Magnetic charge propagation in bulk frustrated materials has yielded a paradigm-shift in science, allowing the symmetry between electricity and magnetism to be studied. Recent work is now suggesting magnetic charge dynamics upon the spin-ice surface may have important implications in determining the ordering and associated phase space. Here we detail a 3D artificial spin-ice, a 3D nanostructured array of magnetic islands which captures the exact geometry of bulk systems, allowing field-driven dynamics of magnetic charge to be directly visualized upon the surface. Using magnetic microscopy, we observe vastly different magnetic charge dynamics along two principle directions. These striking differences are found to be due to the surface-termination and associated coordination which yields different energetics and interaction strengths for magnetic charges upon the surface.
Artificial ices enable the study of geometrical frustration by design and through direct observation. However, it has proven difficult to achieve tailored long-range ordering of their diverse configurations, limiting both fundamental and applied research directions. We designed an artificial spin structure that produces a magnetic charge ice with tunable long-range ordering of eight different configurations. We also developed a technique to precisely manipulate the local magnetic charge states and demonstrate write-read-erase multifunctionality at room temperature. This globally reconfigurable…
We report on a theoretical investigation of artificial spin-ice dipolar arrays, using a nanoisland geometry adopted from recent experiments [A. Farhan \textit{et al.}, Nature Phys.\ \textbf{9} (2013) 375]. The number of thermal magnetic string excitations in the square lattice is drastically increased by a vertical displacement of rows and columns. We find large increments especially for low temperatures and for string excitations with quasi-monopoles of charges $\pm 4$. By kinetic Monte Carlo simulations we address the thermal stability of such excitations, thereby providing time scales for t…
Magnetricity- the magnetic equivalent of electricity- was recently verified experimentally for the first time. Indeed, just as the stream of electric charges produces electric current, emergent magnetic monopoles have been observed to roam freely (generating magnetic current) in geometrically frustrated magnets known as spin ice. However, this is realized only by considering extreme physical conditions as a single crystal of spin ice has to be cooled to a temperature of $0.36 K$. Candidates to overcome this difficulty are artificial analogues of spin ice crystals, the so-called artificial spin…
Magnetization dynamics in an artificial square spin-ice lattice made of Ni80Fe20 with magnetic field applied in the lattice plane is investigated by broadband ferromagnetic resonance spectroscopy. The experimentally observed dispersion shows a rich spectrum of modes corresponding to different magnetization states. These magnetization states are determined by exchange and dipolar interaction between individual islands, as is confirmed by a semianalytical model. In the low field regime below 400 Oe a hysteretic behavior in the mode spectrum is found. Micromagnetic simulations reveal that the ori…
Magnonics, which employs spin-waves to transmit and process information, is a promising venue for low-power data processing. One of the major challenges is the local control of the spin-wave propagation path. Here, we introduce the concept of writable magnonics by taking advantage of the highly flexible reconfigurability and rewritability of artificial spin ice systems. Using micromagnetic simulations, we show that globally switchable spin-wave propagation and the locally writable spin-wave nanochannels can be realized in a ferromagnetic thin film underlying an artificial pinwheel spin ice. Th…
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,…
In order to explain and predict the properties of many physical systems, it is essential to understand the interplay of different energy-scales. Here we present investigations of the magnetic order in thermalised artificial spin ice structures, with different activation energies of the interacting Ising-like elements. We image the thermally equilibrated magnetic states of the nano-structures using synchrotron-based magnetic microscopy. By comparing results obtained from structures with one or two different activation energies, we demonstrate a clear impact on the resulting magnetic order. The …
In this theoretical investigation we address the effect of defects on thermal excitations in square-lattice dipolar arrays. The geometry of the nanomagnets, adopted from recent experiments [A. Farhan \textit{et al.}, Nature Phys.\ \textbf{9} (2013) 375], allows for thermal activation at room temperature. It turns out that excitations can efficiently be pinned at defects. Furthermore, ferromagnetic strings of nanomagnets that connect a pair of defects can be produced; their lengths are closely related to the distance of the defects and the topology of the magnetic ground-state configuration. We…
In 1935, Pauling estimated the residual entropy of water ice with remarkable accuracy by considering the degeneracy of the ice rule {\it solely at the vertex level}. Indeed, his estimate works well for both the three-dimensional pyrochlore lattice and the two-dimensional six-vertex model, solved by Lieb in 1967. The case of honeycomb artificial spin ice is similar: its pseudo-ice rule, like the ice rule in Pauling and Lieb's systems, simply extends a degeneracy which is already present in the vertices to the global ground state. The anisotropy of the magnetic interaction limits the design of i…
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…
Strongly-interacting artificial spin systems are moving beyond mimicking naturally-occurring materials to emerge as versatile functional platforms, from reconfigurable magnonics to neuromorphic computing. Typically artificial spin systems comprise nanomagnets with a single magnetisation texture: collinear macrospins or chiral vortices. By tuning nanoarray dimensions we achieve macrospin/vortex bistability and demonstrate a four-state metamaterial spin-system 'Artificial Spin-Vortex Ice' (ASVI). ASVI can host Ising-like macrospins with strong ice-like vertex interactions, and weakly-coupled vor…
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.
