Half-Heusler Alloys as Promising Thermoelectric Materials by Alexander A. Page A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Physics) in the University of Michigan 2017 Doctoral Committee: Professor Ctirad Uher, Chair Assistant Professor Emmanouil Kioupakis Associate Professor Lu Li Associate Professor Pierre Ferdinand P. Poudeu Professor Anton Van der Ven, University of California Santa Barbara Alexander A. Page [email protected] ORCID iD: 0000-0002-8783-3659 © Alexander A. Page Dedication To my mother, father, and brother. ii Table of Contents Dedication ........................................................................................................................... ii Table of Contents ............................................................................................................... iii List of Figures ................................................................................................................... vii List of Tables ..................................................................................................................... xi Abstract ............................................................................................................................. xii Chapter 1 Introduction to Thermoelectrics ......................................................................... 1 1.1 Waste heat ............................................................................................................. 1 1.2 The thermoelectric effects......................................................................................... 2 Chapter 2 Half Heusler Alloys as Thermoelectric Materials .............................................. 6 2.1 The half-Heusler and full-Heusler crystal structure .................................................. 6 2.2 First principles studies of half-Heusler alloys .......................................................... 8 2.3 Synthesis of half-Heusler alloys ............................................................................. 11 Chapter 3 Density Functional Theory Calculations .......................................................... 14 3.1 Density functional theory ........................................................................................ 14 3.1.1 Schrödinger's Equation .................................................................................... 15 iii 3.1.2 The Hohenberg-Kohn theorem ........................................................................ 15 3.1.3 The Kohn-Sham equations ............................................................................... 16 3.1.4 Exchange-Correlation approximations ............................................................ 18 3.2 Ab Initio simulation methods.................................................................................. 20 3.2.1 General Methods .............................................................................................. 20 3.2.2 Density of states and band structure calculations ............................................ 23 3.2.3 Phonon dispersion calculations ........................................................................ 24 3.2.4 Migration barriers and activation energy calculations: .................................... 25 Chapter 4 Ab-initio Phase Diagrams ................................................................................. 28 Chapter 5 Phase Separation of Full-Heusler Nanostructures in Half-Heusler Thermoelectrics and Vibrational Properties from First-principles Calculations .............. 35 5.1 Introduction ............................................................................................................. 35 5.2 Configurational formation energies ........................................................................ 38 5.3 Pseudo-binary phase diagrams ................................................................................ 41 5.4 Vibrational properties of MNiSn and MNi2Sn compounds .................................... 44 5.4.1 Half-Heusler vibrational properties ................................................................. 44 5.4.2 Full-Heusler vibrational properties .................................................................. 46 5.4.3 Phonon density of states and thermal properties.............................................. 49 5.5 TiNi2Sn instabilitiy ................................................................................................. 52 5.5.1 Energies of distorted cells ................................................................................ 52 iv 5.5.2 Origins of TiNi2Sn instability .......................................................................... 56 5.6 Conclusions ............................................................................................................. 59 Chapter 6 Origins of Phase Separation in Thermoelectric (Ti,Zr,Hf)NiSn half-Heusler Alloys from First Principles .............................................................................................. 61 6.1 Introduction ............................................................................................................. 61 6.2 Methods................................................................................................................... 64 6.3 Pseudo-binary MNiSn systems ............................................................................... 