The dynamics of magnetisation field $\mathbf{m}$ is governed by the Landau-Lifshitz-Gilbert (LLG) equation
$$\frac{d\mathbf{m}}{dt} = \underbrace{-\gamma_{0}(\mathbf{m} \times \mathbf{H}_\text{eff})}_\text{precession} + \underbrace{\alpha\left(\mathbf{m} \times \frac{d\mathbf{m}}{dt}\right)}_\text{damping},$$where $\gamma_{0}$ is the gyromagnetic ratio, $\alpha$ is the Gilbert damping, and $\mathbf{H}_\text{eff}$ is the effective field. It consists of two terms: precession and damping. In this exercise, we will explore some basic properties of this equation to understand how to define it in simulations.
We will study the simplest "zero-dimensional" case - macrospin. In the first step, after we import necessary modules (oommfc and discretisedfield), we create the mesh which consists of a single finite difference cell.
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import oommfc as oc
import discretisedfield as df
%matplotlib inline
# Define macro spin mesh (i.e. one discretisation cell).
p1 = (0, 0, 0) # first point of the mesh domain (m)
p2 = (1e-9, 1e-9, 1e-9) # second point of the mesh domain (m)
cell = (1e-9, 1e-9, 1e-9) # discretisation cell size (m)
mesh = oc.Mesh(p1=p1, p2=p2, cell=cell)
Now, we can create a micromagnetic system object.
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system = oc.System(name="macrospin")
Let us assume we have a simple Hamiltonian which consists of only Zeeman energy term
$$\mathcal{H} = -\mu_{0}M_\text{s}\mathbf{m}\cdot\mathbf{H},$$where $M_\text{s}$ is the saturation magnetisation, $\mu_{0}$ is the magnetic constant, and $\mathbf{H}$ is the external magnetic field. For more information on defining micromagnetic Hamiltonians, please refer to the Hamiltonian tutorial. We apply the external magnetic field with magnitude $H = 2 \times 10^{6} \,\text{A}\,\text{m}^{-1}$ in the positive $z$ direction.
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H = (0, 0, 2e6) # external magnetic field (A/m)
system.hamiltonian = oc.Zeeman(H=H)
In the next step we can define the system's dynamics. Let us assume we have $\gamma_{0} = 2.211 \times 10^{5} \,\text{m}\,\text{A}^{-1}\,\text{s}^{-1}$ and $\alpha=0.1$.
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gamma = 2.211e5 # gyromagnetic ratio (m/As)
alpha = 0.1 # Gilbert damping
system.dynamics = oc.Precession(gamma=gamma) + oc.Damping(alpha=alpha)
To check what is our dynamics equation:
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system.dynamics
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Before we start running time evolution simulations, we need to initialise the magnetisation. In this case, our magnetisation is pointing in the positive $x$ direction with $M_\text{s} = 8 \times 10^{6} \,\text{A}\,\text{m}^{-1}$. The magnetisation is defined using Field class from the discretisedfield package we imported earlier.
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initial_m = (1, 0, 0) # vector in x direction
Ms = 8e6 # magnetisation saturation (A/m)
system.m = df.Field(mesh, value=initial_m, norm=Ms)
Now, we can run the time evolution using TimeDriver for $t=0.1 \,\text{ns}$ and save the magnetisation configuration in $n=200$ steps.
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td = oc.TimeDriver()
td.drive(system, t=0.1e-9, n=200)
How different system parameters vary with time, we can inspect by showing the system's datatable.
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system.dt
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However, in our case it is much more informative if we plot the time evolution of magnetisation $z$ component $m_{z}(t)$.
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system.dt.plot("t", "mz");
Similarly, we can plot all three magnetisation components
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system.dt.plot("t", ["mx", "my", "mz"]);
We can see that after some time the macrospin aligns parallel to the external magnetic field in the $z$ direction. We can explore the effect of Gilbert damping $\alpha = 0.2$ on the magnetisation dynamics.
In [11]:
system.dynamics.damping.alpha = 0.2
system.m = df.Field(mesh, value=initial_m, norm=Ms)
td.drive(system, t=0.1e-9, n=200)
system.dt.plot("t", ["mx", "my", "mz"]);
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