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Intrinsic magnetic topological insulator MnBi2Te4 is believed to be an axion insulator in its antiferromagnetic ground state. However, direct identification of axion insulators remains experimentally elusive because the observed large resistance and the vanishing Hall effect, while indicating the onset of the axion field, is inadequate to distinguish the system from a trivial normal insulator. Using Green&#39;s function method, we theoretically demonstrate the quantized topological magnetoelectric current in a tunnel junction with atomically thin MnBi2Te4 sandwiched between two metalic contacts, which is a smoking-gun signal that unambiguously confirms antiferromagnetic MnBi2Te4 to be an axion insulator. Our predictions can be verified directly by experiments.&lt;br&gt;&lt;br&gt;

**References** Yu-Hang Li and Ran Cheng, Identifying Axion Insulator by Quantized Magnetoelectric Effect in Antiferromagnetic MnBi2Te4 Tunnel Junction. arXiv:2111.11500
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MMM 2022

Identifying Axion Insulator by Quantized Magnetoelectric Effect in Antiferromagnetic MnBi2Te4 Tunnel Junction

Intrinsic magnetic topological insulator MnBi2Te4 is believed to be an axion insulator in its antiferromagnetic ground state. However, direct identification of axion insulators remains experimentally elusive because the observed large resistance and the vanishing Hall effect, while indicating the onset of the axion field, is inadequate to distinguish the system from a trivial normal insulator. Using Green's function method, we theoretically demonstrate the quantized topological magnetoelectric current in a tunnel junction with atomically thin MnBi2Te4 sandwiched between two metalic contacts, which is a smoking-gun signal that unambiguously confirms antiferromagnetic MnBi2Te4 to be an axion insulator. Our predictions can be verified directly by experiments. 

**References** Yu-Hang Li and Ran Cheng, Identifying Axion Insulator by Quantized Magnetoelectric Effect in Antiferromagnetic MnBi2Te4 Tunnel Junction. arXiv:2111.11500
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/d04580c1885317400c8cf89b99efc944.png)

technical paper

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Rare earth (RE)â€“transition metal (TM) ferrimagnet is receving of great interest [1], because the unique coupling of RE and TM moments enables the antiferromagnetic spin dynamics [2]. Since magnetic moments of RE material originates primarily from the 4f-electrons which locates far below the Fermi level while that of TM arises from the 3d-electrons near the Fermi level [3], magnetic moment configuration at different energy levels should be studied more carefully. However, such energy-level dependent approaches have largely been ignored in spintronic researches. Here we present the experimental evidences for the energy-level-dependent distinct magnetic moment configuration in ferrimagnetic TbCo alloys. Four different approaches such as Hall measurement, VSM measurement, TR-MOKE measurement, THz emission measurement, were used to study the magnetic moment at different energy levels. We found that the magnetic moments at deeper energy level are rather easily altered by the external magnetic field than those at Fermi level, suggesting that the magnetic moment responds differently depending on the energy level. Further investigation on the temperature dependence shows that the Tb moment exhibits the spin glass-like freezing possibly induced by the random anisotropy of Tb. Our results reveal the important energy-level dependent characteristics of RE-TM ferrimagnets and therefore pave the way towards a better understanding of antiferromagnetic spin dynamics in RE-TM ferrimagnets

