Improving Thermoelectric Figure of Merit Through Materials Engineering: Minimizing Thermal Conductivity Via Lone Pairs and Intro

Improving Thermoelectric Figure of Merit through Materials Engineering: Minimizing Thermal Conductivity via Lone Pairs and Introducing Resonant Levels to Increase Power Factor

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Michele D Nielsen

Graduate Program in Mechanical Engineering

The Ohio State University

2014

Dissertation Committee:

Professor Joseph P. Heremans, Advisor Professor Roberto Myers Professor Walter Lempert Professor Yann Guezennec

Michele D Nielsen

2014

Abstract

Thermoelectric devices offer a lot of value to an ever growing demand on the energy market.

These devices are able to provide scalable, steady state heating and cooling when provided with power, or they can be used to recover waste heat to convert to electrical power. The efficiency of the device to perform these functions is primarily limited by the thermoelectric material properties, which are ultimately summarized by the thermoelectric figure of merit, zT.

In this work, two approaches are taken to optimize zT: the use of lone pair electrons to minimize lattice thermal conductivity, and resonant levels to increase the power factor. In the first approach, we establish low thermal conductivities, in the range of 0.4-1 W/mK, for a variety of I-

V-VI 2 compounds including a newly established extension to alkali based compounds. In a collaboration between experiment and theory, we determined the root effect of lone pairs on this class of compounds. The knowledge gained from this particular study can then be extended to other classes of compounds to determine which materials can be expected to have low thermal conductivity. In the second approach, we explore several promising systems to seek an effective resonant level, that is, one which increases the density of states in such a way as to increase the

Seebeck coefficient above the Pisarenko relation. In the process, we discover a resonant effect in

PbTe:Ti that allows for robust production methods. We also discover an effective resonant level in CoSb 3:Al that results in a two-fold increase in Seebeck coefficient over literature values at relatively high carrier concentration. Additionally, we were able to provide some insight into a material system, PbTe:Cr, that had previously been misconstrued as an effective resonant level. ii

Acknowledgements

The primary acknowledgement for the completion of my work goes to my advisor, Dr. Joseph

Heremans. His knowledge of the field and ability to pass on knowledge in a meaningful way was essential to these projects. I have learned so much while studying in his group. I would also like to thank Vladimir Jovovic and Christopher Jaworski for their help in teaching me the fundamentals early on and also my entire lab group for many wonderful scientific discussions throughout the years. Additionally, I would like to thank my family for their continual support and encouragement throughout the years.

This work was a result of a collaboration between many other groups, each of which are described here:

I-V-VI 2 Thermal Conductivity Study: This work was supported as part of the Center for

Revolutionary materials for Solid State Energy Conversion, an Energy Frontiers Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award no. DE-SC0001054. Calculations were done by Vidvuds Ozolins from UCLA using resources of National Energy Research Scientific Computing Center (NERSC) which is supported by the Office of Science of the U.S. Department of Energy under Contract no. DE-

AC02-05CH11231. Helpful discussions with D.T. Morelli and C.M. Jaworski also made this work possible. iii

PbTe:Ti study: Partial support from the joint NSF/DOE program on thermoelectrics NSF-

CBET1048622 and from the Air Force Office of Scientific Research MURI FA9550-10-1-0533.

This work was done as a collaboration with Department of Thermoelectric Systems, Fraunhofer

Institute for Physical Measurement Techniques.

PbTe:Cr Study: Sample preparation, transport and magnetic measurements at the Ohio State

University were supported by ZT: Plus, Azusa, CA, and by the joint U.S. National Science

Foundation / Department of Energy Program on Thermoelectricity, NSF-CBET-1048622. The

NMR work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences,

Division of Materials Sciences and Engineering and performed at the Ames Laboratory, which is operated for the U.S. Department of Energy by Iowa State University under Contract No DE-

AC02-07CH11358.

CoSb 3 Studies: Financial support for this investigation was provided by the U.S. Department of

Energy (DOE)-U.S.–China Clean Energy Research Center (CERC-CVC) under the Award No.

DE-PI0000012

Vita

2003 Monroeville High School 2008 B.S. Mechanical Engineering, Ohio State University 2010 M.S. Mechanical Engineering, Ohio State University 2010 to present Graduate Research Associate, Department of Mechanical Engineering, The Ohio State University Publications

Hui, S., Nielsen, M.D., Homer, M., Medlin, D.C., Tobola, J., Salvador, J., Heremans, J.P., Pipe, K., Uher, C., “Influence of substituting Sn for Sb on the thermoelectric transport properties of CoSb 3-based skutterudites” J. Appl. Phys. 115, 103704, 2014 – 2013 Impact Factor 2.210

Evola, E.G., Nielsen, M.D., Jaworski, C.M., Jin, H., Heremans, J.P., “Thermoelectric transport in Indium and Aluminum-doped Lead Selenide” J. of App. Phys., 2014, 115, 053704. – 2012 Impact Factor 2.064

Nielsen, M.D., Ozolins, V., Heremans, J.P. “Lone pair electrons minimize thermal conductivity” Energy Environ. Sci., 2013, 6, 570-578. – 2012 Impact Factor: 11.65

Jaworski, C.M., Nielsen, M.D., Wang, H., Girard, S.N., Cai, W., Porter, W.D., Kanatzidis, M.G., Heremans, J.P., “Valence-band structure of highly efficient p-type thermoelectric PbTe-PbS alloys”, Phys. Rev. B, 2013, 87, 045203 – 2012 Impact Factor: 3.767

Nielsen, M.D., Levin, E.M., Jaworski, C.M., Schmidt-Rohr, K., Heremans, J.P., “Chromium as a resonant donor impurity in PbTe” Phys. Rev. B, 2012, 85, 045210 – 2012 Impact Factor: 3.767

Konig, J.D., Nielsen, M.D., Gao,Y., Winkler, M., Jacquot, A., Bottner, H., Heremans, J.P., “Titanium forms a resonant level in the conduction band of PbTe” Phys. Rev. B, 2011, 84, 205126 – 2012 Impact Factor: 3.767

Chen, Y., Nielsen, M.D., Gao, Y.B., Zhu, T.J., Zhao, X.B., Heremans, J.P., “SnTe – AgSbTe2 thermoelectric alloys”, Advanced Energy Materials 2011, 2 58-62 - 2012 Impact Factor: 10.043

Fields of Study Major Field: Mechanical Engineering

Table of Contents

Abstract ...... ii

Acknowledgements ...... iii

Vita ...... v

List of Figures ...... viii

List of Tables ...... xi

Chapter 1: Fundamentals of Thermoelectricity ...... 1 Atomic Structure ...... 2 Band Structure ...... 8 Seebeck Coefficient ...... 11 Electrical Conductivity ...... 13 Transverse Effects ...... 17 Magnetism ...... 19 Thermal Conductivity ...... 20 Phonons ...... 22 Electrons ...... 30 Peltier Effect ...... 32 Joule Heating ...... 32 Material Efficiency ...... 33 Thermoelectric Figure of Merit ...... 33 Devices ...... 35 Applications ...... 40 Automotive Heating/Cooling ...... 41 Body Energy Harvesting ...... 42 Radioisotope TEG...... 43 Portable Remote Power Source ...... 43 vi

Sensor Applications ...... 44 State of the Art ...... 44 Thermal Conductivity ...... 46 Resonant Levels ...... 48

Chapter 2: Measurement Methods and Objectives ...... 50 Test Methodology ...... 50 Error analysis ...... 53 Analysis Methods ...... 54 Method of Four Coefficients ...... 54 Objectives ...... 56

Chapter 3: Highly Polarizable Anharmonic Crystalline Materials – I-V-VI 2 Compounds .. 58 Methods...... 61 Synthesis & measurements ...... 61 Computation ...... 63 Vibrational properties ...... 63 Experimental ...... 68 Conclusions ...... 80 Doping Studies ...... 81 Off-Stoichiometric Silver Antimony Telluride ...... 81

NaSbSe 2 ...... 91

LiSbSe 2 ...... 97

Chapter 4: Resonant Levels ...... 101 Pb Chalcogenides ...... 101 PbTe:Cr ...... 101 PbTe:Ti ...... 112

CoSb 3 ...... 119

CoSb 3:Zn ...... 122

CoSb 3:Al ...... 124

Overall Conclusions ...... 131

References ...... 133

vii

List of Figures

Figure 1. Energy loss sources 2009. 1 Quad =10 15 BTU = 293,071,083,330 kWh...... 1 Figure 2. Energy diagram for two atoms brought together from infinite separation...... 3 Figure 3. Example of 1D unit cell (left) and 2D unit cell (right) where yellow "atoms" represent the repeating unit and the blue "atoms" represent the rest of the crystal ...... 5 Figure 4. Simple cubic structure ...... 6 Figure 5. Crystallographic direction demonstrated on the left, Miller indices or Crystallographic planes shown on the right ...... 7 Figure 6. Schematic of band formation when atoms are brought into close proximity...... 9 Figure 7. Schematic representation of Seebeck Coefficient ...... 11 Figure 8. Transverse measurement setup ...... 18 Figure 9. Basics of thermal conductivity measurement ...... 21 Figure 10. 1D Chain of atoms in equilibrium (top), out of equilibrium (bottom) ...... 22 Figure 11. Comparison between Einstein (left) and Debye (right) density of states frequency distribution...... 27 Figure 12. Relation between frequency, ω, and wave vector, k, given by Einstein (blue) and Debye (green) assumptions ...... 28 Figure 13. Simple TE device configuration ...... 35 Figure 14. Thermoelectric efficiency compared to other technologies for cold temperature of 300K...... 38 Figure 15. Crossover of TEG efficiency compared to large scale technology ...... 39 Figure 16. 19 BSST TE seat heater/cooler ...... 41 Figure 17. BSST Zone thermal control ...... 42 Figure 18. Classic figure of merit ...... 45 Figure 19. State of the art material figure of merit (2006) ...... 46

Figure 20. Thermal conductivity of PbTe structure reduced by mass disorder. I-V-VI 2 compounds have even lower thermal conductivity...... 47 Figure 21. Generic Pisarenko relation ...... 48

Figure 22. Resonant impurity levels from literature, a)PbTe: Tl, b) Bi 2Te 3: Sn ...... 49

viii

Figure 23. PPMS TTO ...... 51 Figure 24. Cryostat experimental setup ...... 53 Figure 25. Process for developing and testing materials ...... 56

Figure 26. Red solid lines: Calculated phonon dispersion of rocksalt-based AgBiSe 2. Blue dashed lines: calculated phonon dispersion for a lattice constant expanded by 1.7%...... 65

Figure 27. Comparison of the calculated polarization current density in NaSbSe2 under an applied electric field, and when Se atoms are displaced by phonons...... 68 Figure 28. X-ray diffraction data for Na and Ag based compounds The dots are reference angles from the International Center for Diffraction Data PDF database...... 69 Figure 29. Isobaric specific heat C of the compounds indicated; the trigonal (Tri) and rocksalt (R- S) versions of AgBiSe 2 have the same C within the measurement uncertainties...... 70 Figure 30. Typical temperature-dependent x-ray spectra of measured cubic compounds. This measurement is of Ag 0.366 Sb 0.558 Te ...... 73 Figure 31. Temperature dependent lattice parameter was used to calculate the linear thermal expansion coefficients...... 74

Figure 32. a) Low temperature thermal conductivity of various I-V-VI 2 compounds, b) comparison of to measured ...... 75 min Figure 33. Lattice parameter for Ag 1-xNa xSbTe 2 alloys as a function of composition...... 78

Figure 34. Thermal conductivity of NaSbTe 2, Ag 0.73 Sb 1.1 Te 2 and their alloys, (a) as function of temperature, and (b) as a function of composition at T=377 K...... 79 Figure 35. Electrical conductivity of some of the measured compounds...... 80 Figure 36. An area on the modified ternary diagram that indicates samples that show no exothermic reaction in the specified temperature range (blue circles) is surrounded by an area of where small amounts of second phase is indicated (green triangles). Larger variations of composition show an increasing amount of second phase present (orange squares, red circles) .. 84

Figure 37. Latent heat trace reveals an exothermic reaction at 417K in AgSbTe 2 due to Ag 2Te. Off Stoichiometric composition Ag 0.336 Sb 0.558 Te showed no reaction throughout the temperature range...... 85 Figure 38. Seebeck coefficient and resistivity of measured off-stoichiometric compositions ...... 86 Figure 39. Seebeck coefficient shown as a function of Sb and Te content with Ag held constant at 0.366 moles. Color markers are determined through qualitative latent heat trace analysis. Blue shows no sign of phase transition through the selected temperature range. Increasing amount of second phase is indicated by green, yellow, and red respectively...... 87 Figure 40. Resistivity as a function of Sb and Te content with Ag held constant at 0.366 moles. Color markers are determined through qualitative latent heat trace analysis. Blue shows no sign of phase transition through the selected temperature range. Increasing amount of second phase is indicated by green, yellow, and red respectively...... 88 Figure 41. Thermal conductivity data from select compounds ...... 89

Figure 42. Resistivity measurement of undoped NaSbSe 2 with exponential fit of low temperature data...... 93 ix

Figure 43. Seebeck and resistivity various doped NaSbSe 2 samples ...... 95 Figure 44. Fitted resistivity curves with calculated activation energies for each of the doped NaSbSe 2 samples ...... 96

Figure 45. XRD peaks of LiSbSe 2 with unidentified minor phases ...... 98

Figure 46. Seebeck, Resistivity, and Power Factor for three attempts at undoped LiSbSe 2 ...... 99 Figure 47. Thermopower or Seebeck coefficient, electrical resistivity, Nernst-Ettingshausen coefficient, and Hall coefficient of Pb 1-xCr xTe. Inset shows carrier concentration for x=0.25%...... 103 Figure 48. Calculated position of the Fermi level with respect to band edge, effective carrier density-of-states mass, and scattering parameter for Pb Cr Te. The dashed line is the DOS- 1-x x mass for the conduction band of PbTe: no significant increase is observed ...... 105 Figure 49. Pisarenko plot (Seebeck coefficient versus carrier concentration) for PbTe:Cr at 300 K and 100 K (inset). Solid lines are calculated for the conduction band of PbTe...... 106 Figure 50. Temperature dependencies of the magnetization of PbTe doped with 0.25, 0.5, 1, and 2 % Cr measured in a 3 kOe magnetic field. The dashed line is an order-parameter law fit to the 2%, with a Curie temperature of T C ≈ 335 K ...... 107

Figure 51. Magnetization of Pb 1-xCr xTe doped with x = 0.25, 0.5, 1, and 2 at. % Cr measured at various temperatures in a magnetic field up to 70 kOe. The inset to the x = 0.25% frame shows as dashed lines Brillouin function fits to the magnetization assuming a 3d 4 Cr 2+ configuration using N par-Cr as only fitting parameter...... 109 Figure 52. Low temperature electrical conductivity of PbTe:Ti ...... 113 Figure 53. Hall carrier concentration of PbTe:Ti ...... 114 Figure 54. Seebeck coefficient of PbTe:Ti ...... 115 Figure 55. Pisarenko plot from Ref 80 demonstrating no increase in S ...... 116 Figure 56. Carrier concentration as a function of Ti effusion cell temperature. 80 ...... 118

Figure 57. Preliminary electronic band contributions calculations CoSb 3-xZn x, x=0.3% ...... 123

Figure 58. Transport properties of CoSb 3-xZn x ...... 124 Figure 59. Calculated band structure with Al as an impurity ...... 125

Figure 60. Transport properties for CoSb 3-xAl x ...... 127 Figure 61. Pisarenko relation with all available p-type literature values included ...... 129

List of Tables

Table 1. Scattering parameter associated with scattering mechanism ...... 55 Table 2. Summary of heat treatment used in this study ...... 62

Table 3. Experimental and Calculated parameters of several I-V-VI2 compounds...... 72 Table 4. Qualitative analysis of exothermic reaction in latent heat trace. Silver composition was held constant at 0.366...... 83

Table 5. List of unsuccessful attempts at doping NaSbSe 2 ...... 94 Table 6. PbTe: Ti sample properties and identifiers ...... 113

Chapter 1: Fundamentals of Thermoelectricity

The increasing demand for energy combined with the negative effects of CO 2 production are driving the need for energy processes with increased efficiencies as well as waste heat recovery, particularly for transportation, industrial applications, residential, and commercial applications.

Figure 1 shows that in 2009, over 57% of generated energy was lost in the form of waste heat.

One technology of high interest for waste heat recovery is thermoelectric (TE) devices, which are able to directly convert a temperature gradient to electrical power. These devices are small, lightweight, and completely silent making them a favorable option for transportation and other mobile applications.

Figure 1. Energy loss sources 2009 1. 1 Quad =10 15 BTU = 293,071,083,330 kWh.

Thermoelectric devices are also able to provide direct heating and cooling when a current is passed through. The same benefits hold for TE devices for temperature control and because of this, TE devices have also been used in automobiles to replace HVAC systems and as personal climate control devices.

TE devices are mainly limited by the TE material efficiency, characterized by the thermoelectric figure of merit, zT, which is a function of Seebeck coefficient, S, electrical conductivity, σ, and thermal conductivity, . An increase in zT would lead to more capabilities with waste heat recovery and climate control. As a result, various approaches to optimizing zT have been explored such as anharmonicity engineering (this work), nano-structuring, and resonant levels

(this work), among others.

Atomic Structure

A material’s properties stem from the atomic makeup as well as the interactions between the bonded atoms. Using the Bohr atomic model, rooted in quantum mechanics, we understand the basic atom to be equal numbers of protons and neutrons comprising the nucleus, then quantized electrons in orbit around the nucleus. 2 Each shell, or orbit, is able to hold a specific number of electrons, and when it is completely full, it can be considered as part of the core, which also includes the nucleus. Any unfilled shells are valence orbitals. Electrons fill shells starting at the lowest unoccupied energy level, so valence electrons are at a higher energy than electrons that exist in the core. The valence electron configuration determines the type of bond that will occur between different atoms, therefore, understanding the valence electrons is an important step to understanding and determining material properties.

The Lennard-Jones potential allows us to examine some material properties. Considering two atoms, that are otherwise isolated, brought together from an infinite distance, we can examine the change in potential energy and begin to understand some intrinsic properties of a material. The atoms will exert both an attractive force, F A(r) and a repulsive force, F R(r) on each other in opposing directions. When the total force on an atom is zero, it will be in equilibrium and its position is r 0. This state corresponds to the lowest energy configuration in Figure 2 and directly

determines the interatomic spacing in a material.

Figure 2. Energy diagram for two atoms brought together from infinite separation.

The shape of the E P(r) curve near r 0 can help determine many material properties. The magnitude

of the depth of the energy well is an indication of the melting temperatures: Deep wells

correspond to high melting temperatures whereas shallow wells indicate lower melting temps. In

Figure 2, the black dotted line represents the equilibrium position for a perfectly symmetric curve, 3 the red dashed line shows the preferred atomic position for the given curve. With increased energy, such as an increase in temperature, the interatomic distance increases, which can be demonstrated by measuring the coefficient of thermal expansion, α.

Hooke’s law says that for low values of stress and strain, they are linearly related to each other through the materials modulus of elasticity, E. The modulus is also known as Young’s Modulus in this region. Lower modulus materials are considered softer and more flexible, whereas higher modulus materials are stiffer. The idea of the modulus of elasticity can be extended down to the stress-strain relationship at the atomic level. It relates the interatomic forces to the atomic positions

(1) ∝ The stiffness of a material can be determined based on this curve as well. Materials with a very narrow symmetric curve, and therefore a very small expansion with temperature, are very stiff, whereas materials with a wider asymmetric curve, and a very large expansion are more flexible. 2

Ultimately, E implies anharmonic atomic position behavior. This can have strong effects on the material’s electrical and thermal properties.

Atoms form a physical connection between each other through either chemical or physical bonds.

There are three different kinds of chemical bonds which include ionic, covalent, and metallic.

Ionic bonding occurs when a metallic atom donates an electron to a non-metallic atom, which in turn creates ions. Ionic compounds generally have high meting temperatures because they have very strong bonds, therefor very large bond energies. Covalent bonding is characterized by atoms sharing electrons. Metallic bonding creates free electron space with a negative charge that attracts the positively charged atomic cores. Physical bonds, also known as Van der Waals, arise

4 from dipoles and are extremely weak compared to chemical bonds. This is demonstrated with low melting temperatures in inert gases whose primary bonds are Van der Waals. 2

All materials are composed of atoms that are either organized in a periodic manner, or randomly distributed. Materials which have a notable periodic atomic distribution, or long range order, are called crystalline, whereas materials that exhibit no notable periodic distribution are called amorphous. The properties of a material depend greatly on its basic atomic structure, so the differentiation between these two possibilities is very important when examining material properties.

At the very basic level, crystalline solids have a unit cell that can be used to calculate material properties. The unit cell is the smallest, most basic structure that is repeated to form a periodic structure, or lattice. Examples of 1D and 2D unit cells in their periodic structures are shown in

Figure 3. Further extension to 3 dimensions can be visualized for crystalline solids.

