Spin-polarised-current-driven domain wall#
Author: Marijan Beg, Weiwei Wang
Date: 18 Mar 2016
This notebook can be downloaded from the github repository, found here.
Problem specification#
The simulated sample is a 1D nanowire cuboid with \(L=1000 \,\text{nm}\) length with finite difference discretisation \(d_{x} = d_{y} = d_{z} = 2 \,\text{nm}\).
The material parameters (similar to permalloy) are:
exchange energy constant \(A = 1.3 \times 10^{-11} \,\text{J/m}\),
magnetisation saturation \(M_\text{s} = 8.6 \times 10^{5} \,\text{A/m}\),
uniaxial anisotropy constant \(K=5 \times 10^{4} \,\text{J/m}^{3}\) with \((0, 0, 1)\) easy-axis,
Gilbert damping \(\alpha = 0.5\).
After the system is relaxed to a domain wall, a spin-polarised current with \(J=1 \times 10^{12} \text{A/m}^{2}\) density is applied in the positive x direction \((1, 0, 0)\).
Simulation functions#
[1]:
import matplotlib.pyplot as plt
import numpy as np
from fidimag.micro import Sim, UniformExchange, UniaxialAnisotropy
from fidimag.common import CuboidMesh
%matplotlib inline
We start by defining parameters and a function for initialising the system so that it relaxes to the domain wall.
[2]:
Ms = 8.6e5 # magnetisation saturation (A/m)
A = 1.3e-11 # exchange energy constant (J/m)
alpha = 0.5 # Gilbert damping
gamma = 2.211e5 # gyromagnetic ration (m/As)
K = 5e4 # uniaxial anisotropy constant (J/m**3)
J = 1e12 # spin-polarised current density (A/m**2)
beta = 1 # STT parameter
def init_m(pos):
x = pos[0]
if x < 200:
return (1, 0, 0)
elif 200 <= x < 300:
return (0, 1, 1)
else:
return (-1, 0, 0)
Using this function, we create a new function which relaxes the system to its equilibrium (domain wall) state according to the problem specification.
[3]:
def relax_system(mesh):
# Create a simulation object.
sim = Sim(mesh)
# Set simulation parameters.
sim.driver.set_tols(rtol=1e-8, atol=1e-8)
sim.driver.alpha = alpha
sim.driver.gamma = gamma
sim.Ms = Ms
# Add energies to the system.
sim.add(UniformExchange(A=A))
sim.add(UniaxialAnisotropy(K))
# Initialise the system.
sim.set_m(init_m)
# Relax the system and save the state in m0.npy
sim.driver.relax(dt=1e-14, stopping_dmdt=0.01, max_steps=5000,
save_m_steps=None, save_vtk_steps=None)
np.save('m0.npy', sim.spin)
A plot of the system’s magnetisation can be created using the following convenience function.
[4]:
def plot_magnetisation(components):
plt.figure(figsize=(8, 6))
comp = {'mx': 0, 'my': 1, 'mz': 2}
for element in components:
data = np.load(element[0])
data.shape = (-1, 3)
mc = data[:, comp[element[1]]]
# The label is the component and the file name
plt.plot(mc, label=element[1])
plt.legend()
plt.xlabel('x (nm)')
plt.ylabel('mx, my')
plt.grid()
plt.ylim([-1.05, 1.05])
Finally, we create a function for driving a domain wall using the spin-polarised current. All npy and vtk files are saved in the **{simulation_name}_npys** and **{simulation_name}_vtks** folders, respectively.
[5]:
def excite_system(mesh, time, snapshots):
# Specify the stt dynamics in the simulation
sim = Sim(mesh, name='dyn', driver='llg_stt')
# Set the simulation parameters
sim.driver.set_tols(rtol=1e-12, atol=1e-14)
sim.driver.alpha = alpha
sim.driver.gamma = gamma
sim.Ms = Ms
# Add energies to the system.
sim.add(UniformExchange(A=A))
sim.add(UniaxialAnisotropy(K))
# Load the initial state from the npy file saved in the realxation stage.
sim.set_m(np.load('m0.npy'))
# Set the spin-polarised current in the x direction.
sim.driver.jx = J
sim.driver.beta = beta
# The simulation will run for x ns and save
# 'snaps' snapshots of the system in the process
ts = np.linspace(0, time, snapshots)
for t in ts:
sim.driver.run_until(t)
sim.save_vtk()
sim.save_m()
Simulation#
Before we run a simulation using previously defined functions, a finite difference mesh must be created.
