Session L5: Frustrated Magnetism: Artificial Spin Ice

Nearest neighbor correlations in perpendicular artificial spin ice arrays in the presence of an applied field

By studying the field dependent magnetization switching process in perpendicular artificial spin ice arrays arrays, we hope to gain insight in to the dynamical properties of interacting spin systems. To this end, we have used diffraction-limited Kerr imaging to study lithographically patterned arrays of single domain, nanoscale islands of Co/Pt multilayers. We can tune the interaction strength and introduce geometric frustration in to the patterned systems by changing the lattice spacing and geometry of the arrays. Using MOKE microscopy we are able to optically resolve, spatially isolate, and extract the switching field of each island in an array in the presence of an external field. These switching fields allow us to calculate the magnetization and nearest neighbor spin-spin correlation throughout a hysteresis loop. These quantities help us determine the effect of increased interactions and geometric frustration on the switching process of dipole coupled arrays. Funded by DOE.   

Dimensionality Reduction in Artificial Spin Ice

Over the past ten years, square and hexagonal arrays of single-domain nanomagnets, known as artificial spin ice, have been used to study the microscopic properties of geometrical frustration. Here we describe the fabrication of a new type of artificial spin ice, the tetris lattice. The ground state configuration of the nanomagnets' moments was determined with photoemission electron microscopy. This lattice is designed such that its vertices (small clusters of nanomagnets) cannot all simultaneously achieve their ground state. As a consequence, the lattice decomposes into alternating ordered and disordered one-dimensional bands of moments. The disordered bands can be described by a thermal one-dimensional Ising model, underscoring the emergent dimensionality reduction found in this lattice.   

DC Magnetization and FMR results of Fibonacci Distortions on the Honeycomb Artificial Spin Ice

Nanofabrication techniques allow magnetic thin films to be lithographically-patterned into arrays of interacting macro-spins that can be designed to study emergent physical properties. Here we discuss the effects of continuous symmetry breaking on the equilibrium and dynamic magnetic properties of frustrated magnetic metamaterials. We have pattered five Permalloy (Ni$_{\mathrm{0.80}}$Fe$_{\mathrm{0.20}})$ samples of distorted Kagome ASI arrays that are generated by repeated application of a substitution algorithm. This algorithm employs an aperiodic Fibonacci sequence of binary digits that can be mapped into short (d$_{\mathrm{1}})$ and long (d$_{\mathrm{2}})$ distances. This distorts film segment lengths while the width (nominally 70 nm) and thickness (25 nm) remain constant. Additionally, the coordination of each three-fold Kagome vertex is continuously modified via these distortions. Micromagnetic simulations predict the Fibonacci distortions causes jamming of Dirac String propagation. We report DC magnetization and FMR dispersion for different magnitudes of distortion, and compare these results to simulation. Research at University of Kentucky supported by U.S. Nationsal Science Foundation grant no. DMR-1506979.   

Beller Lecture: Artificial Ferroic Systems

In artificial ferroic systems [1], novel functionality is engineered through the combination of structured ferroic materials and the control of the interactions between the different components. I will present two classes of these systems, beginning with hybrid mesoscopic structures incorporating two different ferromagnetic layers whose static and dynamic behaviour result from the mutual imprint of the magnetic domain configurations [2]. Here we have demonstrated a new vortex core reversal mechanism [3], which occurs when it is displaced across domain boundaries with a magnetic field. I will then describe our progress on artificial spin ice, consisting of arrays of dipolar-coupled nanomagnets arranged in frustrated geometries. We have employed photoemission electron microscopy to observe the behaviour of emergent magnetic monopoles in an array of nanomagnets placed on the kagome lattice [4]. We have also created artificial spin ice with fluctuating magnetic moments and observed the evolution of magnetic configurations with time. This has provided a means to study relaxation processes with a controlled route to the lowest-energy state [5]. Recently, we have demonstrated with muon spin relaxation that these magnetic metamaterials can support thermodynamic phase transitions [6], and future directions include the incorporation of novel magnetic materials such as ultrathin magnetic films [7], the investigation of 3D structures [8], as well as the implementation of x-ray resonant magnetic scattering to study magnetic correlations in smaller nanomagnets and at faster timescales [9]. [1] L.J. Heyderman and R.L. Stamps, JPCM (2013); [2] G. Heldt et al., APL (2014); [3] P. Wohlhuter et al., Nature Comms (2015); [4] E. Mengotti et al., Nature Phys. (2011); [5] A. Farhan et al., Nature Phys. and PRL (2013); [6] L. Anghinolfi et al., Nature Comms (2015); [7] V. Kapaklis et al., Nature Nanotech. (2014); [8] C. Donnelly et al., PRL (2015); [9] J. Perron et al., PRB (2013)   

Doped Artificial Spin Ice

We examine square and kagome artificial spin ice for colloids confined in arrays of double-well traps. Unlike magnetic artificial spin ices, colloidal and vortex artificial spin ice realizations allow creation of doping sites through double occupation of individual traps. We find that doping square and kagome ice geometries produces opposite effects. For square ice, doping creates local excitations in the ground state configuration that produce a local melting effect as the temperature is raised. In contrast, the kagome ice ground state can absorb the doping charge without generating non-ground-state excitations, while at elevated temperatures the hopping of individual colloids is suppressed near the doping sites. These results indicate that in the square ice, doping adds degeneracy to the ordered ground state and creates local weak spots, while in the kagome ice, which has a highly degenerate ground state, doping locally decreases the degeneracy and creates local hard regions.   