Energy level

Spinel oxides (AB2O4) are well known materials which possess two important features making them prone to exhibit unusual magnetic properties: (i) competing exchange interactions (JAB, JBB, and JAA) and (ii) the topology of B-sublattice, particularly favorable to the development of geometrical frustration [1,2]. Among the various classes of spinels, ZnFe2O4 is a unique system where the B site cations form a network of tetrahedra sharing their corners (similar to Pyrochlore systems), giving rise to high degree of magnetic frustration. These pyrochlore like structures have great applications in magneto-electronic devices [3,4]. In the current work we focus on the magnetic relaxation dynamics and heat capacity (CP(T)) studies of polycrystalline bulk Zn(Fe1–xMnx)2O4 where the combined effects from dilute substitution of Jahn-Teller active Cu2+ and Mn3+ ions are expected to alter the exchange coupling significantly by lifting the ground state degeneracy. Our results revealed coexistence of ferrimagnetism (FiM) and spin-glass state in Zn0.8Cu0.2FeMnO4 without signatures of standard long-range ordering, as inferred from the ac-susceptibility (χ′ and χ′′) and CP(T,H) studies. From the χ′′(T) (loss spectrum) shown in fig. 1a, we obtain a main cusp ~ 47.2K along with two more anomalies at 8.9K and 63K. Moreover, the differential analysis (∂(χ′T)/∂T vs. T) yields a local minimum at 79K, referred to as the FiM Néel temperature, TFiM. Frequency dispersion noticed in both χ′(T) and χ′′(T) (fig. 1b) provide the signatures of spin-glass state below TFiM. The empirical-scaling-laws like Vogel-Fulcher law and Power-Law (τ=τ0[(T–TSG)/TSG]–zυ) are used to determine the spin freezing temperature TSG=41.8K, the critical exponent zυ=8.9 and the relaxation time τ0=1.5×10–10s resulting in the evidence of cluster-spin-glass like state present in Zn0.8Cu0.2FeMnO4. A detailed dc-field dependent static (χDC) and dynamic susceptibility studies further support the glassy characteristics of the system below TFiM. 
**References** [1] S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. Wildes, A. L. Cornelius, J. D. M. Champion, R. G. Melko, and T. Fennell, Phys. Rev. Lett. 87, 047205 (2001) 
[2] G. Srinivasan and M. S. Seehra, Phys. Rev. B 28, 1 (1983) 
[3] E. Maniv, R. A. Murphy, S. C. Haley, S. Doyle, C. John, A. Maniv, S. K. Ramakrishna, Y. L. Tang, P. Ercius, R. Ramesh, A. P. Reyes, J. R. Long, and J. G. Analytis, Nat. Phys. 17, 525 (2021) 
[4] W. Schiessl, W. Potzel, H. Karzel, M. Steiner, G. M. Kalvius, A. Martin, M. K. Krause, I. Halevy, J. Gal, W. Schäfer, G. Will, M. Hillberg, and R. Wäppling, Phys. Rev. B 53, 9143 (1996) 
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Magnetic Relaxation Dynamics of Frustrated Glassy Tetragonal Spinel Cu0.2Zn0.8FeMnO4

Alex Hamill1, Tao Qu2, Randall H. Victora2, Paul A. Crowell1 
1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, United States,
2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota, United States 
**Abstract Body** The FMR magnon population (hereafter defined as the FMR amplitude c0) can become unstable when driven by microwave power Pa greater than a threshold power PS,such that it undergoes three-magnon splitting to two magnons c±k of half the frequency and equal and opposite wavevectors [1]. We employ time-resolved homodyning spectroscopy to examine this instability over five orders of magnitude in microwave power Pa. We observe an additional threshold power Posc, above which the splitting induces oscillations in c0[2]. At
high powers, we find that these oscillations induce 180° phase shifts in FMR. Our model predicts these phase shifts and that, similarly, the half-frequency magnons undergo 90° phase
shifts. Notably, we find that these phase shifts correspond to reversals in the three-magnon scattering direction between splitting and confluence (see Fig. 1a,b). The scattering direction is found to be determined by |Ψ0|, the absolute phase between FMR and its three-magnon scattering term. These reversals generate pronounced oscillations of the FMR amplitude after turning off the microwave excitation (see Fig. 1b,d). We also observe broadening in the frequency spectra of the oscillations (see Fig. 2a,b). From our micromagnetic simulations, we find that this corresponds to higher-order scattering processes that vary with FMR frequency (see Fig. 2c,d). These findings shed light on the relationship between three-magnon splitting and confluence, the role of phase dynamics in magnon scattering, and the higher-order magnon scattering processes at high microwave powers. 
**References** 
[1] H. Suhl, J. Phys. Chem. Solids, Vol. 1, pp. 209-227 (1957)
[2] T. Qu†, A. Hamill†, R. H. Victora, and P. A. Crowell, arXiv:2204.11969 (2022) 
The Minnesota Supercomputing Institute (MSI) provided resources that contributed to the
research results reported within this abstract. The authors acknowledge support by SMART, a
center funded by nCORE, a SRC program sponsored by NIST. The authors also acknowledge
support by DARPA under Grant W911NF-17-1-0100, MINT at Minnesota, and the NSF
XSEDE through Allocation No. TG-ECS200001. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/2cd20e06470fac11a685d745715202a5.png) 
Fig. 1. (a,b) Numerical solutions of the magnon amplitudes c0 and ck during and after microwave excitation. Color-coding corresponds to |Ψ0|. (c,d) Corresponding experimental data of c0, showing the power dependence of the oscillations and scattering reversals. Data is normalized to the turn-on peak at 46 dB. 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/e93ab1523191603144783a3632406752.png) 
Fig. 2. (a,b) Measured frequency spectra of the oscillations, at various powers Pa relative to Posc, for FMR frequencies of 1.5 GHz (left column) and 2.5 GHz (right column). (c,d) the
corresponding simulated magnon amplitudes at powers below and above the onset of broadening, showing the different higher-order scattering processes at 1.5 GHz and 2.5 GHz.

