France



The interfacial Dzyaloshinskii-Moriya Interaction is a critical property of magnetic thin films for the stabilization of chiral magnetic objects, including skyrmions, for use in future spintronic applications.[1] However, direct measurement of the strength of this interaction remains an experimental challenge.[2] Here, we present a technique for direct extraction of the interfacial Dzyaloshinskii-Moriya Interaction using a tapered c-shape (herein referred to as a nanohorn). A schematic of this idea is shown in figure 1. The nanohorn is designed to provide a driving force for the DW to propagate in the counter-clockwise direction driven by a reduction in the overall length of the DW itself. An in-plane magnetic field coupled with the interfacial DMI provides a driving force for propagation in the clockwise direction. This is due to the Zeeman energy associated with the DW’s internal magnetization and the magnetic field. The balance of these forces results in an energy minimum in wall position that depends directly on the strength of iDMI. In short, the rest position of the DW can be read like the needle of an analog meter to determine the strength of DMI.&lt;br/&gt;&lt;br/&gt;A series of micromagnetic calculations using Mumax3 were performed using the gradient descent approach to determine the energy of the DW as a function of its position along the nanohorn radius. Here, we use material parameters A = 15pJ/m, K&lt;sub&gt;eff&lt;/sub&gt; = 598 kJ/m&lt;sup&gt;3&lt;/sup&gt;, M&lt;sub&gt;s&lt;/sub&gt; = 800 kA/m for a film thickness of 2nm. The width of the magnetic nanohorn ranges from ~100-200nm although larger dimensions are not expected to significantly impact the results. The sides of this horn follow the two polar equations &lt;i&gt;r = a+b&lt;/i&gt;-&lt;i&gt;c&lt;/i&gt;&lt;i&gt;θ, r = a-b&lt;/i&gt;+&lt;i&gt;c&lt;/i&gt;&lt;i&gt;θ&lt;/i&gt;. This produces a shape with a circular arc of radius &lt;i&gt;a&lt;/i&gt; along its center and radial width increasing symmetrically from this arc in a linear rate &lt;i&gt;2c&lt;/i&gt;. We note that there are a range of parameters where there is no energy minima, which is not useful experimentally. However, the degree of horn tapering can be designed to work for material systems with different strengths of DMI. In the presentation, comparison with semi-analytical calculations will be presented, which show minimal differences in the equilibrium wall position likely due to dipolar interactions and slight tilting of the DW. Challenges associated with experimental realization of this technique will be discussed along with ways to mitigate them as evidenced by additional micromagnetic modeling. Most notably, there are dynamic limitations with driving the wall to its equilibrium, which can be addressed through increased statistics (simultaneously examining many nanohorns of different dimensions) and exciting the DW with AC magnetic fields to overcome edge pinning. Prospects for the experimental use of this technique for different materials systems will be discussed.

**References:**

[1] Thiaville, A., Rohart, S., Jué, É., Cros, V. and Fert, A., 2012. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. &lt;i&gt;EPL (Europhysics Letters)&lt;/i&gt;, &lt;i&gt;100&lt;/i&gt;(5), p.57002.&lt;br/&gt;&lt;br/&gt;[2] Lau, D., Pellegren, J.P., Nembach, H.T., Shaw, J.M. and Sokalski, V., 2018. Disentangling factors governing Dzyaloshinskii domain-wall creep in Co/Ni thin films using Pt x Ir 1− x seed layers. &lt;i&gt;Physical Review B&lt;/i&gt;, &lt;i&gt;98&lt;/i&gt;(18), p.184410.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fb0a86c9ee7547974d3ed351125b5c6f.jpg)
Figure 1: Schematic of the coordinate system and example micromagnetic output for a nanohorn with D = 1.6 mJ/m&lt;sup&gt;2&lt;/sup&gt; and μ&lt;sub&gt;0&lt;/sub&gt;H&lt;sub&gt;x&lt;/sub&gt; = 150mT. Orange arrows show the DW internal magnetization direction. Tick marks along the circumference indicate the equilibrium position (θ&lt;sub&gt;MIN&lt;/sub&gt;) of the domain wall for a range of other DMI strengths.

![](https://assets.underline.io/uploads/markdown_image/1/image/ac305d09aeeebe7e5935ed137942f415.jpg)
Figure 2: Domain wall energy vs wall position (θ) normalized to the energy minima for varying DMI strengths.  The lines shown are empirical fits to the micromagnetic energy output. Inset: Equilibrium wall position (θ&lt;sub&gt;MIN&lt;/sub&gt;) vs DMI strength.

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 Magnetic Nano-horns for measurement of the interfacial Dzyaloshinskii-Moriya Interaction

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The interfacial Dzyaloshinskii-Moriya Interaction is a critical property of magnetic thin films for the stabilization of chiral magnetic objects, including skyrmions, for use in future spintronic applications.[1] However, direct measurement of the strength of this interaction remains an experimental challenge.[2] Here, we present a technique for direct extraction of the interfacial Dzyaloshinskii-Moriya Interaction using a tapered c-shape (herein referred to as a nanohorn). A schematic of this idea is shown in figure 1. The nanohorn is designed to provide a driving force for the DW to propagate in the counter-clockwise direction driven by a reduction in the overall length of the DW itself. An in-plane magnetic field coupled with the interfacial DMI provides a driving force for propagation in the clockwise direction. This is due to the Zeeman energy associated with the DW’s internal magnetization and the magnetic field. The balance of these forces results in an energy minimum in wall position that depends directly on the strength of iDMI. In short, the rest position of the DW can be read like the needle of an analog meter to determine the strength of DMI. A series of micromagnetic calculations using Mumax3 were performed using the gradient descent approach to determine the energy of the DW as a function of its position along the nanohorn radius. Here, we use material parameters A = 15pJ/m, Keff = 598 kJ/m3, Ms = 800 kA/m for a film thickness of 2nm. The width of the magnetic nanohorn ranges from ~100-200nm although larger dimensions are not expected to significantly impact the results. The sides of this horn follow the two polar equations r = a+b-cθ, r = a-b+cθ. This produces a shape with a circular arc of radius a along its center and radial width increasing symmetrically from this arc in a linear rate 2c. We note that there are a range of parameters where there is no energy minima, which is not useful experimentally. However, the degree of horn tapering can be designed to work for material systems with different strengths of DMI. In the presentation, comparison with semi-analytical calculations will be presented, which show minimal differences in the equilibrium wall position likely due to dipolar interactions and slight tilting of the DW. Challenges associated with experimental realization of this technique will be discussed along with ways to mitigate them as evidenced by additional micromagnetic modeling. Most notably, there are dynamic limitations with driving the wall to its equilibrium, which can be addressed through increased statistics (simultaneously examining many nanohorns of different dimensions) and exciting the DW with AC magnetic fields to overcome edge pinning. Prospects for the experimental use of this technique for different materials systems will be discussed.

