Small amplitude dipolar oscillations are considered in artificial spin ice on a square lattice in two dimensions. The net magnetic moment of each elongated magnetic island in the spin ice is assumed to have Heisenberg-like dynamics. Each island's magnetic moment is assumed to be influenced by shape anisotropies and by the dipolar interactions with its nearest neighbors. The magnetic dynamics is linearized around one of the ground states, leading to an $8\times 8$ matrix to be diagonalized for the magnetic spin wave modes. Analytic solutions are found and classified as antisymmetric and symmetric with regard to their in-plane dynamic fluctuations. Although only the leading dipolar interactions are included, modes similar to these may be observable experimentally.
Over the past few years, the study of magnetization dynamics in artificial spin ices has become a vibrant field of study. Artificial spin ices are ensembles of geometrically arranged, interacting magnetic nanoislands, which display frustration by design. These were initially created to mimic the behavior in rare earth pyrochlore materials and to study emergent behavior and frustration using two-dimensional magnetic measurement techniques. Recently, it has become clear that it is possible to create artificial spin ices, which can potentially be used as functional materials. In this Perspective,…
Artificial spin ices are ensembles of geometrically-arranged, interacting nanomagnets which have shown promising potential for the realization of reconfigurable magnonic crystals. Such systems allow for the manipulation of spin waves on the nanoscale and their potential use as information carriers. However, there are presently two general obstacles to the realization of artificial spin ice-based magnonic crystals: the magnetic state of artificial spin ices is difficult to reconfigure and the magnetostatic interactions between the nanoislands are often weak, preventing mode coupling. We demonst…
Magnetic frustration effects in artificial kagome arrays of nanomagnets are investigated using x-ray photoemission electron microscopy and Monte Carlo simulations. Spin configurations of demagnetized networks reveal unambiguous signatures of long range, dipolar interaction between the nanomagnets. As soon as the system enters the spin ice manifold, the kagome dipolar spin ice model captures the observed physics, while the short range kagome spin ice model fails.
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 o…
Domain wall propagation dynamics have been studied in nanostructured artificial kagome spin ice structures. A stripline circuit has been used to provide localised pulsed magnetic fields within the artificial spin ice structure. This provides control of the system through electrically assisted domain wall nucleation events. Synchronisation of the pulsed fields with additional global magnetic fields and the use of a focussed magneto-optical Kerr effect magnetometer allows our experiments to probe the domain wall transit through an extended ASI structure. We find that the propagation distance dep…
We introduce FePd$_{3}$ as a new material for studying thermally active artificial spin ice (ASI) systems and use it to investigate both the square and kagome ice geometries. We readily achieve perfect ground state ordering in the square lattice and demonstrate the highest yet degree of monopole charge-ordering in the kagome lattice. We find that smaller lattice constants in the kagome system generally produce larger domains of charge order. Monte Carlo simulations show excellent agreement with our data when a small amount of disorder is included in the simulation.
We have performed a detailed study of thermal annealing of the moment configuration in artificial spin ice. Permalloy (Ni$_{80}$Fe$_{20}$) artificial spin ice samples were examined in the prototypical square ice geometry, studying annealing as a function of island thickness, island shape, and annealing temperature and duration. We also measured the Curie temperature as a function of film thickness, finding that thickness has a strong effect on the Curie temperature in regimes of relevance to many studies of the dynamics of artificial spin ice systems. Increasing the interaction energy between …
In this work we study demagnetization protocols for an artificial spin ice in a triangular geometry. Our results show that a simple hysteresis-like process is very efficient in driving the system to its ground state, even for a relatively strong disorder in the system, confirming previous expectations. In addition, transitions between the magnetized state and the ground state were observed to be mediated by the creation and propagation of vertices that behave like magnetic monopoles pseudo-particles. This is an important step towards a more detailed experimental study of monopole-like excitati…
We report angular-dependent spin-wave spectroscopy on kagome artificial spin ice made of large arrays of interconnected Ni80Fe20 nanobars. Spectra taken in saturated and disordered states exhibit a series of resonances with characteristic in-plane angular dependencies. Micromagnetic simulations allow us to interpret characteristic resonances of a two-step magnetization reversal of the nanomagnets. The dynamic properties are consistent with topological defects that are provoked via a magnetic field applied at specific angles. Simulations that we performed on previously investigated kagome artif…
Artificial spin ices are periodic arrangements of interacting nanomagnets successfully used to investigate 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, and support a large number of programmable magnetic states. 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…
