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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.

MMM 2020

High Order Methods for Computing the Demagnetization Tensor for Periodic Boundaries

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

Granular L10-FePt thin film deposited on (002) textured MgO underlayers have come long way in the support of the present HAMR media areal density capability. However, further reduction of the center-to-center pitch distance between adjacent grains below 6nm is meeting the nucleation density limit of this media/underlayer material combination [1]. Here, we present a novel fabrication method utilizing metal-oxide/nitride materials systems with two-phase separation to develop a pre-defined and robust oxide/nitride template to increase the nucleation density and the columnar growth of L10-ordered FePt grains. The template is directly formed on top of MgO underlayer by co-sputtering metal and oxide/nitride materials, resulting in metal particles embedded in the oxide/nitride matrix. This is followed by the selective etching of the metal nanoparticles from the matrix [2]. Experimental demonstration shows that FePt can be backfilled into the “craters” left by the metal nanoparticles with the correct texture on the MgO underlayer. By comparing the mean pitch distance of FePt grains without and with template layer in Fig. 1, it is shown that the nucleation density of FePt grains is determined by the areal density of the “craters” formed in Au-SiO2 template layer. With the help of such a fabrication method, the material choices for further reducing the pitch distance of the FePt grains in HAMR media are expanded. The several combinations of various matrix materials forming the template and their results on nucleation and growth of FePt will be presented and discussed.References: [1] I. Suzuki, J. Wang, Y. Takahashi, K. Hono, Journal of Magnetism and Magnetic Materials, Vol. 500, 166418 (2020) [2] B. Zhou, C.c. Xu, B. S. D. C. S. Varaprasad, D. Laughlin, J. Zhu, 2019 MMM conference (2019)

Templated Growth of L10-FePt for HAMR Media

Writing tracks narrowly spaced in the crosstrack direction appears to increase the User Areal Density (UAD), but can lead to erasure of the adjacent tracks. This phenomena is called Adjacent Track Erasure (ATE). Our research aims to establish the numerical extent of ATE and introduce efforts to reduce this ATE for different HAMR media. This research can help increase the HAMR storage density by making HAMR media resistant to ATE. The ATE in HAMR is limited to tracks immediately adjacent to the track being overwritten [1]. For a single write, we have previously found that the optimum track spacing between these tracks to maximize the UAD for a noise free reader is 70% of the heat spot FWHM [2]. By implementing micromagnetic simulations, we find the single layer FePt media is more susceptible to ATE as its thickness decreases. The 3nm-6nm high temperature Thermal ECC media [3] has a slightly higher ATE susceptibility compared to the 9nm single layer FePt media. In contrast the 3nm-6nm low temperature Thermal ECC media [4] has a significantly higher ATE susceptibility (Table 1). To reduce the ATE, we change the applied field angle and introduce exchange coupling between grains of the media. The single layer FePt media ATE is impervious to the applied field angle although the ATE reduces as the field angle reduces for the high temperature Thermal ECC media. The ATE reduces for the single layer FePt media and the high temperature Thermal media with optimized exchange coupling (IGC) values. The ATE for the single layer FePt and high temperature Thermal ECC media using these optimization techniques is shown in Fig. 2. Both these optimization techniques however do not sufficiently improve the ATE susceptibility for the low temperature Thermal ECC media. We believe the ATE is proportional to the thermal stability factor at a temperature just below the write temperature. This hypothesis correctly predicts the ATE in the single layer FePt media assuming coherent rotation. For Thermal ECC media, other mathematical techniques that capture incoherent rotation can be used to verify this hypothesis.References: [1] S. Kalarickal, A. Tsoukatos, S. Hernandez, C. Hardie and E. Gage, "Adjacent Track Interference in Heat-Assisted Magnetic Recording: Impact and Implications," in IEEE Transactions on Magnetics, vol. 55, no. 7, pp. 1-4, July 2019 [2] Z.Liu, Y.Jiao and R.H. Victora, “Composite media for high density heat assisted magnetic recording”, Appl. Phys. Lett. 108, 232402(2016) [3] Z.Liu & R.H.Victora (2016). “Composite structure with superparamagnetic writing layer for heat-assisted magnetic recording”, IEEE Transactions on Magnetics, vol. 52, no. 7, pp. 1-4, July 2016 [4] N.A. Natekar, W.Tipcharoen & R.H.Victora, “Composite media with reduced write temperature for heat assisted magnetic recording”, Journal of Magnetism and Magnetic Materials 486 (2019) 165253

