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Skyrmion Spin Ice in Liquid Crystals
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Qubit spin ice
Artificial spin ices are frustrated spin systems that can be engineered, in which fine tuning of geometry and topology has allowed the design and characterization of exotic emergent phenomena at the constituent level. Here, we report a realization of spin ice in a lattice of superconducting qubits. Unlike conventional artificial spin ice, our system is disordered by both quantum and thermal fluctuations. The ground state is classically described by the ice rule, and we achieved control over a fragile degeneracy point, leading to a Coulomb phase. The ability to pin individual spins allows us to demonstrate Gauss’s law for emergent effective monopoles in two dimensions. The demonstrated qubit control lays the groundwork for potential future study of topologically protected artificial quantum spin liquids.
Crystallizing Kagome Artificial Spin Ice
Artificial spin ices are engineered arrays of dipolarly coupled nanobar magnets. They enable direct investigations of fascinating collective phenomena from their diverse microstates. However, experimental access to ground states in the geometrically frustrated systems has proven difficult, limiting studies and applications of novel properties and functionalities from the low energy states. In this work, we introduce a convenient approach to control the competing diploar interactions between the neighboring nanomagnets, allowing us to tailor the vertex degeneracy of the ground states. We achieve this by tuning the length of selected nanobar magnets in the spin ice lattice. We demonstrate the effectiveness of our method by realizing multiple low energy microstates in a kagome artificial spin ice, particularly the hardly accessible long range ordered ground state—the spin crystal state. Our strategy can be directly applied to other artificial spin systems to achieve exotic phases and explore new emergent collective behaviors.
Ice sculpting: An artificial spin ice Tutorial on controlling microstate and geometry for magnonics and neuromorphic computing
Artificial spin ice, arrays of strongly interacting nanomagnets, are complex magnetic systems with many emergent properties, rich microstate spaces, intrinsic physical memory, high-frequency dynamics in the GHz range, and compatibility with a broad range of measurement approaches. This Tutorial article aims to provide the foundational knowledge needed to understand, design, develop, and improve the dynamic properties of artificial spin ice. Special emphasis is placed on introducing the theory of micromagnetics, which describes the complex dynamics within these systems, along with their design, fabrication methods, and standard measurement and control techniques. The article begins with a review of the historical background, introducing the underlying physical phenomena and interactions that govern artificial spin ice. We then explore the standard experimental techniques used to prepare the microstate space of the nanomagnetic array and to characterize magnetization dynamics, both in artificial spin ice and more broadly in ferromagnetic materials. Finally, we introduce the basics of neuromorphic computing applied to the case of artificial spin ice systems with a goal to help researchers new to the field grasp these exciting new developments.
Emergent Spin Dynamics Enabled by Lattice Interactions in a Bicomponent Artificial Spin Ice
Artificial spin ice (ASI) 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 complimentary 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 intra-lattice 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. Here, 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.
Geometric control of emergent antiferromagnetic order in coupled artificial spin ices
Artificial spin ices (ASIs) composed of coupled nanomagnets offer the possibility to create designer geometrical frustration and manipulate inter-nanomagnet interactions. In particular, by using a dimer motif consisting of two strongly coupled single-domain nano magnets as a building block, we can control and realize intriguing antiferromagnetic physical states in which the magnetic charge is not conserved. Here we create a dimer kagome ASI system within which the antiferromagnetic order is controlled by tuning lattice geometry. Different disordered antiferromagnetic phases and the inter-nanomagnet interactions are visualized directly in demagnetized lattices with different lattice parameters. Monte Carlo simulations establish that the collective ground state consists of disordered antiferromagnetic dimers across the lattice when the intra-dimer interaction is dominant. However, for lattices governed by intra-triad-unit interactions, the ground state exhibits a longrange spin-ordered state in which the vertex magnetic charge is uniform across all triads.
Searching for Phase Transitions in Artificial Kagome Spin Ice
Artificial Kagome spin ice is a metasurface consisting of small islands of polarizable magnets. These islands can be oriented by an external magnetic field applied in the same plane as the metasurface. By subsequently applying a relatively weaker field in the opposite direction of the initial field, a portion of the magnets in the lattice can be repolarized or “flipped” the other way. These flips cause instabilities in a previously uniformly aligned lattice, causing the system to interact with itself and induce changes in the overall magnetization of the metasurface. This Summer, my goal was to use this phenomenon to discern whether or not these systems could undergo a phase transition caused only by the initial change of the external magnetic field. Based on previous datasets that indicated that such a transition was observable, my task was to find and reproduce this potential phenomena. This process was conducted by repeating measurements on the system by using an external magnet to align the spin ice and then using Magnetic Force Microscopy (MFM) to probe the surface of the material to understand the changes in the magnetization over time. This report will discuss the characteristics of artificial Kagome spin ice, the methodology used to induce phase transitions, and the image processing methods used to understand the results. Although no reproducible evidence of a phase transition was found during the Summer, we were able to successfully reproduce short-term responses in the spin ice, as well as plot a hysteresis curve of the material.
