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

Onsager's Wien effect on a lattice

The second Wien effect describes the nonlinear, non-equilibrium response of a weak electrolyte in moderate to high electric fields. Onsager's 1934 electrodiffusion theory, along with various extensions, has been invoked for systems and phenomena as diverse as solar cells, surfactant solutions, water splitting reactions, dielectric liquids, electrohydrodynamic flow, water and ice physics, electrical double layers, non-ohmic conduction in semiconductors and oxide glasses, biochemical nerve response and magnetic monopoles in spin ice. In view of this technological importance and the experimental ubiquity of such phenomena, it is surprising that Onsager's Wien effect has never been studied by numerical simulation. Here we present simulations of a lattice Coulomb gas, treating the widely applicable case of a double equilibrium for free charge generation. We obtain detailed characterization of the Wien effect and confirm the accuracy of the analytical theories as regards the field evolution of the free charge density and correlations. We also demonstrate that simulations can uncover further corrections, such as how the field-dependent conductivity may be influenced by details of microscopic dynamics. We conclude that lattice simulation offers a powerful means by which to model and investigate system-specific corrections to the Onsager theory, and thus constitutes a valuable tool for detailed theoretical studies of the numerous practical applications of the second Wien effect.

Degeneracy and criticality from emergent frustration in artificial spin ice

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 defects in a critical manifold and to design degeneracy in artificial magnetic nanoarrays, a task so far elusive.

Artificial ferroic systems: novel functionality from structure, interactions and dynamics

Lithographic processing and film growth technologies are continuing to advance, so that it is now possible to create patterned ferroic materials consisting of arrays of sub-1 μm elements with high definition. Some of the most fascinating behaviour of these arrays can be realised by exploiting interactions between the individual elements to create new functionality. The properties of these artificial ferroic systems differ strikingly from those of their constituent components, with novel emergent behaviour arising from the collective dynamics of the interacting elements, which are arranged in specific designs and can be activated by applying magnetic or electric fields. We first focus on artificial spin systems consisting of arrays of dipolar-coupled nanomagnets and, in particular, review the field of artificial spin ice, which demonstrates a wide range of fascinating phenomena arising from the frustration inherent in particular arrangements of nanomagnets, including emergent magnetic monopoles, domains of ordered macrospins, and novel avalanche behaviour. We outline how demagnetisation protocols have been employed as an effective thermal anneal in an attempt to reach the ground state, comment on phenomena that arise in thermally activated systems and discuss strategies for selectively generating specific configurations using applied magnetic fields. We then move on from slow field and temperature driven dynamics to high frequency phenomena, discussing spinwave excitations in the context of magnonic crystals constructed from arrays of patterned magnetic elements. At high frequencies, these arrays are studied in terms of potential applications including magnetic logic, linear and non-linear microwave optics, and fast, efficient switching, and we consider the possibility to create tunable magnonic crystals with artificial spin ice. Finally, we discuss how functional ferroic composites can be incorporated to realise magnetoelectric effects. Specifically, we discuss artificial multiferroics (or multiferroic composites), which hold promise for new applications that involve electric field control of magnetism, or electric and magnetic field responsive devices for high frequency integrated circuit design in microwave and terahertz signal processing. We close with comments on how enhanced functionality can be realised through engineering of nanostructures with interacting ferroic components, creating opportunities for novel spin electronic devices that, for example, make use of the transport of magnetic charges, thermally activated elements, and reprogrammable nanomagnet systems.

Controlled magnetic reversal in Permalloy films patterned into artificial quasicrystals

We have patterned novel Permalloy thin films with quasicrystalline Penrose P2 tilings and measured their dc magnetization and ferromagnetic resonance absorption. Reproducible anomalies in the hysteretic, low-field data signal a series of abrupt transitions between ordered magnetization textures, culminating in a smooth evolution into a saturated state. Micromagnetic simulations compare well to experimental dc hysteresis loops and ferromagnetic resonance spectra and indicate that systematic control of magnetic reversal and domain wall motion can be achieved via tiling design, offering a new paradigm of magnonic quasicrystals.

