France



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.&lt;br/&gt;&lt;br/&gt;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.&lt;br/&gt;&lt;br/&gt;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&#39;s scientific stack are readily available.&lt;br/&gt;&lt;br/&gt;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.&lt;br/&gt;[2] T. Kluyver &lt;i&gt;et al.&lt;/i&gt; Jupyter Notebooks - a publishing format for reproducible computational workflows. &lt;i&gt;Positioning and Power in Academic Publishing: Players, Agents and Agendas&lt;/i&gt;, 87-90, 2016.&lt;br/&gt;[3] https://mybinder.org/

**Images:**





INTERMAG 2021

 Towards reproducible micromagnetic workflows using Ubermag

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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:**

technical paper

Recent research on deep learning shows that Binary Neural Network (BNN) is able to significantly reduce computation and memory cost with small precision loss [1][2]. By binarizing inputs and weights to +1 or -1 in BNNs, multiply-and-accumulate (MAC) operations can be replaced by XNOR operation in memory. Nonvolatile memory for BNN such as resistive random access memory (RRAM) and magnetic random access memory (MRAM) was reported in [3]-[6]. In this paper, we use spin-toque transfer magnetic random access memory (STT-MRAM) to complete XNOR and single-step convolution in memory. The main contributions include: (1) A STT-MRAM based structure with four transistors and two magnetic tunnel junctions (MTJ) (4T-2M) is proposed for binary convolution and fully connection operations using current mode sense amplifier. (2) The impact of different tunnel magnetoresistance ratio (TMR) and different size of MTJ on accumulating in memory has been explored. (3) By quantizing sensing yield into model, the trade-off between energy and performance is investigated. As shown in Fig. 1 (a), a 4T-2M bit-cell structure represents one synaptic weight, ‘-1’ is represented by two MTJs as parallel (P) and anti-parallel state (AP). The opposite combination is used to represent ‘1’. Two encoded word lines (WLs) represent one input value. To complete the XNOR operation, an offset-canceling current-sampling sense amplifier (OCCS SA) [7] is used (see Fig. 1(b)) to compare the current difference between source line (SL) and SLB. Fig.1 (c) shows the structure of the proposed STT-MRAM block. For convolution operation, the row decoder gated by appropriate timing decodes the multiple WLs to input parallel, and output through path &lt;1&gt; as the result of convolution. For fully connected operation, the row decoder activates the WLs row by row, and output through path &lt;2&gt; with bit-counting as the result of one fully connected. We implement 9 bits XNOR and accumulation to perform 3 × 3 convolution. Fig. 1(d) lists all the combinations in the convolution. The worst case is four ‘-1’ and five ‘1’ accumulation or its opposite combination, which causes the minimum current difference between “5IAP+4IP” and “5IP+4IAP”. Fig. 1(e) illustrates the circuit realization of BNN. When one group of WLs is simultaneously activated, ISL and ISLB will be different depending on the resistance of MTJs on two SLs. A CSA is implemented to generate high level when weighted sum is positive, and low level when weighted sum is negative. The output of the CSA represents the output of the neuron. SPICE-level simulation is performed using a Verilog-A model of STT MTJ and an industrial 28-nm CMOS process. Fig.2 (a) shows the simulation results that the sensing yield in different TMR and MTJ critical dimension (CD) can up to 99.8%. The sensing yield is positively correlated with TMR and negatively correlated with MTJ CD. As Fig.2(b) indicated, smaller MTJ CD leads to higher sensing margin [8]. As shown in Fig. 2(c), a BNN model consisting of two convolutional stages and one fully connected layer as the classifier is designed. Validation results on MNIST handwritten digit dataset is shown in Fig. 2(d). Sensing yield is quantized into model when sensing processes is failed, the results of convolution will be flipped. Lower TMR and larger MTJ CD result in a lower sensing yield and a dramatic decrease in accuracy. When TMR is greater than 2 with 40nm MTJ or TMR is greater than 1.5 with MTJ smaller than 35nm, the accuracy could achieve &gt;95% (&lt;1.8% loss). Fig. 2(e) illustrates the energy saving of single-step convolution can achieve 51.2% compared with that of row by row with a slight decrease in accuracy.

**References:**

[1] M. Courbariaux, et al., arXiv: 1602.02830, 2016. [2] M. Rastegari, et al., arXiv: 1603.05279, 2016. [3] Z. Dong et al., IEEE Transactions on Electron Devices, vol. 66, no. 1, pp. 793-801(2019) [4] L. Chang et al., IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 27, no. 11, pp. 2668-2679(2019) [5] Y. Pan et al., IEEE Transactions on Magnetics, vol. 54, no. 11, pp. 1-5(2018) [6] Y. Zhang et al., IEEE Transactions on Electron Devices, vol. 67, no. 2, pp. 469-473, Feb. 2020. [7] T. Na et al., IEEE Journal of Solid-State Circuits, vol. 52, no. 2, pp. 496-504, Feb. 2017. [8] Zhao, W. S., et al. Microelectronics Reliability, vol. 52, no. 9–10, Sept. 2012.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/946bef43ce91a98b12b19a0fcba53e6c.jpg)
Fig.1 (a) Cell configurations to represent one weight using two MTJs and four transistors. (b) Schematic of OCCS SA. (c) 4T2M based array to complete convolution and fully connection, <1> indicates path 1 and <2> indicates path 2. (d)The list of all the combinations in the convolution. (e) Schematic of the XNOR and accumulate operation as convolution in memory.

![](https://assets.underline.io/uploads/markdown_image/1/image/f8614fa534b410c9fbcbfa4ee258bd1b.jpg)
Fig.2 (a) The sensing yield in different TMR with different MTJ CDs. (b) The resistance distribution of different MTJ CDs. (c) Diagram of the BNN topology used for evaluation with MNIST. (d) The performance of binary neural network with different MTJ CDs and different TMR. (e) Trade-off between energy saving and accuracy loss.

