signiﬁcantly from those under the two more accurate approaches, which are strongly asymmetric about the Fermi level. Taking note of

91 Figure 4.12: a) Calculated temperature-dependent Seebeck coeﬃcient of pristine CoSi under the RTA (red solid lines), the CRTA (blue dashed lines) and eDOS−1 approximation (orange dash-dotted lines) in comparison with experimental measurements [SLM13, RLZ05, SZP09,

SYA07]. (b) Calculated Seebeck coeﬃcient of CoSi1−xAlx at various fractions of Al substitution on Si site at 300 K compared with experiments [LKH04, LRZ05]. The black dashed lines are given as guides to the eye. c) Calculated temperature-dependent electrical resistivity of pristine CoSi in comparison with experiments [SLM13, RLZ05, SZP09, PKS10]. d) Cal- culated temperature-dependent power factor of pristine CoSi compared with experimental measurements [SLM13, RLZ05, SZP09].

92 Eq. 3.1, Eq. 3.3, and Eq. 3.2, and comparing the RTA and CRTA results in Fig. 4.12a, it is conclusive that strong energy-dependence of τ(E) plays the central role in determining the overall transport behavior. CRTA predicts small to zero α because while N 0(E) is negative v0(E) is positive across the flat-and-linear crossing, negating each other’s contribution. When τ 0(E) is accounted for, however, a very large Seebeck coefficient of −80 µV/K or higher is predicted, to the validation of experimental measurements. The experimentally observed hole-doping-induced sign change of α (Al substitution on the Si site) is also predicted and confirmed [see Fig. 4.12b]. It is found that 4% Al substitution is the critical point that induces a sign change of α from negative to positive, again agreeing well with experiments [LKH04, LRZ05]. At high Al concentration (>10%) the calculated α is about one half of the experimental values [LKH04, LRZ05]. This discrepancy is ascribed mainly to the possible distortions in the electronic structure due to heavy doping. Fig. 4.12c compares the calculated and experimental electrical resistivity.

4.3.3 Electron-phonon Scattering

Fig. 4.13b clearly shows the otherwise long electron lifetimes dropping rapidly near the Fermi level. Analysis of the band structure elucidates that it is the abrupt change of eDOS slightly above the Fermi level, owing to the flat band, that leads to preferential scattering of low-energy electrons in the massless bands (energy filtering). In addition, the hole lifetimes are very short due to the much larger hole eDOS near the Fermi level, allowing electrons to dominate transport. These effects create a large gradient in τ(E). The eDOS−1 approximation, despite its simplicity, qualitatively reflects the energy-dependence of τ(E) and thereby the asymmetric behavior of Σ(E). This suggests that e-ph scattering phase space probably dominates over the individual magnitudes of e-ph scattering matrix elements in determining the energy-dependence of carrier lifetimes.

Due to the abrupt change of eDOS, it is crucial to use an accurately determined Fermi level at each temperature by accounting for the Fermi-Dirac distribution. Our calculated resistivity exhibits unusual temperature dependence, that is, it is nearly independent of

93 Figure 4.13: a) Calculated transport distribution function σ(E) under the RTA (red solid lines), the CRTA (blue dashed lines) and the eDOS−1 approximation (orange dash-dotted lines) at 300 K. The light green shaped area indicates the energy window restricted by the energy derivative of Fermi-Dirac distribution. b) Calculated energy-dependent carrier lifetimes at 100 K (blue triangles), 300 K (green squares) and 600 K (red circles) compared with scaled eDOS−1 approximation (orange dash-dotted lines). temperature above 300 K [see Fig. 4.12c]. This contrasts the typical behavior of increasing resistivity with temperature due to intensifying e-ph scattering. To no surprise, the reduction in lifetime with increased temperature is observed, as shown in Fig. 4.13b. However, the Fermi level also shifts towards higher energy due to the broadening Fermi-Dirac distribution. Concomitantly, the mean Fermi velocity increases as more electrons are excited into the Dirac bands, partially counteracting the eﬀect of lifetime reduction. The net result is the nearly temperature-independent electrical resistivity from 300 to 600 K. To further conﬁrm this prediction, experimental measurements on high-quality single crystal CoSi at temperatures above 300 K are suggested.

94 4.3.4 Lessons

The high negative Seebeck coefficient in CoSi arises from the heavily energy-dependent carrier lifetimes due to electron-phonon scattering, an intrinsic mechanism. This is made possible by a heavy band that crosses linear bands near the Fermi level, providing a large phase space for electron-phonon scattering for holes and low-energy electrons, leading to efficient energy filtering. Recently, it has been brought to light that for Dirac-like bands, the electron mean free path (MFP) monotonically decreases with energy, leading to the concept of filtering electrons by long MFP, which automatically filters out low-energy electrons [LZL18]. How- ever, this has required nanostructuring of 10-nm grains in SnTe. Energy filtering by inherent e-ph scattering as in CoSi does not require such an expensive engineering process, another advantage. Nonetheless, some degree of nanostructuring of CoSi would still benefit zT by reducing lattice thermal conductivity, since its zT is severely handicapped by the large bulk

−1 −1 κl that exceeds 10 W m K at 300 K [SM12].

Furthermore, the coexistence of massless and heavy bands in CoSi is reminiscent of flat- and-dispersive,“low-dimensional” band structures in semiconductors (Fe2TiSi and Ba2BiAu). As discussed in the previous sections, such a band structure is not as beneficial as purely dispersive bands for semiconductors. This owes partly to the fact that bipolar transport is not an issue for doped semiconductors, and α is naturally large. Rather, the flat band invites too much scattering of already well-separated majority carriers, a severe detriment to mobility and consequently to the PF. This directly contrasts the case of metals, as showcased by CoSi, for which the categorical need to sustain non-negligible α outweighs the loss of lifetimes due to scattering. Because metals usually have high conductivity and even mobility, a small increase in α may lead to sizable PF. These results and analyses call for renewed efforts in identifying high-performance thermoelectrics among metals.

95 CHAPTER 5

Optimal Electronic Structures for Thermoelectricity

With the lessons learned from the materials studied, and with deeper analyses and considerations of thermoelectric theories, this chapter establishes the optimal electronic structures for both metallic and semiconducting thermoelectrics. To that end, the Mahan-Sofo analysis (introduced in the first chapter) is reviewed and its implications reassessed. A double parabolic band transport model based on Boltzmann transport equations is used to show that the Seebeck coefficient is not a natural function of effective mass.

The essence for metals is to isolate majority carriers from minority carriers in order to generate non-negligible Seebeck coefficients. Due to the continuous electronic states across the Fermi level, such a separation must be achieved by differential behaviors in carrier lifetimes. When one side of the Fermi level experiences heavily preferential scattering, then the other side is allowed to dominate transport not via population (as in doped semiconductors) but via longer lifetimes and higher mobility. To achieve such a phenomenon intrinsically, a band structure must accommodate preferential scattering of minority carriers and low-energy majority carriers in addition to high group velocity for the majority carriers. To that end, localized bands with a significantly higher phase space for electron-phonon scattering on one side of the Fermi level and high-velocity bands on the other constitutes the ideal electronic structure for metallic thermoelectrics.

Semiconductors inherently have band gaps, which play two major roles for thermoelectricity. First, it is a natural filter for minority carriers, and the Seebeck coefficient is allowed to be generally high. Second, because the eDOS at the band edges rise from 0, the Seebeck coefficient is rendered insensitive to the magnitude of the electronic density of states at the

96 band edges for degenerately doped semiconductors. Because bipolar transport is inhibited and the Seebeck coefficient is high by default, the urgent ingredient for high semiconductor thermoelectric performance is high mobility. High mobility closes the gap between the peaks of the Seebeck coefficient and electrical conductivity with respect to doping, thereby offering a solution to the long-standing problem of the anticomplementary nature of the two properties. In all, a set of highly dispersive bands at off-symmetry points in the Brillouin zone (which entails pocket multiplicity) constitutes the ideal band structure for semiconductor thermoelectrics.

5.1 Revisiting the Mahan-Sofo Analysis

It is important to keep in mind that Mahan and Sofo’s 1996 approach was purely mathematical in nature without much consideration of real electronic structures or materials. Eq. 1.17 must be interpreted with caution when applied to reality. While they do discuss non- idealities of real structures in their paper, a general clariﬁcation of the implications of their work in the contexts of real electronic structures and materials is in order. In particular, Eq. 1.17 must be interpreted diﬀerently in semiconducting and metallic environments.

For Eq. 1.17 to be satisﬁed, mathematically, at least one of N(E), v2(E), or τ(E) must

be δE,E† . However, physical considerations forbid the terms from being independently δE,E† . Firstly, v2(E) cannot be a delta function while N(E) is not, because a non-zero value of

2 v (E) categorically cannot occur without some band dispersion around EF, while no band dispersion can arise at all if N(E) has no width. Secondly, provided there is some dispersion, τ(E) cannot be a delta function unless electrons are perfectly scattered everywhere but at single EF. This is next to impossible, if not quite absolutely impossible as the relationship between v2(E) and N(E). Hence, the only plausible way in which τ(E) can be a delta function is if N(E) is also. These two considerations indicate that the only way for Eq.

1.17 to hold is for N(E) = NbδE,E† , reﬂecting one or more (Nb) perfectly localized states, or perfectly ﬂat bands. The factor Nb arises from the fact that eDOS of each band must

97 integrate to 1 (or 2 if spin-degenerate) to conserve the number of electrons:

N Xb Z ∞ Nb = δE,E† dE. (5.1) 1 −∞ This in turn necessarily means that v2(E) = 0 everywhere, and hence Σ(E) = 0. The implication of this is twofold. Firstly, the conductivity is immediately 0, as has also been pointed out by a previous study [ZYC11], Z ∞ ∂f σ = Σ(E) − dE = 0. (5.2) −∞ ∂E Secondly, the Seebeck coeﬃcient is expressible as the following limit as Σ(E) → 0 point-by- point: 1 R ∞ ∂f T −∞ Σ(E)(EF − E) − ∂E dE α = lim ∞ . (5.3) Σ(E)→0 R ∂f −∞ Σ(E) − ∂E dE While this is a non-trivial limit to evaluate generally, because of the widthless Σ(E), the limit can be reformulated as 1 lim Σ(E) = lim Σ0δE,E† , (5.4) Σ(E)→0 n→∞ n where n ≥ 1 is an integer, and evaluate Eq. 5.3 as

1 1 R ∞ ∂f n T −∞ Σ0δE,E† (EF − E) − ∂E dE α = lim ∞ n→∞ 1 R ∂f n −∞ Σ0δE,E† − ∂E dE (5.5) 1 = (E − E†). T F

Because of ﬁnite Seebeck and zero conductivity, the PF would be negligible, and since κl > 0 in real materials, zero PF would lead to zT = 0. Therefore, even if a set of perfectly localized states could exist in real materials, it is not the physically ideal structure for zT or the PF. The fundamental reason for this is, again, that the components of Σ(E) cannot be independently widthless. Instead, given some optimum E†, widthless N(E) would lead to optimum α for the “electronic-part zT ,” α2σ α2 zTe = T = , (5.6) κe L

which assumes κl = 0, where L is the Lorenz number. However, this optimum zTe occurs

only as the PF and κe simultaneously tend to 0, meaning that a ﬁnite κl would enforce zero

98 zT all the same. The lesson is that a simple provision of sharp and high eDOS, while yielding

ﬁnite and optimal Seebeck coeﬃcient for the unrealistic zTe, does not by itself accomplish the holistic goal of delivering high thermoelectric performance.

The utility of localized states is not lost, however, because Σ is related to eDOS not only directly but also indirectly – through group velocities and carrier lifetimes. Not only do τ(E) and v2(E) directly scale σ, but their behaviors can signiﬁcantly alter the magnitude of α as well. Eq. 3.1 instructs that α would also beneﬁt from sharp energy-gradients of τ(E) and hv2(E)i. As discussed, the behaviors of these quantities are tied to the that of

eDOS, so the same interdependencies that render lone localized states near EF useless could help them manifest in a synergetic manner. For one, eDOS is a very good indicator of the phase space for carrier scattering, particularly that by acoustic phonons [WWL17] – so much so that simple τ(E) ∝ N −1(E) approximation often leads to correct characterization of α [XV14, ?, XPZ19]. This indicates that eDOS can form in such a way electrons are preferentially scattered into those states, e.g., by phonons, as to create large τ 0(E). If the states which carriers are scattered from have generally high τ(E) and v2(E), and hence high µ, then simultaneously high σ and α should be achievable, just as in CoSi [XPZ19]. This would require a set of large eDOS near EF (possibly localized states) and dispersive bands elsewhere, which is a structure that can certainly occur in real materials. Furthermore, hv2(E)i0 could as well be high if there is an abrupt change from ﬂat bands to dispersive bands.

