We present mumax+, an extensible GPU-accelerated micromagnetic simulator with a Python user interface, to address the challenges posed by current magnetism research into systems with complex magnetic ordering and interfaces. It is a general solver for the space- and time-dependent evolution of the magnetization and related vector quantities, using finite difference discretization. Here, we present its design and application and discuss features not available in mumax3, such as the modeling of antiferromagnets with magnetoelastic coupling. As an illustration of its capabilities, we use mumax+ to simulate state-of-the-art magnetic systems. Specifically, we demonstrate the current-induced domain wall motion in a polycrystalline antiferromagnet, we simulate the working principle of a strain-driven antiferromagnetic racetrack memory and we reproduce experimentally observed domain structures in a non-collinear antiferromagnet.

We report an experimental investigation of the magnetic microstructure of iron oxide multicore assemblies by means of polarized small-angle neutron scattering (SANS). Guided by a recently developed analytical theory for vortex-state magnetic nanoparticles, we provide a quantitative comparison between the measured and calculated cross sections, revealing signatures that are consistent with vortex-type magnetization configurations at low applied magnetic fields. In particular, the field evolution and the characteristic isotropic ring-type feature of the spin-flip scattering intensity at intermediate momentum transfers are in line with the formation of flux-closure states. The latter are stabilized by the interplay of exchange, Zeeman, and magnetostatic energies. The methodology allows for a statistically significant characterization of vortex states in densely packed nanoparticle systems, thereby complementing surface-sensitive techniques that are commonly limited to the observation of spin structures in individual particles.

In this paper, we demonstrate how altermagnets can be simulated in the recently released micromagnetic simulation package mumax+. We have added a new magnet class for d-wave altermagnets and demonstrate how mumax+ is able to reproduce the analytical solutions for line profiles of the Néel vector and net magnetization for a Bloch domain wall. Next, we show simulation results of the magnon dispersion relation and its dependence on the anisotropic nature of the exchange interaction. Finally, we study the motion of a Néel skyrmion by applying a spin transfer torque. This new feature was implemented by extending the pre-existing code base for antiferromagnetic simulations. The object-oriented design of mumax+ allows for a correct calculation of the magnetostatic field in multi-sublattice systems, a feature that many other micromagnetic simulators lack.

Magnetic particle hyperthermia relies on the efficient conversion of magnetic field energy into heat in biomedical applications, yet the microscopic mechanisms governing heat generation within individual particles remain poorly understood. In this study, AC magnetometry experiments are combined with dynamic micromagnetic simulations to connect microstructural features, magnetization dynamics, and macroscopic heat dissipation. Beyond macroscopic heating metrics, the heat generation is resolved at the intra-particle level, uncovering a heterogeneous landscape of localized ”hot spots” with nanometer spatial and nanosecond temporal resolution. The results demonstrate that grain size acts as a key experimentally tunable parameter, balancing anisotropy disorder and pinning strength, thereby controlling both the magnitude and spatio-temporal distribution of heat release within the particle. In particular, nanoflower architectures composed by larger grains deliver larger heat generation, while the smaller grains offer a deeper intra-particle pinning landscape, which effectively redistributes the heat generation over extended time windows. Together, our results provide a mechanistic framework linking nanoparticle microstructure to magnetic heating and establish design principles for optimizing nanoflowers as magnetic hyperthermia transducers.

Spin disorder, inherent to magnetic nanoparticles, has traditionally been regarded as a detrimental feature, with materials-engineering efforts largely focused on producing ”perfect particles” containing as few defects as possible. Alongside this pursuit of perfection, however, an alternative framework has emerged in recent years that reframes intra-particle disorder as an ”ugly duckling” whose functional potential remains to be unlocked. In this Perspective, we review the emerging concept of disorder engineering in magnetic nanoparticles, identify its current challenges, and outline promising future directions. From a theoretical standpoint, progress requires moving beyond the widely used macrospin approximation, which severely restricts the description of intra-particle degrees of freedom. Micromagnetic modelling, in contrast, treats magnetisation as a continuous vector field and thereby enables (i) the explicit representation of intra-particle degrees of freedom, linking microstructural features to internal magnetisation textures, and (ii) direct correspondence with polarized small-angle neutron scattering, an experimental technique that provides quantitative access to ensemble-averaged magnetic correlations on nanometre length scales. The field must now advance towards falsifiable and uncertainty-aware models with structurally motivated parameters and predictions that can be tested against independent experimental observables. The overarching goal is to establish quantitative relationships between particle structure, intra-particle magnetisation textures, and macroscopic functionality, thereby transforming spin disorder from an elusive hidden variable into an engineerable design parameter.

