United States

We introduce flatspin, a novel simulator for artificial spin ices (ASI), i.e., ensembles of dipole coupled nanomagnets arranged in a deliberately constructed geometry. Understanding the complex behavior of these systems has provided insight into a wide range of phenomena [4]. Modeling the dynamics of ASI and their connection to the specific design has been paramount to the progression of ASI research [5]. However, conventional modeling techniques are often either too computationally costly or do not provide enough insight into the dynamics of the ASI. Monte Carlo methods approximating each magnet as a single dipole are useful in the search for low-energy configurations [1, 6], but capturing state transitions and dynamics observed in real systems is challenging. On the other hand, high-fidelity micromagnetic simulations are restricted to a small number of nanomagnets due to the computational cost. With the relatively small systems available through this approach, it is difficult to capture large-scale emergent behavior, e.g. [6]. With flatspin, we offer a different approach to the challenge of modeling, and aim to capture large-scale ASI behaviors. At the heart of flatspin lies a robust magnetic dipole model and a switching criteria based on a generalized Stoner-Wohlfarth model. GPU acceleration together with a model optimized for speed enables flatspin to simulate realistic dynamics of millions of magnets within practical time frames. We demonstrate the versatility of flatspin by modeling a diverse set of established experimental results. Reproduced results include emergent fine-scale patterns in kagome ASI [3], large-scale domain sizes in square ASI [6], and different magnetic ordering of square and pinwheel ASI [2] (see Fig. 1). Furthermore, magnetization details of pinwheel ASI during field-driven reversal have been reproduced, for the first time, by a dipole model. With its capability to model ASI dynamics at unprecedented speeds, we believe flatspin will be an asset to the ASI community.References: [1] X. Ke, J. Li, C. Nisoli et al., Physical Review Letters., Vol. 101(3), p.037205 (2008) [2] R. Macêdo, G. M. Macauley, F. S. Nascimento et al., Physical Review B., Vol. 98(1), p. 014437 (2018) [3] E. Mengotti, L. J. Heyderman, A. F. Rodríguez et al., Nature Physics., Vol. 7(1), p.68 (2011) [4] S. H. Skjærvø, C. H. Marrows, R. L. Stamps et al., Nature Reviews Physics., Vol. 2(1), p.13–28 (2020) [5] R. L. Stamps, Nature Physics., Vol. 10(9), p.623–624 (2014) [6] S. Zhang, I. Gilbert, C. Nisoli, Nature., Vol. 500(7464), p.553–557 (2013)

MMM 2020

Introducing Flatspin: Enabling Large Scale Dynamical Artificial Spin Ice Simulations

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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technical paper

Today magnetic recording is still the leading technology for mass data storage with data densities beyond 1 Tbit/in2. Pushing the recording density in the terabit regime requires new storage materials, novel recording schemes, and media designs. We propose an approach to magnetic recording upon the basis of thermally induced chemical order-disorder transitions (CODTs) in alloys of transition metals, which exhibit disorder-induced ferromagnetism. [1,2] Here we report on our simulations of CODTs and demonstrate that nanosecond laser irradiation concentrated within a nanoscale spot on the sample surface is able to induce the reversible transitions in Fe-rich FexAl1-x alloys. Figure 1 shows a design of magnetic recording media which can be realized upon that basis by using near-field optical effects and special tools, e.g., a near-field transducer (NFT) [3]. It is crucial for this approach that the bits can be written as erasable nanoscale entities in which the magnitude of magnetization differs from that of encompassing matrix. Therefore, contrary to the existing magnetic memories, the information here can be encoded in not the polarity of magnetization but in its magnitude. As a result, difficult problems in current magnetic recording technologies, e.g., thermal upsets of magnetization and cross-talks between adjacent bits, can be circumvented by using the approach we propose. This work was supported by RFBR (grant #18-02-00827_a), DFG through the SFB 767 at the University of Konstanz, by NECL and FCT support through Project Nos. NORTE-01-0145-FEDER-022096, PTDC/FIS-MAC/31302/2017, and by EXPL/IF/00541/2015, and by NSF (project TG-DMR110090).References: [1] J. Ehrler, M. He, M.V. Shugaev, et al., Laser-rewriteable ferromagnetism at thin-film surfaces. ACS Appl. Mater. Interfaces, Vol.10 (17), p.15232 (2018) [2] N. I. Polushkin, V. Oliveira, R. Vilar, et al., Phase-change magnetic memory: rewritable ferromagnetism by laser quenching of chemical disorder in Fe60Al40 alloy, Phys. Rev. Appl., Vol.10 (2), 024023 (2018) [3] W. Challener, C. Peng, A. Itagi, et al., Heat-assisted magnetic recording by a near-field transducer with efficient optical energy transfer. Nat. Photon., Vol. 3 (4), p. 220 (2009)