64 6.3.1 Pseudo-binary formation energies ................................................................... 64 6.3.2 Pseudo-binary phase diagrams ......................................................................... 66 6.4 Pseudo-ternary Hf1-x-yZrxTiyNiSn system ............................................................... 69 6.4.1 Pseudo-ternary formation energies .................................................................. 69 6.4.2 Pseudo-ternary phase diagram ......................................................................... 70 6.5 Analysis and discussion .......................................................................................... 73 6.5.1 Comparison with experiment ........................................................................... 73 6.5.2 Atomic diffusion in half-Heuslers ................................................................... 74 6.6 Conclusions ............................................................................................................. 81 Chapter 7 Pb-based half-Heusler Alloys........................................................................... 82 7.1 MNiPb compounds.................................................................................................. 82 7.1.1 Introduction ...................................................................................................... 82 7.1.2 Stability of MNiPb ........................................................................................... 83 v 7.2 Solubility limits of Pb in MNiSn1-xPbx ................................................................... 85 7.2.1 Computational results ...................................................................................... 85 7.2.2 Experimental Results ....................................................................................... 86 Chapter 8 Concluding Remarks ........................................................................................ 90 References ......................................................................................................................... 92 vi List of Figures Figure 1. Energy flow diagram of US energy production and consumption. Source: Lawrence Livermore National Laboratory, March 2017. Data is based on DOE/EIA MER (2016) .............. 1 Figure 2. Plot of conversion efficiency vs. hot side temperature. The cold side temperature is taken to be 300 K and efficiencies for ZT = 1, 2, 3, and 4 are shown compared to the Carnot limit. ...... 4 Figure 3. (a) The crystal structure of half-Heusler and (b) full-Heusler alloys. Half-Heusler alloys have composition XYZ, whereas in the full-Heusler structure, the vacancies are filled in with a second atom (Y2-site), making the composition XY2Z. ................................................................. 7 Figure 4. Calculated band structure of ZrNiSn (a) and ZrNi Sn. Band energy is relative to the 2 valence band maximum (b). Unpublished, Page et al. ................................................................... 9 Figure 5. Plot of ZrNiSn formation energy as a function of k-point grid dimension. ................. 23 Figure 6. (a) Example configuration of twenty three atoms A and B (red and blue respectively) in an FCC lattice configuration. (b) Examples of possible singlet and pair clusters on an FCC lattice. ....................................................................................................................................................... 30 Figure 7. Calculated pseudo-binary phase diagram of TiNi Sn from x = 0 to 1. Circles show 1+x calculated phase transition points, between which, a two phase coexistence is stable. The MC method is limited to temperatures below the decomposition point of TiNiSn, above which the phase diagram can no longer be considered pseudo-binary.......................................................... 34 Figure 8. (a) TEM images of HH with excess Ni forming semi-coherent interfaces with the bulk HH matrix. (b) Schematic of the electronic bands across the nanostructure-bulk interface. The potential barrier created at the interface could enable an energy filtering process. (a) and (b) shown with permission from Makongo et al., JACS 133, 18843 (2011). ................................................ 36 Figure 9. Formation energy per formula unit cell relative to the ground states are shown for TiNi Sn (A), ZrNi Sn (B), and HfNi Sn (C). DFT calculated energies are shown as blue 1+x 1+x 1+x diamonds. The Cluster Expansion (CE) predicted energies for the DFT structures and further vii predictions for configurations up to x = 0.1 and x = 0.9 are shown as red dots. The dashed circle in (B) indicates specific configurations discussed in the text. .......................................................... 39 Figure 10. Temperature-concentration pseudo-binary phase diagrams of TiNi Sn (A), ZrNi Sn 1+x 1+x (B), HfNi Sn (C). The small black diamonds represent calculated points along the phase 1+x boundary and the horizontal dashed line indicates experimental melting points for the HH (or decomposition point for TiNiSn). ................................................................................................. 42 Figure 11. Phonon dispersion curves of TiNiSn (A), ZrNiSn (B), and HfNiSn (C) calculated with DFT shown along high symmetry paths. Optical bands are shown in red and acoustic in blue... 47 Figure 12. Phonon dispersion curves of TiNi Sn (A), ZrNi Sn (B), and HfNi Sn (C) calculated 2 2 2 with DFT shown along high symmetry paths. Optical bands are shown in red and acoustic in blue. Imaginary frequencies are shown as negative values. .................................................................. 