**References:**

[1] Thiaville, A., Rohart, S., Jué, É., Cros, V. and Fert, A., 2012. Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. EPL (Europhysics Letters), 100(5), p.57002. [2] Lau, D., Pellegren, J.P., Nembach, H.T., Shaw, J.M. and Sokalski, V., 2018. Disentangling factors governing Dzyaloshinskii domain-wall creep in Co/Ni thin films using Pt x Ir 1− x seed layers. Physical Review B, 98(18), p.184410.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fb0a86c9ee7547974d3ed351125b5c6f.jpg)
Figure 1: Schematic of the coordinate system and example micromagnetic output for a nanohorn with D = 1.6 mJ/m2 and μ0Hx = 150mT. Orange arrows show the DW internal magnetization direction. Tick marks along the circumference indicate the equilibrium position (θMIN) of the domain wall for a range of other DMI strengths.

![](https://assets.underline.io/uploads/markdown_image/1/image/ac305d09aeeebe7e5935ed137942f415.jpg)
Figure 2: Domain wall energy vs wall position (θ) normalized to the energy minima for varying DMI strengths. The lines shown are empirical fits to the micromagnetic energy output. Inset: Equilibrium wall position (θMIN) vs DMI strength.

technical paper

Micromagnetics complements theoretical and experimental methods to support research in magnetism. The Object-Oriented MicroMagnetic Framework (OOMMF) [1] is a micromagnetic simulation tool solving non-linear time-dependent partial differential equations using the finite-difference method. It is probably the most widely used simulation tool in the micromagnetics community. It was written in C++, wrapped with Tcl, and driven through configuration files that follow the Tcl syntax. The typical computational workflow the user must follow to simulate a particular problem is to write a configuration file. After that, the user runs OOMMF by providing the configuration file to the OOMMF executable. When the OOMMF run is complete, results are saved in OOMMF-specific file formats. Finally, the user analyzes the result files. One of the specific goals of a computational micromagnetic study is parameter-space exploration. More precisely, the user repeats the simulation for different values of input parameters by changing them in the configuration file. It is often difficult to automate this, and it is challenging for the user to keep a log of all steps performed in the entire micromagnetic study. Besides, postprocessing and analysis of results is performed outside OOMMF, using techniques and scripts that are mostly developed by the user, or carried out manually. Consequently, it is hard to track, record, and convey the exact simulation procedure. Without this information, resulting publications are generally not reproducible. To address this situation, we developed a Python interface to the OOMMF executable to expose OOMMF to a general-purpose programming language supported by Jupyter. This allows us to conduct micromagnetic simulations from within the Jupyter notebook [2] to capitalize on the benefits of this environment. We developed a set of Python libraries we refer to as Ubermag, which expose the computational capabilities of OOMMF so that it can be controlled from Python. These Python libraries provide a domain-specific language to define a micromagnetic problem. A micromagnetic model, defined using the domain-specific language, is not aware of the particular simulation tool that will perform the actual micromagnetic simulation, and it is only used to describe the model. When a simulation is required, the model is translated into the OOMMF configuration file, the OOMMF executable is called, and the output files are read. By exposing the micromagnetic simulation capabilities to Python and driving the research from Jupyter Notebook, we have available all the benefits of the Jupyter research environment and all tools from Python's scientific stack are readily available. Ubermag and the Jupyter environment simplify the efforts to make micromagnetics publications reproducible. For each figure in the publication, one notebook can be provided. Using Binder [3], the community can inspect and re-run all the calculations in the cloud and make the publication reproducible.

**References:**

[1] M. J. Donahue and D. G. Porter, OOMMF User’s Guide, Version 1.0, Interag. Rep. NISTIR 6376, Natl. Inst. Stand. Technol. Gaithersburg, MD, 1999. [2] T. Kluyver et al. Jupyter Notebooks - a publishing format for reproducible computational workflows. Positioning and Power in Academic Publishing: Players, Agents and Agendas, 87-90, 2016. [3] https://mybinder.org/

**Images:**

 Towards reproducible micromagnetic workflows using Ubermag

Machine learning (ML) entered the field of computational micromagnetics only recently [1,2,3]. The main objective of these new approaches is the automatization of solutions of parameter-dependent problems in micromagnetism such as fast respond curve estimation modeled by the Landau-Lifschitz-Gilbert (LLG) equation. Data-driven models for the prediction of solutions of time- and parameter-dependent partial differential equations require high dimensional training data-structures. ML in this case is by no means a straight-forward trivial task, it needs algorithmic and mathematical innovation. Our work [4] introduces theoretical and computational conceptions of certain kernel-based dimensionality reduction approaches for efficient machine learning of the solution trajectories of LLG via the notion of low-dimensional feature space integration, Fig.1. We will mainly concentrate on computational aspects, specifically, kernel methods will be studied with focus on numerically stable and efficient implementation of kernel ridge regression and kernel principal component analysis via low-rank treatment of certain dense operations. A second line follows neural network autoencoders as dimensional reduction for the training data with focus on smooth latent space variable description suitable for the feature space integration scheme. This will be doable due to a certain modification of contractive regularization that maintains feasibility despite the typically large computational demands coming with such approaches. We verify and compare numerically by means of a NIST standard problem, Fig.2.