We have studied the temperature and magnetic field dependence of the total magnetic moment of large-area permalloy artificial square spin ice arrays. The temperature dependence and hysteresis behavior are consistent with the coherent magnetization reversal expected in the Stoner-Wohlfarth model, with clear deviations due to inter-island interactions at small lattice spacing. Through micromagnetic simulations, we explore this behavior and demonstrate that the deviations result from increasingly complex magnetization reversal at small lattice spacing, induced by inter-island interactions, and de…
Artificial spin ices (ASIs) are designable arrays of interacting nanomagnets that span a wide range of magnetic phases associated with a number of spin lattice models. Here, we demonstrate that the phase of an artificial kagome spin ice can be determined from its initial magnetization curve. As a proof of concept, micromagnetic simulations of these curves were performed starting from representative microstates of different phases of the system. We show that the curves are characterized by phase-specific features in such a way that a pattern recognition algorithm predicts the phase of the initi…
We have studied frustrated kagome arrays and unfrustrated honeycomb arrays of magnetostatically-interacting single-domain ferromagnetic islands with magnetization normal to the plane. The measured pairwise spin correlations of both lattices can be reproduced by models based solely on nearest-neighbor correlations. The kagome array has qualitatively different magnetostatics but identical lattice topology to previously-studied 'artificial spin ice' systems composed of in-plane moments. The two systems show striking similarities in the development of moment pair correlations, demonstrating a univ…
Arrays of suitably patterned and arranged magnetic elements may display artificial spin-ice structures with topological defects in the magnetization, such as Dirac monopoles and Dirac strings. It is known that these defects strongly influence the quasi-static and equilibrium behavior of the spin-ice lattice. Here we study the eigenmode dynamics of such defects in a square lattice consisting of stadium-like thin film elements using micromagnetic simulations. We find that the topological defects display distinct signatures in the mode spectrum, providing a means to qualitatively and quantitative…
We have imaged magnetization textures of permalloy films patterned into Penrose P2 tilings (P2T) using scanning electron microscopy with polarization analysis (SEMPA). P2T film segments have near-uniform, bipolar magnetization, similar to artificial spin ices, but with asymmetric vertex coordination that induces a more complex spin ice behavior mediated by exchange interactions in vertex domain walls. Numerical simulations including long-range dipole interactions agree with SEMPA images of as-grown P2T, and predict a ferromagnetic ground state for a 2D P2T lattice of classical Ising spins.
Artificial square spin ices are structures composed of magnetic elements arranged on a geometrically frustrated lattice and located on the sites of a two-dimensional square lattice, such that there are four interacting magnetic elements at each vertex. Using a semi-analytical approach, we show that square spin ices exhibit a rich spin wave band structure that is tunable both by external magnetic fields and the configuration of individual elements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the edges leading to characteristic excitations; in the p…
Micromagnetic properties of monopoles in artificial kagome spin ice systems are investigated using numerical simulations. We show that micromagnetics brings additional complexity into the physics of these monopoles that is, by essence, absent in spin models: besides a fractionalized classical magnetic charge, monopoles in the artificial kagome ice are chiral at remanence. Our simulations predict that the chirality of these monopoles can be controlled without altering their charge state. This chirality breaks the vertex symmetry and triggers a directional motion of the monopole under an applied…
Systems that exhibit topologically protected edge states are interesting both from a fundamental point of view as well as for potential applications, the latter because of the absence of back-scattering and robustness to perturbations. It is desirable to be able to control and manipulate such edge states. Here, we show that artificial square ices can incorporate both features: an interfacial Dzyaloshinksii-Moriya gives rise to topologically non-trivial magnon bands, and the equilibrium state of the spin ice is reconfigurable with different configurations having different magnon dispersions and…
We report on the study of the thermal dynamics of square artificial spin ice, probed by means of temperature and frequency dependent AC susceptibility. Pronounced influence of the inter-island coupling strength was found on the frequency response of the samples. Through the subsequent analysis of the frequency- and coupling-dependent freezing temperatures, we discuss the phenomenological parameters obtained in the framework of Vogel-Fulcher-Tammann law in terms of the samples microscopic features. The high sensitivity and robust signal to noise ratio of AC susceptibility validates the latter a…
