Topological frustration of artificial spin ice

Vortex Lattices in Active Nematics with Periodic Obstacle Arrays

We numerically model a two-dimensional active nematic confined by a periodic array of fixed obstacles. Even in the passive nematic, the appearance of topological defects is unavoidable due to planar anchoring by the obstacle surfaces. We show that a vortex lattice state emerges as activity is increased, and that this lattice may be tuned from "ferromagnetic" to "antiferromagnetic" by varying the gap size between obstacles. We map the rich variety of states exhibited by the system as a function of distance between obstacles and activity, including a pinned defect state, motile defects, the vortex lattice, and active turbulence. We demonstrate that the flows in the active turbulent phase can be tuned by the presence of obstacles, and explore the effects of a frustrated lattice geometry on the vortex lattice phase.

Defect-induced monopole injection and manipulation in artificial spin ice

Lithographically defined arrays of nanomagnets are well placed for application in areas such as probabilistic computing or reconfigurable magnonics due to their emergent collective dynamics and writable magnetic order. Among them are artificial spin ice (ASI), which are arrays of binary in-plane macrospins exhibiting geometric frustration at the vertex interfaces. Macrospin flips in the arrays create topologically protected magnetic charges, or emergent monopoles, which are bound to an antimonopole to conserve charge. In the absence of controllable pinning, it is difficult to manipulate individual monopoles in the array without also influencing other monopole excitations or the counter-monopole charge. Here, we tailor the local magnetic order of a classic ASI lattice by introducing a ferromagnetic defect with shape anisotropy into the array. This creates monopole injection sites at nucleation fields below the critical lattice switching field. Once formed, the high energy monopoles are fixed to the defect site and may controllably propagate through the lattice under stimulation. Defect programing of bound monopoles within the array allows fine control of the pathways of inverted macrospins. Such control is a necessary prerequisite for the realization of functional devices, e. g. reconfigurable waveguide in nanomagnonic applications.

Artificial spin ice systems offer a promising platform to study the motion of emergent magnetic monopoles, but controlled nucleation of monopoles is challenging. Here the authors demonstrate controlled injection and propagation of emergent monopoles in an artificial spin ice utilizing ferromagnetic defects.

String Phase in an Artificial Spin Ice

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 strings. We show experimentally that the string length follows a simple Boltzmann distribution with an energy scale that is associated with the system's magnetic interactions and is consistent with theoretical predictions. The results demonstrate that string descriptions and associated topological characteristics are not unique to quantum models but can also provide a simplifying description of complex classical systems with non-trivial frustration.

Field-Tunable Interactions and Frustration in Underlayer-Mediated Artificial Spin Ice

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 Kerr effect microscopy to image the demagnetized ground states, we show that the correlation of the demagnetized states depends on the direction of the underlayer magnetization. Further, the relative interaction strength of nearest and next-nearest neighbors varies significantly with the array geometry. We exploit this feature to induce frustration in an inherently unfrustrated square lattice geometry, demonstrating new possibilities for effective geometries in two-dimensional nanomagnetic systems.

Topological Boundary Constraints in Artificial Colloidal Ice

The effect of boundaries and how these can be used to influence the bulk behavior in geometrically frustrated systems are both long-standing puzzles, often relegated to a secondary role. Here, we use numerical simulations and "proof of concept" experiments to demonstrate that boundaries can be engineered to control the bulk behavior in a colloidal artificial ice. We show that an antiferromagnetic frontier forces the system to rapidly reach the ground state (GS), as opposed to the commonly implemented open or periodic boundary conditions. We also show that strategically placing defects at the corners generates novel bistable states, or topological strings, which result from competing GS regions in the bulk. Our results could be generalized to other frustrated micro- and nanostructures where boundary conditions may be engineered with lithographic techniques.

Low-energy states, ground states, and variable frustrations of the finite-size dipolar Cairo lattices

To investigate the influence of geometric frustration on the properties of low-energy configurations of systems of ferromagnetic nanoislands located on the edges of the Cairo lattice, the model of interacting Ising-like magnetic dipoles is used. By the method of complete enumeration, the densities of states of the Cairo pentagonal lattices of a finite number of Ising-like point dipoles are calculated. The calculated ground and low-energy states for systems with a small number of dipoles can be used to solve the problem of searching for the ground states in a system with a relatively large number of dipoles. It is shown that the ground-state energy of the Cairo pentagonal lattices exhibits nonmonotonic behavior on one of the lattice parameters. The lattice parameters can be used to control the degree of geometric frustration. For the studied lattices of a finite number of Ising dipoles on the Cairo lattice in the ground-state configurations, a number of closed pentagons is observed, which are different from the obtained maximum closed pentagons. The magnetic order in the ground-state configurations obeys the ice rule and the quasi-ice rules.

