Kagome Ice Research Articles

Tailoring magnon modes by extending square, kagome, and trigonal spin ice lattices vertically via interlayer coupling of trilayer nanomagnets

In this work high-frequency magnetization dynamics and statics of artificial spin-ice lattices with different geometric nanostructure array configurations are studied where the individual nanostructures are composed of ferromagnetic/non-magnetic/ferromagnetic trilayers with different non-magnetic thicknesses. These thickness variations enable additional control over the magnetic interactions within the spin-ice lattice that directly impacts the resulting magnetization dynamics and the associated magnonic modes. Specifically the geometric arrangements studied are square, kagome and trigonal spin ice configurations, where the individual lithographically patterned nanomagnets (NMs) are trilayers, made up of two magnetic layers of Ni81Fe19 of 30 nm and 70 nm thickness respectively, separated by a non-magnetic copper layer of either 2 nm or 40 nm. We show that coupling via the magnetostatic interactions between the ferromagnetic layers of the NMs within square, kagome and trigonal spin-ice lattices offers fine-control over magnetization states and magnetic resonant modes. In particular, the kagome and trigonal lattices allow tuning of an additional mode and the spacing between multiple resonance modes, increasing functionality beyond square lattices. These results demonstrate the ability to move beyond quasi-2D single magnetic layer nanomagnetics via control of the vertical interlayer interactions in spin ice arrays. This additional control enables multi-mode magnonic programmability of the resonance spectra, which has potential for magnetic metamaterials for microwave or information processing applications.

Nonequilibrium generation of charge defects in kagome spin ice under slow cooling.

Kagome spin ice is one of the canonical examples of highly frustrated magnets. The effective magnetic degrees of freedom in kagome spin ice are Ising spins residing on a two-dimensional network of corner-sharing triangles. Due to strong geometrical frustration, nearest-neighbor antiferromagnetic interactions on the kagome lattice give rise to a macroscopic number of degenerate classical ground states characterized by ice rules. Elementary excitations at low temperatures are defect-triangles that violate the ice rules and carry an additional net magnetic charge relative to the background. We perform large-scale Glauber dynamics simulations to study the nonequilibrium dynamics of kagome ice under slow cooling. We show that the density of residual charge defects exhibits a power-law dependence on the quench rate for the class of algebraic cooling protocols. The numerical results are well captured by the rate equationfor the charge defects based on the reaction kinetics theory. As the relaxation time of the kagome ice phase remains finite, there is no dynamical freezing as in the Kibble-Zurek scenario. Instead, we show that the power-law behavior originates from a thermal excitation that decays algebraically with time at the late stage of the cooling schedule. Similarities and differences in quench dynamics of other spin ice systems are also discussed.

Discrete degeneracies distinguished by the anomalous Hall effect in a metallic kagome ice compound

Discrete degeneracies distinguished by the anomalous Hall effect in a metallic kagome ice compound

Neutron scattering from fragmented frustrated magnets

The fragmentation description is used to analyze calculated neutron scattering intensities from kagome ice and spin ice systems. The longitudinal, transverse, and harmonic fragments produce independent contributions to the neutron scattering intensity. This framework is used to analyze the ordering due to quantum fluctuations in the topologically constrained phase of kagome ice and the monopole crystal phase of spin ice. Here, quantum fluctuations are restricted to the transverse fragment, and they drive the system into a double-$q$ structure in which longitudinal and transverse fragments have a different ordering wave vector. The intensity reduction of the Bragg peaks for the transverse fragments, compared with known classical limits, can be used as a diagnostic tool for quantum fluctuations. Published quantum Monte Carlo data for spin ice in a [111] field are consistent with the proposed protocol.

