Thermoelectric Skutterudites: Why and How High zT Can Be Achieved Citation Tang, Yinglu (2016) Thermoelectric Skutterudites: Why and How High zT Can Be Achieved. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9TD9VB1. https://resolver.caltech.edu/CaltechTHESIS:04192016-013400484 Abstract Thermoelectric materials have been widely studied over the past few decades due to their ability to convert waste heat into useful electricity. Among various thermoelectric materials, skutterudite distinguishes itself in both space and terrestrial applications with its excellent thermoelectric performance, robust mechanical properties, and thermal stability. The thermoelectric excellence of skutterudites is mostly attributed to the low thermal conductivity due to the addition of filler atoms (R) into the void (one per primitive cell Co 4 Sb 12 ). Essential though this is to high zT, the importance of the intrinsic electronic structure in skutterudites is often understated or ignored completely. In this thesis, by combining experimental and computational studies, the electronic origin of high thermoelectric performance of CoSb 3 -based skutterudites is investigated. The high zT was shown to be a direct result of the high valley degeneracy inherent to CoSb 3 , which is further enhanced by band convergence at high temperatures. This successfully explains why the optimum doping carrier concentration in n-CoSb 3 skutterudites is independent on the type of fillers. With the electronic origin of high thermoelectric performance clarified, the thesis moves on to elaborate how to achieve high zT in skutterudite with the aid of phase diagram study. By mapping out the phase regions near the skutterudite phase on the isothermal section of the R-Co-Sb ternary phase diagram, the solubility region of the CoSb 3 skutterudite phase can be determined along with the solubility limit of R, both of which are often determined in stable compositions resulting in a synthesis window. The temperature dependence of the filler solubility is also demonstrated experimentally. This overturns the general understanding that the filler solubility is a single value only dependent on the filler type. The temperature dependence of stable compositions enables easy carrier concentration tuning which allows the optimization of thermoelectric performance. High zT values are achieved in single In, Yb, Ce-CoSb 3 skutterudites. The methodology applied here are not confined to n-CoSb 3 , but can be generalized to any other ternary systems. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Thermoelectrics, skutterudite Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Materials Science Awards: Demetriades-Tsafka-Kokkalis Prize in Benign Renewable Energy Sources or Related Fields, 2016 Thesis Availability: Public (worldwide access) Research Advisor(s): Snyder, G. Jeffrey Thesis Committee: Johnson, William L. (chair) Goddard, William A., III Minnich, Austin J. Snyder, G. Jeffrey Defense Date: 11 December 2015 Non-Caltech Author Email: tangyinglu1517 (AT) gmail.com Funders: Funding Agency Grant Number DOE-Gentherm UNSPECIFIED National Science Council of Taiwan NSC101-3113-P-008-001 Record Number: CaltechTHESIS:04192016-013400484 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04192016-013400484 DOI: 10.7907/Z9TD9VB1 Related URLs: URL URL Type Description http://10.1038/nmat4430 DOI Paper cited in Chapter 3 http://10.1038/ncomms8584 DOI Paper cited in Chapter 5 http://10.1039/C3EE43240H DOI Paper cited in Chapter 5 http://10.1016/j.jmat.2015.03.008 DOI Paper cited in Chapter 5 http://10.1002/adfm.201202571 DOI Paper cited in Chapter 5 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 9681 Collection: CaltechTHESIS Deposited By: Yinglu Tang Deposited On: 09 Mar 2017 22:06 Last Modified: 24 Oct 2020 00:07 Thesis Files Preview PDF - Final Version See Usage Policy. 6MB Repository Staff Only: item control page

i Thermoelectric skutterudites: why and how high zT can be achieved Thesis by Yinglu Tang In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2016 (Defended Dec 11, 2015) ii  2016 Yinglu Tang All rights reserved iii ACKNOWLEDGEMENTS The path to a PhD degree is not easy and many times in the moments that I feel lost and weak, I would practice writing this part of my thesis, as a way of encouraging myself to continue. It is silly, but helpful. Now finally, here I am. I would like to thank Jesus Christ for being my savior. I got to know him in my fourth year and ever since I have been blessed by his unconditional love. I am grateful that he lifts me up through every hardship and difficulty. I would like to thank my parents, for their love and support over the past 27 years and many years to come. I apologize for my constant absence from their lives due to the path I have chosen, and thank them for accepting all without complaint. The completion of this work would be impossible without their support. I hope I can make a better living for them in the future. I would like to thank my advisor Prof. Jeff Snyder. Through the past 5 years, Jeff has been a great mentor to me. He not only gives me research freedom but also ignites my passion and curiosity with his abundant knowledge and sharp research insights, not to mention all those enlightening discussions whenever I encounter problems in my research. I owe him endless thanks for his patience to help me strengthen my confidence throughout the course as well. He knows my potential better than I do myself and I will be grateful forever for his guidance in both research and life. I thank Prof. Sinn-wen Chen and his group at National Tsinghua University for their generous help about phase diagram studies. I learnt a lot from our productive communications (skype, in person, email). I thank Prof. Lidong Chen as well as his group at SICCAS in China for valuable scientific discussions about skutterudites. Former postdocs of the group (Teruyuki, Shiho and Aaron) and my labmates (Nick, Heng and Alex) helped me with instruments when I first started as a graduate student, for which I appreciate greatly their patience. I am grateful to Yuting Qiu and Zach Gibbs for our successful collaborations. Thanks to Yulong Li for transport property measurements, Luis iv Agapito, Guodon Li, and Lily Xi for DFT calculations and Riley Hanus for APT measurements in this work. Dr. Chi Ma at Caltech helped me a lot with microscopy. My student Estelle Sanz was a gift to my first mentor experience. Her intelligence, diligence as well as passion for life inspired me a lot during her four-month stay at Caltech. Prof. Yanzhong Pei at Tongji University, Prof. Tiejun Zhu at Zhejiang University, Prof. Eric Toberer at Colorado School of Mines, and Prof. Holger Kleinke at University of Waterloo offered me incisive suggestions regarding carrier path. I thank DOE-Gentherm and National Science Council of Taiwan for supporting my research. My lab mates are great for building a supportive, relaxing yet productive working environment. It was a pleasure to work with all of them during the past 5 years. Special thanks to Nick, Alex and Alex, Heng and Jie, Tristan and Alice, Hyun-sik and Ok-young, and Saneyuki and Stephen for their constant encouragement. Life will never be the same without you guys around. All my friends along the way, including Yulia, Amanda, Ho-cheng, Chu Hao, Juying, Sijia, Vanessa, Hanhui, Haiyang, Ning Xin, Chenguang, Liu Fan, Siying, Yan Qu, Cindy, Megan, Audrey, Jian Li, Zhichang Liu, Yu Zhang, Min Zhang, Betty, Huijian Li, Qiao Liang, and many more not listed here, I thank you all for being there for me when needed you. v ABSTRACT Thermoelectric materials have been widely studied over the past few decades due to their ability to convert waste heat into useful electricity. Among various thermoelectric materials, skutterudite distinguishes itself in both space and terrestrial applications with its excellent thermoelectric performance, robust mechanical properties, and thermal stability. The thermoelectric excellence of skutterudites is mostly attributed to the low thermal conductivity due to the addition of filler atoms (R) into the void (one per primitive cell Co4Sb12). Essential though this is to high zT, the importance of the intrinsic electronic structure in skutterudites is often understated or ignored completely. In this thesis, by combining experimental and computational studies, the electronic origin of high thermoelectric performance of CoSb3-based skutterudites is investigated. The high zT was shown to be a direct result of the high valley degeneracy inherent to CoSb3, which is further enhanced by band convergence at high temperatures. This successfully explains why the optimum doping carrier concentration in n-CoSb3 skutterudites is independent on the type of fillers. With the electronic origin of high thermoelectric performance clarified, the thesis moves on to elaborate how to achieve high zT in skutterudite with the aid of phase diagram study. By mapping out the phase regions near the skutterudite phase on the isothermal section of the R-Co-Sb ternary phase diagram, the solubility region of the CoSb3 skutterudite phase can be determined along with the solubility limit of R, both of which are often determined in stable compositions resulting in a synthesis window. The temperature dependence of the filler solubility is also demonstrated experimentally. This overturns the general understanding that the filler solubility is a single value only dependent on the filler type. The temperature dependence of stable compositions enables easy carrier concentration tuning which allows the optimization of thermoelectric performance. High zT values are achieved in single In, Yb, Ce-CoSb3 skutterudites. The methodology applied here are not confined to n-CoSb3, but can be generalized to any other ternary systems. vi PUBLISHED CONTENT AND CONTRIBUTIONS Yinglu Tang, Zachary M. Gibbs, Luis Agapito, Guodong Li, Hyun-Sik Kim, Marco Buongiorno-Nardelli, Stefano Curtarolo, G. Jeffrey Snyder, “ Convergence of the Multivalley Bands as the Electronic Origin of High Thermoelectric Performance in CoSb3 Skutterudites” Nature Materials 14, 1223-1228 (2015) DOI: 10.1038/NMAT4430 Y.T. participated in the conception of the work, performed sample synthesis, structural characterization and thermoelectric transport property measurements, confirmed Kane band model effective mass relation and participated in band modelling and writing up the manuscript. Chapter 3 is readapted and reproduced with permission from the copyright holder, Nature Publishing Group. Yinglu Tang, Riley Hanus, Sinn-wen Chen, G. Jeffrey Snyder, “Solubility Design Leading to High zT in Low-Cost Ce-CoSb3 Skutterudites” Nature Communications 6, 7584(1-7) (2015) DOI: 10.1038/ncomms8584 Y.T. contributed to design the study and write the manuscript. Y.T. contributed to sample synthesis, structure characterization and property measurements. Section 5.5 is readapted and reproduced with permission from the copyright holder, Nature Publishing Group. Yinglu Tang, Sinn-wen Chen, G. Jeffrey Snyder, “Temperature Dependent Solubility of Yb in Yb-CoSb3 Skutterudite and its Effect on Preparation, Optimization and Lifetime of Thermoelectrics” Journal of Materiomics 1, 75-84 (2015) DOI:10.1016/j.jmat.2015.03.008 Y.T. contributed to design the study and write the manuscript. Y.T. contributed to sample synthesis, structure characterization and property measurements. Sections 5.2, 5.4 and 5.6 are readapted and reproduced with permission from the copyright holder, The Chinese Ceramic Society. Production and hosting by Elsevier B.V. vii Yinglu Tang, Yuting Qiu, Lili Xi, Xun Shi, Wenqing Zhang, Lidong Chen, Ssu-Ming Tseng, Sinn-wen Chen and G. Jeffrey Snyder “Phase Diagram of In-Co-Sb System and Thermoelectric Properties of In-containing Skutterudites” Energy and Environmental Science 7, 812-819 (2014) DOI: 10.1039/C3EE43240H Y.T. contributed to design the study and write the manuscript. Y.T. contributed to sample synthesis, structure characterization and property measurements. Section 5.3 is readapted and reproduced with permission from the copyright holder, Royal Society of Chemistry. Yuting Qiu, Lili Xi, Pengfei Qiu, James R. Salvador, Jung Y. Cho, Jihui Yang, Xun Shi, Wenqing Zhang, Lidong Chen, Yuan-chun Chen, Sinn-wen Chen, Yinglu Tang, and G. Jeffrey Snyder “Charge-Compensated Compound Defects in Ga-containing Thermoelectric Skutterudites” Advanced Functional Materials 23, 3194-3203 (2013) DOI: 10.1002/adfm.201202571 Y.T. contributed to data analysis and writing of the manuscript. Section 5.3 is readapted and reproduced with permission from the copyright holder, Wiley-VCH Verlag GmbH & Co. KGaA. viii TABLE OF CONTENTS Acknowledgements ............................................................................................ iii Abstract ................................................................................................................ v Published content and contributions .................................................................. vi Table of Contents .............................................................................................. viii List of Figures ..................................................................................................... xi List of Tables .................................................................................................... xvi List of Symbols and Notation .......................................................................... xvii Chapter 1: Introduction ........................................................................................ 1 1.1 Thermoelectric (TE) materials ................................................................ 1 1.2 Skutterudites ............................................................................................ 4 1.3 Summary of research ............................................................................... 5 Chapter 2: Experimental methods ....................................................................... 8 2.1 Summary.................................................................................................. 8 2.2 Synthesis procedures ............................................................................. 10 2.3 Characterization ..................................................................................... 12 2.3.1 Phase and chemical composition identification .......................... 12 2.3.2 Transport property characterization ............................................. 13 2.3.3 Optical property characterization ................................................. 13 2.4 Ab initio DFT calculations .................................................................... 14 Chapter 3: Electronic origin of high zT in n-CoSb3 skutterudites .................... 16 3.1 Summary................................................................................................ 16 3.2 Background introduction ....................................................................... 16 3.3 Results and discussion........................................................................... 17 3.3.1 Multiple conduction band behavior in n-CoSb3 .......................... 17 3.3.2 Linear or kane bands do not increase Seebeck mass ................... 20 3.3.3 Band convergence at high temperatures ...................................... 22 3.4 Conclusion and future work .................................................................. 24 Chapter 4: Role of filler in thermal transport .................................................... 26 4.1 Summary................................................................................................ 26 4.2 Complex phonon modes........................................................................ 27 4.2.1 Rattling - resonant scattering or avoided crossing ...................... 27 4.2.2 Point defect scattering .................................................................. 30 4.2.3 Electron-phonon scattering .......................................................... 35 4.3 Thermal transport calculation ............................................................... 37 4.3.1 Minimum thermal conductivity ................................................... 37 4.3.2 Electronic contribution to thermal conductivity .......................... 39 ix 4.3.3 Callaway model ............................................................................ 40 4.4 Results and discussion........................................................................... 41 4.4.1 Possible electron-phonon scattering in RxCo4Sb12 ...................... 41 4.4.2 Lattice softening due to fillers ..................................................... 44 4.5 Conclusion and future work .................................................................. 47 Chapter 5: Phase diagram studies in n-CoSb3 skutterudites ............................. 48 5.1 Summary................................................................................................ 48 5.2 Methodology.......................................................................................... 49 5.3 Soluble site other than the void ............................................................. 55 5.3.1 Solubility debate of In and Ga ..................................................... 55 5.3.2 DFT calculations of In-CoSb3 systems ........................................ 56 5.3.3 Phase diagram study of In-Co-Sb system .................................... 59 5.3.4 Thermoelectric properties of In-CoSb3........................................ 64 5.3.5 Discussion about of In-CoSb3 ...................................................... 69 5.4 Stable compositions and Vegard’s law ................................................. 70 5.4.1 Solubility debate of Yb ................................................................ 70 5.4.2 Skutterudite lattice expansion due to Yb filling .......................... 71 5.4.3 Stable compositions in ternary phase diagram system ................ 72 5.4.4 Vegard’s law in ternary phase diagram system ........................... 75 5.4.5 Discussion about of Yb-CoSb3 .................................................... 79 5.5 Solubility design strategies.................................................................... 80 5.5.1 Solubility barrier of Ce in Ce-CoSb3 ........................................... 80 5.5.2 Optimum doping of Ce-CoSb3 skutterudites ............................... 82 5.5.3 Ultra-high FFL of Ce in Ce-CoSb3 skutterudites ........................ 83 5.5.4 Thermoelectric properties of optimized Ce-CoSb3 ..................... 87 5.5.5 Discussion about Ce-CoSb3 ......................................................... 89 5.6 Stability of optimized compositions ..................................................... 89 5.7 Conclusion ............................................................................................. 90 Chapter 6: Defect study of intrinsic CoSb3 ....................................................... 91 6.1 Summary................................................................................................ 91 6.2 Literature study ...................................................................................... 91 6.2.1 Bonding chemistry of CoSb3 ....................................................... 91 6.2.2 Synthesis condition and defect type............................................. 92 6.3 Results and discussion ........................................................................... 94 6.3.1 Phase width of CoSb3 ................................................................... 94 6.3.2 Defect type in intrinsic CoSb3...................................................... 99 6.4 Conclusion and future work ................................................................ 102 Chapter 7: Phase diagram study of Ce-Co-Fe-Sb system .............................. 103 7.1 Summary.............................................................................................. 103 7.2 Charge-compensating defects ............................................................. 103 x 7.3 Ce-Fe-Sb isothermal section at 700℃ ................................................ 106 7.4 Co-Fe-Sb isothermal section at 700℃ ................................................ 109 7.5 Phase space of CeyCoxFe4-xSb12 skutterudites at 700℃...................... 113 7.6 n-p change in CeyCo3.25Fe0.75Sb12 skutterudites ................................. 117 7.7 Conclusion and future work ................................................................ 121 Chapter 8: Nano-structuring in bulk skutterudites .......................................... 122 8.1 Summary.............................................................................................. 122 8.2 Experimental results ............................................................................ 122 8.3 Conclusion and future work ................................................................ 125 Chapter 9: Future work .................................................................................... 126 Appendix A: ..................................................................................................... 128 Appendix B: ..................................................................................................... 137 Appendix C: ..................................................................................................... 139 Appendix D: ..................................................................................................... 141 Appendix E: ..................................................................................................... 143 Bibliography .................................................................................................... 145 xi LIST OF FIGURES 1.1 Illustration of Seebeck effect. ......................................................................... …….1 1.2 Illustration of Peltier effect. ..................................................................................... 2 1.3 Optimizing zT through carrier concentration tuning. ............................................... 4 1.4 Crystal structure of (a) unfilled CoSb3 and (b) filled CoSb3 in a unit cell............... 5 2.1 The Co-Sb (Cobalt-Antimony) phase diagram. ...................................................... 9 2.2 SEM photo of a melt-quenched sample with nominal composition CoSb3. ........... 9 3.1: Experimental and theoretical evidence showing multiple conduction bands in n-type CoSb3. .......................................................................................................... 18 3.2 Band non-parabolicity and its effect on the Seebeck coefficient and energy-dependent Seebeck effective mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸). Views .................................... 21 3.3 Band convergence at high temperatures in CoSb3 as shown from optical absorption and thermoelectric figure of merit. ...................................................... 24 4.1 Different mechanisms in reducing lattice thermal conductivity between resonant scattering and avoided crossing on the phonon spectra. ....................................... 30 4.2 Lattice thermal conductivity 𝜅𝜅𝐿𝐿 at room temperature as a function of carrier concentration for Co1-xMxSb3 (M=Ni, Pd, Pt, and Pd+Pt) samples. ...................... 35 4.3 Lattice thermal conductivity versus doping level x with both experimental and calculated results from Klemens model. ................................................................ 42 4.4 Dependence of lattice constant of YbxCo4Sb12 on the Yb actual content x. .......... 43 4.5 Thermal resistivity due to electron-phonon scattering versus measured Hall carrier concentration in YbxCo4Sb12. ...................................................................... 44 4.6 Dependence of elastic moduli on doping content x in YbxCo4Sb12 ....................... 46 5.1 Binary phase diagrams of Yb-Co, Yb-Sb and Co-Sb systems............................... 49 5.2 Selected nominal compositions near target phase CoSb3 for phase region identification in Yb-Co-Sb isothermal section at 700℃. ....................................... 50 5.3 Phase regions near CoSb3 determined from synthesized samples in a section of Yb-Co-Sb isothermal ternary phase diagram at 700℃ (YbSb2-CoSb2-Sb). ..... 52 5.4 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 973K. ................................................................. 52 5.5 Temperature dependence of solubility limit of Yb in Yb-doped skutterudites xii YbxCo4Sb12. ............................................................................................................. 53 5.6 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 873K. ................................................................. 54 5.7 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 1023K. ............................................................... 54 5.8 Temperature dependence of the thermoelectric figure of merit zT for Yb-doped skutterudites targeting the same YbxCo4Sb12 composition with samples annealed at 873K without excess Sb (red stable point) and 1023K with excess Sb (blue stable point). .................................................................................................. 55 5.9 Band structures of CoSb3 and different In-doped systems. ................................... 57 5.10 Calculated Gibbs free energy ( ) as a function of doping content x of (a) Ga in Ga-CoSb3 skutterudite at 923K and (b) In in In-CoSb3 skutterudite at 873K. ... 58 5.11 Formation energies of possible defects as a function of Fermi level at the Co-rich limit in Ga-containing (a) and In-containing (b) skutterudites. ........................... 58 5.12 (a) XRD patterns of the In-containing skutterudites with nominal compositions InxCo4Sb12-x/3 (x=0, 0.075, 0.15, 0.225, 0.30, 0.375, 0.45, 0.60) and corresponding actual compositions x=0, 0.081, 0.106, 0.207, 0.276, 0.265, 0.286, 0.274. (b) Magnification of the XRD patterns. (c) Dependence of lattice parameter on actual indium content x. .................................................................................................. 60 5.13 Electron probe microanalysis (EPMA) indium maps in samples with different nominal compositions. (a) In0.225Co4Sb11.925, (b) In0.30Co4Sb11.90, (c) In0.30Co4.2Sb11.7 and (d) In0.30Co3.8Sb12.1. ......................................................................................... 61 5.14 Proposed phase diagram for In-containing skutterudites at 873K. (a) Full diagram with related binary phases and approximate regions of solubility indicated by red regions. (b) Phase regions enlarged near CoSb3. ................................................. 62 5.15 Temperature dependence of (a) the thermopower S and (b) electrical conductivity σ for samples of InxCo4Sb12-x/3 with different indium impurity content. ............. 65 5.16 (a) Room temperature electron concentration (n) as a function of indium impurity fraction x in In-containing complex compound defect (CCD) skutterudites. (b) Room temperature S as a function of electron concentration for In-containing CCD skutterudites at 300 K. ................................................................................ 66 5.17 (a) Temperature dependence of total thermal conductivity and (b) Room xiii temperature lattice thermal conductivity as a function of the indium impurity fraction in skutterudites. ...................................................................................... 67 5.18 Temperature dependence of thermoelectric figure of merit zT for In-containing skutterudites with complex compound defect (CCD). ........................................ 68 5.19 Repeatability of thermoelectric properties of In-containing skutterudite with complex compound defect (CCD) and x=0.207. ................................................. 68 5.20 (a) Dependence of lattice constant on the actual Yb content, x, in YbxCo4+ySb12+z ; (b) Dependence of the lattice constant on the nominal Yb content, x, in YbxCo4+ySb12+z for samples annealed at 873K. ................................................... 71 5.21 (a) Magnified region of the isothermal section at 973K near CoSb3 of the Yb-Co-Sb ternary phase diagram system with two stable skutterudite compositions (red and blue points) and solubility line (red line). (b) Dependence of the lattice constant derived from X-ray Diffraction data on the Co/Sb ratio for nominal compositions with the same Yb content x = 0.5 marked as empty orange rectangles in (a). ................................................................................................... 73 5.22 Illustration of typical eutectic phase diagram of binary A-B system. ................. 76 5.23 Illustration of isothermal section of ternary In-Co-Sb phase diagram. ............... 76 5.24 Dependence of solubility limit on adjacent phase regions in a ternary phase diagram system. .................................................................................................... 77 5.25 Samples with different nominal Yb content x in YbxCo4Sb12.2 (marked as empty orange rectangles) but the same Sb excess lead to a nonlinear dependence of lattice constant (b) due to the sample traversing different two- and three- phase regions of the phase diagram (a). ......................................................................... 78 5.26 Hall carrier concentration vs number of electrons per primitive cell Co4Sb12. .. 82 5.27 Filling fraction limit (FFL) of Ce in Ce-CoSb3 skutterudites. (a) Skutterudite lattice expansion due to Ce filling. (b) Dependence of FFL on annealing temperature. (c) Dependence of FFL on nominal composition with annealing temperature 973K. ................................................................................................ 83 5.28 APT analysis of the most heavily doped sample Ce0.2Co4Sb12. a) 3D reconstruction of micro-tip containing a grain boundary. b) Concentration profile across the grain boundary and in the grain. ......................................................... 86 xiv 5.29 Transport properties of Ce- and Yb-doped skutterudites. The temperature dependence of : (a) electrical resistivity, (b) Seebeck coefficient, (c) thermal conductivity, (e) thermoelectric figure of merit (zT) are plotted in the temperature range of 300K to 850K. In figure (d) the lattice thermal conductivity with a Lorentz number of 2.0×10-8 V2K-2 is plotted against the filling fraction for various types of fillers. .......................................................................................... 87 5.30 Stability of YbxCo4Sb12 skutterudite with optimum dopant composition x = 0.3. ................................................................................................................... 89 6.1 Bonding illustration in CoSb3 from Dudkin’s bonding model. ............................. 92 6.2 (a) Structure of skutterudite CoSb3 (Co: red; Sb: blue), (b) [CoSb6] octahedron, (c) Sb4 ring. .............................................................................................................. 92 6.3 XRD patterns of CoSb3 samples annealed at 500℃(after hot pressing). ............. 96 6.4 SEM photos of a) Co-rich sample Co4Sb11.97; b) on stoichiometry sample Co4Sb12; c) Sb-rich sample Co4Sb12.17 after annealing at 500℃. .............................. 96 6.5 XRD patterns of CoSb3 samples annealed at 700℃ (ingots or samples after hot pressing). ........................................................................................................... 97 6.6 SEM photos of a) Co-rich sample Co4Sb11.94; b) Co-rich sample Co4Sb11.97; c) on stoichiometry sample Co4Sb12; d) Sb-rich sample Co4Sb12.03; e) Sb-rich sample Co4Sb12.15 after annealing at 700℃. .......................................................................... 97 6.7 XRD patterns of CoSb3 samples annealed at 850℃ (ingots after annealing). ...... 98 6.8 SEM photos of a) Co-rich sample Co4Sb11.94; b) Co-rich sample Co4Sb11.97; c) Sb-rich sample Co4Sb12.03 after annealing at 850℃. ............................................ 98 6.9 Temperature dependent phase width of CoSb3. ..................................................... 99 6.10 Co elemental analysis showing higher Ni impurity level than Fe. .................... 101 6.11 Linear relationship between carrier concentration in CoSb3 and Ni impurity content.................................................................................................................. 102 7.1 Dependence of maximum Ce solubility with Fe amount in CeyCo4-xFexSb12 skutterudite materials. ........................................................................................... 104 7.2 Dependence of carrier concentrations in CeyCo4-xFexSb12 skutterudites on the amount of Co substitution. .................................................................................... 104 7.3 Crossover from p-type to n-type in CeyCoxFe4-xSb12 materials. ........................... 105 xv 7.4 First sketch of Ce-Fe-Sb ternary phase diagram isothermal section at 700℃. .... 107 7.5 SEM photos of samples with nominal composition as a) #1 Ce0.9Fe3.57Sb12.43, b) #2 Ce0.9Fe3.9Sb12.1 and c) #5 Ce1.05Co4.1Sb11.9.................................................... 108 7.6 Second batch of Ce-Fe-Sb samples (#9-12) synthesized based on binary phase diagrams and results from first batch of samples. ................................................ 108 7.7 Isothermal section of Ce-Fe-Sb ternary system at 700℃. .................................... 109 7.8 Reported phase diagram of CoSb3-“FeSb3” system. ........................................... 110 7.9 Proposed isothermal section of Co-Fe-Sb ternary system at 700℃. ................... 111 7.10 SEM photos of samples with nominal composition as a) #1 Co3.4Fe0.3Sb12.6, b) #2 Co4.2Fe0.3Sb11.8, c) #3 Co3.25Fe0.75Sb12, d) #5 Co1.5Fe1.5Sb12, e) #6 Fe4Sb12 and f) #7 Co6Fe6Sb12. ............................................................................................ 112 7.11 Experimental isothermal section of Co-Fe-Sb ternary system at 700℃. .......... 113 7.12 Slices of the isothermal space of Ce-Co-Fe-Sb system with different Co:Fe ratios projected to the Co-Fe-Sb plane as black dashed lines. ..................................... 114 7.13 Proposed phase boundary of CeyCoxFe4-xSb12 skutterudites based on preliminary solubility results from Co-Fe-Sb, Ce-Co-Sb, Ce-Fe-Sb systems. ...................... 114 7.14 Experimentally determined phase space of skutterudite phase with exact stoichiometry (Co+Fe):Sb = 1:3 in Ce-Co-Fe-Sb system at 700℃ ................... 117 7.15 Transport properties of CeyCo3.25Fe0.75Sb12 skutterudites. ................................. 119 7.16 Room temperature transport properties of CeyCo3.25Fe0.75Sb12 skutterudites. ... 120 8.1 Heat treatment parameter determined based on the experimental phase diagram study ....................................................................................................................... 124 8.2 Effect of precipitation time on Yb content in skutterudites after precipitation. . 124 8.3 SEM photos for sample with nominal composition Yb0.40Co4Sb12 precipitated at 600℃ with precipitation times: (a) 2 days; (b) 3 days and (c) 9 days…………..125 xvi LIST OF TABLES 4.1 Ionic radii, atomic mass and rattling frequency of different filler atoms. .... …….28 4.2 Room temperature transport properties of YbxCo4Sb12 skutterudites (with nominal x from 0.0025 to 0.3). ..................................................................... 41 4.3 Theoretical density calculated from EPMA doping content x and XRD-derived lattice parameter a of YbxCo4Sb12 skutterudites. .................................................... 45 4.4 Speed sound measurement data as well as calculated elastic moduli and theoretical density of ball-milled YbxCo4Sb12 skutterudites..................................................... 46 5.1 Selected nominal compositions for investigation of Yb-Co-Sb isothermal section at 700℃. ...................................................................................................... 51 5.2 Nominal indium content and total indium content estimated by EPMA for In-containing skutterudites with different compositions. ...................................... 61 5.3 Not fully charge-compensated compound defect samples (InVF)(2x+δ)/3Co4Sb12-(x-δ)/3(InSb)(x-δ)/3 ............................................................................ 69 6.1 Compositions synthesized with corresponding annealing temperatures. .............. 95 6.2 Phase purity and composition analysis of CoSb3 samples annealed at 500℃. ...... 96 6.3 Phase purity and composition analysis of CoSb3 samples annealed at 700℃. ...... 97 6.4 Phase purity and composition analysis of CoSb3 samples annealed at 850℃. ...... 98 6.5 Room temperature transport data measured on Co4Sbx (x = 11.94 ~ 12.17) samples hot pressed at 700℃. ............................................................................................... 99 6.6 Room temperature transport data measured on Fe, Ni doped CoSb3 samples annealed and hot pressed at 700℃. ....................................................................... 101 7.1 Selected nominal compositions for investigation of Ce-Fe-Sb isothermal section at 700℃. ................................................................................................................. 106 7.2 Selected nominal compositions for investigation of Co-Fe-Sb isothermal section at 700℃. ................................................................................................................. 110 7.3 Nominal compositions of CeyCoxFe4-xSb12 samples for skutterudite phase space investigation in Ce-Co-Fe-Sb quaternary system. ............................................... 115 7.4 EPMA determined compositions for CeyCoxFe4-xSb12 skutterudites. .................. 