\textbf{Systematic Angular Study of Magnetoresistance in Permalloy Connected Kagome Artificial Spin Ice }

Artificial spin ices are nanostructured two-dimensional arrays of ferromagnetic elements, where frustrated interactions lead to unusual collective magnetic behavior. Here we report a room-temperature magnetoresistance study of connected permalloy (Ni$_{81}$Fe$_{19})$ kagome artificial spin ice networks, wherein the direction of the applied in-plane magnetic field is systematically varied. We measure both the longitudinal and transverse magnetoresistance in these structures, and we find certain transport geometries of the network show strong angular sensitivity -- even small variations in the applied field angle lead to dramatic changes of the magnetoresistance response. We also investigate the magnetization reversal of the networks using magnetic force microscopy (MFM), demonstrating avalanche behavior in the magnetization reversal. The magnetoresistance features are analyzed using an anisotropic magnetoresistance (AMR) model. Supported by the US Department of Energy. Work at the University of Minnesota was supported by Seagate Technology, NSF MRSEC, and a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme.   

Spin wave band structure of artificial square ices

Artificial square spin ices are structures composed of magnetic elements located on the sites of a geometrically frustrated, two-dimensional square lattice. Using a semi-analytical approach, we show that square spin ices exhibit a rich spin wave band structure that is tunable both by external magnetic fields and the magnetic state of individual elements. Internal degrees of freedom can give rise to equilibrium states with bent magnetization at the edges of each element, leading to characteristic excitations; in the presence of magnetostatic interactions these form separate bands analogous to impurity bands in semiconductors. Full-scale micromagnetic simulations corroborate our semi-analytical approach. This study shows that the magnon spectra, and therefore group and phase velocities and band gap, can be manipulated by external fields, temperature, or more sophisticated techniques such as using spin torque on individual elements, and suggesting that artificial square spin ices can be used as metamaterials for spin waves. Our results close the gap between the research fields of artificial spin ices and magnonics.   

Lithography patterns and data analysis for topologically frustrated artificial spin ice

Artificial spin ices (ASIs), lattices composed of nanoscale single-domain magnetic islands, have been studied extensively for their insights on frustrated systems. Recently, the square and kagome geometries have received the most attention. We study a variation of the square lattice, where we include one or more edge dislocations in an otherwise perfect arrangement, resulting in topological frustration of the system. We create lithography patterns using a MATLAB script that models the system as a lattice of connected nodes and starts by removing partial rows or columns of elements. We then allow the system to relax, reshaping these patterns with an algorithm that attempts to equalize the angles of the elements at each node and also maintain identical island lengths throughout the lattice. We then analyze experimental Lorentz Transmission Electron Microscopy (TEM) images of these lattices using another program, which manipulates the images in order to find and index all of the individual magnetic islands, and then uses the Lorentz contrast of the element to determine the direction of each island's magnetic moment. These moment directions are then combined to determine the type of each lattice vertex, using the traditional type I-IV notation for square lattices. The script then marks the TEM images to reflect the vertex classification, which allows us to clearly identify chains of type II {\&} III vertices in the Lorentz images. The chains carry net magnetic moment, in a direction defined by the type II vertices, which may then reverse at the type III vertices.   

Topological frustration of artificial spin ice

Dislocations are topological defects ubiquitous in crystalline materials, although they are often neglected in experimental and theoretical studies due to their complexity. Artificial spin ices (ASIs), lithographically patterned arrays of ferromagnetic nanostructures, are highly tunable systems that allow for detailed investigations of frustration by providing exquisite control and insight at the single-spin level. Here, we introduce controlled topological defects into thermally active square ASI lattices and directly observe the resulting spin configurations upon annealing. Whereas a canonical square ASI lattice can support perfect ground state ordering, we find the presence of a dislocation results in extended frustration within the system. Locally, the magnets are unfrustrated, but frustration of the lattice persists due to its topology. A chain of higher energy vertices always originates from each dislocation point and either extends to an edge of our finite crystal, or rarely, to a second dislocation point if it is present in the same crystal. We also simulate our work using a kinetic Monte Carlo technique and find remarkably similar behavior between the simulations and our experiments, with the same types of domain walls and domain patterns as in our experimental samples. Our results indicate that topological defects have non-trivial consequences and should receive more attention in investigations of three dimensional crystals with \textbf{q} $\ne $ \textbf{0} order parameters.   

