Emergent Phenomena in Spin Crossover Systems

Jace Alex Cruddas B.Sc. (Hons)

Candidate’s ORCID

A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in Year School of Mathematics and Physics Abstract

In general, a spin crossover (SCO) system is any complex, material or framework containing two thermodynamically accessible spin-states: one high-spin (HS) and one low-spin (LS). The transition between spin-states is addressable by temperature, pressure, light irradiation, electric and magnetic fields, and chemical environment. The transition itself can be first-order, exhibiting hysteresis, continuous or a crossover. Typically, accompanied by the ferroelastic ordering of spin-states. It can also be part of an incomplete or multi-step transition accompanied by the antiferroelastic ordering of spin-states. In general, any alterations to the structural characteristics of SCO systems can have an effect on their bulk properties and behaviours. Consequently, constructing structure-property relations has traditionally been an extremely challenging task, and one of both great theoretical and experimental interest. Understanding the mechanisms behind these bulk properties and behaviours could lead to the rational design of SCO systems with enhanced applications and the synthesis of novel properties and behaviours. In this thesis we show that a simple, elastic model of SCO systems hosts almost all experimentally reported SCO properties and behaviours. We demonstrate clear structure-property relations that explain these results, derive the mechanisms of multi-step transitions and explain why and how intermolecular interactions play a role. We also propose that a new exotic state of matter could exist in elastically frustrated SCO materials and frameworks. In this phase “spin-state ice”, so-called in analogy to water- and spin- ice, the metal ions lack any kind of long-range order. Instead, local clusters of metal ions follow a local ‘ice rule’. For example on the kagome lattice, each triangle is constrained to have two metal ions in one spin-state and one in the other. The excitations are deconfined quasi-particles, with a fractionalised spin midway between that of the HS and LS states. We show that distinctive signatures of spin-state ice can be measured by neutron scattering, electron paramagnetic resonance, and thermodynamic experiments. Unlike other examples of ices that have been theorized to exist, the unique nature of SCO systems allows for multiple spin-state ice phases to exist on the same lattice with unique properties that can be tuned with external parameters, like temperature and pressure. Declaration by author

This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, signiﬁcant technical procedures, professional editorial advice, ﬁnancial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award.

I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School.

I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. Publications included in this thesis

• [1] Jace Cruddas, and Ben J. Powell, Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Inorg. Chem. Frount., Advance Article, 10.1039/D0QI00799D (2020).

• [2] Jace Cruddas, and Ben J. Powell, Spin-State Ice in Elastically Frustrated Spin-Crossover Materials, J. Am. Chem. Soc. 141, 19790 (2019).

Submitted manuscripts included in this thesis

• [3] Jace Cruddas, and Ben J. Powell, Multiple Coulomb phases with temperature tunable ice rules in pyrochlore spin crossover materials, arXiv:2007.13983 (2020).

Other publications during candidature

• [176] Gian Ruzzi, Jace Cruddas, Rozz H. McKenzie and Ben J. Powell, Equivalence of elastic and Ising models for spin crossover materials, arXiv:2008.08738 (2020).

Contributions by others to the thesis

Elise P. Kenny and Ben J. Powell helped with the proof-reading of this thesis. Ross H. McKenzie provided the argument summarized in Fig. 3.3. All the others contributions made to this thesis are clearly documented on the page preceding the relevant chapter.

Statement of parts of the thesis submitted to qualify for the award of another degree

No works submitted towards another degree have been included in this thesis.

Research involving human or animal subjects

No animal or human subjects were involved in this research. Acknowledgments

I would like to thank Elise Kenny for proof reading my thesis, and Ben Powell, Ross McKenzie, Xiuwen Zhou, Alejandro Mezio, Cameron Kepert, Suzanne Neville, Stephan Rachel, Nic Shannon, Yaroslav Kharkov, Grace Morgan, Timo Nieminen, Gian Ruzzi, Nadeem Muhammad, Nena Batenberg and Blake Peterson for their helpful conversations. In addition, I would like to extend my thanks to my family, my friends, the bartenders who served me, the bartenders who refused to serve me, and the current and former personnel of the School of Mathematics and Physics for their immeasurable support. Financial support

This work was funded by the Australian Research Council through grant number DP200100305 and an Australian Government Research Training Program Scholarship.

Keywords spin crossover, spin transitions, ice, spin ice, exotic states of matter, fractionalization, quasi-particles, structure-property relations, phase transitions, frustration

Australian and New Zealand Standard Research Classiﬁcations (ANZSRC)

020499 Condensed Matter Physics not elsewhere classiﬁed, 60% 030206 Solid State Chemistry, 20% 030207 Transition Metal Chemistry, 20%

Fields of Research (FoR) Classiﬁcation

0204 Condensed Matter Physics, 60% 0302 Inorganic Chemistry, 40% Contents

Abstract ...... ii

Contents vii

List of Figures ix

List of Tables xii

1 Introduction 1 1.1 Frustration ...... 3 1.1.1 Importance of Frustration ...... 3 1.1.2 Landau theory and phase transitions ...... 4 1.1.3 Unfrustrated systems ...... 7 1.1.4 Frustrated Systems ...... 11 1.2 Microscopic Origin of Spin Crossover Systems ...... 19 1.2.1 A Brief Introduction to Coordination Chemistry ...... 19 1.2.2 Crystal ﬁeld theory ...... 21 1.2.3 Molecular orbital theory ...... 23 1.2.4 Ligand Field Theory ...... 26 1.3 Spin crossover ...... 30 1.3.1 Importance of Spin Crossover ...... 30 1.3.2 Spin Crossover in the Solid state ...... 30 1.3.3 Models of Spin Crossover ...... 32 1.4 Thesis Outline ...... 37

2 Structure–property relationships 41 2.1 Introduction ...... 41 2.2 Model ...... 45 2.2.1 Elastic interactions in materials ...... 48 2.3 Methods ...... 49 2.4 Results and Discussion ...... 50 2.4.1 Nearest- and Next Nearest-Neighbour Interactions ...... 50 vii viii CONTENTS

2.4.2 Third Nearest-Neighbour Interactions ...... 53 2.4.3 Longer Range Interactions ...... 56 2.5 Conclusions ...... 59 2.6 Supplementary Information ...... 60 2.6.1 Expansion of the potential ...... 60 2.6.2 Additional results ...... 61

3 Spin-State Ice 73 3.1 Introduction ...... 73 3.2 Model ...... 78 3.3 Monte Carlo calculations ...... 81 3.4 Results and discussion ...... 81 3.5 Conclusions ...... 86 3.6 Supplementary Information ...... 88 3.6.1 Mapping between spin ice rules and spin-state ice rules ...... 88 3.6.2 Snapshots of Monte Carlo simulations ...... 88

4 Multiple Coulomb phases 95 4.1 Introduction ...... 95 4.2 Model ...... 96 4.3 Results and Discussion ...... 98 4.4 Conclusion ...... 102 4.5 Supplementary Information ...... 104 4.5.1 Monte Carlo simulations ...... 104

5 Conclusion 109

Bibliography 111 List of Figures

1.1 Illustration of spontaneous symmetry breaking transition...... 3 1.2 Cartoon of frustration...... 4

1.3 Cartoon phase diagram of H2O...... 5 1.4 First-order transition...... 6 1.5 Magnetic phases of the Ising model...... 8 1.6 Ising Mean-ﬁeld phase diagrams...... 10 1.7 Illustration the 1h phase and the ice rules...... 12 1.8 Examples of frustrated Lattices...... 13 1.9 Ice rules for equivalent systems...... 14 1.10 Illustration of pinch point singularities in the structure factor...... 17 1.11 Propagation of defects in vertex ice...... 18 1.12 Example of a coordination complex...... 20 1.13 Molecular geometries of coordination complexes...... 20 1.14 Illustration of crystal ﬁeld splitting...... 22 1.15 Illustration of bonding and antibonding orbitals...... 24

1.16 Molecular orbitals for a ML6 complex...... 25

1.17 Possible spin-states for a ML6 complex...... 27 6 1.18 Tanabe-Sugano diagram for a ML6 complex with d ...... 29 1.19 Vibronic levels for a spin crossover complex...... 29 1.20 Examples of spin crossovers and transitions...... 31 1.21 Thermodynamic properties of a non-interacting Ising-like model...... 35 1.22 Thermodynamic properties of an Ising-like model...... 35 1.23 Diagram of the elastic model...... 36

2.1 Antiferroelastic spin-state orderings for the elastic model with non-zero k1, k2, k3 and k5. 42 2.2 Lattice structures of “square” spin crossover materials/frameworks...... 45 2.3 Lennard-Jones potential...... 48

2.4 Zero-temperature phase diagram for the elastic model with non-zero k1 and k2...... 50 2.5 Two-step transitions for the 1n24 and 1n02 families...... 51 2.6 Two-step transitions for the 1n14 family...... 53 ix x LIST OF FIGURES

2.7 Four-step transitions for 1n24 and 1n02 families...... 54 2.8 SCO curves for the third nearest neighbour model...... 55 2.9 SCO curves for the ﬁfth nearest neighbour model...... 57 2.10 Eight-step transitions for the 1n24 and 1n02 families...... 58

2.11 Phase diagram for the elastic model with k1 6= 0...... 62

2.12 Heat capacity for the elastic model with k1 6= 0...... 62

2.13 Phase diagram for the elastic model with k1 > 0 and k2 > 0...... 63

2.14 Heat capacities for the elastic model with k1 > 0 and k2 > 0...... 63

2.15 Phase diagram for the elastic model with k1 > 0 and k2 = −0.2k10...... 63

2.16 Heat capacities for the elastic model with k1 > 0 and k2 = −0.2k10...... 64

2.17 Phase diagram for the elastic model with k1 < 0 and k2 = 1.2|k1|0...... 64

2.18 Heat capacities for the elastic model with k1 < 0 and k2 = 1.2|k1|0...... 64

2.19 Heat capacities for the elastic model with k1 > 0, k2 = −0.9k1 and k3 = 0.5k1...... 65 2.20 Heat capacities for the third nearest neighbour model...... 65

2.21 Heat capacities for the elastic model with k1 < 0, k2 = 1.2|k1|, k3 = −0.5|k1|, k4 = 0, and

k5 = 0.2|k1|...... 66 2.22 Heat capacities for the eight step transition...... 66

2.23 Thermodynamics for the elastic model with k1 < 0 and k2 = 0.6|k1|...... 67

2.24 SCO curves for the elastic model k1 > 0...... 68

2.25 SCO curves for the elastic model with k1 and k2 = 0.1k1...... 69 2.26 Zero-temperature for the third nearest neighbour model...... 70 2.27 Zero-temperature for the ﬁfth nearest neighbour model...... 70

2.28 SCO curves for the elastic model with k1 < 0, k2 = 1.2|k1|, k3 = −0.5|k1| and k5 = 0.2|k1|. 71

3.1 Ice rules for various ice phases...... 75 3.2 Propagation of defects in artiﬁcial spin-ice and spin-state ice...... 77 3.3 Spin-state ice model for the kagome lattice...... 79 3.4 Zero-temperature phase diagram for kagome spin-state ice...... 82 3.5 Snapshots of spin-state ice and the structure factor for kagome spin-state ice...... 83 3.6 Thermodynamics for kagome spin-state ice...... 85 3.7 Equivalence of spin-state ice and spin ice rules and lattices...... 89 3.8 Equivalence of spin-state ice and hardcore dimer/loop rules and lattices...... 90 3.9 Snapshot of the full lattice from which Fig. 3.5a is taken...... 91 3.10 Snapshot of the full lattice from which Fig. 3.5b is taken...... 92

4.1 Spin-state ice model for the pyrochlore lattice...... 97 4.2 Zero-temperature phase diagram for the pyrochlore Coulomb phases...... 98 4.3 Structure factors for the pyrochlore Coulomb phases...... 99 4.4 SCO curves for the pyrochlore Coulomb phases...... 101 4.5 Propagation of defects a pyrochlore Coulomb phase...... 102 LIST OF FIGURES xi

4.6 Heat capacities for the pyrochlore Coulomb phases...... 105 4.7 Phase diagrams for the pyrochlore Coulomb phases...... 106 4.8 Defects in the pyrochlore Coulomb phases...... 106 List of Tables

1.1 Coordination numbers and their corresponding molecular geometries...... 20

2.1 Multi-step transitions reported in experimental literature...... 44

3.1 Possible quasi-particles for spin crossover systems...... 84

xii Chapter 1

Introduction

Quantum mechanics, the structure of the atom and the standard model are all triumphs of physics that have been made by reductionism; the idea that we can understand everything about the universe, by breaking it down into its constituents and understanding the symmetries (laws and forces) that govern them [4]. As the constituents get smaller, an increasing number of symmetries accompany them. However, there (as of yet) is no law of nature that requires that the symmetries of large aggregates of constituents can be inferred from their individual constituents. As P. W. Anderson [5] famously stated “More is different”; when matter comes together it can develop novel properties and behaviours that are qualitatively different from the sum of their parts. For example, on the atomic scale, we know almost everything about the motion of electrons, protons and neutrons. The motion of the particles is described by the many-body non-relativistic Schrodinger¨ equation ∂ i} ψ(r1,r2,...) = Hψ(r1,r2,...), (1.1) ∂t which describes the wave-like nature of particles as a many-body wavefunction ψ(r1,r2,...), where ri denotes coordinate of the ith particle. The probability of ﬁnding a particle at those coordinates is given by the square magnitude of the wavefunction

2 p(r1,r2,r3,...) = |ψ(r1,r2,r3,...)| . (1.2)

With certain caveats, everything about the particle’s statistics is described by the wavefunction; the position, momentum, energy and correlations. However, there is another important aspect to consider; while the protons and neutrons are localised to the atomic nuclei, the wavefunctions of the electrons can overlap and constructively interfere, such that the probability distribution

2 2 2 |ψ1 + ψ2| 6= |ψ1| + |ψ2| , (1.3) allowing for electrons to move between nuclei. This interference, among other things, is responsible for giving matter its novel bulk properties: conductivity, elasticity, rigidity, magnetism, spin crossover and more. Emergence is yet another expression of ‘more is different’. Unlike global symmetries, emergent symmetries have no particular significance in terms of the microscopic Schrodinger¨ equation for 1 2 CHAPTER 1. INTRODUCTION the electrons and nuclei, and thus cannot be inferred from their individual constituents. Emergent symmetries typically accompany emergent particle-like behaviours. The behaviour of these “emergent particles” can differ quite dramatically from the behaviour of electrons, protons and neutrons that make up the material. Depending on the atomic architecture of the materials, these emergent particles can carry a fraction of the charge of an electron [6], behave like magnetic monopoles [7] or be Majorana fermions which are their own antiparticles [8]. Remarkably, fractionalised charges and magnetic monopoles have not been observed in the standard model. They only arise as a result of the emergent behaviour of large aggregates of atoms. Furthermore, the emergent gauge symmetries that govern these particle-like behaviours, do not have any significance in the microscopic picture of the Schrodinger¨ equation, and cannot be simply or directly inferred from it. As opposed to a global symmetry, which is a property of the system, a gauge symmetry is a redundancy in the description of the system [9]. We are far from understanding all the possible bulk and emergent properties of matter, nor do we understand all the mechanisms that could give rise to them. This is due, in part, to the fact that for a system of N particles with k degrees of freedom, the wavefunction has on the order of O(kN) degrees of freedom, making solving Schrodinger’s¨ equation a computationally intractable problem [4, 10]. However, even if we could solve the Schrodinger¨ equation exactly, it would only give us part of the answer. While Schrodinger’s¨ equation tells us everything about the bulk and emergent properties of specific materials, it tells us nothing about how to distinguish the bulk and emergent properties of different materials. In order to distinguish different bulk and emergent patterns of behaviour we need emergent principles; organisational principles that are not tied to scale. An example of an emergent principle is Landau’s theory of spontaneous symmetry breaking. The main idea behind Landau theory is that the development of macroscopic order can be tied to a macroscopic order parameter, φ. This parameter is finite in the ordered phase, and zero in the disordered phase. Its physical solutions can be calculated by minimising over the free energy F. Landau proposed that, close to a phase transition the free energy (per unit volume) can be expanded as a power series in φ. Since, the free energy shares the same symmetry as Schrodinger’s¨ equation, we can simplify this expansion by only considering the terms that preserve the symmetry of Schrodinger’s¨ equation. For a φ → −φ symmetry, the free energy (per unit volume) will only contain terms that are even powers of φ F a a f (φ) = = f + 2 φ 2 + 4 φ 4 + O(φ 8), (1.4) V 0 2 4! where V is the volume and the coefficients ai are a function of temperature. A phase transition can occurs at the critical temperature Tc, if a2 changes sign at Tc and a4 remains positive (to ensure the stability of the free energy). Therefore, we can set a2 = a(T − Tc) and a4 = b > 0. Under these conditions, the free energy has one minima at φ = 0 (for T > Tc) and the same symmetry as the many-body Schrodinger¨ equation (see Fig. 1.1). But, for T < Tc, the free energy develops two minima p or “broken symmetry” solutions at φ± = ± 3!(a/b)(Tc − T) . Therefore, as the system cools, it is forced to choose between the broken symmetry states, breaking the symmetry of the many-body Schrodinger¨ equation. Landau associated this symmetry breaking with the development of long-range order. With this simple phenomenological theory, Landau described the universal properties of matter 1.1. FRUSTRATION 3

Figure 1.1: Schematic of the free energy (Eq. 1.4), indicating a spontaneous symmetry breaking transition at the critical temperature Tc. Above Tc, the system has one minima and the same symmetry as the many-body Schrodinger¨ equation. Below Tc, the system develops two global minima, forcing the system to choose between them, breaking the symmetry of Schrodinger’s¨ equation. near a phase transition, without reference to the mechanism behind the ordering or the microscopic origin of the order parameter. Beyond the characterisation of bulk and emergent properties, emergent principles have allowed for parallels to be drawn between different materials and disparate areas of physics, resulting in countless discoveries. Understanding the mechanisms behind these emergent principles, and the discovery of new ones, has become one of the important challenges of modern physics. In this thesis, we use emergent principles to draw parallels between two seemingly unconnected physical phenomena: frustration and spin crossover. In this chapter, we will review the microscopic properties and emergent principles behind both frustration (in Section 1.1) and spin crossover (in Section 1.3).

1.1 Frustration

1.1.1 Importance of Frustration

A system is said to be frustrated when it is unable to simultaneously minimise over all competing interactions. This phenomenon is at the heart of many emergent properties and behaviours in physics, chemistry and biology [11]. In the extreme cases studied in condensed matter, frustration can occur as a result of chemical disorder or geometry [12]. The prototypical example of frustration is the assignment of arrows to a graph (see Fig. 1.2). If we try to place arrows antiparallel to each other on a square lattice, we can simultaneously satisfy this constraint for every arrow (see Fig. 1.2a). However, if we try to do the same thing on a triangular lattice, we can only place two arrows (one up and one down) antiparallel to each other. The third arrow, which is connected to both, cannot be antiparallel to both (see Fig. 1.2b). Therefore, all the interactions cannot be simultaneously satisﬁed and thus, the system is frustrated. In this case, the system is “geometrically frustrated” as the origin of the competition is the geometry of the lattice. Frustration is widespread in condensed matter. Over the recent decades, 4 CHAPTER 1. INTRODUCTION

Figure 1.2: Cartoon illustrating frustration; the inability of a system to simultaneously minimise over competing interactions. (a) In an unfrustrated system each arrow can be placed antiparallel to its nearest neighbour (indicated by black lines). Whereas, (b) In a frustrated system each arrow cannot simultaneously be placed anti-parallel to its nearest neighbours. the discovery and manipulation of frustrated systems has been one of the most powerful driving forces for the discovery of new and exotic emergent behaviours [7, 11, 12]. In this section we will brieﬂy review Landau theory (Section 1.1.2), and the transitions in unfrustrated 1.1.3) and frustrated spin systems (Section 1.1.4) that are relevant to this thesis.

1.1.2 Landau theory and phase transitions

Nature commonly exhibits two different kinds of phase transitions. The most common kind, a first- order transition, occurs when a discontinuity or singularity occurs in the first-order derivative of the free energy. During this process the system either absorbs or releases a finite amount of energy. For example, the ice/liquid/vapour transitions observed in the phase diagram for H2O (see Fig. 1.3a) are all first-order. The other type of transition, a second-order or continuous transition, is the result of a continuous first-order derivative and a discontinuous second-order derivative. During this process no energy is either released or absorbed. The prototypical example of a second-order transition is the paramagnetic/ferromagnetic transition observed in a ferromagnet. We will give examples of both first and second-order transitions in Section 1.1.3. Both first and second-order transitions can be described by Landau theory. According to Landau’s theory, the (Helmholtz) free energy, F = V f , is a non-analytic function of the order parameter. Where V is the volume and f is the free energy per unit volume. It should be noted that f is an analytic function of the order parameter, F only becomes non-analytic in the thermodynamic limit, V → ∞ as the density remains constant. Since f is a analytic function, it can be expanded as a power series in the order parameter φ, such that F a a a a a f = = f + a φ + 2 φ 2 + 3 φ 3 + 4 φ 4 + 5 φ 5 + 6 φ 6 + O(φ 7), (1.5) V 0 1 2 3! 4! 5! 6! where V is the volume and the coefficients ai are a function of the external parameters. The free energy is assumed to have all the symmetries associated with the order parameter. We will assume the order parameter describes a system with a φ → −φ symmetry, and that the coefficients are a function of 1.1. FRUSTRATION 5

Figure 1.3: Cartoon phase diagram of H2O. The black lines indicate first-order transitions, the black circle indicates a critical point, at which point the transition becomes second order, and the white circle indicates a triple point. The Ice region can be broken into at least seventeen different ice phases. Only the Ice 1h phase is shown here, as it is relevant to Section 1.1.4. This diagram has been adapted using information from Ref. [13]. temperature. In this case, only terms that contain even powers of φ preserve the φ → −φ symmetry. The implicit assumption here, is that for a system to be described by a real scalar order parameter, the properties of the system must be uniform or have a mean state. This assumption breaks down when non-locality, spatial inhomogeneity or fluctuations in the system become important. Landau theory describes a second-order phase transition when all the coefficients associated with odd powers of φ parameters are zero, a2 changes sign at the critical point Tc, and all the remaining coefficients are positive (to ensure stability). Here we set a2 = a(T − Tc), a4 = b > 0, and neglect terms above O(m4). As we have previously shown, minimising the free energy gives   f0, if T > Tc f ≈ (1.6) 3a2 2  f0 − 2b |T − Tc| , if T < Tc

Above Tc, the system has one minima at φ0 = 0 and below Tc, the system develops two global minima p at φ± = ± 3!(a/b)(Tc − T) . The height of the barrier between the two global minima, 3a2 F(φ ) − F(0) = −V |T − T |2, (1.7) + 2b c diverges in the thermodynamic limit, V → ∞ as the density remains constant. Hence, as the system passes below the critical temperature, it is forced to choose between the global minima, spontaneously breaking the φ → −φ symmetry. By taking the appropriate limits, one can see that the ﬁrst derivative is continuous at the critical temperature, while the second derivative has a singularity: ∂F ∂F lim lim − lim = 0, (1.8) + − V→∞ T→Tc ∂T T→Tc ∂T ∂ 2F ∂F2 lim lim − lim → ∞. (1.9) + 2 − 2 V→∞ T→Tc ∂T T→Tc ∂T 6 CHAPTER 1. INTRODUCTION

Figure 1.4: Schematic of a ﬁrst-order transition as indicated by (a) the free energy (per unit site) f and (b) the order parameter φ. Depending on the value of a1, the system either has two or one global minima. Increasing or decreasing a1 demotes a global minima to a local minima. Smoothly varying a1 can result in the system being stuck in a metastable (local minima) state, introducing a hysteresis loop in the order parameter.

When a coefﬁcient associated with an odd power of φ is non-zero, Landau theory describes a

first-order transition. Here we set a1 6= 0 and a3 = 0. By minimising over the free energy, one can find: one critical point for T > Tc and either three or one critical points for T < Tc (see Fig. 1.4a). We will − + denote the region below Tc, with three critical points, as a1 < a1 < a1 and the regions with one critical − + point by a1 < a1 and a1 > a1 . In this region, the state of the system will not only depend on the − values of the coefficients, but also the past history of the system (see Fig. 1.4a). Suppose that a1 < a1 , the system will have one global minima at φ = φ−1 < 0. As the value of a1 is slowly increased above − a1 < a1 < 0, the free energy will develop a local minima at φ = φ1 > 0. Increasing a1 further, will + result in a change of the global minima from φ−1 to φ1 when 0 < a1 < a1 . However, the system can be prevented from moving into the global minima. This occurs when the fluctuations in the order parameter are smaller than the height of the energy barrier. Hence, the local minima is a metastable solution, as it only has a finite lifetime. The local minima will eventually disappear when the energy + barrier turns into a saddle point at a1 = a1 , this is known as a spinodal point. Decreasing a1, when + a1 > a1 , produces the converse result. This memory effect induces a characteristic hysteresis loop in the order parameter (see Fig. 1.4b). By taking the appropriate limits one can see that for T < Tc the first derivative has a singularity:

∂F ∂F lim lim − lim → ∞, (1.10) − + V→∞ a1→0 ∂a1 a1→0 ∂a1

In general, the free energy cannot be probed directly by experiment. Rather, the order of the transition can be obtained by measuring the appropriate choice of response function. For example, the energy, entropy and density are ﬁrst-order derivates of the free energy. And, the speciﬁc heat, incompressibility and susceptibility are all second-order derivatives of the free energy. 1.1. FRUSTRATION 7

1.1.3 Unfrustrated systems

The prototypical example of an unfrustrated system, that contains both first-order and second-order transitions, is the Ising model of ferro- and antiferromagnetism on a bipartite lattice. The previous examples of placing anti-parallel arrows on a lattice, is a specific example of the Ising model of antiferromagnetism. In dimensions (D) > 1, the Ising model undergoes a second-order phase transition from a disordered paramagnetic state (T > Tc) to an ordered ferro- or antiferromagnetic state (T < Tc) at Tc. In 1D, a change in short-range ordering (i.e. a crossover) is observed, but no phase transition. Additionally, the presence of an external field can introduce a first-order transition between ordered magnetic states.

The Ising model consists of discrete binary spin variables, σi = ±1, that depicts the atomic spins or magnetic moments associated with the atoms. The spins are constrained to sit on the vertices of a lattice, indexed by the variable i. The nearest neighbour Ising model consists of only the interactions between spins that are connected via a single edge. For a bipartite lattice, the lattice can be broken up into two sublattices A and B, such that all the nearest neighbours of sublattice A are in sublattice B and vice versa. In the presence of an external ﬁeld, the Ising Hamiltonian is given by

H = −J ∑ σiσ j − µ ∑hσi, (1.11) hi, ji i where J is the strength of the interaction between nearest neighbours, hi, ji indicates the sum runs over nearest neighbours i and j, µ is the strength of the magnetic moment and h is the strength of the applied field. In the absence of a field, the Hamiltonian has a σi → −σi symmetry (Z2 symmetric), i.e. changing the orientation or flipping every spin does not change the total enthalpy. The sign of J dictates the type of ordering. A positive J prefers ferromagnetism (see Fig. 1.5a), the alignment of spins parallel to one another. Whereas, a negative J wants antiferromagnetism (see Fig. 1.5b), the alignment of spins antiparallel to their nearest neighbours. The sign of h will favour the ordering of spins parallel to the direction of the field. For example, when J is positive, the spins prefer to either all point ‘up’ (σi = +1) if h is positive or ‘down’ (σi = −1) when h is negative. When J is negative, both J and h cannot be mutually satisfied and thus, will compete to determine the type of ordering. At finite temperatures, thermal fluctuations will pick out the phase that is preferred by entropy. For the Ising model, this is the random or paramagnetic phase (see Fig. 1.5c). As opposed to the antiferro- or ferromagnetic phases, in which the entropy is finite, in the paramagnetic phase, the entropy S → NkB log(2) as T → ∞. Making the entropy of the paramagnetic phase macroscopic, as it scales with the size of the system. Hence, for a non-zero J, thermal fluctuations will drive a change in ordering from the enthalpically favourable antiferro- or ferromagnetic phases to the entropically favourable paramagnetic phase. This is analogous to the energetic preference of H2O molecules to form a vapour at high temperatures. The nature of this change in ordering can be captured by the statistical properties of the system, all 8 CHAPTER 1. INTRODUCTION

Figure 1.5: Illustration of the magnetic ordering in a (a) ferromagnet, (b) antiferromagnet and (c) paramagnet. of which can be obtained form the partition function

Z = ∑e−βH[ν], (1.12) ν

−1 where ν is a microstate of the system and β = (kBT) . The (Boltzmann) probability of any microstate occurring is 1 P[ν] = e−βH[ν]. (1.13) Z The thermal-equilibrium properties of the response functions can be obtained by taking a weighted (Boltzmann) average over the microstates

1 −βH[ν] hAi = ∑Aµ e , (1.14) Z ν where A is a response function (that depends on external parameters) and h...i indicates a thermal average. For a handful of systems, one of which is the 2D Ising model on a square lattice, the (Helmholtz) free energy can be calculated directly by

F = −kBT logZ. (1.15)

For more complex systems, F can be approximated by taking a mean-field approximation: a coarse- grained description of the system, in which microscopic variables, such as spins, are assumed to take on the same value (in equilibrium) and any large fluctuations around the equilibrium are ignored. Generally, this approximation becomes more accurate in higher dimensions. In Section 1.1.3 we will use mean-field theory to approximate the free energy for the Ising model.

Landau Theory

In order to distinguish the ferro-, antiferro- and paramagnetic phases, we require an appropriate choice of order parameter. For the ferromagnetic phase, a good choice is the magnetisation per site 1 m = N ∑i σi, where N is the number of lattice sites. If the ordering is ferromagnetic m 6= 0, and if the 1.1. FRUSTRATION 9 ordering is either antiferro- or paramagnetic m = 0. For the antiferromagnetic phase, we can use the 1 0 staggered magnetisation per site ms = N ∑i σi , where  0 +σi, if i ∈ A σi = . (1.16) −σi, if i ∈ B

If the ordering is antiferromagnetic ms 6= 0, and if the ordering is either ferro- or paramagnetic ms = 0. For a positive J, the mean-ﬁeld free energy (per site) is a a f = f + a m + 2 m2 + 4 m4 + O(m6), (1.17) 0 1 2 4!

