1D domain wall

Author: Marijan Beg

Date: 26/02/2016

This notebook can be downloaded from the github repository, found here.

Problem specification

The domain wall profile is computed in a one-dimenaional domain of \(L=500 \,\text{nm}\) length.

Material paremeters of the simulated material are:

  • exchange energy constant \(A = 1.3 \times 10^{-11} \,\text{J/m}\),
  • magnetisation saturation \(M_\text{s} = 8.6 \times 10^{5} \,\text{A/m}\),
  • anisotropy constant \(K_{1} = 1.86 \times 10^{5} \,\text{J/m}^{3}\) with \((1, 0, 0)\) axis,
  • anisotropy constant \(K_{2} = -0.92 \times 10^{5} \,\text{J/m}^{3}\) with \((0, 0, 1)\) axis.

Simulation

[1]:
from fidimag.micro import Sim
from fidimag.common import CuboidMesh
from fidimag.micro import UniformExchange, UniaxialAnisotropy

Firstly, the mesh is created.

[2]:
# Mesh dimensions.
L = 500  # diameter (nm)

# Mesh discretisation.
dx = dy = dz = 2  # nm

mesh = CuboidMesh(nx=int(L/dx), ny=1, nz=1, dx=dx, dy=dy, dz=dz, unit_length=1e-9)

The simulation object is created, parameters set, and energies added.

[3]:
# PYTEST_VALIDATE_IGNORE_OUTPUT
Ms = 8.6e5  # magnetisation saturation (A/m)
A = 1.3e-11  # exchange energy constant (J/m)
K1 = 1.86e5  # anisotropy energy constant (J/m**3)
K2 = -0.92e5  # anisotropy energy constant (J/m**3)
alpha = 0.5  # Gilbert damping
gamma = 2.211e5  # gyromagnetic ration (m/As)

# Create simulation object.
sim = Sim(mesh)

# Set simulation parameters.
sim.Ms = Ms
sim.driver.alpha = alpha
sim.driver.gamma = gamma

# Add energies.
sim.add(UniformExchange(A=A))
sim.add(UniaxialAnisotropy(K1, axis=(1,0,0)))
sim.add(UniaxialAnisotropy(K2, axis=(0,0,1)))

# Since the magnetisation dynamics is not important in this stage,
# the precession term in LLG equation can be set to artificially zero.
sim.driver.do_precession = False

In order to obtain the domain wall as the relaxed state, the system must be appropriatelly initialised. In this case, the initial magnetisation is set to be in \((0, 0, 1)\) direction for \(x < 0.4L \,\text{nm}\), in \((-1, 0, 0)\) direction for \(0.4L \,\text{nm} < x < 0.6L \,\text{nm}\), and in \((0, 0, 1)\) direction for \(x > 0.6L \,\text{nm}\).

[4]:
def m_init(pos):
    x = pos[0]

    if x < 0.45*L:
        return (1,0,0)
    elif x > 0.55*L:
        return (-1,0,0)
    else:
        return (0,1,0)

In the next step, the magnetisation is initialised, and the system is relaxed.

[5]:
# NBVAL_IGNORE_OUTPUT
# Initialise the system.
sim.set_m(m_init)

