⬇ Download Markdown  ·  ⬇ Download LaTeX  ·  ⬇ original  ·  view original  ·  CSV: table 1

arXiv:2411.18194v4 [cond-mat.mes-hall] 19 Aug 2025

mumax+ : extensible GPU-accelerated micromagnetics and beyond Lars Moreels 1 , Ian Lateur 1 , Diego De Gusem 1 , Jeroen Mulkers 1 , Jonathan Maes 1 , Milorad V. Miloˇsevi´c 2 , Jonathan Leliaert 1* , Bartel Van Waeyenberge 1 1 DyNaMat, Department of Solid State Sciences, Ghent University, 9000 Ghent, Belgium. 2 NANOlab Center of Excellence & Department of Physics, University of Antwerp, 2020 Antwerp, Belgium.

Abstract

+
We present mumax , an extensible GPU-accelerated micromagnetic simulator
with a Python user interface, to address the challenges posed by current mag-
netism research into systems with complex magnetic ordering and interfaces.
It is a general solver for the space- and time-dependent evolution of the mag-
netization and related vector quantities, using finite difference discretization.
Here, we present its application and design and discuss features not available in
3
mumax , such as the modeling of antiferromagnets with magnetoelastic coupling.
+
As an illustration of its capabilities, we use mumax to simulate state of the
art magnetic systems. Specifically, we demonstrate the current induced domain
wall motion in a polycrystalline antiferromagnet, we simulate the working prin-
ciple of a strain-driven antiferromagnetic racetrack memory and we reproduce
experimentally observed domain structures in a non-collinear antiferromagnet.
Keywords: antiferromagnetism, ferrimagnetism, ferromagnetism, finite difference,
GPU, magnetoelasticity, micromagnetism, non-collinear antiferromagnetism, simulation

*Corresponding author:

*Corresponding author:
[email protected]
1

1 Introduction

Micromagnetic calculations have been an indispensable tool for the development of the field of modern magnetism. Especially the availability of open-source [1–7] and GPU-accelerated software packages [5, 8–12] boosted research by enabling efficient numerical studies of nano- and micrometer-sized magnetic systems, complementary to experimental investigations. More recently, research efforts have shifted to more com- plex magnetic systems such as antiferromagnets [13] and 2D magnetic heterostructures (Van der Waals systems) [14]. Different ways of manipulating the magnetization are being explored as well, e.g. by applying strain [15] or electric fields [16]. In order to model these modern materials, software tools must go beyond the stan- dard micromagnetic theory; preferably, they should be easily extensible to include other physical phenomena of interest. With this in mind, we designed mumax+ , a ver- satile GPU-accelerated micromagnetic simulation package. It calculates the space- and time-evolution of nano- to microscale magnets using a finite difference scheme. The program has been designed independently of mumax3 [5] and features a Python [17] based user interface. This allows users to directly use common packages such as NumPy [18], SciPy [19] and Matplotlib [20] to process and view their data directly. The Python module wraps the main part of the code, which is written in C++ [21] and CUDA [22]. The latter allows calculations to be performed in parallel on a GPU, reducing simulation time compared to CPU-based codes. The extensible framework of mumax+ is in stark contrast to the design of mumax3 . This extensibility provides a plethora of new possibilities, many of which are brought together in the implementation of (non-collinear) antiferromagnets with full support for magnetoelastic interactions. Therefore, in this paper we will demonstrate the appli- cation of these two features. It will place emphasis on the practical and contemporary importance in the research field of modern magnetism. We examine the domain wall motion in a polycrystalline antiferromagnet under the influence of an applied current. Furthermore, we show the working principle of a strain-driven magnetic racetrack in NiO and, lastly, reproduce nitrogen-vacancy mag- netometry images of the non-collinear antiferromagnet Mn3 S n. In the final section, we provide a brief description of the design of the software package along with an overview of the implemented physics. The development of mumax+ is motivated by the “plus” mindset, i.e. versatility and extensibility, combined with user-friendliness, in favor of performance and speed. Our intention is not to replace mumax3 , but rather to offer a platform where additional functionalities can be easily developed and used. All capabilities of the code have been thoroughly tested against the micromagnetic standard problems [23], mumax3 [5] and analytical results. These tests will not be repeated here, but can be found freely in the GitHub repository (https://github.com/ mumax/plus), along with the complete open-source code under the GPLv3 license. Details about dependencies, installation instructions and the API can also be found in the repository. 2

2 Demonstrations

This section is devoted to the showcasing of the implementation of antiferromagnets and elastodynamics, which is not present in mumax3 [5, 24]. More details about the used methods and implementations, as well as some key differences with mumax3 , are explained in section 3.

2.1 Current-driven domain wall mobility in a polycrystalline

Understanding the mobility properties of domain walls in various materials has been an important field of study for the development of future spintronic devices [25– 28]. We can use mumax+ to investigate such dynamics in antiferromagnets. Here, we study the current-induced motion of a domain wall in an antiferromagnetic strip with polycrystalline structure. The domain wall is stabilized by interfacial DMI and can be moved by applying a spin-polarized current in the direction of the strip, effectively causing a spin transfer torque as formulated by Slonczewski [29]. Following mumax3 , we are able to define multi-grain geometries inside a material by generating a Voronoi tessellation. The ferromagnetic and antiferromagnetic exchange constants are then reduced at the different grain boundaries [30]. We simulate a strip with a width of 200 nm and a thickness of 10 nm. The strip is discretized into cells of size 2×2×10 nm3 and we define grains with an average diameter of 10 nm. This simulation setup is illustrated in Fig. 1a and described in more detail in the caption of Fig. 1. We monitor the velocity of the domain wall in a time frame of 0.5 ns for different values of the current density and for different reductions of the exchange constants at the various grain boundaries. Results can be found in Fig. 1b. We see a distinct decrease in the slope of the velocity curves for decreasing values of the intergrain exchange along with a stronger domain wall pinning at lower current densities. These results are very similar to those reported in Refs. [30, 32], where similar simulations based on permalloy have been performed. The dashed curve in Fig. 1b represents the model derived in Ref. [31] for the ideal case without grains. Our simulation data for a 0 % reduction of the exchange constants, effectively disregarding any grain boundaries, agree very well with this model.