Realization of artificial ice systems for magnetic vortices in a superconducting MoGe thin film with patterned nanostructures

We report an anomalous matching effect in MoGe thin films containing pairs of circular holes arranged in such a way that four of those pairs meet at each vertex point of a square lattice. A remarkably pronounced fractional matching was observed in the magnetic field dependences of both the resistance and the critical current. At the half matching field the critical current can be even higher than that at zero field. This has never been observed before for vortices in superconductors with pinning arrays. Numerical simulations within the nonlinear Ginzburg-Landau theory reveal a square vortex ice configuration in the ground state at the half matching field and demonstrate similar characteristic features in the field dependence of the critical current, confirming the experimental realization of an artificial ice system for vortices for the first time.

Direct observation of thermal relaxation in artificial spin ice

We study the thermal relaxation of artificial spin ice with photoemission electron microscopy, and are able to directly observe how such a system finds its way from an energetically excited state to the ground state. On plotting vertex-type populations as a function of time, we can characterize the relaxation, which occurs in two stages, namely a string and a domain regime. Kinetic Monte Carlo simulations agree well with the temporal evolution of the magnetic state when including disorder, and the experimental results can be explained by considering the effective interaction energy associated with the separation of pairs of vertex excitations.

Frustrated colloidal ordering and fully packed loops in arrays of optical traps

We propose that a system of colloidal particles interacting with a honeycomb array of optical traps that each contain three wells can be used to realize a fully packed loop model. One of the phases in this system can be mapped to Baxter's three-coloring problem, offering an easily accessible physical realization of this problem. As a function of temperature and interaction strength, we find a series of phases, including long range ordered loop or stripe states, stripes with sliding symmetries, random packed loop states, and disordered states in which the loops break apart. Our geometry could be constructed using ion trap arrays, BEC vortices in optical traps, or magnetic vortices in nanostructured superconductors.

Thermal phase transitions in artificial spin ice

We use the sixteen-vertex model to describe bidimensional artificial spin ice. We find excellent agreement between vertex densities in 15 differently grown samples and the predictions of the model. Our results demonstrate that the samples are in usual thermal equilibrium away from a critical point separating a disordered and an antiferromagnetic phase in the model. The second-order phase transition that we predict suggests that the spatial arrangement of vertices in near-critical artificial spin ice should be studied in more detail in order to verify whether they show the expected space and time long-range correlations.

Magnetic dipole and higher pole interaction on a square lattice

We have studied the magnetic interaction of circular magnetic islands with a dipole character on a square lattice. The square pattern consists of lithographically prepared polycrystalline PdFe islands, 150 nm in diameter and a periodicity of 300 nm. Below the Curie temperature at 260 K, the islands are in a single domain state with isotropic in-plane magnetization. Below 160 K, there is an onset of interisland interaction that leads to a change of the shape of the hysteresis, an increase of coercivity, and a development of in-plane anisotropy. Photoemission electron microscopy with circularly polarized incident x rays tuned to the L3 edge of Fe confirms the increasing correlation of the magnetic islands and the formation of elongated chains, as predicted by Vedmedenko et al. [Phys. Rev. Lett. 95, 207202 (2005)] for contributions from pole interactions of higher order than the dipolar one. Neighboring chains are found to be irregularly oriented either parallel or antiparallel.

Creating an artificial two-dimensional Skyrmion crystal by nanopatterning

A Skyrmion crystal typically arises from helical spin structures induced by the Dzyaloshinskii-Moriya interaction. Experimentally its physical exploration has been impeded because it is a rarity and is found only within a narrow temperature and magnetic field range. We present a method for the assembly of a two-dimensional Skyrmion crystal based upon a combination of a perpendicularly magnetized film and nanopatterned arrays of magnetic vortices that are geometrically confined within nanodisks. The practical feasibility of the method is validated by micromagnetic simulations and computed Skyrmion number per unit cell. We also quantify a wide range in temperature and field strength over which the Skyrmion crystal can be stabilized without the need for any intrinsic Dzyaloshinskii-Moriya interactions, which otherwise is needed to underpin the arrangement as is the case in the very few known Skyrmion crystal cases. Thus, our suggested scheme involves a qualitative breakthrough that comes with a substantial quantitative advance.