The origin of double-step magnetization reversal processes, so-called wasp-waist magnetization hysteresis loops, in single magnetic phase 3D cobalt ferrite nanoassemblies is still poorly understood. So far, this behavior has been mainly attributed to the co-existence of hard-soft magnetic phases and spin canting in nanoparticles. Here, we demonstrate the wasp-waisted magnetization loops in single-phase flower-like Co0.82Fe2.18O4 nanoassemblies that were synthesized by modifying the ligand chemistry. Combining magnetization hysteresis loops at different concentrations, degrees of dipolar interactions, and temperatures, energy dispersive X-ray and in-field Mössbauer spectroscopy, and small-angle neutron scattering, we propose that a combination of a strong dipolar field and spin disordered nano-building blocks, leading to soft magnetic phase at the grain boundaries, accounts for this anomalous and abrupt drop in magnetization in nanoassemblies. The nanoassemblies have a porous nanostructure with nanogaps between their nano-building blocks, as revealed from electron microscopy investigations. Small-angle neutron scattering studies reveal spin disorder at the surface and interfaces of the nano-building blocks. The strong dipolar field at the ensemble level is only achieved when particle colloidal suspensions are dried from high particle concentrations, indicating concentration-dependent nature of this behavior. Single-core nanoparticles with a comparable chemical composition, effective size, coercive field, and magnetization, but with a coherent crystal structure, do not reveal this peculiar behavior even at highest concentrations. This finding demonstrates the role that the assembly of nanoscale building blocks plays to give rise to this peculiar magnetization. Our study introduces organic capping ligands as a novel means to tune magnetization processes in nanoparticles and to initiate new applications, a novel role for organic ligands beyond giving nanoparticles colloidal stability.

The oxidation of magnetite to maghemite is a naturally occurring process that leads to the degradation of the magnetic properties of magnetite nanoparticles. Despite being systematically observed with traditional macroscopic magnetization measurement techniques, a detailed understanding of this process at the microscale is still missing. In this study, we track the evolution of the magnetic structure of magnetite nanoparticles during their oxidation to maghemite through numerical micromagnetic simulations. To capture realistic interparticle effects, we incorporate dipolar interactions by modeling the nanoparticles arranged in chains. Our computational results are benchmarked against experimental data from magnetotactic bacteria, studied over a time scale of years. To resolve the magnetization at the interface between both oxide phases, we propose spin-polarized small-angle neutron scattering (SANS), an experimental technique capable of probing magnetization textures at nanometer length scales. By analyzing the pair-distance distribution function extracted from numerically-computed SANS cross sections, we identify distinct signatures of magnetic disorder. Specifically, our findings suggest that the magnetization from the non-oxidized core region varies smoothly across the (structurally sharp) interface into the oxidized shell. The existence of such a diffuse magnetic interface may account for the superior magnetic properties of partially oxidized magnetite nanoparticles compared to fully converted maghemite samples.

We present Hotspice, a Monte Carlo simulation software designed to capture the dynamics and equilibrium states of Artificial Spin Ice (ASI) systems with both in-plane (IP) and out-of-plane (OOP) geometries. An Ising-like model is used where each nanomagnet is represented as a macrospin, with switching events driven by thermal fluctuations, magnetostatic interactions, and external fields. To improve simulation accuracy, we explore the impact of several corrections to this model, concerning for example the calculation of the dipole interaction in IP and OOP ASI, as well as the impact of allowing asymmetric rather than symmetric energy barriers between stable states. We validate these enhancements by comparing simulation results with experimental data for pinwheel and kagome ASI lattices, demonstrating how these corrections enable a more accurate simulation of the behavior of these systems. We finish with a demonstration of `clocking' in pinwheel and OOP square ASI as an example of reservoir computing.

