Exercise: Domain wall pair conversion

Problem description

We want to simulate a domain wall conversion in a two-dimensional thin film sample with:

• exchange energy constant \(A = 15 \,\text{pJ}\,\text{m}^{-1}\),

• Dzyaloshinskii-Moriya energy constant \(D = 3 \,\text{mJ}\,\text{m}^{-2}\),

• uniaxial anisotropy constant \(K = 0.5 \,\text{MJ}\,\text{m}^{-3}\) with \(\hat{\mathbf{u}} = (0, 0, 1)\) in the out of plane direction,

• gyrotropic ratio \(\gamma = 2.211 \times 10^{5} \,\text{m}\,\text{A}^{-1}\,\text{s}^{-1}\), and

• Gilbert damping \(\alpha=0.3\).

Please carry out the following steps:

1. Create the following geometry with discretisation cell size \((2 \,\text{nm}, 2 \,\text{nm}, 2 \,\text{nm})\).

   

2. Initialise the magnetisation so that when relaxes, a domain pair is present in the narrower part of the geometry.

3. Relax the system. Is a domain wall pair contained in the constrained part?

4. Apply the spin polarised current in the positive \(x\) direction with velocity \(\mathbf{u} = (400, 0, 0) \,\text{m}\,\text{s}^{-1}\), with \(\beta=0.5\).

5. Evolve the system over \(0.2 \,\text{ns}\). What did you get? [1]

References

[1] Zhou, Y., & Ezawa, M. (2014). A reversible conversion between a skyrmion and a domain-wall pair in a junction geometry. Nature Communications 5, 8. https://doi.org/10.1038/ncomms5652

Solution

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

Ms = 5.8e5  # saturation magnetisation (A/m)
A = 15e-12  # exchange energy constant (J/)
D = 3e-3  # Dzyaloshinkii-Moriya energy constant (J/m**2)
K = 0.5e6  # uniaxial anisotropy constant (J/m**3)
u = (0, 0, 1)  # easy axis
gamma0 = 2.211e5  # gyromagnetic ratio (m/As)
alpha = 0.3  # Gilbert damping

system = mm.System(name='dw_pair_conversion')
system.energy = mm.Exchange(A=A) + mm.DMI(D=D, crystalclass='Cnv_z') + mm.UniaxialAnisotropy(K=K, u=u)
system.dynamics = mm.Precession(gamma0=2.211e5) + mm.Damping(alpha=alpha)

p1 = (0, 0, 0)
p2 = (150e-9, 50e-9, 2e-9)
cell = (2e-9, 2e-9, 2e-9)

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

def Ms_fun(pos):
    x, y, z = pos
    if x < 50e-9 and (y < 15e-9 or y > 35e-9):
        return 0
    else:
        return Ms

def m_init(pos):
    x, y, z = pos
    if 30e-9 < x < 40e-9:
        return (0.1, 0.1, -1)
    else:
        return (0.1, 0.1, 1)

system.m = df.Field(mesh, nvdim=3, value=m_init, norm=Ms_fun, valid="norm")

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

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

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

Running OOMMF (ExeOOMMFRunner)[2023/10/23 15:51]... (0.5 s)

ux = 400  # velocity in x direction (m/s)
beta = 0.5  # non-adiabatic STT parameter

system.dynamics += mm.ZhangLi(u=ux, beta=beta)

td = oc.TimeDriver()
td.drive(system, t=0.2e-9, n=200)

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

Running OOMMF (ExeOOMMFRunner)[2023/10/23 15:51]... (2.1 s)

As a result, we got a skyrmion formed in the wider region.

We can use the micromagneticdata package to examine the time drive and create an interactive plot.

import micromagneticdata as md

data = md.Data(system.name)

# Use only the last time drive
time_drive = data[-1]

time_drive.hv(
    kdims=["x", "y"],
    scalar_kw={"clim": (-Ms, Ms), "cmap": "coolwarm"},
)