Exploring the phase diagram of 3D artificial spin-ice
Artificial spin-ices consist of lithographic arrays of single-domain magnetic nanowires organised into frustrated lattices. These geometries are usually two-dimensional, allowing a direct exploration of physics associated with frustration, topology and emergence. Recently, three-dimensional geometries have been realised, in which transport of emergent monopoles can be directly visualised upon the surface. Here we carry out an exploration of the three-dimensional artificial spin-ice phase diagram, whereby dipoles are placed within a diamond-bond lattice geometry. We find a rich phase diagram, consisting of a double-charged monopole crystal, a single-charged monopole crystal and conventional spin-ice with pinch points associated with a Coulomb phase. In experimental demagnetised systems, broken symmetry forces formation of ferromagnetic stripes upon the surface, forbidding the lower energy double-charged monopole crystal. Instead, we observe crystallites of single magnetic charge, superimposed upon an ice background. The crystallites are found to form due to the distribution of magnetic charge around the 3D vertex, which locally favours monopole formation.
Imaging the magnetic nanowire cross section and magnetic ordering within a suspended 3D artificial spin-ice
Artificial spin-ice systems are patterned arrays of magnetic nanoislands arranged into frustrated geometries and provide insight into the physics of ordering and emergence. The majority of these systems have been realized in two-dimensions, mainly due to the ease of fabrication, but with recent developments in advanced nanolithography, three-dimensional artificial spin ice (ASI) structures have become possible, providing a new paradigm in their study. Such artificially engineered 3D systems provide new opportunities in realizing tunable ground states, new domain wall topologies, monopole propagation, and advanced device concepts, such as magnetic racetrack memory. Direct imaging of 3DASI structures with magnetic force microscopy has thus far been key to probing the physics of these systems but is limited in both the depth of measurement and resolution, ultimately restricting measurement to the uppermost layers of the system. In this work, a method is developed to fabricate 3DASI lattices over an aperture using two-photon lithography, thermal evaporation, and oxygen plasma exposure, allowing the probe of element-specific structural and magnetic information using soft x-ray microscopy with x-ray magnetic circular dichroism (XMCD) as magnetic contrast. The suspended polymer–permalloy lattices are found to be stable under repeated soft x-ray exposure. Analysis of the x-ray absorption signal allows the complex cross section of the magnetic nanowires to be reconstructed and demonstrates a crescent-shaped geometry. Measurement of the XMCD images after the application of an in-plane field suggests a decrease in magnetic moment on the lattice surface due to oxidation, while a measurable signal is retained on sub-lattices below the surface.
Artificial spin ice: Paths forward
Artificial spin ice systems are lithographically prepared arrays of interacting nanoscale magnetic moments with collective behavior resulting from the chosen array geometry. These many-body systems are unusual in that their simple constituent elements can be configured to design their interactions with exquisite control. They can then be probed experimentally over a remarkably large range of time scales and length scales, including imaging of the individual moments. The study of artificial spin ice has broadened well beyond the original focus on systems with connection to spin ice and ice models, and now the term is used in reference to many other structures that exhibit a much wider range of physical phenomena. Here in this perspective, we review progress in the field of artificial spin ice since its inception more than a decade ago. We then discuss prospects for future directions – continuing the exploration of nanomagnetism, statistical spin models, and the potential for technological devices.
Formation of Complex Spin Textures in Thermally Demagnetized La 0.7 Sr 0.3 Mn O 3 Artificial-Spin-Ice Structures
Artificial spin ices (ASIs) have traditionally been designed such that each nanomagnet possesses a single-domain magnetic configuration that is assumed to be minimally perturbed by interisland dipolar interactions. Using x-ray photoemission electron microscopy to perform magnetic domain imaging, we study thermally demagnetized La 0.7 Sr 0.3 MnO 3 -based brickwork ASI arrays and showed that complex spin textures (CSTs) can be stabilized through an appropriate selection of nanoisland width and interisland spacing. While the width dependence can be explained through the dominance of shape anisotropy in isolated nanoislands, the ASIs we investigate demonstrate a complex dependence on both the nanoisland width and interisland spacing. Micromagnetic simulations reveal that interisland dipolar interactions play a role in the formation of CSTs, which are composed of single- and double-vortex states. Furthermore, energy analysis of the simulations provides an understanding of the system energetics that arises from a delicate balance between intraisland effects (i.e., shape anisotropy and exchange energy) and interisland effects (i.e., dipolar interactions between nearest-neighbor nanoislands).
Field-Tunable Interactions and Frustration in Underlayer-Mediated Artificial Spin Ice
We report that 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.
Magnetic charge propagation upon a 3D artificial spin-ice
Magnetic charge propagation in spin-ice materials has yielded a paradigm-shift in science, allowing the symmetry between electricity and magnetism to be studied. Recent work is now suggesting the spin-ice surface may be important in mediating the ordering and associated phase space in such materials. Here, we detail a 3D artificial spin-ice, which captures the exact geometry of bulk systems, allowing magnetic charge dynamics to be directly visualized upon the surface. Using magnetic force microscopy, we observe vastly different magnetic charge dynamics along two principal directions. For a field applied along the surface termination, local energetics force magnetic charges to nucleate over a larger characteristic distance, reducing their magnetic Coulomb interaction and producing uncorrelated monopoles. In contrast, applying a field transverse to the surface termination yields highly correlated monopole-antimonopole pairs. Detailed simulations suggest it is the difference in effective chemical potential as well as the energy landscape experienced during dynamics that yields the striking differences in monopole transport.