Switchable X-Ray Orbital Angular Momentum from an Artificial Spin Ice

Artificial spin ices (ASI) have been widely investigated as magnetic metamaterials with exotic properties governed by their geometries. In parallel, interest in x-ray photon orbital angular momentum (OAM) has been rapidly growing. Here we show that a square ASI with a patterned topological defect, a double edge dislocation, imparts OAM to scattered x rays. Unlike single dislocations, a double dislocation does not introduce magnetic frustration, and the ASI equilibrates to its antiferromagnetic (AFM) ground state. The topological charge of the defect differs with respect to the structural and magnetic order; thus, x-ray diffraction from the ASI produces photons with even and odd OAM quantum numbers at the structural and AFM Bragg conditions, respectively. The magnetic transitions of the ASI allow the AFM OAM beams to be switched on and off by modest variations of temperature and applied magnetic field. These results demonstrate ASIs can serve as metasurfaces for reconfigurable x-ray optics that could enable selective probes of electronic and magnetic properties.

Emergent Spin Dynamics Enabled by Lattice Interactions in a Bicomponent Artificial Spin Ice

Artificial spin ice (ASI) networks are arrays of 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 complementary lattice sites in a square ASI to modulate the spin-wave properties effectively. We show that the interaction between the two sublattices results in unique spectra attributed to each sublattice, and we observe inter- and intralattice dynamics facilitated by the distinct magnetization properties of the two materials. The dynamic properties are systematically studied by angular-dependent broadband ferromagnetic resonance and confirmed by micromagnetic simulations. We show that combining materials with dissimilar magnetic properties enables the realization of a wide range of two-dimensional structures, potentially opening the door to new concepts in nanomagnonics.

Ferromagnetic resonance in single vertices and 2D lattices macro-dipoles of elongated nanoelements: measurements and simulations

We report broadband ferromagnetic resonance measurements of the in-plane magnetic field response of three- and four-fold symmetric vertices formed by non-contacting permalloy nano-ellipses together with extended lattices constructed from them. Complementing the experimental data with simulations, we are able to show that, as far as the most intense FMR responses are concerned, the spectra of vertices and lattices can largely be interpreted in terms of a superposition of the underlying hysteretic responses of the individual ellipses, as elemental building blocks of the system. This property suggest that it is possible to understand the orientation of the individual magnetic dipole moments in a dipole network in terms of dynamic measurements alone, thereby offering a powerful tool to analyze the alignment statistics in frustrated systems that are exposed to various magnetic histories.

Geometrical Frustration and Planar Triangular Antiferromagnetism in Quasi-Three-Dimensional Artificial Spin Architecture

We present a realization of highly frustrated planar triangular antiferromagnetism achieved in a quasi-three-dimensional artificial spin system consisting of monodomain Ising-type nanomagnets lithographically arranged onto a deep-etched silicon substrate. We demonstrate how the three-dimensional spin architecture results in the first direct observation of long-range ordered planar triangular antiferromagnetism, in addition to a highly disordered phase with short-range correlations, once competing interactions are perfectly tuned. Our work demonstrates how escaping two-dimensional restrictions can lead to new types of magnetically frustrated metamaterials.

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.

Elevated effective dimension in tree-like nanomagnetic Cayley structures

Using state-of-the-art electron-beam lithography, Ising-type nanomagnets may be defined onto nearly any two-dimensional pattern imaginable. The ability to directly observe magnetic configurations achieved in such artificial spin systems makes them a perfect playground for the realization of artificial spin glasses. However, no experimental realization of a finite-temperature artificial spin glass has been achieved so far. Here, we aim to get a significant step closer in achieving that goal by introducing an artificial spin system with random interactions and increased effective dimension: dipolar Cayley tree. Through synchrotron-based photoemission electron microscopy, we show that an improved balance of ferro- and antiferromagnetic ordering can be achieved in this type of system. This combined with an effective dimension as high as d = 2.72 suggests that future systems generated out of these building blocks can host finite temperature spin glass phases, allowing for real-time observation of glassy dynamics.

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.

Spontaneous Magnetic Superdomain Wall Fluctuations in an Artificial Antiferromagnet

Collective dynamics often play an important role in determining the stability of ground states for both naturally occurring materials and metamaterials. We studied the temperature dependent dynamics of antiferromagnetically ordered superdomains in a square artificial spin lattice using soft x-ray photon correlation spectroscopy. We observed an exponential slowing down of superdomain wall motion below the antiferromagnetic onset temperature, similar to the behavior of typical bulk antiferromagnets. Using a continuous time random walk model we show that these superdomain walls undergo low-temperature ballistic and high-temperature diffusive motions.

Geometric frustration in ordered lattices of plasmonic nanoelements

Inspired by geometrically frustrated magnetic systems, we present the optical response of three cases of hexagonal lattices of plasmonic nanoelements. All of them were designed using a metal-insulator-metal configuration to enhance absorption of light, with elements in close proximity to exploit near-field coupling, and with triangular symmetry to induce frustration of the dipolar polarization in the gaps between neighboring structures. Both simulations and experimental results demonstrate that these systems behave as perfect absorbers in the visible and/or the near infrared. Besides, the numerical study of the time evolution shows that they exhibit a relatively extended time response over which the system fluctuates between localized and collective modes. It is of particular interest the echoed excitation of surface lattice resonance modes, which are still present at long times because of the geometric frustration inherent to the triangular lattice. It is worth noting that the excitation of collective modes is also enhanced in other types of arrays where dipolar excitations of the nanoelements are hampered by the symmetry of the array. However, we would like to emphasize that the enhancement in triangular arrays can be significantly larger because of the inherent geometric incompatibility of dipolar excitations and three-fold symmetry axes.

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.

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.

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.

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.