 In-MRAM Processing Elements with Single-Step Convolution for Binary Neural Network

Unconventional computing utilizing stochastic behavior to effectively solve complex problems, i.e., "probabilistic computing," has gathered much attention [1]. Stochastic magnetic tunnel junction (MTJ) is a promising device to compose the probabilistic bit, a fundamental unit of the probabilistic computer, with efficient as well as individual control of its stochasticity. The average duration of MTJ dwelling in each state, i.e., relaxation time τ, determines the critical indices of the computer's performance, e.g., accuracy and computing time. As a proof of concept, integer factorization has been demonstrated with stochastic MTJs with a perpendicular easy axis (p-MTJs) [2], whose typical τ is on the order of milliseconds. Meanwhile, τ of MTJs with an in-plane magnetic easy axis (i-MTJs) is reported down to sub-microseconds [3]. Here we experimentally and theoretically address the τ of nanomagnets in low-thermal-stability MTJs and elucidate the physical mechanism causing the different fluctuation timescale between p- and i-MTJs. A stack structure, Ta (5.0)/ PtMn (20)/ Co (2.6)/ Ru (0.90)/ CoFeB (2.4)/ MgO/ CoFeB (2.1)/ Ta (5.0)/ Ru (5.0) (thickness in nm), is deposited on a thermally oxidized Si substrate by dc/rf magnetron sputtering. Both bottom (reference) and top (free) CoFeB layers have an in-plane easy axis. Effective perpendicular anisotropy field μ0HKeff of the top CoFeB layer is -0.46 T (negative sign indicates in-plane easy axis). The stack is processed into elliptic MTJs with geometry averaged diameter D of 55 nm. Typical TMR ratio is 110% and RA = 32 Ωμm2. Figure 1(a) shows the measurement circuit and Fig. 1(b) shows time-averaged 〈R〉 as a function of in-plane magnetic field H along a long axis of MTJs, indicating superparamagnetic behavior with zero coercivity. Figure 1(c) shows a measured random telegraph noise (RTN) at μ0H = -7.9 mT. τ averaged for P and AP states τave = (τPτAP)0.5, where τP (τAP) is the τ to stay at P (AP) state, is 19 ns, which is about two orders shorter than that ever reported in MTJs [3]. We then numerically calculate τ of p- and i-MTJs based on a stochastic Landau-Lifshitz-Gilbert (LLG) equation including a thermal field [4]. We define magnetic energy with HKeff and in-plane anisotropy field HK,in as E = MSv(HKeff+HK,insin2φ)sin2θ/2, with MS, v, θ, and φ being the spontaneous magnetization, the volume of nanomagnet, the polar angle, and the azimuthal angle, respectively. We set the same Δ for both systems; μ0HKeff = 10 mT, α = 0.02, and thermal stability factor Δ = HKeffMSv/2kBT = 3.8 (kBT is thermal energy) are assumed for the p-MTJ, and μ0HKeff = -0.46 T, μ0HK,in = 10 mT, α = 0.02, and Δ = HK,inMSv/2kBT = 3.8 for the i-MTJ. Figures 2(a) and 2(d) show the directions of magnetization, a torque from perpendicular magnetic anisotropy, that by damping, and that by the thermal fluctuation calculated by E and LLG equation for p- and i-MTJs, respectively. In the p-MTJs, the torque from HKeff directs in the azimuthal direction, and the only torque directing in the polar direction to induce magnetization switching is the thermal fluctuation field. Accordingly, the magnetization precesses with perpendicular anisotropy field and switches from one energy stable direction to the other with a random fluctuation field, as shown in the magnetization trajectory in Fig. 2(b). In this case, τ is found to be well described by the random-walk under damping [4,5] as τ = π0.5Δ-0.5(2αγμ0HKeff)-1exp(Δ). τ in this condition is calculated as ≈ 0.8 ns, which is consistent with the calculated RTN shown in Fig. 2(c). On the other hand, as shown in Figs. 2(d) and 2(e), the directions of the torques in i-MTJs are different from those in the perpendicular system due to the difference in the landscape of energy potential. τ is calculated to be 20 ns [Fig. 2(f)], which is close to the experimentally obtained τ, and several tens of times shorter than that calculated for the perpendicular system even with the same Δ. We numerically simulate τ as a function of Δ and clarify that the values are well described by τ = π(αγμ0Δ)-1[HKeff(HKeff+HK,in)]-0.5exp(Δ), which is analytically derived with Fokker-Planck equation [4,6] under Δ » 1 and -HKeff » HK,in. In conventional understanding, a decrease of HKeff decreases Δ, leading to shorter τ for p-MTJs. However, our analysis indicates that it increases the prefactor of exp(Δ), in proportion to HKeff -1.5. This dilemma determines a lower limit of τ as τmin = MSvln2(αγμ0kBT)-1, at which magnetization dynamics is dictated by the thermal fluctuation. For example, τmin is 151 ns for a p-MTJ with D = 60 nm, thickness of 1 nm, MS = 1.4 T, and α = 0.02 at room temperature. For i-MTJs, one can be free from the dilemma because the prefactor is mainly determined by HKeff whereas exp(Δ) is determined by HK,in, leading to (πΔ)-0.5MS|HKeff|v/4kBT times shorter τ than that with p-MTJs for the same Δ. This ratio reasonably explains the different τ observed for p-MTJs and i-MTJs, indicating i-MTJs with μ0|HKeff| ~ 1 T can achieve τ down to nanoseconds. This work is in part supported by JST-CREST No. JP1082016, and JSPS Nos. 19J12206, 19J12926, 19H05622, 20H02178, and 19KK0130.