116 xvii LIST OF SYMBOLS AND NOTATIONS S or 𝛼𝛼: Seebeck coefficient (thermopower) Π: Peltier coefficient K: Thompson coefficient 𝜎𝜎: electrical conductivity 𝜅𝜅: thermal conductivity 𝜅𝜅𝑒𝑒 : electronic contribution to thermal conductivity 𝜅𝜅𝑙𝑙 : lattice contribution to thermal conductivity 𝜅𝜅𝑏𝑏 : bipolar contribution to thermal conductivity L: Lorentz number zT: thermoelectric figure of merit 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 : maximum thermoelectric efficiency D: mass diffusion coefficient 𝑙𝑙𝐷𝐷 : diffusion length 𝑡𝑡𝐷𝐷 : diffusion time 𝑁𝑁𝑉𝑉 : number of valley degeneracy 𝐸𝐸𝑔𝑔 : energy band gap Δ𝐸𝐸: energy offset e: electron charge 𝑘𝑘𝐵𝐵 : Boltzmann constant ℏ: reduced Planck constant r: scattering parameter xviii n: carrier concentration 𝑟𝑟𝐻𝐻 : Hall factor 𝑅𝑅𝐻𝐻 : Hall coefficient 𝜇𝜇𝐻𝐻 : Hall mobility 𝑛𝑛𝐻𝐻 : Hall carrier concentration 𝜇𝜇𝐷𝐷 : drift mobility 𝑚𝑚𝑆𝑆∗ : density of states effective mass obtained from Seebeck measurements using the single parabolic band (SPB) model (referred to as the Seebeck mass) 𝑚𝑚0∗ : band-edge effective mass 𝑚𝑚𝑃𝑃∗ :energy-dependent effective mass derived from the electron momentum 𝑚𝑚𝐶𝐶∗ : conductivity mass 𝜆𝜆: correction factor in Mott relation for Seebeck in kane band model B: quality factor Edef: deformation potential Cl: elastic constant in longitudinal direction Ω: atomic volume Θ: Debye temperature 𝑤𝑤𝐷𝐷 : Debye frequency 𝑣𝑣𝑠𝑠 : speed of sound 𝑉𝑉𝑔𝑔 : group velocity 𝑤𝑤0 : resonant frequency 𝜏𝜏𝑟𝑟 : resonant scattering relaxation time Γ: scattering parameter for point defect scattering 𝜏𝜏𝑃𝑃𝑃𝑃 : point defect scattering relaxation time 𝜏𝜏𝑒𝑒𝑒𝑒 : electron-phonon scattering relaxation time xix 𝑣𝑣𝑒𝑒 : velocity of electron 𝜆𝜆𝑒𝑒 : mean free path of electron d: mass density 𝜏𝜏𝐵𝐵 : boundary scattering relaxation time 𝜏𝜏𝑈𝑈 : Umklapp scattering relaxation time K, G: bulk and shear modulus respectively 𝜀𝜀𝐹𝐹 : Fermi level q: charge state of the point defect 𝐸𝐸𝑉𝑉 : valence band maximum ec : number of available valence electrons of the cation element bc : number of cation-cation two-electron bonds and nonbonding lone pair electrons. ba : number of anion-anion two-electron bonds 𝜂𝜂: reduced electrochemical potential ℰ: electron energy 𝜇𝜇: chemical potential Φ: electrostatic potential E: electric field H: magnetic field j: electrical current density f (k, r): electron distribution function of momentum k and position r TE: thermoelectric PGEC: phonon glass electron crystal FFL: filling fraction limit xx SPB: single parabolic band SKB: single kane band SEM: scanning electron microscopy XRD: X-ray diffraction EPMA: electron probe microprobe EDS: energy dispersive spectroscopy WDS: wavelength dispersive spectroscopy APT: atomic probe topographic HP: hot pressing BM: ball milling MMS: melt-melt-spinning DRIFTS: diffuse reflectance infrared fourier transform spectroscopy PAW: projector augmented wave VASP: Vienna ab initio simulation package PBE: Perdew-Burke-Ernzerhof GGA: generalized gradient approximation DFT: density functional theory DOS: density of states JDOS: joint density of states CB2: secondary conduction band ADP: atomic displacement parameter PDOS: partial density of phonon states NIS: nuclear inelastic spectroscopy xxi INS: inelastic neutron scattering EXAFS: extended X-ray fine absorption measurements CCCD: charge-compensated compound depects BTE: Boltzmann transport equation 1 Chapter 1 Introduction 1.1 Thermoelectric (TE) materials Thermoelectricity was discovered and developed in Western Europe by academic scientists in the 100 years between 1820 and 1920 1. The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa. By the 1950s, generator efficiencies had reached 5% and cooling from ambient to below 0℃ was demonstrated. Many thought thermoelectric would soon replace conventional heat engines and refrigeration. However, by the end of the 1960s the pace of progress had stagnated and many research programs were dismantled. Since 1970, the need for reliable, remote power sources enabled niche applications for thermoelectric materials such as space exploration missions (Voyager, Curiosity etc.) even while conventional processes are more efficient. Interest in thermoelectricity renewed in the 1990s with the influx of new ideas such as nano-scale engineered structures. The global need for alternative sources of energy has revived interest in commercial applications and in developing inexpensive and environmentally benign thermoelectric materials.1 There are three separately identified thermoelectric effects: the Seebeck effect, the Peltier effect, and the Thompson effect. Seebeck effect can be observed when electric voltage is produced between a pair of dissimilar materials with their junction subjected to a different temperature. Seebeck effect is illustrated in Figure 1.1. Figure 1.1 Illustration of Seebeck effect. An open circuit voltage V is generated in the presence of a temperature difference ∆𝑇𝑇 in a pair of n- and p- thermoelectric materials. Thermal energy 𝑄𝑄ℎ can then be partly converted to work W. 2 The ratio between the voltage and the temperature difference is called Seebeck coefficient S, which is also called thermopower 𝛼𝛼. As shown in Eq. 1.1, whereas denotes the voltage difference denotes the temperature difference across the material. The minus sign is added such that for p-type material, the Seebeck coefficient is positive; while for n-type material, the Seebeck coefficient is negative. (Eq. 1.1) Peltier effect refers to the presence of heating or cooling at the junction between a pair of dissimilar materials when electric current is passed through. Peltier effect is illustrated in Figure 1.2. Figure 1.2 Illustration of Peltier effect. The temperature difference across a pair of n- and pthermoelectric materials is generated in the presence of an electric current 𝑖𝑖 + . The “Hot” side is attached to a heat sink so that it remains at ambient temperature, while the “Cold” side goes below room temperature. Electrical work W is consumed to move thermal energy 𝑄𝑄 from the cold side to the hot side. The ratio between the heat absorption/creation rate ( Peltier coefficient ) and the electric current (I) is called the , as shown in Eq. 1.2. (Eq. 1.2) While the Seebeck coefficient represents the entropy carried per unit charge, the Peltier coefficient represents heat carried per unit charge. These two coefficients are interrelated by thermodynamics. For time-reversal symmetric materials, Peltier coefficient is simply the Seebeck coefficient times the absolute temperature (known as the second Kelvin relationship shown in Eq. 1.3). (Eq. 1.3) 3 Thompson effect can be observed in a single material with both electric current and temperature gradient applied. The heat absorption/creation rate is proportional to both the electric current and the temperature gradient. The proportionality constant is called Thompson coefficient (K), which relates to both the Seebeck coefficient and the temperature dependence of Peltier coefficient, as shown in Eq.1.4. This equation is also called the first Kelvin relationship. By substituting Eq. 1.3 into Eq. 1.4, we get the simple form relating all three coefficients, as shown in Eq. 1.5. (Eq. 1.4) (Eq. 1.5) The Thomson coefficient is unique among the three main thermoelectric coefficients because it is the only one directly measurable for individual materials. The Peltier and Seebeck coefficients can only be easily determined for pairs of materials. Thermoelectric effects enable thermoelectric materials in many applications such as power generation and refrigeration. The thermoelectric performance of materials is characterized by the dimensionless figure of merit, zT. (Eq. 1.6) where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, and κ is the total thermal conductivity. There is no theoretical upper limit in zT 2, and as zT 𝑇𝑇 −𝑇𝑇𝐶𝐶 approaches infinity, the thermoelectric efficiency 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 approaches the Carnot limit ( 𝐻𝐻𝑇𝑇 𝐻𝐻 1.7). 𝜂𝜂𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑇𝑇𝐻𝐻 −𝑇𝑇𝐶𝐶 1+√𝑧𝑧𝑧𝑧−1 𝑇𝑇𝐻𝐻 1+√𝑧𝑧𝑧𝑧+ 𝑇𝑇𝐶𝐶 ) (Eq. (Eq. 1.7) 𝑇𝑇𝐻𝐻 For thermoelectric materials to be competitive in efficiency compared to traditional energy converters such as internal combustion engines or vapor-compression refrigerators, zT > 3 (about 20-30% efficiency with 𝑇𝑇𝐻𝐻 ≥ 2) 𝑇𝑇𝐶𝐶 is required.3 However, zT > 2 is rarely achieved in bulk 4 thermoelectric materials. Good thermoelectric materials should possess large Seebeck coefficient (large voltage difference), high electrical conductivity (to minimize Joule heating due to electrical resistance), and low thermal conductivity (to minimize heat loss). Due to the inter-correlating nature among these variables (i.e., high electrical conductivity would also lead to high electronic thermal conductivity, large Seebeck coefficient also happens at the band edge which leads to small electrical conductivity), increasing zT is still a major challenge in the field of thermoelectrics. The optimum carrier concentration range for thermoelectric performance is in the heavily doped semiconductor regime, e.g., 1019~1021 cm-3, as shown in Figure 1.3 4. Figure 1.3 Optimizing zT through carrier concentration tuning. Maximizing the efficiency (zT) of a thermoelectric involves a compromise of thermal conductivity (κ; plotted on the y axis from 0 to a top value of 10 Wm–1K–1) and Seebeck coefficient (α; 0 to 500 μVK–1) with electrical conductivity (σ; 0 to 5,000 Ω–1cm–1).4 1.2 Skutterudites Among various state-of-the-art thermoelectric materials, skutterudite is a promising candidate for mid-temperature range (400℃ - 600℃) applications with excellent thermoelectric properties, good mechanical properties, and thermal stability. Skutterudite was named after Skutterud, a small mining town in Norway, where a CoAs3-based mineral was discovered in 1845. Compounds with the structure identical to CoAs3 have since become known as Skutterudites. Binary skutterudites have the general formula MX3, where M stands for one of the group 9 transition metals Co, Ir, or Rh, and X represents P, As, or Sb. Binary skutterudites form with all 9 possible combinations of the M and X elements and crystalize in the body-centered-cubic structure in the space group (#204) of 5 Im . The unit cell M8X24 contains 32 atoms in eight MX3 blocks, as shown in Figure 1.4a. There are three different atomic positions: Co sites (8c), Sb sites (24g), and void sites (2a). (a) (b) Figure 1.4 Crystal structure of (a) unfilled CoSb3 and (b) filled CoSb3 in a unit cell. Study of binary skutterudites for thermoelectric purposes started in the 1950s, and soon it was realized that even though they have high electrical conductivity (104 Sm-1) and large Seebeck coefficient (hundreds of 𝜇𝜇VK-1), their high thermal conductivity (10Wm-1K-1) limits their application as thermoelectric materials5-7. In the 1970s, it was discovered8 that skutterudites have one void (with radius r = 1.89Å)9 per primitive cell (RxM4X12, 0 < x < 1), which allows foreign atoms R to actively fill the structure. These weakly bound atoms can “rattle” around their equilibrium positions. The addition of fillers not only largely decreases the lattice thermal conductivity but also could contribute to n-type doping. This makes skutterudite a perfect example of the phonon glass electron crystal (PGEC) concept, proposed by Slack in 1994 10. Since then, the search for filler type and multiple filling in filled skutteurdites has been a major research trend. Up to date the choice of fillers has been among alkali elements (Na11, K12), alkali earth elements (Ba13, Ca14, Sr15), group 13 elements (Ga16, In17), and rare earth elements (La18, Ce19,20, Nd21, Eu22, Yb23,24) etc. In n-type CoSb3-based skutterudites, the highest zT achieved is 1.3 and 1.9 for single and multiple filling respectively at 850K 11,17,20,23,25. In p-type skutteurdites, the highest zT reported is around 1.3 at 850K, significantly lower compared to its n-type counterpart 26,27. 1.3 Summary of research My research focuses on n-type CoSb3-based skutterudites. In Chapter 2, a detailed description of experimental methods is provided. Chapter 3 and 4 answer the question of why high zT can be achieved in n-type CoSb3-based skutterudites, regardless of the various filler types. High zT in 6 skutterudites is often attributed to the addition of the filler atoms and subsequent reduction in thermal conductivity due to alloying disorder and the complex phonon modes of the filler atom28-30. Although low thermal conductivity is essential to high zT, the importance of the intrinsic electronic structure in skutterudites is often understated or ignored completely, especially given that the optimum doping carrier concentration is independent on the filler type. In Chapter 3, the electronic origin of high thermoelectric performance in n-type CoSb3-based skutterudites is discussed based on experimental and computational results, which is the convergence of a secondary multi-valley conduction band to the primary conduction band at high temperatures. In Chapter 4, the role of fillers in decreasing lattice thermal conductivity is discussed, including resonant scattering, avoided crossing mechanism, point defect scattering, umklapp scattering, and electron-phonon scattering. The maximum thermoelectric performance of a material depends on the charge carrier concentration, which is often controlled by the solubility limit of fillers; therefore the study of the solubility limit of fillers is essential. There has been a lot of debate concerning the solubility limit of various fillers such as Ga, In, Yb, and Ce, etc. When a foreign atom R is added to the CoSb3 system, a phase diagram study of these ternary R-Co-Sb systems is necessary in determining the solubility limit. However this was rarely performed previously, and instead the solubility limit (or filling fraction limit when the filler R only goes to the void position) was mostly considered to be a single value only dependent on the filler type. The necessity and methodology of integrating phase diagram study in skutterudites is elaborated in Chapter 5 to clarify the previous solubility debates with examples of Ga-Co-Sb, In-Co-Sb, Yb-Co-Sb, and Ce-Co-Sb systems. The existence of solubility direction as well as the dependence of the solubility limit on the phase region and annealing temperature should refresh previous understanding in the skutterudite community. These knowledge serves as guidance in reliable precise doping control, which alleviates the influence of synthesis uncertainty on thermoelectric performance and could potentially benefit large-scale commercialization of high zT skutterudites. In Chapter 6, the defect study of intrinsic CoSb3 is discussed combining bonding chemistry, binary phase diagram study, and experimental results from carrier concentration characterization. In Chapter 7, the phase diagram study of quaternary Ce-Co-Fe-Sb system is discussed which sheds light on optimizing p-type (Co,Fe)Sb3-based skutterudites. In Chapter 8, the possibility of nanostructured precipitation driven by temperature dependent filler solubility was demonstrated with experimental results. Lastly, possible future work is discussed in Chapter 9. 7 Appendix A gives a detailed derivation from Boltzmann transport equation to thermoelectric properties with a single parabolic band (SPB) approximation. Appendix B gives the equations for TE properties with a SPB model and acoustic phonon scattering, which is the most common electron scattering mechanism for skutteurdites above 300K. Appendix C gives the equations for TE properties with a SPB model in both non-degenerate and degenerate limits. Appendix D gives the equations to calculate thermoelectric properties with a multi-band model. Appendix E compares the Mott relation derived from both single parabolic band model and kane band model and explains why linear band does not benefit thermoelectric performance. 8 Chapter 2 Experimental methods 2.1 Summary Synthesis of single-phase homogeneous skutterudites is a non-trivial task due to the complexity of the equilibrium phase diagram of binary Co-Sb system. As can be seen from the Co-Sb phase diagram in Figure 2.1, when the temperature cools down, CoSb (γ-phase, metal) forms congruently first from a melt with nominal composition CoSb3, then CoSb2 (δ-phase, semiconductor) and CoSb3 (ε-phase, semiconductor) form peritectically at 936℃ and 874℃, respectively. The skutterudite phase CoSb3 is formed in a peritectic reaction from a solid CoSb2 phase and a liquid at 874℃. Because of the slow kinetics, this reaction can hardly be completed during a quenching process. Figure 2.2 shows the Scanning Electron Microscopy (SEM) photo of a melt-quenched sample with nominal composition CoSb3. As shown in the photo, the CoSb2 phase forms around the CoSb phase whereas the CoSb3 skutterudite phase is barely formed after melting and quenching. The remaining Sb phase confirms the incomplete peritectic reaction of CoSb3 formation. In order to get a single-phase homogeneous skutterudite phase, long time annealing (usually a week) after melting and quenching is thus needed. Besides the traditional melt-annealing method, solid-state reaction such as mechanical alloying (ball milling)31,32 is also widely used. The ball milling process can greatly reduce the grain size and consequently the diffusion length. (Eq. 2.1) where D is the mass diffusion coefficient. From equation 2.1 we can see that by reducing the diffusion length , the diffusion time required could be greatly reduced due to its square dependence on the diffusion length. Thus subsequent annealing time could be largely shortened after ball milling. The same principle works for the melt-spinning method33. In a melt-spinning method, by ejecting the molten charge onto a cold rotating drum of copper, an ultra-fast solidification of the melt happens which yields materials with a very fine grain structure. These methods (melt-annealing method, mechanical alloying or ball milling31,32, melt-spinning method33, high pressure assisted synthesis34, etc.) all result in polycrystalline skutterudites. Single 9 crystalline skutterudites are also synthesized and studied by many researchers (gradient freeze technique5,6, flux-assisted growth35, chemical vapor deposition36, etc.). Readers can refer to the referenced papers and the review paper by Prof Ctirad Uher for more details37. Figure. 2.1 The Co-Sb (Cobalt-Antimony) phase diagram38. Sb CoSb 20um CoSb2 Figure 2.2 Scanning Electron Microprobe (SEM) photo of a melt-quenched sample with nominal composition CoSb3. 10 The choice of synthesis method depends on the goal of research. For electronic band structure study such as Shubnikov-de Haas effect measurement or a detailed study of the crystal structure of a material by techniques such as Bragg diffraction, single crystalline skutterudite is preferred. However, for thermoelectric property investigation, polycrystalline skutterudite is sufficient due to its cubic structure and isotropy nature resulting from various crystalline orientations. Among polycrystalline skutterudite synthesis methods, ball milling and melt-spinning can largely reduce grain size, thus decreasing the lattice thermal conductivity. However, these processes risk throwing the nominal composition off the desired CoSb3 stoichiometry due to possible contaminants (such as Fe, oxygen) and sublimation of volatile Sb, which leads to easy formation of secondary phases. To determine precisely the solubility region of skutterudites such that pure skutterudites can be synthesized for thermoelectric investigation, we adopt the melt-annealing method in this study in order to avoid external compositional perturbations and get thermodynamically stable materials. Samples after melting and quenching are annealed at a specific temperature below the peritectic temperature for one week to ensure thermodynamic homogeneity. Phase analysis is performed on an annealed and quenched ingot to avoid possible compositional drift due to any post-annealing process. 2.2 Synthesis procedures There are two types of samples in this study. One is related to determination of isothermal sections of either ternary systems (Yb-Co-Sb, In-Co-Sb, etc.) or quaternary systems (Ce-Co-Fe-Sb). Nominal compositions of these samples were chosen based on a preliminary phase diagram study with the knowledge of related binary or ternary phase diagrams, which can be quite different from the skutterudite stoichiometry (Co: Sb = 1: 3). The other type of samples is related to an n-type doping study. Samples with skutterudite stoichiometry (nominal compositions YbxCo4Sb12.012, with x ranging from 0.0025 to 0.3) were synthesized with 0.1at% excess of Sb added to compensate for possible Sb evaporation. High-purity elements (Co, Fe (99.95%, slug), Sb (99.9999%, shot), and Ce (99.9%, rod), Yb (99.9%, ingot) etc.) purchased from Alfa Aesar were used as raw materials. The samples were sealed in carbon-coated fused silica tubes under vacuum. The silica tubes were heated slowly up to 923K in 4 hours, held at 923K for 2 hours, then heated up to 1373K in 6 hours, held at this temperature for 12 hours, and then quenched in water to room temperature. The 2-hour saturation at 11 923K was performed such that Sb (with melting temperature 904K) in liquid form will promote mass diffusion and reaction. Samples were then annealed at temperatures ranging from 773K to 1123K for 7 days. After annealing samples were quenched in water to room temperature again. The resulting ingots were taken out of the ampoule and taken to X-ray diffraction (XRD) and SEM, Electron Probe Microanalysis (EPMA) for phase identification and chemical composition determination. Note that the amount of fillers in annealed and quenched samples is “frozen” from high temperature annealing and thus could be indicative of their corresponding high temperature solubility. Ingots were then hand ground into fine powders and consolidated by rapid hot pressing (HP) at 973K for 1h under a pressure of about 60MPa, yielding fully dense bulk samples39. High density (> 95% of the theoretical density of CoSb3) was achieved in all hot pressed samples. Hot pressed samples were sealed in fused tubes under vacuum for further annealing at the same annealing temperatures as before for 7 days again to erase the temperature effect of the hot pressing process before thermoelectric properties were measured. Several tips: 1. Ampoules should be baked prior to the loading the elements in order to clean out possible contaminants (dust, water, etc.). Close attention needs to be paid to possible fissures, which should be gotten rid of by applying the torch, otherwise they may lead to ampoule breakage during the quenching process later. 2. A baked ampoule can be carbon-coated using acetone. Note that if the order is reversed (to bake the already carbon-coated ampoule instead) the carbon coating will disappear upon baking. After rinsing with water for several times, carbon-coated ampoules could be stored in a desiccator to get rid of the remaining water inside of the ampoule. 3. All starting elements are chosen to be in either slug, rod, or shot form. Powder form is avoided such that oxygen contamination is minimized. For elements easy to oxidize such as Yb and Ce, usage of brush or electronic drill is recommended to clean off the oxidized layer before cutting and weighing. Also when using an electronic drill, pay attention to the drilling speed such that no flame comes out due to the reactive nature of these elements. 4. When loading the elements into the ampoule, the element with highest melting point (Co in most samples) is loaded first, whereas the element with lowest melting point (Sb in most samples) is loaded last. This can assure that the whole sample is wrapped in liquid Sb after Sb melts and maximizes the contact and reaction between elements. 12 5. When sealing ampoules, the length of fused ampoule should be controlled to be short enough (not too short to burn the fingers holding it, also usage of cotton gloves are required for protection during sealing) such that even if the temperature distribution in the furnace is not entirely homogeneous, the precipitation of volatile Sb on the colder part of the ampoule can be minimized. 6. Before breaking the ampoule to get the ingot out, always check the vacuum inside the ampoule first by placing the torch close to the top of ampoule. If the ampoule wall curves in (in the appearance of a dip), then it is in good condition. Otherwise, the sample has to be discarded. 2.3 Characterization 2.3.1 Phase and chemical composition identification Ingots after annealing were cut and characterized by room temperature X-ray diffraction (XRD), with data collected on a Panalytical X’Pert Pro diffractometer equipped with Cu Kα radiation to check phase purity and lattice constant. Microstructures of the annealed samples were checked with a ZEISS 1550VP Field Emission Scanning Electron Microscope (SEM). Quantitative elemental analyses of the annealed samples were performed with a JEOL JXA-8200 electron probe microanalysis (EPMA) using an accelerating voltage of 15KeV and a current of 25nA in a WDS mode and averaged over 10 randomly selected locations in the skutterudite phase. Atomic Probe Topographic (APT) measurements were conducted to check possible dopant segregation on sample with x = 0.20 Ce doping content (nominal composition Ce0.5Co4Sb12)20 on a Cameca LEAP-4000X Si equipped with a picosecond UV laser (wavelength 355 nm). Microtip samples of the nominal composition Ce0.5Co4Sb12 were prepared using a dual-beam focused–ion beam microscope (FEI Helios Nanolab) equipped with a micromanipulator (similar to the lift-out method)40. Microtips with a diameter of ~100 nm were fabricated to contain a grain boundary and the last step of the tip sharpening process utilized a low voltage and current (5 kV, 16 pA) Ga+ ion beam to minimized Ga implantation in the sample (Ga content of the region analyzed was <0.01 at%). The sample was maintained at 30K and laser energy of 10 pJ pulse-1 was used at a pulse rate of 250 Hz with a target evaporation rate of 0.5% atom pulse-1. The primary ions detected were Co2+, Co+, Ce3+, Ce2+, Sb3+, Sb2+, and Sb+. Very small amounts of Sb2+ and CoCe+, Ga+, and O2+ were also detected. Ions were detected using a 2D microchannel plate detector with a detector 13 efficiency of 50%. This detection efficiency is the same for all ions evaporated. The data collected were analyzed and a 3D reconstruction was created using the program IVAS v.3.6.6. 2.3.2 Transport property characterization Electrical transport properties for published data, including electrical conductivity (σ) and Seebeck coefficient (S) were measured using the ZEM-3 (ULVAC) apparatus under a helium atmosphere from 300 to 850 K. Some unpublished data are measured on our home-built high temperature Hall measurement system41 and high temperature Seebeck measurement system42. Thermal conductivity (κ) was calculated using 𝜅𝜅 = 𝑑𝑑𝐷𝐷𝑇𝑇 𝐶𝐶𝑃𝑃 , with the thermal diffusivity 𝐷𝐷𝑇𝑇 measured along the cross- plane direction by the laser flash method (Netzsch LFA 457) under argon flow with the Cowan model plus pulse correction. The density of the samples was measured using the geometrical method. The specific heat capacity 𝐶𝐶𝑃𝑃 was determined using the Dulong-Petit law 𝐶𝐶𝑃𝑃 = 3𝑘𝑘𝐵𝐵 per atom throughout the temperature range 300K to 850K. The in-plane Hall coefficient (RH) was measured using the Van der Pauw method in a magnetic field up to 2 T 41. Hall carrier concentration (nH) was then estimated to be equal to 1/RHe assuming a single type of carrier, where e is the elementary charge. The Hall carrier mobility ( ) was calculated according to the relation = RHσ. The estimated measurement uncertainties are listed as follows: 5% for electrical resistivity, 7% for Seebeck coefficient, 5% for thermal diffusivity, and 1% for density. The data precision (reproducibility) is smaller than the accuracy, leading to zT values within the range of ±0.2. 2.3.3 Optical property characterization Optical band gap was measured on an intrinsic CoSb3 powder sample (p-type, with carrier concentration 2E17cm-3). Diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) measurements were performed on a Nicolet 6700 FTIR Spectrometer fitted with a Harrick Praying Mantis Diffuse Reflectance attachment and a low-temperature stage (Harrick CHC). The spectral range of the instrument was from 0.05 to 0.8 eV. The Kubelka Munk function, F(R), was obtained from the measured diffuse reflectance (R), 𝐹𝐹(𝑅𝑅) = (1−𝑅𝑅)2 2𝑅𝑅 , which is known to be proportional to the absorption coefficient (𝛼𝛼) ratioed to the scattering coefficient (K). 14 2.4 Ab initio DFT Calculation For Ga-Co-Sb and In-Co-Sb systems, Lily Xi from SICCAS in China performed the first-principle calculations. The details of her calculation are as follows: All the calculations were carried out using the projector augmented wave (PAW) method, as implemented in the Vienna ab initio Simulation Package (VASP). The Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) for the exchange-correlation potential was used for all the calculations, and computational details9 can be found in our earlier publications. All calculations of pure and gallium/indium doped CoSb3 skutterudites were carried out on a supercell (2×2×2 primitive cell) with a total of 128 atoms and 8 voids. A 3×3×3 Monkhorst-Pack uniform k-point sampling was used for energy calculations of the supercell. Different configuration structures were considered and the one that had the lowest energy was used for further analysis. For Yb-Co-Sb sytem, Luis Agapito from University of North Texas performed both the DFT calculation and ultrafine evaluation of Fermi surfaces. The details of his calculation are described as follows. The positions of the 16 atoms in the CoSb3 unit cell (with conventional lattice parameter of 9.07 Å) are relaxed using norm-conserving pseudopotentials and the Perdew-Burke-Ernzerhof (PBE)43 density functional, as implemented in the ab initio package Quantum Espresso44. The plane-wave basis set is defined by an energy cutoff of 270 Ry. Although PBE is generally known for the systematic underestimation of the band gap of semiconductors, it can usually predict the correct topology of the bands, which is desired in this study; for such cases, the PBE electronic structure can simply be corrected by an energy shift of the unoccupied manifold via the scissor operator. Previous literature also suggests that the specific band gap value is extremely sensitive to the Sb positions and lattice parameter45 and that the exact functional is less important. The obtained theoretical band gap using PBE density functional agrees well with experimental optical band gap in this study. A 9x9x9 Monkhorst-Pack sampling of the reciprocal space is sufficient to converge the DFT wavefunctions. However, ultrafine k-meshes are needed in order to obtain smooth isosurfaces (180x180x180 k-points in the reciprocal unit cell) and resolve the low-energy features of the DOS (100x100x100, with low smearing energy of 0.02 eV). Highly accurate real-space tight-binding 15 Hamiltonian matrices are built by projecting the DFT Bloch states onto a “small” set of atomic orbitals (4p, 3d, 4s for Co; and 5p, 5s for Sb) while filtering out states of low projectability46. Reciprocal- and real-space Hamiltonian matrices are obtained by Fourier transformation and then diagonalized at each point of the ultrafine mesh to obtain the eigenenergies. The resulting tightbinding and the actual DFT values are numerically equivalent for all practical purposes. We used the parallel implementation of the method available in the WanT code47. XCrySDen48 is used for visualizing the isosurfaces. 16 Chapter 3 Electronic origin of high zT in n-CoSb3 skutterudites 3.1 Summary N-type filled skutterudites RxCo4Sb12 are excellent thermoelectric materials owing to their high electronic mobility and high effective mass combined with low thermal conductivity associated with the addition of filler atoms into the void site. The favorable electronic band structure in n-type CoSb3 is typically attributed to three-fold degeneracy at the conduction band minimum accompanied by linear band behavior at higher carrier concentrations, which is thought to be related to the increase in effective mass as the doping level increases. Using combined experimental and computational studies, we show instead that a secondary conduction band with 12 conducting carrier pockets (that probably converge with the primary band at high temperatures) is responsible for the extraordinary thermoelectric performance of n-type CoSb3 skutterudites. A theoretical explanation is also provided as to why the linear (or Kane-type) band feature is not beneficial for thermoelectrics. This chapter is reproduced with permission obtained from the published paper: Nature Materials DOI:10.1038/NMAT4430 49. Section 3.2 gives a background introduction. Section 3.3 shows the results and discussion. Conclusion and future work are in Section 3.4. 3.2 Background introduction Among the best thermoelectric materials are n-type filled skutterudites based on CoSb3. The addition of filler atoms, for example Yb, into a void site (YbxCo4Sb12) can lead to high zT by reducing the thermal conductivity while simultaneously doping the material (adding electrons as charge carriers) 6,50. High zT values (greater than 1) have been reported for both single-element filling (Na11, Ba13, In17, Ce20, and so on) and multiple filling (In+Ce51, Sr+Ba+Yb52, Ba+La+Yb53). High zT in skutterudites is most often attributed to the addition of the filler atoms and subsequent reduction in thermal conductivity due to alloying disorder and the complex phonon modes of the filler atom28-30. Although low thermal conductivity is essential to high zT, the importance of the intrinsic electronic structure in skutterudites is often understated or ignored completely. It has become increasingly apparent that complex band structures -- including: multi-valley Fermi surfaces54,55, convergence of 17 bands (PbTe55, PbSe56, Mg2Si57), or even threads of Fermi-surface-connecting band extrema58 -- are key features of many good thermoelectric materials because the thermoelectric quality factor, B, is proportional to Nv 54,59, the number of degenerate valleys in the electronic structure (or pockets of Fermi surface). While most common semiconductors or metals have simple Fermi surfaces with one or three pockets, thermoelectric materials with zT > 1 often have Nv = 6 or more54,56. Researchers have shown that CoSb3 has very light bands, making a very small (0.05 - 0.22 eV) direct band gap45,60. The single, light valence band has an approximately linear (E ~ k rather than the usual parabolic 𝐸𝐸 = ℏ2 𝑘𝑘 2 ) dispersion. The light effective mass explains the high mobility observed 2𝑚𝑚∗ in lightly doped p-type CoSb3 6 and is beneficial to zT 61, but it also makes the thermopower (magnitude of the Seebeck coefficient) decrease more quickly as the material is doped. The conduction bands in n-type CoSb3 skutterudites are also very light, with one of the three bands mirroring the linear valence band. The n-type thermopower, however, remains high at high doping, where the linear band concept has been used to explain the apparent increase in effective mass6,62-65. This high thermopower at high doping is essential for achieving the high zT in all n-type skutterudites. Here we show that this essential feature of the electronic structure cannot be due to the linear band, but instead is due to a new band (or bands) with high valley degeneracy of Nv = 12 or more. 3.3 Results and discussion 3.3.1 Multiple conduction band behavior in n-CoSb3 Here we shall describe the transport properties of YbxCo4Sb12 using a rigid band approximation66,67, meaning that the conduction band structure does not significantly change with doping (Yb content) from that of unfilled CoSb3. Experimentally, similar electronic properties are obtained whether CoSb3 is doped through filling or by substitution on the Co or Sb sites6,62,64,65 (with optimum thermoelectric performance of the order of 1020 cm-3 or 0.5 electrons per unit cell)53,67 in accordance with the rigid band model. Filled YbxCo4Sb12 is shown theoretically to be an essentially rigid band up to x=0.25 (0.5 electrons per unit cell67). In other thermoelectric materials, such as PbTe, rigid band models have been used successfully and have been confirmed theoretically68. 18 Figure 3.1 Experimental and theoretical evidence showing multiple conduction bands in n-type CoSb3. (a) Pisarenko plot of Seebeck coefficient (S) vs. Hall carrier concentration (nH) at 300K. The solid black line (three-band model) represents the prediction of a semi-empirical model with two conduction bands plus one valence band. Orange and red dashed lines show the expected S vs nH behavior for single parabolic bands with masses equal to the two individual conduction bands. The data of Yang et al69 on Yb-filled CoSb3 are included for comparison. (b) DFT calculated electronic band structure and density of states (DOS) for CoSb3. (c) Fermi surface calculation for Fermi level 0.11eV above the conduction band minimum showing the 12 pockets of the second conduction band CB2 observed as a valley between 𝛤𝛤 − 𝑁𝑁. (d) Room temperature optical absorption measurement with estimated joint density of states from DFT showing two distinct transitions. Multiple band effects that are responsible for the exceptionally high zT in n-type CoSb3 are observable through several methods, both experimental and theoretical, as presented in Figure 3.1. A clear demonstration of complex band behavior is seen in the doping-dependent Seebeck coefficient (Pisarenko plot at 300K) shown in Figure 3.1a. In the degenerate limit the relationship between S, 𝑚𝑚𝑆𝑆∗ and n can be described by: 𝑆𝑆 = 2 2 2𝑘𝑘𝐵𝐵 𝑇𝑇 𝜋𝜋 3 ∗ � � 𝑚𝑚𝑆𝑆 (1 + 𝑟𝑟) 3𝑒𝑒ℏ2 3𝑛𝑛 (Eq. 3.1) where e is the electron charge, kB is the Boltzmann constant, ℏ is the reduced Planck constant, r is the scattering parameter (r = 0 for acoustic phonon scattering, which is fairly common above 300K and is most appropriate for CoSb3), and 𝑚𝑚𝑆𝑆∗ is the density of states (DOS) effective mass obtained from Seebeck measurements using the single parabolic band (SPB) model (which will be referred to as the Seebeck mass herein). At low carrier concentrations (nH less than ~1 × 1019 cm-3), CoSb3 19 shows light mass behavior (𝑚𝑚𝑆𝑆∗ ~0.7 𝑚𝑚𝑒𝑒 ); as the carrier concentration increases, the system transitions to a much heavier mass, requiring 𝑚𝑚𝑆𝑆∗ = 4.8 𝑚𝑚𝑒𝑒 in the heavily doped regime (nH larger than 3 × 1020 cm-3). By considering two conduction bands70-72 plus one valence band in a threeband transport model (with a conduction band offset Δ𝐸𝐸 of ~0.08 eV), we capture the behavior of both the lightly and the heavily doped regions (black line in Figure 3.1a). The calculated electronic band structure is consistent with an increasing 𝑚𝑚𝑆𝑆∗ , that becomes gradually heavier with doping, if we consider not only the primary conduction bands (𝛤𝛤 point), but also the bands higher in energy (labelled CB2 in Figure 3.1b) for heavily doped CoSb3. The ab initio density functional theory (DFT) calculated band structure (Figure 3.1b) shows a direct band gap at the 𝛤𝛤 point (𝐸𝐸𝑔𝑔,𝛤𝛤−𝛤𝛤 = 0.23 𝑒𝑒𝑒𝑒), which yields a triply degenerate (Nv = 3) conduction band edge. However, as a result of heavy doping and relatively light bands at the conduction band minimum, the Fermi level quickly moves up the conduction band, allowing a large population of electrons to form in the secondary conduction band (CB2). Calculations show that this secondary conduction band minimum (CB2) exists about 0.11eV above the conduction band minimum along 𝛤𝛤 − 𝑁𝑁, and that the Fermi level (EF) should be well within CB2 with 0.5 electrons/unit cell67,73 (EF reaches CB2 minimum with 𝑛𝑛𝐻𝐻 ≈ 2 × 1020 𝑐𝑐𝑚𝑚−3 at T = 300K). The iso-energy Fermi surface for an energy level just at CB2 (Figure 3.1c) has a high degeneracy with 12 isolated pockets73. Only PbTe, which reaches a zT of ~2 54, has such a high value of Nv; it has been shown that this high Nv plays a crucial role in the high Seebeck values and zT. In CoSb3, at higher energies these 12 pockets join at corners along 𝛤𝛤 − 𝐹𝐹 (Nv = 24, 0.013 eV above CB2) 73 and then to the 𝛤𝛤 Fermi surface before closing up an opening at 𝛤𝛤 − 𝐻𝐻 (Nv = 6, 0.034 eV above CB2). In addition to thermoelectric transport and DFT calculations, multi-band features in CoSb3 can be directly observed by infrared (IR) optical absorption. Optical absorption edge spectra for a nearly intrinsic sample of CoSb3 (𝑛𝑛𝐻𝐻 = 1.7 × 1017 𝑐𝑐𝑚𝑚−3 ) show two distinct features (Figure 3.1d). The lower-energy (~0.2eV optically, 0.23eV from DFT) transition can be associated with the direct, 𝛤𝛤 − 𝛤𝛤 transition, and the second transition (~0.3eV optically, 0.34eV from DFT) indicates the onset of a 𝛤𝛤 − 𝐶𝐶𝐵𝐵2 transition. Although direct transitions have been shown to exhibit more than ten times the strength of absorption of the indirect transitions74, the transition rate is also proportional to the DOS. Because both 𝑉𝑉𝑉𝑉𝛤𝛤 and 𝐶𝐶𝐶𝐶𝛤𝛤 have very low DOS in comparison to CB2, the 𝛤𝛤 − 𝛤𝛤 transition, despite the fact that it is a direct transition, may occur with a lower intensity than the 𝛤𝛤 − 𝐶𝐶𝐵𝐵2 transition. We roughly estimate the strength of all transitions by calculating the joint density of states (JDOS) from the DFT band structure in the inset of Figure 3.1d, which weights both direct 20 and indirect transitions equally. The JDOS agrees with the observation of two slopes from optical data. Historically, optical measurements in the skutterudite system have been limited, showing an optical band gap for CoP3 of 0.45 eV; no optical gap had been found for CoSb3 or CoAs375. This was probably because the lowest photon energy that they had measured was 0.4 eV. Other optical measurements have focused on very low energies (<0.1 eV) to probe optical phonons76, just missing the frequency range important for interband transitions. 