Spin-Orbital Order and Condensation in 4d and 5d Transition Metal Oxides

Transcription

1 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

2 c Copyright by Christopher Svoboda 2017

3 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 electronic systems. Iridium compounds in the d 5 configuration (Ir 4+ ) 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 d 1, d 2, and d 4 electron counts. We first cover double perovskites A 2 BB O 6 where the B ion is in either the d 1 and d 2 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 freedom play a crucial role in resolving some outstanding puzzles in these compounds including why ferromagnetism is common in d 1 but not d 2 and why the d 1 ferromagnets have negative Curie-Weiss temperatures. Then we cover a broad class of compounds in the d 4 configuration. 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 to the d 4 double perovskite Ba 2 YIrO 6. 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

4 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 Valdés Aguilar, 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 iii

5 Vita December, B.S., Missouri University of Science and Technology, Rolla, MO May, M.S., The Ohio State University, Columbus, OH Publications C. Svoboda, M. Randeria, N. Trivedi. Orbital and spin order in spin-orbit coupled d 1 and d 2 double perovskites. arxiv: C. Svoboda, M. Randeria, N. Trivedi. Effective magnetic interactions in spin-orbit coupled d 4 Mott insulators. Phys. Rev. B 95, (2017) T. T. Mai, C. Svoboda, M. T. Warren, T.-H. Jang, J. Brangham, Y. H. Jeong, S-W. Cheong, R. Valdés Aguilar. Terahertz Spin-Orbital Excitations in the paramagnetic state of multiferroic Sr 2 FeSi 2 O 7. Phys. Rev. B 94, (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 SrRu 2 O 6. Phys. Rev. B 92, (R) (2015) L. Demkó, S. Bordács, T. Vojta, D. Nozadze, F. Hrahsheh, C. Svoboda, B. Dóra, H. Yamada, M. Kawasaki, Y. Tokura, I. Kézsmárki. Disorder Promotes Ferromagnetism: Rounding of the Quantum Phase Transition in Sr 1 x Ca x RuO 3. Phys. Rev. Lett. 108, (2012) C. Svoboda, D. Nozadze, F. Hrahsheh, and T. Vojta. Disorder correlations at smeared phase transitions. Europhys. Lett. 97, (2012) iv

6 Fields of Study Major Field: Physics v

7 Table of Contents Page Abstract ii Acknowledgments iii Vita iv List of Figures viii Chapters 1 Introduction Historical Motivation General Overview Summary of Results Transition Metal Oxides Materials and the Hubbard Model Crystal Field Coulomb Interactions Spin-Orbit Coupling Band Limit versus Mott Limit d 1 and d 2 Double Perovskites Introduction d 1 Double Perovskites Model Zero Temperature Mean Field Theory Finite Temperature Mean Field Theory Simplified Model at Finite Temperature d 2 Double Perovskites Model Zero Temperature Mean Field Theory Finite Temperature Mean Field Theory Discussion d 4 Mott Insulators Introduction Model vi

8 4.3 Exact diagonalization Effective Magnetic Hamiltonian N orb = N orb = N orb = Orbital Frustration Triplon Condensation Overview of the Mechanism Results Local Interactions versus Condensation Materials and Experiments Conclusions Magnetic Condensation in Ba 2 YIrO Introduction Absence of Condensation Application to Experiments Bibliography Appendices A Calculation Details for Transition Metal Oxides A.1 t 2g Orbital Angular Momentum A.2 Multi-Orbital Hubbard Interaction B Calculation Details for d 1 and d 2 Double Perovskites B.1 d 1 Superexchange B.2 µ eff enhancement and T o for d 1 model B.3 Susceptibility in the Simplified Model B.4 Projection to j = 3/ C Calculations Details for d 4 Mott Insulators C.1 Effective Hamiltonian C.2 Condensation Formalism C.3 Condensation from Spin-Orbital Superexchange vii

9 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] 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 S i,x S j,x along red bonds, S i,y S j,y along green bonds, and S i,z S j,z along blue bonds d orbitals on the transition metal site (purple) are degenerate under spherical symmetry. When this symmetry is reduced to O h due to the presence of neighboring ions (red), the d orbitals split into lower energy t 2g and higher energy e g states. The energy difference, CF, is the crystal field splitting After crystal field splitting, spin-orbit coupling further splits the t 2g orbitals with spin degeneracy into lower energy j = 3/2 states and higher energy j = 1/2 states 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. The result of this perturbation may either result in the original spin configuration or a configuration with reversed spins (a) Crystal lattice for double perovskite A 2 BB O 6. (b) The simplified tightbinding model takes hopping between xy orbitals (purple) on B sites within an xy plane. Similarly, zx orbitals are active in zx planes, and yz orbitals are active in yz planes viii

10 3.2 (a) FCC lattice decoupled into four inequivalent sites shown by four different colors. (b) The orbital ordering pattern driven by both J SE and V constrains the direction of orbital angular momentum. The magnetization operator is shown as M = 2S L. (c) The zero temperature phase diagram shows phases where the spin S and orbital L moments in each plane are collinear and the moments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing orbital repulsion V between sites reduces the minimum strength of Hund s coupling required to induce FM. (d) Mean field values for the bottom sites (black, yellow) are shown as a function of temperature. The n yz orbital (red) has the largest occupancy followed by the xy orbital (blue). (e) With J SE = 0, we calculate the orbital ordering temperature T o and effective Curie moment enhancement µ eff for different values of V Typical susceptibility, χ = 1 3 (χ xx + χ yy + χ zz ), and inverse susceptibility are plotted against temperature. The susceptibility curves are shown both without the correction due to covalency, γ = 1, and with the correction, γ = We have chosen J SE = 0 and left V finite to illustrate the consequence of high temperature orbital order on the susceptibility. By choosing J SE = 0, we show that although T c = 0 while T o 0, the fitted Curie- Weiss temperature appears to be negative. Note that a single Curie-Weiss fit cannot span the entire range below T o (a) The canted ferromagnetic solution to equation (3.6) is shown. (b) For J > 0 and K = 0, susceptibility along [110], [110], and [001] is shown for the antiferromagnet with φ = π/4. (c) For J > 0 and K > 0, the susceptibility diverges at T c. The canting angle satisfies π/4 < φ < 0. Note that the Curie-Weiss law still holds at temperatures well above J and K, and the Curie-Weiss intercept is still negative (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent the most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature phase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and FM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in each region. Increasing orbital repulsion V moves the phase boundary between AFM 4-sublattice phase and the AFM [110] phase down to favor the AFM 4-sublattice phase. The phase boundary between the AFM 4-sublattice phase and the FM [100] phase moves up in favor of the AFM 4-sublattice phase Characteristic inverse susceptibility (blue/green) and orbital occupation (purple) curves are plotted against temperature for the three phases in Fig. 3.5: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility is averaged over all three directions, χ 1 = 3(χ xx + χ yy + χ zz ) 1, and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured above each plot ix

11 4.1 (a) The single site total angular momentum is zero in both the jj and LS coupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (4.3) pitting spin-orbit coupling λ against superexchange J SE where λ is the spin-orbit coupling energy scale and J SE is the superexchange energy scale with z being the coordination number. Starting with a van Vleck phase with no atomic moments at large λ we find a triplon condensate at k = π for all values of the Hund s coupling J H /U. The intermediate regime where λ zj SE has not been explored. At large J SE we obtain effective magnetic Hamiltonians that have isotropic Heisenberg spin interactions (antiferromagnetic for small J H /U and ferromagnetic for large J H /U) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and large J SE /λ regimes (a) The N orb = 2 model is an approximation of oxygen mediated electron hopping between t 2g orbitals in a simple cubic lattice. Both d xy and d yz orbitals participate in hopping along the y direction. (b) The N orb = 1 model is an approximation of direct hopping between t 2g orbitals on the face of a face-centered cubic lattice. The d xy orbitals are most relevant for hopping in the xy plane The Hamiltonian in (4.1) is solved for a two-site system. The local total angular momentum squared on one site, Ji 2, is plotted for small and large values of Hund s coupling, J H /U = 0.1 and J H /U = 0.2, for the three types of hopping matrices used in the text. (a-b) Hopping using N orb = 3 produces sizable local moments. For small Hund s coupling, the local moment gradually forms as t is turned on. For large Hund s coupling, there is an abrupt formation of large local moments due to an energy level crossing. (cd) Hopping using N orb = 2 produces qualitatively similar behavior to the N orb = 3 case. (e-f) Hopping using N orb = 1 produces negligible moments (a) The virtual process leaves the first site in a low spin, S = 1 2, configuration and results in antiferromagnetic superexchange. (b) The virtual process leaves the first site in a high spin, S = 3 2, configuration and results in ferromagnetic superexchange Energy eigenvalues of the two-site superexchange Hamiltonian (4.5) are plotted for (a) N orb = 3 using (4.6), (b) N orb = 2 using (4.8), and (c) N orb = 1 using (4.10). In addition to a spin-af ground state, a spin-f ground state can be favored when Hund s coupling is large in both the N orb = 3 and N orb = 2 models Expectation values of different angular momentum correlators are plotted for the two-site effective Hamiltonian in (4.3) using the three different N orb models with the parameterization λ = cos θ, t 2 /U = sin θ, J H /U = 0.1 and λ/j H = 1. The N orb = 3 model features full rotational symmetry while the N orb = 2 and N orb = 1 models only have one axis of rotational symmetry to make the z correlators different than the x and y correlators. The effect of increasing J H /U is to push the crossing point from spin-af to spin-f behavior further left in these plots x

12 4.7 Orbital frustration is graphically illustrated for the N orb = 2 model. The orbitals shown on the vertices of the plaquette are the doubly occupied orbital on each site in a square lattice. Once the first bond, labeled as 1, is chosen to be of a particular type, either (a) AF or (b) F, the next bonds, labeled as 2, are immediately fixed by this choice. The result is that the last bond on the plaquette, labeled as 3, then takes a configuration which is neither the most energetically favorable AF bond nor the most energetically favorable F bond The triplon condensation mechanism is graphically illustrated. When there exists a triplon excitation on a site, superexchange can move the excitation to neighboring sites. This effective hopping causes the triplon s energy to disperse in k-space. When superexchange becomes large enough, condensation of triplon excitations occurs as the bottom of the triplon band becomes lower in energy than the original J i = 0 level Energy levels of the effective Hamiltonian H SE +H SOC appearing in equations (4.18) and (4.19) with the parameterization λ = cos θ and J SE = sin θ. The levels are labeled by their good quantum numbers. In the θ = 0 limit, the eigenstates of spin-orbit coupling are used, and, in the θ = π/2 limit, the eigenstates of the spin-orbital superexchange Hamiltonian are used. The interpretations of the states are discussed in the main text The GGA band structure without spin-orbit coupling is shown. A tightbinding model with t 2g Wannier orbitals is fit to the three bands pictured The triplet excitation spectrum is plotted using a typical value for the spinorbit gap in 5d oxides of λ/2 = 200 mev B.1 Examples of superexchange processes are shown graphically. Of the three t 2g orbitals shown by three levels, the active orbitals along a particular bond direction are highlighted in green. (a) Ferromagnetic spin exchange occurs when only one site contains an electron on the active orbital. The virtual d 2 state is in an S = 1 configuration. (b) Antiferromagnetic spin exchange occurs with the overlap of half filled orbitals. The virtual d 2 state is in a total S = 0 configuration C.1 The flow of angular momentum is graphically shown where ingoing arrows are incoming angular momentum and outgoing arrows are outgoing angular momenta. Wigner-3j symbols and Clebsch-Gordan coefficients are vertices with three legs while the scalar contraction of four Wigner-3j symbols (right) is a Wigner-6j symbol [2, 3]. (a) Equation (C.17) is shown in graphical form for the T ζ part of the equation. A J = 0 state is decomposed into its L = 1 and S = 1 components which are acted on by the (L i ) l m and (S i ) s σ operators. The resulting L = 1 and S = 1 are combined together to give a J = 1 state with quantum number ζ = m + σ. (b) The projection of equation (C.16) to (C.18) conserves angular momentum. Equation (C.19) will appear similarly except that 1m and 1m add to yield LM instead xi

13 Chapter 1 Introduction 1.1 Historical Motivation Two sets of topics have come to dominate the study of electronic systems in condensed matter: strong correlations and strong spin-orbit coupling. Strong correlations in electronic systems refers to situations where interactions between particles drive the system away from the Fermi liquid regime. Phenomenology associated with strong correlations typically includes magnetism, high-temperature superconductivity, Mott physics, and more. Many of the most recent advancements have been in areas involving strong spin-orbit coupling. The associated phenomenology includes topological insulators, topological superconductors, and Weyl and Dirac semimetals. While each of these subjects has drawn an enormous amount of interest individually, systems which combine these two aspects have come to the forefront of condensed matter research in the last decade. [1] From a materials perspective, there are many routes to realizing electronic systems with strong correlations and strong spin-orbit coupling. Strong correlations typically occur in compounds containing atoms with valence d or f shells, ie. transition metals and rare earth elements. Strong spin-orbit coupling intrinsically occurs in materials with heavy nuclei, including materials with d and f shell atoms. Materials with heavy transition metal ions then naturally combine the two aspects. The most well known class of transition metal compounds are the transition metal oxides which consistently support strongly correlated phenomena. While magnetism has been a staple of research in transition metal oxides, only recently have the effects of strong spin- 1