[6]:
L = 2000 # nm
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)
Now, the system is relaxed and the domain wall equilibrium state is obtained, saved, and later used in the next stage.
[7]:
#NBVAL_IGNORE_OUTPUT
relax_system(mesh)
#1 t=1e-14 dt=1e-14 max_dmdt=6.47e+04
#2 t=2e-14 dt=1e-14 max_dmdt=6.48e+04
#3 t=3.05e-14 dt=1.05e-14 max_dmdt=6.48e+04
#4 t=4.09e-14 dt=1.05e-14 max_dmdt=6.47e+04
#5 t=5.14e-14 dt=1.05e-14 max_dmdt=6.47e+04
#6 t=6.19e-14 dt=1.05e-14 max_dmdt=6.45e+04
#7 t=7.23e-14 dt=1.05e-14 max_dmdt=6.43e+04
#8 t=8.28e-14 dt=1.05e-14 max_dmdt=6.41e+04
#9 t=9.33e-14 dt=1.05e-14 max_dmdt=6.38e+04
#10 t=1.04e-13 dt=1.05e-14 max_dmdt=6.35e+04
#11 t=1.23e-13 dt=1.91e-14 max_dmdt=6.29e+04
#12 t=1.42e-13 dt=1.91e-14 max_dmdt=6.21e+04
#13 t=1.61e-13 dt=1.91e-14 max_dmdt=6.11e+04
#14 t=1.8e-13 dt=1.91e-14 max_dmdt=6e+04
#15 t=1.99e-13 dt=1.91e-14 max_dmdt=5.88e+04
#16 t=2.18e-13 dt=1.91e-14 max_dmdt=5.74e+04
#17 t=2.37e-13 dt=1.91e-14 max_dmdt=5.6e+04
#18 t=2.56e-13 dt=1.91e-14 max_dmdt=5.44e+04
#19 t=2.75e-13 dt=1.91e-14 max_dmdt=5.28e+04
#20 t=3.09e-13 dt=3.37e-14 max_dmdt=5.03e+04
#21 t=3.43e-13 dt=3.37e-14 max_dmdt=4.7e+04
#22 t=3.77e-13 dt=3.37e-14 max_dmdt=4.59e+04
#23 t=4.01e-13 dt=2.47e-14 max_dmdt=4.51e+04
#24 t=4.26e-13 dt=2.47e-14 max_dmdt=4.42e+04
#25 t=4.51e-13 dt=2.47e-14 max_dmdt=4.31e+04
#26 t=4.75e-13 dt=2.47e-14 max_dmdt=4.2e+04
#27 t=5e-13 dt=2.47e-14 max_dmdt=4.07e+04
#28 t=5.25e-13 dt=2.47e-14 max_dmdt=3.92e+04
#29 t=5.49e-13 dt=2.47e-14 max_dmdt=3.77e+04
#30 t=5.74e-13 dt=2.47e-14 max_dmdt=3.61e+04
#31 t=5.99e-13 dt=2.47e-14 max_dmdt=3.45e+04
#32 t=6.37e-13 dt=3.85e-14 max_dmdt=3.23e+04
#33 t=6.76e-13 dt=3.85e-14 max_dmdt=2.96e+04
#34 t=7.14e-13 dt=3.85e-14 max_dmdt=2.75e+04
#35 t=7.52e-13 dt=3.85e-14 max_dmdt=2.64e+04
#36 t=7.91e-13 dt=3.85e-14 max_dmdt=2.52e+04
#37 t=8.29e-13 dt=3.85e-14 max_dmdt=2.46e+04
#38 t=8.68e-13 dt=3.85e-14 max_dmdt=2.4e+04
#39 t=9.06e-13 dt=3.85e-14 max_dmdt=2.32e+04
#40 t=9.45e-13 dt=3.85e-14 max_dmdt=2.23e+04
