This dissertation details studies of three different frustrated magnet families: NaCaNi$_2$F$_7$, Er$_3$Ga$_5$O$_{12}$, and ErMgGaO$_4$, using a variety of techniques, including magnetization, specific heat, neutron scattering and muon spin rotation/relaxation ($\mu$SR). NaCaNi$_2$F$_7$ belongs to the fluoride pyrochlores family. In this thesis, we study the effect of the randomness on the A site on the magnetic property of B site. This chemical disorder randomness was caused by the mixture of two different elements from Group 1 and 2 (Na and Ca). DFT calculation and computation simulations indicates a possible non-centred $F-\mu -F$ muon stopping site. Zero field (ZF) and longitudinal field (LF) $\mu$SR shows that the $Ni^{2+}$ spins undergo spin freezing into a disordered ground state below 4K, with persistent spin dynamics to our lowest temperature 75 mK. We obtained high quality single crystal of Er$_3$Ga$_5$O$_{12}$ by utilizing the optical floating-zone (OFZ) technique. We performed inelastic neutron scattering measurements to determine the crystalline electric field (CEF) Hamiltonian, eigenvalues and eigenvectors, indicating an Ising-like anisotropy ground state which is also consistent with our specific heat experiment with entropy approaching $Rln(2)$. All seven crystalline electric field excitations from the ground state Kramers doublet were identified. In addition, Rietveld refinement of neutron powder diffraction data reveal that Er$_3$Ga$_5$O$_{12}$ orders into the $\Gamma_3$ magnetic structure, with ordered magnetic moment $\mu_{ord} = 5.24(4)~\mu_{B}$, in agreement with $\mu_{Ising}$ = 5.61 $\mu_{B}$ from our deduced CEF Hamiltonian. Our $\mu$SR measurements also reveals the presence of fluctuating local fields on the muon timescale, indicating exotic slow spin dynamics in the ground state. The objective of the work on ErMgGaO$_4$ was to identify a new quantum spin liquid candidate. Single crystals of ErMgGaO$_4$ were obtained through (OFZ) technique as well, and the structure was refined and confirmed by powder X-ray diffraction and Laue diffraction after a repeated refinement of our synthesis process. Susceptibility measurements reveal no evidence of a magnetic transition down to 0.5 K, in agreement with specific heat measurement which exhibit no anomalies which would have been evidence of an ordering transition. ZF-$\mu$SR measurements reveal no sign of coherent long range order or spin freezing down to 25 mK, while LF-$\mu$SR measurements shows persistent spin dynamics at 25 mK. Our observations provides evidence for a quantum spin liquid ground state in this compound. / Thesis / Doctor of Philosophy (PhD)

In this thesis, we use Muon Spin Rotation, Relaxation, and Resonance (μSR) as a probe for three frustrated pyrochlore systems; Gd2ScNbO7 (GSNO), Nd2ScNbO7 (NSNO) and Sm2Ti2O7 (STO), as well as the type-I superconductor BeAu. We grew all of the pyrochlore samples at McMaster using the Optical Floating Zone method. We make use of Direct Current (DC) and Alternating Current (AC) susceptibility, powder x-ray diffraction and Laue x-ray diffraction to characterize our samples. We make use of AC susceptibility measurements to explore the dynamics of the classical spin-ice Dy2Ti2O7 (DTO) and find that the system acts as a supercooled magnetic liquid, analogous to glassforming dielectric liquids. We find GSNO is a dense spin-glass based on our μSR and AC susceptibility measurements. NSNO is a moment fragmentation candidate where spin-ice, as well as all-in all-out magnetic ordering, are observed simultaneously. Our μSR measurements on this material show a strong similarity to another moment fragmentation candidate, Nd2Zr2O7, suggesting NSNO may be in a similar state. STO is a closely related compound that fully orders into a magnetic state which we study using μSR. We find subtle evidence of this magnetic transition along with persistent spin dynamics which we suggest has a common, but as of yet unexplained, origin as other frustrated pyrochlores measured in μSR. Finally, we use μSR to measure the temperature dependence of the critical field in the type-I superconductor BeAu. Using an ellipsoid of BeAu and a pressure cell, we study the magnetic properties of the sample under pressure. / Thesis / Candidate in Philosophy / In this thesis, we use Muon Spin Rotation, Relaxation, and Resonance (μSR) as a probe for three frustrated pyrochlore systems; Gd2ScNbO7 (GSNO), Nd2ScNbO7 (NSNO) and Sm2Ti2O7 (STO), as well as the type-I superconductor BeAu. We grew all of the pyrochlore samples at McMaster using the Optical Floating Zone method. We make use of Direct Current (DC) and Alternating Current (AC) susceptibility, powder x-ray diffraction and Laue x-ray diffraction to characterize our samples. We make use of AC susceptibility measurements to explore the dynamics of the classical spin-ice Dy2Ti2O7 (DTO) and find that the system acts as a supercooled magnetic liquid, analogous to glassforming dielectric liquids. We find GSNO is a dense spin-glass based on our μSR and AC susceptibility measurements. NSNO is a moment fragmentation candidate where spin-ice, as well as all-in all-out magnetic ordering, are observed simultaneously. Our μSR measurements on this material show a strong similarity to another moment fragmentation candidate, Nd2Zr2O7, suggesting NSNO may be in a similar state. STO is a closely related compound that fully orders into a magnetic state which we study using μSR. We find subtle evidence of this magnetic transition along with persistent spin dynamics which we suggest has a common, but as of yet unexplained, origin as other frustrated pyrochlores measured in μSR. Finally, we use μSR to measure the temperature dependence of the critical field in the type-I superconductor BeAu. Using an ellipsoid of BeAu and a pressure cell, we study the magnetic properties of the sample under pressure.