2.2 Strain-driven antiferromagnetic racetrack memory

Previous studies have demonstrated the possibility to move domain walls in both ferromagnets and antiferromagnets through the application of surface acoustic waves or strain gradients [33–35]. Building on this concept, we use mumax+ to investigate the controlled movement of domain walls in antiferromagnetic NiO under the influence of strain caused by the application of traction. The simulation consists of 512×128×1 cubic cells with a side length of 3 nm. The magnet has an easy anisotropy axis along the length axis of the magnet. This stabilizes two out-of-plane domain walls, separating domains in which the N´eel vector lies along the easy axis. Four small notches are introduced from 300 nm onward with 150 nm spacing, which serve as pinning sites for 3

(a) 0 100 200 ⊗⊙ 0 100 200 300 400 500 x (nm) 0 100 200 y ( n m ) (b) 0.0 0.5 1.0 1.5 Current density J (108 A/cm2 ) 0.0 0.5 1.0 1.5 V e l o c i t y v ( k m / s ) Model 0 % 50 % 75 % 90 % Fig. 1: Simulation setup (a) and mobility curves (b) of a polycrystalline antiferromag- net. Each sublattice has a uniaxial out-of-plane anisotropy with a strength of 64 kJ/m3 , a saturation magnetization of 400 kA/m, interfacial DMI strength of 0.11 mJ/m2 and a damping constant of 0.1 [31]. The homogeneous antiferromagnetic exchange con- stant is set at −50 pJ/m. The top panel in (a) shows the relaxed N´eel vector of a polycrystalline antiferro- magnetic strip. The domain wall, in which the N´eel vector lies along the positive x direction, separates two out-of-plane domains. The ferromagnetic and inhomoge- neous antiferromagnetic exchange constants, 10 pJ/m and −10 pJ/m respectively, are reduced by 75 % at the grain boundaries. The bottom panel in (a) shows the different grains inside the material, generated using a Voronoi tessellation [30] with an average grain diameter of 10 nm. Panel (b) shows the simulated domain wall velocity in an antiferromagnetic strip under the influence of an applied current density, with each curve corresponding to a different percentage reduction in exchange constants at the grain boundaries. The dashed curve shows the analytical model derived in Ref. [31] for the monocrystalline case. the domain walls. The material parameters used in the simulation can be found in Table 1. Initially two domain walls are placed on the first and third notch. In order to move both domain walls one notch along the strip, we generate an elastic out-of-plane shear pulse by applying one period of a sinusoidal out-of-plane shear traction wave with a frequency of 8 GHz and an amplitude of 0.3 GPa at a boundary. The resulting dynamics are shown in Fig. 2. The induced strain, approximately 0.14 % in amplitude, detaches the domain wall from the first notch via the Villari effect [36] and transports it along the length axis of the magnet. The elastic pulse, traveling with a velocity of about 4 km/s, overtakes the domain wall which gets pinned at the second notch. The pulse then detaches the second domain wall which is transported to the final notch in the same manner. This effectively simulates the underlying working principle of a traction-mediated magnetic racetrack [37]. This concept can be extended to various domains with varying sizes in a single antiferromagnetic strip where each domain corresponds to a series of 0 or 1 bits. The domain walls can be shifted one notch at a time by the controlled application of traction pulses. 4

Table 1: The NiO simulation parameters used for the strain-driven
domain wall motion discussed in Section 2.2, which apply to both sub-
lattices. The magnetic parameters are based on Refs. [38–40], with the
exception of the interfacial DMI, which is chosen to stabilize the domain
wall. The (magneto)elastic parameters are taken from Ref. [41], aside from
the stiffness coefficient of the Rayleigh damping. The latter is chosen to
yield a damping ratio of about 5 % for shear waves with a wavelength of
24 nm.
Magnetic parameters (Magneto)elastic parameters
3

Magnetic parameters Saturation magnetization MS 425 kA/m Ferromagnetic exchange constant A1 1 ∼ 4.4 pJ/m Homogeneous antiferro- magnetic exchange A0 ∼ −46 pJ/m Lattice constant a 0.42 nm Uniaxial anisotropy 85.7 kJ/m3 Gilbert damping α 2.1 × 10− 4 Interfacial DMI 0.7 mJ/m2

(Magneto)elastic parameters First magnetoelatic coupling constant B1 −15 MJ/m3 Second magnetoelastic coupling constant B2 −8.5 MJ/m3 Mass density ρ 6853 kg/m3 Stiffness tensor component C1 1 330 GPa Stiffness tensor component C1 2 60 GPa Stiffness tensor component C4 4 110 GPa Stiffness coefficient of the Rayleigh damping 0.1 ps