#41 t=9.83e-13 dt=3.85e-14 max_dmdt=2.15e+04
#42 t=1.02e-12 dt=3.85e-14 max_dmdt=2.15e+04
#43 t=1.06e-12 dt=3.85e-14 max_dmdt=2.13e+04
#44 t=1.1e-12 dt=3.85e-14 max_dmdt=2.1e+04
#45 t=1.14e-12 dt=3.85e-14 max_dmdt=2.06e+04
#46 t=1.18e-12 dt=3.85e-14 max_dmdt=2.01e+04
#47 t=1.21e-12 dt=3.85e-14 max_dmdt=1.94e+04
#48 t=1.25e-12 dt=3.85e-14 max_dmdt=1.87e+04
#49 t=1.29e-12 dt=3.85e-14 max_dmdt=1.79e+04
#50 t=1.33e-12 dt=3.85e-14 max_dmdt=1.71e+04
#51 t=1.37e-12 dt=3.85e-14 max_dmdt=1.63e+04
#52 t=1.41e-12 dt=3.85e-14 max_dmdt=1.54e+04
#53 t=1.44e-12 dt=3.85e-14 max_dmdt=1.49e+04
#54 t=1.48e-12 dt=3.85e-14 max_dmdt=1.47e+04
#55 t=1.52e-12 dt=3.85e-14 max_dmdt=1.44e+04
#56 t=1.56e-12 dt=3.85e-14 max_dmdt=1.41e+04
#57 t=1.6e-12 dt=3.85e-14 max_dmdt=1.37e+04
#58 t=1.64e-12 dt=3.85e-14 max_dmdt=1.33e+04
#59 t=1.68e-12 dt=3.85e-14 max_dmdt=1.28e+04
#60 t=1.71e-12 dt=3.85e-14 max_dmdt=1.24e+04
#61 t=1.75e-12 dt=3.85e-14 max_dmdt=1.2e+04
#62 t=1.79e-12 dt=3.85e-14 max_dmdt=1.19e+04
#63 t=1.83e-12 dt=3.85e-14 max_dmdt=1.17e+04
#64 t=1.87e-12 dt=3.85e-14 max_dmdt=1.16e+04
#65 t=1.91e-12 dt=3.85e-14 max_dmdt=1.13e+04
#66 t=1.94e-12 dt=3.85e-14 max_dmdt=1.11e+04
#67 t=1.98e-12 dt=3.85e-14 max_dmdt=1.09e+04
#68 t=2.02e-12 dt=3.85e-14 max_dmdt=1.06e+04
#69 t=2.06e-12 dt=3.85e-14 max_dmdt=1.03e+04
#70 t=2.1e-12 dt=3.85e-14 max_dmdt=1e+04
#71 t=2.14e-12 dt=3.85e-14 max_dmdt=9.71e+03
#72 t=2.18e-12 dt=3.85e-14 max_dmdt=9.49e+03
#73 t=2.21e-12 dt=3.85e-14 max_dmdt=9.38e+03
#74 t=2.25e-12 dt=3.85e-14 max_dmdt=9.25e+03
#75 t=2.29e-12 dt=3.85e-14 max_dmdt=9.09e+03
#76 t=2.33e-12 dt=3.85e-14 max_dmdt=8.92e+03
#77 t=2.37e-12 dt=3.85e-14 max_dmdt=8.74e+03
#78 t=2.41e-12 dt=3.85e-14 max_dmdt=8.54e+03
#79 t=2.44e-12 dt=3.85e-14 max_dmdt=8.34e+03
#80 t=2.48e-12 dt=3.85e-14 max_dmdt=8.13e+03
#81 t=2.54e-12 dt=5.79e-14 max_dmdt=7.86e+03
#82 t=2.6e-12 dt=5.79e-14 max_dmdt=7.53e+03
#83 t=2.66e-12 dt=5.79e-14 max_dmdt=7.21e+03
#84 t=2.71e-12 dt=5.79e-14 max_dmdt=6.9e+03