The Aharanov-Bohm effect, magnetic monopoles and reversal in spin-ice lattices

The proof of the Aharonov-Bohm (AB) effect has been one of the most important experiments of the last century and used as essential evidence for the theory of gauge fields. In this article, we look at its fundamental relation to the Dirac monopole and string. Despite the Dirac string being invisible to the AB effect, it can be used to study emergent quasiparticles in condensed matter settings that behave similar to the fundamental monopoles and strings between them. We utilize phase-imaging method based on the AB effect to study the ordering in a one-model system - that of frustrated spin ice - to understand the ordering processes that occur during a magnetic field reversal cycle. The reversal is linked to the propagation of monopole defects linked by flux channels, reminiscent of Dirac strings. Monopole interactions govern the defect densities within the lattice. Furthermore, we exploit these interactions to propose a new ordering method in which high degrees of ground-state ordering can be achieved in a frustrated system.

Spectral analysis of topological defects in an artificial spin-ice lattice

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 quasistatic and equilibrium behavior of the spin-ice lattice. Here, we study the eigenmode dynamics of such defects in a square lattice consisting of stadiumlike 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 quantitatively analyze monopoles and strings that can be measured experimentally.

The non-random walk of chiral magnetic charge carriers in artificial spin ice

The flow of magnetic charge carriers (dubbed magnetic monopoles) through frustrated spin ice lattices, governed simply by Coulombic forces, represents a new direction in electromagnetism. Artificial spin ice nanoarrays realise this effect at room temperature, where the magnetic charge is carried by domain walls. Control of domain wall path is one important element of utilizing this new medium. By imaging the transit of domain walls across different connected 2D honeycomb structures we contribute an important aspect which will enable that control to be realized. Although apparently equivalent paths are presented to a domain wall as it approaches a Y-shaped vertex from a bar parallel to the field, we observe a stark non-random path distribution, which we attribute to the chirality of the magnetic charges. These observations are supported by detailed statistical modelling and micromagnetic simulations. The identification of chiral control to magnetic charge path selectivity invites analogy with spintronics.

A robust and facile approach to assembling mobile and highly-open unfrustrated triangular lattices from ferromagnetic nanorods

A simple and widely applicable approach to assemble long-range two-dimensional mobile arrays of functionalized nickel nanorods with tunable and "highly open" lattice structures is presented. The magnetic assembly of uniformly oriented nanorods in triangular lattices was achieved by a phase separation of the surface confined yet mobile vertical nanorods driven by a gradient magnetic field. In contrast to known approaches, the unfrustrated lattices can be further locked in place allowing for the removal of the applied magnetic field and processing without disrupting the initial order with different symmetries precisely assembled and locked in their position on the same substrate. We suggest that the tunable assemblies of magnetic nanorods provide a versatile platform for downstream handling of open lattice arrays for eventual device integration.

Domain wall interactions at a cross-shaped vertex

The interaction of two domain walls (DWs) at a cross-shaped vertex fabricated from two ferromagnetic nanowires has been experimentally investigated. Both magnetostatically repulsive and attractive interactions have been probed. It is found that in the repulsive case, a passing DW may directly induce the depinning of another that is already pinned at a vertex. This effect can be qualitatively described by considering only simple, magnetostatic-charge-based arguments. In the attractive case, however, asymmetric pinning is found, with complete suppression of depinning possible. This observed effect is contrary to simple charge-based arguments and highlights the need for full micromagnetic characterization of the DW interactions in more complex systems.