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 (…
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 present results of ferromagnetic resonance (FMR) experiments and micromagnetic simulations for a distorted, 2D Kagome artificial spin ice. The distorted structure is created by continuously modulating the 2D primitive lattice translation vectors of a periodic honeycomb lattice, according to an aperiodic Fibonacci sequence used to generate 1D quasicrystals. Experimental data and micromagnetic simulations show the Fibonacci distortion causes broadening and splitting of FMR modes into multiple branches, which accompany the increasing number of segment lengths and orientations that develop with…
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…
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…
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…
Although initially introduced to mimic the spin-ice pyrochlores, no artificial spin ice has yet exhibited the expected degenerate ice-phase with critical correlations similar to the celebrated Coulomb phase in the pyrochlore lattice. Here we study a novel artificial spin ice based on a vertex-frustrated rather than pairwise frustrated geometry and show that it exhibits a quasicritical ice phase of extensive residual entropy and, significantly, algebraic correlations. Interesting in its own regard as a novel realization of frustration in a vertex system, our lattice opens new pathways to study …
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…
We show that in colloidal models of artificial kagome and modified square ice systems, a variety of ordering and disordering regimes occur as a function of biasing field, temperature, and colloid-colloid interaction strength, including ordered monopole crystals, biased ice rule states, thermally induced ice rule ground states, biased triple states, and disordered states. We describe the lattice geometries and biasing field protocols that create the different states and explain the formation of the states in terms of sublattice switching thresholds. For a system prepared in a monopole lattice s…
Geometric frustration emerges when local interaction energies in an ordered lattice structure cannot be simultaneously minimized, resulting in a large number of degenerate states. The numerous degenerate configurations may lead to practical applications in microelectronics, such as data storage, memory and logic. However, it is difficult to achieve extensive degeneracy, especially in a two-dimensional system. Here, we showcase in-situ controllable geometric frustration with massive degeneracy in a two-dimensional flux quantum system. We create this in a superconducting thin film placed underne…
Artificial spin ice (ASI) are arrays on nanoscaled magnets that can serve both as models for frustration in atomic spin ice as well as for exploring new spin-wave-based strategies to transmit, process, and store information. Here, we exploit the intricate interplay of the magnetization dynamics of two dissimilar ferromagnetic metals arranged on complimentary lattice sites in a square ASI to effectively modulate the spin-wave properties. We show that the interaction between the two sublattices results in unique spectra attributed to each sublattice and we observe inter- and intra-lattice dynami…
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…
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 symmetr…
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…
Magnetic monopole unpairing as a function of external magnetic fields is presented as a fingerprint of this emergent quasiparticles freedom in a two-dimensional artificial spin ice system. Such freedom, required for example for further application in magnetricity, is only possible due to ground-state degeneracy, which causes a decreasing of the string energy in rectangular geometries, designed to allow highest equidistance among nanomagnets. We show by simulations that spin correlation in different rectangular artificial spin ices evolves from antiferromagnetic ordered magnetic structure to a …
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 present a numerical study on a disordered artificial spin-ice system which interpolates between the long-range ordered square ice and the fully degenerate shakti ice. Starting from the square-ice geometry, disorder is implemented by adding vertical/horizontal magnetic islands to the center of some randomly chosen square plaquettes of the array, at different densities. When no island is added we have ordered square ice. When all square plaquettes have been modified we obtain shakti ice, which is disordered yet in a topological phase corresponding to the Rys F-model. In between, geometrical f…
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, …
Analyzing the magnetic structure factor of a field demagnetized artificial square ice, qualitative deviations from what would predict the square ice model are observed. Combining micromagnetic and Monte Carlo simulations, we demonstrate that these deviations signal the presence of interactions between nanomagnets that extend beyond nearest neighbors. Including further neighbor, dipolar-like couplings in the square ice model, we find that the first seven or eight coupling strengths are needed to reproduce semi-quantitatively the main features of the magnetic structure factor measured experiment…
Artificial spin ice arrays of micromagnetic islands are a means of engineering additional energy scales and frustration into magnetic materials. Despite much progress in elucidating the properties of such arrays, the `spins' in the systems studied so far have no thermal dynamics as the kinetic constraints are too high. Here we address this problem by using a material with an ordering temperature near room temperature. By measuring the temperature dependent magnetization in different principal directions, and comparing with simulations of idealized statistical mechanical models, we confirm a dy…