Figure 3. Example of 1D unit cell (left) and 2D unit cell (right) where yellow "atoms" represent the repeating unit and the blue "atoms" represent the rest of the crystal

The dimensions of a unit cell, or lattice parameters, as well as the bond angles, are used to categorize the type of crystal structure. Materials that have more than one stable crystal structure are called polymorphic. Although there are many different possible crystal structures, this work is primarily focused on materials that adopt cubic lattices. Such structures are characterized by a single lattice parameter, a, the same in all three dimensions, and all bond angles at 90 (Figure 4). ᵒ

Figure 4. Simple cubic structure

In order to help describe crystal properties with regard for direction, a crystallographic direction is defined by a position vector through the origin and is denoted by square brackets as shown in

Figure 5. Crystallographic planes are defined by Miller indices in a similar manner, but use round brackets instead. Miller planes are defined by the reciprocal of where they intersect the defined axis. Negative directions are noted by a bar over the number. Both vectors and planes can be translated to a different position without having to be redefined due to the periodicity of the lattice.

Figure 5. Crystallographic direction demonstrated on the left, Miller indices or Crystallographic planes shown on the right

Crystalline materials can be grown as a single crystal with no grain boundaries and a perfect or near-perfect repeating lattice structure, or they can be grown as a polycrystal with regions of perfect crystals butted up against each other. In the case of a polycrystalline material, the smaller

crystals are separated by grain boundaries. For crystal structures that are not the same in all three

directions (anisotropic), care must be taken to determine along which crystallographic direction

the measurement is being taken because properties will vary in each direction. In a cubic

structure, this is less often a concern.

Defects that occur in a crystal can cause a variation in properties. Some defects occur naturally,

such as vacancies, interstitials, and substitutional defects. Impurity atoms are sometimes

introduced on purpose, which can result in any of the previously listed defects. If the energy of

formation, E f, of a specific defect in a system is known, it is possible to calculate the defect concentration, N v, at a specific temperature, T.

(2)

NA and k B represent Avogadro’s number and Boltzmann’s constant. Material properties include density, ρd, and molecular weight, A. Ef and T have the largest effect in determining the defect concentration in a material. Large energies of formation exponentially decrease the possibility of that particular defect occurring. Energy of formation can be calculated theoretically and can give insight to which impurity atoms are most likely to be soluble in a base material.

In order to create a solid solution, that is, a base material which has homogeneously incorporated impurity atoms and maintains the same crystal structure, a few considerations must be made.

Hume-Rothery rules dictate that the size of impurity atoms must be within 15% of the base material atoms size in order to prevent distortions in the lattice structure. 2 For increased likelihood of solubility, the crystal structures of the impurity and base materials should be the same. The amount of solubility of the impurity atom in the host material will be very limited if there is a mismatch in the structures. A solid solution is more likely to occur with a small electronegativity difference as well.

Band Structure

When a single atom exists independent of outside effects, its electrons energy levels are quantized with core electrons being at a much lower energy than the outer valence electrons. When there is no external source of energy, the electrons will remain in the available states that have the lowest energy. Each state allows for two electrons with opposing spin, per Pauli’s exclusion principal.

When several atoms are in close proximity of each other, such as when they are bonded to form a material, electron energy states begin to spread in such a way as to produce energy bands, or a range of occupiable energy levels. The size of the bands depend on the electron energy state and relative atomic distance.

Figure 6 shows the effect of bringing atoms into close proximity of each other. The figure shows only two representative atoms, however, the band formation is more representative of an extremely large number of atoms. If there were only two atoms, only two states would need to exist in each band. In this representative figure, the equilibrium position for the atoms would be the green atoms. In this case, the highest energy band is the conduction band and the lowest energy band is the valence band. The lowest energy shells remain unaffected for longer because they are tightly bound to the atom, whereas the highest energy shells begin to interact with each other at greater distances because they are further away from the atomic core.

Figure 6. Schematic of band formation when atoms are brought into close proximity.

It is common practice to characterize material properties in terms of electronic band structure. A plot of electronic band structure has x-axis for electron wave vector, k, and the y-axis is the corresponding energy dependence. Various points along the x-axis denote points of high symmetry in the Brillouin zone, which is defined as reciprocal space of the lattice vectors.

Semiconductors in particular are characterized by a separation between lower energy valence bands and higher energy conduction bands. The separation is a forbidden energy region termed

Band gap and the width of this region is the energy gap, E g.

Band structure is determined partially by the crystal structure of the base material. Introduction of impurities or defects can cause local impurity levels, impurity bands, and in larger concentrations, changes in the shape and/or position of the material’s band structure. At absolute zero, the highest occupied energy state is denoted as the Fermi level, EF. The valence band is described by the band with the highest occupied states. The conduction band is the lowest energy band with unoccupied states.

All of the electronic transport properties are established based on the band structure, along with the highest occupied energy state, EF. When thermal energy is introduced (T>0K), the probability of an electron to be at the Fermi level changes through an effect called thermal smearing.

Introduction of thermal energy excites electrons to a higher energy level, leaving behind holes.

For materials with very small band gaps, electrons can be excited from the valence band to the conduction band very easily, creating a two carrier system. Further discussion of band structure and its effect on material properties is provided in each of the following sections.

Seebeck Coefficient

The Seebeck coefficient, S, otherwise known as thermopower, is a measure of the amount of entropy transported per charge carrier. The effect was first observed by Thomas Seebeck in

1821. When a material is subjected to a temperature difference, ΔT, a voltage, ΔV, is generated.

Thermopower is the magnitude of the voltage generated divided by the imposed temperature gradient:

∆ (3) ∆

More specifically, this is derived from the fact that an electric field, E x, is developed when a one- dimensional temperature gradient is imposed.

(4) Schematically, Figure 7 shows how the Seebeck coefficient is represented. Charge carriers tend

to diffuse towards the cold end of the material, which creates a potential gradient. In the case of

electrons being the dominant carrier, the electrons diffuse to the cold end and the voltage

generated is negative, therefor the thermopower is negative. When holes are the dominant

carrier, thermopower is positive.

Figure 7. Schematic representation of Seebeck Coefficient

The relationship of entropy, s, to the Seebeck coefficient, S, becomes clear when one considers a single free electron in a loss-free system, that is, fully reversible (ds=0). Gibbs free energy, g, defines the electrochemical potential, µe, as follows:

(5) +−

Where u is internal energy, v is specific volume, and T is temperature. µ e can also be defined by measurement as

(6) Considering a differential version of these equations, with assumptions of negligible density change, reversibility, constant pressure, and negligible internal energy change, we can simplify and reduce the equation to state

(7) − which demonstrates how Seebeck coefficient is essentially a measure of entropy carried by a charge carrier.

When a system has two different carriers, such as contributions from two different bands or holes and electrons at the same time, the Seebeck coefficient can be expressed as follows.

(8) where subscripts denote a carrier. It is clear from this discussion that an optimally designed thermoelectric material will be of one charge type – that is, either n-type (electrons) or p-type

(holes). When materials have both holes and electrons, their effects cancel each other, which results in a lower effective voltage.

This is a good reason why narrow band gap semiconductors can present a challenge when trying to optimize zT: electrons that are thermally excited to the conduction band leave behind holes, creating a two carrier system. Larger band gap systems can withstand more thermal energy without allowing the reducing effect on Seebeck coefficient. In a single carrier system, Seebeck coefficient is typically linearly increasing in magnitude with temperature, however, the onset of thermally excited carriers is demonstrated by a reduction in magnitude. This is sometimes described by the Seebeck “turning over”.

Electrical Conductivity

Electrical conductivity is the result of movement of charge carriers and varies greatly in different types of materials. Metals are extremely conductive, insulators are electrically insulating, and undoped semiconductors fall in a range in between. In semiconductors, intrinsic conduction occurs when electrons are excited from a filled band to one which has available energy states.

Extrinsic conduction occurs when electrons from an impurity level are excited to available energy states.

As a general rule, materials follow Ohm’s law, which describes the linear relationship between applied voltage, V, resistance, R, and current, I, as

(9) The electrical conductivity of a material is defined as the current density, J, divided by the applied electric field, . ε

(10) The current density, by definition, is simply the current normalized by the area, A, through which it is passing 13

(11) and likewise, the electric field is defined by the voltage over a specified length, l

(12) ε Electrical conductivity is the inverse of resistivity, ρ, which is directly related to the resistance of a material by

(13) ρ We can see that the electrical conductivity equation is then simply a restatement of Ohm’s law.

When an electric field is applied to a material, it acts as a force to accelerate charge carriers.

When the free carriers are in motion, they experience scattering events that limit their net velocity along the direction of the applied field, or drift velocity, v d. This is related to the resistance of the material, as an increase in scattering events will result in a higher resistance. The drift velocity, vd, is directly related to the electric field by the intrinsic material property, carrier mobility, µ.

(14) Some materials also have contributions from ionic conduction. This occurs when ions are able to move through a lattice, which requires both an ion which is able to overcome a potential barrier to move to its next position, and an available position or vacancy. 3 This type of conduction is common in materials containing transition metals such as copper and can be detrimental to the operation of a thermoelectric device.

The electronic band structure is ultimately what determines how conductive a material is, among other properties. As described previously, a concept of quantum mechanics, an atom exists as a nucleus and electron shells. Each shell has subshells designated by letters s, p, d, and f and each subshell has a corresponding number of available states (s->1, p->3, d->5, f->7). 2 Atoms that are closely spaced create an atmosphere where electrons interact with each other. As electrons can

14 only allow two electrons (spin up & spin down) to occupy an energy level, per Pauli’s exclusion principle, the allowable energy states widen to create an allowable energy band. The energy bands encompass the allowable energy states and energy states not contained within these bands are forbidden. As these energy bands are formed, the valence electrons of the atoms will fill the lowest available energy states first.

At absolute zero, the highest occupied energy state is denoted as the Fermi level, E F. The valence band is described by the band with the highest occupied states. The conduction band is the lowest energy band with unoccupied states. Metals have an overlap between the valence and conduction bands. Insulators have a large forbidden “band gap”, E g, between the valence and conduction band. Semiconductors have a much smaller gap between the two bands. When an electron is situated in a band that is full, it is difficult for the electron to move, that is, it has a low mobility, µ. When an electron is situated in a band that has available energy states, such as in the conduction band, its mobility is higher. Metals have a very high electrical conductivity because the electrons are able to move relatively freely in the conduction band. On the other hand, insulators have a full valence band with low mobility and the next available energy states are at a much higher energy, as band gaps for insulators are typically several electronvolts (eV).

Semiconductors are a middle ground: they have a somewhat narrow band gap, so while there are no energy states immediately available, they are easily attainable with a small increase in energy.

The increase in energy can be in a variety of forms including heat, light, and other electromagnetic waves. Another way to imagine these situations is by examining the different type of bonds that are typically associated with each type of material. A large band gap material typically has ionic or stronger covalent bonds which results in an electron that requires more energy to dissociate from the atom, therefore a larger Eg. On the other end of the spectrum, metallic bonding creates electrons that are not strongly bound to the atoms, resulting in

15 overlapping bands which leads easily to conduction. Covalent bonding falls in between and is typical for semiconductors where a small E g occurs.

The ability of a material to conduct electricity depends not only on the mobility of the carriers, but also on the number of carriers, n (electrons or p for holes), that are available:

(15) || where e is the charge of an electron.

There are two types of charge carriers in a semiconducting system: the electron, and the less familiar “hole”. When an electron is excited out of its base energy state, an empty energy state with a net positive charge remains. Because holes are positively charged, they move in the opposite direction as electrons when an electric field is imposed (in the same direction as the electric field, as opposed to electrons which move in the opposite direction). This can be imagined as an empty electron position propagating through a material as a result of electrons continually filling the empty position and leaving behind a new empty position somewhere else.

Where both holes and electrons are present, electrical conductivity will have contribution from both:

(16) || + || For intrinsic semiconductors, those which have a full valence band and empty conduction band at

0K, the number of electrons contributing to conduction, n, is equal to the number of holes, p, that were left behind when they were excited to the conduction band.

Ioffe Regal criterion establishes a minimum electronic conductivity based on the carrier concentration. As a general understanding, it implies that any electrons with a mean free path less than one interatomic distance are localized and will not transfer a charge through the

16 medium. 4 For carriers that are not localized, there is then a minimum scattering length, which implies a minimum conductivity.

Transverse Effects

Hall coefficient and Nernst effect is useful for analyzing information about the charge carriers.

They are transverse effects that result from an externally applied magnetic field, H. For the following descriptions of Hall and Nernst coefficient, Figure 7 can be referenced. For Hall coefficient, a current is applied and the standard convention is followed in that electrons “flow” in the direction opposite of the current direction. It is assumed that one directional electron flow is the case and that the result of the applied current is an electric field, E x. A magnetic field, which is uniform across the entire sample, is applied, H z. As a result of these, an internal force is established to attain equilibrium: the Lorentz force, F L. This particular force causes carriers to deflect from their path along x, and accumulate on one side, depending on the charge sign, along the y-axis, which establishes a voltage, V y. In this case, V y will be referred to as V H because it is a result of what we call the Hall effect. The representative equations for this process is as follows.

(17) × This equation can be simplified under the assumptions that the velocity of the electrons as well as the magnetic field are uniform and in one direction, as well as that they are perpendicular to each other: F L=qv dBz. The resulting electric field is then

(18) ×

The Hall coefficient, R H, is defined as

(19) 17 where n is the number of carriers (n for electrons, p for holes), and q is the carrier charge. In a very simple parabolic model with one type of carrier, the hall pre-factor, P H=1, however, this can

vary greatly depending on band structure. The Hall field that is established is simply defined as a

function of the measured hall voltage, V H, and the width of the sample, w.

(20) We can substitute all of the known terms to establish that

∗ (21)

Figure 8. Transverse measurement setup

In the case of Nernst coefficient, N, a temperature gradient is established across the sample, in the

same way as a measurement for Seebeck coefficient: It is considered to be a one-dimensional

thermal gradient in the x-direction. As a result of the thermal gradient, carriers diffuse to the cold end with velocity v d. A magnetic field is applied in the negative z-direction. Nernst is essentially 18 the same setup as Hall, except that the carrier movement is induced by a thermal gradient instead of an applied current. With the same assumptions as above, including the assumption that the thermal gradient is one-dimensional and uniform along the x-axis, we see similarities among the equations that describe this process:

(22) ∗ ∗ The temperature gradient needs to be established for this measurement. Equation 22 can be simplified as follows

∗ (23) ∗∆∗

An experimental configuration of Hall and Nernst measurement as well as previously discussed

Seebeck coefficient and electrical conductivity is shown in Chapter 2 (Figure 22) It must be noted here that the measured Nernst coefficient is adiabatic, N a, where reported data undergoes a correction to demonstrate isothermal Nernst, N. The isothermal Nernst can be calculated from N a as shown in Ref. 5.

Magnetism

Diamagnetic behavior is exhibited in materials which show very weak magnetization opposing the direction of any applied magnetic field, and no magnetization when there is no applied magnetic field present. The strength of the magnetism will vary proportionally to the strength of the applied field, but is so weak that it is easily overlooked when another magnetic effect is present in a material.

Paramagnetic effects are demonstrated in materials that have randomly oriented permanent atomic dipole moments. Without an applied magnetic field, H, the dipole moments have a net

B~0. Paramagnetism, like diamagnetism, is a relatively weak effect, however, the magnetic field created adds to the applied field, unlike diamagnetism.

Ferromagnetism is a strong magnetic effect that is characterized by a magnetic moment without an applied magnetic field. This is a result of electron spin alignment or of orbital magnetic moments on valence electrons of d- or p-metals.

Magnetic effects are weakened with an increase in temperature, as thermal energy contributing to atomic motion disrupts magnetic alignment. The Curie temperature, T c, which varies depending on magnetic characteristics of the material, is the temperature at which magnetization is expected to go to zero; that is, the thermal excitation completely disrupts ferromagnetic alignment, and above T c a ferromagnetic material becomes paramagnetic. Thermomagnetic testing can sometimes be conducted to establish the presence of certain magnetic materials by using the unique T c to identify a material.

Thermal Conductivity

Thermal conduction is a transfer of energy from higher energy particles to lower energy particles as a result of some interaction between the particles. 6 The rate of heat flow per unit area, q, through a material is related to the thermal conductivity, κ, of a material by a generalized equation 7, “Fourier’s law”,

(24) −∇ Where T is the temperature gradient in the material and κ is always positive. This general ∇ equation can be directionalized for non-isotropic materials:

(25) − , − , − 20

The rate of heat flow, Q, through a material can then be described by

(26) − ∇ Where A is the cross sectional area, and l is the length over which the temperature gradient exists.

This leads to a method of measurement of thermal conductivity. In a parallel-piped shaped material such as in Figure 9, a heat flux can be introduced at one end, the temperature gradient measured in the middle, and with known geometry, κ can be calculated.

Q Q

T, l ∇

Figure 9. Basics of thermal conductivity measurement

Area of the sample can be simply calculated by multiplying the width, w, by the thickness, th.

For this setup, it is assumed that there is no heat loss along the length of the sample and that the temperature is uniform across a given cross section. If the assumption is that the thermal conductivity changes linearly with respect to temperature, then the calculated thermal conductivity is that of the material at temperature T+ T/2. This can also be approximately ∇ correct for small T even in temperature regimes where κ is changing rapidly. ∇

There are several mechanisms through which heat can be transferred through a medium. In metals, the dominant mechanism is the charge carriers. In insulators, the dominant mechanism is phonons. In semiconductors, a mixture of these two mechanisms are responsible for heat transfer. The magnitude of the contribution of each of these mechanisms changes with temperature, and characteristic behaviors can be identified over specific temperature ranges.

Phonons

A phonon is quantized energy that describes a lattice vibration. Phonons can best be visualized as lattice vibrations. If a material at absolute zero was supplied with heat at one location, the individual atoms at higher energy in the material would begin to vibrate. Increasing the temperature directly translates to increased vibrational amplitude. The intrinsic nature of a crystalline solid to have a periodic structure, combined with the atoms being directly bonded to each other, is the idea behind how phonons propagate through a material. When one atom starts vibrating, it has a tendency to have a direct effect on neighboring atoms by passing along energy/movement. When this motion is coupled in a long string of atoms, the harmonic motion forms displacement waves, in which energy is passed. 8 It is easy to understand when considering a 1D chain of molecules, and the idea translates just as easily to three dimensions. Figure 10 demonstrates movement of a chain composed of balls (atoms) and springs (bonds). The atoms move as a result of thermal energy and cause the bonds to stretch or compress, which corresponds to a change in stiffness.

Figure 10. 1D Chain of atoms in equilibrium (top), out of equilibrium (bottom)

In a perfectly defect free system with zero losses, this can be thought of as harmonic motion.

However, perfectly harmonic systems do not exist, or we would be able to observe an infinite

22 conductivity. Because there are no materials that have an infinite thermal conductivity, there exists at least some degree of anharmonic motion in all material systems.

The ability of a wave to propagate though a solid is directly related to the material density and elastic modulus, and departure from harmonic behavior is a direct result of local deviations in these two material properties as compared to the bulk properties. 8 Local changes in density can be caused by change in bond length or atomic mass. Anharmonicity is directly related to the change in elastic constant with respect to atomic displacement. If the displacement of an atom from its resting position was directly proportional to its restoring force, that is if it obeyed

Hooke’s Law, then the motion would be harmonic. 8 However, if the restoring force is not simply proportional to the atomic displacement (or otherwise a function of displacement), a displacement wave from one phonon would create a situation where a second phonon would experience a different elastic modulus and would therefore be scattered. As the temperature increases, the atomic displacement also increases, creating a larger anharmonic effect. We can explore this concept more by referring back to Figure 2 and exploring the equations that define a spring system. In a simply harmonic system, Hooke’s law can be expressed as F=-kx, where F is the restoring force, k is the spring constant, and x is the displacement (x=r-r0 in reference to Figure

2). In a perfectly symmetrical potential well, k will be constant, and the result is perfectly harmonic motion. We then see , where k is constant. In reality, deviations from , 0 potential well symmetry result in k as function of x such that . The measure , ≠ 0 of anharmonicity can be done through measurements of elastic properties and also temperature variation of thermal expansion.

The Debye Temperature, ΘD, is defined as ℏ , where is Plank’s constant, k B is Θ ℏ Boltzmann’s constant, and is the maximum phonon frequency in a material. At max 23 temperatures higher than the Debye Temperature, ΘD, thermal conductivity is inversely

8 th proportional to temperature. At temperatures ~1/20 ΘD, the thermal conductivity is at its maximum. 8 As the temperature is decreased further, a decrease in phonon density correlates to a strong reduction in thermal conductivity. Changes in phonon density and phonon-phonon interactions (PPIs) have a strong effect on the thermal conductivity, as do defects. These effects are discussed in more detail later.

In crystals, at temperatures greater than ΘD, Umklapp (resistive) processes are expected to create the relationship of κL~1/T. As the temperature is decreased, the phonon mean free path becomes increasingly large and is only limited by crystallographic defects or the size of the crystal. At the point that the phonon mean free path, l, reaches a critical dimension, l=l crit , where l crit is the average distance between defects or the physical dimensions of the crystal, the thermal conductivity will be maximum. Possible phonon scattering defects include, but are not limited to, atoms with different mass than the lattice host atoms, vacancies, interstitial atoms, and dislocations. The temperature where this occurs can vary dramatically and is estimated to be somewhere between 1/20 and ½ of ΘD. For amorphous materials, a periodic crystal structure is not considered to exist, so local defect structure dominates the behavior of thermal conductivity, and this temperature dependent peak does not occur. As the temperature is decreased further, the thermal conductivity will drop to zero following the decrease in phonon density. In this

3 temperature regime, κL~T , and is directly related to the increase in specific heat.

