Towards Fundamental Understanding of Thermoelectric Properties in Novel Materials Using First Principles Simulations Artem R

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June 2018 Towards Fundamental Understanding of Thermoelectric Properties in Novel Materials Using First Principles Simulations Artem R. Khabibullin University of South Florida, [email protected]

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Towards Fundamental Understanding of Thermoelectric Properties in Novel Materials Using

First Principles Simulations

Artem R. Khabibullin

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Applied Physics Department of Physics College of Art and Science University of South Florida

Major Professor: Lilia M. Woods, Ph.D. George S. Nolas, Ph.D. Ivan I. Oleynik, Ph.D. Venkat R. Bhethanabotla, Ph.D.

Date of Approval: June 20, 2018

Keywords: bournonites, chalcogenides, clathrates, first principles simulations, thermal conductivity, thermoelectricity

DEDICATION

“Smile more, gentlemen! Serious face doesn't imply intellect: the most stupid things in the world are done with this very face...”

Baron Münchhausen

To my parents, who always wait and love...

ACKNOWLEDGMENTS

Growing in personality and establishing a balanced attitude to life does not only include the years spent on reading and learning, but also strongly depends on the people who are nearby. I would like to express my gratitude to those people who influenced my life and still do it.

First of all, I would like to thank my advisor, Prof. Lilia M. Woods for a great scientific school. I am very grateful for her supportive and attentive attitude during my first years in USF as an international student. Moreover, her diligence and persistence in pursuing success in research became a good model to follow. Thanks to her help and our countless discussions my skills as an independent researcher evolved significantly.

Second of all, I would like to thank Prof. George S. Nolas for a great collaboration that resulted in several research projects where computational methods engage with experimental observations.

I express my gratitude to Prof. Ivan I. Oleynik and Prof. Venkat R. Bhethanabotla for being the committee members and their helpful questions and comments during preparation of this thesis.

I also would like to thank

 Prof. Inna Ponomareva for her helpful and most practical course in Computational

Physics.

 Prof. Casey W. Miller for the experience I got during his course in Measurement &

Instrumentation.

 The USF Research Computing center and personally, Mr. Anthony J. Green, for his

support and careful assistance with my research projects.

 Dr. David L. Morse for his contribution and support in our collaborative project

dedicated to cancer related problems.

 Prof. Sergey Lisenkov and Dr. Tatiana Miti for their support in my teaching duties.  Dr. Mikalai M. Budzevich for his help and support at different stages of my student time.

 Troy Stedman for his very kind help and helpful discussions during our graduate courses.

 The staff personnel from the Physics Department, Daisy Matos, Jimmy Suarez and

Miguel Nieves for their help in taking care of all of my paperwork

I am grateful to Prof. Nail R. Khusnutdinov for his mentoring and support.

I also would like thank Prof. Nail G. Migranov for involving me in the world of science during my undergrad time.

Lastly, my very warm and deep gratitude goes over the ocean, to my parents and sister, who are always in my thoughts and heart. I am very grateful to my wife, for her patience, support and warmth.

Financial support from the US National Science Foundation under Grant No. DMR-1400957 is acknowledged.

TABLE OF CONTENT Acknowledgments ...... iii

List of Tables ...... iv

List of Figures ...... v

Abstract ...... viii

1 Introduction ...... 10 1.1 Transport theory of thermoelectricity...... 10 1.2 Efficiency of thermoelectrics ...... 16 1.3 Lattice thermal conductivity ...... 19 1.4 Types of materials suitable for TE applications ...... 21

2 Motivation ...... 26

3 Methodology ...... 29 3.1 Basics of Density Functional Theory ...... 29 3.1.1 Hartree-Fock method ...... 29 3.1.2 Thomas-Fermi theory...... 31 3.1.3 Kohn-Hohenberg Theory ...... 32 3.1.4 Exchange-correlation energy ...... 35 3.1.5 DFT+U theory ...... 39 3.2 PAW method ...... 39 3.3 Spin-Orbit Interaction ...... 40 3.4 Van der Waals interactions ...... 41 3.4.1 Interatomic correction method...... 42 3.4.2 Van der Waals Density Functional ...... 43

3.5 Structural properties ...... 44 3.6 Electronic structure properties ...... 45 3.7 Phonon structural properties ...... 48

4 Bi-Sb alloys ...... 52 4.1 Atomic structure properties ...... 54 4.2 Electronic structure properties ...... 57 4.3 Summary ...... 62

5 Bournonite PbCuSbS3 ...... 63 5.1 Crystal structure ...... 65 5.2 Electronic structure ...... 67 5.3 Electron Localization and Chemical Bonding ...... 68 5.4 Lone-Pair Electrons and Their Role in Thermal Conductivity ...... 71 5.5 Summary ...... 73

6 Doped bournonites ...... 74 6.1 Crystal structure ...... 74 6.2 Electronic Structure ...... 77 6.3 Electron Localization and Charge Transfer ...... 78 6.4 Phonon Structure and Thermal Conductivity ...... 83 6.5 Summary ...... 86

7 Clathrates ...... 88 7.1 Tin type II clathrates ...... 89 7.1.1 Structure and stability ...... 89 7.1.2 Electronic structure ...... 93 7.1.3 Electron localization function...... 95 7.1.4 Phonons ...... 96 7.1.5 Summary ...... 99 7.2 Arsenic type I clathrate...... 101

7.2.1 Structural properties ...... 102 7.2.2 Electronic structure ...... 106 7.2.3 Summary ...... 109

8 Quaternary chalcogenides: Cu2CdSnTe4 and (Ag, Cu)2ZnSnSe4 ...... 111

8.1 Cu2CdSnTe4 ...... 111

8.2 Ag2ZnSnSe4 and Cu2ZnSnSe4 ...... 113

8.2.1 Polaronic transport in Ag2ZnSnSe4 ...... 113 8.2.2 Structural properties ...... 116 8.2.3 Electronic properties ...... 118 8.2.4 Summary ...... 121

9 Conclusion and future outlook ...... 122

References...... 125

Appendices...... 139 Appendix 1 ...... 140 Appendix 2 ...... 141

iii

LIST OF TABLES

Table 5.1: Characteristic distances between inequivalent atoms in the unit cell for PbCuSbS3 and

Sb2S3 ...... 66

Table 6.1: Experimental and calculated a, b, c lattice constants for PbCuSbS3, PbCu0.75Ni0.25SbS3

and PbCu0.9Zn0.1SbS3 ...... 76

Table 6.2: Experimental and calculated via simulations bond angles between different Pb and Sn cations and chalcogen anion S atoms ...... 77

Table 6.3: Vibrational properties of PbCuSbS3, PbCu0.75Ni0.25SbS3 and PbCu0.875Zn0.125SbS3 ...... 84

Table 7.1: Lattice constants, nearest neighbor bonds, and formation energies for the studied clathrates...... 92

Table 7.2: Direct energy gap at high symmetry points in the Brillouin zone, transverse, 푣푇퐴, and longitudinal, 푣퐿퐴, speed of sound along characteristic directions, total mode Grüneisen parameter, 훾, and Debye temperature, 휃퐷 for several type II Sn clathrates...... 95

Table 7.3: Formation energy, lattice constants and averaged values of atomic distances calculated via different approximations and their experimental referral values...... 103

Table 8.1: Lattice parameters and tetragonal distortion η for Cu2CdSnTe4 ...... 112

Table 8.2: Lattice structure parameters for both phases of the Ag2ZnSnSe4 and Cu2ZnSnSe4 ...... 118

LIST OF FIGURES

Figure 1.2.1 A material under standard thermoelectric conditions ...... 16

Figure 1.3.1 A schematic representation of different methods for minimizing the lattice thermal conductivity ...... 19

Figure 1.4.1 A schematic representation of a synergistic approach for materials science discoveries...... 26

Figure 3.1.1 A schematic representation of solving the Kohn-Sham equations self-consistently...... 34

Figure 3.5.1 Typical computational results showing total energy minimization as a function of the lattice constant ...... 45

Figure 3.6.1 Optimum DOS for a hypothetical thermoelectric and multi valley region with the hole and electron pocket formations ...... 46

Figure 3.6.2 The ELF calculated for Cs8Na16Si136 clathrate-II...... 48

Figure 3.7.1 Phonons and phonon density of states calculated via FDM for the hypothetical type

II Sn136 clathrate material...... 49

Figure 4.1.1 The supercells of Bi1-xSbx and different perspectives of the Fermi surface of Bi ...... 54

Figure 4.1.2 Rhombohedral lattice parameters and angle 휑 between electron and hole pockets in

reciprocal space as functions of concentration for the Bi1-xSbx alloys...... 55

Figure 4.1.3 Hexagonal lattice parameters as functions of concentration for the Bi1-xSbx alloys...... 56

Figure 4.2.1 Fermi energy differences between Bi1-xSbx and Bi as a function of Sb concentration...... 57

Figure 4.2.2 Total DOS as a function of energy scaled by the Fermi level Ef for Bi, Bi0.875Sb0.125,

and Bi0.75Sb0.25...... 58

Figure 4.2.3 DOS with contributions from s and p-states for Bi1-xSbx with x=6.25% and DOS with

contributions from s and p-states for Bi1-xSbx with x=25% ...... 59

Figure 5.1.1 Crystal lattice views of PbCuSbS3 along the crystallographic a axis ...... 65

Figure 5.2.1 Energy band structure for PbCuSbS3 and Sb2S3 ...... 67

Figure 5.3.1 Calculated electron localization function in various projections for PbCuSbS3 and

Sb2S3 ...... 69

Figure 5.4.1 Measured temperature dependent thermal conductivity for PbCuSbS3 and Sb2S3 ...... 71

Figure 6.1.1 Lattice structures of PbCuSbS3, PbCu0.75Ni0.25SbS3 and PbCu0.875Zn0.125SbS3 ...... 75

Figure 6.2.1 Calculated total and projected DOS for PbCuSbS3, PbCu0.75Ni0.25SbS3 and

PbCu0.875Zn0.125SbS3...... 78

Figure 6.3.1 Calculated electron localization function for PbCuSbS3, PbCu0.75Ni0.25SbS3 and

PbCu0.875Zn0.125SbS3...... 78

Figure 6.3.2 Calculated charge transfer for PbCuSbS3, PbCu0.75Ni0.25SbS3 and

PbCu0.875Zn0.125SbS3 ...... 81

Figure 6.3.3 Schematics of the projections for the Bader charge analysis along the S3-X-S4 and S1-X-S2 atomic chains in pure bournonite...... 82

Figure 6.4.1 Calculated phonon band structure and PDOS for PbCuSbS3, PbCu0.75Ni0.25SbS3 and

PbCu0.875Zn0.125SbS3 ...... 84

Figure 6.4.2 Temperature dependent κ of PbCuSbS3, PbCu0.75Ni0.25SbS3 and PbCu0.9Zn0.1SbS3 ...... 85

Figure 7.1.1 Representation of type II Sn clathrates ...... 91

Figure 7.1.2 Band structure and DOS for Sn136, Cs8Ba16Ga40Sn96 and Xe8Sn136...... 94

Figure 7.1.3 Calculated ELF in for Sn136, Cs8Ba16Ga40Sn96, and Xe24Sn136 ...... 96

Figure 7.1.4 Phonon structure properties for Sn136, Cs8Ba16Ga40Sn96, Xe8Sn136 and Xe24Sn136 ...... 97

Figure 7.2.1 A projected view of clathrate-I Ba8Cu16As30 along the 100 direction ...... 102

Figure 7.2.2 Electronic structure for Ba8Cu16As30 and As46...... 106

Figure 7.2.3 Projected density of states for Ba8Cu16As30 and As46...... 108

Figure 7.2.4 ELF calculated for Ba8Cu16As30 and As46 on different planes...... 109

Figure 8.1.1 Unit cell representation of Cu2CdSnTe4 ...... 111

Figure 8.1.2 The energy band structure of Cu2CdSnTe4…………...... 112

Figure 8.2.1 Temperature dependent 푘 and 휌 for Ag2ZnSnSe4, Ag2.1Zn0.9SnSe4 and

Ag2.3Zn0.7SnSe4 ...... 114

Figure 8.2.2 Stannite and kesterite structures of Ag2ZnSnSe4 ...... 117

Figure 8.2.3 Total DOS for Ag2ZnSnSe4 and Cu2ZnSnSe4 in both structure types, obtained via DFT-HSE hybrid functional ...... 119

Figure 8.2.4 The calculated ELF for KS Ag2ZnSnSe4 and for KS Cu2ZnSnSe4 ...... 120

vii

ABSTRACT

Thermoelectric materials play an important role in energy conversion as they represent environmentally safe and solid state devices with a great potential towards enhancing their efficiency. The ability to generate electric power in a reliable way without using non-renewable resources motivates many experimentalists as well as computational physicists to search and design new thermoelectric materials. Several classes of materials have been identified as good candidates for high efficient thermoelectrics because of their inherently low thermal conductivity. The complex study of the crystal and electronic structures of such materials helps to reveal hidden properties and give fundamental understanding, necessary for the development of a new generation of thermoelectrics.

In the current thesis, ab-initio computational methods along with experimental observations are applied to investigate several material classes suitable for thermoelectric applications. One example are

Bi-Sb bismuth rich alloys, for which it is shown how structural anomalies affect the electronic structure and how inclusion of the Spin-Orbit coupling is necessary for this type of materials. Another example are bournonite materials whose low thermal conductivity is attributed to distortions and interactions associated with lone-electron s2 pair distributions. In addition, it is shown how doping with similar atoms can affects electronic structure of these materials leading to changes in their transport properties.

Clathrate materials from the less studied type II Sn class are also investigated with a detailed analysis for their structural stability, electronic properties and phonons. These systems are considered with partially substituted atoms on the framework and different guests inside. The effect upon insertion of Noble gases into the cage network is also investigated. In addition, the newly synthesized As based cationic material is also studied finding novel structure-property relations. Another class of materials, quaternary chalcogenides, have also been studied. Because of their inherently low thermal conductivity and

viii semiconducting nature their transport properties may be optimized in a favorable way for thermoelectricity.

The present work provides an in-depth study of structural and electronic properties of several classes of materials, which can be used by experimentalists for input and guidance in the laboratory.

1 INTRODUCTION

1.1 Transport theory of thermoelectricity

Thermoelectricity is a phenomenon capturing the direct conversion of heat into electricity and vice versa. For several decades, TE energy conversion has been explored as an alternative way to reduce our dependence on fossil fuels and contribute to effective solutions of a climate control. According to recent statistical data1–5, about two thirds of the energy used for human needs is wasted as heat. This trend is even more prominent in the automotive industry where only about 12-30% of the used fuel is utilized effectively. TE devices also find practical applications in other areas of human life such as for geothermal and solar energy conversion (a long-term power generation), for the space industry and as portable medical devices6–17. Such devices rely on a solid-state mechanism of energy conversion and they have several other advantages, such as being reliable since they do not have moving parts, environmentally safe and having low-cost production.17

TE devices have undergone through several evolutionary stages. Since the discovery and initial characterization of this phenomenon in the late 19th century18–21, the 1950’s is the first era of attempting to use thermoelectrics in practical devices. Most studies at the time have focused on mainly two materials:

22,23 Bi2Te2 as first proposed by Goldsmid and PbTe based materials developed by Ioffe in the USSR . The latter found its applications in the first TE generator implemented in a portable radio powered by heat from a kerosene lamp24. The efficiency of such materials was very low (~2-3% of the Carnot efficiency).

After two decades of further investigations, the performance slightly improved but still was in a single digit range of 4-5%. The lack of progress in increasing the underlying efficiency can largely be attributed to limited understanding of the internal materials properties. The world arms race and military tension in the 70’s-80’s prompted a further interest in thermoelectricity. It has been recognized that the efficient TE

10 devices could be used in advanced military technologies such as a heat missing control, silent heating and air conditioning, military vest cooling and quiet electric power generation25. In the 90’s, a series of publications revived the interest to thermoelectricity. In these works, Mildred Dresselhaus suggested a new concept according to which the efficiency of TE device could be significantly (two-three times as much) enhanced by lowering the materials dimensionality26,27. In practice, devices relying on

Dresselhaus’ ideas are able to reach 11-15% of the Carnot efficiency. To this point, it is believed that better understanding of materials and their properties can increase further the efficiency of TE devices reaching levels of 20-35%4. A notable progress has been made for some classes of materials based on the ideas of Glenn Slack. According to his concept an ideal TE should possess the properties of phonon-glass electron crystals (PGEC)18, which enables carrying electricity as a crystal and conducting heat as an amorphous glass. To date, there are two classes of materials closely satisfying the Slack’s ideas - skutterudites and clathrates, but yet their efficiency is still in the range of 5-11%28–30. A deeper understanding of the nature of TE transport in conjunction with advanced computational methods may allow further design and synthesis of the materials suitable for high efficient thermoelectricity.

Thermoelectricity was discovered in 1821 by Thomas Seebeck who observed that a compass needle is deflected when two ends of coupled dissimilar conductors are under different temperatures19,20.

Later, Hans Ørsted gave a more detailed explanation that a compass needle deflects due to the current induced in a loop, but not due to the induced magnetic field as Thomas Seebeck thought. This effect is quantified by the Seebeck coefficient defined as

훼 = 푉/∇푇, (1.1.1) where V is the induced voltage in response to an applied temperature gradient ∇푇. The Seebeck coefficient is also known as thermopower and it is usually measured in 휇푉/퐾.

In 1834, Jean Peltier observed another related phenomenon, opposite to the Seebeck effect21. He supplied a source of electricity across two coupled dissimilar conductors and discovered that heat is

11 released at one joint of the two conductors and cooled at another one. In 1838, Emil Lenz established the relation

푞 = 휋퐼 (1.1.2)

in which a released heat 푞 is proportional to the current supplied 퐼 to a material through a quantity

휋 called the Peltier coefficient, measured in 푊⁄퐴 or 푉. It was found that this effect is more potent in

푝 − 푛 junction semiconductors due to the hole/electron recombination/generation.

In 1851, William Thomson (known later as Lord Kelvin) discovered another related phenomenon31. In his experiment a heat is released/absorbed if a temperature gradient has a direction opposite/same to the applied current. Mathematically it is described by

푞 = 훽퐼∇푇, (1.1.3)

where 푞 – is the portion of heat released/absorbed, 퐼 – is the applied current, ∇푇 – is the applied temperature gradient and 훽 – is the Thomson coefficient, measured in 푉/퐾. Additionally Thomson established relations connecting all the above coefficients. For a standard thermocouple consisting of n- and p-type metals (퐴 and 퐵 respectevetly), the relation might be written as

훼퐴퐵 = 휋퐴 − 휋퐵⁄푇 and 푑훼퐴퐵⁄푑푇 = (훽퐴 − 훽퐵)⁄푇. (1.1.4)

Note, that the Thomson coefficient 훽 is the only one directly measurable for individual materials.

Although these experimental effects were known for decades, the first theoretical description of

TE transport was proposed in the early 20th century and it originates from thermodynamics. The idea is based on the concept of entropy introduced by Rudolf Clausius and developed later by Max Plank in

187932. In 1931 Lars Onsager introduced the definition of forces and fluxes in the framework of linear non-equilibrium thermodynamics33,34. In the 40’s Herbert Callen developed Onsager’s theory for isotropic media35. Sybren de Groot extended it further in application to TE devices in the 50’s36.

According to the Onsager-de-Groot-Callen theory, the evolution of a TE system is driven by a minimal production of entropy 푆 such that

휕푆/휕푡 = ∑푖 푱푖 ∙ 푭푖 (1.1.5)

where the fluctuation of any flux 푱푖 undergoes a restoring force 푭푖 to equilibrium. In a quasi-equilibrium regime each flux is given as a linear combination of the restoring forces with a reciprocal relation

퐿푖푘 = 퐿푘푖. In application to thermoelectricity the linear coupling between fluxes and forces can be written as

1 − 훁(휇 ) 푱푁 퐿11 퐿12 푇 푒 [ ] = [ ] [ 1 ], (1.1.6) 푱푄 퐿21 퐿22 훁 ( ) 푇

where 푱푁 is the current flux, 푱푄 – is the heat flux, 휇푒 – is the electrochemical potential and 푇 – is the temperature. The linear coefficients 퐿푖푘 are defined uniquely through the thermodynamic equations of continuity for the given fluxes and can be written as

푇 푇2 푇3 퐿 = 휎, 퐿 = 퐿 = 휎푆, 퐿 = 휎푆2 + 푇2푘 , (1.1.7) 11 푒2 12 21 푒2 22 푒2 푒푙

where 휎 is the electrical conductivity, 푘푒푙 is the electronic part of thermal conductivity and 푒 is the elementary charge. As can be seen, the above linear relations (Eq. 1.1.6 and 1.1.7) allow extracting the

37 material properties expressed via 휎 and 푘푒푙 .

Further development in TE transport theory has been achieved via a microscopic level of understanding38 initiated by Drude’s ideas for non-interacting particles39. In this model, electrons interact randomly with stationary (“frozen”) ions whereas the long-range interactions between electrons and ions as well as the interactions between electrons are omitted. The model was realized via classical theory of gases, however its main disadvantage is that scattering processes in crystals are taken into account incorrectly.

The development of quantum and statistical mechanics allowed further progress in TE theory.

Such an approach is based on a statistical description of a thermodynamic system in a quasi-equilibrium state37,38. The key ingredient of this theory is the Boltzmann Transport Equation (BTE)38, which describes the average time evolution of carriers via non-equilibrium Fermi-Dirac (NEFD) function 푓(풓, 풌, 푡)

휕푓 휕푓 휕푓 휕푓 = −풗푘 − 풌̇ + ( ) , (1.1.8) 휕푡 휕풓 휕풌 휕푡 푐표푙푙

where 풗푘 is the group velocity for in a 푘 state, 풌 is the wave vector and 풓 is the radius vector. In fact, the

BTE is equivalent with the continuity equation firstly introduced by Joseph Liouville and adopted later for non-equilibrium thermodynamics by Ludwig Boltzmann. The above equation describes carrier transport through the diffusion term 풗푘 휕푓⁄휕풓 associated with collective motion of the carriers due to an applied temperature gradient, the drift term 풌̇ 휕푓⁄휕풌 associated with external forces, and the term

(휕푓⁄휕푡)푐표푙푙 associated with the scattering sources.

In general, solving the BTE is challenging. The standard approach is to consider Eq. (1.1.8) under steady-state condition i.e. 휕푓⁄휕푡 = 0 and to replace the scattering term by an approximation. In such approach the system is being considered to be in quasi-equilibrium with the collision term approximated

0 0 as (휕푓⁄휕푡)푐표푙푙 ~ (푓 − 푓)⁄휏, where 푓 is the Fermi-Dirac function in thermodynamical equilibrium and

휏 is the average relaxation time. Normally, in real systems there are more than one source of scattering thus all possible contributions have to be taken into account. Typically, this is achieved using

Mattheissen’s rule,

1 1 = ∑ , (1.1.9) 휏 푙 휏푙 where 푙 counts the number of carrier scattering mechanisms in the system due to the presence of impurity

(doping), nanoparticle inclusion, lattice vibration, electron-electron interactions, and others.

At near equilibrium, the NEFD function takes the form of

0 휕푓 퐸푘−휇 푓 = 휏 (− ) 풗푘 (−훁푇 − 훁휇 + 푭), (1.1.10) 휕퐸푘 푇

where 퐸푘 is the energy of a system in a 푘 state and 휇 is the chemical potential. Using Onsager-Callen’s approach the TE fluxes in Eq. (1.1.6 and 1.1.7) can be expressed through the Fermi-Dirac function

(1.1.10) as the sums over 푘 states

1 1 1 푱 = ∑ 풗 푓 , 푱 = ∑ 퐸 풗 푓 , 푱 = ∑ (퐸 풗 − 휇) 푓 , (1.1.11) 푁 Ω 푘 푘 푘 퐸 Ω 푘 푘 푘 푘 푄 Ω 푘 푘 푘 푘 where Ω is the unit volume in 푑풌푑풓 space. They are linearly related to the forces in Eq. (1.6) through the

0 휕푓 푁 coefficients 퐿푖푘, which can be further obtained from ∫ 푑퐸 ∑(퐸) (− ) (퐸 − 휇) , where 푁 = 0 for 퐿11, 휕퐸푘

1 푁 = 1 for 퐿 , 푁 = 2 for 퐿 and ∑(퐸) = ∑ 휏 풗 풗 훿(퐸 − 퐸 ) is the average diffusivity per unit 12 22 Ω 푘 푘 푘 푘 푘 energy and volume. Then the material properties for an isotropic and homogeneous system, such as 훼, 휎 and 푘푒푙 are determined by the microscopic nature of the electronic structure of a material and relaxation time 휏 as

휕푓 (퐸) 휎 = 푒2 ∫ 휏 (퐸)푔(퐸) (− 0 ) 푑퐸, (1.1.12) 휕퐸

푒 휕푓 (퐸) 훼 = ∫ 휏 (퐸)푔(퐸) (− 0 ) (퐸 − 휇)푑퐸, (1.1.13) 휎푇 휕퐸

1 휕푓 (퐸) 푘 = ∫ 휏 (퐸)푔(퐸) (− 0 ) (퐸 − 휇)2푑퐸 − 푇휎훼2, (1.1.14) 푒푙 푇 휕퐸

2푑2풌 where 푔(퐸) = ∫ 푣2훿(퐸 − 퐸 ) is the density of electron states (DOS) per unit volume. It is clear that (2휋)3 푘 the atomistic nature of the material is reflected in 푔(퐸) and 휏(퐸) as shown in Eqs. (1.1.12, 1.1.13,

1.1.14).

1.2 Efficiency of thermoelectrics

Figure 1.2.1 A material under standard thermoelectric conditions, where the current 푰 is induced due to the difference in temperature between the cold (TC) and the hot (TH) junctions. The efficiency of TE generators depends on the specific boundary condition and materials properties as shown below. In general the efficiency of a thermodynamical system is given by40

푊 휂 = , (1.2.1) 푄 where 푊 is the output work and 푄 is the heat supplied to the system. In application to a standard TE material shown in Figure 1.2.1, equation (1.2.1) may be rewritten as22

퐼푉퐿 휂 = 2 , (1.2.2) 푘∆푇+퐼훼푇퐻−1/2퐼 푅푀

here, 퐼푉퐿 is the power dissipated in the load resistor, 푘∆푇 is the heat originating from electron-phonon scattering in thermoelectric material between the cold (푇퐶) and hot (푇퐻) junctions, 퐼훼푇퐻 is the heat

2 induced by the electrical current, transferred to the cold junction and 1/2퐼 푅푀 is the Joule heating, where

푅푀 is the resistance of thermoelectric material. For a simple TE material the current 퐼 and voltage 푉퐿 can be written as

훼∆푇 훼∆푇푅퐿 퐼 = and 푉퐿 = . (1.2.3) 푅퐿+푅푀 푅퐿+푅푀

Then, Eq. (1.2.2) may be further expressed as

2 (훼∆푇) 푅퐿 휂 = 2 2 2 2 . (1.2.4) 푘∆푇(푅퐿+푅푀) +(푅퐿+푅푀)훼 ∆푇푇퐻−1/2훼 ∆푇 푅푀

Now, the maximum efficiency for the TE module can be estimated by introducing the 푅퐿⁄푅푀 ratio.

Firstly, set 푅퐿⁄푅푀 to 푚 and rewrite (1.2.4) as a function of 푚

( ) ∆푇 푚 휂 푚 = 푘푅 (푚+1)2 1 ∆푇 . (1.2.5) 푇퐻 푀 ( 2 +(푚+1)− ) 푇퐻훼 2푇퐻

2 Note, that in Eq. (1.2.5), ∆푇⁄푇퐻 is the Carnot efficiency (휂퐶푟) and 훼 ⁄푘푅푀 determines the figure of merit defined below. Then, (1.2.5) may be modified according to

( ) 푚 휂 푚 = 휂퐶푅 (푚+1)2 1 ∆푇 . (1.2.6) ( +(푚+1)− ) 푍푇퐻 2푇퐻

2+푍(푇 +푇 ) Secondly, the maximum value of 휂(푚) might be achieved when 휕휂⁄휕푚 = 0 for 푚 = √ 퐻 퐶 , 2 where 푇̅ = (푇퐻 + 푇퐶)/2 is the average temperature between the cold and the hot junctions. Then, substituting the above value of 푚 in Eq. (1.2.6), it is easy to get the famous expression for the efficiency18,41

√1+푍푇̅−1 휂 = 휂퐶푟 푇 . (1.2.7) √1+푍푇̅+ 푐⁄ 푇ℎ

The dimensionless quantity 푍푇̅ in (1.2.7) is the figure of merit associated with material properties, defined as

훼2휎 푍푇̅ = 푇̅. (1.2.8) 푘

Here, the thermal conductivity consists of two parts 푘 = 푘푒푙 + 푘푙푎푡, where 푘푙푎푡 is the term that originates from lattice vibrations and 푘푒푙 is the electronic contribution to thermal conductivity. Eqs. (1.2.7) and

(1.2.8) show that in order to convert energy efficiently using thermoelectricity, certain material properties are desirable. This includes high 휎 to maintain a high charge current, high 훼 to maintain a high voltage drop, and low 푘 to maintain a temperature gradient. Larger 푍푇̅ values result in more efficient TE devices.

