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We predict staggered-field-stabilized collinear, spiral, cone, skrymion crystal (SkX), and square crystal (SC) of vortices and antivortices phases in antiferromagnets (see figure). In SkX and SC phases, magnons experience emergent gauge fields, which leads to magnon-mediated topological spin responses. The underlying physics can be qualitatively understood by considering magnon Landau levels induced by the fictitious magnetic flux. In the long wavelength limit, the unevenly spaced energy bands can be described by the relativistic Klein-Gordon equation. As we propose, these Landau levels can be also realized in antiferromagnetic magnonic topological insulator where the fictitious flux is induced by inhomogeneous DMI. A smoking gun experiment observing the Hofstadter&#39;s butterfly of magnons in AFM could establish the connection between emergent electromagnetism and topological properties of antiferromagnetic magnons.

MMM 2020

Magnon Landau Levels and Topological Spin Responses in Antiferromagnets

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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We predict staggered-field-stabilized collinear, spiral, cone, skrymion crystal (SkX), and square crystal (SC) of vortices and antivortices phases in antiferromagnets (see figure). In SkX and SC phases, magnons experience emergent gauge fields, which leads to magnon-mediated topological spin responses. The underlying physics can be qualitatively understood by considering magnon Landau levels induced by the fictitious magnetic flux. In the long wavelength limit, the unevenly spaced energy bands can be described by the relativistic Klein-Gordon equation. As we propose, these Landau levels can be also realized in antiferromagnetic magnonic topological insulator where the fictitious flux is induced by inhomogeneous DMI. A smoking gun experiment observing the Hofstadter's butterfly of magnons in AFM could establish the connection between emergent electromagnetism and topological properties of antiferromagnetic magnons.

poster

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

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)

Introducing Flatspin: Enabling Large Scale Dynamical Artificial Spin Ice Simulations

Droplet solitons are magnetization fluctuations that preserve their shape as they precess with uniform frequency ω=0. They satisfy a delicate balance between anisotropy and exchange interactions, and decay in the presence of dissipation. To prevent this, a spin polarized current σ can be applied via a nanocontact of radius ρ*. The magnitude of the current can be increased to induce switching between uniform precession at the ferromagnetic resonance frequency (ω=1), and a stable precession at a frequency larger than the Zeeman frequency (ω=0, in zero applied field). In the absence of dissipation, conservative solitons of frequency ω0 are described by a function Θ(ρ;ω0), where Θ is the angle of the magnetization with the easy axis and ρ is the distance to the center of the nanocontact1. We introduce an effective energy ξ that quantifies the work done (against damping and spin torque) to create a fluctuation of arbitrary shape Θ(ρ). We show that, for specific values of σ, some conservative soliton solutions are saddles of ξ. This allows us to calculate activation barriers Δξ between uniform precession at the ferromagnetic resonance and stable solitons. We present results of Δξ as a function of σ for a variety of nanocontact radii ρ* and spin-torque anisotropy parameters ν. We present micromagnetic simulations with a modified Spin-Transfer-Torque extension in OOMMF which uses the effective energy ξ introduced here. In this way, the difference of energies between transition states and metastable configurations can be used to estimate annihilation rates of droplet solitons.References: [1] Gabriel D. Chaves-O’Flynn and D. L. Stein, Phys. Rev. B 101, 184421 (2020) [2] M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010).

Thermal activation barriers for creation and annihilation of magnetic droplet solitons in the presence of spin transfer torque

The magnetodynamic properties of a square thin-film micromagnet (2μm×2μm×16nm) with a flux-closure magnetic ground state were investigated by micromagnetic simulations. The system was excited with an applied magnetic field, displacing the vortex core sufficiently far from its equilibrium position to result in a nonlinear relaxation upon removal of the field. We focus on the edge defects that are created in the process of the vortex core displacement, and carry a half-integer winding number. The edge topological defects are seen to perform quasi-periodic exchange explosions and a resulting emission of short-wavelength spin waves upon travelling back to the equilibrium position. The exchange explosions of the edge topological defects are investigated and explained in terms of vortex/antivortex creation and the formation of a Bloch point. This finding could prove useful to the development of nanoscale devices for periodic generation of high-frequency microwave radiation.References: 1. S. D. Slöetjes, E. Folven, and J. K. Grepstad, Physical Review B Vol. 101, p. 014450 (2020) 2. R. Hertel, and C. M. Schneider, Physical Review Letters Vol. 97, p. 177202 (2006) 3. M. Noske et al., Physical Review B Vol. 91, p. 014414 (2015) 4. S. D. Slöetjes et al., Applied Physics Letters Vol. 112, p. 042401 (2018)

Exchange Explosions of Topological Edge Defects in a Square Micromagnet

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)