Scattering Reversals and Higher Order Interactions in Three

Rare-earth iron garnets, a ferrimagnetic insulator, have received attention for their many properties including small damping and tunable magnetization and angular momentum compensation temperatures. Most research efforts have been focused on garnet thin films grown on the (111) plane because of their perpendicular magnetic anisotropy (PMA), and the in-plane anisotropy, however neglected, can be important for our understanding of the switching behavior and the energetics of magnetic textures. In-plane anisotropy can be conveniently explored in garnet thin films on (110) planes. Here we combine a theoretical and experimental analysis of the anisotropy landscape for Europium iron garnet (EuIG) epitaxially grown on (110) Gadolinium Gallium garnet (GGG) using pulsed laser deposition. 

The angular dependence of anisotropy is calculated by combining the individual contributions including magnetocrystalline anisotropy, magnetoelastic anisotropy, shape anisotropy and growth-induced anisotropy, which may contribute for non-ideal stoichiometry. This predicts an easy axis out of plane along [110], with hard and intermediate axes in plane along [001] and [1-10] respectively. The anisotropy landscape is probed with vibrating sample magnetometry (VSM) and spin-Hall magnetoresistance (SMR) measurements. The SMR result is simulated and fitted to a Stoner-Wohlfarth model. The experimental measurements yield the expected hard and easy axes but the model requires additional multi-domain consideration to capture the hysteresis behavior when the external field is swept along the intermediate in-plane axis. Understanding the in-plane anisotropy effect, especially the unexpected hysteresis in in-plane direction for a PMA film could have help with the interpretation of the switching behavior. 
**References:** 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/63c6aed2fe471bacc1d3440ce0a33059.png) 
Fig. 1 (a) VSM measurement along the in-plane easy (IP 35, [1-10]), in-plane hard (-55, [001]) and out-of-plane (OP, [110]) axes. (b) Left: Principal anisotropy axis definition in EuIG and setup for Stoner Wohlfarth simulation. Right: 3D polar plot for anisotropy landscape in EuIG. (c) SMR measurement for field swept along the hard anisotropy axis, with fit (red). (d) SMR measurement for field swept along the intermediate anisotropy axis.

Anisotropy Landscape Analysis in (110) Rare earth Iron Garnet Films

Enormous research activities to develop advanced superparamagnetic iron oxide nanoparticles (SPIONPs) enabling to generate higher AC heat induction temperature (TAC) or specific loss power (SLP) at the biologically safe and physiologically tolerable range of AC magnetic field (HAC) (HAC,safe < 1.8 × 109 A m-1 s-1, fappl < 120 kHz, Happl < 190 Oe (12.5 A m-1)) have been intensively made in the past two decades for magnetic hyperthermia application [1-3]. According to the reports made so far, a high magnetic softness (or high magnetic moment m) and a high initial susceptibility (χ0) at the DC magnetic field were revealed to be critical parameters in enhancing the TAC (SLP) due to the correspondingly enhanced Néel relaxation time constant (τN), which is the same as the effective relaxation time constant (τeff), τN ≈ τeff , in solid-state SPIONPs. However, MnFe2O4 SPIONPs with high magnetic moment usually show low TAC (SLP) compared to magnetically semi-hard CoFe2O4 SPIONPs with a relatively low m [4,5]. 
In this work, our research efforts primarily focused on exploring the physical and chemical reasons why MnFe2O4 SPIONPs with high DC magnetic softness show an extremely low TAC (SLP) by systematically investigating the physical role of magnetically semi-hard Co2+ cation addition in enhancing TAC (SLP) of solid (CoxMn1-x)Fe2O4 SPIONPs. According to the experimentally and theoretically analyzed results, it was clearly demonstrated that the enhancement of magnetic anisotropy (Ku)-dependent AC magnetic softness including the Néel relaxation time constant τN (≈ τeff), and its dependent out-of-phase susceptibility (χ″) are the most dominant parameters to enhance the TAC (SLP). This clarified result strongly suggests that the development of new design and synthesis methods enabling to significantly enhance the Ku by improving the crystalline, shape, stress (magnetoelastic), thermally-induced, and exchange anisotropies is the most critical to enhance the TAC (SLP) of SPIONPs at the HAC,safe (particularly at the lower fappl < 120 kHz) for clinically safe magnetic nanoparticle hyperthermia. 
**References:** [1] Joshi et al, J. Phys. Chem. C, 113, 17761-17767 (2009) 
[2] Lee et al, Nat. Nanotechnol., 6, 418-422 (2011) 
[3] Sathya et al, Chem. Mater., 28 1769-1780 (2016) 
[4] Mameli et al, Nanoscale, 8 10124-10137 (2016) 
[5] Albino et al, J. Phys. Chem. C, 123, 6148-6157 (2019)