**References:**

[1] A. Kovacs, J. Fischbacher, H. Oezelt, M. Gusenbauer, L. Exl, F. Bruckner, D. Suess, T. Schrefl. Learning magnetization dynamics, Journal of Magnetism and Magnetic Materials 491 (2019): 165548. doi: http://10.1016/j.jmmm.2019.165548 [2] L. Exl, N.J. Mauser, T. Schrefl, D. Suess. Learning time-stepping by nonlinear dimensionality reduction to predict magnetization dynamics, Communications in Nonlinear Science and Numerical Simulation 84 (2020): 105205. doi: 10.1016/j.cnsns.2020.105205 [3] L. Exl, J. Fischbacher, A. Kovacs, H. Oezelt, M. Gusenbauer, K. Yokota, T. Shoji, G. Hrkac, and T. Schrefl, Magnetic microstructure machine learning analysis, Journal of Physics: Materials, 2 (2019): 014001. doi: 10.1088/25157639/aaf26d [4] Exl, L., Mauser, N. J., Schaffer, S., Schrefl, T., & Suess, D. (2020). Prediction of magnetization dynamics in a reduced dimensional feature space setting utilizing a low-rank kernel method. arXiv preprint arXiv:2008.05986. https://arxiv.org/pdf/2008.05986.pdf

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7cd58bb91ad23869dce8b0b6893d6b77.jpg)
Fig.1: Time-integration scheme in low-dimensional feature space.

![](https://assets.underline.io/uploads/markdown_image/1/image/548c264f8d7c23ee60cf93515379b54d.jpg)
Fig.2: Snapshots for comparison of simulated and predicted magnetization states for varying ranks r.

 Machine learning methods for the prediction of micromagnetic magnetization dynamics

The exchange interaction between two magnetic materials may lead to a symmetry breaking manifesting itself as a unidirectional anisotropy induced in one layer (usually ferromagnetic material F) by another one less susceptible to an external magnetic field (usually antiferromagnetic material AF). The unidirectional anisotropy in the F layer leads to a hysteresis loop shifted along the magnetic field axis and the phenomenon is called exchange bias (EB). Despite its seeming simplicity, the EB has an intricate microscopic mechanism, which has been approached by many models of different complexity dealing with perfect or imperfect interface, polycrystalline structure of F and AF, and bulk spin configuration in the AF layer [1]. For real-life applications e.g. sensors, memory cells or spintronic devices, the number of significant parameters affecting coercivity and EB field of the F layer could become too large to be accounted by simplified models, so computer simulation becomes a method of choice. The increasing complexity leads to more realistic and universal models but makes it difficult to evaluate their performance and limitations, study interconnections between model’s parameters, and make a fast prediction on how the system with a given set of parameters would behave. In this study we approach this problem by applying methods of machine learning to a dataset obtained by a micromagnetic model representing a polycrystalline bilayer film consisting of F and AF layers. As AF state cannot be readily set in the continuous media approach, we adopted the model described in [2] by substituting AF with F material, for which contributions to Zeeman and magnetostatic energies were disabled. The example of the simulated hysteresis loop along with the geometry is shown in Fig. 1. Micromagnetic simulation was performed using MuMax3 software with GPU acceleration [3]. As an income several material and structural parameters have been chosen including saturation magnetization (same for F and AF), uniaxial magnetic anisotropy constants of F and AF, exchange constants in F and AF, attenuation coefficient of exchange constant between F and AF, thickness of the F layer, and average crystallites size (8 variables in total). As a result of approximately 1800 hysteresis loops simulations, for each configuration (each parameter has been varied randomly within a realistic range) two outcome parameters have been extracted: coercivity and EB field. The model has been validated with a smaller set of simulations where single parameter has been varied while fixing others at realistic values typical for a popular FeNi/FeMn system. Principal component analysis (PCA) allowed identifying readily input parameters having the strongest positive and negative correlations with the output parameters. Correlation coefficients for saturation magnetization, the thickness of the F layer, and the attenuation coefficient were significant for both coercivity and EB field, whereas the coefficient for AF anisotropy constant was among the largest only for EB field. Both PCA and T-SNE (t-distributed stochastic neighbor embedding) data visualizations suggested the possibility of building a data approximator. As EB field and coercivity are known to have complex nonlinear dependencies on some parameters, we chose the gradient boosting, which is a suitable algorithm for the current task. In this work Catboost open source library based on decision trees was used. The dataset was split into two parts for training the model and testing it using the root mean square error (RMSE) metric. The trained model could predict EB field for a test set of 500 examples rather well with RMSE = 0.04, whereas higher RMSE = 0.13 was achieved for coercivity, which could also be considered a good result. To demonstrate the applicability of the model for fast search of optimal coercivity and EB field, we employed Broyden–Fletcher–Goldfarb–Shanno and differential evolution algorithms realized in SciPy library to find sets of input parameters giving the largest values of output parameters. This approach allowed us to identify several areas of the highest coercivity and EB field values exceeding those in the training set, which were confirmed by additional micromagnetic simulations. Although the RMSE was large, the model can be further improved by including more examples of computer simulation to resolve the areas of interest. In conclusion, machine learning algorithms were successfully used for in-depth analysis of a bilayer micromagnetic model with exchange bias. The model capable of predicting values of coercivity and exchange bias using material and structural parameters as an input was successfully created and trained. The proposed approach was demonstrated to be suitable for building optimization algorithms allowing to find areas of interest in multidimensional space of income parameters giving the desired values coercivity and EB field. The demonstrated approach can be useful for the analysis of computer models as well as real systems with exchange bias. This work was supported by the Russian Science Foundation (project No 19-72-00141).

**References:**

[1] F. Hellman, A. Hoffmann, Y. Tserkovnyak et al., Rev. Mod. Phys. Vol. 89, a.n. 025006 (2017) [2] J. De Clercq, A. Vansteenkiste, M. Abes et al., J. Phys. D: Appl. Phys., Vol. 49, a.n. 435001 (2016) [3] A. Vansteenkiste, J. Leliaert, M. Dvornik et al. AIP Advances, Vol. 4, a.n. 107133 (2014)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/a36a3f75ae7126384b7de6c275129056.jpg)
Figure 1. Hysteresis loop simulated using a micromagnetic model of a bilayer with exchange bias consisting of F and AF layers. In the inset a slice of the geometry with Voronoi tessellation is shown. The colors are given to differentiate the crystallites.