However, these transport properties are typically interrelated4,28,42. Taking into account the Wiedemann-

Franz law and the Pisarenko’s curve the TE properties of a typical material can be written as

⁄ 8휋2푘2 휋 2 3 훼 = 퐵 푚∗ ( ) , 휎 = 푛푒2휏⁄푚∗, 푘 = 휎퐿 푇, (1.2.9) 3푒ℎ2 3푛 푒푙 푁

∗ where 푘퐵 is the Boltzmann constant, ℎ is the Plank constant, 푚 is the effective mass, 푛 is the carrier

22,43 concentration and 퐿푁 is Lorentz number . As seen from Eqs. (1.2.8) and (1.2.9), it is not possible to change one of them without influencing the others in some disadvantageous way. Increasing 훼 inversely affects 휎 and decrease in 푘푒푙, while reducing 휎 with little or no improvement on 푍푇̅.

The above relations impose practical limitations on how much 푍푇̅ can be improved. However, over the past few decades some progress has been made in improving TE performance. One of the two commonly accepted methods is to increase 푚∗ through doping, as seen from Eq. (1.2.9) it directly affects

훼 and thus should increase the overall efficiently of a material. In such situation, the valence and conduction bands of the host material should resonate with the energy level originating from the impurity

∗ 2⁄3 ∗ and create a multi-valley region with effective mass 푚 = 푁 푚푏, where 푁 is the number of degenerate

∗ valleys and 푚푏 is the effective mass of the original valley. However, this approach deteriorates carrier mobility. Another method is to improve the thermopower by increasing the carrier mobility 휇 = 푒휏⁄푚∗ and finding some routes to reduce thermal conductivity. The best candidates up to date satisfying this criteria are heavily doped single type (푛-type or 푝-type) semiconductors with an appropriate bandgap in the order of 1 eV, since metals and insulators typically have low values of 휎⁄푘. Nevertheless, their

4,28,42 efficiency remains relatively low (8-11% of the Carnot efficiency) due to the high 휎, 푘푒푙 and low 훼 .

1.3 Lattice thermal conductivity

Figure 1.3.1 A schematic representation of different methods for minimizing the lattice thermal conductivity, klat.

While 훼, 휎 and 푘푒푙 are interrelated quantities, the second term in the thermal conductivity 푘푙푎푡 is somewhat independent. This term plays a crucial role for materials operating at low/medium temperature

44–47 ranges . Minimizing 푘푙푎푡 is one of the most popular strategies in enhancing 푍푇̅. The most popular routes of achieving this goal are illustrated in Figure 1.3.1. Method (A) reflects identifying materials with strong anharmonic phonon-phonon scattering processes associated with the lattice structure. For example, this situation might be realized in clathrates and skutterudites28–30,48 whose cages are filled with guest atoms. The guest atoms behave as rattles inside the voids and thus scatter phonons. Using complex structures with many-atom components can also be used for minimizing 푘푙푎푡 (shown as Method (B)). For example, materials with glass-like conductivity, mass fluctuations, anisotropic effective masses, or large mass mismatch between the different constituents promotes enhanced phonon scattering 4,28. Method (C) reflects scattering phonons in structures with nano-grained inclusion. However, nano-inclusions scatter electrons as well and thus may lead to reduction in electrical conductivity28.

Calculating thermal conductivity is a challenging problem since phonon scattering processes are difficult to capture qualitatively and quantitatively. One of the popular approaches is the relaxation time approximation (RTA) in which phonon scattering rates from different processes are taken as independent contributions. In the framework of the well-established Debye-Callaway model, the total scattering time can be found as44,49,50

4 3 2 3 2 1⁄휏푝ℎ = 1⁄휏퐷 + 1⁄휏푁 + 1⁄휏푈 + 1⁄휏퐵 = 퐴휔 + 퐵1푇 휔 + 퐵2푇 휔 + 푣̅푝ℎ/푙푑, (1.3.1)

where 휔 is the phonon frequency, 푣̅푝ℎ is the average group velocity for three acoustic modes (two transverse – TA and one longitudinal - LA), 푙푑 is the dimensional size of the sample and 퐴, 퐵1, 퐵2 are empirical parameters. The first term in the above expression originates from phonon scattering from impurities, the second term describes Normal phonon scattering, the third term describes Umklapp phonon scattering and the fourth term originates from scattering at boundaries. The Umklapp process is

51 described by 휏푈 ∝ 퐸푥푝[휃퐷/푎푇] (푎 is the empirical parameter) which shows that this type of scattering is dominant at high temperatures. At low temperature regime the acoustic branches of phonons dominate the thermal conductivity, such that 푘푙푎푡 consists of three terms corresponding to one longitudinal (LA) and two transverse acoustic phonon modes (TA)44,49

푘푙푎푡 = 푘푇퐴1 + 푘푇퐴2 + 푘퐿퐴. (1.3.2)

푡ℎ The partial conductivity 푘푙푎푡,푖 for each 푖 acoustic phonon mode in Eq. (1.3.2) can be calculated as

푖 4 푥 2 휃 ⁄푇휏푝ℎ(푥)푥 푒 (∫ 퐷,푖 푑푥) 휏푖 (푥)푥4푒푥 0 휏푖 (푒푥−1)2 1 3 휃퐷,푖⁄푇 푝ℎ 푁 푘푙푎푡,푖 = 퐶푖푇 ∫0 푥 2 푑푥 + 푖 4 푥 (1.3.3) 3 (푒 −1) 휃 ⁄푇 휏푝ℎ(푥)푥 푒 ∫ 퐷,푖 푑푥 0 푖 푖 푥 2 [ 휏푁휏푈(푒 −1) ]

4 2 3 where 휃퐷,푖 is the Debye temperature, 푥 = ℏ휔/푘퐵푇, 퐶푖 = 푘퐵⁄2휋 ℏ 푣푝ℎ,푖, 휔 is the phonon frequency,

푖 푣푝ℎ,푖 is the group velocity, 휏푝ℎ- is the phonon relaxation time at which phonons are distributed to the equilibrium state.52

According to Morelli and Slack the Normal and the Umklapp scattering rates can be found as52–54

3 2 2 1 푘퐵훾퐿퐴푉 푘퐵 2 5 퐿퐴 = 2 5 ( ) 푥 푇 휏푁 (푥) 푀ℏ 푣퐿퐴 ℏ

4 2 1 푘 훾 푉 푘 = 퐵 푇퐴1/푇퐴2 퐵 푥푇5 (1.3.4) 푇퐴1/푇퐴2 푀ℏ3푣5 ℏ 휏푁 (푥) 푇퐴1/푇퐴2

2 2 1 ℏ훾 푘퐵 2 3 −휃퐷,푖⁄3푇 푖 = 2 ( ) 푥 푇 푒 휏푈(푥) 푀푣푖 휃퐷,푖 ℏ where 푉 is the volume per atom, 푀 is the average mass of an atom per a primitive unit cell and 훾 is the

Grüneisen parameter, which essentially is a measure of anharmonicity and related to thermal conductivity

2 55,56 as 푘푙푎푡 ~ 1⁄훾 .

Also, phonon scattering in crystals might be approximated via a kinetic model of gases57. In the framework of this model, 푘푙푎푡 is given by

1 푘 = 퐶 푣̅ 푙 (1.3.5) 푙푎푡 3 푉 푝ℎ

where 퐶푉 is the heat capacity at constant volume 푉, 푣̅푝ℎ is the average group velocity of acoustic phonons and 푙 is the phonon mean free path. When 푙 is comparable with the dimension of nano-inclusions the phonons are scattered more effectively and may lead to a significant reduction of 푘푙푎푡. Although this is an approximate method of describing lattice vibrations, it gives a good qualitative representation for complex structures 58.

Recent attempts have been made towards obtaining thermal conductivity from first principles59–61.

The idea is based on exact solution of the linearized phonon Boltzmann transport equation (PBTE). The

PBTE is solved self-consistently via iterative procedure for the phonon distribution function through the phonon scattering rates (PSRs). The PSRs are derived through the harmonic and anharmonic interatomic force constants (IFCs) calculated via density functional perturbation theory (DFPT). The only inputs are needed for this procedure are IFCs. This approach gives an accurate description of the harmonic and anharmonic interatomic forces semiconductors and insulators59,60.

1.4 Types of materials suitable for TE applications

There are no known fundamental limitations on how large 푍푇̅ can be, and yet for the past several years the maximum 푍푇̅ of the materials used in commercial devices is about 1.0 − 1.5 for all applicable

21 temperature ranges. However, such values of 푍푇̅ are not enough to compete with traditional power generators currently at use and a figure of merit greater than 2.5 (~40% of the Carnot efficiency) is desired.4,28,42

A key problem is how to influence the intrinsic material properties in a favorable way. It is clear that low 푘 and high 훼 and 휎 are required in both 푛 and 푝 type of semiconducting materials, and these requirements are counter-exclusive for most existing materials. Recent achievements in advanced materials synthesis, characterization, and theoretical modeling, however, have demonstrated that it is possible to obtain larger 푍푇̅ in the laboratory28,42,62,63. In addition, other important criteria become relevant, such as an applicable temperature range, toxicity of materials, and production cost. Therefore, thermoelectrics used for commercial purposes should be environmentally safe, available at competitive prices, and possess high 푍푇̅ in a desired temperature range.

The types of materials used for thermoelectricity have gone through several stages. The first generation is tellurium based alloys developed in the late 50's and they still generate great scientific

4,22,23,28 interest. The best known and studied materials are Bi2Te3 and PbTe . Despite their modest 푍푇̅~1

(only 4%-5% of the Carnot efficiency), they are used up to date in small TE modules because of their temperature ranges of operation ~300 − 400 퐾 and ~600 − 900 퐾 for Bi2Te3 and PbTe, respectively.

Materials based on Si-Ge alloying have also become useful for high temperature intervals of operation28,64. During the 60’s-90’s, the most common method to enhance efficiency in such alloys has been a creating point defects through doping with Se, Ge, Sn and Sb. The attractive properties of this class of compounds stem from the structural disorder in terms of atomic mass fluctuations leading to large

-1 -1 phonon scattering, and thus reducing 푘푙푎푡 as low as 1. 5 Wm K . However, such type of the scattering significantly reduces the carrier mobility 푛 and limits 푍푇̅. The best commercial material up to date is (Bi1-

28 xSbx)(Se1-yTey)3 family for which the maximum achieved value of 푍푇̅ is still about unity .

In the 90’s, new concepts have been suggested for enhancing efficiency. One approach is based on the ideas of M. Dresselhaus in 1991, as discussed previously25–27. According to this theory, 푍푇̅ can be significantly increased upon the reduction of dimensionality of the materials from 3D bulk to 2D surface structures by nanostructuring, leading to quantum confinement effects. It is believed that holes or electrons limited by the material dimension can significantly enhance thermopower. Additionally, the presence of interfaces in 2D layered materials should further advantageously affect 푍푇̅ by reducing 푘푙푎푡 due to scattering at the boundaries. Some results were shown for Bi2Te3-Sb2Te2 quantum well superlattices with 푍푇̅ found about 2.4, almost twice greater as compared to their bulk counterparts. For

PbTe-PbSeTe quantum dot superlattices 푍푇̅ has been found as ~1.6, which is almost five times greater as compared to their the bulk counterpart (푍푇̅ ~0.3)4,42. This is achieved mainly due to reduction in thermal conductivity because of the presence of boundaries. However, the quantum confinement effect in such materials has not been proven yet and such research is still in progress28. Although many investigations on

Te-Pb based materials are ongoing, the toxicity of such elements should be taken into account. According to the European Union directives “Waste from Electrical and Electronic Equipment (WEEE)” and

“Restriction of Hazardous Substances (RoHS)”, the use of heavy elements in automotive industry will be prohibited from 201965,66. Thus, new materials have to be included into consideration to create environmentally friendly thermoelectrics.

Another approach is based on identifying materials with strong anharmonicity for strong phonon scattering. Specifically, the ideas proposed by Glenn Slack and collaborators18 suggest that the ideal TE must have a glass like thermal conductivity along with high charge carrier mobility. Such materials are named Phonon Glass Electron Crystals (PGEC). Skutterudites and clathrates have been proven to have a unique crystal structure and considered as promising candidates for PGEC. Both materials have a cage like structure hosting guest atoms which can “rattle” inside. This process causes a damping effect on phonons and leads to significant reduction of the lattice thermal conductivity. At the same time, the power factor might be enhanced through doping in the cage network29,30,67. The performance of materials with

23 strong anharmonicity might be enhanced by exploring systems with large molecular weight, complex crystal structure, anisotropic and weak chemical bonding and other unique structural characteristics28,67.

The most studied skutterudites are Co-Sb based structures due to their high mobility and low

2 electrical resistivity. For example, CoSb3 has high power factor 휎훼 (30 mW/cmK) and high Seebeck coefficient 훼 (200 mV/K), however it also possesses high thermal conductivity which results in low 푍푇̅

68,69. It is proposed that filling the voids in skutterudite structures with guest atoms would reduce thermal conductivity as it leads to scattering of phonons. Recent results have shown a promising trend in reducing

70,71 thermal conductivity for the filled structurers Yb0.2Co4Sb12.3 and Ce0.1La0.2FeCo3Sb12 , which been found to be about 1.8 W/mK at 700-800 K. In addition, the TE performance of skutterudites might be enhanced through doping and nanoparticle inclusion. Summarizing the most recent results achieved for skutterudites shows that the best 푍푇̅ can be obtained only in a high temperature range (750-900 K) with maximum values of 1.0-1.4.

The most studied clathrates are type I and type II composed of tetrahedrally coordinated Si, Ge and Ge atoms from the IVA periodic group 48,72,73. Similar to skutterudites, in their voids they can accommodate large ions such as Cs or Ba in order to reduce the thermal conductivity. Doping on the cage sites with Al and Ga from the IIIA periodic group, for example, may allow changing the transport properties in a favorable way 58,74. Also, the interesting idea of hosting Noble gases inside the clathrate

75,76 structures might be useful in practice . The best 푍푇̅ up to date is achieved for Ba8Ga16Ge30 type I clathrate with the value of 1.4 at 900 K 77–79.

Other mechanisms related to the intrinsic structural properties can be used as routes to tailor 푍푇̅ in desirable ways. For example, complex structures containing atoms from the VI and V periodic group such as AgSbTe2 and PbCuSbS3 have been shown to have thermal conductivities 0.5-0.7 W/mK in 200-

600 K temperature range 25,80,81. Such low thermal conductivity originates from the Sb lone pair s2

24 electrons that causes electrostatic interaction between other neighboring Sb atoms causing anharmonicity in lattice vibrations.

Other, more chemically complex chalcogenides (quaternary structures) constitute sets of materials that may be suitable for thermoelectricity. For example, recent reports show that Cu2ZnSnS4

(CZTS) and Cu2ZnSnSe4 (CZTSe), representatives of the I2-II-IV-VI4 class of materials (I – Cu, Ag; II –

Zn, Cd; IV – Si, Ge, Sn, Pb; VI – S, Se), have good TE properties . Such materials are direct band gap semiconductors and widely known for their photocatalysis and photovoltaic applications due to their direct band gap ~1.0 − 1.5 eV82. The possibility to change their electrical properties via doping and maintain their intrinsically low thermal conductivity ~1.0 − 1.5 Wm-1K-1 makes them very attractive in applications to thermoelectricity. Additionally, their atomic structure allows making cation and anion substitutions, directly affecting 푛 and 푘 in advantageous ways.

2 MOTIVATION

Figure 2.1 A schematic representation of a synergistic approach for materials science discoveries.

The recent progress in thermoelectricity outlined above clearly shows that materials play a crucial role in designing devices for efficient energy conversion. Specifically, the internal electronic and phonon properties governed by the atomic structure and underlying scattering mechanisms are key factors and must be understood qualitatively and quantitatively. Thus it is imperative to search for new materials and build basic scientific knowledge of classes of systems to outline general trends. At the same time, large scale devices must deal with other practical issues, such as synthesis and processing cost. Additionally,

26 materials containing toxic compositions can make them unsafe for large-scale utilization. It appears that there are different aspects related to thermoelectric research and applications, which require a unified approach involving experimental and theoretical efforts.

Computer simulations play an important role in the search of new materials, property optimization, and guiding experiments. With increasing computational power and resources as well as building large scale codes the importance of computer simulations in materials science, and specifically in thermoelectricity, has a continuous upward trend. First principles simulations based on density functional theory (DFT) as implemented in state of the art codes, such as Abinit83–85, Quantum Espresso86, and

VASP87–89 are indispensable in determining stable atomic structures, electronic properties, phonon band structures, and transport characteristics. An interactive approach in which first principles simulations are utilized in collaboration with experiments to validate the results from the calculations as well as to provide a useful feedback to guide further experiments has proven to be a winning strategy in thermoelectric materials optimization.

The main motivation of this research is to discover novel fundamental structure-property relations as well as new materials suitable for thermoelectric energy conversion using first principles simulations in a synergistic approach, as shown schematically in Figure 2.1. Our computational efforts are useful not only from a basic scientific perspective, but they also help with reducing cost and time in the experimental laboratory.

In this research several classes of materials are investigated, such as Bi1-xSbx alloys, bournonites, clathrates and quaternary chalcogenides. The Bi rich Bi1-xSbx materials have been known as the best thermoelectric materials at low temperature regimes, and in this work the delicate balance between atomic structure upon Sb doping and their electronic properties is investigated via DFT simulations.

Anharmonicity is an important factor for phonon-phonon scattering and it is associated with the intrinsic structure of the lattice. Identifying such processes is beneficial to identify materials with low thermal

27 conductivity. Here I focus on clathrates, a class of materials where anharmonicity comes from guest rattling inside the empty voids, and bournonites, a class of materials where anharmonicity originates from stereochemically active s2-pair electrons. In collaboration with Dr. Nolas’ experimental team, several types of type II clathrates and doped bournonites are investigated. In addition, several representatives of the quaternary chalcogenide family of materials are also studied from first principles, where novel transport mechanisms and unusual structure-property relations are discovered. Quaternary chalcogenides have been shown to be useful for photovoltaics, however their low thermal conductivity makes them suitable for thermoelectricity as well.

3 METHODOLOGY

3.1 Basics of Density Functional Theory

3.1.1 Hartree-Fock method

A primary goal in ab-initio simulation is to determine the ground state energy of a many-electron system of the investigated material. In general, the unit or conventional cells of real atomic systems

(crystals) have a very high number of degrees of freedom which makes integration of the Schrödinger equations practically impossible. Historically, there are several theories developed in helping to overcome these issues.

The first approximation, known as the Born-Oppenheimer (BO) approximation, rests on the fact that ions are much heavier than electrons and thus may be considered as frozen particles90–92. Then, the many-body Schrödinger in the presence of such a “frozen” background is given by

2 2 2 ℏ 2 푍퐽푒 1 푒 [− ∑푖=1 ∇푖 + ∑푖=1 ∑퐽=1 + ∑푖=1 ∑푗<푖 ] Ψ = 퐸Ψ, (3.1.1) 2푚 |풓풊−푹푱| 2 |풓풊−풓풋| where the first term describes the kinetic energy of 푁 electrons in the system, the second term describes the Coulomb interactions between the electrons (풓풊) and the nuclei (푹푱) and the third term describes the

Coulomb electron-electron interaction (풓풊, 풓풋). Here, Ψ = Ψ(풓ퟏ, 푠1, … , 풓푵, 푠푁) is the many-electron wave function, where 풓풊 is the spatial radius vector and 푠푖 is the spin of an i-th electron of the many-electron system. It is assumed that electrons do not interact each other i.e. third term in Eq. (3.1.1) is zero. Under this approximation the solution of Eq. (3.1.1) simplifies and Ψ is found as the product of the one-electron wave function ψ푗

Ψ(풓ퟏ, 푠1, … 풓푵, 푠푁) = ∏푗 ψ푗(풓풊, 푠푖), (3.1.2)

This expression is known as a Hartree product93. Unfortunately, such representation does not satisfy the antisymmetry rule, i.e. exchanging of two electrons does not change the sign of the wave function, which is a fundamental drawback of the approximation. To get a better description, another approximation is suggested, known as a Hartree-Fock approximation94, in which the many-electron wavefunction Ψ is

95 found as the Slater determinant of a matrix of ψ푗

1 Ψ(풓 , 푠 , … 풓 , 푠 ) = det [ψ (풓 , 푠 )] (3.1.3) ퟏ 1 푵 푁 √푁 푗 풊 푖

The chosen set of ψ푗 has to be orthonormal and it is further used to construct the electronic density 휌(풓) of the investigated system over 푁 electrons

푁 2 휌(풓) = ∑푗=1|ψ푗(풓)| (3.1.4)

Then, the potential in Eq. (3.1.1) can be written as

푉(풓) = 푉퐸푋(풓) + 푉퐻(풓), (3.1.5)

where 푉퐸푋 is the Coulomb interactions between the electrons and the nuclei, 푉퐻 is the Coulomb interaction (Hartree potential) between pairs of the electrons at points 풓 and 풓′ replaced by

푒2 휌(풓)휌(풓′) 푉 (풓) = ∫ 푑풓푑풓′ (3.1.6) 퐻 2 |풓−풓′|

The Hartree-Fock approximation also has several drawbacks96. The first problem arises from the construction of the many-electron wave function Ψ, especially for real systems. Practically, the algorithm of finding the appropriate set of orbitals (solutions of the Hartree-Fock equations) for medium size systems is not efficient and very slow. In addition, the number of the calculated parameters for finding the

30 ground state energy is incredibly high. Suppose the parameter 퐷 is the degree of freedom of an electron and 푁 is the number of electrons in the system, then the number of the calculated parameters, 푀 is found to be

푀 = 퐷3푁 (3.1.7)

For example, a system consisting of 푁 = 50 electrons with minimum degree of freedom 퐷 = 3 (spin is not included) requires calculating 푀 = 3150 ≈ 1075 parameters. Obviously, such value of 푀 is very

"heavy" for modern computational facilities and medium systems consisting of 100-300 electrons are beyond the scope of current computational facilities. In addition, there is a practical limitation of storing such a big data required for the many-electron wave function.

The second main problem comes from the fact that the potential in Eq. (3.1.5) does not include the exchange-correlation effects between electrons, which is a serious theoretical omission. New theoretical concepts based on the Thomas-Fermi (TF) theory remedy these issues in the following way: firstly, the many-electron system is considered through its electron density; secondly, such consideration includes exchange-correlation effects into the functional.

3.1.2 Thomas-Fermi theory

Shortly after Schrodinger’s theory97, Thomas98 and Fermi99,100, suggested considering the many- electron system as a gas of electrons in a nuclear background. The key point of their idea is replacing the many-electron wave function with its electron density. The total energy functional according to TF theory is described by

2 5 3 2 1 휌(풓)휌(풓′) 퐸 [휌(풓)] = (3휋 )3 ∫[휌(풓)]3 푑풓 + ∫ 푉 (풓)휌(풓)푑풓 + ∫ 푑풓푑풓′, (3.1.8) 푇퐹 10 퐸푋 2 |풓−풓′| where the first term is the electron kinetic contribution for non-interacting electrons, the second term is the contribution from the electron-nuclei interactions and the third term is the Hartree contribution for

101 interacting electrons. Minimization 퐸푇퐹[휌(풓)] leads to the following condition for the electron density :

1 휌(풓′) (3휋휌(풓))2/3 + ∫ 푑풓′ + 푉 − 휆=0. (3.1.9) 2 |풓−풓′| 퐸푋

3 3 Here, 휌(풓) = 푝퐹(풓)⁄3(ℏ휋) is the density determined by the Fermi momentum 푝퐹 with maximum value

3/2 2 of 휌 = (2퐸퐹) ⁄3휋 , where 퐸퐹 is the Fermi energy and 휆 is the chemical potential. If 푉(풓) = 푉퐸푋 +

휌(풓′) ∫ 푑풓′ then the electron density 휌(풓) can be rewritten as |풓−풓′|

휌(풓) = 23/2⁄3휋2 (휆 − 푉(풓))3/2. (3.1.10)

This theory accurately describes the nuclei with infinite charges but does not work for systems with valence electrons and as a result does not properly describe electron correlations102–104. Despite this limitation it is a useful approximation and an important initial step towards Density Functional Theory

(DFT).

3.1.3 Kohn-Hohenberg Theory

TF theory motivated Valter Kohn and Pier Hohenberg to create a more advanced theory where the key ingredient is the electronic density of the investigated system105–107. The origins of modern

Density Functional Theory are based on two theorems published in 1964105:

i. The ground-state energy from Schrödinger’s equation is a unique functional of the electron

density.

ii. The electron density that minimizes the energy of the overall functional is the true electron

density corresponding to the full solution of the Schrödinger equation.

From (i) it follows that any observable of the quantum system can be obtained from the electron density of this system. From (ii) it follows that the electron density obeys the variational principle in the same manner as the wave function.

The Kohn-Hohenberg functional 퐸[휌(풓)] of the total energy of the system with interacting electrons is determined by

퐸[휌(풓)] = 푇[휌(풓)] + ∫ 휌(풓)푉퐸푋(풓)푑풓 + 퐸퐻[휌(풓)] + 퐸푋퐶[휌(풓)], (3.1.11)

where 푇[휌(풓)] is the electron kinetic contribution for non-interacting electrons, ∫ 휌(풓)푉퐸푋(풓)푑풓 is the contribution from the electron-nuclei interactions, 퐸퐻[휌(풓)] is the Hartree contribution for interacting electrons and 퐸푋퐶[휌(풓)] is the exchange-correlation functional, the key term in DFT. According to the

Kohn-Hohenberg approach 퐸[휌(풓)] in Eq. (3.10) has to be minimized in order to determine the true electron density 휌(풓). It is realized via the method of Lagrange multipliers, which leads to

휕푇[휌(풓)] 휌(풓′) 휕퐸 [휌(풓)] + 푉 [휌(풓)] + ∫ 푑풓 + 푋퐶 = 휆휌(풓), (3.1.12) 휕휌(풓) 퐸푋 |풓−풓′| 휕휌(풓) where 휆 is the Lagrange multiplier and might be associated with chemical potential. Eq. (3.1.12) determines the effective potential with its exact electron density 휌(풓):

휌(풓′) 푉 (풓) = 푉 [휌(풓)] + ∫ 푑풓 + 푉 [휌(풓)], (3.1.13) 퐸퐹퐹 퐸푋 |풓−풓′| 푋퐶 where

휕퐸 [휌(풓)] 푉 [휌(풓)] = 푋퐶 . (3.1.14) 푋퐶 휕휌(풓)

Once 푉퐸퐹퐹(풓) is found, the Schrödinger equation for Kohn-Sham orbitals ψ푗 can be constructed

1 (− ∇2 + 푉 (풓)) ψ (풓) = 휀 ψ (풓). (3.1.15) 2 퐸퐹퐹 푗 푗 푗

The solutions of the above equations guarantee that

푁 ∑푗=1 휀푗 = 푇[휌(풓)] + ∫ 푣퐸퐹퐹(풓) 휌(풓)푑풓 (3.1.16) is valid for the given 휌(풓).

Figure 3.1.1 A schematic representation of solving the Kohn-Sham equations self-consistently.

Equations (3.1.13), (3.1.15) and (3.1.16) compose a self-consistent system of the Kohn-Sham equations108 with its solution procedure shown in Figure 3.1.1. Once the electron density 휌0(풓) is found, the ground state energy of the investigated system is given by

1 휌(풓)휌(풓′) 퐸 = ∑푁 휀 + 퐸 [휌(풓)] − ∫ 푑풓푉 (풓)휌(풓) − ∫ 푑풓푑풓′ (3.1.17) 푇푂푇 푗=1 푗 푋퐶 푋퐶 2 |풓−풓′|

In the above expression 퐸푋퐶[휌(풓)] and 푉푋퐶(풓) = 휕퐸푋퐶[휌(풓)]⁄휕휌(풓) are unknown, thus one must utilize suitable approximate methods to facilitate further calculations. There are several state of the

34 art quantum-mechanical computational packages with several types of approximations for the exchange- correlation functionals allowing self-consistent solution of the Kohn-Sham equations. These include

ABINIT83, Quantum-Espresso86, and SIESTA109 among others. Calculations for the present research are carried out via Vienna Ab-Initio Simulation Package (VASP)87, thus details about the computations are given below in the framework of this code.