The Role of Co2+ Cation Addition in Enhancing the AC Heat Induction Power of (CoxMn1 x)Fe2O4 Superparamagnetic Nanoparticles

The structurally coupled interaction between a piezoelectric and ferromagnetic magnetostrictive material can be exploited to develop magnetoelectric systems which have potential to rival current technologies with reduced power consumption and size. In a magnetoelectric multiferroic system the bilateral control of the magnetic and piezoelectric response is dictated by the material properties, where a soft magnetic material with large magnetostriction facilitates the large voltage-induced strain in the piezo/ferroelectric layer. Thin film deposition techniques utilized for device fabrication often complicate the optimization of the interaction between layers due to inherent stresses occurring during deposition which are often deleterious to the magnetoelectric functionality. While incorporation of metalloid substitutions, such as in (Fe0.5Co0.5)1-xCx, and (Fe0.5Co0.5)1-xBx are able to reduce the inherent coercive field and retain a high magnetostrictive and piezomagnetic properties the as-deposited stress states are rarely discussed [1,2]. To this end we have studied the incorporation of Hafnium in to an Fe50Co50 matrix with a focus on correlation between the film stress and magnetic properties. In these sputter-deposited thin films we find a relationship among the magnetic and stress properties as a function of doping percentage. With increasing Hf composition the system exhibits a minimum coercive field occurring at a crossover from a compressive to tensile stress state. Scanning electron microscopy (SEM) and X-ray diffraction (XRD) of the (Fe0.5Co0.5)1-xHfx system reveal the magnetic softening is also correlated to emergence of an amorphous phase with reduced grain size for these sputter-deposited films. This ternary alloy presents an additional avenue for optimization of artificial magnetoelectric systems and has applications for multiple types of microelectromechanical devices. 
**References:** [1] J. Wang, Phys. Rev. Applied 12, 034011 (2019) 
[2] J. Lou, Appl. Phys. Lett. 91, 182504 (2007)