 Combined micromagnetic simulation and machine learning approach to analysis of polycrystalline bilayer system with exchange bias

Smartphone-based indoor localization methods draw increased interest and are frequently employed for position estimation of users inside enclosures like malls, conference halls, and crowded venues. Magnetic sensing has recently attracted considerable attention due to its pervasiveness and autonomy, introducing two main positioning approaches utilizing magnetic sensors. The first approach is identifying the magnetic field characteristics of the surrounding infrastructure as signatures, thus creating a specific magnetic fingerprint. This scheme does not require additional hardware deployment, while interferences and anomalies from indoor infrastructure and objects greatly improve magnetic field discernibility. Yet, this approach requires knowledge of magnetic maps or computationally expensive calculations for applying simultaneous localization and mapping (SLAM). Furthermore, some environments may cause strong electromagnetic interference that differs from the underline magnetic fingerprint, leading to localization errors. The second approach uses active magnetic flux transmitters as markers, which is energy consuming and requires expensive transmitter units. In our recent work, we introduced the Permanent Magnets Superstructure for Indoor Localization method (PMSIL). This approach allows simple, low-power, and robust smartphone localization in unknown-magnetic-fingerprint locations by deploying permanent magnets in pre-known locations. Practically, we embed small-volume-large-moment permanent magnets in given locations inside the building and arrange them in a single specific geometric configuration. This results in a supervised super-structure magnetic signature pattern, constituting unambiguous magnetic environments. The localization algorithm learned this unique pattern during the training stage using artificial intelligence (AI) methods and detected it in ongoing data streams during real-time localization by applying binary classification. Thus, localization is based on smartphone motion rather than on the static positioning of the magnetometer. In this study, we propose an extended version of the PMSIL method, called Multiclass-PMSIL (M-PMSIL), in which magnets are arranged in a variety of configurations in pre-known locations. These constellations generate a bank of distinct patterns that are learned by the multiclass AI-based algorithms during training and recognized on-the-fly during localization. This extension permits covering a broader area in which the user can be localized. As M-PMSIL is landmark-based, we draw comparisons to three competing state-of-the-art approaches for indoor localization with landmark-based magnetism mechanisms. We also refer to localization methods based either on spatial-temporal sequence matching or on fusion with motion. The database is recorded by a user strolling back and forth with a hand-held smartphone across a 100 m long and 3 m wide in-door corridor. Along this corridor, three neodymium magnets with different magnetic moment norms are positioned in a row, with 3 m separating between them. During the experiment, these magnets are permuted in all 6 possible combinations and every new combination is relocated along the corridor, creating 6 distinct magnetic patterns. For the sake of generalization, recordings are made in different walking paces between 0.5 to 1.5 m/s, and different crossing distances from the magnets of 0.5 m, 1m, and 1.5 m. A snippet of data recorded by the smartphone indoor can be seen in Fig. 1, including 2 of the 6 magnet constellations used in this study. The training set includes 40 minutes of randomly selected recorded data that contains 10 passes across each of 6 different magnetic patterns, while the test set contains 20 minutes of data. The 6 classes, i.e., magnetic permutations to be localized, are balanced for both disjoint sets. In this study, 6 AI-based architectures are used for localization. Their performance is compared in Fig. 2 using the receiving operating characteristic (ROC) curve, allowing analysis of the trade-off between true positive rate (TPR) and false-positive rate (FPR) in various operation points. We may observe that the long short-term memory (LSTM) network produced the best classification results, followed by the gated recurrent unit (GRU) and recurrent neural network (RNN) models. The least performant methods are the fully connected deep neural network (DNN), and support vector machine (SVM), with and without principal component analysis (PCA). To quantify the multiclass classification accuracy of the methods, we derive the maximal sum of TPR and FPR from the ROC curve. LSTM shows 95% accuracy, while GRU and RNN reach 90% and 87%, respectively. DNN shows 82%, SVM-PCA method reaches 80%, and SVM produces 75% accuracy. Our approach achieves mean localization error (MLE) lower than 1 m, supersedes competing landmark-based methods, as UnLoc and MapCraft achieved MLE between 1-2 m. The IODetector was able to obtain classification accuracy as high as 82% indoor, which falls short to the 95% obtained by the LSTM. Additional methods, rooted in spatial-temporal sequence matching or on fusion with motion, also show lower MLE and classification accuracy compared to our study.

**References:**

[1] S. He and K. Shin, ACM Comput. Surv., Vol. 50, p.1-37 (2017) [2] K. Watanabe, K., Hiroi, K., Kamiyama, et al., Journal of Information Processing., Vol. 27, p.10–24, (2019) [3] E. Weiss and R. Alimi, Low-Power and High-Sensitivity Magnetic Sensors and Systems. Artech House, (2018) [4] T. N. Sainath, O. Vinyals, A. Senior, et al., IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp.4580–4584, (2015) [5] J. N. Mandrekar, J. Thorac. Oncol., vol. 5, pp. 1315–1316, (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ccecf57041a3306dc63b5f676accee9b.jpg)
Fig 1: Norm value of three-axial magnetic signals recorded with a smartphone. Rectangles confine measurements of three magnets positioned in a row, in two distinct magnet constellations.

![](https://assets.underline.io/uploads/markdown_image/1/image/995c407af3fee196a6aae9cf4d4d9671.jpg)
Fig 2: Multiclass classification performance of six AI methods.

 Multiclass Permanent Magnets Superstructure for Indoor Localization using Artificial Intelligence