**References:**

[1] P. R. Feynman, Int. J. Theor. Phys. Vol. 21, p.467-488 (1982) [2] W. A. Borders, A. Z. Pervaiz, S. Fukami, et al., Nature Vol. 573, p.390-393 (2019) [3] B. Parks, A. Abdelgawad, T. Wong, et al., Phys. Rev. Appl. Vol. 13, p.014063 (2020) [4] W. F. Brown Jr., Phys. Rev. Vol. 130, p.1677-1686 (1963) [5] T. Taniguchi, and H. Imamura, Phys. Rev. B Vol. 85, p.184403 (2012) [6] T. Taniguchi, Y. Utsumi, and H. Imamura, Phys. Rev. B Vol. 88, p.214414 (2013)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d931fe36020e94e2cb208537fe219b0d.jpg)
Fig. 1 (a) Measurement circuit for time-averaged resistance 〈R〉 and random telegraph noise (RTN). (b) In-plane magnetic field μ0H vs. 〈R〉. (c) RTN signal measured as transmitted voltage VT.

![](https://assets.underline.io/uploads/markdown_image/1/image/3dbfad6da1f9511ba98d94f34ea33a69.jpg)
Fig. 2 (a)(d) Directions of the torques. (b)(e) Calculated magnetization trajectories and (c)(f) time vs. their easy axis component. The conditions for (a)-(c) perpendicular and (d)-(f) in-plane easy axes are shown.

 Superiority of In-Plane Easy-Axis Stochastic Nanomagnet for Shorter Relaxation Time

The bottleneck between data storage and processing on conventional computing hardware is seen as a significant limitation to many forms of machine learning algorithms [1]. One route to solve this issue is to employ physical devices that can naturally mimic brain-like behaviour, known as neuromorphic computing. Magnetic materials are already widely used for long-term data storage, but ongoing work seeks to make use of their potential as both working memory and computing architectures. Particularly, devices based on magnetic domain walls (DWs) can perform logic operations [2] and can readily store information [3]. However, the stochasticity of DW pinning limits the feasibility of creating technologically viable devices. Here, we demonstrate how stochasticity can be changed from a technologically inhibitive behaviour into an integral property of machine learning algorithms that could be used in specialised neuromorphic devices. We first present experimental measurements that demonstrate the feasibility of tuning stochastic processes by applying external stimuli. Focused magneto-optic Kerr effect measurements were used to probe the pinning of DWs at notch-shaped defect sites in 400 nm wide Permalloy nanowires (Fig. 1(a)). DWs were injected into the nanowires and propagated to the defect sites using a field parallel to the nanowire (Hx). The probability of the DWs being pinned at the defect sites was found to depend sigmoidally on the magnitude of a field transverse to the nanowire length (Hy), (Fig. 1(b)). We apply this stochastic behaviour in two possible contexts, as a stochastic neuron for a binary network or as a stochastic synapse for feed forward network, where we model the pinning behaviour based on the experimental results. In the first case, we have modelled these DW wires as neurons in a Boltzmann machine, which exploits their randomness in order to sample a defined, or trained, probability distribution. We apply this to the task of integer factorisation [4], where the connectivity of the neurons is defined so that the binary representation of the integer factors are the most probable states. Such a device inherently exploits the tunable stochasticity of the system to solve complex problems. In the second case, we have explored the properties of feedforward artificial neural networks where the nanowires acted as binary stochastic synapses (BSS). These synapses are individually random but the output can be improved by repeatedly sampling the network to include redundancy. We have developed both mean-field and stochastic based learning methods, where for low repeats of the synapse the stochastic learning is seen to outperform the mean-field based method. It is found that for a single cycle of the network it can be trained to recognise handwritten digits with error rate of 17%. Repeated sampling of the network decreased the error rate to 8% for 128 cycles - close to mean field performance (Fig. 2). Following this we have expanded our network to become a multi-layer perceptron with both inhibitory and excitatory synapses. The additional layer increases the pathways through the network thus reducing the impact of the stochasticity on the classification. For the same classification test the error rate is seen to reduce to 5.5% with a hidden layer of 500 neurons. In conclusion, the pinning of DWs in nanowires can be tuned by applying a magnetic field transverse to the wire and the resultant pinning probability has been used as both stochastic neurons and synapses. As a neuron, we have demonstrated its ability to perform integer factorisation as part of a Boltzmann machine. While as a stochastic synapse, we have been able to train single and multi-layer networks to classify handwritten digits with close to digital network performance. Our work illustrates how the intrinsic stochasticity of DW devices can be harnessed to provide tunable bespoke hardware for machine learning tasks.

**References:**

[1] M. M. Sabry Aly et al., Proceedings of the IEEE, Vol. 107, no. 1, p. 19 (2019) [2] D. A. Allwood et al., Science, Vol. 309, Issue 5741, p. 1688 (2005) [3] R. L. Stamps et al., J. Phys. D. Appl. Phys., Vol. 47, p. 333001 (2014) [4] W. A. Borders et al., Nature, Vo. 573, p. 390 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/488de22bf5135fdfd14a1e4eeaa1d155.jpg)
Fig. 1: (a) Schematic of a notched magnetic nanowire for use as a synapse with a parallel driving field (Hx) and a tunable transverse depinning field (Hy). (b) Pinning probability as a function of transverse field, showing a sigmoidal dependence.

![](https://assets.underline.io/uploads/markdown_image/1/image/0c39f92c401e21de2e6218da03982401.jpg)
Fig. 2: The validation error rate of handwritten digits using a single layer network with binary stochastic synapses (BSS) for varying number of cycles of the network. Lines and points show the training using the mean-field or stochastic methods respectively.