2.3 Stray field imaging of the non-collinear antiferromagnet

Mn 3 Sn

The non-collinear antiferromagnet Mn3 S n is a prototypical material in the research field of antiferromagnetic spintronics [42–45]. Owing to the interplay between exchange, DMI and magnetocrystalline anisotropy, it exhibits a small net magnetic moment [43, 45], which makes direct imaging of magnetic structures more evident than in collinear antiferromagnets [46–48]. Li et al. [47] imaged magnetic domain structures using nitrogen-vacancy (NV) magnetometry measurements of Mn3 S n films. Here, we demonstrate the simulation of the measured magnetostatic field using mumax+ by considering an antiferromagnet with three distinct exchange-coupled sublattices. We define a square 500×500×30 nm3 non-collinear antiferromagnetic film discretized into cells of size 1×1×30 nm3 . Grains with an average diameter of 40 nm are generated using a Voronoi tessellation [30]. Each grain has an easy axis which lies either out-of-plane, perpendicular to the film’s surface, or in a random in-plane direction [47]. Furthermore, the exchange coupling between neighboring grains is reduced to 10% of the intragrain value. To reproduce the NV scans shown in Ref. [47], we calculate the out-of-plane component of the magnet’s stray field 60 nm above the film’s surface. The results, along with the simulation parameters, can be found in Fig. 3. The agreement between Figures 3a and 3b, along with the order of magnitude of the stray field showcases the ability of mumax+ to simulate non-collinear antiferromagnetic systems. 5

0.00 0.35 y ( µ m ) t = 0.07 ns t = 0.18 ns t = 0.22 ns t = 0.26 ns t = 0.34 ns 0.3 0.6 0.9 x (µm) 0.00 0.35 y ( µ m ) 0.3 0.6 0.9 x (µm) 0.3 0.6 0.9 x (µm) 0.3 0.6 0.9 x (µm) 0.3 0.6 0.9 x (µm) O u t - o f - p l a n e d i s p l a c e m e n t ( n m ) 0 0.5 Fig. 2: An elastic wave is generated through the application of a traction pulse at the left edge at 0 µm, and subsequently propagates through the antiferromagnetic NiO film. This wave mobilizes the domain walls, which are initially pinned at the notches at 300 nm and 600 nm, as discussed in Section 2.2. The top row shows the out-of-plane elastic displacement, while the bottom row shows the N´eel vector of the antiferromagnet. The in-plane domains, oriented in the directions indicated by the arrows, are separated by out-of-plane domain walls, shown in black or white, whose N´eel vectors lie into, and out of the plane of the film, respectively. Different snapshots after the introduction of the pulse at t = 0 ns are shown. The domain walls have moved to the second and fourth notches at 450 nm and 750 nm respectively, after passage of the pulse.

3 Methods

The mumax+ module is written in C++ and CUDA for GPU support and can be compiled using either single or double precision. The module is wrapped in Python and can be imported in an input script like any other package. The object-oriented character of Python and C++ allows for a very flexible and transparent use of classes and accompanying instances. For example, each magnetic instance has their own set of material parameters and multiple magnets from different magnet classes can simulta- neously coexist. This contrasts with mumax3 , where the user has to resort to defining a limited number of regions in order to create inhomogeneities in material parame- ters [5]. Other advantages of the use of a Python-based interface are prominent, e.g. the use of NumPy’s [18] intuitive array indexing logic to set material parameters in different subsets of cells. These benefits are illustrated in the various examples in the GitHub repository. In the remainder of this section we outline the physics included in mumax+ with an emphasis on the differences compared to mumax3 . A more detailed description of the different functionalities and features can be found in the API.

3.1 Micromagnetism

For each magnet defined in the simulation world, mumax+ solves the dynamic equation ∂m ∂t = τ , (1) 6

(a) 0 200 400 x (nm) 0 100 200 300 400 500 y ( n m ) −6 −2 0 2 6 B z ( G ) (b) Fig. 3: The out-of-plane component of the stray field of a Mn3 S n thin film, as dis- cussed in section 2.3. Panel (a) shows the calculated field using mumax+ after letting a non-collinear antiferromagnet relax from a random state. Each sublattice has a saturation magne- tization of 1 MA/m, uniaxial anisotropy constant of 5 MJ/m2 (based on Ref. [43]), a Gilbert damping constant of 0.01 and a typical lattice constant of 3.5 ˚A. The ferro- magnetic, inhomogeneous and homogeneous antiferromagnetic exchange constants are 10 pJ/m, −15 pJ/m and −25 pJ/m respectively. The homogeneous DMI vector is set at 10 MJ/m3 in the direction perpendicular to the film, while the interfacial DMI has a value of 3.5 mJ/m2 [47]. Panel (b) shows an NV measurement image of a Mn3 S n thin film with a thickness of 30 nm, measured ∼ 60 nm above the film’s surface. The scale bar denotes 0.5 µm. Reprinted (adapted) with permission from Li, S. et al. (2023). Nanoscale Magnetic Domains in Polycrystalline Mn3Sn Films Imaged by a Scanning Single-Spin Mag- netometer. Nano Letters, 23(11), 5326–5333. https://doi.org/10.1021/acs.nanolett. 3c01523. Copyright 2023 American Chemical Society. where m is the magnetization normalized with respect to the saturation magnetization MS and τ is the sum of the Landau-Lifshitz-Gilbert (LLG) torque [49] and any addi- tional spin-transfer torques as formulated by Zhang and Li [50] or Slonczewski [29], which are implemented in the same way as in mumax3 [5]. Several embedded Runge- Kutta methods with the option to use adaptive time stepping [51–55] are available for time integration. A list can be found in Table 2. The LLG torque is given by τ LLG = − γ 1 + α2 [ m × Be ff + α (m × (m × Be ff) ) ] , (2) with γ the gyromagnetic ratio and α the dimensionless Gilbert damping con- stant [49, 56]. The different field terms which play a role and add up to the effective field Be ff are calculated based on the current magnetization state m ≡ m(r, t). These fields include an external field, the uniaxial or cubic anisotropy field, the exchange field, a demagnetizing field, stray fields from other magnets, the field induced by the Dzyaloshinskii-Moriya interaction, a thermal field and the magnetoelastic field. Of these, the external field, anisotropy fields and thermal field [57] are implemented in 7

the same way as in mumax3 . Their details will not be repeated here, and we refer the interested reader to their thorough description in Ref. [5]. The remainder of this section highlights new features which are not present in mumax3 . While our focus is on ferromagnets and collinear antiferromagnets, the underlying formalism naturally generalizes to ferrimagnets and non-collinear anti- ferromagnets. The former is achieved by considering two different sublattices in an antiferromagnet, while the latter can be done by adding a third sublattice with the same interaction terms as the first two. Table 2: Runge-Kutta solvers in mumax+ together with their convergence order and error estimate order which is used for adaptive time stepping. Method Convergence Error order estimate order Heun (RK12) 2 1 Bogacki-Shampine (RK23) 3 2 Cash-Karp 5 4 Fehlberg (RKF45) 5 4 Dormand-Prince (RK45) 5 4

