Skyrmion in a disk¶

In this tutorial, we compute and relax a skyrmion in an interfacial-DMI material in a confined disk like geometry.

In [1]:

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

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

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))

The mesh we defined is:

In [3]:

mesh.mpl()

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

In [4]:

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

system.energy = (
    mm.Exchange(A=1.6e-11)
    + mm.DMI(D=4e-3, crystalclass="Cnv_z")
    + mm.UniaxialAnisotropy(K=0.51e6, u=(0, 0, 1))
    + mm.Demag()
    + mm.Zeeman(H=(0, 0, 2e5))
)

Disk geometry is set up by defining the saturation magnetisation (norm of the magnetisation field). For that, we define a function:

In [5]:

Ms = 1.1e6


def Ms_fun(pos):
    """Function to set magnitude of magnetisation: zero outside cylindric shape,
    Ms inside cylinder.

    Cylinder radius is 50nm.

    """
    x, y, z = pos
    if (x**2 + y**2) ** 0.5 < 50e-9:
        return Ms
    else:
        return 0

And the second function we need is the function to define the initial magnetisation which is going to relax to skyrmion.

In [6]:

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)


# create system with above geometry and initial magnetisation
system.m = df.Field(mesh, nvdim=3, value=m_init, norm=Ms_fun, valid="norm")

The geometry is now cylindrical:

In [7]:

system.m.norm.hv(kdims=["x", "y"])

Out[7]:

and the initial magnetsation is:

In [8]:

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.

In [9]:

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

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

Running OOMMF (ExeOOMMFRunner)[2023/10/23 16:07]... (0.5 s)

In [10]:

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

In [11]:

system.m.hv(kdims=["x", "y"])

Out[11]:

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

In [12]:

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