4 f = −k T ( ) a = − Tc h a = k Tc (T −T ) a = k Tc > where 0 B log 2 , 1 T µ , 2 B T c and 4 B 2T 3 0. From Landau theory we know that, in the absence of a field, the transition between the ferromagnetic (T < Tc) and paramagnetic phases (T > Tc) is second-order (see Fig. 1.6b). Furthermore, we know that a non-zero field will result in a first-order transition for T < Tc. For a non-zero h ∝ a1, raising the temperature does not result in a phase transition, rather the system varies smoothly (crosses over) from the ferromagnetic to paramagnetic states. The full finite temperature phase diagram is shown in Fig. 1.6a.

For a negative J, the free energy (per site) is a function of two order parameters, ms and m. Minimising over both results in the ﬁnite temperature phase diagram shown in Fig. 1.6c.

Correlation Functions

As we previously mentioned, Landau theory can break down when fluctuations around the critical point, spatial inhomogeneity or non-locality becomes important. A more general measure that can account for these effects, is the statistical correlations between the spins. In the paramagnetic phase, the orientation of the individual spins is random, whereas in the ferro- and antiferromagnetic phases, the orientation of each spin strongly influences the orientation of their neighbours. This influence can be directly captured by the appropriate choice of correlation functions. For the Ising model, the fluctuations around the critical point can be captured by the spin-spin correlation function

C(R) = hσ(r)σ(r + R)i − hσ(r)ihσ(r + R)i (1.18) = h(σ(r) − hσ(r)i)(σ(r + R) − hσ(r + R)i)i, where R is the distance between the sites. C(R) averages over the ﬂuctuations about the thermal average at distance R. If either the spins are statistically independent or perfectly ordered, then C(R) → 0 as R → ∞. However, near the phase transition the spin-spin correlation function takes the form 1 C(R) ∼ e−R/ξ (1.19) Rd−2+η for R 1 and |Tc − T| Tc, where d is the spatial dimension, η is a constant that depends on the system and ξ is the correlation length – the distance at which the spins remain correlated. In the paramagnetic state one would expect that ξ is ﬁnite. Whereas, in a perfectly ordered crystal, the spins 10 CHAPTER 1. INTRODUCTION

Figure 1.6: Mean-field phase diagram for the bipartite Ising model with (a) positive J and (b) negative J. Black lines indicate first-order transitions, red dashed lines indicate spinodal lines, second-order transitions are shown by the blue lines, critical points are depicted by a black circle and multicritical points are denoted by white circles. (c) The magnetisation per site m, for h = 0 and J > 0, showing a second-order transition from the ferromagnetic phases (T < Tc) to the paramagnetic phase (T > Tc) at T = Tc. (d) Staggered magnetisation per site ms, for h = 0 and J < 0, showing a second-order transition from the antiferromagnetic phases (T < Tc) to the paramagnetic phase (T > Tc) at T = Tc. remain correlated at an infinitely large distances thus, ξ → ∞. Consequently, as the system passes through the critical point the correlation length diverges. At this point, C(R) decays algebraically as 1 C(R) = . (1.20) Rd−2+η The qualitative change in the behaviour of C(R) around the critical point can be associated with the development of order. In the absence of any kind of ordering, the correlation function (Eq. 1.19) decays exponentially. Whereas, the presence of short- or long- range order at low temperatures will result in a qualitative change in the behaviour of the correlation function around the transition. We will see in Section 1.1.4 that systems can develop local order without undergoing a phase transition. This is evidenced by an algebraic decay of the correlation function. The spin-spin correlation function can be probed experimentally by scattering experiments. For example, the structure factor (the Fourier transform of the spin-spin correlation function),

1 −iq·Ri j S(q) = ∑hσiσ jie , (1.21) N i, j 1.1. FRUSTRATION 11 can be obtained from neutron scattering experiments, where Ri j is the distance between sites i and j. The correlation functions can also give information about the equilibrium properties of the response functions. Using the ﬂuctuation dissipation theorem, we can write the response functions in terms of the equilibrium correlation functions. For example, for R = 0, the magnetisation per site and the magnetic susceptibility per site (χ) can be written as

∂ f 1 m = − = ∑hσii, (1.22) ∂h N i ∂ 2 f 1 χ = − 2 = ∑(hσiσki − hσiihσki). (1.23) ∂h NkBT i,k

1.1.4 Frustrated Systems

In frustrated (non-bipartite) systems, competing interactions can suppress the critical temperature. The resultant ground state can best be described as a “cooperative” paramagnet. The cooperative paramagnet has many similarities to the high temperature paramagnetic phase, just as the liquid phase does to the vapour phase. Similar to a paramagnet, it is macroscopically degenerate and lacks any kind of long-range order. Consequently, it cannot be described by Landau theory. However, in some cases, such as in an ice phase, the degenerate number of states can be described by an emergent gauge field with spin-spin correlations that decay algebraically. In this section we will show that frustration, in “icy” systems, results in apparent violations of the third law of thermodynamics (Section 1.1.4), emergent gauge fields (Section 1.1.3), and deconfined quasi-particle excitations (Section 1.1.4).

Ground State Degeneracy

The macroscopic ground state degeneracy of the cooperative paramagnet manifests in a residual (non-zero) entropy at T = 0. This stands in stark contrast to the predictions made by the third law of thermodynamics [14]. Experiments measuring this apparent violation date back as far as the 1936. A series of measurements by Giauque and Stout [14] found that the 1h phase of water ice possessed a residual entropy of S0 = 0.82 ± 0.05 Cal/(deg·mol) as far down as 15 K. The residual entropy of the 1h phase is a consequence of the mismatch in symmetry between the lattice and the local interactions. The binding energy of a H2O molecule is so strong that, as the oxygen atoms crystallise, the structure of the H2O molecules left essentially unchanged. In the crystalline state, the four-fold coordinated O2− ions form the vertices of a wurtzite crystal structure (see Fig. 1.7a). The H+ ions (protons), are constrained to sit along the edges of the lattice. Naively, one would expect that the H+ ions would sit midway between the O2− ions, minimising the electrostatic interactions between 2− + the protons and the O ions. However, the molecular structure of H2O constrains two of the four H ions to form covalent bonds with O2− ion. The length of which 1.01A˚ , is less than half of the distance between neighbouring O2− ions, 2.75A˚ . The other two H+ ions form hydrogen bonds with the O2− ion. The possible ground state conﬁgurations are then governed by the so-called Bernal-Fowler “ice 12 CHAPTER 1. INTRODUCTION

Figure 1.7: Balls and sticks diagram illustrating the (a) Ice 1h phase. In this phase, the O2− ions (red circles) sit on the vertices of a wurtzite crystal structure and the H+ ions (white circle) sit along 2− the edges. As a result of the large binding energy of a H2O molecule, each O ion forms two covalent and two hydrogen bonds with the adjacent H+ ions. (b) For each vertex, there are six possible conﬁgurations they obey these so-called ice rules. For a wurtzite lattice, there is a macroscopic number of ways in which the ice rules can be satisﬁed.

rules” [15]: for every O2− ion, two protons are constrained to sit at the hydrogen bonding distance, 1.74A,˚ and two are constrained to sit at the covalent bonding distance, 1.01A˚ (see Fig. 1.7b).

The macroscopic number of ways in which the protons can be arranged to obey the ice rules on a wurzite lattice lead Pauling [16], in 1935, to the conclusion that the residual entropy of the 1h phase arose from the disorder of the protons. His argument considered N O2− ions that are constrained to sit on the vertices of a four fold coordinated lattice, with 2N protons sitting along the edges. Since, each proton has two possible positions, there are 22N possible states. For each vertex only 6 of the 24 possible states obey the ice rules (see Fig. 1.7b). Assuming there is no correlation between the vertices, an upper bound for the ground state degeneracy can be estimated by

6 N 3N Ω ≤ 22N = . (1.24) 16 2

This gives a residual entropy of So = Rlog(3/2) ≈ 0.81 Cal/(deg·mol). Which is in very good agreement with Giauque and Stout’s measurements. The residual entropy of square ice was later calculated by Lieb [17], in 1967, who found it to be So = (3/2)Rlog(4/3) ≈ 0.86 Cal/(deg·mol). However in practice, no violations of the third law are observed. Even in Giauque and Stout’s original paper, they pointed out that, at low temperatures the system settles into a single ground state, freezing out the proton degrees of freedom. Since the 1930’s, a handful of materials with a residual entropy have been reported [18–26]. However, none these been reported to violate the third law. It still remains an open question whether violations of the third law can occur. 1.1. FRUSTRATION 13

Figure 1.8: Geometrically frustrated (a) triangular, (b) kagome (2D analogue of the pyrochlore lattice) and (c) chequerboard lattices. The triangular and kagome lattices are intrinsically frustrated as they are composed of equilateral triangles. The chequerboard and pyrochlore (not shown) lattices are intrinsically frustrated as they are composed of tetrahedron.

Equivalent Models

A lot of insight into ice phases can be gained from studying equivalent models with the same low temperature behaviour. Here we will only review a handful of these models and materials. For a thorough review see Ref. [18]. A notable example is the geometrically frustrated antiferromagnetic Ising model. In 1950, Wannier [27] showed that the antiferromagnetic Ising model on a (frustrated) triangular lattice (see Fig. 1.8a) resulted in disordered ground state with a residual entropy of 0.323NkB. The connection to the 1h phase of water ice was later made by Anderson, in 1956, in a study of ordering in spinels [28]. The models proposed by Anderson could be mapped onto an antiferromagnetic Ising model on the (frustrated) pyrochlore lattice (see Fig. 1.8b for the 2D analogue of the pyrochlore lattice). Anderson calculated the residual entropy and noted that the ground state conﬁgurations could be described by the ice rules; each tetrahedron must contain two ‘up’ and two ‘down’ spins (see Fig. 1.9a). He further noted the possibility of an ice phase occurring with three up spins and one down, and vice versa. The ice rules naturally arise in the antiferromagnetic Ising model on a geometrically frustrated lattice. The Hamiltonian can be written as

J 2 µ H = −J ∑ σiσ j − µ ∑hσi ≈ − ∑Lα − ∑hLα , (1.25) hi, ji i 2 α 2 α where the sum over α runs over the local clusters, Lα is the total spin on a local cluster. For the triangular and kagome lattices, the local clusters are triangles. For the pyrochlore and chequerboard lattices, the local clusters are tetrahedron. The mapping between unfrustrated (Eq. 1.11) and frustrated Ising models (Eq. 1.25) consists of re-summing over the spin variables, such that

Lα = ∑ σi, (1.26) i∈α where the sum runs over all spins sitting on the local cluster α. 14 CHAPTER 1. INTRODUCTION

Figure 1.9: Equivalent ice rules for the (a) water ice 1h phase, (b) geometrically frustrated antiferromagnetic Ising model and (c) the vertex model/spin ice. In (a) and (c) the binary variables are constrained to sit along the bonds whereas, in (b) they sit on the sites of the lattice. The white arrows indicate the orientation of the magnetic moments associated with the rare earth ions in spin-ice.

A positive J favours the parallel alignment of spins whereas, a negative J favours the local constraint

Lα = 0. This local constraint or symmetry is the ice rule. For the pyrochlore and chequerboard lattices, this rule constrains each tetrahedron to contain two ‘up’ (σi = +1) and two ‘down’ (σi = −1) spins

(see Fig. 1.9b). In general, for any geometrically frustrated lattice, the constraint Lα = 0 produces a residual entropy. In a ﬁnite ﬁeld, the frustrated Ising Hamiltonian (Eq. 1.25) can be rewritten as

J µh2 H ≈ − ∑ Lα + . (1.27) 2 α J For a negative J, the enthalpically favourable state will be picked out by the strength and sign of µh/J. Large values of |µh/J| prefer the parallel alignment of spins, with the orientation determined out by the sign of h. For smaller values of |µh/J|, it becomes favourable to have one spin on each cluster align antiparallel to the others. Similarly, for very low values of |µh/J|, it is favourable to have half the spins align antiparallel to the other half. For the pyrochlore and chequerboard lattices, in order of decreasing µh/J, it is enthalpically favourable when each tetrahedron contains four down, three down and one up, two down and two up, one down and three up, and four up spins. Since, there are a macroscopic number of ways in which the pyrochlore and chequerboard lattices can accommodate all but the maximally polarised cases, sweeping µh/J can alter the rules. However, in the years since Anderson’s original paper, no example of a cooperative paramagnetic phase has been observed in a Ising antiferromagnet. This is thought to be a result of magnetocrystalline anisotropy accompanying the spins in antiferromagnets [19]. Extensions of the Ising model, that sidestep this problem by having a ferromagnetic nearest neighbour interaction with longer-range antiferromagnetic interactions, have also been shown to give rise to ice rules and a residual entropy [29]. However, as of yet, no materials with these interactions have been found. A more realistic model that 1.1. FRUSTRATION 15 approximates CsNiCrF6 [19, 23, 30, 31], a material known to host an ice phase [23], is the classical Heisenberg antiferromagnet on a frustrated lattice

J 2 H = −J ∑ Si · S j ≈ − ∑|Lα | , (1.28) hi, ji 2 α

2 where J is the interaction strength, Si is a classical spin vector (|Si| = +1) and Lα = ∑i∈α Si.A negative J favours the parallel alignment of spins. Whereas, a positive J is happy when the angles of o 2 the spins on each local cluster sums to 0 , i.e. |Lα | = 0. However, the closest realisations of the 1h phase come from a class of magnetic vertex models. Which consist of a regular periodic graph, with binary spin variables that are constrained to sit along the bonds and point into and out of each vertex. Each vertex conﬁguration is then assigned a weight, which can be used to determine the thermodynamic (Boltzmann) probability of that state occurring. For an appropriate choice of weights one can obtain ice rules. Some mappings between vertex models and Ising models have been proposed [32]. A physical realisation of a vertex ice model is the rare earth pyrochlore magnets Ho2Ti2O7, Dy2Ti2O7 and Ho2Sn2O7 [19] and their artiﬁcial counterparts [25, 26]. In the so-called “spin ices”, the Ho3+ or Dy3+ ions form the vertices of a pyrochlore lattice. The presence of the oxide ions modify the quantum ground state of the magnetic ions, constraining their spins to align along crystallographic h111i axis (i.e. pointing into and out of the tetrahedron). In spin-ices, the interactions between the spins are approximated by a ferromagnetic nearest neighbour exchange and a long-range dipole-dipole interaction

2 " # J µ0µ Si · S j 3(Si · ri j)(S j · ri j) H = ∑ Si · S j + ∑ 3 − 5 , (1.29) 2 hi, ji 4π i< j ri j ri j where the spins Si are constrained to sit along the h111i axis, J is the interaction strength (note the sign difference to previous Hamiltonians), µ is the strength of the dipole-dipole interaction and µ0 is the the magnetic constant. The two in, two out ice phase (see Fig. 1.9c) arises as a result of the dipole-dipole interactions suppressing the ordering temperature [19, 22, 24, 33–40]. This becomes more intuitive when the Hamiltonian (Eq. 1.29) is written in the form

ν µ Lα L H ≈ 0 L2 + 0 β . (1.30) ∑ α ∑ r 2 α 4π α<β αβ

h 2 i The details of this calculation can be found in Ref. [39]. Here J + 4 1 + 4 µ0µ and a is the ν0 ∝ 3 3 3 4πa3 distance between the spins. Evidently, even when the interactions between the spins are ferromagnetic

(J < 0) a strong dipole-dipole interaction can still enforce the ice rules (ν0 > 0). Artificial ices are 2D analogues of spin-ice, consisting of coupled single domain nanomagnets [25, 26]. Intuitively, the ice rules for the vertex/spin ice model and the geometrically frustrated antiferromagnetic Ising model can be mapped by, Si∈α = ηα σixî, where xî is a h111i basis vector (see Fig.

1.9b-c) and ηα = ±1 is sublattice dependent. Analogously to the bipartite lattice, the local clusters on a frustrated lattice can typically be broken up down into two sublattices A and B, such that ηα = 1 on sublattice A and ηα = −1 on sublattice B. On the pyrochlore and chequerboard lattices, sublattice A 16 CHAPTER 1. INTRODUCTION consists of all tetrahedron pointing up and B consists of all tetrahedron pointing down. Generally, a (typically non-bijective) mapping can be deﬁned between any two systems that obey the ice rules. As we will see in the Section 1.1.4, this mapping becomes exact when the models are coarse-grained.

Spin-Spin Correlations

The existence of ice rules has specific consequences for the spin-spin correlations. This can be shown by coarse-graining the magnetisation. The Ising and Heisenberg variables can be mapped onto a field B with the coarse-grained magnetisation B(r) = 1 S , where D is a domain centred at the position r and V is the corresponding VD ∑i∈D i D volume of the domain. The size of D is chosen to be larger than the size of the local cluster and adequately large enough to capture the spatial fluctuations. This mapping is chosen such that ice rules correspond to as many flux variables pointing in as out of each vertex. The number of possible configurations, for a fixed B, that obey the ice rules, N(B), will then be peaked at B(r) = 0. Increasing (or decreasing) values of B will introduce anisotropy, decreasing N(B). For an appropriate choice of domain size the correlations between the local B fields will be weak. Therefore, in accordance with the central limit theorem N(B) ∝ exp(−KV|B|2/2), where K is an empirical parameter that controls the width of the distribution and V is the total volume. The specific entropy is then given by

log(N(B)) K 2 s(B) = lim = s0 − |B| (1.31) V→∞ V 2 and consequently, the free energy of the system is KT Z F[{B(r)}] = F + d3r|B(r)|2. (1.32) 0 2 The second contribution to the free energy is purely entropic. Notably, this has the same form as the free energy of an electrostatic (ε0 = K) or magnetostatic (µ0 = 1/K) classical gauge ﬁeld theory. The relationship between ice phases and classical gauge ﬁeld theories can be strengthened further by coarse-graining the ice rules. Which correspond to the divergence-free condition

∇ · B(r) = 0, (1.33) which is identical to Gauss’s law in a vacuum. A natural consequence of this is the possibility of assigning a gauge vector A(r) to B(r), and identifying an emergent classical gauge field, such that B(r) = ∇ × A(r). In particular, this field corresponds to the deconfined phase of a lattice gauge field theory, the so-called Coulomb phase. Since the gauge symmetry it is not a symmetry of the full Schrodinger¨ equation, it is necessarily emergent. While, high energy excitations can break this symmetry, all low-energy processes preserve it [9]. A necessary consequence of this description is the existence of emergent gauge charges. We will delve into this further in section 1.1.4. The existence of the emergent gauge field has direct experimental consequences. In Fourier space the divergence-free condition corresponds to removal of the transverse component of the magnetisation q · B(q) = 0. Projecting out the transverse component of the magnetisation results in the structure 1.1. FRUSTRATION 17

Figure 1.10: Structure factors for the ice phase on a square lattice, demonstrating pinch point singularities (or bow-tie structures) at q = 0, characteristic of an ice phase. factor 1 q q S (q) = hB (−q)B (q)i ∝ δ − µ ν . (1.34) µν µ ν K µν q2 The first term corresponds to the solution for a paramagnet, the second term arises as a consequence of imposing the ice rules. At q = 0 the structure factor becomes singular (without diverging). This singularity gives rise to a characteristic path dependence of the structure factor. Along the path qx = 0 Sxx = 0 whereas, along the path qy = 0 Sxx = 1. Graphically, this corresponds to pinch point (or bow-tie) singularities in the structure factor (see Fig. 1.10). Consequently, neutrons scattering experiments can be used to directly probe the existence of the emergent gauge field. Fourier transforming Eq. 1.34 back into real space produces a correlation function that decays algebraically around the critical point, for sufficiently large separation distances 1 hB (r)B (r + R)i ∝ δ − 3x ˆ xˆ , (1.35) µ ν KV µν µ ν where V is the total volume. Notably, this is the approximate form of a dipole-dipole interaction. Except, that it is purely a consequence of entropic forces. In Section 1.1.4 we will show that this result has important consequences for the confinement properties of the gauge charges. It is important to note that coarse-graining the lattice has removed all the underlying details about the lattice and spin variables. Consequently, this result is very general and can be applied to any system that has a degenerate ground state governed by ice rules.

Excitations

At finite temperatures, spin flips can be thermally activated. Since each spin sits on the corners of two neighbouring clusters, flipping a spin corresponds to the violation of the ice rules on both clusters (see Fig. 1.11b). Owing to the numerous ways in which the ice rules can be enforced, flipping a spin (with the opposite orientation) on an adjacent cluster can restore the ice rules on that cluster and create a new defect on another (see Fig. 1.11c). Continuing this process of periodically flipping spins, with opposing orientations, on adjacent clusters, can propagate the defects (see Fig. 1.11d). This is a low energy process because it preserves the number of defects, as opposed to high energy processes, like flipping two spins with the same orientation on a single cluster. The only way for the ice rules to be 18 CHAPTER 1. INTRODUCTION

Figure 1.11: Illustration of the propagation of defects in a vertex ice phase on a square lattice. (a) In the ground state each vertex obeys the ice rules: two spins are constrained to point into and out of each vertex. (b) Flipping a spin violates the ice rules on the neighbouring vertices, creating two defects. (c) Flipping an adjacent spin (of the opposite orientation) can restore the ice rules on one vertex and violate them on an adjacent vertex, separating the defects. (d-k) Continuing this process of flipping spins of opposing orientation propagates the defects, until they eventually occupy the same vertex and annihilate one another (i). restored everywhere is for the two ends of the long string of flipped spins to converge on the same cluster (see Fig. 1.11e). Consequently, the defects are said to be topologically protected. Furthermore, it only takes a finite amount of energy to move the defects infinitely far apart, thus the violations are said to be ‘deconfined’ in the language of quantum field theory. Hence, the defects correspond to free independent quasi-particles. Since, it costs a finite amount of energy to create the quasi-particles, there is gap to the excitations and thus, the excitations are massive. Furthermore, the quasi-particles are said to be fractionalised, as each defect carries half of the properties of a constituent (degree of freedom) of the system. Depending on the system, the fractionalised degree of freedom can be either magnetic charge (magnetic monopoles) [7, 19, 39, 40] or spin [29]. For water ices, the energy of the O-H bond is sufficiently large that that any violations of the ice −1 + rule, i.e the creation of HO and H3O molecules, are energetically unfavourable [19]. In spin ices and artificial spin ices, flipping a spin, creates a magnetic dipole with each pole sitting on adjacent cluster. Hence, the defects necessarily correspond to magnetic monopoles [40], which have been directly observed in artificial spin ices [41]. In the coarse-grained approach, the defects correspond to violations of the divergent-free constraint, ∇ · B 6= 0. In the language of classical gauge field theory, the violations correspond to gauge charges. When the gauge charges become separated over sufficiently large distances, the entropic dipole-dipole correlations give rise to a purely entropic [39] Coulomb law

Q Q U ∼ 1 2 . (1.36) 4π|r1 − r2| 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 19

Hence, the name Coulomb phase.

Concluding Remarks

The discovery of new systems with ice phases has been one of the great theoretical and experimental challenges of the last couple of decades. The construction and manipulation of these systems has, and will continue to, give fundamental insight into emergent properties and behaviours. In this thesis we will propose a new type of ice phase, the so-called “spin-state ice” phase, that exists in elastically frustrated spin crossover materials and frameworks.

1.2 Microscopic Origin of Spin Crossover Systems

Spin crossover (SCO) is most commonly reported in near-octahedral coordination complexes, but is also observed in organometallic compounds, metal-metal bonded species and inorganic salts [42,43]. In this section, we will focus on explaining the mechanisms behind SCO in (near) octahedral coordination complexes. While the same theory presented here can be used to explain how SCO arises in most systems, SCO is a widespread phenomenon. For a through review of the systems that exhibit SCO see [42–46].

1.2.1 A Brief Introduction to Coordination Chemistry

Coordination complexes typically consist of a central transition metal ion (a metal with a partially ﬁlled d-shell), surrounded by several ligands, as shown in Fig. 1.12. A ligand can be any atom, ion or molecule usually, with one or more lone electron pairs. The bond between the metal and ligands is a coordinate covalent or dative bond; a bond formed from the donation of two electrons from the same atom. Usually, with the ligand as the electron donor. Depending on whether one or many of the atoms in the ligand form a coordination covalent bond with the metal, the ligand is said to be either monodentate (one bond) or polydentate (many bonds). m A coordination complex is usually denoted by [M(L1)n1 (L2)n2 ···] where M denotes the central metal ion, Lp denotes the type of ligand, np is the number of Lp type ligands and m is the oxidation state. Typically, coordination complexes are mononuclear. However they can be polynuclear, held together through either metal-metal bonds or bridging ligands. The chemistry of coordination materials and frameworks is often more varied with multiple distinct complexes forming the unit cell, for example m [(Ma(L1)n1 (L2)n2 )(Mb(L3)n3 (L4)n4 )] , where Ma and Mb denote different metals. The chemistry of coordination complexes is usually dominated by the s- and p- orbitals of the ligands and the d- orbitals of the metal. This limits number of bonding and non-bonding electrons pairs to nine (ﬁve d-orbitals, three p-orbitals, one s-orbital). Consequently, the number of coordination bonds is limited to the number of empty s-, p- and d- orbitals. Although uncommon, f -orbitals can sometimes contribute to bonding, resulting in coordination complexes with more than nine ligands. The arrangement of ligands depends on the type of metal and the type, symmetry and position of the 20 CHAPTER 1. INTRODUCTION

3− Figure 1.12: (a) Balls (atoms) and sticks (bonds) diagram and (b) structural formula for [Fe(CN)6] . The nitrogen atoms associated with the [CN]− groups sit on the vertices of an octahedron surrounding the central Fe3+ cation. Each of the six nitrogen atoms form a coordinate covalent bond with the central Fe3+ cation. The ligands are monodentate since each [CN]− molecule forms a single covalent bond with the metal.

Figure 1.13: Balls and sticks diagrams for the molecular geometries mentioned in Fig. 1.1 ligands. However typically, for transition metals, only the bonding electron pairs contribute to the molecular geometry. This results in complexes having a regular geometry that depends only on the number of coordination bonds (the coordination number), see Table 1.1.

Coordination number Molecular Geometry Example + 2 linear (Fig. 1.13a) [Ag(NH3)2] 2− 3 trigonal planar (Fig. 1.13b) [Cu(CN)3] 4 tetrahedral (Fig. 1.13c) [Ni(Cl)4] 2− 4 square planar (Fig. 1.13d) [Ni(CN)4] 2− 5 trigonal bipyramidal (Fig. 1.13e) [Co(Cl)5] 3− 5 square pyramidal (Fig. 1.13f) [Ni(CN)5] 3+ 6 octahedral (Fig. 1.13g) [Fe(CN)6] 6 trigonal prismatic (Fig. 1.13h) [W(CH3)6] 2− > 6 more complicated geometries [Re(H)9] Table 1.1: Coordination numbers and their corresponding molecular geometries.

The presence of the ligands has a large influence on the structural, spectral and magnetic properties of the complex. For example, coordination complexes have a remarkable diversity of colours that arise from electronic transitions that are either d-d transitions or charge transfer bands. d-d transitions 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 21 occur as a result of an electronic transition between d-orbitals, and charge transfer bands arise from the transfer of charge between the metal and ligands. Since the ligands are predominantly s- and p- orbitals, the d-d transitions must occur as a result of the splitting of the d-orbitals of the metal. This splitting has a significant effect on the magnetic properties of the complex. For example, complexes 3− with six electrons in the d-orbitals can either be paramagnetic (e.g. [Fe(Cl)6] ) or diamagnetic (e.g. 3− [Fe(CN)6] ). There are a host of theories that attempt to explain the influence of the ligands on the metal: crystal field theory, molecular orbital theory and valence bond theory. Interestingly, each of these theories correctly depict certain aspects of a more complete theory. The current incarnation of this “more complete theory” is ligand field theory. However, ligand field theory is quite abstract by nature and does not lend itself to intuition. Instead we will begin by applying a combination of crystal field theory (in Section 1.2.2) and molecular orbital theory (in Section 1.2.3) to explain both the electrostatic and covalent nature of the bonding. We will then go on to explain the basics of Ligand field theory in Section 1.2.4.

1.2.2 Crystal ﬁeld theory

Crystal field theory (CFT) was first developed to explain the influence of the ligands on the d-orbitals of a transition metal ion. According to CFT the bonding between the metal and surrounding ligands arises from the electrostatic attraction between them. Consequently, CFT is typically only applicable when ligands are negatively charged ions or molecules with lone electron pairs. In these cases, the electrostatic field (formally, crystal field) generated by the ligands can be approximated by a set of point charges or dipoles. Where, negative ions are replaced by negative point charges and molecules with lone electron pairs are replaced by dipoles. Since the symmetry of the crystal field will be lower than the symmetry of the d-orbitals, the presence of the ligands will break the symmetry of the d-orbitals. Consequently, electrons occupying orbitals with lobes closer to the surrounding charges will experience a greater electrostatic force, resulting in a splitting of the orbitals, ∆, relative to the average distance between the orbital lobes and the surrounding point charges or dipoles, see Fig. 1.14. Where ∆ measures the influence of the ligands on the d-orbitals of the central metal ion. The essential ideas of CFT are most intuitively understood using an example. Here we consider 3− 3+ [Fe(CN)6] , shown in Fig. 1.14b. According to CFT, the Fe ion can be approximated by a point charge, with charge q = +3e, and the surrounding [CN]−1 ligands can be approximated by q = −e point charges. The influence of the electrostatic field generated by the point charges on the 3d-orbitals of the Fe3+ ion can be qualitatively understood without preforming any computations. All that is required is the symmetry of the ligands and the position of the ligands relative to the d-orbitals. For 3− −1 [Fe(CN)6] , the six [CN] ligands sit along the Cartesian x,y,z axes, forming the vertices of an octahedron. Relative to the ligands, the dz2 and dx2−y2 orbitals have lobes projecting along the same axes whereas, the dxy, dxz and dyz orbitals have lobes projecting between axes of the ligands (see Fig.