3.3.2 Linear or kane bands do not increase Seebeck mass The unexpectedly high thermopower at high carrier density in YbxCo4Sb12 observed in the Pisarenko plot of Figure 3.1a can also be represented as an increase in 𝑚𝑚𝑆𝑆∗ , the Seebeck effective mass, as a function of carrier concentration, as shown in Figure 3.2a. This apparent increase in effective mass with doping has previously been attributed to non-parabolic (Kane) bands6,62-65, rather than multi-band effects. A non-parabolic dispersion (shown in Figure 3.2b, Eq. 3.2) can arise as a result of interaction between the valence and conduction bands, common in narrow gap semiconductors, which can be described using 𝑘𝑘 ∙ 𝑝𝑝 perturbation theory77. Such bands are often described by a dispersion relation where the band is approximately parabolic near the band edge (at E = 0, with band-edge effective mass 𝑚𝑚0∗ ), but they become more linear as the electron energy, E, becomes large relative to the band gap Eg. 𝐸𝐸 �1 + 𝐸𝐸 ℏ2 𝑘𝑘 2 � = 2𝑚𝑚∗ 𝐸𝐸𝑔𝑔 0 (Eq. 3.2) where k is the electron wave vector. For non-parabolic bands, effective mass is not well defined because there are different definitions for 𝑚𝑚∗, based on different classical relationships. In non- parabolic transport theory, the commonly used energy-dependent effective mass derived from the electron momentum is: 𝑑𝑑𝑑𝑑 −1 2𝐸𝐸 = 𝑚𝑚0∗ (1 + ) 𝑑𝑑𝑑𝑑 𝐸𝐸𝑔𝑔 𝑚𝑚𝑃𝑃∗ (𝐸𝐸) = ℏ2 𝑘𝑘 � � (Eq. 3.3) This is a different definition than that found in most solid-state physics textbooks, 𝑚𝑚∗ = −1 𝑑𝑑 2 𝐸𝐸 � , 2 𝑑𝑑𝑘𝑘 ℏ2 � which relates the effective mass to the band curvature directly. Many common semiconducting materials are reported to show 𝑚𝑚∗ (𝐸𝐸) increasing with energy (Fermi level, EF) including InSb78, InAs79, GaAs79, PbTe80, and others in a variety of different measurements including: electrical susceptibility (measured using optical reflectance)80,81, Shubnikov-De Haas/De Haas-Van Alphen oscillations36, Faraday rotation78,79, and combined galvano thermomagnetic 21 measurements (Seebeck and Nernst coefficients)82. In general these measurements are analyzed based on the dispersion relation Eq. 3.2 to derive the 𝑚𝑚𝑃𝑃∗ (𝐸𝐸) of Eq. 3.3. However, we will show that such an increasing trend with energy should not be expected on the Seebeck Pisarenko plot, and instead a decrease in Seebeck mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) is predicted in the case of a non-parabolic band with a dispersion given by Eq. 3.2. Figure 3.2 Band non-parabolicity and its effect on the Seebeck coefficient and energy-dependent Seebeck effective mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸). a) Effective mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) derived from Seebeck coefficient and Hall effect measurements. The solid black line (three-band model) represents the prediction of a semiempirical model with two conduction bands plus one valence band. Orange and red dashed lines indicate the band masses of the two individual conduction bands. The data of Yang et al. 69 is included for comparison. b) Parabolic and Kane band dispersions with the same band-edge effective mass (𝑚𝑚𝑆𝑆∗ (𝐸𝐸 = 0)). c) Seebeck Pisarenko plot for both Kane and Parabolic bands, illustrating that 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) actually decreases for Kane bands at high carrier concentration. For thermoelectric materials we define the Seebeck mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) as the DOS effective mass that would give the measured Seebeck coefficient with the measured 𝑛𝑛𝐻𝐻 using a SPB model1 (that is, Eq. 3.1 in the degenerate limit). In CoSb3, this carrier-concentration-dependent DOS effective mass 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) is observed to increase with 𝑛𝑛𝐻𝐻 in both n-type (Figure 3.2a) and p-type materials; this increase is commonly attributed to band non-parabolicity using Eq. 3.2 and 3.3.6,36,62-65 However, we must realize that these two distinct definitions of effective mass, 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) and 𝑚𝑚𝑃𝑃∗ (𝐸𝐸), are qualitatively different. For example, the degenerate limit of the Seebeck coefficient with a Kane 22 dispersion relation (the equation analogous to Eq. 3.1 using the Kane dispersion, Eq. 3.2) can be expressed as83: 𝑆𝑆 = 2 2 2𝑘𝑘𝐵𝐵 𝑇𝑇 𝜋𝜋 3 � � 𝑚𝑚∗𝑃𝑃 (𝐸𝐸)(1 + 𝑟𝑟 − 𝜆𝜆) 3𝑒𝑒ℏ2 3𝑛𝑛 where an additional correction factor 𝜆𝜆 = (Eq. 3.4) 4𝐸𝐸 𝐸𝐸 �1+ � 𝐸𝐸𝑔𝑔 𝐸𝐸𝑔𝑔 2 2𝐸𝐸 �1+ � 𝐸𝐸𝑔𝑔 has been added to the equation for that of a parabolic band (Eq. 3.1). Thus, 𝑚𝑚𝑆𝑆∗ , as used in thermoelectric studies, does not necessarily increase with doping or Fermi level as the momentum mass does. In fact, when r = 0, as is commonly found in thermoelectric materials, the 𝑚𝑚𝑆𝑆∗ mass in a Kane band should actually decrease according to: 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) = 𝑚𝑚0∗ 1+ 2𝐸𝐸 𝐸𝐸𝑔𝑔 (Eq. 3.5) which is derived by substituting the expression for 𝜆𝜆 into Eq. 3.4 and comparing the result to the SPB result (Eq. 3.1). Additional details regarding this derivation are included in the Appendix E. The effect of this relation can be seen in Figure 3.2c, which shows a Seebeck coefficient that is lower for the Kane band (𝑚𝑚𝑆𝑆∗ (𝐸𝐸) decreased). In other words, even though 𝑚𝑚𝑃𝑃∗ (𝐸𝐸) increases with energy in a Kane band, the Seebeck coefficient and 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) should actually decrease relative to that of a parabolic band as shown in Figure 3.2c. This may be surprising because both 𝑚𝑚𝑃𝑃∗ (𝐸𝐸) and 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) are described as a density of states effective mass and often implicitly expected to exhibit the same trends. Instead, Eq. 3.5 shows that the Kane band dispersion and linear bands in general do not increase 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) or benefit thermoelectric performance relative to a parabolic band with the same band-edge effective mass. For CoSb3, Eq. 3.5 demonstrates that the increasing 𝑚𝑚𝑆𝑆∗ in Figure 3.2a is not evidence of Kane-type behavior, but rather that multiple conduction bands are necessary to explain the properties of CoSb3. 3.3.3 Band convergence at high temperatures We have shown that a second band is required to explain the room-temperature transport and optical properties. However, it is at high temperatures where the thermoelectric performance of CoSb3 excels. At high temperatures, we show that these exceptional properties are probably the result of band convergence, as indicated by optical absorption edge measurements that show that the two conduction bands approach each other, leading to convergence at 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐 ≈ 800 ± 100 𝐾𝐾 (with effective convergence, i.e., Δ𝐸𝐸 < 1𝑘𝑘𝐵𝐵 𝑇𝑇, for T > 500 K). This band convergence further increases the effective valley degeneracy to 𝑁𝑁𝑣𝑣 ≈ 12 − 15. The optical absorption measured from 23 20 to 400 °C clearly shows that the strong 𝛤𝛤 − 𝐶𝐶𝐵𝐵2 absorption decreases in onset energy with temperature (Figure 3.3a). The extrapolated absorption edges (Figure 3.3b) indicate that the primary (𝛤𝛤 − 𝛤𝛤, direct) transition does not shift much with temperature (and actually is overtaken by free carrier absorption at high temperatures), whereas the secondary band (𝛤𝛤 − 𝐶𝐶𝐵𝐵2 ) shows a clear temperature-dependent decrease in energy at a rate of ~-2.0×10-4 eVK-1. As the two bands become closer in energy, both bands will contribute significantly to thermoelectric transport and improve the thermoelectric quality factor and zT in the same way that band convergence enables high zT in p-type PbTe54. The high zT in YbxCo4Sb12 can now be shown to be a direct result of the high valley degeneracy inherent to CoSb3, which is further enhanced by band convergence at high temperatures. Figure 3.3c shows the carrier-concentration-dependent zT for a series of Yb-doped samples at 800 K along with the calculated results of a three-band model (two conduction bands and one valence band). From this plot, we can see the benefits that having a second conduction band allows, resulting in a significantly higher zT than the primary conduction band at 𝛤𝛤 can provide alone. If we consider both the primary (𝐶𝐶𝐵𝐵𝛤𝛤 , Nv=3) and the secondary band (CB2, 𝑁𝑁𝑣𝑣 = 12) in the context of band engineering and the quality factor 𝐵𝐵 = 2 2𝑘𝑘𝐵𝐵 ℏ 𝑁𝑁𝑣𝑣 𝐶𝐶𝐿𝐿 ∗ 𝑇𝑇 (𝑚𝑚𝛤𝛤∗ ≈ 0.42, 𝑚𝑚𝐶𝐶𝐵𝐵 ≈ 2.88 at 800 K), we 2 2 𝜅𝜅𝐿𝐿 3𝜋𝜋 𝑚𝑚𝑐𝑐∗ 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑 determine that 𝐵𝐵𝐶𝐶𝐶𝐶2 is about four times that of 𝐵𝐵𝐶𝐶𝐶𝐶2 (as indicated by the much larger maximum zT in Figure 3.3c at 800 K). Because 𝐶𝐶𝐵𝐵𝛤𝛤 and CB2 are very near converged at high temperatures (Δ𝐸𝐸 ≈ 0 for 800 K), the overall quality factor is enhanced by the presence of the second band, as both bands can be thought to conduct in parallel, thereby increasing the electrical conductivity without being detrimental to the Seebeck coefficient (in the limit of converged bands 𝐵𝐵𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝐵𝐵𝐶𝐶𝐶𝐶𝐶𝐶 + 𝐵𝐵𝐶𝐶𝐶𝐶2 84). Thus both bands are contributing to the high thermoelectric performance. 24 Figure 3.3 Band convergence at high temperatures in CoSb3 as shown from optical absorption and thermoelectric figure of merit. (a) Temperature dependent optical absorption for CoSb3 from 20 to 400°C. (b) Temperature dependent band gap for the direct (𝐸𝐸𝑔𝑔,𝛤𝛤−𝛤𝛤 ) and indirect (𝐸𝐸𝑔𝑔,𝛤𝛤−𝐶𝐶𝐵𝐵2 ) transitions indicating band convergence at 800 ± 100 𝐾𝐾. Error bars represent the range of extrapolations obtained for both the primary and secondary transitions. (c) zT at 800K vs carrier concentration nH measured at 300K compared with that predicted with the model (solid black line). Colored lines labeled 𝐶𝐶𝐵𝐵𝛤𝛤 and 𝐶𝐶𝐵𝐵2 represent the zT that could have been attained by the primary and secondary conduction bands, respectively, with the valence band. Yang et al.’s data shows the measured zT for comparison. 69 3.4 Conclusion and future work A unified picture explaining the extraordinary thermoelectric properties of skutterudites has emerged from a combined analysis of transport measurements, optical absorption, and theory on a series of YbxCo4Sb12 samples. The primary light conduction bands at Γ have a reasonable thermoelectric quality factor, but they are significantly aided by a secondary conduction band, 25 which has extremely high valley degeneracy Nv = 12. At high temperatures these two bands converge, enabling the extraordinarily high zT > 1 that is observed in many CoSb3 based skutterudites with a variety of filling and doping elements. The contribution of band convergence to the performance at high temperatures suggests that band engineering methods59 to converge the two conduction bands at lower temperatures would improve the low temperature zT. The analysis of linear and kane bands presented here suggests that non-parabolic band dispersions do not lead to an increase in thermopower (Seebeck coefficient) and are not beneficial to thermoelectric performance. The investigation of the electronic origin of high zT in n-type CoSb3-based skutterudites is rooted in the successful synthesis of single-phase Yb-doped skutterudites, which benefits from a thorough phase diagram study elaborated in Chapter 5. The methodology described in this chapter can be generalized and applied to study other systems as well, i.e., p-type skutterudites and half heuslers. For p-type skutterudite, this will enable us to find out why it is less competent compared to its ntype counterpart. For the latter, pure phase half heusler material is difficult to be synthesized and thus its electronic origin of TE performance is hardly investigated. 26 Chapter 4 Role of filler in thermal transport 4.1 Summary The unique crystal structure of skutterudites allows the addition of fillers into the void site. Besides the doping effect that fillers have on electrical transport, the influence of fillers on thermal transport is also prominent and quite complex. The lattice thermal conductivity can be expressed by a simplified version of the Callaway model85,86: 1 𝑤𝑤 𝜅𝜅𝑙𝑙𝑙𝑙𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = ∫0 𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝑣𝑣 (𝑤𝑤)𝑉𝑉𝑔𝑔2 (𝑤𝑤)𝜏𝜏(𝑤𝑤)𝑑𝑑𝑑𝑑 3 (Eq. 4.1) where 𝐶𝐶𝑣𝑣 is the heat capacity, 𝑉𝑉𝑔𝑔 the group velocity, 𝜏𝜏 the scattering time and 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 is related to Debye temperature Θ = 𝑣𝑣𝑠𝑠 (ℏ⁄𝑘𝑘𝐵𝐵 )(6𝜋𝜋 2 ⁄Ω)1/3 by 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 = 𝑤𝑤𝐷𝐷 = speed of sound and Ω is the atomic volume. 𝑘𝑘𝐵𝐵 Θ = ℏ 𝑣𝑣𝑠𝑠 (6𝜋𝜋 2 ⁄Ω)1/3. 𝑣𝑣𝑠𝑠 is By adding atoms into the skutterudite cell, the heat capacity 𝐶𝐶𝑣𝑣 increases because the number of atom per unit cell increases, which leads to the increase of thermal conductivity. The guest atoms might also influence the stiffness of the crystal structure, and thus change the group velocity 𝑉𝑉𝑔𝑔 (as discussed in Section 4.4.2). The scattering time could be influenced the most, through various scattering mechanisms associated with the filler atom. The widely acknowledged rattling effect of fillers is an important factor that requires consideration, albeit its microscopic nature remains unclear (discussed in Section 4.2.1). Point defect scattering could be significant due to partial filling as discussed in Section 4.2.2. Large carrier concentration due to the doping can potentially scatter phonon propagation as well (electron-phonon scattering, discussed in Section 4.2.3). Increased Umklapp scattering due to large anharmonicity associated with the fillers can also influence scattering time. In this chapter, first the complex phonon modes associated with the fillers (resonant scattering/avoided crossing, point defect scattering, and electron-phonon scattering) will be discussed in Section 4.2. Section 4.3 describes how thermal transport properties are calculated as guidance in material design including calculation of minimum thermal conductivity (Section 4.3.1), 27 electronic contribution to thermal conductivity (Section 4.3.2) and theoretical calculation of thermal conductivity using Callaway model (Section 4.3.3). In Section 4.4 a few attempts in revealing the underlying mechanisms of lattice thermal conductivity reduction in RxCo4Sb12 are presented, with the possibility of electron-phonon scattering (Section 4.4.1) and lattice softening (Section 4.4.2) discussed. Lastly, possible future work is mentioned in Section 4.5. 4.2 Complex phonon modes 4.2.1 Rattling – resonant scattering or avoided crossing The decrease of lattice thermal conductivity in RxCo4Sb12 skutterudites has been largely attributed to the “rattling” effect of filler atom R. Due to the size difference between the void and filler atoms (see Table 4.1)9,87,88, fillers are under-constrained and weakly bound, which allows them to “rattle” in the void. The concept of fillers as “rattlers” was first proposed by Slack89. Rattlers are supposed to scatter phonon propagation effectively such that the material behaves as a glass whereas the electron conduction is mostly unaffected as it is in a crystal. Materials with such characteristic features are known as PGEC (phonon glass electron crystal). Even though the concept of rattling has been widely accepted in the skutterudite community, the microscopic mechanism of rattling, as to be more precise, the interaction between guest atom phonon modes and host atom phonon modes, is still unclear and often debated. The first and most common explanation is resonant scattering by guest atom vibrations. In this mechanism, a localized mode of the guest atom (Einstein or so-called rattling mode) within the acoustic frequency range is introduced which is uncorrelated to the phonon modes of the framework described by the usual Debye model. The rattling mode can be approximated as a quantized harmonic oscillator, with a single-frequency 𝑤𝑤0 = �𝐾𝐾�𝑀𝑀, where K is the force constant which depends on the bonding strength between R and Sb atoms and M is the rattler mass. A few Einstein frequencies calculated along [100] direction for different fillers with comparison to experimental results are listed in Table 4.1 9,87,88. As we can see from this equation, the heavier the rattler and the smaller its ionic radius, the lower the rattling frequency would be. The scattering mechanism due to the guest atom vibrations is called resonant scattering, in which only the lattice phonons with the 28 energy similar to that of the local mode introduced by guest atoms would get scattered. The resonant scattering relaxation time can be expressed as: 𝑤𝑤 2 𝜏𝜏𝑟𝑟−1 = 𝐶𝐶0 (𝑤𝑤 2 −𝑤𝑤2 )2 (Eq. 4.2) 0 where 𝐶𝐶0 is a constant that is proportional to the concentration of rattlers, and 𝑤𝑤0 is the frequency of Einstein mode of rattlers. Since the majority of heat is transferred by acoustic phonons with relatively low frequencies, the lower the rattling frequency, the more acoustic phonons are scattered. This can explain why Yb is such good filler in decreasing lattice thermal conductivity. Evidence of resonant scattering has been found in many experiments. Large atomic displacement parameter (ADP) of fillers was observed from Rietveld refinements of X-ray diffraction data37. The existence of single-frequency Einstein modes was also found either from specific heat data, or from peaked response close to the supposed Einstein mode frequency in the partial densities of phonon states (PDOS) from either nuclear inelastic spectroscopy (NIS)90 or time-of-flight inelastic neutron scattering (INS). Moreover, as the void size increases, the decrease in the lattice thermal conductivity and Einstein temperature (vibration frequency) also supports the resonant scattering mechanism37,91. Table 4.1 Ionic radii, atomic mass, and rattling frequency of different filler atoms. Coordination for all fillers is chosen to be eight except for Ga (coordination = 6). Filler type Na K Ca Sr Ba Eu Yb Ga In Ce La Void Charge +1 +1 +2 +2 +2 +2 +2 +3 +3 +3 +3 - Ionic Atomic radius (Å) Mass 1.18 1.51 1.12 1.26 1.42 1.25 1.14 0.62 0.92 1.14 1.16 1.89 22.99 39.10 40.08 87.62 137.33 151.96 173.05 69.72 114.82 140.12 138.91 - 𝒘𝒘𝟎𝟎_𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 (cm-1) 𝒘𝒘𝟎𝟎_𝒆𝒆𝒆𝒆𝒆𝒆 113 142 91 94 59 43 55 68 - 40 55 55 - (cm-1) 29 The resonant scattering mechanism has been challenged when evidence of avoided crossing mechanism was discovered in recent years. In the resonant scattering picture, the rattling mode is considered to be a vibration motion lacking phase coherence compared to the host motion. However recent studies from both experimental work (the lower reduced mass compared to rattler mass from Extended X-ray fine absorption measurements (EXAFS)91, inelastic neutron29, and nuclear inelastic scattering90) and theoretical calculations92 cast doubt on this assumption. From inelastic neutron scattering data, Koza et al.29 showed coherency of coupling between the localized guest mode and host lattice mode in Ce and La-filled Fe4Sb12 skutterudites. This interaction between the localized rattling mode and the host acoustic modes could lead to avoided crossing on the phonon spectra and thus effectively lower group velocity, largely scattering the acoustic phonons and reducing thermal conductivity. In addition, this coupling also leads to an increased anharmonicity for the rattling mode manifested as a higher Gruneisen parameter, which will contribute to more Umklapp scattering, as we will discuss later. Recently it was also discovered that the anharmonicity associated with phonon hybridization can decrease as the phonons modes decouple at high pressure, meaning the rattling mode can be switched off at high pressure90. The different origin in decreasing lattice thermal conductivity between resonant scattering and avoided crossing can be understood as follows. From Eq. 4.1 we know that the lattice thermal 1 𝑤𝑤 conductivity can be expressed as 𝜅𝜅𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 = ∫0 𝑚𝑚𝑚𝑚𝑚𝑚 𝐶𝐶𝑣𝑣 (𝑤𝑤)𝑉𝑉𝑔𝑔2 (𝑤𝑤)𝜏𝜏(𝑤𝑤)𝑑𝑑𝑑𝑑; while resonant 3 scattering reduces the scattering time near the rattling mode frequency 𝜏𝜏(𝑤𝑤) (Figure 4.1a and blue curve in 4.1d), the avoided crossing due to coupling of localized guest atom and host lattice modes leads to a large reduction in the group phonon velocity 𝑉𝑉𝑔𝑔2 (𝑤𝑤) near the guest vibration mode frequency, as illustrated in Figure 4.1b, 4.1c and orange curve in 4.1d 93. 30 Figure 4.1 Different mechanisms in reducing lattice thermal conductivity between resonant scattering and avoided crossing on the phonon spectra. a) The resonant scattering model targets phonons near 𝑤𝑤0 . b) BvK phonon dispersions for a stiff framework (m1, k1) and loosely bound guest atoms (m2, k2). Increased k2 stiffness results in increased coupling (extent of avoided crossing) between the framework and guest modes. c) The avoided crossing reduces 𝑉𝑉𝑔𝑔2 (𝑤𝑤) in the vicinity of 𝑤𝑤0 . d) 𝜅𝜅𝑠𝑠 (𝑤𝑤)for an empty BvK framework, using Umklapp and boundary scattering terms (curve A). Including resonant scattering reduces 𝜅𝜅𝑠𝑠 (𝑤𝑤) near 𝑤𝑤0 (curve B). If instead the effect of coupling on 𝑉𝑉𝑔𝑔2 (𝑤𝑤) is accounted for, a similar reduction is observed (curve C). Reproduced with copyright permission obtained © Royal Society of Chemistry [2011] (http://pubs.rsc.org/en/Content/ArticleLanding/2011/JM/C1JM11754H#!divAbstract). From Figure 4.1 we can see that both the resonant scattering effect (reduction of scattering time 𝜏𝜏(𝑤𝑤)) and the effect of avoided crossing on the phonon spectra (reduction of group velocity 𝑉𝑉𝑔𝑔2 (𝑤𝑤)) will result in the similar reduction of lattice thermal conductivity. It is thus difficult to distinguish between the two effects. Temperature dependent thermal conductivity measurements are not enough but rather frequency-dependent measurements of 𝑉𝑉𝑔𝑔2 (𝑤𝑤) and 𝜏𝜏(𝑤𝑤) are required to unravel the intertwined effects on 𝜅𝜅𝐿𝐿 . 4.2.2 Point defect scattering By partially filling R into the skutterudite structure (RxCo4Sb12, 0 < x < 1), due to the mass and size differences between R and the void, point defects are introduced into the system, which can effectively scatter phonon propagation and thus decrease lattice thermal conductivity. The relaxation time due to point defect scattering can be expressed as: −1 𝜏𝜏𝑃𝑃𝑃𝑃 = Ω 𝜔𝜔4 Γ 4𝜋𝜋𝜈𝜈𝑠𝑠 3 (Eq. 4.3) where 𝛺𝛺, 𝜈𝜈𝑠𝑠 , and Γ are the average atomic volume (volume per atom), the average lattice sound velocity, and the disorder scattering parameter, respectively. Γ can be estimated from experimental 31 lattice thermal conductivity values (as shown below from a Klemens model): the larger the scattering parameter, the lower the lattice thermal conductivity. According to Callaway and Klemens94-96, if we consider a combined scattering mechanism including Umklapp scattering for the pure crystal without disorder, and extra point defect scattering process for the crystal with disorder, then the ratio of 𝜅𝜅𝐿𝐿 of the crystal with disorder to that without disorder, 𝜅𝜅𝐿𝐿𝑃𝑃 is: 𝜅𝜅𝐿𝐿 arctan(𝑢𝑢) = 𝜅𝜅𝐿𝐿𝑃𝑃 𝑢𝑢 𝑢𝑢2 = 𝜋𝜋2 𝜃𝜃𝐷𝐷 𝛺𝛺 𝑃𝑃 𝜅𝜅𝐿𝐿 𝛤𝛤𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 ℎ𝜈𝜈𝑠𝑠 2 (Eq. 4.4) (Eq. 4.5) where 𝑢𝑢, ℎ, and 𝛤𝛤𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 are the disorder scaling parameter, the Planck constant, and the experimental disorder scattering parameter, respectively. Thus, 𝛤𝛤𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 can be derived from 𝜅𝜅𝐿𝐿 measurements using Eq. 4.4 and 4.5. It can then be compared to calculated values, as detailed in the following. Γ can be separated into two components: 𝛤𝛤 = 𝛤𝛤𝑀𝑀 + 𝛤𝛤𝑆𝑆 (Eq. 4.6) where the scattering parameters 𝛤𝛤𝑀𝑀 and 𝛤𝛤𝑆𝑆 represent mass and strain field fluctuations due to the introduction of point defects (e.g., R into the void), respectively. (Eq. 4.7) where 𝜀𝜀 is an empirical fitting parameter related to Gruneisen parameter and elastic properties. 𝑎𝑎 is the lattice constant of pure alloy whereas Δ𝑎𝑎 is the difference in lattice constant between the alloy with point defects and the pure alloy. When there is more than one constituent element in the compound (as in the case of skutterudites RxCo4Sb12), there is some discrepancy regarding the definition of 𝛤𝛤𝑀𝑀 . According to Klemens: (Eq. 4.8) where ∆M is the mass difference between two species on the same site and M is the molar mass of the compound. In the case of skutterudite compound, with Yb filling, we are comparing YbxCo4Sb12 to unfilled Co4Sb12, ∆M = M Yb= 173.04 and = M xM YbCo 4 Sb12 + (1 − x) M= Co 4 Sb12 173.04* x + 4*58.93 + 12*121.76 32 So Γ M = x(1 − x)  ∆M  = x(1 − x)     2  M  173.04   x + + 173.04* 4*58.93 12*121.76   2 (Eq. 4.9) Later, Yang97 proposed a different way of calculating 𝛤𝛤𝑀𝑀 which weighs the influence of mass fluctuation on each site by its degeneracy. To be more specific, it takes into account not only the number of atoms on a specific site but also the mass contrast between the specific site and the rest of the sites. For example, the chemical composition of a material can be expressed as A1c1A2c2A3c3A4c4…Ancn, where the Ai are crystallographic sublattices in the structure and the ci are the relative degeneracies of the respective sites. In this context, the skutterudite compound YbxCo4Sb12 has n=3, A1=Yb, A2=Co, A3=Sb, and c1=1, c2=4, c3=12. In general there will be several different types of atoms that occupy each sublattice, and the kth atom of the ith sublattice has mass 𝑀𝑀𝑖𝑖𝑘𝑘 , radius 𝑟𝑟𝑖𝑖𝑘𝑘 , and fractional occupation 𝑓𝑓𝑖𝑖𝑘𝑘 . The average mass and radius of atoms on the ith sublattice are: M i = ∑ fi k M ik (Eq. 4.10) ri = ∑ f i k ri k (Eq. 4.11) k k The mass fluctuation scattering parameter is then given by: 2 M  ci  i  Γ M ,i ∑ i =1 M ΓM =  n     ∑ ci   i =1  n (Eq. 4.12) where the mass fluctuation scattering parameter for the ith sublattice is:  M ik  Γ= f ∑k i 1 − M  M ,i i   2 k (Eq. 4.13) and the average atomic mass of the compound is: n ∑c M i i M = i =1 n    ∑ ci   i =1  (Eq. 4.14) In the case of skutterudite compound YbxCo4Sb12, there is no mass fluctuation on sites A2 (Co) and A3 (Sb). = M2 M = (Co) 58.93 , = M 3 M= ( Sb) 121.76 . 33 Γ M ,3 = 0 Consequently Γ M ,2 = For site A1 (Yb/void), M 1 = 173.04* x f1Yb M 1Yb + f1void M 1void = 2 f Γ M= ,1 Yb 1  M 1Yb   M 1void  void 1 −  + f1 1 −  M1  M1    2 2 173.04  0    = x 1 −  + (1 − x) 1 −   173.04* x   173.04* x  ( x − 1) 2 1− x 1− x )= = + (1 − x) = (1 − x)(1 + x x x 2 n M = ∑c M i M 1 + 4 M 2 + 12 M 3 173.04* x + 4*58.93 + 12*121.76 = 1 + 4 + 12 17 i =  n   ∑ ci   i =1  i =1 2 2 M  M  ci  i  Γ M ,i  1  Γ M ,1 ∑ i =1 M   M  = ΓM = 17  n   ∑ ci   i =1  n 2     173.04* x  173.04* x + 4*58.93 + 12*121.76    17  1− x = x 17 (Eq. 4.15) 2 173.04   17   (1 − x) x 173.04* 4*58.93 12*121.76 + + x   Now if we compare Eq. 4.15 to Klemens’s result in Eq. 4.9, we can see that there is clearly a difference of 17 times which equals to the total number of atoms per compound formula. If we modify Yang’ formula such that the degeneracy of each site is still taken into account but with a less complicated form, we get 𝛥𝛥𝛥𝛥 2 𝑀𝑀 𝛤𝛤𝑀𝑀 = 𝑥𝑥(1 − 𝑥𝑥)𝑓𝑓 � � � (Eq. 4.16) This formula is quite similar to Klemens’s formula (Eq. 4.8) except with an extra term f and a � , which stand for the fraction of substituted lattice sites and the average different definition for 𝑀𝑀 atomic mass per number of atoms in the compound (rather than the previous molar mass for the compound) respectively. ΔM still holds for the mass difference between the host and impurity atom. In the case of skutterudite compound, ∆M = M Yb= 173.04 and 34 = M [ xM YbCo 4 Sb12 + (1 − x) M= 173.04* x + 4*58.93 + 12*121.76] /17 Co 4 Sb12     1 173.04  ∆M  Γ M = x(1 − x) f  x(1 − x)    = + +  M  17 So  173.04* x 4*58.93 12*121.76  17   2 2 173.04   = 17 x(1 − x)    173.04* x + 4*58.93 + 12*121.76  (Eq. 4.17) 2 Compare Eq. 4.9, Eq. 4.15 and Eq.17, we get (Eq. 4.18) Let us take Cu2ZnGeSexS4-x for another example. Using Klemens model, we have: A = Cu2ZnGeSe4 B = Cu2ZnGeS4 Cu2ZnGeSexS4-x = (x/4)*Cu2ZnGeSe4 + (1-x/4)*Cu2ZnGeS4 4 ( M Se − M S )  x x  ∆M  x x  G M , Klemens =(1 − )    =(1 − )  4 4  M  4 4  x * M Se + (4 − x) * M S + M Zn + M Ge + 2* M Cu  2 2 (Eq. 4.19) x x  ( M Se − M S )  1 (1 − )   4 4  M  4 2 = (Eq. 4.20) (Eq. 4.21) Compare Eq. 4.19, Eq. 4.20, and Eq. 4.21, and we get (Eq. 4.22) 35 Thus the modified equation for calculating gives the same result as Yang’s formula, yet it is easier to implement. From now on, we will only compare Yang’s formula and Klemens’s formula. In a more general case, for compound A1c1A2c2A3c3A4c4…Ancn, the relationship between the calculated scattering parameters from Klemens’s formula (Eq. 4.8) and Yang’s formula (Eq. 4.12) will be: (Eq. 4.23) with the ith element being substituted and Ai position having a Ci degeneracy. The difference between Yang’s formula and Klemens’ formula is that Yang’s formula takes the degeneracy of each site as the weighing factor in calculating scattering parameter, which makes more sense because doping on the Co site (CoxNi1-xSb3) should result in a smaller scattering parameter from doping on the Sb site (Co(SbxTe1-x)3) even when the doping content x is the same. Since is always larger than , the scattering parameter of Yang’s formula is always larger than Klemens’, which leads to a smaller lattice thermal conductivity. 4.2.3 Electron-phonon scattering In binary CoSb3 skutterudites, even adding a few percent of dopant (Fe, Pd, Pt, Ni, Te etc.) can result in large reduction in the lattice thermal conductivity. This was first reported by Dudkin and Abrikosov, who observed a 50% reduction of the lattice thermal conductivity at 300K in CoSb3 doped with 10at% Ni, 10at%Fe or 0.25at% Te 98. Later the influence of different dopant on lattice thermal conductivity reduction was confirmed and explored by many researchers, such as Caillat7, Stokes99, Anno100 etc. A plot with lattice thermal conductivity of CoSb3 versus the carrier concentration of n-type dopants (Ni, Pd, Pt) is shown in Figure 4.2, with data exacted from Anno’s paper100 (1999). 36 Figure 4.2. Lattice thermal conductivity 𝜅𝜅𝐿𝐿 at room temperature as a function of carrier concentration for Co1-xMxSb3 (M=Ni, Pd, Pt, and Pd+Pt) samples. The dashed lines present the calculations based on the Debye model. Reproduced with permission from [Anno, H., Matsubara, K., Notohara, Y., Sakakibara, T. & Tashiro, H. Effects of doping on the transport properties of CoSb3. Journal of Applied Physics 86, 3780, doi:10.1063/1.371287 (1999).]. Copyright [1999], AIP Publishing LLC. A Debye model including the Umklapp scattering, point defect scattering and electron-phonon scattering is used to compare with experimental value. It was concluded that the point defect scattering and electron-phonon scattering are the main processes that result in the reduction of lattice thermal conductivity of n-CoSb3 at room temperature. The relative strength of these two scattering mechanisms depends on the type of dopant. For example, point defect scattering is more dominant for Pd and Pt dopants due to the large difference in atomic mass and size compared to Co, whereas for Ni as dopant, the electron-phonon scattering plays a more crucial role due to the similar atomic mass and size between Ni and Co. Electron-phonon scattering occurs when the doping concentration is high and the temperature is low. This electron-phonon scattering is often ignored at high temperatures because it gets overwhelmed by Umklapp scattering. The relaxation time for electron-phonon scattering is expressed101,102 as Eq. 4.24-25 when the wavelength of electrons is shorter than the wavelength of phonons: −1 𝜏𝜏𝑒𝑒𝑒𝑒 = 𝐶𝐶𝑒𝑒𝑒𝑒 𝜔𝜔2 where 𝐶𝐶𝑒𝑒𝑒𝑒 = 4𝑛𝑛𝑚𝑚∗ 𝜈𝜈𝑒𝑒 𝜆𝜆𝑒𝑒 15𝑑𝑑𝑣𝑣𝑠𝑠 2 (Eq. 4.24) (Eq. 4.25) Here 𝑛𝑛 is the electron concentration, 𝑚𝑚∗ is the effective electron mass, 𝜈𝜈𝑒𝑒 is the electron velocity, 𝜆𝜆𝑒𝑒 is the mean free path of the electrons, and 𝑑𝑑 is the mass density, 𝑣𝑣𝑠𝑠 is the speed of sound. In the region where the electron-phonon scattering dominates, the lattice thermal conductivity increases proportionally with temperature 𝜅𝜅𝐿𝐿 ∝ 𝑇𝑇. While it is reasonable to expect a carrier 37 1 concentration dependent 𝜅𝜅𝐿𝐿 , the exact relation 𝑤𝑤𝐿𝐿 = ∝ 𝑛𝑛𝑎𝑎 remains unclear with different 𝜅𝜅𝐿𝐿 4 3 values of 𝑎𝑎(= 𝑜𝑜𝑜𝑜 2) reported103,104. Shi et al103 made an estimate of the coefficient 𝐶𝐶𝑒𝑒𝑒𝑒 : 𝑛𝑛4/3 ∗2 𝑑𝑑𝑑𝑑𝑑𝑑 𝑚𝑚 𝑘𝑘𝐵𝐵 𝑇𝑇 𝐶𝐶𝑒𝑒𝑒𝑒 ∝ 𝐸𝐸 2 (Eq. 4.26) which is based on the acoustic phonon scattering approximation and a single parabolic band model 1 1 2 𝑛𝑛3 in the degenerate limit (𝑣𝑣𝑒𝑒 = 𝑣𝑣𝐹𝐹 , 𝜆𝜆𝑒𝑒 = 𝑣𝑣𝑒𝑒 𝜏𝜏𝑒𝑒 = 𝑣𝑣𝑒𝑒 𝜏𝜏0 𝐸𝐸 −2 , 𝐸𝐸 = 2 𝑚𝑚∗ 𝑣𝑣𝐹𝐹2 , 𝐸𝐸 ∝ 𝑚𝑚∗ ). The obtained 4 relation 𝐶𝐶 ∝ 𝑛𝑛3 holds when the other parameters are either constant (such as 𝑇𝑇) or independent of carrier concentration such as 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑 , 𝑚𝑚∗. Eq. 4.26 explains why the electron-phonon scattering is more evident in heavily doped samples and at low temperatures. In filled skutterudites, due to the donor nature of filler atoms, the n-type carrier concentration can be quite high (1019 ~ 1021 cm-3). So besides the well-known “rattling” effect and point defect scattering effect of fillers, the electron-phonon scattering should also play an important role in the decrease of lattice thermal conductivity, as it does in binary skutterudites. Unfortunately, it has been challenging to separate the effect of electron-phonon scattering from other scattering processes with the uncertainty of carrier concentration dependence of 𝜅𝜅𝐿𝐿 . The difficulty in getting experimental data with pure electron-phonon scattering also makes it hard to confirm. 4.3 Thermal transport calculation 4.3.1 Minimum thermal conductivity It is important for us to know the minimum thermal conductivity of skutterudites when trying to decrease the lattice thermal conductivity. There are two frequently used formulas for minimum thermal conductivity calculation. The one proposed by Cahill105 was initially for amorphous materials and based on a physical picture of a modified Einstein model, which consists of a random walk of energy between uncorrelated localized oscillators of varying sizes and frequencies. The dominant energy transport is between nearest neighbors. The minimum phonon mean free path is one half of its wavelength (relaxation time equals to one half of one period of vibration). The Cahill minimum thermal conductivity can be expressed as: (Eq. 4.27) 38 where the summation is over the three sound modes (one longitudinal and two transverse modes), and Ω represents the average volume per atom, 𝑣𝑣𝑖𝑖 is the sound velocity for the longitudinal and transverse modes, and Θ𝑖𝑖 = 𝑣𝑣𝑖𝑖 (ℏ⁄𝑘𝑘𝐵𝐵 )(6𝜋𝜋 2 ⁄Ω)1/3 is the Debye temperature for each polarization in degrees K. Equation 4.27 contains no free parameters, since both Ω and 𝑣𝑣𝑖𝑖 are known. In the high temperature limit, , so the equation becomes: (Eq. 4.28) Slack89 proposed a different formula to calculate minimum thermal conductivity based on the notion that heat is carried by waves, and for this they must live longer than one period of vibration. The minimum mean free path is thus one wavelength instead of one half according to Cahill. According to Slack, the minimum thermal conductivity at high temperatures can also be estimated using a Debye approximation as: (Eq. 4.29) (Eq. 4.30) (Eq. 4.31) where 𝑁𝑁 is the number of atoms per primitive cell. The optic contribution is calculated from a sum of 3(𝑁𝑁 − 1) equally spaced optic modes. For CoSb3, 𝑣𝑣𝑙𝑙 = 4743𝑚𝑚/𝑠𝑠, 𝑣𝑣𝑡𝑡 = 2830𝑚𝑚/𝑠𝑠 according to speed of sound measurement, see Section 4.4.2. The average velocity can be calculated from: (Eq. 4.32) 3132m/s. The Debye temperature Θ can be calculated from: we get Θ = 𝑣𝑣(ℏ⁄𝑘𝑘𝐵𝐵 )(6𝜋𝜋 2 ⁄Ω)1/3 With Ω = 39 (Eq. 4.33) 𝑎𝑎 3 9.03483 = ∗ 10−30 𝑚𝑚−3 = 23.05 ∗ 10−30 𝑚𝑚−3, Θ = 327.8K. Both the speed of sound 2𝑁𝑁 2∗16 measurement data and the Debye temperature match well with literature data (𝑣𝑣𝑙𝑙 = 4590𝑚𝑚/𝑠𝑠, 𝑣𝑣𝑡𝑡 = 2643𝑚𝑚/𝑠𝑠, Θ = 307K) 6. With the values obtained we get the minimum thermal conductivity for CoSb3 skutterudites: =0.204+0.232 (W/m/K)=0.436 W/m/K W/m/K (Eq. 4.34) (Eq. 4.35) These values (Eq. 4.34 and Eq. 4.35) should serve as a guide in search of lower lattice thermal conductivity. Notice there is some discrepancy between the values obtained from both methods. The Cahill method surprisingly gives a higher thermal conductivity with only half of the wavelength used in the Slack model. 4.3.2 Electronic contribution to thermal conductivity The data from thermal diffusivity measurements 𝐷𝐷𝑇𝑇 gives us experimental estimations of thermal conductivity 𝜅𝜅 (𝜅𝜅 = 𝑑𝑑𝐷𝐷𝑇𝑇 𝐶𝐶𝑃𝑃 , 𝐶𝐶𝑃𝑃 = 3𝑘𝑘𝐵𝐵 , 𝑑𝑑 𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑). 𝜅𝜅 is the sum of electronic contribution to the thermal conductivity 𝜅𝜅𝑒𝑒 and lattice thermal conductivity 𝜅𝜅𝑙𝑙 . In order to probe the role of fillers in reducing lattice thermal conductivity, we have to separate 𝜅𝜅𝑒𝑒 from 𝜅𝜅 first. 𝜅𝜅𝑒𝑒 can be expressed as in Eq. 4.36: 𝜅𝜅𝑒𝑒 = 𝜅𝜅𝑒𝑒,𝑒𝑒 + 𝜅𝜅𝑒𝑒,ℎ + 𝜅𝜅𝑏𝑏 (Eq. 4.36) where 𝜅𝜅𝑒𝑒,𝑖𝑖 denotes the electronic thermal conductivity from the carrier type i. 𝜅𝜅𝑏𝑏 denotes the electronic thermal conductivity from bipolar conduction, which usually occurs when the band gap 𝐸𝐸𝑔𝑔 is small and temperature 𝑇𝑇 is high enough that intrinsic conduction can happen (𝑘𝑘𝐵𝐵 𝑇𝑇 > 𝐸𝐸𝑔𝑔 ) with both electrons and holes excited in pairs. The values of 𝜅𝜅𝑒𝑒,𝑖𝑖 is large in heavily doped semiconductor and can be easily calculated using the 40 Wiedemann-Franz law. The Wiedemann-Franz relationship states that the electronic contribution from species i is related to the electrical conductivity of species i (𝜎𝜎𝑖𝑖 ), temperature T, and the corresponding Lorenz number 𝐿𝐿𝑖𝑖 via: In a single parabolic band model with carrier scattering time of the form (Eq. 4.37) : (Eq. 4.38) For acoustic phonon scattering, 𝑟𝑟 = −1/2, so the Lorenz number can be expressed as: (Eq. 4.39) For electronic thermal conductivity due to bipolar conduction, (Eq. 4.40) For more details, please refer to Appendix A-D. 4.3.3 Callaway model Once the electronic thermal conductivity is exacted from the total thermal conductivity, the lattice contribution can be estimated experimentally. A Callaway model can be used to calculate theoretical lattice thermal conductivity, taking into account of various phonon scattering mechanisms. By comparing results from Callaway model to experimental values, underlying phonon scattering mechanisms can be revealed. This is quite useful in accessing the role of fillers in reducing the lattice thermal conductivity. The Callaway model is based on a Debye approximation, with which the lattice thermal conductivity can be calculated as: (Eq. 4.41) (Eq. 4.42) 41 is the boundary scattering relaxation time, which is determined by the speed of sound in the material 𝑣𝑣𝑠𝑠 and average grain size 𝐿𝐿. (Eq. 4.43) is the Umklapp scattering relaxation time. Empirically it can be expressed as: (Eq. 4.44) where , ω is phonon frequency, 𝑀𝑀 is the average mass of a single atom, and γ is the Grüneisen parameter that characterizes the anharmonicity of lattice vibration. There are two types of phonon-phonon scattering mechanisms: Umklapp scattering and Normal scattering. Both processes involve three phonons. The difference is that in Normal scattering process, the total phonon momentum is conserved and is generally considered as having no contribution to the thermal resistance, whereas in Umklapp scattering process, with increasing phonon momentum of the incoming wave vectors ( ), the outgoing wave vector can point out of the Brillouin zone, which can be equivalently transferred into a phonon wave vector inside the Brillouin zone but with a smaller magnitude ( , is a reciprocal lattice vector). This leads to a change in the phonon momentum, and results in energy loss and adds thermal resistivity. Umklapp scattering is the main mechanism for thermal resistance at high temperatures for materials with low defect concentration. As discussed previously, (section 4.2.1); denotes empirically the resonant scattering time: 𝜏𝜏𝑟𝑟−1 = 𝐶𝐶0 −1 denotes the point defect scattering time 𝜏𝜏𝑃𝑃𝑃𝑃 = 𝑤𝑤 2 (𝑤𝑤 2 −𝑤𝑤02 )2 Ω 𝜔𝜔4 Γ (section 4.2.2); 4𝜋𝜋𝜈𝜈 3 −1 denotes the electron phonon scattering time 𝜏𝜏𝑒𝑒𝑒𝑒 = 𝐶𝐶𝑒𝑒𝑒𝑒 𝜔𝜔2 (section 4.2.3). Note that the nature of rattling remains unclear, given that both the relaxation time reduction (due to resonant scattering) and the group velocity reduction (because of the avoided crossing in the phonon spectra) can lead to similar reduction in lattice thermal conductivity (Figure 4.1). However here we are using the resonant scattering for simplicity. 