14 Figure 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]. orbit coupling drawn significant interest. The iridates (iridium compounds with oxygen; see [4] for a review) were perhaps the first class of transition metal oxides studied where the effects of spin-orbit coupling were fully appreciated due to some pivotal observations. First was the experimental observation that Sr 2 IrO 4 is a Mott insulator. By comparison, Sr 2 RhO 4 is metallic, and the bandwidth W was expected to increase moving from Rh to Ir. But reducing the bandwidth is key to achieve a Mott insulator as the Coulomb energy scale U needs to dominate. Density functional theory calculations could only achieve an insulating ground state if spin-orbit coupling was included [5]. This turned Sr 2 IrO 4 from effectively a three band system with five electrons (t 5 2g configuration) to an effective halffilled j = 1/2 band with a narrow bandwidth. Due to this effect, Sr 2 IrO 4 was dubbed a spin-orbit assisted Mott insulator. The importance of spin-orbit coupled j = 1/2 bands on the magnetic properties of the iridates was quickly realized. In contrast to spin s = 1/2 systems, the j = 1/2 systems are an entangled mixture of spin and orbital characters. Spins are insensitive to the anisotropies of 2

15 Figure 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 S i,x S j,x along red bonds, S i,y S j,y along green bonds, and S i,z S j,z along blue bonds. crystal structures but orbital degrees of freedom are explicitly anisotropic. These spin-orbit coupled j = 1/2 Mott insulators then allow for highly anisotropic magnetic interactions which would otherwise not be possible in pure spin systems. While these anisotropic interactions do have measurable effects on the magentism in Sr 2 IrO 4, they do not play a fundamentally important role. Advancements in the study of these highly anisotropic magnetic models actually came from iridate materials with honeycomb geometries, such as those found in Li 2 IrO 3 and Na 2 IrO 3. Nominally, honeycomb lattices are simple because they are bipartite and support conventional Néel order from antiferromagnetic Heisenberg interactions. Yet the calculation for the effective magnetic interactions between nearest-neighbor sites with j = 1/2 moments showed there was no Heisenberg term due to a cancellation of superexchange paths in edge-sharing octahedra [6]. Instead, only bond-dependent Ising interactions result with three types of interactions: S i,x S j,x, S i,y S j,y, and S i,z S j,z. See Figure 1.2. For example, along an x-bond, the interaction only involves the x components of the j = 1/2 pseudo-spin. The importance of the resulting model is that it had been solved 3

16 exactly by Kitaev and shown to have a quantum spin liquid ground state [7]. This prompted the experimental search to realize the Kitaev model in transition metal oxides. The j = 1/2 iridates have provided many opportunities for new types of magnetism due to strong spin-orbit coupling, yet these materials in the t 5 2g configuration represent only a fraction of the possibilities. The central theme of this work is to explore the possibilities of other electron counts, t n 2g, in the context of transition metal oxides. Due to the anisotropic nature of orbital degrees of freedom, these strongly spin-orbit coupled systems will result in unexpected magnetic order in the other electron counts. 1.2 General Overview In this work, we study two classes of strongly spin-orbit coupled transition metal oxides. However, there is a large amount of background material required to begin work on magnetism in these compounds. For this reason, the next chapter, Chapter 2, provides background information required for the later chapters. This amounts to a standalone introduction to some of the general aspects of transition metal oxides. We will begin with a phenomenological description of the Hubbard model and how it applies to transition metal oxides. Unlike the one band (single orbital) Hubbard model, the presence of multiple orbitals per lattice site brings in new terms in the Hamiltonian and new possibilities for ordering. Next we discuss crystal field effects and how symmetry can be used to understand them. Then we review how short-ranged Coulomb interactions act in multi-orbital systems. Spin-orbit coupling is then introduced as an important relativistic correction, especially in 4d and 5d systems. Finally, we compare how magnetism occurs in the band limit versus the Mott limit to set the stage for why our results in later chapters are non-trivial. Readers who are already familiar with these basic aspects of transition metal oxides may skip the chapter without loss of continuity. Next we address the two main topics of research presented in this document: (1) d 1 and d 2 double perovskites and (2) d 4 Mott insulators. Typically, each transition metal site in the crystal lattice contains a magnetic moment due to electron spin degrees of freedom, 4

17 electron orbital degrees of freedom, or both. Then nearby magnetic moments interact through spin-spin interactions which causes magnetic order at some ordering temperature T c. In the case of the j = 1/2 iridates on a honeycomb lattice, we just showed an example where this paradigm breaks down. Even though there are preexisting pseudo-spin 1/2 magnetic moments and the honeycomb lattice geometry supports ordering, the novel nature of superexchange interactions between j = 1/2 moments prevents magnetic order at all temperatures. In the classes of materials considered here, we will indeed find magnetic order, but the paradigm for how to achieve magnetic order will be altered. Here we give four key changes to this paradigm from our investigation of these (1) d 1 and d 2 double perovskites and (2) d 4 Mott insulators. First, the d 4 systems we consider do not nominally possess preexisting magnetic moments, and we explore how magnetic moments may be generated and also order order magnetically. Second, in both double perovskites and d 4 Mott insulators, the interactions between sites do not take the form of spin-spin interactions. Instead, since orbital degrees of freedom are also involved, we must consider spin-orbital superexchange, and this type of superexchange simultaneously supports both ferromagnetic and antiferromagnetic interactions between sites. Third, the d 1 and d 2 double perovskites we consider support both magnetic order at some temperature T c and orbital order at another temperature T o. When spin-orbit coupling is involved, these two types of orders are tied together, and orbital interactions play just as important of a role in determining the magnetic order as the magnetic interactions do. Fourth, although strong Hund s coupling is usually a viable way to achieving ferromagnetic interactions and consequently ferromagnetic order, spin-orbit coupling and orbital geometry play a crucial role in determining when ferromagnetism is allowed in spin-orbital systems. 1.3 Summary of Results In Chapter 3, we consider Mott insulators with either one or two electrons in the t 2g (d) shell. While there are potentially many compounds to study, we focus on the strongly spinorbit coupled double perovskites A 2 BB O 6 with B magnetic ions in either d 1 or d 2 electronic 5

18 configuration and non-magnetic B ions. The reason for looking at these compounds is that there are several experimental puzzles which have not yet been resolved by theory. These puzzles include the predominance of ferromagnetism in d 1 versus antiferromagnetism in d 2 systems, the appearance of negative Curie-Weiss temperatures for ferromagnetic materials, and the size of effective magnetic moments. As we will show, the resolutions to these puzzles lie in the orbital degrees of freedom. We develop and solve a microscopic model with both spin and orbital degrees of freedom within the Mott insulating regime at finite temperature using mean field theory. The interplay between anisotropic orbital degrees of freedom and spin-orbit coupling results in complex ground states in both d 1 and d 2 systems. Although the models for these two electron counts are similar, their zero temperature phase diagrams are quite different. We show that the ordering of orbital degrees of freedom in d 1 systems results in coplanar canted ferromagnetic and 4-sublattice antiferromagnetic structures. In d 2 systems we find additional collinear antiferromagnetic and ferromagnetic phases not appearing in d 1 systems. At finite temperatures, we find that orbital ordering driven by both superexchange and Coulomb interactions may lead to both deviations from Curie-Weiss law both at high temperature due to orbital ordering and at low temperature due to the anisotropy induced by orbital order. Our results from Chapter 3 are immediately applicable to experiments on 5d 1 and 5d 2 double perovskites. Despite calculations by another group [8] which suggest ferromagnetism should also be common in 5d 2 double perovskites, the calculations presented here clearly show that ferromagnetism is favored in 5d 1 systems but not 5d 2 systems. We go further to provide a clear explanation for this general trend in terms of the constraints on magnetic interactions imposed by orbital configurations. Others groups have previously canted ferromagnetic phases [9 11] in models for d 1 spin-orbit coupled double perovskites. However, we go further to show how orbital order resolves the paradox of having a material with a diverging magnetic susceptibility (typically associated with a ferromagnet) yet a negative Curie-Weiss temperature (typically associated with an antiferromagnet). Finally, we propose that the orbital order found in the canted ferromagnetic phase may also be present in antiferromagnetically ordered compounds as well. 6

19 In Chapter 4, we discuss the next class of materials which are transition metal oxides with four electrons in the t 2g shell. The rationale for studying these transition metal oxides with the t 4 2g electronic configuration is that they are expected to be nonmagnetic atomic singlets. In the weakly interacting regime, spin-orbit coupling creates a situation with a fully filled j = 3/2 band and an empty j = 1/2 band which leads to the absence of magnetism. In the Mott insulating regime, the total L = 1 and S = 1 angular momenta anti-align on every site to give a total J = 0 on every site and the absence of magnetic moments. Starting with the full multi-orbital electronic Hamiltonian, we calculate the low energy effective magnetic Hamiltonian which contains isotropic superexchange spin interactions but anisotropic orbital interactions. By tuning the ratio of superexchange to spin-orbit coupling J SE /λ, we obtain a phase transition from nonmagnetic atomic singlets to novel magnetic phases. The phase transition is always to an antiferromagnetic ground state which is surprising since large Hund s coupling typically induces ferromagnetic interactions and ferromagnetic order. Spin-orbit coupling plays a non-trivial role in generating a triplon condensate of weakly interacting excitations at antiferromagnetic ordering vector k = π, regardless of whether the local spin interactions are ferromagnetic or antiferromagnetic. In the large J SE /λ regime, the localized spin and orbital moments produce anisotropic orbital interactions that are frustrated or constrained even in the absence of geometric frustration. Our results from Chapter 4 help resolve a conflict in the literature about whether ferromagnetic condensation will occur in strongly spin-orbit coupled d 4 Mott insulators. In contrast to Ref. [12], we show that symmetry actually requires that ferromagnetic interactions turn into antiferromagnetic interactions as spin-orbit coupling increases. This highly unexpected result shows that rotationally invariant single-site effects can effectively negate the sign of magnetic interactions in spin-orbital systems, a feature which is impossible in spin-only systems. The immediate consequence for experiments is that most d 4 Mott insulators with strong spin-orbit coupling will not show ferromagnetism even if Hund s coupling is large. In Chapter 5, we apply the triplon condensation formalism from Chapter 4 to the material Ba 2 YIrO 6. Using density functional theory, we calculate tight-binding parameters in 7

20 the effective triplon Hamiltonian. The calculation shows that despite the experimental observation of magnetism in this compound, it is unlikely the result of superexchange induced condensation. Lastly, we comment on other possible sources of magnetism in Ba 2 YIrO 6. Our results from this chapter complement a DMFT study by Ref. [13] and provides a simpler explanation for why 5d 4 double perovskites with only magnetic B ions will not likely develop magnetism due to superexchange induced condensation. 8

21 Chapter 2 Transition Metal Oxides 2.1 Materials and the Hubbard Model We begin by considering a simple starting Hamiltonian for non-relativistic electronic systems in first quantized notation. It consists of the kinetic energy, an external potential, and electron-electron interaction. H = i p 2 i 2m e + i V ( r i ) + 1 U( r i, r j ) (2.1) 2 i j Here the sum over i is over all electrons in the system. In this starting example, the interaction term is simply the Coulomb interaction, U( r i, r j ) = e2 r i r j, and the principle source of the external potential comes from the positively charged nuclei of the ions. For simplicity, motion of the nuclei is not considered. At even a simple level like this, no closed form solution to this many-body problem is possible, and approximations are needed. The first approximation is to restrict our calculations to only the electrons which are most relevant for determining the properties of solid-state materials systems. For an isolated atom in free space with many electrons, most of the electrons will be highly localized due to the strength of the Coulomb interaction from the nuclear charge, and they will not contribute to most observable properties of the atom. Only the highest energy electrons will play a role. Thus the first approximation is that only the valence electrons on each ion will be important, and we should seek a new effective model for just the valence electrons. As the starting example contained the sum of two single-electron terms and an electron- 9