Phonon Interactions

In the field of thermoelectricity, minimized thermal conductivity is a desired property. One way to effectively change thermal conductivity is through phonon interactions. Phonons can interact

24 with other phonons, electrons, or magnons, and depending on how they interact, an increase or a decrease in thermal conductivity may be observed.

This work focuses on the reduction of thermal conductivity due to phonon-phonon interaction

(PPI) which depends both on the frequency of the phonon collisions as well as the momentum change during each collision. The momentum, p, of a phonon is described by

(27) ℏ where is reduced Plank’s constant and k is the propogation wave vector in the first Brillouin ℏ zone. The phonon wave vector, k, has a maximum defined by half of the reciprocal lattice vector, g.

Phonon energy, εI, is directly proportional to the phonon frequency, ωi.

ε = ω (28) i ℏ i

The maximum energy of a phonon is defined by k BΘi, where Θi is the zone edge energy of mode i. It should be noted here that ΘD is the mode average Debye temperature, as determined by specific heat measurement, that takes into account all phonon modes, not just the acoustic (heat carrying) modes. Θi represents an individual mode of which three emerge as important for discussion in this work; two transverse and one longitudinal acoustic mode. There are three normal modes of vibration, and interactions between these modes can help explain potential phonon scattering outcomes. In three-phonon process, it is possible for one phonon to experience a collision that results in two phonons, or two phonons to collide and form a single phonon.

Collisions that occur between two phonons k 1 and k 2 will result in one of two possible outcomes: a normal process or an Umklapp process. Where g is defined as a reciprocal lattice vector, normal processes occur when |k 1+k 2|=|k 3|<|g/2| where energy and momentum are conserved and only the modal distribution is changed.7,8 This is the expected outcome when the temperature is

25 less than some fraction of the heat carrying modes’ Debye temperature. With increasing temperature, the number of phonons with |k| approaching |g/2| increases. Phonon-phonon interactions are then more likely to occur, such that |k 1+k 2|=|k 3|>|g/2|. This case that is described results in phonon wave vector, k 3, that falls outside of the first Brillouin zone and therefore results in a loss in momentum as |k +k |=|k +g| where a net momentum, p = g. These resistive 1 2 3 g ℏ processes are called Umklapp. 7

Grüneisen parameter, γ, is a measure of the change in phonon frequency, , with change in volume, V. An individual mode, i, has .7 Deviation from perfectly − ln ln harmonic lattice vibrations is described through the use of γ, and an estimate of the lattice thermal

2 conductivity, κl, can be given in terms of 1/ γ , indicating that the larger the anharmonicity, the lower the lattice thermal conductivity is expected to be. In a perfectly harmonic crystal with no other scattering mechanism, thermal conductivity, κ, would be infinite and γ would be zero, however, a typical value of γ is up to ~1.

An equation that has been determined from various sources to give a decent estimate of thermal conductivity above temperatures where Umklapp processes are dominant, is given by equation

29. The relationship of lattice thermal conductivity to the mode averaged Debye temperature and

Grüneisen parameter is made more apparent here 7:

(29) ≈ > 3 where M a is the average atomic weight, and a is the volume of one unit cell divided by the number of atoms in the unit cell. A is a less certain parameter which incorporates constants, and ℏ kB as well as information from the crystal structure. This relationship was developed with different values of A by several different people. 7

The complicated nature of phonons makes predictive calculations quite complicated. Two models will be breifly discussed here: Einstein and Debye. The Einstein model assumes that all phonons exist at one frequency which is more akin to optical (high frequency) phonons. The

Debye model assumes a distribution of the phonon density that goes as ω2 with a cutoff frequency. The differences between these two models are demonstrated qualitatively in Figure 11

Figure 11. Comparison between Einstein (left) and Debye (right) density of states frequency distribution.

The Einstein model uses a single atom model and assumes that all atoms vibrate at the same

frequency. This assumption neglects interaction with neighboring atoms. As temperature is

decreased, the number of phonons decreases with Boltzmann statistics, so the Einstein model

predicts that the specific heat will be reduced exponentially with phonon population, ≈

.9 The Debye model, however, takes into account interactions with neighboring atoms, and correctly predicts the specific heat to vary with T 3.

Figure 12 shows a simplified phonon dispersion relation with each, the Einstein and Debye model assumptions. The group velocity, . As a result, we see that high frequency modes (optical modes), have a low group velocity, and therefore are not efficient heat carriers. These are

modes that are well characterized by the Einstein model. On the other hand, the Debye model

more closely characterizes lower frequency modes (acoustic modes).

Figure 12. Relation between frequency, ω, and wave vector, k, given by Einstein (blue) and Debye (green) assumptions

The Callaway expression, which makes use of the Debye model, approximates thermal conductivity as

(30) ℏ

7 where τc is the relaxation rate of both normal and resistive processes combined.

Pure Einstein modes would not conduct any heat. The Cahill model extends the Einstein model to add interactions with only nearest neighbors to estimate the minimum thermal conductivity as given in equation 31.10 This calculation of lattice thermal conductivity is more accurate over a wider range of temperatures as well.

3/1 2  π    T  Θi /T x3exdx  κ =   k n 3/2 c    min B ∑ i   ∫ x 2  6  i   Θi  e −1   0 ()  (31) Here, n is the number of atoms per unit cell, c is the group velocity, and subscript i refers to a particular heat-carrying mode. The total lattice thermal conductivity is obtained by integrating over energy the amount of heat carried by each phonon mode. In the same way as the Io e– ﬀ Regel limit explains minimum electrical conductivity for electrons, the minimum lattice thermal conductivity k min can be the limit for lattice thermal conductivity: Minimum thermal conductivity occurs when phonons are scattered at approximately one interatomic distance. In rocksalt structures, studied here, the mean free path of one interatomic distance, l=a/2. This analogy is not

10 quantitative, and an appropriate calculation of kmin for temperatures corresponding to resistive processes is given by equation 31.

A commonly used approximation given in equation 32 is

(32) is derived from kinetic gas theory, where c v is the specific heat, v is the sound velocity, l is the mean free path (mfp). Substituting the interatomic distance, a, for the mfp would result in

, but this overestimates min in this work. The cause for this is that the integral over the phonon modes that is measured in c di ers from the integral in equation 31. The specific heat v ﬀ approximation from Debye theory neglects the effects of optical phonon modes (including any

29 interactions with acoustic modes) as well as induced polarization effects.

Electrons

The previous discussion of phonon contribution to thermal conductivity describes the predominant thermal conduction mechanism for semiconductors and insulators: Lattice thermal conductivity. When there are a large number of charge carriers present, such as in metals or highly doped semiconductors, another source of heat conduction becomes important: Electronic thermal conductivity, κ . The total thermal conductivity of a material is expressed as e

κ=κL+κe (33) Wiedemann-Franz law describes a ratio between the electrical conductivity and the thermal conductivity in metals as follows 11

(34)

This ratio is also rooted in Drude theory with the following assumptions: electrons are confined to a crystal due to the electrostatic force of the positive ion cores; this force is constant throughout the entire crystal; and electrons have negligible effect on other electron movement.11 Electrons are the dominant heat carrying mechanism in metals, therefore the ratio is constant. This treatment of electrons is part of classical theory where electrons are treated like particles in a perfect gas. The relationship of electrical conductivity and temperature was described by Lorentz at high temperatures. At high temperatures, using Fermi-Dirac statistics and quantum theory, the electronic thermal conductivity of a degenerate metal takes the form8:

(35)

This expression takes into account that only electrons within a small energy band of the Fermi level (within ~2kT) can conduct heat or electricity, thus the need to use Fermi-Dirac statistics.

Electron Interactions

A parallel can be drawn between phonons and electrons when examining their ability to transfer heat at temperatures well below ΘD. As the temperature is decreased, the electron mean free path increases until defect or crystal boundary scattering occurs. We can then write the different components of electrical conductivity using Matthiessen’s rule:

(36) + + ⋯ The subscripts denote the scattering mechanism by which the conductivity is limited. In the case of σlattice , vibrations in the lattice are responsible for scattering. As established previously, the electrical conductivity is directly proportional to the electronic portion of the thermal conductivity, so Matthiessen’s rule can be extended to represent thermal conductivity components in the same manner. Generally speaking, we do not expect the defect concentration to change as the temperature varies, so σdefect can be expected to remain relatively constant, although there are exceptions for instance with the use of nano-particles. However, σlattice depends on the intensity of the lattice vibrations, therefore, we should expect to see this value to decrease with increasing temperature following what is known as the Bloch-Grüneisen law. At temperatures much higher than θD, we can use the Lorentz number to model electronic thermal conductivity, which allows us to estimate lattice thermal conductivity when total thermal conductivity is measured.

(37) 2.44 ∗ 10 Ω Large deviations from Lorentz number can be observed in non-metallic materials, so this is not a steadfast rule. 31

Phonons may also be scattered by electrons when a large electron concentration is present, such as in metals.

Peltier Effect

The Peltier coefficient, Π, helps determine the amount of heat that is released or absorbed at a junction,12 however it should be noted that it is in fact a bulk effect that results from the heat that is carried through a material by electrons.

(38) Π ∗ I The amount of heat that is supplied/removed is a function of both the Peltier coefficient and the amount of current, I, and direction of flow, that is supplied to the device.

Additionally, Π has been defined for a simple device consisting of an n and p leg as:

(39) Π S − ST where S has been defined previously and the subscripts p and n represent the p-type and n-type material respectively. Because optimal cooling ability will occur with a large Π, it is important for each material to have Seebeck coefficients which are large (where p-type S is positive, and n- type S is negative).

Joule Heating

In any system, there are irreversible losses that cause inefficiencies. For thermoelectrics, Joule heating can be a large source of loss. Joule heating is a non-reversible effect that will work against the Peltier cooling effect. It is defined as follows:

(40) ∗ I

Joule heating depends on the resistance, R, of a material. As a result, there is an optimal current,

I, that will provide most efficient operation.

Material Efficiency

Thermodynamic efficiency, , of a system is a ratio of power output, P , to power input, P . In out in the case of thermoelectrics, power in is supplied as heat, Q in .

(41)

The output power of a material can be expressed in terms of Seebeck voltage minus losses due to

Joule heating (internal resistance/losses) and the thermal input can be expressed in terms of

Peltier and thermal conduction terms resulting in the following expression for efficiency 13

(42) where subscripts H and C represent hot and cold sides, l is the length of the thermoelectric element. This equation can be further simplified with the use of the figure of merit, zT, which is described in more detail in the next section.

∆ √ (43) ∙ √

In this equation, T is the average temperature between T H and T C. It becomes apparent that the efficiency is expressed in terms of Carnot efficiency = T/T H. Carnot ∆

Thermoelectric Figure of Merit

The thermoelectric materials figure-of-merit (FOM), zT , is defined as

(44) where is the Seebeck coefficient, σ is the electrical conductivity, and κ is the thermal conductivity. κ is the sum of the lattice, electronic, and ambipolar contributions.

As a summary of the previous sections, the Seebeck coefficient is the voltage measured over a temperature gradient, and is strongly dependent on the electronic band structure of the material and doping level (number of carriers). Metals typically have very low S (large number of electrons, or carriers) and insulators typically have a large S. The electrical conductivity is the inverse of the bulk resistivity of the material and depends of the density of free charge carriers in the material. The conductivity is very high for metals, very low for insulators, and has an intermediate value for semiconductors. The thermal conductivity primarily consists of two major components, lattice ( κl) and electronic ( κe). The lattice portion of thermal conductivity depends primarily on the phonon structure of the bulk material and the temperature. The electronic thermal conductivity is proportional to the electrical conductivity.

As can be seen from the above description, a major challenge in creating a high zT material lies in the interdependencies of the properties. Typically, an increase in Seebeck coefficient corresponds to a decrease in electrical conductivity and increasing the electrical conductivity causes an increase in the electronic portion of thermal conductivity. This interplay of properties resulted in materials being limited to zT ~1 for much of the history of thermoelectrics. zT also depends on temperature, with a temperature range where zT is maximum. By using equation 37, we can form an expression for zT that forms the basis of this research:

(45)

This equation represents our ability to optimize the figure of merit by reducing the ratio of / . L e Here, the means of this reduction is anharmonicity engineering.

Devices

A conventional TE device is composed of alternating p-type and n-type materials sandwiched between two thermally insulating materials with an electrically conductive material connecting the two legs together in series. The p and n type legs are thermally in parallel and electrically in series. Charge carriers (holes and electrons) condense to the cold side of the TE materials, which creates a voltage across the device and induces a current. There are several factors which enter into the selection of materials for a TE device, including the designed operating temperature range of the device. Both p and n-type materials should have the device operating temperature be close to their zT maximum. Additionally, both TE materials should ideally have similar zT values, as the device figure of merit will be limited by the lowest material figure of merit.

Figure 13. Simple TE device configuration

The most common parameter used to describe the performance of thermoelectric materials and thermoelectric devices is the figure of merit. The device figure of merit, ZT , is determined by the material figure of merit, zT , as well as the thermal and electrical losses due to device contacts.

The effects of thermal losses and electrical contact resistance cause the device figure of merit to be less than the figure of merit for the materials.

Thermoelectric devices excel as heating/cooling devices due to their light weight, silent operation, and quick response times. Compared to conventional cooling systems (refrigeration, forced convection, etc.), this response time is extremely fast, which gives TE devices an advantage in quick performance when that is a demand of the application. The downside is that

TE devices have a much lower efficiency than traditional refrigeration systems.

The device figure of merit, ZT , is given by:

(46)

SD is the effective Seebeck coefficient for the device based on the Seebeck coefficient of the p and n-type materials over a specified temperature range,

(47) − R is the effective resistance for the device based on the resistance of the p and n-type materials as well as contact resistance and other electrical losses,

(48) + + is the effective thermal conductivity of the device based on the thermal conductivity of the p and n-type materials as well as conductive and radiative thermal losses due to the connections and device setup

(49) + + 36

In these equations n is the number of legs in the device. 13

From these equations it is apparent that the best thermoelectric device will have minimal thermal and electrical losses. More significantly, these equations show the figure of merit of the device must be less than the figure of merit of the materials, due to the additional resistance and thermal losses in the device.

When designing a device for a particular application, the amount of power or cooling that can be generated is described as follows 14 :

(50) ≈ ∆

Where P is the amount of power, A TE is the area of the thermoelectric material, and l is the length of the TE legs. Therefore, the power density (P/A) can be adjusted by making the legs shorter or longer.

The thermoelectric device figure of merit, ZT, can be used to approximate a maximum temperature gradient across a device for cooling applications as well. 15

(51) ∆

where Tc is the temperature of the cold side.

The primary application areas for thermoelectric devices are energy harvesting and waste heat recovery. For energy harvesting applications, TE technology must compete against other technologies such as piezoelectrics, electrodynamics, and photovoltaics, while for waste heat recovery TE devices compete against heat engines, such as Rankine or Stirling.

The selection of the appropriate technology for waste heat recovery depends on many factors including the available temperature difference, the required power, weight, desired efficiency,

37 cost per Watt, mass per Watt, installation and maintenance cost, reliability, etc. For waste heat recovery, the most critical factor in selecting a technology is the available temperature difference, as the device efficiency is a function of Carnot efficiency, which is directly related to ∆T.

Figure 14 gives an example of the maximum efficiency of a TE device as a function of temperature for several different ZT values, compared to the efficiency of several other common heat recovery cycles. Thermoelectric generators may have slightly lower efficiency when compared to other technologies, however, the devices have many advantages over traditional waste heat recovery systems, which include long life cycles requiring little to no maintenance, solid state operation with no moving parts, quiet operation, an extremely high power per mass or weight, as well as its ability to directly create electricity from heat. These advantages allow them to remain competitive in some applications.

Figure 14. Thermoelectric efficiency compared to other technologies for cold temperature of

300K 16

Figure 15 qualitatively shows the size dependence of efficiency for a TE device and generic thermal engine 1616 . The TE device is able to approach the theoretical limit for devices producing as little as 0.1W, while the thermal engine only becomes very efficient for sizes greater than

100W. The ‘crossover’, where the TE becomes less efficient, occurs on the order of 100W.

Figure 15. Crossover of TEG efficiency compared to large scale technology

The relatively low efficiency of TEGs puts them at a disadvantage when compared to large scale traditional power generation systems such as Rankine cycles. High ZT, however, is not the only important factor for TE devices. As an example, the auto industry could benefit from slightly lower ZT devices that are less expensive. Some of this work focuses on reducing material cost to lower overall device cost, although, some work to improve manufacturing process would also benefit this technology.

Applications

TE devices have been studied for many applications, including vehicle and industrial waste heat recovery, body energy recovery, and personal heating and cooling, among other applications.

This section reviews a few application areas.

Transportation Waste Heat Recovery

Automotive waste heat recovery has been a topic of extreme interest in recent years. Several companies including Toyota, GM, Ford, and BMW are pursuing some form of TE heat recovery systems, while Honda and BMW have also both pursued a Rankine steam engine for waste heat recovery.

Internal combustion engines, such as those used in automobiles, typically have a somewhat low efficiency in the range of 30-40%. A large percentage of the chemical energy is lost in the form of waste heat, which is transferred primarily through the cooling system and the exhaust, the second of which reaches much higher temperatures. The introduction of thermoelectric waste heat recovery system in automobiles presents an interesting engineering challenge in that any energy recovery system has to be light weight enough to not negate its fuel economy savings, quiet enough to not disturb the vehicle operator or those around it, cheap enough to recover its cost in fuel savings, low maintenance, and non-toxic per government regulations. Thermoelectric generators could provide a good fit for this as they are solid state, light weight, and quiet. The challenges to the device manufacturers currently are to increase the efficiency and decrease the cost, both of which can be attempted by altering the semiconductors properties.

Recently, ATEGs have been installed into test vehicles for several manufacturers. Current reports (2011) describe automotive thermoelectric generators (ATEGs) designed for waste heat

40 recovery with ~500W peak electricity formed from heat that would have otherwise been ejected to the environment.17

Automotive Heating/Cooling

Heating/cooling devices are also being developed for automotive applications. Heated and cooled seats (Climate Control Seats, CCS) were first introduced by Amerigon in 1999.

Thermoelectric devices provide temperature control to automobile seats in nearly 40 different vehicles. 18 There are multiple benefits of the CCS Modular system including being inconspicuous, zonal temperature regions allowing individual passenger comfort, high reliability, and relatively low power usage (40-60W per seat). 19

Figure 16. 19 BSST TE seat heater/cooler

Benefits of this system design also include operation while the engine is not running (not possible with a typical HVAC system where a compressor needs to be run), individual zone control of temperature, and use of sensors to determine which vents can be turned off.20 Use of TE devices eliminates the need for traditional oversized vehicle HVAC systems and aims to reduce the

41 weight of personal climate control systems. The use of TE devices also eliminates the need for harmful refrigerants such as R-134 which has a Global Warming Potential (GWP measures the

21 climate impact over a standard 100 year span) that is 1300 times larger than CO 2. This becomes increasingly important as new regulation is imposed causing restrictions of certain refrigerants for cooling in automobiles. 22

Figure 17. BSST Zone thermal control

Body Energy Harvesting

One of the areas commonly discussed with TE devices is harvesting body energy to power sensors or other devices, eliminating the need to carry batteries. Devices that have very low and constant power requirements are ideal for body energy harvesting. Seiko produced the first thermoelectric wristwatch in 1998. 23 The watch was powered by a temperature gradient of 1.5K from heat produced by a body, and produced 22 mW of electrical power. 24

Although the use of body energy harvesting to power devices is enticing, some understanding of device capability demonstrates its limited use. Even with a full TE body suit (assuming 1.75m 2

42 area and ∆T=1.5K), the power produced is still less than the power needed to run a low power

LED (~3W). Body energy harvesting will remain a means to provide low power levels to sensors and similar devices, but will not be a viable source for active or power-hungry devices because of the very small temperature gradient.

Radioisotope TEG

One of the most successful applications of TE devices has been in radioisotope thermoelectric generation (RTEG). RTEGs work by using the heat emitted by radioactive decay for the hot side of the TE device. The lifetime of this device is mostly dependent on the half-life of the chosen radioactive element as well as the lifetime of the chosen thermocouples. Plutonium-238 is a commonly used material with a half-life of ~85-90 years.

RTEGs have been in use for over 50 years in space missions, and have been used by NASA for over 30 years. They have provided power to over 24 missions including lunar and mars landing missions, as well as several other planetary explorations (Jupiter, Saturn, Uranus, and Neptune). 25

RTEGs are preferred to solar arrays for space missions due to their small size, reliability, light weight, and ability to provide power when no solar source is available.

Portable Remote Power Source

TE devices have the potential to provide low amounts of power in locations that have no access to the grid or other power sources. In many cases, the hot source is provided by an open fire. Other small advances include several devices that are designed for camping or 3rd world locations where there is no access to typical power sources. They take advantage of the use of heat from burning wood or other fuels to produce small amounts of electricity.