42 4.4 Results and discussion 4.4.1 Possible electron-phonon scattering in RxCo4Sb12 A series of Yb-doped CoSb3 skutterudites were synthesized with nominal doping level x = 0.005 to 0.3049. Their room temperature transport data are shown in Table 4.2. Table 4.2. Room temperature transport properties of YbxCo4Sb12 skutterudites (with nominal x from 0.0025 to 0.3). Nominal x 0.0025 0.005 0.0075 0.01 0.025 0.05 0.075 0.1 0.2 0.25 0.3 EPMA x 0.002 0.003 0.009 0.019 0.047 0.071 0.1 0.2 0.25 0.25 nH (cm-3) 6.8E+17 3.5E+18 5.7E+18 7.4E+18 2.1E+19 4.6E+19 7.37E+19 1.24E+20 2.87E+20 4.26E+20 5.08E+20 Seebeck (µVK-1) -369.2 -320.8 -306.4 -298.8 -241.8 -196.8 -172.8 -152.4 -120.6 -102.2 -95.0 Mobility (cm2V-1s-1) 150.3 79.4 82.4 80.1 98.1 86.0 78.5 69.4 36.4 39.5 35.9 Resistivity (10-5Ωm) 61.16 22.27 13.34 10.54 3.08 1.57 1.08 0.73 0.60 0.37 0.34 κ (Wm-1K-1) 8.68 6.85 6.93 5.59 4.81 3.86 4.02 2.99 3.63 3.64 𝜅𝜅𝐿𝐿 (Wm-1K-1) 8.66 6.82 6.89 5.44 4.50 3.39 3.32 2.07 2.1 1.95 The experimental lattice thermal conductivity was calculated using Eq. 4.39. Now two methods can be applied to calculate the lattice thermal conductivity. The first is the Callaway model descried in Section 4.3.3. The other is using the Klemens model introduced in Section 4.2.2. The latter conveniently allows us to use the experimental value of the lattice thermal conductivity of pure (or very lightly doped) skutterudites as input. By comparing the ratio of lattice thermal conductivity between heavily doped CoSb3 and the pure one, the effect of point defect scattering can be estimated. If we consider only the contribution of mass contrast between the void and Yb filler atom (Γ𝑀𝑀 ) to the scattering parameter Γ, the modeling result is shown in Figure 4.3a. Note here that we are using Yang’s formula (Eq. 4.12) for Γ𝑀𝑀 as discussed in Section 4.2.2, which will give a lower estimated lattice thermal conductivity. Yet the modeling result still highly overestimates the lattice thermal conductivity, so in the next step the effect of strain (Eq. 4.7) is also considered. 43 (a) (b) Figure 4.3. Lattice thermal conductivity versus doping level x with both experimental and calculated results from Klemens model. The green line represents the modeling results with a) only mass effect considered and b) both mass and strain effects considered (Γ𝑀𝑀 calculated from Yang’s = 5E5). The blue dot represents experimental value (x from EPMA formula, Γ𝑠𝑠 with measurements listed in Table 4.2). To assess the strain induced by the addition of fillers, the influence of filler atom on the lattice constant is studied, as shown in Figure 4.4. A linear dependence can be obtained, (Eq. 4.45) 44 Figure 4.4. Dependence of lattice constant of YbxCo4Sb12 skutterudites on the Yb actual content x. With this relationship (Eq. 4.45) we can calculate the scattering parameter from strain effect (Eq. 4.7). However, we cannot get a reasonable fit unless is as large as 5E5 (Figure 4.3b). This value is not physical, and thus other effects must be contributing besides the point defect scattering. Among common scattering mechanisms (discussed in Section 4.3.3), Umklapp scattering is not prominent at room temperature. The grain size is on the same level for all samples so no additional boundary scattering rises from filled samples compared to pure skutterudite sample. What remains is the rattling effect and electron-phonon scattering. We investigate the possibility of electron-phonon scattering due to filler addition first. By subtracting the thermal resistance due to point defect scattering from mass effect (the strain effect is ignored because of its small magnitude), we get the good linear dependence of remaining thermal resistance on the carrier concentration shown in Figure 4.5. (Eq. 4.46) This is possibly due to electron phonon scattering the exponent coefficient obtained here as discussed in section 4.2.3. However, is smaller than previously reported values (4/3 or 2) . This might be due to the carrier concentration dependence of 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑 , 𝑚𝑚∗. The increasing 103,104 dependence of 𝑚𝑚∗ on carrier concentration can counteract with the influence of carrier concentration on the scattering constant Cep, thus leading to a lower dependence. Figure 4.5. Thermal resistivity due to electron-phonon scattering versus measured Hall carrier concentration in YbxCo4Sb12. The blue dot represents experimental value (nH listed in Table 4.2). The green line represents modeling result from Klemens model with only mass effect considered. More research is required both on more precise theoretical estimation of electron-phonon scattering and on experimental evidence as well. Before we can conclude about the electron-phonon 45 scattering, it would be difficult to assess the effect of resonant scattering and/or group velocity reduction due to fillers. 4.4.2 Lattice softening due to fillers By adding a filler atom in the void site, not only it will change the density of skutteurdites, but also it may change the elastic modulus by either softening or stiffening the framework. Both will contribute to the change of sound velocities. The longitudinal and transverse sound velocities can be expressed in equations 4.47 and 4.48, respectively. (Eq. 4.47) (Eq. 4.48) where K, G, and d represent the bulk and shear moduli, and theoretical density, respectively. The theoretical density can be calculated using EPMA determined doping content x and lattice parameter a derived from XRD measurements, as shown in Eq. 4.49. Note that the factor 2 equals the number of primitive cells per unit cell. For samples listed in Table 4.2, their calculated theoretical densities are listed in Table 4.3. (Eq. 4.49) where , , and are molar masses for the Yb, Co, and Sb atom, respectively, and is the Avogadro’s constant. Table 4.3. Theoretical density calculated from EPMA doping content x and XRD-derived lattice parameter a of YbxCo4Sb12 skutterudites (with nominal x from 0.0025 to 0.3)49. Nominal EPMA x x 0.0025 0.005 0.002 0.0075 0.003 0.01 0.009 0.025 0.019 0.05 0.047 0.075 0.071 a (Å) 9.035 9.035 9.036 9.036 9.037 9.040 9.041 d (g cm-3) 7.642 7.644 7.643 7.648 7.652 7.668 7.685 46 0.1 0.2 0.25 0.3 0.1 0.2 0.25 0.25 9.044 9.050 9.055 9.057 7.699 7.753 7.785 7.788 From Table 4.3, we can extract a linear relationship between the theoretical density d and EPMA doping content x: (Eq. 4.50) By substituting Eq. 4.50 into Eq. 4.49, we can get a relationship between the lattice parameter and EPMA doping content x. Here a is in Å and M is in g/mol. (Eq. 4.51) With x value from Eq. 4.51. By using Eq. 4.50 again we can get the theoretical density of any sample that has only one of the two variables a and x measured. Unfortunately the speed sound measurements were not performed on this set of samples (YbxCo4Sb12 skutterudites, with nominal x from 0.0025 to 0.3). However, there is indeed another set of YbxCo4Sb12 samples synthesized by ball milling whose XRD (all samples are phase-pure according to XRD analysis) and speed sound measurements were performed but without EPMA results. To solve this problem, we can either use Eq. 4.51 or Eq. 4.45 to get an estimate of the EPMA doping content x from XRD-derived lattice parameter a. The results obtained from both methods are close to each other. The measured speed sound velocities as well as calculated elastic moduli (Eq. 4.47 and Eq. 4.48) and theoretical densities (Eq. 4.50) for this set of ball-milled samples are listed in Table 4.4. Table 4.4. Speed sound measurement data as well as calculated elastic moduli and theoretical density of ball-milled YbxCo4Sb12 skutterudites (with nominal x from 0 to 0.6). EPMA values x listed are calculated using Eq. 4.51 from lattice constant a. Nominal x 0 0.2 0.3 0.4 0.45 0.5 EPMA_calc x 0.165 0.222 0.281 0.310 0.350 a (Å) 9.035 9.049 9.053 9.058 9.060 9.063 d (g cm-3) 7.638 7.737 7.769 7.803 7.819 7.842 (m/s) 4743 4600 4534 4469 4447 4418 (m/s) 2830 2764 2700 2672 2646 2643 (GPa) 90.22 86.47 83.74 83.19 82.07 81.39 (GPa) 61.18 58.52 56.49 55.84 54.78 54.41 47 0.6 0.382 9.066 7.860 4343 2622 81.00 53.79 A decreasing linear trend in elastic moduli (both K and G) with doping content xEPMA can be observed in Figure 4.6. For the maximum doping x = 0.38, the decrease reaches about 10%. This clearly indicates the softening nature of fillers on the framework. Whether it is due to lattice expansion or increased electron screening remains unclear, and more research effort is required to reach a solid conclusion. The decrease in speed sound velocities with doping is another proof that supports the avoided crossing mechanism in reducing thermal conductivity by fillers. Figure 4.6. Dependence of elastic moduli on doping content x in YbxCo4Sb12. 4.5 Conclusion and future work It can be concluded that in RxCo4Sb12, besides the point defect scattering due to partial filling, the role of fillers in reducing lattice thermal conductivity possibly involves some electron-phonon scattering as well. To precisely determine the extent of this scattering such that we can comment on the effect of “rattling”, more research effort needs to be performed. Synthesis of Ni or Te-doped CoSb3 samples is proposed to study the carrier concentration dependence of thermal resistance due to electron-phonon scattering, since the effect of rattling is excluded in this case. Assuming the carrier concentration dependence of thermal resistance does not vary much in unfilled and filled CoSb3, this shall allow us to understand better the effect of rattling on thermal conductivity reduction, whether it is there and if it does, to what extent. It is also encouraged to measure the speed of sound velocities on Ni or Te-doped CoSb3 samples. The closeness in lattice constant 48 between undoped and doped samples will tell more about the mechanism in lattice softening (decrease of elastic moduli) with doping. 49 Chapter 5 Phase diagram studies in n-CoSb3 skutterudites 5.1 Summary In Chapter 3 the electronic origin of high thermoelectric performance of CoSb3 skutterudties was investigated. In Chapter 4 the role of fillers in thermal transport was also discussed. So far we have gained some insight of physics as for why high zT can be achieved in skutterudites, the next step would be how we can synthesize high zT skutterudites from a materials scientist perspective. To optimize both electronic and thermal transport properties, precise doping control is required. This leaves us a few questions that need to be addressed. Firstly, when filler atom R is added to the CoSb3 matrix, is the void the only soluble site? If not what are the other possibilities? Secondly, what is the solubility limit (equals to commonly known filling fraction limit (FFL) when the void is the only soluble site) for each type of filler? Is it a single value? Thirdly, how can we achieve precise doping control with large tolerance of synthesis uncertainty in a repeatable manner? This is important for large-scale commercialization of skutterudites in the future. Because solubility limit is a variable that exists at thermodynamic equilibrium, all the questions mentioned above cannot be answered without a careful and complete phase diagram study. By adding R into CoSb3, a ternary system is under study, which leads to the breakdown of previous experiences in determining solubility limit in binary systems (impurity detection, Vegard’s law etc.). In the following sections, Section 5.2 shall describe in great details the methodology of how to apply phase diagram study in studying skutterudites. Sections 5.3 to 5.5 lays out the experimental results of several ternary systems (R-Co-Sb, with R = Ga, In, Yb, Ce), which answers the questions listed above. The results presented here are mainly reproduced based on the following published papers with copyright permission acquired: Advanced Functional Materials 23, 3194-3203 (2013, Copyright Wiley-VCH Verlag GmbH & Co. KGaA.)16; Energy and Environmental Science 7, 812-819 (2014, http://pubs.rsc.org/en/Content/ArticleLanding/2014/EE/c3ee43240h#!divAbstract)17; Journal of Materiomics 1, 75-84 (2015, http://www.sciencedirect.com/science/article/pii/S235284781500009X)50; 50 Nature Communications 6, 7584(1-7) (2015, http://www.nature.com/ncomms/2015/150720/ncomms8584/full/ncomms8584.html)20. The readers are welcome to refer to these papers for more details. Section 5.6 discusses the influence of precipitation on the stability and lifetime performance of optimized compositions. Section 5.7 is the conclusion and discusses possible future work. 5.2 Methodology The goal of phase diagram study of these R-Co-Sb ternary systems is to map out their isothermal sections at various temperatures lower than 874℃ (peritectic temperature of CoSb3 decomposition)106. From these isothermal sections, not only phase regions near the skutterudite phase can be revealed, but also solubility of R as well as its temperature dependence can be extracted. To lower the experimental burden, only a subsection with CoSb3 phase included of the full isothermal section is studied, e.g., YbSb2-CoSb2-Sb rather than Yb-Co-Sb. The methodology is explained in nine steps with the Yb-Co-Sb ternary system illustrated as an example. Step one: Gather binary phase diagram information Knowledge of binary phase diagrams is required, as shown in Figure 5.1. Most binary phase diagrams are available on the website: http://www1.asminternational.org/asmenterprise/apd/. Occasionally there are more than one phase diagram available for the same system. Either the most updated phase diagram determined from experimental work is preferred or it is recommended to choose the ones determined by the same group (Okamoto in this case) for systems in the study to keep it consistent. 51 Figure 5.1 Binary phase diagrams of Yb-Co, Yb-Sb and Co-Sb systems106-108. Green arrow indicates the temperature dependence. Step two: Select a temperature for isothermal section mapping Samples will be annealed at the selected temperature for isothermal section mapping to reach thermodynamic equilibrium. As a result, neither should it exceed the maximum temperature of phase stability of the target phase, nor should it be too low such that thermodynamic equilibrium is hard to reach on an economic time scale. Temperatures between 500℃ and 850℃ are in the good range. Here we chose 700℃ for an example. Step three: Determine nominal compositions for sample synthesis. Get binary phases from binary phase diagrams (R-Co, R-Sb, Co-Sb) at the chosen temperature 700℃, which are represented as red dots along axis on the Yb-Co-Sb isothermal section in Figure 5.2. Estimate phase regions around target phase CoSb3. Boundaries of phase regions are estimated and shown as black lines in the YbSb2-CoSb2-Sb section. Nominal compositions are chosen for sample synthesis, which are represented as black circles in the highlighted orange oval, as shown in Figure 5.2. These black circles are selected to scatter among different phase regions. The nominal compositions are listed in Table 5.1 to give an idea how off-stoichiometry they are. 52 Figure 5.2 Selected nominal compositions near target phase CoSb3 for phase region identification in the Yb-Co-Sb isothermal section at 700℃. Table 5.1 Selected nominal compositions for investigation of Yb-Co-Sb isothermal section at 700℃. Nominal #1 Yb0.25Co3.8Sb12.2 #2 Yb0.25Co4Sb12 #3 Yb0.25Co4.2Sb11.8 #4 Yb0.5Co4Sb12 #5 Yb0.5Co4.2Sb11.8 #6 Yb0.5Co3.9Sb12.1 #7 Yb0.5Co3.8Sb12.2 Estimated phase region YbxCo4Sb12 + liquid Sb YbxCo4Sb12 YbxCo4Sb12 + CoSb2 YbxCo4Sb12 + CoSb2 + YbSb2 YbxCo4Sb12 + CoSb2 + YbSb2 YbxCo4Sb12 + YbSb2 YbxCo4Sb12 + YbSb2 + liquid Sb Step four: Sample synthesis. Synthesize samples with selected nominal compositions in step three in a melting and annealing method. The annealing temperature has to be the same as selected isothermal section temperature, which is 700℃. The annealing time is chosen to be one week to ensure thermodynamic equilibrium and homogenization of filler distribution. For more synthesis details please refer to Chapter 2 (Section 2.2). 53 Step five: Phase identification and chemical composition characterization. Check the vacuum inside the ampoule. If it is good then proceed to break the ampoule and take the ingot out. Cut the ingot in halves. Take one half to X-ray diffraction and the other half to scanning electron microprobe for phase identification. The actual filler content in the matrix skutterudite phase is determined by electron probe microanalysis. Samples should be characterized in one single experimental routine to avoid any unwanted discrepancies in experimental conditions, such as stability of the electron beam, sample surface contamination, etc. The thermodynamic equilibrium is achieved when the standard deviation of filler content is less than the precision limit of EPMA, which is confirmed in all samples from experimental results. Step six: Determine phase region relations. With determined compositions of the equilibrium phases for each nominal composition, the phase regions near CoSb3 can be determined, as shown in Figure 5.3. The solubility limit of R in CoSb3 can be found in the composition on the furthest point of CoSb3 phase region extending towards R (red dot in Figure 5.4). The solubility limit of Yb in YbxCo4Sb12 at 700℃ is determined to be x = 0.44±0.01. YbSb2 0.3 0.1 0.3 +Y Co Sb 3 0.1 Co Sb 3 +C oS 2 b+ bS b 2 Yb + Sb 2 0.2 liq ui d 0.2 CoSb3+ liquid CoSb2 liquid CoSb3+CoSb2 0.7 CoSb3 0.8 0.0 0.9 1.0 Sb Figure 5.3 Phase regions near CoSb3 determined from synthesized samples in a section of Yb-Co-Sb isothermal ternary phase diagram at 700℃ (YbSb2-CoSb2-Sb). The red line represents the solubility region of CoSb3. A magnification of the triangle area formed by green dashed lines is shown in Figure 5.4, 5.6, and 5.7 at different temperatures (973K, 873K, and 1023K, respectively). 54 #1 Yb0.25Co3.8Sb12.2 #1 Yb0.192Co4Sb12 Yb T = 973K #2 Yb0.25Co4Sb12 0.20 #2 Yb0.215Co4Sb12 #3 Yb0.25Co4.2Sb11.8 #3 Yb0.302Co4Sb12 0.08 0.72 3 Co Sb +Y bS b 2 + liq ui d bSb 2 b bS CoSb3+CoSb2 CoS b3 +Y +Y 2 Sb +C o 3 Sb Co 0.28 Co #5 Yb0.5Co4.2Sb11.8 #5 Yb0.441Co4Sb12 2 0.24 #4 Yb0.5Co4Sb12 #4 Yb0.438Co4Sb12 0.04 #6 Yb0.5Co3.9Sb12.1 #6 Yb0.260Co4Sb12 #7 Yb0.5Co3.8Sb12.2 #7 Yb0.255Co4Sb12 CoSb3+ liquid 0.76 0.80 Sb Figure 5.4 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 973K. The solid red line indicates the void filling in the direction (YbVF)xCo4Sb12. The red point represents the stable composition (with maximum solubility x = 0.44±0.01) of the skutterudite phase obtained when the nominal composition is in the Co-rich three-phase region (YbSb2, CoSb2, and YbxCo4Sb12). The blue point represents the stable composition of the skutterudite phase obtained when the nominal composition is in the Sb-rich three-phase region (YbSb2, liquid Sb, and YbxCo4Sb12). The nominal sample compositions are shown as open symbols. Solid symbols represent the composition of the skutterudite majority phase using the experimental EPMA value for Yb content, and positioned on the red line. The nominal (open symbols) and experimental compositions (solid symbols) are connected with a dotted line. Step seven: Determine temperature dependence of solubility. Repeat step 2-6 for different annealing temperatures to get the temperature dependence of solubility. Note that since the phase relations do not change drastically with temperature and our interest is mainly about solubility, nominal compositions can be chosen in the phase regions that will produce stable compositions of skutterudites (discussed later in Section 5.4). Maximum solubility is often related to stable compositions and this can allow us to minimize experimental load. The temperature dependence for Yb solubility in CoSb3 is shown in Figure 5.5 for example. As we can see in Figure 5.5, as temperature increases, Yb solubility increases with about five times difference (x = 0.11±0.02 at 773K and x = 0.49±0.02 at 1073K). This large variance gives us another degree of freedom to tune the carrier concentration and thus allows us to optimize thermoelectric properties. 55 Yb actual content x in YbxCo4Sb12 0.5 #1 Yb0.25Co3.8Sb12.2 #4 Yb0.5Co4Sb12 #5 Yb0.5Co4.2Sb11.8 0.4 #7 Yb0.5Co3.8Sb12.2 0.3 0.2 0.1 0.0 750 800 850 900 950 1000 1050 1100 T(K) Figure 5.5 Temperature dependence of solubility limit of Yb in Yb-doped skutterudites YbxCo4Sb12. The red line indicates the temperature dependence of the actual Yb content in the stable skutterudite composition represented as a red point (maximum solubility) in Figure 5.4. The blue line shows the temperature dependence of the actual Yb content in the stable skutterudite composition represented as a blue point in Figure 5.4. The dashed green line is the actual Yb content (x = 0.3) targeted for optimized thermoelectric properties, which crosses the solubility lines at 873K and 1023K, respectively. Step eight: Determine synthesis parameters for optimized compositions. Choose an annealing temperature at which the solubility line crosses the optimized composition line. Then select corresponding phase region such that the resulting stable compositions will produce optimized doping content. For example, if the annealing temperature in the Yb-Co-Sb system is chosen to be 873K, then the nominal composition (red triangular and diamond samples in Figure 5.6) should stay in the Co-rich three-phase region (YbSb2, CoSb2, and YbxCo4Sb12). If the annealing temperature is chosen to be 1023K, then the nominal composition (blue circle sample in Figure 5.7) should stay in the Sb-rich three-phase region (YbSb2, liquid Sb, and YbxCo4Sb12). 56 #1 Yb0.25Co3.8Sb12.2 T = 873K Yb #1 Yb0.130Co4Sb12 0.20 #4 Yb0.5Co4Sb12 #4 Yb0.288Co4Sb12 #5 Yb0.5Co4.2Sb11.8 0.08 #5 Yb0.314Co4Sb12 #7 Yb0.5Co3.8Sb12.2 2 0.24 d ui liq + Yb0.36Co4Sb12 b bS Yb0.40Co4Sb12 b Co S 3 +Y Co 0.28 CoSb3+CoSb2 Co 0.04 2 Sb 3 +C o Sb 2 CoS b3 +Y +Y bSb 2 bS b #7 Yb0.125Co4Sb12 Yb0.33Co4Sb12 CoSb3+ liquid 0.72 0.76 Sb 0.80 Figure 5.6 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 873K. Samples with nominal compositions YbxCo4Sb12 (x = 0.33, 0.36, 0.40) are in Co-rich three-phase region and annealed at 873K, which gives desired optimum Yb content x = 0.30 (red dot). #1 Yb0.25Co3.8Sb12.2 T = 1023K Yb #1 Yb0.209Co4Sb12 0.20 #4 Yb0.5Co4Sb12 #4 Yb0.463Co4Sb12 #5 Yb0.5Co4.2Sb11.8 0.08 2 #7 Yb0.274Co4Sb12 3 + Y bS Yb0.33Co4Sb12.2 0.04 CoSb CoS b3 +Y #7 Yb0.5Co3.8Sb12.2 b2 + li bSb 2 Sb Yb 2 b+ oS +C 3 Sb Co #5 Yb0.384Co4Sb12 quid 0.24 0.28 CoSb3+CoSb2 Co 0.72 CoSb3+ liquid 0.76 0.80 Sb Figure 5.7 Magnification of phase regions near CoSb3 of isothermal section of Yb-Co-Sb ternary phase diagram system at 1023K. Sample with nominal composition YbxCo4Sb12.2 (x = 0.33) is in Sbrich three-phase region and annealed at 1023K, which gives desired optimum Yb content x = 0.30 (blue dot). Step nine: TE performance characterization. Characterize thermoelectric properties of samples with optimized compositions synthesized in step eight. As shown in Figure 5.8, the thermoelectric performance does not vary much among different compositions. Thus it enables the production of optimized skutterudites at an easily reproducible and commercial scale. 57 Figure 5.8 Temperature dependence of the thermoelectric figure of merit zT for Yb-doped skutterudites targeting the same YbxCo4Sb12 composition with samples annealed at 873K without excess Sb (red stable point) and 1023K with excess Sb (blue stable point). The dark symbols are reference data from Zhao et al109. 5.3 Soluble site other than the void 5.3.1 Solubility debate of In and Ga One important criterion in determining which site to go to for foreign atom R in CoSb3 sktterudites is the size and electronegativity difference between R and possibly soluble sites. The void site (radius 1.89Å)9 is usually large enough for most alkali, alkaline earth, group 13, and rare earth elements to go in, as shown in Table 4.1 87. Due to the electro-positivity of alkali, alkaline earth, and rare earth elements, it is impossible for them to substitute Sb. The large size difference between Co3+ and these filler ions also hinders them from substituting Co. As a result, the void site is the only soluble site for all the atoms listed in Table 4.1 except for two elements, that is, group 13 elements Ga and In. The soluble sites as well as doping nature for both Ga and In generate a lot of debate. Theoretically, a selection rule based on the electronegativity difference between Sb and R110 was proposed by Shi et al. It was predicted that Ga and In could not be filled into the void site. Experimentally, it was assumed that group 13 elements occupy the void position exclusively and donate electrons, resulting in n-type ternary materials. Tl was expected to donate one electron in CoSb3 111, but for In and Ga it is not clear how many electrons are donated from these void-filling atoms.112-115 Several studies have shown GaSb- and InSb- containing composites were formed and concluded that Ga and In have virtually no solubility in CoSb3.51,116 There was yet another report claiming that In could 58 possibly substitute for Sb in CoSb3 and hardly fill into the void position.117 Finally, since Ga3+ and In3+ are similar in size to Co3+, it may also be possible for these atoms to substitute for the transition metals. The close electronegativity values among In (1.78), Ga (1.81), Sb (2.05), and Co (1.88) also support the possibility of both Co and Sb substitution. Complicating this debate is the possibility that there is no dominant single defect type in skutterudites with group 13 impurities, but that compound defects form as commonly found in wide-gap oxides. Amphoteric impurities were also observed in a few thermoelectric materials, in which one single-type impurity forms both donor and acceptor defects.118 If both defects exist, even separately, and form in equal amounts the material will be nearly fully compensated with few free carriers, as found in Ga-doped CoSb3 skuterudites.113 5.3.2 DFT calculations of In-doped CoSb3 systems Before taking on experimental study, DFT calculations were applied to help navigate the research direction. Two types of calculations were performed, the first being the band structures of CoSb3 and different Ga/In-doped systems. Due to the similarity between band structures of Ga- and Indoped systems, only the results on In-doped systems are shown in Figure 5.9. The second is the formation energies of possible defects as a function of both doping level and Fermi level. Results are shown in Figure 5.10 and 5.11. Co32Sb96 1.0 0.3 Energy (eV) Energy (eV) 0.5 0.0 -0.5 -1.0 Γ Co32Sb95InSb 0.6 0.0 -0.3 H N Γ P -0.6 Γ H N Γ P 59 InVFCo32Sb95InSb 0.8 0.6 (InVF)2Co32Sb95InSb 0.3 Energy (eV) Energy (eV) 0.4 0.0 0.0 -0.3 -0.4 Γ H N Γ P -0.6 Γ H N Γ P Figure 5.9 Band structures of CoSb3 and different In-doped systems. The dashed lines are the Fermi levels. Figure 5.9 shows the band structures of pure CoSb3, Co32Sb95InSb, InVFCo32Sb95InSb, and (InVF)2Co32Sb95(InSb). Results show that In doping at different sites has different effect on band structures. Pure CoSb3 is a semiconductor with small band gap. One In atom at the Sb substitution site generates a deficiency of two electrons and one In atom at the void-filling position adds one extra electron compared to pure CoSb3. As a result, one In substituting for Sb leads to imbalanced charge and pushes up one Sb-based band above the Fermi level, which brings the system to an unstable state at no charge compensation. For the same reason InVFCo32Sb95InSb is also hardly stable whereas (InVF)2Co32Sb95(InSb) is charge-balanced and is thus a semiconductor. Intuitively InxCo4Sb12-x/3, equivalently (InVF)2x/3Co4Sb12-x/3(InSb)x/3, is expected to be a stable skutterudite phase. Similar results are also observed in Ga-doped CoSb3, namely, (GaVF)2x/3Co4Sb12-x/3(GaSb)x/3, is expected to be a stable skutterudite phase as well. The charged defects from In-containing CoSb3, such as InCo, InSb substitutions in the Co-Sb framework, InVF on the crystal void site, and their combinations, with different charge states q are calculated by ab initio methods. The defect formation Gibbs free energy per impurity can be written as follows: (Eq. 5.1) where is the total energy of a 2×2×2 supercell InyCo32-wInwSb96-zInz. is the total energy of supercell of bulk CoSb3. y is the filling fraction of In in crystal voids, w is the substituting fraction at Co sites, and z is the substituting fraction at Sb sites. is chemical potential for R ( ), q is the charge state of the point defect, is the 60 Fermi level, referring to the valence band maximum Ev in CoSb3, and ΔV is the correction term to align the reference potential in the defect supercell with that in the perfect supercell. Figure 5.10(a) shows the calculated for (GaVF)xCo4Sb12-x/2(GaSb)x/2 at 923 K possesses the lowest values and stays negative at about x ≤0.25. The formation energy (-0.116 eV) of GaSb, a known secondary phase in Ga-contained CoSb3, is also shown in Figure 5.10(a), and it becomes lower than for (GaVF)xCo4Sb12-x/2(GaSb)x/2 after x ≥ 0.08, indicating that (GaVF)xCo4Sb12- x/2(GaSb)x/2 is the most stable phase when x is less than 0.08. In other words, the maximum filling fraction for Ga in Co4Sb12 should be around 0.08, and the total Ga atoms in a unit of Co4Sb12 should be around 0.12 (=1.5x) at 923 K. Similarly, a solubility of In in a unit of Co4Sb12 at around 0.27 (=1.5*0.18, x = 0.18 is the filling fraction limit) at 873K is predicted. (a) (b) 0.6 1100K 0.5 873K ∆G (eV) 0.4 0.3 0.2 0.1 InSb 0.0 InxCo4Sb12-x/2Inx/2 InxCo4Sb12 Co4Sb12-xInx -0.1 -0.2 0.0 0.2 0.4 0.6 x 0.8 Figure 5.10 Calculated Gibbs free energy ( ) as a function of doping content x of (a) Ga in GaCoSb3 skutterudite at 923K and (b) In in In-CoSb3 skutterudite at 873K. (a) (b) Figure 5.11 Formation energies of possible defects as a function of Fermi level at the Co-rich limit in Ga-containing (a) and In-containing (b) skutterudites. The zero Fermi level corresponds to the top of the valence band, and the width of shadowed area indicates the energy gap. The number for labeling is the charge state of the point defect. 61 Figure 5.11 shows the defect formation energy as a function of Fermi level at the Co-rich limit, where the system is connected to a Co reservoir. Ga-containing skutterudites form stable chargecompensated compound defects (CCCD) (GaVF)2x/3Co4Sb12-x/3(GaSb)x/3 when the Fermi level falls in the gap area. The formation energy of the single filling defect (GaVF)xCo4Sb12 is always very high. In contrast, In-containing skutterudite favors In-filling into the void at relatively low Fermi level instead, and dual-site In occupancy can form as the Fermi level increases. In the gap area 0~0.17eV (shadow in Figure 5.11), Ga always forms the charge compensated defect, and In forms either a single filling, or a dual-site defect with or without charge compensation, at different Fermi levels which likely corresponds to different experimental conditions such as the amount of indium impurity or the stoichiometric ratio of Co/Sb. The results from theoretical study above can be summarized as follows. Firstly, unlike for other fillers, the void is NOT the only soluble site for both In and Ga. Secondly, Ga or In-doping in CoSb3 based skutterudites results in both void filling and Sb substituting, which leads to compound defects (charge-compensated in the Ga case and not fully charge-compensated in the In case). Thirdly, the solubility limit is predicted to be 0.12 for Ga in a Co4Sb12 unit at 923K and 0.27 for In at 873K. The solubility region can then be estimated to be mainly directed along the direction of (RVF)2x/3Co4Sb12-x/3(RSb)x/3 to CoSb3 (R = Ga, or In). This is different from the commonly known direction of (RVF)x Co4Sb12 to CoSb3 when the void is the only soluble site for alkali, alkaline earth, and rare earth element fillers. The experimental phase diagram study is accordingly designed as shown in Section 5.3.3 below. 5.3.3 Phase diagram study of In-Co-Sb system Due to the similarity between Ga and In-doped CoSb3, we will only take the In-doped CoSb3 system as an example for phase diagram study illustration. Samples with nominal compositions along or off the solubility direction (RVF)2x/3Co4Sb12-x/3(RSb)x/3 to CoSb3 are synthesized. Two sets of samples were prepared by a process of melting followed by high-temperature annealing at 873K for 7 days. One set of samples has nominal compositions InxCo4Sb12-x/3 (or equivalently (InVF)2x/3Co4Sb12-x/3(InSb)x/3 where InVF and InSb stand for void-filling, and Sb-substituting In atom, respectively) with x changing from 0.075 to 0.60. The other set is In0.3Co4-ySb11.9+y (equivalently (InVF)0.20Co4-ySb11.9+y(InSb)0.10) with y changing from -0.20 to 0.20. High-purity elements Co 62 (99.99%, shot), Sb (99.9999%, piece), and In (99.999%, shot) were used as raw materials. XRD, SEM, and EPMA were performed to identify phases and determine chemical compositions. :InSb :CoSb2 InxCo4Sb12-x/3 x=0.000 x=0.081 x=0.000 x=0.106 x=0.081 x=0.207 x=0.106 x=0.276 50 60 70 20  40 2θ (deg)  30 x=0.274 25 30   20  x=0.274 x=0.286     x=0.286 x=0.265   x=0.207 x=0.276 x=0.265  Intensity (arb. units)   InxCo4Sb12-x/3 (b) (a) 35 40 2θ (deg) (c) Figure 5.12 (a) XRD patterns of the In-containing skutterudites with nominal compositions InxCo4Sb12-x/3 (x=0, 0.075, 0.15, 0.225, 0.30, 0.375, 0.45, 0.60) and corresponding actual compositions x=0, 0.081, 0.106, 0.207, 0.276, 0.265, 0.286, 0.274. (b) Magnification of the XRD patterns. (c) Dependence of lattice parameter on actual indium content x. Filled symbols are from this work; open squares denote data from Grytsiv et al119. Figure 5.12(a, b) shows the X-ray diffraction results of annealed InxCo4Sb12-x/3 samples with actual compositions x, as determined from electron probe microanalysis. All major diffraction peaks are indexable to the skutterudite phase, as identified using JCPDS File No. 65-3144 for CoSb3. However, in samples with the nominal indium content x higher than 0.30 (lower three patterns), small amount of impurity phases (secondary phases other than skutterudite phase), especially InSb and CoSb2, are always observed, which indicates that those samples are outside the single-phase region. Figure 5.12(c) plots the dependence of the lattice parameter calculated from the XRD data on the actual indium content x. The dependence in the current work agrees with the data of Grytsiv et al119. 63 As x increases, the lattice parameter increases and becomes saturated when nominal x ≥ 0.30, strongly supporting that the solubility limit of indium is around 0.30. (a) (b) (c) (d) Figure 5.13 Electron probe microanalysis (EPMA) indium maps in samples with different nominal compositions. (a) In0.225Co4Sb11.925, (b) In0.30Co4Sb11.90, (c) In0.30Co4.2Sb11.7, and (d) In0.30Co3.8Sb12.1. Electron probe microanalysis (EPMA) measurements were carried out at ten different locations on the skutterudite phase region in samples and the average total indium contents are listed in Table 5.2. The EPMA maps show that there is a significant volume fraction of In-rich impurity phases in the In0.3Co4-ySb11.9+y samples with compositions that are off the Co/Sb ratio of the CCCD sample (InVF)0.20Co4Sb11.90(InSb)0.10 (see Figure 5.13c and 5.13d). However, in CCCD samples, the indium is nearly completely homogeneously distributed (see Figure 5.13a and 5.13b) with few secondary phases and the solubility limit of indium is estimated to be around x=0.27±0.01 according to EPMA data. When the indium amount exceeds 0.27 in CCCD samples, secondary phase such as liquid and CoSb2 is observed in SEM, which is consistent with XRD data presented above. The sensitivity of formation of secondary phases to the ratio of Co/Sb in samples strongly supports the hypothesis that impurity indium atoms form compound defects in CoSb3 by occupying both the void and Sb sites. Table 5.2 Nominal indium content and total indium content in the skutterudite phase estimated by EPMA for In-containing skutterudites with different compositions. Boldface indium contents correspond to the same skutterudite phase composition In0.27±0.01Co4Sb11.9 represented by a magenta circular symbol in Figure 5.14b. Samples In0.075Co4Sb11.975 In0.15Co4Sb11.95 Nominal In content 0.075 0.15 EPMA In content 0.081±0.004 0.106±0.016 64 In0.225Co4Sb11.925 In0.30Co4Sb11.90 In0.375Co4Sb11.875 In0.45Co4Sb11.85 In0.60Co4Sb11.80 In0.30Co3.8Sb12.1 In0.30Co3.9Sb12 In0.30Co4.1Sb11.8 In0.30Co4.2Sb11.7 In0.20Co4Sb11.8 In0.20Co4Sb12 0.225 0.30 0.375 0.45 0.60 0.30 0.30 0.30 0.30 0.20 0.20 0.207±0.013 0.276±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.265±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.286±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.274±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.086±0.005 0.148±0.011 0.287±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.261±𝟎𝟎. 𝟎𝟎𝟎𝟎𝟎𝟎 0.084±0.016 0.150±0.020 In id Liquid +C b3 oS +C uid Liquid+CoSb2+CoSb3 Liq oSb+ C u Liq oSb 2 Liq uid +C oS b Liquid+CoSb2 Liq uid +C oS b+ Co Liq uid +C o (a) Liquid+CoSb3+Sb Co CoSb CoSb2 CoSb3 In0.05Co4Sb11.975In0.025 In0.081Co4Sb11.973 In0.10Co4Sb11.95In0.05 In0.106Co4Sb11.96467 In0.15Co4Sb11.925In0.075 In0.207Co4Sb11.931 In0.20Co4Sb11.9In0.10 In0.276Co4Sb11.908 In0.25Co4Sb11.875In0.125 In0.265Co4Sb11.91167 In0.30Co4Sb11.85In0.15 In0.286Co4Sb11.90467 In0.3Co3.8Sb12.1 In0.086Co4Sb11.97133 In0.3Co3.9Sb12 In0.148Co4Sb11.95067 In0.3Co4.1Sb11.8 In0.287Co4Sb11.90433 In0.3Co4.2Sb11.7 In0.261Co4Sb11.913 In0.2Co4Sb11.8 In0.084Co4Sb11.972 In0.2Co4Sb12 In0.15Co4Sb11.95 In 0.23 (b) 0.05 0.24 0.26 0.27 Sb 3 +Co oSb 2 id+C Liqu 0.25 0.04 0.03 Liquid+CoSb3 0.02 Co Sb 2 +C oS b 0.01 Liquid+CoSb3+Sb 3 CoSb3+Sb 0.28 Co 0.72 0.00 0.73 0.74 0.75 0.76 0.77 Sb Sb Figure 5.14 Proposed phase diagram for In-containing skutterudites at 873K. (a) Full diagram with related binary phases and approximate regions of solubility indicated by red regions. (b) Phase regions enlarged near CoSb3. Because of the stability of indium in dual sites (CCCD compositions) the region of indium solubility is extended as a red line in the direction of (InVF)2x/3Co4Sb12-x/3(InSb)x/3 up to x=0.27. The solid purple line indicates the void filling in the direction (InVF)xCo4Sb12 up to x=0.27. The magenta circular symbol represents the composition of skutterudite phase In0.27Co4Sb11.9 obtained when the nominal composition is in the three-phase region (liquid, CoSb2 and In-CoSb3). The nominal sample compositions are shown as open symbols. Solid symbols represent the 65 composition of the skutterudite majority phase using the experimental EPMA value for indium content which is below 0.27 for all samples, and positioned on the red CCCD line. The nominal (open symbols) and experimental compositions (solid symbols) are connected with a dotted line. The samples with nominal CCCD compositions when x<0.30 appear to be within the solubility limit, the closed symbols of which are shifted away from the red solubility line according to the observed carrier concentration as explained later. The proposed phase diagram of the In-containing skutterudites at 873K is shown in Figure 5.14a, and is based on general knowledge of the binary phase diagrams. The related binary phases of In, Co, and Sb are shown on the axes including the known regions of solid or liquid solubility for the binary compounds and a rough estimate of the solubility of the ternary compounds. The exact regions of solid solubility of CoSb2 and CoSb3 are not known but they are enlarged for clarity in Figure 5.14a. The white regions of Figure 5.14a designate regions that are expected to contain two or three phases in equilibrium at 873K. The region near CoSb3 is enlarged in Figure 5.14b. Our experimental data suggest that the region of indium solubility extends toward the CCCD samples, (InVF)2x/3Co4Sb12-x/3(InSb)x/3, with an estimated maximum x value around 0.27. When x exceeds 0.27, the CCCD samples contain an In-dual-sitedoped skutterudite phase with indium doping level around x=0.27 and impurity phases InSb and CoSb2. When the nominal composition is off the Co/Sb ratio of CCCD (InVF)2x/3Co4Sb12-x/3(InSb)x/3 samples, significant impurity phases form from x= 0.30 samples. When the nominal composition is slightly Sb-rich (In0.3Co4-ySb11.9+y, y>0), the nominal composition is in a two-phase region between In-CoSb3 and liquid In-Sb where both phases have variable indium content in this region. According to the phase rule, the degrees of freedom F=C-P+0=3-2=1 where C means the number of components and P means the number of phases, 0 indicates both temperature and pressure are fixed. Thus this two-phase region has an additional degree of freedom to describe the tie line between which phases with variable In content form. We observe that the tie line connecting the nominal composition and the skutterudite phase is quite steep, which leads to a much lower In content in the skutterudite phase with an In-rich liquid In-Sb impurity phase. It is reasonable that In prefers the high entropy of the liquid rather than the solid skutterudite phase. In contrast to the Sb rich side, when the nominal composition is slightly Co-rich (In0.3Co4-ySb11.9+y, y<0), it is in a three-phase region between three phases with fixed compositions, approximately liquid InSb, In0.27Co4Sb11.9 and CoSb2. The compositions in a three-phase region are fixed because of the phase rule, F=C-P+0=3-3=0. The composition of the liquid InSb (In0.53Sb0.47) is fixed near the composition of the InSb phase that is stable up to 798.7K. Because the phase boundary between two-phase region and three-phase region must be a straight line, the point of contact (magenta 66 circular symbol) between the skutterudite phase and the phase triangle must have relatively high indium content. Because of the phase rule all the compositions in this three-phase region will produce the same skutterudite composition with high actual indium content (In0.27Co4Sb11.9). This was confirmed by the EPMA measurements to be x=0.27±0.01 as shown in Table 5.2. The steep slope of the tie line on the Sb-rich side of the stable phase (two phase region: liquid and CoSb3) results in relatively lower In content in the skutterudite phase, when compared to the fixed composition In0.27Co4Sb11.9 obtained from samples with nominal compositions on the Co-rich side with sufficient indium content (three-phase region: liquid, CoSb2, and CoSb3). This indicates that being slightly Co-rich instead of Sb-rich is more favorable in order to achieve high indium content in the skutterudite phase. Moreover, it also sheds light on why there is a solubility difference when the skutterudite phase is in coexistence with different impurity phases119. As the phase width appears to be quite narrow, small variations of composition will lead to different two or three-phase equilibrium regions on the phase diagram which can explain why different attempts to produce the same material may in fact produce very different skutterudite materials with different filling fraction, and therefore different thermoelectric properties. In addition the Co-rich and Sb-rich sides of the single phase region (red line in figure 5.14b) will have different temperature dependent characteristics and could lead to different precipitation products as found in Zn-Sb120. For example, solid solubility is typically temperature dependent and can be used to form nano-scale coherent precipitates121, which may explain the observance of InSb in some samples51,122,123 but not others124127 . 