Tutorial 05: Cubic anisotropy energy term

Cubic anisotropy energy density is computed as

\[w_{ca} = -K_{1} [(\mathbf{m} \cdot \mathbf{u}_{1})^{2}(\mathbf{m} \cdot \mathbf{u}_{2})^{2} + (\mathbf{m} \cdot \mathbf{u}_{2})^{2}(\mathbf{m} \cdot \mathbf{u}_{3})^{2} + (\mathbf{m} \cdot \mathbf{u}_{1})^{2}(\mathbf{m} \cdot \mathbf{u}_{3})^{2}]\]

where \(\mathbf{m}\) is the normalised (\(|\mathbf{m}|=1\)) magnetisation, \(K_{1}\) is the cubic anisotropy constant, and \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are the anisotropy axes. Cubic anisotropy energy term tends to align all magnetic moments parallel or antiparallel to one of the three anisotropy axes.

In oommfc, \(\mathbf{m}\) is a part of the magnetisation field system.m. Therefore, only cubic anisotropy constant \(K_{1}\) and axes \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) should be provided as input parameters to uniquely define the energy term. All parameters can be constant in space or spatially varying.

Spatially constant \(K_{1}\) and \(\mathbf{u}\)

Let us start by assembling a simple simple simulation where neither \(K\) nor \(\mathbf{u}\) vary in space. The sample is a “one-dimensional” chain of magnetic moments.

import discretisedfield as df
import micromagneticmodel as mm

import oommfc as oc

p1 = (-10e-9, 0, 0)
p2 = (10e-9, 1e-9, 1e-9)
cell = (1e-9, 1e-9, 1e-9)

region = df.Region(p1=p1, p2=p2)
mesh = df.Mesh(region=region, cell=cell)

The mesh is

mesh.k3d()

The system has a Hamiltonian, which consists of only cubic anisotropy energy term.

K = 1e5  # cubic anisotropy constant (J/m**3)
u1 = (0, 0, 1)  # cubic anisotropy axis
u2 = (0, 1, 0)  # cubic anisotropy axis
system = mm.System(name="cubicanisotropy_constant_K_u")
system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)

We are going to minimise the system’s energy using oommfc.MinDriver later. Therefore, we do not have to define the system’s dynamics equation. Finally, we need to define the system’s magnetisation (system.m). We are going to make it random with \(M_\text{s}=8\times10^{5} \,\text{Am}^{-1}\)

import random

import discretisedfield as df

Ms = 8e5  # saturation magnetisation (A/m)


def m_fun(pos):
    return [2 * random.random() - 1 for i in range(3)]


system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)

The magnetisation, we set is

system.m.k3d.vector(color_field=system.m.z)

Now, we can minimise the system’s energy by using oommfc.MinDriver.

md = oc.MinDriver()
md.drive(system)

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... (0.4 s)

We expect that now all magnetic moments are aligned parallel or antiparallel to the anisotropy axes.

system.m.k3d.vector(color_field=system.m.z)

Spatially varying \(\mathbf{K}_{1}\)

There are two different ways how a parameter can be made spatially varying, by using: 1. Dictionary 2. discretisedfield.Field

Dictionary

In order to define a parameter using a dictionary, regions must be defined in the mesh. Regions are defined as a dictionary, whose keys are the strings and values are discretisedfield.Region objects, which take two corner points of the region as input parameters.

p1 = (-10e-9, 0, 0)
p2 = (10e-9, 1e-9, 1e-9)
cell = (1e-9, 1e-9, 1e-9)
subregions = {
    "region1": df.Region(p1=(-10e-9, 0, 0), p2=(0, 1e-9, 1e-9)),
    "region2": df.Region(p1=(0, 0, 0), p2=(10e-9, 1e-9, 1e-9)),
}
region = df.Region(p1=p1, p2=p2)
mesh = df.Mesh(region=region, cell=cell, subregions=subregions)

mesh.k3d.subregions()

Let us say that there is no cubic anisotropy (\(K=0\)) in region 1, whereas in region 2 it is \(K=10^{5} \,\text{Jm}^{-3}\). K is now defined as a dictionary:

K = {"region1": 0, "region2": 1e5}

The system object is

system = mm.System(name="cubicanisotropy_dict_K")
system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)
system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)

Its magnetisation is

system.m.k3d.vector(color_field=system.m.z)

After we minimise the energy

md.drive(system)

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... (0.2 s)

The magnetisation is as we expected.

system.m.k3d.vector(color_field=system.m.z)

discretisedfield.Field

Let us define the spatailly varying uniaxial anisotropy, so that

$K(x, y, z) = :nbsphinx-math:`left`{

\right. $

The value of K for the spatially varying anisotropy is set using a Python function.

def K_fun(pos):
    x, y, z = pos
    if x <= 0:
        return 0
    else:
        return 1e5

The uniaxial anisotropy parameters are

K = df.Field(mesh, nvdim=1, value=K_fun)

The system is

system = mm.System(name="cubicanisotropy_field_u")
system.energy = mm.CubicAnisotropy(K=K, u1=u1, u2=u2)
system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)

and its magnetisation is

system.m.k3d.vector(color_field=system.m.z)

After the energy minimisation, the magnetisation is:

md.drive(system)
system.m.k3d.vector(color_field=system.m.z)

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:33]... (0.2 s)