3.1.4 Exchange-correlation energy

One of the main problems in DFT is to determine the unknown 퐸푋퐶 term, which consists of kinetic energy 푇[휌(풓)] of interacting electrons, kinetic energy 푇0[휌(풓)] of non-interacting electrons and

Coulomb interaction 푈푋퐶[휌(풓)] of electrons with exchange correlations of holes,

퐸푋퐶[휌(풓)] = 푇[휌(풓)] − 푇0[휌(풓)] + 푈푋퐶[휌(풓)]. (3.1.18)

th There are several approximation developed for 퐸푋퐶[휌(풓)]. In the 60 , Kohn and Hohenberg introduced the Local Density Approximation (LDA), such that 96,110

퐿퐷퐴 퐻푂푀 퐸푋퐶 [휌(풓)] = ∫ 휌(풓)휀푋퐶 (휌(풓))푑풓, (3.1.19)

퐻푂푀 where 휌(풓) is essentially the probability of finding a particle at distance 풓, 휀푋퐶 is the exchange- correlation energy a homogenous electron gas with density 휌. The exchange contribution was introduced by Dirac and named the “Slater” exchange

퐿퐷퐴 4/3 퐸푋 [휌(풓)] = −푐푋 ∫ 휌(풓) 푑풓, (3.1.20)

2 3 3 1/3 3 3 1/3 where 푐 = ( ) is used for LDA and 푐 = ( ) is used in LSDA, here “S” stands for 푋 24/3 2 4휋 푋 2 4휋

“Spin”. The correlation contribution is usually parametrized from Monte Canto simulations111.

Historically, the first estimation for small 푟푠 was done by E. Wigner

퐻푂푀푂 0.44 퐸퐶 [휌(풓)] = − , (3.1.21) 푟푠+7.8

38 where 푟푠 is the Seitz electron radius (the mean interelectronic spacing) .

The LDA describes relatively well the electronic and atomic structures of many metals and transition metals96,110,112. The ionization, bond-dissociation and binding energy are calculated with the accuracy of

10-15%113–115. Bond lengths and atomic structures are calculated a bit better, with accuracy of 1-2% when compared with experimental data. However, LDA poorly describes energies of the systems with strongly correlated effects, typically originating from interaction of d-,f- shell electrons leading to underestimation of the energy band gap112. In addition, LDA is not suitable for non-homogeneous structures.

From the mid-80s DFT has been attracting more popularity due to the introduction of a new approach in the description of the electron density. The Generalized Gradient Approximation (GGA) is used in the structures where electron density is not homogenous115–117. It is achieved by adding the functional dependence on the gradient of 휌(풓):

퐺퐺퐴 퐺퐺퐴 퐸푋퐶 [휌(풓)] = ∫ 휌(풓)휀푋퐶 (휌(풓), ∇휌(풓))푑풓 (3.1.22)

In more detail, the exchange-correlation functional was taken as the additional correction to the LDA functional. Kohn and Hohenberg added the gradient of the electron density expanded in a Taylor series

퐺퐺퐴 퐿퐷퐴 2 2 퐸푋퐶 [휌(풓), ∇휌(풓)] = 퐸푋퐶 + ∫ 퐷1[휌(풓)]∇휌(풓)푑풓 + ∫ 퐷2[휌(풓)](∇ 휌(풓)) 푑풓 + ⋯, (3.1.23)

where 퐷푖 – are universal functionals of 휌(풓). For calculations in crystal structures the GGA approximation gives better results for the ground state energy as compared to LDA. GGA also achieves better description of bandgaps of semiconductors and insulators when compared with experimental data.

There are two ways to construct the GGA functionals. The first approach is based on the functional in the limits of low and high electron density. The second approach is semi-empirical and it is based on varying the parameters of the functional. In 1988, Becke suggested the GGA functional (B88) where the exchange part is modified as116:

휌(풓)3푥2 퐸퐵88[휌(풓)] = 퐸퐿퐷퐴[휌(풓)] − 푐 ∫ 푑풓, (3.1.24) 푋 푋 푋 1+6푠ℎ−1(푥)

−4/3 −4 where 푥 = 휌(풓) |∇휌(풓)|, 푐푋 is chosen to be 42 × 10 in order to reproduce the known atomic energies of inert gases. In 1991 Perdew and Wang suggested the most reliable functional (PW91) for

GGA113. Later, in 1996 Perdew, Burke and Ernzerhof (PBE) modified PW91 to PBE functional118. In

1999 this functional was further modified to RPBE (Revised PBE) functional in which atomization energies are improved for small molecules and molecules on metallic surfaces119. The most commonly used functionals up to date are B88, PBE and Lee-Yang-Parr (LYP)120.

The main problem with the above LDA or GGA methods is that they are local given that the electron interactions i.e. 퐸푋퐶[휌(풓)] on 휌(풓) and ∇휌(풓) calculated at 풓. In contrast with non-local (exact) exchange, the local exchange energy does not contain integration over all spatial coordinates 풓 and results in artificial electron self-correlation effect added to 퐸푋퐶[휌(풓)]. This artificial addition cannot be eliminated with “pure” LDA or GGA and finally leads to inaccuracy in the calculations. Practically, semi- local potentials cannot describe systems with long-range interactions. The exchange energy between electrons with the same spin (Fermi correlation) dominates and accounts for 85-95% of the total exchange-correlation energy 퐸푋퐶. That is why modifying the exchange part is an important issue for functional hybridization. Inclusion of non-local exchange (Hartee-Fock potential) to the semi-local (GGA or LDA) allows eliminating/reducing the self-correlation effect and gives better results for systems with dissociating molecules or for systems containing transition metals. In general, the hybrid functionals can be represented as (휌(풓) is omitted in the formulas below for simplicity):

퐸푋퐴퐶푇 푁푑 퐷퐹푇 퐸푋퐶 = 퐸푋 + 퐸퐶 + 퐸퐶 , (3.1.25)

퐸푋퐴퐶푇 where 퐸푋 is the Hartee-Fock contribution calculated exactly by the integration of the exchange

푁푑 energy density, 퐸퐶 is the long-range Coulomb interaction, responsible for lowering probability when

퐷퐹푇 two electrons with opposite spins are infinitely close each other and 퐸퐶 is the correlation contribution

37 obtained via standard DFT methods (usually by parametrization). Depending on the particular method,

Eq. (3.1.25) is modified with different parameters. The simplest example of hybrid functional is the linear combination introduced by Becke in 1993120

퐵93 퐸푋퐴퐶푇 퐷퐹푇 퐷퐹푇 퐸푋퐶 = 퐶0퐸푋 + (1 − 퐶0)퐸푋 + 퐸퐶 , (3.1.26)

퐷퐹푇 퐿퐷퐴 where 퐸푋 = 퐸푋 and 퐶0 = 0.5. Another popular functional is the hybridization between LDA, B88,

Vosko-Wilk-Nusain correlation functional III (VWN3) and LYP functionals120

퐵3퐿푌푃 퐿퐷퐴 퐵88 푉푊푁3 퐿푌푃 퐸푋퐶 = 0.8퐸푋 + 0.2퐸푋 + 0.72∆퐸퐶 + 0.19퐸퐶 + 0.81퐸퐶 . (3.1.27)

The parameters 퐶푖 in Eq. (3.1.27) are found by using a fit procedure with experimental data.

Another type of functionals, the range separated hybrid functionals (HSE), improves the description of the asymptotic (−1/푟) behavior and the lack of derivative discontinuity ∇휌(풓) in the systems with non-homogenous electron density. The key point is to separate the exchange electron- electron interaction into short range and long range contributions. For example, in the HSE functional used in the current research the long-ranged part of the Fock interaction is replaced by the GGA

퐺퐺퐴,퐿푅 121–124 counterpart 퐸푋

퐻푆퐸 푆푅 퐺퐺퐴,푆푅 퐺퐺퐴,퐿푅 퐺퐺퐴 퐸푋퐶 = 1/4퐸푋 (휇) + 3/4퐸푋 (휇) + 퐸푋 (휇) + 퐸퐶 , (3.1.28)

푆푅 퐺퐺퐴,푆푅 where 퐸푋 is short-ranged part of the Fock interaction, 퐸푋 is the short-ranged part of the interaction

퐺퐺퐴 approximated via GGA and 퐸퐶 is the electronic correlation obtained from GGA. The functional used for GGA is PBE. Note, that all exchange terms in Eq. (3.1.28) depend also on the parameter 휇 which defines the range-separation. It has been shown that the optimum range value for 휇 is 0.2-0.3 Å.

Although a skillful usage of hybrid HF/DFT describes electronic structure correctly (in most cases), other issues may also be important for practical applications125–127. Because of the large number of valence electrons (d-, f- orbitals) and complexity of a given structure, such calculations are very time

38 consuming or even impossible to perform. In application to thermoelectricity the hybrid method is very desirable but the structures of interest are usually complex (different types of atoms, doping) and often contain heavy elements (presence of partially filled d- orbital) which means that the HF/DFT has limited benefits currently.

3.1.5 DFT+U theory

As explained above, structures with localized d- and f- electrons induce strong on-site Coulomb interactions and are not described correctly via standard DFT methods (GGA or LDA). In order to treat

128,129 this issue, an additional Hubbard-like term 푈퐸퐹퐹 is added to the total energy

1 퐸 = 퐸 + ∑ 푈 푇푟(푔푗 − 푔푗푔푗), (3.1.29) 퐷퐹푇+푈 퐷퐹푇 2 푗 퐸퐹퐹 where 푔푗 is the atomic orbital occupation matrix calculated self-consistently with the Kohn-Sham eigenvectors. The Hubbard-like term 푈퐸퐹퐹 consists of two independent parameters introduced by

Liechtenstein and Dudarev

푈퐸퐹퐹 = 푈 − 퐽, (3.1.30) where 푈 is the on-site Coulomb parameter and 퐽 is the on-site exchange parameter which are usually obtained either from experiment or by substitution from computation. The DFT+U method is popular in applications to strongly correlated materials and polaron-like transport130–133 for example, where 퐽 is usually set to zero and 푈 is parametrized. In addition, DFT+U might be used as a “light” substitution for more “heavy” HSE approach134–136, in which the range separation in the Coulomb interactions is factually replaced by the 푈 parameter.

3.2 PAW method

First-principle calculations implemented in VASP are based on the projector augmented-wave

(PAW) method proposed by Blöchl137. In the framework of this method the all-electron wavefunction

(Ψ퐴퐸) is derived from Ψ푃푆 by linear transformations as

퐴퐸 푃푆 퐴퐸 푃푆 푃푆 푃푆 |Ψ ⟩ = |Ψ ⟩ + ∑푖(|휑푖 ⟩ − |휑푖 ⟩) ⟨푝푖 |Ψ ⟩, (3.2.1)

푃푆 퐴퐸 where Ψ are the variational quantities expanded in the plane waves, 휑푖 are the partial wave functions

푃푆 obtained from the Schrödinger equation for the spherical case near the atomic 푖-th nucleus, 휑푖 are the

퐴퐸 푃푆 partial waves identical to Ψ in vicinity between the PAW spheres. The projector functions 푝푖 define

푃푆 푃푆 푃푆 variational coefficients 푐푖 as 푐푖 = ⟨푝푖 |Ψ ⟩. There are some requirements for 푝푖 , such as orthogonality

푃푆 푃푆 푃푆 ⟨푝푖 |휑푗 ⟩ = 훿푖푗 and smoothness of 휑푗 inside the PAW spheres. The above wave functions define charge densities which are further used for constructing the exact PAW Kohn-Sham density functional138.

More details about PAW method in its implementation in VASP can be found in references137–139.

3.3 Spin-Orbit Interaction

For solids containing heavy atoms, layered structures, oxides and others complex structures spin- orbit coupling (SOC) may be very important. SOC is taken as a 푳 ∙ 푺 relativistic correction originating from the interaction between electron spin 푺 and its orbital angular momentum 푳. The SOC contribution to the total energy 퐸 is

1 1 푑푈 퐸 = (푳 ∙ 푺), (3.3.1) 푆푂퐶 2푚2푐2 푟 푑풓 where 푈 is the nucleus potential. The significance of SOC can be demonstrated using the hydrogen atom model, for which the total energy is

2 휇0푍푒 퐸 = 퐸푛 + 2 3 (푳 ∙ 푺), (3.3.2) 8휋푚푒푟

where 휇0 is the magnetic constant, 푍 is the atomic number, 푚푒 is mass of electron and 푟 is the electron radius. If the 푳 ∙ 푺 product is expressed through the eigenvalues of spin, orbital and total angular momentum then the total energy can be expressed as140

훽 (푛,푙) 퐸 = 퐸 + 푠 [푗(푗 + 1) − 푙(푙 + 1) − 푠(푠 + 1)], (3.3.3) 푛 2

40 where 푗 is the total angular momentum quantum number, 푙 is the angular momentum (azimuthal quantum number), 푠 is the spin number with

4 2 푍 휇0푔푠휇퐵 훽푠(푛, 푙) = 3 3 , (3.3.4) 4휋푎0푛 푙(푙+1/2)(푙+1)

where 푔푠 is the spin 푔-factor, 푎0 is the Bohr radius and 휇퐵 is the Bohr magneton.

From Eq. (3.3.4) two useful conclusions can be made. The influence of SOC becomes greater when the atomic number is large. The effect of SOC gets smaller when 푛 increases. In a crystal lattice, the effect of

SOC can be tracked as splitting of the bands at high symmetry points.

퐴퐸 SOC is implemented in DFT calculations as the part of the all-electron wave function 휑푖 of the

PAW Hamiltonian (see Sec.3.2).141,142 For general case of magnetism (non-collinear spin polarization) the

SOC contribution is

2 훼훽 ℏ 퐴퐸 1 푑푉(푟) 퐴퐸 푃푆 푃푆 퐻푆푂퐶 = 2 ∑푖,푗⟨휑푖 | |휑푗 ⟩|푝푖 ⟩𝝈훼훽푳푖푗⟨푝푗 |, (3.3.5) (2푚푒푐) 푟 푑푟

where 푉(푟) is the spherical part of the AE potential within the PAW sphere, 𝝈훼훽 is the Pauli spin matrices and 푳푖푗 is the angular momentum operators. Indexes 훼, 훽 stand for spin +1/2 and -1/2 respectively, describing non-collinear magnetism.

3.4 Van der Waals interactions

For materials containing layered constituents or having voids in the crystal, non-covalent interactions may be important for the structural stability and electronic properties. Interactions characterizing such materials are typically of van der Waals nature, which originate from non-local electron correlations and they are not taken into account in local LDA or semi-local GGA-DFT applications. The long-range dispersive correction 퐸퐷퐼푆푃 is included as an addition to the Kohn-Sham energy 퐸퐾푆: 퐸퐷퐹푇−퐷퐼푆푃 = 퐸퐾푆 + 퐸퐷퐼푆푃. Recent efforts have resulted in several approximations for the dispersive part.

3.4.1 Interatomic correction method

The first popular approach for taking van der Waals interaction into account is the interatomic

143,144 correction method in which 퐸퐷퐼푆푃 is calculated in using a Lennard-Jones pairwise potential :

1 푁푎푡 푁푎푡 푇푅 퐶6푖푗 퐶8푖푗 퐶10푖푗 퐸퐷퐼푆푃 = − ∑푖=1 ∑푗=1 ∑퐿,푖≠푗 [ 6 푓6(푅푖푗,퐿) + 8 푓8(푅푖푗,퐿) + 10 푓10(푅푖푗,퐿) + ⋯ ], (3.4.1) 2 푅푖푗 푅푖푗 푅푖푗 where the summations are over all pairs of 푖, 푗 atoms in the systems and all possible 퐿 translations along in 3퐷 space, 푅푖푗 is the interatomic distance, 퐶□푖푗 is the dispersion coefficients and 푓□(푅푖푗,퐿) is the Fermi- like damping function introduced to eliminate near-singularities.

There are several DFT implementation for vdW corrections modeled by Eq. (3.4.1). In 2006

145 Grimme developed DFT-D2 method in which 퐸퐷퐼푆푃 is calculated as

1 푁푎푡 푁푎푡 푇푅 퐶6푖푗 퐸퐷퐼푆푃 = − ∑푖=1 ∑푗=1 ∑퐿,푖≠푗 6 푓6(푅푖푗,퐿), (3.4.2) 2 푅푖푗 with the dumping function derived as

푠6 푓6(푅) = , (3.4.3) 1+퐸푥푝[−푑(푅⁄푅푇)]

where 푑 is the damping parameter determined empirically, 푅푇 is the sum of atomic vdW radii and 푠6 is the global scaling factor optimized uniquely for each functional (0.75 for PBE, 1.2 for BLYP and 1.05 for

B3LYP). Several modifications, including DFT-D3 and DFT-D3 with Becke-Johnson damping, are also available in VASP along with the DFT-D2 method146–148.

An advanced modification of DFT-D is the Tkachenko-Scheffler approach (DFT-TS), developed in 2009149 and implemented in VASP in 2012150. The DFT-TS improves the previous approach by adding the dipole polarizability to Eq. (3.4.2), where the dispersion coefficients and the damping functions are

푆퐶 now polarizability dependent variables. The dipole polarizability 훼푝 (푖휔) is frequency dependent and derived from

푆퐶 푇푆 푇푆 푁 푆퐶 훼푝 (푖휔) = 훼푝 (푖휔) − 훼푝 (푖휔) ∑푝≠푞 휏푝푞훼푞 (푖휔), (3.4.4)

푇푆 where 휏푝푞 is the dipole interaction tensor and 훼푝 is the self-consistent frequency-dependent polarizability for atom 푝 derived as

훼푃[휌(풓)] 훼푃(푖휔) = 2, (3.4.5) 1+(휔⁄휔푃[휌(풓)])

where 훼푃[휌(풓)] is the static polarizability of an atom, 휔푃[휌(풓)] is the corresponding excitation frequency derived self-consistently. However the DFT-TS method fails to describe ionic solids. A modified method

(DFT-TS/HI) based on the Bultinck iterative algorithm solves this problem151,152. In this algorithm the neutral atoms are replaced with ions taken along with charge density calculated for molecular volumes.

3.4.2 Van der Waals Density Functional

The second popular approach is conceptually different from the ones mentioned above. The vdW interaction is not added as an additional dispersion term 퐸퐷퐼푆푃 to the total energy functional, but it is included as a non-local correlation functional to the exchange energy functional 퐸푋퐶. The idea was proposed by Dion153 in 2004 in which a nonlocal functional is given as

퐺퐺퐴 퐿퐷퐴 푁퐿 퐸푋퐶 = 퐸푋 [휌(풓)] + 퐸퐶 [휌(풓)] + 퐸퐶 [휌(풓)], (3.4.6)

푁퐿 where 퐸퐶 [휌(풓)] is a nonlocal term

1 퐸푁퐿[휌(풓)] = ∫ ∫ 휌(풓)훷(푞, 푞′, 푟̃)휌(풓′)푑풓푑풓′ (3.4.7) 퐶 2 where 푟̃ = |풓 − 풓′| and 훷(푞, 푞′, 푟̃) is the function with values 푞 and 푞′ calculated for a function

′ 푞표[휌(풓), |∇휌(풓)|] at points 풓 and 풓′. The kernel function 훷(푞, 푞 , 푟̃) depends only on two variables 푝 and

푝’ derived as 푝 = |풓 − 풓′|푞 and 푝′ = |풓′ − 풓|푞′ respectively. For long range separation (non-locality) the kernel function takes the following asymptotic form

퐶 훷(푞, 푞′, 푟̃) = − , (3.4.8) 푑2푑′2(푑2+푑′2)

43 where 퐶 = 12(4휋/9)3푚푒4.

The main problem in Dion’s approach is the large demand on computing resources due to the double integral (3.4.7) over the spatial variables. In 2009, Roman-Perez and Soler suggested154 replacing the double integral (3.4.7) by its Fourier transformation which resulted to

1 퐸푁퐿 = ∆Ω2 ∑ ∑ 휃 휃 훷 (푟 ), (3.4.9) 퐶 2 훼훽 푖푗 훼푖 훽푗 훼훽 푖푗 where ∆Ω is the volume per grid point needed to replace the spatial integrals by sums in a uniform grid of points, 휃훼푖 is identical to 휌푖(풓)푃훼[푞0(휌푖(풓), ∇휌푖(풓))], where 푃훼 is a cubic polynomial. The proposed vdW

Density Functional (vdW-DF) method accurately describes vdW interactions for arbitrary geometries, however it overestimates equilibrium interatomic separation and underestimates the strength of hydrogen like bonds. In 2010, Langreth and Lundqvist et al. proposed155 the second version of vdW functional

(vdW-DF2) in which a more accurate GGA functional (PW86) is employed.

3.5 Structural properties

The first step in DFT calculations is the structural optimization96,110. In most of the cases the initial structure can be adopted from available experimental data. For good results, the DFT energy 퐸푇푂푇 can be calculated as a function of the lattice constants with typical results displayed in Figure 3.4.1 for a cubic system with one lattice constant, where the calculations can be performed for a sequence of the lattice constant 푎푚푖푛, … , 푎푚푎푥. The structure of interest undergoes a relaxation procedure (ion relaxation) at constant volume and constant shape. The optimal lattice constant 푎0 must have the lowest energy

퐸푇푂푇(푎). Usually, 푎0 is found as a fitting parameter from the applied curve. The simplest and rough approach to extract 푎0 is to rewrite 퐸푇푂푇(푎) as a truncated Taylor series

2 퐸푇푂푇(푎) ≅ 퐸푇푂푇(푎0) + 훼(푎 − 푎0) + 훽(푎 − 푎0) (3.5.1) where

푑퐸 1 푑2퐸 훼 = 푇푂푇 | and 훽 = 푇푂푇 | . (3.5.2) 푑푎 푎=푎0 2 푑푎2 푎=푎0

When 푎0 is the lattice parameter at which 퐸푇푂푇(푎0) is minimum then 훼 = 0, and Eq.(3.4.1) becomes

2 퐸푇푂푇(푎) ≅ 퐸0 + 훽(푎 − 푎0) (3.5.3)

The numerical data has to be fit to Eq. (3.5.3) represented as the violet quadratic curve shown in

Figure 3.5.1 for which 퐸0 and 푎0 are the fitting parameters. The minimum value of Eq. (3.5.3) is achieved for the optimum value 푎0.

Applying a quadratic fit is not always an accurate approximation since it does not always Figure 3.5.1 Typical computational results showing reproduce the overall shape of the numerical data. total energy ETOT minimization as a function of the lattice constant 풂. Purple curve – quadratic fitting, The most accurate method up to date is the Birch- red curve – BM fitting.

Murnaghan equation156,157 of state (BM) given by

3 2 9푉 퐵 푎 2 푎 2 푎 2 퐸 (푎) = 퐸 + 0 0 {[( 0) − 1] 퐵′ + [( 0) − 1] [6 − 4 ( 0) ]} (3.4.4) 푇푂푇 0 16 푎 0 푎 푎

where 푉0 = 푉(푎)|푎=푎0 is the equilibrium volume, 퐵0 is the bulk modulus calculated as

2 ′ ′ (푉푑 퐸푡표푡⁄푑푉)|푃=0 and 퐵0 = (휕퐵⁄휕푃)|푃=0 where 푃 is the pressure. Here, 퐸0, 퐵0, 퐵0 ,푉0 and 푎0 are the fitting parameters for the red curve shown in Figure 3.5.1.

3.6 Electronic structure properties

The electronic properties studied in this research are investigated in terms of DOS and energy band structures. In application to thermoelectricity it is desirable to have high carrier concentration in the valence band near the Fermi level 퐸퐹 as shown by the sharp and localized peak in Figure 3.6.1a. A

45 material of interest should be a semiconductor with a gap of ~0.15 − 0.25 eV158,159. The electron effective mass derived as 푚∗ = ℏ2/ (푑2퐸⁄푑푘2) can be obtained from the band structure and should be small enough to facilitates high carrier mobility. Another desirable feature is the multi-valley region in the energy band structure, as shown in Figure 3.6.1b, where the bands form multiple extrema in the valence and conduction regions. This is an indication of multiple hole or electron pocket formations leading to an increased 푍푇 160. Such design can be achieved by 푛-type or 푝-type doping in semiconductors. However mixed 푛 − 푝 type systems have lower 푆 because of cancellation of thermal currents161. Similar characteristics are present in systems with high symmetry, such as clathrates and quaternary chalcogenides.

Understanding of chemical bonds present in materials is also important and can be realized via the Electron Localization Function (ELF) introduced by Becke and Edgecombe162. Briefly, the ELF is a

Figure 3.6.1 (a) Optimum DOS for a hypothetical thermoelectric (푬푭 is the Fermi energy). (b) Multi valley region with ′ the hole and electron pocket formations. 푬푭 corresponds to n-type of a material, while 푬푭 corresponds to p-type of a material. measure of probability of finding electron in the vicinity of another electron with the same spin. The dimensionless ELF function is given by

2 −1 퐸퐿퐹 = (1 + (퐷⁄퐷0) ) , (3.6.1)

where 퐷0 corresponds to a uniform electron gas defined as

3 2 5 퐷 = (6휋2) ⁄3휌(풓) ⁄3. (3.6.2) 0 5

푁 Here 휌(풓) = ∑푖 휓푖(풓)휓푖(풓′) is the single-electron density matrix calculated at the 풓-point and reference

풓′-point, 휓푖 are the Kohn-Sham orbitals determined from the self-consistent method shown in Figure 3.1,

푁 is the number of electrons. Also 퐷 is the Pauli kinetic energy density given by

1 (∇휌(풓))2 퐷 = ∑푁|∇휓 (풓)|2 − , (3.6.3) 푖 푖 4 휌(풓) where the first term is the kinetic energy density of all 푁 electrons in the system. When the electron pairs are localized, 퐷 tends towards zero and 퐸퐿퐹 → 1. For non-localized electrons, 퐷 → ∞ leading to

퐸퐿퐹 → 0. In practice, the ELF can be viewed as a colored map of the electron density calculated via DFT

163 methods . For example, Figure 3.6.2 demonstrates the calculated ELF for Cs8Na16Si136 clathrate type II, synthesized by Bobev and Sevov164, simulated later by Khabibullin et al76. The colors range from blue to red. The red color corresponds to perfectly localized electrons (퐸퐿퐹 = 1), the green color corresponds to uniformly disturbed electrons (homogenous electron gas, 퐸퐿퐹 = 0.5) and the blue color (퐸퐿퐹 = 0) corresponds to the regions with no localized electrons. In the case of clathrates, it is well known that the cage network is formed from covalently bonded Si atoms interacting in ionic manner with Cs and Na guest atoms165. The calculated ELF confirms the covalent character of bonding between Si atoms seen as red perfectly localized regions (Figure 3.6.2a). The blue regions at the Na sites shows the

Figure 3.6.2 The ELF calculated for Cs8Na16Si136 clathrate-II. (a) 2D data display of the ELF, Cs atom is not in the chosen lattice plane (cross sectional). The measured level of the ELF is shown on the left scale. (b) The chosen 76 lattice plane for the ELF representation in the Cs8Na16Si136 unit cell (160 atoms per cell). Data taken from Ref. .

complete electron donation to network Si atoms, further confirming the ionic type of bonding between guest and cage atoms (Na pseudopotential used in the present DFT calculations contains 1 valence electron).

3.7 Phonon structural properties

Vibrational properties and structural stability are analyzed via first principle phonon calculations with a finite displacement method (FDM) 166. The key idea of this method is to consider lattice dynamics in the framework of harmonic approximation. The potential energy 푈 of a system is an analytical function depending on equilibrium positions 푹푗,훼 and displacements from these positions 풅푗,훼, where 푗, 훼 denote a particular atom and the Cartesian 푥, 푦, 푧 directions. Since 푈(푹, 풅) is analytical, it can be expanded in a

Taylor series

휕푈 1 푗,푗′ 푈 = 푈0 + ∑푗,훼 풅푗,훼 + ∑푗,훼,푗′,훼′ 풅푗,훼 훷훼,훼′풅푗′,훼′ + ⋯ , (3.7.1) 휕풅푗,훼 2

휕푈 푗,푗′ 푗,푗′ where 푈0, , 훷훼,훼′ and etc. are the zeroth, first, second and n-th order force constants. 훷훼,훼′ is the 휕풅푗,훼 matrix of force constants given as

푗,푗′ 휕2푈 훷훼,훼′ = . (3.7.2) 휕풅푗,훼휕풅푗′,훼′

At equilibrium all stress forces are nearly zero and the first term vanishes. Under harmonic approximation the second order term in Eq. (3.7.1) is considered and others are neglected due to their negligible contribution in the perturbation series. Such approximation allows extracting the information about lattice vibrations (phonons) from the eigenvalue problem as follows:

푗,푗′ 2 퐷훼,훼′(풌)풆풌푗 = 휔풌푗풆풌푗 , (3.7.3)

2 푗,푗′ where phonon frequencies 휔풌푗 are the eigenvalues of the dynamical matrix 퐷훼,훼′ defined as

′ 1 푗,푗 푗,푗′ −푖풌푹훼 퐷 ′(풌) = ∑훼 훷훼,훼′푒 (3.7.4) 훼,훼 √푚푗푚푗′

with 푚푗, 푚푗′ being the atomic masses and

푒−푖풌푹훼 - a polarized wave of lattice vibrations.