5.3.4 Thermoelectric Properties of In-CoSb3 5.3.4.1 Electrical Transport The temperature dependence of the Seebeck coefficients and electrical conductivities of InxCo4Sb12x/3 (EPMA determined x=0.081, 0.106, 0.207, 0.276) is shown in Figure 5.15a and 5.15b, respectively. All the samples show negative Seebeck coefficients throughout the whole temperature range, indicating n-type semiconductor behavior. With increasing temperature, the magnitude of the Seebeck coefficient increases first and starts decreasing after reaching a peak, which is possibly induced by the bipolar effect. Meanwhile, the electrical conductivity (Figure 5.15b) decreases smoothly as temperature increases. As In content increases, the electrical conductivity increases and the magnitude of Seebeck coefficient decreases, which suggests that some of the indium acts as an n-type dopant to increase the carrier concentration. 67 (a) (b) Figure 5.15 Temperature dependence of (a) the thermopower S and (b) electrical conductivity σ for samples of InxCo4Sb12-x/3 with different indium impurity content. Intrinsic CoSb3 is a valence precise semiconductor where the doping effect of defects can be understood with Zintl chemistry128. Ga-containing CoSb3 skutterudites (GaVF)2x/3Co4Sb12-x/3(GaSb)x/3 form charge-compensated compound defects which also makes them valence precise semiconductors. An indium atom at the void filling site (InVF) shows an effective charge state +1 and contributes one extra electron to the doped system. Indium atoms at the Sb-substitutional site accept two electrons. Thus, (InVF)2x/3Co4Sb12-x/3(InSb)x/3 should also be a valence precise semiconductor as demonstrated in Figure 5.9. However, the relatively high electron concentrations (as measured by Hall effect) and their dependence in the In-containing CoSb3 samples compared to Ga-containing CoSb3 samples suggests that the indium defects are not entirely charge compensated, with some excess indium as electron donor defects most likely InVF (Figure 5.16a). Indeed Figure 5.11b predicts that one electron donor defects from InVF are low energy defects particularly at low Fermi levels. Detailed calculations of all defects are necessary for quantitative discussion129 and are described in detail elsewhere130. The mixed defect types (complex compound defects) may also lead to a slight change of the direction of the single-phase region (red line) in Figure 5.14b. For excess indium donor defects in the void filling site, the single-phase region would move toward the purple line (InxCo4Sb12). As shown in Figure 5.16b the S-n dependence in indium-containing complex compound defect samples follows the extrapolated curve of normal filled skutterudites11,12,22,53,131. Figure 5.16b further indicates that due to the In or Ga-rich secondary phases on the grain boundary (as observed in SEM), the GxCo4Sb12 (G=In, or Ga) samples have very low electron concentrations and large 68 thermopowers from low G content in the skutterudite phase113,132. That all the data fall on the same curve in Figure 5.16b also provides strong evidence that the band structure in In-containing skutterudites is virtually unchanged near the Fermi level because the same trend is observed in the normally filled as well as in the Ga-containing skutterudites. This is fully consistent with our band structure calculations shown in Figure 5.9 and discussions in Section 5.3.2. (a) (b) Figure 5.16 (a) Room temperature electron concentration (n) as a function of indium impurity fraction x in In-containing complex compound defect (CCD) skutterudites. The solid line represents the theoretical curve calculated by using n=2x/a3 where a is the lattice constant of CoSb3. (b) Room temperature S as a function of electron concentration for In-containing CCD skutterudites at 300 K. The x values are estimated by EPMA for CCD samples and CCCD samples from Qiu et al’s work16 while they are the nominal values for the single-type defect samples from references (Harnwunggmoung et al113 and Du et al132). The solid line represents the general trend for single filled-CoSb3 with those typical electropositive filler species taken from literature11,12,22,53,131. 5.3.4.2 Thermal Transport Figure 5.17 shows the temperature dependence of total thermal conductivity (κ) of the samples (InVF)2x/3Co4Sb12-x/3(InSb)x/3 with nominal charge-compensated compound defects. In the whole temperature range investigated, the total thermal conductivity for all samples are lower than 3.5 Wm-1K-1. As usual, κL is obtained by subtracting the electronic contribution from the total thermal conductivity using the Wiedemann-Franz law. We adopted a Lorentz number of 2.0×10-8 V2K-2 to simplify the estimation of electronic thermal conductivity, and this value is also consistent with the experimentally estimated value based on our previous work on skutterudites53. All In-containing samples present strongly reduced κL as compared to binary CoSb3 with κL ~ 10 Wm-1K-1 at 300 K. Compared with the observed dependence of κL on impurity filling fraction in the normal alkaline- or 69 alkaline-earth-filled CoSb3, κL reduction in the In-containing samples has a strong dependence on impurity content, as shown in Figure 5.17b. At In content as low as 0.106, κL could be reduced to 2.62 Wm-1K-1, which is even lower than that for GaxCo4Sb12-x/3 (x=0.15). Interestingly, the lattice thermal conductivity for the In-containing systems is close to the κL for the normal rare earth-filled CoSb3. (a) (b) Figure 5.17 (a) Temperature dependence of total thermal conductivity and (b) Room-temperature lattice thermal conductivity as a function of the indium impurity fraction in skutterudites. The κL trends for alkaline- (solid line), alkaline-earth- (dashed-dotted line), and rare-earth- (dashed line) fillings are shown for comparison. The black triangles are for the indium-containing skutterudites with complex compound defects in the present work. The blank triangles are for Ga-containing skutterudites with charge-compensated compound defects16. And the blank circles are for the group 13 element-doped skutterudites113,132 using the as-reported compositions, presumably assumed to form a single-type defect. The dramatic reduction of lattice thermal conductivity in samples with complex compound defects could perhaps be explained by the dual-site occupation of indium impurities and therefore a dualcharacter phonon scattering mechanism. The indium impurity at the Sb-substitutional site establishes a point defect with size and bonding (in addition to slight mass) mismatch from the host Sb atoms, leading to the scattering of high-frequency lattice phonons. The indium atoms at the void sites behave as a typical filler species and scatter long wavelength phonons via resonant scattering and avoided crossing mechanisms. Therefore, the effective reduction of κL implies that a broad spectrum of lattice phonons could be scattered in the In-containing skutterudites with a complex compound defect. The relatively high indium doping content, when counting indium impurities at both the void and Sb sites, is important as it controls the magnitude of the phonon scattering effect. 5.3.4.3 Thermoelectric Figure of Merit 70 Figure 5.18 shows the temperature dependence of dimensionless thermoelectric figure of merit (zT) which are consistent with prior reports133. The maximum zT value in InxCo4Sb12-x/3 samples reaches 1.2 (at 750 K) with indium content x =0.276. The complex compound defect containing InxCo4Sb12x/3 samples presented in this work are also stable and have good repeatability of thermoelectric properties during the measurements, as shown in Figure 5.19. Figure 5.18 Temperature dependence of thermoelectric figure of merit zT for In-containing skutterudites with complex compound defect (CCD). The x values are the same as the ones in Figure 5.15 and 5.17. Figure 5.19 Repeatability of thermoelectric properties of In-containing skutterudite with complex compound defect (CCD) and x=0.207. 5.3.4.4 Not fully charge-compensated compound defects 71 Compared with (GaVF)2x/3Co4Sb12-x/3(GaSb)x/3 samples, the compound defect samples of the indium system have relatively larger carrier concentrations, which may be due to the defects not being fully charge-compensated as discussed in Section 5.3.4.1. If we write the compositions in a way containing both single filling defects ((InVF)δ) and dual-site compound defects ((InVF)2(x-δ)/3 and (InSb)(x-δ)/3) , such as (InVF)(2x+δ)/3Co4Sb12-(x-δ)/3(InSb)(x-δ)/3, where δ can be calculated using the measured Hall carrier concentration nH and lattice constant a: nH = 2δ , then we can get the value of a3 filling fraction (2x+δ)/3 with x being the content of indium determined by EPMA, as shown in Table 5.3. Table 5.3. Not fully charge-compensated compound defect samples (InVF)(2x+δ)/3Co4Sb12-(x-δ)/3(InSb)(xδ)/3 Samples In0.075Co4Sb11.975 In0.15Co4Sb11.95 In0.225Co4Sb11.925 In0.30Co4Sb11.9 nH (1020 cm-3) 0.48 0.93 2.35 1.74 δ 0.013 0.026 0.065 0.048 x 0.081 0.106 0.207 0.276 (2x+δ)/3 0.058 0.079 0.160 0.200 If we plot the compositions (InVF)(2x+δ)/3Co4Sb12-(x-δ)/3(InSb)(x-δ)/3 obtained in Table 5.3 in the enlarged ternary phase diagram in Figure 5.14b, we can see that the maximal solubility line will slightly change its direction to somewhere between the direction of (InVF)2x/3Co4Sb12-x/3(InSb)x/3 – CoSb3 and the direction of (InVF)xCo4Sb12 – CoSb3, which corresponds to a certain thickness of the solubility region and a ratio of void-filling/Sb-substituting of indium larger than two (with complete charge compensation this ratio is two). 5.3.5 Discussion about In-CoSb3 Two sets of samples were prepared by a process of melting followed by long-term high-temperature annealing, (InVF)2x/3Co4Sb12-x/3(InSb)x/3 with x changing from 0.075 to 0.30 and In0.30Co4-ySb11.9+y with y changing from -0.20 to 0.20. XRD and EPMA analysis indicate that when x is less than 0.27, pure single phase skutterudites can be obtained, whereas if x is larger than 0.27 or the composition is off the direction of (InVF)2x/3Co4Sb12-x/3(InSb)x/3 then secondary phases start to form. The maximum indium solubility limit is experimentally confirmed to be around x=0.27, which is in agreement with computational prediction. Thermoelectric properties were measured and a maximal zT value of 1.2 was obtained at 750K when x = 0.27. The relatively larger carrier concentrations compared to Gacontaining skutterudites indicate the possibility of not fully charge-compensated compound defects, 72 meaning single filling defects may coexist with dual-site compound defects, which may slightly change the direction along which the solubility region is aligned. 5.4 Stable compositions and Vegard’s law In Section 5.2 we mentioned the concept of stable compositions, now we will discuss in details what stable compositions are, how they can be both misleading and powerful in guiding solubility design. We will take the Yb-Co-Sb system as an example for illustration. 5.4.1 Solubility debate of Yb Among various kinds of filled skutterudite materials, Yb-doped skutterudites have shown good thermoelectric properties due to the heavy mass and small radius of the Yb atom compared to other fillers, and have been selected by many automotive companies, such as General Motors, as the most promising thermoelectric material for waste heat recovery134. Nolas et al. first reported a zT value of 1.0 for Yb0.19Co4Sb12 at 600K135. Composites containing Yb-filled CoSb3 and well-distributed Yb2O3 particles were synthesized by Zhao et al. with zT of 1.3 for Yb0.25Co4Sb12/Yb2O3 at 850K109. Later zT of 1.3 for Yb0.285Co4Sb12.268 at 800K was reported by Li et al 136. Despite the good thermoelectric performance of Yb-doped skutterdites, there has been debate about the Yb solubility limit in skutterudite phase. Because of the size and chemical difference between Yb and Co or Sb atoms, substitution of Yb for Co or Sb is not likely. Instead, the Yb atoms fit nicely into the void filling site, and thus the solubility limit discussed here corresponds to the filling fraction limit (FFL). Theoretically, the filling fraction limit was calculated by Mei et al.137 to be 0.30 for Yb at 0K; Shi et al.138 studied the temperature dependence of filling fraction limits of fillers in CoSb3 skutterudites using a plane-wave density functional method. By combining the formation energy of inserting an impurity R into the intrinsic void of CoSb3 and that of secondary phases RSb2 and CoSb2, it is found that theoretically the filling fraction limit has a linear dependence on temperature, with x = 0.2 when R = Yb at 1000K which is close to the experimental value but the temperature dependence was not demonstrated experimentally. Dilley et al. synthesized YbxCo4Sb12 by induction heating samples at 1050℃ for 12min then annealing at 600℃ for 35h 139. Lattice constant values were obtained from powder X-ray diffraction data and plotted against nominal x, from which Yb solubility limit about x = 0.22 was extrapolated. Xia et al. synthesized YbxCo4Sb12 73 with nominal x up to 0.4 by melting samples at 1077℃ for 10 hours and subsequent annealing at 800℃ for 7 days and observed no impurity phases140. Addition of oxygen to the system results in less filling due to the formation of Yb oxides141. In addition to traditional preparation methods (melting and long-time annealing), non-equilibrium synthesis methods were also adopted and gave different solubility limits, such as x = 0.29 by melt-melt-spinning (MMS) method by Li et al.136 and x = 0.5 by ball milling by Yang et al 24. In this study, the equilibrium isothermal sections of the ternary phase diagram of Yb-Co-Sb system at various temperatures are proposed in the same way as for the Ga-Co-Sb system and In-Co-Sb system, which are based on knowledge of the related binary phase diagrams and the microstructure of the samples16,17,106-108. The filling fraction limit derived from the ternary phase diagram and especially its temperature dependence naturally explains the solubility debate and the confusion concerning the various reports of maximum solubility of Yb in the skutterudite phase. It also leads to a strategy to synthesize optimized, high zT material. 5.4.2 Skutterudite lattice expansion due to Yb filling Figure 5.20 (a) Dependence of lattice constant on the actual Yb content, x, in YbxCo4Sb12 for samples annealed at various temperatures in this study and samples in references109,142. (b) Dependence of the lattice constant on the nominal Yb content, x, in YbxCo4+ySb12+z for samples annealed at 873K in this study and samples in reference by Dilley et al139. The red dashed line shows the increase of lattice constant followed by a plateau when x is larger than 0.30 in Co-rich samples which indicates the maximum filling fraction limit at 873K (red point in Figure 5.4). The blue dashed line shows another plateau of lattice constant indicating another stable composition Yb0.13Co4Sb12 (blue point in Figure 5.4) in Sb-rich samples. 74 Lattice constants were derived from powder X-ray diffraction data and plotted against Yb actual content determined from EPMA, as shown in Figure 5.20(a). We can see that there is a good linear relationship between the actual Yb content and lattice constant of the skutterudite phase, which agrees with the literature data and is consistent with Yb going into the same site across the whole range (up to x=0.5), presumably the void site109,142. Lattice constants were also plotted against the nominal Yb content in order to compare with Dilley’s data139 in Figure 5.20(b). The two plateaus of lattice constants in this study indicate there are two stable skutterudite compositions at 873K (explained in Section 5.4.3 from Gibbs phase rule). The red dashed line shows the increase of lattice constant followed by a plateau when x is larger than 0.30 when the nominal composition is in a Corich three-phase region. Here the Yb content in the skutterudite phase reaches the maximum filling fraction limit at 873K (red point in Figure 5.4a). The blue dashed line shows another plateau of lattice constant when the nominal composition is in an Sb-rich three-phase region where the skutterudite phase has a stable composition with lower Yb content (Yb0.13Co4Sb12, blue point in Figure 5.4a). The lattice constants in this study (Figure 5.20(b)) are well within the same range from reference Dilley’s data139. Moreover, for the x =0.44 sample from Dilley’s data which has larger lattice constant than other samples, it has YbSb2 and CoSb2 impurities according to the XRD data, which agrees with our conclusion that the skutteridte phase with maximum filling fraction limit coexists with CoSb2 and YbSb2 and is thus in a Co-rich three-phase region. 5.4.3 Stable compositions in ternary phase diagram system (a) Yb T=973K 0.22 0.06 0.26 Co Sb 3 +C 2 bSb CoS b3 +Y oS 2 b+ Yb Sb 2 0.24 0.04 CoSb3 +YbSb2 + liquid Sb 0.02 CoSb3+CoSb2 CoSb3+ liquid Sb 0.28 Co 0.72 0.74 0.76 Sb 75 Lattice constant (a(A)) (b) 9.08 Stable composition 1 (Red, Co-rich) 9.07 9.06 9.05 9.04 Stable composition 2 (Blue, Sb-rich) -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 y in Yb0.5Co4-ySb12+y Figure 5.21 (a) Magnified region of the isothermal section at 973K near CoSb3 of the Yb-Co-Sb ternary phase diagram system with two stable skutterudite compositions (red and blue points) and solubility line (red line). Scattered points (empty magenta circles) indicate possible synthesis error near the stable composition (red point) in the three-phase region of CoSb3, CoSb2, and YbSb2. (b) Dependence of the lattice constant derived from X-ray Diffraction data on the Co/Sb ratio for nominal compositions with the same Yb content x = 0.5 marked as empty orange rectangles in (a). The analysis of all samples is consistent with the proposed phase diagram for Yb-doped skutterudites shown in Figure 5.21(a), which is based on the knowledge of the related binary phase diagrams106-108. Our experimental data suggest that the region of Yb solubility extends toward the void filling samples, (YbVF)xCo4Sb12, up to the maximum solubility point. For example, at 973K the estimated maximum x value is 0.44 as shown in Figure 5.21(a) (red point). When x exceeds 0.44, the sample Yb0.5Co4Sb12 (empty orange rectangle surrounded by empty magenta circles) contains a skutterudite phase with Yb doping level around x = 0.44 and impurity phases YbSb2 and CoSb2. When the nominal composition with x = 0.5 is off the Co/Sb ratio of void filling samples (YbVF)xCo4Sb12 (being either Co-rich or Sb-rich), more impurity phases form in the samples. When the nominal composition contains higher Yb content than its solubility limit, it is in a threephase region between YbxCo4Sb12, YbSb2, and CoSb2. According to the phase rule, there is F=CP+0=3-3=0 degree of freedom where C is the number of components, P the number of phases, and the 0 indicates both temperature and pressure are fixed. Thus the compositions of the three equilibrium phases in a three-phase region are fixed and all the nominal compositions in this threephase region (such as all the scattered empty magenta circles in Figure 5.21(a)) will produce the same skutterudite composition with the same actual Yb content (red point in Figure 5.21(a)). This is 76 indicated by a plateau of the lattice constant marked as stable composition 1 from Figure 5.21(b). At 973K it was also confirmed by the EPMA measurements to be x = 0.44± 0.01(Yb0.44Co4Sb12). When the nominal composition is slightly Sb-rich (for example Yb0.5Co3.9Sb12.1 at 973K), it moves into a two-phase region between YbxCo4Sb12 and YbSb2. Due to the phase rule, the degree of freedom F= C-P+0=3-2=1, which means that the compositions of phases at equilibrium in this twophase region are not fixed, but vary depending on the exact starting composition within this region. As a result, the Yb content in the skutterudite phase decreases as the nominal composition becomes more Sb rich, which is confirmed by the decrease of lattice constant in Figure 5.21(b). However, when the nominal composition is even richer in Sb (for example Yb0.5Co3.8Sb12.2 at 973K), the nominal composition moves into another three-phase region between YbxCo4Sb12, YbSb2 and liquid Sb. Thus the compositions of phases at equilibrium are fixed again and all the nominal compositions in this three-phase region will produce the same skutterudite composition with the same actual Yb content (Yb0.26Co4Sb12) albeit at a much lower Yb content (blue point in Figure 5.21(a)). This was confirmed by a lower plateau of lattice constant marked as stable composition 2 in Figure 5.21(b) and the EPMA measurements confirmed this composition to be x = 0.26±0.02. In the process of synthesizing samples, exact element control is not cost-practical and thus the influence of impurity elements from low-cost starting material will lead to variations in the Co/Sb ratio and Yb content as well, shown as scattered empty magenta circles in Figure 5.21(a). If these scattered circles spill out of the three-phase region and enter into the two-phase region of CoSb3 and YbSb2, there will be a sharp decrease of Yb content in the skutterudite phase, resulting in a sharp decrease in lattice constant as well, which is illustrated in Figure 5.21(b). In order to have precise control of the Yb content and thus the carrier concentration in the skutterudite phase, nominal compositions can be easily designed to remain in a three-phase region to make sure that the skutterudite phase has a stable composition. The existence of these stable compositions as well as their temperature dependence can thus be applied to facilitate the synthesis of optimized skutterudites for low-cost mass production. 77 5.4.4 Vegard’s law in ternary phase diagram system In materials science, Vegard's law is the empirical rule that describes the relationship between the lattice parameter of a solid solution of two constituents and those of the pure constituents. According to Vegard, the lattice parameter of a solid solution should be approximately equal to a linear combination of the two constituents’ lattice parameters at the same temperature given that both pure constituents have the same crystal structure 143: (Eq. 5.2) Here, is the lattice parameter of the solid solution, and are the lattice parameters of the pure constituents A and B, respectively, and x is the molar fraction of B in the solution. In our case, A is the unfilled CoSb3, whereas B is the fully filled YbCo4Sb12, which does not exist in reality due to the solubility limit of Yb. For partially filled YbxCo4Sb12, it can be regarded as: (Eq. 5.3) Thus we should be able to extract a linear relationship between the lattice parameter of partially filled YbxCo4Sb12 and the filling fraction x, as shown in Eq. 5.4. (Eq. 5.4) This equation is often applied to determine the filling fraction limit (FFL) of fillers. In an experimental perspective, FFL is determined when the lattice parameter deviates from the Vegard’s law as nominal filling amount x increases. Mostly this deviation is observed as a plateau131,144; however, sometimes it is also observed as a linear behavior with a different slope19,145,146. This method is valid for binary phase diagram systems, but not always true for ternary phase diagram systems. In a binary system, as shown in Figure 5.22 below, when increasing the amount x of B in A at a given temperature, the nominal composition experiences two phase regions, first a single-phase region of the solid solution , then a two-phase region of the solubility limit (purple point). In a single-phase region of of freedom F= C-P+0=2-1=1; the lattice parameter of and when the amount of x exceeds , due to Gibbs phase rule, the degree changes linearly with the addition of B (x). In the two-phase region, the degree of freedom F= C-P+0=2-2=0; the lattice parameter of phase at equilibrium is fixed due to the fixed amount of B in phase indicated by the purple point (solubility limit of B in A). Thus a linear relationship followed by a plateau will be observed between the lattice parameter of phase and nominal content x of B. 78 Figure 5.22 Illustration of typical eutectic phase diagram of binary A-B system. The green line represents the isothermal line. The orange section represents the solubility region of B in A. The purple dot represents the solubility limit of B in A at the selected temperature. Figure 5.23 Illustration of isothermal section of ternary In-Co-Sb phase diagram. The green triangle represents the isothermal section. The orange area represents the solubility region of R-CoSb3. The purple boundary represents the solubility limit of R in CoSb3 at the selected temperature. However, in a ternary system, e.g., R-Co-Sb, it is more complicated with an additional degree of freedom due to the thickness of the solubility region of RxCo4Sb12, as shown in Figure 5.23. In a ternary system, the isothermal section is a surface rather than a line in a binary system, so is the solubility region. The solubility limit also changes from a single value (represented by purple dot in Figure 5.22) in a binary system to a boundary represented by the purple contour of the orange solubility region in Figure 5.23. The experimentally determined solubility limit is thus largely dependent on the phase regions adjacent to skutterudite phase where the nominal compositions are, as illustrated in Figure 5.24. For the Vegard’s law to be applicable in a ternary system, the nominal compositions should be along the solubility direction, e.g., (InVF)2x/3Co4Sb12-x/3(InSb)x/3 – CoSb3, so it 79 can be considered as in a pseudo-binary system. If nominal compositions are off the solubility direction, e.g., (InVF)xCo4Sb12 – CoSb3 for In-Co-Sb system (purple solid line in Figure 5.14b) and (YbVF)xCo4Sb12.2 – Co4Sb12 for Yb-Co-Sb system (yellow dashed line in Figure 5.25a), then deviations from the Vegard’s law can result from nominal compositions entering either two-phase (2-p in Figure 5.24) or three-phase region (3-p’ in Figure 5.24) other than the one that will give the maximum solubility limit (xmax in 3-p, Figure 5.24). As a result, the solubility limit observed will be smaller than the maximum solubility limit, because its determination is misled by slope change or false plateau in the relationship between lattice parameter and nominal filler content x. Figure 5.24 Dependence of solubility limit on adjacent phase regions in a ternary phase diagram system. Triangles enclosed in yellow lines are three-phase regions (on the Co-rich side: 3-p; on the Sb-rich side: 3-p’), with x3-p and x3-p’ indicating respectively the solubility limit of R in equilibrium skutterudite phase when nominal compositions are in these phase regions. A two-phase region (2-p) is located right above the Sb-rich three-phase region. The solubility limit of R in equilibrium skutterudite phase when nominal compositions are in this two-phase region is represented by the purple dot marked at x2-p. The maximum solubility limit xmax is represented by the purple dot that is the furthest point along the elongated direction (solubility direction). Figure 5.25 serves as a good example to illustrate the inapplicability of Vegard’s law when the nominal compositions are off the solubility direction in a ternary Yb-Co-Sb system. It demonstrates that reaching a plateau of lattice constant could be because of the existence of stable compositions and thus does not necessarily imply reaching the solubility limit. 80 (a) Yb T=973K 0.22 0.06 Co Sb 3 +C CoS b3 +Y oS 2 bSb b+ 2 Yb Sb 2 0.24 0.26 0.04 CoSb3 +YbSb2 + liquid Sb 0.02 CoSb3+CoSb2 CoSb3+ liquid Sb 0.28 Co 0.72 0.74 Sb 0.76 CoSb3 Lattice constant (a(A)) (b) 9.07 9.06 Stable composition 2 (Blue, Sb-rich) 9.05 0.2 Stable composition 1 (Red, Co-rich) Easily mistaken as solubility limit 0.3 0.4 0.5 0.6 0.7 x in nominal YbxCo4Sb12.2 Figure 5.25 Samples with different nominal Yb content x in YbxCo4Sb12.2 (marked as empty orange rectangles) but the same Sb excess lead to a nonlinear dependence of lattice constant (b) due to the sample traversing different two- and three- phase regions of the phase diagram (a). Excess of Sb is often added when synthesizing CoSb3 skutterudites due to the high volatility of Sb. In samples with even a slight Sb excess (depicted in Figure 5.25(a)), increasing the nominal Yb content will lead to samples in the Sb-rich three-phase region of YbxCo4Sb12, YbSb2, and liquid Sb first, then samples in the two-phase region of YbxCo4Sb12 and YbSb2 before reaching samples in the Co-rich three-phase region of YbxCo4Sb12, YbSb2, and CoSb2. Thus the actual Yb content in Yb filled CoSb3 as measured by the lattice constant of the skutterudite phase will stop increasing and be constant for a period (first plateau in Figure 5.25(b), or blue dashed line in Figure 5.20(b)) while the nominal composition is in the Sb-rich three-phase region and the resulting skutterudite phase has stable composition (blue point in Figure 5.25(a)). This Yb content at which lattice constant stops increasing can be easily mistaken as the Yb solubility limit139. However, as seen from Figure 81 5.25(b), further Yb addition can change the Yb actual content in the skutterudite phase dramatically as the nominal composition moves away from the Sb-rich three-phase region and into the two-phase region, and consequently the lattice constant would resume the increasing trend. As the nominal composition moves into the Co-rich three-phase region, the actual Yb content reaches the maximum solubility (red point in Figure 5.25(a)) and thus the lattice constant reaches a plateau again (second plateau in Figure 5.25(b) or red dashed line plateau in Figure 5.20(b)). An overall investigation of the phase relations near the targeted phase is thus needed in order to make a conclusion about solubility limit because: observing a plateau in lattice constant is insufficient in determining the solubility limit in a ternary system, nor is observing impurity phases sufficient (the nominal composition might just be in a two- or three-phase region). 5.4.5 Discussion about Yb-CoSb3 The equilibrium isothermal sections of the ternary phase diagram of Yb-Co-Sb system at various temperatures are studied. XRD, SEM, and EPMA analysis indicates that there is a three-phase region on the Co-rich side of Yb-doped skutterudites (YbxCo4Sb12, CoSb2, and YbSb2) which results in a stable skutterudite composition YbxCo4Sb12 with maximum Yb content (any nominal composition in this three-phase region will produce a skutterudite phase with the same Yb content, for example x = 0.11±0.02 at 773K and x = 0.49±0.02 at 1073K), and another three-phase region (YbxCo4Sb12, YbSb2 and liquid Sb) on the Sb-rich side with a stable composition YbxCo4Sb12 with relatively lower Yb content (for example x = 0.13±0.01 at 873K and x = 0.34±0.01 at 1073K). The existence of these stable compositions can be misleading when applying Vegard’s law to interpret solubility limit in ternary systems while samples are off the solubility direction. Full phase diagram investigation is thus necessary to avoid such misinterpretation. At higher temperatures, the equilibrium Yb content in both stable compositions is also higher, which can be used to form nanoscale precipitates in the matrix skutterudite phase (discussed in Chapter 8). Skutterudite material with the optimized thermoelectric composition can be produced from a range of nominal compositions with appropriate annealing, which has great potential in enabling low-cost large-scale production. Figure of merit zT values greater than 1.1 were obtained at and above 800K in all samples targeting YbxCo4Sb12 with x = 0.3. The highest zT value reaches 1.3 at 850K in Yb0.30Co4Sb12. 82 5.5 Solubility design strategies So far we have demonstrated the possibility of soluble sites other than the void site for fillers (Section 5.3), e.g., Sb substitution on the CoSb3 matrix, as a result of which the solubility direction can be shifted away from common direction along (InVF)xCo4Sb12 – CoSb3 but towards the Co-rich side. The difference of solubility in binary and ternary phase diagram systems is also discussed (Section 5.4). It has to be noted that the dependence of observed solubility on the phase regions adjacent to the skutterudite phase as well as the temperature dependence of solubility might be the reason for many solubility debate. Three solubility design strategies could then be proposed regarding high performance thermoelectric materials design. First of all, stay on the right solubility direction. Secondly, choose the phase region that will result in the maximum filler solubility limit. Thirdly, vary the annealing temperature. Among the strategies mentioned, the large temperature dependence of solubility especially expands our choice of optimized compositions for thermoelectric application, which can be helpful in changing thermoelectric materials with dopant solubility barrier into promising ones instead. We will take Ce-doped CoSb3 as an example for illustration in this section. 5.5.1 Solubility barrier of Ce in Ce-CoSb3 The intermediate temperature range (400-600℃) at which filled skutterudites have optimum thermoelectric performance makes this material a perfect candidate for automotive industry applications. Due to the heavy mass and small radius of the Yb atom compared to other fillers, Yb-CoSb3 skutterudites have excellent thermoelectric properties (zT = 1.3 at 850K for Yb0.3Co4Sb12)50 and have been selected by many automotive companies, such as General Motors, Ford, and BMW, as the most promising thermoelectric material for waste heat recovery147. However, the deficiency of the rare earth element Yb (abundance in earth’s crust 3ppm) could potentially be a problem in large-scale commercialization. Unlike Yb, Ce is the most earth abundant element of the rare earths (abundance in earth’s crust 68ppm)148. In fact, there are huge resources of cerium because it is more abundant than other commonly known elements like lead (14ppm), tin (2ppm), silver (0.07ppm), tellurium (0.005ppm), and gold (0.001ppm). The price of cerium is also more competitive compared to that of ytterbium. For large quantities, the difference is a factor of 10 (Ce: $5000/ton, Yb: $50000/ton 149 ), placing Ce in the category of 83 other commonly used elements such as Ni. With its larger earth abundance and fairly low cost, use of Ce would help the commercialization of n-type skutterudites for automotive industry only if the thermoelectric properties of Ce-doped skutterudites can match those of Yb-doped skutterudites. Interestingly, single filled Ce-CoSb3 skutterudites have been considered as unsuitable thermoelectric materials and there has yet been little report of high temperature thermoelectric properties of pure Ce-CoSb3 skutterudites. Ce has been considered to be a good auxiliary filler in multiple filled skutterudites, such as in (In, Ce)-CoSb3 skutterudites (zT of 1.43 for In0.2Ce0.15Co4Sb12 at 800K)51 or (Ba, Ce)-CoSb3 skutterudites (zT of 1.26 for Ba0.18Ce0.05Co4Sb12.02 at 850 K)150. Ce has not typically been considered as single filler. This is most likely due to its reported small filling fraction limit x = 0.09 in CexCo4Sb1219, which would prevent Ce-containing skutterudites to achieve the optimum carrier concentration for n-type skutterudites.53 However, the filling fraction limit of Ce was concluded19 based on the observance of a plateau in the lattice constant against nominal Ce content, as well as observation of impurity phases. These phenomena, as we have discussed in Section 5.4, are not sufficient proof to conclude about solubility limit in a ternary system. Thus the filling fraction limit of Ce is worth re-investigation by solubility design using an equilibrium phase diagram approach. This approach introduces both the annealing temperature and nominal composition as variables in determining the filling fraction limit, which leads to the possibility of enhancing filling fraction of fillers thermodynamically rather than consider it as a fixed single value. Using this phase diagram approach, a temperature dependent solubility like the Yb in Yb-CoSb3 sheds light on optimizing Ce-CoSb3.50 To increase the filling fraction limit of Ce in CoSb3 skutterudites, charge-compensational doping such as Fe or Mn-doping has been largely applied which can dope the material from n-type to ptype.28,151-154 In this study, single-filling Ce-CoSb3 skutterudites without charge-compensational doping are prepared under equilibrium conditions and their thermoelectric properties are measured up to 850K. In other works, our focus is on the intrinsic solubility limit of Ce in CoSb3 system without any compensational effect. 