# Relax the system to its equilibrium state.
sim.driver.relax(dt=1e-13, stopping_dmdt=0.01, max_steps=5000, save_m_steps=None, save_vtk_steps=None)
step=1, time=1e-13, max_dmdt=2.97e+04 ode_step=0
step=2, time=2e-13, max_dmdt=2.79e+04 ode_step=2.13e-14
step=3, time=3e-13, max_dmdt=2.6e+04 ode_step=3.28e-14
step=4, time=4e-13, max_dmdt=2.41e+04 ode_step=3.28e-14
step=5, time=5e-13, max_dmdt=2.21e+04 ode_step=5.31e-14
step=6, time=6e-13, max_dmdt=2.02e+04 ode_step=5.31e-14
step=7, time=7e-13, max_dmdt=1.83e+04 ode_step=5.31e-14
step=8, time=8e-13, max_dmdt=1.66e+04 ode_step=5.31e-14
step=9, time=9e-13, max_dmdt=1.5e+04 ode_step=8.16e-14
step=10, time=1e-12, max_dmdt=1.36e+04 ode_step=8.16e-14
step=11, time=1.1e-12, max_dmdt=1.22e+04 ode_step=8.16e-14
step=12, time=1.2e-12, max_dmdt=1.1e+04 ode_step=8.16e-14
step=13, time=1.3e-12, max_dmdt=9.97e+03 ode_step=8.16e-14
step=14, time=1.4e-12, max_dmdt=9.01e+03 ode_step=8.16e-14
step=15, time=1.5e-12, max_dmdt=8.16e+03 ode_step=8.16e-14
step=16, time=1.6e-12, max_dmdt=7.4e+03 ode_step=8.16e-14
step=17, time=1.7e-12, max_dmdt=6.73e+03 ode_step=8.16e-14
step=18, time=1.8e-12, max_dmdt=6.44e+03 ode_step=8.16e-14
step=19, time=1.9e-12, max_dmdt=6.27e+03 ode_step=8.16e-14
step=20, time=2e-12, max_dmdt=6.1e+03 ode_step=8.16e-14
step=21, time=2.1e-12, max_dmdt=5.93e+03 ode_step=8.16e-14
step=22, time=2.2e-12, max_dmdt=5.76e+03 ode_step=8.16e-14
step=23, time=2.3e-12, max_dmdt=5.58e+03 ode_step=8.16e-14
step=24, time=2.4e-12, max_dmdt=5.41e+03 ode_step=8.16e-14
step=25, time=2.5e-12, max_dmdt=5.24e+03 ode_step=8.16e-14
step=26, time=2.6e-12, max_dmdt=5.07e+03 ode_step=8.16e-14
step=27, time=2.7e-12, max_dmdt=4.91e+03 ode_step=8.16e-14
step=28, time=2.82e-12, max_dmdt=4.73e+03 ode_step=1.23e-13
step=29, time=2.95e-12, max_dmdt=4.55e+03 ode_step=1.23e-13
step=30, time=3.07e-12, max_dmdt=4.37e+03 ode_step=1.23e-13
step=31, time=3.19e-12, max_dmdt=4.2e+03 ode_step=1.23e-13
step=32, time=3.32e-12, max_dmdt=4.04e+03 ode_step=1.23e-13
step=33, time=3.44e-12, max_dmdt=3.88e+03 ode_step=1.23e-13
step=34, time=3.56e-12, max_dmdt=3.74e+03 ode_step=1.23e-13
step=35, time=3.69e-12, max_dmdt=3.6e+03 ode_step=1.23e-13
step=36, time=3.81e-12, max_dmdt=3.47e+03 ode_step=1.23e-13
step=37, time=3.93e-12, max_dmdt=3.35e+03 ode_step=1.23e-13
step=38, time=4.06e-12, max_dmdt=3.23e+03 ode_step=1.23e-13
step=39, time=4.18e-12, max_dmdt=3.12e+03 ode_step=1.23e-13
step=40, time=4.3e-12, max_dmdt=3.02e+03 ode_step=1.23e-13
step=41, time=4.43e-12, max_dmdt=2.92e+03 ode_step=1.23e-13