The above procedure is implemented in the Phonopy code166 with DFT inputs from

VASP with the following steps: at the first step, small atomic distortions are created as a sequence of input structural files (POSCAR files); at the second step, at constant volume, the stress forces are calculated for each created Figure 3.7.1 Phonons (left panel) and phonon density of distortion in the structural file; at the third step states (right panel) calculated via FDM for the hypothetical type II Sn136 clathrate material. the dynamical matrix is calculated from where

(Eq. 3.7.3) the frequencies (phonons) of interest are extracted. Once phonons are determined, it is possible to obtain the phonon density of states (PDOS)

1 푔(휔) = ∑ 훿(휔 − 휔 ) (3.7.5) 푁 풌,푗 풌,푗 with characteristic results for the PDOS shown in Figure 3.7.1. Phonon analysis helps understanding the structural stability of the system. The stable or metastable structure strictly requires that all phonons must have real frequencies.167 Imaginary modes indicate an unstable structural state or a phase transition.

One can also calculate the speed of propagation of acoustic phonons (group velocity) 푣푝ℎ,푘 along different k-directions using the slope of the phonon dispersion

푣푝ℎ,푘 = 휕휔(풌)/휕풌, (3.7.6)

1 which is relevant for estimating lattice thermal conductivity given as 푘 = 퐶 푣̅ 푙, where 퐶 (also 푙푎푡 3 푉 푝ℎ 푉 calculated via FDM) is the constant volume heat capacity, 푣̅푝ℎ is the average group velocity and 푙 is the mean free path (see Sec.1.3, Eq.1.3.5). Thermal conductivity 푘 is also related to the degree of anharmonicity in a crystal, as suggested by Slack and Berman55,56. Specifically, its lattice contribution

푘푙푎푡 to the total 푘 can be written as

1/3 3 2 푘푙푎푡 = 퐴푛 휃퐷푀푎⁄푇훾 , (3.7.6) where 퐴 is a constant derived from experiment and/or other approximations, 푛 is the volume per atom,

푀푎 is the averaged atomic mass per unit volume, 휃퐷 is the Debye temperature. The above expressions describe thermal transport in insulators and semiconductors in medium-high temperature range (0 <

휃퐷 < 푇). The quantity 훾 is the macroscopic Grüneisen parameter which is essentially a measure of phonon anharmonicity and can be given as

훾 = ∑풌푖 훾풌푖 퐶풌푖 /퐶푉, (3.7.7)

166,168 where 퐶풌푖 is the mode contribution to the heat capacity, 훾풌푖 is the mode Grüneisen parameter (MPG) defined as

훾풌푖 (푉) = −[푉/휔풌푖(푉)][휕휔풌푖(푉)⁄휕푉], (3.7.8) where 푖 is the band index and 푉 is the volume of a unit cell. Usually MGP has a positive value but it can be negative for certain materials suggesting negative thermal expansion.169,170

4 Bi-Sb ALLOYS

Many materials have been reported to be good thermoelectrics at high temperatures, however, Bi and Bi1-

171–174 xSbx alloys are considered to be the best candidates for the low temperature range. Although the Bi type of materials have been investigated for a long time, these systems continue being a source for discoveries of novel properties in their bulk and surfaces phases. For example, recent studies have shown

175 that Bi1-xSbx are excellent candidates to observe 2D and 3D topological insulator states. Dirac valley polarization of the transport properties of Bi upon the application of an external electric field has also been demonstrated.176,177

Although there have been several first principles studies investigating the electronic structure of

178–180 bulk and surface Bi , such studies are lacking for the Bi1-xSbx alloys. It is worth to point out that understanding and quantifying the sensitivity of the electronic structure properties as a function of doping is crucial to analyze many experimental results for their transport.181 Surprisingly, there is limited number of systematic experimental studies for the basic properties of these alloys. Nevertheless, Refs.182–184 report nonlinear dependence as a function of doping in their transport characteristics. This behavior is associated with anomalies in the physical structure of the alloys.185,186

The models currently in use for the electronic structure of the Bi1-xSbx compounds rely on either semi-empirical or 풌 ∙ 풑 tight-binding approaches, which contain a number of input parameters matched with existing experimental data.175,183,187 Although such phenomenological approaches are easy to use, they are not parameter-free and they cannot predict the lattice changes and associated electronic structure nonlinearities upon doping.

Elemental Bi and Sb and the lightly doped Bi1-xSbx alloys (푥 < 0.25) have a rhombohedral lattice structure with a bisectrix and trigonal axes forming a mirror plane bisecting the entire lattice. It appears that the anisotropy of this pseudocubic structure in terms of lattice parameters can be affected by the degree of doping. The spin orbit coupling (SOC) is also significant and it is important to obtain the correct electronic characteristics. The purpose of this research188 is by using first principles calculations to present systematic investigation of the electronic structure properties of Bi rich Bi1-xSbx alloys for

(0 < 푥 < 0.25) in terms of lattice structure parameters, Fermi levels and surfaces, and density of states

(DOS) evolution as a function of 푥. Comparison with the electronic structure of elemental Bi is used to underline common and specific features for these narrow gap materials. The obtained results are compared with available experimental data.

The atomic and electronic structure properties of Bi and Bi1-xSbx alloys (0 < 푥 < 0.25) are investigated via DFT simulations (See Sec.3) using the VASP package.87,189 In this code, the Kohn-Sham equations are solved using the PAW method within a plane wave basis set and periodic boundary conditions. The exchange-correlation energy is taken into account using the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE).118 Ionic relaxation was performed with 200 eV energy cutoff and force and total energy difference criteria of 10−2 eV/Å and 10−4 eV, respectively. During the relaxation process, the cell was allowed to change its shape and volume. The tetrahedron method with Blöchl corrections for accurate k-point integration was used.137 We consider ten values for the concentration with

푥 = 0, 0.00781, 0.0104, 0.0139, 0.0185, 0.0278, 0.0416, 0.0625, 0.125 and 0.25. For this purpose, in the supercells generated using the Materials Project software,190 a Bi atom is substituted by a Sb atom for the corresponding concentrations. Also, the XCrySDen visualization package was used to visualize the

Fermi-surface.191 The SOC is quite important for these materials because of presence of heavy Bi and Sb atoms. Although this brings substantial computational cost, the spin-orbit correction is necessary in order to describe the properties of these alloys correctly.

4.1 Atomic structure properties

The rhombohedral 퐴7 structure is a common crystal phase for Bi and Bi1-x Sbx alloys (0 < 푥 <

0.3).185,186 This type of lattice has trigonal symmetry and it can be viewed as a distorted cubic lattice with a strain along the [111] direction and displacement of the atoms along the same direction. For Bi, there are two atoms in the unit cell, which can be described by the lattice constant 푎 and the rhombohedral

Figure 4.1.1 (a) The supercell of Bi1-xSbx. One Sb atom (yellow) substitutes Bi atom. The characteristic vectors a, b and c and characteristic angles 휶, 휷 and 휸 for the rhombohedral lattice are denoted. A side view perspective of the 퐁퐢 puckered layers is also given. (b) Different perspectives of the Fermi surface of Bi are shown. The angle 흋 between the L and T pockets in the reciprocal basis 품풊 is also shown. The Fermi surface was drawn with isolevel of 4.45 eV. Figure is taken from Ref.188

angle 훼 (Figure 4.1.1a). A side view, parallel to mirror plane shows that the crystal contains puckered layers of atoms with characteristic distances ratio d1/d2 determining the separation between the layers (Figure 4.1.1a).

The DFT calculations upon relaxation without SOC result in 푎 = 4.804 Å, α = 57.263° and

푑1⁄푑2 = 0.881. When the spin-orbit correction is included the relaxed structure is found to have

푎 = 4.845 Å, α = 57.283°, 푑1⁄푑2 = 0.882. These values are in good agreement with reported experimental data192. The supercell for the alloys is constructed by repeating the unit cell in different directions and substituting one Bi atom with a Sb one, as shown in Figure 4.1.1 For example, a cell consisting of 53 Bi atoms and one Sb atom corresponds to 푥 = 0.0185. The particular location of the Sb

54 atom does not affect the calculation results as various substitution positions for a given x were tested and found to yield the same outcome. The inclusion of SOC preserves the rhombohedral lattice and the characteristic lattice parameters 푎, 훼 and 푑1⁄푑2 are found to be dependent on the degree of alloying.

Figure 4.1.2 shows the obtained results for the crystal structure parameters as a function of concentration.

It is clear that the lightly doped alloys (푥 < 5%) experience nonlinearities in 푎 and 훼 vs. 푥 functions.

When the SOC is not included, the minimum for 푎 at 푥 = 0.0185 is accompanied by a minimum in 훼.

The maximum for 푎 at 푥 = 0.0416 corresponds to a minimum in 훼 with a slight shift along the horizontal axis. Decreasing 훼 for small 푥 indicates larger distortion along the trigonal axis for the rhombohedral crystal since α = 60° corresponds to a simple cubic lattice. Figure 4.1.2a also shows that the SOC increases the magnitude of the lattice constant without affecting the characteristic behavior. On the other hand, the angle 훼 is mainly affected for 푥 > 5% concentrations.

It is interesting to see that the nonlinearities in a and α are associated with nonlinearities in the puckered atomic chains (Figure 4.1.2), which are held together by van der Waals interactions.

Figure 4.1.2 (a, b) Rhombohedral lattice parameters, (c) d1/d2 (Figure 4.1.1) and (d) angle 흋 between electron and hole pockets in reciprocal space (Figure 4.1.1) as functions of concentration for the Bi1-xSbx alloys. SOC is included. Figure is taken from Ref.188

Smaller 푑1⁄푑2 ratio corresponds to the smaller distance between nearest neighboring layers. The SOC does not change the dependence vs. the concentration, however it decreases the magnitude of 푑1⁄푑2. The van der Waals interaction is strongly affected by the interatomic distance separation (푅) as the London

−6 dispersion shows 푅 dependence. Thus the smaller distance 푑1 indicates enhanced attraction between the chains. A recent scheme for Bi layer exfoliation has been proposed as means to observe surface

193 topological insulator phases . Thus, the smaller distance 푑1 indicates enhanced attraction between chains. A recent scheme for Bi layer exfoliation has been proposed as the means to observe surface topological insulator phases194. The obtained results may be of interest to such studies as they show for which concentration the attraction is enhanced or inhibited.

The majority of experimental works have been directed mainly towards quantifying the transport properties of these alloys, especially in the 푥 > 7% range where the best TE performance is Figure 4.1.3 Hexagonal lattice parameters 풂풉 and 풄풉 as functions of concentration for the Bi1-xSbx alloys. SOC is included. Figure is taken from Ref.188 observed172–174. It is usually assumed that for the entire range 푥 = 0 − 25%, the lattice parameters satisfy Vegard’s law195. This is an empirical formula from metallurgy which suggests a linear relation of decreasing lattice constants as the alloying concentration is increased.

A limited number of studies, however, have examined how the lattice parameters evolve as a function of concentration. Measurements via X-ray diffraction show that Bi1-xSbx have nonlinearities at smaller Sb concentrations182,184. These results are given for lattice parameters corresponding to a hexagonal lattice – 푎ℎ, 푐ℎ. Relating the rhombohedral description to the equivalent hexagonal one can be

2 2 2 done via the expressions - 푎 = 1/3√3푎ℎ + 푐ℎ and 푠푖푛 (훼⁄2) = 3⁄√3 + (푐ℎ⁄푎ℎ) . In Figure 4.1.3, the hexagonal parameters are displayed with nonlinear characteristics in the 푥 < 7% range corresponding to the ones for the rhombohedral lattice in Figure 4.1.2. The 푎ℎ, 푐ℎ vs. 푥 behavior is very similar to the experimental results in Ref.184. The location of the extremum points is practically the same, however the

56 magnitude of the lattice constants from Figure 4.1.3 is somewhat larger. We attribute this difference to the fact that the measurements are performed at room temperature, while the DFT calculations are at 0K.

The deviations in the lattice structure from the Vegard’s law are accompanied with anomalies in the transport properties of Bi1-xSbx. The minima of 푎ℎ and 푐ℎ at 푥 = 1.85% (Figure 4.1.3) correlate with the minimum in the electrical conductivity reported in Refs.182,183 to be 푥~2%. This is also accompanied by an inflection point in the Seebeck coefficient for measurements at room temperature. The maximum 푐ℎ at 푥~4.16% (Figure 4.1.3b) can be associated with the reported maximum value for the conductivity and minimum value for the Seebeck coefficient at 푥~3.8% 182,183. One notes that although the magnitude of the measured transport properties in Refs.182,183 changes as a function of temperature, the location of their characteristic minimum and maximum points as a function of concentration is fairly robust in terms of its temperature dependence.

Note that nonlinearities in the lattice structure parameters have been observed in other IV-VI semiconducting compounds, including additional systems containing Bi and Te196. Other studies197 have shown deviations from Vegard’s law in metallic alloys. These findings together with computations indicate that Vegard’s law may not be applicable for alloys with very small concentrations.

4.2 Electronic structure properties

The electronic structure evolution upon concentration of the Bi rich alloys is also investigated using the DFT ab initio simulations. The structural anomalies for 푥 < 5% affect the Fermi level as seen in Figure 4.4.1 The local

Fermi energy differences minimum in Ef is associated with the global maximum in 푎 Figure 4.1.4 between Bi1-xSbx and Bi as a function of Sb 188 and the global minimum in α in Figure 4.3.2 at 푥 = 4.16%. concentration. Figure is taken from Ref.

The almost linear increase in the Fermi level for 푥 > 4.16% indicates that these alloys have n-types carriers with increasing concentration.

The distortion from the simple cubic structure in Bi and B1-xSbx results in overlapping between valence and conduction bands which leads to formation of carrier pockets198,199. Figure 4.1.1b shows the Fermi surface of Bi with six electronic half ellipsoidal pockets at the L- points and two hole pockets at the T- points. The triangular regions having threefold symmetry represent the hole Figure 4.2.1 (a) Total DOS as a function of energy scaled by the pockets at the H-points. As the Sb Fermi level Ef for Bi, Bi0.875Sb0.125, and Bi0.75Sb0.25. The SOC is included; (b) Orbital DOS for the p-orbitals for Bi0.973Sb0.027 with concentration is increased, the angle 휑 and without the SOC. Figure is taken from Ref.188

(between the L-electron and T-hole pockets) changes as a function of 푥. Using the relations between the lattice and reciprocal basis196, 휑 as a function of concentration is given in Figure 4.1.2d, which shows that

휑 experiences maximum at 푥 = 4.16%. The nonlinear 휑 vs. 푥 behavior correlates with the structural anomalies of the alloys (Figure 4.1.2, 4.1.3). The calculations also show that the SOC increases the magnitude of the angle for 푥 > 5% concentrations.

We further investigate the density of states (DOS) of these alloys to determine the characteristic behavior as a function of Sb doping. In Figure 4.2.2a, the total DOS is displayed for three cases. Similar features for all systems are found throughout the displayed region. The deviations in DOS as compared to pure Bi at E-Ef=0 can be seen in the insert. The energy region −6 eV < E < 6 eV is determined primarily by the p-orbitals, while the E < −6 eV is dominated by the s-states of these alloys. Figure 4.2.2b, which corresponds to the p-orbital projected DOS for Bi0.973Sb0.027, also shows the strong effect of the SOC. It

58 is clear that without the inclusion of this relativistic correction DOS is nonzero at Ef. The characteristic minimum in DOS at the Fermi level is shifted towards higher energy (~60 meV higher). The effect from the SOC is especially strong for the p-orbital in the range around Ef. At the same time the s-states are practically unaffected by this correction. This is understood by realizing that since the SOC is determined by the 푳 ∙ 푺 contribution, for the s-states the total angular momentum is 푳 = 0.

We also present the orbital projected DOS for 푥 = 6.25% and 푥 = 25% for each atom in Figure

4.2.3. For the smaller concentration, the Sb contribution in the Fermi energy region is negligible as its p- states are practically zero. The Sb s-states however are responsible for the formation of a small peak at

E = −7.98 eV overlapped by a larger Bi peak. This is followed by a gap ~0.391 eV before the formation of the rest of the s-states mainly from Bi.

For the larger concentration, the Sb s-states become noticeable as can be seen from Figure 4.2.3b.

The Sb s-states are more visible as well with the first peak at E = −8.49 eV being the most pronounced.

One notes, that there is a reduced s-states gap ~0.221 eV displaced towards lower energies as compared to the case of x = 6.25%. Clearly the SOC correction affects the p-region mainly. The obtained results

Figure 4.2.2 (a) DOS with contributions from s and p-states for Bi1-xSbx with x=6.25% (top two panels); (b) DOS with contributions from s and p-states for Bi1-xSbx with x=25% (top two panels). The bottom panels show the s-state regions for both concentrations. Figure is taken from Ref.188

59 show that although its contribution is relatively small (always on the order of 50 − 60 meV), its inclusion makes a significant difference to the DOS around Ef, similar to the situation displayed in Figure 4.2.2b.

At this point, it is not possible to obtain the energy band structure for the Bi-Sb alloys using the

VASP code. Since we construct supercells for the simulations, the bandstructure of the first Brillouin zone of the normal cell becomes folded, which results in the appearance of many flat bands. A recent scheme to obtain the unfolded band structure was proposed based on the construction of Wannier functions200. This procedure, however, is still not available for all materials as it involves careful construction of the Wannier basis with proper spectral weights.

A practical way based on a tight binding scheme, where parameters are adjusted to reproduce experimental data is more useful for the evolution of the energy bands closest to the Fermi level176,177,201.

This is essentially a tight binding model where each parameter is taken to have the same dependence on the concentration 푥 − 푡(푥) = 푥푡푆푏 + (1 − 푥)푡퐵푖, where 푡푆푏, 푡퐵푖 are the parameters for Sb and Bi201.

We can use the results from this approach and correlate with the ab initio simulations results obtained here. According to available experiments and the tight-binding model, there is a small gap at the electron L-point for Bi, which starts decreasing as x is increased from zero175–177,197. The hole T-point band, which maximum is above the minimum of the L-conduction band originally, starts going down.

There is closure of the L-point gap at 푥~4%. This behavior is reproduced by the tight-biding model, which is a result from the balance between the concentration dependent parameters. Comparing with the results, one notes that there is a corresponding behavior of the structure parameters of the alloys at

푥 = 4%. The rhombohedral lattice constant has a maximum, while the corresponding angle has a minimum – Figure 4.1.2a, b. This is also consistent with the maximum of ch from the hexagonal description (Figure 4.3.3b), which corresponds to the interlayer distance (Figure 4.1.1).

In accordance with experiments and the tight-binding model, further increasing of 푥 results in an increase of the energy gap between the L-bands and at 푥~7%, the minimum of the T-point band is at the

60 same level as the minimum of the L-band, which corresponds to a semimetal-semiconductor transition175–

177,198,199. Comparing with the lattice structure studied here, we find that many of the parameters of these alloys do not show extrema for this concentration, except for 푎ℎ which has a maximum.

As 푥 is increased, the tight-binding model in accordance with experiments shows that the energy gap between the L-bands increases again and at 푥~7%, the maximum of the T-point band is at the same level as the minimum of the L-band resulting in a semimetal-semiconductor transition174,176,177. We find that many of the structure parameters of this alloy do not show extrema for this concentration. The exception is 푎ℎ which has a maximum at 푥 = 6.25% (Figure 4.1.3a). This shows that the atoms become closer as ahis a measure of the interatomic distance in the hexagonal lattice representation.

Note, that the tight-binding description with many concentration dependent parameters is able to reproduce some experimental data, and it has been utilized to predict some novel topological insulator properties176,177,201. The calculations show that the interpretation of the electronic structure for the Bi-Sb alloys has to be expanded in order to accommodate the lattice structure parameters. The DFT ab initio results show that there is clear correlation between the lattice as a function of 푥 and transport properties as a function of 푥 182,183. Of special interest is the anomalous behavior at the very low concentration point of

푥 = 1.85%. The tight-binding model does not predict any unusual behavior, thus it cannot accommodate the experimental conductivity minimum and Seebeck coefficient maximum in this range183. The obtained results, however, show that the rhombohedral lattice constant as well as the hexagonal constant have a local minima values (Figure 4.1.2, 4.1.3). The above findings strongly suggest that the nonlinear behavior in the transport in these Bi rich Bi1-xSbx alloys is driven by changes in the lattice structure as a function of concentration. Therefore, the fundamental description of the properties of these systems using a tight- binding model has to accommodate the lattice structure evolution as well.

4.3 Summary

The atomic and electronic structure properties for bulk Bi rich Bi1-xSbx alloys have been considered with several concentrations via DFT ab initio methods. It is shown that the lattice structure experiences anomalies as a function of concentration which correlate with available experimental data for their transport properties. This can be considered as establishing a clear connection between atomic structure evolution and relevant transport. The calculated DOS enables understanding characteristic behavior in terms of orbital and site projected contributions. The obtained results show that the Sb contribution is most significant in the valence s-state region, while the range around the Fermi level is mainly consistent of the Bi p-state.

Correlations are established between the structural changes of the Bi1-xSbx alloys and the band structure evolution as a function of concentration calculated via a tight-binding model. It is shown that not all of the atomic structure anomalies calculated here correspond to corresponding transitions in the tight- binding model. Since the overall result has similar characteristic behavior as available data, it is suggested that the tight-binding picture needs revisions in order to accommodate the very low concentration functionalities. We conclude by noting that future developments in first principles codes will resolve the issue of band structure unfolding for large supercells in the near future with sufficient accuracy, thus the energy band structure can be computed via self-consistent methods and compared with the frequently used, but parameter-dependent tight binding results.

5 BOURNONITE PbCuSbS3

To obtain low 푘 materials many different approaches, such as nanostructuring in bulk202–

205, alloying solid solutions with local anisotropic structural disorder,206–209 and phonon mode softening via cage structure filling210,211, have been employed. In all cases the structure-bonding relationships within a given materials class is crucial in understanding the underlying mechanism(s) associated with low 푘. Very recently, Skoug and Morelli212 have shown a

2 correlation between lone-pair electrons (s pair in group 15) and 푘푚푖푛 (the minimal thermal conductivity) in ordered crystalline chalcogenides. The distortion that lone-pair electrons of group

15 elements and neighboring chalcogen atoms is directly related to low 푘 by inducing unusually high lattice anharmonicity. Natural minerals such as tetrahedrites, as well as modified tetrahedrites, possess intrinsically low 푘 due to Sb lone s2 pair electrons213,214 and have been considered promising thermoelectric materials. Doping, impurities and other factors are not expected to affect significantly anharmonic processes as they are intrinsic to the particular material. However, in some cases, demonstrated later in next, they can tune electronic properties that affect thermal conductivity. This feature is especially useful for thermoelectricity as it presents pathways to optimize the electronic properties by methods which do not typically affect this inherently low lattice 푘.

The lone pair electrons / low thermal conductivity correlation is an important step in identifying systems for the potential applications described above. A microscopic understanding of the lone pair (s2) electrons and the mechanisms responsible for enhanced anharmonicity, however, is missing. To advance in this important direction, suitable systems must be investigated in detail in order to determine underlying relationships between the lattice structure and the

63 transport properties. Here, bournonite (PbCuSbS3) and stibnite (Sb2S3) are identified that can serve as a test case for such a detailed investigation. The focus is on the role of the stereochemically active lone s2 pairs, and their interactions, as a mechanism in obtaining low 푘 materials.

In this section, first principles simulations together with experimental efforts are undertaken to reveal how similar structures, which possess lone s2 pair electrons, can have different thermal transport properties. The evolution of structure-property relations for PbCuSbS3 and its parent material, Sb2S3 is considered. The measured 푘 is found to be significantly smaller in

2 PbCuSbS3 case and this is attributed to the stereochemically active lone s pair electrons. The structural transformation associated with deriving bournonite from stibnite has important implications, especially for the electrostatic interactions between neighboring s-electron distribution as well as their electronic structure properties. To get a broader view on how doping might thermal conductivity, the doped bournonites with Ni and Zn are also considered. The detailed analysis of their electronic properties, chemical bonding and vibrational properties along with experimental measurements is provided. This study81 not only elucidates novel features of the role of lone s2 pair interactions, but it is also the first detailed investigation of the bournonite compound.

For the electronic structure calculations using VASP, the exchange-correlation energy was calculated via the Perdew-Burke-Ernzerhof (PBE) functional118 and the ionic relaxation was performed with 500 eV and 300 eV cutoff for bournonite and stibnite, respectively. The force and total energy difference relaxation criteria were 10-2 eV/Å and 10-4 eV/Å, respectively. The cell was allowed to change shape and volume during the structural relaxation with 11 x 11 x 11 and 11

81 x 21 x 11 k-point meshes for PbCuSbS3 and Sb2S3, respectively. The tetrahedron integration method with Blöchl correction was used for the self-consistent calculations on the same k-grid.

The spin-orbit correction was also taken into account. The VESTA software package was used to

64 perform the crystal structure, the ELF function and charge density difference215. The charge distribution is calculated by using Bader charge analysis216–218.

5.1 Crystal structure

The experimental X-ray diffraction measurements

(PRXD) for CuPbSbS3 indicate single-phase bournonite219 crystallized in the non- centrosymmetric orthorhombic phase, space group Pmn21, with two crystallographically Figure 5.1.1 Crystal lattice views of (a) PbCuSbS3 along the crystallographic a axis; Some Pb-S and Sb-S and the Cu-S were omitted independent 2a positions for for clarity (for a full view see Figure 1b. (b) Sb2S3 along the crystallographic b axis. Grey – Pb, blue – Cu, brown – Sb, and yellow – S. lead, one (4b) for copper, two Figure is taken from Ref.81

(2a) for antimony, and four (two 2a and two 4b) for sulfur. The lattice parameters for this orthorhombic structure were first estimated and then refined. The initial positional positions for all the atoms in the structure were from previous data after standardization220,221. The crystal structures of PbCuSbS3 and Sb2S3 have orthorhombic lattices, however PbCuSbS3 has the Pmn21

222 space group, while Sb2S3 has the Pnma space group . The projections of the two crystals in

Figure 5.1.1 show that these systems are comprised of weakly bound quasi-1D chains consisting of Sb sharing polyhedra along selected directions, crystallographic a axis for PbCuSbS3 and crystallographic b axis for Sb2S3.

The PbCuSbS3 crystal structure can be derived from that of Sb2S3 by considering the cation cross-substitution in binary systems to obtain ternary and quaternary materials223–226. The

3+ Sb2S3 - PbCuSbS3 relationship can be understood by realizing that one of Sb cations in the Sb2S3

2+ 1+ 6+ 2+ 1+ 3+ lattice is kept while the other is replaced by Pb Cu , thus Sb2 in Sb2S3 becomes Pb Cu Sb

Table 5.1 Characteristic distances in Å between inequivalent atoms in the unit cell for PbCuSbS3 (top panel, (a)) and Sb2S3 (bottom panel, (b)). Average coordination tetrahedral angles are also given (X=S atom). Spaces with dash mean that these particular bonds are not possible. Experimental data for Sb2S3 are taken from Ref. 26. Experimental values are denoted as I; calculated values are denoted as II

S1 S2 S3 S4 Lattice parameters and angles (a) I II I II I II I II I II a 7.810 7.854 Cu 2.22(2) 2.284 2.34(2) 2.351 2.34(2) 2.341 2.40(2) 2.400 b 8.150 8.280 c 8.700 8.900 Pb1 2.84(2) 2.845 3.02(3) 3.168 2.71(2) 2.832 3.49(2) 3.541 X-Pb1-X ≈96.1º ≈95.3º 2.93(2) 2.864 Pb2 - - 2.90(3) 2.843 3.39(2) 3.406 X-Pb2-X ≈97.1º ≈97.5º 3.14(2) 3.273 Sb1 - - 2.44(2) 2.514 2.46(2) 2.489 - - X-Sb1-X ≈95.3º ≈95.4º Sb2 2.56(3) 2.480 - - - - 2.38(2) 2.489 X-Sb2-X ≈93.4º ≈94.3º

S1 S2 S3 Lattice parameters and angles (b) I II I II I II I II a 11.282 11.440 b 3.830 3.866 Sb1 2.537 2.571 2.514 2.557 - - c 11.225 11.316 X-Sb1-X ≈91.1º ≈91.0º 2.850 2.876 X-Sb2-X ≈85.8º ≈85.8º Sb2 - - 2.675 2.702 2.449 2.456 X-Sb2-X (planar) ≈89.7º ≈89.7º

in PbCuSbS3. In the crystal structure Pb replaces Sb in the polyhedra while Cu are in the tetrahedral interstitial sites. As a result, the 1D units become smaller and rotate so that (110)

PbCuSbS3 diagonals correspond to the crystallographic a and b directions of Sb2S3 (Figure

5.1.1).227 The characteristic experimental and calculated distances between the inequivalent Sb atoms in the unit cell are listed in Table 5.1. Sb2S3 has two crystallographic sites where Sb atoms reside. The Sb1 atoms form trigonal pyramid with one S1 and two S2 atoms on the ends of the 1D ribbons. The Sb2 is in the center of the ribbon and is surrounded by two S2 and three S3 atoms forming square pyramids with a common base.222

There are two inequivalent Sb and four inequivalent S atoms in the PbCuSbS3 unit cell with relevant distances also given in Table 5.1. This leads to two types of Sb-based trigonal pyramids. One type is formed by Sb1, S2 and two S3 atoms and the other is formed by Sb2, S1 and two S4 atoms. There are also two crystallographic sites where Pb atoms reside in the unit cell.