84 5.5.2 Optimum doping of Ce-CoSb3 skutterudites 1E22 nH (cm-3) 1E21 Na K Ba La Yb BaYb BaYbLa Ce Ce - this work 1E20 1E19 1E18 0.0 0.2 0.4 0.6 0.8 1.0 Number of electrons per primitive cell Figure 5.26 Hall carrier concentration vs number of electrons per primitive cell Co4Sb12.1113,19,50,53,155-157 The number of electrons per primitive cell is calculated as the sum of qi*xi for partially filled (R1)x1(R2)x2…(Rn)xnCo4Sb12, where qi is the effective charge state and xi is the filling fraction of the ith filler R in multiple-filled skutterudites. The solid line represents the theoretical curve calculated by using n = 2x/a3 where x is the total number of electrons per primitive cell and a is the lattice constant of CoSb3. The red double arrow represents the estimated high power factor region (see discussions below) and the red dashed line represents the number of electrons per primitive cell limited by the previously thought Ce filling fraction limit19. Unfilled CoSb3 is an intrinsic semiconductor with carrier concentration less than 1018 cm-3. Rare earth filler atoms in the void sites, are nearly perfect n-type donors. The n-type carrier concentration is essentially determined by the number of filling atoms and the ionic charge state. In CoSb3 a Yb filler has a valence charge +2 while Ce is +3 which is a lower oxidation state than often observed in oxides but common for Sb compounds.158 Figure 5.26 demonstrates the trend of the carrier concentration versus the number of electrons per primitive cell, for single, double or triple-filled skutterudites. To achieve high power factor of n-type skutterudites 0.4 to 0.6 electrons per Co4Sb12 unit is required, as indicated by the red double arrow.67 The carrier concentration of Ce-CoSb3 samples in the previous literature study19 is limited by the reported low filling fraction limit (x = 0.09 in CexCo4Sb12 at 973K), consequently these samples are far from the high power factor region. In order to optimize the thermoelectric properties of Ce single filled CoSb3 skutterudites, a higher filling fraction of Ce is needed. As we shall show later, the filling fraction limit of Ce can be as high as x = 0.20 at 1123K through solubility design, which is not only more than twice the reported value x = 0.09 but now allows for the optimization of Ce- 85 CoSb3. After optimization, the four samples in this work fall in the high power factor region (nominal Ce content x = 0.17, 0.18, 0.20, 0.21, and actual Ce content measured on hot pressed pellets are x = 0.14, 0.14, 0.16, 0.15, respectively) and all of them have zT values reaching above 1.0 above 750K (see Figure 5.29e). 5.5.3 Ultra-high FFL of Ce in Ce-CoSb3 skutterudites Figure 5.27 Filling fraction limit (FFL) of Ce in Ce-CoSb3 skutterudites. (a) Skutterudite lattice expansion due to Ce filling. (b) Dependence of FFL on annealing temperature. Red dashed line is a guide for the eye. Error bars represent the standard deviation of filling fraction limit determined from EPMA measurements (c) Dependence of FFL on nominal composition with annealing temperature 973K. The phase region enclosed by red lines is a Co-rich three-phase region, which is related to the maximum solubility limit. The phase region enclosed by blue lines is an Sb-rich three-phase region, which is related to a lower solubility limit detected by Morelli et al 19. The lattice expansion due to filling is an easy and effective way of characterizing the amount of fillers actually going into the void site in the skutterudite cell. Here lattice constants were derived from powder X-ray diffraction data and the actual Ce content was determined from EPMA. The 86 lattice constant shows a simple linear dependence on the actual Ce content x in CexCo4Sb12, which agrees with the literature data and is consistent with Ce going into the same site across the whole range (up to x=0.20), presumably the void site. Samples annealed at higher temperatures show both higher Ce content by EPMA as well as higher lattice constants. The partial filling of Ce in the void site increases the entropy due to the increase of disorder of the material, and thus it can be expected that the equilibrium solubility should increase measurably with temperature. While this is known theoretically138, the filling fraction limit (FFL) is often considered to be a temperature independent quantity; to date, there has been no report on the temperature dependence of filling fraction limit of Ce, let alone using this to design filler solubility in skutterudites. Here we show that the Ce filling fraction limit (determined from red point in Figure 5.27c) increases significantly with temperature (see Figure 5.27b). It increases to about 0.197±0.007 at 1123K and is more than twice the value reported from Morelli et al19 (x = 0.09). This ultra-high filling fraction limit is verified using Atom Probe Tomography (APT) and is discussed later. This presents an opportunity for optimizing thermoelectric properties of CeCoSb3 skutterudites. The filling fraction limit x = 0.120 at 973K is also 25% higher than the reported literature value x = 0.09, which can be explained due to the presence of separate 3-phase regions in the Ce-Co-Sb system with different Ce solubilities in CexCo4Sb12. Figure 5.27c shows a magnified region of an isothermal section at 973K near CoSb3 of the CeCo-Sb ternary phase diagram. There are three two-phase regions and three three-phase regions near CoSb3. The stable compositions (marked as red and blue points in Figure 5.27c) were confirmed by the EPMA measurements to be x=0.120± 0.005 for the red stable composition (Ce0.120Co4Sb12) and x=0.086± 0.007 for the blue stable composition (Ce0.086Co4Sb12) at 973K. When the nominal composition is slightly Co rich (in addition to sufficient Ce, e.g. Ce0.5Co4.2Sb11.8, red empty triangle) the sample is in the three-phase region between CoSb2, CeCoSb3159 and CexCo4Sb12 with x = 0.120± 0.005 (red point). Also if the nominal composition is slightly Sb rich (e.g. Ce0.5Co3.9Sb12.1, yellow empty triangle) the sample is in the three-phase region between CeSb2, CeCoSb3, and CexCo4Sb12 with, coincidentally, the same x = 0.120± 0.005 (red point) as found in the cobalt rich three-phase region. In both cases the equilibrium Ce content in skutterudite phase will reach the same filling fraction limit x = 0.120± 0.005 at 973K (red 87 point) as indicated by tie lines (red dashed lines). Note that these three-phase regions (one with CeSb2, CeCoSb3 and CexCo4Sb12; the other with CoSb2, CeCoSb3 and CexCo4Sb12) are different from the previous phase diagram study of Ce-Co-Sb system at 400℃.159 No similar ternary phase such as CeCoSb3 is found in the Yb-Co-Sb or In-Co-Sb system, but the Ga-Co-Sb system has a ternary phase (Co3Ga2Sb4) as well.160 However, when the nominal composition is very antimony rich, such as used by Morelli19 the samples are in another three-phase region between CexCo4Sb12, CeSb2, and liquid Sb (for example Ce0.2Co3.9Sb12.1 and Ce0.15Co3.9Sb12.1, marked as green and blue empty triangles respectively). Here the Ce content in the skutterudite phase remains constant with x=0.087± 0.007(blue point) as indicated by tie lines (green and blue dashed lines). Note that two nominal compositions (Ce0.10Co4Sb12.2 and Ce0.15Co4Sb12.2) from Morelli’s data19 fall into this three-phase region of CexCo4Sb12, CeSb2, and liquid Sb and one (Ce0.20Co4Sb12.2) falls into the two-phase region of CexCo4Sb12 and CeSb2. The observation of CeSb2 impurity phase in those samples confirms they are in a different field from the Co-rich three-phase region, which gives the higher Ce content. The blue stable composition point coincides with the reported filling fraction limit x = 0.09, which is about 25% less of the filling fraction limit reported in this study for the same annealing temperature. As a result the first solubility design strategy in this study for enhancing the filling fraction limit of Ce in CoSb3 is: focus on the Co-rich three-phase region rather than the lower Sbrich three-phase region. From the analysis of the temperature dependent filling fraction limit (Figure 5.27b) we can also predict that the Ce content in the skutterudite phase can be controlled through the annealing temperature when the nominal composition is in one of the three-phase regions, which has proven to be a practical strategy to optimize thermoelectric performances in Yb-CoSb3 system (Section 5.2) 50. This solubility design strategy is also applied in the study here for enhancing the filling fraction limit of Ce in CoSb3: increase the annealing temperature. Atomic Probe Tomographic (APT) measurements on CexCo4Sb12 skutterudite was also performed to confirm the ultra-high filling fraction of x= 0.20. APT is particularly effective at chemically and structurally characterizing materials in 3D on the nanometer length scale. It provides accurate information about atomic concentrations and can identify even the smallest nanoparticles.161 Small nanoparticles of Yb2O3 and InSb were observed in some Yb and In 88 containing skutterudites, respectively.51,109,162 Such small particles might go unnoticed in the SEM and would contribute to the EPMA signal. Here APT was utilized to verify the Ce content and homogeneity in the CexCo4Sb12 majority phase for sample annealed at 1123K with nominal composition Ce0.5Co4Sb12. According to EPMA, the Ce content in the majority skutterudite phase should be x = 0.20 or 1.2 at%. Figure 5.28a contains a 3D reconstruction of the skutterudite phase at and around a grain boundary. As can be seen in Figure 5.28b, atomic concentrations were consistent and homogeneous away from the grain boundary. No nano-precipitates of Ce-rich phases were observed. Atomic concentrations were averaged over a 30 nm diameter cylinder, from 20 to 60 nm on the scale bar in Figure 5.28a, and the Ce content was measured to be 1.4 ± 0.1 at%. This agrees with the values obtained by EPMA (x=0.20 corresponding to Ce content of 1.2 at%) and supports our conclusion that such a high filling fraction of Ce is indeed uniformly distributed in the skutterudite phase. A uniform accumulation of Ce at the grain boundary was also observed, accompanied by a reduction in Co. Such atomic segregation is frequently observed at grain boundaries.161 Figure 5.28 APT analysis of the most heavily doped sample Ce0.2Co4Sb12. a) 3D reconstruction of micro-tip containing a grain boundary. Ce atoms are displayed in red; Sb and Co atoms omitted for clarity. b) Concentration profile across the grain boundary and in the grain. The black dashed lines show values measured by EPMA, and the error bars represent the statistical error, �𝑐𝑐(1 − 𝑐𝑐)/𝑛𝑛, where 𝑐𝑐 is the concentration and 𝑛𝑛 is the number of atoms detected in each data point. 89 5.5.4 Thermoelectric properties of optimized Ce-CoSb3 skutterudite (b) -100 Seebeck (uV K-1) -120 -140 -160 -180 -200 300 400 500 600 700 800 900 T(K) (c) 3.8 3.6 K(W m-1 K-1) 3.4 3.2 3.0 2.8 2.6 2.4 300 400 500 600 700 800 900 T(K) Figure 5.29 Transport properties of Ce- and Yb-doped skutterudites. The temperature dependence of : (a) electrical resistivity, (b) Seebeck coefficient, (c) thermal conductivity, and (e) thermoelectric figure of merit (zT) are plotted in the temperature range of 300K to 850K. In figure (d) the lattice thermal conductivity with a Lorentz number of 2.0×10-8 V2K-2 is plotted against the filling fraction for various types of fillers. x denotes the nominal doping level of fillers. The thermoelectric properties can be optimized when the filling fraction x of filler R (with valence q) in RxCo4Sb12 satisfies qx = 0.4~0.6.53,67 From Figure 5.27 we can see that when nominal compositions are in the Ce-rich three-phase regions with an annealing temperature at 90 1073K and 1123K, the Ce actual content in skutterudite phase is about x = 0.17 and 0.20 respectively. With Ce valence state being +3, qx for these materials would be in the optimum region. Thus a set of samples with nominal compositions CexCo4Sb12 (x = 0.17, 0.18, 0.20, and 0.21) were prepared for thermoelectric evaluation, of which the first two samples were annealed at 1073K and the latter two were annealed at 1123K. The temperature dependence of transport properties of these samples is shown in Figure 5.29. Moreover, the transport data of two Ybdoped skutterudites with doping level x = 0.20 and 0.30 are also plotted for the reason of comparison.49,50 With Yb valence state being +2, qx for these two samples are right on the boundary of the optimum region. All the samples show negative Seebeck coefficients throughout the whole temperature range, indicating the expected n-type semiconductor behavior. With increasing temperature, both the electrical resistivity and the magnitude of the Seebeck coefficient increase, which is typical behavior for heavily doped semiconductors. It has to be noted that for the Ce-doped samples, the values of electrical resistivity, seebeck coefficient and thermal conductivity all fall between the values of the two Yb-doped samples, which is in agreement with Figure 5.26. κL is obtained by subtracting the electronic contribution from the total thermal conductivity using the WiedemannFranz law. We adopted a Lorentz number of 2.0×10-8 V2K-2 to be consistent with previous work on skutterudites, and the lattice thermal conductivity calculated accordingly has less than 5% difference from the value calculated with the Lorentz number determined from a single parabolic band model.53,163 All Ce-doped samples in this work show strongly reduced κL as compared to binary CoSb3 with κL ~ 10 Wm-1K-1 at 300 K. As apparent from Figure 5.29d, Ce is as effective in reducing lattice thermal conductivity as Yb, with both much more effective than alkali and alkaline earth element fillers. Figure 5.29e shows the continuously increasing zT indicative of the higher doping level than Qiu’s sample156. All Ce-CoSb3 samples in this work have zT values higher than 1.0 at and above 750K, with the maximum zT value in Ce-doped skutterudite samples reaching 1.3 with nominal Ce content x = 0.17 and x = 0.18 at 850K (with composition Ce0.14Co4Sb12 for both samples determined from EPMA), which is 30% higher than that of previously reported literature value for Ce single filled skutterudites (zT = 1.0 for Ce0.11Co4Sb12 at 850K)156. These zT are similar to those for single filled skutterudites13,24,50,135, suggesting that Ce is a suitable earth abundant and low cost replacement for other types of fillers such as Yb, which makes the Ce single-filled 91 skutterudites a more promising candidate for large-scale commercialization. The success of solubility design in Ce-CoSb3 system is a good example that shows its potential in the optimization of thermoelectric properties in any other ternary systems with doping barriers, which allows wide applicability of this strategy. 5.5.5 Discussion about Ce-CoSb3 Due to the low cost of Ce as a single filler, Ce-CoSb3 skutterudites could have potential for scaleup commercialization in fields such as the automotive industry, if their thermoelectric performance can be optimized. Ce single-filled skutterudites in previous literature studies have been under-doped and fall out of the high power factor region due to the assumed low filling fraction limit of Ce (x=0.09). Here by conducting an equilibrium phase diagram study, we show that the filling fraction limit has a large temperature dependence and the value can reach as high as x=0.20 at 1123K. This clear enhancement gives a solubility value more than twice the one reported previously and thus allows optimizing Ce-CoSb3 skutterudites. The ultra-high Ce content is confirmed by APT measurements. This surprisingly high filling fraction limit at high annealing temperatures breaks the solubility barrier and enables the Ce-CoSb3 skutterudites to be optimized with zT of 1.3 for Ce0.14Co4Sb12 at 850K, which is among the highest reported zT values for singlefilled skutterudites and makes it an excellent substitute for Yb-doped skutterudites for waste heat recovery applications. 5.6 Stability of optimized compositions Figure 5.30 Stability of YbxCo4Sb12 skutterudite with optimum dopant composition x = 0.3. During operation (time t > 0), dopant precipitation is predicted in region 2 where the skutterudite phase is neither thermodynamically stable (as in region 3) nor kinetically stable (as in region 1). 92 The temperature dependent solubility limit predicts the precipitation of impurity phases at intermediate temperature, as shown schematically in Figure 5.30. Such temperature dependent solubility and precipitation has been observed in other ternary systems such as Na-doped PbTe as well164. In heavily doped SiGe, the precipitation and re-dissolution of dopants during hightemperature transport measurements was also reported165. The rate and size of the precipitates will depend on the temperature and time. As the optimum carrier concentration corresponds to approximately x = 0.3, Yb0.3Co4Sb12 material above 873K will be stable indefinitely. The Yb0.3Co4Sb12 material at low temperatures will be thermodynamically unstable but kinetically stable because precipitation will require the Yb diffusion, which will be low at low temperatures. In the intermediate temperature range, some precipitation of impurity phases is expected (see more discussion in Chapter 8). Reduction of the Yb content will also reduce the carrier concentration, which will raise the electrical resistivity of the material. Despite the formation of impurity phases, the maximum performance of the material may not be dramatically reduced because a lower carrier concentration at low temperature may actually increase the zT at that temperature4. Such a temperature dependent doping effect was found in Ag doped n-type PbTe-Ag2Te nanocomposites to improve the thermoelectric performance166. The large difference in solubility limit at different temperatures could also be used to form nano-scale precipitates with controlled size, the results of which are also presented in Chapter 8121,167. 5.7 Conclusion In ternary phase diagram systems, a few concepts were cleared. Observing impurity phases are not sufficient in determining solubility limit, because the nominal composition might be just in a twoor three-phase region. This also applies for reaching a plateau of lattice constant against nominal filling content x with the plateau due to the existence of stable compositions. Three solubility design strategies are proposed based on the study of various ternary systems (In or Ga-Co-Sb in Section 5.3, Yb-Co-Sb in Section 5.4, Ce-Co-Sb in Section 5.5): firstly, stay on the correct solubility direction; secondly, aim at the right phase region; thirdly, vary annealing temperature. These strategies can be generalized to any ternary system with solubility barrier of doping elements. 93 Chapter 6 Defect study in intrinsic CoSb3 6.1 Summary The defect chemistry in intrinsic CoSb3, to our most interest, whether intrinsic CoSb3 is p-type or ntype, has been reported with large discrepancy. These reports are based on different types of materials (single-crystal or polycrystalline), which are synthesized with various methods (high temperature melting and annealing or ball milling, etc.) from different types of starting materials (in powder or ingot form, element purity). In this chapter, with a set of standard synthesis procedures, we performed a detailed Co-Sb binary phase diagram study combined with carrier concentration measurements in search of fundamental defect chemistry in CoSb3 compounds. A literature review is presented in section 6.2 with previous reports. Experimental results and discussion are presented in section 6.3. Section 6.4 is the conclusion and future work. 6.2 Literature study 6.2.1 Bonding chemistry of CoSb3 Skutterudite is a type of Zintl compound in which the total number of valences from both cations and anions is zero. According to Zintl chemistry128, the formal valence of a cation and an anion can be calculated following the general valence rule168. As given in Eq. 6.1, available valence electrons of the cation element and is the number of is the number of cation-cation two- electron bonds and nonbonding lone pair electrons. Similarly, the anion formal valence is given by Eq. 6.2, where is the number of anion-anion two-electron bonds. (Eq. 6.1) (Eq. 6.2) Binary CoSb3 compounds are diamagnetic semiconductors, so there should be no unpaired spin in the bonding scheme. For each Sb atom with electronic configuration 5s25p3, it is bonded to its four nearest neighbors: two Sb atoms and two Co atoms. The Sb-Sb bonding has Sb-Sb-Sb bonding angle of 90°, which forms Sb4 rings shown in Figure 6.2c. The remaining three valence electrons 94 participate in the Co-Sb bonding. According to Eq. 6.2, Sb has -1 state (5+2-8 = -1). The distance between the Co atoms is too large for there to by any significant Co-Co bonding. Co atom with electronic configuration 4s23d7 is thus only bonded to Sb atoms in an octahedral coordination (N = 6). For each Co atom, 3 electrons occupy the d2sp3 hybrid orbitals that are the essence of the Co-Sb bonding. The Co atom is in a 3+ state (9-6 = 3) with the remaining 6 electrons paired up leading to a low-spin d6 state. The electronegativity of Sb (2.05) is close to that of Co (1.88), so these bonds have little ionic character. The bonding scheme is shown in Figure 6.1. The structure of [CoSb6] octahedral and Sb4 ring is shown in Figure 6.2. Figure 6.1 Bonding illustration in CoSb3 from Dudkin’s bonding model169. Figure 6.2. (a) Structure of skutterudite CoSb3 (Co: red; Sb: blue), (b) [CoSb6] octahedron, (c) Sb4 ring. © IOP Publishing. Reproduced with permission. All rights reserved.31 6.2.2 Synthesis condition and defect type Various attempts were made by previous researchers to synthesize either polycrystalline (meltannealing method, mechanical alloying (ball milling)31,32, high pressure assisted synthesis34, meltmelt-spinning method33 etc.) or single crystalline skutterudites (gradient freeze technique5,6, fluxassisted growth35, chemical vapor deposition36 etc.). 95 The influence of synthesis methods and conditions on the defect type in CoSb3 compounds is undeniable. Single crystal CoSb3 can only be grown from Sb-rich melt (90 ~ 97 at% Sb) due to the phase diagram limitation38, which is usually p-type5,6,35,36. However for polycrystalline CoSb3, inconsistent results are reported among various groups. Both p- and n-type intrinsic CoSb3 are reported with even the same synthesis method. Confusing though these data are, some results are still enlightening and worthy of attention. Sharp et al33 found that for samples synthesized in a melt-annealing method followed by cold/hot pressing, samples that are cold pressed show p-type while the ones hot pressed with the same starting powder composition show n-type. Deficiency of Sb due to hot pressing was speculated; however, there was no measured stoichiometry data to confirm. Similarly, Liu et al32 reported the sintering temperature can play a large role in the defects. For samples synthesized by mechanical alloying and SPS, when the sintering temperature is at 300℃, the samples show p-type, whereas when the sintering temperature is increased to 600℃, the samples show n-type. The EDS investigation shows a change in the Co:Sb ratio from 1:3.0 to 1:2.7 with increasing SPS temperature from 300℃ to 600℃, which was accounted for the p-n change. Liu et al31 then studied the effect of Sb compensation on the point defects of CoSb3. It was found when excess Sb was increased from zero to 8 at%, the Seebeck coefficient at room temperature changes from −430 to 40μVK−1for samples synthesized with a fixed MA-SPS synthesis route. Sb-vacancy was concluded as the reason for the n-type behavior in Sb-deficient sample. Nakagawa et al170 tried to study the effect of stoichiometry (with nominal Sb:Co = 2.6~3.4) on thermoelectric properties of CoSb3 using a melt-annealing method. Transport properties are measured on slices of annealed ingot to avoid influence of high temperature sintering process. However no clear trend was obtained. The presence of large quantities of secondary phases (CoSb2 for Co-rich samples, Sb for Sb-rich samples) complicates the investigation. Recently, a study171 on the effect of CoSb2 and Sb on the electronic properties of CoSb3 was revealed by a multi‐ band Hall effect analysis. While CoSb2 increases the charge carrier density, the influence of the highly mobile charge carriers introduced by elemental Sb is worth noting as well. By applying a high pressure (1.5GPa-3.5GPa) and high temperature (950K) assisted synthesis starting with stoichiometric Co and Sb powders, Su et al34 reported that all samples synthesized are 96 p-type, more interestingly, it is found that the carrier concentration can be a factor of 10 less when the pressure increases. This stress-dependent defect concentration is attributed to anti-site defects, whose formation energy may vary with the synthesis pressure. However, there was no experimental data to confirm this hypothesis. Excess Sb was observed in the lower pressure sample, which disappears for higher-pressure samples. This can be explained by the high volatility of Sb. From the literature review we can conclude that a few factors are important in the experimental defect study of intrinsic CoSb3: 1) The form and purity of starting material. Powder raw elements tend to introduce O2 up to 1000ppm into the system. Fe or Ni impurities, which are easy to be found in Co, can contribute to p- or n-type doping as well. 2) If we compare the most common synthesis methods, traditional melt-annealing and mechanical alloying (ball milling), the latter is more prone to introduce impurities such as oxygen or Fe (if the vial is made of steel). 3) Consolidation conditions are important. The high sintering temperature can alter the stoichiometry to be Sb-deficient in CoSb3, resulting an n-type behavior. The increasing sintering pressure leads to a decreasing p-type carrier concentration, the reason to which remains unclear. 4) When considering the influence of off-stoichiometry (either Co-rich or Sb-rich) on defects, often large amount of secondary phases is introduced as well. The influence of these secondary phases is not fully clear yet. It is thus required to synthesis of Co-rich/Sb-rich samples with little secondary phases. 6.3 Results and discussion 6.3.1 Phase width of CoSb3 With the discussion in Section 6.2.2, we orient our research as follows: 1) Powders are avoided for all starting materials. High-purity elements with slug or shot forms are chosen and purchased from Alfa Aesar (Co (99.95%, slug), Sb (99.9999%, shot)). 2) Traditional melt-annealing method rather than ball milling is used to avoid contamination. Prior to hot pressing, ingots obtained from annealing are hand grinded instead of ball milled for the same reason. 3) The sintering method was chosen to be hot pressing with a fixed set of parameters (700℃, 60MPa, and 1 hour) to minimize 97 variance due to sintering conditions. 4) Investigated compositions are chosen to have a stoichiometry Co:Sb= 4 : (12+x) close to 1:3 (x = -0.06~0.17) in order to minimize quantities of secondary phases. With all these synthesis conditions determined, we hope to synthesize CoSb3 in both Co-rich and Sb-rich conditions with the least amount of secondary phase, which shall allow us to study the defect type in CoSb3 without influences from these impurities. Binary CoSb3 is often considered to be a line compound up to 874℃. However, we challenge this by trying to measure the temperature dependent phase width of CoSb3. 21 Samples with various compositions are synthesized in a similar manner except with different annealing temperatures (See Chapter 2, Section 2.2 for more detail). The synthesized compositions as well as their annealing temperatures are shown in Table. 6.1, where × denotes the indicated composition is synthesized and annealed at the corresponding temperature. Table 6.1 Compositions synthesized with corresponding annealing temperatures. Annealing Co4Sb11.94 Co4Sb11.97 Co4Sb12 × × × × × × × × × × × × × × Co4Sb12.03 Co4Sb12.13 × × × × Co4Sb12.15 Co4Sb12.17 T(℃) 850 800 700 650 600 500 × × × The phase purity of annealed ingots was checked with SEM and XRD. The stoichiometry of the CoSb3 phase was confirmed by EPMA. SEM photos and XRD patterns for samples annealed at 500℃, 700℃, and 850℃ (on ingots or hot pressed samples) were chosen to show in Figures 6.36.8. The secondary phases (mostly determined from SEM unless otherwise specified from XRD) and EPMA compositions (on annealed ingots) are summarized and shown in Tables 6.2-6.4. After hot pressing, the phases were again checked by XRD; however, unfortunately, the stoichiometry of CoSb3 phase on hot pressed samples was not double-checked. As we shall discuss in Section 6.3.2 later, this might cause problems in analyzing the correct defect type. For all the samples, the EPMA compositions for CoSb3 phase confirm a stoichiometry of Co:Sb close to 1:3 after annealing and before hot pressing. The phase width can be determined from secondary phase identification for each annealing temperature. 98 Table 6.2 Phase purity and composition analysis of CoSb3 samples annealed at 500℃. Nominal EPMA Secondary Composition composition phase Co4Sb11.97 Co4Sb12 Co4Sb12.17 12.05±0.06 12.05±0.08 12.06±0.06 CoSb2 Sb Sb Figure 6.3 XRD patterns of CoSb3 samples annealed at 500℃ (after hot pressing). (a) (b) (c) Figure 6.4 SEM photos of a) Co-rich sample Co4Sb11.97; b) on stoichiometry sample Co4Sb12; c) Sbrich sample Co4Sb12.17 after annealing at 500℃. As we can see from Table 6.2, the phase region of CoSb3 at 500℃ should be between the compositions Co4Sb11.97 and Co4Sb12. The Sb secondary phase for nominal composition Co4Sb12 (as can be seen in Figure 6.4b) disappears after hot pressing in Figure 6.3 (no Sb peak on the green curve), which is consistent with previous findings34. Table 6.3 Phase purity and composition analysis of CoSb3 samples annealed at 700℃. Nominal EPMA Secondary Composition composition phase 99 Co4Sb11.94 Co4Sb11.97 Co4Sb12 Co4Sb12.03 Co4Sb12.15 12.04±0.07 12.03±0.04 12.05±0.04 12.06±0.09 12.00±0.09 CoSb2 (XRD) None None Sb Sb Figure 6.5 XRD patterns of CoSb3 samples annealed at 700℃ (ingots or samples after hot pressing). (a) (b) (d) (c) (e) Figure 6.6 SEM photos of a) Co-rich sample Co4Sb11.94; b) Co-rich sample Co4Sb11.97; c) on stoichiometry sample Co4Sb12; d) Sb-rich sample Co4Sb12.03; e) Sb-rich sample Co4Sb12.15 after annealing at 700℃. As we can see from Table 6.3, the phase region of CoSb3 at 700℃ should be between the compositions Co4Sb11.94 and Co4Sb12.03. The CoSb2 phase for nominal composition Co4Sb11.94 is invisible in SEM but is detected by XRD. The Sb secondary phase for nominal composition Co4Sb12.03 (as can be seen in SEM) disappears after hot pressing in XRD. 100 Table 6.4 Phase purity and composition analysis of CoSb3 samples annealed at 850℃. Nominal EPMA Secondary Composition composition phase Co4Sb11.94 Co4Sb11.97 Co4Sb12.03 12.05±0.04 11.95±0.08 12.04±0.08 CoSb2 (XRD) CoSb2 (XRD) Sb Figure 6.7 XRD patterns of CoSb3 samples annealed at 850℃ (ingots after annealing). (a) (b) (c) Figure 6.8 SEM photos of a) Co-rich sample Co4Sb11.94; b) Co-rich sample Co4Sb11.97; c) Sb-rich sample Co4Sb12.03 after annealing at 850℃. As we can see from Table 6.4, the phase region of CoSb3 at 850℃ should be between the compositions Co4Sb11.97 and Co4Sb12.03. If we combine all the information collected at each temperature, a rough estimate of the temperature dependent phase width of CoSb3 can be depicted in Figure 6.9. As we can see, the phase width (where green circles are in) is very narrow throughout the temperature range investigated, with (∆𝑥𝑥)𝑚𝑚𝑚𝑚𝑚𝑚 < 0.1 for Co4Sb12+x. Due to the accuracy of EPMA measurements, it is hard to distinguish 101 experimentally if the skutteurdite phase is Sb rich or Co rich. Appearance of secondary phases can serve as a guide, but this is true only on annealed samples because of possible modification on the stoichiometry due to high temperature sintering. This work is useful in guiding researchers to synthesize off-stoichiometry CoSb3 with least secondary phases. Figure 6.9 Temperature dependent phase width of CoSb3. Pink downward triangle symbols represent samples in a two-phase region of CoSb2 and CoSb3. Green circle symbols represent samples in a single-phase region of pure CoSb3. Purple upward triangle symbols represent samples in a two-phase region of CoSb3 and Sb. The orange solid line represents the peritectic temperature for CoSb3 874℃. 6.3.2 Defect type in intrinsic CoSb3 In this section, we will see that even though we designed the research with much care, there is still some influence from both the starting elements and synthesis condition on the defect type, which adds complexity to the interpretation of the results. Table 6.5 shows the room temperature transport data measured on samples hot pressed at 700℃ with a pressure of 60MPa and pressing time of 1h. Table 6.5 Room temperature transport data measured on Co4Sbx (x = 11.94 ~ 12.17) samples hot pressed at 700℃. Annealing T (℃) 500 600 Sample nH Resistivity Mobility Seebeck Density -3 Composition (cm ) (mOhm/cm) (cm²/Vs) (uV/K) (g/cm3) Co4Sb11.97 Co4Sb12 Co4Sb12.17 Co4Sb11.97 -8.6E+17 -3.7E+17 -1.1E+18 -4.5E+17 181 94 116 125 -41 -182 -49 -111 -300 -398 -285 -337 7.41 7.37 7.41 7.45 102 650 700 800 850 Co4Sb12 Co4Sb12.17 Co4Sb11.97 Co4Sb12 Co4Sb11.94 Co4Sb11.97 Co4Sb12 Co4Sb12.03 Co4Sb12.15 Co4Sb11.94 Co4Sb11.97 Co4Sb12 Co4Sb12.03 Co4Sb12.13 Co4Sb11.94 Co4Sb11.97 Co4Sb12.03 -4.3E+17 -4.6E+17 -5.2E+17 -4.3E+17 -6.0E+17 -3.9E+17 -3.8E+17 -3.7E+17 -4.2E+17 -5.9E+17 -3.8E+17 -3.8E+17 -5.1E+17 -5.2E+17 -6.2E+17 -5.3E+17 -4.3E+17 160 95 79 76 76 94 124 94 78 73 89 99 98 77 79 80 94 -91 -141 -151 -194 -136 -171 -135 -179 -192 -145 -185 -168 -125 -157 -128 -148 -154 -338 -357 -381 -386 -380 -398 -380 -422 -380 -361 -394 -396 -423 -357 -352 -376 -394 7.40 7.43 7.28 7.31 7.28 7.41 7.46 7.39 7.44 7.32 7.40 7.28 7.32 7.44 7.14 7.26 7.33 From the bonding chemistry of CoSb3, one would expect that CoSb3 with Co-rich stoichiometry would lead to n-type whereas Sb-rich stoichiometry would lead to p-type material. Yet Table 6.5 shows that all the samples synthesized show n-type behavior. This is quite confusing, and interesting. Several possible reasons are speculated to explain these data: 1. Impurity doping in starting Co element. Even though we chose a slug form of cobalt with 99.95% purity, the Co elemental analysis reveals a higher Ni impurity level compared to Fe (as shown in Figure 6.10), which might account for the unusual constant n-type doping. 103 Figure 6.10 Co elemental analysis showing higher Ni impurity level than Fe. To test our hypothesis, four compositions are synthesized with annealing temperature 700℃ and stoichiometry Co4Sb12.03Mx (M = Fe or Ni, x = 0.002 and 0.004, which correspond to 500ppm and 1000ppm holes/electrons, respectively). The Co:Sb ratio and annealing temperature were chosen such that according to the phase diagram in Figure 6.9, the resulting samples should contain little Sb secondary phase and stay slightly Sb-rich (p-type expected according to bonding chemistry). XRD measurements on annealed ingot slices confirmed that all four samples contain little amount of Sb secondary phase. If the influence of the impurity (Fe, Ni) is small then all four samples should be ptype. However this is not the case. The room temperature transport results show p-type for M = Fe, yet n-type for M = Ni on samples sliced from annealed ingot, as shown in Table 6.6. This shows that though as small as 500pm the Ni impurity level is, it could still change completely the defect type. If we plot the carrier concentration versus the Ni content, a perfect linear relationship confirms this point; see Figure 6.11. Table 6.6 Room temperature transport data measured on Fe, Ni doped CoSb3 samples annealed and hot pressed at 700℃. Data for sample Co4Sb12.03 annealed at 700℃ is also shown for comparison. Sample Impurity nH Resistivity Mobility Seebeck -3 Composition level (cm ) (mOhm/cm) (cm²/Vs) (uV/K) Co4Sb12.03Fe0.002 Co4Sb12.03Fe0.004 Co4Sb12.03 Co4Sb12.03Ni0.002 476 h 976 h 24 e 524 e 7.1E+19 6.7E+18 -3.7E+17 -1.1E+18 1 1 94 41 87 679 -179 -145 32 62 -422 -369 104 Co4Sb12.03Ni0.004 1024 e -1.7E+18 28 -133 -324 Figure 6.11 Linear relationship between carrier concentration in CoSb3 and Ni impurity content. 2) Influence of hot pressing on stoichiometry. This should be further confirmed by performing EPMA analysis on hot pressed samples, to doublecheck the stoichiometry. If there is indeed a large amount of Sb deficiency due to the high temperature sintering, similar to the one (Co:Sb=1:2.72) reported previously by Liu et al32, it should well manifest itself on EPMA analysis. However, if the effect of hot pressing on altering the stoichiometry is actually not as significant, due to the accuracy limit of EPMA, it would hard to determine if the material is either Co-rich or Sb-rich. In any case the transport properties should be better performed on annealed ingots to eliminate the possible influence from hot pressing. 6.4 Conclusion and future work Experimental study of the defect type was performed on samples synthesized with the guidance of phase diagram study. The phase width of CoSb3 as well as its temperature dependence was discovered, with (∆𝑥𝑥)𝑚𝑚𝑚𝑚𝑚𝑚 < 0.1 for Co4Sb12+x between 500℃ and 850℃. The excess Ni impurity doping as well as the possible influence of hot pressing might alter the intrinsic defect type of CoSb3. Starting elements with higher purity are needed. Transport properties are better performed on annealed ingots without hot pressing. The possible effect of oxygen on doping is worth studying as well. More research is required to reach a solid conclusion about the intrinsic defect type in both Co and Sb-rich CoSb3. 105 Chapter7 Phase diagram study of Ce-Fe-Co-Sb quaternary system 7.1 Summary So far this thesis is mainly about n-type CoSb3 based skutterudites. To enable thermoelectric application, both n- and p- legs are required, preferably from the same kind of material so that the difference in thermal expansion coefficient could be minimized to avoid detrimental strain and mechanical failure. This chapter is about the phase diagram study of the Ce-Co-Fe-Sb quaternary system, which is motivated by the need of high zT p-type skutterudite materials. From Chapter 5 we already discussed the potential of Ce-CoSb3 as good n-type thermoelectric material. If we substitute Co by Fe, the holes introduced by Fe (1 hole/Fe atom) will dope the material p-type. By tuning the ratio between n-type dopant Ce and p-type dopant Fe, we can change the CeyCoxFe4-xSb12 material to be either n-type or p-type, as discussed in Section 7.2. The thermoelectric properties of these p-type materials, however, are often not promising. In search of optimized compositions for p-type CeyCoxFe4-xSb12 material, we first investigate the phase space of skutterudite phase in this Ce-Co-Fe-Sb quaternary system. Experimental determination of isothermal sections of Ce-Fe-Sb and Co-Fe-Sb ternary systems at 700℃ is presented in Section 7.3 and 7.4 respectively. Together with previously determined Ce-Co-Sb isothermal section at 700℃, these determined ternary phase diagram isothermal sections contribute to the partial determination of phase space of CeyCoxFe4-xSb12 skutterudites as presented in Section 7.5. With a fixed Co:Fe ratio, n-p change of semiconducting type can be realized by varying the amount of Ce alone (Section 7.6). Though the thermoelectric performance of these materials needs to be further optimized, it can be potentially useful in making both n- and p- thermoelectric legs with similar compositions. Section 7.7 is the conclusion and a look at future work. 7.2 Charge-compensating defects In CeyCo4-x FexSb12 materials, Ce acts as a donor with each Ce atom giving three electrons to make the material n-type whereas Fe acts as an acceptor with each Fe atom contributing one hole to make it p-type. This charge-compensating effect can increase the Ce solubility limit, which was reported 106 previously by many researchers19,154,172. Previous results are combined and shown in Figure 7.1. A clear increasing trend of maximum Ce solubility could be observed with the increasing amount of Co substitution by Fe. Figure 7.1 Dependence of maximum Ce solubility with Fe amount in CeyCo4-xFexSb12 skutterudite materials19,154,172. The carrier concentrations for each sample are also extracted and shown in Figure 7.2. Figure 7.2 Dependence of carrier concentrations in CeyCo4-xFexSb12 skutterudites on the amount of Co substitution19,154,172. The horizontal orange solid line represents carrier concentration of zero. The vertical black dashed line corresponds to x(Co) = 3.25. The carrier concentrations of two samples (green triangle symbols) lined on the dashed line of x(Co) = 3.25 are marked by orange arrows. The upper sample with nominal composition Ce0.20Co3.25Fe0.75Sb12 is p-type with carrier concentration 2.7E19 cm-3, while the lower sample with nominal composition Ce0.30Co3.25Fe0.75Sb12 is n-type with carrier concentration -4.8E20 cm-3. As we can see, closer to the Co-rich side, materials are mostly n-type, whereas closer to the Fe-rich side, materials are mostly p-type. Note for a fixed cobalt content x(Co) = 3.25, both n and p type materials are observed with different Ce doping levels. This allows a good opportunity to study 107 skutterudite materials with similar composition yet different semiconducting type (as shown in Section 7.6); however, in the literature study19 high temperature thermoelectric properties were not measured. For each fixed Co content, the largest amount of carriers that can be generated by Ce filling is 3*y(Ce) electrons with y being the solubility limit of Ce for that specific x(Co) (blue dashed line) . The green line represents the amount of holes (x) for Co substitution by Fe in CeyCo4-xFexSb12. Then mathematically the material can be determined to be either n-type or p-type by comparing these two quantities (blue and green lines) for a fixed Co content. To the Fe rich side (left region to the orange vertical line), the green line is higher than the blue dashed line, so materials show p-type behavior. To the Co-rich side (right region to the orange vertical line), the blue dashed line is higher than the green line, so materials show n-type behavior. The n-p crossover happens at about x(Co) = 2.75~3.25. Only with x(Co) > crossover value can both n- and p-type materials with the same Co:Fe ratio be obtained, which is in agreement with the experiment result19. However this value is largely dependent on the experimentally determined Ce filling solubility limit. As we have learnt from the case study of Ce-CoSb3 system, the solubility limit can often be misinterpreted. So in the following sections (Section 7.3-7.5) we set out to re-investigate the phase region of skutterudites in Ce-Fe-CoSb system before trying to optimize thermoelectric properties. Figure 7.3 Crossover from p-type to n-type in CeyCoxFe4-xSb12 materials19,154,172 when 𝑥𝑥 ≈ 2.75 − 3.25. 