Within this talk we show the importance of accurate implementations of the RKKY interactions for antiferromagnetically coupled ferromagnetic layers with thicknesses exceeding the exchange length. In order to evaluate the performance of different implementations of RKKY interaction, we develop a benchmark problem by deriving the analytical formula for the saturation field of two infinitely thick magnetic layers that are antiparallelly coupled (Fig. 1). Without external field, the magnetization in the layers is parallel to the easy axis (y-direction) in the +y and -y direction. When a field in the x-direction is applied, a partial domain wall is formed (Fig. 1). We find that there exists a critical field that fully saturates the structure in the x-direction which is given by, Hx,crit = 2 K1/Js [1+Jrkky2/(AK1)], where Jrkky is the RKKY coupling strength, K1 the anisotropy constant and A exchange constant [1]. This benchmark problem shows that state-of-the-art implementations in commonly used finite-difference codes lead to errors of the saturation field that amount to more than 20% for mesh sizes of 2 nm which is well below the exchange length of the material. In order to improve the accuracy, we develop higher order cell based (see Fig. 2 - paper, cell O1 and O2) and nodal based finite-difference codes (see Fig. 2 - paper, node) that significantly reduce the error compared to state-of-the-art implementations (see Fig. 2 - mumax) [2]. For the second order cell based and first order nodal based finite element approach, the error of the saturation field is reduced by about a factor of 10 (2% error) for the same mesh size of 2 nm. 

**References** 
[1] D. Suess, S. Koraltan, F. Slanovc, F. Bruckner, C. Abert, "Accurate finite-difference micromagnetics of magnets including RKKY interaction -- analytical solution and
comparison to standard micromagnetic codes", https://arxiv.org/abs/2206.11063 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5b0630c6640f4a9ee05ac6fd7125be33.png) 

Fig. 1: Analytic benchmark problem of two infinitely extended layers, that are antiferromagentically coupled via Jrkky.

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/5afd4b7f7745e1b47296276e4f0a8a2e.png) 

Fig. 2: Convergence towards the analytic solution of Hx =5 x 2 K1/Js for different micromagnetic codes as function of discretization length dz. It is shown that standard finite difference
codes (e.g. mumax) that uses zero order methods for the RKKY interaction show significant errors. The developed method (paper,cell O2) shows significantly better convergence.

Accurate finite difference micromagnetics of magnets including RKKY interaction

Soft magnetic composites(SMCs) consist of ferromagnetic particles, in the micrometer-range which are embedded in an electrically insulating material, in order to reduce the eddy-currents in the high-frequency regime. We want to utilize numerical simulations in order to optimize such materials in terms of power-loss and permeability. The problem one faces is that the samples have macroscopic sizes, however if one would want to resolve these materials on the micro-scale, it would limit the total simulation size to a few micrometers. Introducing a truncated geometry or cut-off would also not solve this problem as demagnetization effects would come into play for any finite sized sample. However due to the torus-like shape of the sample, the magnetic flux closure reduces the strayfield to zero. To overcome these issues we developed and implemented our own method for calculating the micromagnetic strayfield with periodic boundary conditions(PBCs)[1]. PBCs allow us to simulate an infinite sample and therefore avoid the demagnetization effects. A complete magnetic simulation framework including PBCs will be presented with which the spatially- and time-resolved magnetization response to an external field is simulated. The model will be validated among other things by reproducing experimental setups that have successfully reduced the power-loss of SMCs. Suetsuna et al.[2] have shown that the power-loss reduces by roughly half when the sample is exposed to a magnetic field during annealing, because an aligned anisotropy direction is induced. This was replicated in the simulation and as can be seen in figure 1 the simulated power-loss Pm decreases by half when an aligned anisotropy is applied, without altering the relative permeability mur. 

**References** 
[1] Bruckner, Florian, et al. "Strayfield calculation for micromagnetic simulations using true periodic boundary conditions." Scientific Reports 11.1 (2021): 1-8.
[2] Suetsuna, Tomohiro, et al. "Soft magnetic composite containing magnetic flakes with in-plane uniaxial magnetic anisotropy." Journal of Magnetism and Magnetic Materials 473 (2019):
416-421. 