Their coordination is such that Pb1 atom is surrounded by six S atoms forming a highly distorted octahedron while Pb2 is surrounded by seven S atoms in a mono-capped trigonal prism. The

66 average coordination angles for the various shapes are also given in Table 5.1. The obtained results show that a good agreement between the experimental and calculated lattice parameters and coordination angles is obtained. In obtaining PbCuSbS3 lattice from Sb2S3 one finds that the

Cu atom takes positions in the tetrahedral void regions. This results in breaking the bonds in the square S pyramidal base in Sb2S3, therefore the unit (a-c projection) becomes broken.

Consequently the PbCuSbS3 1D-unit becomes smaller with a b-c projection (Figure 5.1.1).

5.2 Electronic structure

To further understand these materials and to analyse their relation in terms of other properties, we present results from the ab initio simulations of their electronic structure. Figure

5.2.1 shows the band structures of PbCuSbS3 and Sb2S3 along select symmetry lines in the

Brillouin zone. Note, that spin-orbit coupling (SOC) significantly affects the energy bands of

PbCuSbS3. If this relativistic effect is not taken into account, PbCuSbS 3 is a semiconductor with a direct band gap (Eg) at the  point with Eg = 0.686 eV. With SOC the lowest conduction band decreases with relatively large splitting in the S-Y and Y- directions. PbCuSbS3 is thus an indirect band gap semiconductor in the -Y path with Eg = 0.385 eV. The direct gap at  is also

Figure 5.2.1 Energy band structure for (a) PbCuSbS3 and (b) Sb2S3 without (top) and with (bottom) spin orbit interaction, respectively. Calculated total (black line) and projected density of states with included spin orbit 81 coupling for (c) PbCuSbS3 and (d) Sb2S3. Figure is taken from Ref.

67 reduced to 0.445 eV. Sb2S3 is a semiconductor with an indirect energy gap in the -Z path and the role of the SOC is mainly to reduce this value. The obtain energy gap Eg = 1.311 eV with no SOC and Eg = 1.254 eV with SOC. For both materials the band structures are very similar. There are features with many-valley character (local minima) along the highly symmetric directions of the

Brillouin zone, therefore, many potential transitions are possible. PbCuSbS3 is a smaller gap indirect semiconductor as compared to the parent Sb2S3, which is attributed to the symmetry breaking effect of the Cu and Pb atoms.

The calculated DOS shows that the Cu d-states and S p-states are the main contributors in the highest valence region while the p-states of Pb and Sb make up the conduction band edge

(Figure 5.2.1c). For Sb2S3 the highest valence region contains the p-states of S and Sb. Some hybridization between the Sb s-states and the p-states is also found (Figure 5.2.1d). Below the

Fermi level there are three groups of well separated bands in both materials. These features are indicative of the quasi-1D lattice structures and subsequent transport due to the weakly bound ribbons in the structure (Figure 5.1.1).228

The DOS in the valence regions around (-10, -8) eV and (-14, -12) eV are mostly composed of the s-states of the constituent atoms (Figure 5.2.1c, d). While bournonite has two peaks from the s-states of Pb (~-8.2 eV) and Sb (~-9 eV) for the first region, the corresponding

DOS for stibnite is composed almost entirely by the Sb s-states (-9.1, -6.2) eV. Also, the s-states from S are mostly situated in the second region of (-14, -12) eV for both materials with a small peak from the Sb atoms at (-13.6 eV) for the bournonite material.

5.3 Electron Localization and Chemical Bonding

The bonding in the materials is further analysed through the calculated ELF, electronic structure and bond lengths. The results for the ELF are shown in Figure 5.3.1 for different zone projections. The bonding in the PbCuSbS3 lattice is of rather complex character. The atom-atom length analysis from Table 5.1 shows that almost all distances are larger than the threshold for a

Figure 5.3.1 Calculated electron localization function in various projections for: (a) and (b) PbCuSbS3; (c) and (d) 2 Sb2S3. The shown projections are chosen to represent the different types of bonding and lone s pair electron characteristics. Figure is taken from Ref.81

covalent bonding set by the sum of the covalent radii (2.47Å for Sb-S, 2.55Å for Pb-S, 1.86Å for

Cu-S). The bournonite Sb-S=2.480 and 2.489Å bonds, however, are close to the threshold of

2.47Å covalent radii sum, thus one may infer that a weak covalent electron coordination is

possible. This is confirmed by the calculated ELF results shown in Figure 5.3.1b, c and d, where

this partial covalent bonding for the case of Sb2-S4 and Sb1-S3 is displayed. There is an

associated p-electron hybridization, which can be traced in the DOS in the (0, 4.4) eV region,

according to Figure 5.2.1c. At the same time, the Cu atoms couple with adjacent S atoms in a

primarily ionic character as shown in Figure 5.3.1b. Specifically, the S1 atom accepts two valence

s-electrons from the two Cu atoms reflected in DOS in the region (-5.8, -4.3) eV, where the Cu s-

states and the S p-states can be found. The Cu atom also donates three d-electrons to the other

neighbouring S-atoms, which can also be found in the (-4.0,0) eV DOS region. For Sb2S3 the

bonding is also of mixed character with weak covalently shared electrons, as already analysed

in229,230, and also shown in the displayed ELF results in Figure 5.3.1e, f and g.

The analysis regarding the covalent/ionic character agrees well with the simple bond

valence and Madelung site potentials calculations shown in Table A1 (See Appendix 1). The

iconicity of these materials, on the other hand, can also be estimated from the relation 푓푖 =

1/2(1 − cos(휋(퐸퐴푆 /퐸푉퐵)), where EAS is the energy gap between the two lowest valence bands

230 and EVB is the total valence band width . We find that the PbCuSbS3 ionicity factor is 푓푖 ≅

0.134 and it is mainly due to the Pb and Cu donating electrons to the neighboring S atoms. Sb 2S3

231 iconicity is calculated to be 푓푖 ≅ 0.089 and it is primarily due to the Sb2 and S3 atoms .

From the cation-anion relations in PbCuSbS3, the oxidation states of each atom, and the type of bonding discussed above, one finds that the Cu s and d-electrons, the Pb p-electrons, and most of the Sb p-electrons (some sharing due to the weak covalency is discussed above) are transferred to the p-shell of the S atom. As a result, Pb and Sb become closed shelled and are

2 characterized with the formation of lone s pair electrons. For Sb2S3 p-electrons are transferred from the Sb to the S atoms, which also leads to a closed shell Sb with a lone s2 pair electrons. The shown projections in Figure 5.3.1 clearly depict the lobe-like extensions around the Sb atoms in both materials, which are classic signatures of lone s2 pair electrons232,233. It is important to note, however, that the relative orientation of the s-clouds in bournonite and stibnite is different. The

Sb1 and Sb2 lobes in PbCuSbS3 share a common plane of symmetry with directly opposite orientation, however, the Sb s-clouds in Sb2S3 do not share such a common plane (Figures 5.3.1a and e).

The characteristics of Pb lone s2 pair electrons are determined by the coordination of the

Pb atoms. Since Pb1 is surrounded by 6 S and Pb2 by 7 S, the s-electron cloud becomes spherical- like in form. Its slight protrusion indicates that the electronic distribution is slightly off-set due to the short extension of the s-pair electrons as opposed to the electronic Sb s-pair being far from the atom with clear localization preference (Figure 5.3.1a and b). We also note that the lobe-like representation of the anion atoms is apparent in the ELF; the localization of the S s-pair electrons in PbCuSbS3 is stronger and the s-clouds share a plane of symmetry similar to the Sb s-electrons.

For Sb2S3 the localization trend is less pronounced and the s-clouds of S do not share such a symmetry coordination.

5.4 Lone-Pair Electrons and Their Role in Thermal Conductivity

The experimental results for 푘 for both

PbCuSbS3 and Sb2S3 below 300 K are shown in

Figure 5.4.1 The 푘 values decrease rapidly with increasing temperature. Both materials show relatively low thermal conductivity at 300 K, however 푘 for PbCuSbS3 is lower than that of

Sb2S3. The inset figure shows high and low temperature 푘 data for PbCuSbS3. As shown in the inset figure, there is good agreement between the Figure 5.4.1 Measured temperature dependent thermal conductivity for PbCuSbS3 (red circle) and Sb2S3 (blue triangle). The inset shows the entire temperature range data high and low temperature data for PbCuSb3. This 81 for PbCuSb3. Figure is taken from Ref. is an indication of the homogeneity of the polycrystalline specimens. The 푘 values decrease rapidly up to 100 K for both materials, indicative of

Umklapp scattering of phonons typical of crystalline solids234. Above 100 K, 푘 decreases gradually with increasing temperature. The inset figure also shows that there is no “upturn” in the data at high temperatures indicating that there is little or no bipolar contribution to 푘 in the measured temperature range.

The measured thermal conductivity can be understood by analysing the lattice structure-electronic structure relationship between PbCuSbS3 and Sb2S3. PbCuSbS3 can be derived from Sb2S3 via the cation substitution method, a widely used approach to obtain derivative compounds from binary ones.223–226 In general, the derivatives can have very different structure and other properties as compared to the parent compound. Here, however, there are many similarities in the lattice and electronic structure of Sb2S3 and

PbCuSbS3, this being quite beneficial in understanding their phonon characteristics. Although 푘 for both materials is relatively low, 푘 for PbCuSbS3 is lower despite its lower energy gap, according to the electronic structure calculations (Figure 5.2.1). Both materials have cation atoms with lone s2 pair

71 electrons (Sb for Sb2S3; Sb and Pb for PbCuSbS3). These are stereochemically active electronic distributions characterized by atomic s-p hybridization (Figure 5.2.1). The presence of such lone pair

212 electrons has been associated with low values of 푘 which is the case for PbCuSbS3 and Sb2S3. In addition, a correlation between the coordination number (CN) of the cation atom and the average angles

X-M-X is also found (M is a cation atom possessing lone pair electrons and X is a neighboring atom)212.

The X-M-X angle should be very close to 90º for CN ≥ 6, 95~96º for CN = 4 or 5, and > 99º for CN = 3 with 푘 tending to decrease with an increase in the average bond angle. This increase is related to the induced distorted coordination environment from the lone s2 pairs facing outward, as suggested by

235 valence shell electron pair repulsion theory . The Sb atoms for PbCuSbS3 and Sb2S3 have a coordination number CN=3 with average angles less than the angle of 99º (Table 5.1). For the Pb atoms, the CN ≥ 6 and the respective angles are found to be greater than the 90º values, however (Table 5.1). Therefore, the larger average angle X-M-X / low κ trend is not very clear for PbCuSbS3 and Sb2S3.

The ab initio results also indicate that there are other factors stemming from the lone pair s2 electrons and contributing to the lower 푘 of PbCuSbS3 as compared to Sb2S3 (푘 = 0.81 W / m∙K at 300 K and 0.48 W / m∙K at 600 K for PbCuSbS3 and 푘 = 1.16 W / m∙K at 300 K for Sb2S3). The lobe-like s- electron distribution is present for Sb atoms in both materials, however, their orientation is different. The

2 fact that Sb lone s electron clouds in PbCuSbS3 share a common symmetry plane with lobes directly from each other (Figure 5.3.1) indicates that their electrostatic repulsion is the strongest. For the Sb s- electron lobes in Sb2S3 this electrostatic repulsion is weakened due to their orientation lacking a common symmetry plane and extending in different directions, not across from each other as in PbCuSbS3 (Figure

5.3.1). We further note that the mutual orientation of the s2 pair electrons of the Pb is very similar to the case of the Sb atoms. The almost circular electronic clouds across from each other (Figure 5.3.1) are also repelled electrostatically.

The electrostatic repulsion between the s2 pair electrons from neighboring atoms can contribute as anharmonic scattering thus limiting the thermal conduction. Since this effect is enhanced in PbCuSbS3, 푘

2 is lower as compared to Sb2S3. Even though both materials have atoms with inert lone s pair electrons, 푘 for the compound with the smaller energy gap (PbCuSbS3) has lower values due to the stronger effect of the mutual electrostatic repulsion. A distinct picture of the overall transport in these related materials emerges. The electronic conduction is primarily along the quasi-1D chains as the process is more conducting in bournonite due to the Cu and Pb contributions. The distorted coordination environment and the electrostatic repulsion from the inert s electrons from Sb and Pb are present in directions perpendicular to the chains. This decoupling between the anharmonic phonon scattering process and the electronic conduction can be beneficial in enhancing the thermoelectric properties of a material, for example.

5.5 Summary

The presented study indicates that PbCuSbS3 possesses two types of stereochemically active lone- pair electrons (6s2 from Pb and 5s2 from Sb) resulting in low 푘 values above 200 K. The properties of this compound were also compared to that of Sb2S3 from which PbCuSbS3 can be derived via cation substitution. This comparative analysis is instrumental in understanding the role of the lone-pair electrons in the transport properties of these materials. The similarities and differences discussed herein are especially beneficial for understanding the electrostatic repulsion between the s-electron distributions of neighboring atoms as a source of anharmonic phonon scattering. The obtained results show that such interaction is responsible for the lower 푘 for PbCuSbS3, despite its smaller energy gap, when compared to

Sb2S3. This approach of utilizing the presence of stereochemically active lone-pair electrons and manipulating their interactions to obtain low 푘 materials can be expanded to other compositions,212,236 allowing for future investigations on materials for phase-change data storage, thermal barriers, and thermoelectric applications. Another important question in this regard is how doping would change the electronic and thermal properties of a given material.

6 DOPED BOURNONITES

Here we investigate the important question as to how doping changes the electronic and thermal properties of the pure bournonite material237. For example, doping on the Cu, Pb, or Sb atoms will have different effects in the orientation and distortion of the lone s2 electron pairs. Depending on the doping site, the inherent anharmonicity of the doped materials may be enhanced or inhibited. In addition, doping may induce changes in the thermal conductivity due to the carrier contribution, which must be understood to manage the control of properties. Simulations based on density functional theory (DFT) are also employed to investigate the atomic, electronic, and phonon structures of these bournonites. The detailed results for the total and partial density of states due to charge carriers and phonons help us understand how the transport evolves as a result of different types of doping. The extensive charge transfer analysis and electron localization show the different types of bonding which is further related to the transport properties of the materials. Comparisons between the experimental and computational data are further used to understand changes in the thermal conductivity upon doping for these materials.

6.1 Crystal structure

The crystal structures of the investigated bournonite materials, shown in Figure 6.1.1, have an orthorhombic lattice specified by the 푃푛푚21 space group. The atomic structures for the undoped and doped bournonites are analyzed in terms of lattice constants, bond lengths, and angles. In Table 6.1 the measured and calculated lattice constants and nearest neighbor bond lengths are shown. The DFT-GGA calculations indicate that Ni and Zn doping results in changes of the lattice constants as compared to the ones of the pure bournonite, such that the symmetry remains the same, but the orthorhombic lattice acquires different dimensions along the characteristic directions. This can be correlated with different interatomic distances between Ni, Cu, Zn and their coordination environment. The calculations further

Figure 6.1.1 Lattice structures of (a) PbCuSbS3; (b) PbCu0.75Ni0.25SbS3; (c) PbCu0.875Zn0.125SbS3. Figure is taken from Ref.237 indicate that the inclusion of the SOC correction for the pure bournonite leads to reduction of the 푎 lattice constant and increase of the 푏, 푐 constants when compared to the standard GGA results. For the

PbCu0.75Ni0.25SbS3 system, however, the SOC leads to decreasing of all lattice constants when compared to those obtained by GGA. Experimental findings show a slight reduction along b and c directions for the doped bournonites and an additional increase along a direction for PbCu0.875Ni0.125SbS3. The computed and measured lattice constants are in close agreement with each other. The pure bournonite unit cell consists of 24 atoms, which includes two inequivalent Sb (2a) and four inequivalent S atoms (two are in

2a and two are in 4b sites) forming two types of distinct trigonal pyramids. There are also one Cu atom

(4b) and two inequivalent Pb atoms (2a) such that Pb1 is surrounded by six S atoms and Pb2 is surrounded by seven S atoms. Doping with 25% of Ni on the Cu site in PbCu0.75Ni0.25SbS3 is done by substituting one of the four Cu atoms in the unit cell with a Ni atom. To generate the structure of the experimental PbCu0.9Zn0.1SbS3 one must construct a large supercell, which puts significant limitations on obtaining simulation results for the lattice and electronic structures. Thus instead of 10% Zn doping, we consider PbCu0.875Zn0.125SbS3 which is obtained by constructing a (1x1x2) super cell in which one of the eight Cu atoms is substituted with Zn atom. We expect that even though there might be some bond length differences between the experimental and computational results, the electronic and vibrational properties are going to be very similar due to the very similar compositions.

Table 6.1 Experimental (Exp.) and calculated a, b, c lattice constants in Å for (i) PbCuSbS3, (ii) PbCu0.75Ni0.25SbS3, (iii) PbCu0.9Zn0.1SbS3, (from experiment) and PbCu0.875Zn0.125SbS3 (from calculations, denoted with *). Interatomic distances between the different cations and anions in Å for these systems are also give. Calculated values obtained via standard methods (GGA) and with relativistic spin orbit corrections (GGA+SOC) are given.

Lattice constants, (Å) Lattice i ii iii constants Exp. GGA GGA+SOC Exp. GGA GGA+SOC Exp. GGA a 7.810 7.854 7.814 7.810 7.879 7.867 7.817 7.832 b 8.150 8.280 8.340 8.144 8.175 8.171 8.148 8.336 c 8.700 8.900 8.999 8.686 8.667 8.639 8.681 8.972 Characteristic interatomic distances, (Å) Cations Types S1 S2 S3 S4 i ii iii i ii iii i ii iii i ii iii Cu GGA 2.284 2.310 2.32* 2.351 2.340 2.38* 2.341 2.341 2.31* 2.400 2.395 2.37* SOC 2.320 2.308 - 2.374 2.354 - 2.336 2.323 - 2.358 2.382 - Exp. 2.22(2) 2.08(8) 2.30(6) 2.34(2) 2.31(8) 2.34(6) 2.34(2) 2.36(9) 2.33(7) 2.40(2) 2.28(9) 2.40(8) Ni GGA - 2.443 - - 2.488 - - 2.269 - - 2.204 - SOC 2.438 2.484 2.272 2.203 Exp. 2.74(6) 2.54(10) 2.52(9) 2.39(9) Zn GGA - - 2.326* - - 2.359* - - 2.364* - - 2.388* SOC - - - - Exp. 2.30(6) 2.34(6) 2.33(7) 2.40(8) Pb1 GGA 2.845 2.832 2.861* 3.168 3.200 3.117* 2.832 2.935 2.887* 3.541 3.412 3.728* SOC 2.858 2.826 - 3.156 3.220 - 2.796 2.867 - 3.618 3.313 - Exp. 2.84(2) 2.94(11) 2.85(9) 3.02(2) 3.12(11) 3.05(8) 2.71(2) 2.53(7) 2.83(6) 3.49(2) 2.53(7) 2.83(6) Pb2 GGA - - - 3.168 2.880 2.994* 3.406 3.287 3.325* 2.864 2.984 3.079* SOC 2.865 2.870 - 3.400 3.418 - 2.852 2.979 - Exp. 3.02(3) 2.77(11) 2.82(9) 3.92(2) 3.28(9) 3.33(9) 2.93(2) 2.72(9) 2.87(6) Sb1 GGA - - - 2.514 2.517 2.503* 2.489 2.500 2.499* - - - SOC 2.517 2.515 - 2.496 2.496 - Exp. 2.44(2) 2.45(9) 2.52(7) 2.46(2) 2.54(8) 2.50(6) Sb2 GGA 2.480 2.519 2.522* ------2.489 2.491 2.506* SOC 2.481 2.519 - 2.490 2.581 - Exp. 2.56(3) 2.46(9) 2.45(7) 2.38(2) 2.33(8) 2.39(6)

A comparison between the different bond lengths can be obtained from the reported data in Table

6.1. We find that in general for the pure bournonite and the doped compounds there is good agreement between the experimental values and the ones obtained using standard DFT-GGA. The greatest disagreement between computed and experimental interatomic distances is observed for

PbCu0.75Ni0.25SbS3. Namely, for Cu-S1, Ni-S1, Ni-S3, Pb1-S3 and Pb-S4. Such a discrepancy might be related to an additional non-trivial electrostatic interaction which is not included in standard DFT method and arising from interaction between Ni and heavy Sb2 atoms (more details will be provided in the next section).

The different bond angles are also summarized in Table 6.2. The experimental data shows that S-

Sb1-S and most of the S-Pb-S angles decrease upon doping with the exception of S4-Pb2-S2, which does

Table 6.2 Experimental (Exp.) and calculated via simulations (GGA and SOC) bond angles between different Pb and Sn cations and chalcogen anion S atoms

PbCu0.9Zn0.1SbS3 (Exp.) PbCuSbS3 PbCu0.75Ni0.25SbS3 S-Pb-S PbCu0.875Zn0.125SbS3 (GGA) Exp. GGA GGA+SOC Exp. GGA GGA+SOC Exp. GGA S3-Pb1-S3 95.17 93.45 94.87 92.93 91.67 91.93 92.96 94.72 S1-Pb1-S3 82.23 83.70 85.23 81.08 84.43 84.13 81.11 82.43 S4-Pb2-S4 94.96 95.47 96.44 93.82 93.95 93.54 93.89 95.32 S4-Pb2-S2 83.23 85.54 86.30 83.24 84.64 84.41 83.22 83.08 S-Sb-S Exp. GGA SOC Exp. GGA SOC Exp. GGA S2-Sb1-S3 92.29 93.63 95.09 90.54 93.11 93.14 90.57 93.70 S3-Sb1-S3 101.44 97.07 95.54 96.08 95.67 95.71 96.13 95.11 S1-Sb2-S4 93.21 94.10 95.48 94.00 93.34 93.03 93.97 92.98 S4-Sb2-S4 93.85 94.10 95.15 98.25 91.04 90.41 98.33 92.98

not change significantly, and S-Sb2-S angles, which increase upon doping. At the same time, Table 6.2 shows that the corresponding experimental angles have similar values when comparing PbCu0.75Ni0.25SbS3 and PbCu0.9Zn0.1SbS3. The results from the simulations preserve these trends. There is a good agreement between the corresponding angles for the experimental PbCu0.9Zn0.1SbS3 and computational

PbCu0.875Zn0.125SbS3 compositions.

6.2 Electronic Structure

The calculated total and projected carrier densities of states (DOS) are presented in Figure 6.2.1.

We find that the pure bournonite (Figure 6.2.1a-e) is a semiconductor with an energy gap of ~0.686 eV obtained via DFT-GGA. The DOS around the Fermi level 퐸퐹 is composed mainly of Cu-d and S-p states in the valence region (Figure 6.2.1c, e) and Pb-p and Sb-p states in the conduction region (Figure 6.2.1b, d). The localized peak at 퐸~ − 9 eV is primarily due to the Sb-s and Pb-s states and the peak at 퐸~ − 12 eV is due to the S-s states81. Including SOC reduces the band gap to ~0.385 eV and it also results in some shifts in the peak structure due to the S-s states. We find that the pure bournonite is a p-type material, which is in agreement with previous studies238.

The current DFT calculations show that doping with 25% of Ni results in shifting the highest valence band closer to the Fermi level, which indicates that PbCu0.75Ni0.25SbS3 is a p-type semiconductor

Figure 6.2.1 Calculated total and projected DOS for (a, b, c, d, e) PbCuSbS3, (g, h, i, j, k, l) PbCu0.75Ni0.25SbS3 and (m, n, o, p, q, r) PbCu0.875Zn0.125SbS3. (f) Total DOS calculated via standard GGA for (1) PbCuSbS3, (2) PbCu0.75Ni0.25SbS3 and (3) PbCu0.875Zn0.125SbS3. The figure legends with “SOC” mean that SOC is added to calculations. Figure is taken from Ref.237 with an energy gap of 0.301 eV (Figure 6.2.1g). The Ni d-states strongly hybridize with the Cu-d and S-p states as evident from Figure 6.2.1i, k and l. Taking SOC into account increases the band gap to 0.590 eV, showing that this relativistic correction has an opposite effect as compared to the pure bournonite. Doping with 12.5% of Zn shifts the conduction band closer to the Fermi level, which might characterize the material as an n-type semiconductor with an energy gap of 0.555 eV (Figure 6.2.1m). The main effect from Zn is a localized peak at ~ − 7.5 eV in the valence region, while the compositional structure of the rest of the DOS is similar to the one of pure bournonite. The shifts of the localized peak structures due to the Sb-s and Pb-s states (~ − 9 eV) and the S-s states (~ − 13 eV) upon doping can also be found in

Figure 6.2.1n, p and q.

6.3 Electron Localization and Charge Transfer

The electronic structure of the materials is further characterized by calculating the ELF shown in

Figure 6.3.1 for different lattice planes. The cation-anion lattice planes are chosen in order to track the electron localization upon the doping along c direction containing S1-X-S2 atomic chain (Figure 6.3.1b,d, and f) and along a direction containing S3-X-S4 atomic chain (Figure 6.3.1a,c, and e), where X is Cu

Figure 6.3.1 Calculated Electron Localization Function (ELF) for (a, b, g) PbCuSbS3, (c, d, h) PbCu0.75Ni0.25SbS3 and (e, f, i) PbCu0.875Zn0.125SbS3. The projections shown were chosen to represent different types of bonding. Figure is taken from Ref.237 substituted with Ni or Zn. In addition, the lattice plane containing Sb and Pb is chosen to track distribution of lone electron pairs from the two opposing Sb1-Pb1-Sb1 and Sb2-Pb2-Sb2 atomic chains

(Fig. 6.3.1g, h and i). The calculated ELF for all three structures shows the interaction between Cu, (Ni,

Zn) and nearest S neighbors as ionic with covalent features. Particularly, the blue areas at and around Cu,

Ni and Zn indicate a complete electron donation to the nearest S anions, but the corresponding turquoise areas are indicative of a partial electron donation. This means that there is some covalency between cations and anions, which is seen in the red localized regions between the Cu, (Ni, Zn) and S atoms. The

ELF of the lattice planes with two Sb1-Pb1-Sb1 and Sb2-Pb2-Sb2 atomic chains clearly show that the electron clouds due to the lone s2 electron pairs have lobe-like distributions around Pb and Sb atoms and are situated by facing each other. This symmetric orientation promotes an electrostatic repulsion, which contributes towards enhancing the anharmonicity in the lattice which leads to lower thermal conductivity81.

Figure 6.3.1 further shows that there are some specific differences in the ELF of

2 PbCu0.75Ni0.25SbS3 when compared to pure bournonite and the Zn-doped material. While the s -electron

79 clouds from the two chain are positioned symmetrically from each other, the electron lone pair clouds around Sb2 and Pb1 are distorted and the degree of symmetry of the relative electron distribution is reduced, as seen in Figure 4d and h. This distortion may be related to the characteristic distances involving the Sb and Pb atoms. From the calculations, we find that Cu-Sb2=3.400 Å in PbCuSbS3.

Substituting Cu with Zn results in an increased Zn-Sb2=3.624 Å in PbCu0.875Zn0.123SbS3, however substituting Cu with Ni results in the much reduced Ni-Sb2=2.600 Å in PbCu0.75Ni0.25SbS3. Similar trend is found for the calculated bonds involving Pb, such that Cu-Pb1=3.259 Å in PbCuSbS3 but upon doping we find Zn-Pb1=3.696 Å in PbCu0.875Zn0.123SbS3 and Ni-Pb1=2.912 Å in PbCu0.75Ni0.25SbS3.

In addition to the ELF distribution, we also calculate the charge transfer in the unit cell of each compound by considering the difference ∆휌X−cell = 휌푡표푡 − (휌푋 + 휌푡표푡−푋), where 휌푡표푡 is the total charge density, 휌푋 is the charge density of atom X, and 휌푡표푡−푋 is the charge density of the unit cell without the X atom. The results, given in Figure 6.3.2a and b for pure bournonite, show that some electron charge, donated from the Cu atom is located in the regions surrounding the neighboring S atoms, while some of the donated charge is also concentrated on the S atoms, which indicates ionic-covalent type of bonding.