Models of magnetism show complex collective behaviour which arises from simple interactions among microscopic degrees of freedom. Upon cooling from high temperatures conventional magnets typically undergo a phase transition to a magnetically ordered phase due to microscopic interactions which favour an ordered state. In frustrated magnets however, competing microscopic interactions place non-trivial constraints on the allowed configurations at low temperature, without selecting a unique ordered state. In this thesis, we investigate the collective behaviour of a paradigmatic frustrated magnet, the classical Heisenberg model on the pyrochlore lattice with antiferromagnetic nearest neighbour interactions. Within a self consistent Gaussian approximation, we derive analytic expressions for correlation functions which match Monte Carlo simulations extremely well at all temperatures. We study the precessional dynamics of the model and provide a comprehensive description of the dynamics by constructing an analytically tractable stochastic model by extending the self-consistent Gaussian approximation to include dynamics. We relate these results to other highly constrained models. Real experimental systems often have features that go beyond the phenomenology afforded by the simplest models; we investigate the effects of further neighbour interactions on paramagnetic spin correlations, and propose further neighbour interactions as the mechanism underlying experimentally observed patterns of scattering in frustrated spinel compounds. In the dynamics linearized around a ground-state, the macroscopic degeneracy of the classical model leads to modes with zero frequency. Small perturbations stabilize ordered states and lift such zero modes to finite frequency. The ordered state has two widely separated energy scales both of which affect the dynamics, the leading scale of nearest neighbour exchange coupling, and the much smaller scale of the perturbation which relieves the frustration. We investigate the interplay between these widely different energy scales in setting the mode frequencies in states ordered by weak interactions.

Complex lanthanide oxide systems are known to host novel phases of matter, while also providing functionality for practical applications. In this dissertation, the structural, magnetic and magnetocaloric properties of three families of lanthanide oxides have been studied with the dual aims of investigating the magnetic behaviour and identifying promising magnetic refrigerants for cooling to temperatures currently accessible using non-renewable liquid He. The thesis presents a two-part study of the magnetic and magnetocaloric properties of the geometrically frustrated lanthanide garnets, where the magnetic $Ln^{3+}$ form corner-sharing triangles. First, the family of garnets $Ln_3A_2X_3$O$_{12}$, $Ln$ = Gd, Tb, Dy, Ho, $A$ = Ga, Sc, In, Te, $X$ = Ga, Al, Li are investigated. Changes to the single-ion anisotropy of the magnetic ion as well as variations in the chemical pressure radically alters the nature of magnetic ordering, the degree of frustration and the magnetocaloric performance. In the second part, the garnets $Ln_3A$Ga$_4$O$_{12}$, $Ln$ = Gd, Tb, Dy, Ho, $A$ = Cr, Mn, are studied. Introducing additional spins significantly reduces the frustration in the garnet lattice. Low temperature powder neutron diffraction of Ho$_3$MnGa$_4$O$_{12}$ reveals concomitant ordering of Ho$^{3+}$ and Mn$^{3+}$ moments below the ordering temperature, $T_N$ = 5.8 K. The magnetocaloric performance of $Ln$_3CrGa$_4$O$_{12}$, $Ln$ = Gd, Dy, Ho, greatly surpasses that of the parent $Ln_3$Ga$_5$O$_{12}$ at $T$ = 2 K. The final results chapters in the thesis describe the magnetism and magnetocaloric effect in the lanthanide orthoborates, $Ln$BO$_3$ , $Ln$ = Eu, Gd, Tb, Dy, Ho, Er, Yb and the lanthanide metaborates, $Ln$(BO$_2$)$_3$, $Ln$ = Pr, Nd, Gd, Tb. The magnetic $Ln^{3+}$ form slightly distorted edge-sharing triangular layers in $Ln$BO$_3$. Unique magnetic features are observed, including short-range ordering and spin reorientation transitions depending on the single-ion anisotropy of the $Ln^{3+}$. The $Ln$BO$_3$ are also efficient magnetocalorics in the liquid helium temperature range. The lanthanide metaborates contain one-dimensional chains of magnetic lanthanide ions. Bulk magnetic measurements show features consistent with low-dimensional magnetism, such as magnetisation plateaux at one-third of the saturation magnetisation for Nd(BO$_2$)$_3$ and Tb(BO$_2$)$_3$ in a field of 14 T. This thesis provides insight into the fundamental magnetic properties of complex lanthanide oxide systems and also demonstrates strategies for identifying new magnetocaloric materials: both through chemical control of the structure of well-known magnetocalorics and by studying materials that have not been explored previously. The results pave the way for further in-depth investigations and finding new magnetic coolants based on complex lanthanide oxide systems.

Order-by-disorder is the phenomenon of the selection of a long-ranged ordered state by fluctuations in a many-body system. This mechanism, at first sight, seems paradoxical, since fluctuations (disorder) intuitively tend to suppress the order. However, when ObD happens, disorder works in favour of a particular ordered phase. Order-by-disorder can happen where an accidental degeneracy occurs in classical or mean field treatment of a system. These degeneracies, which are not due to exact symmetries of the system, can then be lifted by quantum or thermal fluctuations. The ObD phenomenon is ubiquitous in condensed matter systems with competing or frustrated interactions. Traditionally, the ObD is studied at $T = 0^+$ where the ground state of the system can be selected by quantum fluctuations. The study of ObD at temperature regimes near criticality, $T \lesssim Tc$ where transition happens from the paramagnetic phase to an ordered phase, however, have not received as much attention. In this thesis, we study the ObD phenomenon in three dimensional frustrated sys- tems close to criticality. We consider 3-component classical Heisenberg spins on pyrochlore lattice and FCC lattice. In the former, the spins interact via a Hamiltonian that can include the most general nearest-neighbour symmetry allowed bilinear interactions, long- range magnetostatic dipole-dipole interaction and second and/or third nearest-neighbour exchange interactions. However, in the latter, the Hamiltonian only consists of long-range magnetostatic dipole-dipole interactions. These two systems, correspond to insulating rare- earth pyrochlore oxides and rare-earth FCC salts. The mean field treatment shows, that accidental $O(4)$ and $U(1)$ symmetries emerge in two different regions of the parameter space of the Hamiltonian of pyrochlore system. While in the FCC system, an accidental $O(3)$ symmetry emerges at the mean field level. We show that fluctuations break these symmetries by respectively introducing cubic (in 4-vector and 3-vector models) and hexag- onal anisotropies to the free-energy of the system. To study these system beyond mean field approximation, we use Monte Carlo simulations, spin wave theory and we develop the E-TAP method which is an extended version of the method originally proposed by Thouless, Anderson and Palmer to study fluctuations in spin glasses.

This work is the result of two separate lines of study into the family of frustrated rare-earth pyrochlores with ferromagnetic interactions. The first is an examination of Yb2Ti2O7 and Ho2Ti2O7 through small angle neutron scattering (SANS) techniques. The sensitivity to anisotropic ferromagnetic correlations of SANS makes it an ideal tool to investigate the anomalous scattering of Yb2Ti2O7 , and to take a closer look into the low Q region of the spin ice Ho2Ti2O7 , where long range dipolar effects modify magnetic scattering. We show that in Yb2Ti2O7 the ferromagnetic order observed by other researchers coexists with short range HHH correlations to 0.03 K. We identify a new feature in Yb2Ti2O7 , a medium range (on the order of 100 ̊A) ferromagnetic correlation which appears to correlate well with the systems heat capacity anomaly. In Ho2Ti2O7 , we observe isotropic magnetic scattering in the low Q region that correlates to the system heat capacity anomaly. The second research project involves the system Ho2Sn2−xTixO7 with x=0,0.5,1,1.5,2. We use SQUID magnetometry and inelastic neutron scattering to examine the effects of B-site disorder on the spin ice system. We find some discrepancies with previous generation instrument results for the crystal electric field (CEF) Hamiltonian, and find results consistent with our previously hypothesized picture of the effects of B-site disorder on the CEF levels. We also observe increased spin dynamics in the disordered compounds, which is consistent with recent theory work predicting a transition to a quantum spin liquid phase in disordered non-kramers spin ice compounds. / Thesis / Master of Science (MSc)

Thesis advisor: Michael J. Graf / Quantum spin liquids (QSL)’s have been one of the most hotly researched areas ofcondensed matter physics for the past decade. Yet, science has yet to unconditionally identify any one system as harboring a QSL state. This is because QSL’s are largely defined as systems whose electronic spins do not undergo a thermodynamic transition as T→0. Quantum spin liquids remain fully paramagnetic, including dynamical spin fluctuations, at T=0. As a result, distinguishing a QSL system from a conventionally disordered system remains an outstanding challenge. If a system spin freezes or magnetically orders, it cannot be a QSL. In this thesis I present published experiments I have performed on QSL candidate materials. By using muon spin rotation (μSR) and AC magnetic susceptibility I have evaluated the ground states of several candidates for the absence of long-range magnetic disorder and low-temperature spin-fluctuations. For the systems which order or spin-freeze, my research provided key knowledge to the field of frustrated magnetism. The systems I studied are as follows: The geometrically frustrated systems NaYbO2 and LiYbO2; the Kitaev honeycomb systems Cu2IrO3 and Ag3LiIr2O6; and the metallic kagome system KV3Sb5. Each of these systems brought new physics to the field of frustrated magnetism. NaYbO2 is a promising QSL candidate. LiYbO2 harbors an usual form of spiral incommensurate order that has a staggered transition. Cu2IrO3 has charge state disorder that results in a magnetically inhonogenious state. Ag3LiIr2O6 illustrates the role structural disorder plays in disguising long-range magnetic order. And finally, KV3Sb5 isn’t conventionally magnetic at all; our measurements ruled out ionic magnetism and uncovered a type-II superconductor. Our measurements on KV3Sb5 stimulated further research into KV3Sb5 and it’s unconventional electronic states. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.

The extended kagome lattice, composed of alternating kagome and triangular layers, provides a novel geometry for frustrated magnetism. In this thesis, we study the properties of Heisenberg spins with nearest-neighbour antiferromagnetic interactions on this lattice. In common with many other models of frustrated magnets, this system has highly degenerate classical ground states. It is set apart from other examples, however, by the strong interlayer correlations between triangular layer spins. We study the implications of such correlations in both the statics and dynamics. We characterise classical ground states using a flux picture for a single layer of kagome spins, a theoretical description that sets geometrical bounds on correlations. We quantify the divergent but sub-extensive ground state degeneracy by a Maxwellian counting argument, and verify this calculation by analysing the energy eigenvalues of numerical ground states. We explore the ground state connectedness but do not reach firm conclusions on this issue. We use the self-consistent Gaussian approximation (SCGA) to calculate static spin correlations at finite temperature. The results of these calculations agree well with elastic neutron scattering experiments. We derive an expression for the effective interlayer interaction between kagome spins by integrating out the triangular lattice spins. We use linear spinwave theory to compute the spin excitation spectrum numerically. This shows encouraging similarity with inelastic neutron scattering data on a single-crystal YBaCo$_4$O$_7$ sample, for a wide range of wavevector and frequency. This agreement shows that our spin model is a reasonable description of the physics, and suggests that this numerical technique might be useful for other geometrically frustrated magnets. We study the dynamics analytically using the stochastic SCGA recently developed for the pyrochlore lattice. For technical reasons, we apply this technique on a related model, the stacked kagome lattice, rather than on the extended kagome lattice itself. From this we find slow relaxation at low temperature, with a rate ~ T² compared to the faster ~ T scaling for the pyrochlore. Strikingly, in simulations of the dynamics on the extended kagome lattice by numerical integration of the semiclassical equations of motion, we find two different relaxation rates. Kagome layer spins relax more quickly than the triangular layer spins, having ~ T.