22 electron interaction, our new effective Hamiltonian should contain both a quadratic piece serving as the effective kinetic energy and a quartic piece serving as the new electron-electron interactions. Additionally, we know two things which will greatly assist in the simplification of the model. First, the potential V is periodic in space due to the crystalline nature of solids, and the new Hamiltonian must express this periodicity in some way. Second, the valence electrons in the d shells of these transition metal ions tend to be localized to their ions. We then consider a Hamiltonian where the relevant single-particle states that an electron may occupy are localized near transition metal ions. These single particle states are expressed using fermion creation and annihilation operators. Let c i,α be a fermion creation operator for an electron localized to the α-th state on site i, and the annihilation operator c i,α is defined similarly. Then effective model will take the form below. H = ij t ij,αβ c i,α c j,β + U ijkl,αβγδ c i,α c j,β c k,γ c l,δ (2.2) αβ ijkl αβγδ The quadratic term is the so called tight-binding term. (The minus sign in front is convention.) It acts as an effective kinetic energy giving the electrons an energy dispersion in k-space. The quartic term is then the effective Coulomb interaction between fermionic states. There are two questions that immediately arise. The first question is obvious: how precisely is this effective model obtained? Typically in solid-state physics, density functional theory (DFT) tools are used to approximate the t values. The effective single-electron Wannier states states are computed within DFT, and then the matrix elements of t are determined by computing an integral involving the Wannier states, ψ i,α V ψ j,β. [14] While this method has been successfully used for decades to give good estimates of the t values, obtaining accurate U values is still a current subject of research today. Unfortunately, this limitation is one of several major impediments to microscopically determining the properties of strongly correlated systems in solid-state physics. This leads to the second question: what physical principles could be used to determine the effective model in lieu of microscopically exact calculations? Our first assumption is 10

23 that the single-electron wavefunctions decay quickly in space, and only the overlaps between wavefunctions on atoms nearby will be important in determining t. The next assumption is that the long-ranged Coulomb interaction is screened so that the short-range part of the Coulomb interaction is most relevant, although this assumption breaks down in the Mott limit. The simplest non-trivial model is where t is only nonzero between nearest neighbor sites and U is only relevant on-site. This is referred to as a Hubbard model, and the simplest Hubbard model is given below. H = t ij σ=, (c i,σ c j,σ + h.c. ) + U i n i, n i, (2.3) Here ij refers to all pairs nearest-neighbor pairs of sites, and the number operator is defined as n i,σ = c i,σ c i,σ. This simple Hubbard model is arguably the single most important model in all of condensed matter physics. Here each lattice site only possesses four types of states: unoccupied 0, spin-up, spin-down, or double occupancy. However, in addition to the spin degree of freedom, orbital degrees of freedom will need to be introduced in order to correctly describe most transition metal oxides. 2.2 Crystal Field In the limit of a single atom, the single-electron states on that atom are described by atomic orbitals. Since the isolated atom is in a spherically symmetric environment, we can label the single-electron states by orbital angular momentum. Equivalently, the single-electron states can be labeled by the type of orbital corresponding to that angular momentum. (ie. s corresponds to L = 0, p corresponds to L = 1, etc.) For the transition metals, the valence electron shell consists of L = 2 or d orbitals so that there are 5 singleelectron orbital degrees of freedom per atom. While it is clear that this works for an isolated atom, ions in a crystal are not in spherically symmetric environments. In transition metal oxides, the transition metal ions are surrounded by neighboring oxygen ions, and the presence of these ions can change the relative energies of the d orbitals on the transition metal ion. The most common 11

24 Figure 2.1: d orbitals on the transition metal site (purple) are degenerate under spherical symmetry. When this symmetry is reduced to O h due to the presence of neighboring ions (red), the d orbitals split into lower energy t 2g and higher energy e g states. The energy difference, CF, is the crystal field splitting. situation in oxides is where the transition metal ion is surrounded by six oxygens forming an octahedral complex shown in Figure 2.1. Also shown in Figure 2.1 is the splitting of d states due to this octahedral crystal field. This occurs due to bonding of the transition metal ion s d orbitals with the neighboring oxygen p orbitals. Since two orbitals labeled as e g extend directly toward the neighboring oxygen sites, the resulting σ-bonding/antibonding is strong. However since the three orbitals labeled as t 2g extend outward at 45 degrees from the neighboring p sites, their π-bonding is weak compared to that of the e g orbitals. This difference in bonding strength causes a difference in the energy shift of the two types of d orbitals. The result is that the effective antibonding e g orbitals move higher in energy than the antibonding t 2g orbitals. The difference in energy between states is octahedral crystal field splitting of d-orbitals. The effect considered so far is essentially a linear correction in perturbation theory due to bonding/antibonding. What are the effects of higher order corrections? Or what about other effects which could split otherwise degenerate orbitals? It would be extremely difficult 12

25 to track down every microscopic detail. Clearly a more economical approach is needed. Group theory provides the framework to resolve these problems. In the present problem, we can identify that although the transition metal ion is not in a spherically symmetric environment, it is in an environment with octahedral symmetry. More specifically, the point group symmetry associated with transition metal site is O h (full octahedral symmetry). Both the e g orbitals and t 2g orbitals are labeled as such because they correspond to the E g and T 2g irreducible representations of the O h point group. [15] We can then be sure that as long as O h symmetry is present, the e g and t 2g levels will not further split. However, there are two notes about using group theory to determine the results. First, it cannot determine whether the e g or t 2g level is higher in energy. This can only be accomplished through calculation. Second, the resulting e g and t 2g wavefunctions from the crystal field splitting do not necessarily correspond to atomic orbitals. The only requirement is that the resulting wavefunctions share a minimum set of symmetry properties with their atomic orbital counterparts. We have just shown how group theory can be applied to understand why degenerate orbitals split when symmetry is reduced. It is easy to imagine that crystal field splitting will alter other properties as well. For example, the effective Hubbard model in equation (2.2) will now involve fewer degrees of freedom since some orbitals become energetically irrelevant. More importantly, the tight-binding parameters t ij,αβ and the Coulomb interaction parameters U ijkl,αβγδ in the effective Hubbard model may also change. In principle, there are well established methods to reliably estimate tight-binding parameters using electronic structure calculations which makes the tight-binding term less challenging to deal with. However, the effective Coulomb interactions should be handled up-front. 2.3 Coulomb Interactions A recurring theme in transition metal oxides is the importance of strong correlations due to Coulomb interactions. In the chapters to come, we will frequently be considering materials where the e g orbitals are higher enough in energy to project out, and the t 2g 13

26 orbitals are the relevant orbital degrees of freedom. Here we derive the effective on-site Coulomb interaction for t 2g orbitals for use in the later chapters. [16] In other words, we derive the Hubbard interaction for triply degenerate t 2g orbitals, a multiorbital version of the Hubbard interaction (ie. U i n i, n i, ) appearing in the single orbital Hubbard model in equation (2.3). Coulomb interactions between different sites will be treated in later chapters as necessary. In this derivation, let the indices (a, b, c) be an arbitrary permutation of the t 2g orbital labels (yz, zx, xy). The corresponding wavefunctions are denoted as φ a, φ b, and φ c. Since the orbitals in a solid do not necessarily correspond to atomic orbitals, we can only assume a minimal set of requirements about the t 2g orbitals. The assumptions are then that these orbitals are real-valued (ie. φ a = φ a), and they transform as the T 2g irreducible representation of the O h point group. This immediately gives some important symmetry properties for t 2g orbitals. For example, let σ a be the a-plane mirror symmetry (ie. σ xy is a mirror about the xy plane). Then σ a φ a = φ a and σ a φ b = φ b for a b. Now we consider the matrix elements of the Coulomb interaction between t 2g states. Let αβ refer to a two electron state where the first electron is in state α and the second electron is in state β. Then a generic matrix element between the two electron states αβ and γδ is given below. αβ V γδ = dr 1 dr 2 φ α (r 1 )φ β (r 2 )V (r 1 r 2 )φ γ (r 1 )φ δ (r 2 ) (2.4) Although it is easy to calculate matrix elements using V (r 1 r 2 ) = e 2 / r 1 r 2, the Coulomb interaction between electrons is screened by other electrons in the solid. This way, the effective Coulomb interaction may differ from the bare Coulomb interaction. While the exact form may be material dependent, we may still proceed using symmetry arguments. The screened Coulomb interaction V must still be invariant under the symmetries of the Hamiltonian. As an example, V (r 1 r 2 ) should at least be invariant under the mirror plane operation σ a. Given the symmetries that the interaction must obey, we will be able to determine which matrix elements are generally zero and which matrix elements are not. Since there are only three types of orbitals, there can be at most three mutually distinct 14

27 indices appearing in the matrix elements αβ V γδ. This results in four types of cases. First, consider the case where all indices are identical. This is the direct Hubbard interaction (where both electrons are on the same orbital but have opposite spins) and is defined as aa V aa U. Second, consider the case where three indices are the same and one index is different: aa V ab. This must be identically zero since it is odd under the mirror plane operation σ b. More generally, the same reasoning holds for any permutation of the four indices (a, a, a, b). Third, consider the case where two of the indices are the same and the other two are different, aa V bc. Again this matrix element must be identically zero due to mirror symmetry, and all permutations of the four indices must also be zero. Fourth, consider situations where there are two pairs of indices which are the same. They are defined in the following way: ab V ab U, ab V ba J, and aa V bb J. However, since the wavefunctions φ(r) are real valued, it immediately follows that J = J. This leaves a total of three undetermined Coulomb parameters: U, U, and J. Given the definitions of the Coulomb parameters, we may now write down the secondquantized form of the on-site Coulomb interaction (ie. Hubbard interaction). Let φ ασ be a fermion creation operator for a state in orbital α and spin σ. H Hubbard = ασ 1 βσ 2 γσ 3 δσ 4 φ ασ 1 φ βσ 2 V γσ3,δσ4 ασ 1,βσ 2 φ δσ4 φ γσ3 (2.5) The matrix elements between two-particle fermionic states are given by V γσ 3,δσ 4 ασ 1,βσ 2 = { (ασ 1 )(βσ 2 ) V (γσ 3 )(δσ 4 ) } (2.6) using antisymmetric wavefunctions AB} = 1 2 ( AB BA ). The result simplifies to the following. V γσ 3,δσ 4 ασ 1,βσ 2 = αβ V γδ δ σ1 σ 3 δ σ2 σ 4 αβ V δγ δ σ1 σ 4 δ σ2 σ 3 (2.7) By explicitly writing out the Hubbard interaction in terms of the Coulomb parameters, the 15

28 following form results. H Hubbard = U a n a n a + U n a n b + (U J) 1 2 n aσ n bσ a b a b σ (2.8) J a b φ a φ b φ b φ a + J a b φ a φ a φ b φ b While this form is very general, it isn t practical to leave three independent parameters characterizing merely one type of term in a model Hamiltonian. The most common fix is to assume spherical symmetry as in the case of an isolated transition metal ion, which amounts to setting U = U 2J. [16] Given that the result must be spherically symmetric, the on-site Coulomb interaction can be rewritten in a form explicitly showing this. This is accomplished using a total number operator N, total spin operator S, and total orbital angular momentum operator in the t 2g orbital subspace L = (L x, L y, L z ). N = a σ n aσ (2.9) S = 1 2 a L α = i β γ φ aστ σσ φ aσ (2.10) σ σ ε αβγ φ βσ φ γσ (2.11) σ Here τ is a vector of Pauli matrices and ε is the Levi-Civita tensor. Note there is a commonly used mapping between t 2g orbital indices and Cartesian indices (xy z, yz x, and zx y). This is in fact due to the similarity between t 2g orbital angular momentum and p orbital angular momentum. However, the t 2g orbital angular momentum operators satisfy the negative of the usual commutation relations for angular momenta: L L = il. (Note that is factored out of all angular momentum operators in this text.) A derivation of the t 2g orbital angular momentum operators is provided in Appendix A.1. Given the N, S, and L operators, we may now write the spherically symmetric form of the on-site Coulomb interaction. The derivation is given in Appendix A.2. H Hubbard = (U 3J H ) 1 2 N(N 1) J HN 2J H S J HL 2 (2.12) 16

29 Since the symbol J appears in many contexts in magnetism, we have instead used the symbol J J H for Hund s coupling. 2.4 Spin-Orbit Coupling So far we have built up our effective Hamiltonian for t 2g orbitals using only nonrelativistic terms. This non-relativistic approach is a good approximation for materials with only light nuclei. Transition metal ions have heavy nuclei that generate strong electric fields which lead to relativistic corrections to the Hamiltonian. In general, there are three types of corrections: kinetic, Darwin, and spin-orbit coupling. The kinetic term and Darwin terms can be absorbed into the tight-binding model we choose to work with. However, the spin-orbit coupling term acts in a non-trivial way. This is because it reduces the symmetry of our Hamiltonian by explicitly coupling the orbital and spin parts together which would have otherwise been independent. H SO = e 2 1 m 2 c 2 r V r S L (2.13) Here V is the electric potential the electrons experience due to the positively charged nucleus. For atomic orbitals, the expectation value of 1 r V r may be used, and the terms to the left of S L are lumped into an overall positive constant, λ, which characterizes the strength of spin-orbit coupling for a set of degenerate orbitals. This single-particle effect is most accurately expressed in the form below. H SO = λ m,m c m,σ S σ,σ L m,mc m,σ (2.14) σ,σ Since λ is positive, we would naively expect that states where the orbital angular momentum was anti-aligned with the spin would be the lowest in energy, and the states where they were aligned would be the highest in energy. For t 2g orbitals, this is exactly opposite to the correct result as shown in Figure 2.2. This is due to the fact that the commutation relations for t 2g angular momentum operators imply they are reversed with respect to the mathematical angular momentum operators for an L = 1 subspace. (See Appendix A.1.) 17