3.1.1 Exchange field

The exchange field present in an antiferromagnet acting on sublattice s under the influence of the other sublattice s′ is described in Ref. [31] and given by B( s) exch = 2A( s) 1 1 M (s) S ∇2 m( s) + A1 2 M (s) S ∇2 m( s′ ) + 4A0 M (s) S a2 m( s′ ) . (3) The first term in Eq. (3) accounts for the standard ferromagnetic exchange interaction. This is the only term present when simulating ferromagnetic systems and is imple- mented as in mumax3 [5]. The second term contains the inhomogeneous exchange stiffness constant A1 2 < 0 J/m and describes a similar antiferromagnetic exchange interaction between sublattices. The last term accounts for the homogeneous exchange interaction between antiferromagnetically coupled spins within the same simulation cell. This interaction is described by a homogeneous exchange constant A0 < 0 J/m and a lattice parameter a.

3.1.2 Magnetostatic field

The calculation of the magnetostatic field is similar to the method used in OOMMF
[1]. It is done by using the analytical result [58] below a certain cutoff radius and the
asymptotic result [59] above that radius. In the case of antiferromagnets, the vector
sum of the spins within a simulation cell is computed prior to evaluating the resulting
field.
8

3.1.3 Dzyaloshinskii–Moriya interaction and boundary conditions

In mumax+ , the user can set the 9 independent components of the antisymmetric rank- 3 tensor ˆD (Di jk = −Di kj ) of the Dzyaloshinskii–Moriya interaction (DMI), which encompasses the DMI strengths and chiral properties of the considered lattice. Since the DMI tensor is determined by the crystallographic and magnetic symmetries of the material, each sublattice in an antiferromagnet can have its own crystallographic environment and thus its own DMI tensor. Furthermore, a third DMI tensor is used to describe the intersublattice interaction. Generally, the energy density due to DMI in an antiferromagnet can be written as [60, 61] E = s,s′ i,j,k D( s,s′ ) ijk m( s) j ∂i m ( s′ ) k , (4) where the indices (i, j, k) are spatial, (s, s′ ) denote the different sublattices and it is assumed that D( s,s′ ) ijk = D( s′ ,s) ijk . The ferromagnetic expression is obtained by considering s = s′ . The tensorial character of the DMI also comes into play when calculating the Neumann boundary conditions. Given the structure of the exchange field in Eq. (3), the boundary conditions for a sublattice s in an antiferromagnet are given by [31, 60] 2A( s) 1 1 ∂n m ( s) = A1 2 ∂n m ( s′ ) × m( s) × m( s) + ˆD( s) · n ⊗ m( s) , (5) where n is the surface normal and a term proportional to the inhomogeneous exchange constant A1 2 takes the interaction with the other sublattice into account. This term is not present when considering ferromagnets. The Neumann boundary conditions as given in Eq. (5) are the default in mumax+ . Alternatively, open or periodic boundary conditions can be used. The cho- sen boundary conditions will not only affect the DMI field calculations, but also the ferromagnetic and antiferromagnetic inhomogeneous exchange calculations. Periodic boundary conditions also affect the calculation of the magnetostatic fields. Apart from the inhomogeneous interaction described by Eq. (4), mumax+ also considers a homogeneous DMI between spins within the same simulation cell, given by the energy density [62, 63] E = d · m( s) × m( s′ ) , (6) where s and s′ denote two distinct sublattices and d is the DMI vector. The latter lies parallel to the principal axis of the material [62]. 9

3.2 Elasticity

Apart from the magnetodynamic equation (1), mumax+ can also solve the second order elastodynamic equation for cubic crystal symmetries [64, 65] ρ ∂2 u ∂t2 = ft ot , (7) with u the elastic displacement, ρ the mass density and ft ot the total body force density acting on the magnet. The implementation is mostly based on a previous extension of mumax3 [66], except that the elastic force fe l does not use second order differences of the elastic displacement. Rather, fe l is computed as the numeric divergence of the stress ˆσ, while the strain is calculated with the numeric gradient of the displacement. In this way, traction and traction-free boundary conditions [67] were easily implemented. These impose ˆσ · n = t at the boundary, where n is the normal to the boundary and t is the applied traction. Both spatial derivative calculations utilize a fourth order accurate central difference scheme [68] in the bulk material. Lower order schemes are used close to and at the boundaries. A second difference between the mumax3 extension described in Ref. [66] and our implementation, is the addition of viscous damping, where an extra term, proportional to the strain rate, is added to the total stress tensor [69–71]. This tends to dampen high frequency oscillations [72].