At the microscopic scale, magnetic systems can be described by the formalism of micromagnetics, which is used for e.g. micromagnetics of rare-earth efficient permanent magnets [1]. In a micromagnetic model, the governing equations must be discretized and represented by a number of interactions taking place between the discretized moments in the system. These interactions can be described by different magnetic fields, such as an exchange- and an anisotropy field. Of these, it is the calculation of the demagnetization field that is the most time consuming part [2]. There are in general three methods that have been applied to calculate the demagnetization field: finite difference-based fast Fourier transform methods, tensor-grid methods and finite-element methods. All methods have inherent advantages and shortcomings, e.g. the finite difference Fourier-based approaches are extremely fast but unfortunately require that the magnetic moments lie on a regular grid [3], while the finite element method allows easy meshing of complex geometries but also requires that an environment outside the object of interest is modelled. Here we present a novel open-source micromagnetism framework called MagTense that uses an analytically exact approach to calculate the full demagnetization tensor in real space for computations [4]. This ensures that the calculated magnetic field is correct, and at the same time allows for easy meshing of complex geometries. In MagTense the discretization approach used subdivides the region Ω into N "tiles", adjacent to each other. The magnetization field is then represented by a vector of 3N components, three for each tile. The demagnetization field generated by each tile is computed analytically by assuming that the tile is uniformly magnetized. The analytical expression of the demagnetization tensor is known for many different tiles, such as the rectangular cuboid (rectangular prism) [5] and the tetrahedron [6], thus allowing great flexibility to build meshes for any geometry. The total demagnetization field at any point is obtained by summing the individual contributions from all the tiles in the magnet domain Ω. Besides the calculation of the demagnetization field, the other components of the effective field must also be calculated, i.e. the exchange-, external- and anisotropy field. For the case of a Cartesian mesh of rectangular cuboids the exchange interaction field can easily be computed by applying standard finite difference schemes. However, methods are also available for the case of unstructured polyhedral meshes. Using MagTense we have calculated the solution to the μmag standard micromagnetic problems 2, 3 and 4 and find that the MagTense framework accurately predicts the solution to each of these [7]. For example for standard problem 3, we get side length at the crossover energy between the vortex and flower state of 8.477, which is in line with two of the published μmag results. We have also implemented the MagTense framework to support both CPU and GPU calculations, where in the latter the demagnetization calculations are performed on the GPU using the CUDA framework. On μmag standard problem 4, we see a decrease in total simulation time of a factor of 100 for the GPU implementation. To demonstrate the use of MagTense on a sample with multiple grains we show in Fig. 1 the magnetization evolution of a thin slice of a multi-grain material starting from a random state. The structure has been generated using a centroidal Voronoi tessellation. The hard magnetic phase in the five grains is characterized by a strong anisotropy, corresponding to the direction indicated by the thick black arrows, while the grain-boundary is soft-phase with isotropic properties. Inside the grains we observe the emergence of magnetic domains aligned to the easy axis, while at the grain-boundary we see the formation of magnetic vortexes. To simulate realistic grain samples, we have also utilized a sintering modeling framework based on kinetic Monte Carlo [8] to generate fully realistic microstructures of sintered powder compacts. We then subdivide these grain structures into magnetic tiles for simulating the magnetic properties of the complete sample. Shown in Fig. 2 (a) is an example of multi-grain sample generated using the sintering modeling framework. The colours indicate different grains, gray corresponding to the grain boundary soft-phase. The thickness of the grain boundary phase can be specified as an input parameter. Fig. 2 (b) and 2(c) show the tetrahedral and Cartesian meshes, respectively, applied to this geometry. The colours indicate the distance of a point from the nearest grain boundary, which in both cases was used to control the resolution of the mesh. In this way the mesh is finer at the boundary between grains which is where the nucleation is expected to occur during a field-reversal process.

**References:**

[1] J. Fischbacher, A. Kovacs, M. Gusenbauer, H. Oezelt, L. Exl, S. Bance, and T. Schrefl., J. Phys. D: Appl. Phys, 51(19):193002, (2018). [2] C. Abert, L. Exl, G. Selke, A. Drews, T. Schrefl, Journal of Magnetism and Magnetic Materials,326:176–185, (2013). [3] A. Vansteenkiste, Ben Van de Wiele, Journal of Magnetism and Magnetic Materials, 323(21):2585–2591, (2011). [4] R. Bjørk and K. K. Nielsen, Magtense - a micromagnetism and magnetostatic framework,doi.org/10.11581/DTU:00000071, https://www.magtense.org, (2019). [5] R. Bjørk, E. B. Poulsen, K. K. Nielsen, A. R. Insinga, “MagTense: a micromagnetic frameworkusing the analytical demagnetization tensor”, submitted, (2021). [6] A. Smith, K. K. Nielsen, D. V. Christensen, C. R. H. Bahl, R. Bjørk, J. Hattel. Journal ofApplied Physics, 107(10):103910, (2010). [7] K. K. Nielsen, A. R. Insinga, R. Bjørk, IEEE Magnetics Letters, 10:8918242, (2019). [8] V. Tikare, M. Braginsky, D. Bouvard, A. Vagnon, Comp. Mat. Sci., 48:317-25, (2010).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/21f2425d866e8b1eb830bc6b71276f78.jpg)
Spontaneous evolution of magnetic domains in a multi-grain structure. The color indicate the local direction of magnetization.

![](https://assets.underline.io/uploads/markdown_image/1/image/368893183811e0823acd0ef821e19e69.jpg)
Simulated sintered powder sample and corresponding adaptive meshes.