Chemical order-disorder transitions in intermetallic alloys for high-density magnetic recording

Micromagnetic modelling enables the accurate simulation of large magnetic systems with a significantly reduced computational cost compared to atomistic simulations. Through micromagnetic modelling it is possible to simulate systems consisting of thousands of grains over a range of timescales from nanoseconds to years. Here we present an open-source multi-timescale micromagnetic code, called MARS, consisting of the key solvers Landau-Lifshitz-Gilbert (LLG), Landau-Lifshitz-Bloch (LLB) and Kinetic Monte Carlo (kMC). The included solvers have been rigorously tested and are bundled into an overarching solver class to enable the creation of simulations which can utilise any of the available solvers at run-time. This combination of solvers also enables multi-timescale simulations with dynamic switching between solvers for specified timescales. The granular systems used are generated via a Voronoi construction and can consist of multiple or single layers. Each layer is assigned its own material with separate material configuration files. MARS is designed to use atomistic parameterisation for the material parameter inputs, obtained from software such as VAMPIRE [1]. MARS includes numerous bundled simulation types including but not limited to: FORC, FMR, HAMR writing. Fig.1 shows the result of a HAMR writing simulation of a pseudorandom bit sequence via the LLB solver followed by the read back of the system after a week via the kMC solver. Such simulations provide an easy method to test a system’s suitability for data writing and long term storage. A detailed model of the HAMR write head is provided including custom field and temperature profiles, skew angle and near-field transducer to write head spacing.References: [1] R. Evans, W. Fan and P. Chureemart, et al. Journal of Physics: Condensed Matter, Vol. 26, p103202 (2014)

Models of Advanced Recording Systems (MARS) - A Multi-Timescale Micromagnetic Code for Granular Film Simulations

The data storage industry is currently migrating to Heat Assisted Magnetic Recording (HAMR) technology for the next generations of hard drives. HAMR allows the use of high anisotropy media by introducing laser pulses to lower the coercivity to writable levels at elevated temperatures. However, the long-term stability of written data remains a problem, especially when it may be subjected to high temperatures during the writing or rewriting of neighbouring tracks. A simple model is desirable for the evaluation of energy barriers for complex media. Concerning the long term and controlled stability of the recorded information, we developed a numerical model which allows us to perform energy barrier calculations for multispin systems. Our work is based on the Lagrange multiplier method of finding points of extremum for functions subject to equality constraints. While this approach has been previously used in conjunction with the Landau-Lifshitz-Gilbert equation [1-2], we were able to integrate the Lagrange optimisation strategy in a direct minimisation tool. Making use of our model, we investigate several spin chain systems, mapping the minimum energy paths (MEPs) and extracting the corresponding energy barriers, while characterising the specific reversal mechanisms, ranging from the extremes of coherent rotation and domain wall formation (see Fig.1). Furthermore we use our description to explore the exchange spring behaviour and the reduction of the switching field and energy barrier via the hard/soft coupling [3].References: [1] D.A. Garanin and H. Kachkachi, Phys. Rev. Lett. 90, 65504,(2003). [2] R. Yanes, O. Chubykalo-Fesenko, H. Kachkachi, Phys. Rev. B 76, 064416 (2007) [3] K. Yu. Guslienko, O. Chubykalo-Fesenko, O. Mryasov, Phys. Rev. B 70, 104405 (2004)