1.14a). Therefore, an electron in either of the dx2−y2 or dz2 orbitals should experience a greater repulsive force than an electron in any of the dxy, dxz or dyz orbitals. Furthermore since, the dz2 orbital is a linear 22 CHAPTER 1. INTRODUCTION

Figure 1.14: (a) Illustration of the atomic d-orbitals. The blue (+) and orange (−) shaded regions indicate the phase of the electronic wavefunction. The presence of ligands surrounding a transition metal ion will split the d-orbitals of the metal into distinct orbital sets. (b) In absence of any ligands a metal cation will have ﬁve 3d orbitals. (c) If the surrounding ligands are placed uniformly over a sphere, then an electron occupying any orbital will, on average, experience the same electrostatic force. (d) However, for a complex with a non-spherical geometry, the electrons will experience an unequal force depending on which orbital lobe they occupy. For a complex with an octahedral molecular 3− geometry, like [Fe(CN)6] , an electron occupying the dx2−y2 or dz2 orbitals will be, on average, closer to the negative point charges and hence, experience a greater repulsive force than an electron in any of the dxy, dxz or dyz orbitals. Consequently, the interactions between the surrounding charges and 3+ the electrons will split the 3d-orbitals of the Fe cation into the eg (dx2−y2 , dz2 ) and t2g (dxy, dxz and dyz) orbital sets. Relative to energy of the individual atoms, the eg orbitals will increase in energy by 0.6∆oct and the t2g orbital decrease by 0.4∆oct, where E(eg) − E(t2g) ≡ ∆oct.

combination, of equal parts of dx2−z2 and dy2−z2 orbitals, the dz2 and dx2−y2 should experience the same repulsive force. Similarly, the dxy, dxz and dyz orbitals will experience the same repulsive force. Consequently, the interactions between the point charges and the electrons will split the 3d-orbitals of 3+ the Fe cation into two orbital sets: the eg (dx2−y2 , dz2 ) and t2g (dxy, dxz and dyz) orbital sets, where E(eg) > E(t2g). It should be noted that eg and t2g are symmetry labels. In general, complexes like 3− [Fe(CN)6] are not perfectly symmetric. The presence of polydentate ligands, unequal bond lengths, spin-orbit coupling or structural distortions such as Jahn-teller distortions will slightly and sometimes dramatically lift the degeneracy of the orbitals [47]. The magnitude of ∆ can then be calculated with the Schrodinger¨ equation. The full details of the calculations are given in ref. [47]. For brevity, we will only show the relevant details. Consider an electron (occupying a d-orbital lobe) in an electrostatic ﬁeld generated by a metal ion with charge ZMe

(where Z is the atomic number) surrounded by six point charges, with charge −ZLe, sitting along the Cartesian x,y,z axes, forming the vertices of an octahedron. The corresponding Schrodinger¨ equation 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 23 is given by 2 6 ! } 2 − ∇ +VM(r) + ∑ VL(r,Ri) φ(r) = εφ(r), (1.37) 2m i=1 where r = (x,y,z) is the electron coordination, Ri is the position vector of the ith point charge, VM(r) 2 is the potential energy due to the nucleus of the metal cation, VL(r,Ri) = ZLe /|Ri − r| is the potential energy of the electrons due to the ith point charge, φ(r) is the single electron wavefunction and ε is the energy eigenvalue. Since the symmetry of the surrounding point charges is octahedral, the potential due to the point charges is

6 6 2 2 2 6ZLe 35ZLe 4 4 4 3 4 ∑ VL(r,Ri) = ∑ ZLe /|Ri − r| = + 5 x + y + z − r , (1.38) i=1 i=1 R 4R 5 where R is the distance between the metal ion and a ligand. The ﬁrst term in VL(r) represents the energy associated with the core electrons of the metal ion, raising the energy of all the orbitals equally. The second term shifts the energy relative to the location of the d electron. Symmetry arguments show that the presence of the surrounding point charges splits the d-orbitals into eg and t2g orbital sets.

Consequently, ∆ can be calculated from taking the energy difference between an electron in an eg and t2g orbital, 5Z e2hr4i ∆ ≡ E(e ) − E(t ) = L , (1.39) oct g 2g 3R5 4 where ∆oct is the orbital splitting due to an octahedral molecular geometry and hr i is mean value of the forth power of the radial distance between the orbital lobe and the metal ion. The strength of ∆oct depends on the type of metal and the type, position and symmetry of the surrounding ligands. Notably −5 ∆oct ∝ R , decreasing for increasing metal-ligand bond lengths. Relative to the atomic energy of the atoms, the eg orbitals will increase in energy by 0.6∆oct and the t2g orbitals decrease by 0.4∆oct.

Consequently, the t2g orbitals are stabilised while, the eg orbitals are destabilised. In general, since CFT neglects the ﬁnite size of the atomic orbitals and the covalent nature of the bonding, its successes are limited to explaining d-d transitions in systems where CFT is appropriate. A full explanation of the spectral and magnetic properties of coordination complexes requires both a treatment of the electrostatic and covalent nature of the bonding.

1.2.3 Molecular orbital theory

Molecular orbital theory (MOT) was developed as a way of explaining covalent bonding in molecules. In MOT, the electrons are not assigned to individual atoms orbitals, but “molecular orbitals” that are delocalized over the molecule. More accurately, MOT approximates the many-electron wave function as a product of single electron wavefunctions called molecular orbitals. Each molecular orbital can be thought of as an average ﬁeld of the nuclei and electrons. Essentially, MOT has the same theoretical foundation as the Hartee-Fock method [47]. Consequently, MOT is most accurate for molecules with strong covalent bonding. The molecular orbitals can be classiﬁed as either bonding, anti-bonding or non-bonding. Bonding and anti-bonding molecular orbitals are created from the superposition of atomic orbitals with a 24 CHAPTER 1. INTRODUCTION

Figure 1.15: Molecular orbitals (b,d) for a H2 molecule with the corresponding electron density (c,f) and energy level diagrams (g). When the hydrogen 1s orbitals constructively interfere (a) they form a 2 bonding molecular orbital ψB(r1,r2) causing an increase in the electron density |ψB(r1,r2)| in the inter-proton regime (c). When the orbitals destructively interfere (d) they form an antibonding orbital 2 ψAB(r1,r2) creating a node in the electron density |ψAB(r1,r2)| in the interproton regime (f). The bonding orbital (g middle; bottom) has a lower energy and higher molecular stability than the free atomic orbitals (g left and right). Conversely, the antibonding orbital (g middle; top) has a higher overall energy and lower molecular stability. The orbitals are filled according to the aufbau filling principle. The white circles indicate the location of the protons. similar energy and the same symmetry; one from the constructive interference (bonding) and one from destructively interference (anti-bonding) of the atomic orbitals (see Fig. 1.15). The anti-bonding ∗ orbitals are usually distinguished by an asterisk, for example eg. The atomic orbitals that do not have a symmetric counterpart are called non-bonding orbitals. For bonding orbitals, the constructive interference of the atomic orbitals will result in an increase in the concentration of the electron density in the region between the nuclei (see Fig. 1.15c). The increased electron density will attract the atoms closer together, strengthening the bond between the atoms and increasing the molecular stability. Whereas, for anti-bonding orbitals, the destructive interference of the atomic orbitals will decrease the electron density in the inter-nuclei regime and shift it further away from nuclei (see Fig. 1.15f). This decrease in the electron density increases the Coulomb repulsion between the atoms, pushing them away from each other, weakening the bond between them, and decreasing the molecular stability. Evidently, certain physical properties like the strength, length and stability of the metal-ligand bond will depend on the population of the bonding and anti-bonding orbitals. Typically, this can be quantified by the bond order N − N bond order = BO ABO , (1.40) 2 where Nx is the population of electrons in the bonding (x = BO) and antibonding (x = ABO) orbitals. The bond order indicates the number of free electron pairs and more loosely corresponds to the strength of the bond between electrons. The stronger the electronic interactions are, the more tightly held together the molecules are. And thus, a shorter metal-ligand bond distance and higher molecular stability. 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 25

Figure 1.16: Molecular orbital energy level diagrams for ML6, showing (a) the inﬂuence of σ-bonding and (b-d) π-bonding on the molecular orbitals. The energy difference between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) strongly depends on the type and strength of the π-bonding, decreasing for stronger (b) π-donation and increasing for stronger (d) π-acceptance. (cf. (c) with (b) and (d))

In the MOT description of octahedral molecular complexes, there are a host of different bonding, anti-bonding and non-bonding orbitals (see Fig. 1.16a). For a perfectly octahedral molecular complex, MOT arrives at the same conclusion as CFT. The highest occupied molecular orbitals (HOMO) will ∗ be the t2g orbital set and the lowest unoccupied molecular orbitals (LUMO) will be the eg orbital set. ∗ However, in the MOT description, the eg and t2g orbitals are formed from the bonding between the metal and ligands. The stability of t2g orbitals depends strongly on the nature of the σ- and π- bonding (see Fig. 1.16b-d). σ-bonding being the strongest type of bond, formed from the head-on overlap of atomic orbitals. And, π-bonding being typically weaker, formed from the side-on overlap of atomic orbitals.

In the absence of any π-bonding, the t2g orbitals will be non-bonding (see Fig. 1.16c). Whereas, if the π-bonding orbitals of the ligand have a similar energy and the same symmetry as the non-bonding t2g orbitals, then either the metal or ligand will donate an electron pair to form a π-bond. When the ligand donates the electron pair, it is referred to as π-donation, and when the metal denotes the electron pair, it is referred to as π-acceptance. This bond splits the non-bonding t2g orbitals into bonding t2g ∗ and anti-bonding t2g orbitals. Depending on whether the ligands are π-donors (see Fig. 1.16b) or ∗ π-acceptors (see Fig. 1.16d), the HOMO will be either the t2g or t2g orbital sets, respectively. Since the ∗ t2g orbitals will have a higher energy than the non-bonding t2g orbitals, π-donation will decrease ∆oct (see Fig. 1.16b), and π-acceptance will increase ∆oct (see Fig. 1.16d). Generally, ∆oct decreases the stronger the π-donation is and increases the stronger the π-acceptance is. This trend has been observed empirically and is referred to as the spectroscopic series [48]. Alongside this trend, metal-to-ligand (and ligand-to-metal) charge transfer can be used to explain the remarkable diversity of colours that arise in coordination complexes. 26 CHAPTER 1. INTRODUCTION

MOT also goes some way to explaining the magnetic properties of coordination complexes. Complexes with strong π-acceptance are filled according to Hund’s rule; maximise the spin multiplicity. Whereas, complexes with strong π-donation are filled according to the aufbau principle; fill each orbital set in order of decreasing stability, maximising the spin multiplicity of each. We can illustrate this phenomenon without going into the microscopic details of the calculation by considering two energy scales: ∆ and P – the energy required for an electron to form an electron pair. If ∆ > P, the electrons will preferentially form electron pairs instead of occupying higher energy orbitals. Whereas, if ∆ < P, it becomes preferable for the electrons to occupy higher energy orbitals instead of forming electron pairs. Depending on the number of d electrons, the different regimes correspond to either the same or different spin-states, see Fig. 1.17. For an octahedral molecular complex with d4-d7, shown in Fig. 1.17, the ground state can be either the high-spin (HS) state if ∆ < P or the low-spin (LS) state if ∆ > P. The HS state is filled according to Hund’s first rule and the LS state is filled according to 4− 6 0 aufbau filling principle. For example, the ground state of [Fe(CN)6] is t2geg (LS; S = 0) and the 2+ 4 2 ground state of [Fe(H2O)6] is t2geg (HS; S = 2). Since the HS state arises from π-acceptance and the LS state stems from π-donation, the orbital splitting for HS complexes, ∆HS, will be smaller than the orbital splitting for LS complexes, ∆LS. Complexes that have either no π-bonding, weak π-donor ligands or a mixture of π-donor and π-acceptor ligands can sit within an intermediate regime, in which ∆HS < P < ∆LS [48]. In general, since the HS state has a higher population of anti-bonding orbitals than the LS state, the HS state will have a higher bond length and spin multiplicity, and a lower bond strength and molecular stability. A secondary effect of this is a dramatic increase in the entropy of the HS state over the LS state. Evidently, the higher spin and orbital multiplicity of the HS state will result in a higher overall electronic entropy than the LS state. Less obviously, the lower molecular stability in the HS state is associated with a softening of the vibrational modes and a dramatic increase in the vibrational entropy [44, 49]. For complexes in the ∆HS < P < ∆LS region, the significant differences in bond length and entropy between the HS and LS states allows for small external perturbations to drive a spin crossover or spin transition [42]. There are hundreds of known complexes that meet this criterion [42]. Most commonly the molecular geometry is (near) octahedral with a central Fe2+, Fe3+ or Co2+ ion, or less commonly the central metal is Co3+, Cr2+, Mn2+ or Mn3+ [42, 44]. For the more common metals ions there are numerous possible choices of ligands that can result in a spin crossover and spin transition [44]. By combining the MOT description of orbitals with quantum mechanics we can obtain a detailed depiction of this intermediate regime.

1.2.4 Ligand Field Theory

Ligand ﬁeld theory (LFT) was ﬁrst created as a way of combining CFT and MOT to create a more complete version of MOT that includes electrostatic repulsions. It is a purely microscopic description of the coordination complexes that in addition to, considering metal-ligand bonding and electrostatic 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 27

∗ Figure 1.17: Depending on the strength of orbital splitting, ∆oct, and spin pairing energy, P, the eg 4 7 and t2g orbitals of an octahedral molecular complex with d -d can be filled according to either (a) Hund’s rule; maximise the spin multiplicity or (b) the aufbau principle; fill the orbitals in order of increasing stability, maximising the spin multiplicity of each. repulsion, also considers the electron-electron interactions amongst the d electrons. In LFT the nd orbitals of a transition metal ion, in a perfectly octahedral complex, are split into ∗ ∗ eg and t2g orbital sets. The orbital splitting between the eg and t2g orbital sets is referred to as the ligand field splitting parameter 10Dq. Here we use the “differential quanta” 10Dq instead of ∆ to distinguish LFT from CFT and MOT. The strength 10Dq depends on the nature of the metal ion and the nature and symmetry of the surrounding ligands. In particular, the strength of 10Dq depends on the relative population of the bonding and anti-bonding orbitals (bond order) and the metal-ligand bond distance. The former is a consequence of the metal-ligand bonding while the later is predominantly a −n consequence of electrostatic repulsion. More explicitly 10Dq ∝ rM−L, where rM−L is the metal-ligand bond length and n ∈ 5,6 [42]. This relationship highlights an important structure-property relationship for spin crossover systems; spin crossovers and spin transitions are most commonly observed in complexes with the largest structural differences between the HS and LS states. We can illustrate this by considering the ratio LS n 10Dq rHS HS = , (1.41) 10Dq rLS LS HS An increase in rHS/rLS leads to an increase in 10Dq /10Dq . In turn, this leads to a larger regime in which spin crossovers and spin transitions can occur. If we restrict our analysis to only the d-orbitals, we can write down a microscopic Hamiltonian for an octahedral coordination complex. While the whole Hamiltonian (see Ref. [50]) contains multiple competing interactions, the essential physics can be obtained from

H = HCF + Hel−el, (1.42) where HCF is the splitting of the d-orbitals due to the electrostatic ﬁeld and Hel−el is the Coulomb 28 CHAPTER 1. INTRODUCTION interaction between the d electrons. In the second quantised basis, the Hamiltonian can be written as

d † 1 d † † H = A d dm + I d d dm dm , (1.43) ∑ ∑ m1m2 m1σ 2σ 2 ∑ ∑ m1m2m3m4 m1σ1 m2σ2 3σ2 4σ1 m1m2 σ m1m2m3m4 σ1σ2

d† d m Ad [50] where mσ is a creation operator for a electron with spin σ in orbital , m1m2 is the coefficient d Id for the crystal field potential amongst the electrons and m1m2m3m4 is a Coulomb matrix element amongst the d electrons. From symmetry arguments, we can write down the the first term (the electrostatic interaction) in Eq. 1.43 as † HCF = ∑ εγ dγσ dγσ (1.44) γ,σ

∗ † where γ labels the orbital set, i.e. eg and t2g, dγσ is a creation operator for a d electron with spin σ in

∗ the orbital set γ and εγ is the energy of the orbital set γ. Such that εeg − εt2g ≡ 10Dq. Similarly, we can write down the second term (the electron-electron interactions) in the Hamiltonian as 1 † † H = I , , d d d d , (1.45) el−el 2 ∑ ∑ γ1σ1 γ2σ2;γ3σ2 γ4σ1 γσ1 γ2σ2 γ3σ2 γ4σ1 γ1γ2γ3γ4 σ1σ2 d where Iγ1σ1,γ2σ2;γ3σ2,γ4σ1 is given by a combination of the elements of I . For the purposes of diago- nalising the Hamiltonian it is convenient to express I in terms of the Racah parameters A, B and C. A is the total average interelectron repulsion, and both B and C correspond to individual d electron repulsion. Provided the number of electrons is fixed, the A dependence is the same for all states, and we can approximate C ∼ B/4, leaving only B to characterise the strength of the interactions [47]. The value of B is obtained empirically, typically from the spectrum of the metal ion [47]. The resultant Hamiltonian can be partially diagonalised analytically, however most solutions are obtained numerically [50]. The numerical solutions are usually expressed (for a fixed number of d electrons) as an energy transition, or formally, a Tanabe-Sugano diagram (see Fig. 1.18). The Tanabe-Sugano diagram is fully characterised by E/B, 10Dq/B and the total number of electrons in the metal ion (see Fig. 1.18), where E is the energy eigenvalue. ∗ In general, the ground state can be understood by filling the eg and t2g orbitals with n electrons. 1 0 When n = 1 the ground state corresponds to the sixfold degenerate t2geg state. Similarly, for n = 2 0 3 0 6 2 2,3,8,9 the grounds states are t2geg (ten fold degenerate), t2geg (four fold degenerate), t2geg (three fold 6 3 degenerate) and t2geg (four fold degenerate) respectively. However, for n = 4 − 7 there are two distinct types of ground states. For the n = 6 case shown in Fig. 1.18 the ground state can either be the fifteen 4 2 5 6 0 1 fold degenerate t2geg ( T2g; HS) state for 10Dq B or the non-degenerate t2geg ( A1g; LS) state when 10Dq B. At the point 10Dq = 2.5B + 4C ∼ 19B the system undergoes a discontinuous change in quantum mechanical ground state [42]. Using the nomenclature defined in the previous section, the spin pairing energy is P = 19B. We can gain a further understanding for what happens at 10Dq = P by considering the adiabatic potential energy surfaces for the HS and LS states, shown in Fig. 1.19. Here we have assumed that the vibronic modes for each electronic state can be approximated as a simple harmonic oscillator. 1.2. MICROSCOPIC ORIGIN OF SPIN CROSSOVER SYSTEMS 29

Figure 1.18: Tanabe-Sugano diagram for a transition metal ion with d6 electrons in an octahedral molecular geometry, showing only the relevant low lying energy states. The diagram shows the energy eigenvalues, E, as a function of the ratio of the ligand ﬁeld splitting parameter, 10Dq, and the Racah parameter B. The energy levels are repositioned relative to the energy of the ground state, resulting in 5 a sharp discontinuity at 10Dq = 19B = P, in which the lowest energy state changes from the T2g (HS) 1 (for 10Dq < P) state to the A1g (LS) (for 10Dq > P) state. The straight lines correspond to electronic states, the curved lines are a consequence of the mixing between multiple electronic states.

Figure 1.19: Energy level diagram showing the lowest vibronic levels of the LS and HS states of an octahedral complex as the ratio of 10Dq and P is varied.

For 10Dq < P the enthalpy difference between the lowest vibronic levels of the HS and LS states is 0 0 0 ∆EHL = EHS − ELS < 0 (see Fig. 1.19a), consequently the quantum mechanical ground state is HS. 0 For 10Dq = P the ground state is either HS or LS (∆EHL = 0, see Fig. 1.19b). And, for 10Dq > P the 0 ground state is LS (∆EHL > 0, see Fig. 1.19c). It should be noted that the point at which the potential wells cross is never the lowest energy state, the quantum mechanical ground state is always either HS or LS. Based on Fig. 1.19, the condition for the thermal occupation of the higher energy spin-state 0 (spin crossover) becomes clear: we require that ∆EHL ∼ kBT. However, this definition requires some refinement, since it does not account for the entropy differences between the spin-states. Due to 30 CHAPTER 1. INTRODUCTION the larger electronic and vibrational entropy of the HS state over the LS state, the HS will remain energetically favourable at all temperatures if it is the ground state. Whereas, if the ground state is LS, then there exists a critical temperature at which the HS state becomes energetically favourable, and a spin crossover or spin transition can occur. This transition can occur as a result of changes in 0 temperature, pressure, light irradiation, external fields and chemical environment when ∆EHL ∼ T∆S.

1.3 Spin crossover

1.3.1 Importance of Spin Crossover

In the solid-state, the local molecular distortions caused by the metal ions changing spin state couple to the long-range elastic interactions, which can lead to ﬁrst-order transitions with hysteresis, indicating signiﬁcant interactions between the metal centres. Consequently, over recent years, SCO systems have emerged as both an experimental and theoretical playground of bulk properties. On the applied side, the unique functionality of SCO systems makes them highly sought after for their numerous applications as smart molecular devices including: reversible high density memory, multi-bit electronics, quantum bits, optical displays and nanoscale sensing [43, 44, 52–59]. On the fundamental side, the remarkable chemical versatility and variability of these systems allows for the manipulation of both the local chemistry and molecular structure to produce a diverse range of bulk properties and behaviours. In this section we will present a micro-review of the bulk properties of SCO systems (in Section 1.3.2) and the attempts to model them (in Section 1.3.3).

1.3.2 Spin Crossover in the Solid state

The distinct differences between structural, spectral and magnetic properties of the HS and LS states of a transition metal ion in a SCO system allows for spin transitions and crossovers to be detected by changes in the vibrational spectroscopy, magnetic susceptibility, heat capacity, crystal structure and Mossbauer¨ spectroscopy [44, 49, 51]. Due to the pronounced changes in the magnetic properties of the system, and the accuracy in which the experiments can be done, the most common measurement is the temperature dependent magnetic susceptibility χ(T). Generally, either Evans NMR method or SQUID magnetometry is used [42]. As long as the Curie-Weiss law holds we can define χT ∼ C, where C is a constant that depends on the spin-state. This relationship follows from defining the effective magnetic moment µeff as s 3kBχT µeff = 2 , (1.46) NAµB where kB is the Boltzmann constant, T is temperature, NA is Avogadro’s number and µB is the Bohr magneton of an electron. For first row transition metals, we can assume orbital angular momentum is small compared to the spin angular momentum and thus

2 2 g s(s + 1) = µeff ∝ χT, (1.47) 1.3. SPIN CROSSOVER 31

Figure 1.20: Heating (red) and cooling (blue) SCO curves (the fraction of high spins, nHS, vs. temperature, T) for (a) a crossover, (b) a one-step transition with hysteresis, (c) a continuous crossover, (d) a two-step transition with two ﬁrst-order transitions, (e) a two-step transition with one ﬁrst and one second-order transition and (f) an incomplete one-step transition with hysteresis.

2 where g is the gyromagnetic ratio. Consequently, C = gNAµB s(s + 1). 3kB In the solid state the molar contributions of the HS and LS states to the magnetic susceptibility

χHST ∼ CHS and χLST ∼ CLS can easily be derived allowing for the total magnetic susceptibility to be deﬁned as

χT ∼ CHSnHS +CLS(1 − nHS), (1.48)

2 where C = gNAµB s (s + 1) for x ∈ {HS,LS} and n is the fraction of metal ions in the HS state. x 3kB x x HS Consequently, plotting nHS as a function of temperature is a good measure of the cooperativity – the extent to which the electronic and structural changes of a transition metal ion are propagated throughout the lattice. For cases in which χHS and χLS are not known or cannot be measured with great accuracy, the cooperativity can still be measured by plotting χT vs T. Typically the plot of χT vs T is referred to as the SCO or spin-transition curve. The most commonly reported SCO curves are either crossovers (see Fig. 1.20a) or first-order transitions (see Fig. 1.20b). Although in general, SCO curves can also be continuous (see Fig. 1.20c), a multi-step process with many transitions (see Fig. 1.20d-e) or incomplete (see Fig. 1.20f) [44]. An incomplete SCO curve refers to a SCO system with a low-temperature ground state with nHS 6= 0 or a high-temperature phase nHS 6= 1. Incomplete transitions and multi-step transitions often display antiferroelastic ordering of spin-states (see Fig. 1.20f). A detailed review of multi-step transitions and antiferroelastic phases reported in experiments on SCO materials and frameworks is given in Chapter 2, also see Refs. [42, 45, 46, 51]. The cooperative interactions between the metal ions are a result of elastic interactions [44, 49, 51]. The local structural distortions caused by metal ions changing spin-state, creates a local elastic strain that, couples to the long-range elastic interactions, generating a build up of stress over the material [51]. Strong cooperative interactions typically occur when there is a direct covalent interaction between 32 CHAPTER 1. INTRODUCTION the metal ions, a large structural difference between the spin-states and the lattice and chemical environment is flexible enough to accommodate the molecular distortions [43]. How the structural changes occur on the molecular level does not seem to matter, for example whether the change occurs as result of an increase in metal-ligand distance or ligand distortion [43]. Typically, while the local molecular distortions are usually totally symmetric, the structural distortions of the lattice can either involve the expanding or contracting of the lattice or a significant spatial rearrangement of the metal ions as the transition occurs [51].

Less is known about the role intermolecular interactions play in determining the cooperative behaviour. Though hydrogen bonding, aromatic π − π bonding and van der Waals interactions the presence of guest molecules [60], solvent atoms [61] and anions [62] have been shown to strongly modify the collective SCO behaviours [51]. For example, Murphy et al. [60] showed that changing the number and type of guest molecules changed the number of reported transitions from one-step to two- and three- step. Although in general, any changes to the structural properties of the atomic architecture, including changes in polymorphism [63], can alter the bulk properties and behaviours [51].

Very little is known about the role spin-spin interactions play in determining the bulk properties and behaviours of spin crossover systems. One would expect that these effects would be most pronounced in di-nuclear and multi-nuclear systems. Estimates of the strength of these interactions for tetra- and decanuclear molecular materials range from 4 − 6 K [64], two orders of magnitude lower than estimates of the strength of the elastic interactions 100 − 400 K [1]. Consequently, the effects of spin-spin interactions are typically ignored in the temperature regimes where spin crossover occurs.

Deriving structure-property relations and determining the mechanisms that govern them, has become one of the great challenges of the SCO community. Overcoming this challenge would allow for the rational design of SCO systems with enhanced applications and novel bulk properties and behaviours.

1.3.3 Models of Spin Crossover

A variety of microscopic (and macroscopic) models have been proposed to explain the emergent properties reported in SCO systems: Ising-like, ligand field, vibronic and elastic models [65]. The ligand field and vibronic models consider a purely microscopic approach to the Hamiltonian, focusing on the coupling between the electronic levels and the intramolecular vibrations. While these models are quite interesting in their own right and present a more realistic option for specific material based calculations and out of equilibrium dynamics, they require quantum-chemistry calculations to make predictions. Hence, more general, simpler models could greatly aid in constructing structure-property relations and making predictions. In this section we will review the Ising-like and elastic models, which present a simpler method of modelling SCO systems. For a comprehensive review of SCO models see Ref. [65]. 1.3. SPIN CROSSOVER 33

Non-interacting SCO models

In order to model the many-body physics of SCO systems, it is convenient to map the spin-states of k k the metal ions onto a set of binary pseudo-spin variables: σi = +1 for a HS state and σi = −1 a LS state. The index i labels the metal ion and k labels the degeneracy of the spin-states. The degeneracy of the HS state k ranges from 1 to gHS and the degeneracy of the LS state k ranges from 1 to gLS. Using the pseudo-spin notation, we can write down a Hamiltonian for a solution of non-interacting SCO molecules as k H0 = ∑H(σi ), (1.49) i k k where H(σi ) is the enthalpy of a metal ion in spin-state σi . We can take advantage of the binary nature of the problem and deﬁne the mapping

H(+1) + H(−1) H(+1) − H(−1) H(σ k) = + σ k. (1.50) i 2 2 i

H(+1)+H(−1) For ease of notation we will set H = 2 and ∆H = H(+1)−H(−1). Using this mapping, the Hamiltonian can be written as

∆H k ∆H k H0 = NH + ∑σi ≈ ∑σi , (1.51) 2 i 2 i where N is the number of metal ions. Using the methodology proposed by Wajnflasz and Pick [66] we can “integrate out” the spin-state degeneracy. The benefit of this method is that it allows for an intuitive interpretation of the thermodynamics. The partition function for the Hamiltonian defined in Eq. 1.51 is given by

− ∆H k Z = ∑ e β 2 ∑i σi , (1.52) k {σi }

k where {σi } denotes a possible conﬁguration of spin-states and β = 1/(kBT). Since, the energy (Eq. 1.51) is independent of k, we can sum over the degeneracy of the spin-states. Consequently, we can write the partition function as ∆H −β ∑ σi Z = ∑ e 2 i W({σi}), (1.53) {σi} where {σi} denotes a possible conﬁguration of pseudo-spins and W({σi}) counts the total spin-state degeneracy. For the simplest case, in which we only consider the electronic degeneracy, we can write

W({σi}) down as

NHS NLS W({σi}) = (gHS) (gLS) , (1.54) where NHS and NLS are the number of metal ions in the HS and LS states respectively. A more convenient way of writing this expression is 1 + σi 1 − σi W({σi}) = ∏ gHS + gLS , (1.55) i 2 2 34 CHAPTER 1. INTRODUCTION as it allows us to deﬁne the mapping

1 gHS 1 + σi 1 − σi √ σi log g + g = g g e 2 gLS , (1.56) HS 2 LS 2 HS LS and reabsorb the degeneracy back into the Hamiltonian. Consequently, the partition function takes the form g N 1 HS −β 2 ∆H−kBT log g ∑i σi Z = (gHSgLS) 2 ∑ e LS , (1.57) {σi} with the effective “Ising-like” Hamiltonian 1 Heff = (∆H − T∆S)∑σi, (1.58) 2 i

gHS where ∆S = kB log . As previously mentioned, due to the higher population of anti-bonding orbitals gLS in the HS state, the HS state will have a larger spin, orbital and vibrational multiplicity, thus ∆S > 0. The suffix “-like” is added since the different spin-states have different enthalpies and entropies. The striking feature of this model is that the mismatch in the degeneracy of the spin-states gives rise to a temperature dependent field. The sign of the effective field, heff = ∆H −T∆S, picks out the enthalpically ∆H ∆H favourable state. For T < ∆S the majority LS state is favoured whereas, for T > ∆S the majority HS ∆H state is preferred. At the point T1/2 = ∆S , exactly half of the spin-states are HS and the other half are ∆H in the LS state. The strength of ∆S plays a pivotal role in determining the thermodynamic behaviour of the system (see. Fig. 1.21). For all parameter values, a crossover is observed from the low-temperature short-range ordered LS phase (or HS phase) to a high temperature trivial phase. The strength of ∆S/kB strongly influences the “abruptness” of the transition and the value of nHS in the high temperature phase. ∆S/k ∆S/k As T → ∞, the value of nHS → e B /(1 + e B ). For a solution of non-interacting complexes, the change in entropy of a metal ion can be written down as ∆S = ∆Svib +∆Selec +∆Sorb. Where Svib, Selec and Sorb are the vibronic, electronic and orbital contributions to the entropy, respectively. For materials and frameworks Svib is typically the largest contribution [49]. Here we only consider the electronic contribution, which can be straightforwardly calculated using quantum theory. When the central metal has d6, the HS state has the total spin S = 2 and the LS state has the total spin S = 0, such that HS LS HS LS ∆Sspin = Sspin − Sspin = kB log5 and ∆Sorb = Sorb − Sorb = kB log3 consequently, ∆S ∼ kB log15 [49] and nHS → 15/16 ≈ 0.938.