 Simulating sintered magnets using the full demagnetization tensor with MagTense

The numerical studies of magnetization dynamics in micromagnetic systems with spatially distributed magnetization crucially rely on the computation of magnetostatic fields generated by the ferromagnet. The solution of the magnetostatic problem can be written as convolution of an appropriate Green’s function with the magnetization vector field. Since the Green’s function is long-range, after discretization of the problem in N cells, the direct numerical computation of the convolution integral has a computational cost scaling as N2. In this work, we use, as it is often done in micromagnetics, the Fast Fourier Transform (FFT) method to reduce the computational cost scaling to N logN. We consider micromagnetic systems with thin-film geometry where the magnetization can be considered constant across the thickness of the magnet. In these situations, only the field averaged across the thickness is relevant to the magnetostatic interactions and both the magnetization and the averaged field inside the magnet depend only on two spatial variables. The usual approach to deal with this type of problem is to use FFT to compute the discretized version of the convolution integral relating the field and the magnetization [1]. The dual approach to the usual one, is based on the discretization of the integral relation between the field (averaged across the thickness) and the magnetization in the reciprocal space (k-space). The connection in the k-space is given by the Fourier transform of Green function of the magnetostatic problem which, in case of infinite thin-films, can be written in a special form taking into account the thin-film geometry [2]. The analysis in the k-space has been extensively used in connection with linear spin wave dynamics in thin films working as magnonic waveguide, where all the relation of practical use (e.g dispersion relations), are easily obtained by algebraic manipulations. When the dimensions of the thin film are finite, the reconstruction of the magnetostatic field with the infinite film Green's function, is altered by the presence of replicas of the thin film in the ordinary space due to the sampling operation in the k-space. This effect is usually smoothed by using zero-padding, which increases the distance between two interacting replicas weakening their coupling. In this work, a fast convolution algorithm in the k-space able to remove the influence of replicas in the reconstruction of the magnetostatic field in thin films of finite size, is proposed and applied to study linear spin wave dynamics. It is based on the use of the Fourier transform of truncated Green function, which has been recently used to compute volume potentials defined on free-space in a finite (truncated) domain [3]. Such method, guarantees a regularization of the Green function by cutting off the interaction in physical space beyond the domain of interest and this in turn produces spectral accuracy [5] for the Fourier transform evaluation by FFT algorithm. The accuracy of the proposed method is studied by comparing the results with analytical calculations and with those obtained by fast convolution with the infinite thin film Green's function as a function of mesh size. For example, in the figure, the computation of the demagnetizing factor along the x direction for a thin film of dimensions Lx = 300nm, Ly = 100nm and Lz = 3nm as a function of the mesh dimension (kxmax = kymax = kmax) and discretization step-size (Δkx = Δky = Δk) in the k-space is considered. With solid lines, the curves in the infinite film approximation (infin.) are indicated, while curves with symbols results from the proposed approach (trunc.). The horizontal dotted line, represents the analytical value of the demagnetizing factor evaluated with Newell’s formula [6]. Computation with Δk&lt; π/300 nm-1, for the duality property of the Fourier transform, are equivalent to use zero padding in the ordinary space. This produces a substantial improving of estimation accuracy with respect to the analytical value, in case of infinite film approximation. As a benchmark of the proposed approach, it results that the curves computed with the truncated Green function, show a weak dependence from Δk for Δk&gt;π/R and collapse into the same one for Δk&lt;π/R, where R is the radius of a disk region which defines the compact support of the truncated Green function. This, as a confirmation of the fact, that when the region that exceeds the support of the truncated Green's function is zero padded, there is no effect on the computation of the magnetostatic field. Moreover, for sufficient value of kmax the asymptotic value tends towards analytical one, since when Δk&lt;π/R the influence of replicas on the magnetostatic field computation is removed.

**References:**

[1] R.D. McMichael, M.J. Donahue, D.G. Porter, J.Eicke, Journal of Applied Physics, vol. 85, pp. 5816-5818, (1999). [2] V. Scalera et al., IEEE Transactions on Magnetics, doi: 10.1109/TMAG.2020.3014383. [3] F. Vico, L. Greengard, M. Ferrando, Journal of Computational Physics, Vol. 323, p. 191-203 (2016). [4] B.A. Kalinikos, A.N.Slavin, J. Phys. C: Solid State Phys. Vol. 19, p.7013-7033 (1986). [5] L.N. trefethen, SIAM, Philadelphia, (2000). [6] A.J. Newell, W. Williams, D.J. Dunlop, J. Geophysical Research, Vol. 98, pp. 9551-9555 (1993).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0c3a6bfd333a895ba9655d5de5798653.jpg)
Comparison of the demagnetizing factor computed analytically with Newell’s formula [6] and numerically with k-space FFT convolution with the infinite film Green's function (infin.) and with the truncated Green's function (trunc.), for different mesh dimension kmax and step-size Δk in the k-space.

 Magnetostatic field computation in thin films based on k-space fast convolution with truncated Green's function

Magnetic materials play a major role in many spintronic and other technological applications such as magnetic storage, logic devices, magnetic field sensors and magnetic random access memory. Materials with strong intrinsic ferromagnetic (FM) order above room temperature, such as the transition metals Fe, Ni or Co and their alloys, are rather unusual among the magnetic materials known today and there is still the need for new functional materials with magnetic order above room temperature. In the past two decades, a method of creating artificial ferromagnetic materials has emerged and a multitude of so-called defect-induced ferromagnets were reported. Since the first prediction of an artificial ferromagnetic material with transition temperature above 300~K, based on Mn doped ZnO appeared twenty years ago, the field has substantially evolved. First, it was realized that doping with magnetic impurities was not at all necessary in order to induce a robust FM order in the non-magnetic host matrix, rather all kinds of lattice defects were at the origin of the measured magnetic signals. This realization promised great possibilities to construct new functional magnetic materials, as any non-magnetic material could potentially host a certain kind of defect, turning it into an artificial ferromagnet. The hunt was on and the result was a plethora of reports ranging from oxide, nitride, carbon-based, 2D van der Waals and many more materials showing signals of ferromagnetism upon introducing all kinds of nominally non-magnetic defects. One of the most promising and versatile methods for introducing these defects is the irradiation with non-magnetic ions, owing to the availability of ion sources ranging over the whole periodic table and energies from few eV to hundreds of MeV. Although many experiments were accompanied by theoretical studies, such as electronic structure calculations based on density functional theory (DFT), the search was mostly guided by blind trial and error and a brute force approach. It is therefore not very surprising that most of the reported materials only showed very tiny magnetic signals, which soon led to debates about the nature of the effect and raised the question of whether this route could eventually lead to a robust magnetic order above room temperature, comparable with intrinsic ferromagnets. Furthermore, the measurement of the magnetization of such artificial ferromagnetic samples turns out to be quite difficult due to the inherent uncertainty of the magnetic volume, leading to largely underestimated values in the literature. Considering the enormous amount of host material candidates and lattice defects, a more systematic search method and better selection criteria are highly needed. In this presentation, a computational scheme for the systematic discovery of candidate artificial magnetic materials that can be created by ion irradiation is introduced. The scheme is based on first principle calculations, guided by experimental constraints, automatically restricting the potential defects to those accessible experimentally and can readily be implemented for high throughput material discovery. We further provide a method to determine the defect distribution created within the host lattices, allowing to obtain accurate magnetization values. We demonstrate the predictive power of the scheme by comparing systematic experimental investigations to the calculations and provide corrections to the magnetization, that reach values as high as those of typical intrinsic ferromagnets. We show that the method is applicable to a wide range of host materials and ion irradiation parameters (100 eV - 200 MeV) and yields artificial functional magnetic materials, e.g. in TiO2, SiC and CeO2 hosts, in a controllable way, ranging from thin magnetic films down to 2D magnetic monolayers. The artificial magnetic materials can be tailored to application requirements, exhibiting properties like, e.g., strong perpendicular magnetic anisotropy.

