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Simulating altermagnets using mumax +
Lars Moreels, 1 Nicolai Bechler, 2 Bartel Van Waeyenberge, 1 Jonathan Leliaert, 1 and Jan Masell 2 1) DyNaMat, Department of Solid State Sciences, Ghent University, 9000 Ghent, Belgium 2) Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology (KIT), 76049 Karlsruhe, Germany
(*Electronic mail: [email protected])
In this paper, we demonstrate how altermagnets can be simulated in the recently released micromagnetic simulation package mumax + . We have added a new magnet class for d-wave altermagnets and demonstrate how mumax + is able to reproduce the analytical solutions for line profiles of the Néel vector and net magnetization for a Bloch domain wall. Next, we show simulation results of the magnon dispersion relation and its dependence on the anisotropic nature of the exchange interaction. Finally, we study the motion of a Néel skyrmion by applying a spin transfer torque. This new feature was implemented by extending the pre-existing code base for antiferromagnetic simulations. The objectoriented design of mumax + allows for a correct calculation of the magnetostatic field in multi-sublattice systems, a feature that many other micromagnetic simulators lack.
Altermagnets 1,2 have recently been identified as a new class of magnetic order, where the breaking of time-reversal symmetry is combined with compensated magnetic moments of antiferromagnetic order. This emerging research field in condensed matter physics has offered many promising results for the use of altermagnets in spintronic and magnonic applications 1,3,4 . Numerical studies of altermagnets often utilize ab initio, atomistic DFT calculations 5–9 or micromagnetic modeling of spin dynamics 10–12 by exchange-coupling two ferromagnetic layers, representing two sublattices. However, when using the latter method, the magnetostatic interactions are calculated incorrectly. This is an important shortcoming, because in contrast to antiferromagnets, stray fields of altermagnetic textures are not always negligible 10,11,13 . This demonstrates the need for updated micromagnetic tools to properly simulate altermagnets. In the current work, we present the implementation, verification, and usage of a new Altermagnet class in the micromagnetic simulation package mumax +14 , a recently released GPU-accelerated finite-difference solver. First, we outline the foundation and implementation of the added physics. Next, we verify its correctness by matching the simulation results to theoretical models for the static domain wall profile and the magnon dispersion relation. Finally, we simulate the motion of a Néel skyrmion under the influence of a spin transfer torque.
Extensibility is the core design feature of mumax +14 , meaning the package is built to facilitate easy addition of new physics in order to keep up with the quickly evolving research field of magnetism. By treating a ferromagnet as an instance from a Ferromagnet class, mumax + is able to couple multiple ferromagnets behind the scenes, effectively creating multi-sublattice systems, such as antiferromagnets, ferrimagnets and non-collinear antiferromagnets containing multiple coinciding ferromagnetic sublattices. Exchange-coupled sublattice spins reside in the same simulation cell, allowing for a correct calculation of the magnetostatic field. This makes mumax + the right platform for the implementation of d-wave altermagnets.
The implementation of altermagnets builds upon the one of antiferromagnets, and contains an additional anisotropic exchange field, which arises from a broken symmetry in the local environment of magnetic sublattice moments 6,10,15 . The resulting spin order has d-wave symmetry in both real and reciprocal space 5 . In a micromagnetic framework, the anisotropic nature of the exchange interaction boils down to a different value of the exchange constant along the different principal axes of the simulation grid. In general, one can define the symmetric, positive definite matrix A to capture the spin stiffness 16 . The exchange field H ex is then calculated with A i j ∂ i ∂ j m (s) , where m (s) is the magnetization field of a single ferromagnetic sublattice s normalized to the saturation magnetization, ∂ i denotes a spatial partial derivative along the principal direction i (i, j = x, y) of the simulation grid and a summation over repeated indices is implied. An important observation is that the basis in which A is diagonal, does not necessarily coincide with the simulation grid directions. Since mumax + calculates derivatives along the latter, one can in general rotate the matrix A from its eigenbasis using a standard rotation matrix R(φ ). Explicitly, the matrix to be used in the calculation of H ex is given by
where A 1 and A 2 are the diagonal elements of A in its eigenbasis (i.e., its eigenvalues). The full exchange field in a d-wave altermagnet is then calculated in mumax + using φ A
i j ∂ i ∂ j m
(s) and the user is able to set the values of A
1 , A 2 and φ as shown in this minimal working example Simulating altermagnets using mumax + 2
from mumaxplus import World, Grid, Altermagnet
The d-wave symmetry of the altermagnetic spin ordering implies that the eigenbasis of A is rotated by 90 ◦ between the two sublattices, thereby interchanging the roles of A 1 and A 2 . This is done automatically behind the scenes. It should also be noted that the matrix in Eq. (1) is diagonal in every orthogonal basis if A 1 = A 2 , as it reduces to the case of isotropic exchange interaction. When the exchange matrix eigenbasis coincides with the simulation grid, φ = 0 and the rotated matrix in Eq. (1) reduces to a diagonal form, leading to an exchange field given by
where e i (i = x, y, z) denote the principal directions along the simulation grid and C is a constant depending on the magnetocrystalline anisotropy, the homogeneous exchange field (between sublattice spins in the same simulation cell) and the exchange matrix in Eq. (1).
