SAF Skyrmions Results - Micromagnetics

Some simulation results are shown with the same input as in the initial notebook with the only difference is the change in the spacer height and the cell size. The first simulation was done with a really “big” spacer of 2 nm; however for a system Co-Ru-Co the value for the RKKY coefficient doesnt correspond to the value of the spacer used:

Figure 1:Coupling coefficient for Co/Ru/Co layers from Bloemen *et al.* (1994)

According to Bloemen et al. (1994) the value for the RKKY coefficient corresponding to $\sigma=-0.3 \, \text{mJ/m}^2$ should be for an spacer of around 0.8 nm, so this results are shown for an spacer of height 0.8 nm and cell size of 3x3x0.2 nm

The following script was used to create the images shown at the bottom of the notebook for the results of the simulation.

saf_skyrmions_analysis.py

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from results_analysis import find_omf_file, visualize_field

RESULTS_PATH = "../results"
IMAGES_PATH = "../images/saf_skyrmion-results-0.8nm"

def create_plots():
  Ds = range(150, 375, 75)
  Mss = range(260, 440, 20)

  system_name = 'saf_skyrmion'

  for D in Ds:
    for Ms in Mss:
      dirname = f'saf_skyrmion-{D}nm-{Ms}kA_m'
      drive_path = f"{RESULTS_PATH}/{dirname}/{system_name}/drive-0/"
      omf_file_path = find_omf_file(drive_path)
      visualize_field(
        f"{drive_path}/{omf_file_path}",
        D=D,
        Ms=Ms,
        images_path=IMAGES_PATH
      )

if __name__ == '__main__':
  create_plots()

The source code can be checked at the repo for scripts. The following cells show the topological phases obtained from the simulation. For visualization, one just has to click the button to use the binder server linked to this repo and use interactively this notebook.

Simlation Results¶

Simulation results for the following conditions are shown:

$K=0.1\times 10^{6}$ J/m $^3$
$\sigma_{RKKY}=-0.3 \times 10^{-3}$ J/m $^2$
$DMI=0.5\times 10^{-3}$ J/m $^2$

from IPython.display import display, Image

import ipywidgets as widgets
import os

Ds = range(150, 825, 75)
Mss = range(260, 460, 20)

image_options = {}
for D in Ds:
    for Ms in Mss:
        label = f'D={D} nm and Ms={Ms} kA/m'
        image_name = f'skyrmion-{D}nm-{Ms}kA_m.png'
        image_t_name = f'skyrmion-{D}nm-{Ms}kA_m-tranversal.png'
        
        image_options[label] = [image_name, image_t_name]

image_folder = "../images/saf_skyrmion-0.8nm-z_0.8nm-results"

def show_image_pair(option):
    image_files = image_options[option]
    for img_file in image_files:
        img_path = os.path.join(image_folder, img_file)
        display(Image(img_path, width=800))
        
widgets.interact(show_image_pair, option=list(image_options.keys()))

For this simulation the phase diagram was plotted according to the topological constant $k$ for each field

Then the simulations were repeteaded, but now the value for DMI interaction was set to $DMI=1.0 \times 10^{-3}$ J/m $^2$

Ds = range(150, 750, 75)
Mss = range(260, 460, 20)

image_options = {}
for D in Ds:
    for Ms in Mss:
        label = f'D={D} nm,  Ms={Ms} kA/m'
        image_name = f'skyrmion-{D}nm-{Ms}kA_m.png'
        image_t_name = f'skyrmion-{D}nm-{Ms}kA_m-tranversal.png'
        
        image_options[label] = [image_name, image_t_name]

image_folder = "../images/saf_skyrmion-0.8nm-z_0.8nm-d_1-results"

def show_image_pair(option):
    image_files = image_options[option]
    for img_file in image_files:
        img_path = os.path.join(image_folder, img_file)
        try:
            display(Image(img_path, width=800))
        except:
            print('404')
        
widgets.interact(show_image_pair, option=list(image_options.keys()))

For this simulation the phase diagram was plotted according to the topological constant $k$ for each field

We can plot the skyrmion size differences too for two

radii_150nm = {
    260: 30,
    280: 30,
    300: 33,
    320: 36,
    340: 36,
    360: 39,
    380: 42,
    400: 45,
    420: 48,
    440: 51
}

radii_300nm = {
    260: 81,
    280: 87,
    300: 90,
    320: 93,
    340: 96,
    360: 96
}
radii_450nm = {
    260: 147,
    280: 156,
    300: 159,
    320: 162
}

import matplotlib.pyplot as plt

magnetisation_values = range(260, 380, 20)

fig, ax = plt.subplots(figsize=(8,8))
ax.plot(radii_150nm.keys(), radii_150nm.values(), marker='v', ls='--', label=r'$D=150nm$', ms=10)
ax.plot(radii_300nm.keys(), radii_300nm.values(), marker='D', ls='--', label=r'$D=300nm$', ms=10)
ax.plot(radii_450nm.keys(), radii_450nm.values(), marker='^', ls='--', label=r'$D=450nm$', ms=12)
ax.set_xlabel('Ms [kA/m]')
ax.set_ylabel('r [nm]')
ax.set_title('Skyrmion size (radius) vs Magnetization')
ax.legend()

import micromagneticdata as md



fig, ax = plt.subplots(figsize=(8,8))

options = {
    300: {
        'marker': 's',
        'color': 'r'
    },
    450: {
        'marker': 'D',
        'color': 'g'
    }
}

for D in [300, 450]:
    total_energy = {}
    for Ms in range(260, 460, 20):
        data = md.Data(name='saf_skyrmion', dirname=f'../results/saf_skyrmion-0.8nm/saf_skyrmion-{D}nm-{Ms}kA_m')
        total_energy[Ms] = data[0].table.data.E
    ax.plot(total_energy.keys(), total_energy.values(), marker=options[D]['marker'], c=options[D]['color'], ls='--', label=rf'$D={D}nm$', ms=10)

ax.set_xlabel('Ms [kA/m]')
ax.set_ylabel('E [J]')
ax.set_title('Relaxation Energy vs Magnetization')
ax.legend()

data[0].table.data.columns

Index(['max_mxHxm', 'E', 'delta_E', 'bracket_count', 'line_min_count',
       'conjugate_cycle_count', 'cycle_count', 'cycle_sub_count',
       'energy_calc_count', 'E_exchange', 'max_spin_ang_exchange',
       'stage_max_spin_ang_exchange', 'run_max_spin_ang_exchange', 'E_rkky',
       'E_uniaxialanisotropy', 'E_demag', 'DMI_Cnv_z:dmi:Energy', 'iteration',
       'stage_iteration', 'stage', 'mx', 'my', 'mz'],
      dtype='object')

References¶

Bloemen, P. J. H., van Kesteren, H. W., Swagten, H. J. M., & de Jonge, W. J. M. (1994). Oscillatory interlayer exchange coupling in Co/Ru multilayers and bilayers. Physical Review B, 50(18), 13505–13514. 10.1103/physrevb.50.13505

Micromagnetics

SAF Skyrmions