3.3 Magnetoelasticity

Magnetoelasticity for ferromagnets with cubic crystal symmetry is implemented in the same way as described in Ref. [66]. It is the combination of magnetostriction [36] and the Villari effect or inverse magnetostriction [36]. In the former the magnetization affects the elastic displacement by applying a magnetoelastic force given by fm el = 2B1 mx ∂ x m x m y ∂y my mz ∂z mz + B2 mx (∂y my + ∂z mz ) + my ∂y mx + mz ∂z mx m y (∂x m x + ∂z mz ) + mx ∂ x m y + mz ∂z my mz (∂x m x + ∂y my ) + mx ∂ x m z + my ∂y mz , (8) while in the Villari effect the elastic strain affects the magnetization by generating an effective field given by Bm el = − 2 MS B 1 εx xm x ε y y my εz z mz + B2 εx y my + εx z mz ε x y mx + εy z mz εx z mx + εy z my . (9) In both Eq. (8) and Eq. (9) B1 and B2 are the magnetoelastic coupling constants and ˆε is the strain tensor. For antiferromagnets, magnetoelasticity is implemented similarly to how it is reported in Ref. [73]. Both sublattices have a shared elastic displacement, velocity, stiffness, strain and body force, while each ferromagnetic sublattice s has its own magnetoelastic coupling constants B( s) 1 and B( s) 2 , magnetization m( s) and saturation 10

magnetization M (s) S . Each sublattice then creates a magnetoelastic body force f (s) mel, which is added to the total body force. Conversely, each sublattice magnetization is affected by a unique magnetoelastic field B( s) mel created by its own magnetization m( s) and the shared strain.

4 Conclusions and outlook

As a demonstration of the capabilities of mumax+ that are not present in mumax3 , we first presented the current-driven domain wall mobility in a polycrystalline antiferro- magnet. Furthermore, the addition of magnetoelastics has provided a way to simulate the working principle of a racetrack memory in NiO driven by strain through the appli- cation of traction. Finally, we were able to reproduce nitrogen-vacancy magnetometry images in the non-collinear antiferromagnet Mn3 S n. We presented the micromagnetic model implemented in mumax+ and highlighted the key differences from its predecessor mumax3 . These advancements include a Python-based interface, support for calculations involving (non-collinear) antiferro- magnets and ferrimagnets, and the integration of magnetoelasticity across all magnet types in the package. The development of mumax+ is ongoing and more functionalities are expected to be added in future releases. Some possibilities include adaptive time stepping appro- priate for magnetoelastic simulations, a more general stiffness tensor and piezoelectric coupling. Since mumax+ is open-source and built with an object-oriented design for easy extensibility, anyone can fork the repository (https://github.com/mumax/plus) and enhance the module with additional features.

Acknowledgments

We acknowledge financial support from the SHAPEme project (EOS ID 400077525)
from the FWO and F.R.S.-FNRS under the Excellence of Science (EOS) program.
J.L. was supported by the Fonds Wetenschappelijk Onderzoek (FWO-Vlaanderen)
with postdoctoral fellowship No. 12W7622N.
We would like to thank Oleh Kozynets for his software engineering additions to
the code.

References

[1] Donahue, M.: OOMMF User’s Guide, Version 1.0. - 6376, National Institute of
Standards and Technology, Gaithersburg, MD (1999). https://doi.org/10.6028/
NIST.IR.6376
[2] Vansteenkiste, A., Van de Wiele, B.: MuMax: A new high-performance micro-
magnetic simulation tool. Journal of Magnetism and Magnetic Materials 323(21),
2585–2591 (2011) https://doi.org/10.1016/j.jmmm.2011.05.037
[3] Vansteenkiste, A., Van de Wiele, B., Dvornik, M., Lassalle-Balier, R., Rowlands,
G.: mumax2 (2012). https://github.com/mumax/2
11
[4] Scholz, W., Fidler, J., Schrefl, T., Suess, D., Dittrich, R., Forster, H., Tsiantos,
V.: Scalable parallel micromagnetic solvers for magnetic nanostructures. Com-
putational Materials Science 28(2), 366–383 (2003) https://doi.org/10.1016/
S0927-0256(03)00119-8 . Proceedings of the Symposium on Software Development
for Process and Materials Design
[5] Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia-Sanchez, F.,
Van Waeyenberge, B.: The design and verification of MuMax3. AIP Advances
4(10), 107133 (2014) https://doi.org/10.1063/1.4899186
[6] Fischbacher, T., Franchin, M., Bordignon, G., Fangohr, H.: A systematic approach
to multiphysics extensions of finite-element-based micromagnetic simulations:
Nmag. IEEE Transactions on Magnetics 43(6), 2896–2898 (2007) https://doi.
org/10.1109/TMAG.2007.893843
[7] MicroMagnetics Team: Magnum.FE: Finite-Element Micromagnetic Simulation.
https://github.com/micromagnetics/magnum.fe. Accessed: 2024-12-04 (2024)
[8] Chang, R., Li, S., Lubarda, M.V., Livshitz, B., Lomakin, V.: FastMag: Fast micro-
magnetic simulator for complex magnetic structures (invited). Journal of Applied
Physics 109(7), 07–358 (2011) https://doi.org/10.1063/1.3563081
[9] Kakay, A., Westphal, E., Hertel, R.: Speedup of FEM Micromagnetic Simulations
With Graphical Processing Units. IEEE Transactions on Magnetics 46(6), 2303–
2306 (2010) https://doi.org/10.1109/TMAG.2010.2048016
[10] Lepadatu, S.: Boris computational spintronics—High performance multi-mesh
magnetic and spin transport modeling software. Journal of Applied Physics
128(24), 243902 (2020) https://doi.org/10.1063/5.0024382
[11] Drews, A., Selke, G., Krueger, B., Abert, C., Gerhardt, T.: MicroMag-
num: Micromagnetic Simulation Framework. https://github.com/MicroMagnum/
MicroMagnum. Accessed: 2024-12-04 (2024)
[12] MicroMagnum Team: Magnum.FD: Finite-Difference Micromagnetic Simulation.
https://github.com/micromagnetics/magnum.fd. Forked from MicroMagnum,
Accessed: 2024-12-04 (2024)
[13] Baltz, V., Manchon, A., Tsoi, M., Moriyama, T., Ono, T., Tserkovnyak, Y.: Anti-
ferromagnetic spintronics. Rev. Mod. Phys. 90, 015005 (2018) https://doi.org/
10.1103/RevModPhys.90.015005
[14] Geim, A.K., Grigorieva, I.V.: Van der Waals heterostructures. Nature 499(7459),
419–425 (2013) https://doi.org/10.1038/nature12385 . Publisher: Nature Pub-
lishing Group. Accessed 2024-12-02
12