 Machine Learning with Stochastic Magnetic Domain Wall Based Neurons and Synapses

Magnetic tunnel junctions consist of two ferromagnetic layers separated by a thin insulating layer. The relative orientation of the two layers' magnetization has two stable configurations: parallel or antiparallel (P or AP). For use as a binary non-volatile memory in magnetic random access memory (MRAM), the retention time in each of the two configurations needs to be on the order of several years. This corresponds to energy barriers larger than 40 kT (k is the Boltzmann constant, and T=300 K is room temperature). If the energy barrier becomes smaller than 12 kT, thermal fluctuations induce random switching of the magnetization between the P and the AP states with a mean timescale between 10 µs and 1 ms. Such fluctuating, low-energy-barrier tunnel junctions are referred to as superparamagnetic tunnel junctions [1]. Recently, superparamagnetic tunnel junctions have become attractive as compact and energy-efficient devices for applications that require truly random variables [2,3], including innovative computing schemes [4,5,6,7,8]. Through adapted circuit co-design approaches, we previously showed that those devices are promising for implementing artificial neural networks with reduced energy consumption [9]. In cognitive computing, these superparamagnetic tunnel junctions can be used to emulate biological neurons [10, 11]. Here, we investigate simple ways of coupling these devices to extend these innovative computing schemes. To mimic neuron-like interconnections, it is important to develop simple schemes to couple such devices. Significant work has been done to develop coupling strategies for magnetic systems such as harmonic spin-torque nano-oscillators, which are useful for cognitive computing [12,13,14,15], but there is no equivalent scheme for superparamagnetic tunnel junctions. In this work, we experimentally demonstrate a simple electrical coupling method between two superparamagnetic tunnel junctions. We couple two stochastic magnetic junctions by placing them in parallel in a simple electrical circuit. We use the fact that their switching rate can be controlled deterministically by the applied current. They couple because their stochastic electrical transitions change the current flowing through the other device, changing its switching rate. In our circuit, the total applied current in each nanojunction is shared and depends on the joint resistance states of the nanojunctions. Depending on the current state of the system, transitions to certain joint states are favored while others are less probable. This coupling mechanism leads to non-zero cross-correlation between the stochastic switching events of the two fluctuating tunnel junctions, demonstrating their mutual coupling. We show experimentally that the cross-correlation becomes large when the switching time scales of the two superparamagnetic tunnel junctions are such that they both have roughly a 50 % probability to be in the parallel or antiparallel state. We also demonstrate that the cross-correlation increases when the total current applied to the system increases. We reach 18 % for the largest Pearson correlation value measured in the system. We find good agreements between numerical simulations of a comparable four-state Markov state process and our experiments. As we do not use sophisticated circuits for the coupling purpose, our approach opens new paths to compact and energy-efficient implementations of scaled-up arrays of coupled superparamagnetic tunnel junctions for new cognitive computing schemes.

**References:**

[1] W. H. Rippard et. al. PRB, Vol 84, 064439 (2011). [2] B. Parks et. al. AIP Advances, Vol 8, 055903 (2018). [3] D. Vodenicarevic et. al. Phys. Rev. Appl, Vol 8, 054045 (2020). [4] A. Mizrahi et. al. Nature Communications, Vol 9, 1533 (2018). [5] K. Camsari et. al. Appl. Phys. Rev, Vol 6, 011305 (2019). [6] K. Camsari et. al. Phys. Rev. X, Vol 7, 031014 (2017). [7] A. Sengupta et. al. IEEE Trans. on Electron Devices, Vol 63, 2963 (2016). [8] W. A. Borders et. al. Nature, Vol 573, 390 (2019). [9] M. W. Daniels, et al. Phys. Rev. Appl, Vol 13, 034016 (2020). [10] W. Gerstner et. al. Neuronal Dynamics, Cambridge Univ. Press (2014). [11] J. Grollier et. al. Nature Electronics, Vol 3, 360 (2020). [12] W. H. Rippard et. al. PRL, Vol 95, 067203 (2005). [13] S. Kaka et. al. Nature, Vol 437, 389 (2005). [14] A. A. Awad, Nat. Phys, Vol 13, 292 (2017). [15] M. Romera†, P. Talatchian† et al. Nature, Vol 563, 230-234 (2018).

**Images:**

 Mutually coupled superparamagnetic tunnel junctions

The end of Moore’s Law for complementary metal-oxide semiconductor (CMOS) technology has prompted the search for low-power computing, resulting in several promising proposals based on magnetic logic. One approach aims at tailoring arrays of nanomagnetic islands in which the magnetostatic interactions constrain the equilibrium orientation of the magnetization to embed logical functionalities [1]. Despite the realization of several proofs of concepts of such nanomagnetic logic, it is still unclear what the advantages are compared to the widespread CMOS designs, due to their need for clocking and/or thermal annealing for which fast convergence to the ground state is not guaranteed. In fact, it seems increasingly evident that “beyond CMOS” technology will require a fundamental rethinking of our computing paradigm. In this respect, digital memcomputing machines have been proposed [2]. These machines consist of self-organizing logic gates (SOLGs) which are able to dynamically “self-organize” into their logically correct states irrespective of whether the signal is applied to the traditional input terminals, or the traditional output terminals. This type of terminal-agnostic logic thus allows for reversed computing. Here [3], we introduce a novel concept of SOLGs that employs stray-field coupled nanomagnetic islands to perform terminal-agnostic logic: nanomagnetic self-organizing logic gates. We thereby focus on the design of a functionally complete NAND gate with which any Boolean circuit can be constructed in a bottom-up approach. Our strategy to build such a nanomagnetic SO-NAND gate relies on two main properties: balancedness and dynamic error suppression (DES) – see Figure 1. For reversible logic to work, the logically correct states should be occupied equally in thermodynamical equilibrium (and not only after a relaxation process into the ground states, as balancedness was originally defined [4]). This contrasts the balanced NAND gate proposed in Ref. [5]. To allow for larger circuits, the probability of being in a logically incorrect state should suppressed. To this end, our DES scheme periodically checks the logical correctness of the gate/coupling and applies additional fields on each island of the gate/coupling if necessary. Our DES scheme is thus based on local information, i.e. the magnetization state of a single gate or coupling. This locality is in sharp contrast to the biasing used in p-bit logic inversion, for which the logical behavior of a collection of super-paramagnetic tunnel junctions (sMTJ) is fully imposed by globally-determined time-varying spin currents [6]. Owing to its locality, our approach does not suffer from scaling issues. To show the usefulness of our SO-NAND gate, we use it as a building block to construct a self-organizing two-bit multiplier – see Figure 2. Our SO-2BM is capable of decomposing any output number into its (prime)factors, finding each possible decomposition without clearly favoring nor penalizing any solution. This fact indicates that the balancedness remains largely conserved throughout the whole system, which allows the extension to larger systems in order to solve more complex problems.