The charge transfer involving Ni in PbCu0.75Ni0.25SbS3 is quite different. Figure 6.3.2c and d shows that

Ni accepts charge by depleting some of the charge donated by Cu around the S atoms. One finds a lobe- like charge distribution around Ni (Figure 6.3.2d), which is indicative of a lone s2-electron pair distribution and a covalent character of the established Ni-Sb2 chemical bonding. The charge transfer involving Zn in PbCu0.875Zn0.125SbS3 is similar as the one between Cu and S in the undoped bournonite, however the regions of depleted and gained electrons are bigger and some excess charge is found around the Sb2 site, as seen in Figure 6.3.2e and f. The charge transfer mechanisms can also be understood from basic chemistry. In the case of the undoped bournonite Cu with its [Ar]3d84s1 electron configuration

8 2 donates the 4s electron to the nearest S sites. In PbCu0.75Zn0.25SbS3, Ni having [Ar]3d 4s attracts electrons from the neighboring atoms in a covalent manner in order to fill its partially filed 3d shell. In

Figure 6.3.2 Calculated charge transfer between one chosen Cu (Ni, Zn) and all other atoms in the unit cell for (a, b) PbCuSbS3, (c, d) PbCu0.75Ni0.25SbS3 and (e, f) PbCu0.875Zn0.125SbS3. The charge density difference is shown as yellow and blue isosurfaces that mean gained and lost electrons respectively. The chosen level of an isosurface is 0.003 e/Bohr3. The lattice perspectives are chosen to show charge transfer along S3-X-S4 atomic chain (a, c, e) and along S1-X-S2 atomic chain (b, d, f), where X denotes Cu, Ni and Zn respectively. Figure is taken from Ref.237

10 2 PbCu0.875Zn0.125SbS3, however, the closed shell Zn atom with [Ar]3d 4s does not promote charge transfer leading to an “excess” of electrons, which may explain the n-type of conduction found from the

DOS calculations (Figure 6.2.1m).

The charge transfer for individual atoms can further be quantified using Bader analysis216–218.

Figure 6.3.3a displays the lattice structure of the undoped bournonite along with schematics of the atomic chains along the a and c directions where the charge transfer mainly occurs. The Bader population is shown for the considered systems and Figure 6.3.3b indicates that one Cu atom donates -0.32e of charge by giving to S1, S2, Pb2, and Sb2 in the unit cell of the undoped bournonite. In the case of

PbCu0.75Ni0.25SbS3, shown in Figure 6.3.3c, Ni accepts +0.22e of charge and the total loss of charge along the S3-Ni-S4 chain is -0.41e, which is much greater than the -0.02e lost charge along the S3-Cu-S4 chain

Figure 6.3.3 (a) Schematics of the projections for the Bader charge analysis along the S3-X-S4 (green color) and S1-X-S2 (brown color) atomic chains in pure bournonite. Bader charges associated with (b) 푿=Cu in PbCuSbS3, 푵 푩 (c) 푿=Ni in PbCu0.75Ni0.25SbS3, and (d) 푿=Zn in PbCu0.875Zn0.125SbS3 are shown. Charge differences ∑풊=ퟏ ∆풆풊 and ∆풆̅ (defined in the text) are given in the table. Figure is taken from Ref.237 in the undoped bournonite. We note that such a loss of charge along S3-Ni-S4 may play a key role in enhancing the p-type transport in PbCu0.75Ni0.25SbS3. In the case of PbCu0.875Zn0.125SbS3, shown in Figure

6.3.3d, Zn donates -0.48e to all surrounding atoms except S1.

The Bader analysis also involves calculating the net change in charge populations for a particular

푛 퐵 푛 퐵 퐵 type of atoms in the unit cell given as ∑푖=1 ∆푒푖 = ∑푖 (푒푖 − 푒푖,푋), where 푛 is the number of atoms of the

퐵 same type in the unit cell, 푒푖 is the charge of the i-th atom when atom 푋 is not present in the unit cell, and

퐵 푒푖−푋 is the charge of the i-th atom when atom 푋 is present in the unit cell. Using this quantity one can track how the charge associated with a particular atom, 푋, is distributed among the rest of the atoms in the different materials. Specifically, the net change upon Ni doping results in significant loss of net charge at the S sites (-0.253e) further showing enhanced p-type conductivity. The net charge difference at the Pb

82 and Sb atoms upon Zn doping, on the other hand, indicates that the transferred charge is mainly concentrated at the Pb (0.102e) and the Sb (0.061e) sites as compared with the same sites in PbCuSbS3 and PbCu0.75Ni0.25SbS3. Such a concentrated charge distribution results in enhanced electrostatic interaction between two opposing lattice planes with the Sb1-Pb1-Sb1 and Sb2-Pb2-Sb2 atomic chains

(Figures 6.3.1g, h and i). One can also consider the average charge distribution per atom ∆푒̅ =

푛 ∑푖 (푒푖,푖푛 − 푒푖,푐표푚)/푛, where 푒푖,푖푛 is the number of valence electrons of the i-th atom as supplied initially by the VASP pseudopotential and 푒푖,푐표푚 is the computed number of valence electrons for the same atom after relaxation (results given in Figure 6.3.3). This data gives further details of the charge donating- accepting balance within each compound. It is interesting to observe that Cu donates the most charge in the undoped bournonite lattice and the least amount of charge in the Zn-doped material. Pb and Sb are also donors, however, ∆푒̅ is smaller and of the same order for the doped bournonites. The average charge accepted by Ni and charge donated by Zn is also given in the table of Figure 6.3.3.

6.4 Phonon Structure and Thermal Conductivity

Results from our calculations for the phonon band structure and phonon DOS are shown in Figure

6.4.1. The absence of imaginary branches shows that the considered compositions are dynamically stable, although some numerical errors are evident around the Г-point for PbCu0.875Zn0.125SbS3 (Figure 6.4.1c).

The overall composition of the peaks below 3 THz has similar structure for all materials, however the calculated acoustic velocities along Г-X and Г-Y shown in Table 6.3 are different. The 푣푇퐴1,2 and 푣퐿퐴 velocities in both doped materials are generally larger than the ones for the undoped bournonite, except for the velocities along the Г-Y direction for the Zn-doped material, which are slightly smaller. The

Debye temperatures associated with each sound velocity are calculated by239

1/3 ℏ 2 푁 휃푖 = 푣푖 (6휋 ) , (6.4.1) 푘퐵 푉

where 푖 stands for particular acoustic mode (transverse or longitudinal), 푣푖 is the speed of sound, 푁 is the number of atoms per unit cell, 푉 is the volume of the unit cell. Although, Debye temperature calculated

Figure 6.4.1 Calculated phonon band structure and PDOS for (a) PbCuSbS3, (b) PbCu0.75Ni0.25SbS3 and (c) 237 PbCu0.875Zn0.125SbS3. Figure is taken from Ref. by Eq. (6.4.1) is not practically useful as it is an ideal approximation for harmonic crystals at low temperatures. For complex structures the Debye values are the temperature dependent quantities240,241 and need more fundamental investigations. Note that, 휃푇퐴1,푇퐴2 are in good agreement with previously reported experimental values of 157 K for the pure bournonite81.

From the phonon band structure and DOS, one finds that the result of Ni and Zn doping is primarily seen in the (3, 5) THz range (Figure 6.4.1b and c). The coupling of Ni with Sb and S in the

(3.25, 4.5) THz range is stronger as compared with Zn in PbCu0.875Ni0.125SbS3. In addition, the PDOS for

PbCu0.875Ni0.125SbS3 is less and more spread in the acoustic range (0, 3) THz in comparison with other structures. This is a sign of less flattening of optical phonons and might characterize greater thermal conductivity58.

Table 6.3 Vibrational properties of chosen compounds: PbCuSbS3 (i), PbCu0.75Ni0.25SbS3 (ii) and PbCu0.875Zn0.125SbS3 (iii), TA1, TA2 and LA denote two transverse and one longitudinal acoustic modes respectively

Computed PbCuSbS3 PbCu0.75Ni0.25SbS3 PbCu0.875Zn0.125SbS3 quantities Г-X Г-Y Г-X Г-Y Г-X Г-Y 푣푇퐴1, (m/s) 1498.23 1521.32 1515.20 1536.24 1634.04 1506.18 푣푇퐴2, (m/s) 1505.27 1551.19 1642.93 1684.56 1581.51 1550.83 푣퐿퐴, (m/s) 3102.48 2632.84 3147.53 2672.91 3190.05 2312.20 휃푇퐴1, (K) 154.01 156.38 158.06 160.26 167.73 154.60 휃푇퐴2, (K) 154.73 159.45 171.39 175.73 162.34 159.19 휃퐿퐴, (K) 318.92 269.72 328.35 278.73 327.45 237.34

The experimental results for the thermal conductivity can be examined in the context of the DFT calculations for the various properties. As discussed earlier, the presence of stereochemically active lone pair electrons is associated with low

휅 values due to coordination distortions and electrostatic int0eractions from the electron distribution81,212,213. Our ELF Figure 6.4.2 Temperature dependent κ of PbCuSbS3 (circle), PbCu0.75Ni0.25SbS3 (triangle up), and PbCu0.9Zn0.1SbS3 (triangle results displayed in Figure 6.3.1 explicitly down). The inset is the high temperature κ of three specimens. Figure is taken from Ref.237 show that the undoped and doped bournonites have lobe-like distributions associated with lone pair electrons from Sb and Pb atoms. Also, there is a correlation between the coordination number of cations and their angles indicative of the distortion in the network, which further contributes towards low 푘81. Specifically, the average S-Sb-S angles should be > 99° since the Sb coordination number is 3 and the average S-Pb-S angles should be

~90° since the Pb coordination number is 6 or greater81. Our results in Table 6.2 show that the average

° ° ° experimental S-Sb1-S angle is 96.87 for PbCuSbS3, 93.31 for PbCu0.75Ni0.25SbS3, and 93.31 for

° ° PbCu0.9Zn0.1SbS3. The average experimental S-Sb2-S angle 93.53 for PbCuSbS3, 96.13 for

° ° PbCu0.75Ni0.25SbS3, and 96.15 for PbCu0.9Zn0.1SbS3. These results show that all S-Sb-S angles are < 99 , however the doping atom reduces the angle involving Sb1, while increases the angle involving Sb2. The type of dopant does not affect these characteristic values since S-Sb1-S and S-Sb2-S average angle are very close for the two doped compositions. Comparing the average S-Pb-S angles shows that S-Pb1-S is

° ° ° 88.7 for PbCuSbS3, 87.01 for PbCu0.75Ni0.25SbS3, and 87.04 for PbCu0.9Zn0.1SbS3 and S-Sb2-S is

° ° ° 89.1 for PbCuSbS3, 88.53 for PbCu0.75Ni0.25SbS3, and 88.56 for PbCu0.9Zn0.1SbS3. One notes that all

85 average angles involving Pb are close to 90°, however they are slightly reduced due to the Ni and Zn dopants.

The presence of stereochemically active lone pair electrons and the lattice distortion due to the coordination number – average angle correlations can be related to the low thermal conductivity values in all considered materials. Nevertheless, Figure 6.4.2 shows that at 푇 = 300 퐾 푘 = 0.79 W/m∙K for

PbCuSbS3, 푘 = 0.87 W/m∙K for PbCu0.75Ni0.25SbS3 and 푘 = 0.73 W/m∙K for PbCu0.9Zn0.1SbS3. Given that S-Sb-S and S-Pb-S angles in the doped bournonites are very similar, such considerations alone cannot explain the fact that 푘 for PbCu0.75Ni0.25SbS3 is 0.14 W/m∙K greater than 푘 for PbCu0.9Zn0.1SbS3.

The origin of this difference is related to the lone electron pair mutual distribution and the electronic contribution of the thermal conductivity 푘푒. Our previous discussions of the electron charge transfer and

Bader analysis (Figure 6.3.2 and 6.3.3) show that Ni accepts electron charge especially along the S3-Ni-

S3 containing chain, which strengthens the conducting properties of PbCu0.75Ni0.25SbS3 meaning that that

푘푒 increases. In addition, the reduced symmetry of the mutually tilted lone pair distribution in Figure

6.3.1h indicates that the electrostatic repulsion is weakened as compared to PbCuSbS3 in Figure 6.3.1g.

This reduced anharmonicity may result in higher thermal conductivity associated with the lattice contribution. In the case of PbCu0.9Zn0.1SbS3, the Zn has a closed electron shell and it donates charge to the surrounding S atoms. This is interpreted as 푘푒 being reduced as compared to 푘푒 for PbCuSbS3.

6.5 Summary

In this study, the atomic structure, transport properties, and charge transfer characteristics in doped bournonites are studied using experimental and computational methods. Our detailed analysis shows that doping with 25% of Ni creates covalent type of interaction between Ni and Sb2. As a result, the p-type conductivity is enhanced and the anharmonicity due to the distortion from the lone electron cloud is reduced leading to an increased thermal conductivity in PbCu0.75Ni0.25SbS3. On the other hand, the closed 3d-shell Zn atom reduces the conduction properties of PbCu0.875Zn0.125SbS3 and the enhanced

86 electrostatic interaction between the lattice planes containing Sb1-Pb1-Sb1 and Sb2-Pb2-Sb2 atomic chains results in smaller thermal conductivity when compared to the undoped bournonite. The detailed description of atomic bonds, angles, and charge transfer indicates that transport mechanisms in bournonite and its doped composition is complex. Doping with various atoms can affect the electronic and phonon dynamics and synergistic experimental and computational efforts are needed to understand effective ways of thermoelectric control in a given class of materials.

7 CLATHRATES

The group 14 clathrates were discovered over five decades ago242–244. The structure of these materials, which features a relatively open tetrahedral network that provides an effective host for guest atoms or molecules, underpins many of the interesting properties they display245, for example, glasslike thermal conductivity, superconductivity in sp3 bonded solids, magnetism, and heavy atom tunnelling in the crystalline state72,73,210,246–250. These unique physical properties are the reason clathrates continue to be of interest for technological applications, including thermoelectricity72,210,246–248, photovoltaics30, ultra- hard materials30, and magnetic refrigeration251.

Exploring clathrates of different compositions shows that the frameworks can be categorized by symmetry of the constituent polyhedra, which can encapsulate different types of guests, typically with atoms from group 1 or 2 252. It is also possible that the guest occupants are methane or noble gas atoms, which weakly interact with the framework. An interesting possibility of putting gaseous atoms inside the inorganic clathrate voids can lead to new pathways for property tuning. Another possibility to design clathrate properties is to utilize mixed cage materials.

Depending on the electric charge of the framework, the clathrates can be neutral, anionic or cationic, with the latter being much less investigated. There are only a few reports on the synthesis and properties of clathrates with primarily group 15 atoms in the framework. These type of clathrate may also

211 reduce thermal conductivity, as indicated by the report on orthorhombic phosphorus based clathrate .

However, a fundamental understanding of their structure-property relationships, including the potential for altering their electrical properties, are equally important253. The goal of this research is to advance a fundamental understanding of type II Sn-based clathrates and type I As based clathrate.

7.1 Tin type II clathrates

In this section the ab initio calculations is presented for the electronic and phonon properties of several Sn type II clathrates, including empty cage Sn136, compositions with partial Ga substitution on the framework, and compounds of Sn136 filled with inert Xe atoms.

The atomic and electronic structures for the above structures are utilized via the VASP package87–

89. The exchange-correlation energy is calculated with the PBE functional118. The unit cells for the considered materials take advantage of the inherent symmetries of each composition. Ionic relaxation is performed with 367 eV cutoff. The force and total energy difference relaxation criteria are 10−4 eV/Å and

10−8 eV, respectively. The cell is allowed to change shape and volume during the structural relaxation with 9 × 9 × 9 k-mesh. The tetrahedron integration method with Blöchl correction is used for the self- consistent calculations on the same k-grid254. Also, the VESTA software package is utilized to perform the crystal structure and the electron localization function215.

The phonon properties in terms of vibrational density of states (vDOS) and phonon dispersion spectra for the considered structures are calculated with the PBE functional using the Phonopy package166.

The finite atomic displacements with an amplitude of 0.01 Å are introduced in the simulated structures.

The atomic forces within the supercell are calculated using VASP followed by phonon frequency calculations from the dynamical matrix represented in terms of the force constants. The mode Grüneisen parameter, defined as 훾푖 = −휕ln (휔푖)/휕 ln(푉) for each phonon mode with frequency 휔푖 (푉 is the volume of the lattice), is also calculated. For this purpose, phonon calculations for the equilibrium volume and two additional volumes that are slightly larger and smaller, are calculated in order to compute the logarithmic derivate in 훾푖.

7.1.1 Structure and stability

Type II clathrates are described by the general chemical formula A8B16X136 (A=Cs, Rb; B=Na, K,

Ba; X=Si, Ge or Sn). Such materials have the 퐹푑3̅푚 cubic space group with 136 tetrahedrally

89 coordinated atoms (X) in a unit cell with two types of cages: eight hexakaidecahedra [51264] and sixteen dodecahedra [512]30. Guest atoms can be accommodated inside each cage, where two types of atoms can occupy the different polyhedra. It is also possible that either [51264] or [512] are occupied or the same type of atoms are hosted by all cages. Clathrates with partially substituted cages are feasible as well (YmX136- m), where charge transfer from the guests to the cage is balanced.

The focus of this investigation are type II Sn-based clathrates. Although Ba16Ga32Sn104 with Ga and Sn being on the framework have been synthesized almost three decades ago255, only recently have researchers started investigating these materials. Structural data and transport property measurements of compounds synthesized by various methods have been reported showing intriguing structure-property relations256–260. First principles simulations on a handful of materials have also been performed showing much reduced band gaps and/or transfer to metallic states upon alkaline and earth-alkaline atoms cage insertion261,262. The phonon dynamics for a few materials has also been reported263,264.

To obtain broader and more systematic insight into the basic science of these systems the following systems Sn136, Ga40Sn96, Cs8Ba16Ga40Sn96, K2Ba16Ga30Sn106, K8Ba16Ga340Sn96, Xe8Sn136,

Xe16Sn136, and Xe24Sn136 are considered. The variety of these compositions gives an excellent opportunity to compare and contrast properties of clathrates with empty cages, partially substituted framework, and cages hosting various guests. Using the inherent symmetries of the 퐹푑3̅푚 space group the conventional

265 unit cell of the empty X136 system can be reduced to a primitive unit cell with 34 atoms, X34 . Similar four-fold reduction can be applied to A8B16X136 where the Wyckoff positions for the cage sites are 96g,

32e and 8a, while the Wyckoff’s positions for the guest atoms are 16c and 8b. Reducing the number of atoms is especially useful for better computational efficiency. The Ga substitution in the mixed cage systems, however, leads to many possibilities where the Ga atoms could reside. Considering Ga40Sn96, for

34 8 example, there are (10) ≈ 10 ways of possible arrangements. For many Ga concentrations the reduction to a primitive cell is not possible, but for a handful of cases (Ga8Sn128, Ga32Sn104, and Ga40Sn96), the construction of a primitive cell can be done. To study these types of arrangements in what follows the

Figure 7.1.1 Representation of type II Sn clathrates. (a) Empty cage Sn136; (b) A snapshot of Ga40Sn96 during the relaxation process showing significant distortion; (c) Cs8Ba16Ga40Sn96 in which all cages are filled (Cs – red, Ba - blue); (d) The two characteristic polyhedra of Cs8Ba16Ga40Sn96 with random Ga distribution with no direct Ga-Ga bond are shown. Figure is taken from Ref.76 advantage of the primitive unit cell of Ga40Sn96 is taken into account. Further, the following possibilities of the Ga atoms arrangements is considered: (i) a substitution by placing the Ga atoms in symmetric 32e and 8a Wyckoff positons; (ii) a random substitution with some direct Ga-Ga bonds; (iii) a random substitution with no direct Ga-Ga bond.

Figure 7.1.1 displays some of the studied structures after the ab initio relaxation process. It is found that the randomly substituted Ga40Sn96 is not stable. A snapshot of the computational process at some intermediate step is given in Figure 7.1.1b showing distortion of the lattice when compared to the

Sn136 system (Figure 7.1.1a). The Ga40Sn96 eventually becomes severely twisted which leads to breaking of the crystal lattice. The calculations indicate that the guest atoms in the [512] and [51264] cages remove this distortion and stabilize the structure. Figure 7.1.1c displays this situation for Cs8Ba16Ga40Sn96 and

Figure 7.1.1e shows a larger view of the two types of polyhedra with Cs and Ba in each center. It is also found that the Xe8Sn136, Xe16Sn136, and Xe24Sn136 have the same structure as the one in Figure 7.1.1a with

Xe atoms residing in the center of the respective polyhedra (not shown). The structural parameters together with the formation energies for several systems are also given in Table 7.1. The formation energy

푛 is defined as 퐸Δ = 퐸푇 − ∑푖=1 퐸푖, where 퐸푇 is the total energy for the compound and 퐸푖 is the energy of the constituent atoms (n – total number of atoms in the cell). According to Table 7.1, the empty caged Sn clathrate has the smallest lattice constant and nearest neighbor distance, and it is the most stable compound. Inserting alkaline and earth-alkaline in cages containing Ga leads to increasing 푎 and

Table 7.1 Lattice constants, nearest neighbor bonds, and formation energies for the studied clathrates. The Ga substitution in the cages is denoted as: (i) Wyckoff positions; (ii) random positions with some direct Ga-Ga bonds; (iii) random with no direct Ga-Ga bonds. The formation energy per atom in the unit cell is calculated using 푬횫 = 풏 푬푻 − ∑풊=ퟏ 푬풊 (푬푻 – total energy; 푬풊 – energy for constituent atoms).

Material a=b=c, (Å) Sn-Sn, (Å) Ga-Sn, (Å) 퐸∆, (eV/atom) Sn136 17.285 2.779 - -3.672 Cs8Ba16Ga40Sn96 (i) 17.389 - - -3.285 Cs8Ba16Ga40Sn96 (ii) 17.376 - - -3.290 Cs8Ba16Ga40Sn96 (iii) 17.375 2.931 2.732 -3.300 K2Ba14Ga30Sn106 (ii) 17.485 2.922 2.749 -3.379 K8Ba16Ga40Sn96 (i) 17.387 2.929 2.727 -3.261 Xe8Sn136 17.435 2.849 - -3.449 Xe16Sn136 17.456 2.898 - -3.211 Xe24Sn136 17.463 3.053 - -3.047

decreasing 퐸Δ. One also finds an increased average Sn-Sn distance when compared to Sn136. The average

Sn-Ga distance, on the other hand, is obtained to be about 0.2 Å smaller than the respective to Sn-Sn separation for each material, which is attributed to the smaller Ga atom. Table 7.1 further shows that the location of the Ga atom on the framework affects the lattice constant and structural stability. In particular, it is obtained that Cs8Ba16Ga40Sn96 has the lowest energy 퐸Δ when Ga atoms are randomly distributed with no direct Ga-Ga bond ((iii) case) when compared with the (i) and (ii) cases. These findings are in agreement with results for type I Ge clathrates, where Ge atoms with randomly distributed Ga atoms with no direct Ga-Ga bonds also result in a more stable state266

Comparing Cs8Ba16Ga40Sn96 and K8Ba16Ga40Sn96 one sees that the lattice constants are very similar, but the first material is more stable by 0.024 eV/atom. On the other hand, it is realized that the

257 recently studied K2Ba14Ga30Sn96 (for which it is not possible to use symmetric Wyckoff positions in the unit cell) is more stable by 0.118 eV/atom than the K8Ba16Ga40Sn96, for which all cages are occupied by the guests.

It is also important to note that Ga distribution on the framework affects the location of the guests. In the case of lack of direct Ga-Ga bonds (iii), Cs and Ba are found in the center of each polyhedron, as shown in Figure 7.1.1d. In the other two cases (i) and (ii), however, direct Ga-Ga bonds

92 imply clustering of Ga atoms on the framework, which also leads to displacing of the guest atoms from the center. Specifically for the Cs8Ba16Ga40Sn96 studied here, it is found that the displacements for the Cs atoms are 푑1 = 0.29 Å ((i) case) and 푑1 = 0.63 Å ((ii) case), while the displacements for the Ba atoms are

푑2 = 0.2 Å ((i) case) and 푑2 = 0.426 Å ((ii) case), as shown in Figure A2.1 (see Appendix 2). Similar off-center displacements have been reported in recent experiments concerning type-II Si mixed cage clathrates with vacancies in the framework, where off-center rattling has been considered as an additional factor towards reduction of the thermal conductivity257. Note, that the electrostatic interaction has an important role for these off-center displacements. Following the Zintl-Klemm concept, charge transfer balance between the framework and guest atoms can be achieved and as a result, the Coulomb attraction induces a displacement of the guests towards the clustered Ga atoms in the framework (see Appendix 2).

The lattice constant-energy relation in systems with Xe atoms encapsulated in the cages is also interesting. Previous authors have shown that type I clathrates cannot accommodate rare-earth gas atoms

(with the exception of Ar) due to the relatively small cages75,267,268. Nevertheless, this becomes possible for the type II Sn clathrates, as shown here. The lattice constants for all Xe based clathrates are larger than

12 4 12 the one for Sn136, and they increase as one populates the [5 6 ] (Xe8Sn136), the [5 ] (Xe16Sn136), and both type of polyhedra (Xe24Sn136). The same behavior is noted for the nearest neighbor bonds with 3.03 Å being the largest Sn-Sn distance for Xe24Sn136. The trend for 퐸Δ is in the opposite direction. The obtained results show that the Xe residing in the larger polyhedra, which have the smallest 푎, results in the most stable composition, Xe8Sn136, among the studied Xe filled clathrates.

7.1.2 Electronic structure

The electronic structure of the different materials in terms of energy bands, density of states

(DOS) and electron localization function (ELF) calculations is also investigated. Figure 7.1.2a displays the energy bands and DOS for the empty cage Sn136. This is a semiconductor with a direct gap of 0.327 eV at the L-point and a second direct gap of 0.669 eV is found at the Г point. The obtained gap at the L- point here is smaller by 0.135 eV than the one found via DFT-LDA in261, which is expected due to the

Figure 7.1.2 Band structure and DOS for (a) Sn136, (b) Cs8Ba16Ga40Sn96 with random Ga substitution and no direct Ga-Ga bond, (c) Xe8Sn136. The DOS panels show total and projected contributions. All calculations are done using the primitive unit cell of the materials. Figure is taken from Ref.76 well-known fact that LDA methods systematically underestimate energy gaps. Characteristic energy gaps at different high symmetry points in the band structure for Sn136 and several other materials are shown in

Table 7.2. The Sn136 DOS in the vicinity of (0, -2.43) eV is primarily composed of p orbitals, followed by a rather large gap in the valence region. The region of (-3.85, -5.47) eV in DOS is characterized by s and p orbitals due to the sp3 – hybridization in the framework (see Appendix 2, Figure A2.2a). The band structure profile of Sn136 is similar to the one of Si136 and Ge136, which are reported to have energy gaps of

1.24-2.4 eV and 0.7-0.8 eV, respectively58,269,270 (Figure A2.2c in see Appendix 2). Figure 7.1.2b displays the electronic structure of Cs8Ba16Ga40Sn96 with random Ga atom organization for which symmetry is used to reduce the conventional cell to a primitive unit cell ((iii) case). This DOS shows that placing guest atoms inside the voids has a significant effect on the overall behavior. Specifically, the Ga atoms together with mainly the Ba atoms lead to shifting of the conduction bands downward, which results in a much reduced energy gap at the L-point (0.171 eV). This maybe explained via the Zintl concept, which reflects

271 the presence of uncompensated charges in the filled cage clathrate . The DOS for Cs8Ba16Ga40Sn96 for the (i) and (ii) cases are very similar to the one in Figure 7.1.2b, however, there is no energy gap found at the Fermi level indicating a metallic state for such structural compositions. Analyzing the obtained DOS in Figure 7.1.2b shows that the valence region (0.0, -2.8) eV is composed of p orbitals from Ga and Sn atoms. The distinct peak localized in the (-2.8, -3.2) eV range is due to the contribution from p orbitals of

Sn atoms, while sp3 hybridization signatures are observed in (-3.5, -5.5) eV (region not shown). The orbital hybridization between Ba and nearest Ga atoms occurs in (-2.82, -3.2) eV and (0, 3.43) eV mainly

Table 7.2 Direct energy gap at high symmetry points in the Brillouin zone, transverse,풗푻푨, and longitudinal, 풗푳푨, speed of sound along characteristic directions, total mode Grüneisen parameter, 휸, and Debye temperature, 휽푫 for several type II Sn clathrates are shown.