Ising-like models

In their seminal paper, Wajnflasz and Pick [66] reported that an Ising-like model with a phenomenological nearest neighbour Ising interaction can model both a spin crossover and a first-order spin transition with hysteresis. Using the Wajnflasz and Pick method of integrating of the spin-state degeneracy, outlined in Section 1.3.3, the effective-interacting Hamiltonian can be expressed as 1 Heff = −J ∑ σiσ j + (∆H − T∆S)∑σi. (1.59) hi, ji 2 i Which corresponds precisely to a pseudo-spin Ising model with a temperature dependent longitudinal field. Similar to the non-interacting case studied in Section 1.3.3, the temperature dependence of 1.3. SPIN CROSSOVER 35

Figure 1.21: SCO curves for the non-interacting Wajnﬂasz and Pick model. The difference in entropy between HS and LS states, ∆S, plays an important role in determining the “abruptness” of the crossover and the value of nHS in the high temperature trivial phase.

Figure 1.22: Mean-field phase diagrams for the Wajnflasz and Pick model with overlayed SCO curves for fixed parameters. Both ∆S/kB and ∆H/(kBTc) strongly influence the nature of the transition. For increasing values of ∆H/(kBTC) (or decreasing values of ∆S/kB) a first-order transition with hysteresis (T1/2 < Tc), a continuous transition (T1/2 = Tc) and a crossover (T1/2 > Tc) are observed. The hysteresis loop is indicated by the negative gradient of the nHS curve. Black lines indicate first-order transitions, red dashed lines indicate spinodal lines and black circles indicate critical points.

the effective Hamiltonian is a result of the mismatch in entropy between the HS and LS states. The ﬁnite phase diagram can be obtained by taking a mean-ﬁeld approximation, as shown in Fig. 1.22.

Depending on the strength of J ∝ Tc, ∆S and ∆H, the transition can be ﬁrst-order with hysteresis, continuous or a crossover. A common measure of the cooperativity is the parameter ∆H/J. For large ∆H/J, the single molecule physics dominates resulting in a crossover. Whereas, for small ∆H/J the transition is ﬁrst-order with hysteresis, indicating strong cooperative interactions between the metal ions. As the system passes through the critical point, the transition becomes continuous. Similar to the non-interacting case, ∆S plays a large role in determining the nature of the transition and the value of nHS in the the high-temperature trivial phase. 36 CHAPTER 1. INTRODUCTION

Figure 1.23: A diagrammatic representation of the elastic model, showing the parameters used in Eq. 1.61.

Aside from the fact that the model neglects the structural distortions of the molecules and the long-range interactions important to understanding SCO systems, the downside is that the interactions are added in phenomenologically. However, it was later shown by Slichter and Drickamer [67] that, in a mean-field approximation, the parameters defined in the Wajnflasz and Pick model could be mapped onto a macroscopic model with experimentally measurable parameters. This later lead Sorai and Seki [68,69] to the important discovery that the vibronic contribution to the entropy was much larger than the electronic contribution. Typically, for materials with a central Fe2+ ion, the vibrational contribution is the largest, such that ∆S ∼ kB4log5 (nHS → 625/626 ≈ 0.998 as T → ∞) [49]. However, due to the inability of the Ising-like model to predict volumetric changes or allow for structural phase transitions, recent work has turned to the more realistic models, in particular elastic models.

Elastic models

As previously stated, cooperative spin-transitions occur as a result of long-range elastic interactions coupling to the local structural distortions caused by metal ions changing spin-state. The simplest model that captures this behaviour is the elastic model, sometimes referred to as the atom-phonon model. This approach is based on the work by Nasser et al. in 2001 [70], who modelled a SCO crystal as a network of springs connecting neighbouring molecules. The spin-states of the metal ions are mapped onto a set of binary pseudo-spin variables and each molecule is replaced by a hard sphere with a pseudo-spin dependent radius, R(σ), where R(σi = +1) = rHS and R(σi = −1) = rLS. Each sphere has both position and pseudo-spin degrees of freedoms. Here we will use the same approach introduced by Nasser et al., but for pedagogical reasons we will use a different notation. Using the Wajnﬂasz and Pick method of integrating out the spin-state degeneracy, we can deﬁne the single molecular contribution to the Hamiltonian as

1 H0 = (∆H − T∆S)∑σi. (1.60) 2 i 1.4. THESIS OUTLINE 37

The cooperative interactions are then given by

k 2 k H = 1 r − R(σ ) + R(σ ) + 2 (r − η [R(σ ) + R(σ )])2 , (1.61) 1 2 ∑ i j i j 2 ∑ ik ik i k hi, ji1 hi,ki2 where hi, jin indicates that the sum runs over all nth nearest neighbour pairs, kn is the elastic constant for an nth nearest neighbour, ri j is the instantaneous distance between nearest neighbours measured from the centre of both metal ions and ηik is the geometric distance between the first- and second- nearest neighbours. Both first- and second- nearest neighbour interactions are required to stabilise the lattice from collapsing. Angular elastic interactions can also be used to stabilise the lattice [71]. Models including the second-nearest neighbour interaction have been able to reproduce one-step transitions, crossovers and both incomplete and two- step transitions (with hysteresis) accompanied by the antiferroelastic order of the spin-states on square lattices [72]. Similar to the Ising-like model, the 2 cooperativity can be measured by the parameter ∆H/(k1δ ), where δ is the difference in the metal- 2 ligand bond length between the HS and LS states. For large values of ∆H/(k1δ ), the single molecular 2 physics dominates resulting in a crossover. For small values of ∆H/(k1δ ), the elastic interactions dominate, resulting in one- or two- step transitions with hysteresis. Most notably, this result agrees with experiment on the origin of the cooperativity in SCO systems; a strong direct covalent interaction between the metal ions (large k1) and a large structural difference between the spin-states (large δ) leads to strong cooperativity. The value of ∆S influences the nature of the transition and determines the value of nHS in the high-temperature trivial phase. However, there are downsides to the elastic model. Historically, the approach to solving this Hamiltonian involves using numerical methods to find both the dynamical and spin-state equilibriums [70–72]. The need to find both equilibriums significantly increases the computational time required to simulate the thermodynamics of the system. Furthermore, since the positional degrees of freedom are continuous, the parameter space of the system is infinite. To greatly simplify the process of constructing structure-property relations and making predictions, new methods are required to reduce the size of the parameter space.

1.4 Thesis Outline

In this thesis we have endeavoured to explain the breadth of bulk properties and behaviours that has been reported experimentally in SCO systems. Using a simple, elastic model we have constructed structure-property relations, derived the mechanisms of multi-step transitions and explained why and how intermolecular interactions play a role. Furthermore, we have predicted new emergent properties and behaviours, in particular, a new type of ice phase. In Chapter 2 we show that an Ising-like model with a long-range interactions can be obtained from an elastic model. We show that this model hosts thirty-six different antiferroelastic phases and up-to eight step transitions. We demonstrate clear structure-property relations that explain these results, in terms of the competition and cooperation of through-space and through-bond interactions. Using 38 CHAPTER 1. INTRODUCTION these structure-property relations we explain the role intermolecular interactions play in modifying the observed SCO behaviours. In Chapter 3 we propose a new state of matter “spin-state ice” – so-called in analogy to water ice and spin ice – that arises in elastically frustrated SCO materials and frameworks. We use the elastic model on the kagome lattice to show that no long-range ordering exists down to low temperatures; instead the ground states follow the local ice rule that each triangle must contain two HS states and one LS or vice versa. We show that at finite temperatures ices phases exist over a wide temperature regime, and support deconfined quasi-particles excitations that carry a fractionalised spin midway between that the HS and LS states. In Chapter 4 we show that the elastic model, on the pyrochlore lattice, can give rise to multiple spin- state ice phases. We show that the unique nature of SCO systems allows for the unprecedented ability to straightforwardly tune between different ice phases with external parameters, such as temperature and pressure. In Chapter 5 we will comment on future prospects of this research. 1.4. THESIS OUTLINE 39 The following publication has been incorporated as Chapter 2. For ease of reading figures have been restructured to fit the thesis format. 1. [1] Jace Cruddas, and Ben J. Powell, Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks, Inorg. Chem. Frount., Advance Article, 10.1039/D0QI00799D (2020).

Contributor Statement of contribution % Jace Cruddas writing of text 60 proof-reading 50 theoretical derivations 90 numerical calculations 100 preparation of ﬁgures 95 initial concept 50 Ben J. Powell writing of text 40 proof-reading 50 theoretical derivations 10 numerical calculations 0 preparation of ﬁgures 5 initial concept 50 supervision, guidance 100

Powell made Figure 2.3. Chapter 2

Structure–property relationships and the mechanisms of multistep transitions in spin crossover materials and frameworks

Spin crossover frameworks and molecular crystals display fascinating collective behaviours. This includes multi-step transitions with hysteresis and a wide variety of long-range ordered patterns of high-spin and low-spin metal centres. From both practical and fundamental perspectives it is important to understand the mechanisms behind these collective behaviours. We study a simple model of elastic interactions and identify thirty six different spin-state ordered phases. We observe spin-state transitions with between one and eight steps. These include both sharp transitions and crossovers, and both complete and incomplete spin crossover. We demonstrate structure-property relationships that explain these differences. These arise because through-bond interactions are antiferroelastic (favour metal centres with different spin-states); whereas, through-space interactions are typically ferroelastic (favour the same spin-state). In general, rigid materials with longer range elastic interactions lead to transitions with more steps and more diverse spin-state ordering, which explains why both are prominent in frameworks.

2.1 Introduction

Spin crossover (SCO) materials typically consist of a transition metal ion with a partially occupied d-shell surrounded by several ligands. Depending on the physical environment (temperature, pressure, magnetic ﬁeld, exposure to light, etc.) these materials can exist in either high spin (HS) or low spin (LS) states. [44] Metal-ligand bond lengths in the HS state are around 10 % longer than in the LS state. [44] In the solid state, long-range elastic interactions between metal centres couple to the local structural distortions caused by individual metal centres changing their spin-state. This causes a wide range of different thermodynamic behaviours, including ﬁrst-order transitions with hysteresis, incomplete 41 42 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.1: (A-T) Truncated snapshots of the antiferroelastic spin-state orderings with majority or half LS metal centres (blue circles) observed in our Monte Carlo simulations on a 60×60 lattice. Equivalent phases are found with majority HS (orange circles). For each state the fraction of HS ions, nHS, and the Bragg wave vector, ~q, are indicated. By analogy to magnetic order phase A is known as Neel´ order and phase B is referred to as stripe order. transitions, crossovers, and up to eight-step transitions. [44–46, 73] Many different long-range ordered patterns of HS and LS metal centres, collectively known as antiferroelastic phases (Fig. 2.1; Table 2.1), have been observed. [60, 74–109] This leads one to ask what mechanism is responsible for these collective effects? and can one predict what other behaviours might exist? Beyond the fundamental interest in these questions, SCO materials and frameworks have been sought after for their many potential applications including high-density reversible memory [43,44,52,53,55–57], actuators [58], ultrafast nanoscale switches, [43,44,53,55–57] sensors [44,54,56,57] and displays [43,44,52,55–57]. Understanding the mechanisms that control the collective behaviours of SCO materials could signiﬁcantly enhance their potential to be engineered for speciﬁc applications. Two-step transitions can arise as a result of molecular multistability, from crystallographic differences between metal centers or as a result of interactions between the spin-states of the metals. [51] Interaction-driven two-step transtions have been studied extensively on the basis of empirical models. [109–113] Recently, it has been suggested that the competition between elastic interactions or ‘elastic frustration’ is crucial for understanding two-step transitions and antiferroelastic order in SCO materials. [2,71,72] For example, a mismatch between the equilibrium bond lengths of nearest and next nearest-neighbour bonds, has been shown to induce two-step transitions on the square lattice. [72] 2.1. INTRODUCTION 43

Similarly, it has been argued that in phenomenological Ising-like models cooperative nearest and next-nearest neighbour interactions are required to stabilize two-step transitions and antiferroelastic order in SCO materials. [109–113] However, significantly less is known about the mechanisms of multi-step transitions. The question of what mechanism is responsible for three-, four- and higher-step transitions, and the diversity of antiferroelastic orders reported remains open. In particular, the link between antiferroelastic order and the topology of the lattice remains largely unexplored. Multi-step transitions have been reported for a variety of different molecular materials and frameworks on square arrays including Hofmann-type molecular frameworks [45, 46, 60, 74–90], other coordination polymers [91–95] and molecular crystals held together by supramolecular interactions [46, 96–108]. Hofmann-type frameworks that contain one metal species, M, that is SCO active and another, M0, that is diamagnetic have proved a particularly interesting playground for antiferroelasticity (Table 2.1). [45] We will discuss two families of Hofmann-type frameworks with the general formulae 0 0 0 0 0 [M(L)nM (L )4] and [M(L)n{M (L )2}2], where L is the ligand within the plane, L is the ligand connecting layers, and n = 1 or 2 for bridging and monodentate ligands respectively. Henceforth we will refer to these as the 1n14 and 1n24 families respectively. Prototypical examples are [Fe(pz)Pt(CN)4] and [Fe(pz){Au(CN)2}2]. [45] These crystals have importantly different topologies, Figs. 2.2a,b. Nevertheless, in both families the SCO active M sites form simple square sublattices. Antiferroelasticity has been observed in a number of other coordination polymers where the metal centres form square sublattices, (Fig. 2.2c; Table 2.1). For example, the 1n02 family of materials with 0 the formula [M(L)n(L )2], e.g., [Fe(azpy)2(NCS)2](Fig. 2.2c). [94] Antiferroelastic order is also found in square lattice supramolecular crystals where molecules are predominantly bound via weak interactions (Fig. 2.2d; Table 2.1). [96–108] Therefore, a thorough theoretical study of the square lattice combined with a detailed comparison with the extensive experimental literature is an ideal starting point to establish structure-property relations for SCO materials. This is the goal of this chapter. There is a strong correlation between material structure and the collective SCO behaviours (Table 2.1). A particularly striking example is the antiferroelastic order in states with equal numbers of HS and LS ions. In the 1n14 family stripe ordering (Fig. 2.1b) is commonly reported, [60,74–81] whereas in the 1n24 [83–86, 88–90] and 1n02 [91–95] families Neel´ order (also known as checkerboard order, Fig. 2.1a) is prevalent. In the 1n14 family (Fig. 2.2a) there are covalent bonds connecting the second and fifth nearest neighbour M sites but only weak through-space interactions connecting nearest, third and fourth neighbours. Whereas, for the 1n24 and 1n02 families (Fig. 2.2b) there are covalent bonds connecting the nearest and third nearest neighbour M sites and through-space interactions connecting the second, fourth and fifth nearest neighbours. We will show below that through-bond interactions favour neighbouring metal centres with different spin-states (one high spin and the other low spin); whereas through-space interactions favour neighbouring metal centres with the same spin-states (both high 44 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Family Material Ref. nHS plateaus Phases Figure 1 1n14 [Fe(thtrz)2Pd(CN)4]·EtOH, H2O [74, 75] 0, 2 , 1 LS, B, HS 2.6e 1 1n14 {Fe(Hppy)2[Pd(CN)4]}·H2O [76] 0, 2 , 1 LS, B, HS 2.6e 1 1n14 {Fe(Hppy)2[Pt(CN)4]}·H2O [76] 0, 2 , 1 LS, B, HS 2.6e 1 1n14 {Fe(bpb)[M(CN)4]}·2naph [77] 0, 2 , 1 LS, B, HS 2.6e 1 1n14 [Fe(trz−py)2{Pt(CN)4}]·3H2O [78] 2 , 1 B, HS 2.6d 1 1n14 [Fe(proptrz)2Pt(CN)4]·2H2O [79] 2 , 1 B, HS 2.6d 1 1n14 [Fe(proptrz)2Pd(CN)4]·2H2O [79] 2 , 1 B, HS 2.6d 1 1n14 [Fe(bztrz)2(Pd(CN)4)]·(H2O,EtOH) [60] 2 , 1 B, HS 2.6d 1 1 1n14 [Fe(bztrz)2(Pd(CN)4)]·3H2O [60] 4 , 2 , 1 XL, B, HS - 1 1 1n14 [Fe(bztrz)2(Pd(CN)4)]·∼ 2H2O [60] 0, 4 , 2 , 1 LS, XL, B, HS - 1 1 5 1 1 3 1n14 [Fe(Hbpt)Pt(CN)4]· 2 Hbpt· 2 CH3OH· 2 H2O [80] 4 , 2 , 4 , 1 XL, B, XH, HS - 2 1 1 1n14 [Fe(dpsme)Pt(CN)4]· 3 dpsme·xEtOH·yH2O [81] 0, 3 , 2 , 1 LS, RL?, B, HS 2.8j 1 2 5 1n14 [Fe3(saltrz)6(Pt(CN)4)3]8(H2O) [82] 0, 6 , 3 , 6 , 1 LS, ??, KH, ??, HS S17c 5 1 1n24 {Fe(3−Fpy)2[Au(CN)2]2} (form 1; 10 Pa) [46, 83] 2 , 1 A, HS 2.5d 1 1n24 {Fe(3−Fpy)2[Au(CN)2]2} (form 1; 0.18 GPa) [46, 83] 0, 2 , 1 LS, A, HS 2.5f 1 1n24 {Fe(3−Fpy)2[Au(CN)2]2} (form 1; 0.26 GPa) [46, 83] 0, 2 , 1 LS, A, HS 2.5g 1 1n24 {Fe(3−Fpy)2[Au(CN)2]2} (form 2) [46, 84] 0, 2 , 1 LS, A, HS 2.5f 1 1n24 {Fe(DMAS)2[Au(CN)2]2} [85] 0, 2 , 1 LS, A, HS 2.5g 1 1 2 1n24 [Fe(bipytz)(Au(CN)2)2]·x(EtOH) [86] 0, 3 , 2 , 3 , 1 LS, DL, A, DH, HS 2.8d 2 1n24 {Fe(DEAS)2[Ag(CN)2]2} [46, 85] 3 , 1 DH, HS 2.8c 1 3 1n24 [Fe(isoq)2{Ag(CN)2}2] [87] 4 , 4 , 1 FL,FH, HS 2.9a 1 1 3 1n24 [Fe(dpoda){Ag(CN)2}2]·1.5naph [88, 89] 0, 4 , 2 , 4 , 1 LS, FL, A, FH?, HS 2.9a 1 1 3 1n24 [Fe(4−abpt){Ag(CN)2}2]·2DMF·EtOH [90] 0, 4 , 2 , 4 , 1 LS, FL, A, FH, HS 2.9a 0 1 1n02 [Fe(4,4 −bipy)2(NCS)2]·4CHCl3 [91] 0, 2 , 1 LS, A, HS 2.5e 2 1 1n02 ∞[Fe(2,3−bpt)2] [92] 0, 2 , 1 LS, A, HS 2.5g 2 1 1n02 ∞[Fe(2,3−Mebpt)2] [92] 0, 2 , 1 LS, A, HS 2.5g 1 1n02 Fe(bpe)2(NCS)2 ·3(acetone) [93] 0, 2 , 1 LS, A, HS 2.5g 1 1n02 Fe2(azpy)4(NCS)4(PrOH) [94] 0, 2 , 1 LS, A, HS 2.5g 1 † 1n02 [Fe(bdpt)2] [95] 0, 2 , 1 LS, A, HS 2.5e-g 1 † 1n02 [Fe(bdpt)2]·MeOH [95] 0, 2 , 1 LS, A, HS 2.5e-g 1 † 1n02 [Fe(bdpt)2]·EtOH [95] 0, 2 , 1 LS, A, HS 2.5e-g 1 molecular β−[FeLBr(dca)2] [96] 0, 2 , 1 LS, B, HS 2.6e 1 molecular [Fe(nsal2trien)]SCN [97] 0, 2 , 1 LS, B, HS 2.6f 1 molecular [Fe(2−pic)3]Cl2 ·EtOH [98] 0, 2 , 1 LS, B, HS 2.6g 1 molecular [Fe(tdz−py)2(NCS)2] [99] 0, 2 , 1 LS, B, HS 2.6f 1 molecular [Mn(3,5−ClSal2(323))]NTf2 [100] 0, 2 , 1 LS, B, HS 2.6f 3 molecular [Fe(salpm)2]ClO4 ·0.5EtOH [101] 0, 4 , 1 LS, CH, HS 2.8a,g 1 3 molecular [Fe(bmpzpy)2][BF4]2 ·xH2O [46, 102] 2 , 4 , 1 B, KH, HS 2.9b 2 molecular [Fe(bpmen)(NCSe)2] [103] 0, 3 , 1 LS, RH, HS 2.8f 2Me 1 molecular [FeH2L ]−(PF6)2 [104, 109] 0, 2 , 1 LS, S, HS 2.9c 1 2 molecular [Fe(H−5−Brthsa−Me)(5−Br−thsa−Me)]·H2O [105] 0, 3 , 3 , 1 LS, TL,TH, HS 2.9d Table 2.1: Summary of the antiferroelastic order that have been been reported experimentally in the three families of frameworks discussed here and in molecular crystals where the molecules form an approximately square lattice. Letters A-T correspond to the phases shown in Fig. 2.1. Subscripts H and L indicate the majority spin state. HS and LS indicate the ferroelastic phases. The ﬁgure listed in the rightmost column is one that shows the same qualitative behaviour, i.e. the same spin-state phases, and both number and order of the transitions. Fits to the experimental data have not been attempted. A ? after the name of the phase indicates that the assignment is uncertain and a ?? indicates that the ordering is unclear, see the original literature for details. The X phase has alternating stripes of width 3 and 1 in the vertical or horizontal direction (e.g. -LS-LS-LS-HS-LS-LS-LS-HS-; similar to R, which has stripes of width 2 and 1 in the vertical or horizontal direction, e.g. -LS-LS-HS-LS-LS-HS-). The X phase is not found in the model described here but is found if anisotropy is introduced into the model, for example, by allowing the k1 in the x direction to be different from the † k1 in the y direction. indicates that the nature of the transitions changes as pressure is applied. Here thtrz = N-thiophenylidene- 4H-1,2,4-triazol-4-amine, Hppy = 4-(1H-pyrazol-3-yl)pyridine, Hbpt = 4,4’-(1H-1,2,4-triazole-3,5-diyl) dipyridine, trz-py = 4-(2- pyridyl)-1,2,4,4H-triazole, bztrz = (E)-1-phenyl-N-(1,2,4-triazol-4-yl)-methanimine, dpsme = 4,4’-di(pyridylthio)methane, MeOH = methanol, EtOH = ethanol, saltrz = (E)-2-((((4H-1,2,4-triazol-4-yl)imino)methyl)phenol), proptrz = (E)-3-phenyl-N-(4H-1,2,4-triazol- 4-yl)prop-2-yn-1-imine, bpb = bis(4-pyridyl)butadiyne, naph = naphthalene, bipytz = 3,6-bis(4-pyridyl)-1,2,4,5-tetrazine, isoq = iso- quinoline, dpoda = 2,5-di-(pyridyl)-1,3,4-oxadiazole, py = pyridine, 4-abpt=4-amino-3,5-bis(4-pyridyl)-1,2,4-triazole, DMAS = 4’- dimethylaminostilbazole, DEAS = 4’-diethylaminostilbazole, 2,3-bptH = 3-(2-pyridyl)-5-(3-pyridyl)-1,2,4-triazole, 2,3-MebptH = 3- (3-methyl-2-pyridyl)-5-(3-pyridyl)-1,2,4-triazole, bpe = 1,2-bis(4’-pyridyl)ethane), azpy = trans-4,4-azopyridine, PrOH = propanol, bdpt = 3-(5-bromo-2-pyridyl)-5-(4-pyridyl)-1,2,4-triazole, tpa = tris(2-pyridylmethyl)amine, nsal2trien is obtained by condensation of triethylenetetramine and 2 equiv. of 2-hydroxy-1-naphthaldehyde, pic = picolylamine, tdz-py = 2,5-di-(2-pyridyl)-1,3,4-thiadiazole, salpm = 2-((pyridin-2-ylmethylimino)methyl)phenolate, bmpzpy = 2,6-bis3-methylpyrazol-1-ylpyridine, bpmen = N,N’-dimethyl- 2Me N,N’-bis(2-pyridylmethyl)-1,2-ethanediamine, H2L = bis[N-(2-methylimidazol-4-yl)methylidene-3-aminopropyl]ethylenediamine, and H2-5-Br-thsa-Me = 5-bromosalicylaldehyde methylthiosemicarbazone 2.2. MODEL 45

0 0 0 0 Figure 2.2: The in-plane structures of (a) the 1n14, [M(L)nM (L )4], and (b) 1n24, [M(L)n{M (L )2}2], 0 families of Hofmann frameworks, (c) the 1n02, [M(L)n(L )2], family of coordination polymers, and (d) a simple square supramolecular crystal. In each case the elastic interactions, kn, between nth nearest neighbour SCO active M sites (orange circles) are marked. Non-SCO-active ions (M0, black circles) and in-plane ligands (L0, black lines) are also shown. All four classes of materials are described by the same model (Eq. 2.11), but with different magnitudes and signs of the kn due to the topological differences shown here. spin or both low spin). This leads to important predicted structure-property relationship for SCO frameworks. For example, this explains why Neel´ order is found in 1n24 and 1n04 frameworks whereas stripe order is found in 1n14 frameworks. A detailed comparison with the experimental literature is given in Table 2.1. This demonstrates that the antiferroelastic orders previously observed in square lattice SCO materials can be understood from these structure-property relations or straightforward extensions of them.

2.2 Model

We start from the free energy difference, ∆G1, of a single metal centre undergoing spin crossover (SCO): ˜ 1 ∆G1 H0 = ∑(∆H1 − T∆S)σi ≡ ∑σi, (2.1) 2 i 2 i where ∆H1 = HH − HL and ∆S = SH − SL are the enthalpy and entropy differences between isolated HS and LS metal centres respectively. Following Wajnﬂasz and Pick [66] we have absorbed the single ion entropy difference into the local Hamiltonian. The entropy difference arises from changes in three microscopic terms: the spin and orbital degeneracies are different in the HS and LS states, and the vibrational entropy also changes due to the softening of the vibrational modes in the HS state. 2+ For example, for Fe metal centres in an octahedral complex, ∆Sspin = kB ln5 and ∆Sorb = kB ln3.

Typically, the vibrational contribution is larger, such that ∆S ∼ 4∆Sspin. [49] Therefore, in all the 46 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS calculations presented below we set ∆S = 4kB ln5. H˜ 0 describes spin crossover in non- or weakly- interacting complexes, e.g., in solution. For ∆H1 < 0 the HS state is thermodynamically stable at all temperatures. While, for ∆H1 > 0 a LS state is realised at low temperatures, gradually undergoing a crossover to a HS state with equal numbers of HS and LS molecules at T1/2 = ∆H1/∆S. In the solid state, cooperative elastic interactions between metal centres can lead to first-order phase transitions and hence hysteresis. If the neighbouring metal centres are connected through strong covalent bonds then one expects that the potential is close to its minimum and thus the harmonic approximation is reasonable. This justifies modelling the material as a network of springs. However, when multiple interactions are present they may be competing such that, it may not be possible to minimize all of the interactions simultaneously – i.e., there is frustration in the system. Alternatively the interactions may be incommensurate. Paez-Espejo et al. [72] described this by introducing a ‘frustration parameter’, which measures the extent to which different interactions are minimized by different structures. A limitation with the approach presented by Paez-Espejo et al. is that it implicitly assumes the harmonic approximation, which is only valid if all interactions to be near their minima. However, weaker interactions, particularly through-space interactions, may be far from their minima. Here we introduce an approach that removes that difficulty.

We consider an arbitrary interaction, Vi j(r), between two SCO active M sites, i and j. In a crystal the equilibrium structure minimises the total free energy, which includes the sum of Vi j(r) over all pairs i, j. Thus, if the interactions are frustrated there is no guarantee that that any particular Vi j(r) is minimised in the equilibrium structure.

The equilibrium separation between nearest neighbour metal centres in the HS phase, rH, is found experimentally to be larger than that in the LS phase, rL. We linearly interpolate between these two lengths:

r0 = R + δ(σi + σ j), (2.2) where R = (rH + rL)/2 and δ = (rH − rL)/4. For simplicity we assume that Eq. (2.2) holds in 2 antiferroelastic phases as well. On noting that σi = 1 because σi = ±1, we can write any pairwise symmetric function of the spin-states of the metal ions, σi and σ j, in the form f (σi,σ j) = A + B(σi +

σ j) +Cσiσ j, where A,B and C are constants. Thus, an arbitrary pairwise potential between metal centers can be expanded as

1 2 V (r,σ ,σ ) =g (r) + h (r)r + η R − δ(σ + σ ) + k (r)r + η R − δ(σ + σ ) , (2.3) i j i j i j i j i j i j 2 i j i j i j √ √ √ where ηi j = ηn = 1, 2 ,2, 5 ,2 2 ,... is the ratio of distances between the nth and 1st nearest- neighbour distance on the undistorted square lattice.