To validate our implementation, we create an instance from the Altermagnet class:
from mumaxplus import World, Grid, Altermagnet
The different material parameters in mumax + are defined as class properties 14 . Consequently, they can be set for each created instance individually. Here we choose some prototypical values:
Simulating altermagnets using mumax + 3
FIG. 1: Domain wall profiles of the z-component of the sublattice magnetizations of an altermagnet with uncoupled sublattices.
The double arrows and dashed lines indicate twice the domain wall widths ∆ 1 and ∆ 2 for sublattice 1 and 2 respectively, obtained from fitting the Walker solution Eq. (4). The exchange matrix eigenvalues are chosen as A 1 = 50 pJ/m = 10 A 2 to
clearly visualize the difference in the domain wall widths. Further details are explained in Section II B.
FIG. 2: Profiles of the Néel vector (top) and the net magnetization (bottom) of an altermagnet as modeled by
Gomonay et al. 10 (dashed line) and simulated using mumax + (points) for a Bloch domain wall (ϕ = −π/2) with
width ∆. More details are explained in Section II B.
More precisely, magnons with large momenta k (small wavelengths) are mostly localized on individual sublattices 10 and therefore will experience two different exchange couplings in some directions, leading to a different effective magnon velocity. This lifts the degeneracy usually present in uniaxial antiSimulating altermagnets using mumax + 4 ferromagnets and creates two separate branches in the dispersion relation. Gomonay et al. 10 derived an analytical expression for the magnon frequency ω in terms of the wave vector k starting from a spin Hamiltonian in reciprocal space. This result can also be obtained by linearizing the Landau-LifshitzGilbert equation and imposing a wave-like perturbation on the ground state. We arrived at propagation. The magnons are excited by an external field pulse with a sinc time profile in the middle of the simulation domain.
where the + and − sign correspond to the two different frequency branches. The first term represents the antiferromagnetic magnon frequency with an averaged exchange stiffness
where γ is the gyromagnetic ratio, k the norm of the wave vector k, K is the uniaxial anisotropy constant, and a is a fixed typical lattice constant 14 . The antiferromagnetic exchange stiffness constant A 12 < 0 J/m describes exchange between nearest-neighboring spins from different sublattices, while the antiferromagnetic exchange constant A 0 < 0 J/m describes a homogeneous exchange interaction between sublattice spins in the same simulation cell 14 . An external field H ext lifts the degeneracy by causing branch splitting with a magnitude of ω ext = γ H ext . The anisotropic character of altermagnetism causes an additional splitting, given by
Here, k x and k y are the components of the wave vector k along the principal directions of the simulation grid. The expression derived in Ref. 10 corresponds to Eq. (8) for the case of φ = 45 ◦ . In this case, magnons traveling along one of the principal axes of the simulation grid will experience an averaged exchange stiffness that is the same for both sublattices, which can also be seen by the diagonal elements in Eq. (1). This leads to a degenerate dispersion and no branch splitting. To further validate our implementation of d-wave altermagnets, we use the above dispersion as a second test case. Here, we simulate a 1D altermagnet using mumax + .