Energy barrier calculations in multispin systems for magnetic recording purposes

Synthetic ferrimagnets and antiferromagnets with perpendicular anisotropy have attracted great interest as they allow for stabilization of room-temperature skyrmions [1], fast domain wall motion [2], and all-optical switching of magnetization [3]. Here we study Ni3Pt/Ir/Co structures in the vicinity of the compensation point temperature TM, where the magnetizations of the two layers have the same magnitude and are anti-parallel to each other. By measuring element-specific hysteresis loops using x-ray magnetic circular dichroism (XMCD) and wide-field magneto-optical Kerr effect (MOKE) microscopy, we explore the magnetization reversal and domain structure of the Ni3Pt and Co layers (Fig. 1). We show that the change of the magnetic state upon increasing the temperature through the compensation point is a continuous transition, associated with formation of a domain structure at remanence. Furthermore, we show that increasing the magnetic field sweep rate between saturation and remanence changes the mechanism of the magnetization reversal from propagation of domain walls to nucleation of reverse domains. Finally, we demonstrate how the layers can be switched by optical laser pulses, with the assistance of a small external magnetic field (down to 1 mT). The Ni3Pt/Ir/Co structures are promising candidates for all-optical control of magnetism in rare-earth-free synthetic ferrimagnets.References: [1] W. Legrand et al., Nature Materials 19, 34-42 (2020). [2] S.-H. Yang et al., Nature Nanotechnology 10, 221-226 (2015). [3] J. W. Liao et al., Advanced Science 1901876 (2019).

Layer-selective magnetization reversal and optical switching in Ni3Pt/Ir/Co synthetic ferrimagnets

In finite-difference micromagnetics the self-magnetostatic field is computed via the demagnetization tensor N, where H(r2) = –N(r2-r1) M(r1). For simulations with periodic boundary conditions, N is replaced with the periodic variant NPBC, which is the infinite sum of the non-periodic tensor offset by periodic dimension width W, NPBC(r)=∑∞k=-∞ N(r+kW). Truncating the sum at k=±M yields a poor approximation to NPBC because the convergence rate is only O(1/M2) (Fig. 1). This can be improved to O(1/M4) by replacing the summation tails with an integral of the dipole approximation [1], but even so roughly 200 terms are needed to obtain single precision accuracy. This work studies high-order approximations to NPBC using the Euler-Maclaurin formula [2]: NPBC(r) = ∑M k=-M N(r+kW) + (∫-MW -∞+ ∫ ∞MW) F(r+sW) ds + (F(r-MW)-F(r+MW))/2 + ∑Pk=2 Bk/k! (F(k-1)(r-MW) - F(k-1)(r+MW)) where Bk are Bernoulli numbers and F is a high-order asymptotic approximate to N. High accuracy in NPBC can be obtained with M as small as 5 or 10. This improvement is especially pertinent for 2D PBC, where the truncation convergence drops from O(1/M2) to O(1/M), and the number of N images within distance MW increases from O(M) to O(M2). The Euler-Maclaurin formula requires integrals and derivatives of F. F has the form ∑ijkn cijkn xiyjzk/Rn, and can be represented as a list of quintuples, (c,i,j,k,n). Derivatives of F have the same form and can be computed via a simple rule, admitting implementation as a handful of routines operating on quintuples rather than hard-coded explicit formulae. Integrals of F can be handled similarly. The result is a compact code that is simple to write and maintain that allows run-time selection of method order. The last is important because optimal method order and cut-off value M vary with discretization cell aspect ratio. The presentation will detail NPBC error estimates for 1D and 2D PBC, method order and cut-off value M selection, and estimated computation times.References: [1] K.M. Lebecki, M.J. Donahue, and M.W. Gutowski, Journal of Physics D-Applied Physics, Vol. 41, p. 175005 (2008). [2] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.27, Eq. 2.10.1. F. W. J. Olver, A. B. Olde Daalhuis, et al., eds.

