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 disk sample with \(r=50 \,\text{nm}\) edge length and \(10\,\text{nm}\) 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 \(\alpha = 0.05\) 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
import matplotlib.pyplot as plt
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
r = 50e-9  # Radius of the thin nano magnetic disk (m)
thickness = 10e-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 = 0.05  # 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, _ = point
    c = 1e9  # (1/m)
    return (-c*y, c*x, 0.1)

# Defining the geometry of the material as a circular disk
def Ms_func(point):
    x, y, _ = point
    if x**2 + y**2 <= r**2:
        return Ms
        return 0

# Sample's centre is placed at origin
region = df.Region(p1=(-r, -r, -thickness/2), p2=(r, r, thickness/2))
mesh = df.Mesh(region=region, cell=(5e-9, 5e-9, 10e-9))

system.m = df.Field(mesh, nvdim=3, value=m_init, norm=Ms_func, valid='norm')

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 oc  # Micromagnetic Calculator

md = oc.MinDriver()

Running OOMMF (ExeOOMMFRunner)[2023/10/23 16:17]... (0.4 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 \(H=3.4 \times 10^{4}\,\text{Am}^{-1}\) in the positive \(x\)-direction to displace the vortex core. We do that by adding the Zeeman energy term to the energy equation:

H = (3.4e4, 0, 0)  # an external magnetic field (A/m) += mm.Zeeman(H=H)
Running OOMMF (ExeOOMMFRunner)[2023/10/23 16:17]... (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 \(20\,\text{ns}\) and save the magnetisation in \(500\) steps.

[8]: -= mm.Zeeman(H=H)

td = oc.TimeDriver(), t=20e-9, n=500, verbose=2)
Running OOMMF (ExeOOMMFRunner)[2023/10/23 16:17] took 12.4 s

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


Data analysis#

Spatially averaged data#

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

[10]:[['t', 'mx', 'my', 'mz', 'E']].head()
t mx my mz E
0 4.00e-11 4.18e-01 5.03e-02 1.82e-02 3.94e-18
1 8.00e-11 4.27e-01 1.92e-01 1.07e-02 3.89e-18
2 1.20e-10 4.26e-01 2.63e-01 3.90e-02 3.86e-18
3 1.60e-10 3.05e-01 2.82e-01 7.84e-03 3.83e-18
4 2.00e-10 2.57e-01 3.97e-01 2.66e-02 3.80e-18

We can now plot the average \(m_{x}\), \(m_{y}\) and \(m_{z}\) 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', 'mz'])

Spatially resolved data#

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

import micromagneticdata as mdata

data = mdata.Data(
drive_number date time driver n_threads t n
0 0 2023-10-23 16:17:40 MinDriver None NaN NaN
1 1 2023-10-23 16:17:40 MinDriver None NaN NaN
2 2 2023-10-23 16:17:41 TimeDriver None 2.00e-08 5.00e+02

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)})

We can now compute winding number using operators from discretisedfield:

\[S = \frac{1}{4\pi}\iint q \,\,\text{d}x\text{d}y = \frac{1}{4\pi}\iint\mathbf{m}\cdot\left(\frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y~}\right)\text{d}x\text{d}y\]
import math

m = system.m.orientation.sel('z')
S ="x").cross(m.diff("y"))).integrate() / (4 * math.pi)

The winding number is commonly used and there is a predefined function in To get more accurate results we use a different numerical method than just “naively” evaluating the integral.

[15]:'z'), method='berg-luescher')

We can also plot the topological charge density in an interactive plot.

data[-1].register_callback(lambda f:'z'))).hv(kdims=['x', 'y'])

Trajectory of the vortex core#

We can compute the trajectory of the vortex core via the center of mass of the topological charge:

\[\mathbf{R} = \frac{ \int \mathbf{r} \rho(\mathbf{r}) d^2\mathbf{r}}{\int \rho(\mathbf{r}) d^2\mathbf{r}}.\]
rho ='z'))
r = system.m.sel('z').mesh.coordinate_field()
R = (r*rho).integrate()/rho.integrate()
array([6.24115027e-12, 4.54330909e-12])

Now, we need to find the center of the vortex at each time step, this can be achieved by taking the data from last drive.

def compute_vortex_centre(drive):
    x_coords = []
    y_coords = []

    r = drive[0].sel('z').mesh.coordinate_field()

    for m in drive:
        tcd ='z'))
        centre_of_mass = (r*tcd).integrate()/tcd.integrate()

    return pd.DataFrame({'t':['t'], 'pos x': x_coords, 'pos y': y_coords})
pos_pol_plus = compute_vortex_centre(data[-1])

We can now plot the vortex trajectory on top of the initial configuration.

fig, ax = plt.subplots()
data[-1][0].orientation.sel('z').mpl(ax=ax, scalar_kw={'clim': (0, 1)})
ax.plot(pos_pol_plus['pos x']*1e9, pos_pol_plus['pos y']*1e9, c='yellow')
[<matplotlib.lines.Line2D at 0x7ff4e412a500>]

Finally, let us delete all simulation files: