Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands

Modal Frustration and Periodicity Breaking in Artificial Spin Ice

Here, an artificial spin ice lattice is introduced that exhibits unique Ising and non-Ising behavior under specific field switching protocols because of the inclusion of coupled nanomagnets into the unit cell. In the Ising regime, a magnetic switching mechanism that generates a uni- or bimodal distribution of states dependent on the alignment of the field is demonstrated with respect to the lattice unit cell. In addition, a method for generating a plethora of randomly distributed energy states across the lattice, consisting of Ising and Landau states, is investigated through magnetic force microscopy and micromagnetic modeling. It is demonstrated that the dispersed energy distribution across the lattice is a result of the intrinsic design and can be finely tuned through control of the incident angle of a critical field. The present manuscript explores a complex frustrated environment beyond the 16-vertex Ising model for the development of novel logic-based technologies.

Finite Length Effects on Switching Mechanisms in Chains of Magnetic Particles

Periodic systems of magnetic nanoparticles are now of interest as they support GHz spin waves. Their equilibrium configurations, switchable with the external magnetic field, are crucial for such applications. We study infinite and finite chains of particles of two shapes (i) ellipsoidal and (ii) rectangular stripes with long axes perpendicular to the chain axis. A variable magnetic field is applied parallel to the long axes. Micromagnetic simulations are compared with the corresponding discrete spin models (Stoner-Wohlfarth model, S-W). An antiferromagnetic configuration is the ground state for all the systems at vanishing field but a ferromagnetic configuration occurs when the field is strong enough. The switching of the infinite chains to the reversed ferromagnetic configuration proceeds directly for the ellipsoids and by an intermediate configuration, in which the magnetization within the particle is non-uniform, in the case of the stripes. The non-uniform configurations are well represented by tilted states in S-W model. Important differences are found in the finite analogs: the switching of ellipsoids becomes multistage and starts from the innermost particles relatively well reproduced with S-W model, whereas the reversal of the stripes, starts from the outermost particles and has no analog in S-W model. Practical consequences of the findings are discussed.

Magnon Modes of Microstates and Microwave-Induced Avalanche in Kagome Artificial Spin Ice with Topological Defects

We investigate spin dynamics of microstates in artificial spin ice (ASI) in Ni_{81}Fe_{19} nanomagnets arranged in an interconnected kagome lattice using microfocus Brillouin light scattering, broadband ferromagnetic resonance, magnetic force microscopy, x-ray photoemission electron microscopy, and simulations. We experimentally reconfigure microstates in ASI using a 2D vector field protocol and apply microwave-assisted switching to intentionally trigger reversal. Our work is key for the creation of avalanches inside the kagome ASI and reprogrammable magnonics based on ASIs.

Machine Learning Magnetic Parameters from Spin Configurations

Hamiltonian parameters estimation is crucial in condensed matter physics, but is time- and cost-consuming. High-resolution images provide detailed information of underlying physics, but extracting Hamiltonian parameters from them is difficult due to the huge Hilbert space. Here, a protocol for Hamiltonian parameters estimation from images based on a machine learning (ML) architecture is provided. It consists in learning a mapping between spin configurations and Hamiltonian parameters from a small amount of simulated images, applying the trained ML model to a single unexplored experimental image to estimate its key parameters, and predicting the corresponding materials properties by a physical model. The efficiency of the approach is demonstrated by reproducing the same spin configuration as the experimental one and predicting the coercive field, the saturation field, and even the volume of the experiment specimen accurately. The proposed approach paves a way to achieve a stable and efficient parameters estimation.

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 a priori 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.

Energetic analysis of disorder effects in an artificial spin ice with dipolar interactions

We study the effect of quenched disorder in square artificial spin ice by means of numerical simulations. We introduce disorder in the length of magnetic islands using two kinds of distributions: Gaussian and uniform. As the system behavior depends on its geometrical parameters, we focus on studying it in the proximity of the ice regime which is quite difficult to thermalize both in experiments and simulations. We show how length disorder affect the antiferromagnetic and (locally) ferromagnetic ordering, by inducing the system, in the case of weak disorder, to intermediate or mix states. Moreover, in the case of strong disorder, ferromagnetic plaquettes prevail regardless of whether the mean length of the islands corresponds to an antiferromagnetic ordering.

Effects of magnetic monopoles charge on the cracking reversal processes in artificial square ices

In this paper we perform nanofabrication of square artificial spin ices with different lattice parameters, in order to investigate the roles of vertex excitation on the features of the system. In particular, the character of magnetic charge distribution asymmetry on the vertices are observed under magnetic hysteresis loop experiments. We then compare our results with simulation using an emergent Hamiltonian containing objects such as magnetic monopoles and dipoles in the vertices of the array (instead of the usual Hamiltonian based on the dipolar interactions among the magnetic nanoislands). All possible interactions between these objects are considered (monopole-monopole, monopole-dipole and dipole-dipole). Using realistic parameters we observe very good match between experiments and theory, which allow us to better understand the system dynamics in function of monopole charge intensity.

The history of spin ice

This review is a study of how the idea of spin ice has evolved over the years, with a focus on the scientific questions that have come to define the subject. Since our initial discovery of spin ice in 1997, there have been well over five thousand papers that discuss it, and in the face of such detail, it must be difficult for the curious observer to 'see the wood for the trees'. To help in this task, we go in search of the biggest insight to have emerged from the study of spin ice. On the way, we identify highlights and outstanding puzzles, and celebrate the inspirational role that Roger Cowley played in the early years.

Topology Restricts Quasidegeneracy in Sheared Square Colloidal Ice

Recovery of ground-state degeneracy in two-dimensional square ice is a significant challenge in the field of geometric frustration with far-reaching fundamental implications, such as realization of vertex models and understanding the effect of dimensionality reduction. We combine experiments, theory, and numerical simulations to demonstrate that sheared square colloidal ice partially recovers the ground-state degeneracy for a wide range of field strengths and lattice shear angles. Our method could inspire engineering a novel class of frustrated microstructures and nanostructures based on sheared magnetic lattices in a wide range of soft- and condensed-matter systems.