Speed of sound along chosen k-path, (m/s) Direct energy gap, (eV) 훾 Г-L Г-X Material 휃 , (K) [0,2] [2,∞] 퐷 L Г X W 푣 푣 푣 푣 푇푅 퐿퐴 푇푅 퐿퐴 THz THz

Sn136 0.327 0.669 1.430 1.411 1448.75 3220.49 1450.96 3221.87 1.42 1.54 174.2 Cs8Ba16Ga40Sn96 0.171 0.547 1.166 0.949 1646.27 2927.79 1647.44 2926.85 1.28 1.49 199.3

Xe8Sn136 0.607 1.022 1.722 1.695 1491.12 3139.17 1490.47 3138.52 1.42 1.52 180.4 Xe24Sn136 0.430 0.541 1.124 0.968 1688.16 3127.53 1687.55 3126.44 1.11 1.42 206.0

due to the Ba s and Ga p states. The interaction between Cs and nearest Ga atoms can be traced as hybridization between Cs s and Ga p states in (-2.31, -1.91) eV and (0.5, 4.02) eV. DOS for

K2Ba14Ga30Sn106 (ii) (given in Figure A2.2d, Appendix 2), shows an essentially semimetallic state. The strong hybridization between the Ga and Sn state are responsible for the steep peak around -0.5 eV, while the peak around 1.5 eV is an admixture between Ba, Ga, and Sn. The electronic structure Xe8Sn136 in

Figure 7.1.2c shows that the effect of the Xe encapsulation is small on the overall DOS and energy bands composition. It is found that in the depicted energy range, the energy bands and corresponding peaks in

DOS are primarily due to the Sn atoms. Nevertheless, the energy gap at the L-point is 0.607 eV, which is significantly larger than the L-gap of Sn136. The Xe atom has a significant contribution in the (-4,-5) eV valence region characterized by a localized peak feature, as evident in Figure A2.2b.

7.1.3 Electron localization function

The nature of chemical bonds and charge transfer characteristics is further investigated by calculating the ELF. The results, shown in Figure 7.1.3 for several clathrates, depict the type of bonding in these systems. The red cigar-shaped regions in Figure 7.1.3a indicate the covalently shared electrons between the atoms in the Sn136 network, which is very similar as the type of bonding found in Si136.

Introducing Ga into the framework, changes the ELF landscape. In Figure 7.1.3b the ELF is given for

Cs8Ba16Ga40Sn96, which shows that while Sn-Sn have covalent character (similar for Ga-Ga, but not shown in Figure 7.1.3), the pronounced red region in vicinity of Sn indicates polar covalent bonding for

Ga-Sn. The interaction between the framework and the Cs and Ba guests is determined to be of ionic

Figure 7.1.3 Calculated ELF in the [111] plane for: (a) Sn136, (b) Cs8Ba16Ga40Sn96, and (c) Xe24Sn136. Figure is taken from Ref.76 character. Ionic bonding between guests from the first two columns of the periodic table and the framework is typical for other clathrates as can be seen in Figure A2.3 displaying ELF for Cs8Na16Si136

(see Appendix 2). Finally, the ELF results for Xe24Sn136 are given in Figure 7.1.3c and they are very

6 similar as the ELF for Xe24Sn136 and Xe24Sn136 (not shown). Due to the full 3p shell structure of Xe, no chemical bonding with the framework is expected.

7.1.4 Phonons

The calculated vibrational properties of the Sn clathrates include the phonon density of states vDOS, the phonon band structure, and the mode Grüneisen parameter. The calculations rely on obtaining the dynamical matrix 퐷(푞), which is then diagonalized within the harmonic approximation to obtain the phonon eigenmodes and eigenvectors. The total and partial vDOS are given in Figure 7.1.4 (bottom panels) for several compounds, while the corresponding phonon band structures are shown in Figure A2.5

(see Appendix 2). The phonon bands allow us to extract the acoustic sound velocities along different k- path directions in the Brillouin zone for the studied materials (given in Table 7.2). The mode Grüneisen parameter 훾푖, calculated within the quasi-harmonic approximation (where frequencies are volume dependent but temperature effects are ignored), is also shown in Figure 7.1.4 (top panels).

Examining the vDOS for Sn136, given in Figure 7.1.4a, shows that acoustic phonons reside in the

(0,1) THz range with sound velocities along the Г-L and Г-X paths given in Table 7.2. The peak centered

~ 1.5 THz results from several flat optical modes associated with the vibration of the framework. Similar

Figure 7.1.4 Phonon structure properties for: (a) Sn136, (b) Cs8Ba16Ga40Sn96, (c) Xe8Sn136, and (d) Xe24Sn136. In each case, top panels correspond to the mode Grüneisen parameter, and the bottom panels correspond to the phonon density of states (showing results for total and partial density of states). Figure is taken from Ref.76 flat modes are responsible for another sharp peak in the (5, 5.5) THz range. The phonon band from acoustic and optical branches along characteristic directions can be also examined in the Sn136 phonon band structure, given in Figure A2.5a (see Appendix 2). The vDOS for Cs8Ba16Ga40Sn96 in Figure 7.1.4b has similarities with the results for the empty cage Sn136 material. The peaks in the (1, 2) and (5, 5.5) THz regions are still present, but their sharpness and magnitude are reduced. Figure 7.1.4b shows that the vibrations of the Ba atoms, located in the smaller polyhedra, together with Sn and Ga contributions from the network determine the vDOS in the (1, 2) THz range, and the peak in the (5, 5.5) THz region is broadened due to the Ga-Sn admixture. Comparing with the results for Sn136, it is seen that there is an additional relatively sharp feature centered ~0.5 THz in the vDOS, which entirely due to the vibration of

97 the Cs atoms in the larger cages. The corresponding acoustic and flat optical phonon bands are displayed in Figure A2.5b. A prominent peak located in (0.5, 0.8) THz region from the vibrations of guests in the larger cages is also present in the vDOS for Xe8Sn136 and Xe24Sn136 (Figure 7.1.4c and d), however with much more pronounced sharpness, which is also reflected in the corresponding flat bands in Figure

A2.5c,d. The rattling of the Xe atoms in the smaller cages results in two peaks centered at ~1.6 and ~2

THz with very little admixture from the framework (Figure 7.1.4d). The presence of guest atoms also affects the sound velocities. Table 7.2 shows that the most dramatic changes are found for

Cs8Ba16Ga40Sn96, where along the Г-L direction 푣푇푅 is increased by 197 m/s, while 푣퐿퐴 is decreased by

292 m/s when compared with the corresponding values for Sn136. The least dramatic changes are calculated for Xe8Sn136, where for the Г-L path 푣푇푅 is increased by 42 m/s, while 푣퐿퐴 is decreased by 81 m/s when compared to Sn136. Very similar values are found for the Г-X direction. The Debye temperatures (averaged for along Г-L and Г-X paths) for the studied materials are also calculated and the results are given in Table 7.2. One finds that 휃퐷 is the smallest for the empty cage material and

휃퐷increases when the lattice constant of the material increases. Note, further, that the Debye temperature is smaller than the one for clathrates with Ge or Si frameworks as reported in the literature, nevertheless,

30 the values in Table 7.2 are of similar order as experimental 휃퐷 for type II Sn clathrates .

The mode Grüneisen parameter gives further information about the motion of the atoms in the studied materials. It quantifies the stiffness of the bonds as affected by the vibration of the atoms. Figure

7.1.4 (top panels) shows that 훾푖 has similar characteristics for all materials especially for frequencies larger than 2 THz where 훾푖 ≅ 푐표푛푠푡. By examining the 휔 < 2 THz, one finds that there is localized sharp negative peak at 휔~0.5 THz corresponding to the Ba atoms vibrations in Cs8Ba16Ga40Sn96 (larger scale of Figure 7.1.4b is given in Figure A2.4, see Appendix 2). Some negative 훾푖 points in a more spread out range are seen for Sn136, Xe8Sn136, and Xe24Sn136 in the (0.5, 2) THz region. The mode Grüneisen parameter for the studied type II Sn clathrates exhibit similar features as 훾푖 for Si types of clathrates, however the magnitude is larger and the range of negative 훾푖 for Si type II systems is much more

98 extended272. Let us note that a negative Grüneisen parameter characterizes negative thermal expansion of a given material, meaning that the material experiences contraction instead of expansion when temperature is raised. It has been suggested that negative thermal expansion may be present at

272 temperatures below room temperature due to the overwhelmingly negative 훾푖 for Si clathrates for low

휔, however, given the mostly positive 훾푖 in Figure 7.1.4, it is concluded that such a phenomenon is unlikely for the Sn clathrates.

The Grüneisen parameter is a quantity that is indicative of the magnitude of the lattice thermal

2 conductivity 휅퐿. Specifically, in the high temperature regime 휅퐿~1/훾 , where 훾 is the total Grüneisen parameter, indicates that in order to achieve low thermal conductivity, the Grüneisen parameter of the material must be large. Within the quasiharmonic approximation, one can show that at lower temperatures

훾 = ∑ 퐶푉,푖훾푖 / ∑ 퐶푉,푖, where 퐶푉,푖 is the specific heat at constant volume for each mode. At higher temperatures, however, 퐶푉,푖 approach the limiting case of the total specific heat 퐶푉, thus 훾 = ∑ 훾푖 /푁 , where 푁 is the number of vibrational modes273. The so-calculated 훾 for the low and high 휔 are shown in

Table 7.2. It is interesting to see that 훾 in the [2,∞] THz range (corresponding to higher temperatures) has very similar values for all materials, even though 훾 for Sn136 is the largest. For the [0,2] THz range, however, the Grüneisen parameters for Sn136 and Xe8Sn136 are the same, which is larger by ~10% and

21% as compared to the calculated values for Cs8Ba16Ga40Sn96 and Xe24Sn136.

7.1.5 Summary

In this chapter, using ab initio DFT methods, the systematic studies for the basic electronic and phonon properties of type II Sn clathrates are performed. It is found that guest atoms are important for the energetic and structural stability of the materials especially for those with frameworks with Ga atom admixtures. Typically alkaline or earth-alkaline atoms result in reduced energy band gap and in some cases a transition to a metallic state can be achieved. The semiconducting type of electronic structure can be preserved by inserting inert gas atoms, such as Xe, whose accommodation becomes feasible due to the

99 large enough Sn cages. The calculated phonon band structure also shows that vibrations from the two different types of polyhedra and the framework have specific signatures. In the vDOS, a distinct peak originating from guest atoms vibrating in the larger, [51264], polyhedra is found in the low frequency regime. While this feature has little influence from the framework, the guest vibrations in the smaller,

[512], polyhedra tend to experience significant coupling from the network. The lack of interaction between the Xe atoms and the Sn framework also results a Grüneisen parameter that is very similar to the one for

Sn136.

The DFT investigation reveals interesting structure-property relations in type II Sn clathrates especially in clarifying the stability role of the guest atoms in the different polyhedra and their effects in the electronic and vibrational band structures. The obtained results might serve as a motivation to experimentalists to explore the synthesis of Sn clathrates with inert gases as possible routes to achieve a favorable energy band gap for thermoelectricity while taking advantage of guest atom rattling for a reduced thermal conductivity.

100

7.2 Arsenic type I clathrate

In this section the DFT ab-initio investigation is performed for clathrate-I Ba8Cu16As30 synthesized by

Prof. Nolas’ group274. The first principles calculations together with experiment are employed to determine the atomic structure and electronic properties of the synthesized material. The structure is experimentally confirmed to be Pm3̅n (No. 223) with a=10.4563(3) Å. It is found that the smallest number of direct Cu-Cu bonds are preferred, while some of the guest atoms have off-center equilibrium positions. The calculated Fermi surface, electron localization and charge transfer, as well as a comparison with the results for elemental As46, provide insight into the fundamental properties of this cationic- framework clathrate-I material.

The electronic structure was calculated according to the DFT approach as implemented in the VASP package, which relies on a projector-augmented wave method with plane-wave basis sets and periodic boundary conditions189,189. The exchange-correlation energy was taken into account via the Perdew-

Burke-Ernzerhof (PBE) functional within the local density approximation (LDA) and the generalized

118 gradient approximation (GGA) . The unit cell was constructed according to the symmetry group for type

I clathrates. Ionic relaxations were performed with an energy cutoff of 374 eV and 380 eV for As46 and

-4 -4 Ba8Cu16As30, respectively. The total energy difference and force relaxation criteria were 10 eV and 10 eV/Å, respectively. During the relaxation process the unit cell was allowed to change its shape and volume. Tetrahedron integration with Blöchl corrections were used for self-consistent calculations with a

9 × 9 × 9 k-mesh. Relativistic effects originating from the spin-orbit coupling (SOC) were also taken into account with a 7 × 7 × 7 k-mesh sampling. In VASP the SOC is calculated using non-collinear magnetism where the valence electrons are accounted for with a variational method and scalar relativistic eigenfunctions. Although SOC calculations require bigger computational resources, it is found that such relativistic effects have important consequences in the energy band structure of these materials. Also, the

VESTA package was used for the crystal structure and the ELF determination215. The XCrySDen program was used for Fermi surface visualization191.

101

7.2.1 Structural properties

Clathrate-I Ba8Cu16As30 crystallizes in the cubic 푃푚3̅푛 symmetry (No. 223) with a lattice parameter a=10.4563(3) Å, which yields a unit-cell volume of 1143.23(5) Å3. Figure 7.2.1a gives the packing diagram of this structure viewed along the [100] direction that illustrates the alternating arrangements of the (Cu/As)20 and (Cu/As)24 polyhedra, with the structure outline drawn through the average positions of As and Cu. In this figure, the Ba atoms have been omitted for clarity. Figure 7.2.1b highlights the two distinct types of polyhedra that form the cubic unit cell of Ba8Cu16As30: the dodecahedron (20-atom polyhedra with 12 pentagonal faces, (Cu/As)20) and the tetrakeidecahedron (24- atom cage with 12 pentagonal and 2 hexagonal faces, (Cu/As)24). Each unit cell contains two (Cu/As)20 dodecahedra and six (Cu/As)24 tetrakaidecahedra. The 6c site is only present in the (Cu/As)24 dodecahedra. The dodecahedra can be thought of as linked via the interstitial 6c positions. The framework structure is built from the tetrahedral network of As and Cu atoms. The As and Cu atoms are essentially adjacent to each other. In the figure, the bonds are drawn through the average (Cu/As) positions. Copper substitutes for As in the framework on all three crystallographic sites. On the 24k site, the Cu/As ratio is

Figure 7.2.1 (a) A projected view of clathrate-I Ba8Cu16As30 along the 100 direction. The positions of the Ba atoms inside the (As/Cu)20 and (As/Cu)24 polyhedra are shown in (b). The 6c, 16i and 24k framework sites are labeled. Figure is taken from Ref.274

102

Table 7.3 Formation energy ∆E, lattice constants (Å) and averaged values of atomic distances (Å) calculated via different approximations and their experimental referral values.

Calculated values Ba8Cu16As30 As46 -4.037 (C1, GGA) Formation energy ∆E, -4.059 (C2, GGA) -4.119 (GGA) [eV/atom] -4.158 (C3, GGA) -4.128 (GGA)* -4.176 (C3, GGA)* 10.766 (C1, GGA) 10.871 (C2, GGA) 11.076 (LDA) 10.362 (C3, LDA) Lattice constant a, [Å] 11.383 (GGA) 10.659 (C3, GGA) 11.372 (GGA)* 10.682 (C3, GGA)* 10.456 (Exp.) 2.439 (LDA) 2.493 (GGA) As-As, [Å] 2.449 (GGA)* 2.428 (Exp.) 2.387 (LDA) 2.455 (GGA) 8 bonds/cell As-Cu, [Å] 2.422 (GGA)* As(16i)-As(16i): 2.732 (LDA) 2.425 (Exp.) 2.909 (GGA) 2.904 (GGA)* 2.529 (LDA)

2.680 (GGA) Cu-Cu, [Å] 6 bonds/cell 2.503 (GGA)* As(24k)-As(24k): 2.587 (LDA) 2.426 (Exp.) 2.666 (GGA) 3.396 (LDA) 2.658 (GGA)* 3.487 (GGA) Ba1-As, [Å] 3.438 (GGA)* 36 bonds/cell 3.412 (Exp.) As(6c)-As(24k): 2.527 (LDA) 3.683 (LDA) 2.628 (GGA) 3.770 (GGA) Ba2-As, [Å] 2.622 (GGA)* 3.780 (GGA)*

3.733 (Exp.) 48 bonds/cell 3.351 (LDA) As(16i)-As(24k): 2.415 (LDA) 3.436 (GGA) Ba1-Cu, [Å] 2.446 (GGA) 3.444 (GGA)* 2.449 (GGA)* 3.424 (Exp.) 3.738 (LDA) 3.856 (GGA) Ba2-Cu, [Å] 3.759 (GGA)* 3.728 (Exp.)

approximately 28/72, which is a smaller ratio than that of the 16i site (35/65); however, Cu is the major element on the 6c site with a Cu/As ratio of 60/40. The 6c crystallographic site is located on the hexagonal face of the larger tetrakaidecahedra (Figure 7.2.1b) where As and Cu co-occupy the same position without splitting. The stoichiometry of the compound was refined to be Ba8Cu15.9(2)As30.1(2), which is essentially Ba8Cu16As30. The preferential 6c site occupancy for Cu in Ba8Cu16As30 agrees well with previous reports on other clathrate-I compositions with framework-substitutions, such as

103

275 276 277 278 279 Ba8Cd8Ge38 , Cs8Cd4Sn42 , Ba8Mn2Ge44 , and Cs8Zn4Sn42 , and Cs8Zn18As28 , thus, 6c site selection may not be size dependent with respect to the substituents.

From experiment, it is found that the guest atoms Ba1, inside (Cu/As)20, resides at 2a (0, 0, 0), while Ba2 is disordered inside (Cu/As)24; instead of having the coordinate (0, ¼, ½) (at the 6d site) it is at

(0, 0.2548(6), 0.4820(2)) with the site occupancy of ¼ (the 24k site). In Figure 7.2.1b, the “disordered”, or split, (Cu/As) sites that form the framework are shown individually. In addition the synchrotron X-ray refinements support the disorder of Ba2 in the tetrakaidecahedra. This inference is confirmed by the computational results, as will be described in the next section.

To fully investigate the structural features of Ba8Cu16As30, as well as compare them with that of elemental clathrate-I As46, the structural and electronic properties of Ba8Cu16As30 were investigated using

DFT calculations by considering different unit-cell configurations with respect to the location of the Cu atoms. There are over 1011 possible arrangements for Cu and As in the clathrate-I framework. Here, the representative configurations are considered, in which three types of distinct Cu positions are taken. In the first system (C1), Cu is located at 16i resulting in eight direct Cu-Cu bonds in the framework. In the second (C2), there are four direct Cu-Cu bonds in the framework and in the third (C3) there are two direct

Cu-Cu. In all three configurations, the characteristic Cu/As ratios are Cu3/As3=0.5/0.5 (6c);

Cu4/As4=0.375/0.625 (16i) and Cu5/As5=0.29/0.71 (24k). Note, that these ratios are very close to those from the experimental data, however, an exact match is impossible at this stage due to the limitation of computational resources.

The stability of the structure was determined by calculating the formation energy 퐸∆ = 퐸푇 −

푁 ∑푖=1 퐸푖, where 퐸푇 is the total energy and 퐸푖 is the energy of the different constituent atoms in the unit cell

(푁 in total). The calculated formation energies (results via GGA only) and lattices (results via GGA and

LDA) are given in Table 7.3. As shown in this table, C3 is the most stable configuration with a formation energy lower by 0.12 and 0.10 eV/atom as compared to C1 and C2, respectively. Note, that the SOC is an important factor in the energy stability, such that 퐸∆ is further lowered by 0.02 eV/atom and 0.01 eV/atom

104 for Ba8Cu16As30 (C3) and As46, respectively. Essentially, the smallest number of direct Cu-Cu bonds diminishes the distortion in the cages, which is favourable for the structural and energetic stability of the material. Similar trends have been found in mixed cage Ga-Ge clathrate-I and Ga-Sn clathrate-II

76,266 compositions . In addition, 퐸∆ for As46 was calculated to be higher by 0.04 eV/atom in comparison to

C3, which indicates a decreased energetic stability of the elemental clathrate as compared to Ba8Cu16As30.

The computed lattice constant for C3 is in good agreement with experiment (Table 7.3), with the best agreement achieved for the DFT-LDA calculations. It is observed that the lattice constant for As46 is

~7% larger than that of Ba8Cu16As30, which is attributed to the large As(16i)-As(16i) bond, as shown in

Table 7.3. It is interesting to note that at these particular crystallographic positions the As atoms have unbound electron pairs each positioned directly across from each other, as discussed later in terms of electron localization. A summary of the structural parameters for the most stable Ba8Cu16As30 case are given in Table 7.3, where the average atomic distances calculated under different DFT approximations are shown. The atomic distances obtained via LDA are generally smaller than those obtained from GGA, however, the SOC reduces the GGA bond lengths. For example, the Cu-Cu average distance using GGA is larger than the experimental value by more than 10%, while the inclusion of SOC reduces this difference to about 3%. Note, that the GGA-SOC bond lengths agree with most of the experimental results, except for the LDA As-As and Ba2-Cu bonds which compare best with the experimental values.

Nevertheless, in all cases the difference between computation and experiment is ≲ 3%. It is further noted that some of the guest atoms are displaced from the centers of their corresponding cages. Specifically, the calculations indicate that Ba2 in (Cu/As)24 is off center by 0.27 Å. This is in good agreement with the experimental value of 0.2 Å, considering the difference between the experimental and theoretical Cu/As ratios. The average atomic bonds for the empty-cage clathrate As46 are also shown in Table 7.3. The

As(16i)-As(16i), As(24k)-As(24k) and As(6c)-As(16i) distances are larger on average by 0.39Å, 0.18Å and 0.13Å, respectively, than the analogous distances for Ba8Cu16As30.

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7.2.2 Electronic structure

The Cu-As bonds are in tetrahedral coordination so that each atom forming the cages (Cu and As)

requires four electrons for the network. There are, therefore, 184 electrons per formula unit. Given that Ba

provides two electrons to the network, and that Cu and As are expected to provide 1 and 5 valence

electrons, respectively, the total number of electrons per unit formula is 182. This charge imbalance

indicates that Ba8Cu16As30 should have p-type metallic behaviour. The DFT simulations confirm this

simple argument based on the Zintl rule; the band structure of Ba8Cu16As30 obtained via DFT-GGA is

indeed metallic as shown in Figure 7.2.2a. For comparison, the band structure for the empty-cage As46 is

also displayed in Figure 7.2.2c, and is similar to that reported in a thesis280. There are several crossings at

EF for both materials from conduction and valence bands. Including SOC causes the bands to shift and

removes some of the degeneracy in the band structure for both materials, as shown in Figures 7.2.2b and

Figure 7.2.2 Electronic structure for Ba8Cu16As30 calculated via (a) GGA and (b) GGA+SOC, and As46 calculated via (c) GGA and (d) GGA+SOC. Fermi surface in the Brillouin zone obtained via GGA with SOC accounted for (e) Ba8Cu16As30 and (f) As46. The blue and green regions correspond to contributions from the valence and conduction bands, respectively, crossing the Fermi level. Figure is taken from Ref.274

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To further understand the electronic behavior of these clathrates, their Fermi surfaces were calculated and are shown in Figures 7.2.2e and f. For Ba8Cu16As30, there are well-defined pockets around the R-points in the Brillouin zone that originate from the conduction bands crossing EF, while the pockets around the X-points are due to the valence bands crossing EF (Figure 7.2.2e). Typically, such pockets are indicative of a semi-metallic-like behavior. The Fermi surface for As46 has pipe-like features centered at the Г point with closed spherical surfaces at the R point, the origin of which is from the valence bands

(blue regions in Figure 7.2.2f). The hollow toroidal surfaces with openings at the R-points represent the conduction band contribution (green regions in Figure 7.2.2f). The Fermi surface in Figure 7.2.2f indicates that As46 is a stronger metal as compared to Ba8Cu16As30.

The role of SOC can also be investigated by comparing the total Density of States (DOS) for the two systems. Specifically, the sharp peak at the Fermi level of As46 shifts towards the valence region upon including SOC, as evident from Figure 7.2.2c and d. Such an effect, although more pronounced, is also observed in Ba8Cu16As30 (Figures 7.2.2a and b). One can conclude that this relativistic correction results in reduced DOS at EF, indicating weaker metallic behavior, especially in Ba8Cu16As30, when compared with the results obtained without including SOC in the calculations. The projected density of states

(PDOS) given in Figure 7.2.3 shows the character of the contributing orbitals in different energy regions.

The E > - 6 eV region is mainly comprised of Cu-d and As-p states (Figures 7.2.3a and b) whereas below this region there is a gap of about 1.5 eV followed by Ba-p and As-s states occupying the E < - 9.5 eV range. The PDOS for As46 has only As-p states for E > -5.5 eV and As-s states for E < -9.5 eV, as shown in Figures 7.2.3c and d.

Further, the chemical bonding in these materials by calculating the ELF is investigated. The results for Ba8Cu16As30 are shown in Figures 7.2.4a and b. In these figures the yellow regions between the

As atoms, separated by 2.449 Å, correspond to shared electrons, which is a signature of covalent bonding.

The Cu-As bonds (2.422 Å), on the other hand, can be characterized as polar-covalent. The red regions shift towards As indicating electron depletion of Cu. It is found that the Cu atoms shift towards the As

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Figure 7.2.3 Projected density of states for Ba8Cu16As30 calculated via (a) GGA and (b) GGA+SOC, and As46 calculated via (c) GGA and (d) GGA+SOC. Figure is taken from Ref.274 sites, which is explained by As having larger electronegativity thus leading to attraction of the less electronegative Cu. The concentric ELF rings around Ba and the shared ellipsoid regions between Ba and

Cu (Figures 7.2.4a and b) indicate that there may be donation of electrons from Ba to Cu. The calculated

ELF for As46 shows electronic localization, which is the evidence of the metallic character of the tetrahedrally bonded As atoms in the clathrate framework (Figures 7.2.4c and d). Interestingly, all As atoms at the 16i site have lobe-like regions facing each other, which is a signature of a lone-pair electron distribution (Figure 7.2.4c).

Investigating charge transfer is also important in understanding the chemical bonding in

Ba8Cu16As30. The charge density difference ∆휌Cu−As between Cu and As atoms making up the cages and the charge density difference ∆휌Ba−CuAs between Ba and the Cu-As network is calculated. Specifically,

∆휌Cu−As = 휌Cu16As30 − (휌Cu16 + 휌As30), where 휌Cu16As30 is the charge density of a Cu-As network, and

∆휌Ba−CuAs = 휌Ba8Cu16As30 − (휌Ba8 + 휌Cu16As30 ), where 휌Ba8Cu16As30 is the charge density of

Ba8Cu16As30 and 휌Ba8 is the charge density of Ba atoms.

The obtained results are given in Figure 7.2.4 where the turquoise isosurfaces around the Cu atoms indicate that electrons are being transferred ionically to the regions localized between Cu and As

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Figure 7.2.4 ELF calculated for (a, b) Ba8Cu16As30 and (c, d) As46 on different planes. The distances d from the origin for each plane are also shown in the figures. The ELF isosurface level is 0.870. (e, f) Charge transfer between As (green) and Cu (blue) on the CuAs framework. Ba is not shown. (g, h) Charge transfer between Ba (emerald) and the CuAs framework. The electron accumulation and depletion are shown as yellow and turquoise isosurfaces, respectively, with an isosurface value of 0.006 e/Bohr3 for all cases. Figure is taken from Ref.274 atoms (Figures 7.2.4e and f). These locations (yellow isosurfaces) are slightly shifted towards the As sites, further indicating a polar-covalent character of bonding between Cu and As (see also Figures 7.2.4e and f). For the charge transfer involving Ba, Figures 7.2.4g and h show small yellow and turquoise isosurface points between guest atoms and the Cu-As network indicating weak coupling. The obtained isosurfaces are much smaller than those in Figures 7.2.4e and f.

7.2.3 Summary

In this section the DFT ab-initio methods were applied for investigation of the synthesized a single-crystal cationic clathrate-I Ba8Cu16As30. The computational results confirmed the experimental observations of the structural disorder of Ba2 inside the larger cages. Spin-orbit coupling was needed in

109 order to obtain better agreement with the experimental results. This spin-orbit correction also results in band shifts and partial removal of degeneracy, leading to weaker metallic behavior when compared to the results obtained without taking it into account. The type of bonding within the framework and between the guest and framework of Ba8Cu16As30 is relatively complex, as revealed by the electron localization and charge transfer calculations.