5.2 Revisiting the Role of Eﬀective Mass

The aim of this section is to clarify that a smaller eﬀective mass is more beneﬁcial for the power factor and zT in conventional thermoelectrics. To show this, a double parabolic transport model is constructed to which Boltzmann transport equations in the relaxation time approximation are applied (Eqs. 2.8∼2.12). In treating τ, acoustic phonon scattering, polar-optical scattering, and ionized impurity scattering processes are approximately taken into account

99 As the name suggests, the model consists of one parabolic valence band and one parabolic conduction band, separated by a gap of 0.3 eV. While a simple model, it provides general insights to the variation of thermoelectric properties with respect to important physical parameters such as effective mass, doping, prevailing scattering mechanism, temperature and dielectric constant. It is an improvement upon single parabolic band model as it properly captures the bipolar effects due to the opposing band, which is of particular importance for the Seebeck coefficient and the Lorenz number. The model is quite relevant for high- performance thermoelectrics such as Ba2BiAu, whose conduction band of interest is in fact close to being parabolic. To cover a wider range of m, the conduction band is kept lighter than the valence band.

Ionized impurity scattering is treated with the standard Brooks-Herring theory (Eqs. 2.44∼2.46). Doping concentration (impurity concentration) is approximated by

3 2 ±E ∓E 2πmkBT F CBM,VBM n = e kBT (5.7) h2

Acoustic phonon scattering lifetime and mobility are estimated using the deformation potential theory [BS50], under which the lifetime is given by the following:

− 1 E 2 τdef ∝ 3 . (5.8) m 2 T ∆2

Polar optical scattering lifetime is approximated according to the following [NSG01]:

1 E 2 τpol ∝ 1 . (5.9) m 2 T

Constants are again ignored. This formulation captures the fact that τeph (and particularly

τdef) is proportional to the inverse density of states for parabolic bands. ∆ denotes the deformation potential constant, which is adjust separately for the valence band and the conduction band. The valence band ∆ is set to be ten times smaller than the conduction band ∆ to make τdef comparable to τii and τpol for holes, and also to provide holes as much “advantage”. Once the lifetimes at the electron-phonon limit and ionized-impurity-limit are calculated, the overall τ is obtained by Matthiesen’s rule. Then σ, µ, α, the power factor, κe,

100 a c e

b d f

Figure 5.1: (Color online) a) The double parabolic set-up. b) Electron and hole mobilities at various scattering regimes versus respective effective masses. c) The p-type and n-type Seebeck coefficients versus respective effective masses. d) The p-type and n-type power factors versus respective effective masses. e) The power factor versus the Fermi level. f) The Lorenz number versus the Fermi level, where the green horizontal line indicates the Wiedemann-Franz value. The quantities are computed at various scattering regimes – electron-phonon, ionized impurity, and the two processes at competition. All units are arbitrary except for that of the Lorenz number. For b, c, and d, the Fermi levels for holes and electrons are fixed to their band edges for all me-mh pairs. Also me and mh are varied such that

Eint is kept consistent for all pairing of valence and conduction bands (nh ≈ 180 ne). This way, doping concentration necessary to place the Fermi level at each band extremum has a consistent reference frame. For e and f, me and mh are kept ﬁxed as EF is varied.

101 and the Lorenz number, each limited by electron-phonon or ionized impurity or in between, are calculated.

−1 For a parabolic band, Brooks-Herring theory suggests that τii decreases as m if energy is referenced to the band minimum (no constant shift term). If energy is referenced diﬀerently, then the m−1 trend holds in the small-m limit but gradually transitions to an increasing

1 −1 m 2 · log (m) dependence as m grows larger. Therefore, it is clear that small m is favorable

for long τii and high µii. Meanwhile, τeph and µeph decrease monotonically respectively as

−1 − 3 −2 − 5 m ∼ m 2 and m ∼ m 2 as expected for acoustic phonon and polar-optical scattering

processes, which decrease more sharply with m than µii. As a result, whether in the electron- phonon limit, the ionized impurity limit, or in between, the overall µ almost always increases with decreasing m throughout a realistic range of m, as evidenced by Fig. 5.1b. Electrons with very small m are likely limited by ionized impurity scattering due to very weak electron- phonon scattering, but become increasingly dominated by the latter was m grows large. Holes, which are designed to be heavier than electron, have generally much lower mobility than electrons. Holes would be limited only by electron-phonon scattering no matter the eﬀective mass if the valence band ∆ were more comparable to the conduction band ∆.

Now let us turn our attention to the Seebeck coefficient. Fig. 5.1c clearly shows that α is independent of m, regardless of the scattering regime. This result demystifies a common belief that large m leads to high α since large m means large electronic density of states. While it is true that high Seebeck coefficients have often been associated with large effective masses, evidences (Eqs. 3.2, 5.5 and Fig. 5.1c) suggest that the Seebeck coefficient is an m-independent function of energy relative to the Fermi level particularly for parabolic bands.

In further clariﬁcation, the following expression of the Seebeck coeﬃcient is often coined for a single parabolic band in the degenerate or metallic limit,

2 2 2 8π k π 3 α = B mT , (5.10) 3eh2 3n and in the non-degenerate case, the following expression, 3 !! k 5 2 2πmk T 2 α = B + s + log B , (5.11) e 2 n h2

102 where n is the carrier concentration, m is effective mass, h is Planck’s constant, kB is Boltz- mann constant, and s is a scattering-dependent constant. These formulations seem to suggest that a large m, or a very flat band, is beneficial for the Seebeck coefficient. In reality, this is not the case upon examining the roots of the above formulations. The original form of Eq. 5.10 is in fact π2k2 T α = B , (5.12) 3eEF which is the direct result of applying Eq. 3.1 to a parabolic band. Only by replacing the Fermi level with 2 h2 3n 3 E = (5.13) F 8m π does one retrieve Eq. 5.10. Clearly, the dependence of α on m arises purely through the Fermi level’s relationship to carrier concentration. Likewise, the original form of Eq. 5.11 depends only on the Fermi level,

k 5 E α = B + s − F , (5.14) e 2 kBT whereby inserting the non-degenerate expression for EF results in Eq. 5.11. Because the

† † ultimate goal in thermoelectrics is to seek certain optimal EF that leads to α that maximizes the power factor or zT , Eqs. 5.12 and 5.14 provide a clearer guidance. They simply state

† † that, in achieving EF, n happens to be larger for a larger m and smaller for a smaller m. Because n is a variable that is more or less tunable at will via doping, it is clear that a larger m does not inherently lead to a higher α. As long as m and n are optimally paired, all values of m are equally capable of generating a given value of α. If the Seebeck coefficients were equal between a flat band and a dispersive band, the latter would certainly attain the higher power factor due to much higher group velocity and longer lifetimes, as seen in Fig. 5.1d. Higher mobility for smaller m and the m-independent Seebeck coefficient lead to the conclusion that the PF purely benefits from a smaller effective mass.

Finally, Fig. 5.1f shows that the Lorenz number (L) is mostly constant with a slight negative deviation from LWF in the direction of the lighter band. First, this suggests the validity of using LWF to estimate κe. for bands that are nearly parabolic, such as the

103 conduction band of Ba2BiAu. Plus, the combination of light band and negative deviation from LWF is ideal for both high power factor and high zT . Since the contribution κe. is

signiﬁcant in compounds with high PF and low κl, such as Ba2BiAu, the negative deviation

from LWF would lead to higher zT than expected. The model clearly established that, for a single parabolic band opposed by another parabolic band, a smaller m is always beneﬁcial for the power factor and zT .

5.3 Deduction of Optimal Electronic Structures

5.3.1 Semiconductors

Fe2TiSi and Ba2BiAu have also shown that ﬂat-and-dispersive bands do not provide an inherent advantage for α relative to a set of purely dispersive bands. The Mott formula (Eq. 3.1) in fact explains this observation: what improves α is N 0(E)/N(E), not just N 0(E) or N(E). For a given energy-gradient N 0(E), it is in fact more advantageous that N(E) is small. However for semiconductors, the eDOS at the band edge necessarily rises from zero, which in turn means that N 0(E) ∝ N(E) at the band edge, and it does not matter how large the eDOS is for the sake of the Seebeck coeﬃcient. Therefore, at a semiconductor band edge, a set of dispersive bands that exhibit high mobility is critical.

Our discussion of the semiconductor case has a very similar well-known concept, that

3 the weighted mobility, µm 2 Np, must be as high as possible [NSG01, PXA11, ZLF17], where m is the eﬀective mass. If µ is limited by acoustic phonon scattering, as it often is, the

− 5 3 −1 weighted mobility varies as m 2 m 2 = m , implying that dispersive bands with smaller m

are more beneficial, and the weighted mobility ought to be increased through Nb [PLW12]. Also, high Fermi surface complexity and high band anisotropicity ratio have been associated with high PF [GRL17, WWL17], though the anisotropicity cannot be too strong (e.g. flat- and-dispersive) as to invite harmful scattering events. In addition, when all pockets are close to being degenerate if not completely by symmetry, α suffers minimally if at all from the

multi-pocket eﬀect (Eq. 1.19) as all pockets deliver the same σp as weights. Note also that

104 the overall α remains the same no matter how many pockets there are as long as they all deliver the same σp – again a proof that having a large eDOS at a semiconductor band edge is not a necessary ingredient for α.

From an overarching perspective, the search for the next-generation semiconductor thermoelectrics must target highly dispersive bands with small m at off-symmetry points in the Brillouin zone and aligned at the band edge. These band structures benefit µ and σ while preserving high α. A qualitative schematic of such a band structure is given in Fig. 5.2a. It ought to be mentioned that while pocket multiplicity is generally deemed beneficial for the PF, the associated increase in intervalley scattering could outweigh the benefit by reducing carrier lifetimes and mobilities [XTL18, NV16]. Comparison of Sr2BiAu and Ba2BiAu indicates that thermoelectric performance would not necessarily increase monotonically with pocket multiplicity [?]. At low temperatures, the additional eDOS due to more dispersive pockets does not translate to additional scattering processes, and intravalley scattering remains the only relevant process. Accordingly, the additional pocket only works in favor of the PF by increasing carrier concentration per Fermi level, thereby enhancing σ. Given that there cannot be indefinitely many energy-aligned pockets to saturate the Brillouin zone, high pocket multiplicity is purely beneficial for the PF at low temperatures. At higher temperatures, on the other hand, intervalley scattering starts to interfere, reducing lifetimes and mobilities enough to result in lower overall PFs. There would then exist an optimum pocket multiplicity for a given set of pocket effective masses that ideally compromises high mobility and carrier concentration. In general, the more dispersive the pockets and the lower the temperature, the more unambiguously beneficial the pocket multiplicity.

5.3.2 Metals

The class of thermoelectric materials that is hampered by insufficiently high α is metals. In metals, due to the absence of band gaps, bipolar transport is unchallenged, leading to very small α in general. The flip side of the coin is that high σ and often high µ as well are guaranteed. The priority with metals then is to generate non-negligible Seebeck coefficient by

105 a b

Figure 5.2: (Color online) a) The ideal band structure for semiconductor thermoelectrics. b) The ideal band structure for metallic thermoelectrics. Both schematics are qualitative and from the perspective of promoting n-type performance.

opening a channel by which one type of carrier can dominate transport over the other. This requirement is identical to the theoretical stipulation that Σ(E) and its components sharply

vary across EF such that one carrier either dominates N(E) to outweigh the other through population, or dominates v2(E) or τ(E) to claim dominance through faster transport. Owing to continuous bands across EF in metals, the ﬁrst two terms in the bracket of Eq. 3.1, which are directly subject to the electronic structure, often do not contribute enough. One is then forced to rely on a large magnitude of the third, lifetime term – but how? This requires a heavily preferential scattering mechanism for minority carriers and low-energy majority carriers. In the extreme, a “lifetime gap” could be formed. Given an enough gap-width, a lifetime gap would function just as well as a band gap to separate a dominant charge carrier type. The dominant carriers generated as such would be majority carriers not in the sense of population but in the sense of transport. All the same, these majority carriers would lead to non-negligible α. When coupled with high µ and σ, even mildly high α can result in very high PF in metallic compounds.

