Vortex dynamics

Exercise

We want to simulate the vortex dynamics in a two-dimensional disk sample with $d=100 \,\text{nm}$ diameter and $5\,\text{nm}$ thickness with:

magnetisation saturation $M_\text{s} = 8 \times 10^{5} \,\text{A}\,\text{m}^{-1}$,
exchange energy constant $A = 13 \,\text{pJ}\,\text{m}^{-1}$,
gyrotropic ratio $\gamma = 2.211 \times 10^{5} \,\text{m}\,\text{A}^{-1}\,\text{s}^{-1}$, and
Gilbert damping $\alpha=0.2$.

Please carry out the following steps:

Initialise the system so that $(m_{x}, m_{y}, m_{z}) = (-Ay, Ax, 10)$, where $A = 10^{9}\,\text{m}^{-1}$.
Minimise the system's energy. What state did you obtain?
Apply an external magnetic field of $H = 10^{4} \,\text{A}\,\text{m}^{-1}$ in the positive $x$ direction and relax the system. Did the vortex core move in the positive $y$ direction?
Turn off an external magnetic field and simulate the vortex dynamics for $t = 5 \,\text{ns}$ and save magnetisation in $n = 500$ steps. Plot all three components of magnetisation as a function of time.