Artificial kagome spin ice: dimensional reduction, avalanche control and emergent magnetic monopoles

Artificial spin-ice systems consisting of nanolithographic arrays of isolated nanomagnets are model systems for the study of frustration-induced phenomena. We have recently demonstrated that monopoles and Dirac strings can be directly observed via synchrotron-based photoemission electron microscopy, where the magnetic state of individual nanoislands can be imaged in real space. These experimental results of Dirac string formation are in excellent agreement with Monte Carlo simulations of the hysteresis of an array of dipoles situated on a kagome lattice with randomized switching fields. This formation of one-dimensional avalanches in a two-dimensional system is in sharp contrast to disordered thin films, where avalanches associated with magnetization reversal are two-dimensional. The self-organized restriction of avalanches to one dimension provides an example of dimensional reduction due to frustration. We give simple explanations for the origin of this dimensional reduction and discuss the disorder dependence of these avalanches. We conclude with the explicit demonstration of how these avalanches can be controlled via locally modified anisotropies. Such a controlled start and stop of avalanches will have potential applications in data storage and information processing.

Magnetic dipole configurations on honeycomb lattices: effect of finite size and boundaries

Artificial dipolar spin-ice patterns have attracted much attention recently because of their rich configurations and excitations in the form of Dirac strings connecting magnetic monopoles. We have analysed the distribution of excitations in the form of strings and vertices carrying magnetic charges Q=±3q in honeycomb artificial spin-ice patterns. Two types of patterns are compared, those that terminate with open hexagons and those with closed hexagons. The dipole configurations and the frequency of spin-ice rule-violating Q=±3q vertices depend slightly on the boundary conditions of the pattern. Upon rotation of the patterns by 2π in a coercive magnetic field of 500 Oe, complete reversibility of the charge and string configuration is observed.

Macroscopic magnetic frustration

Although geometrical frustration transcends scale, it has primarily been evoked in the micro- and mesoscopic realm to characterize such phases as spin ice, liquids, and glasses and to explain the behavior of such materials as multiferroics, high-temperature superconductors, colloids, and copolymers. Here we introduce a system of macroscopic ferromagnetic rotors arranged in a planar lattice capable of out-of-plane movement that exhibit the characteristic honeycomb spin ice rules studied and seen so far only in its mesoscopic manifestation. We find that a polarized initial state of this system settles into the honeycomb spin ice phase with relaxation on multiple time scales. We explain this relaxation process using a minimal classical mechanical model that includes Coulombic interactions between magnetic charges located at the ends of the magnets and viscous dissipation at the hinges. Our study shows how macroscopic frustration arises in a purely classical setting that is amenable to experiment, easy manipulation, theory, and computation, and shows phenomena that are not visible in their microscopic counterparts.

Focused electron beam induced deposition: A perspective

BACKGROUND

Focused electron beam induced deposition (FEBID) is a direct-writing technique with nanometer resolution, which has received strongly increasing attention within the last decade. In FEBID a precursor previously adsorbed on a substrate surface is dissociated in the focus of an electron beam. After 20 years of continuous development FEBID has reached a stage at which this technique is now particularly attractive for several areas in both, basic and applied research. The present topical review addresses selected examples that highlight this development in the areas of charge-transport regimes in nanogranular metals close to an insulator-to-metal transition, the use of these materials for strain- and magnetic-field sensing, and the prospect of extending FEBID to multicomponent systems, such as binary alloys and intermetallic compounds with cooperative ground states.

RESULTS

After a brief introduction to the technique, recent work concerning FEBID of Pt-Si alloys and (hard-magnetic) Co-Pt intermetallic compounds on the nanometer scale is reviewed. The growth process in the presence of two precursors, whose flux is independently controlled, is analyzed within a continuum model of FEBID that employs rate equations. Predictions are made for the tunability of the composition of the Co-Pt system by simply changing the dwell time of the electron beam during the writing process. The charge-transport regimes of nanogranular metals are reviewed next with a focus on recent theoretical advancements in the field. As a case study the transport properties of Pt-C nanogranular FEBID structures are discussed. It is shown that by means of a post-growth electron-irradiation treatment the electronic intergrain-coupling strength can be continuously tuned over a wide range. This provides unique access to the transport properties of this material close to the insulator-to-metal transition. In the last part of the review, recent developments in mechanical strain-sensing and the detection of small, inhomogeneous magnetic fields by employing nanogranular FEBID structures are highlighted.