108 7.3 Ce-Fe-Sb isothermal section at 700℃ The isothermal section of Ce-Fe-Sb ternary system is determined following the methodology described in details in Section 5.2. To decrease the workload, only a section (CeSb2-FeSb2-Sb) was considered assuming the phase relations around CoSb3 would not involve phases beyond this region. A sketch was first drawn based on binary phase diagram informations173-175 and previous literature study154, shown in. Note the large amount of excess Sb in the nominal composition Ce1.05Fe4Sb12.2 used in literature should set it in an Sb-rich phase region. Eight nominal compositions were proposed to a first batch synthesis to determine phase regions, as listed in Table 7.1. They are also shown illustratively on the proposed isothermal section on Figure 7.4, from which we can see that the nominal compositions of our synthesis are dispersed around the previously determined skutterudite composition (Ce0.98Fe4Sb12) and scattered in different estimated phase regions. To identify three-phase regions, at least two nominal compositions are selected in each region to confirm the equilibrium stable compositions. Table 7.1 Selected nominal compositions for investigation of Ce-Fe-Sb isothermal section at 700℃. Bold FeSb2 phase was confirmed to be FeSb phase after characterization of the samples #1-8. Sample name Ce-Fe-Sb #1 Ce-Fe-Sb #2 Ce-Fe-Sb #3 Ce-Fe-Sb #4 Ce-Fe-Sb #5 Ce-Fe-Sb #6 Ce-Fe-Sb #7 Ce-Fe-Sb #8 Nominal composition Ce0.9Fe3.57Sb12.43 Ce0.9Fe3.9Sb12.1 Ce0.9Fe4Sb12 Ce1.05Fe4Sb12 Ce1.05Fe4.1Sb11.9 Ce1.05Fe4.2Sb11.8 Ce1.05Fe3.9Sb12.1 Ce1.05Fe3.8Sb12.2 Estimated phase region CexFe4Sb12 + liquid Sb CexFe4Sb12 + FeSb2 + liquid Sb CexFe4Sb12 + FeSb2 + liquid Sb CexFe4Sb12 + FeSb2 + CeSb2 CexFe4Sb12 + FeSb2 + CeSb2 CexFe4Sb12 + FeSb2 + CeSb2 CexFe4Sb12 + CeSb2 + liquid Sb CexFe4Sb12 + CeSb2 + liquid Sb Ce-Fe-Sb #9 Ce-Fe-Sb #10 Ce-Fe-Sb #11 Ce-Fe-Sb #12 Ce0.7Fe4.65Sb11.35 Ce0.7Fe4.5Sb11.5 Ce0.7Fe4.1Sb11.9 Ce0.7Fe3.9Sb12.1 CexFe4Sb12 + FeSb2 + FeSb CexFe4Sb12 + FeSb2 + FeSb CexFe4Sb12 + FeSb2 + liquid Sb CexFe4Sb12 + FeSb2 + liquid Sb 109 Figure 7.4 First sketch of Ce-Fe-Sb ternary phase diagram isothermal section at 700℃. Blue empty circle represents the nominal composition Ce1.05Fe4Sb12.2 in literature154. The corresponding skutterudite phase compositon Ce0.98Fe4Sb12 is shown as a blue dot. Proposed nominal compositions for the first batch 8 samples were shown as empty black circles. After phase purity and chemical composition characterization of this batch of samples, we found that the experimental results confirm on a large part our initial estimation except for two inconsistencies. The first is the appearance of FeSb phase that was out of our scope in the first place. For samples #4-6, the equilibrium phases are CeSb2 and FeSb, not the CeSb2 and FeSb2 phases that we had estimated (Table 7.1). This is shown in the SEM photo (Figure 7.5c). XRD analysis of these samples (#4-6) is in agreement with SEM/EDS analysis. This suggests that the phase diagram should take into account the FeSb phase as well, and thus we enlarge the scope of our investigation to the (CeSb-FeSb-Sb) region instead. With this modification, we made a second batch of the sample (Ce-Fe-Sb #9-12). Their nominal compositions are added to Table 7.1 and shown in the enlarged isothermal section in Figure 7.6. The second inconsistency is that the determined skutterudite phase composition Ce0.96±0.02Fe4Sb12 in this study is slightly different from the previous literature value Ce0.98Fe4Sb12. This compositional change is not huge yet its influence is subtle by slightly shifting the phase boundaries downward towards the Co-Sb base line. It could well explain why the estimated phase region of sample #1 and #2 are off. As shown in Figure 7.5a, sample #1 with nominal composition Ce0.9Fe3.57Sb12.43 is 110 actually in a three-phase region (CexFe4Sb12 + CeSb2 + liquid Sb) rather than the estimated twophase region (CexFe4Sb12 + liquid Sb). Figure 7.5b shows sample #2 with nominal composition Ce0.9Fe3.9Sb12.1 actually in a two-phase region (CexFe4Sb12 + liquid Sb) rather than the estimated three-phase region (CexFe4Sb12 + FeSb2 + liquid Sb). Figure 7.5 SEM photos of samples with nominal composition as a) #1 Ce0.9Fe3.57Sb12.43, b) #2 Ce0.9Fe3.9Sb12.1 and c) #5 Ce1.05Co4.1Sb11.9. The bright white impurity phase (marked by orange arrow) is determined with EDS to be CeSb2, the light white phase is Sb and the dark grey phase is FeSb. Figure 7.6 Second batch of Ce-Fe-Sb samples (#9-12) synthesized based on binary phase diagrams and results from the first batch of samples. Note in this batch that the nominal content of Ce is much lower (x=0.7) compared to the previous synthesis. This is to make sure that samples could fall in the phase region (the one that involve FeSb phase especially) we would like to investigate and have large quantities of impurity phases to facilitate phase identification as well. 111 The results on these two batches of samples are self-consistent. #9 and #10 shows impurity phases of FeSb and FeSb2 perfectly matching the phase diagram relations. With all the information combined we are ready to determine the isothermal section (shown in Figure 7.7) with the Ce solubility region in skutterudites being 0.917±0.008 < x < 0.961±0.018 in CexFe4Sb12. Note the lower solubility limt for Ce filling, meaning a minimum amount of Ce is required to stabilize the material. This is for the first time that the solubility of Ce is reported to have a width in Ce-filled Febased CexFe4Sb12 skuterudites, which is quite different from the single value x = 0.98 reported earlier154 and request some thought in synthesis. Ce 0.20 Ce0.961Fe4Sb12 Ce0.917Fe4Sb12 0.15 Ce0.7Fe3.9Sb12.1 Ce0.7Fe4.1Sb11.9 Ce0.7Fe4.5Sb11.5 Ce0.7Fe4.65Sb11.35 0.25 b2 eS +C id qu 0.35 Fe 0.65 Ce 12 Sb 4 + Ce1.05Fe3.9Sb12.1 + li Ce Fe x Sb b 12 Ce1.05Fe3.8Sb12.2 2 0.30 6 0.9 Ce0.9Fe3.9Sb12.1 Sb 1 Fe 4 1 .96 Sb 2 Fe b+ S Fe Ce0.9Fe3.57Sb12.43 1 Ce 0 2 0.10 .96 Ce 0 Sb 1 Fe 4 b+ eS F + Sb 2 Ce + liqui d Sb Ce1.05Fe4Sb12 0.05 Ce1.05Fe4.1Sb11.9 Ce1.05Fe4.2Sb11.8 Ce0.917Fe4Sb12 + FeSb + liquid Sb 2 S Fe 4 1 liquid 0.70 0.75 0.80 0.00 Sb Figure 7.7 Isothermal section of Ce-Fe-Sb ternary system at 700℃. 7.4 Co-Fe-Sb isothermal section at 700℃ Due to serious deficiency of electrons, FeSb3 phase does not exist in thermodynamic equilibrium form. However, these is indeed some solubility of Fe in CoSb3, which makes it p-type material. This has been reported with x = 0.8 in FexCo4-xSb12 176. As we can see from the phase diagram in Figure 7.8 below, the phase region changes when the amount of substitutional Fe increases. Upon x = 0.8, the three-phase region (liquid + (Co,Fe)Sb3 + Sb) forms for x between (0.8, 2.48); for x between 112 (2.48, 3.38), two-phase region of (liquid + (Co,Fe)Sb2) forms; and lastly, another three-phase region (liquid + (Co,Fe)Sb2 + Sb) forms for x between (3.38, 4). The three-phase regions with the co-existence of liquid phase and Sb phase are quite confusing because at 700℃ Sb is also in liquid form and there seems to be no miscibility gap between the Sb and liquid phase. These results are important for later comparison to our own data. Figure 7.8 Reported phase diagram of CoSb3-“FeSb3” system177. D02 means the (Co,Fe)Sb3 phase. C18 means the (Co,Fe)Sb2 phase. Isothermal section of Co-Fe-Sb ternary system is proposed based on binary phase diagram information106,174 and literature results177,178. Proposed isothermal section of Co-Fe-Sb ternary system is shown in Figure 7.9. Nominal compositions for sample synthesis are listed in Table 7.2. Table 7.2 Selected nominal compositions for investigation of Co-Fe-Sb isothermal section at 700℃. Bold phases denote that besides these phases, unexpected phases also show up for these samples (see detailed discussion in text). Samples Ce-Co-Fe-Sb #5-1 and #6-1 are proposed to investigate if a three-phase region will form on the Co-rich side. Samples (Co, Fe)Sb2 #1-4 are proposed to study the possible phase segregation between CoSb2/FeSb2 and the stable composition Co0.33Fe0.67Sb2 . Sample name Co-Fe-Sb #1 Co-Fe-Sb #2 Co-Fe-Sb #3 Co-Fe-Sb #4 Co-Fe-Sb #5 Co-Fe-Sb #6 Co-Fe-Sb #7 Nominal composition Co3.4Fe0.3Sb12.6 Co4.2Fe0.3Sb11.8 Co3.25Fe0.75Sb12 Co2Fe2Sb12 Co1.5Fe1.5Sb12 Fe4Sb12 Co6Fe6Sb12 Estimated phase region (Co, Fe)Sb3 + liquid Sb (Co, Fe)Sb3 + (Co,Fe)Sb2 (Co, Fe)Sb3 (Co, Fe)Sb3 + (Co, Fe)Sb2 + liquid Sb (Co, Fe)Sb3 + (Co, Fe)Sb2 + liquid Sb FeSb2 + liquid Sb (Co, Fe)Sb2 Ce-Co-Fe-Sb #5-1 Ce0Co2.5Fe1.5Sb8.5 (Co, Fe)Sb3 + (Co,Fe)Sb2 113 Ce-Co-Fe-Sb #6-1 Ce0Co3Fe1Sb9 (Co, Fe)Sb3 + (Co,Fe)Sb2 (Co, Fe)Sb2 #1 (Co, Fe)Sb2 #2 (Co, Fe)Sb2 #3 (Co, Fe)Sb2 #4 Co0.30Fe0.70Sb2 Co0.33Fe0.67Sb2 Co0.36Fe0.64Sb2 Co0.40Fe0.60Sb2 FeSb2 + Co0.33Fe0.67Sb2 Co0.33Fe0.67Sb2 CoSb2 + Co0.33Fe0.67Sb2 CoSb2 + Co0.33Fe0.67Sb2 Figure 7.9 Proposed isothermal section of Co-Fe-Sb ternary system at 700℃. The solubility region of skutterudite phase Co4-xFexSb12 was represented by a red solid line with solubility limit of Fe x = 0.80. 177 After phase purity and chemical composition characterization of these samples, we found a few interesting discoveries. First of all, the solubility limit is x = 0.68 in Co4-xFexSb12, which is 15% less than the previously reported value. Secondly, stable compositions for equilibrium phases Co4-xFexSb12 and CoxFe1-xSb2 are observed in samples #3-5, which are Co3.32Fe0.68Sb12 and Co0.33Fe0.67Sb2 respectively. Thirdly, the solubility regions along CoSb3-“FeSb3” line are single-phase skutterudite region (Co, Fe) Sb3, three-phase region (Co3.32Fe0.68Sb12 + Co0.33Fe0.67Sb2 + liquid Sb) and two-phase region (FeSb2 + liquid Sb). This is drastically different from the previous experimental results at 700℃ but more close to its lower temperature results such as 550℃. Error in temperature measurements might be a possible reason for this discrepancy. All the findings above combined can explain why the estimated phase region for sample #3 Co3.25Fe0.75Sb12 is a single-phase skutterudite phase region, yet what we observed is a three-phase region (Co3.32Fe0.68Sb12 + Co0.33Fe0.67Sb2 + liquid Sb). Co0.33Fe0.67Sb2 phase was observed on SEM and confirmed by EPMA, and Sb phase was observed on XRD. 114 The determined phase region for samples #1, 2, 4, 5, and 6 matches what we estimated. Some of the SEM photos are shown in Figure 7.10. However, for sample #7, a small amount of skutterudite phase Co4-xFexSb12 was observed (see Figure 7.10f) while we expect a single-phase region of (Co, Fe)Sb2. This must be due to the compositional shift during synthesis, as we can see from the obtained phase diagram in Figure 7.11. (a) (b) (c) (d) (e) (f) Figure 7.10 SEM photos of samples with nominal composition as a) #1 Co3.4Fe0.3Sb12.6, b) #2 Co4.2Fe0.3Sb11.8, c) #3 Co3.25Fe0.75Sb12, d) #5 Co1.5Fe1.5Sb12, e) #6 Fe4Sb12, and f) #7 Co6Fe6Sb12. The impurity phases are detected by EDS and confirmed by EPMA. The discovery of the stable composition Co0.33Fe0.67Sb2 is quite interesting. The author could not find any published report on the crystal structure of this ternary compound. Since the two end phases CoSb2 and FeSb2 have completely different crystal structures179,180 and lattice parameters, it is unlikely for them to form solid solution across the whole range10. To investigate if phase segregation will happen, in another way, to find out if a three-phase region ((Co, Fe)Sb3 + Co-rich (Co,Fe)Sb2 + Co0.33Fe0.67Sb2) will form on the Co-rich side of skutterudite phase (see Figure 7.12), two more samples with nominal compositions Ce0Co2.5Fe1.5Sb8.5 and Ce0Co3Fe1Sb9 (compositions added to Table 7.2) are synthesized. Results confirm that both of them are in a Co-rich two-phase region (Co, Fe)Sb3 + (Co,Fe)Sb2. So no phase segregation between Co-rich (Co,Fe)Sb2 and Co0.33Fe0.67Sb2 seems to happen for Fe amount up to 0.53 in FexCo1-xSb2. Four more samples with nominal compositions close to stable composition Co0.33Fe0.67Sb2 (FexCo1-xSb2, x from 0.60 to 0.70, see Table 7.2) are synthesized to study the possible segregation between stable composition 115 Co0.33Fe0.67Sb2 and end phases (Co,Fe)Sb2. Synchrotron XRD measurements were performed on these samples. Results are still under analysis so no conclusion can be made at this moment. Based on the results from phase identification and compositional analysis, the experimental isothermal section of Co-Fe-Sb ternary system is determined, as shown in Figure 7.11. Figure 7.11 Experimental isothermal section of Co-Fe-Sb ternary system at 700℃. 7.5 Phase space of CeyCoxFe4-xSb12 skutterudites at 700℃ It would be really demanding to study the whole phase diagram of Ce-Co-Fe-Sb quaternary system at 700℃. Since the phase regions we are interested in are around CoSb3, we can narrow our investigation to the region near Sb enclosed by the three ternary systems (Ce-Co-Sb, Ce-Fe-Sb and Co-Fe-Sb), which would spare us lots of time and energy. With results from sections 7.3, 7.4, and 5.5.3 and the assumption that the smooth phase boundary is almost linear, we can now set out to determine the phase space of skutterudite phase CeyCoxFe4-xSb12 in Ce-Co-Fe-Sb system at 700℃. As a first step, a series of CeyCoxFe4-xSb12 samples with exact skutterudite stoichiometry (Co+Fe):Sb =1:3 but different Co:Sb ratios (x varying from 0 to 4, each represented by a black dashed line in Figure 7.12) were synthesized, with varying amount of Ce for each selected ratio. Nominal 116 compositions (listed in Table 7.3) are chosen to be near the phase boundary, with Ce content both below and above the estimated solubility limit for each chosen Co:Fe ratio as shown in Figure 7.13. Note that there are two solubility limits for each chosen Co:Fe ratio for the same reason we discussed in Section 7.3. Figure 7.12 Slices of the isothermal space of Ce-Co-Fe-Sb system with different Co:Fe ratios projected to the Co-Fe-Sb plane as black dashed lines. Ce (not shown here) is perpendicular to the CoSb-FeSb-Sb phase diagram. 1.0 CeyCoxFe4-xSb12 5.5e21 5.0e21 Ce-Fe-Sb y_min y_max 4.4e21 0.8 2.68e20 1.78e20 0.6 Bao XF Tang Morelli 3.4e21 6.0e20 y(Ce) 1.30e20 6.78e19 0.4 2.9e19, -4.3e21, -1.9e21 2.43e19 0.2 0.0 1.1e20 3.4e19 -2.6e21 1.0e20 -3e20 -6e20 Ce-Co-Sb 1.1e20 Co-Fe-Sb 0 1 2 3 4 x(Co) Figure 7.13 Proposed phase boundary of CeyCoxFe4-xSb12 skutterudites based on preliminary solubility results from Co-Fe-Sb, Ce-Co-Sb, Ce-Fe-Sb systems. Each dark circle represents a sample to make. Red and blue circles represent the maximum and minimum solubility limit of Ce in 117 CeyCoxFe4-xSb12 skutterudites determined from this study. Literature data are presented in empty triangles with carrier concentration labeled near each data point. Note here that the stoichiometry of (Co+Fe):Sb is fixed to 1:3. Table 7.3 Nominal compositions of CeyCoxFe4-xSb12 samples for skutterudite phase space investigation in the Ce-Co-Fe-Sb quaternary system. Sample name #1-1 #1-2 #1-3 #1-4 #1-5 #1-6 #2-1 #2-2 #2-3 #2-4 #2-5 #2-6 #3-1 #3-2 #3-3 #3-4 #3-5 #3-6 #4-1 #4-2 #4-3 #4-4 #4-5 #4-6 #5-1 #5-2 #5-3 #5-4 #5-5 #5-6 #5-7 #6-1 #6-2 #6-3 #6-4 #6-5 #6-6 #7-1 #7-2 #7-3 #7-4 Ce 0.6 0.7 0.8 0.9 0.95 1 0.45 0.55 0.65 0.8 0.9 1 0.3 0.4 0.5 0.7 0.8 0.9 0.2 0.3 0.4 0.6 0.7 0.8 0 0.1 0.2 0.3 0.5 0.6 0.7 0 0.05 0.1 0.4 0.5 0.6 0 0.25 0.35 0.45 Co 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1.5 1.5 1.5 1.5 1.5 1.5 2 2 2 2 2 2 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3 3 3 3 3 3 3.5 3.5 3.5 3.5 Sb 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 8.5 12 12 12 12 12 12 9 12 12 12 12 12 12 12 12 12 Fe 3.5 3.5 3.5 3.5 3.5 3.5 3 3 3 3 3 3 2.5 2.5 2.5 2.5 2.5 2.5 2 2 2 2 2 2 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1 1 1 1 1 1 0.5 0.5 0.5 0.5 118 The results are summarized in the experimentally determined phase space shown in Figure 7.14. Towards the Fe-rich part, the minimum Ce solubility limit is higher than expected, meaning the skutterudite phase space is narrower. Other than that, the solubility region has a boundary that not only overlaps roughly with our estimation but also is in agreement with literature data19,154,172. The EPMA determined compositions for CeyCoxFe4-xSb12 skutterudites are listed in Table 7.4. Table 7.4 EPMA determined compositions for CeyCoxFe4-xSb12 skutterudites. x(Co) nominal 1.5 1.5 1.5 1.5 2 2 2 2 2 2 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3 3 3 3 3 3 3.5 3.5 3.5 3.5 x(Co) epma -1.53 1.57 1.46 2.45 2.23 2 1.95 1.99 1.97 3.65 2.69 2.59 2.46 2.5 2.43 2.5 3.81 3.08 3.09 3.06 3.04 2.92 -3.43 3.5 3.49 x(Co) std -0.09 0.17 0.1 0.09 0.06 0.15 0.02 0.06 0.11 0.02 0.24 0.17 0.1 0.08 0.06 0.12 0.02 0.1 0.08 0.11 0.16 0.14 -0.13 0.09 0.06 y(Ce) nominal 0.5 0.7 0.8 0.9 0.2 0.3 0.4 0.6 0.7 0.8 0 0.1 0.2 0.3 0.5 0.6 0.7 0 0.05 0.1 0.4 0.5 0.6 0 0.25 0.35 0.45 y(Ce) epma -0.68 0.7 0.76 0.3 0.38 0.45 0.62 0.6 0.65 -0.14 0.22 0.33 0.48 0.5 0.51 -0.03 0.06 0.32 0.37 0.42 -0.22 0.21 0.24 y(Ce) std -0.04 0.09 0.05 0.04 0.03 0.03 0.03 0.03 0.05 -0.05 0.02 0.05 0.03 0.03 0.05 -0.04 0.04 0.05 0.08 0.07 -0.02 0.05 0.05 119 1.0 CeyCoxFe4-xSb12 5.5e21 5.0e21 Ce-Fe-Sb y_min y_max 4.4e21 0.8 2.68e20 1.78e20 0.6 Bao XF Tang Morelli 3.4e21 6.0e20 y(Ce) 1.30e20 6.78e19 0.4 2.9e19, -4.3e21, -1.9e21 2.43e19 0.2 0.0 1.1e20 3.4e19 -2.6e21 1.0e20 -3e20 -6e20 Ce-Co-Sb 1.1e20 Co-Fe-Sb 0 1 2 3 4 x(Co) Figure 7.14 Experimentally determined phase space of skutterudite phase with exact stoichiometry (Co+Fe):Sb = 1:3 in Ce-Co-Fe-Sb system at 700℃. Each blue dot represents a measured value from EPMA listed in Table 7.4. Figure 7.14 is only a slice of the phase space of CeyCoxFe4-xSb12 with exact skutterudite stoichiometry. To map out the full solubility region, samples similar to those listed in this section (Table 7.3) need to be synthesized with both Sb-rich and Co-rich stoichiometry instead, namely, synthesis of series of CeyCoxFe4-xSb12+z samples with z values either larger or smaller than 0 are needed. After the full phase region is mapped out, carrier concentration needs to be measured for each sample. Ultimately, this one-to-one mapping between carrier concentration and phase space location will enable us to synthesize optimized p-type skutterudite material with the guidance of experimental phase diagram map. 7.6 n-p change in CeyCo3.25Fe0.75Sb12 materials As we have discussed in the end of Section 7.5, the phase diagram space is still not fully mapped out yet. However, based on the current results, a p-n semiconducting type change can be expected as the amount of Ce increases for x(Co) = 3.25 in CeyCoxFe4-xSb12 samples19. It would thus be interesting to compare their high temperature thermoelectric properties. Most importantly, by fixing the Co content, we are eliminating the influence of Co:Fe ratio on the band structure, which shall give us some insight on the electronic origin for n- and p-type conduction in CeyCo3.25Fe0.75Sb12 skutterudites. 120 A set of samples with y(Ce) = 0, 0.1, 0.2, 0.3, 0.4, 0.5 are synthesized with sample named from #3-1 to #3-6. Only samples with y = 0.1, 0.2, and 0.3 are pure. For y = 0 sample, (Co,Fe)Sb2 and liquid Sb phases are observed as impurity phases, which is consistent with results from Section 7.3. For y = 0.4, 0.5 samples, both CeSb2 and Ce(Co,Fe)Sb3 impurity phases are observed. The solubility limit is estimated to be: y = 0.32±0.03. The transport properties of these samples are measured up to 850K in our home-built high-temperature Hall system41 and high-temperature Seebeck system42, with data shown in Figure 7.15. Ce content measured from EPMA measurement for all samples is shown in the legend. The room temperature transport properties are also shown in Figure 7.16. The thermoelectric performance of this set of samples are not promising, with maximum zT = 0.5 at 700K for y = 0.4 and 0.5 samples. Due to the discrepancy between our home built Seebeck system and commercial ZEM-3 system, the current data would be 15% less of measured Seebeck values on ZEM-3 systems, which will result in 20% less of zT. However, even with that into account, the zT value (maximum 0.6) is still not promising, more research needs to be performed to optimize these concentrations for both p- and n-type applications. Even though the zT values of this set of samples are not great, there still are lots of interesting physics that request attention. From Figure 7.15a we can see that all samples except #3-4 show typical heavily doped semiconductor behavior, with increasing resistivity as temperature increases. Sample #3-4 shows intrinsic semiconductor behavior, as can be seen from large resistivity (Figure 7.15a) and low carrier concentration (Figure 7.16a). Note that for this sample, the carrier concentration shows p-type, while the Seebeck coefficient is n-type, which might be due to the large difference in mobility between electrons and holes in CexCo3.25Fe0.75Sb12 and the different dependence on mobilities for Hall coefficient and Seebeck coefficient. As we can see from Appendix D, when two types of carriers coexist, the total Hall coefficient is more sensitive to the highly mobile carrier (holes in the case of skutterudites) compared to Seebeck coefficient. Samples #3-1, #3-2 and #3-3 show p-type carrier concentration (Figure 7.16a), positive Seebeck coefficient (Figure 7.16d) and high mobility (Figure 7.16b). All these evidence indicate that holes are the main conducting carrier type in these samples. For samples #3-5 and #3-6, the magnitude of room temperature carrier concentrations (Figure 7.16a) are unreasonably high, which could be due 121 to the fact that two types of carriers are conducting so the equation for calculating assuming single type carrier no longer holds. The large Seebeck coefficient in n-type samples (#3-4 and #3-5) compared to p-type samples greatly benefit both the power factor and the zT even though their mobility is relatively lower compared to p-type samples. (a) (b) (c) (d) (e) (f) 122 Figure 7.15 Transport properties of CeyCo3.25Fe0.75Sb12 skutterudites. The temperature dependence of: (a) electrical resistivity (with all six samples), (b) electrical resistivity (magnification a)), (c) Seebeck coefficient, d) thermal conductivity, (e) power factor, and (f) thermoelectric figure of merit (zT) are plotted in the temperature range of 300K to 850K. In figure c, 181 the numbers close to each curve is the energy band gap calculated from . (a) (b) (c) (d) (e) (f) Figure 7.16 Room temperature transport properties of CeyCo3.25Fe0.75Sb12 skutterudites. (a) Hall carrier concentration, (b) mobility, (c) electrical resistivity, (d) Seebeck coefficient, e) thermal 123 conductivity, and (f) power factor are plotted with Ce content y determined from EPMA measurement. 7.7 Conclusion and future work This chapter mainly discusses the phase diagram study of Ce-Co-Fe-Sb system at 700℃. By combining results of isothermal sections of Ce-Fe-Sb, Co-Fe-Sb and Ce-Co-Sb ternary phase diagrams, the phase regions near skutterudite phase CeyCoxFe4-xSb12 are investigated. This leads to the determination of phase space of skutterudite phase on the stoichiometric section CeyCoxFe4xSb12. A few attempts were made to synthesis both n- and p-type material with the same Co:Sb ratio, in search of good thermoelectric materials paired with similar thermal expansion coefficients. Though the obtained thermoelectric performance is not satisfying, with further study of complete mapping between the carrier concentration and the phase space, it is optimistic to obtain optimized p-type compositions with the guidance of the phase diagram in the future. 124 Chapter 8 Nano-structuring in bulk skutterudites 8.1 Summary From the equation for we can see that the lattice thermal conductivity is the only parameter that allows independent tuning. Nano-structuring, as a means of bringing in nanoscale features to effectively scatter phonon prorogation thus reduce lattice thermal conductivity, has been applied in a lot of systems to improve thermoelectric properties. One mechanism is reducing the grain size to nano-scale. The introduced large density of grain boundaries leads to enhanced grain boundary scattering. This is easily realized by ball milling. Another mechanism is to introduce nano-scale foreign phases. Many efforts have been tried such as by adding more electron conductive nanoparticles or introducing filler oxides by deliberate exposure to oxygen109. The kinetics of these foreign phases, however, is hard to control. Here we introduce another method of nano-structuring, which is precipitation of nano-scale secondary phases according to temperature dependent filler solubility based on phase diagrams. As we have shown in Chapter 5, the solubility limit of filler atoms R in CoSb3 system is not a single value but rather has a large dependence on the annealing temperature. Thus by simply combining various heat treatments, the kinetics of precipitation can be controlled such that nano-scale secondary phases may be introduced to lower lattice thermal conductivity, and even more importantly, these phases can be dissolved back into the matrix skutterudite phase in case a “reset” is needed (e.g., when precipitate size grows too large). Experimental details and results are shown in Section 8.2 and a conclusion and future work in Section 8.3. 8.2 Experimental results Yb-doped CoSb3 were chosen to perform this study with nominal compositions of YbxCo4Sb12 (x = 0.30, 0.33, 0.36 and 0.40). Samples were synthesized with similar procedures as described in Chapter 2 and 5. There are five steps. First, the raw elements are melted at 1100℃ into liquid phase to maximize the mass diffusion and exchange. After 12 hours at 1100℃, the melt is quenched in water. Secondly, an annealing process at 750℃ for a week follows, which allows the filler content 125 in skutterudites to achieve its solubility limit at 750℃ and be homogenously distributed. Thirdly, the homogenized sample is quenched with water again, in order to freeze the content of Yb allowed at 750℃. Fourthly, the sample is annealed once again for 1 week, but at a lower temperature (600℃). The solubility limit at 600℃ is lower compared to that at 750℃ due to the previously experimentally determined temperature dependence of solubility limit (Section 5.2)23. This will drive the filler atoms out of skutterudite phase and precipitate in forms of secondary phases such as YbSb2 and CoSb2. Precipitation time at 600℃ was held for 1, 2, 3, and 9 days for each sample YbxCo4Sb12, respectively. For each composition, samples held at 600℃ for various precipitation times were different pieces cut from the same homogenized (at 750℃) ingot, so the only variable is the precipitation time. This shall spare us trouble if the samples are indeed homogeneous. Lastly, samples after precipitation are quenched to freeze any possible microstructural features for SEM, EDS and EPMA analysis. The heat treatment parameters (nominal composition and heat treatment temperatures) are chosen based on the phase diagram, as shown in Figure 8.1. The larger the temperature difference between the homogenization temperature (750℃) and precipitation temperature (600℃), the larger the driving force for precipitation. According to nucleation and growth theory, this would promote more nucleation, resulting in smaller-size precipitation, which would possibly match our expectation of nano-scale secondary phases. Larger temperature difference will also lead to larger solubility difference. According to the lever rule, when the precipitation temperature decreases, for the same starting composition, the amount of secondary phases will increase, which would also facilitate our experimental observation. However, the homogenization temperature should not exceed the peritectic temperature of CoSb3 (874℃), neither should the precipitation temperature be too low such that kinetics hinders the precipitation to actually happen. The nominal compositions are chosen to have an Yb content between the two solubility limits corresponding to selected temperatures. 126 Figure 8.1 Heat treatment parameter determined based on the experimental phase diagram study (as described in Chapter 5, Section 5.2 and 5.4). The green dashed lines represent the homogenization and precipitation temperatures, 750℃ and 600℃, respectively. The orange dashed lines represent the solubility of Yb in CoSb3 skutterudite phase at each temperature. The red arrow indicates the solubility difference between the two selected temperatures. After precipitation, the amount of Yb in skutterudite phase was measured with EPMA for each composition, as shown in Figure 8.2. Figure 8.2 Effect of precipitation time on Yb content in skutterudites after precipitation. Each solid line represents a nominal composition. Figure 8.2 does not show clear trend except for the data set circled in red, which exhibit a decreasing trend of Yb content in skutterudite phase as precipitation time increases. This indicates that the amount of secondary phases (especially YbSb2) increases with precipitation time. However, SEM photos of these three samples are quite confusing. Although YbSb2 phases did show up after 3 days’ precipitation, we do not see any YbSb2 phases at all after 9 days’ precipitation, as shown in Figure 8.3. One possible reason might be the inhomogeneity in “homogenized” sample even after 1week treatment at 750℃. To find out the reason, it is recommended to redo this set of synthesis. 127 Moreover, the size of the precipitates after 3 days’ precipitation is on the order of 2-10 micrometer. Most of the precipitates observed are YbSb2 phase, while the CoSb2 phase is rarely observed. The reason for this phenomenon remains unclear. Although the effective scattering length for precipitates in skutterudites is not fully clear167, the observed precipitate size is much larger if we are aiming for nano-scale precipitates in general, and thus a smaller increment on the precipitation time and or a lower precipitating temperature is recommended. The observed precipitates are mostly distributed along the grain boundaries. No in-grain precipitation is observed, which makes sense due to the high level of defects along grain boundaries and lower energy barrier. (a) (b) (c) Figure 8.3 SEM photos for sample with nominal composition Yb0.40Co4Sb12 precipitated at 600℃ with different precipitation times: (a) 2 days, (b) 3 days, and (c) 9 days. 8.3 Conclusion and future work Although this work is not complete, it still reveals the possibility of introducing precipitates into skutterudite system based on temperature dependent solubility of fillers. To resume this work, it is recommended to re-investigate the heat treatment parameters, from possibly higher homogenization temperature, lower annealing temperature to shorter precipitation time. More experiments need to be designed in a systematic way to avoid inconsistency. 128 Chapter 9 Future work In this thesis, RxCo4Sb12 skutterudite has been studied successfully from two main perspectives. First of all, why high thermoelectric performance is achievable in filled skutterudites was discussed, including both the electronic and thermal transport origin. The electronic origin study sets a good example to find beneficial electronic band structure features of intrinsic CoSb3 via both experiments and computation. This methodology can be generalized and applied to other systems in the future. It also prompts more interest on a closer investigation of the bonding chemistry of CoSb3. As the temperature dependent band offset in conduction bands is speculated to be due to different temperature dependence of related bonding features. This work is undergoing with synchrotron data acquired at Argonne National laboratory just recently. More investigation is on the way. The role of fillers on reducing lattice thermal conductivity is still open to debate, which leaves large room for future study. Either Te or Ni doped CoSb3 can be synthesized to study the effect of electron-phonon scattering on phonon transport, especially the coefficient of dependence of the thermal resistance on the carrier concentration. With such information obtained, the role of fillers in terms of resonant scattering or avoided scattering can be more clearly distinguished. The second part of this thesis elaborates on how high thermoelectric performance can be achieved through precise doping control to obtain optimized thermoelectric compositions. Phase diagram study has been proved to be a necessary and helpful method in this case. Future work can evolve around re-investigation of systems that were reported to have solubility barriers. As we have shown with Ce-CoSb3, this could possibly lead to the overturn of such conclusions. Defect study of intrinsic CoSb3 is to be continued. The effect of oxygen in CoSb3 defect chemistry is quite interesting and worthy of more attention. Calculation shows that the addition of oxygen will not result in any change in the defect type, yet we have one sample that shows n-type behavior with 129 oxygen addition. Continued efforts are needed with special care for synthesizing conditions as detailed in Chapter 6. Phase diagram study of quaternary system Ce-Co-Fe-Sb is mostly done, with the phase space of skutterudites on the (Co+Fe):Sb = 1:3 stoichiometry section determined at 700℃. To obtain a full phase space, similar work as described in Section 7.5 needs to be performed for both Sb-rich and Co-rich stoichiometry. To enable discovery of optimum compositions for p-type thermoelectric skutterudites, mapping of the carrier concentration of samples synthesized across the skutterudite phase space is needed. More detailed analysis of the CeyCo3.25Fe0.75Sb12 samples is needed to probe possible multi-band conduction behavior in both p- and n-type CeyCo3.25Fe0.75Sb12 skutterudite, which can be realized by magnetic field dependence of Hall effect measurements. Nano-structuring through secondary phase precipitation in skutterudites show some promises, yet more efforts are needed to engineer the amount and size of these precipitates such that they are effective in lowing lattice thermal conductivities. It is also recommended to study the influence of common secondary phases such as CoSb2, Sb and RSb2 etc. on transport properties of intrinsic CoSb3. This not only can serve as a reference in elucidating the benefits of nano-precipitates on thermal transport, but also is enlightening in the defect study of intrinsic CoSb3. 130 Appendix A From Boltzmann Transport Equation to Thermoelectric Properties182-184 Boltzmann transport equation (BTE) describes how the distribution function of a thermodynamic system changes with time when it is subjected to external forces. For an electron gas, the distribution function 𝑓𝑓(𝑘𝑘, 𝑟𝑟, 𝑡𝑡) tells us how the electrons are distributed in momentum space and from this information all of the transport properties can be evaluated. At equilibrium, the distribution function is the Fermi-Dirac function: (Eq. A.1) where is the energy, with is the reduced electrochemical potential chemical potential (equals to Fermi energy at 0K) and the the electrostatic potential. With the presence of external fields (electric field or temperature gradient), the electron distribution 𝑓𝑓(𝑘𝑘, 𝑟𝑟, 𝑡𝑡) will be subjected to two effects: one is the driving force of the external field and the other is the dissipative effect of scattering. Note that in the case of thermoelectric materials, the temperature gradient is often small such that the temperature dependence of chemical potential can be neglected, and so can the effect of diffusion on the distribution function. This can be expressed by Eq. A.2 and A.3 where s denotes “scattering process”. (Eq. A.2) (Eq. A.3) In a steady state, there is no net change in the distribution function, . By comparing Eq. A.2 and Eq. A.3, we get: (Eq. A.4) With a relaxation time approximation, the effect of scattering process on restoring the distribution function can be described in a linear equation, shown in Eq. A.5. The scattering time means the 131 amount of time required to restore the equilibrium distribution function 𝑓𝑓0 (𝑘𝑘, 𝑟𝑟) which deviates to because of the external field. (Eq. A.5) Substitute Eq. A.5 into Eq. A.4, and we get: (Eq. A.6) It is normally considered the distribution function is not too much away from the equilibrium, so for simplicity we can replace and by and , respectively. With the assumption of a single parabolic band model, In a semi-empirical model of conduction, Table. A.1 Relation between transport effect and generalized force. Transport effect Acting force Generalized Electrical conductivity E -eE Hall effect RH E,H -eE-ev×H Seebeck effect S ∇T -e(∇μ/e-∇ϕ) = Thermal conductivity κ ∇T -e(∇μ/e-∇ϕ) = 1. Carrier concentration n 132 (Eq. A.7) 2. Electrical conductivity σ As shown in Table A.1, the external force is the electrical field E and the temperature gradient ∇T is 0. (no electric current at equilibrium) =0 since there is no temperature gradient; , (Eq. A.8) is along one direction, whereas the kinetic energy directions equivalently. is distributed among three 133 Note for acoustic phonon scattering r = -1/2 By partial integration, we get the expression of 𝜎𝜎: (Eq. A.9) Thus we have the expression for electrical conductivity with the assumption that there is only one predominant scattering mechanism. 3. Hall coefficient RH The derivation of hall coefficient RH is similar to that of electrical conductivity except with a different force F= -eE-ev×H Since , in a steady state, (a) Lattice constant ( a(A) ) since 9.07 9.06 9.05 9.04 9.03 YbxCo4+ySb12+z 773K 823K 873K 923K 973K 1023K 1073K ref [4] ref [19] Lattice constant ( a(A) ) 134 (b) 9.07 Co-rich this work 873K Sb-rich this work 873K Co-rich ref[8] 873K ref [8] 873K 9.06 9.05 9.04 9.03 YbxCo4+ySb12+z 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0 Yb actual content x Yb nominal content x 135 (Eq. A.10) 4. Hall factor rH (Eq. A.11) 5. Drift mobility (Eq. A.12) When r=0, which is independent of energy. 