One such device, Hatsuden-Nabe thermoelectric charger by TES NewEnergy, is used to charge devices such as cell phones in remote areas where there is a campfire available. This device utilizes the temperature difference between the boiling water and the fire and is estimated to take

3-5 hours to charge a typical iPhone. 26 Similar applications can be envisioned where a fuel is burned to create a heat source and TE devices use the temperature differential to charge batteries or other electronics.

One example of a unique thermoelectric technology is a portable ‘mosquito catcher’ by Marlow.

The ‘mosquito catchers’ use a propane tank and a small flame to provide heat to a thermoelectric

27 device which powers a small fan, blowing the mosquitoes into CO 2 where they suffocate.

Sensor Applications

One other application commonly discussed for thermoelectrics is sensors, either using the thermoelectric effect to measure temperature, or using the thermoelectric effect to power the sensor. Much of the current interest is driven by a desire to install sensor networks within the heating, ventilation, and air conditioning (HVAC) system of buildings. It is cost prohibitive to run wires to all the sensors, TE devices can be used to power small sensors inside ductwork and other areas where sufficient ∆T is provided. For some applications, the TE devices can also be used as heat flow sensors that can indicate both the magnitude and the direction of heat flow. 28

State of the Art

A standard device that can demonstrate the state of the market is one which uses Bi 2Te 3 as the thermoelectric material, which has a material zT of ~1.1 and a device efficiency ZT ~0.7. A new material would either need to have an overall device efficiency greater than 0.7 or a lower cost.

The interdependence of properties of thermoelectric materials has resulted in materials being limited to zT ~1 for much of the history of thermoelectrics. Classical values of zT are shown in

Figure 18 and Figure 19. zT also depends on temperature, and typically includes a narrow temperature range where zT is maximum. Several materials since 2001 have been reported to have zT >1.4 due to various technologies including superlattice structures, nanoscale inclusions, and resonant levels (Figure 19)29 , however, nanomaterials have met difficulty in scalability and bulk prepared materials are not yet commercially available. 16 A high zT of 2.2 at 900K has recently been reported on PbTe doped with SrTe nanocrystals and Na by making use of multiscale crystal structures.30 One potential drawback is that many of the most efficient materials, such as PbTe, are potentially hazardous or expensive. Pb based compounds face regulations that will eventually eliminate their use and Te based compounds are costly due to its relatively low abundance.

Figure 18. Classic figure of merit

Figure 19. State of the art material figure of merit (2006)

Recent work shows robust zT approaching 1 at temperatures greater than 700K for an earth abundant (cheap) naturally occuring tetrahedrite mineral, 31 which introduces a whole range of opportunity for low cost thermoelectric materials. This mineral has promising zT due to strongly anharmonic phonons resulting in low thermal conductivity.

A recent paper examined the maximum achievable value of zT for a single material by writing the electrical conductivity, thermopower, and thermal conductivity as integrals of transport distribution function, which was then varied to maximize zT . Using reasonable values for the group velocity and the thermal conductivity, it was shown the maximum zT occurs for a delta- function shaped transport distribution function, with a maximum value of zT ~14 32 , although in practice, this is not a foreseeable value.

Thermal Conductivity

There are many different ways to reduce thermal conductivity including introducing different defects and creating closely spaced boundary layers. Another approach to finding low thermal 46 conductivity materials is to explore very complex crystal structures, as more complexity typically leads to more phonon scattering. The use of nanoparticles has been shown to cause a reduction in thermal conductivity 33 , however high cost of large scale production and long term thermal

stability still presents an issue for their commercial use.

Figure 20. Thermal conductivity of PbTe structure reduced by mass disorder. I-V-VI 2 compounds have even lower thermal conductivity.

34 In Figure 20 , we see the difference between PbTe, AgInTe 2, and I-V-VI 2 compounds such as

AgSbTe 2 and AgBiSe 2. The reduction of thermal conductivity in AgInTe 2 can be partially

attributed to the introduction of mass disorder when compared to PbTe. A phonon will

experience more scattering events as a result of the differing atomic masses. The I-V-VI 2

compounds have similar mass disorder to that of AgInTe 2, however, they exhibit dramatically lower thermal conductivity comparatively. Anharmonicity in the bonds has been determined to be the source of the low thermal conductivity, and since this is an intrinsic mechanism, it is robust and relatively unaffected by grain size, defects, etc. Although the lone pair electrons are

47 commonly cited as the cause of low thermal conductivity, the origin of the anharmonicity is what is being explored in this work.

Resonant Levels

The thermoelectric figure of merit can be increased by decreasing the thermal conductivity or by increasing the power factor, S 2σ. As explained previously, many thermoelectric properties are interdependent, which is described by the generic Pisarenko relation as shown in Figure 21. As

σ=nqµ, where n is the charge carrier concentration per unit volume, q is the charge of the carrier and µ is mobility, we see the corresponding relationship between decreasing S and increasing σ.

Figure 21. Generic Pisarenko relation

The Seebeck coefficient is directly related to the electronic band structure in such a way that it can be influenced by introducing a resonant impurity level 35 . The Seebeck coefficient can see a very large increase at a given carrier concentration. The density of states (DOS), g(E), gives an approximation to the number of available electronic states in a given material per unit volume at a specific energy. For a basic case with one type of carrier, equation 52 can be used to understand the relationship between the Seebeck coefficient and various other parameters, including DOS.

(52) It is clear from this relationship that an increase in the density of states at a given carrier concentration could cause an increase in S.

At present, resonant levels have been proven successful in both PbTe with Tl as the impurity 36 ,

37 and Bi 2Te 3 with Sn as the impurity as shown in Figure 22. The Pisarenko relation varies slightly with different scattering parameters, λ, so some deviation from the calculated values can be expected depending on what the dominant scattering mechanism is. Figure 22b demonstrates the variation in Seebeck coefficient for different dopants in Bi 2Te 3 from λ=1/2 to λ=1 representing varying degrees of different scattering mechanisms.

Figure 22. Resonant impurity levels from literature, a)PbTe: Tl, b) Bi 2Te 3: Sn

A considerable increase in S for both these systems above the Pisarenko relation is observed as a result of an increase in density of states.

Chapter 2: Measurement Methods and Objectives

Here we describe how the measurements were performed through various experimental setups and discuss the possible sources of error. Any variations from these methods are reported in individual studies.

Test Methodology

Powder X-ray diffraction (XRD) data was taken on all samples to confirm correct crystal structure and to determine lattice parameter. Using Braggs law and the peak angles, θ, from the

XRD data, we can use the formula

(53) where d is a plane spacing parameter, n in an integer, and is the radiation source wavelength. In this work, all systems are cubic structure, so we can solve for lattice spacing using

(54) √

In this equation, a is the lattice parameter, and h, k, and l are the Miller indices of the Bragg plane for which the radiation source reflects. These are well established in literature. Temperature dependent XRD measurements from room temperature to 770 K are used to obtain linear thermal expansion coefficients, α, for some of the compounds .

Parallelepiped-shaped samples, with dimensions ~2x2x5mm 3, were cut from ingots for thermal conductivity measurements between 2 and 200 K, although some variation in geometry occurred..

Low-temperature thermal conductivity measurements were carried out using a quasi-static heater and- sink technique in a Quantum Design Physical Properties Measurement System (PPMS). Low

50 temperature Seebeck coefficient and resistivity were also taken using the Quantum Design PPMS as shown in

Figure 23. Geometrical error can give absolute values of error up to 10%. Error bars, shown in the data plots, increase slightly for temperatures below 4K where additional error can occur due to very small temperature gradients and imprecision in thermometry. Error bars also increase above 100K due to unavoidable radiative losses that are difficult to account for. Thermal conductivity data taken above 200K on the PPMS is not reported due to large uncertainty on the emissivity of the sample and our inability to correct for the error.

The specific heat, C, of all samples is measured using a quasi-adiabatic calorimeter with an accuracy of approximately 2%. The sample size for all specific heat tests was ~50mg.

Disc shaped samples ~10mm in diameter and 1mm in thickness are used to obtain the high- temperature values of κ. A flash diffusivity method is used, with an Anter Flashline 3000, to

obtain thermal diffusivities, D. Using equation 55,

κ =C ρdD (55)

we are able to obtain our experimental high temperature thermal conductivity values, where ρd is

the room temperature value of density unless otherwise noted in text. Density measurements are

performed either through geometrical method or Archimedes method.

Figure 23. PPMS TTO

S, ρ, R H, and N measurements are taken in stepped intervals from 78K up to 630K using a conventional liquid nitrogen cryostat. Samples are cut into parallelepipeds with dimensions of approximately 1mm x 1mm x 5mm for cryostat measurements. Wires, with diameter 0.025mm, are spot welded, when possible, to the sample and silver epoxy is added for mechanical stability.

In materials with very low thermal conductivity, the thermal shock imposed with spot welding can cause cracking, so wires were added simply using epoxy. The diameters of the wires are intentionally kept small to minimize the effect of Joule heating. Two copper-constantan thermocouples are attached to one side of the sample to measure the temperature at two distinct points. Voltage measurements are taken from the two copper wires to determine Seebeck coefficient. A resistance heater is connected to the top of the sample to provide a temperature gradient for Seebeck and Nernst measurements. The sample is attached to a thermally and electrically insulated base. Current wires are attached on both the top and the bottom of the sample and a brass plate is attached to assure uniform thermal contact. Resistivity is measured between the two copper thermocouple wires while a current is applied on the sample. Hall and

Nernst measurements are taken using the copper wire from the thermocouple and a wire placed on the opposite side of the parallelpiped while a magnetic field is imposed in a transverse direction. The magnetic field is varied typically in the range of -1.5T to 1.5T at each temperature point. Each measurement is taken after the sample reaches steady state. The complete cryostat material setup is as shown in Figure 24.

Figure 24. Cryostat experimental setup

Error analysis

The effect of a 1.5T magnetic field on the sensitivity of the type-T, copper-constantine thermal couples was tested and found to be less than 3% from 80K to 400K. No correction is applied in any of the reported data.

The error on Seebeck coefficient is mostly due to small amounts of surface contact heating and is estimated to be approximately 3%.

Electrical resistivity is measured using a standard 4-wire AC measurement, where the main source of error is the geometrical uncertainty. Small innacuracies in sample dimension can lead to an estimated error of up to 10% depending on dimensions. High resistance samples were measured using a DC source and have a slightly higher uncertainty due to small amounts of internal heating upon application of current. 53

Nernst error is estimated to be ~5% and Hall error between 5 and 10%. The standard correction is applied to measured adiabatic Nernst coefficient to establish isothermal Nernst from the adiabatically measured coefficient. 38

Analysis Methods

Various analysis methods can be used to help interpret the data gathered from the various experiments that we run. The “Method of Four Coefficients” is one that is employed frequently, especially in lead chalcogenide systems. An outline is given below to describe the method, however, a more detailed analysis as well as additional information on ways to handle exceptions to the assumptions, can be found in literature. 39,5

Method of Four Coefficients

In the case where a parabolic band assumption can be made, we know that the density of states,

DOS, is proportional to the square root of the energy dispersion

(56) ∝

We can use a scaling coefficient τ0 to define electron relaxation time, τe as

(57) where is the scattering parameter. Some literature sources describe the exponent of the energy term in equation 57 as simply “ ” rather than “ -1/2”, so that the ½ is incorporated into the scattering parameter. Care must be taken to differentiate between the two possibilities when discussing scattering. In this text, we will use notation as defined in equation 57.

For lead chalcogenides, the Fermi surfaces are ellipsoids of revolution. Longitudinal electron wave vector, k l, along the [111] direction, and two transverse wave vectors, k t, normal to k l, describe ellipsoid directions. The electron dispersion relation can now be described 54

ℏ ℏ (58) 1 + ∗ + ∗ where m * is the effective mass of the electrons and ℏ is Plank’s constant. We can define a DOS

∗ ∗ ∗ effective mass ∗ where N D is the degeneracy number. For lead chalcogenides, there are four ellipsoids of revolution so N D4.

Next we define the density of states, gE as

∗ √ 59) ℏ ′

Where ′ is simply the derivative of : . For this non-parabolic case, the relaxation time is reformulated as

60)

Four equations representing measured parameters, σ, R H, S, and N are given as a function of

, m *, T, and Fermi-Dirac distribution function, f in reference 39. These equations are d o solved simultaneously to establish , Fermi energy, E , m *, and µ. Table 1 explains the scattering F d parameter expected with the most common electron scattering types.

Table 1. Scattering parameter associated with scattering mechanism Scattering parameter, λ Scattering mechanism

0 Acoustic phonon

½ Neutral impurity

1 Polar optical phonon

2 Ionized impurity

Objectives

We use a systematic approach to understanding material systems. When possible, collaboration with theoreticians is a preliminary step. Materials are developed through a cyclical series of steps as demonstrated in Figure 25.

Synthesis

Sample Re-design Preparation

Analysis Measurement

Figure 25. Process for developing and testing materials

An idea is found based on theoretical calculations or other inspiration. After the idea is conceived, a literature review is conducted to gather information and to assure that efforts are not being duplicated. After these initial steps a cycle of material synthesis, sample preparation, measurement, data interpretation, and re-design occurs. The material is made through one of various available methods such as powder processing or melting. The materials are then cut and prepared for testing by affixing various wires or probes. The desired properties are then tested and the results are analyzed. The test results help guide us to establish the appropriate changes to 56 the materials. In the case of a doping study, the base compound is synthesized and tested, a dopant is added, then after the test results are analyzed, it is decided on whether more or less dopant should be added, or if another step is necessary. In most cases the results are discussed with colleagues and new results are published when it is deemed an appropriate contribution to the field. A summary of specific steps and details of the experimental setup for this research is contained in the following sections as it pertains to each separate study.

Chapter 3: Highly Polarizable Anharmonic Crystalline Materials – I-V-VI 2 Compounds

Developing crystalline dielectric solids with minimal lattice thermal conductivity κL improves energy conservation because it enables the development of higher-efficiency thermoelectric materials, refractories, and thermal barrier coatings such as those that insulate turbine blades or cylinder walls from the combustion gases in internal combustion engines. Reducing the κL of solids has typically been achieved by alloying, by increased phonon scattering on nanostructures, by selecting solids with soft phonon modes, or introducing “rattling” phonon modes. Recently, anharmonicity and strong phonon-phonon interactions (PPI's) have been identified as the origin of low thermal conductivity in several specific materials, 34,40 ,41,42,43 representing a new paradigm for lowering lattice thermal conductivity. As discussed previously, the anharmonicity of interatomic bonds is intrinsic to the material, therefore it is relatively unaffected by impurity levels, crystallinity, grain size, and other structural variations. As a result, it presents a particularly attractive strategy for designing low- κL thermoelectric materials. Previous to this work, generalizing these ideas to other materials has been difficult to do by traditional trial-and- error approaches.

In this study, we focus on explaining the root cause of PPIs through theoretical and experimental work on materials in which anharmonicity and the associated PPI's are the main source of resistance to heat conduction. Using first-principles density-functional theory (DFT) calculations

(conducted by Vidvuds Ozolins of UCLA) of the phonon frequencies, in combination with experimental synthesis and measurements, we demonstrate that lone s 2 pair electrons on the group

V atoms (As, Sb, Bi) can create very anharmonic bonds in a whole class of simple high-symmetry crystalline solids. We consider rocksalt-based ABX 2 compounds with A = Cu, Ag, Au, Na, K,

Rb, Cs as the group I elements, with B =As, Sb, or Bi as the group V elements, and X = S, Se, or

Te as the group VI elements as a case study.

Using all possible combinations, sixty-three such ABX 2 compounds are established. This list was extended to 72 compounds by including A=Tl. First-principles DFT calculations show that many of these compounds have dynamically unstable phonon branches and are therefore expected to either adopt non-cubic crystal structures, or to be thermodynamically unstable – that is they do not exist in ABX 2 form. The compounds that are marginally stable are found to have highly anharmonic acoustic (heat carrying) phonons.

Our systematic approach to understanding the effect of lone pairs on thermal conductivity utilizes a strong collaboration between first-principles density functional theory (DFT) calculations of the phonon frequencies and experimental synthesis and measurements. We demonstrate through calculations that lone s 2 pair electrons on the group V atoms (As, Sb, Bi) can create very strong polarizability which leads to experimentally confirmed strong anharmonicity in rock-salt ABX 2 compounds, which are simple high-symmetry crystalline solids. Additionally, we experimentally show that non-rocksalt (distorted) ABX 2 compounds do not demonstrate the strong anharmonicity that leads to a minimized . L

The physical origin of anharmonicity in the ABX 2 compounds can be traced to the existence of stereochemically active s 2-orbitals of the group V element, as can be understood by comparing

34 2 1 the thermal conductivity of AgSbTe 2 with that of AgInTe 2. In AgInTe 2, all of the 5s 5p valence electrons of indium participate in forming sp 3 hybridized bonds with tellurium, resulting in a 59

3 zincblende-related crystal structure. In AgSbTe 2, sp hybridized bond formation is more difficult because of the valence configuration of Sb. The 5s electrons in the 5s 25p 3 configuration are at a much lower energy than the p electrons. Accordingly, the crystal coordination changes from tetrahedral in AgInTe 2 to octahedral in AgSbTe 2. The group V element becomes trivalent and forms polar covalent bonds with the chalcogen by sharing its p electrons, while the s 2 electron pair forms an isolated ("lone pair") band, which couples to and is repelled from the valence band of predominantly noble metal d and chalcogen p character. As a consequence, the s-electron shell of the group V element is easily deformed by lattice vibrations, resulting in a strong anharmonicity due to the nonlinearity of the electronic response. It is well understood that lone pair effects may lead to structural instabilities, which is also demonstrated by multiple stable phases indicated in phase diagrams for various ABX 2 compounds. We show that in the rocksalt- based ABX 2 compounds considered here, the competition between the covalent bonding and lone pair repulsion in some cases leads to crystal instability. In the cases where the rocksalt structure is stable, this effect leads to high acoustic mode Grüneisen parameters , defined for each phonon i mode as the negative of the logarithmic derivative of its frequency with respect to the crystal i volume V:

(61) − As previously discussed, the Grüneisen parameter is a strong indication of PPI’s. Large leads to a situation where atoms in an anharmonic solid are displaced by a wave, resulting in a change of elastic properties of the medium as seen by another wave vary along the path of the first. This variation in elastic properties results in strong interactions and ultimately phonon scattering.

Because quantifies by how much the stiffness of the bonds is affected by the interatomic i distances, the PPI probability is proportional to 2. i

When the phonon modes that carry heat are the modes with high , they strongly lower lattice i thermal conductivity. The same principle applies when the phonons that carry heat collide with phonons that have high . The lone pair can also affect phonon modes that are not primarily heat i carrying modes, such as in the case of lead salts. PbTe has lone pair electrons that affects one of the optical modes by lowering the energy of the modes, resulting in interaction with an acoustic heat carrying mode. The end result of the interaction is a lower thermal conductivity, but not to the same degree that the thermal conductivity is lowered when it directly affects an acoustic mode such as in the I-V-VI compounds. To experimentally prove the effects of high on the 2 i reduction of thermal conductivity, where subscript i is an acoustic mode, we choose several of the

72 modeled compounds with a large range of values for experimental verification. We study large-grained polycrystals of AgSbSe 2, AgBiSe 2, Ag 0.73 Sb 1.12 Te 2 (the phase stable version of the theoretical compound AgSbTe 2 – see doping studies later), NaSbSe 2, NaBiTe 2, and NaSbTe 2. We also have recently added thermal conductivity measurements of NaBiS 2 and LiBiS 2. We show that the anharmonicity in these compounds can be so intense that phonon scattering reduces to L the amorphous limit in materials with the highest . Structural disorder is known to exist in min nominal AgSbTe compounds and is sometimes invoked as the mechanism behind its low To 2 L. address this point, we add here a study of Ag xNa 1-xSbTe 2 alloys and later a study of variation of stoichiometry in silver antimony telluride, and show that varying defect concentration and alloy scattering does not affect . This further illustrates that the phonon scattering mechanism is L likely intrinsic.

Methods

Synthesis & measurements

Polycrystalline samples of all compounds were prepared by loading elements in appropriate amounts into vacuum-sealed quartz ampoules, with graphite liners for the Na-compounds. The 61 ampoules, containing the elements, were heated at a rate of 2 K/min to approximately 50K above the compounds liquidous temperature. The samples were then cooled at -1 K/min to the anneal temperature noted in Table 2.

Table 2. Summary of heat treatment used in this study Composition Anneal Time (days) Anneal Temperature (K)

AgSbSe 2 3 773

Ag 0.73 Sb 1.12 Te 2 3 773

NaSbTe 2 1 788

NaSbSe 2 1 853

NaBiTe 2 1 808

AgBiSe 2 7 842

We prepared Ag 0.73 Sb 1.12 Te 2 rather than AgSbTe 2 because that composition is stable over a wide temperature range without precipitating second-phase particles. Samples of AgBiSe 2 were subjected to two different heat-treatments. AgBiSe 2 can be prepared in either rocksalt or trigonal structure, the first of which is thermodynamically stable above 560K and the second is stable from 393K to 560K. 44 Trigonal samples were obtained by water quenching from above the liquidous, cubic samples were obtained through a long anneal in the thermodynamically stable temperature regime. After samples were brought to room temperature, the crystal structure is phase stable – that is, it retains its crystal structure for a length of time much longer than required for this study.