Analysis of Adjacent Track Erasure for the HAMR Media

Heat-assisted magnetic recording (HAMR) is the most promising candidate for next generation data storage. Previous study suggests that user areal density in HAMR with composite media can reach up to 4.7 Tb/in2 [1]. On the other hand, shingled magnetic recording has been adopted to keep the growth in user areal density for current state-of-art perpendicular magnetic recording. In this work, Heat-assisted Shingled Magnetic Recording (HSMR) is studied. It is known that adjacent track erasure can degrade signal-to-noise ratio (SNR) in conventional HAMR especially when the number of adjacent track write increases [2]. In HSMR, since there will be only one adjacent track write due to the shingled-style writing, HSMR should have better performance than conventional HAMR under the same cross-track density. As a consequence of transition curvature in HAMR [3], the shape of the remaining track after shingled writing is a half arc (Fig. 1). Thus, the read head should also be rotated by a certain angle (θ) to capture the effect from the arc shape of the transition. Table 1 shows the SNR for different track pitch and rotation angle of the read head for a heat spot with peak temperature of 850 K and full width at half maximum (FWHM) of 30 nm. It can be seen that SNR is enhanced by around 1 dB if the read head is rotated an optimal amount. Bit error rate is then calculated with a 1-D equalizer and an improved Viterbi detector [4]. It is shown that for a three-read head arrays, the user areal density can approach up to 5 Tb/in2.References: 1 Z. Liu, Y. Jiao, and R.H. Victora, Applied Physics Letters 108, 232402 (2016) 2 S. Kalarickal, A. Tsoukatos, S. Hernandez, C. Hardie, and E. Gage, IEEE Transactions on Magnetics 1 (2019) 3 J.-G.J. Zhu and H. Li, IEEE Transactions on Magnetics 53, 1 (2017) 4 Y. Wang, M.F. Erden, and R.H. Victora, IEEE Magnetics Letters 3, (2012)

Rotated Read Head for High Density Heat-Assisted Shingled 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

The ultrafast optical control of the magnetic order rapidly emerged as a promising approach in ultrafast magnetism. Yet, more than a decade after the discovery of complete and deterministic magnetization switching induced by femtosecond circularly polarized laser pulses1, many fundamental questions remain unanswered. In particular, the issue of the relative contribution of pure thermal or nonthermal effects induced by the laser pulses is crucial for the understanding of the magnetization reversal mechanism. Several theoretical models have been developed for multipulse all-optical helicity-dependent switching (AO-HDS) without reaching a consensus. Using circular polarization, transfer of light angular momentum and the emergence of an effective magnetic field via the inverse Faraday effect (IFE), a nonthermal optomagnetic effect, may occur2. In contrast, it has been argued that AO-HDS could originate from a purely heating mechanism3. In this case, the asymmetry would arise from the difference in light absorption between left- and right-circular polarization due to the magnetic circular dichroism (MCD). By conducting helicity-dependent ultrafast magnetization dynamics in a CoTb ferrimagnetic alloy that exhibits AO-HDS, we are able to quantitatively determine the MCD and resolve its role in the switching mechanism4. We find a MCD of ~1% from time-resolved Kerr effect measurements (Fig. 1a). Unequivocal interpretation of the sign of the dichroism is provided by performing AOS and, for the first time in a ferrimagnet, fs laser-induced domain wall motion experiments (Figs 2a-2b). We demonstrate that AOS occurs when the magnetization is initially in the most absorbent state, according to the light helicity. Moreover, we evidence that the MCD creates a thermal gradient that drives a domain wall toward hotter regions. Our experimental results are in agreement with the purely thermal models of AOS reported in the literature.References: 1C. D. Stanciu et al., PRL 99, 047601 (2007) 2T. D. Cornelissen et al., APL 108, 142405 (2016) 3J. Gorchon et al., PRB 94, 020409(R) (2016) 4Y. Quessab et al., PRB 100, 024425 (2019)

Resolving the role of magnetic circular dichroism in multishot helicity-dependent all-optical switching INVITED

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)

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High Order Methods for Computing the Demagnetization Tensor for Periodic Boundaries

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