Spin-Orbital Order and Condensation in 4d and 5d Transition Metal Oxides DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Christopher Svoboda, M.S., B.S. Graduate Program in Physics The Ohio State University 2017 Dissertation Committee: Professor Nandini Trivedi, Advisor Professor Mohit Randeria Professor Jay Gupta Professor Linda Carpenter c Copyright by Christopher Svoboda 2017 Abstract Strong correlations and strong spin-orbit coupling are important areas of research in condensed matter physics with many open questions. Transition metal oxides provide a natural way to combine these strong correlations and strong spin-orbit coupling in elec- tronic systems. Iridium compounds in the d5 configuration (Ir4+) have received most of the focus in this area for the last decade, yet there remains much more unexplored territory with other electron counts. Here we explore the magnetism in several classes of 4d and 5d Mott insulating transition metal oxides with d1, d2, and d4 electron counts. We first cover 1 2 double perovskites A2BB'O6 where the B' ion is in either the d and d configuration and the other ions are nonmagnetic. We develop and solve magnetic models with both spin and orbital degrees of freedom within mean field theory. The anisotropic orbital degrees of free- dom play a crucial role in resolving some outstanding puzzles in these compounds including why ferromagnetism is common in d1 but not d2 and why the d1 ferromagnets have negative Curie-Weiss temperatures. Then we cover a broad class of compounds in the d4 configura- tion. Despite the fact that Hund's rules dictate the ground state should be nonmagnetic, we find that superexchange may induce magnetic moments and magnetic ordering through the condensation of triplon excitations. We find condensation occurs only at ~k = ~π which generates antiferromagnetic order in the models we consider, and strong Hund's coupling does not induce ferromagnetism in the large spin-orbit coupling limit even though it induces ferromagnetic interactions in the absence of spin-orbit coupling. We then apply our results 4 to the d double perovskite Ba2YIrO6. Although experimental observations indicate the material possesses magnetic moments, we show that these magnetic moments are likely not due to condensation induced by superexchange. ii Acknowledgments I am grateful to my graduate advisor, Professor Nandini Trivedi, for the support she provided throughout the entire course of my PhD. I would like to thank Professor Mohit Randeria for his wisdom and perspective on several projects. I would also like to thank our experimental collaborators, Professor Patrick Woodward, Professor Jiaqiang Yan, and Professor Rolando Vald´esAguilar, for their insights and their commitment to our multidisciplinary endeavors. Finally I would like to thank my undergraduate advisor, Professor Thomas Vojta, for the opportunities he provided during my undergraduate years. I acknowledge the support of the Center for Emergent Materials, an NSF MRSEC, under Award Number DMR-1420451. iii Vita December, 2011 . B.S., Missouri University of Science and Technology, Rolla, MO May, 2015 . M.S., The Ohio State University, Colum- bus, OH Publications C. Svoboda, M. Randeria, N. Trivedi. \Orbital and spin order in spin-orbit coupled d1 and d2 double perovskites". arXiv:1702.03199 C. Svoboda, M. Randeria, N. Trivedi. “Effective magnetic interactions in spin-orbit coupled d4 Mott insulators". Phys. Rev. B 95, 014409 (2017) T. T. Mai, C. Svoboda, M. T. Warren, T.-H. Jang, J. Brangham, Y. H. Jeong, S-W. Cheong, R. Vald´esAguilar. \Terahertz Spin-Orbital Excitations in the paramagnetic state of multiferroic Sr2FeSi2O7". Phys. Rev. B 94, 224416 (2016) W. Tian, C. Svoboda, M. Ochi, M. Matsuda, H. B. Cao, J.-G. Cheng, B. C. Sales, D. G. Mandrus, R. Arita, N. Trivedi, J.-Q. Yan. \High antiferromagnetic transition temperature of a honeycomb compound SrRu2O6". Phys. Rev. B 92, 100404(R) (2015) L. Demk´o,S. Bord´acs,T. Vojta, D. Nozadze, F. Hrahsheh, C. Svoboda, B. D´ora,H. Yamada, M. Kawasaki, Y. Tokura, I. K´ezsm´arki. \Disorder Promotes Ferromagnetism: Rounding of the Quantum Phase Transition in Sr1−xCaxRuO3". Phys. Rev. Lett. 108, 185701 (2012) C. Svoboda, D. Nozadze, F. Hrahsheh, and T. Vojta. \Disorder correlations at smeared phase transitions". Europhys. Lett. 97, 20007 (2012) iv Fields of Study Major Field: Physics v Table of Contents Page Abstract........................................... ii Acknowledgments..................................... iii Vita............................................. iv List of Figures ...................................... viii Chapters 1 Introduction ..................................... 1 1.1 Historical Motivation............................... 