X-ray diffraction (XRD) data on all samples are reported in the experimental section; all index to literature values of rocksalt structures, except for trigonal AgBiSe 2. The lattice parameter for

Ag xNa 1-xSbTe 2 alloys was found to follow Vegard’s law, suggesting formation of solid solution.

The samples appeared to be large-grained polycrystals, with grain sizes on the order of 10-100

µm for the Na and Li compounds and 0.1 to 1 mm for the Ag compounds.

Thermal conductivity measurements were conducted as described in the measurements section in

Chapter 2. The error bars shown in the experimental data include thermometry imprecision below 4K and radiative heat losses above 100K. Data is not reported above 200K due to uncertainties associated with radiative loss and emissivity of the samples.

Computation

To visualize the electronic origins of lone pair polarization effects, we use the polarization density, which describes the microscopic current in response to an external perturbation (atomic displacement or external electric field). Polarization density is calculated by Vidvuds Ozolins at

UCLA using the DFT linear response method.

Vibrational properties

Since only a few of the 72 possible ABX 2 compounds have been studied experimentally, one needs to determine the type of atomic ordering on the cation sites of the rocksalt lattice. For

AgSbTe 2 and AgBiTe 2, cubic D4-type cation ordering is predicted to persist at all temperatures below the melting point, 45 and here we assume that other noble metal (A=Cu, Ag, Au) compounds will have the same ordering as AgSbTe 2. The calculated ordering energies suggest that the other compounds, if they are stable in rocksalt-based structures, could also be ordered

Our experimental x-ray data collected on the synthesized compounds does not contain any superstructure peaks that would indicate the presence of cation ordering. The reasons for this apparent discrepancy are not clear. This difference is expected to be insignificant for the calculated elastic constants, acoustic velocities and Grüneisen parameters, since the latter are relatively insensitive to cation ordering.

Calculated phonon dispersions of 72 compounds described previously were calculated and several of the ABX 2 compounds have dynamically unstable phonon branches (imaginary frequencies) at the calculated equilibrium lattice parameters. These are compounds with mostly lighter group V and VI elements (CuSbSe 2, NaAsS 2, RbAsS 2, etc.) that either do not exist at the

ABX 2 stoichiometry or assume non-rocksalt based crystal structures. The crystal structures of those that have been synthesized are found to be either not rocksalt or resemble highly distorted variants of rocksalt. The structural distortion relieves a large share of the repulsion associated with the lone s 2 pair of group V element, the phonon properties of the structurally distorted compounds are not expected to be particularly anomalous and have not been considered in this work. For CuBiSe 2, AgAsSe 2, and AgSbSe 2 slightly unstable phonons at the Brillouin zone boundary are predicted; available experiments list them as disordered rocksalt at high temperatures. We hypothesize that the harmonically unstable short-wavelength modes are anharmonically stabilized at high temperatures. The calculated phonon dispersion relations and

Grüneisen parameters for all stable compounds are given in Ref. 49. Compounds with the highest Grüneisen parameters and strongest anharmonic effects lie on the boundary between the dynamically stable and dynamically unstable compounds, representing cases where the repulsion between the lone s 2 pairs and valence electrons is barely stabilized by covalent bonding forces.

The highest Grüneisen parameters are found in compounds where A is a noble metal. However, many of the noble metal compounds are predicted to have metallic band structures and therefore not expected to be promising thermoelectrics. Additionally, these elements are costly and are subject to ionic conduction.

Figure 26. Red solid lines: Calculated phonon dispersion of rocksalt-based AgBiSe 2. Blue dashed lines: calculated phonon dispersion for a lattice constant expanded by 1.7%.

We begin by discussing typical phonon anomalies found in rocksalt-based ABX 2 compounds.

The calculated phonon dispersion curves of AgBiSe 2 are shown in Figure 26 at two lattice parameters differing by 1.7%. The expansion can be representative of either normal lattice motion or thermal expansion. Unstable imaginary frequencies are shown as negative values. The negative value at the L-point indicates that the compound is not stable at this volume and may distort or melt to lower the total energy. Upon lattice expansion, the frequency of the transverse acoustic (TA) branch at the zone-boundary L-point drops from 25 cm -1 to an unstable imaginary value of i32 cm -1, resulting in an abnormally high mode Grüneisen parameter of more than 100.

The frequencies of the optical phonons at the L-point are also very sensitive to changes in the lattice parameter and the lower-lying branches have Grüneisen parameters that are similar in magnitude to those of the TA mode. Since the intensity of phonon-phonon scattering scales as 2 i (i individually taken for both the heat carrying phonon and the phonon it interacts with), phonon scattering is very frequent and the phonon mean free path is expected to be very short. The

65 thermodynamic average (defined as ), which includes contributions from all phonon 〈 〉 modes and frequencies weighed by the mode-dependent heat capacity function, is found to have a less anomalous, but still a very high value of 2.5 at 300 K.

We next turn to the electronic origins of the phonon anomalies in Figure 26. The effect of lone- pair electrons on the anharmonicity of the crystal is best illustrated in the calculated polarization density contour plots for NaSbSe 2 in Figure 27, showing the electronic response to two types of perturbations. In this figure, Sb atoms are shown as orange spheres, Se as yellow spheres, and Na as light blue spheres In Figure 27 (a) & (b), the perturbations are applied external fields E along

[ ] and [111], respectively, and the macroscopic response of the crystal. The calculations 110 suggests strong polarization effects are due to Sb s- Se p bonds. In Figure 27b, the contour values are half those shown in Figure 27(a) because the dielectric constant along the [111] direction is less than half of the value found along the [ ] direction. There seems to be little polarization 110 of the Sb s-Se p bonds. In Figure 27 (c) and (d), the perturbations are the displacements of Se ions, and the macroscopic response is described by the Born effective charge tensor.46 Figure 27c shows the same type of displacement as that of the Te atoms in the anomalous TA mode in Figure

26. The induced polarization density extends all the way to include the lone pair s orbital on Sb, illustrating that these electrons are nearly 100% polarized by the displacement. Figure 27d shows the polarization in response to a Se displacement along the [111] direction shows no unusual behavior.

The behavior of the Sb s-electrons when the electric field E is along the [ ] axis (Figure 27a) 110 is markedly different from the electronic response when E is along the three-fold symmetric [111] direction (Figure 27b), showing a much higher polarization of the antimony s 2 orbitals for E along the [ ]. Accordingly, the dielectric constant is double the value of . 110 110 110

Furthermore, in response to a displacement of a selenium ion along the [ ] direction (Figure 110 27c), which is also involved in the anomalous TA mode in Figure 26, the induced polarization density encompasses the lone s 2 pair orbitals on antimony. As a result, the calculated Born effective charge of Se for a displacement along the [ ] direction is twice as large as its formal 110 valence. In contrast, Figure 27d shows that for a selenium displacement along the [111] direction, the induced polarization does not include the lone pair orbitals on Sb, and the calculated Born effective charge of Se is normal. This analysis clearly shows that the antimony lone pair electrons are strongly polarizable along the [ ] direction perpendicular to the trigonal 110 axis, resulting in a sensitivity of the Se-Sb bond stiffness to Se displacements along that direction and serving as the physical mechanism for the high .

a b

c d

Figure 27. Comparison of the calculated polarization current density (green blob) in NaSbSe 2 under an applied electric field, and when Se atoms are displaced by phonons.

Experimental

XRD results are reported in Figure 28, where all samples indexed to literature indicated rocksalt structures except for trigonal AgBiSe 2 which was made intentionally to compare and contrast behavior between rocksalt and non-rocksalt structure.

AgBiSe2 tri

AgBiSe cubic 2

AgSbSe 2 ) . U . A (

t Ag0.73Sb1.12Te2 i s n e t n I

NaSbSe2

NaSbTe 2

NaBiTe 2

20 30 40 50 60 70 Angle (2θ)

Figure 28. X-ray diffraction data for Na and Ag based compounds The dots are reference angles from the International Center for Diffraction Data PDF database.

Figure 29. Isobaric specific heat C of the compounds indicated; the trigonal (Tri) and rocksalt (R- S) versions of AgBiSe 2 have the same C within the measurement uncertainties.

The specific heat of all samples is reported as a function of temperature in Figure 29, and numerical values (C 300K ) of C at T=300 K are given in Table 3. The trigonal and the rocksalt versions of AgBiSe 2 have the same specific heat. The experimental C(T) follow Debye laws, with characteristic T 3-behaviour at low temperatures up to approximately 15 K. The fitted Debye temperatures , based on C, are determined with an accuracy of +5 K, and also shown in Table D 3. values are much larger than the calculated zone-edge Debye temperatures , because D i optical phonons contribute to the heat capacity, and the dispersion curves for some acoustic branches are non-monotonic, which reduces the zone edge temperature to below what would be

70 expected if they were monotonic (as in the Debye approximation). At T = 300 K, the heat capacities reach the classical Dulong-Petit values (C DP ). These are calculated for each compound and shown in Table 3, where they can be compared to C 300K . Calculated and literature (for

AgSbSe 2 and Ag 0.73 Sb 1.1 Te 2) values of bulk modulus, B, are also reported in Table 3.

Θ Θ 300K (J/gr K) (K) ) -1 K 0.03 0.314 165 -1 ± (100K) C

κ κ 0.3 0.52 Measured

γ γ ± 3.72.1 0.42 0.252 0.54 143 0.207 144 a b 45 129 B (GPa) ) m (W -1 1 - 1.5

± K

α α -6 (10 i,TA2 compounds. Θ Θ 2 VI - V - i, TA1 I

Θ Θ i, LA

Θ Θ DP (J/gr K) (K) (K) (K) ) Calculated -1 K -1 (100K) C min (W m (W κ κ

γ γ B 4435 1.734 1.6 0.5678 1.5 0.44 0.32967 3.5 0.35 114 0.24974 2.3 0.43 94 76 0.21 2.5 0.44 0.257 56 63 89 0.47 62 0.206 22.5 40 85 65 0.21 47 26.7 56 53 52 - 26 56 24.5 1.8 - 30.8 0.74 1.9 0.26 0.63 31.8 146 0.202 - 136 2.9 0.63 0.212 135 (GPa) 2 . Experimental and Calculated parameters of several of parameters Calculated and . Experimental 2 2 2 2 2 Te 3 1.1 Sb 0.73 Table NaSbSe AgBiSe Material NaBiTe AgSbSe NaSbTe

Table 3 further gives the experimental high-temperature (T >> ) linear thermal expansion D coefficient α determined from XRD measurements, and the thermodynamic average Grüneisen

parameters are also given. Estimated errors are ±5 K in and ±0.002 J g -1 K-1 in C . D 300K

Figure 30 shows the temperature dependence of the x-ray diffraction peaks. The two different plots are the same data visualized in different ways. XRD peaks were measured at 35̊ C then at

100̊ C degree intervals from 100̊ C to 500̊ C. The top plot shows darkened lines interpolating between the different measured peaks.

Figure 30. Typical temperature-dependent x-ray spectra of measured cubic compounds. This measurement is of Ag 0.366 Sb 0.558 Te

The lattice parameter was calculated at each temperature based on the x-ray diffraction data.

Using equations 53 & 54, isotropic linear thermal expansion, αLTE , was calculated based on the linearly increasing lattice parameter. The values are reported in Table 3. The individual molar 73 volume established from calculated lattice parameter is plotted as a function of temperature in

Figure 31.

Figure 31. Temperature dependent lattice parameter was used to calculate the linear thermal expansion coefficients.

The experimentally derived was calculated by

(62) where B is bulk modulus, and V is the volume of a unit cell. Literature values of the bulk moduli

47 48 were found only for AgSbSe 2 and AgSbTe 2 : the agreement with the first-principles calculated

average Grüneisen parameters is good in those cases. In the other cases we use the calculated B,

and the agreement is less satisfactory, which can be attributed to the well-known tendency of the

LDA calculations to underestimate lattice parameters and overestimate bulk moduli.

Figure 32. a) Low temperature thermal conductivity of various I-V-VI 2 compounds, b) comparison of to measured min

Figure 32 shows a) low temperature thermal conductivity measurements for various I-V-VI 2 compounds as well as b) a comparison between measured and calculated using equation min 31. The thermal conductivity of all compounds below 200 K is attributable to phonons. It is probable that there is a small contribution of radiative heat losses above 100 K, and their possible effect is included in the error bars shown in Figure 32b. The trigonal AgBiSe 2 compound has a thermal conductivity that behaves in a way that is typical for crystalline semiconductors, with a maximum in (T) around T=35+5 K. Below 20 K, the thermal conductivity increases with increasing T, reflecting the increase in phonon density with increasing temperature, as

75 demonstrated in C(T). Above 40 K, its thermal conductivity is limited by Umklapp processes and

(T) decreases with increasing temperature). Around 35K, various defects (vacancies, isotopes) contribute to phonon scattering near the maximum. The measurement reveals a high-T dependence that tends to a T -1 law, and a low-T dependence that tends to a T3 law. All rocksalt compounds have a (T) that is unusual for large-grained crystalline dielectrics, with a plateau at L high-T and a shallow maximum or a hump near 5K. We suggest that phonon-phonon interactions remain frequent in AgSbSe 2 at all temperatures, but that at T > 30K, they are resistive Umklapp

Processes and at T < 30 K, they are non-resistive Normal Processes that give excess conduction.

This could provide an explanation for why, at T>30K, κ(T) mimics (T), while below this min temperature, there appears to be a “hump” in the data.

In contrast to the normal behavior of trigonal AgBiSe 2, the other rocksalt-based compounds all exhibit thermal conductivities that have unusual characteristics for large-grained crystals of dielectric solids. First, anomalously low values of (T) from 0.4 to 0.75 W m -1 K-1 are found at temperatures above 35 K. Second, thermal conductivities are relatively temperature independent.

Finally, an excess contribution to (T) appears in several compounds at temperatures below approximately 10 K, whereas no abnormal phonon modes are observed in the heat capacity. We point out that compounds with higher Grüneisen parameters have more pronounced anomalies in Figure 32, and use NaSbTe (calculated =1.6 in Table 1) to compare with AgSbSe ( =3.4). 2 2

The calculated values of (T) for AgSbSe and NaBiTe determined using equation 31 are min 2 2 demonstrated in the figure on the right with dotted lines. We chose these two materials to demonstrate in comparison to measure because AgSbSe has the highest of all the min 2 compounds synthesized in this study and NaBiTe was among the lowest. was calculated 2 min using the theoretical values for the acoustic Debye frequencies along the [011] axes i (corresponding to the transverse acoustic mode which is most effected by the lone pair electrons). 76

These values are available for reference in Table 3. At temperatures of approximately 20 K and above, the thermal conductivity of AgSbSe reaches (T), which explains why the expected 2 min behavior of decreasing (T) with increasing T does not hold. In contrast, NaBiTe has a thermal 2 conductivity value roughly twice the theoretical minimum at 100K. Numerical values of the min measured thermal conductivities and theoretical minimum thermal conductivities are given min in Table 3 for all compounds at T=100 K. The difference between and generally scales min with the Grüneisen parameter , which suggests that phonon Umklapp scattering is responsible for the low values of thermal conductivity. The excess in conduction at temperatures below 20 K is understood by considering the difference between Umklapp and Normal processes, which were previously described in Chapter 1. The polarizability of the lone-pair electrons and acoustic mode

Grüneisen parameters are mostly temperature-independent, in other words, the phonon-phonon interaction matrix elements depend only on the interatomic potential, not on the temperature.

Therefore, the probability of phonon-phonon interactions, due to the lone pairs, is not expected to have a temperature-dependence which could explain the excess thermal conductivity at low temperatures (T < 20K). However, as the temperature decreases below one half of the acoustic

Debye temperature , an increasingly large number of phonons have wave vectors shorter than i one half the distance to the Brillouin zone boundary. As a result, at T < /2 most phonon- i phonon interactions are non-resistive Normal processes. The number of Umklapp processes decreases with temperature below some fraction of the acoustic zone edge temperature, which leads to a decreasing effect on minimizing thermal conductivity. This ultimately leads to additional heat conduction compared to the case where T > /2. i

An alloy system with compositions between Ag 0.73 Sb 1.12 Te 2 and Ag xNa 1-xSbTe 2 were synthesized and tested. XRD tests confirmed that the lattice constant followed Vegard’s law as shown in

Figure 33, suggesting that the compounds are solid solutions.

Figure 33. Lattice parameter for Ag 1-xNa xSbTe 2 alloys as a function of composition.

Figure 34 shows thermal conductivities of Ag 0.73 Sb 1.12 Te 2 and Ag xNa 1-xSbTe 2 alloys (with x=0,

0.5, and 0.8) from 300 to 570 K. For Ag 0.73 Sb 1.12 Te 2 at T>500 K ambipolar electronic contribution

sets in and we observe an increase in thermal conduction due to this extra heat transfer

mechanism. Figure 34a shows that the thermal conductivity, (T) is between 0.45 and 0.65 W m - 1 K-1 and slightly decreases with temperature. Figure 34b shows the composition-dependence of at 377 K, illustrating a minimal variation with x. Reduction in thermal conductivity is usually observed in alloys due to the scattering of phonons by the mass difference between atoms and by local strain fields, however, in this case we do not observe this effect. We conclude that alloy scattering does not further reduce the thermal conductivity below the values in the pure end compounds, reinforcing the conclusion that intrinsic phonon-phonon interactions limit to the theoretical lower limit . min

Figure 34. Thermal conductivity of NaSbTe 2, Ag 0.73 Sb 1.1 Te 2 and their alloys, (a) as function of temperature, and (b) as a function of composition at T=377 K.

The electrical resistivity of most compounds studied experimentally is shown in Figure 35.

NaSbSe 2 is too electrically insulating to be measured in this particular experimental setup, however, the resistivity was measured at higher temperatures in a doping study discussed later: It showed semiconducting behavior, but exhibited a very large band gap. Ag 0.73 Sb 1.12 Te 2 is a very narrow-gap semiconductor. All of the electrical properties imply that the density of native defects, or at least the extent to which they act as donors or acceptors, is limited to an extremely small amount.

Figure 35. Electrical conductivity of some of the measured compounds.

Conclusions

In summary, ab initio calculations link the polarizability of lone-pair electrons to phonon instabilities in ABX 2 compounds and explain the extremely high Grüneisen parameters of

marginally stable rocksalt compounds. We demonstrate how anharmonicity leads to phonon-

phonon interactions that limit the lattice thermal conductivity, through Umklapp processes, to a

value reaching the amorphous limit in cases where the Grüneisen parameter is large. Thermal

conductivity reductions by this mechanism are intrinsic to the nature of the chemical bond in the

solid. Since they do not depend on grain size or defects, they are expected to be robust and

insensitive to variations in synthesis or processing techniques. The last point, in particular, is

demonstrated in the next section on doping studies. Because the valence band structures of all

ABX 2 compounds studied here are similar and quite favorable for thermoelectric applications,

several of them promise to be good thermoelectric materials once doping techniques are

developed. More generally, we suggest that lone s 2 pair electrons are likely to be responsible for 80 the low thermal conductivity observed experimentally in many other classes of dielectric crystals, and can be exploited in the ab initio design of new materials with tailored thermal properties, including in refractory oxides (X=O).

Doping Studies

Doping studies were conducted on several of the I-V-VI 2 compounds that exhibit intrinsically limited thermal conductivity. The first study discussed involves varying the composition of off- stoichiometric silver antimony telluride in order to establish the effect of defect composition and second phases on thermal conductivity. Focus was then shifted to two compounds that replace the traditional expensive transition metal, Ag, with a much less expensive and lighter weight alkali metals, Na and Li. Additionally, tellurium is a commonly used, yet expensive element, that we aim to replace with selenium in these doping studies. We explore sodium and lithium based compounds and show both success and failure in our attempts at doping these large gap semiconductors.

Off-Stoichiometric Silver Antimony Telluride

AgSbTe 2 is among a class of compounds that shows minimal thermal conductivity as a result of strong anharmonicity resulting from the unbonded lone pair of s 2 electrons from antimony. 49 As a result, AgSbTe 2 has been a material of interest in the thermoelectric field. While there is an extensive body of literature on this system, 50,51 many of the ternary Ag-Sb-Te and pseudobinary

Sb 2Te 3-Ag 2Te phase diagrams are contradictory. It has been shown experimentally that stoichiometric AgSbTe 2 is a compound that is not phase stable, and therefore could never be used in any practical applications where thermal cycling could occur. AgSbTe 2 is a semiconductor with a very small energy gap Eg ~ 7.6 3 meV. 50 Due to this small gap (~100K) being less than ±

81 the thermal excitation energy at room temperature, AgSbTe 2 is a two carrier system having both holes and electrons. Native defects produce a large excess of holes with concentrations p>> n; however, the mobilities are very different, µe>> µh. This causes the material to have an unusual combination of positive thermopower and negative Hall coefficient. Extrinsic p-type doping has proven effective: zT =1.2 is reported at 400K on AgSbTe 2 when doped with small amounts of

52 Na 2Se. Even with this zT , the material currently cannot be used at higher temperature due to a

417K phase transition caused by a parasitic Ag 2Te phase that precipitates in the stoichiometric material.