step=42, time=4.55e-12, max_dmdt=2.83e+03 ode_step=1.23e-13
step=43, time=4.74e-12, max_dmdt=2.77e+03 ode_step=1.86e-13
step=44, time=4.92e-12, max_dmdt=2.71e+03 ode_step=1.86e-13
step=45, time=5.11e-12, max_dmdt=2.64e+03 ode_step=1.86e-13
step=46, time=5.3e-12, max_dmdt=2.58e+03 ode_step=1.86e-13
step=47, time=5.48e-12, max_dmdt=2.52e+03 ode_step=1.86e-13
step=48, time=5.67e-12, max_dmdt=2.46e+03 ode_step=1.86e-13
step=49, time=5.86e-12, max_dmdt=2.4e+03 ode_step=1.86e-13
step=50, time=6.04e-12, max_dmdt=2.34e+03 ode_step=1.86e-13
step=51, time=6.23e-12, max_dmdt=2.29e+03 ode_step=1.86e-13
step=52, time=6.41e-12, max_dmdt=2.24e+03 ode_step=1.86e-13
step=53, time=6.6e-12, max_dmdt=2.18e+03 ode_step=1.86e-13
step=54, time=6.79e-12, max_dmdt=2.13e+03 ode_step=1.86e-13
step=55, time=6.97e-12, max_dmdt=2.09e+03 ode_step=1.86e-13
step=56, time=7.26e-12, max_dmdt=2.03e+03 ode_step=2.82e-13
step=57, time=7.54e-12, max_dmdt=1.97e+03 ode_step=2.82e-13
step=58, time=7.82e-12, max_dmdt=1.9e+03 ode_step=2.82e-13
step=59, time=8.1e-12, max_dmdt=1.85e+03 ode_step=2.82e-13
step=60, time=8.38e-12, max_dmdt=1.79e+03 ode_step=2.82e-13
step=61, time=8.67e-12, max_dmdt=1.74e+03 ode_step=2.82e-13
step=62, time=8.95e-12, max_dmdt=1.69e+03 ode_step=2.82e-13
step=63, time=9.23e-12, max_dmdt=1.64e+03 ode_step=2.82e-13
step=64, time=9.51e-12, max_dmdt=1.61e+03 ode_step=2.82e-13
step=65, time=9.8e-12, max_dmdt=1.59e+03 ode_step=2.82e-13
step=66, time=1.01e-11, max_dmdt=1.56e+03 ode_step=2.82e-13
step=67, time=1.04e-11, max_dmdt=1.54e+03 ode_step=2.82e-13
step=68, time=1.06e-11, max_dmdt=1.51e+03 ode_step=2.82e-13
step=69, time=1.11e-11, max_dmdt=1.48e+03 ode_step=4.23e-13
step=70, time=1.15e-11, max_dmdt=1.45e+03 ode_step=4.23e-13
step=71, time=1.19e-11, max_dmdt=1.42e+03 ode_step=4.23e-13
step=72, time=1.23e-11, max_dmdt=1.39e+03 ode_step=4.23e-13
step=73, time=1.28e-11, max_dmdt=1.36e+03 ode_step=4.23e-13
step=74, time=1.32e-11, max_dmdt=1.33e+03 ode_step=4.23e-13
step=75, time=1.36e-11, max_dmdt=1.3e+03 ode_step=4.23e-13
step=76, time=1.4e-11, max_dmdt=1.27e+03 ode_step=4.23e-13
step=77, time=1.45e-11, max_dmdt=1.25e+03 ode_step=4.23e-13
step=78, time=1.49e-11, max_dmdt=1.22e+03 ode_step=4.23e-13
step=79, time=1.53e-11, max_dmdt=1.2e+03 ode_step=4.23e-13
step=80, time=1.57e-11, max_dmdt=1.18e+03 ode_step=4.23e-13
step=81, time=1.61e-11, max_dmdt=1.16e+03 ode_step=4.23e-13
step=82, time=1.68e-11, max_dmdt=1.15e+03 ode_step=6.4e-13
step=83, time=1.74e-11, max_dmdt=1.13e+03 ode_step=6.4e-13
step=84, time=1.81e-11, max_dmdt=1.11e+03 ode_step=6.4e-13