High Order Methods for Computing the Demagnetization Tensor for Periodic Boundaries

Micromagnetic simulations are employed for predicting the behavior of magnetic materials from their microscopic properties, using the Landau-Lifshitz-Gilbert Equation dM / dt = - γ M × Heff - γ α (M × (M × Heff)) / Ms, with magnetization M, gyromagnetic ratio γ, dampening coefficient α and effective field Heff, Heff = Hexch + Hdem + Hext + Hanis. Here Hexch is the exchange field, Hdem the demagnetization field, Hext an applied field and Hanis the anisotropy field. By far the most time consuming part of these calculations consists of calculating the demagnetization field, Hdem. Standard approaches compute Hdem as the convolution of magnetization with the dipolar interaction kernel using fast-Fourier transforms[1][2][3]. The framework MagTense[4], by contrast, calculates the demagnetization field analytically from a mesh of computational elements, which can be prisms[5] or tetrahedrons[6], as shown in Fig. 1. Such unstructured grids present new challenges for computing the differential operator in the exchange field Hexch. Here we investigate differential operators on general unstructured polyhedral meshes[7] and apply them to the micromagnetics standard problems[8] using MagTense. Specifically, we employ a weighted least squares interpolation[9] and a potentially novel method for calculating the Laplacian directly. The results are analyzed in terms of accuracy and speed and compared to those obtained with prism meshes.References: [1] M.J. Donahue and D.G. Porter, Interagency Report NISTIR 6376, National Institute of Standards and Technology, Gaithersburg, MD (1999) [2] A. Vansteenkiste and B. Van de Wiele, Journal of Magnetism and Magnetic Materials 323, (21), 2585 - 2591, (2011) [3] A. Kakay, E. Westphal and R. Hertel, IEEE Transactions on Magnetics 46, (6), 2303-2306, (2010) [4] K.K. Nielsen and R. Bjørk, https://doi.org/10.11581/DTU:00000071 (2019) [5] A. Smith, K.K. Nielsen and D.V. Christensen, J. Appl. Phys. 107 (10), 103910 1-8, (2010) [6] K. K. Nielsen, A. R. Insinga and R. Bjørk, IEEE Magnetics Letters 10, 1-5, (2019) [7] E. Sozer, C. Brehm and C.C. Kiris, Gradient Calculation Methods on Arbitrary Polyhedral Unstructured Meshes for Cell-Centered CFD Solvers, 52nd Aerospace Sciences Meeting (2014) [8] https://www.ctcms.nist.gov/∼rdm/mumag.org.html (Visited June 26, 2020) [9] D. Pérez-Grande, O. Gonzalez-Martinez and P. Fajardo, Applied Sciences 6 (2016)

Micromagnetic Exchange Field Calculations for Unstructured Meshes

We use micromagnetic simulations based on the stochastic Landau–Lifshitz–Gilbert equation to calculate dynamic magnetic hysteresis loops at finite temperatures that are invariant with simulation cell size. As a test case, we simulate a magnetite nanorod, the building block of magnetic nanoparticles that have been employed in preclinical studies of hyperthermia [1]. With the goal to effectively simulate loops for large iron-oxide-based systems at relatively slow sweep rates on the order of 1 Oe/ns or less, we modify and employ a previously derived renormalization group (RG) approach for coarse-graining [2]. The scaling algorithm is shown to produce nearly identical loops over several decades in the model cell volume [3]. We also demonstrate sweep-rate scaling involving the Gilbert damping parameter that allows orders of magnitude speed-up of the loop calculations.References: [1] C. L. Dennis et al., Nanotechnology Vol. 20, p.395103 (2009) [2] G. Grinstein and R. H. Koch, Phys. Rev. Lett. Vol. 90, No. 20, p.207201 (2003) [3] R. Behbahani, M. L. Plumer and I. Saika-Voivod, J. Phys. Condens. Matter, Vol. 32, No.35LT01 (2020)