We interpolate Vi j(r,σi,σ j) by introducing a new function V˜i j(r) deﬁned such that V˜i j(r −rHηi j) =

Vi j(r,1,1), V˜i j(r − rLηi j) = Vi j(r,−1,−1), and V˜i j(r − Rηi j) = Vi j(r,1,−1) = Vi j(r,−1,1). We show in Section 2.6 that

1 2 V (r,σ ,σ ) = f (r) + δη h (r)(σ + σ ) + k (r)r − Rη − δη (σ + σ ) , (2.4) i j i j i j i j i j i j 2 i j i j i j i j 2.2. MODEL 47 where fi j(r) = gi j(r) + hi j(r)(r − Rηi j). The functions fi j(r), hi j(r), and ki j(r) to leading order in

δηi j are

fi j = Vi j(Rηi j), (2.5a) (1) hi j = Vi j Rηi j , (2.5b) (2) ki j = Vi j Rηi j , (2.5c)

(n) n n where V (x) ≡ (∂ V(r)/∂r )|r=x. The term proportional to hi j has the same functional form as H˜ 0 so we define ∆H = ∆H1 + ˜ 4δηi j ∑ j hi j and ∆G = ∆H − T∆S. Thus, we replace H0 by 1 ∆G H0 = ∑(∆H − T∆S)σi ≡ ∑σi. (2.6) 2 i 2 i This reflects the change in the lattice contribution to the enthalpy when spin states change. However, it does not materially affect the calculations reported here as we do not calculate ∆H for specific materials. The term proportional to ki j is simply a harmonic interaction between the metal centres. Thus, the elastic interactions between metal centres can be represented by a network of springs even if the individual interactions are far from their minima. To investigate spin-state transitions we model the elastic interactions between SCO active sites of the simple square lattice as a network of springs, illustrated in Fig. 2.2. We consider elastic interactions between nth nearest-neighbour metal centres

k 2 H = n r − η R + δ(σ + σ ) (2.7) n 2 ∑ i, j n i j hi, jin where kn is the spring constant between nth nearest-neighbour, hi, jin indicates the sum runs over all nth nearest-neighbours, and ri, j is the instantaneous distance between sites i and j. We include interactions up to mth nearest-neighbour metal centres (we study m = 1 − 5). Thus, the total Hamiltonian is m H = H0 + ∑ Hn. (2.8) n=1 We solve this model in the ‘symmetric breathing mode approximation’, [2] i.e., we assume that for all nearest neighbours, ri, j = x, and that the topology of the lattice is not altered by the changes in the spin-states. The instantaneous lattice distance can then be integrated out by minimizing with respect to x. For this purpose it is useful to consider the expanded and simpliﬁed form of the total Hamiltonian - ∆G m k H = σ + n η2{(x − R)2 − 2δ(x − R)(σ + σ ) + 2δ 2(σ σ + 1)}, (2.9) 2 ∑ i ∑ 2 ∑ n i j i j i n=1 hi, jin where N is the number of M sites. Minimising over Eq. 2.9 yields a critical point at the position 1 x = R + 2δ ∑σi. (2.10) N i

2 2 N m 2 In order to ensure the critical point remains a minimum, we require that (∂ H)/(∂x ) = 2 ∑n=1 knznηn ∝ 2 m 2 J∞ > 0, where J∞ = δ ∑n=1(knznηn ) is the long-range strain and zn is the coordination number for 48 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.3: The Lennard-Jones potential V(r) between a pair of of molecules separated by a distance (2) r. Near the minimum of the potential the second derivative, Vi j (r0), is positive (red curve), thus we expect the elastic interaction, ki j, to be strong and positive. Whereas, at larger distances (blue, green, grey and pink curves) the second derivative becomes negative and drops off with increasing distance. Therefore, we expect k < 0 for through-space interactions away from the potential minimum. nth nearest neighbours. Outside this region the crystal is dynamically unstable and we do not study parameters in that regime below. Substituting 2.10 back into Eq. 2.9 yields the effective Ising-Husimi- Temperley model in a longitudinal ﬁeld

m J ∆G H ≈ J σ σ − ∞ σ σ + σ , (2.11) ∑ n ∑ i j N ∑ i j 2 ∑ i n=1 hi, jin i, j i

2 2 where, Jn = knηn δ is the effective interaction between of the nth nearest-neighbour metal centres. The long-range strain has equal strength between all metal centres, distributing the impact of local molecular volume changes due to spin-state transitions over the lattice.

2.2.1 Elastic interactions in materials

For nearby metal centres joined via (networks of) covalent bonds one expects the metal-metal separation to be close to the minimum of the potential. Hence, one expects that the spring constant, kn, is large and positive (i.e., an antiferroelastic interaction). For through-space interactions a separation larger than 2 2 the minimum of the potential leads to a negative (ferroelastic) spring constant, kn ' ∂ Vi j(r)/∂r |r=R, as illustrated in Fig. 2.3.

As through-space interactions can be antiferroelastic or ferroelastic, one might ﬁnd that both k1 > 0 and k2 > 0 for some materials in any family of coordination polymers and supramolecular crystals. We will see below that in this case the long-range strain dominates and the SCO transition is always one step. The three families of frameworks that we consider embody two distinct topologies with the framework of our model, Fig. 2.2. In the 1n14 family k2 and k5 are through-bond whereas k1, k3 and k4 are through-space. Therefore, one expects k2 > k5 > 0 and k2 > |k1|, but it is reasonable to expect that in many materials k1 < k3 < k4 < 0. In contrast in the 1n24 and 1n02 families k1 and k3 are through-bond whereas k2, k4 and k5 are through-space. Therefore, one expects k1 > k3 > 0 and k1 > |k2|, but in many materials one will ﬁnd that k2 < k4 < k5 < 0. We show below that these 2.3. METHODS 49 differences are responsible for the different antiferroelastic orders observed in the 1n14, 1n24, and 1n02 families.

2.3 Methods

To construct the zero-temperature phase diagrams we analytically compared the energies of the spin- state phases found in the Monte Carlo calculations (on a variety of lattice sizes) alongside a catalogue of possible phases. To investigate the thermodynamic properties of Hamiltonian we employ Monte Carlo methods using single spin-flips and periodic boundary conditions with N = 60 × 60 metal ions for the lattices. For each data point we take a measurement every Monte Carlo step for N measurements, after equilibrating for 10 Monte Carlo steps. Each Monte Carlo step consists of N Monte Carlo attempts. We choose a 60×60 lattice for our Monte Carlo simulations because it is commensurate with every phase observed in the zero-temperature phase diagram for the parameters studied. We will show below that increasing the range of interactions often increases the size of the unit cell for the antiferroelastic phases and as such, requires a larger grid in our simulations. For each parameter set we performed three separate calculations: heating, cooling and parallel tempering. For the heating calculation, we initialize the simulation at the lowest temperature studied 2 (T = 0.01|k1|δ /kB) in the T = 0 ground state predicted by analytic calculations; for higher temperature data points we seed the simulation with the spin-state output from the previous data point. Conversely, for the cooling run we initialize the calculations at the highest temperature studied in a random configuration, then use the resultant output state as a seed for the next data point. When using single spin-flip Monte Carlo the transitions can become frozen out at low temperatures leading to exaggerated predictions of the transition temperatures. We employ parallel tempering to find the lowest free energy state. For the parallel tempering calculations we initialize the simulation in a random configuration. 1 N 1 The total heat capacity, cV = cV + cV , consists of both the single body contribution, cV , and the N many body contribution, cV . The many-body contribution comes from the fluctuations in the true N 2 2 enthalpy cV = (hE i − hEi )/(NkBT), where E = H + (1/2)T∆S∑i σi. The single body contribution 1 1 comes from the contribution of the single molecules to the entropy cv = T(∂S /∂T) = T∆S(∂nHS/∂T). Where, we have used a Savitzky-Golay filter [114] to fit nHS and thus calculate ∂nHS/∂T. The total heat ∆S capacity can also be calculated from the fluctuations in the enthalpy as (∂H)/(∂T) − 2 (2nHS − 1) = 1 N cV + cV . We identify the critical behaviour by calculating the first (number of high spins, energy) and second order (susceptibility, heat capacity) derivatives of the free energy. The presence of first order derivatives can be further distinguished by the presence of spinodal lines. To distinguish a second order transition from an abrupt crossover we calculate the first and second derivatives of the susceptibility and heat capacity using a Savitzky-Golay filter to fit the data, we then calculate the height and location of the peaks. To distinguish the singularities from fluctuations we remove all low frequency peaks and keep only the peaks with a high frequency. We then compare the number, location and density of the 50 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.4: Zero temperature phase diagrams with nearest (k1) and next nearest (k2) neighbour elastic interactions for (a) the 1n24 and 1n02 families, and (b) the 1n14 family. peaks. This process is repeated with various frequency cutoffs. If all six plots contain a high frequency peak at the same location (within a small error margin), then the peak is classiﬁed as a second order transition. Otherwise, the transition is said to be a crossover.

2.4 Results and Discussion

2.4.1 Nearest- and Next Nearest-Neighbour Interactions

To understand the zero-temperature phase diagrams, Fig. 2.4, it is helpful to consider which ordering patterns minimise the interactions individually. The nearest-neighbour interaction is minimised by

Neel´ (also known as checkerboard) order (Fig. 2.1a) for k1 > 0 and by HS or LS order, depending on the sign of ∆G, for k1 < 0. We will henceforth refer to the HS and LS order as ferroelastic phases.

The second nearest-neighbour interaction is minimised by stripe order (Fig. 2.1b) for k2 > 0 and by either ferroelastic or Neel´ order for k2 < 0. The elastic interactions cooperate for k1 > 0 and k2 < 0, which are both minimised by Neel´ order (also known as checkerboard order), but are frustrated for any other parameters. Both the long range strain and the single ion enthalpy favour ferroelasticity. These effects compete 2 2 with the short-range elastic interactions to determine the ground state. If J∞/(|k1|δ ) and |∆H/(|k1|δ )| 2 are large then the ground state is either LS or HS, determined by the sign of ∆G. While if J∞/(|k1|δ ) 2 and |∆H/(|k1|δ )| are small then either Neel´ or stripe ordering is thermodynamically stable.

Purely ferroelastic interactions (any family)

With only nearest neighbour interactions, k1 > 0, the HS, LS and Neel´ phases are degenerate at ∆G = 0 (Fig. 2.4), and thermal ﬂuctuations stabilise short range Neel´ correlations at elevated temperatures (Fig. 2.24 and section 2.6.2). This can lead to a two step transition (Fig. 2.24d) but does not lead to a spontaneously broken symmetry or long-range order. Furthermore, the intermediate plateau is observed only in an extremely narrow temperature range. Experimentally, many different antiferroelastic phases with true long-range order have been found and these phases can be stable over relatively broad 2.4. RESULTS AND DISCUSSION 51

Figure 2.5: (a) Typical phase diagram for the next nearest neighbour model of the 1n24 and 1n02 families with k1 > 0, k2 < 0 (here k2 = −0.2k1). Shading indicates the fraction of high-spin M sites, nHS ∼ χT where χ is the susceptibility, calculated via parallel tempering Monte Carlo. The (black solid) lines of ﬁrst order transitions become (black dashed) lines of second order transitions at critical points (black dots). The red (blue) dashed lines mark the limits of metastability cooling (resp. heating), cf. Fig. 2.15, and hence show the width of the hysteresis. Individual materials have ﬁxed ∆H, white lines correspond to panels (b-h), where the fraction of high spins (blue; cooling, red; heating, black; equilibrium) is plotted (see Fig. 2.16 for the corresponding heat capacities). temperature ranges. This suggests that longer range elastic interactions are vitally important for multistep transitions.

Considering next nearest neighbour interactions, k1 > 0 and k2 > 0 is possible for any of the lattices shown in Fig. 2.2 provided that the minima of both interactions are roughly commensurate with a square lattice. At T = 0 the LS and HS states are separated by a first order phase transition at ∆G = 0, Figs. 2.4a and 2.25a. The frustration entirely suppresses the Neel´ order found at finite temperatures for k2 = 0 (cf. Figs. 2.24 and 2.25). For constant ∆H, which represents individual materials, we find a single step transition (Fig. 2.25). Which can be sharp and first order, second order, or a crossover, depending on the relative strengths of the elastic interactions and the single ion entropy (see section 2.6.2). Consistent with this prediction, single step transitions are common and observed in all of the families of materials discussed here.

Antiferroelastic interactions in the 1n24 and 1n02 families

One expects that many materials in 1n24 and 1n02 families will have ferroelastic next nearest neighbour interactions (k2 < 0, which requires k1 > 2|k2|). This leads to a much richer range of behaviours, Fig. 2.5. The Neel´ phase is stable at T = 0 and there are two lines of first order transitions ending at two critical points joined by a line of second order transitions, Fig. 2.5a. For individual materials with fixed ∆H this leads to seven thermodynamically distinct behaviours. Generically, there is a two step transition from HS to Neel´ to LS as the temperature is lowered. Each step can be either a crossover, a first order transition or a second order transition. 52 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

2 If the single ion enthalpies of the HS and LS states are finely balanced (small |∆H|/(|k1|δ )) then there is a first order, one step, incomplete transition between the Neel´ ordered phase and the HS phase (Figs. 2.5b-c and 2.16b-c). An incomplete one-step transition is even observed when ∆H is small and negative, Figs. 2.5b and 2.16b. This is remarkable as the single ion free energy, ∆G, favours the HS state at all temperatures. This transition is a truely collective effect driven by the system’s need to minimise the energy of the elastic interactions, which are strong in this regime, in order to minimise the total free energy of the system. In some cases the hysteresis is sufficiently broad that straightforwardly cooling the system does not achieve the true low temperature ground state. This is observed experimentally and has been called ‘hidden hysteresis’. [115] Nevertheless it may be possible to prepare the ground state either via the reverse LIESST effect or by applying and subsequently adiabatically releasing pressure. 2 For larger ∆G/(|k1|δ ) both steps are straightforwardly observable and show significant hysteresis 2 (Figs. 2.5e and 2.16e). The width of the hysteresis loops decrease as ∆G/(|k1|δ ) increases. The low temperature step always displays wider hysteresis than the high temperature step. On further increasing 2 ∆G/(|k1|δ ) first the high temperature step passes through the critical point and becomes second order, with the low temperature step remaining first order and hysteretic (Figs. 2.5f and 2.16f). As ∆H is further increased the lower temperature step passes through the critical point and both transitions become crossovers (Figs. 2.5g and 2.16g). There are no parameters for which the high temperature step is first order and the low temperature step is a second order (cf. Fig. 2.5a). For sufficiently 2 large ∆G/(|k1|δ ) the elastic interactions become unimportant and the temperature dependence of nHS begins to resemble a single crossover (Figs. 2.5h and 2.16h). The distinction between a second order 1 transition and a crossover regimes is much clearer in the heat capacity, Fig. 2.16, than in nHS.

In all of these cases, the phase with nHS ' 1/2 is Neel´ ordered. The Neel´ phase is found experimentally in many materials in the 1n24 [83–86, 88–90] and 1n02 [91–95] families, and also in supramolecular crystals [106,108] (see Table 2.1). Just as we report here, Neel´ ordering is observed experimentally both as the intermediate spin-state in two-step transitions, [83–85, 91–93, 95, 106] and as the low temperature phase in incomplete one-step transitions [83, 85, 91, 94].

Antiferroelastic interactions in the 1n14 family

One expects that for many 1n14 materials k1 < 0 and k2 > |k1|/2. This leads to antiferroelastic states with stripe order (Figs. 2.4b, 2.6 and S13). Apart from this important difference, the thermodynamic behaviour is extremely similar to that predicted for the 1n24 and 1n02 families (k1 > 0, k2 < 0; Figs.

2.5 and 2.16). Both the Neel´ and stripe phases have nHS = 1/2. In both cases we observe two lines of ﬁrst order transitions ending in two critical points (compare Fig. 2.6a to 2.5a). This gives rise to the same range of possible behaviours for individual materials (ﬁxed ∆H) as we found on the 1n24 family and 1n02 families: compare Figs. 2.5 and 2.16 to 2.6 and 2.18. In contrast to the purely ferroelastic case (k1 > 0 and k2 > 0) when one of the interactions is antiferroelastic the relative magnitudes k1 and

1 This is because nHS is a ﬁrst derivative of the free energy, whereas heat capacity is a second derivative of the free energy. Therefore, ∂nHS/∂T contains similar information to the heat capacity. 2.4. RESULTS AND DISCUSSION 53

Figure 2.6: (a) Typical phase diagram for the next nearest neighbour model of the 1n14 family with k1 < 0 and k2 > 0 (here k2 = 1.2|k1|; see also Fig. 2.17). (b-f) The fraction of high spins, nHS (see Fig. 2.18 for the corresponding heat capacities). Symbols have the same meanings as in Fig. 2.5. In contrast to the 1n24 and 1n02 families (Fig. 2.5) the antiferroelastic order is striped (Fig. 2.1b) rather than Neel´ (Fig. 2.1a). Otherwise the behaviours are extremely similar.

k2 has an important consequences for the phases diagram, most notably how stable the antiferroelastic phase is, compare Figs. 2.6 and 2.23. Stripe order has been observed in many 1n14 frameworks [60, 74–81] and also in supramolecular crystals [96–100, 107], see Table 2.1. Just as we ﬁnd in our calculations stripe ordering is found as the low temperature phase in an incomplete ﬁrst-order transition [60, 78, 79] and as the intermediate phase in two-step transitions [74–78, 96–100]. Neel´ ordering is commonly found as an intermediate spin-state in the 1n24 and 1n02 families while, stripe ordering is common in the 1n14 family. Our calculations therefore give a clear explanation for this structure-property relationship. In the 1n24 and 1n02 families one expects the k1 interactions to be strong and antiferroelastic (positive) as they are through-bond, but the k2 interactions to be weaker and possibly ferroelastic (negative) as they are through-space – this favours Neel´ order. Conversely, in a 1n14 family one expects the k1 interactions to be weaker and possibly ferroelastic (negative) as they are through-space, but the k2 interactions to be strong and antiferroelastic (positive) as they are through-bond – this favours stripe order. Phase diagrams for two-step transition have previously been reported for a phenomenological Ising-like model [109] and a Landau theory. [112] Many of the same qualitative features can be observed. However, our explanation of the structure-property relationship described above is new.

2.4.2 Third Nearest-Neighbour Interactions

The third nearest-neighbour interaction, k3, is through-bond in the 1n24 and 1n02 families, but through- space in the 1n14 family, Fig. 2.2. We therefore expect that it will be antiferroelastic (k3 > 0) for most materials in the 1n24 and 1n02 families but it may be ferroelastic (k3 < 0) for members of the 1n14 54 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.7: (a) Typical slice of the phase diagram for the third nearest neighbour model of the 1n24 and 1n02 families with k1 and k3 > 0, and k2 < 0 (here k2 = −0.9k1 and k3 = 0.5k1). (b-i) The fraction of high spins, nHS (see Fig. 2.19 for the corresponding heat capacities). For simplicity we only show the parallel tempering Monte Carlo predictions. Symbols have the same meanings as in Fig. 2.5.

family. Thus k3 could have signiﬁcantly different effects on the two different classes of materials. In supramolecular crystals for non-zero k1, k2 and k3 any one or any pair of elastic constants may be 2 negative so long as the lattice remains dynamically stable (J∞ = 4δ (k1 + 2k2 + 4k3) > 0). Interestingly, in addition to the phases that minimize the energy of any single elastic interaction, elastic frustration introduces additional phases into the zero-temperature phase diagram, Fig. 2.26. We ﬁnd 13 states that are thermodynamically stable at T = 0 in extended regions of parameter space: HS,

LS, Neel,´ stripe, CH,CL,DH,DL, E, G, RH,RL and S, cf. Fig. 2.1, where the subscript indicates the majority spin-state.

For materials in the 1n24 and 1n02 families one expects k1 > 0 and k3 > 0, and in many materials one may find k2 < 0. A typical slice of the finite temperature phase diagram for this parameter regime 2 is shown in Fig. 2.7a. This phase diagram has stable states consistent with plateaus at nHS = 1 (HS), 3 1 1 (DH), 2 (E or G, which are degenerate), 3 (DL), or 0 (LS). Here we have chosen an example where 1 the third nearest neighbour interaction has changed the order at nHS = 2 . But, note that, for other parameters we find similar phase diagrams where the Neel´ order remains (cf. Figs. 2.8d,g and 2.26). Considering lines of constant ∆H corresponding to individual materials, Figs. 2.7b-j and 2.19b-j, we observe a rich range of behaviours. For small ∆H we find incomplete one-, two-, and three-step transitions (Figs. 2.7b-d and 2.19b-d), where all the transitions are first order. Note that again we find incomplete transitions driven purely by the elastic interactions with ∆H < 0, i.e., when the single ion free energy favours HS ions at all temperatures. For moderate ∆H a complete four step 2.4. RESULTS AND DISCUSSION 55

Figure 2.8: Examples of the wide range of behaviours found in the third nearest neighbour model. Here we study parameters relevant to (a) the 1n14 family, (b-e) the 1n24 and 1n02 families and (a-j) supramolecular lattices. Only the parallel tempering Monte Carlo simulations are shown. The intermediate spin-state phases are labelled on the plots (cf. Fig. 2.1). The corresponding heat capacities are shown in Fig. 2.20. transition is observed as four first order transitions (Figs. 2.7e and 2.19e). As ∆H is increased the high temperature transitions successively become second order and then crossovers (Figs. 2.7e-j and 1 2.19e-j). Interestingly, for large ∆H the E or G phase (nHS = 2 ) is suppressed by thermal fluctuations and we see either one crossover and two phase transitions (one first and one second order; Figs. 2.7h and 2.19h) or a single crossover where the remains a sharp drop in nHS at low temperature, but no hysteresis (Figs. 2.7i and 2.19i). Eventually, for large ∆H the effects of the elastic interactions are negligible and there is a single smooth crossover from HS to LS (Figs. 2.7j and 2.19j). The increased complexity caused by these multistep transitions means that the heat capacity (Fig. 2.19) and ∂nHS/∂T are more sensitive probes of the number and location of the transitions/crossovers. Alternatively, one could employ the techniques that are directly sensitive to spatial symmetry breaking that accompanies the development of long-range spin-state order; for example, x-ray scattering (see Fig. 2.1). However, this requires monitoring the changes in the Bragg peaks as temperature varies [109,112] – and such experiments are not commonplace. While Fig. 2.7 is typical of the rich behaviour of the third nearest neighbour model, the behaviours displayed there are far from exhaustive of this model. In Figs. 2.8b-e (and Figs. 2.20b-e) we show some additional examples of two-, three-, and four-step transitions with k1 > 0, k2 < 0 and k3 > 0 (relevant to the 1n24 and 1n02 families). Importantly the antiferroelastic order can vary with the Neel´ 56 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS phase competing with the E and G phases at nHS = 1/2. D phases are often stable for nHS = 1/3 or 2/3 in this parameter regime. Although we only show a single temperature trace for each case we emphasise that for each case varying ∆H results in the same range of thermodynamic behaviours as we saw in Fig. 2.7, with incomplete transitions, first order transitions, second order transitions, and crossovers observed. Some of the phenomenology found here has been reported in experimental literature, for example

Clements et al. [86] reported a four-step transition with intermediate plateaus at nHS = 2/3, 1/2, and

1/3 in [Fe(bipytz)(Au(CN)2)2]·x(EtOH), which is in the 1n24 family. They demonstrated that the antiferroelastic order in these plateaus is DH,Neel,´ and DL respectively. This is precisely the behaviour shown in Fig. 2.8d (and Fig. 2.20d). Table 2.1 shows that similar agreement is found with many other experiments.

For the 1n14 family one expects k2 > 0 and, in many materials, k1 < 0 and k3 < 0. In this regime we observe only one- and two-step transitions. The latter with intermediate stripe order similar to the transitions reported in Fig. 2.6. However, one also expects that in some materials k1 > 0. In this regime we ﬁnd a four-step transition with intermediate nHS = 1/4 (CL), 1/2 (stripe) and 3/4 (CH) plateaus (Figs. 2.8a and 2.20a).

Many materials in the all families will have k1 > 0, k2 > 0, and k3 > 0. In this case the long-range strain (J∞) dominates over the elastic interactions. This allows only one-step transitions. For supramolecular crystals the only constraint on the parameters is that the lattice is stable (i.e., 2 that J∞ = 4δ (k1 + 2k2 + 4k3) > 0). This allows for an even greater range of possibilities. Some of these are demonstrated in Figs. 2.8a-j and 2.20a-j. We see four-step transitions with a variety of different antiferroelastic orders and fractions of HS in the intermediate plateaus. Similar to the previous 2 cases studied, varying ∆H/(|k1|δ ) can lead to incomplete transitions, ﬁrst order transitions, second order transitions and crossovers. Several of these behaviours have been reported in the experimental literature. For example, two step transitions with intermediate stripe phases have been reported by Hang et al. [96], Vieira et al. [97], Chernyshov et al. [98], Klingele et al. [99], Fitzpatrick et al. [100] and a two-step incomplete transition with a low temperature stripe and intermediate CH (nHS = 0.75) antiferroelastic states has been reported by Matasumoto et al. [107]. Overall the third nearest neighbour model suggests that longer range elastic interactions lead to a wider range of behaviours. This includes transitions with a greater number of steps and a greater variety of antiferroelastic order.

2.4.3 Longer Range Interactions

Fourth nearest neighbour interactions are through-space in the 1n14, 1n24 and 1n02 families, thus one might expect it to be weak in all classes of materials. However, k5 is through-space in the 1n24 and 1n02 families but through-bond in the 1n14 family. Thus, we expect k5 to be positive in 1n14 family materials. Therefore, to test our proposal that longer range interactions generically increase the number of plateaus in SCO materials we study our model with k4 = 0 and non-zero k1, k2, k3 and k5. 2.4. RESULTS AND DISCUSSION 57

Figure 2.9: Examples of the wide range of behaviours found in the ﬁfth nearest neighbour model. Here we have selected behaviours observed experimentally, see Table 2.1. The intermediate spin-state phases are labelled on the plots (cf. Fig. 2.1).

For materials in the 1n14 family one expects k2 > 0 and k5 > 0, but for many materials k1 < 0 2 1 and k3 < 0. This leads to stable states consistent with plateaus at nHS = 1 (HS), 3 (RH), 2 (stripe), 1 3 (RL), or 0 (LS), Fig. 2.28a. Other than the change in the antiferroelastic order the behaviours are extremely similar to those shown in Fig. 2.7 where we also find plateaus at the same HS fractions. The five phases are separated by four first order lines ending at critical points where a line of second order 2 transitions begin. Once again, for smaller values of ∆H/(|k1|δ ) the transitions are sharp and first order, broadening for increasing values until the transitions become second order and then crossovers (see Figs. 2.28b-j and 2.21b-j).

The inclusion of the k5 interaction also allows for a large number of possible antiferroelastic states and an extremely rich phase diagram, Fig. 2.27. At T = 0 we have identiﬁed 36 possible ground states that are stabilised in extended regions of the phase diagram, Fig. 2.1. For parameters relevant to

1n14 family (k2 > 0 and k5 > 0) the HS, LS, stripe (B), C, J, K, R, S, and I phases are predicted. For parameters relevant to 1n24 and 1n02 families (k1 > 0 and k3 > 0) the HS, LS, Neel´ (A), stripe (B), C, D, E, F, G, L, M, N, O, and P phases are found. Once again, this is consistent with experimental literature (Table 2.1), where K [82] and R [81] phases have been reported for materials in the 1n14 family; and D [85, 86] and F [87–90] phases have been observed in materials in the 1n24 family. A few examples of the possible dependence of nHS as one sweeps the temperature are given in Fig. 2.9. These examples have been selected for their relevance to the existing experimental literature, cf. Table 2.1, and are far from exhaustive. If anisotropy is added to the model – either via crystallographically distinct metal centers or through anisotropic interactions, e.g., allowing k1 to be different in the x and y directions – then even more phases are found. A particularly important example in the X phase, which consists of alternating stripes of width 3 and 1 in the vertical or horizontal direction, e.g. -LS-LS-LS-HS-LS-LS-LS-HS-. This is similar to the R phase (Fig. 2.1), which has stripes of width 2 and 1 in the vertical or horizontal direction (-LS-LS-HS-LS-LS-HS-). The fact that the X phase is observed in several members of the 1n24 family (Table 2.1) suggests that the interplay between anisotropy and elastic interactions may play an important role in determining which spin-state orders are observed. Thus, most of the phenomenology reported in experimental literature is thus reproduced in our calculations (see Table 2.1 and Fig. 2.9). For example four step transitions with FL (nHS = 0.25), A 58 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.10: (a) Typical slice of the ﬁnite temperature phase diagram for interactions up to ﬁfth nearest neighbours as expected in the 1n24 and 1n02 families with k1, k2 and k3 > 0, and k5 < 0 (here k2 = 0.3k1, k3 = 0.07k1 and k5 = −0.16k1). Lines and dots have the same meanings as in Fig. 2.5a. (b-f) The fraction of high spins, nHS (See Fig. 2.22 for the corresponding heat capacities). Only the parallel tempering predictions are shown.

(0.5), and FH (0.75) phases (Fig. 2.9a) have been reported in 1n24 frameworks, [88–90]; whereas in molecular crystals there have been observations of two step transitions with intermediate stripe (B; nHS = 0.5) and KH (0.75) phases [46, 102] (Fig. 2.9b), two step transitions with an intermediate S 1 2 (0.5) phase [104, 109] (Fig. 2.9c), and a three step transition with intermediate TL ( 3 ) and TH ( 3 ) phases [105] (Fig. 2.9d). The diversity of the phases, Fig. 2.10a, also leads to a very large numbers of steps. For example, in Fig. 2.10b (and Fig. 2.22b) we report an eight-step transition, for parameters relevant to 1n24 and 1n02 families. In the ﬁnal stages of preparing this manuscript we became aware of an eight step transition reported recently by Peng et al. [73] for a 1n24 material. 2 As in the cases described above, increasing ∆H/(|k1|δ ) increases the widths of the transitions, and eventually causes them to become second order and then crossovers. This starts with the highest 2 temperature transitions and works progressively down (see Figs. 2.10b-e). For large ∆H/(|k1|δ ) the crossover becomes extremely smooth as it passes through all nine phases (see Figs. 2.10f and 2.10f).

Thus, nHS, does not show any signiﬁcant differences from a trivial single crossover. However, clear signatures are still observed in the cV and ∂nHS/∂T, see Fig. 2.10. Unless the heat capacity, ∂nHS/∂T, or temperature dependent x-ray scattering are measured then this highly-multi-step crossover could be broadly dismissed as a single broad crossover. This means that extremely multistep SCO crossovers may hidden in plain sight in the literature. It is interesting to note that while, in this two dimensional model increasing the range of interactions on these two-dimensional lattices leads to transitions with an increasing numbers of steps, in the one dimensional Ising model only next nearest neighbour interactions are required to see an inﬁnite number of steps (Devil’s staircase) [116,117]. In contrast one does not expect the model in three dimensions to be dramatically different from the two dimensional model studied here, although that may be an interesting avenue for future study as it would allow a full 2.5. CONCLUSIONS 59 understanding of the interlayer ordering in SCO materials.