![](https://s3.eu-west-1.amazonaws.com/underline.prod/uploads/markdown_image/1/image/1c066203e20a987f28c178a793a3e69b.png) 

Fig. 1 Powerloss Pm and permeability mur were calculated from hysteresis simulations utilizing the SMC model with aligned and random anisotropy.

Simulation of Soft Magnetic Composites with Periodic Strayfield Calculation

The automotion of spin textures, where geometrical gradients are exploited to propagate localized magnetic states without the need of external stimuli, is a promising phenomenon for fast, low-power transmission of magnetic information, as previously demonstrated in planar nanowire systems [1]. Here, we show that automotion is an attractive mechanism in 3D nanowire circuits for the vertical interconnectivity of functional magnetic planes, alternative to the standard current-based spintronic effects normally employed to move spin textures. To showcase this new approach, we have prototyped 3D spiral nanowires by a combination of 3D-printing Focused Electron Beam Induced Deposition and physical vapor deposition methods [2, 3]. The magnetic devices (Fig. 1) are made of Permalloy and are 3 Î¼m tall, 150 nm wide, presenting curvature gradients of 0.09 Î¼m-2. Due to the high directionality of the physical deposition process, they have average thickness gradients of -5.3nm/Î¼m; this very large thickness gradient results to be the dominating mechanism responsible for 3D domain wall automotion, as micromagnetic simulations indicate. We have directly imaged the automotion of domain walls in the 3D spirals using â€œshadowâ€ X-ray photoelectron microscopy [4]. In these experiments, we nucleate a pair of domain walls at different locations within the spirals using external magnetic fields, and directly observe the evolution of the system as the field is reduced until reaching zero value (Fig. 2). The experiments show the successful automotion of the walls along the 3D interconnectors, and its competition with the intrinsic pinning present in the devices at localized regions. This work thus proposes and successfully demonstrates the automotion of spin textures as a viable effect to create 3D interconnected magnetic devices [5].

Domain Wall Automotion for Three Dimensional Magnetic Interconnectivity

The extent of magnetic ordering in highly-frustrated thermally active artificial kagome spin ice is limited by the lowest achievable blocking temperature, as the moments freeze before ordering is achieved [1]. A more substantial degree of magnetic ordering can be achieved by either lowering the blocking temperature of the individual nanomagnets or by increasing the critical transition temperature of the system. First, we pursue an approach of introducing interfacial Dzyaloshinskii-Moriya interactions (DMI), which lowers the blocking temperature by a factor of five, while keeping the transition temperature unchanged [2]. We demonstrate that a seven-ring kagome structure consisting of 30 nanomagnets with DMI can be thermally annealed into its ground state. Furthermore, the spin-ice correlations extracted from extended kagome lattices are found to exhibit the quantitative signatures of long-range charge-order, thereby giving experimental evidence for the theoretically predicted continuous transition to a charge-ordered state. Second, we find that slight modifications of the nanomagnets at the vertices in the artificial kagome ice allow us to control the energy landscape which leads to an increase of the critical temperatures [3]. These results provide paths for the tuning of magnetic ordering in artificial spin systems.

