Artificial spin ices consist of nanomagnets arranged on the sites of various periodic and aperiodic lattices. They have enabled the experimental investigation of a variety of fascinating phenomena such as frustration, emergent magnetic monopoles and phase transitions that have previously been the domain of bulk spin crystals and theory, as we discuss in this Review. Artificial spin ices also show promise as reprogrammable magnonic crystals and, with this in mind, we give an overview of the measurements of fast dynamics in these magnetic metamaterials. We survey the variety of geometries that have been implemented, in terms of both the form of the nanomagnets and the lattices on which they are placed, including quasicrystalline systems and artificial spin systems in 3D. Different magnetic materials can also be incorporated to modify anisotropies and blocking temperatures, for example. With this large variety of systems, the way is open to discover new phenomena, and we complete this Review with possible directions for the future. Artificial spin ices are metamaterials displaying fascinating phenomena arising from the collective behaviour of nanoscale magnets. We review recent developments in terms of emergent magnetic monopoles, phase transitions, dynamics and geometries, and discuss future directions for research and potential applications.

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Advances in artificial spin ice

Using mean-field theory, exact diagonalizations, and SU(3) flavor theory, we have precisely mapped out the phase diagram of the S = 1 bilinear-biquadratic Heisenberg model on the triangular lattice in a magnetic field, with emphasis on the quadrupolar phases and their excitations. In particular, we show that ferroquadrupolar order can coexist with short-range helical magnetic order, and that the antiferroquadrupolar phase is characterized by a remarkable 2/3 magnetization plateau, in which one site per triangle retains quadrupolar order while the other two are polarized along the field. Implications for actual S=1 magnets are discussed.

Quadrupolar phases of the S=1 bilinear-biquadratic Heisenberg model on the triangular lattice

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.

https://iopscience.iop.org/article/10.1088/1361-648X/ab3e78/pdf

Magnetization dynamics in artificial spin ice.

Over the past few years, the study of magnetization dynamics in artificial spin ices has become a vibrant field of study. Artificial spin ices are ensembles of geometrically arranged, interacting magnetic nanoislands, which display frustration by design. These were initially created to mimic the behavior in rare earth pyrochlore materials and to study emergent behavior and frustration using two-dimensional magnetic measurement techniques. Recently, it has become clear that it is possible to create artificial spin ices, which can potentially be used as functional materials. In this perspective, we review the resonant behavior of spin ices in the GHz frequency range, focusing on their potential application as magnonic crystals. In magnonic crystals, spin waves are functionalized for logic applications by means of band structure engineering. While it has been established that artificial spin ices can possess rich mode spectra, the applicability of spin ices to create magnonic crystals hinges upon their reconfigurability. Consequently, we describe recent work aiming to develop techniques and create geometries allowing full reconfigurability of the spin ice magnetic state. We also discuss experimental, theoretical, and numerical methods for determining the spectral response of artificial spin ices and give an outlook on new directions for reconfigurable spin ices.

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Dynamics of reconfigurable artificial spin ice: Toward magnonic functional materials

Complex behavior in soft materials comes from the interactions between its components and geometrical constraints associated with lattice structures. Frustration is one classical example of such complex behavior leading to a plethora of phases and new phenomena. In this Colloquium the case of the ice rule is discussed and how it reflects in other soft materials.

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Colloquium: Ice rule and emergent frustration in particle ice and beyond

Systems of interacting nanomagnets known as artificial spin ice1–4 have allowed the design, realization and study of geometrically frustrated exotic collective states5–10 that are absent in natural magnets. We have experimentally measured11,12 the thermally induced moment fluctuations in the Shakti geometry of artificial spin ice. We show that its disordered moment configuration is a topological phase described by an emergent dimer-cover model13 with excitations that can be characterized as topologically charged defects. Examination of the low-energy dynamics of the system confirms that these effective topological charges have long lifetimes associated with their topological protection, that is, they can be created and annihilated only as charge pairs with opposite sign and are kinetically constrained. This manifestation of classical topological order14–19 demonstrates that geometrical design in nanomagnetic systems can lead to emergent, topologically protected kinetics that can limit pathways to equilibration and ergodicity. Experiments on the Shakti geometry of artificial spin ice show that its low-energy excitations are topologically protected, and that an emergent classical topological order influences the ergodicity and equilibration of this nanomagnetic system.

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Classical topological order in the kinetics of artificial spin ice

Artificial spin-ice systems are magnetic metamaterials consisting of nanomagnet arrays that can be designed to study exotic magnetic states not found in natural materials. Typically, these arrays are modelled as interacting binary macrospins that can only be in an up or down state and are described by the Ising model. These materials have demonstrated ordering transitions, but only via a spontaneous symmetry-breaking mechanism. We have designed and studied a quadrupole artificial spin-ice system consisting of interacting plaquettes of coupled single-domain nanomagnets that can be interpreted as a composite, ternary variable. After annealing this system in an external magnetic field, we observe both a ferroquadrupolar and an antiferroquadrupolar phase, with an apparent first-order phase boundary and a coexistence regime. The phase diagram of this material is reminiscent of a model used to describe phase coexistence in the superfluid transition of 4He with 3He impurities. These results illustrate how composite magnetic objects realize exotic statistical physics models beyond the Ising model. Artificial spin-ice materials are usually described by spins that are either up or down. Here, a new type of spin ice is fabricated where the spins can be in one of three states with different coexisting phases separated by a first-order transition.

Alan Albrecht

Papers

Classical topological order in the kinetics of artificial spin ice

Field-induced phase coexistence in an artificial spin ice