**References:**

[1] A. Imre, G. Csaba, L. Ji et al., Science, Vol. 311, p. 205-208 (2006). [2] F. Traversa and M. Di Ventra, Chaos, Vol. 27, p. 023107 (2017). [3] P. Gypens, J. Leliaert, M. Di Ventra et al., arXiv preprint arXiv:2012.12721 (2020). [4] J. Lust and B. Dixon, Inf. Sci., Vol. 113, p. 193-204 (1999). [5] P. Gypens, J. Leliaert and B. Van Waeyenberge, Phys. Rev. Appl., Vol. 9, p. 034004 (2018). [6] W. Borders, A. Pervaiz, S. Fukami et al., Nature, Vol. 573, p. 390-393 (2019). [7] J. Leliaert, A. Vansteenkiste, A. Coene et al., Med. Biol. Eng. Comput., Vol. 53, p. 309-317 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7d77a99461c449a155c9bf16943ac1b1.jpg)
Fig. 1: Schematic representation of a self-organizing NAND gate consisting of four nanomagnetic islands with perpendicular anisotropy. The probability distribution is obtained with the macrospin simulation tool Vinamax [7] which numerically solves the Landau-Lifshitz-Gilbert equation taking thermal fluctuations into account. Logically incorrect states are suppressed by a dynamic error suppression (DES) scheme, while still preserving the balancedness of the gate. Prel is the probability relative to the most likely logical state.

![](https://assets.underline.io/uploads/markdown_image/1/image/bdd9a7d3218e170d709dc62ca31af92c.jpg)
Fig. 2: Schematic representation of a self-organizing two-bit multiplier and the relative probability distribution obtained with Vinamax for an output fixed to 0. Prel is the probability relative to the most likely logical state. The four output islands of the SO-2BM, labeled from a to d, are fixed such that they represent a certain number (0, 1, 2, 3, 4, 6, or 9) and the system is allowed to explore its state space. The magnetization of the input islands, corresponding to the solution of the computation, are labeled from A to D. All states of which one of the inputs is logically inconsistent are aggregated into the "??"-state.

 Nanomagnetic Self-Organizing Logic Gates

The rate of thermally activated transitions can generally be described by the Arrhenius law, as k=f0 exp(-ΔE/kBT). In the previous expression, ΔE is the internal energy barrier, and f0 is a prefactor that encompasses both dynamical and entropic contributions. In this work, we show how the activation entropy can play a crucial role in thermally activated magnetic transitions. To that end, we study two types of transitions involving solitons either as the stable state or the transition state, namely, skyrmion annihilation, and domain-wall mediated magnetization reversal in nanodisks. The aforementioned transitions pertain to the class of rare events, i.e., they are unlikely to arise in the course of a Langevin simulation, because the mean waiting time between events is much larger than the integration timestep. In this case, one possible approach is to use a form of reaction rate theory to calculate the energy barrier and the Arrhenius prefactor. To this end, we apply Langer’s theory [1], which is an extension of the Kramers method to many dimensions. Alternatively, one may use a path sampling method such as forward flux sampling [2] in order to sample the transition path ensemble from the system’s dynamics and extract a transition rate. In this work, we employ both methods in conjunction to study the thermal stability of magnetic skyrmions and nanodisks, for which the two methods show a good agreement. Magnetic skyrmions are particle-like, two-dimensional, non-collinear solitonic spin textures with a nontrivial topology. In ultrathin films and multilayers, small skyrmions are stabilized by an interfacial form of the Dzyaloshinskii-Moriya interaction (DMI). Skyrmion-based designs currently appear very promising for novel spintronics devices, e.g., for racetrack memories and logic gates, reservoir computing, or probabilistic computing [3]. Isolated skyrmions exist as metastable excitations of the collinear magnetic ground state and can be longed-lived. For applications, a stability of up to 10 years at room temperature is a technological requirement, and the ability to accurately predict that stability is therefore crucial. We find that skyrmions can be stabilized not only by an internal energy barrier, but also by a large activation entropy. This stems from the fact that the metastable skyrmion state possesses a large entropy compared to the transition state, because skyrmions exhibit stable internal modes of deformation [4]. When applying a destabilizing magnetic field perpendicular to the skyrmion core, the skyrmion lifetime decreases quickly with the field amplitude. This results from a shrinking of the skyrmion, which simultaneously reduces the the energy barrier, as well as the activation entropy through the suppression of the stable internal modes (Fig. 1) [5]. Next, we compute the mean waiting times between thermally activated domain-wall mediated magnetization reversals in a perpendicularly magnetized nanodisk. We use parameters that resemble that of a free CoFeB layer, as typically used in magnetoresistive random access memories [6]. By varying the perpendicular anisotropy and the interfacial DMI, we find that the Arrhenius prefactor can take extreme, seemingly non-physical values up to 1021 Hz, which is orders of magnitude beyond the typically assumed value of 109 Hz, and vary drastically as a function of material parameters. We show that the prefactor behaves like an exponential of the energy barrier, which stems from a linear variation of the activation entropy with the energy barrier (Fig. 2) [7]. This phenomenon, known as the Meyer-Neldel rule, or entropy-enthalpy compensation, has been reported across diverse fields of the natural sciences for over a century—in semiconductors, chemical reactions, biological death rates, etc [8]. In magnetism, it was only recently observed in the decay of the skyrmion lattice [9]. This result implies that the transition state, which, in the case of the magnetization reversal, is a domain wall found in the center of the disk, tend to possess a very large entropy compared to the collinear ground state. This is verified by the existence of gapless, low energy modes propagating along domain walls [10]. These findings suggest that the Arrhenius prefactor, which is typically taken as an attempt frequency within the characteristic timescale of a magnetic system's dynamics, may in some cases contain a very large entropic contribution which should not be neglected. This seems to be particularly true in magnetic transitions involving solitons.