Development of a Hafnium Iron

High-performance magneto-dielectrics with high permeability and permittivity, low magnetic and dielectric losses, and high cutoff frequency are promising candidate materials for next-generation communication applications, owing to their significant advantages for antenna miniaturization and bandwidth enhancement. Due to limited resonant frequencies and large natural resonance linewidths, conventional microwave ferrites and their composites, which are usually used over the frequency range from several hundred MHz to several GHz, do not meet the requirements for 5G communications.1-3 A unique type of hexaferrites named Mg-Zn 18H hexaferrite, whose crystal structure can be interpreted as the structure of Y-type hexaferrites with intercalation of a three-layered hexagonal barium titanate block into the middle of the T-block as shown in Fig. 1a, has been synthesized, and its microwave properties have been comprehensively investigated for the first time.4 The Mg-Zn 18H hexaferrite shows an extremely low damping coefficient of 0.1-0.2, indicating concentrated magnetic loss within a narrow frequency band and a narrow ferrimagnetic resonance linewidth of 486-660 Oe. Owing to this remarkably low damping coefficient, the Mg-Zn 18H hexaferrite possesses an excellent magnetic loss tangent as low as 0.06 and a small dielectric loss tangent less than 0.006 over 2-4 GHz, representing superior magneto-dielectric performance among the reported microwave ferrites for the S band applications as shown in Fig. 1b. Moreover, excellent thermal stability for the damping coefficient is observed to be 0.0004 K-1 over the temperature variation of most practical applications. When employed as the substrate of a 3.6-GHz patch antenna, the Mg-Zn 18H hexaferrite shows a miniaturization factor of ~5 and significant bandwidth improvement of ~50-110% over conventional dielectrics. These results imply the great technological potential and commercial value of the 18H hexaferrites for 5G communications. 
**References:** 1V. G. Harris, IEEE Trans. Mag. 48, 1075 (2012). 
2V. G. Harris, A. Geiler, Y. Chen, S. Yoon, M. Wu, A. Yang, Z. Chen, P. He, P. V. Parimi, X. Zuo, C. E. Patton, M. Abe, O. Acher, and C. Vittoria, J. Magn. Magn. Mater. 321, 2035 (2009). 
3Q. Li, Y. Chen, C. Yu, K. Qian, V. G. Harris, J. Am. Ceram. Soc. 103, 5076 (2020). 
4Q. Li, Y. Chen, C. Yu, L, Young, J. Spector, and V. G. Harris, Acta Mater. 231, 117854 (2022). 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/b2b57c67782800adf088a5c437a9d3be.png) 
(a) Schematic illustration of the Mg 18H hexaferrite unit cell with the stacking subunit blocks denoted. (b) Performance of magneto-dielectric materials for microwave applications. The figure of merit (FOM) are indicated by the dashed lines.

Emerging Magneto Dielectric Materials for 5G Antenna Miniaturization: 18H Hexaferrites

Tunable perpendicular magnetic anisotropy (PMA) in thin film magnetic devices provides an isotropic in-plane permeability, necessary for integrated microwave and mm-wave devices and an opportunity to break inversion symmetry that allows implementations of non-reciprocal devices like circulators and isolators. In ferrimagnetic alloys like GdxCo1-x, this is usually achieved by changing the composition such that the magnetic properties lie on either side of the composition-dependent magnetic compensation at room temperature, as shown in Fig. 1. Different compositions achieved in this work by changing the sputtering power of the Gd target show that magnetic compensation window lies between x = 28-32, and the effective anisotropy diverges about the compensation point. 

One less explored pathway is the incorporation of oxygen, which until now has been ruled out as a detriment to the ferrimagnetic properties [1]. Here, we show that preferential oxygenation of GdxCo1-x through a reactive sputtering process, such that the cation (Gd and Co) to oxygen anion ratio is 1:1, increases the effective anisotropy energy density by two orders of magnitude compared to the corresponding alloys without oxygen. X-ray photoelectron spectroscopy reveals that strong PMA with an optimal oxygen flow results in a Co-deficient stoichiometry of Gd21Co28O51, which suggests that strong ferrimagnetic order likely arises from a superexchange-like coupling between Gd and Co via the non-magnetic O [2]. This can produce a spontaneous magnetization in the out-of-plane direction due to weaker exchange interactions within the alloy in a heavy-metal/ferrimagnet heterostructure for PMA. Even greater anisotropy fields on the order of 11 kOe are achieved in a super-lattice with 10 repetitions of the GdxCo1-x heterostructure unit-cell, which corresponds to an operating frequency of 30 GHz. Therefore, not only does oxygenation increase PMA and provide tunable anisotropy to amorphous Gd-Co films, but the oxygen incorporation itself also increases the resistivity of the film, making it even more suitable for that frequency range. 
**References:** [1] K. Ueda, A. J. Tan, and G. S. D. Beach, AIP Advances, vol. 8, no. 12, p. 125204, Dec. 2018. 
[2] R. J. Gambino and J. J. Cuomo, Journal of Vacuum Science and Technology, vol. 15, no. 2, pp. 296–301, Mar. 1978. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/0641d84a745d89c9edb1bbc17e16653a.png) 
Fig. 1. Saturation Magnetization and Effective Anisotropy Energy Density for varying GdxCo1-x compositions for x = 18-39.