**References:**

L. Botsch, P. D. Esquinazi, C. Bundesmann et al., Under review (2021) P. D. Esquinazi, W. Hergert, M. Stiller et al., Physica Status Solidi B, Vol. 257, p. 1900623 (2020) M. Stiller, A. T. N'Diaye, H. Ohldag et al., Physical Review B, Vol. 101, p. 014412 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/70bddfc09a928593fcc11b71f82a5651.jpg)
Computational scheme for the discovery of artificial magnetic materials

![](https://assets.underline.io/uploads/markdown_image/1/image/673a09062f57898c005553896fcb9e44.jpg)
Magnetic phase diagram of an artificial 2D ferromagnetic phase emerging in a TiO2 host by low energy ion irradiation, calculated using the scheme (Figure 1) and realized experimentally.

 Toward a Systematic Discovery of Artificial Functional Magnetic Materials

Devices based on magnetic domain wall (DW) motion are among the most promising candidates for future spintronic data storage, logic, and neuromorphic devices. One critical issue that hinders DW motion is the presence of structural inhomogeneities such as interface intermixing, interface roughness, or crystalline texture, that cause a distribution of magnetic properties in the material. As a result, domain walls undergo local pinning leading to the so-called creep regime at low driving forces. Even beyond this creep regime, the DW motion remains affected by defects. Thus, to achieve fast and efficient DW motion, good control over the intrinsic defects in the material must be obtained. Ultra-thin CoFeB/MgO-based structures with perpendicular magnetic anisotropy are considered the most promising candidates for the future generation of Magnetic Random Access Memory (MRAM) devices. These films are usually annealed at high temperatures (300-400°C) to crystallize the CoFeB layer and get high Tunnelling Magnetoresistance (TMR). This treatment at high temperatures results in the presence of undesirable intrinsic defects. To address this issue, we have developed a very promising process solution based on post-growth He+ ion irradiation to tailor the structural properties of ultrathin magnetic films at the atomic level and improve their performances [1-4]. The utilization of light ions provides the precise control of inter-atomic displacements through the low energy transfer. The key feature of the technology is the post-growth control of structural properties and the related magnetic properties. To perform these irradiation techniques, we have used a compact ion-irradiation facility (Helium-S® from Spin-Ion Technologies), capable of ultra-fast He+ ion irradiation on 1-inch wafers with energies ranging between 5-30 keV [5]. Here, we have studied the influence of two different processes of crystallization in W-CoFeB-MgO structure on domain wall dynamics. The first process is based on the combination of He+ ion irradiation at low fluences (1014 ions/cm2 to 1015 ions/cm2) with annealing. The second process consists of pure thermal annealing. For both processes, annealing temperatures ranging from RT to 400°C have been used. Our results first demonstrate that ion irradiation can induce full crystallization of the CoFeB-MgO materials at much lower temperatures (&lt;250°C) than the pure annealing process (&gt;350°C). In addition, higher anisotropy values can be obtained, which is consistent with sharper interfaces. Domain wall dynamics measured by Kerr microscopy have been compared for both processes. Strong differences have been evidenced in both the creep (few nm/s) and flow regime (few m/s). In particular, the ion-irradiation-induced crystallization process results in higher domain wall velocities as shown in Figure 1 for the creep regime where structural defects have a strong influence. This is consistent with a much lower density of structural inhomogeneities in the material crystallized by ion irradiation. Finally, we will discuss the crystallization process under ion irradiation and show preliminary results of DW motion in magnetic wires where local ion irradiation can also be used to minimize edge defects. Our results show that He+ ion irradiation is a very versatile tool to enhance magnetic materials and devices.

**References:**

[1] J. Fassbender, D. Ravelosona, and Y. Samson, J. of Phys. D.: Appl. Phys., Vol. 37, no. 16 (2004) [2] T. Devolder, I. Barisic, S. Eimer et al., J. of Appl. Phys., Vol. 113, no. 20, p. 203912 (2013) [3] L. Herrera Diez, F. García-Sanchez, J.-P. Adam et al., Appl. Phys. Lett., vol 107, p. 032401 (2015) [4] X. Zhao, B. Zhang, N. Vernier et al., Appl. Phys. Lett., vol. 115, p.122404 (2019) [5] www.spin-ion.com

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/689f25424d7c3fb6596488bc040d25c7.jpg)
Domain wall velocity as a function of the applied out-of-plane magnetic field in the creep regime for the regular annealing process (400°C, 1h) and the process based on He+ irradiation at lower annealing temperatures.

 Enhancing domain wall motion in W/CoFeB/MgO ultrathin films through He+-irradiation-induced crystallization

Ferromagnetic one-dimensional conduits carry the promises of future high-density and low consumption memories as well as new logic gate paradigms. These systems are indeed ideal for domain walls propagation and spin waves generation, and have also been proposed as field sensor [1]. The emergence of curvilinear magnetism [2] came up with the possibility on dealing with tridimensional consideration of such 1D-conduits (being also of fundamental academic interest as new magnetic textures) : ferromagnetic nanotubes are merging on a single structure the micromagnetism consideration of domain walls and the multi-layered stack needed for spintronic (core-shell structures). Unexpectedly, but technologically ideal, an azimuthal curling of magnetization was initially observed in electroless Co-rich CoNi nanotubes [2], that we attributed to curvature-induced magneto-elastic anisotropy. We report here detailed magnetic and structural information to quantify the effect of anisotropy, and better identify its microscopic origin by analyzing different parameter of tube structure and chemistry as well as using new tools for tube magnetic investigation. CoNiB-based ferromagnetic nanotubes were fabricated by electroless plating in polycarbonate templates with diameters ranging from 160 to 450 nm, wall thicknesses from 20 to 65 nm, and several tens of μm in length. Following template dissolution and dispersion, nanotubes were either contacted using laser lithography for AMR (Anisotropic Magneto-Resistance) measurements, dispersed on suitable TEM (Transmission Electron Microscopy) grids or prepared by means of FIB (Focused Ion Beam) for cross-section analysis. Structural analysis revealed a relative homogeneity in critical parameters such as diameter and tube thickness along the whole object, with expected slight variations towards tubes extremities. We noticed a segregation of Nickel towards the tube surfaces that could be associated to an inhomogeneity of magnetic properties across the tube wall (radial direction) already observed in thin films [4]. This inhomogeneity would imply inverse-magnetostriction (magnetoelastic) related anisotropy, even in the case of an alloy with no expected magnetostriction nor magnetocristaline anisotropy from the nominal composition. Magneto-transport measurements were carried out at varying temperature, applied field magnitude and direction, and showed a material resistivity (at 300K ) below 200 µΩ.cm, still one order of magnitude higher than in the bulk. This has been related to the nanocrystalline nature of the tubes (induced by grain boundaries) and is considered with non curvilinear system (electrodeposited thin films) for magnetostriction investigation. From R(Bext) loops with a magnetic field applied along the tube axis, we confirmed the azimuthal magnetic curling at remanence, inferred the magnetization process and thereby extracted the anisotropy field of the order of Keff = 7.5 kJ.m-3. Such reversal has been then scrutinized through Electron Holography to locally map and quantify the magnetic flux (see Figure) during a magnetic loop. We extended these measurements to various tube parameters to propose a general picture of the curvature-induced anisotropy. These on-going experiments are of fundamental importance to infer for the capacities of such system to experimentally confirm high domain wall velocities (as wall as spin wave propagation) that could offer technological solution for racetrack memories.