Long-range Magnetic Ordering in Artificial Kagome Spin Ice

Dipolar Spin Ices (DSI) are geometrically frustrated magnetic materials, where the moments are arranged on the sites of a pyrochlore lattice [1]. The ground-state obeys the ice-rule, where two magnetic moments point to the center of the tetrahedral unit cell and two point out. If the rule is broken, i.e. one moment is switched, a net magnetic charge is observed at the center of the tetrahedron, referred to as magnetic monopoles. The trace of the switched magnetic moments is referred to as a Dirac string, which has magnetic monopoles at its ends. However, these materials require cryogenic conditions for the stability of monopoles. To study the magnetic monopoles on more suitable platform, artificial spin ices (ASI) consisting of elongated magnetic islands were considered [2]. Their 2D nature lifts the degeneracy of the ground state, as not all ice-rule obeying vertices have equal energies. Due to this differentiation between energetic states of the vertices, the Dirac strings bind the monopoles and store energy, limiting the mobility within the lattice. Here, we combine the advantages of DSI and 2D ASI, and present a 3D ASI lattice [3], where 3D ellipsoids with nearly perfect ising behavior are arranged on an Ice Ih crystal, where the angle between the ellipsoids is always θ = 109.5°, see Fig. 1. This symmetry recovers the lifted degeneracy of the ground state and enables tension-free Dirac strings. Based on the definitions of unbound magnetic monopoles [4], we show that the mobility threshold for magnetic charges is by 2 eV lower than their unbinding energy, while the propagation barrier is constant. The energy of the system does not increase with the length of the Dirac string, showing its tension-free nature. Furthermore, we apply external global fields and steer one magnetic monopole in the desired direction without creating additional magnetic charges, see Fig. 2. Finite temperature micromagnetic analysis reveals that the magnetic monopoles within the 3D ASI lattice are stable and steerable at room temperature.

![](https://assets.underline.io/uploads/markdown_image/1/image/7329a0465c413b41f6b95412aa66de49.jpg)

Fig.1: 3DASI lattice with 3D and top view.

![](https://assets.underline.io/uploads/markdown_image/1/image/ebc78830ac4545aedb5992bd9d0f1124.jpg)

Fig. 2: Averaged magnetization and applied field with respect to time, and the schematic magnetic configurations before and after propagation of monopole.

Tension-free Dirac Strings and Steered Magnetic Charges in 3D Artificial Spin Ice

Skyrmions and antiskyrmions are topologically protected spin structures with opposite vorticities. Particularly in coexisting phases, these two types of magnetic quasiparticles may show fascinating physics and potential for spintronic devices. While skyrmions are observed in a wide range of materials until now antiskyrmions were exclusive to materials with D2d symmetry [1]. In our recent work, we have shown first and second-order antiskyrmions stabilized by magnetic dipole-dipole interaction in Fe/Gd-based multilayers [2]. Using Lorentz transmission electron microscopy imaging, we observed coexisting first and second-order antiskyrmions, Bloch skyrmions, and type-2 bubbles (Fig. 1) and determine the range of material properties and magnetic fields where the different spin objects form and dissipate. Phase diagrams of the spin objects were created in dependence on the magnetic out-of-plane field, temperature, saturation magnetization, and uniaxial magnetic anisotropy. We performed micromagnetic simulations to obtain more insight into the studied system and conclude that low values of saturation magnetization and uniaxial magnetic anisotropy lead to the existence of this zoo of different spin objects and that they are primarily stabilized by dipole-dipole interaction. Further, we investigated the nucleation process of antiskyrmions experimentally and theoretically revealing the necessity of a crossing point of three magnetic stripe domains to form an isolated antiskyrmion with an out-of-plane magnetic field. The previous unobserved second-order antiskyrmions and the disclosed coexistence of antiskyrmions and skyrmions potentially reachable for a whole range of different materials provide great potential for further studies on quasi-particle interactions, spin dynamics as well as for possible future applications in spintronics.

![](https://assets.underline.io/uploads/markdown_image/1/image/50c16dd6974021e442524cdd7ed4e0a3.jpg)

Fig. 1. Underfocused LTEM image of [Fe/Ir/Gd]2/[Fe/Gd]76/[Fe/Ir/Gd]2 showing five different coexisting spin objects (a). Bloch skyrmion with counter- (b) and with clockwise rotation (c), type-2 bubble (d), first-order antiskyrmion (e), and second-order antiskyrmion (f). Zoomed-in spin objects (b-f, I) and simulated LTEM contrast (b-f, II) caused by the theoretical spin textures (b-f, III).