Probing nanoscale fluctuation of ferromagnetic meta-atoms with a stochastic photonic spin Hall effect

The photonic spin Hall effect, a deep subdiffraction-limited shift between the opposite spin components of light, emerges when light undergoes an evolution of polarization or trajectory that induces the geometric phase. Here, we study a stochastic photonic spin Hall effect arising from space-variant Berry-Zak phases, which are generated by disordered magneto-optical effects. This spin shift is observed from a spatially bounded lattice of ferromagnetic meta-atoms displaying nanoscale disorders. A random variation of the radii of the meta-atoms induces the nanoscale fluctuation. The standard deviation of the probability distribution of the spin shifts is proportional to the fluctuation of the meta-atoms. This enables us to detect a five-nanometre fluctuation by measuring the probability distribution of the spin shifts via weak measurements. Our approach may be used for sensing deep-subwavelength disorders by actively breaking the photonic spin symmetry and may enable investigations of fluctuation effects in magnetic nanosystems.

Realization of the kagome spin ice state in a frustrated intermetallic compound

Spin ices are exotic phases of matter characterized by frustrated spins obeying local "ice rules," in analogy with the electric dipoles in water ice. In two dimensions, one can similarly define ice rules for in-plane Ising-like spins arranged on a kagome lattice. These ice rules require each triangle plaquette to have a single monopole and can lead to different types of orders and excitations. Using experimental and theoretical approaches including magnetometry, thermodynamic measurements, neutron scattering, and Monte Carlo simulations, we establish HoAgGe as a crystalline (i.e., nonartificial) system that realizes the kagome spin ice state. The system features a variety of partially and fully ordered states and a sequence of field-induced phases at low temperatures, all consistent with the kagome ice rule.

Two-dimensional magnetic monopole gas in an oxide heterostructure

Magnetic monopoles have been proposed as emergent quasiparticles in pyrochlore spin ice compounds. However, unlike semiconductors and two-dimensional electron gases where the charge degree of freedom can be actively controlled by chemical doping, interface modulation, and electrostatic gating, there is as of yet no analogue of these effects for emergent magnetic monopoles. To date, all experimental investigations have been limited to large ensembles comprised of equal numbers of monopoles and antimonopoles in bulk crystals. To address these issues, we propose the formation of a two-dimensional magnetic monopole gas (2DMG) with a net magnetic charge, confined at the interface between a spin ice and an isostructural antiferromagnetic pyrochlore iridate and whose monopole density can be controlled by an external field. Our proposal is based on Monte Carlo simulations of the thermodynamic and transport properties. This proposed 2DMG should enable experiments and devices which can be performed on magnetic monopoles, akin to two-dimensional electron gases in semiconductor heterostructures.

Heterostructure interfaces have physical properties distinct from bulk materials, providing the basis for many electronic devices. Miao et al. propose a spin ice heterostructure that can host a two-dimensional gas of emergent magnetic monopoles with a net magnetic charge.

Archimedean lattices emerge in template-directed eutectic solidification

Template-directed assembly has been shown to yield a broad diversity of highly ordered mesostructures^1,2, which in a few cases exhibit symmetries not present in the native material^3-5. However, this technique has not yet been applied to eutectic materials, which underpin many modern technologies ranging from high-performance turbine blades to solder alloys. Here we use directional solidification of a simple AgCl-KCl lamellar eutectic material within a pillar template to show that interactions of the material with the template lead to the emergence of a set of microstructures that are distinct from the eutectic's native lamellar structure and the template's hexagonal lattice structure. By modifying the solidification rate of this material-template system, trefoil, quatrefoil, cinquefoil and hexafoil mesostructures with submicrometre-size features are realized. Phase-field simulations suggest that these mesostructures appear owing to constraints imposed on diffusion by the hexagonally arrayed pillar template. We note that the trefoil and hexafoil patterns resemble Archimedean honeycomb and square-hexagonal-dodecagonal lattices⁶, respectively. We also find that by using monolayer colloidal crystals as templates, a variety of eutectic mesostructures including trefoil and hexafoil are observed, the former resembling the Archimedean kagome lattice. Potential emerging applications for the structures provided by templated eutectics include non-reciprocal metasurfaces⁷, magnetic spin-ice systems^8,9, and micro- and nano-lattices with enhanced mechanical properties^10,11.

Emergent Dynamics of Artificial Spin-Ice Lattice Based on an Ultrathin Ferromagnet

We present high-frequency dynamics of magnetic nanostructure lattices, fabricated in the form of "artificial spin-ice", that possess magnetically frustrated states. Dynamics of such structures feature multiple resonance excitation that reveals rich and intriguing microwave characteristics, which are highly dependent on field-cycle history. Geometrical parameters such as dimensions and ferromagnetic layer thickness, which control the interplay of different demagnetizing factors, are found to play a pivotal role in governing the dynamics. Our findings are highlighted by the evolution of unique excitations pertaining to magnetic frustration, which are well supported by static magnetometry studies and micromagnetic simulations.

Magnetization dynamics in artificial spin ice

In this topical review, we present key results of studies on magnetization dynamics in artificial spin ice (ASI), which are arrays of magnetically interacting nanostructures. Recent experimental and theoretical progress in this emerging area, which is at the boundary between research on frustrated magnetism and high-frequency studies of artificially created nanomagnets, is reviewed. The exploration of ASI structures has revealed fascinating discoveries in correlated spin systems. Artificially created spin ice lattices offer unique advantages as they allow for a control of the interactions between the elements by their geometric properties and arrangement. Magnonics, on the other hand, is a field that explores spin dynamics in the gigahertz frequency range in magnetic micro- and nanostructures. In this context, magnonic crystals are particularly important as they allow the modification of spin-wave properties and the observation of band gaps in the resonance spectra. Very recently, there has been considerable progress, experimentally and theoretically, in combining aspects of both fields-artificial spin ice and magnonics-enabling new functionalities in magnonic and spintronic applications using ASI, as well as providing a deeper understanding of geometrical frustration in the gigahertz range. Different approaches for the realization of ASI structures and their experimental characterization in the high-frequency range are described and the appropriate theoretical models and simulations are reviewed. Special attention is devoted to linking these findings to the quasi-static behavior of ASI and dynamic investigations in magnonics in an effort to bridge the gap between both areas further and to stimulate new research endeavors.

Magnetization dynamics of weakly interacting sub-100 nm square artificial spin ices

Artificial Spin Ice (ASI), consisting of a two dimensional array of nanoscale magnetic elements, provides a fascinating opportunity to observe the physics of out-of-equilibrium systems. Initial studies concentrated on the static, frozen state, whilst more recent studies have accessed the out-of-equilibrium dynamic, fluctuating state. This opens up exciting possibilities such as the observation of systems exploring their energy landscape through monopole quasiparticle creation, potentially leading to ASI magnetricity, and to directly observe unconventional phase transitions. In this work we have measured and analysed the magnetic relaxation of thermally active ASI systems by means of SQUID magnetometry. We have investigated the effect of the interaction strength on the magnetization dynamics at different temperatures in the range where the nanomagnets are thermally active. We have observed that they follow an Arrhenius-type Néel-Brown behaviour. An unexpected negative correlation of the average blocking temperature with the interaction strength is also observed, which is supported by Monte Carlo simulations. The magnetization relaxation measurements show faster relaxation for more strongly coupled nanoelements with similar dimensions. The analysis of the stretching exponents obtained from the measurements suggest 1-D chain-like magnetization dynamics. This indicates that the nature of the interactions between nanoelements lowers the dimensionality of the ASI from 2-D to 1-D. Finally, we present a way to quantify the effective interaction energy of a square ASI system, and compare it to the interaction energy computed with micromagnetic simulations.

Stray-Field Imaging of a Chiral Artificial Spin Ice during Magnetization Reversal

Artificial spin ices are a class of metamaterials consisting of magnetostatically coupled nanomagnets. Their interactions give rise to emergent behavior, which has the potential to be harnessed for the creation of functional materials. Consequently, the ability to map the stray field of such systems can be decisive for gaining an understanding of their properties. Here, we use a scanning nanometer-scale superconducting quantum interference device (SQUID) to image the magnetic stray field distribution of an artificial spin ice system exhibiting structural chirality as a function of applied magnetic fields at 4.2 K. The images reveal that the magnetostatic interaction gives rise to a measurable bending of the magnetization at the edges of the nanomagnets. Micromagnetic simulations predict that, owing to the structural chirality of the system, this edge bending is asymmetric in the presence of an external field and gives rise to a preferred direction for the reversal of the magnetization. This effect is not captured by models assuming a uniform magnetization. Our technique thus provides a promising means for understanding the collective response of artificial spin ices and their interactions.

Influence of nanoholes array geometrical parameters on magnetic properties of Dy-Fe antidot thin films

Nanoscale artificially engineered spintronic materials could be used to enlarge the storage density of magnetic recording media. For this purpose, magnetic nanostructures such as antidot arrays exhibiting high uniaxial magnetic anisotropy are new contestants in the field of ultrahigh density magnetic data storage devices. In this context, we focus on the synthesis of nanostructured magnetic materials consisting of Dy-Fe alloyed antidot thin films, deposited onto the surface of nanoporous alumina membranes served as patterned templates. Noticeable variations of in the in-plane magnetic anisotropy have been observed by modifying the layer thickness at both microscopic and macroscopic scales. The microscopic magnetic properties have been locally studied by Nano-MOKE magnetometry. For thinner antidot samples with 15, 20 and 25 nm in thickness, a tri-axial in-plane magnetic anisotropy has been detected. Meanwhile, for thicker antidot samples (40-60 nm of layer thickness), an in-plane uniaxial magnetic anisotropy has been noted. We attribute these changes in the magnetic anisotropy to the strong correlation between the edge-to-edge distance among adjacent nanoholes, W, and the local magnetic anisotropy of antidot samples. The effective magnetic anisotropy exhibits an unexpected crossover from the in-plane to out-of-plane direction due to the increasing of the effective perpendicular magnetic anisotropy with varying the layer thickness of antidot thin films. Therefore, we detected a critical layer thickness, t = 25 nm for the Dy-Fe alloy antidot arrays, at which the appearance of the perpendicular magnetization is observed. Furthermore, an enhancement in the Curie temperature of the antidot arrays compared to the continuous thin films has been obtained. We attribute these effects to the complex magnetization reversal processes and the high thermal stability of the hexagonal structure of antidot arrays. These findings can be of high interest for the development of novel magnetic sensors and for thermo-magnetic recording patterned media based on template-assisted deposition techniques.

Thermally and field-driven mobility of emergent magnetic charges in square artificial spin ice

Designing and constructing model systems that embody the statistical mechanics of frustration is now possible using nanotechnology. We have arranged nanomagnets on a two-dimensional square lattice to form an artificial spin ice, and studied its fractional excitations, emergent magnetic monopoles, and how they respond to a driving field using X-ray magnetic microscopy. We observe a regime in which the monopole drift velocity is linear in field above a critical field for the onset of motion. The temperature dependence of the critical field can be described by introducing an interaction term into the Bean-Livingston model of field-assisted barrier hopping. By analogy with electrical charge drift motion, we define and measure a monopole mobility that is larger both for higher temperatures and stronger interactions between nanomagnets. The mobility in this linear regime is described by a creep model of zero-dimensional charges moving within a network of quasi-one-dimensional objects.

Magnetic quadrupole assemblies with arbitrary shapes and magnetizations

Magnetic dipole-dipole interactions govern the behavior of magnetic matter across scales from micrometer colloidal particles to centimeter magnetic soft robots. This pairwise long-range interaction creates rich emergent phenomena under both static and dynamic magnetic fields. However, magnetic dipole particles, from either ferromagnetic or paramagnetic materials, tend to form chain-like structures as low-energy configurations due to dipole symmetry. The repulsion force between two magnetic dipoles raises challenges for creating stable magnetic assemblies with complex two-dimensional (2D) shapes. In this work, we propose a magnetic quadrupole module that is able to form stable and frustration-free magnetic assemblies with arbitrary 2D shapes. The quadrupole structure changes the magnetic particle-particle interaction in terms of both symmetry and strength. Each module has a tunable dipole moment that allows the magnetization of overall assemblies to be programmed at the single module level. We provide a simple combinatorial design method to reach both arbitrary shapes and arbitrary magnetizations concurrently. Last, by combining modules with soft segments, we demonstrate programmable actuation of magnetic metamaterials that could be used in applications for soft robots and electromagnetic metasurfaces.

Topologically protected superconducting ratchet effect generated by spin-ice nanomagnets

We have designed, fabricated and tested a robust superconducting ratchet device based on topologically frustrated spin ice nanomagnets. The device is made of a magnetic Co honeycomb array embedded in a superconducting Nb film. This device is based on three simple mechanisms: (i) the topology of the Co honeycomb array frustrates in-plane magnetic configurations in the array yielding a distribution of magnetic charges which can be ordered or disordered with in-plane magnetic fields, following spin ice rules; (ii) the local vertex magnetization, which consists of a magnetic half vortex with two charged magnetic Néel walls; (iii) the interaction between superconducting vortices and the asymmetric potentials provided by the Néel walls. The combination of these elements leads to a superconducting ratchet effect. Thus, superconducting vortices driven by alternating forces and moving on magnetic half vortices generate a unidirectional net vortex flow. This ratchet effect is independent of the distribution of magnetic charges in the array.

Selective and fast plasmon-assisted photo-heating of nanomagnets

Thermal relaxation of nanoscale magnetic islands, mimicking Ising macrospins, is indispensable for studies of geometrically frustrated artificial spin systems and low-energy nanomagnetic computation. Currently-used heating schemes based on contact to a thermal reservoir, however, lack the speed and spatial selectivity required for the implementation in technological applications. Applying a hybrid approach by combining a plasmonic nanoheater with a magnetic element, in this work we establish the robust and reliable control of local temperatures in nanomagnetic arrays by contactless optical means. Plasmon-assisted photo-heating allows for temperature increases of up to several hundred kelvins, which lead to thermally-activated moment reversals and a pronounced reduction of the magnetic coercive field. Furthermore, the polarization-dependent absorption cross section of elongated plasmonic elements enables sublattice-specific heating on sub-nanosecond time scales. Using optical degrees of freedom, i.e. focal position, polarization, power, and pulse length, thermoplasmonic heating of nanomagnets offers itself for the use in flexible, fast, spatially-, and element-selective thermalization for functional magnetic metamaterials.

Chirally coupled nanomagnets

Magnetically coupled nanomagnets have multiple applications in nonvolatile memories, logic gates, and sensors. The most effective couplings have been found to occur between the magnetic layers in a vertical stack. We achieved strong coupling of laterally adjacent nanomagnets using the interfacial Dzyaloshinskii-Moriya interaction. This coupling is mediated by chiral domain walls between out-of-plane and in-plane magnetic regions and dominates the behavior of nanomagnets below a critical size. We used this concept to realize lateral exchange bias, field-free current-induced switching between multistate magnetic configurations as well as synthetic antiferromagnets, skyrmions, and artificial spin ices covering a broad range of length scales and topologies. Our work provides a platform to design arrays of correlated nanomagnets and to achieve all-electric control of planar logic gates and memory devices.

Phase transitions in few-monolayer spin ice films

Vertex models are an important class of statistical mechanical system that admit exact solutions and exotic physics. Applications include water ice, ferro- and antiferro-electrics, spin ice and artificial spin ice. Here we show that it is possible to engineer spin ice films with atomic-layer precision down to the monolayer limit. Specific heat measurements show that these films, which have a fundamentally different symmetry to bulk spin ice, realise systems close to the two-dimensional F-model, with exotic phase transitions on topologically-constrained configurational manifolds. Our results show how spin ice thin films can release the celebrated Pauling entropy of spin ice without an anomaly in the specific heat. They also significantly expand the class of vertex models available to experiment.

Superferromagnetism and Domain-Wall Topologies in Artificial "Pinwheel" Spin Ice

For over ten years, arrays of interacting single-domain nanomagnets, referred to as artificial spin ices, have been engineered with the aim to study frustration in model spin systems. Here, we use Fresnel imaging to study the reversal process in "pinwheel" artificial spin ice, a modified square ASI structure obtained by rotating each island by some angle about its midpoint. Our results demonstrate that a simple 45° rotation changes the magnetic ordering from antiferromagnetic to ferromagnetic, creating a superferromagnet which exhibits mesoscopic domain growth mediated by domain wall nucleation and coherent domain propagation. We observe several domain-wall configurations, most of which are direct analogues to those seen in continuous ferromagnetic films. However, charged walls also appear due to the geometric constraints of the system. Changing the orientation of the external magnetic field allows control of the nature of the spin reversal with the emergence of either one- or two-dimensional avalanches. This property of pinwheel ASI could be employed to tune devices based on magnetotransport phenomena such as Hall circuits.

Emergent magnetic monopole dynamics in macroscopically degenerate artificial spin ice

Magnetic monopoles, proposed as elementary particles that act as isolated magnetic south and north poles, have long attracted research interest as magnetic analogs to electric charge. In solid-state physics, a classical analog to these elusive particles has emerged as topological excitations within pyrochlore spin ice systems. We present the first real-time imaging of emergent magnetic monopole motion in a macroscopically degenerate artificial spin ice system consisting of thermally activated Ising-type nanomagnets lithographically arranged onto a pre-etched silicon substrate. A real-space characterization of emergent magnetic monopoles within the framework of Debye-Hückel theory is performed, providing visual evidence that these topological defects act like a plasma of Coulomb-type magnetic charges. In contrast to vertex defects in a purely two-dimensional artificial square ice, magnetic monopoles are free to evolve within a divergence-free vacuum, a magnetic Coulomb phase, for which features in the form of pinch-point singularities in magnetic structure factors are observed.

Experimental and theoretical evidences for the ice regime in planar artificial spin ices

In this work, we explore a kind of geometrical effect in the thermodynamics of artificial spin ices (ASI). In general, such artificial materials are athermal. Here, We demonstrate that geometrically driven dynamics in ASI can open up the panorama of exploring distinct ground states and thermally magnetic monopole excitations. It is shown that a particular ASI lattice will provide a richer thermodynamics with nanomagnet spins experiencing less restriction to flip precisely in a kind of rhombic lattice. This can be observed by analysis of only three types of rectangular artificial spin ices (RASI). Denoting the horizontal and vertical lattice spacings by [Formula: see text] and [Formula: see text], respectively, then, a RASI material can be described by its aspect ratio [Formula: see text]. The rhombic lattice emerges when [Formula: see text]. So, by comparing the impact of thermal effects on the spin flips in these three appropriate different RASI arrays, it is possible to find a system very close to the ice regime.

Spatial and Temporal Correlations of XY Macro Spins

We use nano disk arrays with square and honeycomb symmetry to investigate magnetic phases and spin correlations of XY dipolar systems at the micro scale. Utilizing magnetization sensitive X-ray photoemission electron microscopy, we probe magnetic ground states and the "order-by-disorder" phenomenon predicted 30 years ago. We observe the antiferromagnetic striped ground state in square lattices, and 6-fold symmetric structures, including trigonal vortex lattices and disordered floating vortices, in the honeycomb lattice. The spin frustration in the honeycomb lattice causes a phase transition from a long-range ordered locked phase over a floating phase with quasi long-range order and indications of a Berezinskii-Thouless-Kosterlitz-like character, to the thermally excited paramagnetic state. Absent spatial correlation and quasi periodic switching of isolated vortices in the quasi long-range ordered phase suggest a degeneracy of the vortex circulation.

A tunable magnetic metamaterial based on the dipolar four-state Potts model

Metamaterials, tunable artificial materials, are useful playgrounds to investigate magnetic systems. So far, artificial Ising spin systems have revealed features such as emergent magnetic monopoles^1,2 and charge fragmentation³. Here we present a metasystem composed of a lattice of dipolarly coupled nanomagnets. The magnetic spin of each nanomagnet is constrained to lie along a body diagonal, which yields four possible spin states. We show that the magnetic ordering of this metasystem (antiferromagnetic, ferromagnetic or spin ice like) is determined by the spin states orientation relative to the underlying lattice. The dipolar four-state Potts model explains our experimental observations and sheds light on the role of symmetry, as well as short- and long-range dipolar magnetic interactions, in such non-Ising spin systems.

Ice rule fragility via topological charge transfer in artificial colloidal ice

Artificial particle ices are model systems of constrained, interacting particles. They have been introduced theoretically to study ice-manifolds emergent from frustration, along with domain wall and grain boundary dynamics, doping, pinning-depinning, controlled transport of topological defects, avalanches, and memory effects. Recently such particle-based ices have been experimentally realized with vortices in nano-patterned superconductors or gravitationally trapped colloids. Here we demonstrate that, although these ices are generally considered equivalent to magnetic spin ices, they can access a novel spectrum of phenomenologies that are inaccessible to the latter. With experiments, theory and simulations we demonstrate that in mixed coordination geometries, entropy-driven negative monopoles spontaneously appear at a density determined by the vertex-mixture ratio. Unlike its spin-based analogue, the colloidal system displays a "fragile ice" manifold, where local energetics oppose the ice rule, which is instead enforced through conservation of the global topological charge. The fragile colloidal ice, stabilized by topology, can be spontaneously broken by topological charge transfer.

Full-field imaging of thermal and acoustic dynamics in an individual nanostructure using tabletop high harmonic beams

Imaging charge, spin, and energy flow in materials is a current grand challenge that is relevant to a host of nanoenhanced systems, including thermoelectric, photovoltaic, electronic, and spin devices. Ultrafast coherent x-ray sources enable functional imaging on nanometer length and femtosecond timescales particularly when combined with advances in coherent imaging techniques. Here, we combine ptychographic coherent diffractive imaging with an extreme ultraviolet high harmonic light source to directly visualize the complex thermal and acoustic response of an individual nanoscale antenna after impulsive heating by a femtosecond laser. We directly image the deformations induced in both the nickel tapered nanoantenna and the silicon substrate and see the lowest-order generalized Lamb wave that is partially confined to a uniform nanoantenna. The resolution achieved-sub-100 nm transverse and 0.5-Å axial spatial resolution, combined with ≈10-fs temporal resolution-represents a significant advance in full-field dynamic imaging capabilities. The tapered nanoantenna is sufficiently complex that a full simulation of the dynamic response would require enormous computational power. We therefore use our data to benchmark approximate models and achieve excellent agreement between theory and experiment. In the future, this work will enable three-dimensional functional imaging of opaque materials and nanostructures that are sufficiently complex that their functional properties cannot be predicted.

Ising-like behaviour of mesoscopic magnetic chains

We demonstrate an experimental realization of the short range magnetic order in a one-dimensional Ising chain using fabricated mesospins. We confirm an excellent agreement between the experimental findings and simulations obtained using the original Ising model. In particular, we are able to show that the thermal behaviour of the mesoscopic Ising chain dominates over the thermal behaviour of the individual mesospins themselves, confirming that fabricated mesospins can be viewed as artificial magnetic atoms.

Collective magnetism in an artificial 2D XY spin system

Two-dimensional magnetic systems with continuous spin degrees of freedom exhibit a rich spectrum of thermal behaviour due to the strong competition between fluctuations and correlations. When such systems incorporate coupling via the anisotropic dipolar interaction, a discrete symmetry emerges, which can be spontaneously broken leading to a low-temperature ordered phase. However, the experimental realisation of such two-dimensional spin systems in crystalline materials is difficult since the dipolar coupling is usually much weaker than the exchange interaction. Here we realise two-dimensional magnetostatically coupled XY spin systems with nanoscale thermally active magnetic discs placed on square lattices. Using low-energy muon-spin relaxation and soft X-ray scattering, we observe correlated dynamics at the critical temperature and the emergence of static long-range order at low temperatures, which is compatible with theoretical predictions for dipolar-coupled XY spin systems. Furthermore, by modifying the sample design, we demonstrate the possibility to tune the collective magnetic behaviour in thermally active artificial spin systems with continuous degrees of freedom.

Imaging the magnetic structures of artificial quasicrystal magnets using resonant coherent diffraction of circularly polarized X-rays

Unraveling nanoscale spin structures has long been an important activity addressing various scientific interests, that are also readily adaptable to technological applications. This has invigorated the development of versatile nanoprobes suitable for imaging specimens under native conditions. Here we have demonstrated the resonant coherent diffraction of an artificial quasicrystal magnet with circularly polarized X-rays. The nanoscale magnetic structure was revealed from X-ray speckle patterns by comparing with micromagnetic simulations, as a step toward understanding the intricate relationship between the chemical and spin structures in an aperiodic quasicrystal lattice. Femtosecond X-ray pulses from free electron lasers are expected to immediately extend the current work to nanoscale structure investigations of ultrafast spin dynamics, surpassing the present spatio-temporal resolution.

Tunable and switchable magnetic dipole patterns in nanostructured superconductors

Design and manipulation of magnetic moment arrays have been at the focus of studying the interesting cooperative physical phenomena in various magnetic systems. However, long-range ordered magnetic moments are rather difficult to achieve due to the excited states arising from the relatively weak exchange interactions between the localized moments. Here, using a nanostructured superconductor, we investigate a perfectly ordered magnetic dipole pattern with the magnetic poles having the same distribution as the magnetic charges in an artificial spin ice. The magnetic states can simply be switched on/off by applying a current flowing through nanopatterned area. Moreover, by coupling magnetic dipoles with the pinned vortex lattice, we are able to erase the positive/negative poles, resulting in a magnetic dipole pattern of only one polarity, analogous to the recently predicted vortex ice. These switchable and tunable magnetic dipole patterns open pathways for the study of exotic ordering phenomena in magnetic systems.

Switchable geometric frustration in an artificial-spin-ice-superconductor heterosystem

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 microelectronics¹, such as data storage, memory and logic². However, it is difficult to achieve very high degeneracy, especially in a two-dimensional system^3,4. Here, we showcase in situ controllable geometric frustration with high degeneracy in a two-dimensional flux-quantum system. We create this in a superconducting thin film placed underneath a reconfigurable artificial-spin-ice structure⁵. The tunable magnetic charges in the artificial-spin-ice strongly interact with the flux quanta in the superconductor, enabling switching between frustrated and crystallized flux quanta states. The different states have measurable effects on the superconducting critical current profile, which can be reconfigured by precise selection of the spin-ice magnetic state through the application of an external magnetic field. We demonstrate the applicability of these effects by realizing a reprogrammable flux quanta diode. The tailoring of the energy landscape of interacting 'particles' using artificial-spin-ices provides a new paradigm for the design of geometric frustration, which could illuminate a path to control new functionalities in other material systems, such as magnetic skyrmions⁶, electrons and holes in two-dimensional materials^7,8, and topological insulators⁹, as well as colloids in soft materials^10-13.

Stability Analysis of an Array of Magnets: When Will It Jump?

A bidimensional array of magnets whose magnetic moments share the same vertical orientation, and lying on a planar surface, can be gradually compacted. As the density reaches a threshold, the assembly becomes unstable, and the magnets violently pop out of plane. In this Letter, we investigate experimentally and theoretically the maximum packing fraction (or density) of a bidimensional planar assembly of identical cylindrical magnets. We show that the instability can be attributed to local fluctuations of the altitude of the magnets on the planar surface. The maximum density is theoretically predicted assuming dipolar interactions between the magnets and is in excellent agreement with experimental results using a variety of cylindrical magnets.

Unexpected Phenomenology in Particle-Based Ice Absent in Magnetic Spin Ice

While particle-based ices are often considered essentially equivalent to magnet-based spin ices, the two differ essentially in frustration and energetics. We show that at equilibrium particle-based ices correspond exactly to spin ices coupled to a background field. In trivial geometries, such a field has no effect, and the two systems are indeed thermodynamically equivalent. In other cases, however, the field controls a richer phenomenology, absent in magnetic ices, and still largely unexplored: ice rule fragility, topological charge transfer, radial polarization, decimation induced disorder, and glassiness.

Direct-write of free-form building blocks for artificial magnetic 3D lattices

By the fabrication of periodically arranged nanomagnetic systems it is possible to engineer novel physical properties by realizing artificial lattice geometries that are not accessible via natural crystallization or chemical synthesis. This has been accomplished with great success in two dimensions in the fields of artificial spin ice and magnetic logic devices, to name just two. Although first proposals have been made to advance into three dimensions (3D), established nanofabrication pathways based on electron beam lithography have not been adapted to obtain free-form 3D nanostructures. Here we demonstrate the direct-write fabrication of freestanding ferromagnetic 3D nano-architectures. By employing micro-Hall sensing, we have determined the magnetic stray field generated by our free-form structures in an externally applied magnetic field and we have performed micromagnetic and macro-spin simulations to deduce the spatial magnetization profiles in the structures and analyze their switching behavior. Furthermore we show that the magnetic 3D elements can be combined with other 3D elements of different chemical composition and intrinsic material properties.

Decoherence-Free Interaction between Giant Atoms in Waveguide Quantum Electrodynamics

In quantum-optics experiments with both natural and artificial atoms, the atoms are usually small enough that they can be approximated as pointlike compared to the wavelength of the electromagnetic radiation with which they interact. However, superconducting qubits coupled to a meandering transmission line, or to surface acoustic waves, can realize "giant artificial atoms" that couple to a bosonic field at several points which are wavelengths apart. Here, we study setups with multiple giant atoms coupled at multiple points to a one-dimensional (1D) waveguide. We show that the giant atoms can be protected from decohering through the waveguide, but still have exchange interactions mediated by the waveguide. Unlike in decoherence-free subspaces, here the entire multiatom Hilbert space (2^{N} states for N atoms) is protected from decoherence. This is not possible with "small" atoms. We further show how this decoherence-free interaction can be designed in setups with multiple atoms to implement, e.g., a 1D chain of atoms with nearest-neighbor couplings or a collection of atoms with all-to-all connectivity. This may have important applications in quantum simulation and quantum computing.

Attraction Controls the Entropy of Fluctuations in Isosceles Triangular Networks

We study two-dimensional triangular-network models, which have degenerate ground states composed of straight or randomly-zigzagging stripes and thus sub-extensive residual entropy. We show that attraction is responsible for the inversion of the stable phase by changing the entropy of fluctuations around the ground-state configurations. By using a real-space shell-expansion method, we compute the exact expression of the entropy for harmonic interactions, while for repulsive harmonic interactions we obtain the entropy arising from a limited subset of the system by numerical integration. We compare these results with a three-dimensional triangular-network model, which shows the same attraction-mediated selection mechanism of the stable phase, and conclude that this effect is general with respect to the dimensionality of the system.

Magnetic Characterization of Direct-Write Free-Form Building Blocks for Artificial Magnetic 3D Lattices

Three-dimensional (3D) nanomagnetism, where spin configurations extend into the vertical direction of a substrate plane allow for more complex, hierarchical systems and the design of novel magnetic effects. As an important step towards this goal, we have recently demonstrated the direct-write fabrication of freestanding ferromagnetic 3D nano-architectures of ferromagnetic CoFe in shapes of nano-tree and nano-cube structures by means of focused electron beam induced deposition. Here, we present a comprehensive characterization of the magnetic properties of these structures by local stray-field measurements using a high-resolution micro-Hall magnetometer. Measurements in a wide range of temperatures and different angles of the externally applied magnetic field with respect to the surface plane of the sensor are supported by corresponding micromagnetic simulations, which explain the overall switching behavior of in part rather complex magnetization configurations remarkably well. In particular, the simulations yield coercive and switching fields that are in good quantitative correspondence with the measured coercive and switching fields assuming a bulk metal content of 100 at % consisting of bcc Co 3 Fe. We show that thermally-unstable magnetization states can be repetitively prepared and their lifetime controlled at will, a prerequisite to realizing dynamic and thermally-active magnetic configurations if the building blocks are to be used in lattice structures.

Inner Phases of Colloidal Hexagonal Spin Ice

Using numerical simulations that mimic recent experiments on hexagonal colloidal ice, we show that colloidal hexagonal artificial spin ice exhibits an inner phase within its ice state that has not been observed previously. Under increasing colloid-colloid repulsion, the initially paramagnetic system crosses into a disordered ice regime, then forms a topologically charge ordered state with disordered colloids, and finally reaches a threefold degenerate, ordered ferromagnetic state. This is reminiscent of, yet distinct from, the inner phases of the magnetic kagome spin ice analog. The difference in the inner phases of the two systems is explained by their difference in energetics and frustration.

Confined Chemical Fluid Deposition of Ferromagnetic Metalattices

A magnetic, metallic inverse opal fabricated by infiltration into a silica nanosphere template assembled from spheres with diameters less than 100 nm is an archetypal example of a "metalattice". In traditional quantum confined structures such as dots, wires, and thin films, the physical dynamics in the free dimensions is typically largely decoupled from the behavior in the confining directions. In a metalattice, the confined and extended degrees of freedom cannot be separated. Modeling predicts that magnetic metalattices should exhibit multiple topologically distinct magnetic phases separated by sharp transitions in their hysteresis curves as their spatial dimensions become comparable to and smaller than the magnetic exchange length, potentially enabling an interesting class of "spin-engineered" magnetic materials. The challenge to synthesizing magnetic inverse opal metalattices from templates assembled from sub-100 nm spheres is in infiltrating the nanoscale, tortuous voids between the nanospheres void-free with a suitable magnetic material. Chemical fluid deposition from supercritical carbon dioxide could be a viable approach to void-free infiltration of magnetic metals in view of the ability of supercritical fluids to penetrate small void spaces. However, we find that conventional chemical fluid deposition of the magnetic late transition metal nickel into sub-100 nm silica sphere templates in conventional macroscale reactors produces a film on top of the template that appears to largely block infiltration. Other deposition approaches also face difficulties in void-free infiltration into such small nanoscale templates or require conducting substrates that may interfere with properties measurements. Here we report that introduction of "spatial confinement" into the chemical fluid reactor allows for fabrication of nearly void-free nickel metalattices by infiltration into templates with sphere sizes from 14 to 100 nm. Magnetic measurements suggest that these nickel metalattices behave as interconnected systems rather than as isolated superparamagnetic systems coupled solely by dipolar interactions.