30 crystal field spin-orbit coupling d e g j=1/2 L S t 2g j=3/2 L S Figure 2.2: After crystal field splitting, spin-orbit coupling further splits the t 2g orbitals with spin degeneracy into lower energy j = 3/2 states and higher energy j = 1/2 states. When dealing with strongly correlated systems, we must inherently deal with many-body wavefunctions. For example, when Hund s coupling is strong, the degenerate multi-electron wavefunctions are labeled by total S and total L, not single particle S and L. We should ask how the single-particle spin-orbit coupling operator effectively acts on these multi-electron states? Due to the fact that S and L are spherical tensors, the Wigner-Eckart theorem guarantees that the projection of H SO to the subspace of total S and total L must be composed of spherical tensors of the same rank. The only operators which satisfy this constraint are proportional to the spin-orbit coupling operators for total S and total L, ie. S L. The only difference is that the energy scale λ now acquires a constant prefactor which is equal to the reduced matrix element in the Wigner-Eckart theorem. This prefactor can be calculated for each relevant multiplet of total S and total L. 2.5 Band Limit versus Mott Limit Up to this point, we have referenced many different energy scales: t, CF, U, J H, and λ. Although the relative magnitudes of all of these parameters are important, perhaps the most important ratio is between the tight-binding term, t, and the on-site Coulomb interaction, U. For simplicity, we just consider the single orbital Hubbard model in equation (2.3) where there are exactly N electrons in the system of N sites (ie. half-filling). In the band limit, 18

31 U 0, the Hamiltonian is quadratic and exactly diagonalized by Fourier transform H = k E(k) n k,σ (2.15) σ where E(k) is the energy dispersion of the single-particle electron states in k-space. In the Mott limit, t 0, the problem is already diagonal in the site occupation basis. H = U i n i, n i, (2.16) The solutions in the two limits differ substantially. In the band limit, the total energy is just the sum of the energies of the individual electrons in the system, and the non-degenerate ground state simply consists of filling the lowest N occupied energy levels. In the Mott limit, the total energy explicitly depends on the correlations between electron locations, and the ground states all have exactly one electron per site. Notice that this condition is fulfilled without regard to spin on each site so that the t = 0 limit has an extensive degeneracy of 2 N. Due to the difference in degeneracy, the route to magnetism in these two limits differs substantially. In the band limit, susceptibility in the random phase approximation is given by χ(q) = (gµ B ) 2 χ (0) (q) 1 Uχ (0) (q) (2.17) where χ (0) (q) = p f(p) f(p+q) E(p+q) E(q) is the magnetic magnetic susceptibility of the noninteracting system (with U = 0). The Fermi surface then becomes unstable to magnetism when the generalized Stoner criterion Uχ (0) (q) 1 is satisfied. [17] Unlike in the band limit, the presence of degeneracy in the Mott limit always allows for some type of magnetism to occur for arbitrarily small t. However, when competing interactions, geometric frustration, or low dimensionality is involved, the result is not necessarily classical magnetic order. In the Mott limit, these effects can lead to exotic ground states such as valence-bond crystals, spin liquids, and multipolar order. [1] Aside from degeneracy, there is another crucial difference in the way magnetism forms in these two limits. In the band limit, the details of the Fermi surface determine which q vector 19

32 Figure 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. The result of this perturbation may either result in the original spin configuration or a configuration with reversed spins. first results in an instability to the formation of magnetic ordering. For instance, when U is large enough, an instability occurs at q = 0 resulting in ferromagnetism. For q 0, an instability will first occur if the nesting condition, E(q + k) = E(k), is met on the Fermi surface. So it would appear that the band limit allows for all types of classical magnetic order as long as the Fermi surface is shaped appropriately. This is in stark contrast with the Mott limit at half-filling. When t U, degeneracy in the Mott limit is broken due to perturbative charge fluctuations between nearby sites. These fluctuations are only possible when the spins on these nearby sites are antiparallel due to the Pauli exclusion principle. See Figure 2.3. Since the single orbital Hubbard model in equation (2.3) is invariant under global spin rotations, total spin is conserved. Then the results of these charge fluctuations in perturbation theory must also be rotationally invariant. The effective Hamiltonian from these second order perturbation processes is called a superexchange Hamiltonian, and it has the form given below. ( H (2) SE = J SE S i S j 1 ) 4 (2.18) Here J SE = 4t 2 /U is positive and dictates an antiferromagnetic interactions between spins. Notice that this result is quite general as it is insensitive to the lattice structure, or the value of t. We might ask how ferromagnetism is possible in the Mott limit. The first remark is that 20

33 ferromagnetism can be achieved away from half-filing (ie. the total electron count is either less than or greater than N). In fact, there is a rigorous theorem, the Nagaoka theorem, which proves the ground state is ferromagnetic when exactly one electron is removed from the system at half-filling in the limit that U +. However, even if the possibility of doping is excluded, ferromagnetism can still be achieved in the Mott limit if multiple orbitals are involved. This is one of the central themes covered here, and we will focus on microscopic routes to ferromagnetism next two chapters. 21

34 Chapter 3 d 1 and d 2 Double Perovskites 3.1 Introduction Strong spin-orbit coupling in correlated materials has provided new platforms to study quantum spin liquids and correlated topological phases of matter [1, 4]. Among the strongly spin-orbit coupled materials include 4d and 5d double perovskites of the form A 2 BB O 6. Here we study the special cases where only the B ions are magnetic with a particular focus on cases where the B ion is in the 5d 1 or 5d 2 electronic configuration. Due to large distances between the magnetic B ions, these materials are usually Mott insulators and present a promising class of materials to explore the interplay of spin-orbit coupling and strong correlations. Here we study magnetic spin-orbital models in the Mott regime applicable to this class of double perovskties. These models involve highly anisotropic interactions due to the orbital degrees of freedom. We find that orbital order accompanies magnetic order and may occur at temperatures much higher than the magnetic ordering temperature. This orbital ordering results in deviations from Curie-Weiss behavior both near the orbital ordering temperature and near the magnetic ordering temperature. At low temperatures, we show how the magnetic phases are dictated by the orbital degrees of freedom resulting in unusual magnetic ordering patterns. Our results go beyond previous theoretical works on both 5d 1 and 5d 2 compounds and address several experimental puzzles. To set the stage, we contrast these results with what would be observed in systems 22

35 with other electron counts. In the context of iridates, 5d 5 systems have been extensively studied using pseudospin j = 1/2 models. In particular, double perovskites may offer a route to realizing spin liquids and other interesting phenomena in three dimensions [18 20]. Moving to the next electron count, 5d 4 systems are quite unique since spin-orbit coupling dictates that local moments should be absent and magnetism is forbidden. However several theoretical [12, 13, 21 23] and experimental [24 26] studies have examined the possibility of obtaining local moments in these otherwise nonmagnetic systems. The next electron count, 5d 3, involves half-filled t 2g shells nominally resulting in an effective spin-3/2 model which is expected to be described as a classically frustrated spin system [27, 28]. The 5d 2 and 5d 1 electron counts stand out in that they combine aspects of the former electron counts. First, they possess local angular momenta large enough to support orbital (quadrupolar) order. Second, they possess unquenched orbital degrees of freedom that result in highly anisotropic interactions between magnetic ions [29, 30]. Both of these aspects allow for the orbital degrees of freedom to play a significant role in determining the spin, orbital, and spin-orbital ordering. In the limit of large spin-orbit coupling, the spin S = 1/2 and orbital L eff = 1 angular momenta add to a total angular momenta of j = 3/2. Within the jj-coupling scheme, magnetic moments are identically zero due to cancellation of the spin and orbital moments, M = 2S L = 0. On the other hand, d 2 systems allow for a nonzero moment within the LS-coupling scheme of M = 6 2 µ BJ for total J = 2. However both systems are experimentally observed to be magnetic. In this chapter, we visit several questions surrounding the magnetically ordered phases of the 5d 1 and 5d 2 materials and attempt to self-consistently resolve a number of puzzles described below. d 1 versus d 2. Currently there are many experimental examples of ferromagnetic Mott insulating d 1 systems for the present class of double perovskites, but there are few experimental examples (if any) of ferromagnetism in d 2. Although we start with the same electronic model for both d 1 and d 2 systems, the energetics of the ground states strongly depend on the electron count. This is reflected in how the spin and orbital degrees of freedom order and provides a qualitative understanding for why ferromagnetism has been 23

36 repeatedly observed in d 1 systems while antiferromagnetic interactions tend to dominate in d 2 systems. Entropy. In the case of Ba 2 NaOsO 6, the entropy recovered through the magnetic transition is R ln 2 instead of R ln 4 expected for j = 3/2 systems [31]. This points toward the existence of a second phase transition at higher temperature where the remaining entropy is recovered. In our picture, this second transition comes from the orbital degrees of freedom and anisotropic interactions that accompany them. In particular, we show that the anisotropic interactions result in orbital order that stabilizes exotic magnetic order. The orbital ordering temperature scale is set both by superexchange interactions and by intersite Coulomb repulsion. Orbital ordering may occur at temperatures much higher than the magnetic ordering temperatures if this ordering is primarily determined by orbital repulsion. Susceptibility. The experimental observation of a negative Curie-Weiss temperature in the ferromagnet Ba 2 NaOsO 6 seems to contradict simple mean-field arguments [31]. Recently, non-curie-weiss behavior has been reported in two ferromagnetic d 1 compounds Ba 2 MgReO 6 and Ba 2 ZnReO 6 [32]. In our model, the onset of orbital order causes changes in magnetic susceptibility resulting in non-curie-weiss behavior. For situations where orbital order occurs at temperatures much higher than the magnetic ordering temperature, we show how a negative Curie-Weiss temperature could be extracted for the ferromagnetic phase using simple mean-field calculations. We then use a simplified model to show how anisotropy induced by orbital order can result in a proper negative Curie-Weiss intercept despite diverging susceptibility at the transition temperature. Magnetic Moment. Although the magnetic moment for a j = 3/2 system is nominally zero, reduction of the orbital moment due to covalency allows for a finite moment. However, if orbital order occurs, covalency with oxygen alone does not reproduce the experimentally determined magnitudes of the magnetic moments in d 1 systems. Further corrections are necessary which may arise from dynamical Jahn-Teller effects [33] and more generally with mixing of the j = 3/2 and j = 1/2 states. Lastly, we outline where our calculations stand with respect to other work. Density functional theory studies have so far revealed two important aspects of these compounds. 24

37 First, they have revealed the importance of oxygen covalency in suppressing the orbital moment so that a net moment results [33, 34]. Second, they have pointed out that spin-orbit coupling and hybridized orbitals play a major role in opening a gap within DFT+U [35 37]. Model Hamiltonian approaches have also shed some light on these materials by using spinorbital Hamiltonians [11], projecting spin-orbital Hamiltonians to the lowest energy total angular momentum multiplet [8, 9], lowest energy doublet [38], and other approaches [10]. In both electron counts, Chen et. al. [8, 9] find canted ferromagnetism accompanied by quadrupolar order occupies a majority of parameter space. Additionally they find a novel non-collinear antiferromagnetic phase in d 2 but not d 1. Recently this phase was found in d 1 as the most energetically favorable antiferromagnetic state [11]. Proposals for both valence bond ground states [9, 11] and quantum spin liquids [9, 39] also exist. Our findings are largely compatible with those of Romhányi et. al. [11], and we further provide a clear interpretation of why these orbital ordering patterns occur, how they dictate the magnetic ordering, and then extend our calculations to finite temperature. Like Chen et. al., we find that orbital ordering can occur at temperatures much higher than the magnetic ordering temperature, however, we provide a clear interpretation of the negative Curie- Weiss temperature in d 1 ferromagnets. Furthermore, our spin-orbital approach includes mixing between the j = 3/2 and j = 1/2 states induced by orbital order and intermediate spin-orbit coupling energy scales. Our zero temperature phase diagrams differ from those of Chen et. al. [8, 9], with striking differences in d 2, which we discuss in detail in later sections. The most significant difference is in the energetics of antiferromagnetism versus ferromagnetism in d 2 systems which gives a qualitative explanation for the broadly observed differences in ordering between 5d 1 and 5d 2 compounds. Finally, we do not consider valence bond or spin liquid phases in this chapter although both may be applicable to d 1 and d 2 systems. Many of our findings can be tested using multiple probes. At the orbital ordering temperature, there will be a second order phase transition with a signature in heat capacity as well as changes in the magnetic susceptibility which are relevant for both powder samples and single crystals. NMR has recently found evidence of time-reversal invariant order above 25