Macromagnetic Simulations by Micromagnetic Superposition INVITED

Recently, noncollinear antiferromagnets (AFMs) have become one of the most fascinating classes among AFMs, because they exhibit various peculiar magnetotransport properties include the anomalous Hall effect (AHE).[1] The theoretical study shows that the noncollinear AFM Néel vector can be manipulated by spin current with a realistic current density of 107 A/cm2[2]. The antiperovskite manganese nitride Mn3GaN (MGN) is a good platform to investigate the electrical current manipulation of noncollinear AFMs. In MGN, the Mn atoms form a kagome lattice in the (111) plane. The Neel temperature is reported to be above 380 K[3], which are relatively high for this class of materials. In this presentation, we show the spin-torque switching of noncollinear AFM MGN/Pt bilayers at room temperature[4] as well as AHE. MGN (001) films were prepared on MgO (001) substrates by reactive magnetron sputtering. The AHE versus magnetic field curve for MGN films are shown in Fig. 1. The AHE loop is observed with the same coercive field as the out of plane magnetization curves for MGN films. Therefore, the AHE loops shows the electrical detection of the magnetization for MGN films. The change in the Hall resistivity as a function of current density for MGN(20 nm)/Pt(3 nm) bilayers is shown in Fig. 2. Hall resistivity changes sharply and a clear hysteresis loop is observed at 300 K without an external magnetic field. The change of Hall resistivity is the same as AHE loop shown in Fig. 1, indicating the electrical write–read operation of noncollinear AFM Néel vector. The critical current density is estimated to be 3.6×106 A/cm2, which is one to two orders of magnitude smaller than that for typical FM/HM bilayers[5] and the threshold current for collinear AFMs NiO/Pt spin-torque switching.[6] These results open the pathway to the efficient control of the noncollinear AFM order.References: [1] S. Nakatsuji, N. Kiyohara, and T.Higo, Nature 527, 212-215(2015). [2] Y. Yamane, O. Gomonay, and J. Sinova, Phys. Rev. B 100, 054415 (2019). [3] W. J. Feng, D. Li, Y. F. Dend, and Q. Zhang et al., J. Mater. Sci. 45, 2770 (2010). [4] T. Hajiri, S. Ishino, and K. Matsuura et al., Appl. Phys. Lett. 115, 052403 (2019). [5] L. Liu, O. J. Lee, and T. J. Gudmundsen et al., Phys. Rev. Lett. 109, 096602 (2012). [6] X. Z. Chen, R. Zarzuela, and J. Zhang et al., Phys. Rev. Lett. 120, 207204 (2018).

Electrical switching of Hall resistance in noncollinear antiferromagnetic antiperovskite nitrides

Antiferromagnets (AFMs) are promising spintronic materials compared to ferromagnets, since they potentially enable faster operation, enhanced stability with respect to interfering magnetic fields and higher bit packing density due to the absence of stray fields [1]. However, the use of AFMs in applications requires efficient electrical reading and writing of information. While current-induced switching in NiO/Pt thin film systems has been shown, with various mechanisms proposed [2-4], key information such as the absolute torque strength is missing. Studying MgO//CoO(001)/Pt entails a number of advantages: due to the compressive strain by the substrate, a fourfold in-plane magnetic anisotropy of the CoO layer with two easy axes in the (001) plane is favored and the spin flop field is accessible [5]. Such anisotropy is ideal for applications where the orientation of the Néel order n is read by spin Hall magnetoresistance (SMR) [6,7]. To study current-induced switching, we used 8-arms Hall stars devices with the pulsing arms oriented along the [110] and [-110] easy axes directions (Fig. 1a). Initially n is aligned along [110]. When current pulses are applied along the contacts 3-2 (initial state n ‖ jpulse), the transverse resistance drops after the first pulse, indicating a current-induced 90° n rotation (Fig. 1b). Performing a MR scan with field along 4-1 afterwards (Fig. 1c) yields a field-induced spin flop transition of n back to the initial state (Fig. 1d). By applying current pulses and magnetic fields, we can quantify the current field equivalence of the current-induced torques on the reorientation of n in the CoO, showing that, for the switching of AFMS, currents are much more efficient than magnetic fields.References: [1] V. Baltz et al., Rev. Mod. Phys., 90, 015005 (2018). [2] X. Z. Chen et al., PRL, 120, 207204 (2018). [3] L. Baldrati et al., PRL, 123, 177201 (2019). [4] P. Zhang et al., PRL 123, 247206 (2019). [5] L. Baldrati et al., ArXiv:2003.05923 (2020). [6] H. Nakayama et al., PRL, 110, 206601 (2013). [7] L. Baldrati et al., PRB, 98, 024422 (2018).

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Magnon Landau Levels and Topological Spin Responses in Antiferromagnets

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