#85 t=2.77e-12 dt=5.79e-14 max_dmdt=6.59e+03
#86 t=2.83e-12 dt=5.79e-14 max_dmdt=6.46e+03
#87 t=2.89e-12 dt=5.79e-14 max_dmdt=6.33e+03
#88 t=2.95e-12 dt=5.79e-14 max_dmdt=6.29e+03
#89 t=3e-12 dt=5.79e-14 max_dmdt=6.23e+03
#90 t=3.06e-12 dt=5.79e-14 max_dmdt=6.15e+03
#91 t=3.12e-12 dt=5.79e-14 max_dmdt=6.06e+03
#92 t=3.18e-12 dt=5.79e-14 max_dmdt=5.96e+03
#93 t=3.24e-12 dt=5.79e-14 max_dmdt=5.85e+03
#94 t=3.29e-12 dt=5.79e-14 max_dmdt=5.73e+03
#95 t=3.35e-12 dt=5.79e-14 max_dmdt=5.6e+03
#96 t=3.41e-12 dt=5.79e-14 max_dmdt=5.48e+03
#97 t=3.47e-12 dt=5.79e-14 max_dmdt=5.36e+03
#98 t=3.52e-12 dt=5.79e-14 max_dmdt=5.23e+03
#99 t=3.58e-12 dt=5.79e-14 max_dmdt=5.11e+03
#100 t=3.64e-12 dt=5.79e-14 max_dmdt=4.99e+03
#101 t=3.7e-12 dt=5.79e-14 max_dmdt=4.87e+03
#102 t=3.76e-12 dt=5.79e-14 max_dmdt=4.76e+03
#103 t=3.81e-12 dt=5.79e-14 max_dmdt=4.65e+03
#104 t=3.87e-12 dt=5.79e-14 max_dmdt=4.54e+03
#105 t=3.93e-12 dt=5.79e-14 max_dmdt=4.45e+03
#106 t=3.99e-12 dt=5.79e-14 max_dmdt=4.38e+03
#107 t=4.08e-12 dt=8.74e-14 max_dmdt=4.28e+03
#108 t=4.16e-12 dt=8.74e-14 max_dmdt=4.17e+03
#109 t=4.25e-12 dt=8.74e-14 max_dmdt=4.08e+03
#110 t=4.34e-12 dt=8.74e-14 max_dmdt=4.02e+03
#111 t=4.43e-12 dt=8.74e-14 max_dmdt=3.96e+03
#112 t=4.51e-12 dt=8.74e-14 max_dmdt=3.89e+03
#113 t=4.6e-12 dt=8.74e-14 max_dmdt=3.82e+03
#114 t=4.69e-12 dt=8.74e-14 max_dmdt=3.75e+03
#115 t=4.77e-12 dt=8.74e-14 max_dmdt=3.68e+03
#116 t=4.86e-12 dt=8.74e-14 max_dmdt=3.61e+03
#117 t=4.95e-12 dt=8.74e-14 max_dmdt=3.53e+03
#118 t=5.04e-12 dt=8.74e-14 max_dmdt=3.46e+03
#119 t=5.12e-12 dt=8.74e-14 max_dmdt=3.38e+03
#120 t=5.21e-12 dt=8.74e-14 max_dmdt=3.31e+03
#121 t=5.3e-12 dt=8.74e-14 max_dmdt=3.25e+03
#122 t=5.39e-12 dt=8.74e-14 max_dmdt=3.2e+03
#123 t=5.47e-12 dt=8.74e-14 max_dmdt=3.15e+03
#124 t=5.56e-12 dt=8.74e-14 max_dmdt=3.1e+03
#125 t=5.65e-12 dt=8.74e-14 max_dmdt=3.05e+03
#126 t=5.74e-12 dt=8.74e-14 max_dmdt=3e+03
#127 t=5.82e-12 dt=8.74e-14 max_dmdt=2.96e+03
#128 t=5.91e-12 dt=8.74e-14 max_dmdt=2.91e+03