Tunable geometrical frustration in magnonic vortex crystals

A novel approach to investigate geometrical frustration is introduced using two-dimensional magnonic vortex crystals. The frustration of the crystal can be manipulated and turned on and off dynamically on the timescale of milliseconds. The vortices are studied using scanning transmission x-ray microscopy and ferromagnetic resonance spectroscopy. They are arranged analogous to the nanomagnets in artificial spin-ice systems. The polarization state of the vortices is tuned in a way that geometrical frustration arises. We demonstrate that frustrated polarization states and non-frustrated states can be tuned to the crystal by changing the frequency of the state formation process.

Realization of ground state in artificial kagome spin ice via topological defect-driven magnetic writing

Arrays of non-interacting nanomagnets are widespread in data storage and processing. As current technologies approach fundamental limits on size and thermal stability, enhancing functionality through embracing the strong interactions present at high array densities becomes attractive. In this respect, artificial spin ices are geometrically frustrated magnetic metamaterials that offer vast untapped potential due to their unique microstate landscapes, with intriguing prospects in applications from reconfigurable logic to magnonic devices or hardware neural networks. However, progress in such systems is impeded by the inability to access more than a fraction of the total microstate space. Here, we demonstrate that topological defect-driven magnetic writing-a scanning probe technique-provides access to all of the possible microstates in artificial spin ices and related arrays of nanomagnets. We create previously elusive configurations such as the spin-crystal ground state of artificial kagome dipolar spin ices and high-energy, low-entropy 'monopole-chain' states that exhibit negative effective temperatures.

New Description of Evolution of Magnetic Phases in Artificial Honeycomb Lattice

Artificial magnetic honeycomb lattice provides a two-dimensional archetypal system to explore novel phenomena of geometrically frustrated magnets. According to theoretical reports, an artificial magnetic honeycomb lattice is expected to exhibit several phase transitions to unique magnetic states as a function of reducing temperature. Experimental investigations of permalloy artificial honeycomb lattice of connected ultra-small elements, [Formula: see text] 12 nm, reveal a more complicated behavior. First, upon cooling the sample to intermediate temperature, [Formula: see text] 175 K, the system manifests a non-unique state where the long range order co-exists with short-range magnetic charge order and weak spin ice state. Second, at much lower temperature, [Formula: see text] 6 K, the long-range spin solid state exhibits a re-entrant behavior. Both observations are in direct contrast to the present understanding of this system. New theoretical approaches are needed to develop a comprehensive formulation of this two dimensional magnet.

Emergent dynamic chirality in a thermally driven artificial spin ratchet

Modern nanofabrication techniques have opened the possibility to create novel functional materials, whose properties transcend those of their constituent elements. In particular, tuning the magnetostatic interactions in geometrically frustrated arrangements of nanoelements called artificial spin ice can lead to specific collective behaviour, including emergent magnetic monopoles, charge screening and transport, as well as magnonic response. Here, we demonstrate a spin-ice-based active material in which energy is converted into unidirectional dynamics. Using X-ray photoemission electron microscopy we show that the collective rotation of the average magnetization proceeds in a unique sense during thermal relaxation. Our simulations demonstrate that this emergent chiral behaviour is driven by the topology of the magnetostatic field at the edges of the nanomagnet array, resulting in an asymmetric energy landscape. In addition, a bias field can be used to modify the sense of rotation of the average magnetization. This opens the possibility of implementing a magnetic Brownian ratchet, which may find applications in novel nanoscale devices, such as magnetic nanomotors, actuators, sensors or memory cells.

Realization of Rectangular Artificial Spin Ice and Direct Observation of High Energy Topology

In this work, we have constructed and experimentally investigated frustrated arrays of dipoles forming two-dimensional artificial spin ices with different lattice parameters (rectangular arrays with horizontal and vertical lattice spacings denoted by a and b respectively). Arrays with three different aspect ratios γ = a/b = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{2}}}$$\end{document}2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{3}}}$$\end{document}3 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{4}}}$$\end{document}4 are studied. Theoretical calculations of low-energy demagnetized configurations for these same parameters are also presented. Experimental data for demagnetized samples confirm most of the theoretical results. However, the highest energy topology (doubly-charged monopoles) does not emerge in our theoretical model, while they are seen in experiments for large enough γ. Our results also insinuate that the string tension connecting two magnetic monopoles in a pair vanishes in rectangular lattices with a critical ratio γ = γ_c = \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{3}}}$$\end{document}3, supporting previous theoretical predictions.

Nanoscale control of competing interactions and geometrical frustration in a dipolar trident lattice

Geometrical frustration occurs when entities in a system, subject to given lattice constraints, are hindered to simultaneously minimize their local interactions. In magnetism, systems incorporating geometrical frustration are fascinating, as their behavior is not only hard to predict, but also leads to the emergence of exotic states of matter. Here, we provide a first look into an artificial frustrated system, the dipolar trident lattice, where the balance of competing interactions between nearest-neighbor magnetic moments can be directly controlled, thus allowing versatile tuning of geometrical frustration and manipulation of ground state configurations. Our findings not only provide the basis for future studies on the low-temperature physics of the dipolar trident lattice, but also demonstrate how this frustration-by-design concept can deliver magnetically frustrated metamaterials.

Artificial magnetic nanostructures enable the study of competing frustrated interactions with more control over the system parameters than is possible in magnetic materials. Farhan et al. present a two-dimensional lattice geometry where the frustration can be controlled by tuning the unit cell parameters.