step=85, time=1.87e-11, max_dmdt=1.09e+03 ode_step=6.4e-13
step=86, time=1.93e-11, max_dmdt=1.08e+03 ode_step=6.4e-13
step=87, time=2e-11, max_dmdt=1.06e+03 ode_step=6.4e-13
step=88, time=2.06e-11, max_dmdt=1.04e+03 ode_step=6.4e-13
step=89, time=2.13e-11, max_dmdt=1.03e+03 ode_step=6.4e-13
step=90, time=2.19e-11, max_dmdt=1.01e+03 ode_step=6.4e-13
step=91, time=2.26e-11, max_dmdt=994 ode_step=6.4e-13
step=92, time=2.32e-11, max_dmdt=980 ode_step=6.4e-13
step=93, time=2.42e-11, max_dmdt=969 ode_step=9.91e-13
step=94, time=2.52e-11, max_dmdt=956 ode_step=9.91e-13
step=95, time=2.62e-11, max_dmdt=943 ode_step=9.91e-13
step=96, time=2.72e-11, max_dmdt=929 ode_step=9.91e-13
step=97, time=2.81e-11, max_dmdt=915 ode_step=9.91e-13
step=98, time=2.91e-11, max_dmdt=901 ode_step=9.91e-13
step=99, time=3.01e-11, max_dmdt=887 ode_step=9.91e-13
step=100, time=3.11e-11, max_dmdt=872 ode_step=9.91e-13
step=101, time=3.21e-11, max_dmdt=863 ode_step=9.91e-13
step=102, time=3.31e-11, max_dmdt=854 ode_step=9.91e-13
step=103, time=3.41e-11, max_dmdt=844 ode_step=9.91e-13
step=104, time=3.51e-11, max_dmdt=834 ode_step=9.91e-13
step=105, time=3.61e-11, max_dmdt=823 ode_step=9.91e-13
step=106, time=3.71e-11, max_dmdt=811 ode_step=9.91e-13
step=107, time=3.81e-11, max_dmdt=798 ode_step=9.91e-13
step=108, time=3.9e-11, max_dmdt=786 ode_step=9.91e-13
step=109, time=4.07e-11, max_dmdt=768 ode_step=1.61e-12
step=110, time=4.23e-11, max_dmdt=748 ode_step=1.61e-12
step=111, time=4.39e-11, max_dmdt=730 ode_step=1.61e-12
step=112, time=4.55e-11, max_dmdt=711 ode_step=1.61e-12
step=113, time=4.71e-11, max_dmdt=690 ode_step=1.61e-12
step=114, time=4.87e-11, max_dmdt=669 ode_step=1.61e-12
step=115, time=5.03e-11, max_dmdt=646 ode_step=1.61e-12
step=116, time=5.19e-11, max_dmdt=623 ode_step=1.61e-12
step=117, time=5.35e-11, max_dmdt=600 ode_step=1.61e-12
step=118, time=5.51e-11, max_dmdt=576 ode_step=1.61e-12
step=119, time=5.67e-11, max_dmdt=552 ode_step=1.61e-12
step=120, time=5.83e-11, max_dmdt=530 ode_step=1.61e-12
step=121, time=5.99e-11, max_dmdt=509 ode_step=1.61e-12
step=122, time=6.15e-11, max_dmdt=488 ode_step=1.61e-12
step=123, time=6.31e-11, max_dmdt=468 ode_step=1.61e-12
step=124, time=6.48e-11, max_dmdt=447 ode_step=1.61e-12
step=125, time=6.64e-11, max_dmdt=434 ode_step=1.61e-12
step=126, time=6.8e-11, max_dmdt=422 ode_step=1.61e-12
step=127, time=6.96e-11, max_dmdt=410 ode_step=1.61e-12
step=128, time=7.12e-11, max_dmdt=397 ode_step=1.61e-12
step=129, time=7.28e-11, max_dmdt=384 ode_step=1.61e-12
step=130, time=7.44e-11, max_dmdt=371 ode_step=1.61e-12