38 the magnetic ordering temperature in Ba 2 NaOsO 6 [40]. Resonant X-ray scattering may also provide crucial insights into this hidden order as it is sensitive to orbital occupancy. We show that time-reversal invariant orbital order occurs in both ferromagnetic and antiferromagnetic phases we find, and we suggest that experimental probes which are sensitive to such order should also be pointed at the antiferromagnetic compounds. On the experimental side, many d 1 and d 2 compounds have already been investigated. The 4d 1 compound Ba 2 YMoO 6 shows no long range magnetic order down to 2 K despite having a large Curie-Weiss temperature θ = 160 K and retaining cubic symmetry which leads to the conclusion that the ground state consists of valence bonds [41 45]. Among the 5d 1 compounds are ferromagnetic Ba 2 NaOsO 6 [31, 40, 46, 47], Ba 2 MgReO 6 [32, 48], and Ba 2 ZnReO 6 [32] which is unusual since ferromagnetism in Mott insulators is uncommon. There are two additional twists to the story: first, negative Curie-Weiss temperatures have been observed in these ferromagnets, and, second, Ba 2 LiOsO 6 is antiferromagnetic despite sharing the same cubic structure as Ba 2 NaOsO 6 [46]. The d 2 compounds offer a similar platform to search for unusual magnetism, however experimental studies seem to suggest that antiferromagnetic interactions are dominant in d 2 systems. Phase transitions to antiferromagnetic order are reported in Ca 3 OsO 6 [49], Ba 2 CaOsO 6 [50], Ba 2 YReO 6 [51], and Sr 2 MgOsO 6 [52, 53] while glass-like transitions are reported in Ca 2 MgOsO 6 [52] and Sr 2 YReO 6 [54]. There are also several possible singlet ground states: La 2 LiReO 6 [51], SrLaMReO 6 [55], and Sr 2 InReO 6 [54]. 3.2 d 1 Double Perovskites Here we develop a spin-orbital model for the d 1 double perovskites with magnetic B ions with spin-orbit coupling featuring both spin-orbital superexchange and inter-site Coulomb repulsion between B ions. We then solve the model within mean field theory at both zero temperature and finite temperature. At zero temperature, we find phases with orbital order and show how this ordering restricts the magnetic order. At finite temperature, we examine how orbital order modifies magnetic susceptibility and the Curie-Weiss parameters. 26

39 3.2.1 Model In the presence of cubic symmetry, the magnetic B ions form an FCC lattice and contain one electron in the outermost d shell. The five degenerate levels are split by the octahedral crystal field into the higher energy e g orbitals and lower t 2g orbitals so that the t 2g shell contains one electron. The electronic structure for the t 2g orbitals may be approximated by a nearest neighbor tight-binding model where only one of the three orbitals interacts along each direction. See Fig H TB = t α c i,α,σ c j,α,σ + h.c. (3.1) ij α σ Here the sum over α is over all yz, zx, and xy planes in the FCC lattice. As an example, for B sites in an α = xy plane, the xy orbital on site i overlaps with the xy orbital on site j. Each orbital has four neighboring orbitals of the same kind in its plane giving a total of twelve relevant B neighbors per B site. In addition to the tight-binding term, the unquenched t 2g orbital angular momentum L = 1 results in a spin-orbit coupling on each B ion [4] H SO = λ i L i S i. Here the orbital L = 1 and spin S = 1/2 operators both satisfy the usual commutation relations for angular momentum, L L = i L. The on-site multi-orbital Coulomb interaction is given by H U = i H(i) U where H U H Hubbard from Section 2.3 in (2.12). Here U is the Coulomb repulsion and J H is Hund s coupling [16]. Being in the Mott limit, we calculate the effective spin-orbital superexchange Hamiltonian within second order perturbation theory. The superexchange Hamiltonian is given by the following H SE = J SE 4 α ij α { r1 ( S i S j )(n α i n α j ) 2 +( 1 4 S i S j ) [ r 2 (n α i + n α j ) (r 3 r 2 )n α i n α ]} j (3.2) where J SE = 4t 2 /U is the superexchange strength and r 1 = (1 3η) 1, r 2 = (1 η) 1, and r 3 = (1 + 2η) 1 with η = J H /U [30]. Here the t 2g orbital electron occupation numbers are written as n α i = σ c i,α,σ c i,α,σ. See Appendix B.1 for details. The top line of (3.2) contributes a ferromagnetic (FM) spin interaction which requires that one of the two orbitals 27

40 Figure 3.1: (a) Crystal lattice for double perovskite A 2 BB O 6. (b) The simplified tightbinding model takes hopping between xy orbitals (purple) on B sites within an xy plane. Similarly, zx orbitals are active in zx planes, and yz orbitals are active in yz planes. participating in the interaction is occupied while the other is unoccupied. The bottom line of (3.2) contributes an antiferromagnetic (AFM) spin interaction which is maximized when both orbitals are occupied. The strength of Hund s coupling, J H /U, determines the strength of the two interactions relative to each other. Additionally there is an effective orbital repulsion n α i nα j in the superexchange processes when finite Hund s coupling is considered. Due to the large spatial extent of 5d orbitals from strong oxygen covalency, we include an additional term accounting for the direct Coulomb repulsion between orbitals on different sites. From general symmetry constraints, we can express the repulsion within the xy plane for the t 2g orbitals (yz, zx, xy). H (xy) V = ij xy n α i n β j αβ V 1 V 2 V 3 V 2 V 1 V 3 V 3 V 3 V 4 αβ (3.3) To constrain the number of independent parameters, we take the matrix elements to be determined in the limit of electric quadrupolar interactions [9]. Now let (α, β, γ) be a cyclic permutation of the three t 2g orbitals. The repulsion term for the entire lattice can then be 28

41 expressed in the following form. H V = V α ij α [ 9 4 nα i n α j 4 3 (nβ i nγ i )(nβ j nγ j ) ] (3.4) For example, within the xy plane, a pair of xy orbitals repel each other more than an xy and yz orbital. The total effective magnetic interaction then reads H = H SO + H SE + H V. Of the three parameters, spin-orbit coupling has the largest energy scale λ 0.4 ev for the 5d oxides while superexchange and intersite Coulomb repulsion are taken to have energy scales on the order of tens of mev. For 4d oxides, the spin-orbit energy scale is reduced to ev so that mixing between the j = 3/2 and j = 1/2 states is likely to occur. While our spin-orbit superexchange interaction is calculated in the LS-coupling scheme, recent evidence suggests that the true picture for the 5d oxides lies between the LS and jj limits [56]. We decouple H SE and H V into all possible on-site mean fields, i.e. S i n α i S jn α j S i n α i S jn α j + S in α i S jn α j S in α i S jn α j. Since the FCC lattice is not bipartite, we decouple into four inequivalent sites shown in Fig. 3.2(a) where each set of four inequivalent neighbors forms a tetrahedron. Since the mean fields need not factor into the product of spins and orbitals, S i n α i = S i n α i, there are a total of 15 mean fields per site comprised of three spin operators, three orbital operators, and products of the spin and orbital operators. Applying the constraint that one electron resides on each site, there are 11 independent mean fields per site giving a total of 44 mean fields in the tetrahedron. We then numerically solve for the lowest energy solutions of the mean field equations Zero Temperature Mean Field Theory In the limit where spin-orbit coupling λ is the dominant energy scale, the magnetically ordered phases can be characterized by an arrangement of ordered j = 3/2 multipoles [9]. However, when J SE and λ are comparable, a multipolar description within the j = 3/2 states breaks down and consequently both spin and orbital parts must be considered independently. Furthermore, the orbital contributions come in the forms of both orbital 29

42 (a) (b) (c) 0.20 y 0.18 x Canted FM (d) T c 0.2 M n zx k B T / λ n yz n xy T o (e) kbto / λ = V/λ μ eff (μ B ) 0.04 JH / U 0.16 L S 0.14 AFM V/λ=0 4-sublattice L S J SE / λ Figure 3.2: (a) FCC lattice decoupled into four inequivalent sites shown by four different colors. (b) The orbital ordering pattern driven by both J SE and V constrains the direction of orbital angular momentum. The magnetization operator is shown as M = 2S L. (c) The zero temperature phase diagram shows phases where the spin S and orbital L moments in each plane are collinear and the moments between planes are at approximately 90 degrees due to the orbital ordering pattern. Increasing orbital repulsion V between sites reduces the minimum strength of Hund s coupling required to induce FM. (d) Mean field values for the bottom sites (black, yellow) are shown as a function of temperature. The n yz orbital (red) has the largest occupancy followed by the xy orbital (blue). (e) With J SE = 0, we calculate the orbital ordering temperature T o and effective Curie moment enhancement µ eff for different values of V. 30

43 occupancy n α and orbital angular momentum L. Since n α, L, and S are coupled, there is competition between order parameters which results in non-trivial ordering. The zero temperature phase diagram is shown in Fig. 3.2(c) as a function of the strength of Hund s coupling η = J H /U and superexchange J SE /λ. Large values of η support a canted ferromagnetic (FM) structure while smaller values support an antiferromagnetic (AFM) structure. The spin-1/2 and orbital-1 angular momenta order parameters S and L are shown for each of the four inequivalent sites from Fig. 3.2(a). In both phases, one of the three directions has no ordered angular momenta, e.g. L z = S z = 0, so that both magnetic structures are co-planar. Both phases feature some separation of the ordered spin and orbital moments which increases as a function of J SE /λ. To understand why these magnetic structures emerge, we examine the orbital occupancy order parameters, n α, separately from the magnetic order parameters. In both the FM and AFM phases, there is an orbital ordering pattern pictured in Fig. 3.2(b). The two sites in the lower plane of Fig. 3.2(b) have the yz orbital (red) with the highest electron occupancy while the xy orbital (blue) receives the second highest and the zx orbital receives the lowest (green, not pictured). The two sites in the upper plane have identical ordering except the roles of the yz and zx orbitals are reversed. Qualitatively this orbital ordering pattern is favored by both the H V and H SE terms which pushes electrons onto orbitals that have small overlaps. This allows the electron on a green orbital to hop onto an unoccupied green orbital in the plane directly above or below (and similarly for red orbitals). Since these mechanisms work to suppress the overlap of half filled orbitals, FM interactions may become energetically favorable. A derivation of the mean field solution for H V is provided in Appendix B.2. Once orbital order sets in, the allowed magnetic phases are restricted by the direction of orbital angular momentum. Full orbital polarization is time-reversal invariant and would not allow orbital magnetic order. However Fig. 3.2(d) shows that each site has at least two orbitals with non-negligible occupancy which allows for the development of an orbital moment. Thus the direction of the orbital moment is determined by the direction common to the two planes of occupied orbitals with the overall sign of the direction (e.g. +x or x) 31

44 left undetermined. Figure 3.2(c) shows that the orbital angular momenta between planes are close to 90 degrees apart for both FM and AFM phases. As spin and orbital angular momentum are coupled together, the spin moments will select which direction the orbital moments choose (i.e. +x or x). The decision to enter an FM or AFM state is then determined by the spin interactions characterized both by the strength of η = J H /U and the magnitude of the orbital order parameter. If η is large, then FM spin interactions follow and result in both the spin and orbital degrees of freedom aligning within each xy plane producing a net canted FM structure. If η is small, then AFM spin interactions follow which result in the 4-sublattice AFM structure. We note that the Goodenough-Kanamori- Anderson rules [57 59] are not enough to determine whether FM or AFM is favored since both magnetic structures have the same underlying orbital order. The interplay between spin-orbit coupling and orbital ordering plays a crucial role in tipping the energy scales in favor of one of the two magnetic structures. There are two additional factors that determine if the FM or AFM state is selected. The dominant effect is the degree of orbital polarization. When the strength of orbital repulsion V is increased, the tendency for orbitals to order becomes stronger. This disfavors the overlap of half filled orbitals causing AFM superexchange, and hence promotes FM superexchange. Figure 3.2(c) shows a dramatic shift toward FM when a small V interaction is included. The second effect comes from the separation of spin and orbital degrees of freedom. When J SE becomes comparable to λ, the spin moments can partially break away from the orbital moments tending more toward a regular spin FM state instead of a canted spin FM state. Since a spin AFM state does not benefit from this separation to the same extent, FM becomes increasingly energetically favorable. Dimer phases have been proposed [9, 11] and offer a way to explain the absence of magnetic order in d 1 materials. However when λ/j SE is large, these dimer phases only occur at very small values of η = J H /U [11]. Furthermore, orbital repulsion V acts to further suppress dimerization. Since our focus is on the magnetically ordered phases of these double perovskites, we will not pursue these possibilities in this chapter. 32