Role of Composition and Oxygen in Ferrimagnetic GdxCo1 x Alloys for Microwave/mm

Grain Pattern Media Recording (GPMR) technology is proposed as a candidate to replace Bit Pattern Media (BPM). GPMR media fabrication process shows great promise to resolve the technical and manufacturing issues associated with conventional BPM. Rather than a highly aggressive subtractive process with thick masking layers to pattern magnetic bits, GPMR employs a bottom-up new approach where thin (2nm) and narrow (3nm) radial line pattern is fabricated on top of the interlayer, then FePt granular alloy is deposited using conventional sputter deposition. This Line pattern acts as nucleation constraints to force the newly deposited FePt grains to grow on the interlayer along the radial direction to ensure alignment of grain edges. Fig. 1 shows a illustration of the GPMR magnetic configuration and radial lines. To ensure grains only grow on the interlayer and no grains grow on top of segregant lines, the segregant lines needs to be controlled to be less than ~3 nm. We proposed to use simple sidewall patterning approach which combines nano-imprint lithography with dry plasma etch technique to create < 3 nm width dielectric segregant lines for fabricating GPMR media. Two sidewall processes with different mandrel materials have been developed to successfully fabricate < 3 nm SiO2 lines on MTO surfaces on the disks. The segreant line width is determined by the deposition thickness of SiO2, whereas the segregant line height of 1-3 nm is determined by the CF4 etch process. Fig. 2A shows the sidewall process flow and the top-down SEM image with inserted TEM cross sectional image of 2.8 nm linewidth radial segregant SiO2 line patterns on MTO (pitch= 32 nm). Fig. 2B shows that TEM image of 2 nm FePt grains was deposited on 3 nm SiO2 lines on MTO at 650C (pitch= 160 nm). This is the first successful experimental demonstration of phase segregation of ordered FePt granular media using this newly invented bottoms up approach to form patterned media. 
**References:** [1] T. R. Albrecht et al., in IEEE Transactions on Magnetics, vol. 51, no. 5, pp. 1-42, May 2015, TMAG.2015 
[2] D. S. Kuo et al., 2016 International Conference of Asian Union of Magnetics Societies (ICAUMS), 2016, pp. 1-1, 2016. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/c237bc59556c85d1959b85819b7ccbf5.png) 
Fig. 1 The proposed design with single-grain-row-per-line pitch design for GPMR media 
![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/47e2756cb182410a4b4f2f277fb1a1f8.png) 
Fig. 2(a) The sidewall process and SEM image of 2.8 nm linewidth SiO2 lines on MTO (pitch= 32 nm). Fig. 2(b) shows 2 nm FePt grains deposited on 3 nm SiO2 lines

Fabrication of 3 nm Width Dielectric Segregant Lines for Radially Ordered Grain Pattern Media Recording

Reaching the full potential of spintronics requires modeling to predict the results of spintronic experiments, optimize structures, and design systems. Currently, the gold standard for modeling spintronic structures is micromagnetic simulations. Unfortunately, these simulations can be extremely slow and struggle with modeling accurately the imperfections
and noise inherent to experimental devices. In this work, we introduce an alternative approach: design models for spintronic devices by letting an artificial intelligence automatically
generate a model based on examples of the device behavior. For this, we make use of the recently-proposed formalism of Neural Ordinary Differential Equations (Neural ODE) [1], which
we adapt to the specificities of spintronics [2]. An appropriately-trained Neural ODE can simulate a skyrmion-based device 200 times faster than micromagnetic simulations while being
highly accurate, even in complex situations and for multiple values of materials parameters (magnetic anisotropy, DMI). We also train a Neural ODE based on five milliseconds of
experimental measurements of vortex-based spin-transfer nano oscillators. The Neural ODE could then predict weeks of experimental measurements. We show how to adapt the Neural
ODE, an originally deterministic formalism, to incorporate noise and show that the predictions of the Neural ODE match the experiments very precisely. All these results highlight the
potential of Neural Ordinary Differential Equations for developing spintronic applications. 
**References** 
[1] R. T. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, "Neural Ordinary Differential Equations", Advances in Neural Information Processing Systems, 6571–6583
(2018).
[2] X. Chen, F. A. Araujo, M. Riou, J. Torrejon, D. Ravelosona, W. Kang, W. Zhao, J. Grollier, D. Querlioz, "Forecasting the outcome of spintronic experiments with Neural Ordinary
Differential Equations", Nature Communications 13, 1016 (2022).

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