step=131, time=7.6e-11, max_dmdt=357 ode_step=1.61e-12
step=132, time=7.76e-11, max_dmdt=344 ode_step=1.61e-12
step=133, time=7.92e-11, max_dmdt=331 ode_step=1.61e-12
step=134, time=8.08e-11, max_dmdt=318 ode_step=1.61e-12
step=135, time=8.24e-11, max_dmdt=306 ode_step=1.61e-12
step=136, time=8.4e-11, max_dmdt=293 ode_step=1.61e-12
step=137, time=8.56e-11, max_dmdt=281 ode_step=1.61e-12
step=138, time=8.72e-11, max_dmdt=269 ode_step=1.61e-12
step=139, time=8.88e-11, max_dmdt=258 ode_step=1.61e-12
step=140, time=9.05e-11, max_dmdt=246 ode_step=1.61e-12
step=141, time=9.21e-11, max_dmdt=235 ode_step=1.61e-12
step=142, time=9.37e-11, max_dmdt=225 ode_step=1.61e-12
step=143, time=9.53e-11, max_dmdt=215 ode_step=1.61e-12
step=144, time=9.77e-11, max_dmdt=202 ode_step=2.41e-12
step=145, time=1e-10, max_dmdt=188 ode_step=2.41e-12
step=146, time=1.03e-10, max_dmdt=175 ode_step=2.41e-12
step=147, time=1.05e-10, max_dmdt=162 ode_step=2.41e-12
step=148, time=1.07e-10, max_dmdt=151 ode_step=2.41e-12
step=149, time=1.1e-10, max_dmdt=140 ode_step=2.41e-12
step=150, time=1.12e-10, max_dmdt=129 ode_step=2.41e-12
step=151, time=1.15e-10, max_dmdt=120 ode_step=2.41e-12
step=152, time=1.17e-10, max_dmdt=111 ode_step=2.41e-12
step=153, time=1.19e-10, max_dmdt=103 ode_step=2.41e-12
step=154, time=1.22e-10, max_dmdt=94.9 ode_step=2.41e-12
step=155, time=1.24e-10, max_dmdt=87.7 ode_step=2.41e-12
step=156, time=1.27e-10, max_dmdt=81 ode_step=2.41e-12
step=157, time=1.29e-10, max_dmdt=74.9 ode_step=2.41e-12
step=158, time=1.31e-10, max_dmdt=69.1 ode_step=2.41e-12
step=159, time=1.34e-10, max_dmdt=63.8 ode_step=2.41e-12
step=160, time=1.36e-10, max_dmdt=58.9 ode_step=2.41e-12
step=161, time=1.4e-10, max_dmdt=53.3 ode_step=3.72e-12
step=162, time=1.44e-10, max_dmdt=47.1 ode_step=3.72e-12
step=163, time=1.47e-10, max_dmdt=41.6 ode_step=3.72e-12
step=164, time=1.51e-10, max_dmdt=36.8 ode_step=3.72e-12
step=165, time=1.55e-10, max_dmdt=32.5 ode_step=3.72e-12
step=166, time=1.59e-10, max_dmdt=28.7 ode_step=3.72e-12
step=167, time=1.62e-10, max_dmdt=25.4 ode_step=3.72e-12
step=168, time=1.66e-10, max_dmdt=22.5 ode_step=3.72e-12
step=169, time=1.7e-10, max_dmdt=19.9 ode_step=3.72e-12
step=170, time=1.73e-10, max_dmdt=17.6 ode_step=3.72e-12
step=171, time=1.77e-10, max_dmdt=15.5 ode_step=3.72e-12
step=172, time=1.81e-10, max_dmdt=13.7 ode_step=3.72e-12
step=173, time=1.85e-10, max_dmdt=12.1 ode_step=3.72e-12
step=174, time=1.88e-10, max_dmdt=10.7 ode_step=3.72e-12
step=175, time=1.92e-10, max_dmdt=9.5 ode_step=3.72e-12
