Tutorial 14: Periodic boundary conditions¶

Interactive online tutorial:

In this tutorial, we compute and relax a skyrmion in an interfacial-DMI material thin film using periodic boundary conditions.

In [1]:

import discretisedfield as df
import micromagneticmodel as mm
import oommfc as oc

%matplotlib inline

We define mesh in cuboid through corner points p1 and p2, and discretisation cell size cell. To define periodic boundary conditions, we pass an additional argument pbc. This argument can be any iterable (list, tuple, string, set) containing strings 'x', 'y', and/or 'z'. Let us assume we want the periodic boundary conditions in $x$ and $y$ directions.

In [2]:

region = df.Region(p1=(-50e-9, -50e-9, 0), p2=(50e-9, 50e-9, 10e-9))
mesh = df.Mesh(region=region, cell=(5e-9, 5e-9, 5e-9), pbc="xy")

Now, we can define the system object:

In [3]:

system = mm.System(name="skyrmion")

system.energy = (
    mm.Exchange(A=1.6e-11)
    + mm.DMI(D=4e-3, crystalclass="Cnv")
    + mm.UniaxialAnisotropy(K=0.51e6, u=(0, 0, 1))
    + mm.Zeeman(H=(0, 0, 0.2e5))
)

Ms = 1.1e6


def m_init(pos):
    x, y, z = pos
    if (x**2 + y**2) ** 0.5 < 10e-9:
        return (0, 0, -1)
    else:
        return (0, 0, 1)


# create system with above geometry and initial magnetisation
system.m = df.Field(mesh, dim=3, value=m_init, norm=Ms)

Finally we can minimise the energy and plot the magnetisation.

In [4]:

# minimize the energy
md = oc.MinDriver()
md.drive(system)

# Plot relaxed configuration: vectors in z-plane
system.m.plane("z").mpl()

2020/03/06 15:22: Running OOMMF (skyrmion.mif) ... (1.0 s)

In [5]:

# Plot z-component only:
system.m.z.plane("z").mpl()

In [6]:

# 3d-plot of z-component
system.m.z.k3d_voxels(norm_field=system.m.norm)

Output()

In [7]:

md.delete(system)