This work identifies the composition range of off-stoichiometric Ag 0.5-xSb 0.5+y Te 1±z (or similarly

Ag 1-2x Sb 1+2y Te 2±2z ) compounds that avoid secondary phases and their transitions. Also reported here are data on the thermoelectric properties of these compounds and attempts to optimize the zT through intrinsic defect-doping. 38 This study aims to determine the effects of secondary phases and defect concentration on the intrinsically limited thermal conductivity while establishing an off-stoichiometric composition of silver antimony telluride that does not form minor phases of

Ag 2Te or Sb 2Te 3, which are common second phases in the Ag-Sb-Te system.

Selected compounds in the range of Ag 0.366 Sb 0.48-0.569Te 0.95-1.05 were prepared and subjected to a latent heat trace in a Differential Scanning Calorimeter (DSC) in order to identify any secondary phases. The preparation method included loading high quality pure elements into a quartz ampoule, then placing under a vaccum of 10 -6 torr. The elements were then melted at 950C and slow cooled to 500C where they were annealed for 5 days.

Table 4 qualitatively summarizes the findings concerning the phase purity of the samples.

Several of the samples were found to have a subtle indication of a low temperature phase transition from Ag 2Te and possibly other compositions of silver telluride.

Table 4. Qualitative analysis of exothermic reaction in latent heat trace. Silver composition was held constant at 0.366. [Sb] [Te] Phase Transition Intesity Temperature (K) 0.558 1.05 606 strong 637 strong 0.53 1.05 606 strong 633 strong 0.54 1.03 603 Med 633 Strong 0.558 1.02 443 Light 683 Med 0.58 1 603 Light 693 Light 0.569 1 None - 0.558 1 None - 0.55 1 433 light 533 light 623 light 723 light 0.53 1 553 light 713 med 0.5 1 606 strong 633 strong 0.48 1 606 strong 633 strong 0.563 0.95 443 light 633 light 693 med 0.558 0.95 453 med 688 med 0.53 0.95 423 med 663 med

Samples that exhibited no apparent secondary phases all had Ag:Te ratios of 0.366:1 and were in the range of Ag 0.366 Sb 0.558-0.569 Te. In Figure 36, we present a modified ternary diagram characterizing the samples reviewed with the thermal treatment described in the preparation of these samples. By conducting a qualitative analysis of the exothermic reactions, we find that there is a distinct region of samples that do not exhibit a phase transition and an area encompassing the

83 single phase region in which the latent heat trace revealed minimal amounts of 2 nd phase material.

They are denoted by the blue triangles and green circles respectively. Other samples were found to have a larger indication of phase transitions from different compositions of silver telluride as well as antimony telluride. These samples are indicated by orange squares and red triangles on the phase diagream. We see that the single-phase samples form a region that (i) substantially deviates from the perfect stoichiometry, highlighted by the blue diamonds, towards Ag-poor and

Sb-rich compositions, and (ii) extends away from the Ag 2Te-Sb 2Te 3 psuedobinary line (dashed line).

Figure 36. An area on the modified ternary diagram that indicates samples that show no exothermic reaction in the specified temperature range (blue circles) is surrounded by an area of where small amounts of second phase is indicated (green triangles). Larger variations of composition show an increasing amount of second phase present (orange squares, red circles)

Figure 37 shows a latent heat trace comparison between that of the stoichiometric and off- stoichiometric compounds. A large exothermic reaction is observed at 417K in AgSbTe 2 which is credited to the formation of Ag 2Te. The single phase off-stoichiometric compound shows no exothermic reaction throughout the temperature range, which indicates the absence of Ag 2Te as well as Sb 2Te 3.

-0.2 )

s -0.4 t i n U

r -0.6 A (

w o l -0.8 F

t a e

H -1

-1.2 300 400 500 600 700 Temperature (K)

Figure 37. Latent heat trace reveals an exothermic reaction at 417K in AgSbTe 2 (red) due to Ag 2Te. Off Stoichiometric composition Ag 0.336 Sb 0.558 Te (blue) showed no reaction throughout the temperature range.

Thermopower and resistivity were measured on the samples at room temperature; the results, shown in Figure 38, display a somewhat characteristic trend based on Sb:Te ratio. Thermopower and resistivity generally decrease while decreasing antimony and increasing tellurium from the identified single phase region.

Figure 38. Seebeck coefficient and resistivity of measured off-stoichiometric compositions

Seebeck coefficient and resistivity is clearly strongly dependent on defect and minor phase compositions. The data is cut off at or below the phase transition temperature if the minor phase caused disturbance in the data at a transition temperature. Samples with decreased amounts of antimony showed a decrease in thermopower and resistivity, suggesting that the doping native

86 defect is inversely related to the Sb concentration; in turn, this suggests that the p-type doping defect might be Sb antisites, which, as in Bi 2Te 3, act as acceptors.

We observe from S(T) at higher temperatures that electrons start to become the dominant carrier; this bipolar conduction reduces the thermopower. Resistivity also starts to decrease in the same temperature region due to the much higher mobility of the electrons compared to holes. 52

Figure 39 shows Seebeck coefficient data points with a polynomial interpolation as a function of

Sb and Te content with Ag held constant at 0.366 moles. As Sb and Te conent varies away from the single phase region indicated by the blue markers, a decrease in Seebeck coefficient is evident indicating intrinsic doping is occurring.

Figure 39. Seebeck coefficient shown as a function of Sb and Te content with Ag held constant at 0.366 moles. Color markers are determined through qualitative latent heat trace analysis. Blue shows no sign of phase transition through the selected temperature range. Increasing amount of second phase is indicated by green, yellow, and red respectively.

Figure 40 shows resistivity as a function of Sb and Te content with Ag held constant at 0.366 moles. In general, decreasing the Sb content causes a reduction in resistivity, so we may determine a sample composition with optimal power factor as the single phase composition with the least amount of antimony.

Figure 40. Resistivity as a function of Sb and Te content with Ag held constant at 0.366 moles. Color markers are determined through qualitative latent heat trace analysis. Blue shows no sign of phase transition through the selected temperature range. Increasing amount of second phase is indicated by green, yellow, and red respectively.

Thermal diffusivity was measured from 300K in an Anter Flashline 3000 and the corresponding thermal conductivity values are shown in Figure 41. At high temperature, the thermal conductivity increases strongly above that of the lattice, by more than can be estimated using only the Wiedemann-Franz relation. Consistent with previous work on AgSbTe 2, a strong ambipolar

thermal conductivity is expected to become dominant and can explain the data points at 500 K

and above, reaching 1 W m -1-K-1 that fall far above the usual plateau at 0.55 – 0.6 W m -1-K-1 at

T<400K. 88

) K

m 2 / W (

y t i v i

t 1 c u d n o

C 0.5

l a m r e h T 0.2 200 300 400 500 600 700 800 Temperature (K)

Figure 41. Thermal conductivity data from select compounds

Changes in resistivity of over an order of magnitude are seen as a result of only minor changes in composition, which is an indication of substantial changes in defect composition. In our previous work, we establish the mechanism for low thermal conductivity to be intrinsic and therefor robust to changes in defects, doping, etc. That is why it is not surprising to find that despite the large variations in both Seebeck coefficient and resistivity, within the error of the measurement, variations in thermal conductivity exist only with the addition of electronic thermal conduction and ambipoloar conduction contributions above 500K.

Figure 42. Figure of Merit for off-stoich single phase Ag 0.366 Sb 0.558 Te

The figure of merit for undoped Ag 0.366 Sb 0.558 Te exceeds 0.5 above 440K and maintains this value until 580K. This single phase material exhibits similar zT, throughout the entire temperature range, as some previous reports for undoped AgSbTe 2. This material, however, has the advantage of not showing a phase transition and is therefore presumably single phase and more reliable.

Attempts to extrinsically dope this material could increase the zT further, however, it will be met with challenges in maintaining phase purity.

Conclusions

Materials with the nominal stoichiometry of AgSbTe 2 form a material with compositions including off-stoichiometric Ag-Sb-Te and a second phase Ag 2Te which has a low temperature phase transition at ~417K. XRD analysis is not always sufficient to identify the phase instability 90 in this compound, and a latent heat trace needs to be conducted in order to determine composition. Through latent heat trace analysis, a single phase region was identified with compositions in the range of Ag 0.366 Sb 0.558-0.569 Te (or similarly Ag 0.672 Sb 1.116-1.138 Te 2) and compositions in this immediate range may also be single phase.

Transport properties of these compounds are very sensitive to variations in composition which results in strong intrinsic doping. However, thermal conductivity remains relatively constant with variations resulting only from increased electrical conductivity and ambipolar conduction at high temperatures.

Use of an off-stoichimetric composition in materials that contain Ag, Sb, and Te could be extended to improve materials such as TAGS and LAST resulting in better thermal cycling.

NaSbSe 2

The motivation behind investigating NaSbSe 2 has been previously noted. I-V-VI 2 compounds have a robust intrinsically low thermal conductivity and a favorable valence band structure.

Additionally, alkali based compounds have the potential to have high zT similar to those of noble transition metal or Te based compounds, but at a much lower cost. The added challenge of this compound, however, is the synthesis, since alkali metals react with quartz and make laboratory synthesis much more difficult.

Collaboration with Vidvuds Ozolins of UCLA provided a preliminary look at the difficulties of doping this system and determined that Ca, Pb, Mg, and Sr could be potential p-type dopants when substituted for Sb. Many of the other potential dopants are predicted to have issues such as causing distortions, acting as an n-type or amphoteric dopant, or having large formation energies which corresponds to low/no solubility.

Ingots are typically formed by melting elements in a quartz ampoule, however, Na reacts with quartz even at relatively low temperatures. As a result, two different synthesis methods were developed, the latter of which was used for this doping study due to the fairly short syntheses time. The first method mimics the standard synthesis procedure, but introduces graphite liners that create a barrier between the elements and the quartz ampoules. The raw elements are loaded into the graphite liners, then subsequently the quartz tubes, all while under an Argon environment. The tubes are then evacuated and sealed. The heat treatment is as described in

Table 2. The second method makes use of a pre-made Na 2Se, which is much more stable than Na alone, although it still is handled under Argon. Appropriate amounts of Na 2Se, Sb, Se, and dopant are placed into a high energy ball mill for three sets of 90 minutes each. The resulting powder shows single phase NaSbSe 2 with XRD peaks matching literature values. The powder is then lightly cold pressed into a pellet and subjected to Spark Plasma Sinter (SPS), where it forms a sintered pellet. Although samples were cut in appropriate size and shape for testing, most of the samples showed mechanical instability by crumbling. This suggests that the optimal temperature/pressure profile may not have been attained, or that the sample composition may be such that large grain boundaries are being formed – possibly with Na rich regions. Further testing is required to determine the root cause.

Figure 43. Resistivity measurement of undoped NaSbSe 2 with exponential fit of low temperature

data.

A DC electrical resistivity measurement was made on undoped NaSbSe2 in order to understand and gather basic information about the compound. Figure 43 shows the resistivity plotted on a log scale as a function of 1000/T. The trend demonstrates typical semiconductor behavior and we are able to fit the lowest temperature range to equation 63 to establish the band gap.

(63)

The lowest temperature range corresponds best to the intrinisic temperature regime. The energy gap was found to be E g~0.71eV, which is quite large for a thermoelectric material. Although a large gap ensures that intrinsic two carrier conduction will not be an issue, the large gap makes the material difficult to dope. Several different dopants were attempted. As summary of materials that did not initially work is provided in Table 5. Some of the doped samples were

93 extremely resistive beyond the means of our equipment to measure. Additionally, some of the materials crumbled so badly that there was no possibility of having a piece of sufficient size/shape to measure.

Table 5. List of unsuccessful attempts at doping NaSbSe 2

Measurement Composition Difficulty Extremely Na 1.02 Sb 0.98 Ca 0.02 Se 2 insulating NaSb 0.98 Al 0.02 Se 2

NaSb 0.98Mg 0.02 Se 2

Na 1.05 Sb 0.95Se 2

Mechanically NaSb 0.98 Ni 0.02 Se 2 unstable NaSb 0.98 Co 0.02 Se 2

NaSb 0.98 Fe 0.02 Se 2

The samples that were able to be measured included NaSb 1-xAxSe 2 where x=0.02 and A=Sn, Pb, and Ca, as well as NaSb 1.01 Se 1.99 . The results are summarized in Figure 43 and compared to the results for undoped NaSbSe 2.

Figure 44. Seebeck and resistivity various doped NaSbSe 2 samples

Attempting to increase Sb content to place on the Se site resulted in a higher resistivity and put this material on the edge of our current cryostat measurement capabilities. Sn as a dopant had almost negligible effect with only slightly reduced resistivity at temperatures below 330K. Pb and Ca both showed somewhat promising results as predicted by calculations. With just 2% impurity, Pb and Ca both lowered the resistivity by an order of magnitude or more over the entire temperature range while still maintaining very high Seebeck coefficient. Even with these favorable property changes and the very low thermal conductivity, the zT of these materials is basically negligible. A much larger reduction in resistivity is needed in order to establish

NaSbSe 2 as a possibility for use in a TE device. Even so, this preliminary doping study was able to help establish possible dopants/alloy systems that can provide large property changes with minor amounts of dopants added.

Using the electrical data gathered from these samples, we were able to fit the data to approximate the location of the impurity level with respect to the valence band, known as the activation

95 energy, E a. Smaller activation energies should indicate a better ability to act as a dopant and that

trend is clear when comparing Figure 44 and Figure 45.

Figure 45. Fitted resistivity curves with calculated activation energies for each of the doped NaSbSe 2 samples

Calcium was the most effective dopant and has the lowest E a=0.3eV. Increasing the

concentration of Ca is the most promising next step.

Conclusions

NaSbSe2 is a relatively large gap semiconductor that is difficult to synthesize and dope.

However, this material and others like it hold promise for good TE materials due to their favorable p-type band structure and intrinsically low thermal conductivity. Further study of this system should involve work on creating a better temperature profile for synthesis, increasing the amount of Ca and Pb to determine any additional doping capability, and possibly alloying with another system to reduce the band gap.

LiSbSe 2

LiSbSe 2 has all of the same apparent advantages as NaSbSe 2 with the added benefit that Li is just slightly less reactive than Na, which theoretically makes it easier to prepare samples. The obvious disadvantage is that since Li is such a small atom, it tends to diffuse into quartz resulting in ampoules that cannot hold up under high temperatures. In order to help prevent Li from weakening the quartz ampoules, a graphite coating or liner was used during synthesis. While Li also diffuses into graphite, the liner helps slow the diffusion or Li into the quartz ampoule long enough for heat treatment. Additional difficulties arise from accidental formation of Li 2Se, where energy is rapidly released. As a result, very slow heating is required to better control the reactions. Li based compounds can also be prepared using SPS when pre-cursor compounds

Li 2Se or Li 2S are used.

This work will act as a starting ground for Li based compounds and will provide basic information about the system that will make future doping studies easier. In our first few attempts at synthesizing the compound, we met many challenges that resulted in repeated failures.

After altering our methods, we were able to create three Li based rock-salt samples, two with unidentified minor phases present, and a third with broadened high θ peaks in XRD analysis. The third sample with broadened peaks resulted from the Li oxidizing before it was placed under 97 vacuum. Surprisingly, of the three samples, this one resulted in the least amount of minor phase formation and the broadened peaks are likely a result of some O on Se sites.

Figure 46. XRD peaks of LiSbSe 2 with unidentified minor phases

Second phases present a problem only when they exhibit a phase transition or do not hold up to thermal cycling. Second phases are sometimes introduced purposely to dope the base compound.

Here, we find that the second phase acts as a dopant and introduces a large number of carriers into the system. Transport properties are shown in Figure 47.

Figure 47. Seebeck, Resistivity, and Power Factor for three attempts at undoped LiSbSe 2

The sample that was oxidized before the melt step is labelled “oxidized”. The other two are labelled according to their approximate carrier concentration. The sample labelled “Low p” had a measured Hall coefficient that corresponded to carrier concentration on the order of 10 12 -10 15 cm -

3, which is too low to get reliable Hall measurements and establish carrier concentration. Both of these samples had high electrical resistivity due to their low carrier concentration, and demonstrated intrinsic conductivity with temperature. The last sample had carrier concentration

~10 20 cm -3 and demonstrated a very low electrical resistivity with metallic behavior (increasing with temperature). An optimum power factor, PF~18 was attained as a result of the increased carrier concentration. Although this PF is almost comparable to AgSbTe2 with zT=1.4 at 400K 52 ,

99 preliminary results indicate that >5W/mK due to the large contribution of electronic thermal conductivity. As a result, zT is very low.

Conclusions

This short study has established that it is possible to dope LiSbSe 2 in such a way as to attain excellent power factors. The data has provided boundaries for tuning for ideal carrier concentration and the optimum carrier concentration in this system was bracketed, but not yet achieved. Through trial and error we have also developed a relatively good synthesis technique that will prove useful in future doping studies.

100

Chapter 4: Resonant Levels

As summarized in Chapter 1, resonant levels (RLs) can be used to increase the thermoelectric figure of merit by improving the power factor, PF=S 2σ. An impurity can introduce a distortion in the local band structure near the Fermi level. This distortion is characterized by an increase in

Seebeck coefficient as described by the Mott relation. The work presented here includes work on both PbTe as well as CoSb 3. The first study was a collaboration with Fraunhofer Institute for

Physical Measurement Techniques, where PbTe was doped with titanium. Reports of PbTe doped with chromium acting as a low temperature RL spurred the second study presented. The third study presented was an attempt to improve thermoelectric properties of p-type CoSb 3 to more closely match the success of n-type skutterudites.

Pb Chalcogenides

PbTe:Cr

Chromium has been previously reported to be a resonant impurity in PbTe at 100meV above the conduction band minimum at low temperature and also has been reported to cause Fermi level pinning. 53,54,55,56 The purpose of this particular study was to determine if the impurity electrons are sufficiently delocalized to cause an increase in Seebeck coefficient. We compare the behavior of PbTe:Cr to an effective resonant level, PbTe:Tl, and a system exhibiting Fermi level pinning,

PbTe:Ti. Here we present thermomagnetic and thermoelectric transport properties. We also

101 summarize the results of nuclear magnetic resonance testing ( 125 Te) performed by Ames laboratory on our samples as a concerted effort to understand the Pb 1-xCr xTe system.

All transport properties were measured on parallelepipeds in a conventional flow liquid nitrogen cooled cryostat as described in Chapter 2. The Hall coefficient ( RH) of the sample with x=0.25 % was also measured in Quantum Design PPMS down to 2 K. We report here on four transport properties; electrical resistivity ( ρ) and RH were measured using an AC bridge while thermopower

(or Seebeck coefficient S) and adiabatic transverse Nernst-Ettingshausen coefficient (“Nernst” or

N) were measured using the static heater and sink method. Since the samples have a magnetic response (see following section), we sweep the Hall resistance and Nernst voltage over the entire

-15 kOe to + 15 kOe field range to check for anomalous effects, but the curves remain linear in H: the Hall and Nernst coefficients ( RH and N) are then taken to be the low field slopes of the Hall resistance and Nernst voltage, respectively.

The transport coefficients S, ρ, N , and RH from 80 K to 400 K are shown in Figure 48. S and RH for all samples are negative confirming that Cr is an electron donor. Electrical resistivity shows typical metallic behavior with more than an order of magnitude increase, over this T range, with a positive temperature coefficient. The sample with x=0.25% has the highest ρ, which first decreases, then saturates when the amount of Cr is further increased. Increasing Cr concentration leads to a smaller RH, or larger carrier density, at 300 K, which is reflected in a smaller thermopower. With T, RH exhibits unique behavior: it increases in magnitude with increasing temperature in the range of 100-350 K. At T <80 K and T >350 K, RH remains relatively temperature independent. The trends in RH correlate well with S, and we also cross-check the

125 relation between RH and the Knight shift in Te NMR (see later), suggesting that the change in

RH represents a change in carrier concentration and not a change in the Hall prefactor or an anomalous Hall effect. As temperature is increased, three regions of nearly linear S behavior with 102 different slopes emerge: a low and high temperature region of S with lower slope than that of the mid temperature range. The decreasing carrier density between 100 K and 350 K coincides with the steeper slope of S as expected from the Mott formula. The behavior of RH and S in relation to the Cr impurity level will be discussed quantitatively in the next paragraph. The Nernst coefficient is large, shows a similar T-dependence for all samples, and has no significantly systematic relation to Cr content.

Figure 48. Thermopower or Seebeck coefficient, electrical resistivity, Nernst-Ettingshausen coefficient, and Hall coefficient of Pb 1-xCr xTe. Inset shows carrier concentration for x=0.25%.

