Periodic boundary conditions

In this tutorial, we compute and relax a skyrmion in an interfacial-DMI (Cnv) material in a part of an infinitely large thin film.

import oommfc as mc
import discretisedfield as df
import micromagneticmodel as mm

We define mesh in cuboid through corner points p1 and p2, and discretisation cell size cell.

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), bc='xy')

Here bc="xy" means that we have periodic boundary conditions along the x and y directions. The mesh we defined is:

mesh.mpl()

Now, we can define the system object by first setting up the Hamiltonian:

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

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

system.energy

$- A \mathbf{m} \cdot \nabla^{2} \mathbf{m}+ D ( \mathbf{m} \cdot \nabla m_{z} - m_{z} \nabla \cdot \mathbf{m} )-K (\mathbf{m} \cdot \mathbf{u})^{2}-\mu_{0}M_\text{s} \mathbf{m} \cdot \mathbf{H}$

Now, we need to define a function to define the initial magnetisation which is going to relax to skyrmion.

def m_init(pos):
    """Function to set initial magnetisation direction:
    -z inside cylinder (r=10nm),
    +z outside cylinder.
    y-component to break symmetry.

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

system.m = df.Field(mesh, nvdim=3, value=m_init, norm=1.1e6)

The initial magnetsation is:

system.m.sel('z').mpl()

/home/mlang/miniconda3/envs/ubermagdev310/lib/python3.10/site-packages/matplotlib/quiver.py:645: RuntimeWarning: divide by zero encountered in scalar divide
  length = a * (widthu_per_lenu / (self.scale * self.width))
/home/mlang/miniconda3/envs/ubermagdev310/lib/python3.10/site-packages/matplotlib/quiver.py:645: RuntimeWarning: invalid value encountered in multiply
  length = a * (widthu_per_lenu / (self.scale * self.width))

Finally we can minimise the energy and plot the magnetisation.

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

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

Running OOMMF (ExeOOMMFRunner)[2023/10/23 16:08]... (0.4 s)

# Plot z-component only:
system.m.z.sel('z').mpl()

Finally we can sample and plot the magnetisation along the line:

system.m.z.line(p1=(-49e-9, 0, 0), p2=(49e-9, 0, 0), n=20).mpl()

Finally let us compute the skyrmion number

\[S = \frac{1}{4\pi}\int\mathbf{m}\cdot\left(\frac{\partial \mathbf{m}}{\partial x} \times \frac{\partial \mathbf{m}}{\partial y}\right)dxdy\]

import math
m = system.m.orientation.sel('z')

1/(4*math.pi) * (m.dot(m.diff('x').cross(m.diff('y')))).integrate()

array([-0.92414447])

import discretisedfield.tools as dft
dft.topological_charge(system.m.sel('z'))

-0.9241444720743723

dft.topological_charge(system.m.sel('z'), method='berg-luescher')

-1.0001397845194369