Dipolar-Stabilized First and Second-Order Antiskyrmions in Ferrimagnetic Multilayers

In this work, we present an analytical solution of the magnetic field for a uniformly magnetized cylinder tile (= cylinder volume element, see Fig. 1). The derivation is based on the method of magnetic surface charge densities for arbitrary Cartesian magnetization directions and cylinder tile geometries. In contrast to previous works [1,2], all resulting expressions have been reduced to analytical forms containing only elementary operations, trigonometric functions and elliptic integrals. All special cases (singularities and indefinite expressions) are discussed, allowing an exact evaluation of the magnetic field (up to numerical errors) at any point in space. The results are validated by comparison with finite elements solutions and numerical integration. It is further demonstrated how vectorized implementation of all special functions enables effective computational times in few tens of microseconds on simple mobile CPUs (e.g. Intel i5-8365U). An efficient implementation has been integrated in version 4 of the open-source Python-package Magpylib [3]. Possible applications of the derivations are the analytical study of the Halbach cylinder [4] (see Fig. 2), rotary encoder geometries [5,6], electrical machines [7], or any other magnetic geometry consisting at least partially of cylinder tiles. Furthermore, since a full cylinder can be completely subdivided into cylinder tiles, it allows the study of demagnetization effects while maintaining the rotational symmetry using the magnetostatic moment method [8].

![](https://assets.underline.io/uploads/markdown_image/1/image/bc3e0d7162c81197735b0ceeae63edec.jpg)

Geometry of cylinder tile with parametrization in cylindrical coordinates. Six surfaces contribute to the magnetic field.

![](https://assets.underline.io/uploads/markdown_image/1/image/3b7aea7898e7ab1db5d67b69b6e94d45.jpg)

a) Cross-section through a Halbach cylinder array consisting of twelve cylinder tiles with different magnetization directions. The magnetic field is evaluated along the dashed blue line. b) Field for different numbers of subdivided cylinder tiles as a function of the radial coordinate r.

Full Analytical Solution for the Magnetic Field of Uniformly Magnetized Cylinder Tiles

Structural inversion symmetry breaking in low-dimensional magnetic systems determines their electronic and magnetic properties at interfaces [1,2]. Asymmetrically sandwiched magnetic films can provide strong perpendicular magnetic anisotropy and Dzyaloshinskii-Moriya interactions (DMI), which is necessary for prospective memory and logic devices based on chiral non-collinear magnetic textures, e.g. skyrmions [3,4], skyrmion bubbles and chiral domain walls (DWs) [5]. The device performance is determined by the static and dynamic micromagnetic parameters [6,7]. In particular, the speed of a DW-based racetrack memory is defined by both the strength of the external driving, e.g. magnetic field or spin-polarized current, and internal magnetic parameters, e.g. the DMI constant and damping parameter [6,7]. The necessity of having a strong DMI in asymmetrically sandwiched magnetic structures requires the utilization of ultrathin (in the range of 1 nm) magnetic films, which implies the polycrystalinity and compromized structural quality of the layer stack. Structural imperfections in addition to the spin-pumping mechanism [8,9], that arises due to the proximity of a ferromagnetic material with a heavy-metal, lead to a substantial enhancement of the magnetic damping parameter of ultrathin films compared to bulk. Accessing this parameter typically requires dynamic experiments on the motion of DWs in confined geometries, which are usually done in the creep regime due to the pronounced pinning. Here, we demonstrate both experimentally and theoretically the presence of tilted DWs in statics in perpendicularly magnetized asymmetric //CrOx/Co/Pt layer stacks with surface-induced DMI, Fig. 1. We show that in such systems there are two possible theoretical mechanism for the appearance of titled DWs: (I) A unidirectional tilt could appear in equilibrium as a result of the competition between the DMI and additional in-plane easy-axis anisotropy, which breaks the symmetry of the magnetic texture and introduce tilts [10]. (II) A static DW tilt could appear due to the spatial variation of magnetic parameters, which introduce pinning centers for DWs. A moving DW can be trapped in a tilted state after the external driving field is off. Based on these theoretical approaches, we perform a statistical analysis of the DW tilt angles obtained in staticts after the external magnetic field used for the sample demagnetization was off. We found that the second approach corresponds better to the experimental observations and allows to determine self-consistently the range of DW damping parameters and DMI constants for the particular layer stack. Using two reference fields, which provide two characteristic tilt angles, allow us to retrieve the range of DMI strength D ≥ 0.8 mJ/m2 and DW damping parameters α ≥ 0.1. The upper limit for the DMI constant agrees with an independent transport-based measurement giving D = 0.90±0.13 mJ/m2, which further refines our estimate of the damping parameter α = 0.13±0.02. This value lies in a typical DW damping range for the Co-based asymmetrical layer stacks, that are obtained from dynamic experiments [11,12]. Thus, the combination of the proposed method with standard metrological techniques opens up opportunities for the quantification of both static and dynamic micromagnetic parameters based on static measurements of the DW morphology.