103

We use the method of four coefficients 39 with the four measured transport properties (S, ρ, N , and

RH) and calculate Fermi level relative to the bottom of the conduction band ( EF), the mobility, the density of states (DOS ) or, alternatively, a DOS-effective mass ( m* ) which includes the effect of the four-fold degeneracy of the conduction band pockets, and the scattering parameter ( λ) defined as the exponent of the power law that characterizes the energy dependence of the relaxation time

λ − /1 2 * τ = τ 0 E . The parameters EF, m and λ are shown in Figure 49 for Pb 1-xCr xTe. The principal result is that the Fermi level is calculated to move toward the bottom of the conduction band with increased temperature. Indeed, for x=0.25% the Fermi level actually moves into the band gap around 350 K with a linear temperature dependence of 0.31 meV K -1. The slope of the

-1 EF(T) is somewhat less steep at higher x, -0.23 meV K for x=2%. This explains the flattening of

the Hall coefficient at T>300 K. Assuming that the Cr level indeed pins EF, then the conclusion is that the Cr impurity level moves toward the conduction band edge with increasing temperature, and that the number of electrons that Cr can donate drops, although this is counteracted by the smearing of the Fermi surface. Note that this behavior of the Cr is identical to that of the In level in PbTe, 57 and that it has several consequences. Firstly, like In, Cr is expected to act as a trap in

PbTe at T>350 K, casting doubt on the results of Paul et al 58 . Secondly, since the movement between 80 and 350 to 400 K of the Cr or In levels vis-à-vis the conduction band edge mirrors

35 half the change of the energy gap, Error! Bookmark not defined. this implies that these levels have almost no T-dependence when referred to the middle of the gap at the L-point of the Brillouin zone. This suggests that the temperature dependence is a result of the temperature dependence of the energy gap, rather than the impurity level moving.

104

Figure 49. Calculated position of the Fermi level with respect to band edge, effective carrier density-of-states mass, and scattering parameter for Pb Cr Te. The dashed line is the DOS- 1-x x mass for the conduction band of PbTe: no significant increase is observed

The scattering parameter, λ, is within the experimental accuracy, constant for all samples and equal to that in Pb-rich PbTe,59 indicating that the dominant scattering mechanisms involve

105 acoustic and polar optical phonons, with no hint of resonant scattering. The DOS-effective mass is not increased over the lowest conduction band of PbTe as depicted by the dashed line. Indeed,

Figure 50 contains the Pisarenko relation (solid line) at T=300 K as calculated for the conduction

band of PbTe. Despite the doping level change of Cr and observed Fermi level pinning we do not

see increased thermopower at any given carrier concentration, neither at 300 K nor at 100 K

(inset) as demonstrated in Figure 50 .

Figure 50. Pisarenko plot (Seebeck coefficient versus carrier concentration) for PbTe:Cr at 300 K and 100 K (inset). Solid lines are calculated for the conduction band of PbTe.

Magnetism

Doping of binary group IV tellurides with transition metals may form a dilute magnetic semiconductor (DMS) 60 when the solubility is sufficient. However, if the solubility of transition metal in a particular telluride is low, second phases form that confuse the magnetic response, so that it is necessary to supplement magnetic measurements with others, such as transport and

106

NMR studies in this work. Indeed, depending on the properties of the second phase, parameters of the major phase measured in experiments can be inaccurate, e.g. the carrier concentration

61 obtained from RH or magnetization. The following magnetic measurements of PbTe doped with Cr must be integrated with the other data to yield an understanding of the physical picture.

Figure 51. Temperature dependencies of the magnetization of PbTe doped with 0.25, 0.5, 1, and 2 % Cr measured in a 3 kOe magnetic field. The dashed line is an order-parameter law fit to the 2%, with a Curie temperature of T C ≈ 335 K 107

Figure 51 shows temperature dependencies of the magnetization of Pb 1-xCr xTe doped with x=0.25

%, 0.5 %, 1 %, and 2 % (in atomic percent) of Cr measured in a 3 kOe magnetic field in a

Quantum Design Physical Properties Measurement System’s Vibrating Sample Magnetometer.

Figure 52 shows their magnetization M(H) measured versus magnetic field at various temperatures between 2.4 and 305 K and in magnetic fields up to 70 kOe. The raw data agree with those reported in Ref. 62 for Pb 1-xCr xTe alloys measured in a 4 kOe magnetic field. The magnetization of all samples is positive and much larger than expected for diamagnetic PbTe 63 , which can be attributed to paramagnetic Cr ions embedded into the PbTe matrix or in other Cr-Te compounds.

Between 2.4 and 300 K, the magnetization curves of Pb 1-xCrxTe samples show several features, which can be considered in three regions: (i) below ~30 K, (ii) at ~170 K, and (iii) around 305 K.

The behavior of the magnetization at ~170 K can be attributed mostly to the presence of Cr 2Te 3,

64 which exhibit ferromagnetic order below the Curie temperature TC = 170 K. However, Cr 2Te 3 has a complicated magnetic structure: its magnetization decreases with lowering of temperature, which can be associated with an increase of non-collinearity of Cr magnetic moments 64 or due to

65 antiparallel alignment of spins that belong to Cr I, Cr II and Cr III ions located on different sites .

The field-dependent magnetization of PbTe containing x=1 % and 2% Cr (Figure 52) confirms the presence of the ferromagnetic phase at low temperatures; in addition it shows sharp saturation in low magnetic field, ~ 2 kOe, at 305 K. Because Cr 2Te 3 at 305 K is expected to be paramagnetic 64 , we suggest that the observed saturated magnetization should be attributed to another more Cr-rich ferromagnetic phase. As there is a solid-solubility range to the Cr side of

Cr 3Te 4 in the Cr-Te phase diagram, generating compounds with Tc between 317 K and 340

66,67,68 K , we will label this composition Cr 3+ δTe 4. An order-parameter fit with a Tc~335K fits the 108 data for x=2% well. Finally, the presence of neither Cr 3+ δTe 4 nor Cr 2Te 3 explains the increase below ~30 K, so the Curie-Weiss-like behavior of the magnetization of all four samples below

~30 K is attributed to paramagnetic Cr ions located on Pb sublattice (established from NMR data) in the Pb 1-xCr xTe solid solution. We do not see magnetic evidence for the presence of a minor

Cr 5Te 8 phase, as the Tc of that compound varies between 180-230K, depending on the exact Cr-

Te ratio. 69

Figure 52. Magnetization of Pb 1-xCr xTe doped with x = 0.25, 0.5, 1, and 2 at. % Cr measured at various temperatures in a magnetic field up to 70 kOe. The inset to the x = 0.25% frame shows as dashed lines Brillouin function fits to the magnetization assuming a 3d4 Cr 2+ configuration using N par-Cr as only fitting parameter. 109

The paramagnetic component due to the isolated Cr atoms in the PbTe lattice dominates both the

T-dependence and the field dependence of the magnetization of the x=0.25 % Cr sample. In the temperature range 50 ≤ T ≤ 250 K the moment is less T-dependent than a pure Curie-Weiss law, suggesting that sample has a minor amount of ferromagnetic phase present. Hence, the total magnetization of PbTe:0.25% Cr measured in experiment at 305 K can be shown as Mexp = Mpar +

Mdia + Mfer where Mpar contains contributions from paramagnetic Cr ions in Pb 1-xCr xTe solid solution, and from paramagnetic ions in Cr 2Te 3, which possesses paramagnetic properties above

64 ~170 K. As the ferromagnetic component of the magnetization due to Cr 3+ δTe 4 in this x=0.25 % sample at 100 K to 300 K is small and constant in H >15 kOe, Mfer can be subtracted first. The experimentally observed negative magnetization is a result of the dominance of the diamagnetic component from the PbTe matrix. The diamagnetic susceptibility can be calculated next from the linear part of the magnetic field dependence of the magnetization at 305K to be χdia = Mdia /H = -

2.4×10 -7 emu (per weight in gram and field in Ørsted), smaller in absolute value than that

-7 63 measured for PbTe, χdia = -3.4×10 emu. The resulting Mpar (T) fits a 1/ T dependence, with a

-6 -1 paramagnetic susceptibility of χpar /T = 1.46×10 emu K . After subtracting the field dependent

Mdia and Mfer from the 2 K and 15 K magnetization field sweeps, we fit the concentration Npar-Cr of paramagnetic Cr atoms to the experimental data with Brillouin functions (Figure 52 inset) using S=J=2, L=0 for 3 d4 Cr 2+ with its orbital moment L quenched, which results in a net moment

2+ of 4.9 µB. The fit works well, confirming that the paramagnetic atoms are indeed Cr , and gives

19 -3 Npar-Cr = 2.6×10 cm (in mole fraction, xpar-Cr = 0.19%), a value that is not significantly different from the concentration of free electrons in that sample below 50K. Simple electron counting rules do not explain the donor action of Cr 2+ on the Pb sublattice of PbTe, and indeed the

110 presence of Cr 3+ has been postulated in the past, 60 so we suggest that detailed band structure calculations are necessary to shed light on this point.

125 Te NMR of PbTe:Cr

The magnetic data above are now complemented with a summary of the 125 Te NMR data. Our results are consistent with Ref 70, showing the presence of microinclusions in PbTe:{Sn,Cr} alloys. Although we sweep magnetic field to check for non-linearities in Hall resistivity measurements, given the possibility of an anomalous Hall effect in similar Cr chalcogenides (for

71 example Sb 2-xCr xTe 3) the magnitude of the carrier concentration as determined from RH will be confirmed here. Finally, the presence of paramagnetic Cr ions and ferromagnetic phases in PbTe can induce paramagnetic effects 72 in 125 Te NMR peak position and relaxation times.

The results indicated a positive Knight shift, which indicates a moderately high n-type carrier concentration of ~7x10 18 cm -3,73 consistently with the carrier concentration measured by Hall effect at 300 K. The intensities of Cr-induced peaks did not double from with chromium concentration increase from x= 0.25% to x=0.5%, which suggests that the solubility of Cr in PbTe is x< 0.4%. Indication of long-range effects of the ferromagnetic phases (Cr 3+ δTe 4 and Cr 2Te 3) were also present in samples with higher concentrations of Cr.

Conclusions

Cr forms a resonant donor level in the conduction band of PbTe, about 100 meV above the band edge at 0 K. Like the indium level, the Cr level moves into the gap as temperature reaches room temperature, probably due to the temperature dependence of the band gap itself. All studied

2+ PbTe:Cr samples contain paramagnetic Cr ions on the Pb sublattice of Pb 1-xCr xTe solid solution and small fraction of ferromagnetic phases. The secondary phases are most likely Cr 2Te 3 and

111

Cr 3+ δTe 4 with Curie temperature of 170 and 317 K, respectively. While it is not evident from simple electron counting rules how Cr 2+ substituted for Pb 2+ might act as a donor, 125 Te NMR shows signals that confirm the solubility of Cr in PbTe lattice up to 0.3%, and also confirm the existence of secondary phases above the solubility limit.

PbTe:Ti

A previous study of Ti as a dopant in PbTe suggested that Ti may act as a resonant level in

PbTe.74 We demonstrate that Ti causes Fermi level pinning in PbTe, which is comparable to Fe in HgSe. 75 For Fermi level pinning to occur, the general idea is that the carrier concentration, n, increases with increasing impurity concentration until a critical carrier concentration, n crit , is attained, after which n remains constant for a range of increasing impurity concentration..

PbTe:Ti is different from the HgSe:Fe system in that after the pinned region, carrier concentration starts to increase with increasing impurity. Sample preparation, by molecular beam epitaxy, and high temperature measurements were conducted at Fraunhofer Institute for Physical Measurement

Techniques by Jan D. Konig. 76

Pb 1-xTi xTe thin film samples were prepared using Ti effusion from 1250C to 1500C. The thermoelectric properties were measured in the standard way (See Chapter 2) for temperatures from 80K to 420K. High temperature Seebeck coefficient was measured with T=10K. High ∆ temperature resistivity and Hall coefficient was measured using van-der-Pauw method. The standard assumption for PbTe is that R H=1/ne and that is the assumption that we use here to establish the carrier concentration with respect to the cell effusion temperature, proportional to the number of added Ti atoms.

112

Figure 53. Low temperature electrical conductivity of PbTe:Ti

Electrical conductivity measurements of the various samples show metallic behavior as a function of temperature. Only low temperature data that was measured at OSU is shown in these plots.

High temperature data can be seen in Ref. 76.

Table 6. PbTe: Ti sample properties and identifiers Carrier conc. (cm -3) T [Ti] ( oC) Identifier

6x10 18 1330

1x10 19 1340

1x10 20 1500

1.2x10 20 1500

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Increasing the cell effusion temperature increases the electrical conductivity up until about

1330 oC, at which point only minor changes in conductivity occurs through 1450 oC. Above this

temperature, the conductivity begins to increase again.

Figure 54. Hall carrier concentration of PbTe:Ti

The Hall carrier concentration in Figure 54 is calculated in the standard way with n=1/eR H. The carrier concentration remains relatively constant over the entire range of temperatures measured.

From equation 15 we can deduce that the change of conductivity with temperature is primarily a result of changing mobility. Seebeck coefficient is shown in Figure 55 and displays very typical linearly increasing behavior up through about 400K, and as shown in Ref 76, continues this trend through ~550K.

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Figure 55. Seebeck coefficient of PbTe:Ti

The Seebeck coefficient as a function of carrier concentration is plotted with the Pisarenko relation in Figure 56. For an effective resonant level, we would expect to observe an increase in

Seebeck coefficient over the Pisarenko relation established as a result of the band structure. We see that Ti does not show any increases in Seebeck coefficient, however, Ti in PbTe does demonstrate a different resonant effect.

115

Figure 56. Pisarenko plot from Ref 80 demonstrating no increase in S

The assumption is that Ti substitutes for Pb. When Ti is in its Ti 2+ state, it is electrically neutral and adds no extra carriers. Titanium can also substitute as Ti 3+ and Ti 4+ , in which case it will act

2+ (2+k)+ - 3+ 4+ as a donor as described by Ti ↔Ti +ke , where k=1 for Ti , and k=2 for Ti . When

NTi

extra electrons from the Ti atoms remain localized, and therefor do not contribute to conduction

or carrier concentration. As more Ti atoms are added, N Ti >N crit occurs, which happens after the

Ti impurity level is filled, therefor E F>E Ti . A generalized model that is based on chemical

116 equilibrium equations for buffer solutions can be demonstrated in equation 64, which applies to both systems, Ti in PbTe and Fe in HgSe. In this model, M can be Ti or Fe.

(64) ↔ +

When E F

Through the use of Gibbs Free energy principles and chemical equilibrium equations, we establish a generalized equation representative of the Fermi level pinning effect:

(65) where E represents the impurity donor energy level, in this case, E . is the electron affinity, o D Ti i is the impurity ionization energy, and E p is the quasiparticle binding energy. This shows a relationship for the number of carriers, dependent on the ionization properties of the dopant and energy at which the impurity level exists.

117

Figure 57. Carrier concentration as a function of Ti effusion cell temperature. 76

As the cell effusion temperature increases to just over 1325 oC, the electrons likely localize on the

3d levels. As the effusion temperature exceeds 1450 oC, the large number of carriers become delocalized, which forms an impurity band. Carriers in the newly formed impurity band can then contribute to conduction and carrier concentration.

Fermi level pinning could prove to be a valuable asset to the semiconductor manufacturing process. Akin to how chemical buffering solutions can fix the pH of a solution, the resonant effect establishes a wide range of impurity concentrations that results in similar properties over the entire range. Typically, even just a small amount of variation in impurity concentration can drastically alter the properties of a material. This makes it necessary for semiconductors to be produced using expensive high purity starting elements. Establishing an effect that allows some

118 variation of impurity concentration with little effect on transport properties could be a valuable tool if a material with high zT had E F~E D

Conclusion

Ti in PbTe acts as a resonant level, showing Fermi level pinning as the resonant effect, however there is no increase in Seebeck coefficient. Ti forms an impurity level ~52meV above the bottom of the conduction band. A model was developed analogous with the buffer solution, which describes the carrier concentration as a result of the amount of Ti impurity. Fermi level pinning has merit for manufacturing processes if it can be tuned to the desired doping level.

CoSb 3

Skutterudites are compounds composed of group IXB atoms (Co, Rh, and Ir) forming a simple cubic structure, and group VA 3 pnictide atoms (primarily Sb and As) forming rings inside of 3 of every 4 empty cubes. The remaining cube remains empty and is an asset to our ability to tune the material properties. CoSb 3 has been experimentally determined to be electrically conductive and possesses a large thermal conductivity, the latter of which presents a challenge to thermoelectric materials as the optimal figure of merit has a low and high , high electrical conductivity, Seebeck coefficient. A common method for reducing thermal conductivity, for example, is to introduce impurity atoms that act as fillers. These fillers then introduce a “rattling” mode which

77 results in a strong reduction in thermal conductivity. N-type doping of CoSb 3 has been very successful, reaching zT values >1 reported from several groups and the N-type material is the leading candidate for use in TE waste heat recovery systems. P-type doping has had some advances in zT, but is generally much less successful than n-type doping due to the light hole bands attributed to Sb. Thermoelectric devices require use of both p and n type materials with

119 similar mechanical and electrical properties, which makes finding a good p-type skutterudite a challenging yet worthy task. A promising method for improving p-type properties is to introduce an effective resonant level into the energy levels occupied by the light hole band, thereby increasing the Seebeck coefficient without strongly effecting other transport properties.

Band structure calculations have shown that the valence band consists of a single non-degenerate light valence band at the Г-point of the Brillouin zone. There is approximately 0.57eV gap between conduction band at Г and a second, heavy valence band also at Г, due to Co d-levels at

P78 and a direct gap at Г of 0.80eV from the conduction band. One study describes the light

78 valence band of CoSb 3 as parabolic near the band maxima and linear elsewhere, and another study states that the two-band Kane model can be used to describe the light valence band near the maximum.79 Singh and Pickett describe the parabolic region of the valence band corresponding to a maximum hole concentration of ~3x10 16 cm -3 and the remainder of the single band dispersion

2 1/3 78 can be approximated as S=-(2 πkB T/3e α)( π/3n) where α=3.10 eV Å. This is the dispersion used later in this work in the Pisarenko relation to compare Seebeck values at varying carrier concentrations.

Theoretical examination has predicted the band gap to vary from 0.05eV 78,79 to 0.22eV 79 with bracketed values being obtained by one study, which used both the experimental values as well as the minimum energy atomic positions. The difference in the atomic positions between experimental and minimum energy were extremely small, which demonstrates the potential energy gap sensitivity to changes in atomic position. The band gap energy was estimated to be

70-80meV through use of magnetic susceptibility measurement. 80 In another experimental study by Mandrus, the narrow bandgap was found to be 50 meV and a secondary gap of 0.31eV was found, however their low temperature data implied ionized impurity scattering in contrast to

Singh and Picketts calculation using acoustic phonon scattering. 81 120

The transport properties have been considered to be strongly influenced by the grain size. 80 Other studies have concluded that the band gap is strongly influenced by changes in the position of Sb rings. 79

The single non-degenerate band, at Г-point, that establishes the p-type characteristics of this material is a light hole band which is generally not conducive to high power factors because very small effective mass leads to the very small Seebeck coefficient. Here, we hope to use band structure engineering to introduce an impurity in the valence band that will cause an increase in the DOS near the Fermi level (EF) which can increase the Seebeck coefficient without significantly impacting other material properties. 35 Three impurities were tried, Sn, Zn, Al and

Ga. The work on the latter two is started here and will be completed as part of another thesis (M.

Adams). The work on Sn-doped material was published and is not reported here as it was the primary content of another thesis (S. Hui, UM).

Methods

High purity starting elements of Co (99.95% pure), Sb (99.9999% pure), and Al (99.999% pure) were weighed to the correct stoichiometry under Argon. Samples were placed into a high energy ball mill for three sets of 90 minutes, then cold pressed into a low density pellet. The pellet was then subjected to SPS where the pressure was 2.6kN throughout the process and the temperature was raised to 450 over 4 minutes, raised to 500 over 4 minutes, then held constant for 4 minutes.

The total synthesis process lasted less than 6 hours for each sample.

Transport measurements were conducted as described in Chapter 2. Hall coefficient was used to establish hole carrier concentration, p, assuming 1 as the Hall prefactor such that R H=1/pe. This assumes single carrier parabolic bands, which we know to not be completely correct, but will

121 provide us with the ability to compare our data to literature values. Mobility, µH, is determined using the relationship µH=σ/pe.

Part of the remainder of the pellet was ground into powder and was used for x-ray diffraction to establish correct crystal structure and determine if any noticeable (>3% secondary phases were present).

CoSb 3:Zn

A preliminary calculation conducted by Januz Tobola at AGH University in Poland, revealed that introducing Zn in place of Sb may cause an increase in density of states above what is expected without the impurity. Figure 58 shows this preliminary calculation and the contribution of each element to the total DOS. The notable contribution is from Zn d states. For x=0.3%, the Fermi level is calculated to be almost in the impurity states. The placement of the impurity energy in the band structure is ideal for forming a resonant level, so this study was conducted to determine the resonant effects, if any.

