Dimensionality Reduction in Artificial Spin Ice

Nature Physics, 2015

Reducing the dimensionality of a physical system can have a profound effect on its properties, as in the ordering of low-dimensional magnetic materials 1 , phonon dispersion in mercury chain salts 2 , sliding phases 3 , and the electronic states of graphene 4. Here we explore the emergence of quasi-one-dimensional behavior in two-dimensional artificial spin ice, a class of lithographically-fabricated nanomagnet arrays used to study geometrical frustration 5-7. We extend the implementation of artificial spin ice by fabricating a new array geometry, the so-called tetris lattice 8. We demonstrate that the ground state of the tetris lattice consists of alternating ordered and disordered bands of nanomagnetic moments. The disordered bands can be mapped onto an emergent thermal one-dimensional

Nature, 2006

Frustration, defined as a competition between interactions such that not all of them can be satisfied, is important in systems ranging from neural networks to structural glasses. Geometrical frustration, which arises from the topology of a well-ordered structure rather than from disorder, has recently become a topic of considerable interest 1 . In particular, geometrical frustration among spins in magnetic materials can lead to exotic lowtemperature states 2 , including 'spin ice', in which the local moments mimic the frustration of hydrogen ion positions in frozen water 3-6 . Here we report an artificial geometrically frustrated magnet based on an array of lithographically fabricated single-domain ferromagnetic islands. The islands are arranged such that the dipole interactions create a two-dimensional analogue to spin ice. Images of the magnetic moments of individual elements in this correlated system allow us to study the local accommodation of frustration. We see both ice-like short-range correlations and an absence of long-range correlations, behaviour which is strikingly similar to the low-temperature state of spin ice. These results demonstrate that artificial frustrated magnets can provide an uncharted arena in which the physics of frustration can be directly visualized.

Applied Physics Letters, 2014

Artificial spin ice is a frustrated magnetic two-dimensional nano-material, recently employed to study variety of tailor-designed unusual collective behaviours. Recently proposed extensions to three dimensions are based on self-assembly techniques and allow little control over geometry and disorder. We present a viable design for the realization of a three-dimensional artificial spin ice with the same level of precision and control allowed by lithographic nano-fabrication of the popular two-dimensional case. Our geometry is based on layering already available two-dimensional artificial spin ice and leads to an arrangement of ice-rule-frustrated units which is topologically equivalent to that of the tetrahedra in a pyrochlore lattice. Consequently, we show, it exhibits a genuine ice phase and its excitations are, as in natural spin ice materials, magnetic monopoles interacting via Coulomb law.

Springer Series in Solid-State Sciences, 2018

Artificial Spin Ices are two dimensional arrays of magnetic, interacting nano-structures whose geometry can be chosen at will, and whose elementary degrees of freedom can be characterized directly. They were introduced at first to study frustration in a controllable setting, to mimic the behavior of spin ice rare earth pyrochlores, but at more useful temperature and field ranges and with direct characterization, and to provide practical implementation to celebrated, exactly solvable models of statistical mechanics previously devised to gain an understanding of degenerate ensembles with residual entropy. With the evolution of nano-fabrication and of experimental protocols it is now possible to characterize the material in real-time, real-space, and to realize virtually any geometry, for direct control over the collective dynamics. This has recently opened a path toward the deliberate design of novel, exotic states, not found in natural materials, and often characterized by topological properties. Without any pretense of exhaustiveness, we will provide an introduction to the material, the early works, and then, by reporting on more recent results, we will proceed to describe the new direction, which includes the design of desired topological states and their implications to kinetics.