[15] Bandyopadhyay, S.: Perspective: There is plenty of room for magnetic strain- tronics in the analog domain. npj Spintronics 2(1), 1–15 (2024) https://doi. org/10.1038/s44306-024-00018-3 . Publisher: Nature Publishing Group. Accessed 2024-12-02 [16] Bandyopadhyay, S., Atulasimha, J., Barman, A.: Magnetic straintronics: Manip- ulating the magnetization of magnetostrictive nanomagnets with strain for energy-efficient applications. Applied Physics Reviews 8(4), 041323 (2021) https: //doi.org/10.1063/5.0062993 . Accessed 2024-12-05 [17] Van Rossum, G., Drake, F.L.: Python 3 Reference Manual. CreateSpace, Scotts Valley, CA (2009) [18] Harris, C.R., Millman, K.J., Walt, S.J., Gommers, R., Virtanen, P., Cournapeau, D., Wieser, E., Taylor, J., Berg, S., Smith, N.J., Kern, R., Picus, M., Hoyer, S., Kerkwijk, M.H., Brett, M., Haldane, A., R´ıo, J.F., Wiebe, M., Peterson, P., G´erard-Marchant, P., Sheppard, K., Reddy, T., Weckesser, W., Abbasi, H., Gohlke, C., Oliphant, T.E.: Array programming with NumPy. Nature 585(7825), 357–362 (2020) https://doi.org/10.1038/s41586-020-2649-2 [19] Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Courna- peau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, ˙I., Feng, Y., Moore, E.W., Vander- Plas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P., SciPy 1.0 Contributors: SciPy 1.0: Fundamental Algorithms for Scientific Com- puting in Python. Nature Methods 17, 261–272 (2020) https://doi.org/10.1038/ s41592-019-0686-2 [20] Hunter, J.D.: Matplotlib: A 2d graphics environment. Computing in Science & Engineering 9(3), 90–95 (2007) https://doi.org/10.1109/MCSE.2007.55 [21] ISO/IEC: ISO/IEC 14882:2024 - Programming Languages – C++. ISO/IEC Standard (2020). https://www.iso.org/standard/83626.html [22] NVIDIA Corporation: NVIDIA CUDA C Programming Guide. Version 12.6 (2024) [23] µMAG organization: µMAG Micromagnetic Modeling Activity Group (2021). https://www.ctcms.nist.gov/∼ r dm/mumag.org.html Accessed 2024-11-25 [24] Joos, J.J., Bassirian, P., Gypens, P., Mulkers, J., Litzius, K., Van Waeyen- berge, B., Leliaert, J.: Tutorial: Simulating modern magnetic material systems in mumax3. Journal of Applied Physics 134(17) (2023) [25] Allwood, D.A., Xiong, G., Faulkner, C.C., Atkinson, D., 13