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8 QUATERNARY CHALCOGENIDES

Quaternary chalcogenides considered in the current chapter have 퐼4̅2푚 stannite (ST) or 퐼4̅ kesterite (KS) phases. The similarity between these phases sometimes causes difficulties in proper determination of crystal symmetry using standard XRD approach281. The difference is in the atomic arrangements between metal atoms at specific crystallographic sites. The structures are formed from 2D anion-cation layers having insulating and conducting behavior. The Figure 8.1.1 Unit cell representation of Cu2CdSnTe4. The Cu atoms are represented by red circles at the 4d possibility to substitute anions and cations for crystallographic site (0,1/2,1/4), the Cd atoms are represented by gray circles at the 2a crystallographic site achieving a higher ZT motivates in synthesis of (0,0,0), the Sn atoms are represented by blue circles at the 2b crystallographic site (0,0,1/2), and the Te atoms are represented by yellow circles at 8i crystallographic site new QSs for which a deep understanding of (0.245,0.245,0.129). The dotted lines show two different metaltellurium layers composed of [Cu2Te2] and 282 electronic structure is needed. [CdSnTe2]. Figure is taken from Ref.

8.1 Cu2CdSnTe4

One such example is Cu2CdSnTe4 synthesized in ST-phase for the first time by Prof. Nolas’ group for which the DFT electronic structure simulations are performed282. The calculated lattice constants with inclusion of SOC are found in good agreement with the experimental values (see Figure

8.1.1 and Table 7.4). The DFT electronic structure analysis within the experimental measurements revealed this material as a semimetal. Temperature dependent 훼 and resistivity 휌 indicate strong hole conductivity, while the 푘 values had a temperature dependence typical of dielectric materials. This

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Table 8.1 Lattice parameters and tetragonal distortion η for Cu2CdSnTe4

Cu2CdSnTe4 a, (Å) c, (Å) η=c/2a GGA 6.2889 12.5159 0.995 GGA+SOC 6.3099 12.5812 0.997 Experiment 6.198(1) 12.256(3) 0.989

Figure 8.1.2 (a) The energy band structure along a characteristic path of the Brillouin zone; SOC is included in the calculated results. The corresponding total and site projected DOS is displayed for the same energy range as in the band structure. (b) The Brillouin zone with several characteristic points. The Fermi surface for the highest valence (purple region) and lowest conduction bands (pipe-like blue/red regions) is shown. Figure is taken from Ref.282

conductive behavior was supported by the fact that DOS at the Fermi level is non-zero as well as there is a formation of the hole pockets in conduction band (Figure 8.1.2a). Additionally the Fermi surface was calculated representing the pipe-like energy regions along which the carrier transport facilitates (Figure

8.1.2b). At room temperature, 푘 for Cu2CdSnTe4 is higher (5.5 W/m∙K) than that of Cu2CdSnSe4 (2.8

283 284 W/m∙K) and Cu2ZnSnSe4 (2.7 W/m∙K) . Although Te is heavier than Se, the reason for this may be found in the crystal structure (Figure 8.1.1). The crystal structure of the selenides are composed of

283,284 electrically conducting layers and electrically insulating layers . The [Cu2Q2] layers (with Q = S, Se or Te) are the same for all these compounds; however, only the [MSnQ2] layers (with M = Zn or Cd) differ. In Cu2CdSnSe4 and Cu2ZnSnSe4, for example, there is a relatively large mass difference between the metal atoms and Se or between metals in the [MSnSe2] layer, thereby providing strong mass

112 fluctuation scattering of phonons. In the case of Cu2CdSnTe4, there is a relatively small mass difference between Cd or Sn and Te. Nevertheless, since 푘 ≈ 푘푒 at elevated temperatures, this material is likely to be a good candidate thermoelectric material, if its electrical properties can be optimized appropriately.

8.2 Ag2ZnSnSe4 and Cu2ZnSnSe4

285 The next focus is on Ag2ZnSnSe4 and Cu2ZnSnSe4 . This particularly family of QCs can be tuned via doping to optimize their desired properties such as optimum band gap suitable for TE applications. At the same time such class of materials has inherently low 푘 due to the complex lattice structure (for more details see Section 1.4).

Although Ag2ZnSnSe4 and Cu2ZnSnSe4 can be both in similar phases (KS or ST), a fundamental understanding of the structure-property relations due to these different structure types continues to be investigated. Since the differences between the various phases are rather small in terms of specific bond lengths and formation energies, resolving the crystal structures for a particular quaternary chalcogenide has proven to be difficult with standard methods286, thus electronic structure simulations must accompany experimental studies to understand the structure and properties of these chalcogenides. In addition, despite their structural similarity, Ag2ZnSnSe4 exhibits transport properties consistent with a polaronic- type behavior, unlike that of Cu2ZnSnSe4.

8.2.1 Polaronic transport in Ag2ZnSnSe4

Figure 8.2.1a and b show the temperature dependent 푘 and 휌 measurements, respectively, for

Ag2ZnSnSe4, Ag2.1Zn0.9SnSe4, and Ag2.3Zn0.7SnSe4. Due to the relatively high values for all specimens, the lattice contribution to 푘 is dominant. As shown in Figure 8.2.1a, 푘 decreases with higher Ag content in the entire measured temperature range. This is expected since a higher Ag concentration on the Zn site will lead to stronger alloy scattering of phonons given the fact that Ag has a much larger mass than does Zn.

This effect is minimized near room temperature, as seen in Figure 8.2.1a by the overlapping data points, as Umklapp scattering becomes the dominant phonon scattering mechanism in the thermal transport. The

113 room temperature 푘 values (5.3-5.6 W/m∙K) are about twice that for bulk Cu2ZnSnSe4 (2.7 W/m∙K) and

287,288 nanostructured Cu2ZnSnSe4 (0.9 W/m∙K) .

The results for the resistivity (Figure 8.2.1b) are highly atypical for quaternary chalcogenides of similar composition. For each material, there is an anomalous peak below room temperature which decreases in magnitude with increased Ag content. This well- localized peak-like behavior in the resistivity is very different when compared to experimental resistivity values for other quaternary chalcogenides289–291. For example, Cu2ZnSnS4, Cu2ZnSnSe4, and Cu2CdSnSe4 exhibit a decreasing 휌 with increasing temperature below Figure 8.2.1 Temperature dependent (a) 풌 and (b) room temperature, consistent with their semiconducting 𝝆 for Ag2ZnSnSe4 (up triangle), Ag2.1Zn0.9SnSe4

38,289–291 (diamond), and Ag2.3Zn0.7SnSe4 (down triangle). electronic structure . Doped quaternary The solid lines are fits using the two-component model, as described in the text. The inset of (b) is chalcogenides, such as Cu2+xCd1-xSnS4, are reported to the S of Ag2.3Zn0.7SnSe4 fitted as described in Ref. 38 with an ~0.9 ratio of the number of carriers to have a broad maximum in their conductivities at higher the number of sites. Figure is taken from Ref.285 temperatures that is attributed to the materials becoming degenerate semiconductors283. Chalcopyrites, such as CuInSe2, may exhibit similar transport behavior, however, the reported minimum in the conductivity vs. temperature measurements is much broader and appears at higher temperatures292. This temperature dependence can be attributed to the contribution of both p-type and n-type carriers to the conduction process in these materials. Room temperature 휌 values are 13, 12, and 0.4 Ω∙cm for

Ag2ZnSnSe4, Ag2.1Zn0.9SnSe4, and Ag2.3Zn0.7SnSe4, respectively. Furthermore, in the case of Ag2+xZn1- xSnSe4 the 휌 peak is not associated with a phase transition, as confirmed by temperature dependent XRD results that showed no phase change before and after the peak temperature. This type of anomaly in the resistivity has been associated with polaronic type conduction293.

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The formation of polarons is based on trapping of carriers due to strong carrier-phonon coupling.

At low temperature these can form a conductive band with a heavy mass, according to Holstein’s model294. As temperature increases this band becomes disrupted as the overlap between the neighboring phonon clouds decreases. At high temperatures inelastic emission and absorption of phonons dominate and polarons become localized. While the low temperature regime corresponds to an itinerant polaron band, the high temperature regime is characterized by semiconducting behavior due to polaron hopping via thermally activated inelastic processes. The competition between these two regimes results in a peak in ρ vs temperature293,295. This type of transport can be described by a model taking into account the coexistence of the localized and itinerant polarons expressed as

−1 푓(푇) 1−푓(푇) 1 휌(푇) = [ + ] , 푓(푇) = 푘 (푇−푇 )/Δ (8.2.1) 휌푖푡(푇) 휌ℎ표푝(푇) 푒 퐵 0 +1

where 휌푖푡(푇) is the resistivity for the itinerant mechanism and 휌ℎ표푝(푇) is the one for the thermally activated hopping. These are given according to

퐸푎/푘퐵푇 2 휌ℎ표푝(푇) = 퐴ℎ푇푒 , 휌푖푡(푇) = 휌0 + 퐵푖푡푇 , (8.2.2)

where 퐴ℎ, 퐵푖푡, 휌0, 퐸푎, and ∆ are constants to be determined from experimental data and 푇0 is the temperature where the resistive peak appears.

To better understand the influence structure has on the transport properties of these materials, the first principles calculations are performed in order to determine the characteristic properties of the underlined physical and electronic structure for both the kesterite and stannite crystal structures. For this purpose, a comparative study for Ag2ZnSnSe4 and Cu2ZnSnSe4 is presented. These two quaternary chalcogenides are compositionally and structurally similar, as it will be discuss below. The crystal lattice of Cu2ZnSnSe4 and Ag2ZnSnSe4, for both the stannite (퐼4̅2푚) or kesterite (퐼4̅) crystal structures, are only a few meV different in energy, as has been previously reported296 and are corroborated by the DFT calculations below.

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Note, that standard experimental XRD methods are not able to resolve the two structures286, therefore the investigation of their crystal structures rely on the DFT calculations for this particular compositions. However, standard DFT calculations using local density approximation (LDA) or generalized gradient approximations (GGA) do not correctly capture the electronic structure. DFT-LDA or GGA are not precisely accurate in situations where the energy band gap results from d-states and p- states hybridization, which is the case for these quaternary chalcogenides. Furthermore, DFT within LDA or GGA does not describe adequately the interatomic separations, or bond lengths, which also leads to significantly underestimated energy gaps297,298. The hybrid functional HSE06 in VASP solves these problems as the calculated energy gaps are in the same range as that obtained by experiment, as described in what follows.

Here the ab-initio simulations for the kesterite (KS) and stannite (ST) phases for both

Ag2ZnSnSe4 and Cu2ZnSnSe4 are performed. The main goal is to understand the microscopic description of the structural and electronic properties of these systems which show markedly different transport;

291 polaronic-like behavior for Ag2ZnSnSe4 and typical semiconducting resistivity for Cu2ZnSnSe4 . Note, that due to the large computational cost involved in the simulations with the HSE06 functional, calculations for the doped specimens were not feasible at this time.

8.2.2 Structural properties

The conventional cells for both phases of Ag2ZnSnSe4 and Cu2ZnSnSe4 materials are shown in

Figure 8.2.2 with the interatomic distances obtained from the DFT calculations also denoted. The primitive unit cell of the KS phase is characterized by the 퐼4̅ symmetry with Wyckoff positions described as 2a (0, 0, 0) and 2c (0, 0.5, 0.25) for the Ag(Cu) atoms and 2b (0.5,0.5,0), 2d (0.5,0,0.25) and 8g (0.756,

0.756, 0.872) for the Zn, Sn and Se atoms, respectively. For the ST phase, the symmetry is 퐼4̅2푚 with

Wyckoff position 4d (0, 0.5, 0.25) for the Ag(Cu) and 2a (0,0,0), 2b (0,0,0.5) and 8i (0.238, 0.238, 0.129) for the Zn, Sn and Se atoms, respectively. The structural unit cell characteristics for both phases can be described by two Cu(Ag)2Se4 and SnZnSe4 tetrahedral slabs with conducting and semiconducting

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Figure 8.2.2 (a) Stannite and (b) kesterite structures of Ag2ZnSnSe4; (c) stannite and (d) kesterite structures of Cu2ZnSnSe4. The denoted interatomic distances are obtained from the DFT-HSE calculations. Figure is taken from Ref.285 properties, respectively. The lattice constants obtained from the calculations and XRD are summarized in

Table 8.1. The DFT calculations show that the KS structure is more stable by 0.02 eV as compared to ST

Ag2ZnSnSe4, while KS Cu2ZnSnSe4 is more stable by 0.01 eV as compared to its ST counterpart. It is further noted that the a, b, and c lattice parameters obtained from XRD of the synthesized specimens are in better agreement with the lattice parameters corresponding to the KS structure for both materials, thus one may conclude that this is the preferred phase for both compositions.

There is a significant change between a = b for the KS and ST Ag2ZnSnSe4 with largely varying c/a ratio; however, the lattice parameters change to a much lesser degree when comparing KS and ST

Cu2ZnSnSe4 with the c/a ratio being almost constant. The remarkable increase along a = b for the KS

Ag2ZnSnSe4 as compared to that for the ST phase can be explained by considering the ionic radii of the various atoms. Specifically, comparing the Zn-Se (Figure 8.2.2a) and Ag-Se (Figure 8.2.2b) atomic distances for the ST and KS phases, which directly relate the a = b lattice parameters. The smaller ionic radius of Zn (~0.60 Å) together with the ionic radius of Se (~1.98 Å) results is a Zn-Se bond length of

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Table 8.2 Lattice structure parameters (Å) for both phases of the Ag2ZnSnSe4 and Cu2ZnSnSe4 materials obtained from the simulations.

Ag2ZnSnSe4 Cu2ZnSnSe4

ST a=b=5.72 c=12.10; c/a≈2.12 a=b=5.58; c=11.21; c/a≈2.01

KS a=b=6.09; c=11.70; c/a≈1.92 a=b=5.77; c=11.51; c/a≈1.99

Experimental Results a=b=5.99 c=11.45; c/a≈1.91 a=b=5.69 c=11.34; c/a≈1.99

2.44Å, while the larger ionic radius of Ag (~1.02 Å) yields a bigger Ag-Se bond length of 2.65 Å.

Consequently, the ab plane for the KS phase (Figure 8.2.2b) becomes expanded as compared to the ab plane of the ST phase (Figure 8.2.2a). The overall comparison of Ag2ZnSnSe4 and Cu2ZnSnSe4 in both phases shows that the role of Ag is to “expand”, although non uniformly, the structure along all directions

(also reflected in the larger lattice constants, as discussed earlier). The a=b value for ST Ag2ZnSnSe4 increases by ~2.5% as compared to a=b for ST Cu2ZnSnSe4, while the relative increase in c is ~7.9%. As a result of this anisotropic stretching, the Se-Se distance between two adjacent layers (along c-direction) is found to be 4.23 Å in Ag2ZnSnSe4, while the corresponding Se-Se distance is 3.80Å for Cu2ZnSnSe4

(Figures 8.2.2a, c). A similar overall expansion of the Ag2ZnSnSe4 unit cell with respect to the

Cu2ZnSnSe4 unit cell is also found for the KS structure. In this case, however, the expansion along a=b is

5.5% but only 1.6% along the c direction, as shown in Table 8.1 and Figures 8.2.2b and d. Other characteristic interatomic distances also increase in Ag2ZnSnSe4 as compared to Cu2ZnSnSe4.

8.2.3 Electronic properties

The calculated electronic structure is presented in Figure 8.2.3 with the orbital projected density of states (DOS) for both materials also shown. Both materials are semiconductors with energy gaps 퐸푔, at the Г-point and 퐸푔 = 1.43 eV for KS Ag2ZnSnSe4 and 퐸푔 = 1.16 eV for ST Ag2ZnSnSe4. Experimentally, the obtained values for 퐸푔 are 1.28 eV, 1.25 eV, and 1.23 eV for Ag2ZnSnSe4, Ag2.1Zn0.9SnSe4, and

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Ag2.3Zn0.7SnSe4, respectively. Since previous reports

82,223,299– confirm Cu2ZnSnSe4 and related materials to be KS

301 , the emphasis here is on Ag2ZnSnSe4. It is important to note that the 퐸푔 values are not correctly reproduced by

DFT calculations utilizing the PBE-PAW potentials297,299,300,302. Taking into account the hybrid functional HSE06, which describes properly the hybridization between p-d states, is necessary to obtain the underlying electronic structure of these systems. The total

DOS for both ST materials (Figure 8.2.3) has a similar appearance around the Fermi level 퐸퐹 with a localized peak at the edge of the valence region and another broader peak at the lowest conduction region. The total DOS for the KS

Ag2ZnSnSe4 shows a similar characteristic behavior, but the highest valence and the lowest conduction bands are Figure 8.2.3 (a) Total DOS for Ag2ZnSnSe4 and Cu2ZnSnSe4 in both structure types, obtained via shifted towards lower energies by eV ~0.36 and ~0.14 eV DFT-HSE hybrid functional. Orbital projected DOS for the individual atoms for (b) KS respectively. Figures 8.2.3b, c, and d show that the Se p- Ag2ZnSnSe4, (c) ST Ag2ZnSnSe4, (d) KS Cu2ZnSnSe4, and (e) ST Cu2ZnSnSe4. Figure is taken from Ref.285 states and metallic atoms d-states (Ag and Cu) contribute significantly right below 퐸퐹; however, the Se contribution is much more prominent in Ag2ZnSnSe4 for both phases. The conduction peak in the 1-2.5 eV region is due to hybridization between the Se p-states and Sn s-states.

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Figure 8.2.4 The calculated ELF with isosurface value of 0.82 along the planes indicated, where (a), (b), (c) and (d) 285 are for KS Ag2ZnSnSe4 and (e), (f), (g), and (h) are for KS Cu2ZnSnSe4. Figure is taken from Ref.

The distinct resistivity measured for Ag2ZnSnSe4 as compared to other structurally similar quaternary chalcogenides is therefore believed to be strongly related to the local changes in the crystal lattice induced by Ag. To obtain a better understanding of these Ag-induced modifications, the ELF plots along various directions for both materials is also shown in Figure 8.2.4. Specifically, Figures 8.2.4a, b, e, and f display similar atomic coordination planes with characteristic Se-Se distances for the KS phases.

Figures 8.2.4c, d, g, and h further show the ELF for Zn-Se; Sn-Se, Ag-Se and Cu-Se display similar ELF features for Ag2ZnSnSe4 and Cu2ZnSnSe4. The Zn-Se bonds are purely ionic for both materials (Figure 8.2.4c, g), while the Sn-Se bonds can be characterized as semi-ionic due to some electron sharing, as shown in Figure 8.2.4d and h. This can be further explained by relating the atom-atom distance with the respective ionic radii. The Sn-Se distance in both materials is ~2.63 Å, less than the sum of the Sn and Se ionic radii (~3.16 Å), thus allowing for some characteristic electron sharing for covalent bonding. The Ag-Se bonding is less localized since some distorted electronic distribution still remains on the Ag atom (Figure 8.2.4c, d), while the iconicity is more pronounced for the Cu-Se bonds (Figure

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8.2.4g, h). This can be explained by the larger Ag which is further apart from the Se atom than would be the Cu atom.

8.2.4 Summary

In this chapter the transport properties of three synthesized Ag2+xZn1-xSnSe4 compositions are investigated. The two-component model was employed and indicates strong electron-phonon coupling in these materials, as corroborated by the DFT ab-initio calculations. The crystal and electronic structures of

Ag2ZnSnSe4 and Cu2ZnSnSe4 were calculated and compared in order to understand how the lattice structure and electronic properties evolve for these materials. These calculations where done for both the kesterite and stannite crystal structures. The obtained results suggest that by replacing Cu with Ag, the bonding results in anisotropic stretching which may encourage stronger interactions between electrons and phonons in Ag2ZnSnSe4. Further computational efforts will be needed to provide even more details on the formation of polarons which will involve simulations of lattice distortions coupled with other modification. Further computational efforts, involving lattice distortions coupled with electron carrier modifications and lattice vibrations, are in progress to provide more details on the formation of polarons and their role on the transport.

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9 CONCLUSION AND FUTURE OUTLOOK

In this work I have utilized first principles simulations towards the understanding of fundamental properties, such as atomic, electronic and vibrational characteristics of several types of materials suitable for thermoelectric applications. Many of the results are presented in the context of experimental measurements for better analysis at the microscopic level. The results from this research also suggest pathways for transport property optimization for thermoelectric applications.

My initial interests were devoted to Bi rich Bi1-xSbx alloys. This class of material is well-known for low temperature thermoelectric applications, and the primary purpose was to test the applied computational methods followed by further comparison with experiment. Surprisingly, for low Sb concentrations, there were only a few experimental results available and no computational studies. The obtained DFT results show that low concentration of Sb anomalously affects the atomic and electronic structures with further reflection on transport properties. Additionally, the simulations show that inclusion of SOC in calculations for the structures with heavy elements has a principle importance.

The next focus was on bournonite materials. The detailed DFT investigation, for the first time, revealed that the presence of lone pair s2 electrons is directly correlated with the inherently low thermal conductivity of these systems. The doped bournonites were also considered, in order to investigate how impurity affects the material properties at the microscopic level. For this study the detailed analysis of the charge transfer was additionally employed in order to track all changes in chemical bonding upon the applied doping. The obtained results in comparison with experiment show that doping with similar metallic atoms affects the thermal conductivity differently and it may change the type of electrical conductivity.

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My next interest was motivated by materials possessing cage-like structures, such as inorganic clathrates with a type II framework. The DFT simulations are applied to several Sn based clathrates, including empty cage Sn136, compositions with partial Ga substitution on the framework, and compounds of Sn136 filled with inert Xe atoms. It is found that cage disorder due to atomic substitution and guest encapsulation affects the fundamental characteristics of these materials in profound ways. It is also determined that the stability of the materials is enhanced by the presence of guests and lack of direct Ga-

Ga bonds in disordered clathrates. Inert Xe atoms provide a unique opportunity to preserve the overall electronic structure of Sn136 and take advantage of the loosely bound guest rattling for enhanced phonon scattering. The calculated energy bands and density of states as well as phonon band structure and mode

Gruneisen parameter enable further analysis of type II Sn clathrates, which shows interesting structure- property relations. In addition to the above family of Sn clathrates, type I arsenic based clathrate was also investigated. Single crystals of clathrate-I Ba8Cu16As30 have been synthesized, and their structure and electronic properties determined using synchrotron-based X-ray diffraction and first principles calculations. Agreement between the experimental and theoretically predicted structures was achieved after accounting for spin-orbit coupling. The calculated Fermi surface, electron localization and charge transfer, as well as a comparison with the results for elemental As46, provided insight into the fundamental properties of this cationic-framework clathrate-I material.

Research efforts are also devoted to the large class of I2-II-IV-VI4 quaternary chalcogenides. The DFT calculations on Ag2ZnSnSe4 for both energetically similar kesterite and stannite structure types are performed in order to compare the results to those of the compositionally similar but well known

Cu2ZnSnSe4. This theoretical comparison is crucial in understanding the type of bonding that results in polaronic type transport for Ag2ZnSnSe4, as well as the structural and electronic properties of the crystal structure types for this class of systems.

In the process of studying quaternary chalcogenides, I have learned that entire classes of stoichiometric multi-component systems can be generated as a result of atomic substitutions accompanied by lattice

123 mutations, in which the overall valence state is preserved and the charge neutrality of a periodic crystal is maintained. Within this approach the family of I2-II-IV-VI4 materials can be obtained starting with a binary II-VI system. First, one can obtain ternary I-III-VI2 compounds by mutating two group II atoms into I and III group of atoms. From there, replacing two group III atoms by one group II and one group IV atoms, one obtains I2-II-IV-VI4. There can be a second round of substitution replacing one group I and one group III atoms at the ternary stage by two groups II atoms resulting in I-II2-III-VI4 materials. This is a much less explored class of materials, which is also composed of naturally abundant constituents. A systematic future investigation of synthetic techniques, property measurements, and first principles calculations of the electronic, phonon, and transport characteristics of this class of materials is important for building basic science of the ever expanding materials library. In addition, in depth simulations of I-

II2-III-VI4 materials in the context of much more explored I2-II-IV-VI4 family will be extremely beneficial to find additional ways of property/structure control for targeted applications.

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APPENDICES

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Appendix 1

Bond Valence calculation

The bond valence sums and Madelung site potentials were calculated with the program

303 EUTAX with the results listed in Table A1. The default values of Rij (Pb-S = 2.55; Sb-S =

2.47; Cu-S = 1.86) for the cation–anion single bonds were used to calculate the bond valence sums. The Madelung potentials were determined by assigning ionic charges for all atoms (Pb2+,

Cu+, Sb3+, S2-). The calculated valence sums for an atom in purely ionic compounds are expected to be close to their valence and typically calculated site potentials are approximately –10 times the formal charge of the ion.2 The calculated Madelung site potentials listed in Table A1 indicate that the bond character of the Pb and Cu-sulfur bonds are close to ionic and Sb-S is more

2+ + 3+ 2- covalent, in nature. These charge valences can also be described as [Pb ][Cu ][Sb ][S ]3, the bond valence sum, as expected for a closed shell diamagnetic semiconductor.

Table A1 Calculated bond valence sums and Madelung site potentials for PbCuSbS3

Atoms Input charge Bond valence sum Madelung site potential (V) Pb1 +2 2.29 -17.6 Pb2 +2 1.74 -15.6 Cu +1 1.16 -13.8 Sb1 +3 3.07 -24.8 Sb2 +3 3.34 -24.2 S1 -2 2.09 17.0 S2 -2 2.26 17.1 S3 -2 2.07 17.2 S4 -2 2.13 18.7

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Appendix 2

Figure A2.1 The two characteristic polyhedra of Cs8Ba16Ga40Sn96 with Ga distribution where direct Ga-Ga bond are formed ((ii) case). The guest Cs (red) and Ba (blue) atoms are shifted from the center of each polyhedron 76 towards the Ga sites by the distances d1=0.473 Å and d2=0.648 Å, respectively. Figure is taken from Ref.

The location of the guest atoms with respect to the equilibrium center of each polyhedron is affected by the Ga distribution on the framework. It is determined that direct Ga-Ga bonds result in Ga clustering on the framework. In such cases, the guest atoms become displaced from the center of each polyhedron, as shown in Figure A2.1 for the (ii) case of Cs8Ba16Ga40Sn96

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Figure A2.2 Density of States for (a) Sn136, (b) Xe8Sn136, (c) Si136 and (d) K2Ba14Ga30Sn106 with (ii) structural arrangement. The partial DOS for each atom is also shown. Figure is taken from Ref.76

Results for the DOS in a larger energy region for Sn136, Xe8Sn136, Si136 and K2Ba14Ga30Sn106 (ii) are shown in Figure A2.2. DOS of Sn136 shows that this material is a semiconductor with an energy gap of 0.33 eV. Inclusion of the noble gas into the Sn network leads to a larger energy gap with value of 0.49 eV. The hybridization between Xe and Sn atoms is seen in the deeper valence band with a peak-like feature at -4 eV. DOS for Si136 shows this clathrate is a semiconductor with 1.36 eV energy gap. DOS of K2Ba14Ga40Sn96 (ii) shows that there is no gap at the Fermi level. Hybridization between Ga, Ba, and Sn states determines the conduction peak, while hybridization between Ga and Sn is responsible for the steep DOS in the highest conduction region.

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Figure A2.3. Electron Localization Function along the [111] plane for Cs8Na16Si136. Some guest and cage atoms 76 are depicted explicitly. The ELF scale is also given. Figure is taken from Ref. .

Results for calculated ELF along the [111] plane for Cs8Na16Si136 are shown in Figure A2.3. The covalent Si-Si bonding is similar as the one for Sn clathrates. The Cs and Na guests are ionically bonded with the Si framework. The evidence of the electron donation from Na+1 to the Si framework is seen as deep blue regions at the Na locations. The blue shared regions between Cs atom and neighboring Si atoms also indicate the full electron transfer from Cs to the Si framework. Such type of chemical bonding is also similar to Sn type II clathrates with alkaline and earth-alkaline guests

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Figure A2.4 Mode Grüneisen parameter (top panel) and vDOS (bottom panel) for 76 Cs8Ba16Ga40Sn96 in a wider frequency region. Figure is taken from Ref.

. The mode Grüneisen parameter for Cs8Ba16Ga40Sn96 in a wider frequency region, given in Figure

A2.4, shows the localized negative peak at 휔~0.5 THz.

Figure A2.5 Phonon band structure for (a) Sn136, (b) Cs8Ba16Ga40Sn96 (i), (c) Xe8Sn136 and (d) 76 Xe24Sn136. Figure is taken from Ref.

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The phonon band structure for the studied materials is given in Figure A2.5. The lowest flat bands correspond to rattling of guest atoms in the large [51264] polyhedral. It appears that the range of these bands is the largest for Cs8Ba16Ga40Sn96 ((~0.3, 0.6) THz) as compared to the highly localized flat bands at ~(0.55, 0.65) THz for Xe8Sn136 and Xe24Sn136. The next section of flat optical bands (~1,2) THz corresponds to vibrations of the guests located in the smaller [512] polyhedral. It is found that in the case of Cs8Ba16Ga40Sn96, the Ba atoms interact much stronger with the cages (Figure A2.5b) as compared to the decoupled Xe atoms for the Xe-filled materials, which results in separated well localized peaks at ~1.65 and ~2 THZ in the vDOS.

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