Tutorial 01: Zeeman energy term#

Zeeman energy density is computed as

\[w_\text{z} = -\mu_{0}M_\text{s}\mathbf{m}\cdot\mathbf{H}\]

where \(\mu_{0}\) is the magnetic constant, \(M_\text{s}\) is the magnetisation saturation, \(\mathbf{m}\) is the normalised (\(|\mathbf{m}|=1\)) magnetisation, and \(\mathbf{H}\) is the external magnetic field. Zeeman energy term tends to align all magnetic moments parallel to the external magnetic field.

In oommfc, \(M_\text{s}\) and \(\mathbf{m}\) are part of the magnetisation field system.m. Therefore, only external magnetic field \(\mathbf{H}\) should be provided as an input parameter to uniquely define the Zeeman energy term. \(\mathbf{H}\) can be constant in space or spatially varying.

Spatially constant \(\mathbf{H}\)#

Let us start by assembling a simple simple simulation where \(\mathbf{H}\) does not vary in space. The sample is a “one-dimensional” chain of magnetic moments.

[1]:

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 system has a Hamiltonian, which consists of only Zeeman energy term.

[2]:

H = (0, 0, 1e6)  # external magnetic field (A/m)
system = mm.System(name="zeeman_constant_H")
system.energy = mm.Zeeman(H=H)

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 \(M_\text{s}=8\times10^{5} \,\text{Am}^{-1}\)

[3]:

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

[4]:

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

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

[5]:

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

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:41]... (0.4 s)

We expect that now all magnetic moments are aligned parallel to the external magnetic field (in the \(z\)-direction).

[6]:

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

Spatially varying \(\mathbf{H}\)#

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.

[7]:

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)

Let us say we want to apply the external magnetic field \(\mathbf{H}\) in region 1 in the \(x\)-direction and in region 2 in the negative \(z\)-direction. H is now defined as a dictionary:

[8]:

H = {"region1": (1e6, 0, 0), "region2": (0, 0, -1e6)}

The system object is

[9]:

system = mm.System(name="zeeman_dict_H")
system.energy = mm.Zeeman(H=H)
system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)

Its magnetisation is

[10]:

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

After we minimise the energy

[11]:

md.drive(system)

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:41]... (0.2 s)

The magnetisation is as we expected.

[12]:

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

discretisedfield.Field#

Let us say that the magnetisation field varies in space as

\[\mathbf{H}(x, y, z) = (c^{2}x, 0, c)\]

where \(c=10^{9}\) and the entire field is normalised with \(H = 10^{6} \,\text{Am}^{-1}\). The value of a spatially varying field is set using a Python function.

[13]:

def H_fun(pos):
    x, y, z = pos
    c = 1e9
    return (c * c * x, 0, c)

The external magnetic field is

[14]:

H = df.Field(mesh, nvdim=3, value=H_fun, norm=1e6)

The system is

[15]:

system = mm.System(name="zeeman_field_H")
system.energy = mm.Zeeman(H=H)
system.m = df.Field(mesh, nvdim=3, value=m_fun, norm=Ms)

and its magnetisation is

[16]:

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

After the energy minimisation, the magnetisation is:

[17]:

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

Running OOMMF (ExeOOMMFRunner)[2023/10/18 12:41]... (0.2 s)