To gain deeper insight into the complex, stable, and robust configurations of magnetic textures, topological characterization has proven essential. In particular, while the skyrmion number is a well-established topological invariant for two-dimensional magnetic textures, the Hopf index serves as a key topological descriptor for three-dimensional magnetic structures. In this paper, we present and compare various methods for numerically calculating the Hopf index, provide implementations, and offer a detailed analysis of their accuracy and computational efficiency. Additionally, we identify and address common pitfalls and challenges associated with the numerical computation of the Hopf index, offering insights for improving the robustness of these techniques.

Objective.Magnetic fluid hyperthermia is a promising adjuvant cancer therapy presently approaching clinical application. The therapeutic effect stems from the heat produced by magnetic nanoparticles (MNPs) administered to the tumor site and exposed to an AC magnetic field applied from outside the body. The objective of our study is to improve the integration of magnetic particle imaging (MPI) and hyperthermia as a theranostic application by allowing a real-time monitoring of local heat generation.Approach.The area of the dynamic hysteresis loop of the MNP is a measure of the heat produced by the MNP. However, depending on the specifics of the measurements, an accurate determination of the dynamic hysteresis loop of MNPs by conventional magnetic particle spectroscopy (MPS) can be hindered due to missing information of the first harmonic. The method presented in this work provides a solution to this problem by extracting the area of the hysteresis loop from measured MPS spectra through the reconstruction of the first harmonic.Main results.The method was tested on three distinct commercial MNP systems and found to be in good agreement with hysteresis loops directly obtained through AC magnetometry, confirming the method's reliability.Significance.This advancement enables accurate real-time monitoring of the energy dissipated as heat by the particles during MPS measurements and thus directly contributes to the development of MPI-guided hyperthermia.

Iron‐oxide nanoflowers (NFs) are one of the most efficient nanoheaters for magnetic hyperthermia therapy. However, the physics underlying the dynamic response of realistic nanoparticles, containing disorder, beyond the single‐domain limit remains poorly understood. Using large‐scale micromagnetic simulations, the magnetization of biocompatible iron‐oxide NFs ( d = 10–400 nm) has been mapped, connecting their microstructure to their macroscopic magnetic response. Above the single‐domain regime ( d > 50 nm), the magnetization folds into a vortex state, within which the coercivity reaches a secondary maximum, not present for nondisordered nanoparticles. The dynamics of the vortex shows two distinct reversal modes: 1) a core‐dominated one, with an increasing coercivity with d ; 2) a flux‐closure‐domains dominated reversal mode, with a decreasing coercivity‐size dependence. The coercivity maximum is located at the transition between both reversal modes and results from the combination of grain anisotropy and grain‐boundary pinning. The results provide the first description of spin textures in iron oxide NFs beyond the macrospin framework, revealing how particles with identical static magnetization exhibit fundamentally distinct dynamics, which result in different macroscopic behavior. By adjusting the grain size, the coercivity “sweet spot” can be tailored, offering a practical route to next‐generation, high‐efficiency nanoheaters.