**References:**

[1] A. Fert, N. Reyren, and V. Cros, Nature Reviews Materials 2, 17031 (2017). [2] R. Wiesendanger, Nature Reviews Materials 1, 16044 (2016). [3] A. N. Bogdanov and D. A. Yablonskii, Zh. Eksp. Teor. Fiz. 95, 178 (1989). [4] S. Woo, K. Litzius, B. Krüger, et al., Nature Materials 15, 501 (2016). [5] S. Emori, U. Bauer, S.-M. Ahn, et al., Nature Materials 12, 611 (2013). [6] C. Garg, S.-H. Yang, T. Phung, et al., Science Advances 3, e1602804 (2017). [7] S. Parkin and S.-H. Yang, Nature Nanotechnology 10, 195 (2015). [8] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, et al., Reviews of Modern Physics 77, 1375 (2005). [9] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Physical Review Letters 101, 037207 (2008). [10] O. V. Pylypovskyi, V. P. Kravchuk, O. M. Volkov, et al., Journal of Physics D: Applied Physics 53, 395003 (2020). [11] J.-M. L. Beaujour, J. H. Lee, A. D. Kent, et al., Physical Review B 74 (2006). [12] A. J. Schellekens, L. Deen, D. Wang, et al., Applied Physics Letters 102, 082405 (2013).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/8589560e9d8ad6beae1b337b7a324f0f.jpg)
Fig. 1. (a) Schematic illustration of a //CrOx/Co/Pt asymmetric multilayer nanostripe and the corresponding magnetic contrast with two tilted DWs. The angle between the DW plane and the y-axis is the DW mechanical tilt angle, χ, while the magnetization inside the DW is canted by the angle, ψ with respect to y-axis. (b) Phase diagram of equilibrium DW morphologies for a ferromagnetic stripe with the width 200 nm extracted from X-ray magnetic circular dichroism photoelectron emission microscopy images after the AC demagnitization. (c) The corresponding distribution of DW counts vs. the tilt angle χ.

 Domain wall damping in ultrathin nanostripes with Dzyaloshinskii-Moriya interaction

Bringing nanomagnetism into the third dimension is of growing interest due to many advantages that three-dimensional (3D) structures provide. The introduction of 3D geometrical effects such as curvature and chirality1, as well as the high density and enhanced device connectivity, lead to many opportunities for new physics and applications2,3. When we combine this promising field of 3D nanomagnetism with the well-established field of spintronics, a number of proposals ranging from the ultra-high-density 3D racetrack memory4 to 3D interconnected memristors for neuromorphic computing5 appear. However, before we can fully exploit the promise of 3D spintronics, significant advances in both the integration of 3D nanomagnets into 2D microelectronic circuits，and the understanding of the influence of a 3D geometry on the magnetotransport properties, are required. The implementation of 3D nanomagnetic circuits faces great challenges regarding the fabrication of a 3D nanostructure and its interconnectivity to the electronic components. Until recently, most magnetotransport experiments have concerned cylindrical nanowires, with complex micromanipulations needed to electrically contact them6. In this work, we employ 3D nano-printing capabilities provided by Focused Electron Beam Induced Deposition (FEBID)7, to directly fabricate a high-quality 3D Cobalt nanobridge with well-defined leads on pre-synthesized electrical contacts (Figure 1a). The SEM image shown in Figure 1b demonstrates the successful integration of this complex, high-aspect-ratio 3D nanostructure onto lithographically patterned electrodes. This printed 3D nanomagnetic circuit allows four-probe transport measurements with which we can probe the magnetotransport properties of the 3D nanostructure - and determine the influence of the three-dimensionality on these properties. To fully characterise the magnetotransport (MT) properties of the 3D magnetic nanobridge, we measured MT hysteresis loops for fields applied in different directions in three-dimensions (Figure 2a, b). These magnetotransport measurements, in combination with macrospin and finite element simulations, reveal that the three-dimensional structure of the nanobridge directly affects the magnetotransport in several ways. Firstly, the non-collinear current path in the 3D geometry results in a complex superposition of different MT effects, leading to an unusual angular dependence of well-known effects such as the Anomalous Hall effect. Secondly, we also identify a significant angular-dependence of the magnetoelectric signal due to the magnon magnetoresistance. This strong angular dependence occurs due to the strong influence of the demagnetising field within the non-collinear regions of the 3D geometry, highlighting the importance of magnetostatics in 3D nanostructures8. These new insights into the influence of a 3D geometry on the magnetotransport effects reported in this work provide the basis for the future study of new spintronic effects in 3D magnetic nanostructures, as well as the realization of 3D spintronic technologies. This methodology shown here combining FEBID 3D printing and standard planar lithography can be extended to more complicated geometries and other materials, opening the door to the fundamental study of new phenomena that exploit the interplay between 3D geometry and magnetotransport and may find applications in the future of areas such as magnonics, frustrated magnetic systems, and magnetic neural networks.

