Thanks to advances in chemical synthesis that enable control over the size, structure, properties, and functionalization, magnetic nanoparticles (NPs) present unique opportunities in areas as diverse as data storage, cancer treatment, and biomedical imaging. While superparamagnetism dominates the properties of magnetic NPs, a quantitative understanding of superparamagnetic blocking in NP assemblies remains elusive. We address this challenge here via comprehensive magnetic characterization and analysis of soft ferromagnetic NP ensembles based on Ni. NPs were synthesized by the injection of a Ni−oleylamine (OAm) complex into 200 °C trioctylphosphine (TOP), with size control achieved via the TOP:OAm ratio, reaction time, and differential centrifugation. X-ray diffraction, electron microscopy, and various spectroscopies reveal polycrystalline/twinned face-centered-cubic Ni NPs with mean diameters from 4 to 22 nm, dispersities down to 10%, and TOP and OAm ligands. Superparamagnetic blocking temperatures are carefully determined, quantitatively accounting for the substantial yet frequently ignored effects of dispersity, resulting in mean blocking temperatures spanning 5 K to >300 K. Even accounting for an ∼1 nm-thick magnetically dead/canted shell (deduced from magnetization) and the temperature dependence of the Ni magnetocrystalline anisotropy, these mean blocking temperatures cannot be quantitatively reproduced. Remarkably, this discrepancy is substantially resolved by accounting for shape anisotropy effects that result from even modest average deviations from spherical shapes. A quantitative understanding of the size-dependent blocking temperature of soft ferromagnetic metallic NP assemblies is thus achieved, with no adjustable fitting parameters, by quantitatively accounting for the size distribution, effective ferromagnetic volume, temperature-dependent magnetocrystalline anisotropy, and random shape anisotropy. While frequently ignored, the characterization of such factors is thus vital, paving the way to quantitative understanding of superparamagnetism in other magnetic NP systems.

The temperature dependence of the spin diffusion length typically reflects the scattering mechanism responsible for spin relaxation. Within nonmagnetic metals it is reasonable to expect the Elliot-Yafet mechanism to play a role and thus the temperature dependence of the spin diffusion length might be inversely proportional to resistivity. In lateral spin valves, measurements have found that at low temperatures the spin diffusion length unexpectedly decreases. By measuring the transport properties of lateral Py/Cu/Py spin valves, fabricated from Cu with magnetic impurities of <1 ppm and ∼ 4 ppm, we extract a spin diffusion length which shows this suppression below 30 K only in the presence of the Kondo effect. We have calculated the spin-relaxation rate and isolated the contribution from magnetic impurities. We find the spin-flip probability of a magnetic impurity to be 34%. Our analysis demonstrates the dominant role of Kondo scattering in spin relaxation, even in low concentrations of order 1 ppm, and hence illustrates its importance to the reduction in spin diffusion length observed by ourselves and others.

We have performed a detailed study of thermal annealing of the moment configuration in artificial spin ice. Permalloy (Ni 80 Fe 20 ) artificial spin ice samples were examined in the prototypical square ice geometry, studying annealing as a function of island thickness, island shape, and annealing temperature and duration. We also measured the Curie temperature as a function of film thickness, finding that thickness has a strong effect on the Curie temperature in regimes of relevance to many studies of the dynamics of artificial spin ice systems. Increasing the interaction energy between island moments and reducing the energy barrier to flipping the island moments allows the system to more closely approach the collective low energy state of the moments upon annealing, suggesting new channels for understanding the thermalization processes in these important model systems.Artificial spin ice systems are two-dimensional arrays of nanoscale elements, typically composed of single domain ferromagnetic islands 1 . These systems have been the subject of extensive study and have provided models for the study of a range of novel collective behaviors 2 . Certain artificial spin ice geometries have well-defined collective magnetic ground states, such as the square lattice 1 , while others have intrinsically disordered and complex ground states, such as the Shakti lattice 3-5 . These low-energy collective states have sparked considerable interest in attempting to realize the lowest energy state of different artificial spin ice lattices 6-10 . One successful approach to collective energy minimization involves annealing the arrays by heating them to temperatures near or above the Curie temperature (T C ) of the ferromagnetic material 11,12 . Upon cooling, the island moments arrange themselves into a low energy state via magnetostatic interactions. Using this method, both long-range-ordered 11-13 and intrinsically disordered ground states 4,14 have been achieved, both in permalloy (Ni 80 Fe 20 ) and in other alloys 9,10 . Notably, the method works well even for geometries known to exhibit slow relaxation toward the low energy state 4 . Given the high T C of permalloy, and its importance as a model material for these systems, we investigated thermal annealing of permalloy artificial spin ice by varying the annealing conditions and the geometry of the islands, with the goal of understanding how to improve the effectiveness of annealing.We fabricated our artificial square spin ice samples on Si wafers coated with a 200-nm-thick layer of Si-N deposited a) Electronic mail: peter.schiffer@yale.edu by low pressure chemical vapor deposition. The nanoislands, with varied lateral dimensions and inter-island gaps indicated below, were produced by electron beam lithography and liftoff as described previously 12 . The total area of all nanoislands in each square ice sample was about 200×200 µm 2 . In order to keep uniformity of all nanoislands, the write field for lithography was set to cover the whole sample. We deposited our samples with va...

One-dimensional strings of local excitations are a fascinating feature of the physical behavior of strongly correlated topological quantum matter. Here we study strings of local excitations in a classical system of interacting nanomagnets, the Santa Fe Ice geometry of artificial spin ice. We measured the moment configuration of the nanomagnets, both after annealing near the ferromagnetic Curie point and in a thermally dynamic state. While the Santa Fe Ice lattice structure is complex, we demonstrate that its disordered magnetic state is naturally described within a framework of emergent strings. We show experimentally that the string length follows a simple Boltzmann distribution with an energy scale that is associated with the system’s magnetic interactions and is consistent with theoretical predictions. The results demonstrate that string descriptions and associated topological characteristics are not unique to quantum models but can also provide a simplifying description of complex classical systems with non-trivial frustration.

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