The origin of double-step magnetization reversal processes, so-called wasp-waist magnetization hysteresis loops, in single magnetic phase 3D cobalt ferrite nanoassemblies is still poorly understood. So far, this behavior has been mainly attributed to the coexistence of hard-soft magnetic phases and spin canting in nanoparticles. Here, we demonstrate the wasp-waisted magnetization loops in single-phase flower-like Co0.83Fe2.17O4 nanoassemblies that were synthesized by modifying the ligand chemistry. Combining magnetization hysteresis loops at different concentrations, degrees of dipolar interactions, and temperatures, energy-dispersive X-ray and in-field Mossbauer spectroscopy, and small-angle neutron scattering, we propose that a combination of a strong dipolar field and spin-disordered nanobuilding blocks, leading to a soft magnetic phase at the grain boundaries, accounts for this anomalous and abrupt drop in magnetization in nanoassemblies. The nanoassemblies have a porous nanostructure with nanogaps between their nanobuilding blocks, as revealed from electron microscopy investigations. Small-angle neutron-scattering studies reveal spin disorder at the surfaces and interfaces of the nanobuilding blocks. The strong dipolar field at the ensemble level is achieved only when particle colloidal suspensions are dried from high particle concentrations, indicating the concentration-dependent nature of this behavior. Single-core nanoparticles with comparable chemical composition, effective size, coercive field, and magnetization, but with a coherent crystal structure, do not reveal this peculiar behavior even at the highest concentrations. Our study introduces organic capping ligands as a means to tune magnetization processes in nanoparticles for applications in magnetic hyperthermia, an unexplored role for organic ligands beyond giving nanoparticles colloidal stability.

Magnetic nanoparticles (MNPs) are emerging as key tools in biomedical and technical applications due to their tunable magnetic properties and responsiveness to external magnetic fields. However, the effectiveness of MNPs in applications such as targeted drug delivery, magnetic imaging and magnetic hyperthermia critically depends on achieving a narrow particle size distribution. Conventional gradient magnetic separation techniques often fall short in delivering high resolution size separation, particularly in the challenging 20 to 200 nm range, where the interplay between Brownian motion and magnetophoretic forces reduces separation precision. Therefore, in this study, we propose an enhanced gradient magnetic separation (GMS) method that superimposes a homogeneous alternating magnetic field onto an inhomogeneous gradient field and makes use of size-dependent magnetization dynamics. The proposed dual-field method is first verified in a simple test case, confirming that the desired separation behavior can principally be achieved. Simulations show that the magnetization ratio between particles of different sizes can be significantly increased beyond the predictions of the Langevin function. By systematically varying offset and alternating field strengths, an optimal combination maximizing this ratio is identified. Additionally, the influence of the alternating field frequency is investigated, showing that separation efficiency improves with increasing frequency up to a saturation point. To translate this behavior into effective spatial separation, particle trajectories are simulated while dynamically optimizing the alternating field strength over time to maximize the travelled distance ratio between large and small particles. The results demonstrate that large particles maintain strong alignment with the field, while smaller particles experience reduced time averaged magnetization, resulting in notably reduced mobility. Additionally, travelled distance ratios between particle sizes increase significantly compared to using a gradient field alone. The introduced dual-field method is also shown to remain effective for various particle sizes and under more realistic conditions where hydrodynamic and magnetic radii differ due to surface coatings. Finally, it is shown that the separation cut-off radius can be chosen arbitrarily, confirming the size independence of the method. These findings demonstrate that the proposed method substantially enhances size based separation, enabling improved control over particle size distributions and potentially advancing biomedical applications.

This study explores the impact of different magnetic driving field waveforms on nanoparticle heating in magnetic hyperthermia. Our research, which shifts the usual focus from individual nanoparticle properties to interacting particle clusters, evidences that square waves induce more uniform and greater heating than sinusoidal waves. The sequential switching observed with sinusoidal waves, which additionally strongly depends on the alignment of the particle cluster with respect to the direction of the field, leads to less uniform heating within and among different clusters. In contrast, a square waveform leads to simultaneous particle switching, thereby homogenizing the heat and potentially mitigating hazardous hot spots. These findings reaffirm the potential advantages for magnetic hyperthermia treatments using non-harmonic field waveforms, offering more uniform heating and the possibility of reducing the applied field exposure.

Recent research has demonstrated that thermal fluctuations on the net zero magnetization of a magnetic nanoparticle (MNP) ensemble can serve as a valuable tool for characterizing the sample's magnetic properties. These spontaneous fluctuations are intrinsically linked to the MNP system's response to small perturbations, as described by the fluctuation-dissipation theorem. We experimentally compare fluctuations and dissipation in both the linear and non-linear response regimes. Notably, a strong correspondence between the power spectral density (PSD) of the fluctuations and the out-of-phase dynamic susceptibility in the linear response regime was observed over a 500-kHz frequency range, facilitating interchangeability between these two characterization methods. This work contributes to the advanced characterization of MNPs for biomedical applications.