CONCLUSION

FEBID has now reached a state of maturity that allows a shift of the focus towards the development of new application fields, be it in basic research or applied. This is shown for selected examples in the present review. At the same time, when seen from a broader perspective, FEBID still has to live up to the original idea of providing a tool for electron-controlled chemistry on the nanometer scale. This has to be understood in the sense that, by providing a suitable environment during the FEBID process, the outcome of the electron-induced reactions can be steered in a controlled way towards yielding the desired composition of the products. The development of a FEBID-specialized surface chemistry is mostly still in its infancy. Next to application development, it is this aspect that will likely be a guiding light for the future development of the field of focused electron beam induced deposition.

Perpendicular magnetization and generic realization of the Ising model in artificial spin ice

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 universality in artificial spin ice behavior independent of specific realization in a particular material system.

Hysteresis and return-point memory in colloidal artificial spin ice systems

Using computer simulations, we investigate hysteresis loops and return-point memory for artificial square and kagome spin ice systems by cycling an applied bias force and comparing microscopic effective spin configurations throughout the hysteresis cycle. Return-point memory loss is caused by motion of individual defects in kagome ice or of grain boundaries in square ice. In successive cycles, return-point memory is recovered rapidly in kagome ice. Memory is recovered more gradually in square ice due to the extended nature of the grain boundaries. Increasing the amount of quenched disorder increases the defect density but also enhances the return-point memory since the defects become trapped more easily.

Magnetic anisotropy of elongated thin ferromagnetic nano-islands for artificial spin ice arrays

The energetics of thin elongated ferromagnetic nano-islands is considered for some different shapes, aspect ratios and applied magnetic field directions. These nano-island particles are important for artificial spin ice materials. For low temperature, the magnetic internal energy of an individual particle is evaluated numerically as a function of the direction of a particle's net magnetization. This leads to estimations of effective anisotropy constants for (1) the easy axis along the particle's long direction, and (2) the hard axis along the particle's thin direction. A spin relaxation algorithm together with fast Fourier transform for the demagnetization field is used to solve the micromagnetics problem for a thin system. The magnetic hysteresis is also found. The results indicate some possibilities for controlling the equilibrium and dynamics in spin ice materials by using different island geometries.

Disorder strength and field-driven ground state domain formation in artificial spin ice: experiment, simulation, and theory

Quenched disorder affects how nonequilibrium systems respond to driving. In the context of artificial spin ice, an athermal system comprised of geometrically frustrated classical Ising spins with a twofold degenerate ground state, we give experimental and numerical evidence of how such disorder washes out edge effects and provide an estimate of disorder strength in the experimental system. We prove analytically that a sequence of applied fields with fixed amplitude is unable to drive the system to its ground state from a saturated state. These results should be relevant for other systems where disorder does not change the nature of the ground state.

Electric dipoles on magnetic monopoles in spin ice

The close connection of electricity and magnetism is one of the cornerstones of modern physics. This connection has a crucial role from a fundamental point of view and in practical applications, including spintronics and multiferroic materials. A breakthrough was a recent proposal that in magnetic materials called spin ice the elementary excitations have a magnetic charge and behave as magnetic monopoles. I show that, besides magnetic charge, there should be an electric dipole attached to each magnetic monopole. This opens new possibilities to study and control such monopoles using an electric field. Thus, the electric-magnetic analogy goes even further than usually assumed: whereas electrons have electric charge and magnetic dipole (spin), magnetic monopoles in spin ice, while having magnetic charge, also have an electric dipole.