Coarse-Graining in Micromagnetic Simulations of Dynamic Hysteresis Loops

An important task in micromagnetics is performing a large ensemble of simulations of a structure with only one or a few changed parameters. An example is computing write error rates in magnetic random-access memories (MRAM), where simulations are needed for different stochastic thermal fields. Such computations, which are done by multiple runs on a single or multiple computing devices. have various numerical overheads related to preprocessing, matrix operations, and excessive data storage. We demonstrate a computational approach for performing a large batch of repetitive micromagnetic simulations on a Graphics Processing Unit (GPU) simultaneously. This batched approach has the following benefits: (i) it utilizes GPUs much more efficiently than the traditional approach of running multiple simulations, including much increased speed and reduced memory consumption, (ii) it hides various numerical overheads, such as preprocessing as well as matrix and time stepping operations, (iii) it makes obtaining results and post-processing much more efficient by allowing computing only required properties, e.g. ensemble averages or switching counts. We extended our FEM based micromagnetic framework FastMag to implement the batched approach in all code components, including pre-processing, matrix operations (Fig. 1a), time stepping with adaptivity to step, order, and decisions of a stopping evens per instance (Fig. 1b). Figure 2 shows major improvements of the computational time and memory consumption for switching simulations of an MRAM cell using three 100-instance batch scenarios on an Nvidia GTX1080 GPU: sequential launching of 100 instances, simulating a 100 x 100 array of sufficiently separated cells (to prevent the strong magnetostatic coupling), batched approach. This work was supported as part of the Quantum Materials for Energy Efficient Neuromorphic Computing, an EFRC funded by the U.S. DOE, Office of Science.

High-performance micromagnetics on GPUs for batched simulations

Since the observation of all-optical switching of magnetization in the ferrimagnetic alloy GdFeCo using femtosecond laser pulses, there has been a significant interest in exploiting this switching process for data recording [1]. The ultrafast speed of the magnetization switching allows to push the writing speeds of magnetic memories towards THz frequencies. A first approach for the exploitation of all-optical switching in spintronic devices was explored by Chen et al. [2] on optically switchable GdFeCo layer integrated within a microsized magnetic tunnel junction pillar. In this talk we will discuss the recent advances on the development of perpendicular magnetic tunnel junctions incorporating [Tb/Co]-CoFeB electrodes whose magnetization can be optically controlled via helicity-independent single-shot switching. The ultrafast optical control of the magnetization in this system gives rise to capabilities to be exploited as a storage layer in a magnetic tunnel junction. Within our results we will highlight a first evidence of helicity-independent all-optical switching in a [Co/Tb] multilayered-based system coupled to CoFeB layers with both ps- and fs-long single laser pulses. We also explore the magneto-optical properties of multilayers and their thermal stability upon different annealing temperatures. The magneto optical response and the perpendicular magnetic anisotropy of our system was achieved even after annealing at 250°C. The laser pulse duration and fluence dependence for the CoFeB/[Tb/Co]5 electrodes were explored using single 70fs and 7ps laser pulses with fluences F = 3.1, 3.5 and 4.0mJ/cm2. As can be seen from Fig.1, images obtained for a laser pulse duration D = 7ps and 70fs, a clear magnetization reversal can be achieved for F = 3.5mJ/cm2. Results also determined that reducing the pulse duration to 70fs, the fluence required to induce thermal demagnetization on the sample increases. Our all-optical switching electrodes FeCoB/Ta/[Tb/Co]N were finally integrated into a magnetic tunnel junction. Electrical evaluation of nanopatterned AOS-MTJ showed TMR ratios up to 36 % depending on the diameter of the junctions and on the number of repetitions of the [Tb/Co] bilayers. The full structure of the junctions consist of: Ta(30Å)/FeCoB(11Å)/MgO(23Å)/FeCoB(13Å)/Ta(2Å)/[Tb(9.5Å)/Co(12.5Å)]5. The TMR values distribution of hundreds of MTJ with different diameters is shown in Fig. 2a. The maximum TMR ratio (36 %) was observed for a 200 nm-diameter junction with a minimum resistance of 6 kΩ. Fig. 2b shows the corresponding Resistance loop extracted from the 200 nm-diameter (yellow) distribution of Fig. 2a. To our knowledge, we are presenting the first study that reports the helicity-independent all-optical switching in a Co/Tb multilayered-based system. The integration of optically switchable electrodes within magnetic tunnel junctions is a key step towards future spintronic technologies. In this direction, the research presented here, related to the nanofabrication of high TMR ratio-magnetic tunnel junction using [Co/Tb]-CoFeB electrodes, are a contribution that can lead to substantial advances in hybrid spintronic-photonic devices and contribute to the development of magnetic tunnel junctions with new functionalities, particularly the ability to control the magnetization of the storage layer with ultrafast laser pulses. This research has received funding from the European Union’s Horizon 2020 research and innovation program under FET-Open Grant Agreement No. 713481 SPICE.References: 1. A. V. Kimel et al. Nat. Mater. 13 225 (2014) 2. J. Y. Chen et al. Phys. Rev. Appl. 7 021001 (2017) 3. L. Avilés-Félix et al. Sci. Rep. 10 5211 (2020). 4. L. Avilés-Félix, et al. AIP Advances 9, 125328 (2019) 5. A. Olivier et al. Accepted in IOP Nanotechnology (2020)