Preferential scattering (ﬁltering) of low-energy carriers occurs when localized states around

EF and/or to one side of it oﬀers high eDOS and hence the necessary phase space for scattering events. Such beneﬁcial localized states can indeed be engineered with resonance levels

106 induced by impurities or defects, of which the Kondo resonance and scattering in YbAl3 is a prime example. Yet, it would be preferable if such a filtering effect arose intrinsically with no external intervention. The first-principles studies of CoSi and YbAl3 at high-T [LFJ17] have shown it can indeed occur intrinsically via electron-phonon scattering given a right electronic structure. In CoSi, a band crossing other high-mobility bands near EF in a flat manner before curving towards the valence bands filters out holes and low-energy electrons via strongly energy-dependent e-ph scattering. In YbAl3 well above the Kondo-resonance temperature regime, the strong electron-phonon scattering invited by localized Yb 4f states just below the Fermi level filters out holes and sustains sizable n-type α.

Therefore, it is conclusive that the ideal band structure for metallic thermoelectrics is one in which a set of localized bands cross linear bands at the Fermi level and then bend to one side of it as to preferentially scatter (minority) carriers on that side and some low- energy (majority) carriers on the other side. A qualitative schematic of such a band structure is given in Fig. 5.2b. Moreover, linear or linear-ish bands are prevalent in metals, whose Seebeck coefficients do not benefit much from group velocities which have no energy-gradient. If flat bands cross linear bands appropriately near the Fermi level, they could create an extra contribution from hv2(E)i0 as well, to further improvement of α.

107 CHAPTER 6

Summary and Outlooks

6.1 Conclusions

The theoretical zT values of Ba2BiAu and Sr2BiAu speak for themselves. The two compounds constitute the ﬁrst instances where successful coexistence of very high power factor and ultralow lattice thermal conductivity leads to unprecedentedly high zT . The supreme

zT values of Ba2BiAu and Sr2BiAu could pave a new plateau in the real-world applications of thermoelectricity upon experimental realization, but this may be difficult as n-doping the compounds look energetically unfavorable [see Appendix]. They nevertheless offer a theoretical proof of concept that zT > 4 and high-performance at cryogenic temperatures is not out of reach within bulk compounds. Their high power factors originate from dispersive band pockets at off-symmetry points with inherent multiplicity. Highly dispersive bands minimize (acoustic) phonon scattering of majority carriers such that the mobility is high, while the naturally energy-aligned pockets sustain high carrier concentration for a given Fermi level. CoSi possesses very high power factors on par with some of the best semiconductor thermoelectrics, indicating that metals are not to be neglected in pursuit of next-generation thermoelectric materials. In CoSi, high Seebeck coefficient is established by naturally occurring energy-filtering of holes and low-energy electrons via electron-phonon scattering accommodated by a flat band crossing linear bands.

The compounds studied here shed light on the critical importance of mechanisms that

are intrinsic to the material. The key to high zT in Ba2BiAu and Sr2BiAu is that high power factor and low lattice thermal conductivity occur both due to intrinsic mechanisms that fully respect the decoupled nature of electronic and phonon transport. The vibrational

108 independence of Au atoms that lead to strong anharmonicity and ultralow κl imposes no effect on electronic transport. In turn, highly dispersive conduction band pockets responsible for high power factor have no effect on lattice dynamics. A simultaneous occurrence of the two would be much more difficult to achieve if the inherent properties were poor and external engineering interventions were required to improve them, which often cannot promote both at the same time. Hence going forward, focusing on intrinsic structures and mechanisms would increase the chance of identifying materials in which both high power factor and low lattice thermal conductivity arise without sabotaging the other.

A closer investigation of the seminal Mahan-Sofo analysis indicates that sharp features in the electronic density of states and localized bands by themselves do not lead to high thermoelectric performance. While they could lead to high Seebeck coefficients and the “electronic-part” zT , their power factors would be negligible. This reduces their relevance in the context of real materials which have non-zero lattice thermal conductivity. For metals, however, well-placed localized states among other dispersive or Dirac-like states would offer an ideal platform for preferential electron-phonon scattering to generate high Seebeck coefficients. For semiconductors the utility of localized band is much more limited due to the presence of band gaps that naturally filters out minority carriers and establish high Seebeck coefficients. Accordingly, the focus with semiconductor thermoelectrics ought to be on attaining high mobility through highly dispersive bands which exhibit high group velocity and lifetimes due to limited phase space for (acoustic) phonon scattering. In light of this, flat-and-dispersive bands are not ideal for semiconductors because it invites too much majority-carrier scattering, as exemplified by both Fe2TiSi and Ba2BiAu. In all, the qualitative schematics of band structures provided in Fig. 5.2 constitute ideal band structures for intrinsic thermoelectric performance, respectively for semiconductors and metals. These also constitute a systematic approach to finding high-performance, high-power-factor thermoelectric materials.

109 6.2 Future Outlooks

While Ba2BiAu and Sr2BiAu are very high-performing materials, the fact that these materials contain gold, one of the rarest and most expensive elements, will limit their mass-scale commercial integration even if they could be synthesized and properly doped. Research efforts for identifying thermoelectric materials exhibiting similar properties but composed of more earth-abundant, cheaper elements are critical going forward. Because earth-abundant elements tend to be light in mass, a main challenge would be to keep lattice thermal conductivity low. The physical properties of Ba2BiAu and Sr2BiAu may serve as important theoretical guidelines in these efforts. The ultralow κl in these compounds arise partly due to the heavy atomic mass that yield soft phonons, but largely due to the decoupling of Au vibration from the other atoms that leads to strong anharmonic rattling. The latter kind of phenomenon is not in principle exclusive to heavy atomic compositions, as it is mainly the result of bonding characteristics and that Au has little contribution to the valence band. In terms of the power factor, the study presented here provides a conclusive evidence that a multitude of highly dispersive pockets is optimal for high-performing semiconductors, and band structure that lead to efficient energy filtering is optimal for high-performing metals. Focusing on materials with similar electronic band structures, density of states, and phonon density of states to these compounds could expedite the overall searching process

Owing to the ever-growing computational power, application of machine learning and “big data” approaches to screen, discover, and inverse-design materials has burgeoned as of late and will make increasingly larger contributions in the future [FZ99, FZ99, BDC18, REA16, LZJ17, CZR18, PK16, SS16, AC16b, HGK16]. Materials Genome Initiative re- ﬂects these developments and looks to capitalize on the combined strengths of computational, theoretical and experimental approaches for rapid materials innovation [Pat11, JPC16, AC16a, GCH17, PJK14]. Such a trend has not taken exception to thermoelectrics [JSP16, GST17, Mad16, CGM13, CLM14, ZHA15, GGO16, GOM16, HAX16]. So far, learning of thermoelectric materials has been mostly based on relating numerous material pheno- types (chemical composition, crystal structure, and electronegativity etc.) to thermoelectric

110 properties. Automated search engines have been developed that can recommend materials with favorable thermoelectric properties based on users’ input chemical composition and space groups, and also in reverse allow screening of materials based on users’ input desired properties [GGO16, GOM16]. On the other hand, the perspective here is that the features directly critical to achieving next-generation thermoelectrics are encoded largely if not entirely in the electronic structure – a rather simple set of data expressible in both numbers and images. Incidentally, graphical pattern search of band structures for desired properties have been implemented for organic materials [BOG18, BGB17], and such an approach could be adopted for inorganic thermoelectrics. Databases such as Materials Project [JOH13] have begun compiling band structure and density of states diagrams and even phonon dispersion diagrams for many compounds. Going forward, the understanding of optimal electronic structures for high thermoelectric performance should enable a machine-learning pathway based on the electronic structure to rapidly screen and identify many more promising thermoelectric materials.

111 APPENDIX A

Stability, Defects, and Doping of Ba2BiAu and Sr2BiAu

Ba2BiAu and Sr2BiAu full-Heusler compounds are theoretically predicted to be highly eﬃ- cient n-type thermoelectric materials with unprecedentedly high zT at both low and high temperatures. For the compounds to experimentally realize their potentials, they ought to have a large region of stability and be n-dopable – desirably to their ideal carrier concentra-

19 −3 tions of ne ≈ 10 cm . Intrinsic point defects, notably vacancies and antisites, could play signiﬁcant roles in determining the intrinsic carrier concentration and dopability of a given compound. Defects may become charged by donating electrons (holes), resulting in intrinsic n(p)-type characteristics, or by trapping electrons (holes), hindering accessible n(p)-doping range. Charge-neutral defects may still alter the band structure of the host compound. In particular for the full-Heusler compounds under investigation, it is critical that the favored defects not only allow n-doping but also preserve the highly dispersive conduction band pockets along Γ − X and at the L-point – the features responsible for the high thermoelectric power factors. These lowermost conduction band pockets are composed almost entirely of Ba/Sr d states, an indication that defects involving Ba/Sr atoms may directly damage realizable thermoelectric performance and are therefore undesirable.

In this ﬁrst-principles study, DFT calculations are implemented to investigate in detail the stability of full-Heusler compounds and formation energies of all vacancies and antisite defects (charged and neutral) that may intrinsically occur in them. Our calculations indicate that Ba2BiAu and Sr2BiAu have similarly large regions of stability in the chemical potential space. In general, defects that involve Au have relatively lower formation energies, while those that do not have very high formation energies. These are indicative of the fact that

Au is very loosely bonded in Ba2BiAu and Sr2BiAu, inducing their strongly anharmonic

112 a b

Figure A.1: (Color online) a) An example of defected 2×2×2 supercell with a Ba/Sr vacancy

(VBa/Sr). b) The supercell’s (2 0 0) lattice plane. The location of the defect is marked with the red squares. Ba/Sr atoms are in green, Bi atoms are in purple, and Au atoms are in, well, gold. vibrational characteristics. In particular, defects involving Ba/Sr are very high in formation energies and would not form – a beneﬁcial trend for preserving the dispersive conduction band pocket. Among vacancies, the Au vacancy has the lowest formation energy, but still too high to practically matter. It is discovered that the BiAu antisite defect has by far the lowest formation energy of all, and low enough to have a practical inﬂuence. In particular, it exhibits negative formation energy when negatively charged in both compounds, and would act as electron traps and hinder n-doping endeavors. Consequently, n-doping of Sr2BiAu seems unlikely at best and impossible at worst.

A.1 Computational Methods

A.1.1 Defect Formation Energies

The standard protocol for calculating point defect formation energies is employed:

D D FH X ∆Ef = E − E − ∆Naµa + q(Ev + EF) + Ecor. (A.1) a

113 Here, ED is the total energy of a defected cell, EFH is the total energy of an undefected full-Heusler cell into which a defect is introduced, q is the charge on the defect, and ∆Na is the excess (positive) or deﬁcient (negative) number of atoms of element a in the defected cell relative to the undefected cell. For instance, if the defect is an AuBi antisite (Au in place of Bi), then ∆NAu = 1 and ∆NBi = −1. EF is a free parameter and represents the Fermi level as counted positively up from Ev, which in turn is the energy required to remove an electron from a given host, or the energy of a hole. Calculation of Ev becomes increasingly accurate with a larger supercell size, converging to the valence-band maximum (VBM) of the undefected cell: FH FH FH lim E (N) − Eq=1(N) = EVBM. (A.2) N→∞

Next, µa is the chemical potential of element a in the full-Heusler compounds, which is bounded by three conditions. 1) In the extreme a-rich condition, pure-a phases would form. This means that

0 µa ≤ µa = 0 eV (A.3) must be obeyed. 2) Binary phases that are in competition with the full-Heusler compounds impose a set of restrictions,

X CP CP Na µa ≤ ∆Ef , (A.4) a one for each competing phase. 3) In a thermodynamic equilibrium under which the full- Heusler compounds form and remain stable, it must be that

X FH FH Na µa = ∆Ef . (A.5) a

These three conditions constrict the set of µa to a space in which the full-Heusler compounds form without decomposing into other phases. The formation energies of the undefected compounds and the competing phases that occur in Eqs. A.4 and A.5 are

FH FH X FH 0 ∆Ef = E − Na Ea a (A.6) CP CP X CP 0 ∆Ef = E − Na Ea a

114 0 where Ea is the per-atom energy of a metallic pure-a phase. All competing phases are extracted from Materials Project [JOH13].