**References:**

[1] M. Stano & O. Fruchart. Magnetic nanowires and nanotubes, Handbook of magnetic materials vol.27, Ed. Ekkes Brück,North Holland (2018). [2] Fernández-Pacheco, A. et al. Three-dimensional nanomagnetism. Nature Communications 8,15756 (2017). [3] Michal Stano et al. Flux-closure domains in high aspect ratio electroless-deposited CoNiB nanotubes. SciPost Physics 5, 38 (2018) [4] Hertel, R. Ultrafast domain wall dynamics in magnetic nanotubes and nanowires. Journal of Physics: Condensed Matter 28, 483002 (2016).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/0f4c0e976dc0d77763c639f89b0fd9f3.jpg)
(Top Left) SEM view of a contacted Nanotube for magneto-transport measurements. Inset display an EDX spectra of the tube for Co/Ni composition. (Top right) Single tube hysteresis loop reconstructed from resistance measurement under applied field along the tube axis (two senses) and associated anisotropy field and saturated magnetization. (Bottom) Cosine of phase image obtained by Electron holography on a magnetized tube extremity: the black and withe lines display the magnetic flux. The phase profile (inset) enables to quantify the local saturated magnetization (phase derivative) knowing the sample thickness.

 Exploration of Magnetic Nanotubes as new spintronic building block

The huge amount of new data generated every day has caused the need to search for new systems to store all this information in the smallest possible areas. S. Parkin proposed a new storage device called racetrack memory, in which magnetic domains are used to store the information in ferromagnetic stripes along its length [1]. In this novel system, the domain walls (DWs) are pinned at artificial defects created along the stripe, and the depinning is achieved by the application of spin-polarized current pulses. These pulses initiate the DWs movement, transporting the bits along the stripe reaching the read/write heads, increasing the speed of reading/writing data. In particular, 3D racetrack memories, made of arrays of vertically aligned nanowires, allow us to greatly increase the information density stored on a device. However, this potential new storage device still has some limitations that prevent its use. One of the biggest drawbacks is the unwanted overheating of the system caused by the current pulses in the physical notches in the stripe, which are required to pin the DWs. This issue could be solved using chemical constraints, which would allow to pin the DWs maintaining the diameter constant along the wire. Considering this idea, a novel configuration based on cylindrical nanowires (NWs) with chemical constraints is proposed in this work. The choice of cylindrical NWs instead of stripes with rectangular section allows, in addition to the use of chemical constraints, to avoid the Walker breakdown thus enabling a faster propagation of information along the device [2]. Furthermore, the manufacturing method of NWs is cheaper than the stripes fabrication method, making the NWs more scalable [3]. In this work, we propose a novel design of a racetrack memory using two functional segments along the NW: a “writing section” and a “storage section” (Fig. 1). The introduction, for the first time, of a “writing section” along the NW eases to write the information directly in the NW, without the needed of an external writing head. The writing section design in the nanowire is made of two magnetic segments with different coercive fields, isolated from the “storage section” by a non-magnetic constraint. The magnetic simulations performed have proved how one can switch the magnetization (bit information) of the “writing section” without modifying the bit recorded in the rest of the NW [4]. The information written in this first section can then be recorded into the racetrack memory wire by applying a spin-polarized current through the device. The magnetic simulations show a successful depinning of the DWs from the chemical constraints achieved by the application of a spin current pulse through the NW, thus moving the magnetic bits in the whole NW without any loss of information (Fig. 2). The success of this novel design has been proved by the design of three different configurations of “writing sections”, obtaining in all three cases a successful recording and bit movement [4]. Arrays of 3 and 7 NWs (hexagonal array) were also simulated to investigate the effect of neighboring magnetic wires in the bit movement of the central wire. These novel NWs can be easily fabricated by the electrodeposition method, a well-known process where it is possible to tune the properties of the nanowires. The subsequent characterization of their properties, as well as their effectiveness to record a new bit, can then be measured by fast domain wall injection [5], a process where it is possible to characterize the nanowires individually. The improvements achieved in the use of cylindrical magnetic NWs as racetrack memories will help to develop a faster magnetic storage device with greater capacity and reliability, fabricated with low-cost and scalable methods. This work was financially supported by projects POCI-01-0145-FEDER-028676 and UIDB/04968/2020 from Portuguese FCT and COMPETE 2020 (FEDER).

**References:**

[1] S. P. Parkin, M. Hayashi, and L. Thomas, Science, vol. 320, p. 90-194 (2008) [2] N. Biziere et al., Nano Letters vol. 13, p.2053 (2013) [3] C. T. Sousa et al., Applied Physics Reviews, vol. 1, p. 031102 (2014) [4] J. Rial and M. P. Proenca, Nanomaterials vol.10, p.2403 (2020) [5] J.L. Prieto, M. Muñoz, and E. Martínez, Physical Review B, vol. 83, p.104425 (2011)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/db3d44ded592293e17fe72f4dc4eebfc.jpg)
Figure 1. Schematic representation of a 3D racetrack memory device using arrays of cylindrical magnetic nanowires, illustrating the writing and storage sections.

![](https://assets.underline.io/uploads/markdown_image/1/image/37d0169422136a20c08b6b94695ee002.jpg)
Figure 2. Domain wall movement along the central nanowire under the application of a spin-polarized current.

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