Petit, D., Cowburn, R.P.: Magnetic domain-wall logic. Science
309(5741), 1688–1692 (2005) https://doi.org/10.1126/science.1108813
https://www.science.org/doi/pdf/10.1126/science.1108813
[26] Baltz, V., Manchon, A., Tsoi, M., Moriyama, T., Ono, T., Tserkovnyak, Y.: Anti-
ferromagnetic spintronics 90(1), 015005 https://doi.org/10.1103/RevModPhys.
90.015005 . Publisher: American Physical Society. Accessed 2025-05-19
[27] Nomoto, T., Arita, R.: Cluster multipole dynamics in noncollinear antiferromag-
nets 2(1), 012045 https://doi.org/10.1103/PhysRevResearch.2.012045 . Accessed
2025-05-14
[28] Leliaert, J., Wiele, B., Vansteenkiste, A., Laurson, L., Durin, G., Dupr´e, L.,
Van Waeyenberge, B.: Influence of material defects on current-driven vortex
domain wall mobility. Phys. Rev. B 89, 064419 (2014) https://doi.org/10.1103/
PhysRevB.89.064419
[29] Slonczewski, J.C.: Current-driven excitation of magnetic multilayers. Journal of
Magnetism and Magnetic Materials 159(1), 1–7 (1996) https://doi.org/10.1016/
0304-8853(96)00062-5
[30] Leliaert, J., Wiele, B., Vansteenkiste, A., Laurson, L., Durin,
G., Dupr´e, L., Van Waeyenberge, B.: Current-driven domain
wall mobility in polycrystalline permalloy nanowires: A numer-
ical study. Journal of Applied Physics 115(23), 233903 (2014)
https://doi.org/10.1063/1.4883297 https://pubs.aip.org/aip/jap/article-
pdf/doi/10.1063/1.4883297/15138111/233903 1 online.pdf
[31] S´anchez-Tejerina, L., Puliafito, V., Khalili Amiri, P., Carpentieri, M., Finocchio,
G.: Dynamics of domain-wall motion driven by spin-orbit torque in antiferromag-
nets. Phys. Rev. B 101, 014433 (2020) https://doi.org/10.1103/PhysRevB.101.
014433
[32] Leliaert, J., Wiele, B., Vansteenkiste, A., Laurson, L., Durin, G., Dupr´e, L.,
Van Waeyenberge, B.: A numerical approach to incorporate intrinsic material
defects in micromagnetic simulations. Journal of Applied Physics 115(17), 17–102
(2014) https://doi.org/10.1063/1.4854956 https://pubs.aip.org/aip/jap/article-
pdf/doi/10.1063/1.4854956/15135090/17d102 1 online.pdf
[33] Chen, F., Ge, X., Luo, W., Xing, R., Liang, S., Yang, X., You, L., Xiong, R.,
Otani, Y., Zhang, Y.: Strain-induced megahertz oscillation and stable velocity
of an antiferromagnetic domain wall. Phys. Rev. Appl. 15, 014030 (2021) https:
//doi.org/10.1103/PhysRevApplied.15.014030
[34] Yu, G., Shi, S., Peng, R., Guo, R., Qiu, Y., Wu, G., Li, Y., Zhu, M., Zhou, H.:
Strain-driven magnetic domain wall dynamics controlled by voltage in multifer-
roic heterostructures. Journal of Magnetism and Magnetic Materials 552, 169229
14
(2022) https://doi.org/10.1016/j.jmmm.2022.169229
[35] Yu, G., He, X., Qiu, Y., Wu, G., Guo, R., Zhu, M., Zhou,
H.: Dynamics of domain wall induced by voltage-controlled
strain-field gradient. AIP Advances 12(3), 035036 (2022) https:
//doi.org/10.1063/9.0000279 https://pubs.aip.org/aip/adv/article-
pdf/doi/10.1063/9.0000279/16461111/035036 1 online.pdf
[36] Chikazumi, S.: Physics of Ferromagnetism. International Series of Monographs
on Physics. OUP Oxford, Oxford (2009)
[37] Parkin, S., Hayashi, M., Thomas, L.: Magnetic domain-wall racetrack memory.
Science (New York, N.Y.) 320, 190–4 (2008) https://doi.org/10.1126/science.
1145799
[38] K¨odderitzsch, D., Hergert, W., Temmerman, W.M., Szotek, Z., Ernst, A., Winter,
H.: Exchange interactions in nio and at the nio(100) surface. Phys. Rev. B 66,
064434 (2002) https://doi.org/10.1103/PhysRevB.66.064434
[39] Chen, F., Ge, X., Luo, W., Xing, R., Liang, S., Yang, X., You, L., Xiong, R.,
Otani, Y., Zhang, Y.: Strain-induced megahertz oscillation and stable velocity
of an antiferromagnetic domain wall. Phys. Rev. Appl. 15, 014030 (2021) https:
//doi.org/10.1103/PhysRevApplied.15.014030
[40] Kampfrath, T., Sell, A., Klatt, G., Pashkin, A., M¨ahrlein, S., Dekorsy, T.,
Wolf, M., Fiebig, M., Leitenstorfer, A., Huber, R.: Coherent terahertz control
of antiferromagnetic spin waves. Nature Photonics 5(1), 31–34 (2010) https:
//doi.org/10.1038/nphoton.2010.259
[41] Azovtsev, A.V., Pertsev, N.A.: Thz and sub-thz antiferromagnetic magnons via
magnetoacoustic resonances excited by picosecond strain pulses in nio. Phys. Rev.
Mater. 8, 044404 (2024) https://doi.org/10.1103/PhysRevMaterials.8.044404
[42] Wang, X., Yan, H., Zhou, X., Chen, H., Feng, Z., Qin, P., Meng, Z., Liu, L.,
Liu, Z.: Noncollinear mn3sn for antiferromagnetic spintronics 28, 100878 https:
//doi.org/10.1016/j.mtphys.2022.100878 . Accessed 2025-05-20
[43] Lee, W.-B., Hwang, S., Ko, H.-W., Park, B.-G., Lee, K.-J., Choi, G.-M.:
Spin-torque-driven gigahertz magnetization dynamics in the non-collinear anti-
ferromagnet mn3sn 20(4), 487–493 https://doi.org/10.1038/s41565-025-01859-7
. Accessed 2025-05-14
[44] Park, P., Oh, J., Uhl´ıˇrov´a, K., Jackson, J., De´ak, A., Szunyogh, L., Lee, K.H., Cho,
H., Kim, H.-L., Walker, H.C., Adroja, D., Sechovsky´, V., Park, J.-G.: Magnetic
excitations in non-collinear antiferromagnetic weyl semimetal mn3sn 3(1), 1–8
https://doi.org/10.1038/s41535-018-0137-9 . Accessed 2025-06-10
15
[45] Nakatsuji, S., Kiyohara, N., Higo, T.: Large anomalous hall effect in a non-
collinear antiferromagnet at room temperature 527(7577), 212–215 https://doi.
org/10.1038/nature15723 . Accessed 2025-05-13
[46] Reichlova, H., Janda, T., Godinho, J., Markou, A., Kriegner, D., Schlitz, R.,
Zelezny, J., Soban, Z., Bejarano, M., Schultheiss, H., Nemec, P., Jungwirth, T.,