2.5 Conclusions

The simple elastic model explored here hosts a rich variety of SCO transitions and intermediate spin-state phases. Almost all of the spin-state ordered phases reported experimentally in square lattice SCO materials are found in our calculations, see Table 2.1. However, we also predict new orderings that have not yet been observed, see Fig. 2.1. We have established clear structure-property relations for the 1n14, 1n24 and 1n02 families of SCO materials. The key point is that through-bond elastic interactions are generically antiferroelastic (described by positive spring constants) whereas through-space elastic interactions can often be ferroelastic (negative spring constants). This provides a natural explanation for the different antiferroelastic phases found in different families of SCO frameworks. In general, increasing the range of interactions results in an increase in the number of observed transitions and spin-state phases. Therefore, the rigidity of framework materials may explain why they are such a rich playground for multistep transitions and antiferroelastic order. Our results show that multistep transitions and complex patterns of antiferroelastic order on lattices with initially equivalent metal ions can be understood in terms of the competition and cooperation between through-bond and through-space interactions. The range of the elastic interactions is a vital criteria for understanding collective phenomena in SCO materials. Our results show that strong through-space interactions are the key requirement for multi-step transitions on lattices with initially equivalent metal ions. This explains why and how the presence of guest molecules, solvent atoms and anions [51] can strongly modify collective SCO behaviours – through hydrogen bonding, aromatic π-π bonding and van der Waals interactions. The inclusion of up to ﬁfth nearest neighbour elastic interactions allows for eight-step transitions for parameters relevant to the 1n24 and 1n02 families. This suggests that transitions with larger numbers of steps could be obtained more readily in the 1n24 and 1n02 families than in the 1n14 family. Due to the abundance of experimental literature on the 1n24, 1n02 and 1n04 families of frameworks we have specialised, in this chapter, to studying the square lattice. However, the same theory presented here is applicable to understand SCO materials on any lattice. [2, 3] Further insight into these trends could be gained by parametrising our model for speciﬁc materials. However, this remains a major challenge. [118] 60 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS 2.6 Supplementary Information

2.6.1 Expansion of the potential

In the main text (Eq. 2.4) we consider a pairwise potential between neighbouring metal cites that depends on the spin spin-states of the metal ions, σi and σ j. This potential is given by

1 2 V (r,σ ,σ ) =g (r) + h (r)r − η R − δ(σ + σ ) + k (r)r − η R − δ(σ + σ ) ,(2.12) i j i j i j i j i j i j 2 i j i j i j

√ √ √ where ηi j = ηn = 1, 2 ,2, 5 ,2 2 ,... is the ratio of distances between the nth and 1st nearest- neighbour distance on the undistorted square lattice.

We interpolate Vi j(r,σi,σ j) by introducing a new function V˜i j(r − r0) deﬁned such that V˜i j(r − rHηi j) = Vi j(r,1,1), V˜i j(r − rLηi j) = Vi j(r,−1,−1), and V˜i j(r − Rηi j) = V˜i j(r,1,−1) = Vi j(r,−1,1). Interpolating yields

˜ ˜ ! Vi j(r − Rηi j + 2δηi j) −Vi j(r − Rηi j − 2δηi j) gi j(r) =V˜i j(r − Rηi j) − r − Rηi j 4δηi j (2.13) ˜ ˜ ˜ ! 1 2 Vi j(r − Rηi j + 2δηi j) − 2Vi j(r − Rηi j) +Vi j(r − Rηi j − 2δηi j) + r − Rηi j 2 , 2 (2δηi j)

! V˜i j(r − Rηi j + 2δηi j) −V˜i j(r − Rηi j − 2δηi j) hi j(r) = 4δηi j (2.14) V˜i j(r − Rηi j + 2δηi j) − 2V˜i j(r − Rηi j) +V˜i j(r − Rηi j − 2δηi j) − (r − Rηi j) 2 , (2δηi j) and

V˜i j(r − Rηi j + 2δηi j) − 2V˜i j(r − Rηi j) +V˜i j(r − Rηi j − 2δηi j) ki j(r) = 2 . (2.15) (2δηi j)

Noting that

∞ ( )n ˜ ˜ (n) 2δηi j V r − Rηi j + 2δηi j = ∑ V r − Rηi j , (2.16) n=0 n! where n (n) ∂ V(r) V (x) ≡ n , (2.17) ∂r r=x 2.6. SUPPLEMENTARY INFORMATION 61 we ﬁnd, with no further approximation, that

∞ ( )2n ˜ 2δηi j ˜ (2n+1) gi j(r) = Vi j(r − Rηi j) − r − Rηi j ∑ Vi j r − Rηi j n=0 (2n + 1)! ∞ ( )2(n−1) 2 2δηi j ˜ (2n) + r − Rηi j ∑ Vi j r − Rηi j (2.18) n=1 (2n)! 2 r − Rηi j = V˜ (r − Rη ) − r − Rη V˜ (1) r − Rη + V˜ (2) r − Rη i j i j i j i j i j 2 i j i j 2 2 2(δηi j) 2 (δηi j) −r − Rη V˜ (3) r − Rη + r − Rη V˜ (4) r − Rη i j 3 i j i j i j 6 i j i j +··· (2.19) ∞ ( )2n 2δηi j ˜ (2n+1) hi j(r) = ∑ Vi j r − Rηi j n=0 (2n + 1)! ∞ ( )2(n−1) 2δηi j ˜ (2n) −2(r − Rηi j) ∑ Vi j r − Rηi j , (2.20) n=1 (2n)! 2 2(δηi j) = V˜ (1) r − Rη − (r − Rη )V˜ (2) r − Rη + V˜ (3) r − Rη i j i j i j i j i j 3 i j i j 2 (δηi j) −(r − Rη ) V˜ (4) r − Rη + ··· (2.21) i j 3 i j i j ∞ ( )2(n−1) 2δηi j ˜ (2n) ki j(r) = 2 ∑ Vi j r − Rηi j (2.22) n=1 (2n)! 2 (δηi j) = V˜ (2) r − Rη + V˜ (4) r − Rη + ··· (2.23) i j i j 3 i j i j

To reach Eqs. 2.4 and 2.5a-2.5c of main text we set fi j(r) = gi j(r) + hi j(r)(r − Rηi j).

To evaluate gi j(r), hi j(r) and ki j(r) we note that V˜i j(r − Rηi j) can be expanded as a function of the potential ∞ (r − R )n ˜ (n) ηi j Vi j(r − Rηi j) = ∑ V (Rηi j) , (2.24) n=0 n! and close to the equilibrium, r ≈ r0, we can make the approximation r − Rηi j ≈ 2δηi j σi + σ j .

Consequently, to leading order in δηi j the only terms that survive are

fi j = Vi j(Rηi j), (2.25a) (1) hi j = Vi j Rηi j , (2.25b) (2) ki j = Vi j Rηi j . (2.25c)

2.6.2 Additional results

In Figs. 2.24a, 2.25a, 2.5a, and 2.6a we show phase diagrams with lines indicating the limits of metastability on heating and cooling at ﬁxed ∆H. The calculations which these lines are based on are shown in (Fig. 2.11, 2.13, 2.15 and 2.17). In Figs. 2.24b-f, 2.25b-d, 2.5b-h, 2.6b-h, 2.7b-j, 2.8a-j, and 2.10b-f of the main text we report the fraction of high spins as temperature varies, the corresponding heat capacities, which provide a more 62 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.11: The HS fraction, nHS, calculated on (a) cooling and (b) heating for the nearest neighbour square lattice model with k1 > 0. Lines and dots have the same meanings as in Fig. 2.12a, where we show the full phase diagram.

Figure 2.12: (a) The phase diagram for nearest-neighbour interactions, k1 > 0. The colours indicate the equilibrium values for the fraction of high-spin metal centres, nHS, calculated via parallel tempering. We find a (black) line of first order transitions that bifurcates at a triple point and ends in two critical points (black dots). There is no spontaneously broken symmetry or long-range order in the region between the two first order lines where nHS ' 1/2. The blue (red) dashed line marks the limit of metastability for on the HS (resp. LS) phases on cooling (resp. heating), see Fig. 2.11. Individual materials have fixed ∆H (not fixed ∆G); the white lines are lines of constant ∆H and their labels correspond to panels (b-f) where the heat capacity, cV , is plotted along these lines (see Fig. 2.24 for the corresponding high spin fractions). In these plots the blue, red and black lines represent the cooling, heating and equilibrium values respectively. sensitive signature of phase transitions and crossovers in SCO materials, are shown in Fig. 2.12b-f, 2.14b-d, 2.16b-h, 2.18b-h, 2.19b-j, 2.20a-j, and 2.22b-f respectively. Lastly, in the main text we show the phase diagram for the next nearest neighbour 1n14 family lattice model with k1 < 1, k2 = 1.2|k1| (Fig. 2.6, see also Fig. 2.18). In Fig. 2.23 we show phase diagram for the nearest neighbour model with k1 < 1, k2 = 0.6|k1| demonstrating the importance of the relative contributions of the magnitudes of k1 and k2 on the thermodynamics.

Nearest-neighbour interactions

In this section, we consider only the nearest-neighbour elastic interaction, k1. That is, we set k2 = k3 = k4 = k5 = 0. The stability of the lattice requires k1 > 0. [177] The phase diagram of this model 2.6. SUPPLEMENTARY INFORMATION 63

Figure 2.13: The HS fraction, nHS, calculated on (a) cooling and (b) heating for the next nearest neighbour square lattice model with k2 = 0.1k1 > 0. Lines and dots have the same meanings as in Fig. 2.12a. See Fig. 2.14a for the full phase diagram.

Figure 2.14: (a) Phase diagram for the next nearest neighbour square lattice model with k2 = 0.1k1 > 0 reprinted from Fig. 2.25a for convenience. Lines and dots have the same meanings as in Fig. 2.12a. (b-d) The heat capacity, cV , (see Fig. 2.25 for the corresponding HS fractions). Blue, red and black lines show data for the cooling, heating and thermal equilibrium predictions, respectively.

Figure 2.15: The HS fraction, nHS, calculated on (a) cooling and (b) heating for the next nearest neighbour square lattice model with k1 > 0 and k2 = −0.2k1, appropriate for the 1n24 and 1n02 families. Lines and dots have the same meanings as in Fig. 2.12a. See Fig. 2.16a for the full phase diagram. 64 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.16: (a) Phase diagram for the next nearest neighbour square lattice model with k1 > 0 and k2 = −0.2k1, appropriate for the 1n24 and 1n02 families, reprinted from Fig. 2.5a for convenience. Lines and dots have the same meanings as in Fig. 2.12a. (b-h) The heat capacity, cV , (see Fig. 2.5 for the corresponding HS fractions). Blue, red and black lines show data for the cooling, heating and thermal equilibrium predictions, respectively.

Figure 2.17: The HS fraction, nHS, calculated on (a) cooling and (b) heating for the next nearest neighbour square lattice model with k1 < 0 and k2 = 1.2|k1|, appropriate for the 1n14 family. Lines and dots have the same meanings as in Fig. 2.12a. See Fig. 2.18a for the full phase diagram.

Figure 2.18: (a) Phase diagram for the next nearest neighbour square lattice model with k1 < 0 and k2 = 1.2|k1|, appropriate for the 1n14 family, reprinted from Fig. 2.6a for convenience. Lines and dots have the same meanings as in Fig. 2.12a. (b-h) The heat capacity, cV , (see Fig. 2.6 for the corresponding HS fractions). Blue, red and black lines show data for the cooling, heating and thermal equilibrium predictions, respectively. 2.6. SUPPLEMENTARY INFORMATION 65

Figure 2.19: (a) Phase diagram for the third nearest neighbour square lattice model with k1 > 0, k2 = −0.9k1 and k3 = 0.5k1, appropriate for the 1n24 and 1n02 families, reprinted from Fig. 2.7a for convenience. Lines and dots have the same meanings as in Fig. 2.12a. (b-j) The heat capacity, cV , (see Fig. 2.7 for the corresponding HS fractions). For simplicity only the parallel tempering results are shown.

Figure 2.20: Temperature dependence of the heat capacity in the third nearest neighbour model with the same parameters for which nHS is plotted in Fig. 2.8. The parameters are reasonable for (a) 1n14 family, (b-e) 1n24 and 1n02 families and (a-j) supramolecular lattices. For simplicity we only show the parallel tempering results. 66 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.21: (a) Phase diagram for the ﬁfth nearest neighbour square lattice model with k1 < 0, k2 = 1.2|k1|, k3 = −0.5|k1|, k4 = 0, and k5 = 0.2|k1|, appropriate for the 1n14 family, reprinted from Fig. 2.28a for convenience. Lines and dots have the same meanings as in Fig. 2.12a. (b-j) The heat capacity, cV , (see Fig. 2.28 for the corresponding HS fractions). For simplicity only the parallel tempering results are shown.

Figure 2.22: (a) Typical slice of the ﬁnite temperature phase diagram all interactions up to ﬁfth nearest neighbours. Lines and dots have the same meanings as in Fig. 2.12a. (b-f) The heat capacity, cV (See Fig. 2.10 for the corresponding fraction of high spins). For simplicity only the parallel tempering predictions are shown. 2.6. SUPPLEMENTARY INFORMATION 67

Figure 2.23: (a) Phase diagrams for the next nearest neighbour model with k1 < 0, k2 > 0 (here we take k2 = 0.6|k1|, other parameters give similar results). Lines and dots have the same meanings as in Fig. 2.12a. The HS fractions calculated on (b) heating and (c) cooling with ﬁxed ∆H are used calculate the limits of stability. Note that in contrast to the results for k1 > 0 and k2 > 0 the relative magnitudes of k1 and k2 has importance consequences for the observed behaviours (see, e.g., Figs. 2.6 and 2.18). (d-j) The fraction of high spins, nHS and (k-q) the corresponding heat capacities. Blue, red and black lines show data for the cooling, heating and thermal equilibrium predictions, respectively. 68 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.24: (a) The phase diagram for nearest-neighbour interactions, k1 > 0. Colours in the phase diagram indicate the fraction of high-spin M sites, nHS ∼ χT where χ is the susceptibility, calculated via parallel tempering. The (black) line of first order transitions bifurcates at a triple point and ends in two critical points (black dots). There is no spontaneously broken symmetry or long-range order in the region between the two first order lines where nHS ' 1/2. The blue (red) dashed line marks the limit of metastability for the HS (resp. LS) phases on cooling (resp. heating), cf. Fig. 2.12, and show the width of the hysteresis. Individual materials have fixed ∆H, white lines correspond to panels (b-f), where the fraction of high spins is plotted (see Fig. 2.11 for the corresponding heat capacities). In these plots the blue, red and black lines represent the cooling, heating and parallel tempering values respectively. is shown in Fig. 2.24a. As individual materials have constant ∆H, rather than constant ∆G, we mark lines of constant ∆H on the phase diagram and report the thermodynamic properties at selected values of ∆H (Figs. 2.24b-f). When the single molecule contribution to the free energy is much greater than the contribution 2 of the cooperative elastic interactions, ∆H k1δ , we observe a gradual crossover (Figs. 2.24f and 2.12f). This thermodynamic behaviour is commonly reported in experiments on weakly cooperative materials. [44] 2 In the opposite regime, ∆H k1δ , we see a sharp one-step transition with hysteresis (Figs. 2.24b and 2.12b). This indicates a first order phase transition, as is commonly observed in bistable SCO materials. [44] Note that in our calculations the first order spin-state transition is not accompanied by a crystallographic phase transition, which is excluded by the symmetric breathing mode approximation. 2 In the intermediate regime, ∆H/(|k1|δ ) ∼ 15−30, we see two-steps. Depending on the magnitude 2 of ∆H/(|k1|δ ) these steps can occur as two first-order transitions with hysteresis (Figs. 2.24c and 2.12c), one first order transition and one crossover (Figs. 2.24d and 2.12d), or two crossovers (Figs. 2.24e and 2.12e). In between these regimes are critical points where the transitions become continuous. The two-step behaviour is a result of the competition between the long-range strain and the elastic interactions. The elastic interactions favour an antiferroelastic phase with Neel´ ordering (Fig. 2.1a), whereas the long-range strain prefers all metal centres to be in the same spin-state. At T = 0 when ∆H = ∆G > 0 the LS phase is realized and for ∆H = ∆G < 0 the HS phase is energetically favourable. However, for T = 0 and ∆H = 0 the Neel´ ordered state is degenerate with the HS and LS states (Figs. 2.6. SUPPLEMENTARY INFORMATION 69

Figure 2.25: (a) Typical phase diagram for the next nearest neighbour square lattice model with k1 and k2 > 0 (here k2 = 0.1k1; see also Fig. 2.13). (b-d) The fraction of high spins, nHS (see Fig. 2.14 for the corresponding heat capacities). Symbols have the same meanings as in Fig. 2.24.

2.24a and 2.12a). At sufﬁciently high temperatures thermal ﬂuctuations stabilize short-range Neel´ correlations for ∆G ≈ 0, Figs. 2.24a and 2.12a. This cannot stabilise a true, long-range ordered, Neel´ phase, but does result in an observed two-step transition, Figs. 2.24d and 2.12d. The same qualitative features of this phase diagram have been also been captured by Chenyshov et al. [112] using a Landau theory. However, with only nearest neighbour interactions present, the Neel´ phase is the only antiferroelastic phase present. Furthermore, the Neel´ phase is observed only in an extremely narrow temperature range. Experimentally, many different antiferroelastic phases have been found and these phases can be stable over relatively broad temperature ranges. This is consistent with the idea that longer range elastic interactions are vitally important for multistep transitions.

Strictly positive next nearest neighbour interactions (any family) k1 > 0 and k2 > 0 is possible for any of the lattices shown in Fig. 2.2 provided that the minima of both interactions are roughly commensurate with a square lattice. At T = 0 the LS and HS states are separated by a first order phase transition at ∆G = 0, Figs. 2.4a and 2.25a. For T > 0 this phase transition remains at ∆G = 0 until it reaches a critical point, where the transition is continuous. Thus, the frustration has entirely suppressed the Neel´ order found at finite temperatures for k2 = 0 (compare Figs. 2.24 and 2.25). Considering lines of constant ∆H, which represent individual materials, we see three distinct thermodynamic behaviours: When the elastic interactions are strong compared to the single molecule 2 physics, ∆H |k1|δ , the transitions are sharp and first-order, the width of the hysteresis is greater the 2 2 smaller ∆H/|k1|δ is (Figs. 2.25b,c and 2.14b,c). If the elastic interactions are weak, ∆H |k1|δ , we find a crossover (Figs. 2.25d and 2.14d). These two regimes are separated by a continuous (or second order) phase transition at the critical point. Consistent with this prediction, single step transitions are common and observed in all of the families of materials discussed here. 70 CHAPTER 2. STRUCTURE–PROPERTY RELATIONSHIPS

Figure 2.26: Slices of the zero-temperature phase diagram with up to third nearest-neighbour elastic interactions k1, k2 and k3 for (a,e) constant ∆H and (b-d,f-h) constant J∞. Grey lines in (a,e) indicate lines of constant J∞. In addition to the phases that are energetically preferred by individual elastic interactions (HS, LS, Neel,´ stripe, C, E and G), frustration introduces additional phases into the zero-temperature phase diagram (D, R and S). E, G indicates that the E and G phases are degenerate.

Figure 2.27: Selected slices of the zero temperature phase diagram with up to ﬁfth nearest-neighbour interactions showing thirty-six distinct phases. In general increasing the number of nearest-neighbour interactions increases the complexity of the phase diagram. 2.6. SUPPLEMENTARY INFORMATION 71

Figure 2.28: (a) Typical phase diagram for the 1n14 family with k1 < 0, k2 > 0, k3 < 0, k4 = 0 and k5 > 0 (here k2 = 1.2|k1|, k3 = −0.5|k1| and k5 = 0.2|k1|). The R phase consists of alternating stripes of width 2 and width 1 (Fig. 2.1). Lines and dots have the same meanings as in Fig. 2.12a. (b-i) The fraction of high spins, nHS (see Fig. 2.21 for the corresponding heat capacities). Only the parallel tempering Monte Carlo predictions are reported. The following publication has been incorporated as Chapter 3. For ease of reading ﬁgures have been restructured to ﬁt the thesis format. 1. [2] Jace Cruddas, and Ben J. Powell, Spin-State Ice in Elastically Frustrated Spin-Crossover Materials, J. Am. Chem. Soc. 141, 19790 (2019).

Contributor Statement of contribution % Jace Cruddas writing of text 50 proof-reading 50 theoretical derivations 90 numerical calculations 100 preparation of ﬁgures 90 initial concept 0 Ben J. Powell writing of text 50 proof-reading 50 theoretical derivations 10 numerical calculations 0 preparation of ﬁgures 10 initial concept 100 supervision, guidance 100

Powell made Figures 3.3a-d. Chapter 3

Spin-State Ice in Elastically Frustrated Spin-Crossover Materials

Molecules with bistable spin states are widely studied because of their importance to the natural world and their potential applications as molecular scale switches. In molecular crystals and framework materials elastic interactions between molecules lead to collective phenomena including hysteresis, multistep transitions, and antiferroelastic order of spin-states. Elastic frustration, the inability to simultaneously minimize competing elastic interactions, plays a key role in many of the most important phenomena in spin crossover materials. Here we use an elastic model to predict that a new phase of matter occurs for bistable molecules on the kagome lattice, which is intrinsically frustrated as it is composed of equilateral triangles. In this phase, which we call ‘spin-state ice’ in analogy to water and spin ices, there is no long-range order of spin-states; instead they follow a local ‘ice rule’ that each triangle must contain two metal centers in one spin-state and one in the other. We show that spin-state ice supports mobile collective excitations that carry a spin midway between the two spin-states of a single metal center but no electrical charge. We show that there are distinctive signatures of spin-state ice in neutron scattering, electron paramagnetic resonance and thermodynamic experiments.

3.1 Introduction

The spin crossover (SCO) of transition metal ions in complexes and frameworks is a prominent route 4 2 to smart molecular materials. [42, 43] SCO, for example between the high-spin (HS; t2geg) to low-spin 6 0 (LS; t2geg) states of Fe(II), provides reversible molecular switches that are addressable by multiple stimuli including temperature, pressure, light irradiation, magnetic field, and chemical environment. Furthermore, SCO materials are multifunctional as spin transitions are accompanied by dramatic changes in the color, volume (up to 15 %), and the magnetic, electrical, and mechanical properties of the materials. [42, 43] This remarkable chemistry has important implications in the natural world from geology [119, 120] to biology. [121] Furthermore, SCO materials have many potential uses including high-density 73 74 CHAPTER 3. SPIN-STATE ICE reversible memories, actuators, ultrafast nanoscale switches, thermometers, barometers, and in displays. [42,43,52–58,122] However, these applications require the engineering of cooperative behaviors in the solid state; the underlying mechanisms of which are still being identified and understood. For example, a key design criteria for applications of bistable molecules in digital electronics is a wide hysteresis at room temperature. [123] Designing materials with this property requires understanding and control of the interactions between molecules and the corresponding structure-property relationships. Multistep phase transitions, which open up the possibility of multiple memory states, depend crucially on the competition between elastic interactions [72]. From a fundamental perspective the existence of long range antiferroelastic ordered states [60, 86, 124, 125] naturally leads one to ask what other collective phenomena might occur in spin crossover materials? A wide variety of empirical and semi-empirical models have been invoked to explain the full range of cooperative phenomena observed experimentally in spin crossover materials, particularly multistep transitions and antiferroelastic ordering. [65] Initially an entirely empirical approach based on Ising-like models was explored. But recently more realistic elastic models have been introduced. These take into account the volume change associated with a change in the spin-state [44] and its interplay with the elastic interactions between metal centers. [71, 72, 126–135] However, such models require the simultaneous calculation of the molecular dynamics and spin-state configuration. Water ice is a far more familiar material, yet it is equally remarkable. The enthalpy of ice (in the Ih phase, commonly found in the biosphere) is minimized if every oxygen atom forms covalent bonds with two hydrogen atoms and hydrogen bonds with another two, Figure 3.1a. This ‘ice rule’ [15] arises from the mismatch between the local chemistry of H2O and the crystal symmetry of ice. Remarkably, Pauling [16] predicted that the protons lack any long-range order. This was later confirmed experimentally. [136] Furthermore, the number of ways of arranging the hydrogen atoms that satisfy the ice rule grows exponentially with the size of the system. This means that the proton disorder contributes a residual or zero-point entropy which Pauling estimated to be S0 = kB ln(3/2) (per formula unit), very close to the best calculations to date. [33] These calculations predict that even if the temperature were absolute zero the entropy would grow with the system size. In other words, the ‘macroscopic degeneracy’ due to the very large number of states with the same enthalpy leads to a violation of the third law of thermodynamics! Nature sidesteps this problem – ice displays many crystallographic phases and neither the Ih phase nor the Ic phase (which is also governed by the ice rule and has similar properties) are thermodynamically stable at the lowest temperatures. [137–139] Similar behaviors have recently been discovered in frustrated magnets known as spin ices. [33, 39] 3+ 3+ The canonical examples are Ho2Ti2O7 and Dy2Ti2O7. In these materials the magnetic Ho or Dy ions are the vertices of a ‘pyrochlore’ lattice, which consists of corner-sharing tetrahedra, Figure 3.1b. The crystal field due to the oxide ion at the center of each tetrahedron modifies the quantum ground state of the magnetic ions so that they are HS and their magnetic moment (spin) is aligned to the h111i crystallographic axis, which connects the centers of neighboring tetrahedra. The interactions between the spins are minimized when every tetrahedron obeys the ice rule that two spins point in, and two spins point out, Figure 3.1b. Comparison of the water and spin ice rules show that they are highly 3.1. INTRODUCTION 75

Figure 3.1: Ice rules in (a) water ice, (b) spin ice, (c) low-spin spin-state ice (LSSI), and (d) high-spin spin-state ice (HSSI). (a) The water ice rule is that each oxygen forms covalent bonds with two hydrogens atoms and hydrogen bonds with another two. Oxygen atoms are represented by blue-gray spheres and hydrogen atoms are colored by purple or green. The blue-gray tetrahedra are constructed by connecting the midpoints between neighboring oxygen atoms. Hydrogens inside the four exterior tetrahedra are colored purple; these atoms form covalent bonds with the four outer oxygen atoms and hydrogen bonds and with either the central oxygen or oxygens that are not shown. The green circles represent hydrogens in the central tetrahedron or tetrahedra that are not shown. These atoms form covalent bonds with either the central oxygen or oxygens that are not shown and hydrogen bonds with the four outer oxygen atoms. (b) In the spin ice ground states two spins point into each tetrahedron and the other two point out. Spins pointing into the four exterior tetrahedra are colored purple, whereas spins pointing out of the four exterior tetrahedra are colored green. Notice that if one were to replace the heads of the arrows by spheres of the same color the arrangement would be identical to that of the hydrogens in panel (a). Thus the color highlights the essential similarity between water and spin ices. (c) In LSSI each triangle of the kagome lattice contains two low-spin (LS; purple) metal ions and one high-spin (HS; green) ion. (d) In HSSI each triangle of the kagome lattice contains two HS ions and one LS ion. In each case there are only three possible arrangements on any triangle. Extended regions of both spin-state ices are shown in Figure 3.5.

analogous, Figure 3.1. This is reﬂected in the similarity of the two materials’ properties. Most notably, the Pauling estimate of the residual entropy of spin ice is also S0 = kB ln(3/2) (per formula unit). But unlike water ice no phase transitions have been observed in Ho2Ti2O7 or Dy2Ti2O7 and how, or indeed whether, the third law is obeyed in the bulk spin ice remains an open question. [140].

