Demo¶

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 $c = 10^{9} \text{m}^{-1}$.