Field angle dependent resonant dynamics of artificial spin ice lattices

Artificial spin ice structures which are networks of coupled nanomagnets arranged on different lattices that exhibit a number of interesting phenomena are promising for future information processing. We report reconfigurable microwave properties in artificial spin ice structures with three different lattice symmetries namely square, kagome, and triangle. Magnetization dynamics are systematically investigated using field angle dependent ferromagnetic resonance spectroscopy. Two distinct ferromagnetic resonance modes are observed in square spin ice structures in contrast with the three well-separated modes in kagome and triangular spin ice structures that are spatially localized at the center of the individual nanomagnets. A simple rotation of the sample placed in magnetic field results in the merging and splitting of the modes due to the different orientations of the nanomagnets with respect to the applied magnetic field. Magnetostatic interactions are found to shift the mode positions after comparing the microwave responses from the array of nanomagnets with control simulations with isolated nanomagnets. Moreover, the extent of the mode splitting has been studied by varying the thickness of the lattice structures. The results have potential implications for microwave filter-type applications which can be operated for a wide range of frequencies with ease of tunability.

X-ray imaging of the magnetic configuration of a three-dimensional artificial spin ice building block

The extension of artificial spin systems to the third dimension offers advances in functionalities and opportunities for technological applications. One of the main challenges facing their realization is the fabrication of three-dimensional geometries with nanoscale resolution. In this work, we combine two-photon lithography with deformation-free pyrolysis and a GdCo coating to create a three-dimensional (3D) tripod structure that represents a building block of an 3D artificial spin ice, surrounded by a two-dimensional magnetic film. We map the three-dimensional magnetic configuration of the structure and its surroundings using soft x-ray magnetic laminography. In this way, we determine the magnetic configuration of the tripod nanostructure to be in the low-energy two-in-one-out spin ice state, observed at the 2D vertex of a kagome ice and predicted for three-dimensional vertices of magnetic buckyball structures. In contrast to isolated vertices, the degeneracy of this state can be lifted by the surrounding film, which also offers a route toward the controlled injection of emergent charges. This demonstration of the building block of a 3D spin system represents the first step toward the realization and understanding of more complex 3D artificial spin systems.

Magnetization process and ordering of the S=12 pyrochlore Heisenberg antiferromagnet in a magnetic field

We study the $S=\frac{1}{2}$ pyrochlore Heisenberg antiferromagnet in a magnetic field. Using large-scale density-matrix renormalization group calculations for clusters with up to 128 spins, we find indications of a finite triplet gap, causing a threshold field to nonzero magnetization in the magnetization curve. We obtain a robust saturation field consistent with a magnon crystal, although the corresponding $5/6$ magnetization plateau is very slim and possibly unstable. Most remarkably, there is a pronounced and apparently robust $1/2$ magnetization plateau where the ground state breaks the rotational symmetry of the lattice, exhibiting oppositely polarized spins on alternating kagome and triangular planes. Reminiscent of the kagome ice plateau of the pyrochlore Ising antiferromagnet known as spin ice, it arises via a much more subtle ``quantum order by disorder'' mechanism.

Tunable critical correlations in kagome ice

We present a comprehensive experimental and theoretical study of the kagome ice Coulomb phase, that explores the fine tuning of critical correlations by applied field, temperature and crystal orientation. The continuous modification of algebraic correlations is observed by polarised neutron scattering experiments and is found to be well described by numerical simulations of an idealised model. We further clarify the thermodynamics of field tuned Kasteleyn transitions and demonstrate some dramatic finite size scaling properties that depend on how topological string defects wind around the system boundaries. We conclude that kagome ice is a remarkable example of a critical and topological state in a real system that may be subject to fine experimental control.

Spin-Wave Dynamics and Symmetry Breaking in an Artificial Spin Ice.

Artificial spin ices are periodic arrangements of interacting nanomagnets which allow investigating emergent phenomena in the presence of geometric frustration. Recently, it has been shown that artificial spin ices can be used as building blocks for creating functional materials, such as magnonic crystals. We investigate the magnetization dynamics in a system exhibiting anisotropic magnetostatic interactions owing to locally broken structural inversion symmetry. We find a rich spin-wave spectrum and investigate its evolution in an external magnetic field. We determine the evolution of individual modes, from building blocks up to larger arrays, highlighting the role of symmetry breaking in defining the mode profiles. Moreover, we demonstrate that the mode spectra exhibit signatures of long-range interactions in the system. These results contribute to the understanding of magnetization dynamics in spin ices beyond the kagome and square ice geometries and are relevant for the realization of reconfigurable magnonic crystals based on spin ices.

Quantum Versus Classical Spin Fragmentation in Dipolar Kagome Ice Ho3Mg2Sb3O14.

A promising route to realize entangled magnetic states combines geometrical frustration with quantum-tunneling effects. Spin-ice materials are canonical examples of frustration, and Ising spins in a transverse magnetic field are the simplest many-body model of quantum tunneling. Here, we show that the tripod-kagome lattice material Ho3Mg2Sb3O14 unites an icelike magnetic degeneracy with quantum-tunneling terms generated by an intrinsic splitting of the Ho3+ ground-state doublet, which is further coupled to a nuclear spin bath. Using neutron scattering and thermodynamic experiments, we observe a symmetry-breaking transition at to a remarkable state with three peculiarities: a concurrent recovery of magnetic entropy associated with the strongly coupled electronic and nuclear degrees of freedom; a fragmentation of the spin into periodic and icelike components; and persistent inelastic magnetic excitations down to . These observations deviate from expectations of classical spin fragmentation on a kagome lattice, but can be understood within a model of dipolar kagome ice under a homogeneous transverse magnetic field, which we survey with exact diagonalization on small clusters and mean-field calculations. In Ho3Mg2Sb3O14, hyperfine interactions dramatically alter the single-ion and collective properties, and suppress possible quantum correlations, rendering the fragmentation with predominantly single-ion quantum fluctuations. Our results highlight the crucial role played by hyperfine interactions in frustrated quantum magnets and motivate further investigations of the role of quantum fluctuations on partially ordered magnetic states.

Bypassing Dynamical Freezing in Artificial Kagome Ice.

Spin liquids are correlated, disordered states of matter that fluctuate even at low temperatures. Experimentally, the extensive degeneracy characterizing their low-energy manifold is expected to be lifted, for example, because of dipolar interactions, leading to an ordered ground state at absolute zero. However, this is not what is usually observed, and many systems, whether they are chemically synthesized or nanofabricated, dynamically freeze before magnetic ordering sets in. In artificial realizations of highly frustrated magnets, ground state configurations, and even low-energy manifolds, thus remain out of reach for practical reasons. Here, we show how dynamical freezing can be bypassed in an artificial kagome ice. We illustrate the efficiency of our method by demonstrating that the apriori dynamically inaccessible ordered ground state and fragmented spin liquid configurations can be obtained reproducibly, imaged in real space at room temperature, and studied conveniently. We then identify the mechanism by which dynamical freezing occurs in the dipolar kagome ice.

Emergent snake magnetic domains in canted kagome ice

We study the two-dimensional kagome-ice model derived from a pyrochlore lattice with second- and third-neighbor interactions. The canted moments align along the local $\langle 111 \rangle$ axes of the pyrochlore and respond to both in-plane and out-of-plane external fields. We find that the combination of further-neighbor interactions together with the external fields introduces a rich phase diagram with different spin textures. Close to the phase boundaries, metastable $\textit{"snake"}$ domains emerge with extremely long relaxation time. Our kinetic Monte Carlo analysis of the magnetic-field quench process from saturated state shows unusually slow dynamics. Despite that the interior spins are almost frozen in snake domains, the spins on the edge are free to fluctuate locally, leading to frequent creation and annihilation of monopole-anti-monopole bound states. Once the domains are formed, these excitations are localized and can hardly propagate due to the energy barrier of snakes. The emergence of such snake domains may shed light on the experimental observation of dipolar spin ice under tilted fields, and provide a new strategy to manipulate both spin and charge textures in artificial spin ice.

Tunneling-induced restoration of classical degeneracy in quantum kagome ice

Quantum effect is expected to dictate the behaviour of physical systems at low temperature. For quantum magnets with geometrical frustration, quantum fluctuation usually lifts the macroscopic classical degeneracy, and exotic quantum states emerge. However, how different types of quantum processes entangle wave functions in a constrained Hilbert space is not well understood. Here, we study the topological entanglement entropy (TEE) and the thermal entropy of a quantum ice model on a geometrically frustrated kagome lattice. We find that the system does not show a $Z_2$ topological order down to extremely low temperature, yet continues to behave like a classical kagome ice with finite residual entropy. Our theoretical analysis indicates an intricate competition of off-diagonal and diagonal quantum processes leading to the quasi-degeneracy of states and effectively, the classical degeneracy is restored.

Anisotropic exchange Hamiltonian, magnetic phase diagram, and domain inversion of Nd2Zr2O7

We present thermodynamic and neutron scattering measurements on the quantum spin ice candidate Nd$_2$Zr$_2$O$_7$. The parameterization of the anisotropic exchange Hamiltonian is refined based on high-energy-resolution inelastic neutron scattering data together with thermodynamic data using linear spin wave theory and numerical linked cluster expansion. Magnetic phase diagrams are calculated using classical Monte Carlo simulations with fields along \mbox{[100]}, \mbox{[110]} and \mbox{[111]} crystallographic directions which agree qualitatively with the experiment. Large hysteresis and irreversibility for \mbox{[111]} is reproduced and the microscopic mechanism is revealed by mean field calculations to be the existence of metastable states and domain inversion. Our results shed light on the explanations of the recently observed dynamical kagome ice in Nd$_2$Zr$_2$O$_7$ in \mbox{[111]} fields.

Competing orders in colloidal kagome ice: Importance of the in-trap motion of the particles

Artificial ice systems have been designed to replicate paradigmatic phenomena observed in frustrated spin systems. Here we present a detailed theoretical analysis based on Monte-Carlo simulations of the low energy phases in an artificial colloidal ice system, a recently introduced ice system where an ensemble of repulsive colloids are two-dimensionally confined by gravity to a lattice of double wells at a one-to-one filling. Triggered by recent results obtained by Brownian dynamics simulations [A. Lib\'al et al, Phys. Rev. Lett. 120, 027204 (2018)], we analyze the energetics and the phase transitions that occur in the honeycomb geometry (realizing the analogue of a spin ice system on a kagome lattice) when decreasing the temperature. When the particles are restricted to occupy the two minima of the potential well, we recover the same phase diagram as the dipolar spin ice system, with a long range ordered chiral ground state. In contrast, when considering the particle motion and their relaxation within the traps, we observe ferromagnetic ordering at low temperature. This observation highlights the fundamental role played by the continuous motion of the colloids in artificial ice systems.

Evidence for dynamic kagome ice

The search for two-dimensional quantum spin liquids, exotic magnetic states remaining disordered down to zero temperature, has been a great challenge in frustrated magnetism over the last few decades. Recently, evidence for fractionalized excitations, called spinons, emerging from these states has been observed in kagome and triangular antiferromagnets. In contrast, quantum ferromagnetic spin liquids in two dimensions, namely quantum kagome ices, have been less investigated, yet their classical counterparts exhibit amazing properties, magnetic monopole crystals as well as magnetic fragmentation. Here, we show that applying a magnetic field to the pyrochlore oxide Nd2Zr2O7, which has been shown to develop three-dimensional quantum magnetic fragmentation in zero field, results in a dimensional reduction, creating a dynamic kagome ice state: the spin excitation spectrum determined by neutron scattering encompasses a flat mode with a six arm shape akin to the kagome ice structure factor, from which dispersive branches emerge.

Magnetotransport in Artificial Kagome Spin Ice

Magnetic nanoarray with special geometries exhibits nontrivial collective behaviors similar to those observed in the spin ice materials. Here we present a novel circuit model to describe the complex magnetotransport phenomena in artificial kagome spin ice. In this picture, the system can be viewed as a resistor network driven by voltage sources that are located at vertices of the honeycomb array. The differential voltages across different terminals of these sources are related to the ice-rules that govern the local magnetization ordering. The circuit model relates the transverse Hall voltage of kagome ice to the underlying spin correlations. Treating the magnetic nanoarray as metamaterials, we present a mesoscopic constitutive equation relating the Hall resistance to magnetization components of the system. We further show that the Hall signal is significantly enhanced when the kagome ice undergoes a magnetic-charge ordering transition. Our analysis can be readily generalized to other lattice geometry, providing a quantitative method for the design of magnetoresistance devices based on artificial spin ices.

Nanostructured complex oxides as a route towards thermal behavior in artificial spin ice systems

We have used soft x-ray photoemission electron microscopy to image the magnetization of single domain La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ nano-islands arranged in geometrically frustrated configurations such as square ice and kagome ice geometries. Upon thermal randomization, ensembles of nano-islands with strong inter-island magnetic coupling relax towards low-energy configurations. Statistical analysis shows that the likelihood of ensembles falling into low-energy configurations depends strongly on the annealing temperature. Annealing to just below the Curie temperature of the ferromagnetic film (T$_{C}$ = 338 K) allows for a much greater probability of achieving low energy configurations as compared to annealing above the Curie temperature. At this thermally active temperature of 325 K, the ensemble of ferromagnetic nano-islands explore their energy landscape over time and eventually transition to lower energy states as compared to the frozen-in configurations obtained upon cooling from above the Curie temperature. Thus, this materials system allows for a facile method to systematically study thermal evolution of artificial spin ice arrays of nano-islands at temperatures modestly above room temperature.

Theory of quantum kagome ice and vison zero modes

We derive an effective $Z_2$ gauge theory to describe the quantum kagome ice (QKI) state that has been observed by Carrasquilla $\textit{et. al.}$ in Monte Carlo studies of the $S = 1/2$ kagome XYZ model in a Zeeman field. The numerical results on QKI are consistent with, but do not confirm or rule out, the hypothesis that it is a $Z_2$ spin liquid. Our effective theory allows us to explore this hypothesis and make a striking prediction for future numerical studies, namely that symmetry-protected vison zero modes arise at lattice disclination defects, leading to a Curie defect term in the spin susceptibility, and a characteristic $(N_{dis} - 1) \ln 2$ contribution to the entropy, where $N_{dis}$ is the number of disclinations. Only the $Z_2$ Ising symmetry is required to protect the vison zero modes. This is remarkable because a unitary $Z_2$ symmetry cannot be responsible for symmetry-protected degeneracies of local degrees of freedom. We also discuss other signatures of symmetry fractionalization in the $Z_2$ spin liquid, and phase transitions out of the $Z_2$ spin liquid to nearby ordered phases.

Magnetic diffuse scattering in artificial kagome spin ice

The study of magnetic correlations in dipolar-coupled nanomagnet systems with synchrotron x-ray scattering provides a means to uncover emergent phenomena and exotic phases, in particular in systems with thermally active magnetic moments. From the diffuse signal of soft x-ray resonant magnetic scattering, we have measured magnetic correlations in a highly dynamic artificial kagome spin ice with sub-70-nm Permalloy nanomagnets. On comparing experimental scattering patterns with Monte Carlo simulations based on a needle-dipole model, we conclude that kagome ice I phase correlations exist in our experimental system even in the presence of moment fluctuations, which is analogous to bulk spin ice and spin liquid behavior. In addition, we describe the emergence of quasi-pinch-points in the magnetic diffuse scattering in the kagome ice I phase. These quasi-pinch-points bear similarities to the fully developed pinch points with singularities of a magnetic Coulomb phase, and continually evolve into the latter on lowering the temperature. The possibility to measure magnetic diffuse scattering with soft x rays opens the way to study magnetic correlations in a variety of nanomagnetic systems.