Interest in spin ices goes beyond the analogy to water ice. In the vacuum one observes dipoles, quadrupoles, and so on of both the electric and magnetic fields. But, while we see electric monopoles, corresponding to the locations of electrically charged particles, there are no magnetic monopoles or magnetic charges. If one cuts a bar magnet in half one has two bar magnets, not separate north and south poles. Yet monopoles are observed in spin ices. [7] These do not arise from any local excitation, but rather from collective excitations of the material, a phenomenon known as ‘fractionalization’. [141] In fact the emergence of magnetic monopoles in spin ice is perhaps the simplest example of fractionalization. It can be understood as follows (see Figure 3.2a-d, where we illustrate the process for honeycomb spin ice): flipping a single spin makes a dipole and two local violations of the ice rule, known as defects (Figure 3.2b). Flipping a neighboring spin moves the defects apart and lengthens the dipole (Figure 3.2c). Continuing this process allows the two defects to move independently through the crystal (Figure 3.2d) – so instead of behaving as the two ends of a dipole they behave as independent 76 CHAPTER 3. SPIN-STATE ICE monopoles. [7] The macroscopic degeneracy of the ground state of spin ice can be described in terms of an ‘emergent gauge field theory’. [18] This may sound like abstract theoretical statement, but it has direct and important experimental consequences. Gauge field theories are the mathematical framework for describing the fundamental forces of nature, including electromagnetism and gravity. Gauge theories are characterized by variables that cannot be measured directly and that can be changed in specific well-defined ways at any point in space, without changing the predictions of the theory. [9] In ices the many ways of obeying the ice rules give the system just this kind of local or gauge symmetry. The fact that the ground state of spin ice can be described by a gauge field theory tells us that (almost) all of the degenerate ground states have the same thermodynamic properties. The gauge field theory of spin ice is emergent in the sense that it is not a symmetry of the full Schrodinger¨ equation for the electrons and nuclei that make up the material. Processes involving high energy excitations will defy the emergent gauge symmetry, but all of the low-energy processes respect the symmetry. [9] It has been shown that the emergent gauge field theory in spin ice is mathematically equivalent to electromagnetism and supports low energy excitations that behave like photons. [18,142] An important open question is do long wavelength quantum fluctuations pick out a ground state or form a quantum spin liquid. [18, 141, 142] Remarkably, the emergent gauge field in spin ice can be observed experimentally. In scattering experiments one measures a structure factor, which quantifies correlations within the material studied. For example, x-ray scattering experiments can measure the (position) structure factor, which quantifies the correlations between the positions of the atoms. In a crystal the positions are highly correlated, leading to Bragg peaks in the structure factor. But measurements of the structure factor, which quantifies diffuse scattering, also provide vital information about the behaviors of disordered materials, such as liquids and glasses. Similarly neutron scattering experiments can measure the spin structure factor. Magnetic order leads to Bragg peaks in the spin structure factor. (See, e.g., Chaikin and Lubensky [143] for a more detailed introduction to scattering experiments and structure factors.) Again, in disordered spin systems, including spin ices, diffuse scattering leads to distinct signatures in the spin structure factor that provide crucial information about the correlations between the spins. In spin ices so-called pinch points are observed [24, 40] in spin structure factor [18, 39] – they are straightforward to predict from the emergent gauge field theory and are thus regarded as the smoking gun evidence for the emergent gauge field and indeed ice behavior. Two dimensional models that display the ice-like behaviors are important in applied mathematics, where are usually known as vertex models. [17, 32] There has been recent experimental progress towards realizing these models in arrays of ferromagnetic nanostructures, which behave as giant spins, and are consequently known as artificial spin ices. [25] Most work has focused on the so-called square and honeycomb artificial spin ices. However, one should note that in this nomenclature the spins are located on the bonds rather than the sites of the lattice as is more usual (e.g., in graphene the carbon atoms are at the sites of the honeycomb lattice). In the present context it is clearer to use the latter convention: in square artificial spin ice the spins lie at the vertices of the checkerboard lattice; 3.1. INTRODUCTION 77

Figure 3.2: Propagation of magnetic monopoles in spin ice (a-d), fractionalized spins in spin-state ice (e-h) and defects in a dimer model (i-l). (a) A state obeying the honeycomb spin ‘ice II’ rule [25]: every site of one sub-lattice (marked by navy blue dots) has one spin pointing in and two spins pointing out; whereas every site on the other sub-lattice (pale blue dots) has two spins pointing in and one spin pointing out. (b) Flipping one spin (highlighted orange) creates a dipole – two sites (highlighted blue) that have three spins pointing out (navy sublattice) or in (pale blue sublattice). (c) Flipping a second spin (also highlighted orange) moves the end of the dipoles further apart. Note that when the end of the dipole moved to the other sublattice it changed from three-in to two-in, one-out, which we highlight by coloring the site red. Both of these states have one more spin pointing in than the ice rule state shown in (a), and so carry the same magnetic ‘charge’. (d) Flipping more spins (all highlighted orange) moves the end of the dipoles further apart. It can be shown [25, 39] that only a finite amount of energy is required to move the two ends of the dipole arbitrarily far apart (similar to the finite ionization energy of an atom or molecule). This means that the two ends are independent particles (formally they are deconfined), known as magnetic monopoles. (e) A section of kagome lattice obeying the LSSI rule: on every triangle the are one HS (green) and two LS (purple) metal centers. (f) Changing the spin-state of one metal center (ringed black) creates two triangles with three LS metals (highlighted blue). (g) Changing the spin-state of another metal center (also ringed black) restores the ice rule on one of those triangles, but makes another triangle with two HS and one LS metals (highlighted in red). (h) Further changes in the spin states (all ringed black) allow the two defects to move independently. As the defect hops between sublattices they change between three LS (blue) and two HS, one LS (red). Nevertheless the topological charge (defined in equation 3.6) is conserved. (i) A hardcore dimer covering: each vertex has a single dimer (solid black line) connecting it to an adjacent vertex. (j) Removing a dimer (highlighted orange) creates two defects (zero dimers per vertex) shown by blue circles. (k) Adding a dimer (highlighted orange) on an adjacent vertex enforces the hardcore dimer rules on one vertex and creates a defect (two dimers per vertex; shown by a red circle) on another; destroying one defect and creating another. (l) The further addition/removal of dimers (all highlighted orange) can propagate the defects. Note that, all of the kagome spin-state ice states can be mapped to the honeycomb spin ice state and honeycomb dimer models by the mappings described in Figures 3.7 and 3.8. 78 CHAPTER 3. SPIN-STATE ICE and in honeycomb artificial spin ice the spins sit on the vertices of the kagome lattice [see Figure 3.7]. The great advantage of artificial spin ices is that the magnetic nanostructures are large enough for their magnetic moments to be directly imaged via magnetic force microscopy. This enables the full classical state of the system to be imaged and its dynamics to be studied in real time. However, achieving thermal equilibrium in these systems is challenging. [25, 144, 145] Furthermore, the square lattice displays long-range magnetic order at room temperature in thermal equilibrium. [25, 144, 145] Therefore, most attention has been focused on studying the non-equilibrium dynamics of artificial spin ices. In this paper we show that spin crossover materials on the kagome lattice can exhibit ice-like properties. In these ‘spin-state ice’ phases the spin states are disordered but obey a local ice rule of two HS and one LS (or two LS and one HS; Figure 3.1c,d) per triangle. We demonstrate that the low-energy states are described by an emergent gauge field. We show that defects in the ice carry spin, with a magnitude that is midway between the spins of the HS and LS states of an individual metal centers – despite no individual metal center being in an intermediate spin-state.

3.2 Model

In a complex or framework the d-orbitals of the ion are split by the ligand ﬁeld, ∆, Figure 3.3a-c. If this splitting is small compared to the electron pairing energy (Hund’s rule coupling) then the ground state is HS, Figure 3.3b. However, if the splitting is large then the ground state is LS, Figure 3.3c. In general there will be an enthalpy difference between the HS and LS states ε = εHS − εLS. Furthermore, the metal-ligand bond length in the HS state is often as much as 10 % larger than that in the LS state [42, 43, 57]. Molecules have higher entropies in their HS states. The most obvious source of this entropy II difference is the spin entropy, Ss = kB ln(2S + 1). For Fe the HS state has spin S = 2 and the LS state HS LS is S = 0, so ∆Ss = Ss − Ss = kB ln5. However, the two spin-states can also have different orbital entropies and the lengthening of the metal-ligand bonds concomitant with the change of spin-state softens the associated vibrational modes in the HS complex, further increasing the entropy difference,

∆S. Typically the vibrational contribution is largest and ∆S ∼ 4∆Ss [42, 68, 69]. To describe the many-body physics of SCO materials it is convenient to label the spin-state of each th metal center by a binary or Ising variable: σi = 1 if the i metal is HS and σi = −1 if it is LS. It is convenient to absorb the entropy of a single ion into the Hamiltonian [66]. This has the side effect of making the effective Hamiltonian for a single molecule appear temperature dependent:

1 H0 = ∑ (ε − T∆S)σi ≡ ∆G1 ∑σi, (3.1) i 2 i where ∆G1 is half the free energy change associated with a single isolated complex switching its spin-state. We note that this resummation is only possible because magnetic interactions between the spins are weak in most SCO materials. If magnetic interactions were large (compared to the energy 3.2. MODEL 79

Figure 3.3: Model of spin-crossover materials. a) When placed in an organic cage the d-orbitals of a 2+ transition metal, such as Fe , are split into eg and t2g levels, separated by the crystal ﬁeld splitting, ∆. b) The high-spin (HS) state for long metal-ligand bonds (small ∆). c) The low-spin (LS) state for short metal-ligand bonds (large ∆). d) The Lennard-Jones potential, V(r) (blue curve) between two molecules separated by a distance r. Near the minimum the second derivative is positive (pink curve), whereas at larger distances the second derivative becomes negative (green curve). Thus, if the nearest neighbour separation x ' r0 we expect k1 > 0 but k2 < 0 and k3 < 0. Away from the minimum there 2 are also linear terms and higher order terms, but, because σi = 1, these simply change the values of ∆G1 and Jn and do not add new terms to the Hamiltonian. e) The kagome lattice, with the three nearest neighbour interactions, k1-k3 and the angular interaction kθ , marked. The through space interaction, k3s is not included in our model. scales relevant to spin crossover) then the entropy of any spin-state with S 6= 0 would be lowered due to magnetic correlations and a more detailed treatment of spin and orbital degeneracy would be required.

Clearly ε < 0 (i.e., εHS < εLS) favors majority HS states at all temperatures. Similarly if ε is large and positive majority LS states arise. However, at T ∼ ε/∆S one expects a crossover between the low-enthalpy, low-entropy LS state and the high-enthalpy, high-entropy HS state. In the solid state, the crossover can become a ﬁrst order transition with hysteresis, indicating signiﬁcant interactions between metal centers. Multi-step transitions are also observed, often with intermediate states displaying ordered patterns of HS and LS metals. [60, 86, 124, 125] In a crystal the change in size of a molecule undergoing a spin-state transition induces elastic interactions on neighbouring molecules [44, 65, 71, 72, 126–135]. We model these interactions as a network of springs,

k 2 H = n r − [R(σ ) + R(σ )] , (3.2) n 2 ∑ i, j i j hi, jin including harmonic angular terms

2 kθ δ 2 Hθ = ∑ [θi jk − θ0] , (3.3) 2 {i, j,k}∈4

th where kn is the spring constant for the spring connecting n nearest neighbor molecules, R(σi) is half the relaxed distance between two nearest neighbour metal-centers both in spin-state σi such that 80 CHAPTER 3. SPIN-STATE ICE

R(1) > R(−1), δ = [R(1) − R(−1)]/2, ri, j is the instantaneous bond distance between sites i and j, th hi, jin indicates that the sum runs over all n nearest neighbors, θi jk is the angle between the metal centers i, j and k, {i, j,k} ∈ 4 indicates that the sum runs over only the interior angles of the elementary ◦ triangles of the kagome lattice, Figure 3.3e, and θ0 = π/3 = 60 . Models including these interactions have been shown to reproduce thermal hysteresis and multi-step SCO transitions [71,72,126,134,135]. On simple chemical grounds one expects angular variations to be more energetically expensive than changes in bond lengths. This expectation is conﬁrmed by ﬁrst principles calculations [71]. In the limit kθ k1 only the breathing modes of the lattice remains active. If one assumes that a and ` triangles remain the same size then ri, j = x for all nearest neighbors i and j. One can then solve the molecular dynamics problem analytically. Minimizing with respect to x and expanding the resulting Hamiltonian yields nmax J H = J σ σ − ∞ σ σ + ∆G σ , (3.4) ∑ ∑ n i j N ∑ i j 1 ∑ i n=1 hi, jin i, j i √ 2 2 th where Jn = 4αn knδ , αn is the distance to the n nearest neighbour in units of x: {αn} = {1, 3 ,2,...},

J∞ = ∑n Jn is a long-range strain interaction, which has equal strength between all sites regardless of their separation, N is the number of lattice sites, and we have neglected a constant term. This model is mathematically equivalent to the Ising–Husimi–Temperley model of magnetism in a longitudinal ﬁeld: the ﬁrst term is the Ising interaction; the second is the Husimi–Temperley interaction; in the third term

∆G1 enters in the same way as a magnetic ﬁeld parallel to the magnetic moments would in a magnetic model. Interestingly, long range ferromagnetic terms have previously been added to empirical models of SCO. [129,130,146] The derivation of eq 3.4 provides a controlled basis for this approximation. Note that this model is derived on very general grounds and should apply to many crystalline materials where molecules (or other subunits) can access two states and the molecule changes its shape or volume when switching between these states. Possible examples include photo-induced changes on individual molecules in a crystal, systems which show thermal charge transfer, such as some Prussian blue analogues, and weakly bonded networks. [147–149] Previous work on the theory of water and spin ices shows that frustration is a key ingredient for ice states. Elastic frustration is also known to play an important role in driving multistep transitions and complex ordering patterns in SCO materials. [72,127,128] This leads us to ask if materials could be grown that host spin-state ices. Lattices where one might best hope to ﬁnd ices are the kagome (Figure 3.3e), triangular, checkerboard (Figure 3.7) and pyrochlore (Figure 3.1b). [25] Of these the most promising from a synthetic perspective is probably the kagome. Indeed, recently crystals of [Co2Fe2] complexes have been grown where these complexes lie at the vertices of a kagome lattice [150]. This system shows a two step SCO transition with macroscopic spin-state disorder in the low temperature phase. However, in this material there are three crystallographically distinct molecular sites, one of which is HS at low temperatures. So the state in this material cannot be the simplest possible spin-state ice described below. Nevertheless these general considerations, Sekine et al.’s work, [150] and recent synthetic progress towards kagome lattice SCO framework, motivates us to study our model on the kagome lattice, Figure 3.3e, with interactions up to third nearest neighbours. There are two distinct 3.3. MONTE CARLO CALCULATIONS 81 third nearest neighbours; we include the through-bond interaction, k3, but neglect the through-space interaction k3s, which one expects to be weaker. Generically, one expects k1 > 0 because the nearest neighbor distance will be close to minimum of the intermolecular interactions. However, one expects the longer range elastic constants k2 and k3 to be negative as long-range interactions generally fall off rapidly. This is shown explicitly for the Lennard-Jones potential in Figure 3.3d. The model remains mechanically stable provided J∞ > 0.

3.3 Monte Carlo calculations

To investigate the ﬁnite temperature properties of our model (eq 3.4) we carried out Monte Carlo simulations on a N = 40 × 40 × 3 site lattice with periodic boundary conditions allowing single pseudospin ﬂip, worm and loop moves [151]. Additionally, we employ parallel tempering, switching between simulations with a Boltzmann probability. For each point we perform 5000 measurements on the 4800 site lattice after initializing for 24,000 moves with 4800 moves between each measurement.

The calculation of the heat capacity, cV , for this model has some subtleties. There are two contributions to the entropy, and thus the heat capacity, (i) the entropy associated with the individual (1) metal ions, S = (2nHS − 1)∆S/2, where nHS is the fraction of metals centers in the HS state; and (ii) the many-body entropy associated with the configuration of Ising pseudospins, S(N). To calculate the single molecule term a Savitzky-Golay filter implementation [114] was used to fit nHS.

In Figure 3.6 both the data (points) and the Savitzky-Golay filtered curve are shown for nHS; the difference is much smaller than the symbol size. The fits were used to calculate ∂nHS/∂T and thence (1) (1) cV = T(∂S /∂T) = T∆S(∂nHS/∂T). The many-body contribution to heat capacity is evaluated from the fluctuations in the physical enthalpy, E = H + T∆S∑i σi, cf. eq 3.1,

2 2 (N) hE i − hEi cV = 2 . (3.5) NkBT

(1) (N) Thus, the physical heat capacity is given by cV = cV + cV .

3.4 Results and discussion

Our central result is that we find extended spin-state ice (SSI) phases both at zero and non-zero temperatures, Figs. 3.4–3.6. At T = 0 we have confirmed that there are no long-range ordered states with a unit cell of 48 or fewer sites that have lower energies than the SSIs in the relevant parts of the phase diagram, Figure 3.4. We find two SSI phases: in the majority LS spin-state ice (LSSI) each triangle contains one HS and two LS metal centers (Figure 3.1c); conversely the majority HS spin-state ice (HSSI) obeys the ice rule that each triangle contains two HS sites and one LS site (Figure 3.1d). Thus, these phases are analogous to the three state ices of the kagome lattice Ising model in a field, which can be mapped to the honeycomb dimer model and have an extensive residual entropy

S = 0.108kB associated with the Ising degrees of freedom [152]. We also ﬁnd HS and LS phases, 82 CHAPTER 3. SPIN-STATE ICE

Figure 3.4: The zero temperature phase diagram for model (3.4) of spin-crossover materials on the kagome lattice for k1 > 0 and k3 = (3/4)k2. HSSI indicates majority high-spin spin-state ice. LSSI indicates majority low-spin spin-state ice. The vertical dashed black line and the points labelled a-c show the parameters explored in Figure 3.6. Note that the zero temperature phase diagram only 2 depends on the ratios k2/k1 and ε/k1δ and not k1, k2, ε, and δ individually.

where almost all metal ions are in the same spin-state and hence almost all metal ions are in the same spin-state within each triangle. Our model (eq 3.4) is closely related to the Ising models in a field, but has important differences. Most importantly, it has been widely recognized that the long-range strain interaction (which is absent in a simple Ising model of magnetism) plays a crucial role in determining the collective spin crossover behavior in the solid state. [65,71,72,126–135] J∞ > 0 favors all metals having the same spin-state, with 1 the majority spin-state selected by the sign of the single ion free energy difference, ∆G1 = 2 (ε −T∆S). For weaker J∞ (k2 and k3 more negative) short-range interactions dominate and can stabilize SSIs, again the majority spin-state is determined by ∆G1. Given these important differences between the two models it is fascinating that the same ice rules that emerge from both. [153]. Snapshots from Monte Carlo simulations reveal states that obey the ice rules over a large temperature range, examples are shown in Figure 3.5a,b (we quantify the range of stability below, see Figure 3.6 in particular). To confirm that the Monte Carlo calculations indeed find SSIs we have computed the q 1 iqq.ri, j spin-state structure factor, Sσσ (q) = N2 ∑i j hσiσ ji − hσiihσ ji e which is shown in Figure 3.5c. This clearly shows the pinch point singularities that are the signature of ice states. [39] In the context of the Ising model these pinch points are well understood in terms of a mapping to an emergent gauge field similar to classical electromagnetism. [18] This theory shows that the pinch points are a direct consequence of the material obeying the ice rule. Pinch points are observed experimentally in neutron scattering experiments on spin ices [24, 40]. However, the situation is more complicated for SSI as the Ising degrees of freedoms are not directly amenable to neutron scattering. Given the large structural changes between the two spin-states it is possible that x-ray scattering could provide a signature of SSI. Particularly, one would expect a significant increase in the diffuse scattering due to the inherent spin-state (and thus structural) disorder in spin-state ice. Alternatively, one could use an external 3.4. RESULTS AND DISCUSSION 83

Figure 3.5: Spin-state ice at non-zero temperatures. a) Snapshot of LSSI. Almost every triangle has two low-spin and one high-spin metal centers, obeying the LSSI rule. There are two exceptions: βLS a triangle with three LS metal centers, where a string of LS metal centers branches, and τLS a triangle with two LS and one HS metal centers, where a string terminates. This snapshot was 2 2 taken at T = 0.4k1δ /kB, with ε = 4k1δ and k2 = −0.139k1. b) Snapshot of melting HSSI. Most triangles still obey the ice rule, but there are significant numbers of defects with βHS outnumbering τHS defects. This increases the fraction of HS metal centers: nHS = 0.689 for the configuration this snapshot is taken from (Figure 3.10); nHS = 2/3 in defectless HSSI. This snapshot was taken at 2 2 T = 0.4k1δ /kB, with ε = 4k1δ and k2 = −0.139k1. The full lattice snapshots from which panels a and b are cropped are shown in Figures 3.9 and 3.10, respectively. c) The spin-state structure factor for SSI, Sσσ displays pinch points at the Brillouin zone boundaries, one of which is circled. These pinch points are the smoking gun evidence that a spin-state ice exists and that it is described by an emergent gauge field [18, 39]. In a magnetic field the spin structure factor, Szz ∝ Sσσ . Thus, the pinch points would be directly detectable via neutron scattering. The structure factor was calculated for 2 2 k2/k1 = −1.39, k3 = (3/4)k2, T = 0.11k1δ /kB and ∆G1 = −1.32k1δ

magnetic field to (partially) align the spins on the HS sites. For a field in the z-direction one then has z z z z z S = S + σiδS, where S = (S + S )/2 and δS = (S − S )/2. It follows straightforwardly that HS LS HS LS q 1 z z z z iqq.ri, j q 2 the spin structure factor, Szz(q) = N2 ∑i j hSi S ji − hSi ihS ji e = Sσσ (q)δS is identical to the spin-state structure factor except for a multiplicative constant. As the spin structure factor is directly measurable via neutron scattering this experiment could provide direct evidence for the existence of SSIs. Monte Carlo simulations also reveal local violations of the ice rules; two such excitations in LSSI are marked in Figure 3.5a. The defects are a triangle with three LS sites (βLS) and another with one LS and two HS sites (τLS). Alternatively, one can view the LSSI as long strings of LS sites; then the defects correspond to branching (βLS) and terminating (τLS) strings. To understand the properties of these excitations it is helpful to consider the system in a magnetic field in the z-direction that polarizes the z 1 z z spins. As each site belongs to two triangles, the LSSI vacuum has a net spin SLSvac = 2 (2SLS + SHS) z 3 z z per triangle. Thus, a βLS-excitation carries a net spin of S = S −S = −δS and the net spin of βLS 2 LS LSvac an -excitation is Sz = 1 (Sz + 2Sz ) − Sz = S. Analogous excitations are found in the HSSI τLS τLS 2 LS HS LSvac δ phase, Figure 3.5b, with Sz = 1 (2Sz +Sz ) per triangle; Sz = δS and Sz = −δS. In contrast a HSvac 2 HS LS βHS τHS z z single pseudospin flip is a ∆SSCO = SHS −SLS = 2δS excitation. Thus the β and τ excitations carry only half the spin associated with a single metal changing its spin-state between HS and LS. We emphasize 84 CHAPTER 3. SPIN-STATE ICE

d4 d5 d6 d7 SHS 2 5/2 2 3/2 SLS 1 1/2 0 1/2 ∆SSCO 1 2 2 1 δS 1/2 1 1 1/2 Table 3.1: Spin states relevant for octahedral complexes. SCO can occur for metal ions with d4-d7 z z electrons. ∆SSCO = SHS −SLS is the magnitude in the change in spin when a single molecule undergoes SCO. δS = ∆SSCO/2 is the spin carried by the fractionalized quasiparticles. that there are no intermediate spin-states in our model. Therefore this ‘fractionalization’ [141] of the excitation is entirely a collective effect, sketched in Fig. 3.2e-h. Depending on the number of d electrons on the metal ions, the τ and β excitations may carry either spin-1/2 (d4 and d7) or spin-1 (d5 and d6), Table 3.1. A full understanding of the defects requires a quantum mechanical description of the spin. For 6 concreteness we specialize to d HSSI, which is characterized by strings of SHS = 2 ions separated by SLS = 0 ions. Although exchange interactions are small compared to the elastic interactions that cause the SSI, they are not negligible in the low T quantum analysis of the spin. For nearest neighbour exchange interactions we have a one-dimensional S = 2 Heisenberg model along the HS strings, which has a Haldane ‘valence bond’ ground state [154]. τHS defects are terminating strings, and give rise to a fractionalized S = 1 states localized at the ends of the strings [154]. βHS defects are branches in the strings and are topologically equivalent to a Y-junction, where again one finds a S = 1 fractionalized excitation localized at the defect [155]. Thus, this quantum analysis is entirely consistent with the classical conclusion that the quasiparticles carry a fractionalized spin δS = ∆SSCO/2. The existence of collective excitations with a spin that is different from that on any individual metal center in (some) SSI phases suggests another experimental approach to directly detect the phase. Resonance experiments, for example, electron paramagnetic resonance experiments, are sensitive to spin nature of unpaired spins. Therefore, they will show dramatic changes on entering the SSI phase. For example, in d6 metals one would expect to see many S = 1 impurities in the SSI phase, which disappear at the phase transition out of this phase. Spin-state transitions of a single ion on a triangle where the ice rule is violated are low energy processes and allow defects to move (as sketched in Figure 3.2e-h). These processes interconvert β and τ defects, Fig 3.2e-h. Thus, the numbers of β and τ defects are not conserved. However, a topological charge Q = ηδS, (3.6) where η = +1 (−1) for a (`) triangles, is conserved [29]. At T > 0 it only costs a finite energy to move at pair of excitations infinitely far apart. Hence, the fractionalized spin-δS excitations are deconfined. [39] For T > 0 the SSI vacuum can become unstable with respect to the spontaneous production of defects. As the single ion free energy will generally favor one spin-state over the other (∆G1 6= 0) equilibrium SSI states generically contain unequal numbers of β and τ quasiparticles as the ice begins 3.4. RESULTS AND DISCUSSION 85

Figure 3.6: Thermodynamic signatures of spin-state ice: heat capacity (left axes; purple), cV ; fraction of HS metal centers (right axes; black), nHS ' χT, where χ is the spin susceptibility; and ∂nHS/∂T (right axes; pink). a) For small enthalpy differences between the two spin-states and ε = εHS −εLS < 0 the ground state is HSSI and there is a phase transition to a high temperature HS phase. Notice that an incomplete spin transition occurs due to cooperative effects even though the free energy of an isolated LS ion is always greater than that of an isolated HS ion. b) For small ε > 0 the ground state is LSSI. Temperature induces a HSSI phase for ∆G1 > 0, which shows some “melting” (cf. Figure 3.5b) as witnessed by the increase in nHS, followed by a transition to the HS phase. c-f) For larger ε > 0 the LS phase occurs at T = 0. As the temperature is increased the system undergoes three phase transitions. Large ε also broadens the transitions. Thus, if only the fraction of HS states is calculated, or if only χT is measured, the SSI phases could be dismissed as a broad single step HS to LS transition or crossover. However, cV and ∂nHS/∂T both show clear signatures of the intermediate phases. In all panels the dashed vertical lines mark the phase transitions, k2 = −0.139k1, ∆S = 4kB ln5, and k3 = (3/4)k2. The arrows marked a and b in panel (c) show the simulations from which the snapshots shown in Figure 3.5a,b where taken. to melt. A snapshot of a melting HSSI, Figure 3.5b, shows many defects but more branches than terminating strings, indicative of a greater fraction of HS ions than for pure HSSI (nHS = 2/3).

The enthalpy difference between the HS and LS states (ε = εHS − εLS) has two important consequences for the finite temperature behaviour of the system, Figure 3.6. We have already seen that ε plays an important role in determining the zero temperature physics (cf. Figure 3.4). This, in turn, helps determine the finite temperature behaviour. For large positive ε (Figure 3.6c-f) we observe four phases. At the lowest temperatures an LS phase that first gives way to a LSSI phase as the temperature is raised. When the temperature is raised sufficiently to change the sign of ∆G1 (cf. eq 3.1) there is a transition from LSSI to HSSI. Finally there is a transition from HSSI to HS. Note that at high temperatures HS and LS metal ions are not equally likely because of their intrinsic entropy difference. At high temperatures (formally as T → ∞) the high-spin fraction becomes very large ∆S/k ∆S/k nHS → e B /(1 + e B ) = 625/626 ' 0.9984 for ∆S = 4kB ln5, the value studied in Figure 3.6. For small positive ε (Figure 3.6b) the LSSI phase is found at T = 0 and the LS phase is not realized 86 CHAPTER 3. SPIN-STATE ICE at any temperature. That is, we find an incomplete spin transition. Otherwise the finite temperature physics is quite similar to large positive ε, with the HSSI and HS phases realized as the temperature is raised. For small negative ε (Figure 3.6a) one has the HSSI phase at zero temperature. No majority LS phases are found at any temperature. Nevertheless, the finite temperature physics is similar to that of positive ε, but with the low temperature majority LS phases removed. This can all be understood straightforwardly as the ε sets the minimum ∆G1, cf. eq 3.1. Nevertheless, there is something very surprising about this result. In this parameter regime the free energy of a single ion is always greater in the HS state. Yet we find an (incomplete) spin transition! This is entirely a cooperative effect driven by the elastic interactions between the ions. For large negative ε (not shown) the HS phase occurs at all temperatures. The breadth of the transition is also highly dependent on enthalpy difference between the HS and LS states. For small |ε| we see sharp, well defined signatures of the phase transitions via both peaks in the heat capacity and steps in the fraction of HS ions. As ε increases these features become increasingly washed out and crossover-like. The magnetic susceptibility, χ, is the most commonly measured property of SCO materials: because of the extremely weak spin interactions typically found in these materials nHS ∝ χT. For the largest ε studied (Figure 3.6f) the SSI phases are confined to a narrow region around ∆G1 = 0 and the changes in nHS occur rather slowly. Thus, if only χT were measured SSI physics could easily be dismissed as a broad single step SCO between the LS and HS phases, which are rather common. On the other hand, there are clear maxima in the specific heats even 2 at ε = 16k1δ , showing that clear thermodynamic signatures of phase transitions or crossovers into, out of, and between the SSI phases remain. These maxima are also found in ∂nHS/∂T, which is a more routine measurement in the spin crossover community. It is interesting to note that this is not just (1) a straightforward consequence of single ion behavior, cV ∝ ∂nHS/∂T. Rather, we find that collective (N) (1) contribution to the heat capacity cV cV in all cases studied: even near the phase transitions. Thus, the critical fluctuations are present in both the single-body and many-body contributions to the entropy.

3.5 Conclusions

Our calculations predict that a new phase of matter, spin-state ice, can exist in spin crossover materials on the kagome lattice. We find spin-state ice as both a ground state and a thermodynamically stable phase at non-zero temperatures. We have demonstrated that this phase has remarkable properties including collective excitations that carry a spin halfway between that of the HS and LS states. These excitations are directly detectable in electron paramagnetic resonance experiments. The ground state is governed by a set of local ‘ice rules’ that give rise to an emergent gauge field and hence distinctive pinch points in both the spin-state and spin structure factors. The latter is directly measurable via neutron scattering experiments. The thermodynamics of spin-state ice also show signatures of this phase. A natural questions arising for our work is: What is the quantum mechanical ground state of 3.5. CONCLUSIONS 87 the elastic Hamiltonian? Quantum effects are large for phonons and molecular vibrations, and have been shown to have a significant effect on spin crossover materials [156]. Therefore, the key question is do quantum fluctuations pick out a unique ground state, thereby respecting the third law of thermodynamics, or do they cause a quantum spin-state liquid? 88 CHAPTER 3. SPIN-STATE ICE 3.6 Supplementary Information

3.6.1 Mapping between spin ice rules and spin-state ice rules

In the field of artificial spin ices it is traditional to name lattices with spins located on the bonds of the lattice. This is natural given the physical structure of artificial spin lattices. However, in the current work we use the locations of the metal atoms to define the vertices of the lattice. In Figure 3.7 we demonstrate how lattices are renamed when moving between these two nomenclatures, in particular we show that the square lattice (in the bond convention) is equivalent to the checkerboard lattice (in the site convention) and that the honeycomb (bond) lattice is equivalent to the kagome site (lattice). Figure 3.7 also demonstrates that spin ice states on the bond lattices map directly onto spin-state ice rules on the site lattices after an appropriate relabelling.