Magnetic Coulomb fields of monopoles in spin ice and their signatures in the internal field distribution

Fractionalization-the breaking up of an apparently indivisible microscopic degree of freedom-is one of the most counterintuitive phenomena in many-body physics. Here we study its most fundamental manifestation in spin ice, the only known fractionalized magnetic compound in 3D: we directly visualize the 1/r(2) magnetic Coulomb field of monopoles that emerge as the atomic magnetic dipoles fractionalize. We analyze the internal magnetic field distribution, relevant for local experimental probes. In particular, we present new zero-field NMR measurements that exhibit excellent agreement with the calculated line shapes, noting that this experimental technique can in principle measure directly the monopole density in spin ice. The distribution of field strengths is captured by a simple analytical form that exhibits a low density of low-field sites-in apparent disagreement with reported muon spin rotation results. Counterintuitively, the density of low-field locations decreases as the local ferromagnetic correlations imposed by the ice rules weaken.

Possible valence-bond condensation in the frustrated cluster magnet LiZn2Mo3O8

The emergence of complex electronic behaviour from simple ingredients has resulted in the discovery of numerous states of matter. Many examples are found in systems exhibiting geometric magnetic frustration, which prevents simultaneous satisfaction of all magnetic interactions. This frustration gives rise to complex magnetic properties such as chiral spin structures, orbitally driven magnetism, spin-ice behaviour exhibiting Dirac strings with magnetic monopoles, valence-bond solids and spin liquids. Here we report the synthesis and characterization of LiZn(2)Mo(3)O(8), a geometrically frustrated antiferromagnet in which the magnetic moments are localized on small transition-metal clusters rather than individual ions. By doing so, first-order Jahn-Teller instabilities and orbital ordering are prevented, allowing the strongly interacting magnetic clusters in LiZn(2)Mo(3)O(8) to probably give rise to an exotic condensed valence-bond ground state reminiscent of the proposed resonating valence-bond state. Our results also link magnetism on clusters to geometric magnetic frustration in extended solids, demonstrating a new approach for unparalleled chemical control and tunability in the search for collective, emergent electronic states of matter.

Emerging chirality in artificial spin ice

Artificial spin ice, made up of planar nanostructured arrays of simple ferromagnetic bars, is a playground for rich physics associated with the spin alignment of the bars and spin texture associated with the magnetic frustration at the bar vertices. The phase diagram is exotic, showing magnetic monopole-like defects and liquid and solid phases of spins arranged in loop states with predicted chiral order. We show that magnetotransport measurements in connected honeycomb structures yield the onset of an anomalous Hall signal at 50 kelvin. The temperature scale can be attributed to the long-range dipolar ice phase. The topological Hall signal arises because chiral loops form at the sample edges, indicating a generic route to exotic states via nanoarray edge structure.

Small-network approximations for geometrically frustrated Ising systems

The study of frustrated spin systems often requires time-consuming numerical simulations. As the simplest approach, the classical Ising model is often used to investigate the thermodynamic behavior of such systems. Exploiting the small correlation lengths in frustrated Ising systems, we develop a method for obtaining first approximations to the energetic properties of frustrated two-dimensional Ising systems using small networks of less than 30 spins. These small networks allow much faster numerical simulations, and more importantly, analytical evaluations of their properties are numerically tractable. We choose Ising systems on the triangular lattice, the kagome lattice, and the triangular kagome lattice as prototype systems and find small systems that can serve as good approximations to these prototype systems. Through comparisons between the properties of extended models and small systems, we develop a set of criteria for constructing small networks to approximate general infinite two-dimensional frustrated Ising systems. This method of using small networks provides a different and efficient way to obtain a first approximation to the properties of frustrated spin systems.