1 1.2 General Overview................................. 4 1.3 Summary of Results ............................... 5 2 Transition Metal Oxides .............................. 9 2.1 Materials and the Hubbard Model ....................... 9 2.2 Crystal Field ................................... 11 2.3 Coulomb Interactions .............................. 13 2.4 Spin-Orbit Coupling............................... 17 2.5 Band Limit versus Mott Limit ......................... 18 3 d1 and d2 Double Perovskites ........................... 22 3.1 Introduction.................................... 22 3.2 d1 Double Perovskites .............................. 26 3.2.1 Model................................... 27 3.2.2 Zero Temperature Mean Field Theory................. 29 3.2.3 Finite Temperature Mean Field Theory ................ 33 3.2.4 Simplified Model at Finite Temperature................ 38 3.3 d2 Double Perovskites .............................. 40 3.3.1 Model................................... 41 3.3.2 Zero Temperature Mean Field Theory................. 41 3.3.3 Finite Temperature Mean Field Theory ................ 44 3.4 Discussion..................................... 46 4 d4 Mott Insulators .................................. 48 4.1 Introduction.................................... 48 4.2 Model....................................... 51 vi 4.3 Exact diagonalization .............................. 54 4.4 Effective Magnetic Hamiltonian......................... 57 4.4.1 Norb = 3.................................. 59 4.4.2 Norb = 2.................................. 60 4.4.3 Norb = 1.................................. 62 4.5 Orbital Frustration................................ 64 4.6 Triplon Condensation .............................. 65 4.6.1 Overview of the Mechanism....................... 66 4.6.2 Results .................................. 68 4.6.3 Local Interactions versus Condensation ................ 70 4.7 Materials and Experiments ........................... 73 4.8 Conclusions.................................... 75 5 Magnetic Condensation in Ba2YIrO6 ...................... 76 5.1 Introduction.................................... 76 5.2 Absence of Condensation ............................ 78 5.3 Application to Experiments........................... 82 Bibliography ....................................... 85 Appendices A Calculation Details for Transition Metal Oxides ............... 92 A.1 t2g Orbital Angular Momentum......................... 92 A.2 Multi-Orbital Hubbard Interaction....................... 94 B Calculation Details for d1 and d2 Double Perovskites ............ 97 B.1 d1 Superexchange................................. 97 1 B.2 µeff enhancement and To for d model ..................... 98 B.3 Susceptibility in the Simplified Model ..................... 100 B.4 Projection to j = 3=2............................... 101 C Calculations Details for d4 Mott Insulators .................. 104 C.1 Effective Hamiltonian .............................. 104 C.2 Condensation Formalism............................. 106 C.3 Condensation from Spin-Orbital Superexchange . 110 vii List of Figures Figure Page 1.1 Conventional band insulators and simple metals are found when spin-orbit coupling and Coulomb interactions are small. When spin-orbit coupling is tuned to be large, the result is still either a band insulator or metal, however, the result may be topologically non-trivial. When Coulomb interactions are tuned to be large, the result is a Mott insulator. Both 4d and 5d transition metal oxides combine both strong spin-orbit coupling and strong correlations. Adapted from reference [1]. ........................... 2 1.2 Honeycomb structure formed out of edge-sharing oxygen (purple) octahedra each enclosing a transition metal site (yellow). The Kitaev model is formed from three types of bond-dependent Ising interactions between sites (yellow) on the honeycomb lattice. The Ising interactions are Si;xSj;x along red bonds, Si;ySj;y along green bonds, and Si;zSj;z along blue bonds........... 3 2.1 d orbitals on the transition metal site (purple) are degenerate under spherical symmetry. When this symmetry is reduced to Oh due to the presence of neighboring ions (red), the d orbitals split into lower energy t2g and higher energy eg states. The energy difference, ∆CF, is the crystal field splitting. 12 2.2 After crystal field splitting, spin-orbit coupling further splits the t2g orbitals with spin degeneracy into lower energy j = 3=2 states and higher energy j = 1=2 states. .................................. 18 2.3 When t = 0 at half-filling, there is one electron per site in the ground state. Perturbing to second order in t U, charge fluctuation is allowed when the spins on nearby sites are antiparallel.