Reviews of Modern Physics, 2013

Frustration-the presence of competing interactions-is ubiquitous in the physical sciences and is a source of degeneracy and disorder, which in turn give rise to new and interesting physical phenomena. Perhaps nowhere does it occur more simply than in correlated spin systems, where it has been studied in the most detail. In disordered magnetic materials, frustration leads to spin-glass phenomena, with analogies to the behavior of structural glasses and neural networks. In structurally ordered magnetic materials, it has also been the topic of extensive theoretical and experimental studies over the past two decades. Such geometrical frustration has opened a window on a wide range of fundamentally new exotic behavior. This includes spin liquids in which the spins continue to fluctuate down to the lowest temperatures; and spin ice, which appears to retain macroscopic entropy even in the low temperature limit where it enters a topological Coulomb phase. In the past seven years a new perspective has opened in the study of frustration through the creation of artificial frustrated magnetic systems. These materials consist of arrays of lithographically fabricated single-domain ferromagnetic nanostructures that behave like giant Ising spins. The nanostructures' interactions can be controlled through appropriate choices of their geometric properties and arrangement on a (frustrated) lattice. The degrees of freedom of the material can not only be directly tuned, but also individually observed. Experimental studies have unearthed intriguing connections to the out-of-equilibrium physics of disordered systems and non-thermal 'granular' materials, while revealing strong analogies to (spin) ice materials and their fractionalized magnetic monopole excitations, lending the enterprise a distinctly interdisciplinary flavor. The experimental results have also been closely coupled to theoretical and computational analyses, facilitated by connections to classic models of frustrated magnetism, whose hitherto unobserved aspects have here found an experimental realization. We review the considerable experimental and theoretical progress in this field, including connections to other frustrated phenomena, and we outline future vistas for progress in this rapidly expanding field. V. True degeneracy, monopoles and more 18 V.a The quest for true degeneracy 18 V.b Monopoles and multipoles 20 V.c Collective physics in honeycomb artificial spin ice 25 VI. Other "artificial" spin systems 27 VII. Future prospects 28 Acknowledgements: 29 References

Journal of Physics: Condensed Matter, 2018

Geometry induced dynamics yields remarkable physical phenomena. In the macrocosms, the curvature of spacetime (geometrodynamics) tells matter how to move (gravity). In the microcosms, an example is the geometrical frustration in magnetic materials, whereas under certain conditions, can lead to the formation of spin liquids, in which the constituent spins still fluctuate strongly down to a temperature of absolute zero. In this work, we would like to explore a geometrical effect in artificial spin ices (ASI). It is well known that, in general, such artificial materials are athermal because they are constructed with elongated nanomagnets containing a large number of atomic spins, generating a big net magnetic moment that need a great amount of energy to flip. Therefore, recently, thermally driven dynamics in ASI materials became an important subject of investigation. We then expand this picture by showing that geometrically driven dynamics in ASI can open up the panorama of exploring distinct ground states and thermally magnetic monopole excitations. Here, it is shown that a particular ASI lattice, whereas four spins meet at every vertex, will provide a richer thermodynamics only due to its geometry. Indeed, for all kinds of planar ASI geometries, with ground states obeying the familiar 'two-in, two-out' ice rule in each vertex, the nanomagnets spin will experience less restriction to flip precisely in a kind of rhombic lattice. This can be observed by analysing only three types of rectangular artificial spin ices (RASI). Denoting the horizontal and vertical lattice spacings by a and b, respectively, then, a RASI material can be described by its aspect ratio γ ≡ a/b. The rhombic lattice emerges when γ = √ 3. So, by comparing the impact of thermal effects on the spin flips in these three appropriate different RASI arrays, it is possible to find the phenomenon we call ASI geometrothermodynamic. The comparison is done among RASI with γ = √ 2, γ = γR = √ 3 and γ = √ 4. The experimental data and the direct imaging of individual nanomagnets and their magnetization are obtained, as a function of temperature, by using Photoemission Electron Microscopy (P EEM) combined with X-ray Magnetic Circular Dichroism (XM CD) and Magneto-optic Kerr effect (M OKE) measurements. Our experimental data corroborates the unusual behavior of the critical temperatures in the RASI materials investigated here, as predicted by our Monte Carlo simulations.

Physical Review B, 2012

In this work we propose and study a realization of an artificial spin ice-like system, not based on any real material, in a triangular geometry. At each vertex of the lattice, the "ice-like rule" dictates that three spins must point inward while the other three must point outward. We have studied the system's ground-state and the lowest energy excitations as well as the thermodynamic properties of the system. Our results show that, despite fundamental differences in the vertices topologies as compared to the artificial square spin ice, in the triangular array the lowest energy excitations also behave as a kind of Nambu monopoles (two opposite monopoles connected by an energetic string). Indeed, our results suggest that the monopoles charge value may have a universal value while the string tension could be tuned by changing the system's geometry, probably allowing the design of systems with different string tensions. Our Monte Carlo results suggest a phase transition in the Ising universality class where the mean distance between monopoles and anti-monopoles increases considerably at the critical temperature. The differences on the vertices topologies seem to facilitate the experimental achievement of the system's ground-state, thereby allowing a more detailed experimental study of the system's properties.

Nature, 2013

Physical Review Letters, 2012

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.

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