Nanomagnetic field-driven thermal mobility of emergent monopoles in artificial spin ice.

Artificial spin ices are nanomagnetic islands confined in 2D and their size means they can be considered as single domain and Ising-like. In the square geometry, each vertex has four nanomagnets which can point either in or out. The lowest energy arrangement consists of two-in and two-out and obeys the so-called `ice-rule'. It is possible to construct an ordered state by tiling such vertices, above which it is possible to have ice-rule-violating excitations known as emergent magnetic monopoles. It is their propagation which has been imaged with a transmission X-ray microscope and, due to a novel on-membrane heater, elevated temperatures applied up to 700 K. Square ice arrays were fabricated on a SiN membrane, the CoFeB islands were 80x250 nm, 10nm thick and had lattice spacings in the 350-500nm. Increased avalanche length and faster string velocities were observed for both stronger interaction and increased temperature. We have also been able to define a magnetic mobility in our systems and observed increased mobility in more interacting systems or elevated temperature. The largest change in the magnetic mobility was found for the most strongly interacting array, increasing by 1.7\pm 0.7mm$^{\mathrm{2}}$A$^{\mathrm{-1}}$s$^{\mathrm{-1}}$for $\Delta $T $\approx $ 30 K.   

Macroscopic Artificial Magnetic Honeycomb Lattice of Thermally Controlled Ultra-Small Bonds

The two-dimensional artificial magnetic honeycomb lattice system is evolving into a new research arena to explore a plethora of novel magnetism that are predicted to occur as functions of temperature and magnetic field: a long-range spin ice, spin liquid, an entropy-driven magnetic charge-ordered state involving topological vortex pairs and a spin-order due to the spin chirality. We have created macroscopic samples of artificial magnetic honeycomb lattices of Cobalt and Permalloy having connected ultra-small elements (bonds), with length scales of sub-10 nm to 30 nm, which have never before been possible. The equivalent energy of the resulting systems is 10-100 K and is thus amenable to both temperature- and field-dependent exploration of novel magnetic phenomena. We have performed detailed magnetic and small angle neutron scattering measurements (SANS) on the newly fabricated honeycomb lattice of Permalloy that show the thermal character of the system. Furthermore, the experimental data reveals the onset of magnetic ordered regimes in temperature that are consistent with the predicted novel phase diagram in artificial honeycomb lattice. Research is supported by U.S. Department of Energy, Office of Basic Energy Sciences under Grant No. DE-SC0014461.   

Real time dynamic behavior of vertex frustrated artificial spin ice.

Artificial spin ice systems comprise two dimensional arrays of nanoscale single domain ferromagnets designed to have frustrated interactions among the moments. By decimating islands from the common square artificial spin ice, one can design lattices with so called `vertex frustration'. In such lattices, the geometry prevents all vertices from occupying local ground states simultaneously. Using Photoemission Electron Microscopy (PEEM), we access the real time thermally induced dynamics of the moment behavior in those lattices. Operating at a proper temperature, the moment direction of each island fluctuates with a sufficiently slow frequency that it can be resolvable by acquiring successive PEEM images. We can extract information regarding the collective excitations of the moments and understand how they reflect the frustration of lattice. Supported by the US Department of Energy, Office of Basic Energy Sciences, Materials Science and Engineering Division under grant no. DE-SC0010778. The work of C.N. was carried out under the auspices of the US Department of Energy at LANL under contract no. DE-AC52-06NA253962. The ALS is supported by the Director, Office of Science, Office of Basic Energy Sciences, of the US Department of Energy under contract no. DE-AC02-05CH11231.   

Monte Carlo simulations of kagome lattices with magnetic dipolar interactions

Monte Carlo simulations of classical spins on the two-dimensional kagome lattice with only dipolar interactions are presented [1]. In addition to revealing the sixfold-degenerate ground state, the nature of the finite-temperature phase transition to long-range magnetic order is discussed. Low-temperature states consisting of mixtures of degenerate ground-state configurations separated by domain walls can be explained as a result of competing exchange-like and shape-anisotropy-like terms in the dipolar coupling. Fluctuations between pairs of degenerate spin configurations are found to persist well into the ordered state as the temperature is lowered until locking in to a low-energy state. Results suggest that the system undergoes a continuous phase transition at $T\approx0.43$ in agreement with previous MC simulations [2] but the nature of the ordering process differs [3]. Preliminary results which extend this analysis to the 3D fcc ABC-stacked kagome systems will be presented [4]. \\ 1. M. S. Holden, M. L. Plumer, I. Saika-Voivod, and B. W. Southern, Phys. Rev. B 91, 224425 (2015). \\ 2. M. Maksymenko, V. R. Chandra, and R. Moessner, Phys. Rev. B 91, 184407 (2015). \\ 3. Y. Tomita, J. Phys. Soc. Jpn. 78, 114004 (2009). \\ 4. V. Hemmati, et al., Phys. Rev. B 86, 104419 (2012)