Vortex dynamics#

In this example, we are going to simulate vortex core dynamics. After creating a vortex structure, we are first going to displace it by applying an external magnetic field. We will then turn off the external field, and compute the time-development of the system, and then be able to see the dynamics of the vortex core.

The sample is a two-dimensional Permalloy square sample with \(d=100 \,\text{nm}\) edge length and \(5\,\text{nm}\) thickness. Its energy equation consists of ferromagnetic exchange, Zeeman, and demagnetisation energy terms:

\[E = \int_{V} \left[-A\mathbf{m}\cdot\nabla^{2}\mathbf{m} - \mu_{0}M_\text{s}\mathbf{m}\cdot\mathbf{H} + w_\text{d}\right] \text{d}V,\]

where \(A = 13 \,\text{pJ}\,\text{m}^{-1}\) is the exchange energy constant, \(M_\text{s} = 8 \times 10^{5} \,\text{A}\,\text{m}^{-1}\) magnetisation saturation, \(w_\text{d}\) demagnetisation energy density, \(\mathbf{H}\) an external magnetic field, and \(\mathbf{m}=\mathbf{M}/M_\text{s}\) the normalised magnetisation field.

The magnetisation dynamics is governed by the Landau-Lifshitz-Gilbert equation consisting of precession and damping terms:

\[\frac{\partial\mathbf{m}}{\partial t} = -\frac{\gamma_{0}}{1+\alpha^{2}}\mathbf{m}\times\mathbf{H}_\text{eff} - \frac{\gamma_{0}\alpha}{1+\alpha^{2}}\mathbf{m}\times(\mathbf{m}\times\mathbf{H}_\text{eff}),\]

where \(\gamma_{0} = 2.211 \times 10^{5} \,\text{m}\,\text{A}^{-1}\,\text{s}^{-1}\) and \(\alpha = 0.2\) is the Gilbert damping.

The (initial) magnetisation field is a vortex state, whose magnetisation at each point \((x, y, z)\) in the sample can be represented as \((m_{x}, m_{y}, m_{z}) = (-cy, cx, 0.1)\), with \(c = 10^{9} \text{m}^{-1}\).

# Some initial configurations
%config InlineBackend.figure_formats = ['svg']  # output matplotlib plots as SVG
import pandas as pd
pd.options.display.max_rows = 5
pd.options.display.float_format = '{:,.2e}'.format

System initialisation#

The Ubermag code for defining the micromagnetic system is:

import discretisedfield as df
import micromagneticmodel as mm

# Geometry
L = 100e-9  # sample edge length (m)
thickness = 5e-9  # sample thickness (m)

# Material (Permalloy) parameters
Ms = 8e5  # saturation magnetisation (A/m)
A = 13e-12  # exchange energy constant (J/m)

# Dynamics (LLG equation) parameters
gamma0 = mm.consts.gamma0  # gyromagnetic ratio (m/As)
alpha = 0.2  # Gilbert damping

system = mm.System(name='vortex_dynamics')

# Energy equation. We omit Zeeman energy term, because H=0. = mm.Exchange(A=A) + mm.Demag()

# Dynamics equation
system.dynamics = mm.Precession(gamma0=gamma0) + mm.Damping(alpha=alpha)

# initial magnetisation state
def m_init(point):
    x, y, z = point
    c = 1e9  # (1/m)
    return (-c*y, c*x, 0.1)

# Sample's centre is placed at origin
region = df.Region(p1=(-L/2, -L/2, -thickness/2), p2=(L/2, L/2, thickness/2))
mesh = df.Mesh(region=region, cell=(5e-9, 5e-9, 5e-9))

system.m = df.Field(mesh, dim=3, value=m_init, norm=Ms)

The system object is now defined and we can investigate some of its properties:

$- A \mathbf{m} \cdot \nabla^{2} \mathbf{m}-\frac{1}{2}\mu_{0}M_\text{s}\mathbf{m} \cdot \mathbf{H}_\text{d}$
$-\frac{\gamma_{0}}{1 + \alpha^{2}} \mathbf{m} \times \mathbf{H}_\text{eff}-\frac{\gamma_{0} \alpha}{1 + \alpha^{2}} \mathbf{m} \times (\mathbf{m} \times \mathbf{H}_\text{eff})$

Energy minimisation#

To carry out micromagnetic simulation, we need to use a micromagnetic calulator. We are going to use OOMMF for this. We can now relax the system in the absence of external magnetic field using energy minimisation driver (MinDriver):

import oommfc as mc  # Micromagnetic Calculator

md = mc.MinDriver()

Running OOMMF (ExeOOMMFRunner)[2022/10/20 15:24]... (0.3 s)

Displacement with magnetic field#

Now, we have a relaxed vortex state, with its core at the centre of the sample. As the next step, we want to add an external magnetic field \(H=10^{4}\,\text{Am}^{-1}\) in the positive \(x\)-direction to displace the vortex core. We do that by adding the Zeeman energy term to the energy equation:

H = (1e4, 0, 0)  # an external magnetic field (A/m) += mm.Zeeman(H=H)
Running OOMMF (ExeOOMMFRunner)[2022/10/20 15:24]... (0.3 s)

Free relaxation#

The vortex core is now displaced in the positive \(y\)-direction. As the last step, we are going to turn off the external magnetic field and simulate dynamics using TimeDriver. We are going to run simulation for \(5\,\text{ns}\) and save the magnetisation in \(500\) steps.

[8]: = (0, 0, 0)

td = mc.TimeDriver(), t=5e-9, n=500, verbose=2)

The final magnetisation state shows that the vortex core has moved back to the sample’s centre.


We can also visualise \(M_{z}\) using an interactive three-dimensional plot.


Data analysis#

The table with scalar data saved during the simulation. Each row corresponds to one of the 500 saved configurations. We only show selected columns.

[11]:[['t', 'mx', 'my', 'mz', 'E']]
t mx my mz E
0 1.00e-11 3.52e-01 7.35e-04 2.32e-02 1.85e-18
1 2.00e-11 3.36e-01 2.52e-03 2.31e-02 1.85e-18
... ... ... ... ... ...
498 4.99e-09 9.90e-06 -2.33e-04 2.29e-02 1.77e-18
499 5.00e-09 2.83e-05 -2.26e-04 2.29e-02 1.77e-18

500 rows × 5 columns

We can now plot the average \(m_{x}\) and \(m_{y}\) values as taken from the table as a function of time to give us an idea of the vortex core position.

system.table.mpl(y=['mx', 'my'])

Finally, we are going to have a look at the magnetisation field at different time-steps using micromagneticdata.

import micromagneticdata as md

data = md.Data(
drive_number date time driver t n
0 0 2022-10-20 15:23:33 MinDriver NaN NaN
1 1 2022-10-20 15:23:34 MinDriver NaN NaN
... ... ... ... ... ... ...
4 4 2022-10-20 15:24:30 MinDriver NaN NaN
5 5 2022-10-20 15:24:30 TimeDriver 5.00e-09 5.00e+02

6 rows × 6 columns

To interactively inspect the time dependent magnetisation, we use data[-1] to refer to the last drive.

data[-1].hv(kdims=['x', 'y'], vdims=['x', 'y'], scalar_kw={'cmap': 'viridis', 'clim': (0, Ms)})