48 Figure 13. Total density of states (DOS) and partial density of states (PDOS) calculated with DFT. Total DOS is shown in black and contributions from each atom are shown in color, green for M = Ti, Zr, or Hf, red for Ni and blue for Sn. Parts (A) through (F) show the DOS for TiNiSn, ZrNiSn, HfNiSn, TiNi2Sn, ZrNi2Sn, HfNi2Sn, respectively....................................................................... 50 Figure 14. Heat capacity per atom is calculated using DFT. TiNiSn results are compared to experimental data of B. Zhong (ref. 92) ....................................................................................... 51 Figure 15. Atomic motions of the X-point TA phonon mode are shown for each atom of the TiNi Sn structure. The phonon mode travels in the cubic [001], out of the page. Black arrows 2 indicate atomic motion with amplitudes magnified by 20x. ......................................................... 53 Figure 16. Energy per formula unit of 2x2x2 supercells is shown as a function of phonon mode displacement amplitude for (A) X-point, (B) K-point, and (C) U-point modes for TiNi Sn. ZrNi Sn 2 2 and HfNi Sn energies are shown in (A) for comparison and have positive curvature along the 2 distortion path, whereas TiNi Sn has negative curvature with energies that drop 1.1 meV below 2 that of cubic TiNi Sn. K and U points show no instabilities along their paths. The horizontal axis 2 measures the displacement of Ni atoms from their equilibrium position. .................................... 54 Figure 17. Contour plot shows the change in formation energy of TiNi Sn structure as a function 2 of X-TA phonon mode amplitudes ε1 and ε2. ............................................................................... 54 Figure 18. Formation energies of TA-X mode distorted TiNi Sn structures relative to the FCC 2 structure are shown for cells of varying lattice parameter a, where a = d*a , and d = 0.98, 0.99, 0 1.00, 1.01, 1.02, 1.04 and a0 = 6.116 Å ......................................................................................... 57 viii Figure 19. Schematic of the FCC structure (A) and the X-TA mode displaced structure (B), viewed along the cubic [001] direction. The distance between atoms is shown next to each bond. Black arrows centered on atoms show the direction of their displacement relative to the FCC structure. ....................................................................................................................................................... 57 Figure 20. Formation energy per formula unit relative to the pure end states for Zr Ti NiSn (a), 1-x x Hf Ti NiSn (b), and Hf Zr NiSn (c). The configurational energies calculated by DFT are shown 1-x x 1-x x as blue diamonds, and CE predicted energies are shown as red dots. .......................................... 66 Figure 21. The temperature-composition phase diagrams for (a) Zr Ti NiSn and (b) Hf 1-x x 1- Ti NiSn. Black dots represent calculated points along the phase boundary. Outside of the x x boundary, a solid solution minimizes the free energy. Inside the boundary, a two phase coexistence minimizes the free energy. ............................................................................................................ 67 Figure 22. Pseudo-ternary DFT formation energies of the Hf Zr Ti NiSn system relative to the 1-x-y x y three pure states. 277 different configurations up to volumes five times the primitive cell were calculated in VASP. All configurations are found to have positive formation energy, confirming no local ordering is stable at zero temperature. ............................................................................ 69 Figure 23. Pseudo-ternary DFT lattice parameters of configurations. The lattice parameters follow Vegard’s law very closely. ............................................................................................................ 70 Figure 24. Pseudo-ternary (Hf Zr Ti )NiSn phase diagrams were calculated at 300 K (a), 500 1-x-y x y K (b), 700 K (c). The solid solution region is shown in color, and the miscibility gap region is shown in white. As the temperature increases, the miscibility gap shrinks, and completely disappears above 900 K. Part (d) shows a summary of phase boundaries calculated from 300 to 800 K. ............................................................................................................................................ 71 Figure 25. (a) Schematic of the migration path in the MNiSn (M = Ti, Zr, Hf) matrix of M-vacancy diffusion. (b), (c), and (d) show possible paths for Ni diffusion in variable Ni rich environments. ....................................................................................................................................................... 77 Figure 26. (top) plot of migration barrier energy along the reaction path for Ti, Zr and Hf atoms in TiNiSn, ZrNiSn, and HfNiSn respectively. Symbols represent calculated points along the reaction path and schematic lines are drawn to guide the eye. (bottom) Plot of the migration barrier energy along the lowest energy path for Ni atoms in stoichiometric ZrNiSn. ............................. 78 ix