Lastly, Ecor is a correction term for ﬁnite-sized supercells with charged defects, which experience some ﬁctitious electrostatic interactions: those between periodic images of the defects due to the periodic boundary condition, and those between a defect and the homogeneous, jellium-like background charge that keeps the overall system neutral. These are corrected by the method of Markov and Payne [MP95],

q2α 2πqQ E = − , (A.7) cor 2L 3L3

where α is the Madelung constant, Q is the quadrupole moment, L is the supercell lattice parameter, and is the dielectric constant of the host compound. Clearly for charge-neutral

cells (q = 0), Ecor vanishes. While more accurate correction schemes have been proposed [FNW09, FNW11, LZ08], they are not employed for several reasons. First, supercells in use are larger in size (L > 16 A)˚ than the size-scale for which more accurate correction is

deemed critical. Second, the very high of Ba2BiAu and Sr2BiAu (respectively 22 and 33 as determined from density-functional perturbation theory [BGC01, GL97]) would screen these interactions from strongly forming over long distances and also improve the validity of Eq. A.7. Third, (probably) the main conclusions drawn from this work would not change with

D a minor betterment of the accuracy of ∆Ef .

A.1.2 Defect Concentration

Once the formation energies of defects are determined, defect concentrations are calculated by D −∆Ef k T nD = nsite · e B (A.8)

where nsite is the concentration of lattice sites where a given defect could form. In doing

D D so, it is assumed that ∆Gf = ∆Ef by neglecting the vibrational entropic and volume contributions to the Gibbs free energy of defect formation.

115 A.1.3 DFT Calculations

Vienna Ab initio Simulations Package (VASP) [KH93, KH94, KF96a, KF96b] is used to perform DFT calculations. SOC is incorporated throughout the entirety of our study.The projector-augmented wave (PAW) pseudopotentials [Bl94] with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional [PBE96] are used. A plane-wave cut-oﬀ energy of 500 eV and a 2 × 2 × 2 k-point mesh are employed throughout all self-consistent calculations. Accurate band gaps of the pure compounds were determined using the Tran-Blaha-modiﬁed Becke-Johnson potential (mBJ) [BJ06, TB09] with SOC incorporated. The results are 0.566 eV for Ba2BiAu and 0.532 eV for Sr2BiAu. In order to make the most conservative estimate of n-dopability, the shift in the gap is applied entirely to the conduction bands.

Undefected and defected supercells that are 2 × 2 × 2 expansions of the fully relaxed conventional cubic unit cells of Ba2BiAu and Sr2BiAu are utilized for energy calculations. Hence created undefected supercells contain N = 128 atoms, of which 64 are Ba/Sr atoms, 32 are Bi atoms, and 32 are Au atoms. Point defects are always placed at the center of a supercell as to maintain the full crystal symmetry of the undefected crystal (space group Fm3m,¯ 48 symmetry operations) and expedite computations. This is allowed for all types of point defects because a full-Heusler crystal structure remains identical upon the exchange of lattice sites between the Ba/Sr atoms and the Bi and Au atoms. A schematic of a defected cell, with an Ba/Sr vacancy at the cell center, is shown in Fig. A.1. For charged supercells, electrons are either removed or added according to the charge.

Relaxation of each defected supercell was performed under three different conditions: no relaxation, relaxation of atomic coordinates only, and relaxation of cell-shape and volume in addition to atomic coordinates. All relaxations were performed with the Methfessel- Paxton’s smearing scheme [MP89] to properly account for metallic characteristics of many of the defected supercells. The total energies of the the final configurations from the three relaxation runs were calculated again with the improved tetrahedron method [BJA94] in order to accurately sample electronic densities of states. The total energies of the three configurations were then compared, upon which the lowest energy configuration was selected

116 D (E in Eq. A.1). Pure elemental metallic states of Ba, Bi2, and Au were also relaxed in

0 geometry for total energy (Ea in Eq. A.6) calculations. All competing binary phases were also relaxed in geometry for the calculation of total energies (ECP in Eq. A.6).

A.2 Results & Discussions

A.2.1 Chemical Potentials & Stability

First studied are the stability Ba2BiAu and Sr2BiAu and the chemical potentials of each element associated with the formation of the compounds. Ba2BiAu has nine potentially competing binary phases, while Sr2BiAu has eleven potentially competing binary phases, all of which have been extracted from the Materials Project database [JOH13]. The ternary phase diagram, shown in Fig. A.2, is constructed using the calculated formation energies.

It indicates that Sr2BiAu may coexist with Sr2Bi3, SrAu, and SrAu2, while Ba2BiAu may coexist with Ba4Bi3, BaAu, and BaAu2. The equilibrium chemical potentials derived using the corresponding phase ﬁelds correspond to approximately µSr ≈ −0.94 eV, µBi ≈ −0.81 eV, and µAu ≈ −0.63 for Sr2BiAu, and µBa ≈ −0.67 eV, µBi ≈ −1.26 eV, and µAu ≈ −0.75

Ba2BiAu. These chemical potentials are subsequently used in the analysis of defect energies. Both compounds are stable in the immediate vicinity of these chemical potentials.

A.2.2 Defects

The fact that all defected supercells converge to the lowest-energy configuration when only atomic coordinates are relaxed (as opposed to when the volume and shape are also relaxed) indicates that the spheres of influence of the defects are confined within the supercells. This validates the size-sufficiency of the supercells used here as far as local effects of a defect are concerned. The small magnitudes of the correction term (Eq. A.7) reflect that high of the two compounds inhibit strong electrostatic interactions between defects and their periodic images, validating the overall size-sufficiency of the supercells used.

One general trend that stands out is that defects that involve Au are low in formation

117 a b

Figure A.2: (Color online) The ternary phase diagrams of a) Sr2BiAu and b) Ba2BiAu. The phase ﬁelds relevant to the the compounds are bounded by red lines.

a b

Figure A.3: (Color online) The formation energies of VAu (solid blue lines) and VBi (dotted red lines) in a) Sr2BiAu and b) Ba2BiAu. The slopes indicate the charges (−2 ∼ +2). The formation energies under the Bi-and-Au-poor condition is lower, and therefore shown.

EF = 0 corresponds to the VBM, and the black vertical lines correspond to the CBM determined using the band gaps calculated with mBJ+SOC.

118 a b

Figure A.4: (Color online) The formation energies of BiAu in a) Sr2BiAu and b) Ba2BiAu. The Bi-and-Au-poor condition (solid red lines) and the Bi-and-Au-rich condition (dotted blue lines) are both shown. The former must be targeted since it leads to higher defect formation energies. EF = 0 corresponds to the VBM, and the black vertical lines correspond to the CBM determined using the band gaps calculated with mBJ+SOC. energies while those that do not are high in formation energies. This is likely because at their sites, Au atoms are very weakly bonded in both compounds. After all, the vibrational independence of Au is precisely the phenomenon that leads to strong anharmonicity and ultralow lattice thermal conductivity in the two compounds. Of all vacancies, VAu exhibits the lowest formation energy. Nevertheless, vacancy formation energies are generally very high, above 3 eV per defect. These lead to negligible intrinsic vacancy concentrations, and it is safe to determine that vacancies will not pose any harm in practice.

By far the most energetically favored of all defects is the BiAu antisite defect in either compound, whose formation energies are shown in Fig. A.4. They are so low in energy and even negative such that a suﬃcient amount would form even at 0 K. Unfortunately,

BiAu renders the compounds naturally p-type when charge-neutral. The charge-neutral defects formation energy is 0.1 eV in Sr2BiAu and 0.2 eV in Ba2BiAu. Worse, near the CBM

119 determined by mBJ+SOC, these defects favor negatively charged states and hence are predicted to act as electron traps (acceptors), signiﬁcantly hampering n-doping. The situation is particularly severe for Sr2BiAu where the defect may accept up to −3 charge. The defect

21 −3 concentration is estimated to be on the order of nD ≈ 10 cm or higher, easily eclipsing

19 −3 the ideal electron doping concentration of ne ≈ 10 cm in either compound. Even without the band gap shift, at the CBM determined by plain PBE+SOC (0.22 eV for Sr2BiAu and

0.31 eV for Ba2BiAu), the negatively charged states attain nearly 0 eV of formation energies.

Therefore, it is determined that n-doping of either (in particular Sr2BiAu) is highly unlikely if not impossible. Fig. A.4 indicates however that the two compounds would be intrinsically p-type due to the much higher formation energy of donor states near the VBM (∆Ef ≈ 0.1

19 −3 eV, nD ≈ 2 × 10 cm at 300 K for Sr2BiAu). This indicates that the two compounds may be serviceable as p-type materials.

A.3 Conclusions

While Ba2BiAu and Sr2BiAu are theoretically capable of ultrahigh n-type thermoelectric performance, they seem unlikely to be realized. Unfortunately n-doping of neither compound seem facorable due to very low formation energy of BiAu, which is a natural p-type substitution. This conclusion calls for further search of high-performance thermoelectrics that share the characteristics of the two compounds but are favorably dopable.

120 REFERENCES

[AC16a] Ankit Agrawal and Alok Choudhary. “Perspective: Data infrastructure for high throughput materials discovery.” APL Mater., 4:053203, 2016.

[AC16b] Ankit Agrawal and Alok Choudhary. “Perspective: Materials informatics and big data: Realization of the fourth paradigm of science in materials science.” APL Mater., 4:053208, 2016.

[AGP56a] I. G. Austin, C. H. L. Goodman, and A. E. Pengelly. “New semiconductors with the chalcopyrite structure.” J. Elec. Chem. Soc., 103:609–610, 1956.

[AGP56b] I. G. Austin, C. H. L. Goodman, and A. E. Pengelly. “Semiconductors with chalcopyrite structure.” Nature, 178:433, 1956.

[AKT15] R. Ang, A.U. Khan, N. Tsujii, K. Takai, R. Nakamura, and T. Mori. “Thermo- electricity generation and electron-magnon scattering in a natural chalcopyrite mineral from a deep-sea hydrothermal vent.” Ang. Chem., 54:12909–12913, 2015.

[AKT17] R. Ang, A.U. Khan, N. Tsujii, K. Takai, R. Nakamura, and T. Mori. “The Role of Zn in Chalcopyrite CuFeS2: Enhanced Thermoelectric Properties of Cu1−xZnxFeS2 with In Situ Nanoprecipitates.” Adv. Energy Mater., 7:1601299, 2017.

[APC14] G. Antonius, S. Poncé, M. Coté, and X. Gonze. “Many-Body effects on the zero-point renormalization of the band structure.” Phys. Rev. Lett., 112:215501, 2014.

[BDC18] Keith T. Butler, Daniel W. Davies, Hugh Cartwright, Olexandr Isayev, and Aron Walsh. “Machine learning for molecular and materials science.” Nature, 559:546– 555, 2018.

[Bec88] Axel D. Becke. “Density-fnnctional exchange-energy approximation with correct asymptotic behavior.” J. Chem. Phys., 38:3098–3100, 1988.

[Bel08] Lon E. Bell. “Cooling, Heating, Generating Power, and Recovering Waste Heat with Thermoelectric Systems.” Science, 321:1457–1461, 2008.

[BGB17] Stanislav S. Borysov, R. Matthias Geilhufe, and Alexander V. Balatsky. “Organic materials database: An open-access online database for data mining.” PLoS One, 12:e0171501, 2017.

[BGC01] Stefano Baroni, Stefano de Gironcoli, and Andrea Dal Corso. “Phonons and related crystal properties from density-functional perturbation theory.” Rev. Mod. Phys., 73:515–562, 2001.

