Double layers of Py / CoFeB

Here, we calculate the dispersion of the Damon-Eshbach (DE) modes in a bilayer made of two different materials. In this system, mirror symmetries are broken, and the dipolar interaction induces a dispersion asymmetry. For more details, see Grassi et al. in Ref. 1. For the present case, we consider a bilayer with 25 nm permalloy and 25 nm CoFeB (without any nonmagnetic spacer).

Sample creation

This can be modeled using the monolayer geometry and by setting inhomogenous material parameters. We start by creating the sample

import tetrax as tx
import numpy as np

t1_Py = 25
t2_CoFeB = 25
total_thickness = t1_Py + t2_CoFeB
res = 2.0


dlayer_50 = tx.Sample(
    tx.geometries.layer.monolayer(total_thickness, res),
    name="Py25CoFeB25"
)

and set the material parameters in both layers.

dlayer_50.gamma = 29.2 * 2 * np.pi * 1e9

Msat_Py = 845e3
Msat_CoFeB = 1270e3
middle = -(total_thickness)/2 + t1_Py
Msat_ = np.piecewise(dlayer_50.xyz.x, [dlayer_50.xyz.y <= middle, dlayer_50.xyz.y > middle], [Msat_Py,Msat_CoFeB])
dlayer_50.material["Msat"] = Msat_

A_Py = 12.8e-12
A_CoFeB = 17e-12
Aex_ = np.piecewise(dlayer_50.xyz.x, [dlayer_50.xyz.y <= middle, dlayer_50.xyz.y > middle], [A_Py,A_CoFeB])
dlayer_50.material["Aex"] = Aex_

Here, the piecewise function of the NumPy package creates a staircase dependent on the material parameters (which are piecewise constant).

dlayer_50.mag = (1,0,0)
dlayer_50.show()

/Users/attilak/micromag/tetrax-tests/fresh_environement_test/venv/lib/python3.13/site-packages/traittypes/traittypes.py:97: UserWarning: Given trait value dtype "float32" does not match required type "float32". A coerced copy has been created.
  warnings.warn(

We see that the difference in saturation magnetization Msat between the two layers is visualized as different arrow lengths in the 3D plot. Note, that dlayer_50.mag is still a unitary vector field \(\mathbf{m}(\mathbf{r})\) of length 1 everywhere. The scaled magnetization vector \(M_\mathrm{s}(\mathbf{r})\mathbf{m}(\mathbf{r})\) can be obtained as dlayer_50.mag_full.

dlayer_50.mag_full

MeshVector([[ 845000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [ 845000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.],
            [1270000.,       0.,       0.]])

The vectors displayed in the 3D plot are simply dlayer_50.mag_full / dlayer_50.material["Msat"].average.

Dispersion calculation for the first N=5 Damon-Eshbach modes

To calculate the first 5 DE modes, we apply an external field perpendicular to the propagation (\(z\)) direction. A homogeneous magnetization in the same direction can be assumed as the equilibrium. So we do not need to relax anymore.

dlayer_50.external_field = (30e-3,0,0)
dlayer_50.mag = (1,0,0)

spectrum = tx.experiments.eigenmodes(dlayer_50, kmin=-50e6,kmax=50e6,num_k=81, num_cpus=-1,num_modes=5)

100%|███████████████████| 41/41 [00:01<00:00, 34.14it/s]

Plotting the spectrum shows hybridization and dispersion asymmetry of the lowest two modes.

spectrum.plot(fscale="G", kscale="u", n=[0,1], renderer="notebook") # remove the renderer argument when running on your computer

References

1. Grassi et al. “Slow-Wave-Based Nanomagnonic Diode”, Physical Review Applied 14, 024047 (2020)