#129 t=6e-12 dt=8.74e-14 max_dmdt=2.86e+03
#130 t=6.09e-12 dt=8.74e-14 max_dmdt=2.82e+03
#131 t=6.17e-12 dt=8.74e-14 max_dmdt=2.78e+03
#132 t=6.26e-12 dt=8.74e-14 max_dmdt=2.73e+03
#133 t=6.35e-12 dt=8.74e-14 max_dmdt=2.69e+03
#134 t=6.44e-12 dt=8.74e-14 max_dmdt=2.65e+03
#135 t=6.52e-12 dt=8.74e-14 max_dmdt=2.61e+03
#136 t=6.61e-12 dt=8.74e-14 max_dmdt=2.57e+03
#137 t=6.7e-12 dt=8.74e-14 max_dmdt=2.53e+03
#138 t=6.78e-12 dt=8.74e-14 max_dmdt=2.49e+03
#139 t=6.87e-12 dt=8.74e-14 max_dmdt=2.45e+03
#140 t=6.96e-12 dt=8.74e-14 max_dmdt=2.42e+03
#141 t=7.05e-12 dt=8.74e-14 max_dmdt=2.38e+03
#142 t=7.13e-12 dt=8.74e-14 max_dmdt=2.35e+03
#143 t=7.22e-12 dt=8.74e-14 max_dmdt=2.31e+03
#144 t=7.31e-12 dt=8.74e-14 max_dmdt=2.29e+03
#145 t=7.4e-12 dt=8.74e-14 max_dmdt=2.27e+03
#146 t=7.48e-12 dt=8.74e-14 max_dmdt=2.25e+03
#147 t=7.57e-12 dt=8.74e-14 max_dmdt=2.23e+03
#148 t=7.66e-12 dt=8.74e-14 max_dmdt=2.21e+03
#149 t=7.75e-12 dt=8.74e-14 max_dmdt=2.18e+03
#150 t=7.83e-12 dt=8.74e-14 max_dmdt=2.16e+03
#151 t=7.92e-12 dt=8.74e-14 max_dmdt=2.14e+03
#152 t=8.01e-12 dt=8.74e-14 max_dmdt=2.12e+03
#153 t=8.1e-12 dt=8.74e-14 max_dmdt=2.1e+03
#154 t=8.18e-12 dt=8.74e-14 max_dmdt=2.08e+03
#155 t=8.27e-12 dt=8.74e-14 max_dmdt=2.06e+03
#156 t=8.36e-12 dt=8.74e-14 max_dmdt=2.04e+03
#157 t=8.45e-12 dt=8.74e-14 max_dmdt=2.02e+03
#158 t=8.53e-12 dt=8.74e-14 max_dmdt=2e+03
#159 t=8.62e-12 dt=8.74e-14 max_dmdt=1.98e+03
#160 t=8.71e-12 dt=8.74e-14 max_dmdt=1.96e+03
#161 t=8.79e-12 dt=8.74e-14 max_dmdt=1.94e+03
#162 t=8.88e-12 dt=8.74e-14 max_dmdt=1.92e+03
#163 t=8.97e-12 dt=8.74e-14 max_dmdt=1.9e+03
#164 t=9.06e-12 dt=8.74e-14 max_dmdt=1.88e+03
#165 t=9.14e-12 dt=8.74e-14 max_dmdt=1.86e+03
#166 t=9.28e-12 dt=1.32e-13 max_dmdt=1.84e+03
#167 t=9.41e-12 dt=1.32e-13 max_dmdt=1.81e+03
#168 t=9.54e-12 dt=1.32e-13 max_dmdt=1.78e+03
#169 t=9.67e-12 dt=1.32e-13 max_dmdt=1.76e+03
#170 t=9.81e-12 dt=1.32e-13 max_dmdt=1.73e+03
#171 t=1e-11 dt=2.01e-13 max_dmdt=1.7e+03
#172 t=1.02e-11 dt=2.01e-13 max_dmdt=1.66e+03