Studies of nanomagnetism using synchrotron-based x-ray photoemission electron microscopy (X-PEEM)

As interest in magnetic devices has increased over the last 20 years, research into nanomagnetism has experienced a corresponding growth. Device applications from magnetic storage to magnetic logic have compelled interest in the influence of geometry and finite size on magnetism and magnetic excitations, in particular where the smallest dimensions reach the important magnetic interaction length scales. The dynamical behavior of nanoscale magnets is an especially important subset of research, as these phenomena are both critical for device physics and profoundly influenced by finite size. At the same time, nanoscale systems offer unique geometries to promote and study model systems, such as magnetic vortices, leading to new fundamental insights into magnetization dynamics. A wide array of experimental and computational techniques have been applied to these problems. Among these, imaging techniques that provide real-space information on the magnetic order are particularly useful. X-ray microscopy offers several advantages over scanning probe or optical techniques, such as high spatial resolution, element specificity and the possibility for high time resolution. Here, we review recent contributions using static and time-resolved x-ray photoemission electron microscopy to nanomagnetism research.

Biotemplated magnetic nanoparticle arrays

Immobilized biomineralizing protein Mms6 templates the formation of uniform magnetite nanoparticles in situ when selectively patterned onto a surface. Magnetic force microscopy shows that the stable magnetite particles maintain their magnetic orientation at room temperature, and may be exchange coupled. This precision-mixed biomimetic/soft-lithography methodology offers great potential for the future of nanodevice fabrication.

Orbital ice: an exact Coulomb phase on the diamond lattice

We demonstrate the existence of an orbital Coulomb phase as the exact ground state of a p-orbital exchange Hamiltonian on the diamond lattice. The Coulomb phase is an emergent state characterized by algebraic dipolar correlations and a gauge structure resulting from local constraints (ice rules) of the underlying lattice models. For most ice models on the pyrochlore lattice, these local constraints are a direct consequence of minimizing the energy of each individual tetrahedron. On the contrary, the orbital ice rules are emergent phenomena resulting from the quantum orbital dynamics. We show that the orbital ice model exhibits an emergent geometrical frustration by mapping the degenerate quantum orbital ground states to the spin-ice states obeying the 2-in-2-out constraints on the pyrochlore lattice. We also discuss possible realization of the orbital ice model in optical lattices with p-band fermionic cold atoms.

Diversity enabling equilibration: disorder and the ground state in artificial spin ice

We report a novel approach to the question of whether and how the ground state can be achieved in square artificial spin ices where frustration is incomplete. We identify two sources of randomness that affect the approach to ground state: quenched disorder in the island response to fields and randomness in the sequence of driving fields. Numerical simulations show that quenched disorder can lead to final states with lower energy, and randomness in the sequence of driving fields always lowers the final energy attained by the system. We use a network picture to understand these two effects: disorder in island responses creates new dynamical pathways, and a random sequence of driving fields allows more pathways to be followed.

Deterministic and non-deterministic switching in chains of magnetic hysterons

This paper presents a fundamental analysis of a single-domain ferromagnetic particles chain hysteresis in perpendicular geometry as a prototype for ultra-high density memories. Due to magnetostatic long range interactions the system has a complex hysteresis but stable features can be found. The loop has a number of deterministic Barkhausen jumps and consequently a number of stable plateaus that could be used in multistate memories. The fundamental elements that sustain this behavior are shown and discussed.

Analysis of a fully packed loop model arising in a magnetic Coulomb phase

The Coulomb phase of spin ice, and indeed the I(c) phase of water ice, naturally realize a fully packed two-color loop model in 3D. We present a detailed analysis of the statistics of these loops: we find loops spanning the system multiple times hosting a finite fraction of all sites while the average loop length remains finite. We contrast the behavior with an analogous 2D model. We connect this body of results to properties of polymers, percolation and insights from Schramm-Loewner evolution processes. We also study another extended degree of freedom, called worms, which appear as "Dirac strings" in spin ice. We discuss implications of these results for the efficiency of numerical cluster algorithms, and address implications for the ordering properties of a broader class of magnetic systems, e.g., with Heisenberg spins, such as CsNiCrF(6) or ZnCr(2)O(4).