121 [BHB09] Kanishka Biswas, Jiaqing He, Ivan D. Blum, Chun-I Wu, Timothy P. Hogan, David N. Seidman, Vinayak P. Dravid, and Mercouri G. Kanatzidis. “High- performance bulk thermoelectrics with all-scale hierarchical architectures.” Na- ture, 489:414–418, 2009. [BHW13] Daniel I. Bilc, Geoffroy Hautier, David Waroquiers, Gian-Marco Rignanese, and Philippe Ghosez. “Markus Meinert and Manuel P. Geisler and Jan Schmalhorst and Ulrich Heinzmann and Elke Arenholz and Walid Hetaba and Michael Stöger- Pollach and Andreas Hütten,1and Günter Reiss.” Phys. Rev. B, 90:085127, 2013. [BHW15] Daniel I. Bilc, Geoffroy Hautier, David Waroquiers, Gian-Marco Rignanese, and Philippe Ghosez. “Low-Dimensional Transport and Large Thermoelectric Power Factors in Bulk Semiconductors by Band Engineering of Highly Directional Elec- tronic States.” Phys. Rev. Lett., 114:136601, 2015. [BJ06] Axel Becke and Erin Johnson. “A simple effective potential for exchange.” J. Chem. Phys., 124:221101, 2006. [BJA94] Peter E. Blöchl, O. Jepsen, and O.K. Anderson. “Improved tetrahedron method for Brillouin-zone integrations.” Phys. Rev. B, 49:3616–3621, 1994. [Bl94] P. E. Blöchl. “Projector augmented-wave method.” Phys. Rev. B, 50:17953– 17979, Dec 1994. [BOG18] Stanislav S. Borysov, Bart Olsthoorn, M. Berk Gedik, R. Matthias Geilhufe, and Alexander V. Balatsky. “Online search tool for graphical patterns in electronic band structures.” Comput. Mater., 4:1–8, 2018. [Bro55] H. Brooks. “Theory of the Electrical Properties of Germanium and Silicon.” Adv. Elec. Elec. Phys., 7:85–182, 1955. [BS50] J. Bardeen and W. Shockley. “Deformation Potentials and Mobilities in Non- Polar Crystals.” Phys. Rev., 80:72–80, May 1950. [BUB59] R. Bowers, R. W. Ure, J. E. Bauerle, and A. J. Cornish. “InAs and InSb as Thermoelectric Materials.” J. Appl. Phys., 30:930–934, 1959. [Cal48] Herbert Callen. “The Application of Onsager’s Reciprocal Relations to Thermo- electric, Thermomagnetic, and Galvanomagnetic Effects.” Phys. Rev,, 73:1949– 1958, 1948. [CAL15] Sergio Conejeros, Pere Alemany, Miquel Llunell, Ibéiode P. R. Moreira, V´ıctor Sáchez, and Jaime Llanos. “Electronic Structure and Magnetic Properties of CuFeS2.” Inorg. Chem., 54:4840–4849, 2015. [CGM13] Stefano Curtarolo, Marco Buongiorno Nardelli Gus L. W. Hart and, Natalio Mingo, Stefano Sanvito, and Ohad Levy. “The high-throughput highway to computational materials design.” Nat. Mater., 12:191–201, 2013.

122 [CLM14] Jes´usCarrete, Wu Li, Natalio Mingo, Shidong Wang, and Stefano Curtarolo. “Finding Unprecedentedly Low-Thermal-Conductivity Half-Heusler Semiconduc- tors via High-Throughput Materials Modeling.” Phys. Rev. X., 4:1019, 2014.

[CNF99] J. L. Cohn, G. S. Nolas, V. Fessatidis, T. H. Metcalf, and G. A. Slack. “Glasslike Heat Conduction in High-Mobility Crystalline Semiconductors.” Phys. Rev. Lett., 82:779, 1999.

[CQ81] D. Chattopadhyay and H. J. Queisser. “Electron Scattering by Ionized Impurities in Semiconductors.” Rev. Mod. Phys., 53:745–768, Oct 1981.

[CWH18] Cheng Chang, Minghui Wu, Dongsheng He, Yanling Pei, Chao-Feng Wu, Xuefeng Wu, Hulei Yu, Fangyuan Zhu, Kedong Wang, Yue Chen, Li Huang, Jing-Feng Li, Jiaqing He, and Li-Dong Zhao. “3D charge and 2D phonon transports leading to high out-of-plane ZT in n-type SnSe crystals.” Science, 360:778–783, 2018.

[CZR18] Kamal Choudhary, Qin Zhang, Andrew C.E. Reid, Sugata Chowdhury, Nhan Van Nguyen, Zachary Trautt, Marcus W. Newrock, and Faical Yannick Con- goand Francesca Tavazza. “Data Descriptor: Computational screening of high-performance optoelectronic materials using OptB88vdW and TB-mBJ for- malisms.” Sci. Data, 5:1–12, 2018.

[DBS98] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton. “Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study.” Phys. Rev. B, 57:1505–1509, 1998.

[DCD58] G. Donnay, L. M. Corliss, J. D. H. Donnay, N. Elliot, and J. M. Hastings. “Sym- metry of magnetic structures: magnetic structure of chalcopyrite.” Phys. Rev., 112:1917–1923, 1958.

[DCT07] Mildred S. Dresselhaus, Gang Chen, Ming Y. Tang, Ronggui Yang, Hohyun Lee, Dezhi Wang, Zhifeng Ren, Jean-Pierre Fleurial, and Pawan Gogna. “New Di- rections for Low-Dimensional Thermoelectric Materials.” Adv. Mater., 19:1043– 1053, 2007.

[FIL15] Alireza Faghaninia, Joel W. Ager III, and Cynthia S. Lo. “Ab initio electronic transport model with explicit solution to the linearized Boltzmann transport equation.” Phys. Rev. B, 91:235123, 2015.

[FNW09] Christoph Freysoldt, J¨orgNeugebauer, and Chris G. Van de Walle. “Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations.” Phys. Rev. B, 102:6402, 2009.

[FNW11] Christoph Freysoldt, J¨orgNeugebauer, and Chris G. Van de Walle. “Fully Ab Initio Finite-Size Corrections for Charged-Defect Supercell Calculations.” Phys. Stat. Solid., 248:1067, 2011.

123 [FZ99] Alberto Franceschetti and Alex Zunger. “The inverse band-structure problem of ﬁnding an atomic conﬁguration with given electronic properties.” Nature, 402:60–63, 1999.

[FZP14] Chenguang Fu, Tiejun Zhu, Yanzhong Pei, Hanhui Xie, Heng Wang, G. Jef- frey Snyder, Yong Liu, Yintu Liu, and Xinbing Zhao. “High Band Degeneracy Contributes to High Thermoelectric Performance in p-Type Half-Heusler Com- pounds.” Adv. Energy Mater., 4:1400600, 2014.

[GAA09] X. Gonze, B. Amadon, P.M. Anglade, J.-M. Beuken, F. Bottin, P. Boulanger, F. Bruneva, D. Caliste, R. Caracas, M. Cot´e, T. Deutsch, L. Genovese, M. Gi- antomassi Ph. Ghosez, S. Goedecker, D. Hamann, P. Hermet, F. Jollet, G. Jo- mard, S. Leroux, M. Mancini, S. Mazevet, M.J.T. Oliveira, G. Onida, Y. Pouil- lon, T. Rangel, G.-M. Rignanese, D. Sangalli, R. Shaltaf, M. Torrent, G. Zrah M.J. Verstraete, and J.W. Zwanziger. “ABINIT : ﬁrst-principles approach to material and nanosystem properties”.” Comput. Phys. Commun., 180:2582–2615, 2009.

[GAB17] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M Buongiorno Nardelli, M Calandra, R. Car, C. Cavazzoni, D. Ceresoli, and M. Cococcioni. “Advanced capabilities for materials modeling with Quantum ESPRESSO.” J. Phys. Con- dens. Matter, p. 465901 (31pp), 2017.

[GBB09] Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Matteo Calandra, Roberto Car, Carlo Cavazzoni, Davide Ceresoli, Guido L Chiarotti, Matteo Cococcioni, Is- maila Dabo, Andrea Dal Corso, Stefano de Gironcoli, Stefano Fabris, Guido Fratesi, Ralph Gebauer, Uwe Gerstmann, Christos Gougoussis, Anton Kokalj, Michele Lazzeri, Layla Martin-Samos, Nicola Marzari, Francesco Mauri, Riccardo Mazzarello, Stefano Paolini, Alfredo Pasquarello, Lorenzo Paulatto, Carlo Sbrac- cia, Sandro Scandolo, Gabriele Sclauzero, Ari P Seitsonen, Alexander Smogunov, Paolo Umari, and Renata M Wentzcovitch. “QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials.” J. Phys. Condens. Matter, p. 395502 (19pp), 2009.

[GBC02] X. Gonze, J.-M. Beuken, R. Caracas, F. Detraux, M. Fuchs, G.-M. Rignanese, L. Sindic, M. Verstraete, G. Zerah, F. Jollet, M. Torrent, M. Mikami A. Roy, Ph. Ghosez, J.-Y. Raty, and D.C. Allan. “First-principles computation of material properties: the ABINIT software project.” Comput. Mater. Sci., 25:478–492, 2002.

[GCH17] M. L. Green, C. L. Choi, J. R. Hattrick-Simpers, A. M. Joshi, I. Takeuchi, S. C. Barron, E. Campo, T. Chiang, S. Empedocles, J. M. Gregoire, A. G. Kusne, J. Martin, A. Mehta, K. Persson, Z. Trautt, J. Van Duren, and A. Zakutayev. “Fulﬁlling the promise of the materials genome initiative with high-throughput experimental methodologies.” Appl. Phys. Rev., 4:011105, 2017.

124 [GCL07] Feliciano Giustino, Marvin L. Cohen, and Steven G. Louie. “Electron-phonon interaction using Wannier functions.” Phys. Rev. B, 76:165108, 2007.

[GGO16] Prashun Gorai, Duanfeng Gao, Brenden Ortiz, Sam Miller, Scott A. Barnett, Thomas Mason, Qin Lv Vladan Stevanovi´c,and Eric S.Toberer. “TE Design Lab: A virtual laboratory for thermoelectric material design.” Comput. Mater. Sci., 112:368–376, 2016.

[Giu17] Feliciano Giustino. “Electron-phonon interactions from ﬁrst principles.” Rev. Mod. Phys., 89:015003, 2017.

[GL97] Xavier Gonze and Changyol Lee. “Dynamical matrices, Born eﬀective charges, dielectric permittivity tensors, and interatomic force constants from density- functional perturbation theory.” Phys. Rev. B, 55:10355–10368, Apr 1997.

[GLC10] Feliciano Giustino, Steven G. Louie, and Marvin L. Cohen. “Electron-Phonon Renormalization of the Direct Band Gap of Diamond.” Phys. Rev. Lett., 105:265501, 2010.

[GO09] T. Goldstein and S. Osher. “The split bregman method for L1-regularized problems.” SIAM J. Imaging Sci., 2:323–343, 2009.

[GOM16] Michael W. Gaultois, Anton O. Oliynyk, Arthur Mar, Taylor D. Sparks, Gre- gory J. Mulholland, and Bryce Meredig. “Perspective: Web-based machine learning models for real-time screening of thermoelectric materials properties.” APL Mater., 4:3213, 2016.

[Gon95a] Xavier Gonze. “Adiabatic density-functional perturbation theory.” Phys. Rev. A, 52:1096–1114, Aug 1995.

[Gon95b] Xavier Gonze. “Perturbation expansion of variational principles at arbitrary order.” Phys. Rev. A, 52:1086–1095, Aug 1995.

[GOS05] Franck Gascoin, Sandra Ottensmann, Daniel Stark, Sossina M. Ha¨ıle,and G. Jef- frey Snyder. “Zintl Phases as Thermoelectric Materials: Tuned Transport Prop- erties of the Compounds CaxYb1±xZn2Sb2.” Adv. Funct. Mater., 15:1860–1864, 2005.

[GRL17] Zachary M. Gibbs, Francesco Ricci, Guodong Li, Hong Zhu, Kristin Persson, Gerbrand Ceder, Geoﬀroy Hautier, Anubhav Jain, and G. Jeﬀrey Snyder. “Ef- fective mass and Fermi surface complexity factor from ab initio band structure calculations.” Comput. Mater., 3:1–7, 2017.

[GRV05] X. Gonze, G.-M. Rignanese, M. Verstraete, Y. Pouillon J.-M. Beuken, R. Cara- cas, M. Torrent F. Jollet, G. Zerah, M. Mikami, Ph. Ghosez, M. Veithen, J.-Y. Raty, V. Olevano, F. Bruneval, L. Reining, R. Godby, G. Onida, D.R. Hamann, and D.C. Allan. “A brief introduction to the ABINIT software package.” Zeit. Kristallogr, 220:558–562, 2005.

125 [GST17] Prashun Gorai, Vladan Stevanovi´c,and Eric S. Toberer. “Computationally guided discovery of thermoelectric materials.” Nat. Rev. Mater., 2:1–16, 2017.

[HAX16] J. He, M. Amsler, Y. Xia, S. S. Naghavi, V. Hegde, S. Hao, S. Goedecker, V. Ozoli¸nˇs,and C. Wolverton. “Ultralow Thermal Conductivity in Full Heusler Semiconductors.” Phys. Rev. Lett., 117:046602, 2016.

[HCY18] Min Hong, Zhi-Gang Chen, Lei Yang, Yi-Chao Zou, Matthew S. Dargusch, Hao Wang, and Jin Zou. “Realizing zT of 2.3 in Ge1−x−ySbxInyTe via Reducing the Phase-Transition Temperature and Introducing Resonant Energy Doping.” Adv. Mater., 30:1705942, 2018.

[HD93] L. D. Hicks and M.S. Dresselhaus. “Eﬀect of quantum-well structures on the thermoelectric Figure of merit.” Phys. Rev. B, 47:12727–12731, 1993.

[Hed65] Lars Hedin. “New Method for Calculating the One-Particle Green’s Function with Application to the Electron-Gas Problem.” Phys. Rev., 139:A797–A823, 1965.

