Physical test scenario 7 - finite 180° degree 1D domain wall including Zeeman interaction

[AdamsMP92]

Motivation

A finite one-dimensional 180-degree domain wall in an applied field provides a semi-analytical reference solution for testing exchange, uniaxial anisotropy, Zeeman energy, boundary constraints, and energy minimisation.

Unlike the zero-field wall, an applied field along the easy axis makes the equilibrium profile asymmetric on a finite interval. The numerical equilibrium state can therefore be compared with a more sensitive reference profile than the isolated hyperbolic-tangent wall.

Physical model

Consider a one-dimensional magnetic system along the $x$-axis with exchange energy density

$$w_{\mathrm{ex}} = A \left| \frac{d\mathbf{m}}{dx} \right|^2,$$

uniaxial anisotropy along the $z$-axis,

$$w_{\mathrm{a}} = K \left(1-m_z^2\right),$$

and a Zeeman contribution from a field parallel to the easy axis,

$$w_{\mathrm{z}} = -\mu_0 M_s H_z m_z.$$

The magnetisation is constrained to form a 180-degree wall between pinned end states,

$$\mathbf{m}(0)=\mathbf{e}_z, \qquad \mathbf{m}(L)=-\mathbf{e}_z.$$

Assuming rotation in the $yz$-plane,

$$\mathbf{m}(x) = \begin{bmatrix} 0\\\ \sin\theta(x)\\\ \cos\theta(x) \end{bmatrix},$$

the finite-domain problem can be written in reduced coordinates

$$\xi = \frac{x-x_0}{\Delta}, \qquad \Delta = \sqrt{\frac{A}{K}},$$

on the interval

$$\xi\in[-w,w],$$

with a reduced field parameter $b$. In these variables the first integral can be written in the form

$$\left(\frac{d\theta}{d\xi}\right)^2 = a +\sin^2\theta +2b\sin^2\frac{\theta}{2}, \qquad b\ge0,$$

or equivalently

$$\left(\frac{d\theta}{d\xi}\right)^2 = a +\sin^2\theta -2b\cos^2\frac{\theta}{2}, \qquad b \lt 0.$$

The integration constant $a$ is determined by the finite-length constraint.

Proposed test setup

A one-dimensional mesh can be used with exchange, uniaxial anisotropy, and a Zeeman field along the easy axis.

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

Ms = 1.4e6
A = 30e-12
K = 520e3

delta = np.sqrt(A / K)
w = 2.0
length = 2 * w * delta
cell = length / 100
x0 = length / 2

region = df.Region(
    p1=(0, 0, 0),
    p2=(length, cell, cell),
)

mesh = df.Mesh(
    region=region,
    cell=(cell, cell, cell),
)

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

H = (0, 0, Hz)

system.energy = (
    mm.Exchange(A=A)
    + mm.UniaxialAnisotropy(K=K, u=(0, 0, 1))
    + mm.Zeeman(H=H)
)

The initial state should contain a continuous transition between $+\mathbf{e}_z$ and $-\mathbf{e}_z$. The end cells should be fixed to preserve the 180-degree wall during relaxation.

For example,

def initial_m(pos):
    x = pos[0]

    mz = 1.0 - 2.0 * x / length
    my = np.sqrt(max(0.0, 1.0 - mz**2))

    return (0.0, my, mz)

The system can then be relaxed using an energy-minimisation driver or strongly damped time evolution.

Semi-analytical reference implementation

The reference profile is obtained by solving the scalar finite-length condition for $a$,

$$\int_0^\pi \frac{d\theta} {\sqrt{ a+\sin^2\theta+2b\sin^2(\theta/2) }} = 2w, \qquad b\ge0,$$

or

$$\int_0^\pi \frac{d\theta} {\sqrt{ a+\sin^2\theta-2b\cos^2(\theta/2) }} = 2w, \qquad b \lt 0.$$

After $a$ has been found, the reference profile is defined implicitly by

$$\xi(\theta) = \int_\theta^\pi \frac{dt} {\sqrt{ a+\sin^2t+2b\sin^2(t/2) }} -w, \qquad b\ge0,$$

or

$$\xi(\theta) = \int_\theta^\pi \frac{dt} {\sqrt{ a+\sin^2t-2b\cos^2(t/2) }} -w, \qquad b \lt 0.$$

The magnetisation components are then

$$m_x(\xi)=0, \qquad m_y(\xi)=\sin\theta(\xi), \qquad m_z(\xi)=\cos\theta(\xi).$$

A local prototype implementation is available in:

python/domain_wall_case3_finite_zeeman.py

For the dimensionless reference parameters used in the prototype,

w = 2.0
b = -0.5

the scalar solve gives approximately

$$a \approx 4.790005\times10^{-2}.$$

Primary assertion

After relaxation, the normalised numerical magnetisation should be compared with the semi-analytical finite-field profile:

m_exact = finite_zeeman_domain_wall(
    x=x_coordinates,
    x0=x0,
    delta=delta,
    w=w,
    b=b,
)

np.testing.assert_allclose(
    m_simulated,
    m_exact,
    rtol=...,
    atol=...,
)

A component-wise error can also be used:

error_my = np.max(
    np.abs(m_simulated[:, 1] - m_exact[:, 1])
)

error_mz = np.max(
    np.abs(m_simulated[:, 2] - m_exact[:, 2])
)

For a coarse 51-point finite-difference prototype, the maximum profile errors are already on the order of $10^{-3}$ to $10^{-2}$. A production Ubermag test can use a finer mesh and choose tolerances based on the numerical driver and discretisation.

Additional checks

The following properties can be tested independently:

$$m_x(x)\approx0,$$ $$\mathbf{m}(0)\approx\mathbf{e}_z, \qquad \mathbf{m}(L)\approx-\mathbf{e}_z,$$

and

$$\lVert\mathbf{m}(x)\rVert\approx1.$$

The field-biased wall is not generally symmetric, so this test should not assert the zero-field relations

$$m_z(x_0-\xi)\approx-m_z(x_0+\xi), \qquad m_y(x_0-\xi)\approx m_y(x_0+\xi).$$

Instead, the asymmetric spatial profile should be compared directly with the semi-analytical reference.

Expected value

This test would validate:

• the exchange energy term;
• uniaxial anisotropy;
• the Zeeman energy term;
• fixed boundary regions;
• relaxation or minimisation in an asymmetric finite-domain wall;
• conversion between physical and reduced domain-wall coordinates;
• and recovery of a semi-analytical equilibrium profile beyond the standard hyperbolic-tangent wall.

This scenario extends the basic 180-degree domain-wall benchmark by checking a field-distorted finite wall with a quantitative reference solution.