Reducing disorder in artificial kagome ice

Artificial spin ice has become a valuable tool for understanding magnetic interactions on a microscopic level. The strength in the approach lies in the ability of a synthetic array of nanoscale magnets to mimic crystalline materials, composed of atomic magnetic moments. Unfortunately, these nanoscale magnets, patterned from metal alloys, can show substantial variation in relevant quantities such as the coercive field, with deviations up to 16%. By carefully studying the reversal process of artificial kagome ice, we can directly measure the distribution of coercivities, and, by switching from disconnected islands to a connected structure, we find that the coercivity distribution can achieve a deviation of only 3.3%. These narrow deviations should allow the observation of behavior that mimics canonical spin-ice materials more closely.

Ignoring your neighbors: moment correlations dominated by indirect or distant interactions in an ordered nanomagnet array

We have studied the moment correlations within triangular lattice arrays of single-domain coaligned nanoscale ferromagnetic islands. Independent variation of lattice spacing along and perpendicular to the island axis tunes the magnetostatic interactions between islands through a broad range of relative strengths. For certain lattice parameters, the sign of the correlations between near-neighbor island moments is opposite to that favored by the pairwise interaction. This finding, supported by analysis of the total correlation in terms of direct and convoluted indirect contributions across multiple pairwise interactions, indicates that indirect interactions and/or those mediated by further neighbors can be tuned to be dominant, with implications for the wide range of systems composed of interacting nanomagnets.

Order by disorder in the antiferromagnetic Ising model on an elastic triangular lattice

Geometrically frustrated materials have a ground-state degeneracy that may be lifted by subtle effects, such as higher-order interactions causing small energetic preferences for ordered structures. Alternatively, ordering may result from entropic differences between configurations in an effect termed order by disorder. Motivated by recent experiments in a frustrated colloidal system in which ordering is suspected to result from entropy, we consider in this paper the antiferromagnetic Ising model on a deformable triangular lattice. We calculate the displacements exactly at the microscopic level and, contrary to previous studies, find a partially disordered ground state of randomly zigzagging stripes. Each such configuration is deformed differently and thus has a unique phonon spectrum with distinct entropy, lifting the degeneracy at finite temperature. Nonetheless, due to the free-energy barriers between the ground-state configurations, the system falls into a disordered glassy state.

Magnetic ordering of systems of nanodisks with quenched positional disorder

The effect of positional disorder in systems of single domain ferromagnetic nanodisks placed on a two-dimensional square lattice is studied by Monte Carlo simulations. Nanodisks are treated as magnetic dipoles pointing along one of the two principal axes of the lattice. Disorder is introduced displacing each nanodisk by (δx, δy) from its regular lattice position, where δx is randomly chosen within the interval 0 ≤ δx ≤ Δ and analogously for δy. Two different regimes are found: for Δ < Δ(0) = 0.18(2) (in units of lattice spacing) a thermally driven transition between a paramagnetic and a dipolar antiferromagnetic phase with a critical exponent α/ν changing continuously with Δ; for Δ ≥ Δ(0) a paramagnetic phase covering the whole T > 0 range. Plots of the spin-glass overlap parameter versus temperature T or lattice size L seem to exclude an equilibrium spin-glass phase in the latter regime.

Artificial kagome arrays of nanomagnets: a frozen dipolar spin ice

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.

Dynamics of magnetic charges in artificial spin ice

Artificial spin ice has been recently implemented in two-dimensional arrays of mesoscopic magnetic wires. We propose a theoretical model of magnetization dynamics in artificial spin ice under the action of an applied magnetic field. Magnetization reversal is mediated by domain walls carrying two units of magnetic charge. They are emitted by lattice junctions when the local field exceeds a critical value Hc required to pull apart magnetic charges of opposite sign. Positive feedback from Coulomb interactions between magnetic charges induces avalanches in magnetization reversal.

Effective temperature in an interacting vertex system: theory and experiment on artificial spin ice

Frustrated arrays of interacting single-domain nanomagnets provide important model systems for statistical mechanics, as they map closely onto well-studied vertex models and are amenable to direct imaging and custom engineering. Although these systems are manifestly athermal, we demonstrate that an effective temperature, controlled by an external magnetic drive, describes their microstates and therefore their full statistical properties.