This thesis presents research of two novel magnetic materials, LiHoF4 and Tb2Ti2O7. Experiments were performed at low temperatures and in an applied magnetic field to study thermal expansion and magnetostriction using a capacitive dilatometer designed during this project. This thesis presents 3 distinct topics. This manuscript begins with a thermodynamic description of thermal expansion and magnetostriction. The design of a capacitive dilatometer suitable for use at ultra-low temperatures and in high magnetic fields is presented. The thermal expansion of oxygen free high conductivity copper is used as a test of the absolute accuracy of the dilatometer. The first material studied using this dilatometer was LiHoF4. Pure LiHoF4 is a dipolar coupled Ising ferromagnet and in an applied transverse magnetic field is a good representation of the transverse field Ising model. An ongoing discrepancy between theoretical and experimental work motivates further study of this textbook material. Presented here are thermal expansion and magnetostriction measurements of LiHoF4 in an applied transverse field. We find good agreement with existing experimental work. This suggests that there is some aspect of LiHoF4 or the effect of quantum mechanical fluctuations at finite temperatures which is not well understood. The second material studied is the spin liquid Tb2Ti2O7. Despite theoretical predictions that Tb2Ti2O7 will order at finite temperature, a large body of experimental evidence demonstrates that spins within Tb2Ti2O7 remain dynamic to the lowest temperatures studied. In addition Tb2Ti2O7 also exhibits anomalous thermal expansion below 20K, giant magnetostriction, and orders in an applied magnetic field. Thermal expansion and magnetostriction measurements of Tb2Ti2O7 are presented in applied longitudinal and transverse fields. Zero-field thermal expansion measurements do not repeat the previously observed anomalous thermal expansion. A large feature is observed in thermal expansion at 100mK, in rough agreement with existing experimental work. Longitudinal and transverse magnetic fields were applied to Tb2Ti2O7. Longitudinal magnetostriction measurements show qualitatively di erent behavior than previous observations. These measurements were taken along di erent crystal axes so direct comparison cannot be made. Thermal expansion measurements in an applied transverse field show evolution with the strength of the applied field. This evolution may relate to an ordering transition, however difficulties in repeatability in a transverse field require that these results be repeated in an improved setup.

Various experiments on Yb₂Ti₂O₇ have shown evidence of strange magnetic behaviour at low temperatures. Specific heat measurements on powder samples of Yb₂Ti₂O₇ show evidence of a sharp peak, indicating the occurence of a first order phase transition. Meanwhile, neutron scattering, Mössbauer absorption, and μSR measurements find no evidence of long range order below the temperature of this phase transition, leaving the nature of the low temperature phase a mystery. Quantifying the magnetic interactions in this material should allow us to better understand the low temperature behaviour of this material. In this study, we fit a symmetry allowed nearest-neighbour bilinear exchange model to quasi-elastic neutron scattering data collected well above the temperature of the experimentally observed phase transition. This neutron scattering data shows evidence of rods of scattering intensity along the ⟨111⟩ crystallographic directions. Neutron scattering probes the correlations between magnetic moments in a material, so fitting an interaction model to the neutron scattering is equivalent to fitting the interactions to the magnetic correlations. These correlations are driven by the interactions between the magnetic moments, so the neutron scattering should give us direct access to the form of these interactions. Using this method we successfully identify an anisotropic nearest-neighbour bilinear exchange model that reproduces the experimentally observed quasi-elastic neutron scattering. With this model we then proceed to compute real space correlation functions, finding that the rods of neutron scattering arise from the presence of strong correlations along nearest-neighbour chains. We also compute the bulk susceptibility and local susceptibility, obtaining very good fits to experiment with no variation of the model determined from the neutron scattering. The success of these calculations provides a further independent confirmation of the success of our interaction model in describing the magnetic interactions in Yb₂Ti₂O₇. Finally, we present a brief summary of ongoing work based on our anisotropic exchange model, including mean field calculations to determine the low temperature ground state of this model and classical Monte Carlo simulations to study the phase transition present in this model. We also discuss potential further studies of this and other models.