45 3.2.3 Finite Temperature Mean Field Theory Orbital Order. We now examine the model at finite temperature. Figure 3.2(d) shows a characteristic order parameter versus temperature curve. At high temperatures all order parameters are trivial and each orbital occupancy takes a value of n yz = n zx = n xy = 1/3. As temperature is lowered, the first transition is to a time reversal invariant orbitally ordered state (see Fig. 3.2(b)) at temperature T o whose scale is set both by V and J SE, including when V = 0. At T o, the entropy released is from orbital degeneracy with the spin entropy remaining. Below the second transition at T c whose energy scale is set only by J SE, time reversal symmetry is broken on each site with the development of magnetic order, and the remaining entropy is released. The fundamental question arises of how large the exchange interaction J SE and repulsion V are in materials systems. Fits to experimental susceptibility show Ba 2 LiOsO 6 and Ba 2 NaOsO 6 have relatively small Curie-Weiss temperatures of θ = 40 K for AFM Ba 2 LiOsO 6 and θ = 32 K for FM Ba 2 NaOsO 6 [46]. (Alternatively θ = 10 K for the FM from a later study [31].) This indicates that J SE in cubic 5d 1 double perovskites is weak. However integrated heat capacity [31] of Ba 2 NaOsO 6 shows an entropy release just short of R ln 2 at T c consistent with the splitting of a local Kramer s doublet with no further anomalies in heat capacity up to 300 K. This suggests T o T c so that V is the most relevant parameter for setting the scale of T o. Magnetic Moment. Since the onset of orbital order necessarily alters the angular momenta available to order and respond to an applied magnetic field, we calculate how the effective Curie-Weiss constant depends on orbital ordering. Using J SE = 0, we calculate the temperature dependent susceptibility within mean field theory as a function of temperature for different values of V/λ. For each value of V/λ we calculate both the orbital ordering temperature T o and the effective Curie moment µ eff = gµ B j(j + 1) from a fit to low temperature inverse susceptibility. Figure 3.2(e) gives numerical results from our mean field theory that shows a linear relationship between T o and µ eff. In the absence of orbital order, the projection of the magnetization operator to the j = 3/2 space is identically zero. 33

46 However once orbital order sets in, the j = 1/2 components of the wavefunction get mixed with the j = 3/2 components. The matrix elements that connect these two j spaces then acquire expectation values and allow the effective Curie moment to become non-zero. An approximate derivation of this relation is provided in Appendix B.2, and the result is given by µ eff 172V δn x µ B /9λ (3.5) where δn x = n yz 1 3 is the deviation of yz occupancy from its high temperature value on the sites where n yz is the most occupied orbital. We note that the temperature dependence of δn x makes the effective moment dependent on temperature. In addition to the perturbative separation of L and S due to mixing of the j states, oxygen covalency has been shown to greatly reduce the orbital contribution to the moment [33, 34]. Here the magnetization operator assumes the form M = 2S γl where γ = and results in an effective Curie moment of 0.60µ B compared to an experimental value of 0.67µ B [46]. However the onset of quadrupolar order within the j = 3/2 states results in a reduction of the nominal 0.60µ B value. In general, the projection of a linear combination of the n yz, n zx, and n xy operators to the j = 3/2 states is (up to a constant shift) a linear combination of the operators Jx 2 Jy 2 and Jz 2. By projecting to the lowest energy doublet induced by these operators, we may calculate the g factors for this pseudo-spin 1/2 space. While the g factors are different in the three cubic directions due to the anisotropic nature of quadrupolar order, the sum of the squares is a constant, and the powder average is g 2 = 1 3 (g2 x + gy 2 + gz) 2 = 3. Then splitting of the j = 3/2 states reduces the Curie moment by a factor of (g 3/4)/( 15/4) = 3/5 which makes the calculated moment 0.47µ B. Mixing between the j = 3/2 and j = 1/2 states brings the calculated moment closer to experimental values. Susceptibility near T o. There are more consequences of orbital ordering that are particularly important for the magnetic susceptibility of this spin-orbital system. The orbital order reduces the symmetry of the system and causes the susceptibility to become anisotropic. Since the orbital ordering pattern tends to push angular momentum into the ordering planes, 34

47 without covalency with covalency χ 2 χ 4 χ T o T o k B T / λ k B T / λ χ T o T o k B T / λ k B T / λ Figure 3.3: Typical susceptibility, χ = 1 3 (χ xx + χ yy + χ zz ), and inverse susceptibility are plotted against temperature. The susceptibility curves are shown both without the correction due to covalency, γ = 1, and with the correction, γ = We have chosen J SE = 0 and left V finite to illustrate the consequence of high temperature orbital order on the susceptibility. By choosing J SE = 0, we show that although T c = 0 while T o 0, the fitted Curie-Weiss temperature appears to be negative. Note that a single Curie-Weiss fit cannot span the entire range below T o. 35

48 the susceptibility is enhanced in these two directions while reduced in the third direction. Although anisotropic susceptibility is expected once cubic symmetry is broken, it is an easy test to determine at what temperature orbital order occurs. Additionally, when orbital order sets in at T o, the effective moment changes as the orbital degrees of freedom tend toward a (partially) quenched state which results in an effective moment that changes with temperature. The non-curie-weiss behavior will be critical when interpreting the observed negative Curie-Weiss temperatures in 5d 1 FM compounds. To show this effect within our mean field theory, we now calculate the susceptibility both without and with the covalency correction γ. For clarity, we set J SE = 0 to isolate the contributions from orbital order from those of magnetic interactions. Figure 3.3 shows that below the orbital ordering temperature, the susceptibility deviates from the Curie-Weiss law. However the data below T o can be fit over a large range to give a negative Curie-Weiss intercept despite the absence of magnetic interactions. In fact the region where the fit works the best is just below T o where the orbital occupation is rapidly changing. To interpret this in a simple way, we will consider the case without covalency where the effective moment for the j = 3/2 states is identically zero. When orbital order occurs, there is mixing between the j = 3/2 and j = 1/2 states proportional to V δn /λ. Here δn refers to a change in orbital occupancy due to orbital order. Then below T o, the effective magnetization operator for the lowest energy Kramer s doublet increases in a way proportional to δn due to the matrix elements between j = 3/2 and j = 1/2. The effective Curie moment goes as the square of magnetization and thus the enhancement is of order δn 2. Since orbital order below T o scales as δn T o T 1/2 within mean field theory, the effective Curie moment gains a contribution scaling as T o T just below T o. At temperatures near T o, the leading correction to susceptibility and inverse susceptibility is linear leading to the appearance of a negative Curie-Weiss intercept. We note, however, that this is not indicative of the physical magnetic interactions. Despite using mean field critical exponents, qualitatively we have understood how deviations from the Curie-Weiss law occur from changing orbital occupancy. Because we have used a simple model consisting of only λ and V with a-priori knowledge of the ideal 36

49 (a) Canted FM (b) J > 0 K = 0 site 1 ϕ y site 2 ϕ y S 2 (gμ B ) 2 / χ x (b) J > 0 K = 0 (c) J > 0 K > 0 S 1 x k B T / J site 2 x ϕ y S 2 (gμ B ) 2 / χ (gμ B ) 2 / χ [110] [110] [001] k B T / J k B T / J Figure 3.4: (a) The canted ferromagnetic solution to equation (3.6) is shown. (b) For J > 0 and K = 0, susceptibility along [110], [110], and [001] is shown for the antiferromagnet with φ = π/4. (c) For J > 0 and K > 0, the susceptibility diverges at T c. The canting angle satisfies π/4 < φ < 0. Note that the Curie-Weiss law still holds at temperatures well above J and K, and the Curie-Weiss intercept is still negative. Curie-Weiss temperature of zero, we have been able to clearly interpret the non-curie-weiss susceptibility. However the fitting procedure must be performed with some caution since both the fit region and the chosen value of χ 0 (temperature independent term) determine the reported θ CW and µ eff. In fact, experimental behavior may deviate even more strongly due to the quantitative details of how orbital occupancies change with temperature. In particular, coupling between orbitals and phonons may be a crucial aspect here [33]. 37

50 3.2.4 Simplified Model at Finite Temperature Canting from Orbital Order. In our microscopic model, we found that orbital order induced deviations from Curie-Weiss behavior that resulted in fitted negative Curie-Weiss constants despite diverging magnetic susceptibility at T c. Although this explanation is self-consistent within the context of the model we study, it depends on the existence of a continuous phase transition for orbital order. However, experiments have not yet found a clear signature of high temperature orbital ordering. We then seek an explanation of the negative Curie-Weiss constant which is insensitive to the particular details of orbital ordering but still incorporates it in a phenomenological way. There are multiple reasons for this approach. Unlike models with only spin degrees of freedom, spin-orbital models are very sensitive to the tight-binding model used, but the results should not sensitively depend on the tight-binding parameters used. Our explanation should also not assume which mechanism lead to the orbital ordering nor require a continuous phase transition at T o. We then develop a minimal model to explain this phenomena. Our zero temperature results showed that the presence of orbital order tended to pin the orbital angular momentum along the axis common to the two most occupied orbitals. This enters as an anisotropy that moments see, although an anisotropy which depends on the xy plane that each B site is in. In the canted FM phase, there are only two inequivalent sites, so our simplified model will consist of exactly two sites labeled as site 1 and site 2 each with effective S = 3/2. The Hamiltonian for this model is then given below. H 12 = JS 1 S 2 K [ (S x 1 ) 2 + (S y 2 )2] (3.6) The sign of J may be positive or negative while the anisotropy K is positive. The parameterization J = sin 2φ and K = cos 2φ where π/4 < φ < π/4 gives the classical ground state with S 1 = (cos φ, sin φ, 0) and S 2 = (sin φ, cos φ, 0). This state is shown in Fig. 3.4(a). Susceptibility near T c. For J > 0 and K = 0, the model is a simple AFM shown in Fig. 3.4(b). When K > 0, susceptibility changes drastically at T c shown in Fig. 3.4(c). The kink at the ordering temperature changes to a divergence which signals a transition 38

51 to a FM state instead of an AFM state. However, the divergence smoothly turns into a Curie-Weiss law as temperature is increased above the J and K energy scales. Only when the temperature approaches T c does the deviation from Curie-Weiss behavior become important. Note that the Curie-Weiss temperature is still negative from fitting the region well above these energy scales. Appendix B.3 gives an example of how this transition from AFM Curie-Weiss behavior to a FM divergence in susceptibility may be calculated in closed form. This simple model contains the essential explanation for why the 5d 1 FM compounds have negative Curie-Weiss temperatures despite a diverging susceptibility. If orbital order occurs and creates a staggered anisotropy between xy planes, then canting immediately follows without further considering anisotropic Ising interactions or exotic octupolar interactions (see Appendix B.4). Positive Curie-Weiss temperatures can be associated with canting in the positive φ direction, and negative Curie-Weiss temperatures can be associated with canting in the negative φ direction. This suggests that Ba 2 NaOsO 6 may be better identified as a canted AFM instead of a FM. To complete the phase diagram for d 1, we can create a simplified Hamiltonian that includes both the canted FM phase and the AFM 4-sublattice phase. This requires 4 inequivalent sites shown on the tetrahedron in Fig. 3.2(a). H = J (S 1 S 2 + S 1 S 4 + S 3 S 2 + S 3 S 4 ) + J (S 1 S 3 + S 2 S 4 ) K[(S x 1 ) 2 + (S x 3 ) 2 + (S y 2 )2 + (S y 4 )2 ] (3.7) Interactions between xy planes are characterized by J, and interactions within an xy plane are characterized by J. For K > 0, the ground states are then given either by a canted phase with two inequivalent sites or the AFM 4-sublattice structure. Again, this shows the magnetic structures are largely dictated by anisotropy from orbital ordering, and the microscopic form of the interactions are less important. Furthermore, to better match the microscopic model, a uniform anisotropy, K i (Sz i )2, may also be introduced to fine-tune orbital occupancies and anisotropic susceptibilities. Chen et. al. [9] claimed negative Curie-Weiss temperatures were achievable in their 39