On the degeneracy of spin ice graphs, and its estimate via the Bethe permanent
The concept of spin ice can be extended to a general graph. We study the degeneracy of spin ice graph on arbitrary interaction structures via graph theory. We map spin ice graphs to the Ising model on a graph and clarify whether the inverse mapping is possible via a modified Krausz construction. From the gauge freedom of frustrated Ising systems, we derive exact, general results about frustration and degeneracy. We demonstrate for the first time that every spin ice graph, with the exception of the one-dimensional Ising model, is degenerate. We then study how degeneracy scales in size, using the mapping between Eulerian trails and spin ice manifolds, and a permanental identity for the number of Eulerian orientations. Furthermore, we show that the Bethe permanent technique provides both an estimate and a lower bound to the frustration of spin ices on arbitrary graphs of even degree. While such a technique can also be used to obtain an upper bound, we find that in all finite degree examples we studied, another upper bound based on Schrijver inequality is tighter.
Revisiting spin ice physics in the ferromagnetic Ising pyrochlore Pr 2 Sn 2 O 7
Pyrochlore materials are characterized by their hallmark network of corner-sharing rare-earth tetrahedra, which can produce a wide array of complex magnetic ground states. Ferromagnetic Ising pyrochlores often obey the “two-in-two-out” spin ice rules, which can lead to a highly degenerate spin structure. Large moment systems, such as Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 , tend to host a classical spin ice state with low-temperature spin freezing and emergent magnetic monopoles. Systems with smaller effective moments, such as Pr 3+ -based pyrochlores, have been proposed as excellent candidates for hosting a “quantum spin ice” characterized by entanglement and a slew of exotic quasiparticle excitations. However, experimental evidence for a quantum spin ice state has remained elusive. Here, in this work, we show that the low-temperature magnetic properties of Pr 2 Sn 2 O 7 satisfy several important criteria for continued consideration as a quantum spin ice. We find that Pr 2 Sn 2 O 7 exhibits two distinct spin-correlation time scales of τ ≥ 10 –4 and ~10 –10 s in the spin ice regime. Our comprehensive bulk characterization and neutron scattering measurements enable us to map out the magnetic field-temperature phase diagram, producing results consistent with expectations for a ferromagnetic Ising pyrochlore. We identify key hallmarks of spin ice physics and show that the application of small magnetic fields (μ 0 H c ~0.5 T) suppresses the spin ice state and induces a field-polarized, ordered spin-ice phase. Together, our work clarifies the current state of Pr 2 Sn 2 O 7 and encourages future studies aimed at exploring the potential for a quantum spin ice ground state in this system.
The concept of spin ice graphs and a field theory for their charges
Originally detected in rare earth pyrochlores, spin ice physics is now being artificially extended to a variety of geometries that control collective behavior and exotic properties, making graph theory their proper framework. We relate spin ice notions, such as ice rule, ice manifold, Coulomb phases, charges, and monopoles, to graph-theoretical notions, such as balance, in/out-degrees, and Euler paths. We then propose a field-theoretical treatment in which topological charges and monopoles are the degrees of freedom, while the binary spins are subsumed in an entropic interaction among charges. We show that for a spin ice on a graph in a Gaussian approximation, the kernel of the entropic interaction is the inverse of the graph Laplacian, and we compute screening functions from the graph spectra as Green operators for the screened Poisson problem on a graph. We then apply the treatment to star graphs, tournaments, cycles, and regular spin ice in different dimensions. Our aim is twofold: to set spin ice physics in a proper graph setting, where only topological rather than geometrical notions hold, and to invite graph theorists to contribute their powerful tools to the field of spin ice.
Persistent dynamic magnetic state in artificial honeycomb spin ice
Abstract Topological magnetic charges, arising due to the non-vanishing magnetic flux on spin ice vertices, serve as the origin of magnetic monopoles that traverse the underlying lattice effortlessly. Unlike spin ice materials of atomic origin, the dynamic state in artificial honeycomb spin ice is conventionally described in terms of finite size domain wall kinetics that require magnetic field or current application. Contrary to this common understanding, here we show that a thermally tunable artificial permalloy honeycomb lattice exhibits a perpetual dynamic state due to self-propelled magnetic charge defect relaxation in the absence of any external tuning agent. Quantitative investigation of magnetic charge defect dynamics using neutron spin echo spectroscopy reveals sub-ns relaxation times that are comparable to the relaxation of monopoles in bulk spin ices. Most importantly, the kinetic process remains unabated at low temperature where thermal fluctuation is negligible. This suggests that dynamic phenomena in honeycomb spin ice are mediated by quasi-particle type entities, also confirmed by dynamic Monte-Carlo simulations that replicate the kinetic behavior. Our research unveils a macroscopic magnetic particle that shares many known traits of quantum particles, namely magnetic monopole and magnon.