**References:**

[1] J. S. Langer, Annals of Physics, Vol. 54, p.258 (1969) [2] R. J. Allen, P. B. Warren, and P. R. ten Wolde, Phys. Rev. Lett., Vol. 94, p.018104 (2005) [3] A. Fert et al., Nat. Nanotechnol., Vol. 3, p.152 (2013) ; Prychynenko et al., Phys. Rev. Appl., Vol. 9, p.014034 (2018) ; D. Pinna et al, Phys. Rev. Appl., Vol. 9, p.064018 (2018) [4] L. Desplat, D. Suess, J.-V. Kim et al., Phys. Rev. B, Vol. 98, p.134407 (2018) [5] L. Desplat, C. Vogler, D. Suess et al., Phys. Rev. B, Vol. 101, p.060403(R) (2020) [6] J. Sampaio, A. Khvalkovskiy, M. Kuteifan et al., Appl. Phys. Lett., Vol. 108, p.112403 (2016) [7] L. Desplat and J-.V. Kim, Phys. Rev. Lett., Vol 125, p.107201 (2020) ; L. Desplat and J-.V. Kim, Phys. Rev. Appl., Vol. 14, p.064064 (2020) [8] A. Yelon, B. Movaghar, and R. Crandall, Rep. Prog. Phys., Vol. 69, p.1145 (2006) [9] J. Wild, T. N. Meier, S. Pollath et al., Sci. Adv., Vol. 3, p.e1701704 (2017) [10] F. Garcia-Sanchez, P. Borys, R. Soucaille, et al., Phys. Rev. Lett., Vol. 114, p.247206 (2015)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3d6013214984a1f3ea7698c4d79f4b7a.jpg)
Arrhenius prefactor for skyrmion collapse as a function of the perpendicular external magnetic field in units of the exchange coupling constant, Jex. The prefactor is computed through Langer's theory, forward flux sampling (FFS), and direct Langevin simulations.

![](https://assets.underline.io/uploads/markdown_image/1/image/1c239dc0374a9439b358445c11e10d48.jpg)
Activation entropy for domain-wall mediated magnetization reversal in a nanodisk at T = 300 K as a function of the activation energy normalized by the exchange constant, A, and the disk thickness d, for variations of the DMI, D, and of the anisotropy, K. The inset shows the corresponding variation of the Arrhenius prefactor, f0.

 Entropic effects and solitons in thermally activated magnetic transitions INVITED

The topological charge or skyrmion number Q is a useful measure to characterize the topology of spin textures such as vortices and skyrmions in two-dimensional systems. When the magnetization field m(r) is projected onto the unit sphere, Q measures the number of times the magnetic moments wrap around this sphere. For vortices and merons, Q = ±1/2, while for skyrmions, Q = ±1. Higher-order half- and full-integer charges are also possible. In numerical micromagnetism, a common approach involves solving the equations of motions for m(r) that is discretized on a regular grid. The underlying assumption is that cell-to-cell variations in m are sufficiently small such that the exchange energy, proportional to the square of the magnetization gradient, remains meaningful. However, issues can arise under certain conditions, such as in the nucleation and annihilation of vortices and skyrmions, or in the stochastic dynamics under random fields, where large cell-to-cell variations in the magnetization can reduce the accuracy of the finite-difference approximations of the spatial derivatives used to estimate Q, which can result in spurious deviations from half- and full-integer values of Q. Here, we will describe a lattice-based implementation for the open source micromagnetics code MuMax3 [1] that allows an accurate determination of Q, particularly under Langevin dynamics used to model finite temperatures and in nucleation and annihilation processes [2]. The implementation is based on the approach of Berg and Lüscher [3], which involves summing over the solid angle subtended by three spins, forming elementary signed triangles, that are constructed from the finite difference cells. The implementation accounts for both periodic boundary conditions and finite-sized systems. A comparison between the finite-difference and lattice-based approaches is given in Fig. 1, where the time variation of the total topological charge is shown for a system hosting a single skyrmion under a temperature of 300 K. The blue circles represent the Q computed using finite-difference derivatives, which exhibit large fluctuations about the expected value of Q = -1 that are about as large as Q itself. Moreover, the time-averaged mean Q does not coincide with -1. The solid red line represents the result of the lattice-based approach, which remains constant near the expected value of -1. More precisely, the result is limited primarily by the single-precision of the floating-point arithmetic used, e.g., we find Q = −1.0000001, −1.0000004, −1.0000008, −1.0000002, and −0.9999996 over 20-ns intervals at 400 K. We also show how the lattice-based approach is vastly superior to the finite-difference approximation in other test cases, which involve skyrmion-antiskyrmion pair nucleation due to spin-transfer torques and skyrmion fluctuations in finite dots. While the results do not necessarily call into question the validity of published work (since the topological charge is often only used as a proxy for magnetization gradients), they do highlight the care with which noninteger values of Q should be interpreted, particularly when processes such as nucleation, annihilation, and thermal fluctuations are at play. This work was partially supported by the Agence Nationale de la Recherche under contract no. ANR-17-CE24-0025 (TOPSKY) and Fonds WetenschappelijkOnderzoek (FWO-Vlaanderen) through Project No. G098917N.

**References:**

[1] A. Vansteenkiste, J. Leliaert, M. Dvornik et al., AIP Adv 4, 107133 (2014). [2] J.-V. Kim and J. Mulkers, IOP SciNotes 1, 025211 (2020). [3] B. Berg and M. Lüscher, Nucl Phys 190, 412 (1981).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d45266e543d925ddaf2fecf323e4aa0c.jpg)
Topological charge, Q, as a function of time for a skyrmion in a 1-nm thick film with periodic boundary conditions at 300 K. Blue circles represent the calculation based on a finite-difference approximation to spatial derivatives, as provided by the package ext_topologicalcharge in MuMax3, while the solid red line represents the calculation with the lattice-based approach, which is provided by the package ext_topologicalchargelattice. The top inset illustrates snapshots of the micromagnetic state.

 On quantifying the topological charge in micromagnetics using a lattice-based approach

Magnetic skyrmions are whirling magnetic structures that can be found on certain magnetic materials [1]. They are promising candidates to become the next generation of information carriers as well as basic elements in ultradense magnetic memories, logic devices or computational systems due to their small size and high mobility. Skyrmions can be stabilized in ferromagnetic ultrathin films with the aid of interfacial Dzyaloshinskii-Moriya (iDM) interaction with a heavy-metal substrate. The same mechanism allows the formation of skyrmions in multilayers with alternate ferromagnets and heavy-metals. The experimental finding of room-temperature skyrmions has boosted the potentiality of skyrmions for applications and, in consequence, the study of their current-driving dynamics at non-zero temperatures. Skyrmionic racetracks are considered one of the most reliable systems to transport skyrmions using the spin-orbit torque produced by a spin-polarized current fed in a heavy-metal substrate. The racetrack borders create a confining potential whose intensity determines the limit speed that a skyrmion can achieve inside a track without being annihilated at the border. As temperature increases, the topological protection of skyrmions is reduced, however, in some systems it is possible to stabilize and transport them. Apart from compromising stability, temperature can turn skyrmion dynamics into stochastic. As temperature increases the dispersion of the skyrmions position becomes larger, behaving like a Brownian particle. This stochastic motion sets new conditions and restrictions on the applicability of racetracks. In particular some binary questions like: Will the skyrmion be annihilated at the edge of the racetrack?, now become probabilistic questions: Which is the probability of a skyrmion to be annihilated at the edge of the racetrack? To address this kind of questions, a deterministic approach has been developed to study the dynamics of skyrmions at non-zero temperatures [2]. By solving the Fokker-Planck equation (FPE) for a skyrmion we can directly determine the probability of finding a skyrmion for a given time and position. In [2], the dynamics of a skyrmion when approaching a pinning potential and when driven along a racetrack was studied. In this work we present a natural continuation to that study: the granularity of the ferromagnet is taken into account while the skyrmion is moving along a racetrack. When nanomagnetic systems are studied from a theoretical point of view, usually we do not consider the granularity of the ferromagnetic materials that form the system, neglecting the influence of the polycrystalline structure and the presence of defects. However, in experiments we cannot avoid them. The granularity and defects difficult the motion of skyrmions at low velocities setting an unpinning velocity, adding an additional ingredient to the transport of skyrmions: they can be pinned in some point of the track [3]. We simulate the granularity of the material setting a random array of pining potentials with random (within a range) intensities. We study how the probability of a skyrmion to reach the end of track of length without being trapped or annihilated (success probability) changes as we vary the density of pining potentials, which is proportional to the density of crystal grains, and the intensity of the pining potentials, which is proportional to the difference of the magnetic properties between grains. In Fig.1 and Fig.2 we show different snapshots for the density of probability of a skyrmion being on a given position, in a simulation where 100 pin potentials are placed randomly. In Fig.2 the average intensity of the pining potentials is higher than in Fig. 1. We can observe in Fig. 1 that the probability of success is close to 1, indicating that the skyrmion will survive to its travel along the track without getting pinned. However, in Fig.2 we observe that there is a considerable probability that the skyrmion gets pined and does not reach the end of the track. In order to build real devices, we need to ensure that the probability of success is as close to 1 as possible. We study how this success probability depends on the skyrmion speed, and granularity of the track at room temperature, finding that the granularity can even help in some cases.

**References:**

[1] Albert Fert, Vincent Cros, João Sampaio, Nat. Nanotechnology, 8, 152–156(2013) [2] Josep Castell-Queralt, Leonardo González-Gómez, Carles Navau, Phys. Rev. B 101, 140404(R) (2020) [3] Barton L. Brown, Uwe C. Täuber, and Michel Pleimling, Phys. Rev. B, 100, 024410 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c29d136d6432c1a8980e5c1679287717.jpg)
Snapshots of the time evolution of the probability density of finding a skyrmion at a certain position. The corresponding snapshot times are, from left to right, 1.5/3/4.5/7.8/13.5 ns. The track length is 1 mm. There are 100 pining sites randomly distributed on the track with moderate intensity. The success probability is close to 1 (higher than when we have no granularity), with a small probability of being annihilated at the edge of the track or being trapped.

![](https://assets.underline.io/uploads/markdown_image/1/image/01ca6d0e939f9fbbd60fcf1caf2580f6.jpg)
Snapshots of the time evolution of the probability density of finding a skyrmion at a certain position. The corresponding snapshot times are, from left to right, 0.6/1.8/3.9/9/15 ns. The track length is 1 mm. There are 100 pining sites randomly distributed on the track with high intensity. The success probability is close to 0 since the trapping probability of the skyrmion is close to 1 (the skyrmion would be most likely trapped at some pinning site).