Integration of single-shot all-optical switching Tb/Co multilayer-based electrodes within perpendicular magnetic tunnel junctions INVITED

Micromagnetic simulations are typically limited to a few micrometers due to the high demand on computing resources. But in applications and experiments the specimen size is usually orders of magnitude larger. While the micromagnetic simulations reproduce trends nicely, the absolute value of the results differ from the experiments [1]. Often the size limitation in the simulations is the reason for this deviation. Normally this is overcome either by artificial scaling [2] or by a reduced-order model [3]. In this work we introduce a type of reduced-order model to bridge the length scale from micromagnetism to experiments. In contrast to the work of Blank [4], we also consider the microstructural features of the magnet. We subdivide the computation of a large sample, e.g. from experimental measurements, into multiple independent feasible-sized subsets. For each subset, the nucleation field is either calculated by micromagnetic simulation or predicted by a trained machine learning model [5] (Fig. 1). The subsets along with their microstructure and nucleation fields are fed into a newly developed python code to spatially reassemble the entire sample and compute the overall hysteresis (Fig. 2). To take the microstructure into account, each subset is further discretized in cubic elements with their own anisotropic easy axis. In each calculation step, the magnetostatic field and the exchange field of the entire sample is calculated based on the magnetic moments of these elements. The two field contributions and an increasing external field are summed up to a total field. Taking this total field into account, we compute the reversible rotations with the Stoner-Wohlfarth model and use the prestored nucleation fields to irreversibly switch the respective subset. We gratefully acknowledge the financial support of the Austrian Science Fund (FWF), Project: I 3288-N36 and the German Research Foundation (DFG), Project:326646134.References: [1] X. Tang, et al., Acta Mater. 144, 884-895 (2018) [2] S. Bance, et al., Acta Mater. 131, 48-56 (2017) [3] L. Exl, et al., J. Phys. Mater. 2, 014001 (2019) [4] R. Blank, J. Magn. Magn. Mater. 101, 1-3, 317-322 (1991) [5] M. Gusenbauer, et al., accepted npj Comput. Mater., (2020)

Introducing Flatspin: Enabling Large Scale Dynamical Artificial Spin Ice Simulations

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Chemical order-disorder transitions in intermetallic alloys for high-density magnetic recording

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