Felser, C., Wunderlich, J., Goennenwein, S.T.B.: Imaging and writing magnetic
domains in the non-collinear antiferromagnet mn3sn 10(1), 5459 https://doi.org/
10.1038/s41467-019-13391-z . Accessed 2025-05-12
[47] Li, S., Huang, M., Lu, H., McLaughlin, N.J., Xiao, Y., Zhou, J., Fullerton, E.E.,
Chen, H., Wang, H., Du, C.R.: Nanoscale magnetic domains in polycrystalline
mn3sn films imaged by a scanning single-spin magnetometer 23(11), 5326–5333
https://doi.org/10.1021/acs.nanolett.3c01523 . Accessed 2025-06-10
[48] Higo, T., Man, H., Gopman, D.B., Wu, L., Koretsune, T., Erve, O.M.J., Kabanov,
Y.P., Rees, D., Li, Y., Suzuki, M.-T., Patankar, S., Ikhlas, M., Chien, C.L., Arita,
R., Shull, R.D., Orenstein, J., Nakatsuji, S.: Large magneto-optical kerr effect
and imaging of magnetic octupole domains in an antiferromagnetic metal 12(2),
73–78 https://doi.org/10.1038/s41566-017-0086-z . Accessed 2025-06-24
[49] Landau, L., Lifshitz, E.: On the theory of the dispersion of magnetic permeability
in ferromagnetic bodies. reproduced in collected papers of ld landau. Pergamon,
New York (1935)
[50] Zhang, S., Li, Z.: Roles of Nonequilibrium Conduction Electrons on the Magne-
tization Dynamics of Ferromagnets. Phys. Rev. Lett. 93, 127204 (2004) https:
//doi.org/10.1103/PhysRevLett.93.127204
[51] Leader, J.J.: Numerical Analysis and Scientific Computation. Pearson Addison
Wesley, Edinburgh (2004)
[52] Bogacki, P., Shampine, L.F.: A 3(2) pair of runge - kutta formulas. Applied Math-
ematics Letters 2(4), 321–325 (1989) https://doi.org/10.1016/0893-9659(89)
90079-7
[53] Cash, J.R., Karp, A.H.: A variable order runge-kutta method for initial value
problems with rapidly varying right-hand sides. ACM Trans. Math. Softw. 16(3),
201–222 (1990) https://doi.org/10.1145/79505.79507
[54] Fehlberg, E., Center, G.C.M.S.F.: Low-order Classical Runge-Kutta Formulas
with Stepsize Control and Their Application to Some Heat Transfer Prob-
lems. NASA technical report. National Aeronautics and Space Administration,
Washington (1969)
[55] Dormand, J.R., Prince, P.J.: A family of embedded runge-kutta formulae. Journal
of Computational and Applied Mathematics 6(1), 19–26 (1980) https://doi.org/
16
10.1016/0771-050X(80)90013-3
[56] Gilbert, T.L.: A Lagrangian formulation of the gyromagnetic equation of the
magnetization field. Phys. Rev. 100, 1243 (1955)
[57] Leliaert, J., Mulkers, J., De Clercq, J., Coene, A., Dvornik, M., Van Waeyenberge,
B.: Adaptively time stepping the stochastic Landau-Lifshitz-Gilbert equation at
nonzero temperature: Implementation and validation in MuMax3. AIP Advances
7(12), 125010 (2017) https://doi.org/10.1063/1.5003957
[58] Newell, A.J., Williams, W., Dunlop, D.J.: A generalization of the demagnetiz-
ing tensor for nonuniform magnetization. Journal of Geophysical Research: Solid
Earth 98(B6), 9551–9555 (1993) https://doi.org/10.1029/93JB00694
[59] Donahue, M.J.: Accurate computation of the demagnetization tensor. 6th Interna-
tional Symposium on Hysteresis Modeling and Micromagnetics (2007). Accessed
2024-11-25
[60] Hals, K.M.D., Everschor-Sitte, K.: New Boundary-Driven Twist States in Systems
with Broken Spatial Inversion Symmetry. Phys. Rev. Lett. 119, 127203 (2017)
https://doi.org/10.1103/PhysRevLett.119.127203
[61] Mahfouzi, F., Kioussis, N.: Bulk generalized Dzyaloshinskii-Moriya interaction in
PT -symmetric antiferromagnets. Phys. Rev. B 106, 220404 (2022) https://doi.
org/10.1103/PhysRevB.106.L220404
[62] Gorobets, O.Y., Gorobets, Y.I.: 3d analytical model of skyrmion-like structures
in an antiferromagnet with dmi. Journal of Magnetism and Magnetic Materials
507, 166800 (2020) https://doi.org/10.1016/j.jmmm.2020.166800
[63] Khoshlahni, R., Qaiumzadeh, A., Bergman, A., Brataas, A.: Ultrafast generation
and dynamics of isolated skyrmions in antiferromagnetic insulators. Phys. Rev.
B 99, 054423 (2019) https://doi.org/10.1103/PhysRevB.99.054423
[64] Graff, K.F.: Wave Motion in Elastic Solids. (1975)
[65] Achenbach, J.D.: Wave Propagation in Elastic Solids. North-Holland Publishing
Company/American Elsevier, Amsterdam (1973)
[66] Vanderveken, F., Mulkers, J., Leliaert, J., Van Waeyenberge, B., Sor´ee, B.,
Zografos, O., Zografos, O., Ciubotaru, F., Adelmann, C.: Finite difference mag-
netoelastic simulator. Open Research Europe 1(35) (2021) https://doi.org/10.
12688/openreseurope.13302.1
[67] Barber, J.R.: Elasticity. Solid Mechanics and Its Applications. Springer, Dor-
drecht (2012)
[68] Sauer, T., Sauer, T.: Numerical Analysis. Always learning. Pearson, London
17
(2012)
[69] Wineman, A.S., Rajagopal, K.R.: Mechanical Response of Polymers: an Intro-
duction. Cambridge university press, Cambridge (2000)
[70] Rajagopal, K.R.: A note on a reappraisal and generalization of the kelvin–voigt
model. Mechanics Research Communications 36(2), 232–235 (2009) https://doi.
org/10.1016/j.mechrescom.2008.09.005
[71] Stachurski, Z.H.: Mechanical behavior of materials. Materials Today 12(3), 44
(2009) https://doi.org/10.1016/S1369-7021(09)70086-0
[72] Semblat, J.F.: Rheological interpretation of rayleigh damping. Journal of Sound
and Vibration 206(5), 741–744 (1997) https://doi.org/10.1006/jsvi.1997.1067
[73] Barra, A., Domann, J., Kim, K.W., Carman, G.: Voltage Control of Antiferromag-
netic Phases at Near-Terahertz Frequencies. Phys. Rev. Appl. 9, 034017 (2018)
https://doi.org/10.1103/PhysRevApplied.9.034017
18