 Theoretical Study of theTransport of Skyrmions at Room Temperature in Granular Racetracks

During the last decade, the increasing demand of energy efficiency for low power electronics oriented a considerable progress in optimizing nanomagnetic and spintronic devices [1]. In this framework, magnetization switching is a fundamental process to investigate, in order to obtain a compromise between energy efficiency and speed in magnetic recording technology. In particular, magnetic random access memories (MRAMs) have proved to represent a valid alternative solution with respect to conventional non volatile electronic memories in terms of speed, scalability and low operation power [2-3]. Concerning the latter aspect, a significantly important issue is the design of memory cells with low enough operation power (e.g. within the IoT paradigm) but capable of reliable switching within a prescribed delay time, which may vary depending on applications (from fractions of ns to seconds and beyond). This means, in terms of nanomagnet physics, that magnetization switching for low energy barriers compared to the energy of thermal fluctuations, in the presence of a small/moderate external excitation (magnetic field/spin-polarized current) becomes a central issue to investigate. This occurs because, when the external current/field pulse is applied, the energy barrier which guarantees the required data retention at zero excitation (usually &gt; 60 kBT) is lowered, and thermal fluctuations may trigger and drive the magnetization reversal process. The analysis of stochastic magnetization dynamics [4] is usually performed by using two approaches which are complementary. On one hand, one can use the so-called Langevin dynamics approach, where the magnetization switching process of a large number of nanomagnet replicas is numerically studied via the stochastic Landau-Lifshitz-Gilbert (LLG) equation and the relevant information are extracted from the ensemble statistics [5-6]. On the other hand, the second approach exploits the fact that the stochastic LLG equation is associated with a drift-diffusion equation, referred to as Fokker-Planck (FP) equation, which describes the time evolution of the magnetization transition probability density function (pdf). In this paper, we numerically solve the FP equation on the unit sphere without any symmetry assumption on the particle. To this end, we have developed two methods: the spectral-collocation (SM) [7] and the FEM [8] methods, respectively. We show that both methods accurately predict the dominant relaxation time by comparing the numerical results with the asymptotic analytical theory [5] valid for large energy barriers ΔG with respect to the thermal energy kBT. The first method consists in choosing a grid of points and expanding the solution using a set of smooth basis functions. The grid on the unit sphere is constructed by using as coordinates the component of magnetization along the z-axis and the azimuthal angle associated to the rotation around the z-axis itself. The grid points are equispaced in the azimuth angle and are distributed as Chebyshev nodes in the z variable. For the FEM, a triangulated unit sphere is considered and it is obtained recursively bisecting the edges starting from a tetrahedral polyhedron [10]. The mesh has vertices pi, i=1,...,Np, which are the nodes where the unknown pdf is located. The results were tested for different triangulations with an increasing number of points in order to verify the convergence of the numerical FEM method. The computational approaches are validated comparing the slowest thermal relaxation time constant for an ensemble of magnetic particles, and the approximate analytical expression given in [5], as shown in Fig.(1). This time constant is of interest because it determines the thermal stability of a magnetic memory. We verify the FEM code in the following conditions: Np=(8194, 65536, 131074) until reaching the desired convergence with respect to the analytical results in the limit of large energy barriers. The SM method shows a high accuracy from small to intermediate energy barriers, whereas for higher energy barriers the method requires an increasing number of grid points, on the unit sphere, limiting the possibility to compute the relaxation time to slightly above 26 kBT, after which the numerical results deviate from the analytical prediction. Conversely the method is accurate for low energy barriers and it achieves machine precision in the limit case of zero energy barrier(pure diffusion on the unit sphere). The FEM method exhibits a lower accuracy in the relaxation time computation, one can see that it covers a broader range of energy barriers up to 35 kBT. The numerical tests presented here run on a PC using MATLAB. In conclusion, the developed methods allow accurate analysis of thermal switching processes for generic particles under different operating conditions depending on the moderately large or small amplitude of the energy barrier in the presence of external excitation. It is possible to show (not reported here due to limited space) that they allow to straightforwardly determine the switching times distributions [9], instrumental to assess the write-error rates of MRAM cells [11], in the intermediate regime between thermal and ballistic switching which is of great interest for applications.

**References:**

[1] B. Dieny et al., Nature Electronics 3 (8), 446-459, (2020). [2] D. Apalkov et al., Proceedings of the IEEE, Vol. 104, No. 10, (2016). [3] S. Bhatti et al., Materials Today, Volume 20, Number 9, (2017). [4] G. Bertotti, I.D. Mayergoyz, C. Serpico, Nonlinear Dynamics in Nanosystems, Elsevier, (2009). [5] W.F. Brown, Physical Review, vol. 130, no. 5, (1963). [6] D.B. Reeves, J.B. Weaver, Applied PhysicsLetters, vol. 107, issue 22, (2019). [7] D. P. Ansalone, C. Ragusa, M. d’Aquino, C. Serpico and G. Bertotti, IEEE Transactions on Magnetics, vol. 45, (2009). [8] N.V. Peskov, Physica B: Condensed Matter, vol. 599, (2020). [9] M. d’Aquino et al., Journal of Magnetism and Magnetic Materials 475, 652-661, (2019). [10] S. Maddali S. Maddali, SphereMesh, https://github.com/siddharth-maddali/SphereMesh, (2021). [11] M. d'Aquino, S. Perna, C. Serpico, Physica B: Condensed Matter, Volume 577, (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c9a9725762d961cc4018595a8d70f838.jpg)
Fig. 1. Comparison between numerical results and analytical approximated values, for the following parameters values: T=300K, α=0.01and keff=0.3569 takes into account shape and crystalline anisotropy, Ms=8 105 A/m and no applied field. The number of unknowns are Np=131074 for the FEM and Np=21632 for SC methods.

 Numerical Solution of the Fokker-Planck Equation by Spectral Collocation and FEM Methods for Stochastic Magnetization Dynamics

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.

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