from mumaxplus import Altermagnet, Grid, World
We choose the same set of prototypical material parameters as in Section II B, apart from the Gilbert damping constant α: we decrease it to the value of 0.001 to facilitate magnon import numpy as np fmax = 1e13 # maximum excitation frequency T = 20e-12 # total simulation time # Put signal at the center of the simulation box
for sub in magnet.sublattices:
sub.bias_magnetic_field.add_time_term(Bt, mask)
We save the sublattice magnetization states at a sampling rate of 2 f max and perform a spatiotemporal Fourier transform on this data so the results can be analyzed in reciprocal space (k, ω). The resulting dispersion relations are shown in Fig. 3, along with the model discussed above and derived in Ref. 10 . The deviation between the model and the simulation data for large values of |k| (small wavelengths) is to be expected, since a micromagnetic theory no longer forms an adequate description at such length scales. Our simulation data also confirm the prediction of mode degeneracy when φ = 45 ◦ , as shown in the bottom panel of Fig. 3. Fig. 4a shows the frequency gap between both branches at a fixed value of |k| for varying angles φ . Gomonay et al. 10 also derived an expression for the relative magnon intensity between both sublattices along the two different branches. We calculated the same quantity, which is shown in Fig. 4b. The simulation data obtained with mumax + agree very well with the model, with a maximal deviation of approximately 1.5% and 2.5% for the data in Fig. 4a and Fig. 4b respectively.
There have been recent reports of altermagnets exhibiting a skyrmion Hall effect induced by the anisotropic nature of the exchange interaction 11,18–20 when applying a Zhang-Li spin transfer torque 21 . However, Bechler and Masell 22 attributed this observed effect to a numerical artefact, scaling with the simulation cell size. They showed that the rounding error induced by the finite-difference implementation of the spin transfer torque in micromagnetic simulators gives rise to a transverse skyrmion velocity, and thus an apparent skyrmion Hall effect. When taking higher order finite-difference terms into account, the skyrmion Hall effect vanishes. Moreover, the observed transverse velocity scales with the cell size of the discretized grid, reducing for smaller cell sizes 22 . We use the scaling relation derived in Ref. 22 as a third showcase of the altermagnet implementation in mumax + . We create an Altermagnet instance as shown in Section II B and initialize a Néel skyrmion:
Simulating altermagnets using mumax + 5
FIG. 3: The dispersion relation of magnons in a 1D altermagnet as described in Section II C. The angle φ between the exchange matrix eigenbasis and the simulation grid is set to 0 ◦ in the top panel and 45 ◦ in the bottom panel. The wave vector k lies along the length axis of the magnet. The dashed lines correspond to the model Eq. (6), while the
intensity plot data is obtained with mumax + .
FIG. 4: Frequency gap ∆ω at k = 0.75 nm −1 (a) and intensity
ratio between both sublattices along the two different branches (b) of the magnon dispersion relation. The model (dashed lines) is described in Section II C and Ref. 10 , while
data points are obtained with mumax + . The angular coordinate in (a) corresponds to the relative angle φ between
the exchange matrix eigenbasis and the simulation grid. Simulation details are described in Section II C, but the total
simulation time T is increased to 200 ps to improve the
frequency resolution.
We have outlined the implementation of a new magnet class in the mumax + simulation package to support the modeling of 2D d-wave altermagnets. This was done by the addition of an anisotropic exchange field term to the pre-existing code base designed to simulate antiferromagnets. The object-oriented design of mumax + allows for multiple sublattice spins to reside within the same simulation cell and, consequently, an accurate calculation of the magnetostatic field. We have showcased this new feature by comparing the Néel and net magnetization profiles of a Bloch domain wall to an analytical model. Next, we derived and showed the magnon dispersion relation in a 1D altermagnet. Finally, we showed the cell size scaling of the skyrmion Hall effect. All three test cases showed excellent agreement between the results of mumax + and analytical models, thereby verifying its implementation. This also establishes the extensible nature of mumax + , where minimal effort was necessary in order to support d-wave altermagnets in a micromagnetic framework. This addition to the package is now able to nurture numerical studies in the emerging research field of altermagnetism.
Simulating altermagnets using mumax + 6
FIG. 5: Skyrmion velocity components in function of the simulation cell size. The dashed lines represent the model in Ref. 22 , datapoints show simulated results using mumax + , and
the crosses represent data taken from Ref. 20 . Simulation
details are given in Section II D.
Further generalizations are to be expected in future release of mumax + , such as support for other exchange symmetries, for example f - and g-wave, and three dimensional altermagnetic simulations.
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