To gain deeper insight into the complex, stable, and robust configurations of magnetic textures, topological characterisation has proven essential. In particular, while the skyrmion number is a well-established topological invariant for 2D magnetic textures, the Hopf index serves as a key topological descriptor for 3D magnetic structures. In this work, we present and compare various methods for numerically calculating the Hopf index, provide implementations, and offer a detailed analysis of their accuracy and computational efficiency. Additionally, we identify and address common pitfalls and challenges associated with the numerical computation of the Hopf index, offering insights for improving the robustness of these techniques.

This contribution gives an overview of recent advances in thermal noise magnetometry (TNM), an emerging method for the characterization of magnetic nanoparticles. This method is unique as it does not rely on measuring the response to an external field excitation. Instead it passively measures the fluctuations in the magnetization of the particle ensemble in thermal equilibrium. Through theoretical and experimental advances, we show how TNM can now be applied to a variety of applications in a practical and accessible manner. Specifically, we show how TNM can be used to study aggregation processes in biological media.

This Tutorial article focuses on magnetic phenomena and material systems that have gained significant importance since the original development of mumax3, but are challenging to simulate for users who rely solely on the originally provided examples. Alongside the physical background, we provide hands-on examples of advanced magnetic systems, including detailed explanations of complete mumax3 input files (13 in total, often showing different ways to achieve things), and highlighting potential pitfalls where applicable. Specifically, we explore two approaches to incorporate spin–orbit torques in mumax simulations, considering the trade-off between versatility and speed. We also examine complex multilayer material stacks, including synthetic antiferromagnets, demonstrating different implementation methods that again vary in speed, versatility, and realism. A key criterion for selecting the optimal simulation strategy is its suitability for modeling systems where the magnetization varies significantly in the third dimension. The material covered in this Tutorial paper includes content developed for the mumax3 workshop presented during the summer of 2020 within the context of the IEEE online spintronics seminar, along with additional new topics. Throughout the explanations, we ensure broad applicability beyond specific examples.

Understanding and predicting the heat released by magnetic nanoparticles is central to magnetic hyperthermia treatment planning. In most cases, nanoparticles form aggregates when injected in living tissues, thereby altering their response to the applied alternating magnetic field and preventing the accurate prediction of the released heat. We performed a computational analysis to investigate the heat released by nanoparticle aggregates featuring different sizes and fractal geometry factors. By digitally mirroring aggregates seen in biological tissues, we found that the average heat released per particle stabilizes starting from moderately small aggregates, thereby facilitating making estimates for their larger counterparts. Additionally, we studied the heating performance of particle aggregates over a wide range of fractal parameters. We compared this result with the heat released by non-interacting nanoparticles to quantify the reduction of heating power after being instilled into tissues. This set of results can be used to estimate the expected heating in vivo based on the experimentally determined nanoparticle properties.

Physical reservoir computing is a computational framework that implements spatiotemporal information processing directly within physical systems. By exciting nonlinear dynamical systems and creating linear models from their state, we can create highly energy-efficient devices capable of solving machine learning tasks without building a modular system consisting of millions of neurons interconnected by synapses. To act as an effective reservoir, the chosen dynamical system must have two desirable properties: nonlinearity and memory. We present task agnostic spatial measures to locally measure both of these properties and exemplify them for a specific physical reservoir based upon magnetic skyrmion textures. In contrast to typical reservoir computing metrics, these metrics can be resolved spatially and in parallel from a single input signal, allowing for efficient parameter search to design efficient and high-performance reservoirs. Additionally, we show the natural trade-off between memory capacity and nonlinearity in our reservoir's behaviour, both locally and globally. Finally, by balancing the memory and nonlinearity in a reservoir, we can improve its performance for specific tasks.