52 (a) (b) 0.20 FM [100] (FM [110]) 0.15 V/λ=0 JH / U AFM 4-sublattice (AFM [100]) y AFM [110] x J SE / λ Figure 3.5: (a) Orbital ordering patterns are shown for each type of magnetic order. Orbitals shown in solid colors represent the most occupied orbitals while orbitals not shown or shown transparently have lower occupancy. (b) The zero temperature phase diagram shows three ground state phases: AFM with moments (anti)parallel to [110], AFM 4-sublattice structure, and FM with moments parallel to [100]. Phases shown in parenthesis (AFM [100], FM [110]) show the next lowest energy phase in each region. Increasing orbital repulsion V moves the phase boundary between AFM 4-sublattice phase and the AFM [110] phase down to favor the AFM 4-sublattice phase. The phase boundary between the AFM 4-sublattice phase and the FM [100] phase moves up in favor of the AFM 4-sublattice phase. model for FM ground states, although this crucial result was not explicitly shown. Marjerrison et. al. [32] have reproduced that model to generate FM with negative Curie-Weiss temperatures, and they find near jump discontinuities in the magnetic susceptibility around T c. Such jump discontinuities are not seen in Ba 2 NaOsO 6, Ba 2 MgReO 6, or Ba 2 ZnReO 6. We remark that a sharp feature in susceptibility occurs as K/J 0, but the sharp jump is smoothed away from this limit. Additionally, Marjerrison et. al. [32] claim to see a pronounced feature in the heat capacity at temperatures just higher than T c in both Ba 2 ZnReO 6 and Ba 2 MgReO 6. We note that this unusually pronounced feature in heat capacity is not present in experimental data on Ba 2 NaOsO 6. 40

53 3.3 d 2 Double Perovskites Here we modify the d 1 spin-orbital model to accommodate two electrons. We then solve the model within mean field theory at both zero temperature and finite temperature. At zero temperature, we find new orbital phases not found in our d 1 phase diagram. For completeness, we show susceptibilities and orbital occupancies at finite temperature Model Our model for d 2 is constructed from the same considerations used in d 1 only changing the electron count. The tight-binding model H TB, the inter-site orbital repulsion H V, and the on-site Coulomb interaction H U are valid for the d 2 model without modification. However spin-orbit coupling and superexchange will change since the total spin and orbital angular momentum on each site are now composed of two electrons. In the Mott limit, Hund s rules are enforced by H U resulting in a total spin S = 1 and total orbital angular momentum L = 1 on each lattice site. Within this space, the spin-orbit interaction takes the form H SO = λ 2 i L i S i. The superexchange Hamiltonian is given by the following H SE = J SE 12 α ij α { r1 (2 + S i S j )(n α i n α j ) 2 +(1 S i S j ) [ (n α i + n α j ) 2 + ( 3 2 r )nα i n α ]} j (3.8) where the definitions of J SE, r 1, and r 3 correspond to those used previously. As before, the top line in (3.8) gives a FM spin interaction when only one of the two interacting orbitals is occupied while the second line gives an AFM spin interaction which is maximized when two half filled orbitals overlap. The total effective magnetic interaction then reads H = H SO +H SE +H V. We decouple H SE and H V into all possible on-site mean fields using four inequivalent sites as before and then solve the mean field equations numerically Zero Temperature Mean Field Theory The zero temperature phase diagram is shown in Fig. 3.5(b) as a function of the strength of Hund s coupling η = J H /U and superexchange J SE /λ. The orbital ordering structures 41

54 corresponding to each magnetic phase are shown in Fig. 3.5(a). Even though the phases are labeled AFM and FM, the complex spin-orbital structures are described in detail below. Similarly to our d 1 treatment, we will again focus on how orbital order dictates magnetic order. At small Hund s coupling, the ground state is an AFM structure with the moment pointing either parallel or antiparallel to [110]. Within an xy plane, the moments point in the same direction, and the moments in neighboring xy planes point in opposite directions as shown in the the phase labeled AFM 110 in Fig. 3.5(b). (Note that we use label [110] for the moment direction and not the structure factor.) To see why this phase occupies such a large region of phase space, we analyze the orbital structure that accompanies it, as shown in Fig. 3.5(a). On each site, one electron moves onto the yz orbital and the other onto the zx orbital. This corresponds to L z = 0 on every site so that orbital angular momentum cannot point in the z direction. In this configuration both occupied orbitals overlap with occupied orbitals on neighboring sites and unoccupied orbitals overlap with other unoccupied orbitals so that AFM superexchange is maximized. These orbitally controlled AFM interactions then take place between planes and not within planes resulting in AFM between planes and consequently FM alignment in each plane. Since this this orbital pattern is compatible with tetragonal distortion, as observed in Sr 2 MgOsO 6 [53], we expect nominally cubic crystal structures to distort. At intermediate Hund s coupling and J SE /λ, we find the AFM 4-sublattice coplanar structure with the same accompanying orbital structure previously found in the d 1 phase diagram. As before, the orbital angular momentum is closely aligned with the directions perpendicular to the occupied orbitals, and the spin and orbital moments tend to separate from each other with increasing superexchange. However there are important differences between this phase in the d 2 and d 1 cases. Although the underlying d 1 and d 2 orbital ordering structures possess the same symmetry, the electron count strongly influences the energetics. As in the d 1 case, we find that this orbital structure supports a canted FM structure in d 2 systems, but this solution to the mean field equations is significantly higher in energy than the other phases shown. In d 1 systems, orbital order would create a situation 42

55 where occupied orbitals overlapped with unoccupied orbitals on other sites. While ordering of the yz and zx orbitals in Figure 3.5(a) is necessary for the canted FM structure, so is the elimination of electron occupancy from the xy orbitals so that AFM interactions do not take place within each plane. This is possible with one electron per site but not with two, and the AFM 4-sublattice magnetic ordering dominates over the canted FM magnetic order in d 2 systems. It is also worth noting that in this region of the phase diagram, the next lowest energy phase is AFM [100] which can become a competitive ground state. For large superexchange and Hund s coupling, we find a FM phase with ordered moments parallel or antiparallel to [100]. This phase is best characterized as a 3-up, 1-down collinear structure where three of the four moments order parallel to each other along the chosen direction and the fourth moment orders anti-parallel to the other three. It is worth noting that the second most energetically favorable phase in this region of the phase diagram is another 3-up, 1-down structure where each moment is either approximately parallel or antiparallel to the [110] direction. The energy difference between the FM [100] and FM [110] phases is negligible and either phase is a suitable ground state. These two phases contain an underlying orbital structure which eliminates the overlap of half filled orbitals between three of the four sites so that FM interactions result between these three sites. However, the fourth site cannot be chosen in the same way as two orbitals must have AFM interactions with the other three sites, and consequently the fourth moment points antiparallel to the other three. When inter-site orbital repulsion H V is included, the phase boundaries shift. The most dramatic effect is the recession of the boundary between AFM [110] and the AFM 4- sublattice structure. This becomes apparent by comparing the orbital configurations of the two phases as the AFM [110] structure maximizes the number of AFM bonds which are penalized by the orbital repulsion. Unlike in the d 1 situation, we find that the inclusion of V does not enhance FM. Again, this is due to the constraints imposed by having two electrons per site. We also note that unlike the AFM [110] and AFM 4-sublattice structures, the FM/AFM [100] orbital structures feature more degenerate choices for orbital configurations. Of the four tetrahedral sites, three of them are able to minimize the repulsion and 43

56 (a) AFM 110 (b) AFM 4-sublattice (c) FM 100 ( χ - χ0 ) -1 (a.u.) n yz,n zx n xy T c nyz, nzx, nxy ( χ - χ0 ) -1 (a.u.) n n xy yz 0.15 n xy n yz n zx n zx T c T o T c T o nyz, nzx, nxy ( χ - χ0 ) -1 (a.u.) nyz, nzx, nxy k BT / λ k BT / λ k BT / λ Figure 3.6: Characteristic inverse susceptibility (blue/green) and orbital occupation (purple) curves are plotted against temperature for the three phases in Fig. 3.5: (a) AFM [110], (b) AFM 4-sublattice, and (c) FM [100]. Susceptibility is averaged over all three directions, χ 1 = 3(χ xx + χ yy + χ zz ) 1, and all sites in the tetrahedra. Orbital occupancies are shown for the site pictured above each plot. allow occupied orbitals to hop to unoccupied orbitals. Since the fourth site cannot minimize repulsion, its orbital occupancies are free and can be different on every tetrahedron in the FCC lattice. However, this degeneracy is then broken by magnetic ordering which selects only orbital configurations which are compatible with the magnetic order. Although we have focused on spin-orbital magnetic order, it is necessary to remark that exotic singlet ground states are also possible. The Kramer s theorem guarantees that trivial ionic singlets will not occur in d 1 systems, and therefore the experimental observation of singlet behavior is an indication of a non-trivial ground state. Such considerations do not apply to d 2, and experimental observations of singlet behavior may arise from trivial local magnetic singlets. Consequently this local non-magnetic singlet possibility must first be ruled out when searching for exotic singlet behavior Finite Temperature Mean Field Theory Here we consider the model at finite temperature. Figure 3.6 shows orbital occupations and inverse magnetic susceptibility as a function of temperature for the three ground state phases from the previous section. At high temperature, the orbitals have a uniform occupancy of n yz = n zx = n xy = 2/3. There is a temperature T o where time-reversal invariant 44

57 order sets in through the orbitals and a second temperature T c where magnetic order sets in. In the case of the AFM [110] phase, Fig. 3.6(a) shows the two ordering temperatures coincide and that the electrons are pushed onto the n yz and n zx orbitals to maximize AFM superexchange. This is different from the orbital ordering previously reported because this ordering maximizes orbital repulsion instead of minimizing it, so this orbital order is entirely driven by AFM superexchange. In this situation, the Curie-Weiss law with a negative Curie-Weiss temperature occurs as expected. The transition to an AFM 4-sublattice structure is shown in Fig. 3.6(b). Above T o susceptibility follows the Curie-Weiss law with a negative Curie-Weiss constant. Below T o the orbital occupancies change along with the inverse susceptibility to deviate from the high temperature behavior. Just below T o, susceptibility may be fit to another Curie-Weiss law with another negative Curie-Weiss constant. Similarly to the d 1 case, there is still deviation from the Curie-Weiss law in this regime, however, the deviations are smaller and so is the enhancement of the effective magnetic moment due to mixing of the J = 2 states with higher energy multiplets. But we note that when J SE = 0, we still find the appearance of a negative Curie-Weiss constant due to non-curie-weiss susceptibility as we did in the d 1 model. Finally, the transition to an FM [100] structure is shown in Fig. 3.6(c). Deviations from the Curie-Weiss law are seen below T o, and the sign of the Curie-Weiss constant can switch from negative to positive depending which region fitted. Unlike the other phases, magnetic order appears at T c with a first-order transition marked by the jumps in orbital occupancy and susceptibility. This arises from competition between having the most energetically favorable orbital structure at high temperature and the most energetically favorable magnetic structure at low temperature. As in the d 1 case, we compare values of the theoretical moments to those from experiment. Oxygen covalency will result in a Curie moment of µ eff = 6(1 γ/2)µ B. Assuming almost half of the moment resides on oxygen, the calculated moment is then µ eff 1.8µ B. This is just short of the experimentally observed moments in Sr 2 MgOsO 6 and Ca 2 MgOsO 6 (1.87µ B ) [52], Ba 2 YReO 6 (1.93µ B ) [51], and La 2 LiReO 6 (1.97µ B ) [51]. 45

58 3.4 Discussion We have studied spin-orbital models for both d 1 and d 2 double perovskites where the B ions are magnetic and have strong spin-orbit coupling. We found several non-trivial magnetically ordered phases characterized both by ordering of the spin/orbital angular momentum and ordering of the orbitals. We emphasize that examination of the spin and orbital degrees of freedom separately gives an enhanced qualitative understanding of the magnetism for this class of spin-orbit coupled double perovskites. Our results allow us to draw many conclusions that can be connected to particular materials and experiments. First, the canted FM (canted AFM) phase may describe the known ferromagnets Ba 2 NaOsO 6, Ba 2 MgReO 6, and Ba 2 ZnReO 6. Experimental fitting inverse susceptibility versus temperature data may strongly depend on the region fitted due to deviations from the Curie-Weiss law. Furthermore, high temperature susceptibility may lead to an incorrect interpretation of the magnetic interactions in these materials. On the other hand, our simplified model shows how canted states due to anisotropy from orbital order can genuinely give negative Curie-Weiss temperatures while retaining a diverging susceptibility at T c. Second, while there is also strong evidence for orbital (quadrupolar) order in the ferromagnet Ba 2 NaOsO 6, we propose the same order may be present in antiferromagnetic Ba 2 LiOsO 6. We found that the orbital structure which allows canted states can also support a unique antiferromagnetic state. This is due to the qualitative interpretation that orbital order constrains the direction of orbital angular momentum which tends to favor one of two types of coplanar structures. Third, we predominantly found antiferromagnetism in d 2 compounds since ferromagnetism was not as favorable due to energetic constraints imposed by orbital occupancies. We take this as an explanation for why so many d 1 ferromagnets exist but few (if any) d 2 ferromagnets exist. Furthermore, although the AFM [110] structure is likely to be found in tetragonal crystals such as Sr 2 MgOsO 6, the unusual 4-sublattice antiferromagnetic structure common to both d 1 and d 2 is likely to be found in cubic d 2 materials due to the large 46