We report broadband spin-wave spectroscopy on kagome artificial spin ice (ASI) made of large arrays of interconnected Ni$_{80}$Fe$_{20}$ nanobars. Spectra taken in saturated and disordered states exhibit a series of resonances with characteristic magnetic field dependencies. Making use of micromagnetic simulations, we identify resonances that reflect the spin-solid-state and monopole-antimonopole pairs on Dirac strings. The latter resonances allow for the generation of highly-charged vertices in ASIs via microwave assisted switching. Our findings open further perspectives for fundamental studi…
Dynamical effects under geometrical frustration are considered in a model for artificial spin ice on a square lattice in two dimensions. Each island of the spin ice has a three-component Heisenberg-like dipole moment subject to shape anisotropies that influence its direction. The model has real dynamics, including rotation of the magnetic degrees of freedom, going beyond the Ising-type models of spin ice. The dynamics is studied using a Langevin equation solved via a second order Heun algorithm. Thermodynamic properties such as the specific heat are presented for different couplings. A peak in…
Artificial spin ices are engineered arrays of dipolarly coupled nanobar magnets. They enable direct investigations of fascinating collective phenomena from their diverse microstates. However, experimental access to ground states in the geometrically frustrated systems has proven difficult, limiting studies and applications of novel properties and functionalities from the low energy states. Here, we introduce a convenient approach to control the competing diploar interactions between the neighboring nanomagnets, allowing us to tailor the vertex degeneracy of the ground states. We achieve this b…
We have performed a systematic study of the effects of field strength and quenched disorder on the driven dynamics of square artificial spin ice. We construct a network representation of the configurational phase space, where nodes represent the microscopic configurations and a directed link between node i and node j means that the field may induce a transition between the corresponding configurations. In this way, we are able to quantitatively describe how the field and the disorder affect the connectedness of states and the reversibility of dynamics. In particular, we have shown that for opt…
In artificial spin ice systems, an interplay of defects and dipolar interactions is expected to play important roles in stabilizing different collective magnetic states. In this work, we investigated the magnetization reversal of individual defective square artificial spin ice vertices where defects break four-fold rotational symmetry of the system. By varying the angle between the applied field and the geometrical axis of the vertices, we observe a change in energy landscape of the system resulting into the stabilization of collective low-energy magnetic states. We also observe that by changi…
We report here the results of micromagnetic simulations of square artificial spin ice (ASI) systems with defects. The defects are introduced by misaligning of a nanomagnet at the vertex. In these defective systems, we are able to stabilize emergent monopole-like state by applying a small external field. We observe a systematic change of dipolar energies of the systems with varying misalignment angle. The fields at which the emergent monopoles are created vary linearly with the dipolar energies of the systems. Our results clearly show that the magnetization reversal of the ASI systems is intric…
The field-induced dynamics of artificial spin ice are determined in part by interactions between magnetic islands, and the switching characteristics of each island. Disorder in either of these affects the response to applied fields. Numerical simulations are used to show that disorder effects are determined primarily by the strength of disorder relative to inter-island interactions, rather than by the type of disorder. Weak and strong disorder regimes exist and can be defined in a quantitative way.
We provide experimental and numerical evidence for thermal excitations within and among magnetic mesospins, forming artificial spin ice structures. At low temperatures, a decrease in magnetization and increase in susceptibility is observed with increasing temperature, interpreted as an onset of thermal fluctuations of the magnetic texture within the mesospins. At elevated temperatures a pronounced susceptibility peak is observed, related to thermally induced flipping of the mesospins and a collapse of the remanent state. The fluctuations, while occurring at distinct length- and energy-scales, …
We investigate spin dynamics of artificial spin ice (ASI) where topological defects confine magnon modes in Ni$_{81}$Fe$_{19}$ nanomagnets arranged on an interconnected kagome lattice. Brillouin light scattering microscopy performed on magnetically disordered states exhibit a series of magnon resonances which depend on topological defect configurations detected by magnetic force microscopy. Nanomagnets on a Dirac string and between a monopole-antimonopole pair show pronounced modifications in magnon frequencies both in experiments and simulations. Our work is key for the creation and annihilat…
We report evidence of spontaneous formation of a heterogeneous network of superdomains in two-dimensional square artificial spin ice nanostructures in externally applied magnetic fields. The magnetic heterogeneity is locally disordered but has a zig-zag texture at longer length scales. Resonant coherent soft-x-ray scattering off such textures give rise to unique internal structure in Bragg peaks. Our result shows that the macroscopic magnetic texture is derived from the microscopic structure of the Dirac strings.