[HGK16] Jason R. Hattrick-Simpers, John M. Gregoire, and A. Gilad Kusne. “Perspec- tive: Composition-structure-property mapping in high-throughput experiments: Turning data into knowledge.” APL Mater., 4:053211, 2016.

[HJT08] Joseph P. Heremans, Vladimir Jovovic, Eric S. Toberer, Ali Saramat, Ken Kurosaki, Anek Charoenphakdee, Shinsuke Yamanaka, and G. Jeﬀrey Snyder. “Enhancement of Thermoelectric Eﬃciency in PbTe by Distortion of the Elec- tronic Density of States.” Science, 321:554–557, 2008.

[HKM16] Ran He, Daniel Kraemer, Jun Mao, Lingping Zeng, Qing Jie, Yucheng Lan, Chunhua Lie, Jing Shuai, Hee Seok Kim, Yuan Liu, David Broido, Ching-Wu Chu, Gang Chen, and Zhifeng Ren. “Achieving high power factor and output power density in p-type half-Heuslers Nb1−xTixFeSb.” Proc. Natl. Acad. Sci., 113:13576–13581, 2016.

[HLG04] Kuei Fang Hsu, Sim Loo, Fu Guo, Wei Chen, Jeﬀrey S. Dyck, Ctirad Uher, Tim Hogan, E. K. Polychroniadis, and Mercouri G. Kanatzidis. “Cubic AgPbmSbTe2+m: Bulk Thermoelectric Materials with High Figure of Merit.” Sci- ence, 303:818–821, 2004.

[HSE03] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. “Hybrid functionals based on a screened Coulomb potential.” J. Chem. Phys., 118:8207–8215, 2003.

[HSE04] Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. “Eﬃcient hybrid density functional calculations in solids: Assessment of the Heyd–Scuseria–Ernzerhof screened Coulomb hybrid functional.” J. Chem. Phys., 121:1187–1192, 2004.

[HT17] Jian He and Terry M. Tritt. “Advances in Thermoelectric Materials Research: Looking back and moving forward.” Science, 357:1–9, 2017.

126 [HWC12] Joseph P. Heremans, Bartlomiej Wiendlochaac, and Audrey M. Chamoire. “Reso- nant levels in bulk thermoelectric semiconductors.” Energy Environ. Sci., 5:5510– 5530, 2012.

[HWZ15] L. Hu, H. Wu, T. Zhu, C. Fu, J. He, P. Ying, and X. Zhao. “Tuning multiscale microstructures to enhance thermoelectric performance of n-type bismuth-telluride- based solid solutions.” Adv. Energy Mater., 5:1500411, 2015.

[JM85] William Jones and Norman H. March. Theoretical Solid State Physics. Dover Books, 1985.

[JOH13] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Perssson. “Commentary: The Mate- rials Project: A materials genome approach to accelerating materials innovation.” APL Mater., 1:1002, 2013.

[JPC16] Anubhav Jain, Kristin A. Persson, and Gerbrand Ceder. “Research Update: The materials genome initiative: Data sharing and the impact of collaborative ab initio databases.” APL Mater., 4:053102, 2016.

[JSP16] Anubhav Jain, Yongsoo Shing, and Kristin A. Persson. “Computational predictions of energy materials using density functional theory.” Nat. Rev. Mater., 1:1–13, 2016.

[JT10] C. Rostgaard K. W. Jacobsen and K. S. Thygesen. “Fully self-consistent GW calculations for molecules.” Phys. Rev. B, 81:085103, 2010.

[KBS07] Susan M. Kauzlarich, Shawna R. Browna, and G. Jeﬀrey Snyder. “Zintl phases for thermoelectric devices.” Dalton Trans., 5:2099–2107, 2007.

[KF96a] G. Kresse and J. Furthm¨uller. “Eﬃciency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set.” Comput. Mater. Sci., 6:15–50, 1996.

[KF96b] G. Kresse and J. Furthm¨uller. “Eﬃcient iterative schemes for ab initio total- energy calculations using a plane-wave basis set.” Phys. Rev. B, 54:11169–11186, Oct 1996.

[KGV14] V. V. Klekovkina, R. R. Gainov, F. G. Vagizov, A. V. Dooglav, V. A. Golo- vanevskiy, and I. N. PenOkov.˜ “Oxidation and magnetic states of chalcopyrite CuFeS2: a ﬁrst-principles calculation.” Opt. and Spec., 116:885–888, 2014. [KH93] G. Kresse and J. Hafner. “Ab initio molecular dynamics for liquid metals.” Phys. Rev. B, 47:558–561, Jan 1993.

[KH94] G. Kresse and J. Hafner. “Ab initio molecular-dynamics simulation of the liquid-metal˘amorphous-semiconductor transition in germanium.” Phys. Rev. B, 49:14251–14269, May 1994.

127 [KHK09] Yoon-Suk Kim, Kerstin Hummer, and Georg Kresse. “Accurate band structures and eﬀective masses for InP, InAs, and InSb using hybrid functionals.” Phys. Rev. B, 80:035203, 2009.

[KLM15] Sang Il Kim, Kyu Hyoung Lee, Hyeon A Mun, Hyun Sik Kim, Sung Woo Hwang, Jong Wook Roh, Dae Jin Yang, Weon Ho Shin, Xiang Shu Li, Young Hee Lee, G. Jeﬀrey Snyder, and Sung Wng Kim. “Dense dislocation arrays embedded in grain boundaries for high-performance bulk thermoelectrics.” Science, 348:109– 114, 2015.

[KOS12] N. Kanazawa, Y. Onose, Y. Shiomi, S. Ishiwata, and Y. Tokura. “Band-ﬁlling dependence of thermoelectric properties in B20-type CoGe.” Appl. Phys. Lett., 100(9), 2012.

[KS65] W. Kohn and L. J. Sham. “Self-consistent equations including exchange and correlation eﬀects.” Phys. Rev., 140:A1133–A1138, Nov 1965.

[LCK14] Wu Li, Jes´usCarrete, Nebil A. Katcho, and Natalio Mingo. “ShengBTE: A solver of the Boltzmann transport equation for phonons.” Comput. Phys. Commun., 185:1747–1758, 2014.

[LEL14] Sangyeop Lee, Keivan Esfarjani, Tengfei Luo, Jiawei Zhou, Zhiting Tian, and Gang Chen. “Resonant bonding leads to low lattice thermal conductivity.” Nat. Commun., 5:3525–3532, 2014.

[LFJ17] Jinghua Lianga, Dengdong Fana, Peiheng Jianga, Huijun Liua, and Wenyu Zhao. “First-principles study of the thermoelectric properties of intermetallic compound YbAl3.” Intermetallics, 87:27–30, 2017. [LHM16] C. W. Li, J. Hong, A. F. May, D. Bansal, S. Chi, T. Hong, G. Ehlers, and O. Delaire. “Orbitally driven giant phonon anharmonicity in SnSe.” Nat. Phys., 11:1063–1069, 2016.

[LKH04] C. S. Lue, Y.-K. Kuo, C. L. Huang, and W. J. Lai. “Hole-doping eﬀect on the thermoelectric properties and electronic structure of CoSi.” Phys. Rev. B, 69:125111, Mar 2004.

[LKJ16] Weishu Liu, Hee Seok Kim, Qing Jie, and Zhifeng Ren. “Importance of high power factor in thermoelectric materials for power generation application: A perspective.” Scrip. Mater., 111:3–9, 2016.

[LLZ10] J. Li, W. Liu, L. Zhao, and M. Zhou. “High-performance nanostructured thermoelectric materials.” NPG Asia Mater., 2:152–158, 2010.

[LLZ12] Fu Li, Jing-Feng Li, Li-Dong Zhao, Kai Xiang, Yong Liu, Bo-Ping Zhang, Yuan- Hua Lin, Ce-Wen Nana, and Hong-Min Zhuc. “Polycrystalline BiCuSeO oxide as a potential thermoelectric material.” Energy Environ. Sci., 5:7188–7195, 2012.

128 [LMX13] X. Lu, D. T. Morelli, Y. Xia, F. Zhou, V. Ozoli¸nˇs,H. Chi, X. Zhou, and C. Uher. “High performance thermoelectricity in earth-abundant compounds based on natural mineral tetrahedrites.” Adv. Energy Mater., 3:342–348, 2013.

[LMX15] X. Lu, D. T. Morelli, Y. Xia, and V. Ozoli¸nˇs. “Increasing the thermoelectric ﬁgure of merit of tetrahedrites by co-doping with nickel and zinc.” Chem. Mater., 27:408–413, 2015.

[LMX16] X. Lu, D. T. Morelli, Y. Xia, and V. Ozoli¸nˇs. “Phase Stability, Crystal Struc- ture, and Thermoelectric Properties of Cu12Sb4S13−xSex Solid Solutions.” Chem. Mater., 28:1781–1786, 2016.

[LNP04] J.La˙zewski,H. Neumann, and K. Parlinski. “Ab initio characterization of magnetic CuFeS2.” Phys. Rev. B, 70:5206, 2004. [LRZ05] C. C. Li, W. L. Ren, L. T. Zhang, K. Ito, and J. S. Wu. “Eﬀects of Al doping on the thermoelectric performance of CoSi single crystal.” J. Appl. Phys., 98(6), 2005.

[LTL13] Jianhui Li, Qing Tan, and Jing-Feng Li. “Synthesis and property evaluation of CuFeS2x as earth-abundant and environmentally-friendly thermoelectric materials.” J Alloys Compd., 551:143–149, 2013.

[LYP88] Chengteh Lee, Weitao Yang, and Robert G. Parr. “Development of the Colic- Salvetti correlation-energy formula into a functional of the electron density.” Phys. Rev. B, 37:785–789, 1988.

[LZ08] Stephan Lany and Alex Zunger. “Assessment of correction methods for the band- gap problem and for ﬁnite-size eﬀects in supercell defect calculations: Case studies for ZnO and GaAs.” Phys. Rev. B, 78:5104, 2008.

[LZJ17] Yue Liu, Tianlu Zhao, Wangwei Ju, and Siqi Shi. “Materials discovery and design using machine learning.” J. Materiomics, 3:159–177, 2017.

[LZL11] W. Liu, Q. Zhang, Y. Lan, S. Chen, X. Yan, Q. Zhang, H. Wang, D. Wang, G. Chen, and Z. Ren. “Thermoelectric Property Studies on Cu-Doped n-type CuxBi2Te2.7Se0.3 Nanocomposites.” Adv. Energy Mater., 1:577–587, 2011. [LZL18] Te-Huan Liu, Jiawei Zhou, Mingda Lia, Zhiwei Ding, Qichen Song, Bolin Liao, Liang Fu, and Gang Chen. “Electron mean-free-path ﬁltering in Dirac material for improved thermoelectric performance.” Proc. Natl. Acad. Sci., 115:879–884, 2018.

[LZQ14] Yulong Li, Tiansong Zhang, Yuting Qin, Tristan Day, G. Jeﬀrey Snyder, Xun Shi, and Lidong Chen. “Thermoelectric transport properties of diamond-like Cu1−xF e1+xS2 tetrahedral compounds.” J. Appl. Phys., 116, 2014.