6. Hall carrier concentration nH (Eq. A.13) 7. Hall Mobility 136 (Eq. A.14) 8. Seebeck coefficient Since Seebeck coefficient is measured when there is no current, we have j=0. The driving force for Seebeck effect is -∇𝜁𝜁 and the definition of Seebeck is S= −∇𝜁𝜁 −𝑒𝑒∇𝑇𝑇 137 (Eq. A.15) 9. Lorentz number L Again the driving force is the -∇𝜁𝜁 and the electric current j is zero. The definition of thermal conductivity is defined by: 138 Thus the Lorenz number can be written as: (Eq. A.16) 139 Appendix B TE properties from SPB model with acoustic phonon scattering From Appendix A, the expressions for thermoelectric property such as electrical conductivity, Seebeck coefficient and thermal conductivity can be calculated with a single parabolic band model (Eq. A.9-A.16), since for skutterudites at room temperatures and above electrons are mostly scattered by acoustic phonons. The scattering time can be expressed by where is energy (not reduced energy), and f or acoustic phonon scattering r = -1/2. We list the equations for TE properties from SPB model and acoustic phonon scattering below. 1. The chemical carrier density (Eq. B.1) 2. The electrical conductivity (Eq. A.9) (Eq. B.2) 3. Hall coefficient RH (Eq. A.10) (Eq. B.3) 4. Hall factor rH (Eq. A.11) (Eq. B.4) 5. Drift mobility 140 (Eq. A.12) (Eq. B.5) 6. Hall carrier concentration nH (Eq. A.13) (Eq. B.6) 7. Hall Mobility (Eq. A.14) (Eq. B.7) 8. Seebeck coefficient (Eq. A.15) (Eq. B.8) 9. Lorentz number L (Eq. A.16) (Eq. B.9) 141 Appendix C TE properties from SPB model in non-degenerate and degenerate limit In a non-degenerate limit, the electrochemical potential well away from the appropriate band edge ( lies well within the forbidden gap <<0). In order to neglect minority carriers, it should also be far away from the opposite band edge. In non-degenerate limit, TE properties can be expressed in equations C.1-C.5. ; ; (Eq. C.1) (Eq. C.2) (Eq. C.3) (Eq. C.4) (Eq. C.5) In a degenerate limit, the electrochemical potential edge and into the band ( is way above the appropriate band >>0). The material is so heavily doped that it shows metallic behavior. In the non-degenerate limit, TE properties can be expressed in equations C.6-C.10. (Eq. C.6) (Eq. C.7) 142 (Eq. C.8) (Eq. C.9) (Eq. C.10) when r=-1/2 (for acoustic phonon scattering), (Eq. C.11) (Eq. C.12) 143 Appendix D TE properties from a multi-band model For a multi-band system, the electric current density and heat flux rate for each band i can be represented as: (Eq. D.1) (Eq. D.2) In terms of electric conduction, with , thus we get the expression for electrical conductivity for a multi-band system: (Eq. D.3) (Eq. D.4) (Eq. D.5) (Eq. D.6) For Seebeck coefficient and thermal conductivity, (Eq. D.7) 144 (Eq. D.8) For a two-band model, we get: (Eq. D.9) If 1 represents the electron band and 2 represents the valence band, then Eq. D.9 is just the thermal conductivity for a material with bipolar conduction term included. 145 Appendix E Mott relation for SPB and SKB with acoustic phonon scattering In Chapter 3, we have shown that the non-parabolic Kane band relation yields a decreasing Seebeck-dependent effective mass as carrier concentration increases. This is significant because it not only shows that the common explanation of Kane bands causing the increasing mass with doping in CoSb3 is inadequate, but also this should have wider implications in terms of nonparabolicity’s contribution to thermoelectric performance. The well-known Mott relation is 185: 𝑆𝑆 = 2 𝜋𝜋2 𝑘𝑘𝐵𝐵 𝑇𝑇 1 𝑑𝑑𝜎𝜎𝐸𝐸 3𝑒𝑒 𝜎𝜎𝐸𝐸 𝑑𝑑𝑑𝑑 Eq. E.1 e2 where σE = 3 𝑣𝑣(𝐸𝐸)2 τ(E)𝐷𝐷(𝐸𝐸) is the energy dependent electronic conductivity186 that depends on the electron drift velocity (v), the scattering time (τ), and the density of states (D). Using the relationships given previously by Young et al.83 we can show that: 𝑣𝑣(𝐸𝐸) = 𝐷𝐷(𝐸𝐸) = 1 𝑑𝑑𝑑𝑑 2 𝑑𝑑𝑑𝑑 −1 = � ∗ � � 𝛾𝛾1/2 ℏ 𝑑𝑑𝑑𝑑 𝑚𝑚0 𝑑𝑑𝑑𝑑 3/2 √2𝑚𝑚0∗ 𝜋𝜋2 ℏ3 1 𝑑𝑑𝑑𝑑 𝛾𝛾1/2 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 −1 ℏ2 k2 where γ(E) = 2m∗ . 𝑑𝑑𝑑𝑑 0 and 𝜏𝜏 = 𝜏𝜏0 𝛾𝛾 𝑟𝑟−2 � � The Mott relation for a general band becomes: 𝑆𝑆 = 2 2 𝑘𝑘𝐵𝐵 𝑇𝑇 𝜋𝜋 2/3 ∗ 𝑑𝑑𝑑𝑑 � � 𝑚𝑚0 𝑑𝑑𝐸𝐸 [𝑟𝑟 + 1 − 𝜆𝜆] 2 3 𝑒𝑒ℏ 3𝑛𝑛 𝑑𝑑𝑑𝑑 −2 𝑑𝑑 2 𝛾𝛾 𝑑𝑑𝑑𝑑 𝑑𝑑𝐸𝐸 2 where 𝜆𝜆 = 2𝛾𝛾 � � Eq. E.2 Eq. E.3 This expression (Eq.E.2) is derived by differentiating the obtained expression for 𝜎𝜎𝐸𝐸 = 1/2 2 23/2 𝑚𝑚0∗ 𝑒𝑒 𝜏𝜏0 𝑑𝑑𝑑𝑑 −2 𝑟𝑟+1 � � 𝛾𝛾 . 3𝜋𝜋2 ℏ3 𝑑𝑑𝑑𝑑 1/2 2 𝑑𝑑𝜎𝜎𝐸𝐸 23/2 𝑚𝑚0∗ 𝑒𝑒 𝜏𝜏0 𝑑𝑑𝑑𝑑 −1 𝑑𝑑𝑑𝑑 −3 𝑑𝑑 2 𝛾𝛾 𝑟𝑟+1 (𝑟𝑟 + 1)𝛾𝛾 𝑟𝑟 − 2 � � = (� � 𝛾𝛾 ) 2 3 𝑑𝑑𝑑𝑑 3𝜋𝜋 ℏ 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝐸𝐸 2 Eq. E.4 which upon substitution into the Mott equation (Eq. E.1) yields the energy dependent Seebeck relation: 𝑆𝑆 = 2 𝜋𝜋2 𝑘𝑘𝐵𝐵 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 −2 𝑑𝑑 2 𝛾𝛾 �𝑟𝑟 + 1 − 2𝛾𝛾 � � � 3𝑒𝑒𝑒𝑒 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑𝐸𝐸 2 Eq. E.5 146 Next, we must first approximate the carrier concentration as a function of Fermi level: 𝑛𝑛 ≈ 3 3 𝐸𝐸𝐹𝐹 22 𝑚𝑚∗ 2 3 ∫0 𝐷𝐷(𝐸𝐸)𝑑𝑑𝑑𝑑 = 3𝜋𝜋2 ℏ03 𝛾𝛾 2 � 𝐸𝐸=𝐸𝐸 𝐹𝐹 which can be solved for 𝛾𝛾(𝑛𝑛) = expression (Eq. E.2). ℏ2 (3𝜋𝜋 2 𝑛𝑛)2/3 and substituted into Eq. E.5 to obtain the final 2𝑚𝑚0∗ In order to obtain the 𝑚𝑚𝑆𝑆∗ that is presented in the main text for Kane band, we substitute 𝛾𝛾 = 𝐸𝐸 + 𝐸𝐸 2 /𝐸𝐸𝑔𝑔 as is commonly used by Ravich et al. 187, 𝑑𝑑𝑑𝑑 𝑑𝑑 2 𝛾𝛾 = 1 + 2𝐸𝐸/𝐸𝐸𝑔𝑔 , and 2 = 2/𝐸𝐸𝑔𝑔 into Eq. E.2. 𝑑𝑑𝑑𝑑 𝑑𝑑𝐸𝐸 Upon substituting in 𝜆𝜆 (Eq. E.3) and simplifying, we obtain 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) = 𝑚𝑚0∗ by inspection. It 1+2𝐸𝐸/𝐸𝐸𝑔𝑔 should be noted that this result only holds for r=0 (acoustic phonon scattering). Alternative scattering mechanisms can show increases in the energy dependent 𝑚𝑚𝑆𝑆∗ 188. While the derivation from the main text assumes a simpler scattering time where the matrix element coupling valence and conduction band state is independent of electron energy, Ravich et al. suggested an alternative form for the acoustic phonon scattering time that includes an energy dependent matrix element 187,189. Upon simplification, the expression for the scattering time can be shown to be: 𝜏𝜏(𝐸𝐸) = −1 2 2 3ℏ 𝐶𝐶𝑙𝑙 𝑁𝑁𝑣𝑣 2𝐸𝐸 2𝐸𝐸 + �1 + � � + 2� ��1 2 𝜋𝜋𝑘𝑘𝐵𝐵 𝑇𝑇𝑇𝑇(𝐸𝐸)𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑 𝐸𝐸𝑔𝑔 𝐸𝐸𝑔𝑔 Eq. E.6 where 𝐶𝐶𝑙𝑙 = 𝜌𝜌𝑣𝑣𝑙𝑙2 (𝑣𝑣𝑙𝑙 is the longitudinal speed of sound) and 𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑 is the acoustic phonon deformation potential. In this case, the equivalent Mott relation can be derived (using Eq. E.1 with the modified 𝜎𝜎𝐸𝐸 ); we find the Seebeck effective mass simplifies to: 𝑚𝑚𝑆𝑆∗ (𝐸𝐸) = 2𝐸𝐸 � 𝐸𝐸𝑔𝑔 2 3𝑚𝑚0∗ �1+ 2𝐸𝐸 � +2 𝐸𝐸𝑔𝑔 �1+ Eq. E.7 which also tends to decrease with energy at high Fermi levels, but initially shows a small increase (of only a few percent). 147 148 BIBLIOGRAPHY 1 Snyder, G. J. <http://www.thermoelectrics.caltech.edu/thermoelectrics/history.html> ( 2 Vining, C. B. The thermoelectric limit zT = 1: Factor or artifact. Proceedings of the 11th International Conference on Thermoelectrics, 223-231 (1992). 3 Harman, T. C., Taylor, P. J., Walsh, M. P. & LaForge, B. E. Quantum Dot Superlattice Thermoelectric Materials and Devices. Science 297, 2229-2232, doi:10.1126/science.1072886 (2002). 4 Snyder, G. J. & Toberer, E. S. Complex thermoelectric materials. Nature materials 7, 105-114 (2008). 5 Morelli, D. et al. Low-temperature transport properties of p-type CoSb3. Physical Review B 51, 9622-9628, doi:10.1103/PhysRevB.51.9622 (1995). 6 Caillat, T., Borshchevsky, A. & Fleurial, J. P. Properties of single crystalline semiconducting CoSb3. Journal of Applied Physics 80, 4442, doi:10.1063/1.363405 (1996). 7 Caillat, T., Borshchevsky, A. & Fleurial, J.-P. Preparation and thermoelectric properties of p- and n-type CoSb3. 316, 58-61, doi:10.1063/1.46835 (1994). 8 W. Jeitschko & Braun, D. LaFe4P12 with Filled CoAs3-Type Structure and lsotypic Lanthanoid-Transition Metal Polyphosphides Acta Cryst. 33, 3401-3406 (1977). 9 Arne, K. & Trond, R. Compounds with the Skutterudite Type Crystal Structure. 3. Structural Data for Arsenides and Antimonides. Acta Chemica Scandinavica A 28, 99-103 (1974). 10 Slack, G. A. CRC Handbook of Thermoelectrics. 407 (CRC Press, 1995). 11 Pei, Y. Z. et al. Improving thermoelectric performance of caged compounds through light-element filling. Applied Physics Letters 95, 042101, doi:10.1063/1.3182800 (2009). 12 Pei, Y. Z. et al. Synthesis and thermoelectric properties of K[sub y]Co[sub 4]Sb[sub 12]. Applied Physics Letters 89, 221107, doi:10.1063/1.2397538 (2006). 13 Chen, L. D. et al. Anomalous barium filling fraction and n-type thermoelectric performance of Ba[sub y]Co[sub 4]Sb[sub 12]. Journal of Applied Physics 90, 1864, doi:10.1063/1.1388162 (2001). 14 Park, K.-H. & Kim, I.-H. Thermoelectric Properties of Ca-Filled CoSb3-Based Skutterudites Synthesized by Mechanical Alloying. Journal of Electronic Materials 40, 493498, doi:10.1007/s11664-010-1400-4 (2010). 15 Bai, S. Q. et al. Thermoelectric properties of n-type Sr x M y Co4Sb12 (M=Yb, Ba) double-filled skutterudites. Applied Physics A 100, 1109-1114, doi:10.1007/s00339-0105711-3 (2010). 16 Qiu, Y. et al. Charge-Compensated Compound Defects in Ga-containing Thermoelectric Skutterudites. Advanced Functional Materials 23, 3194-3203, doi:10.1002/adfm.201202571 (2013). 17 Tang, Y. et al. Phase diagram of In–Co–Sb system and thermoelectric properties of Incontaining skutterudites. Energy & Environmental Science 7, 812, doi:10.1039/c3ee43240h (2014). 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 149 G. S. Nolas, J. L. Cohn & Slack, G. A. Effect of partial void filling on the lattice thermal conductivity of skutterudites. Physical Review B 58, 164-170 (1998). Morelli, D. T., Meisner, G. P., Chen, B., Hu, S. & Uher, C. Cerium Filling and Doping of Cobalt Triantimonide. Physical Review B 56, 7376-7383 (1997). Tang, Y., Hanus, R., Chen, S. W. & Snyder, G. J. Solubility design leading to high figure of merit in low-cost Ce-CoSb3 skutterudites. Nature communications 6, 7584, doi:10.1038/ncomms8584 (2015). V. L. Kuznetsov, L. A. Kuznetsova & Rowe, D. M. Effect of partial void filling on the transport properties of NdxCo4Sb12 skutterudites. J. phys.: Condens. Matter 15, 50355048 (2003). Pei, Y. Z., Bai, S. Q., Zhao, X. Y., Zhang, W. & Chen, L. D. Thermoelectric properties of EuyCo4Sb12 filled skutterudites. Solid State Sciences 10, 1422-1428, doi:10.1016/j.solidstatesciences.2008.01.016 (2008). Tang, Y., Chen, S.-w. & Snyder, G. J. Temperature dependent solubility of Yb in Yb– CoSb3 skutterudite and its effect on preparation, optimization and lifetime of thermoelectrics. Journal of Materiomics 1, 75-84, doi:10.1016/j.jmat.2015.03.008 (2015). Yang, J. et al. Solubility study of Yb in n-type skutterudites YbxCo4Sb12 and their enhanced thermoelectric properties. Physical Review B 80, 115329, doi:10.1103/PhysRevB.80.115329 (2009). Rogl, G. et al. Nanostructuring of p- and n-type skutterudites reaching figures of merit of approximately 1.3 and 1.6, respectively. Acta Materialia 76, 434-448, doi:10.1016/j.actamat.2014.05.051 (2014). Rogl, G. et al. New bulk p-type skutterudites DD0.7Fe2.7Co1.3Sb12−xXx (X=Ge, Sn) reaching ZT>1.3. Acta Materialia 91, 227-238, doi:10.1016/j.actamat.2015.03.008 (2015). Meng, X. et al. Enhanced thermoelectric performance of p-type filled skutterudites via the coherency strain fields from spinodal decomposition. Acta Materialia 98, 405-415, doi:10.1016/j.actamat.2015.07.027 (2015). Meisner, G., Morelli, D., Hu, S., Yang, J. & Uher, C. Structure and lattice thermal conductivity of fractionally filled skutterudites: Solid solutions of Fully Filled and Unfilled End Members. Physical Review Letters 80, 3551-3554 (1998). Koza, M. M. et al. Breakdown of phonon glass paradigm in La- and Ce-filled Fe4Sb12 skutterudites. Nature materials 7, 805-810, doi:10.1038/nmat2260 (2008). Nolas, G. S., Fowler, G. & Yang, J. Assessing the role of filler atoms on the thermal conductivity of filled skutterudites. Journal of Applied Physics 100, 043705, doi:10.1063/1.2245204 (2006). Liu, W.-S., Zhang, B.-P., Li, J.-F. & Zhao, L.-D. Effects of Sb compensation on microstructure, thermoelectric properties and point defect of CoSb3 compound. Journal of Physics D: Applied Physics 40, 6784-6790, doi:10.1088/0022-3727/40/21/044 (2007). Liu, W.-S., Zhang, B.-P., Li, J.-F. & Zhao, L.-D. Thermoelectric property of finegrained CoSb3 skutterudite compound fabricated by mechanical alloying and spark plasma sintering. Journal of Physics D: Applied Physics 40, 566-572, doi:10.1088/00223727/40/2/035 (2007). 150 Sharp, J. W., Jones, E. C., Williams, R. K., Martin, P. M. & Sales, B. C. Thermoelectric properties of CoSb3 and related alloys. Journal of Applied Physics 78, 1013, doi:10.1063/1.360402 (1995). 34 Su, T. et al. Fast Preparation and Low-Temperature Thermoelectric Properties of CoSb3. Journal of Electronic Materials 42, 109-113, doi:10.1007/s11664-012-2259-3 (2012). 35 Mandrus, D. et al. Electronic transport in lightly doped CoSb3. Physical Review B 52, 4926-4931, doi:10.1103/PhysRevB.52.4926 (1995). 36 Arushanov, E., Respaud, M., Rakoto, H., Broto, J. M. & Caillat, T. Shubnikov–de Haas oscillations in CoSb3 single crystals. PHYSICAL REVIEW B 61, 4672 (2000). 37 Ctirad, U. in Semiconductors and semimetals Vol. 69 Ch. 5, (Academic Press, 2001). 38 Okamoto, H. Co-Sb. J. Phase Equilib 12, 244-245 (1991). 39 LaLonde, A. D., Ikeda, T. & Snyder, G. J. Rapid consolidation of powdered materials by induction hot pressing. The Review of scientific instruments 82, 025104, doi:10.1063/1.3534080 (2011). 40 Miller, M. K., Russell, K. F., Thompson, K., Alvis, R. & Larson, D. J. Review of Atom Probe FIB-Based Specimen Preparation Methods. Microscopy and Microanalysis 13, 428436, doi:10.1017/s1431927607070845 (2007). 41 Borup, K. A. et al. Measurement of the electrical resistivity and Hall coefficient at high temperatures. The Review of scientific instruments 83, 123902, doi:10.1063/1.4770124 (2012). 42 Iwanaga, S., Toberer, E. S., LaLonde, A. & Snyder, G. J. A high temperature apparatus for measurement of the Seebeck coefficient. The Review of scientific instruments 82, 063905, doi:10.1063/1.3601358 (2011). 43 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys Rev Lett 77, 3865-3868 (1996). 44 Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. Journal of Physics: Condensed Matter 21, 395502 (2009). 45 Sofo, J. O. & Mahan, G. D. Electronic structure of CoSb3: A narrow-band-gap semiconductor. Phys Rev B 58, 15620-15623, doi:DOI 10.1103/PhysRevB.58.15620 (1998). 46 Agapito, L. A., Ferretti, A., Calzolari, A., Curtarolo, S. & Buongiorno Nardelli, M. Effective and accurate representation of extended Bloch states on finite Hilbert spaces. Physical Review B 88, 165127 (2013). 47 WanT code by A. Ferretti, B. Bonferroni, A. Calzolari, and M. Buongiorno Nardelli, (http://www.wannier-transport.org) 48 Kokalj, A. Computer graphics and graphical user interfaces as tools in simulations of matter at the atomic scale. Computational Materials Science 28, 155-168, doi:http://dx.doi.org/10.1016/S0927-0256(03)00104-6 (2003). 49 Tang, Y. et al. Convergence of multi-valley bands as the electronic origin of high thermoelectric performance in CoSb3 skutterudites. Nature materials advance online publication, doi:10.1038/nmat4430 http://www.nature.com/nmat/journal/vaop/ncurrent/abs/nmat4430.html - supplementaryinformation (2015). 33 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 151 Tang, Y., Chen, S. W. & Snyder, G. J. Temperature Dependent Solubility of Yb in Yb-CoSb3 Skutterudite and its Effect on Preparation, Optimization and Lifetime of Thermoelectrics. Journal of Materiomics 1, 74-83, doi:http://dx.doi.org/10.1016/j.jmat.2015.03.008 (2015). Li, H., Tang, X., Zhang, Q. & Uher, C. High performance In[sub x]Ce[sub y]Co[sub 4]Sb[sub 12] thermoelectric materials with in situ forming nanostructured InSb phase. Applied Physics Letters 94, 102114, doi:10.1063/1.3099804 (2009). Rogl, G. et al. n-Type skutterudites (R,Ba,Yb)yCo4Sb12 (R=Sr, La, Mm, DD, SrMm, SrDD) approaching ZT≈2.0. Acta Materialia 63, 30-43, doi:10.1016/j.actamat.2013.09.039 (2014). Shi, X. et al. Multiple-filled skutterudites: high thermoelectric figure of merit through separately optimizing electrical and thermal transports. Journal of the American Chemical Society 133, 7837-7846, doi:10.1021/ja111199y (2011). Pei, Y. et al. Convergence of electronic bands for high performance bulk thermoelectrics. Nature 473, 66-69, doi:10.1038/nature09996 (2011). Zhao, L. D. et al. All-scale hierarchical thermoelectrics: MgTe in PbTe facilitates valence band convergence and suppresses bipolar thermal transport for high performance. Energy & Environmental Science 6, 3346, doi:10.1039/c3ee42187b (2013). Wang, H., Gibbs, Z. M., Takagiwa, Y. & Snyder, G. J. Tuning bands of PbSe for better thermoelectric efficiency. Energ Environ Sci 7, 804-811, doi:10.1039/C3EE43438A (2014). Liu, W. et al. Convergence of Conduction Bands as a Means of Enhancing Thermoelectric Performance of n-Type Mg2Si1-xSnx Solid Solutions. Phys Rev Lett 108, 166601 (2012). Parker, D., Chen, X. & Singh, D. J. High Three-Dimensional Thermoelectric Performance from Low-Dimensional Bands. Phys Rev Lett 110, 146601 (2013). Pei, Y., Wang, H. & Snyder, G. J. Band engineering of thermoelectric materials. Advanced materials 24, 6125-6135, doi:10.1002/adma.201202919 (2012). Singh, D. J. & Pickett, W. E. Skutterudite antimonides: Quasilinear bands and unusual transport. Phys Rev B 50, 11235-11238 (1994). Pei, Y., LaLonde, A. D., Wang, H. & Snyder, G. J. Low effective mass leading to high thermoelectric performance. Energy & Environmental Science 5, 7963, doi:10.1039/c2ee21536e (2012). Kuznetsov, V. L., Kuznetsova, L. A. & Rowe, D. M. Effect of partial void filling on the transport properties of NdxCo4Sb12 skutterudites. Journal of Physics: Condens. Matter 15, 5035-5048 (2003). Salvador, J. R., Yang, J., Wang, H. & Shi, X. Double-filled skutterudites of the type Yb[sub x]Ca[sub y]Co[sub 4]Sb[sub 12]: Synthesis and properties. Journal of Applied Physics 107, 043705, doi:10.1063/1.3296186 (2010). Anno, H., Matsubara, K., Notohara, Y., Sakakibara, T. & Tashiro, H. Effects of doping on the transport properties of CoSb3. J Appl Phys 86, 3780-3786, doi:doi:http://dx.doi.org/10.1063/1.371287 (1999). Dyck, J. S. et al. Thermoelectric properties of the n-type filled skutterudite Ba[sub 0.3]Co[sub 4]Sb[sub 12] doped with Ni. Journal of Applied Physics 91, 3698, doi:10.1063/1.1450036 (2002). 152 Zhou, A., Liu, L.-s., Zhai, P.-c., Zhao, W.-y. & Zhang, Q.-j. Electronic Structures and Transport Properties of Single-Filled CoSb3. Journal of Electronic Materials 39, 18321836, doi:10.1007/s11664-010-1073-z (2010). 67 Yang, J., Xi, L., Zhang, W., Chen, L. D. & Yang, J. Electrical Transport Properties of Filled CoSb3 Skutterudites: A Theoretical Study. Journal of Electronic Materials 38, 13971401, doi:10.1007/s11664-009-0703-9 (2009). 68 Takagiwa, Y., Pei, Y. Z., Pomrehn, G. & Snyder, G. J. Validity of rigid band approximation of PbTe thermoelectric materials. Apl Materials 1, 011101, doi:Artn 011101 Doi 10.1063/1.4809545 (2013). 69 Yang, J. et al. Solubility study of Yb in n-type skutterudites Yb_xCo4Sb12 and their enhanced thermoelectric properties. Phys Rev B 80, 115329 (2009). 70 Liu, W.-S., Zhao, L.-D., Zhang, B.-P., Zhang, H.-L. & Li, J.-F. Enhanced thermoelectric property originating from additional carrier pocket in skutterudite compounds. Appl Phys Lett 93, -, doi:doi:http://dx.doi.org/10.1063/1.2965123 (2008). 71 Nagamoto, Y., Tanaka, K. & Koyanagi, T. in Thermoelectrics, 1998. Proceedings ICT 98. XVII International Conference on. 302-305. 72 Kajikawa, Y. Refined analysis of the transport properties of Co1−xNixSb3 according to a model including a deep donor level and the second lowest valleys of the conduction band. Journal of Alloys and Compounds 621, 170-178, doi:10.1016/j.jallcom.2014.09.173 (2015). 73 Chaput, L., Pécheur, P., Tobola, J. & Scherrer, H. Transport in doped skutterudites: Ab initio electronic structure calculations. Physical Review B 72, doi:10.1103/PhysRevB.72.085126 (2005). 74 Gnutzman, U. & Clauseck, K. Theory of Direct Optical-Transitions in an Optical Indirect Semiconductor with a Superlattice Structure. Appl Phys 3, 9-14 (1974). 75 Ackermann, J. & Wold, A. The preparation and characterization of the cobalt skutterudites CoP3, CoAs3 and CoSb3. J Phys Chem Solids 38, 1013-1016, doi:http://dx.doi.org/10.1016/0022-3697(77)90203-7 (1977). 76 Kliche, G. & Lutz, H. D. Temperature dependence of the FIR reflection spectra of the skutterudites CoAs3 and CoSb3. Infrared Phys 24, 171-177, doi:http://dx.doi.org/10.1016/0020-0891(84)90066-6 (1984). 77 Kane, E. O. Band Structure of Indium Antimonide. J Phys Chem Solids 1, 249-261 (1957). 78 Smith, S. D., Moss, T. S. & Taylor, K. W. The energy-dependence of electron mass in indium antimonide determined from measurements of the infrared Faraday effect. J Phys Chem Solids 11, 131-139, doi:http://dx.doi.org/10.1016/0022-3697(59)90051-4 (1959). 79 Cardona, M. Electron Effective Masses of InAs and GaAs as a Function of Temperature and Doping. Physical Review 121, 752-758, doi:10.1103/PhysRev.121.752 (1961). 80 Dixon, J. R. & Riedl, H. R. Electric-Susceptibility Hole Mass of Lead Telluride. Physical Review 138, A873-A881, doi:10.1103/PhysRev.138.A873 (1965). 66 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 153 Dionne, G. & Woolley, J. C. Optical Properties of Some Pb1-Xsnxte Alloys Determined from Infrared Plasma Reflectivity Measurements. Phys Rev B 6, 3898-3913 (1972). Harman, T. C. Galvano-thermomagnetic effects in semiconductors and semimetals— IV. mercury selenide. J Phys Chem Solids 25, 931-940, doi:http://dx.doi.org/10.1016/0022-3697(64)90030-7 (1964). Young, D. L., Coutts, T. J., Kaydanov, V. I., Gilmore, A. S. & Mulligan, W. P. Direct measurement of density-of-states effective mass and scattering parameter in transparent conducting oxides using second-order transport phenomena. Journal of Vacuum Science & Technology A 18, 2978-2985, doi:doi:http://dx.doi.org/10.1116/1.1290372 (2000). Pshenaĭ-Severin, D. A. & Fedorov, M. I. Effect of the band structure on the thermoelectric properties of a semiconductor. Phys Solid State+ 49, 1633-1637, doi:10.1134/S1063783407090053 (2007). Callaway, J. Model for Lattice Thermal Conductivity at Low Temperatures. Physical Review 113, 1046-1051, doi:10.1103/PhysRev.113.1046 (1959). Holland, M. G. Analysis of Lattice Thermal Conductivity. Physical Review 132, 24612471, doi:10.1103/PhysRev.132.2461 (1963). Shannon. <http://abulafia.mt.ic.ac.uk/shannon/ptable.php> ( Xi, L., Zhang, W., Chen, L. & Yang, J. Filled skutterudites: from single to multiple filling. Journal of the Korean Ceramic Society 47, 54-60 (2010). Slack, G. A. & Tsoukala, V. G. Some properties of semiconducting IrSb3. Journal of Applied Physics 76, 1665, doi:10.1063/1.357750 (1994). Sergueev, I. et al. Quenching rattling modes in skutterudites with pressure. Physical Review B 91, doi:10.1103/PhysRevB.91.224304 (2015). Cao, D. et al. Evidence for rattling behavior of the filler atom(L)in the filled skutteruditesLT4X12(L=Ce,Eu,Yb;T=Fe,Ru;X=P,Sb)from EXAFS studies. Physical Review B 70, doi:10.1103/PhysRevB.70.094109 (2004). Feldman, J. L. et al. Lattice vibrations inLa(Ce)Fe4Sb12andCoSb3: Inelastic neutron scattering and theory. Physical Review B 73, doi:10.1103/PhysRevB.73.014306 (2006). Toberer, E. S., Zevalkink, A. & Snyder, G. J. Phonon engineering through crystal chemistry. Journal of Materials Chemistry 21, 15843, doi:10.1039/c1jm11754h (2011). Callaway, J. & von Baeyer, H. Effect of Point Imperfections on Lattice Thermal Conductivity. Physical Review 120, 1149-1154, doi:10.1103/PhysRev.120.1149 (1960). Klemens, P. Thermal Resistance due to Point Defects at High Temperatures. Physical Review 119, 507-509, doi:10.1103/PhysRev.119.507 (1960). Klemens, P. G. The scattering of low-frequency lattice waves by static imperfections. Proc. Phys. Soc. A 68, 1113-1128 (1955). Yang, J., Meisner, G. P. & Chen, L. Strain field fluctuation effects on lattice thermal conductivity of ZrNiSn-based thermoelectric compounds. Applied Physics Letters 85, 1140, doi:10.1063/1.1783022 (2004). Dudkin, L. D. & Abrikosov, N. K. Zh. Neorg. Khim. 2, 212 (1957). Stokes, K. L., Ehrlich, A. C. & Nolas, G. S. Thermal Conductivity of Fe-Doped Cosb3 Skutterudites. MRS Proceedings 545, 339, doi:10.1557/PROC-545-339. (1998). 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 154 Anno, H., Matsubara, K., Notohara, Y., Sakakibara, T. & Tashiro, H. Effects of doping on the transport properties of CoSb[sub 3]. Journal of Applied Physics 86, 3780, doi:10.1063/1.371287 (1999). Yang, J. et al. Influence of electron-phonon interaction on the lattice thermal conductivity ofCo1−xNixSb3. Physical Review B 65, doi:10.1103/PhysRevB.65.094115 (2002). Pippard, A. B. On Ultrasonic attenuation in metals. Philos. Mag. 46, 1104-1114 (1955). Shi, X., Pei, Y., Snyder, G. J. & Chen, L. Optimized thermoelectric properties of Mo3Sb7−xTex with significant phonon scattering by electrons. Energy & Environmental Science 4, 4086, doi:10.1039/c1ee01406d (2011). Poujade, A. M. & Albany, H. J. Carrier-Concentration Dependence of ElectronPhonon Scattering in Te-Doped GaSb at Low Temperature. Physical Review 182, 802807, doi:10.1103/PhysRev.182.802 (1969). Cahill, D., Watson, S. & Pohl, R. Lower limit to the thermal conductivity of disordered crystals. Physical Review B 46, 6131-6140, doi:10.1103/PhysRevB.46.6131 (1992). Okamoto, H. Co-Sb (Cobalt-Antimony). J. Phase Equilib. 12, 244-245 (1991). Okamoto, H. in Binary Alloy Phase Diagrams Vol. 2 (ed T. B. Massalski) Ch. 1261-1262, 2 (1990). Okamoto, H. in Binary Alloy Phase Diagrams Vol. 3 (ed T. B. Massalski) 3317-3320 (1990). Zhao, X. Y. et al. Synthesis of Yb[sub y]Co[sub 4]Sb[sub 12]∕Yb[sub 2]O[sub 3] composites and their thermoelectric properties. Applied Physics Letters 89, 092121, doi:10.1063/1.2345249 (2006). Shi, X., Zhang, W., Chen, L. & Yang, J. Filling Fraction Limit for Intrinsic Voids in Crystals: Doping in Skutterudites. Physical Review Letters 95, doi:10.1103/PhysRevLett.95.185503 (2005). Harnwunggmoung, A., Kurosaki, K., Ohishi, Y., Muta, H. & Yamanaka, S. Effects of Tl-filling into the voids and Rh substitution for Co on the thermoelectric properties of CoSb3. Journal of Alloys and Compounds 509, 1084-1089, doi:10.1016/j.jallcom.2010.09.207 (2011). Mallik, R. C., Stiewe, C., Karpinski, G., Hassdorf, R. & Muller, E. Thermoelectric Properties of Co4Sb12 Skutterudite Materials with Partial In Filling and Excess In Additions. Journal of Electronic Materials 38, 1337-1343, doi:10.1007/s11664-009-0663-0 (2009). Harnwunggmoung, A. et al. Thermoelectric properties of Ga-added CoSb3 based skutterudites. Journal of Applied Physics 110, 5, doi:10.1063/1.3606417 (2011). Visnow, E. et al. On the True Indium Content of In-Filled Skutterudites. Inorganic chemistry 54, 7818-7827, doi:10.1021/acs.inorgchem.5b00799 (2015). Zhao, W. et al. Multi-localization transport behaviour in bulk thermoelectric materials. Nature communications 6, 6197, doi:10.1038/ncomms7197 (2015). Xiong, Z., Chen, X., Huang, X., Bai, S. & Chen, L. High thermoelectric performance of Yb0.26Co4Sb12/yGaSb nanocomposites originating from scattering electrons of low energy. Acta Materialia 58, 3995-4002, doi:10.1016/j.actamat.2010.03.025 (2010). 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 155 Sesselmann, A., Dasgupta, T., Kelm, K. & Müller, E. Transport properties and microstructure of indium-added cobalt–antimony-based skutterudites. J. Mater. Res. 26, 1820-1826, doi:10.1557/jmr.2011.102 (2011). Jaworski, C. M., Tobola, J., Levin, E. M., Schmidt-Rohr, K. & Heremans, J. P. Antimony as an amphoteric dopant in lead telluride. Physical Review B 80, doi:10.1103/PhysRevB.80.125208 (2009). Grytsiv, A., Rogl, P., Michor, H., Bauer, E. & Giester, G. In y Co4Sb12 Skutterudite: Phase Equilibria and Crystal Structure. Journal of Electronic Materials 42, 2940-2952, doi:10.1007/s11664-013-2679-8 (2013). Pomrehn, G. S., Toberer, E. S., Snyder, G. J. & van de Walle, A. Entropic stabilization and retrograde solubility in Zn4Sb3. Physical Review B 83, doi:10.1103/PhysRevB.83.094106 (2011). Ikeda, T., Ravi, V. & Snyder, G. Formation of Sb2Te3 Widmanstätten precipitates in thermoelectric PbTe. Acta Materialia 57, 666-672, doi:10.1016/j.actamat.2008.10.005 (2009). GRAFF, J. et al. High-Temperature Thermoelectric Properties of Co4Sb12-Based Skutterudites with Multiple Filler Atoms: Ce0.1InxYbyCo4Sb12. Journal of ELECTRONIC MATERIALS 40, 696-701, doi:10.1007/s11664-011-1514-3 (2011). Ballikaya, S., Uzar, N., Yildirim, S., Salvador, J. R. & Uher, C. High thermoelectric performance of In, Yb, Ce multiple filled CoSb3 based skutterudite compounds. Journal of Solid State Chemistry 193, 31-35, doi:10.1016/j.jssc.2012.03.029 (2012). Zhao, W. et al. Enhanced Thermoelectric Performance in Barium and Indium DoubleFilled Skutterudite Bulk Materials via Orbital Hybridization Induced by Indium Filler. J. AM. CHEM. SOC. 131, 3713-3720 (2009). Zhao, W. Y. et al. Synthesis and high temperature transport properties of barium and indium double-filled skutterudites BaxInyCo4Sb12−z. JOURNAL OF APPLIED PHYSICS 102, 6, doi:10.1063/1.2821364 (2007). Peng, J. Y. et al. Thermoelectric properties of (In,Yb) double-filled CoSb3 skutterudite. JOURNAL OF APPLIED PHYSICS 104, 5, doi:10.1063/1.2975184 (2008). Peng, J. et al. High temperature thermoelectric properties of double-filled InxYbyCo4Sb12 skutterudites. JOURNAL OF APPLIED PHYSICS 105, 5, doi:10.1063/1.3116151 (2009). Toberer, E. S., May, A. F. & Snyder, G. J. Zintl Chemistry for Designing High Efficiency Thermoelectric Materials. Chem. Mater. 22, 624-634, doi:10.1021/cm901956r (2010). Pomrehn, G. S., Zevalkink, A., Zeier, W. G., van de Walle, A. & Snyder, G. J. DefectControlled Electronic Properties in AZn2Sb2 Zintl Phases. Angew. Chem. Int. Ed. 53, 3422-3426, doi:10.1002/anie (2014). Xi, L. et al. Complex doping of group 13 elements In and Ga in caged skutterudite CoSb3. Acta Materialia 85, 112-121, doi:10.1016/j.actamat.2014.11.022 (2015). He, T., Chen, J., Rosenfeld, H. D. & Subramanian, M. A. Thermoelectric Properties of Indium-Filled Skutterudites. Chem. Mater. 18, 759-762 (2006). DU, Y., CAI, K. F., CHEN, S., QIN, Z. & SHEN, S. Z. Investigation on IndiumFilled Skutterudite Materials Prepared by Combining Hydrothermal Synthesis and Hot 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 156 Pressing. Journal of ELECTRONIC MATERIALS 40, 1215-1220, doi:10.1007/s11664-011-1603-3 (2011). Mallik, R. C., Jung, J.-Y., Ur, S.-C. & Kim, I.-H. Thermoelectric Properties of InzCo4Sb12 Skutterudites. METALS AND MATERIALS International 14, 223-228, doi:10.3365/met.mat.2008.04.223 (2008). Meisner, G. P. in Directions in Engine-Efficiency and Emissions Research. Nolas, G. S., Kaeser, M., Littleton, R. T. & Tritt, T. M. High figure of merit in partially filled ytterbium skutterudite materials. Applied Physics Letters 77, 1855-1857, doi:10.1063/1.1311597 (2000). Li, H., Tang, X., Zhang, Q. & Uher, C. Rapid preparation method of bulk nanostructured Yb[sub 0.3]Co[sub 4]Sb[sub 12+y] compounds and their improved thermoelectric performance. Applied Physics Letters 93, 252109, doi:10.1063/1.3054158 (2008). Mei, Z., Zhang, W., Chen, L. & Yang, J. Filling fraction limits for rare-earth atoms in CoSb3: An ab initio approach. Physical Review B 74, doi:10.1103/PhysRevB.74.153202 (2006). Shi, X., Zhang, W., Chen, L. D., Yang, J. & Uher, C. Thermodynamic analysis of the filling fraction limits for impurities in CoSb3 based on ab initio calculations. Acta Materialia 56, 1733-1740, doi:10.1016/j.actamat.2007.12.012 (2008). Dilley, N. R., Bauer, E. D., Maple, M. B. & Sales, B. C. Thermoelectric properties of chemically substituted skutterudites Yb[sub y]Co[sub 4]Sn[sub x]Sb[sub 12−x]. Journal of Applied Physics 88, 1948-1951, doi:10.1063/1.1305837 (2000). Xia, X. et al. High-Temperature Oxidation Behavior of Filled Skutterudites Yb y Co4Sb12. Journal of Electronic Materials 41, 2225-2231, doi:10.1007/s11664-012-2038-1 (2012). Salvador, J. R. et al. Transport and mechanical properties of Yb-filled skutterudites. Philosophical Magazine 89, 1517-1534, doi:10.1080/14786430903019099 (2009). Lamberton, G. A., Tedstrom, R. H., Tritt, T. M. & Nolas, G. S. Thermoelectric properties of Yb-filled Ge-compensated CoSb[sub 3] skutterudite materials. Journal of Applied Physics 97, 113715, doi:10.1063/1.1927702 (2005). Vegard, L. Die Konstitution der Mischkristalle und die Raumfüllung der Atome. Zeitschrift für Physik 5, 17-26 (1921). Li, X. Y. et al. Thermoelectric properties of Te-doped CoSb[sub 3] by spark plasma sintering. Journal of Applied Physics 98, 083702, doi:10.1063/1.2067704 (2005). Wojciechowski, K. T., Toboła, J. & Leszczyński, J. Thermoelectric properties and electronic structure of CoSb3 doped with Se and Te. Journal of Alloys and Compounds 361, 19-27, doi:10.1016/s0925-8388(03)00411-0 (2003). Liu, W.-S., Zhang, B.-P., Li, J.-F., Zhang, H.-L. & Zhao, L.-D. Enhanced thermoelectric properties in CoSb[sub 3-x]Te[sub x] alloys prepared by mechanical alloying and spark plasma sintering. Journal of Applied Physics 102, 103717, doi:10.1063/1.2815671 (2007). Fairbanks, J. W. in Directions in Engine-Efficiency and Emissions Research. Emsley, J. Nature's Building Blocks: An A-Z Guide to the Elements. New Edition 2011 edn, 120, 614 (Oxford University Press Inc., 2011). The Chinese Rare Earth Website, <http://www.zg-xtw.com/> (2015). 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 157 Bai, S. et al. Enhanced Thermoelectric Performance of Dual-Element-Filled Acta Materialia 57, 3135-3139, Skutterudites BaxCeyCo4Sb12. doi:10.1016/j.actamat.2009.03.018 (2009). Zhang, L. et al. Thermoelectric performance of mischmetal skutterudites MmyFe4−xCoxSb12 at elevated temperatures. Journal of Alloys and Compounds 490, 1925, doi:10.1016/j.jallcom.2009.10.033 (2010). Liu, R., Qiu, P., Chen, X., Huang, X. & Chen, L. Composition optimization of p-type skutterudites CeyFexCo4−xSb12 and YbyFexCo4−xSb12. Journal of Materials Research 26, 1813-1819, doi:10.1557/jmr.2011.85 (2011). Tang, X., Zhang, Q., Chen, L., Goto, T. & Hirai, T. Synthesis and thermoelectric properties of p-type- and n-type-filled skutterudite R[sub y]M[sub x]Co[sub 4−x]Sb[sub 12](R:Ce,Ba,Y;M:Fe,Ni). Journal of Applied Physics 97, 093712, doi:10.1063/1.1888048 (2005). Chen, B. et al. Low-temperature transport properties of the filled skutterudites CeFe4xCoxSb12. Physical Review B 55, 1476-1480 (1997). Nolas, G. S., Cohn, J. L. & Slack, G. A. Effect of partial void filling on the lattice thermal conductivity of skutterudites. Physical Review B 58, 164-170 (1998). Qiu, P. et al. Enhancement of thermoelectric performance in slightly chargecompensated CeyCo4Sb12 skutterudites. Applied Physics Letters 103, 062103, doi:10.1063/1.4817720 (2013). Shi, X. et al. Low thermal conductivity and high thermoelectric figure of merit in n-type Ba[sub x]Yb[sub y]Co[sub 4]Sb[sub 12] double-filled skutterudites. Applied Physics Letters 92, 182101, doi:10.1063/1.2920210 (2008). Uher, C. in Semiconductors and Semimetals Vol. 69 (ed T. M. Tritt) Ch. 5, 14-16 (Academic Press, 2000). Luo, R. M., Liu, F. S., Li, J. Q. & Feng, X. W. The isothermal section of the Ce–Co–Sb ternary system at 400°C. Journal of Alloys and Compounds 471, 60-63, doi:10.1016/j.jallcom.2008.03.061 (2009). Gierlotka, W. et al. The Co-Sb-Ga system: isoplethal section and thermodynamic modeling. Metallurgical and Materials Transactions (2015). Kim, Y.-J. et al. Three-Dimensional Atom-Probe Tomographic Analyses of LeadTelluride Based Thermoelectric Materials. Jom 66, 2288-2297, doi:10.1007/s11837-0141155-0 (2014). Ding, J. et al. Creation of Yb2O3 Nanoprecipitates Through an Oxidation Process in Bulk Yb-Filled Skutterudites. Journal of Electronic Materials 42, 382-388, doi:10.1007/s11664-012-2370-5 (2013). Kim, H.-S., Gibbs, Z. M., Tang, Y., Wang, H. & Snyder, G. J. Characterization of Lorenz number with Seebeck coefficient measurement. APL Materials 3, 041506, doi:10.1063/1.4908244 (2015). Aminorroaya-Yamini, S. et al. Rational Design of p-Type Thermoelectric PbTe: Temperature Dependent Sodium Solubility. Journal of Materials Chemistry A 1, 8725-8730 (2013). Vining, C. B. A model for the high-temperature transport properties of heavily doped n-type silicon-germanium alloys. Journal of Applied Physics 69, 331-341, doi:10.1063/1.347717 (1991). 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 158 Pei, Y., May, A. F. & Snyder, G. J. Self-Tuning the Carrier Concentration of PbTe/Ag2Te Composites with Excess Ag for High Thermoelectric Performance. Advanced Energy Materials 1, 291-296, doi:10.1002/aenm.201000072 (2011). Heinz, N. A., Ikeda, T., Pei, Y. & Snyder, G. J. Applying Quantitative Microstructure Control in Advanced Functional Composites. Advanced Functional Materials 24, 21352153, doi:10.1002/adfm.201302899 (2014). Pearson, W. B. The crystal structures of semiconductors and a general valence rule. Acta Crystallographica 17, 1-15, doi:10.1107/S0365110X64000019 (1964). Dudkin, L. D. Sov. Phys.-Tech. Physics 3, 216 (1958). Nakagawa, H. et al. Thermoelectric Properties of CoSb3 prepared by copper mold quenching technique. 15th International Conference on Thermoelectrics 117-121 (1996). Ikeda, M. et al. Multiband Transport in CoSb3 Prepared by Rapid Solidification. ZEITSCHRIFT FUR ANORGANISCHE UND ALLGEMEINE CHEMIE 641, 2020-2028, doi:10.1002/zaac.201500179 (2015). Tang, X. F. et al. Effects of filling atoms Ba, Ce, Y and substituting atoms Fe, Ni on lattice thermal conductivity of RyMxCo4−xSb12. Journal of Alloys and Compounds 394, 259-264, doi:10.1016/j.jallcom.2004.08.105 (2005). Okamoto, H. Ce-Sb (Cerium-Antimony). J. Phase Equilib. 22, 88 (2001). Okamoto, H. in Binary Alloy Phase Diagrams Vol. 2 (ed T. B. Massalski) 1763-1767 (1990). Su, X. & Tedenac, J.-C. Thermodynamic modeling of the ternary Ce-Fe-Sb system: assessment of the Ce-Sb and Ce-Fe systems. CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 30, 455-460 (2006). Zhu, T., Fu, C., Xie, H., Liu, Y. & Zhao, X. High Efficiency Half-Heusler Thermoelectric Materials for Energy Harvesting. Advanced Energy Materials, n/a-n/a, doi:10.1002/aenm.201500588 (2015). Amornpitoksuk, P., Li, H., Tedenac, J.-C., Fries, S. G. & Ravot, D. Experimental determination of phase equilibrium in the Fe–Co–Sb ternary system. Intermetallics 15, 475-478, doi:10.1016/j.intermet.2006.07.004 (2007). Geller, W. Das System Eisen-Kobalt-Antimon. Arch. Eisenhuettenwes. 13, 263-266 (1939/40). Li, J.-Q. et al. Solvothermal synthesis of nano-sized skutterudite Co4-x Fex Sb12 powders. Materials Chemistry and Physics 112, 57-62, doi:10.1016/j.matchemphys.2008.05.017 (2008). Chumak, I. V., Pavlyuk, V. V., Dmytriv, G. S. & Stepen' Damm, Y. Phase equilibria and crystal structure of compounds in the Fe - Zn - Sb system at 570 K. Journal of Alloys and Compounds 307 (2000). Goldsmid, H. J. & Sharp, J. W. Estimation of the Thermal Band Gap of a Semiconductor from Seebeck Measurements. Journal of Electronic Materials 28, 869-872 (1999). Nolas, G. S., Sharp, J. W. & Goldsmid, H. J. Thermoelectrics Basic Principles and New Materials Developments. (Springer, 2001). Ioffe, A. F. Physics of Semiconductors. (Academic Press Inc., 1960). Fistul, V. I. Heavily Doped Semiconductors. (Plenum Press, 1969). 185 186 187 188 189 159 Heremans, J. P. et al. Enhancement of thermoelectric efficiency in PbTe by distortion of the electronic density of states. Science 321, 554-557, doi:DOI 10.1126/science.1159725 (2008). Mizutani, U. Introduction to the electron theory of metals. (Cambridge University Press, 2001). Ravich, Y. I., Efimova, B. A. & Smirnov, I. A. Semiconducting Lead Chalcogenides Vol. 5 (Plenum Press, 1970). Gibbs, Z. M. Band Engineering in Thermoelectric Materials Using Optical, Electronic, and Abinitio Computed Properties PhD thesis, California Institute of Technology, (2015). Wang, H., Pei, Y., LaLonde, A. D. & Snyder, G. J. Weak electron–phonon coupling contributing to high thermoelectric performance in n-type PbSe. Proceedings of the National Academy of Sciences 109, 9705-9709, doi:10.1073/pnas.1111419109 (2012).