We experimentally demonstrate that arrays of interacting nanoscale ferromagnetic islands, known as artificial spin ice, develop reproducible microstates upon cycling an applied magnetic field. The onset of this memory effect is determined by the strength of the applied field relative to the array coercivity. Specifically, when the applied field strength is almost exactly equal to the array coercivity, several training cycles are required before the array achieves a nearly completely repeatable microstate, whereas when the applied field strength is stronger or weaker than the array coercivity, …
Artificial spin ices are arrays of correlated nano-scale magnetic islands that prove an excellent playground in which to study the role of topology in critical phenomena. Here, we investigate a continuum of spin ice geometries, parameterised by rotation of the islands. In doing so, we morph from the classic square ice to the recently studied pinwheel geometry, with the rotation angle acting as a proxy for controlling inter-island interactions. We experimentally observe a change in ground state magnetic order from antiferromagnetic to ferromagnetic across this class of geometries using Lorentz …
We report a novel approach to the question of whether and how the ground state can be achieved in square artificial spin ices where frustration is incomplete. We identify two types of disorder: quenched disorder in the island response to fields and disorder in the sequence of driving fields. Numerical simulations show that quenched disorder can lead to final states with lower energy, and disorder in the driving fields always lowers the final energy attained by the system. We use a network picture to understand these two effects: disorder in island responses creates new dynamical pathways, and …
In this work we propose and study a realization of an artificial spin ice-like system, not based on any real material, in a triangular geometry. At each vertex of the lattice, the "ice-like rule" dictates that three spins must point inward while the other three must point outward. We have studied the system's ground-state and the lowest energy excitations as well as the thermodynamic properties of the system. Our results show that, despite fundamental differences in the vertices topologies as compared to the artificial square spin ice, in the triangular array the lowest energy excitations also…
Artificial spin ice is a special class of engineered lattice of highly shape anisotropic single domain magnetic nanostructures which is used as one of the model systems to study the spin ice behavior observed in pyrochlore oxides. The nanomagnets interact via dipolar interaction which results in correlated magnetization dynamics exhibiting macroscopic spin configuration states. Here, we exploit the interplay of underlying magnetic state and external bias field orientation to study controlled spin wave propagation in square Artificial Spin Ice (sASI) by performing detailed micromagnetic simulat…
Geometrical frustration arised in spin ices leads to fascinating emergent physical properties. Nowadays there is a wide diversity of the artificial structures, mimicking spin ice at the nanoscale and demonstrating some new effects. Most of the nanoscaled spin ices are two dimensional. Ferromagnetic inverse opal-like structures (IOLS) are among inspiring examples of the three-dimensional system exhibiting spin ice behaviour. However detailed examination of its properties is not straightforward. Experimental technique which is able to unambiguously recover magnetization distribution in 3D mesosc…
We report a magneto-optical Kerr effect study of the collective magnetic response of artificial square spin ice, a lithographically-defined array of single-domain ferromagnetic islands. We find that the anisotropic inter-island interactions lead to a non-monotonic angular dependence of the array coercive field. Comparisons with micromagnetic simulations indicate that the two perpendicular sublattices exhibit distinct responses to island edge roughness, which clearly influence the magnetization reversal process. Furthermore, such comparisons demonstrate that disorder associated with roughness i…
We investigate experimentally magnetic frustration effects in thermally active artificial kagome spin ice. Starting from a paramagnetic state, the system is cooled down below the Curie temperature of the constituent material. The resulting magnetic configurations show that our arrays are locally brought into the so-called spin ice 2 phase, predicted by at-equilibrium Monte Carlo simulations and characterized by a magnetic charge crystal embedded in a disordered kagome spin lattice. However, by studying our arrays on a larger scale, we find unambiguous signature of an out-of-equilibrium physics…
Frustrated systems, typically characterized by competing interactions that cannot all be simultaneously satisfied, display rich behaviours not found elsewhere in nature. Artificial spin ice takes a materials-by-design approach to studying frustration, where lithographically patterned bar magnets mimic the frustrated interactions in real materials but are also amenable to direct characterization. Here, we introduce controlled topological defects into square artificial spin ice lattices in the form of lattice edge dislocations and directly observe the resulting spin configurations. We find the p…