3.6.2 Snapshots of Monte Carlo simulations

In Figure 3.5 of the main text we show regions of snapshots of our Monte Carlo simulations. Images of the full state of the simulations from which these are taken are shown in Figures 3.9 and 3.10. 3.6. SUPPLEMENTARY INFORMATION 89

Figure 3.7: Equivalence of lattices and ice rules between the honeycomb (bond/spin ice) and kagome (site/spin-state ice) lattices (a-c); and equivalence of the square (bond/spin ice) and the putative checkerboard (site/spin-state ice) lattices (d-f). (a) In honeycomb artificial ice the spins (arrows in the figure) lie on the bonds of the lattice. Here we show a state that obeys the honeycomb ‘ice II’ rule: [25] every site of one sub-lattice (marked by navy blue dots) has one spin pointing in and two spins pointing out; whereas every site on the other sub-lattice (pale blue dots) has two spins pointing in and one spin pointing out. (b) The same as (a), but we have placed a colored circle on each spin: purple if the arrow points from a pale blue site to a navy site; and green if the arrow points from a navy site to a pale blue site. The interactions between the spins are marked by blue lines. (c) Same as (b) but with the spins and blue dots removed. The kagome lattice structure is evident. If we now interpret purple circles as LS and green circles as HS the HSSI ice rule is obeyed – each blue shaded triangle contains two HS and two LS sites. (d) In square artificial ice the spins lie on the bonds of the lattice. Here we show a state that obeys the 2-in, 2-out ice rule: every site of the lattice (navy and pale blue dots, different shades indicate the two sublattices) has two spins pointing into it and two spins pointing out. (e) As (d), but with colored circles added on each spin: purple if the arrow points from a pale blue site to a navy site; and green if the arrow points from a navy site to a pale blue site. Interactions between sites are marked by blue lines. Diamonds with interactions across the diagonals are shaded blue. (f) As (e) but with the spins and blue dots removed. Interpreting purple (green) circles as LS (HS) a checkerboard spin-state ice rule is obeyed – every diamond shaded blue contains exactly two HS and two LS sites. 90 CHAPTER 3. SPIN-STATE ICE

Figure 3.8: Equivalence of lattices and ice rules between the honeycomb (bond/dimer) and kagome (site/spin-state ice) lattices (a-c); and equivalence of the square (bond/dimer) and the putative checkerboard (site/spin-state ice) lattices (d-f). (a) In the honeycomb dimer model the dimers (solid black lines in the ﬁgure) lie on the bonds of the lattice. Here we show a state that obeys the hardcore dimer rule: [25] every vertex as one dimer connecting it to an adjacent vertex. (b) The same as (a), but we have placed a purple circle on each dimer (and green if no dimer exists). The interactions between the spins are marked by blue lines. (c) Same as (b) but with the spins and blue dots removed. The kagome lattice structure is evident. If we now interpret purple circles as LS and green circles as HS the HSSI ice rule is obeyed – each blue shaded triangle contains two HS and two LS sites. (d) In the square model the dimers lie on the bonds of the lattice. Here we show a state that obeys the loop rules: every vertex has two dimers connecting it to adjacent vertices. The long strings of dimers form closed loops, hence ”loop” model. (e) As (d), but with colored circles added on each spin: purple if a dimer exists (and green if the dimer does not exist). Interactions between sites are marked by blue lines. Diamonds with interactions across the diagonals are shaded blue. (f) As (e) but with the spins and blue dots removed. Interpreting purple (green) circles as LS (HS) a checkerboard spin-state ice rule is obeyed – every diamond shaded blue contains exactly two HS and two LS sites. 3.6. SUPPLEMENTARY INFORMATION 91

Figure 3.9: Snapshot of the full lattice from which Fig. 3.5a is taken. 92 CHAPTER 3. SPIN-STATE ICE

Figure 3.10: Snapshot of the full lattice from which Fig. 3.5b is taken. Note the tendency for the defects to cluster and that there are no triangles containing three LS metal centers, this is typical. 3.6. SUPPLEMENTARY INFORMATION 93 The following publication has been incorporated as Chapter 4. For ease of reading section titles have been added that do not appear in the original publication, ﬁgures have also been restructured to ﬁt the thesis formatting. 1. [3] Jace Cruddas, and Ben J. Powell, Multiple Coulomb phases with temperature tunable ice rules in pyrochlore spin crossover materials, arXiv:2007.13983 (2020).

Contributor Statement of contribution % Jace Cruddas writing of text 60 proof-reading 50 theoretical derivations 100 numerical calculations 100 preparation of ﬁgures 100 initial concept 50 Ben J. Powell writing of text 40 proof-reading 50 theoretical derivations 0 numerical calculations 0 preparation of ﬁgures 0 initial concept 50 supervision, guidance 100 Chapter 4

Multiple Coulomb phases with temperature tunable ice rules in pyrochlore spin crossover materials

Spin crossover molecules have two accessible states: high spin (HS) and low spin (LS). We show that, on the pyrochlore lattice, elastic interactions between SCO molecules can give rise to three spin-state ice phases. Each is a “Coulomb phase” where a local ice-rule can be mapped to a divergence free gauge ﬁeld and the low energy excitations carry a spin fractionalized midway between the LS and HS states. The unique nature of spin crossover materials allows temperature or pressure to change the ice rules allowing straightforward access to Coulomb phases not yet observed in water or spin ices.

4.1 Introduction

In spin ices, such as Dy2Ti2O7 and Ho2Ti2O7, the magnetic Dy (Ho) atoms form a pyrochlore lattice composed of vertex sharing tetrahedra [33,39]. The combination of the crystal field [22,34] and the long-range dipolar interaction [35, 38] constrain the magnetic moments to obey the 2-in/2-out ice rule: two of the spins point in to each tetrahedron and two point out. The number of ways of satisfying the ice rule grows with the size of the system, leading to a large zero-point entropy [16, 22]. The ice rule can be mapped onto a divergence-free flux, analogous to constraints in magnetostatics and electrostatics. Violations of the ice rule carry a fraction of the magnetic spin degree of freedom, behaving effectively as magnetic monopoles with an emergent Coulombic interaction between them [7,24, 40]. Spin ices are therefore said to be in a “Coulomb phase”, which could also arise in frustrated antiferromagnets and some quantum spin liquids [18, 157–159]. Coulomb phases can also arise from other degrees of freedom. Three coexisting Coulomb phases, each obeying 2-in/2-out (or, more strictly, divergence free) ice rules, are observed in the charge, atom displacement, and spin configurations of CsNiCrF6 [23]. In principle ices obeying 1-in/3-out or 3-in/1-out ice rules should also give rise to Coulomb 95 96 CHAPTER 4. MULTIPLE COULOMB PHASES phases [28]. This has not yet been observed, although an ordered phase containing 1-in/3-out and 3-in/1-out tetrahedra has been reported [160]. For the nearest neighbor Ising model on the pyrochlore lattice with collinear spins, Si, then, up to a constant,

2 HI = J ∑ SiS j + B∑Si = J ∑(Lα + B/4J) , (4.1) hi, ji i α where Lα = ∑i∈α Si and α labels the tetrahedra. The ground state is clearly achieved whenever the magnitude of the ‘flux’, Lα + B/4J, is minimized for all α. Ice rules require the same value of Lα on all tetrahedra. Therefore, sweeping the field, B, moves the ground state between different ice rules. However, in spin ices the spins are not collinear and so one cannot simply apply a magnetic field to change the ice-rules in these systems. Recently, it has been proposed that spin-state ice (SSI) phases can occur in spin-crossover materials [2]. In this chapter we show that multiple Coulomb phases arise for spin-crossover materials on the pyrochlore lattice. Furthermore, we show that sweeping temperature alone is sufficient to tune between Coulomb phases obeying three different ice rules. This is a direct consequence of the competition between the single molecule spin crossover behavior and the many-body ice physics. We predict that pinch points singularities, the sine qua non of the Coulomb phase, will be detectable via neutron scattering in small magnetic fields and that the low-energy excitations are deconfined and carry a spin midway between that of the two spin states of a single molecule. Spin crossover (SCO) occurs in transition metal centers in complexes and frameworks when 6 0 4 2 the low-spin (LS; e.g., t2geg, S = 0) and high-spin (HS; e.g., t2geg, S = 2) states have comparable enthalpy. SCO provides a reversible molecular switch, which is addressable by changes in temperature, pressure, light irradiation, magnetic field, and chemical environment [44]. As spin-state changes are accompanied by changes in molecular volume, color, and magnetic susceptibility, SCO materials are intrinsically multifunctional and have been widely explored for applications such as high-density reversible memory, and ultrafast nanoscale switches [44, 57, 161]. However, many questions about the fundamental physics at play in these systems remain open [1, 65, 71, 72, 126–135, 162].

4.2 Model

Changes in the molecular volume accompany spin-state switching due to the (de)population of antibonding eg orbitals in the (LS) HS state: the metal-ligand bond length in the HS state is often ∼ 10 % larger than that in the LS state. In molecular materials and frameworks, the local structural distortions caused by metals changing spin state couple to long-range elastic interactions. It is convenient to introduce a pseudospin label for the spin state of each metal, σi = 1 (−1) if the ith 2 molecule is HS (LS). Because σi = 1 no terms above quadratic order appear in pairwise interactions [1]. Letting RH (RL) be the equilibrium distance between metals in the HS (LS) phases, we can write the equilibrium midpoint between neighboring metals as R + δ(σi + σ j), where, R = (RH + RL)/2 and 4.2. MODEL 97

Figure 4.1: (a) Pyrochlore lattice with the three nearest neighbor interactions (k1, k2, and k3) marked. We neglect through-space interactions between third nearest neighbors, k3s, as we expect these to be much weaker than k3. Ice rules in the (b) H2L2 and (c) H1L3 phases. (d) Near the minimum of 2 2 the potential between neighboring metals ∂ V(r)/∂r ' k1, is positive and large (blue lines). At larger distances the second derivative is negative and decreases in magnitude with increasing distances (orange and green lines). Therefore, one expects k1 > 0 and k2 < k3 < 0 for the pyrochlore lattice. At larger distances the higher order (non-harmonic) contributions to the potential simply renormalize ∆H 2 and kn because σi = 1.

δ = (RH − RL)/2. Hence, m ∆H k 2 H = σ + n r − η R + δ(σ + σ ) , (4.2) 2 ∑ i ∑ 2 ∑ i j n i j i n=1 hi, jin where ∆H = HH − HL is the enthalpy difference between HS and LS metals and kn are the effective spring constants between nth nearest neighbors (see Fig. 4.1), the sum over hi, jin includes all nth √ nearest-neighbors, ri j is the instantaneous distance between sites i and j, and ηn = 1, 3 ,2,... is the ratio of distances between the nth and 1st nearest-neighbor distances on the undistorted pyrochlore lattice. Inspired by recent progress in the synthesis of tetrahedral iron cages [163] we study this model with m = 3 on the pyrochlore lattice (4.1a), where we expect k2 < k3 < 0 (4.1d) and require k1 + 6k2 + 4k3 > 0 for structural stability. There are two contributions to the entropy. The usual many-body entropy described by the configuration of the σi and a single molecule term, ∆S, due to changes in the spin and orbital quantum numbers and vibrational frequencies between the HS and LS states. Below we set ∆S = 4ln5, a typical value of Fe(II) complexes [49]. It is convenient to define ∆G = ∆H − T∆S and absorb this term into the Hamiltonian [66]. We make the ‘symmetric breathing mode approximation’, which neglects asymmetric structural distortions [1, 2], and calculate the finite temperature properties from a combination of single spin-flip Monte Carlo, worm and loop algorithms on a 12 × 12 × 12 × 4 = 6912 lattice except where stated [164]. The weak magnetic interactions in most SCO materials means that the fraction of HS molecules, nHS ∼ χT. Hence the magnetic susceptibility, χ, is commonly used as a diagnostic for the cooperative behavior in the system. Strong cooperative behaviors often give rise to multiple step transitions [1,44,60,86,124,125]. Typically, when the effective interaction between nearest neighbor SCO centers, 2 k1δ , are of order kBT1/2, where T1/2 = ∆H/∆S, the temperature at which one expects nHS = 1/2, is typically 100-400 K [1]. As δ ' 0.05 A˚ this yields k ∼ 4 − 16 × 104 K/A;˚ an estimate in good 98 CHAPTER 4. MULTIPLE COULOMB PHASES

Figure 4.2: a) The zero-temperature phase diagram for k1 > 0 and k3 = 3k2/4. For small |k2| the long- range strain dominates and picks out a ferroelastic state, but large |k2| suppresses this effect allowing the Coulomb phases to emerge. b) The finite temperature phase diagram for k1 > 0, k2 = −0.1k1, and k3 = 0.075k1. The colors of the phase diagram indicate the equilibrium values of the fraction of high spins nHS, calculated via parallel tempering. We find four (black) lines of first order transitions that terminate at critical end points (black circles). The blue and red lines indicate lines of metastability for the cooling and heating calculations, respectively, cf. Fig. 4.7. Hence, the distance between blue and red lines is the width of the hysteresis. Individual materials have a fixed ∆H (white lines), the corresponding HS fractions, nHS, and heat capacities are shown in Fig. 4.4 and Fig. 4.6 [164], respectively.

4 ˚ agreement with values calculated from ﬁrst principles, e.g., k ∼ 8×10 K/A in [Fe(pz)Pt(CN)4]·2H2O (pz=pyrazine) [71]. The intermediate plateaus often display long-range patterns of HS and LS metals. But, several disordered phases have been reported also [150, 165, 166].

4.3 Results and Discussion

We ﬁnd three distinct spin-state ice phases: H3L1, H2L2 and H1L3 [164]. In the ground state of the

HnL4−n phase every tetrahedron contains n HS and 4 − n LS metals (Fig. 4.1b-c). Thus, the H3L1 and

H1L3 phases can be mapped onto the dimer model on a diamond lattice, and the H2L2 phase can be mapped onto the loop model on a diamond lattice [18]. We expect the stability of the ice phases to be underestimated in calculations because the long-range strain, which the favors ferroelastic phases, goes to a constant at large separations in the symmetric breathing mode approximation [1], rather than decaying with a power law as expected physically. In order to verify that the phases at T = 0 are indeed Coulomb phases we have calculated the pseudo-spin structure factor, Fig. 4.3j-l,

1 i~q·r~i j Sσσ (~q) = ∑hσiσ jie . (4.3) N i j

We clearly observe singularities in Sσσ at the Brillouin zone boundary, known as pinch points, which are a direct consequence of the existence of a divergenceless gauge ﬁeld [18]. This conﬁrms that the intermediate plateaus are indeed Coulomb phases. However, directly measuring the pseudospin structure factor is not straightforward. In SCO materials the magnetic correlations between sites are typically negligible. Therefore, the spins are z z described by a Zeeman Hamiltonian, HZ = ∑i µBBSi (σi), where the spin, Si(σi), depends on the 4.3. RESULTS AND DISCUSSION 99

Figure 4.3: (a-c) The pseudospin structure factors, Sσσ , Eq. 4.3, for (a) H1L3, (b) H2L2, and (c) H3L1. All structure factors display pinch points, one of which is circled, at the Brillouin zone boundaries characteristic of a Coulomb phase. The existence of the pinch points demonstrates that the low-energy physics of Coulomb phases are described by a divergence-less gauge field [18]. (d-l) The spin structure factors, SSS, Eq. 4.4, for (d,g,j) H1L3, (e,h,k) H2L2, and (f,i,l) H3L1 at selected magnetic field strengths. For kδ 2 ∼ 100 K pinch points are observed for magnetic fields of < 0.1 T, so only moderate fields are required for the pinch points to be observable via neutron scattering. All structure factors were 2 calculated for k1 > 0, k2/k1 = 4k3/3k1 = −0.1 and kBT/(k1δ ) = 0.01 on a 36 × 36 × 4 × 4 = 20376 2 2 lattice with (a,d,g,j) ∆H = 2k1δ , (b,e,h,k) ∆H = 0 and (c,f,i,l) ∆H = −2k1δ . 100 CHAPTER 4. MULTIPLE COULOMB PHASES spin-state, B is the applied field, and µB is Bohr magneton. The spin structure factor is

1 i~q·r~i j SSS(~q) ≡ ∑hSi(σi) · S j(σ j)ie (4.4a) N i j S (~q) = m2 + m2 − 2m m σσ + S (Bz) + S (Bz)δ(q), (4.4b) + − + − 4 d B z z z where m± = ∑ihSi (±1)i/N, and the expressions for the trivial diffuse, Sd(B ) and Bragg, SB(B ), scattering are

z 2 2 2 Sd(B ) = SL (SL + 1) − m− + SH (SH + 1) − SL (SL + 1) − m+ + m− nHS (4.5) m2 + m2 + 2m m m2 − m2 S (Bz) = + − + − + + − (2n − 1). (4.6) B 4 2 HS This, is directly measurable via neutron scattering and clearly shows the pinch points, Fig. 4.3. Importantly, the fields required are modest. For kδ 2 ∼ 100 K the magnetic fields required to observe pinch points are at < 0.1 T. Applied pressure favors the smaller LS state. Therefore, pressure would allow one to tune between the three different Coulomb phases at low T. 2 The ratio ∆H/(δ k1) not only has important consequences for the low-temperature physics, but 2 also for the high-temperature behavior as well, Figs. 4.2b and 4.4. For ∆H ∼ −(k1δ ) we observe a single first order transition. This is a purely collective phenomena, as the single ion physics always 1 favors the HS state (∆G < 0∀T). Increasing ∆H induces further transitions with plateaus at nHS ' 0, 4 , 1 3 2 , 4 and 1, corresponding to the LS, H1L3,H2L2,H3L1 and HS phases respectively, 4.4b-h. Hence, for a wide range of parameters, it is possible to tune between different Coulomb phases with temperature alone. This can be understood as follows: The single molecule entropy difference between spin-states (∆S) can be absorbed into the Hamiltonian [66]. Thus, ∆G = ∆H − T∆S replaces ∆H in 4.2. This term couples to the pseudospin just as a magnetic field couples to spin in the Ising model, 4.1. That is, the single molecule spin crossover behavior acts as an effective temperature-dependent ‘field’ for the pseudospins. This changes the ice rules as the temperature varies. Due to the large width of the hysteresis loop at low temperatures, simulations of straightforward cooling does not always result in the same low temperature phase as is found by parallel tempering, Fig. 4.4a-g. Similar effects have been observed experimentally in SCO materials that display long-range antiferroelastic order [78,115], and labeled “hidden hysteresis”. The hidden low temperature states can be realized by either photoswitching (i.e., reverse-light induced excited spin-state trapping) or applying and adiabatically releasing a pressure to the system. Hence, it becomes possible to tune between different SSI phases with not only temperature, but pressure and light as well. Thus, spin crossover materials provide a rare opportunity to study the transition between two different topological phases. 2 Further increasing ∆H/(k1δ ) moves the transitions towards and through critical points, where the transition is continuous, and into the crossover regime, Figs. 4.2b and 4.4g-i. The higher temperature 2 transitions become crossovers first as ∆H/(k1δ ) increases until there is eventually a single crossover. This results in significant melting of the SSI phases and the spontaneous production of defects. For 4.3. RESULTS AND DISCUSSION 101

2 Figure 4.4: The fraction of high spins, nHS, for selected values of ∆H/(k1δ ) with k1 > 0 and k2/k1 = 4k3/3k1 = −0.1. The red, blue. and black lines represent heating, cooling and parallel tempering calculations respectively. The parallel tempering calculations return the state with the lowest free energy, while the heating and cooling calculations indicate the limits of metastability (spinodal 1 1 3 lines). We find plateaus at nHS ' 0, 4 , 2 , 4 and 1, corresponding to the LS, H1L3,H2L2,H3L1, and HS phases. Hence, for a wide range of parameters, it is possible to tune between different Coulomb phases with temperature alone. Snapshots of these phases are shown in Fig. 4.8. each SSI phase there are two different types of defects. For a state obeying the spin-state ice rules everywhere changing a metal from a LS to HS state creates a h-defect on both of the tetrahedra connected to the metal center, 4.5a,b. Conversely, changing the spin-state of a metal from HS to LS creates two `-defects. To understand these defects it is helpful to consider a large magnetic field in the z direction, such z z that Si (1) = SH and Si (−1) = SL. The creation of two h defects increase the number of HS metals by 1 one. Thus, each defect carries a spin 2 (SH − SL) ≡ δS. Similarly, the process of creating two ` defects on the connected tetrahedra corresponds to the creation of two quasi-particles with spin −δS. It is important to note that there are no intermediate spin-states in the model. Hence, these defects arise purely as a collective effect and thus correspond to fractionalized quasi-particles with spin midway between HS and LS states. The multiple of ways to satisfy the spin-state ice rules allow defects to propagate, Fig. 4.5. For example, a metal center changing from HS to LS on a tetrahedron containing a h defect restores the ice rules on that tetrahedron and creates a ` defect on the other tetrahedron connected to the metal center, 4.5b,c. Thus, the number of h and ` type defects are not conserved. However, the topological charge Q = κδS is conserved, where κ = ±1 for tetrahedra pointing in the ±z direction. It only takes a finite 102 CHAPTER 4. MULTIPLE COULOMB PHASES

Figure 4.5: Sketch of the propagation of defects in the H2L2 phase. (a) The H2L2 vacuum state, every tetrahedron obeys the 2HS/2LS spin-state ice rule. (b) Changing the spin-state of a LS metal center (highlighted in purple) creates h defects, on both tetrahedra connected to the metal, but does not change the total topological charge, ∑Q = ∑κδS, where, κ = ±1 for tetrahedra pointing in the ±z direction, and the sum runs over all tetrahedra. (c-f) Additional spin-state changes on tetrahedra hosting defects are low-energy process and cause the defects to propagate. This conserves Q, but not the numbers of h or ` defects. Similar processes result in deconfined excitations in both the H1L3 and H3L1 SSI phases. amount of energy to move a pair of defects infinitely far apart. Hence, the fractionalized topological charges are deconfined [18]. h and ` defects carry the opposite spin. Therefore, if δS and −δS topological charges meet on the same tetrahedron they annihilate, restoring the spin-state ice rules. Passing a current through a molecule [167, 168] or mechanically stretching or squeezing its ligands [161,169] can reversibly change its spin-state; a voltage can switch between the two broken symmetry states of an antiferroelastically ordered chain [170]; and strain can induce the motion of domain walls in ordered phases of SCO materials [171]. These experiments suggest that applying a voltage or strain could also induce motion of the defects in SSI Coulomb phases, somewhat analogous to the current induced motion of skyrmions [172, 173]. Either of these effects could make SSI a valuable resource for spintronic applications.

4.4 Conclusion

Our calculations predict that three distinct Coulomb phases arise in spin crossover materials on pryochlore lattices. In each phase the low energy excitations are mobile and deconﬁned, carrying a spin midway between that of the HS and LS states. Realizing Coulomb phases beyond the 2-in/2-out phases in water and spin ices has proven extremely challenging. However, the unique role of the single molecule entropy in spin crossover materials allows temperature and pressure to change the ice rules. The physics of SCO molecules could also allow for the use of pressure and light to manipulate and control the excitations. 4.4. CONCLUSION 103

Important questions arising from this work include: (1) What are the leading quantum mechanical corrections to the Hamiltonian? Exchange is weak but spin-orbit coupling is vital for the transition of a metal from HS to LS [174] and may act like a transverse ﬁeld [156]. (2) What state do quantum mechanical corrections lead to? Is a single classical state selected (order by disorder) or does a “quantum spin-state liquid” emerge? Interestingly, an Ising model in the 3-HS/1-LS or 1-HS/3-LS states with an added transverse ﬁeld term is closely related to the quantum dimer model on the diamond lattice, which is believed host a gapless U(1) spin (state) liquid [175]. 104 CHAPTER 4. MULTIPLE COULOMB PHASES 4.5 Supplementary Information

4.5.1 Monte Carlo simulations

To investigate the properties of the Hamiltonian we employ three different types of calculation: cooling, heating, and parallel tempering. For each type of calculation we use single spin flip, loop and worm algorithms. The loop algorithm aids in relaxing the system when a large number of metastable states, that obey the ice rules, have a slightly higher energy than a long-range ordered ground state that obeys the ice rules. The algorithm works by creating a global loop update that maps between two configurations of spins that preserves the ice rules. This update takes the form of a loop since, the difference between two configurations of pseudo-spins that obey the ice rules can be decomposed into loops of flipped pseudo-spins [151]. The worm algorithm helps relax the system by removing metastable defects. When the ground state is an ice state, the only way to remove a defect is to annihilate it with a defect of the opposite topological charge. The algorithm works by creating a global string update that connects and annihilates the two defects [151]. The loop and worm algorithms are integrated into a single update. This works by initially picking a random spin and calculating the energy required to flip it. We then calculate the Boltzmann probability of this update occurring. If it is accepted then the process stops and the update occurs. If it is not accepted, then a neighbouring spin is picked based on a normalized Boltzmann probability, and the energy required to flip both spins is calculated. Similarly, the probability of this combined update occurring is checked and the process either stops or continues. This process continues for a random length of time. If the string of flipped spins overlaps at any point then the process stops. The path of flipped spins is then decomposed into a loop and a string. The probability of the loop update occurring is then checked (the string update has already been checked). For an ice state a loop update is always accepted as it has no energy cost. All calculations reported here contain three stages: initialization, equilibration, and measurement. During the equilibration and measurement stages we choose a single spin flip update with probability 1 − 2/N or a loop or worm update with probability 2/N. For the cooling calculations we initialize the model in a random configuration at the highest temperature studied. For each data point we initially equilibrate for 10N steps and then take N measurements, each containing N steps. The final state of the model at a given temperature is used as the initial state for the next temperature studied. Heating calculations are analogous to the cooling calculations, but with the initial state chosen to be the T = 0 state (shown in Fig. 4.2a) for the relevant parameters. Therefore around the first order transitions the heating and cooling curves correspond to limits of metastability (i.e., spinodal lines). In terms of the free energy, they represent the point at which the energy barrier between the phases becomes a saddle point. These lines are useful for identifying first order transitions. 4.5. SUPPLEMENTARY INFORMATION 105

2 Figure 4.6: The heat capacity per site cV for the same values of ∆H/(k1δ ) as Fig. 4.4 with fixed values of k1 > 0 and k2 = 4k3/3 = −0.1k1. The black, blue and red lines have the same meaning as 2 Fig. 4.4. Similarly, for increasing values of ∆H/(k1δ ) we observe (in the equilibrium values) (a) a first order-transition, (b) two first order transitions, (c-d) three first order transitions, (e-f) four first order transitions, (f-g) three first order transitions and a crossover, (i) two first order transitions and two crossovers, (j) one first order transition and three crossovers, (k) two crossovers and (l) a crossover. Heating and cooling the system would result in (a) no transitions, (b-c) one first order transition, (d-e) two first order transitions, (f) a three-step transition and (g) two-first order transitions.

At high temperatures updates that change the energy of the system are much more likely that at low temperatures. Parallel tempering calculations can avoid metastability and accelerate the relaxation of the system by switching the spin conﬁgurations of two or more isolated systems. In the parallel tempering calculations we initialize 100 simulations in parallel over a range of temperatures with the same parameters. For each data point we begin with 10N initialization steps. Each step is a spin ﬂip 10 update with probability 1 − N or a parallel tempering update with probability 10/N, where N is the number of lattice sites. The probability of parallel tempering updates being accepted is based on the principle of detailed balance [151]. After initialization, the calculations are run independently. We then equilibrate the system again for 10N steps and then take N measurements with N steps. The parallel tempering calculations therefore allow us to identify the lowest free energy state at every temperature. 106 CHAPTER 4. MULTIPLE COULOMB PHASES

Figure 4.7: The HS fraction, nHS, calculated on (a) cooling and (b) heating for the third nearest neighbor elastic model on the pyrochlore lattice with ﬁxed values of k1 > 0 and k2 = (4/3)k3 = −0.1k1. Lines and dots have the same meanings as in Fig. 4.2b.

Figure 4.8: Truncated snapshots of (a) H1L3, (b) H2L2 and (c) H3L1 for k1 > 0 and k2 = 4k3/2 = −0.1k1. Almost all triangles obey the spin-state ice rules, Fig. 4.1. There are two expectations to the rule; one tetrahedron containing an excess of HS (h) and one containing an excess of LS 2 2 2 (`). The snapshots are taken for (a) ∆H = 10.11k1δ and kBT = 1.12k1δ , (b) ∆H = 5.47k1δ and 2 2 2 kBT = 0.76k1δ and (c) ∆H = 5.47k1δ and kBT = 1.11k1δ . 4.5. SUPPLEMENTARY INFORMATION 107

Chapter 5

Conclusion

In this thesis we have used a simple, elastic model to explain almost all the experimentally reported bulk properties and behaviours in spin crossover (SCO) systems. We have used this model to derive structure-property relations to explain these results and make predictions. In Chapter 2, we showed that the elastic model hosts almost all spin-state phases and SCO behaviours that have been reported experimentally on square and cubic lattices. We demonstrated clear structure-property relations that underpin these results, and predicted new SCO behaviours that have not yet been reported experimentally. This research is the first step toward the rational design of SCO materials and frameworks with enhanced applications. The next step is the further identification of structure-property relations through material-specific, quantum-chemistry calculations. We hope to produce a ‘theoretical cookbook’ of SCO properties and behaviours, allowing for the highly efficient design and synthesis of SCO materials with enhanced applications. Recent advances made by Ohlrich and Powell [118] and Ruzzi [176] et al. have opened up the possibility of using density functional theory in a fast and accurate manner to study SCO systems, pulling the possibility of the rational design of SCO systems into the near future. In Chapter 3 and 4, we proposed a new states of matter, spin-state ice, that could exist in elastically frustrated SCO materials and frameworks. We demonstrated that this state is an ice phase through the identification of pinch point singularities in the neutron scattering structure factor. We demonstrated that the unique properties of SCO systems endowed these systems with interesting bulk properties and emergent particle-like behaviours. Allowing for applications in spintronic and insight into bulk and emergent properties and behaviours. The biggest questions left unanswered by this research is: what are the properties of the quantum mechanical ground state? Does this ground state violate the third law of thermodynamics? This question is quite open as there has been no research published on the collective properties of quantum SCO systems. This question is of high interest as SCO systems have been long sought after for their applications toward quantum computation [59]. Recently, it was shown by my colleagues (Ruzzi, McKenzie, and Powell) [176] and I that an exact mapping between long-range Ising-like and elastic models could be obtained by integrating out the positional degrees of freedom. We showed that the long-range interactions decayed as a power law. 109 110 CHAPTER 5. CONCLUSION

This result has interesting ramifications for the deconfinement properties of the spin-state ice phases and their relative stability. Further work will focus on using the same methodology presented in the paper to study the properties of spin-state ice on frustrated lattices. In this thesis, we have used scale invariant organisational principles to connect two seemingly unconnected areas of physics: spin ice and spin crossover. This work only scratches the surface of the deep underlying connections that intertwine our universe. Through slowly unravelling these connections, we hope to gain a more complete understanding of how emergent phenomena like the flocking birds can behave like electron clouds. Bibliography

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-C. S. Lewis