129 [LZZ16] Yong Liu, Li-Dong Zhao, Yingcai Zhu, Yaochun, Liu, Fu Li, Meijuan Yu, Da- Bo Liu, Wei Xu, Yuan-Hua Lin, and Ce-Wen Nan. “Synergistically Optimizing Electrical and Thermal Transport Properties of BiCuSeO via a Dual-Doping Ap- proach.” Adv. Energy Mater., 6:150243, 2016. [Mad16] Georg K. H. Madsen. “Automated Search for New Thermoelectric Materials: The Case of LiZnSb.” J. Amer. Chem. Soc., 128:12140–12146, 2016. [MCM16] Ruth Martnez-Casado, Vincent H. Y. Chen, Giuseppe Mallia, and Nicholas M. Harrison. “A hybrid-exchange density functional study of the bonding and electronic structure in bulk CuFeS2.” J. Chem. Phys., 114:184702, 2016. [MDR09] A. J. Minnich, M. S. Dresselhaus, Z. F. Ren, and G. Chen. “Bulk nanostructured thermoelectric materials: current research and future prospects.” Energy Environ. Sci., 2:466–479, 2009. [MJH08] D. T. Morelli, V. Jovovic, and J.P. Heremans. “Intrinsically Minimal Thermal Conductivity in Cubic I-V-VI2 Semiconductors.” Phys. Rev. Lett., 101:035901, Nov 2008. [MLZ18] Jun Mao, Zihang Liu, Jiawei Zhou, Hangtian Zhu, Qian Zhang, Gang Chen, and Zhifeng Ren. “Advances in Thermoelectrics.” Adv. in Phys., 67:69–147, 2018. [MMH98] G. P. Meisner, D. T. Morelli, S. Hu, J. Yang, and C. Uher. “Structure and Lattice Thermal Conductivity of Fractionally Filled Skutterudites: Solid Solutions of Fully Filled and Unﬁlled End Members.” Phys. Rev. Lett., 80:3551–3554, 1998. [MP89] M. Methfessel and A.T. Paxton. “High-precision sampling for Brillouin-zone integration in metals.” Phys. Rev. B, 40:3616–3621, 1989. [MP95] G. Makov and M. C. Payne. “Periodic boundary conditions in ab initio calculations.” Phys. Rev. B, 51:4014, 1995. [MS96] G. D. Mahan and J. Sofo. “The Best Thermoelectric.” Proc. Natl. Acad. Sci., 93:7436–7439, 1996. [MS06] Georg K.H. Madsen and David J. Singh. “BoltzTraP. A code for calculating band- structure dependent quantities.” Comput. Phys. Commun., 175:67–71, 2006. [MV97] Nicola Marzari and David Vanderbilt. “Maximally localized generalized Wannier functions for composite energy bands.” Phys. Rev. B, 56:12847–12865, 1997. [MYL08] Arash A. Mostoﬁ, Jonathan R. Yates, Young-Su Lee, Ivo Souza, David Vander- bilt, and Nicola Marzari. “wannier90: A tool for obtaining maximally-localised Wannier functions.” Comput. Phys. Commun., 178:685–699, 2008. [NCS98] G. S. Nolas, J. L. Cohn, G. A. Slack, and S. B. Scuhjman. “Semiconducting Ge clathrates: Promising candidates for thermoelectric applications.” Appl. Phys. Lett., 73:178, 1998.

130 [NGM10] Jesse Noﬀsinger, Feliciano Giustino, Brad D. Malone, Cheol Hwan Park, Steven G. Louie, and Marvin L. Cohen. “EPW: A program for calculating the electron–phonon coupling using maximally localized Wannier functions.” Com- put. Phys. Commun., 55:2140–2148, 2010.

[NHZ13] L.J. Nelson, G.L.W. Hart, F. Zhou, and V. Ozoli¸nˇs. “Compressive sensing as a paradigm for building physics models.” Phys. Rev. B, 87:035125, Jan 2013.

[NOH13] M. D. Nielsen, V. Ozoli¸nˇs,and J.P. Heremans. “Lone pair electrons minimize lattice thermal conductivity.” Energy Environ. Sci., 6:570–578, 2013.

[Nol96] G. S. Nolas. “The eﬀect of rare-earth ﬁlling on the lattice thermal conductivity of skutterudites.” J. Appl. Phys., 79:4002, 1996.

[Nol00] G. S. Nolas. “High ﬁgure of merit in partially ﬁlled ytterbium skutterudite materials.” Appl. Phys. Lett., 77:1886, 2000.

[NOR13] L.J. Nelson, V. Ozoli¸nˇs, S.C. Reese, F. Zhou, and G.L.W. Hart. “Cluster expansion made easy with Bayesian compressive sensing.” Phys. Rev. B, 88:155105, Oct 2013.

[NSG01] G. S. Nolas, J. Sharp, and H. J. Goldsmid. Thermoelectrics. Springer, 2001.

[NV16] Payam Norouzzadeh and Daryoosh Vashaee. “Classication of Valleytronics in Thermoelectricity.” Sci. Rep., 6:1–15, 2016.

[OLC13] Vidvuds Ozoli¸nˇs,Rongjie Laib, Russel Caﬂisch, and Stanley Osher. “Compressed modes for variational problems in mathematics and physics.” Proc. Natl. Acad. Sci., 110:18368–18373, 2013.

[OMS17] A. A. Olvera, N. A. Moroz, P. Sahoo, P. Ren, T. P. Bailey, A. A. Page, C. Uher, and P. F. P. Poudeu. “Partial indium solubility induces chemical stability and colossal thermoelectric ﬁgure of merit in Cu2Se.” Energy Environ. Sci.., 10:1668– 1676, 2017.

[Ons31a] Lars Onsager. “Reciprocal Relations in Irreversible Processes I.” Phys. Rev,, 37:405–426, 1931.

[Ons31b] Lars Onsager. “Reciprocal Relations in Irreversible Processes II.” Phys. Rev,, 38:2265–2279, 1931.

[PAG14] S. Poncé, G. Antonius, Y. Gillet, P. Boulanger, J. Laflamme Janssen, A. Marini, M. Côté, and X. Gonze. “Temperature dependence of electronic eigenenergies in the adiabatic harmonic approximation.” Phys. Rev. B, 90:214304, 2014.

[Pat11] Prachi Patel. “Materials Genome Initiative and energy.” MRS Bull., 36:964–966, 2011.

131 [PBE96] John P. Perdew, Kieron Burke, and Matthias Ernzerhof. “Generalized gradient approximation made simple.” Phys. Rev. Lett., 77:3865–3868, Oct 1996.

[PCS13] David Parker, Xin Chen, and David J. Singh. “High Three-Dimensional Thermoelectric Performance from Low-Dimensional Bands.” Phys. Rev. Lett., 119:146691, 2013.

[Per85] J. P. Perdew. “Density Functional Theory and the Band Gap Problem.” Int. J. Quantum Chem., 28:497–523, 1985.

[PGJ15] S. Ponc´e, Y. Gillet, J. Laﬂamme Janssen, A. Marini, M. Verstraete, and X. Gonze. “Temperature dependence of the electronic structure of semiconductors and insulators.” J. Chem. Phys., 143:1028137, 2015.

[PHM08] B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yang, D. Wang, A. Muto, D. Vashaee, X. Chen, J. Liu, M. Dresselhaus, G. Chen, and Z. Ren. “High-Thermoelectric Performance of Nanostructured Bismuth Antimony Tel- luride Bulk Alloys.” Science, 320:634–638, 2008.

[PJK14] Juan J. de Pablo, Barbara Jones, Cora Lind Kovacs, Vidvuds Ozoli¸nˇs, and Arthur P. Ramirez. “The Materials Genome Initiative, the interplay of experi- ment, theory and computation.” Curr. Opin. Solid State Mater. Sci., 18:99–117, 2014.

[PK16] E. A. Pfeif and K. Kroenlein. “Perspective: Data infrastructure for high throughput materials discovery.” APL Mater., 4:053203, 2016.

[PKS10] Alla E. Petrova, Vladimir N. Krasnorussky, Anatoly A. Shikov, William M. Yuhasz, Thomas A. Lograsso, Jason C. Lashley, and Sergei M. Stishov. “Elastic, thermodynamic, and electronic properties of MnSi, FeSi, and CoSi.” Phys. Rev. B, 82:155124, Oct 2010.

[PLW12] Yanzhong Pei, Aaron D. LaLonde, Heng Wang, and G. Jeﬀrey Snyder. “Low Eﬀective Mass Leading to High Thermoelectric Performance.” Energy Environ. Sci., 5:7963–7969, 2012.

[PMV16] S. Ponce, E. R. Margine, C. Verdi, and F. Giustino. “EPW: Electron–phonon coupling, transport and superconducting properties using maximally localized Wannier functions.” Comput. Phys. Commun., 55:116–133, 2016.

[PWS12] Y. Pei, H. Wang, and G.J. Snyder. “Band engineering of thermoelectric materials.” Adv. Mater., 24:6125–6135, 2012.

[PXA11] Y. Pei, S. Xiaoya, L. Aaron, L. Chen H. Wang, and G.J. Snyder. “Convergence of electronic bands for high performance bulk thermoelectrics.” Nature, 473:66–69, 2011.

132 [PXA14] Y. Pei, S. Xiaoya, L. Aaron, L. Chen H. Wang, and G.J. Snyder. “Band convergence in the non-cubic chalcopyrite compounds Cu2MGeSe4.” J. Mater. Chem. C, 2:10189–10194, 2014.

[PXO19] Junsoo Park, Yi Xia, and Vidvuds Ozoli¸nˇs.“High Thermoelectric Power Factor and Eﬃciency from a Highly Dispersive Band in Ba2BiAu.” Phys. Rev. Appl., 11:014058, 2019.

[PZ81] J. P. Perdew and Alex Zunger. “Self-interaction correction to density-functional approximations for many-electron systems.” Phys. Rev. B, 23:5048–5079, May 1981.

[REA16] Paul Raccuglia, Katherine C. Elbert, Philip D. F. Adler, Casey Falk, Malia B. Wenny, Aurelio Mollo, Matthias Zeller, Sorelle A. Friedler, Joshua Schrier, and Alexander J. Norquist. “Machine-learning-assisted materials discovery using failed experiments.” Nature, 533:73–77, 2016.

[RK71] D. L. Rode and S. Knight. “Electron Transport in GaAs.” Phys. Rev. B, 3:2534– 2541, 1971.

[RKK02] D. M. Rowe, V. L. Kuznetsov, L. A. Kuznetsova, and Gao Min. “Electrical and thermal transport properties of intermediate-valence YbAl3.” J. Phys. D: Appl. Phys., 35:2183–2186, 2002.

[RLZ05] W.L. Ren, C.C. Li, L.T. Zhang, K. Ito, and J.S. Wu. “Eﬀects of Ge and B substitution on thermoelectric properties of CoSi.” Journal of Alloys and Compounds, 392(1–2):50 – 54, 2005.

[SBP16] J. Skelton, L. Burton, S. Parker, A. Walsh, C. Kim, A. Soon, J. Buckeridge, A. Sokol, C. R. A. Catlow, A. Togo, and I. Tanaka. “Anharmonicity in the High- Temperature Cmcm Phase of SnSe: Soft Modes and Three-Phonon Interactions.” Phys. Rev. Lett., 117:075502, 2016.

[SCK09] J. R. Sootsman, D. Chung, and M.G. Kanatzidis. “New and old concepts in thermoelectric materials.” Angew. Chem., 48:8616–8639, 2009.

[SK06] M. Shishkin and G. Kresse. “Implementation and performance of the frequency- dependent GW method within the PAW framework.” Phys. Rev. B, 74:035101, 2006.

[SK07] M. Shishkin and G. Kresse. “Self-consistent GW calculations for semiconductors and insulators.” Phys. Rev. B, 75:235102, 2007.

[SK15] Georgy Samsonidze and Boris Kozinsky. “Electron-phonon-averaged approximation for ﬁrst-principles computations of electron relaxation times and transport properties in semiconductor materials.” arxiv, 2015.

133 [SKU08] Joseph R. Sootsman, Huijun Kong, Ctirad Uher, Jonathan James D Angelo, Chun-I Wu, Timothy P. Hogan, Thierry Caillat, and Mercouri G. Kanatzidis. “Large Enhancements in the Thermoelectric Power Factor of Bulk PbTe at High Temperature by Synergistic Nanostructuring.” Angew. Chem., 47:8618–8622, 2008.

[SL12] M. Stoica and C. Lo. “Electrical transport properties of Co-based skutterudites ﬁlled with Ag and Au.” Phys. Rev. B, 86:115211, 2012.

[SLM13] Hui Sun, Xu Lu, and Donald T. Morelli. “Eﬀects of Ni, Pd, and Pt Substitutions on Thermoelectric Properties of CoSi Alloys.” J. Elec. Mater., 42(7):1352–1357, 2013.

[SM11] Eric Skoug and Donald Morelli. “Role of Lone-Pair Electrons in Producing Mini- mum Thermal Conductivity in Nitrogen-Group Chalcogenide Compounds.” Phys. Rev. Lett., 107:235901, Nov 2011.

[SM12] Hui Sun and Donald T. Morelli. “Thermoelectric Properties of Co1−xRhxSi0.98B0.02 Alloys.” J. Elec. Mater., 41(6):1125–1129, 2012. [SMK07] M. Shishkin, M. Marsman, and G. Kresse. “Accurate Quasiparticle Spectra from Self-Consistent GW Calculations with Vertex Corrections.” Phys. Rev. Lett, 99:246403, 2007.

[SMS17] Jing Shuai, Jun Mao, Shaowei Song, Qinyong Zhang, Gang Chen, and Zhifeng Ren. “Recent progress and future challenges on thermoelectric Zintl materials.” Mater. Phys. Today, 1:74–95, 2017.

[SMV01] Ivo Souza, Nicola Marzari, and David Vanderbilt. “Maximally localized Wannier functions for entangled energy bands.” Phys. Rev. B, 65:035109–1–13, 2001.