59 region of the phase diagram that it occupies. Finally, in light of recent NMR work on Ba 2 NaOsO 6 [40], we highlight an experimental puzzle. Above the magnetic ordering temperature of approximately 7 K, a broken local point group symmetry persists to higher temperatures between 10 K to 15 K. With the assumption that the signal for this broken local point group symmetry corresponds to an orbital (quadrupolar) order parameter, we would conclude that orbital order disappears above 15 K. In fact our microscopic model does allow for situations where T o is close to T c. This could be used to show how antiferromagnetic interactions dominate above T o, and only below T o would the orbital occupancies change to favor ferromagnetism and a diverging magnetic susceptibility at T c. The difficulty with these explanations is that it may be inconsistent with specific heat measurements [31, 32]. This leads us to favor a simplified interpretation of canted AFM over instead of low temperature orbital order. We are then still left with the task of interpreting these seemingly contradictory results. Using just NMR data, negative Curie-Weiss temperatures could be trivially explained using an orbital ordering energy scale near the magnetic ordering temperature. Using just specific heat data, this explanation instead suggests high temperature orbital order. This leaves several open questions. Does the broken local point group symmetry persist above 15 K? At what temperature is the R ln 4 magnetic entropy recovered? Is there a relationship between the orbital order and the negative Curie-Weiss intercept? 47

60 Chapter 4 d 4 Mott Insulators 4.1 Introduction Between weakly correlated topological insulators and strongly correlated 3d transition metal oxides lie 5d compounds that combine both strong spin-orbit coupling and correlations on an equal footing [4]. In contrast to the well studied 5d 5 materials, the effect of strong spin-orbit coupling in 4d 4 and 5d 4 systems have been sparsely studied due to expectations that these will naturally lead to non-magnetic insulating behavior. However there are several experimental counter examples to this notion. The first example is Ca 2 RuO 4 which displays a moment of 1.3µ B [60, 61]. Recently both double perovskite iridates [62 64] and honeycomb ruthenates [65] in the d 4 configuration have been found to show magnetism. It has been argued that partial quenching of the orbital angular momentum from the presence of lattice distortions is the root cause. Very recent developments [13, 24 26] on Ba 2 YIrO 6 have piqued interest on the origin of magnetism in this 5d 4 system because the compound is negligibly distorted and still shows a Curie response. In transition metal oxides with oxygen octahedra, the large crystal field splitting puts d 4 ions into the t 4 2g electronic configuration. For materials with strong spin-orbit coupling, the j = 3/2 band is filled and the j = 1/2 band is empty leading to the conclusion that weakly correlated d 4 materials are non-magnetic. However when Coulomb interactions are strong, a total spin S = 1 and orbital angular momentum L = 1 lead to a total angular momentum J = 0 on every d 4 ion with no magnetism. Thus both jj coupling and LS 48

61 coupling schemes lead to the same conclusion that a single atom is in a J = 0 singlet state and therefore trivially non-magnetic [8] as shown in Fig. 4.1(a). We build on previous work by Khaliullin [21] that proposed an exciton condensation mechanism, more accurately a condensation of J = 1 triplon excitations, to drive the onset of antiferromagnetism in nominally non-magnetic d 4 systems and our previous study [12] showing that ferromagnetic superexchange interactions caused by strong Hund s coupling can precipitate ferromagnetic coupling. In this work we start with the atomic multi-orbital Hamiltonian with intra- and inter-orbital Coulomb interactions and spin-orbital coupling specifically for t 4 2g systems. We next allow hopping between atoms and investigate all cases of orbital geometries the idealized fully symmetric case when all orbitals participate in hopping, as well as more realistic cases suitable for simple cubic and face-centered cubic lattices. For each case, we derive the effective spin-orbital superexchange Hamiltonian which competes with spin-orbit coupling to produce strong deviations from the non-magnetic atomic behavior. These results are obtained both using exact diagonalization on a two-site problem and perturbation theory for the effective magnetic interactions. Tuning the superexchange interactions J SE relative to spin-orbit coupling λ, we first see the formation of local moments followed by a Bose condensation of weakly interacting J = 1 triplet excitations, or triplon condensation. Rather remarkably, regardless of the local spin interactions favoring antiferromagnetic spin superexchange (spin-af) at small Hund s coupling or ferromagnetic spin superexchange (spin-f) at large Hund s coupling, the J = 1 triplons condense at the k = π point. This result that the rotationally invariant spin-orbit coupling can effectively flip the sign of superexchange is unusual and unique to spin-orbital coupled systems. In the opposite regime where J SE dominates, the orbital interactions are frustrated even in the absence of geometric frustration and can potentially lead to orbital liquid phases. Even when λ = 0 and the local spin interactions are simple Heisenberg FM or AFM, the frustrated orbital interactions generate frustration in the spin channel as well, leading to the possibility of ground states with both orbital and spin entanglement on lattices without geometric frustration. This is summarized schematically in Fig. 4.1(b). The chapter is organized in the following way. In Section 4.2 we introduce the lattice 49

62 Hamiltonian used as the basis for the rest of the chapter which includes electron hopping, atomic spin-orbit coupling, and an effective multi-orbital Coulomb interaction that captures Hund s rules. The orbital geometries for hopping used throughout the chapter include both a highly symmetric toy model to be used as a simplified diagnostic tool as well as two other more realistic cases found in perovskites. In Section 4.3 we use exact diagonalization to study a two-site specialization of the problem introduced in Section 4.2. Isobe et. al. [66] has used a similar procedure to study transition metal systems with other electron counts. Local magnetic moments are absent when spin-orbit coupling is large, as expected in the atomic picture, but electron hopping introduces sizeable moments when t λ when two or three orbitals strongly overlap between sites. Although a single orbital overlap can also promote superexchange which competes with spin-orbit coupling, the number of superexchange pathways is limited and local moments do not form for any reasonable ratio of t/λ. In Section 4.4 we derive an effective magnetic Hamiltonian in terms of orbital angular momentum and spin operators using second order perturbation theory. We check that the spin-orbital superexchange Hamiltonian captures both spin-af and spin-f interactions between spins depending on the value of Hund s coupling, and the sum of spin-orbit coupling and the spin-orbital superexchange Hamiltonian reproduce the phases found in exact diagonalization of a two-site system. In Section 4.5, we give a qualitative description of how bond-dependent spin-orbital superexchange results in orbital frustration. However, finding solutions to orbitally frustrated models can be challenging and is outside the scope of the present chapter [30, 67 69]. In Section 4.6 we first review the excitonic condensation mechanism where the Bose condensation of van Vleck excitations gives magnetism to d 4 systems with spin-orbit coupling. Although the condensation mechanism involves approximations to full spin-orbital models derived in the previous section, it gives valuable insight into the behavior at large spin-orbit coupling. Regardless of the nature of local interactions, only AF condensates are supported for the models studied, and we give the the critical superexchange required for AF condensation for the three orbital geometries studied. 50

63 Section 4.7 discusses potential materials realizations and experiments beyond those mentioned in the introduction. 4.2 Model Our model Hamiltonian for t 2g systems 1 is composed of three parts: (i) kinetic part, (ii) Coulomb interaction, and (iii) spin-orbit coupling. H = ij H (ij) t + i H (i) int + i H (i) so (4.1) The general form of the kinetic part H (ij) t = mm σ t (ij) m m c im σ c jmσ + h.c. (4.2) is given in terms of matrix elements t (ij) m m between t 2g orbitals m and m (with values yz, zx, and xy) on sites i and j. The index σ is for spin. We take the on-site Coulomb interaction to be the t 2g interaction Hamiltonian, H int H Hubbard, derived in Section 2.3 and appearing in (2.12). The on-site intra-orbital Hubbard interaction is characterized U and J H characterizes the strength of Hund s coupling. We have chosen to use J H instead of J to avoid confusion with total angular momentum in the next two sections. The atomic spin-orbit coupling has the form given in Section 2.4 and appearing in (2.14). We focus on three special cases of t (ij) m m which differ by the number of orbitals, N orb participating in hopping. N orb = 3: First we consider the orbitally symmetric case where t (ij) m m = tδ m m and all orbitals participate in hopping. While this full rotational symmetry is not usually present in material systems, the N orb = 3 case serves as a diagnostic tool where total angular momentum in the system is conserved and correlation functions have rotational symmetry. N orb = 2: The next case, t (ij) m m = tδ m m (1 δ km ), uses two orbitals participating in hopping, N orb = 2, while one orbital is blocked. The blocked orbital k is determined by 1 This effective model for t 2g orbitals assumes the ligand orbitals have been effectively integrated out, ie. the Mott-Hubbard limit. 51

64 (a) jj coupling LS coupling j = 1/2 S = L = 1 j = 3/2 J = 0 J = 0 (b) JH/U van Vleck PM AF Triplon BEC Spin F Spin AF λ zj SE zj SE λ Figure 4.1: (a) The single site total angular momentum is zero in both the jj and LS coupling schemes. (b) Schematic phase diagram of the spin-orbital model appearing in (4.3) pitting spin-orbit coupling λ against superexchange J SE where λ is the spin-orbit coupling energy scale and J SE is the superexchange energy scale with z being the coordination number. Starting with a van Vleck phase with no atomic moments at large λ we find a triplon condensate at k = π for all values of the Hund s coupling J H /U. The intermediate regime where λ zj SE has not been explored. At large J SE we obtain effective magnetic Hamiltonians that have isotropic Heisenberg spin interactions (antiferromagnetic for small J H /U and ferromagnetic for large J H /U) but the orbital interactions are more complex and anisotropic. We expect novel magnetic phases arising from orbital frustration in the intermediate and large J SE /λ regimes. 52

65 (a) x z y d yz p z d yz d xy p x d xy (b) d xy d xy Figure 4.2: (a) The N orb = 2 model is an approximation of oxygen mediated electron hopping between t 2g orbitals in a simple cubic lattice. Both d xy and d yz orbitals participate in hopping along the y direction. (b) The N orb = 1 model is an approximation of direct hopping between t 2g orbitals on the face of a face-centered cubic lattice. The d xy orbitals are most relevant for hopping in the xy plane. 53

66 the direction of the line connecting sites i and j. This situation is commonly found in simple cubic lattices where t comes from oxygen-mediated superexchange. See Fig. 4.2(a). N orb = 1: The final case, t (ij) m m = tδ m mδ km, only has one orbital contributing, N orb = 1, while two orbitals are blocked and approximates the hopping between nearest-neighbors on a face-centered cubic lattice. The active orbital k is determined by which plane the sites i and j share. See Fig. 4.2(b). 4.3 Exact diagonalization Before analyzing the full lattice problem which will require approximations to be made, it is useful to examine exact results for a pair of interacting sites. We numerically diagonalize (4.1) for a two-site site system, with site labels i and j, to extract the magnetic interactions in the Mott limit. We choose the blocked orbital k to be the xy orbital for the N orb = 2 and N orb = 1 models. Fig. 4.3 gives ground state values of the square of the local total angular momentum, Ji 2, for the two-site specialization of (4.1). For all three types of hopping matrices, small t compared to λ give negligible local moments since spin-orbit coupling keeps each site in a nonmagnetic J i = 0 spin-orbital singlet. For larger values of t, local moments may form from the tendency of superexchange to cause spin and orbital ordering which is incompatible with local spin-orbital singlet behavior on each site. For both N orb = 3 and N orb = 2, this effect is pronounced and requires t/λ 2 at the two-site level. In a lattice, this critical ratio will be reduced due to presence of many neighboring sites contributing to superexchange, hence a smaller hopping t is able to destabilize the atomic singlet. For N orb = 1, the effect is much less pronounced since the number of superexchange paths is limited. When a single orbital is active, the results do not sensitively depend on J H /U, however, the presence of strong Hund s coupling results in qualitatively different behavior for the N orb = 3 and N orb = 2 models. We expect that antiferromagnetic superexchange between spins (spin-af) is responsible for moment formation and can qualitatively be understood in the following way. Each site has a local total spin S i = 1 and local orbital angular momen- 54

67 J 2 i (a) J H /U = 0.1 J H /U = (b) N orb = 3 N orb = 3 (c) (d) t / U N orb = 2 N orb = 2 (e) (f) N orb = 1 N orb = t / λ Figure 4.3: The Hamiltonian in (4.1) is solved for a two-site system. The local total angular momentum squared on one site, Ji 2, is plotted for small and large values of Hund s coupling, J H /U = 0.1 and J H /U = 0.2, for the three types of hopping matrices used in the text. (a-b) Hopping using N orb = 3 produces sizable local moments. For small Hund s coupling, the local moment gradually forms as t is turned on. For large Hund s coupling, there is an abrupt formation of large local moments due to an energy level crossing. (c-d) Hopping using N orb = 2 produces qualitatively similar behavior to the N orb = 3 case. (e-f) Hopping using N orb = 1 produces negligible moments. 55