step=176, time=1.96e-10, max_dmdt=8.41 ode_step=3.72e-12
step=177, time=2e-10, max_dmdt=7.44 ode_step=3.72e-12
step=178, time=2.03e-10, max_dmdt=6.58 ode_step=3.72e-12
step=179, time=2.07e-10, max_dmdt=5.83 ode_step=3.72e-12
step=180, time=2.11e-10, max_dmdt=5.16 ode_step=3.72e-12
step=181, time=2.14e-10, max_dmdt=4.56 ode_step=3.72e-12
step=182, time=2.2e-10, max_dmdt=3.91 ode_step=5.7e-12
step=183, time=2.26e-10, max_dmdt=3.25 ode_step=5.7e-12
step=184, time=2.31e-10, max_dmdt=2.7 ode_step=5.7e-12
step=185, time=2.37e-10, max_dmdt=2.24 ode_step=5.7e-12
step=186, time=2.43e-10, max_dmdt=1.86 ode_step=5.7e-12
step=187, time=2.49e-10, max_dmdt=1.54 ode_step=5.7e-12
step=188, time=2.54e-10, max_dmdt=1.28 ode_step=5.7e-12
step=189, time=2.6e-10, max_dmdt=1.07 ode_step=5.7e-12
step=190, time=2.66e-10, max_dmdt=0.886 ode_step=5.7e-12
step=191, time=2.71e-10, max_dmdt=0.737 ode_step=5.7e-12
step=192, time=2.77e-10, max_dmdt=0.614 ode_step=5.7e-12
step=193, time=2.83e-10, max_dmdt=0.511 ode_step=5.7e-12
step=194, time=2.89e-10, max_dmdt=0.425 ode_step=5.7e-12
step=195, time=2.94e-10, max_dmdt=0.354 ode_step=5.7e-12
step=196, time=3e-10, max_dmdt=0.295 ode_step=5.7e-12
step=197, time=3.06e-10, max_dmdt=0.246 ode_step=5.7e-12
step=198, time=3.11e-10, max_dmdt=0.205 ode_step=5.7e-12
step=199, time=3.2e-10, max_dmdt=0.163 ode_step=8.66e-12
step=200, time=3.29e-10, max_dmdt=0.123 ode_step=8.66e-12
step=201, time=3.37e-10, max_dmdt=0.0936 ode_step=8.66e-12
step=202, time=3.46e-10, max_dmdt=0.071 ode_step=8.66e-12
step=203, time=3.55e-10, max_dmdt=0.0539 ode_step=8.66e-12
step=204, time=3.63e-10, max_dmdt=0.041 ode_step=8.66e-12
step=205, time=3.72e-10, max_dmdt=0.0311 ode_step=8.66e-12
step=206, time=3.81e-10, max_dmdt=0.0236 ode_step=8.66e-12
step=207, time=3.89e-10, max_dmdt=0.018 ode_step=8.66e-12
step=208, time=3.98e-10, max_dmdt=0.0137 ode_step=8.66e-12
step=209, time=4.07e-10, max_dmdt=0.0104 ode_step=8.66e-12
step=210, time=4.2e-10, max_dmdt=0.0073 ode_step=1.36e-11

After the system is relaxed, the magnetisation profile can be plotted.

[6]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

m = np.copy(sim.spin)

m.shape = (-1, 3)

mx = m[:, 0]
my = m[:, 1]
mz = m[:, 2]

x_array = np.arange(0, L, dx)

plt.figure(figsize=(10,6))
plt.plot(x_array, mx, label='mx')
plt.plot(x_array, my, label='my')
plt.plot(x_array, mz, label='mz')
plt.xlabel('x (nm)')
plt.ylabel('<m>')
plt.xlim([0.4*L, 0.6*L])
plt.ylim([-1, 1])
plt.grid()
plt.legend()
[6]:
<matplotlib.legend.Legend at 0x7f234cebbba8>
../_images/ipynb_1d_domain_wall_15_1.png