122

Figure 58. Preliminary electronic band contributions calculations CoSb 3-xZn x, x=0.3%

Transport properties were measured in the normal way and are reported in Figure 59. While there does not appear to be any pattern associated with the transport properties with the amount of impurity added, there is also only a very small variation to note. The resistivities of all of the spark plasma sintered (SPS) samples are nearly the same. Additionally, the carrier concentration only ranges from 1x10 18 -4x10 18 cm -3, which is relatively minor when considering the amount of

dopant added. The undoped sample has carrier concentration 2x10 18 cm -3. As temperature increases to above 150K, S starts to turn over, which is likely a result of thermally excited carriers. As a result, we do not expect a high PF at higher temperatures. The general conclusion drawn from this data is that Zn is not a very effective p-type dopant and it is unlikely that the

Fermi level can be tuned in such a way as to hit the impurity DOS contributions.

123

Figure 59. Transport properties of CoSb 3-xZn x

An additional sample that was formed with the standard melting technique and annealed for over

a week. Data taken on it reinforce our conclusions. Despite the long anneal time, the carrier

concentration of this sample does not exceed 4x10 17 cm -3, though it does exhibit an interesting non-monotonic trend in the electrical resistivity’s temperature dependence.

CoSb 3:Al

A preliminary calculation by Januz Tobola at AGH University in Poland for another potential resonant impurity, Al, is shown in Figure 60 . Each of the individual electronic contributions to 124 the band structure are labeled accordingly. Calculations have described the relevant band contributions, that is, the valence bands which are highest in energy or closest to the Fermi

79 energy, to be contributions from Sb 5p and Co 3d states.

Figure 60. Calculated band structure with Al as an impurity

We know from literature that the valence band, at least at relatively low carrier concentrations, is a light hole band, so we expect it to have small DOS as displayed in Figure 60. We also see from this calculation that introducing Al impurity into the structure introduces an excess DOS contribution from Al 3p states. An important point of trying to create a resonant level is that the 125 distortion in the density of states must be within close range (~k BT) of the Fermi energy. An increase in Al concentration will move E F in the desired direction.

Experimental

X-ray diffraction of CoSb 3 samples that were taken correspond well with literature values. No secondary phases were present based on XRD.

Figure 61 includes transport measurements of CoSb 3-xAl x for 0

~3.3% replacement of Sb. Al is slightly smaller than Sb, so it is not unreasonable to think that Al could be soluble to this doping limit in that regard, however, a limit is reached in the contribution of carriers with respect to concentration of Al.

126

Figure 61. Transport properties for CoSb 3-xAl x

Although Al acts as an effective p-type dopant in CoSb 3, the transport properties for CoSb 3-xAl x do not change monotonically even at low doping levels. An undoped sample of CoSb 3 shown in

18 -3 the study of CoSb 3:Zn shows a carrier concentration of 2x10 cm , which is lower than all of the

Al doped samples. We know Al to be an effective p-type dopant because carrier concentration spans two orders of magnitude with the addition of just 3% Al.

127

The Seebeck coefficient increases nearly linearly with temperature, and does not exhibit the “turn over” that occurred in the Zn doped samples, because of the higher doping levels. This is a promising characteristic for achieving high zT values at high T.

An ingot was also formed for this study with the highest doping concentration. Its trends are also different from the SPS samples (similar to CoSb 3:Zn), in that it demonstrates metallic behavior in electrical resistivity instead of the typical trend of a semiconductor.

The Pisarenko relation helps establish the expected relationship between Seebeck coefficient and carrier concentration. Figure 62 shows the Pisarenko relation established by Singh and Pickett 78 with experimental values from Morelli 82 , Hui 83 , Caillat 84 , Tobola 85 , Park 86 , Zobrina 87 as well as the data from this study. It is notable that below mid 10 18 carrier concentration, the experimental

Seebeck coefficient almost always falls above the theoretical relation, whereas above 10 18 the calculated relation holds well. A vertical line is drawn in the chart to demonstrate this point.

Corresponding well with the theoretical calculations by Tobola in this study, an observed increase occurs in the Seebeck coefficient of the Al-doped material in the range of high 10 18 and low 10 19 cm -3 carrier concentration. The Pisarenko relation is shown with a green line and a trend line is added in blue to show an approximate average for where p-type data points generally occur.

128

Figure 62. Pisarenko relation with all available p-type literature values included

Data points are added for both Zn and Al doping studies. All samples doped with Zn fall within

an expected “normal” range. Al doped samples with carrier concentration less than 6x10 18 cm -3 also fall in the same “normal” region, while samples with higher carrier concentration appear to be more promising. Unfortunately, with the ambiguity of where the real Pisarenko relation should fall and the discrepancy reported with the various CoSb 2.9 Al 0.1 samples, there is not currently enough evidence to draw any definitive conclusions.

Further testing on Al doped samples will be conducted to see if Al is indeed substituting for Sb sites or if they are partially filling voids.

If it is established that Al is an effective level in CoSb 3, then the next steps would include introducing filler atoms to induce “rattling” phonon modes in order to reduce lattice thermal conductivity. A delicate balance will need to be achieved, as atoms that act as fillers typically

129 also act as electron donors, which will begin to negate the effect Al has on increasing carrier concentration, making it more difficult to tune E F to the energy associated with the distorted

DOS.

Conclusions

Two doping studies were conducted on CoSb 3 to establish if a resonant level forms. Preliminary calculations implied that Al and Zn impurity levels may be located in the light hole band within an energy region for which E F may be tuned. Transport properties showed Zn to be a poor p-type dopant and Al to be a very effective p-type dopant. Ambiguities in band structure and solubility of Al in CoSb 3 lead to uncertainties in the final conclusion of this study, however, several samples with carrier concentration ~1.5x10 19 cm -3 not only show improved Seebeck over

Pisarenko relation established by Singh and Pickett, they also show improvement over literature values of several different studies that used p-type dopants other than Al.

130

Overall Conclusions

Thermoelectric devices have many different possible applications that can be better served by cheaper materials with higher zT. Several approaches have been taken to optimize the figure of merit, two of which are studied in detail here.

The first of which is the use of lone pair electrons to minimize lattice thermal conductivity. In this work, we were able to establish the large polarizability of a transverse acoustic phonon mode to be the source of the large Gruneisen parameter, which translates to very low thermal conductivity. The result of this study is that through calculation, we can begin to predict materials that will exhibit low thermal conductivity based on their electronic polarizabilty. We have also demonstrated that the strong phonon-phonon interaction is intrinsic to the materials and relatively unaffected by defects, alloys, etc. Further, we have extended the principles of the low thermal conductivity of transition metal I-V-VI 2 compounds to alkali based materials presenting opportunity for lower cost TE materials to be developed. In the doping studies of alkali based I-

V-VI 2 compounds, we established NaSbSe 2 to be a large band gap semiconductor that is somewhat difficult to dope. LiSbSe 2 demonstrated a promising power factor >16 around 300K.

Further work on this system should be conducted to establish optimal doping level and improve synthesis technique.

The second approach to increasing zT that is explored in this work is resonant levels. We have examined several systems including PbTe:Cr, PbTe:Ti, CoSb 3:Zn, and CoSb 3:Al. Our published

131 work on PbTe:Cr can be expected to clear up confusion that existed on this particular system as we accurately established, through magnetic measurements and a collaboration with a group who performed NMR, a solubility limit and existence of secondary magnetic phases. We also established that Cr is not an effective resonant level in PbTe, that is, it does not increase the

Seebeck coefficient over the Pisarenko relation. In the study of PbTe:Ti, a collaboration with

Fraunhofer Institute, we were able to demonstrate that Ti causes Fermi level pinning and then further establish a representative equation for the effect. In doping studies of skutterudite, CoSb 3, a collaboration with AGH University in Krakow, we found Zn to not be an effective dopant, and

Al to be a possible resonant level. Future work in this area should include an annealing study to establish if it is possible to maintain high Seebeck coefficient while increasing mobility. Such an increase could better demonstrate the expected increase in power factor that one should see with an effective resonant level.

132

References

1 “Energy Informatics”, Lawrence Livermore National Laboratory, https://missions.llnl.gov/energy/analysis/energy-informatics , Jan. 15, 2014 2 W.D. Callister Jr. “Materials Science and Engineering: An Introduction”, John Wiley & Sons, Inc., 5 th ed., 2000 3 R.E. Hummel, “Electronic Properties of Materials”, © Springer Science + Business Media, LLC 2011 4 K. Barnham, D. Vvedensky, “Low-Dimensional Semiconductor Structures: Fundamentals and Device Applications”, Cambridge University Press, 2001 5 E.H. Putley, “The Hall Effect and Semi-Conductor Physics”, New York Dover Publications, 1968 6 M.J. Moran, H.N. Shapiro, “Fundamentals of Engineering Thermodynamics”, John Wiley & Sons, Inc., 2004 7 R. Berman, “Thermal Conduction in Solids” Oxford University Press 1976 8 J.R. Drabble, H.J. Goldsmid, “Thermal Conduction in Semiconductors”, Pergamon Press Ltd., 1961 9 Humel, R.E., “Electronic Properties of Materials”, 4 th ed., Springer Science, New York, 2011 10 D. G. Cahill, S. K. Watson and R. O. Pohl, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 46, 6131. 11 S.L Kakani, A. Kakani, “Material Science” New Age International Publishers 2004 12 F.J. DiSalvo, “Thermoelectric cooling and power generation”, Science 285 , (5428):(703-706) 1999 13 Snyder, G.J., Ursell, T.S., “Thermoelectric Efficiency and Compatibility”, Phys. Rev. Lett., 2003, 91, 148301 14 “Thermoelectrics: The science of thermoelectric” http://www.its.caltech.edu/~jsnyder/thermoelectrics/ 15 J. Snyder, “Thermoelectrics”, http://www.thermoelectrics.caltech.edu/thermoelectrics/engineering.html , (accessed Feb. 28 2014) 16 C.B. Vining, “An Inconvenient Truth about Thermoelectrics”, Nature Materials, 8, 83, 2009 133

17 www1.eere.energy.gove/vehiclesandfuels/pdfs/thermoelectrics_app_2011/Monday/ maranville.pdf 18 http://www.amerigon.com/ccs_seating.php 19 http://www.amerigon.com/pdfs/Broader-Use-of-Thermoelectrics-Systems-in-Vehicles- Oct2008.pdf 20 http://www.amerigon.com/green_technology.php 21 Hundy, Guy, Trott, A.R., Velch, T., Refrigeration and Air-Conditioning , Elsevier © 2008 22 Directive 2006/40/EC of the European Parliament and of the Council of 17 May 2006. 23 http://www.seikowatches.com/heritage/worlds_first.html 24 www.seikowatches.com 25 Spacecraft Power for Cassini, NASA Fact Sheet 26 Japan gadget charges cellphone over campfire . June 2011. 27 http://www.marlow.com/cordless-mosquito-traps 28 Leephakpreeda, T. “Applications of thermoelectric modules on heat flow detection”, ISA Transactions, Volume 51, Issue 2, March 2012, Pages 345–350 29 Tritt, T.M., Subramaniam, M.A. “Thermoelectric Materials, Phenomena, and Applications: A Bird’s Eye View” MRS Bulletin Volume 31, March 2006, Pages 188-194 30 Biswas, K., Kanatzidas, M. G., Nature Volume: 489, Pages: 414–418 Date published: (20 September 2012) 31 Lu, X., Morelli, D.T., Xia, Y., Zhou, F., Ozolins, V., Chi, H., Zhou, X., Uher, C., “Higher Performance Thermoelectricity in Earth-Abundant Compounds Based on Natural Mineral Tetrahedrites”, Adv. Energy Mat. 2013, 3, 342-348. 32 G. D. Mahan and J. 0. Sofo, “The Best Thermoelectric” Proc. Natl. Acad. Sci. Vol. 93, pp. 7436-7439, July 1996 33 Hsu, K. F., Loo, S., Guo, F., Chen, W., Dyck, J.S., Uher, C., Hogan, T., Polychroniadis, E.K., “Cubic AgPb mSbTe 2+m : Bulk Thermoelectric Materials with High Figure of Merit” Science, 2004, 303, 818. 34 Morelli, D. T., Jovovic, V., Heremans, J.P., “Intrinsically Minimal Thermal Conductivity in Cubic I-V-VI 2 Semiconductors”, Phys. Rev. Lett., 101 035901, (2008) 35 J.P. Heremans, B. Wiendlocha, A.M. Chamoire, “Resonant levels in bulk thermoelectric semicondcutors”, Energy & Environ. Sci., 2012, 5, 5510 (2011) 36 J. P. Heremans, V. Jovovic, E. S. Toberer, A. Saramat, K. Kurosaki, A. Charoen Phakdee, S. Yamanaka and G. J. Snyder, Science, 2008, 321, 554. 37 C. M. Jaworski, V. Kulbachinskii and J. P. Heremans,134 Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 233201.

38 V. Jovovic and J. P. Heremans, Phys. Rev. B 77 245204 (2008) 39 J. P. Heremans, C. M. Thrush, and D. T. Morelli, Phys. Rev. B 70, 115334 (2004) 40 G. A. Slack, Solid State Physics, ed. H. Ehrenreich, F. Weitz and D. Turnbull, Academic Press, New York, 1979, vol. 34, p.1. 41 O. Delaire, et al., Nat. Mater., 2011, 10, 614 42 E. J. Skoug and D. T. Morelli, Phys. Rev. Lett., 2011, 107, 235901 43 Y. Zhang, et al., Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 85, 054306 44 J.H. Wernick, S. Geller, and K.E. Benson, J. PHys Chem. Solids. 1958, 7, 240, S. Geller and J. H. Wernick, Acta Crystal., 1959, 12, 46 45 S.V. Barabash, V. Ozolins, and C. Wolverton, Phys. Rev. Lett., 2008,101, 155704; S.V. Barabash and V. Ozolins, Phys Rev. B: Condens. Matter Phys., 2010, 81, 075212 46 J. Kresse and G. Hafner, Phys. Rev. B: Condens. Matter Mater. Phys., 1193, 47, 558; J. Kresse and D. Joubert , Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758 47 R.S. Kumar, R.S., Sekar, A., Jaya, N. V., Natarajan, S., “Synthesis and high pressure studies of the semiconductor AgSbSe 2”, J. Alloys Comp., 1999, 285, 48 48 R.S. Kumar, et al. Phys Rev. B: Condens. Matter Mater. Phys., 2001, 72, 060701 49 Nielsen, M.D., Ozolins, V., Heremans, J.P., Energy & Env. Sci., 2012 DOI: 10.1039/c2ee23391f 50 F. D. Rosi, E. F. Hockings, and N. E. Lindenblad, RCA Reviews 22 , 82 (1961) 51 Josh Sugar, J.D., Medlin, D.L., Journal Mat. Sci., V46, Issue 6, pp 1668-1679 52 Jovovic V., Heremans, J.P. Journal Elec Mat., 2009, V 38, Issue 7, pp 1504-1509 53 B.A. Akimov,, P.V. Verteletskii, V.P. Zlomanov, L.T. Ryabova, O.C.Tananaeva, and N.A. Shirokava, Sov. Phys. Semicond. 24,848, 1990, B.A. Akimov, N.A. Luova, and L.I. Ryabova, Phys. Rev. B. 58, 10430, 1998 54 M.I. Baleva and M. D.Borisova, J. Phys. C: Solid State Phys. 16, L907 (1983); V.D. Vulchev and L.D. Borisova, Phys. Status Solidi, A99,K53 (1987) 55 V.D. Vulchev, L.D. Borisova, and S.K. Dimitrova, Phys. Status Solidi, A97,D79, 1986 56 T.Story, Z. Wilamovski, E. Grodicka, B. Witkovsky, and W. Dobrovolski, Acta Phys. Pol. A84, 773 (1993), E. Grodzicka, W. Dobrovolski, J. Kossut, T. Story, and B Witkovska, J. Cryst. Growth, 138, L034 (1994) 57 V. Jovovic, S. J. Thiagarajan, J. P. Heremans, T. Komissarova, D. Khokhlov, A. Nicorici, , J. Appl. Phys. 103, 053710 1-7 (2008) 58 B. Paul, P.K. Rawat, and P. Banerji, Appl. Phys. Lett. 98, 262101 (2011) 59 J. P. Heremans, C. M. Thrush and D. T. Morelli, J. Appl. Phys. 98 063703 (2005) 60 T. Story, Chapter 6, p. 385, in “Lead Chalcogenides, Physics and Applications”, D. Khokhlov, Ed., Taylor and Francis Books Inc., New York, 2003 61 E. M. Levin, X. W. Fang, S. L. Bud’ko, W. E.135 Straszheim, R. W. McCallum, and K. Schmidt- Rohr, Phys. Rev. B 77, 054418 (2008).

62 E.A. Zvereva, E.P. Skipetrov, O.A. Savelieva, N.A. Pichugin, A.E. Primenko, E.I. Slynko, and V.E. Slynko, J. Phys.: Conf. Series 200 , 062039 (2010) 63 F. T. Hedgcock, P.C. Sullivan, and J.T. Grembowicz, Canadian J. Phys. 64 , 1345 (1986). 64 J. Dijkstra, H. H. Weitering, C. F. van Bruggen, C. Haas, and R. A. de Groot, J. Phys.: Condens. Matter 1, 9141 (1989). 65 S.J. Youn, S.K. Kwon, and B.I. Min, J. Appl. Phys. 101 , 09G522 (2007). 66 Y. Hinatsu, T. Tsuji, and K. Ishida, J. Solid State Chem. 120 , 49 (1995) 67 N. Suzuki, T. Kanomata, R. Konno, T. Kaneko, H. Yamauchi, K. Koyama, H. Nojiri, Y. Yamaguchi, and M. Motokawa, J. Alloys Comp. 290 , 25 (1999). 68 I. Stefaniuk, M. Bester, and M. Kuzma, Rev. Adv. Mater. Sci. 23 , 133 (2010). 69 K. Lukoschus, S. Kraschinski, C. Näther, W. Bensch, R.K. Kremer, J. Sol. State. Chem. 177, 951 (2004) 70 E.P. Skipetrov, N.A. Pichugin, E.I. Slynko, and V. E. Slynko, Low Temp. Phys. 37, 210 (2011) 71 J. S. Dyck, C. Drasar, P. Lost’ak, and C. Uher, Phys. Rev. B 71 , 115214 (2005) 72 J. D. Satterlee, Concepts in Magnetic Resonance 2, 69 (1999). 73 E.M. Levin, B.A. Cook, K. Ahn, M.G. Kanatzidis, K. Schmidt-Rohr, Phys. Rev. B 2009, 80,115211 74 F.F.Sizov, V.V., Teterkin, L.V., Prokof’eva, and E.A. Gurieva, Sov. Phys. Semicond. 14, 1063 (1980); M. N. Vinogradova, E. A. Gurieva, V. I. ZharskiT, S. V. Zarubo, L. V. Prokof’eva,T. T. Dedegkaev, and I. I. Kryukov, ibid. 12, 387 (1978) 75 I. M. Tsidil’kovskii, Sov. Phys. Usp. 35, 85 (1992). 76 J.D. Konig, M.D. Nielsen, Y.B. Gao, M. Winkler, A. Jacquot, H Bottner, J.P. Heremans 77 G.S. Nolas, J.C. Cohn, G.A. Slack, “Effect of partial void filling on lattice thermal conductivity of skutterudites” Phys. Rev. B. 1998, 58, 1 78 D.J. Singh, W.E. Pickett, “Skutterudite antimonides: Quasilinear bands and unusual transport”, Phys Rev B, Volume 50, Number 15, (1994) 79 J.O. Sofo, G.D. Mahan, “Electronic structure of CoSb 3: A narrow-band-gap semiconductor”, Phys Rev. B, Volume 58, Number 23, (1998) 80 H. Anno, H. Tashiro, Y. Notohara, T. Sakakibara, K. Hatada, K. Motoya, H. Shimizu, K. th Matsubara, “Electronic Transport Properties of p-type CoSb 3”, 16 International Conference on Thermoelectrics, (1997) 81 D. Mandrus, A. Migliori, T.W. Darling, M.F. Hundley, E.J. Peterson, J.D. Thompson, “Electronic transport in lightly doped CoSb 3”, Phys Rev B, Volume 52, Number 7, (1995) 82 D.T. Morelli, T. Caillat, J.P. Fleurial, A. Borshchevsky, J. Vandersande, B. Chen, C.Uher, “Low-temperature transport properties of p-type CoSb 3”, Phys Rev B, Volume 51, Number 15, (1995) 83 S. Hui, M.D. Nielsen, M.R. Homer, D.L. Medlin, J. Tobola, J.R. Salvador, J.P. Heremans, K.P. 136 Pipe, C. Uher, J. Appl. Phys. 115, 103704 (2014)

84 T. Caillat, A. Borshchevsky, J.P. Fleurial, “Prperties of single crystalline semiconducting CoSb3”, J. Appl. Phys. 80, 4442 (1996) 85 J. Tobola, K. Wojciechowski, J. Cieslak, J. Leszczynski, “Thermoelectric Properties and Electronic Structure of Sn-Doped CoSb3” 22 nd International Conference on Thermoelectrics (2003) 86 K. Park, S. Ur, I.Kim, S. Choi, W. Seo, “Transport Properties of Sn-doped CoSb3 Skutterudites”, Journal of the Korean Physical Society, Vol. 57, No. 4, pp. 1000-1004, (2010) 87 B.N. Zobrina, L.D. Dudkin, Sov. Phys. Solid State 1, 1688 (1960)

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