Artificial spin ice systems have opened experimental windows into a range of model magnetic systems through the control of interactions among nanomagnet moments. This control has previously been enabled by altering the nanomagnet size and the geometry of their placement. Here we demonstrate that the interactions in artificial spin ice can be further controlled by including a soft ferromagnetic underlayer below the moments. Such a substrate also breaks the symmetry in the array when magnetized, introducing a directional component to the correlations. Using spatially resolved magneto-optical Ker…
Geometrical frustration in magnetic materials often gives rise to exotic, low-temperature states of matter, like the ones observed in spin ices. Here we report the imaging of the magnetic states of a thermally-active artificial magnetic ice that reveal the fingerprints of a spin fragmentation process. This fragmentation corresponds to a splitting of the magnetic degree of freedom into two channels and is evidenced in both real and reciprocal space. Furthermore, the internal organization of both channels is interpreted within the framework of a hybrid spin-charge model that directly emerges fro…
We investigate a set of topological arrangements of individual ferromagnetic islands in ideal and disordered artificial spin ice (ASI) arrays in order to evaluate how aspects of their field-driven reversal are affected by the model used. The set contains the pinwheel and square ice tilings, and thus a range of magnetic ordering and reversal properties are tested. We find that a simple point dipole model performs relatively well for square ice, but it does not replicate the properties observed in recent experiments with pinwheel ice. Parameterization of the reversal barrier in a Stoner-Wohlfart…
One-dimensional strings of local excitations are a fascinating feature of the physical behavior of strongly correlated topological quantum matter. Here we study strings of local excitations in a classical system of interacting nanomagnets, the Santa Fe Ice geometry of artificial spin ice. We measured the moment configuration of the nanomagnets, both after annealing near the ferromagnetic Curie point and in a thermally dynamic state. While the Santa Fe Ice lattice structure is complex, we demonstrate that its disordered magnetic state is naturally described within a framework of emergent string…
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 e…
Magnetization reversal of interconnected Kagome artificial spin ice was studied by the first-order reversal curve (FORC) technique based on the magneto-optical Kerr effect and magnetoresistance measurements. The magnetization reversal exhibits a distinct six-fold symmetry with the external field orientation. When the field is parallel to one of the nano-bar branches, the domain nucleation/propagation and annihilation processes sensitively depend on the field cycling history and the maximum field applied. When the field is nearly perpendicular to one of the branches, the FORC measurement reveal…
Artificial spin ice is a frustrated magnetic two-dimensional nano-material, recently employed to study variety of tailor-designed unusual collective behaviours. Recently proposed extensions to three dimensions are based on self-assembly techniques and allow little control over geometry and disorder. We present a viable design for the realization of a three-dimensional artificial spin ice with the same level of precision and control allowed by lithographic nano-fabrication of the popular two-dimensional case. Our geometry is based on layering already available two-dimensional artificial spin ic…
The properties of natural and artificial assemblies of interacting elements, ranging from Quarks to Galaxies, are at the heart of Physics. The collective response and dynamics of such assemblies are dictated by the intrinsic dynamical properties of the building blocks, the nature of their interactions and topological constraints. Here we report on the relaxation dynamics of the magnetization of artificial assemblies of mesoscopic spins. In our model nano-magnetic system - square artificial spin ice - we are able to control the geometrical arrangement and interaction strength between the magnet…
We use spatially resolved magneto-optical Kerr microscopy to track the complete microstates of arrays of perpendicular anisotropy nanomagnets during magnetization hysteresis cycles. These measurements allow us to disentangle the intertwined effects of nearest neighbor interaction, disorder, and stochasticity on magnetization switching. We find that the nearest neighbor correlations depend on both interaction strength and disorder. We also find that although the global characteristics of the hysteretic switching are repeatable, the exact microstate sampled is stochastic with the behavior of ind…