**References:**

 (1) R. Streubel, P. Fischer and F. Kronast, J Phys D Appl Phys., Vol. 49, p.363001(2016) (2) A. Fernández-Pacheco, R. Streubel and O. Fruchart, Nat Commun., Vol. 8, p.15756 (2017) (3) P. Fischer, D. Sanz-Hernández and R. Streubel, Apl Mater., Vol. 8, p.15756 (2017) (4) S. Parkin, M. Hayashi and L. Thomas, Science., Vol. 320, p.190-194 (2008) (5) A. May, M. Hunt and A. Berg, Commun Phys., Vol. 2, p.13 (2019) (6) M. Schöbitz, A. Riz and S. Martin, Phys Rev Lett ., Vol. 123, p.217201 (2019) (7) L. Skoric, D. Sanz-Hernández and F. Meng, Nano Lett ., Vol. 20, p.184-191 (2019) (8) F. Meng, C. Donnelly and C. Abert, Arxiv., arXiv:2011.09199 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/f82793d4777c5e628b8b37e46d66e2d7.jpg)
Fig.1 (a) 3D nano-printing of the bridge via Focused Electron Beam Induced Deposition (FEBID) on pre-patterned electrical contacts. (b) The SEM image of the as-fabricated nanomagnetic circuit. (c) Four-probe magnetotransport measurement configuration. (d) The SEM image of the top view of the nanobridge.

![](https://assets.underline.io/uploads/markdown_image/1/image/ba14c1bac15d9b830f10e775d14b93fe.jpg)
Fig.2 (a) The schematic shows how the field was applied relative to the 3D nanobridge, θ is the angle between the applied field and the substrate (b) Magnetotransport hysteresis loop obtained from -4 T to 4 T and 4 T to -4 T for each field angle θ, from θ = 0° to θ = 90° (from violet to red).

 Non-planar geometrical effects on the magnetoelectrical signal in a 3D nanomagnetic circuit

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Presentations

Accurate finite difference micromagnetics of magnets including RKKY interaction

Simulation of Soft Magnetic Composites with Periodic Strayfield Calculation

Domain Wall Automotion for Three Dimensional Magnetic Interconnectivity

Long-range Magnetic Ordering in Artificial Kagome Spin Ice

Tension-free Dirac Strings and Steered Magnetic Charges in 3D Artificial Spin Ice

Dipolar-Stabilized First and Second-Order Antiskyrmions in Ferrimagnetic Multilayers

Full Analytical Solution for the Magnetic Field of Uniformly Magnetized Cylinder Tiles

Domain wall damping in ultrathin nanostripes with Dzyaloshinskii-Moriya interaction

Non-planar geometrical effects on the magnetoelectrical signal in a 3D nanomagnetic circuit

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