[SMW96] B. C. Sales, D. Mandrus, and R. K. Williams. “Filled Skutterudite Antimonides: A New Class of Thermoelectric Materials.” Science, 272:1325–1328, 1996.

[SRP15] Jianwei Sun, Adrienn Ruzsinszky, and John P. Perdew. “Strongly Constrained and Appropriately Normed Semilocal Density Functional.” Phys. Rev. Lett., 115:036402, 2015.

[SS16] Ram Seshadri and Taylor D. Sparks. “Perspective: Interactive material property databases through aggregation of literature data.” APL Mater., 4:053206, 2016.

[ST08] J. Snyder and E. Toberer. “Complex thermoelectric materials.” Nat. Mater., 7:105–114, 2008.

[STL15] Hezhu Shao, Xiaojian Tan, Guo-Qiang Liu, Jun Jiang, and Haochuan Jiang. “A ﬁrst-principles study on the phonon transport in layered BiCuOSe.” Sci. Rep., 6:21035, 2015.

134 [SYA07] A. Sakai, S. Yotsuhashi, H. Adachi, F. Ishii, Y. Onose, Y. Tomioka, N. Nagaosa, and Y. Tokura. “Filling dependence of thermoelectric power in transition-metal monosilicides.” In Thermoelectrics, 2007. ICT 2007. 26th International Confer- ence on, pp. 256–259, June 2007.

[SYS11] Xun Shi, Jiong Yang, James R. Salvador, Miaofang Chi, Jung Y. Cho, Hsin Wang, Shengqiang Bai, Jihui Yang, Wenqing Zhang, and Lidong Chen. “Multiple- Filled Skutterudites: High Thermoelectric Figure of Merit through Separately Optimizing Electrical and Thermal Transports.” J. Amer. Chem. Soc., 133:7837– 7846, 2011.

[SZP09] E. Skoug, C. Zhou, Y. Pei, and D. T. Morelli. “High thermoelectric power factor in alloys based on CoSi.” Appl. Phys. Lett., 94(2), 2009.

[TB09] Fabien Tran and Peter Blaha. “Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential.” Phys. Rev. Lett., 102:226401, 2009.

[TCB08] Eric S. Toberer, Catherine A. Cox, Shawna R. Brown, Teruyuki Ikeda, An- drew F. May, Susan M. Kauzlarich, and G. Jeﬀrey Snyder. “Traversing the Metal-Insulator Transition in a Zintl Phase: Rational Enhancement of Thermo- electric Eﬃciency in Yb14Mn1−xAlxSb11.” Adv. Funct. Mater., 18:2795–2800, 2008.

[TCS15] Y. Tang, S. Chen, and G. Jeﬀrey Snyder. “Temperature dependent solubility of Yb in Yb-CoSb3 skutterudite and its eﬀect on preparation, optimization and lifetime of thermoelectrics.” J. Materiomics., 1:75–84, 2015.

[TKS17a] Hirokazu Takaki, Kazuaki Kobayashi, Masato Shimono, Nobuhiko Kobayashi, Kenji Hirose, Naohito Tsujii, and Takao Mori. “First-principles calculations of Seebeck coeﬃcients in a magnetic semiconductor CuFeS2.” Appl. Phys. Lett., 110:072107, 2017.

[TKS17b] Hirokazu Takaki, Kazuaki Kobayashi, Masato Shimono, Nobuhiko Kobayashi, Kenji Hirose, Naohito Tsujii, and Takao Mori. “Thermoelectric properties of a magnetic semiconductor CuFeS2.” Mater. Today Phys., 3:85–92, 2017. [TM13] N. Tsujii and T. Mori. “High thermoelectric power factor in a carrier-doped magnetic semiconductor CuFeS2.” Appl. Phys. Exp., 6:3001–, 2013. [TMI14] N. Tsujii, T. Mori, and Y. Isoda. “Phase stability and thermoelectric properties of CuFeS2-based magnetic semiconductor.” J. Elec. Mater., 43:2371–2375, 2014. [TMS10] Eric S. Toberer, Andrew F. May, and G. Jeﬀrey Snyder. “Zintl Chemistry for Designing High Eﬃciency Thermoelectric Materials.” Chem. Mater., 22:624–634, 2010.

135 [TMT15] N. Tsujii, F. Meng, K. Tsuchiya, S. Maruyama, and T. Mori. “Effect of nanostructuring and high-pressure torsion process on thermal conductivity of carrier-doped chalcopyrite.” J. Elec. Mater., 112:1–6, 2015. [TS06] T.M. Tritt and M.A. Subramanian. “Thermoelectric materials, phenomena, and applications: a birdOs˜ eye view.” MRS Bulletin, 31:188–198, 2006. [TSK74] T. Teranishi, K. Sato, and K. Kondo. “Optical properties of a magnetic semiconductor.” J. Phys. Soc., 36:1618–1624, 1974. [VG15] C. Verdi and F. Giustino. “Fröhlich Electron-Phonon Vertex from First Princi- ples.” Phys. Rev. Lett., 115:176401, 2015. [VSC01] Rama Venkatasubramanian, Edward Siivola, Thomas Colpitts, and Brooks O’Quinn. “Thin-film thermoelectric devices with high room-temperature figures of merit.” Nature, 413:597–602, 2001. [VSM10] C. J. Vineis, A. Shakouri, A. Majumdar, and M.G. Kanatzidis. “Nanostructured thermoelectrics: big efficiency gains from small features.” Adv. Mater., 22:3970– 3980, 2010. [WPE11] Aron Walsh, David J. Payne, Russell G. Egdell, and Graeme W. Watson. “Stere- ochemistry of post-transition metal oxides: revision of the classical lone pair model.” Chem. Soc. Rev., 40:4455–4463, 2011. [WW05] Aron Walsh and Graeme W. Watson. “Influence of the Anion on Lone Pair Formation in Sn(II) Monochalcogenides: A DFT Study.” The Journal of Physical Chemistry B, 109:18868–18875, 2005. PMID: 16853428. [WWL17] Evan Witkoske, Xufeng Wang, Mark Lundstrom, Vahid Askarpour, and Jesse Maassen. “Thermoelectric band engineering: The role of carrier scattering.” J. Appl. Phys., 122:5102, 2017. [WZZ14] H. J. Wu, L-D Zhao, F.S. Zheng, D. Wu, Y.L. Pei, X. Tong, M.G. Kanatzidis, and J. Q. He. “Broad temperature plateau for thermoelectric figure of merit ZT > 2 in phase-separated PbTe0.7S0.3.” Nat. Commun., 5:1–7, 2014. [Xia15] Li-Dong Zhao Xiao Zhang. “Thermoelectric materials: Energy conversion between heat and electricity.” J. Materiomics., 1:92–105, 2015. [Xia18] Yi Xia. “Revisiting lattice thermal transport in PbTe: The crucial role of quartic anharmonicity.” Appl. Phys. Lett., 113:073901, 2018. [XPZ19] Yi Xia, Junsoo Park, Fei Zhou, and Vidivuds Ozoli¸nˇs. “High Thermoelectric Power Factor in Intermetallic CoSi Arising from Energy Filtering of Electrons by Phonon Scattering.” Phys. Rev. Appl., 11:024017, February 2019. [XTL18] Jiazhan Xin, Yinglu Tang, Yintu Liu, Xinbing Zhao, Hongge Pan, and Tiejun Zhu. “Valleytronics in Thermoelectric Materials.” Quantum Mater., 3:1–9, 2018.

136 [XV14] Bin Xu and Matthieu J. Verstraete. “First Principles Explanation of the Positive Seebeck Coeﬃcient of Lithium.” Phys. Rev. Lett., 112:196603, May 2014.

[YC06] J. Yang and T. Caillat. “Thermoelectric materials for space and automotive power generation.” MRS Bulletin, 31:224–229, 2006.

[YON13] Shin Yabuuchi, Masakuni Okamoto, Akinori Nishide, Yosuke Kurosaki, and Jun Hayakawa. “Large Seebeck Coeﬃcients of Fe2TiSn and Fe2TiSi: First-Principles Study.” Appl. Phys. Exp., 6:025504, 2013.

[YYK05] Shigeo Yamaguchi, Takayuki Matsumoto Jun Yamazaki, Nakaba Kaiwa, and At- sushi Yamamoto. “Thermoelectric properties and ﬁgure of merit of a Te-doped InSb bulk single crystal.” Appl. Phys. Lett., 87:201902, 2005.

[ZGY14] M. Zhou, X. Gao, X. Chen Y. Cheng, and L. Cai. “Structural, electronic, and elastic properties of CuFeS2: ﬁrst-principles study.” Appl. Phys. A, 118:1145– 1152, 2014.

[ZHA15] Hong Zhu, Geoffroy Hautier, Umut Aydemir, Zachary M. Gibbs, Guodong Li, Saurabh Bajaj, Jan-Hendrik Pöhls,Danny Broberg, Wei Chen, Anubhav Jain, Mary Anne White, Mark Asta, G. Jeffrey Snyder, Kristin Persson, and Gerbrand Ceder. “Computational and experimental investigation of TmAgTe2 and XYZ2 compounds, a new group of thermoelectric materials identified by first-principles high-throughput screening.” J. Mater. Chem. C, 5:5510–5530, 2015.

[ZHB14] Li-Dong Zhao, Jiaqing He, David Berardan, Yuanhua Lin, Jing-Feng Li, Ce-Wen Nanc, and Nita Dragoe. “BiCuSeO oxyselenides: new promising thermoelectric materials.” Energy Environ. Sci., 7:2900–2924, 2014.

[Zim60] J. M. Ziman. Electrons and Phonons: the theory of transport phenomena in solids. Oxford University Press, 1960.

[ZLC14] Jiawei Zhang, R. Liu, N. Cheng, Y. Zhang, J. Yang, C. Uher, X. Shi, L. Chen, and W. Zhang. “High-performance pseudocubic thermoelectric materials from non-cubic chalcopyrite compounds.” Adv. Mater., 26:3848–3853, 2014.

[ZLF17] T. Zhu, Y. Liu, C. Fu, J. P. Heremans, J. G. Snyder, and X. Zhao. “Compro- mise and Synergy in High-Eﬃciency Thermoelectric Materials.” Adv. Mater., 29:1605884, 2017.

[ZLZ14] L. Zhao, S. Lo, Y. Zhang, H. Sun, G. Tan, C. Uher, C. Wolverton, V. P. Dravid, and M. G. Kanatzidis. “Ultralow thermal conductivity and high thermoelectric ﬁgure of merit in SnSe crystals.” Nature, 508:373–377, 2014.

[ZNX14] F. Zhou, W. Nielson, Y. Xia, and V. Ozoli¸nˇs.“Lattice Anharmonicity and Ther- mal Conductivity from Compressive Sensing of First-Principles Calculations.” Phys. Rev. Lett., 113:185501, Oct 2014.

137 [ZTH16] L. Zhao, G. Tan, S. Hao, J. He, Y. Pei, H. Chi, H. Wang, S. Gong, H. Xu, V. Dravid, C. Uher, G. J. Snyder, C. Wolverton, and M. G. Kanatzidis. “Ultrahigh power factor and thermoelectric performance in hole-doped single-crystal SnSe.” Science, 351:141–144, 2016.

[ZYC11] Jun Zhou, Ronggui Yang, Gang Chen, , and Mildred S. Dresselhaus. “Optimal Bandwidth for High Eﬃciency Thermoelectrics.” Phys. Rev. Lett., 108:226601, 2011.

[ZZG16] W. G. Zeier, A. Zevalkink, Z. M. Gibbs, G. Hautier, M. G. Knatzidis, and G. J. Snyder. “Thinking Like a Chemist: Intuition in Thermoelectric Materi- als.” Angew. Chem., 55:6826–6841, 2016.

[ZZL14] Bin Zhong, Yong Zhang, Weiqian Li, Zhenrui Chen, Jingying Cui, Wei Li, Yuan- dong Xie, Qing Hao, and Qinyu He. “High superionic conduction arising from aligned large lamellae and large ﬁgure of merit in bulk Cu1.94Al0.02Se.” Appl. Phys. Lett., 105:123902, 2014.

[ZZL18] J. Zhou, H. Zhu, T. Liu, Q. Song, R. He, J. Mao, Z. Liu, W. Ren, B. Liao, D. J. Singh, and G. Chen. “Large thermoelectric power factor from crystal symmetry- protected non-bonding orbital in half-Heuslers Nb1−xTixFeSb.” Nat. Commun., 9:1–9, 2018.

138