#173 t=1.04e-11 dt=2.01e-13 max_dmdt=1.63e+03
#174 t=1.06e-11 dt=2.01e-13 max_dmdt=1.59e+03
#175 t=1.08e-11 dt=2.01e-13 max_dmdt=1.57e+03
#176 t=1.1e-11 dt=2.01e-13 max_dmdt=1.54e+03
#177 t=1.12e-11 dt=2.01e-13 max_dmdt=1.52e+03
#178 t=1.14e-11 dt=2.01e-13 max_dmdt=1.5e+03
#179 t=1.16e-11 dt=2.01e-13 max_dmdt=1.47e+03
#180 t=1.18e-11 dt=2.01e-13 max_dmdt=1.45e+03
#181 t=1.2e-11 dt=2.01e-13 max_dmdt=1.43e+03
#182 t=1.22e-11 dt=2.01e-13 max_dmdt=1.41e+03
#183 t=1.24e-11 dt=2.01e-13 max_dmdt=1.38e+03
#184 t=1.26e-11 dt=2.01e-13 max_dmdt=1.36e+03
#185 t=1.28e-11 dt=2.01e-13 max_dmdt=1.34e+03
#186 t=1.3e-11 dt=2.01e-13 max_dmdt=1.32e+03
#187 t=1.32e-11 dt=2.01e-13 max_dmdt=1.3e+03
#188 t=1.34e-11 dt=2.01e-13 max_dmdt=1.28e+03
#189 t=1.36e-11 dt=2.01e-13 max_dmdt=1.26e+03
#190 t=1.38e-11 dt=2.01e-13 max_dmdt=1.24e+03
#191 t=1.4e-11 dt=2.01e-13 max_dmdt=1.22e+03
#192 t=1.42e-11 dt=2.01e-13 max_dmdt=1.21e+03
#193 t=1.44e-11 dt=2.01e-13 max_dmdt=1.19e+03
#194 t=1.46e-11 dt=2.01e-13 max_dmdt=1.17e+03
#195 t=1.48e-11 dt=2.01e-13 max_dmdt=1.15e+03
#196 t=1.51e-11 dt=3.05e-13 max_dmdt=1.14e+03
#197 t=1.54e-11 dt=3.05e-13 max_dmdt=1.12e+03
#198 t=1.58e-11 dt=3.05e-13 max_dmdt=1.1e+03
#199 t=1.61e-11 dt=3.05e-13 max_dmdt=1.08e+03
#200 t=1.64e-11 dt=3.05e-13 max_dmdt=1.06e+03
#201 t=1.67e-11 dt=3.05e-13 max_dmdt=1.05e+03
#202 t=1.7e-11 dt=3.05e-13 max_dmdt=1.03e+03
#203 t=1.73e-11 dt=3.05e-13 max_dmdt=1.01e+03
#204 t=1.76e-11 dt=3.05e-13 max_dmdt=997
#205 t=1.79e-11 dt=3.05e-13 max_dmdt=980
#206 t=1.82e-11 dt=3.05e-13 max_dmdt=965
#207 t=1.85e-11 dt=3.05e-13 max_dmdt=949
#208 t=1.88e-11 dt=3.05e-13 max_dmdt=934
#209 t=1.91e-11 dt=3.05e-13 max_dmdt=920
#210 t=1.94e-11 dt=3.05e-13 max_dmdt=905
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#509 t=2.11e-10 dt=1.81e-12 max_dmdt=224
#510 t=2.13e-10 dt=1.81e-12 max_dmdt=223
#511 t=2.15e-10 dt=1.81e-12 max_dmdt=222
#512 t=2.17e-10 dt=1.81e-12 max_dmdt=220
#513 t=2.19e-10 dt=1.81e-12 max_dmdt=218
#514 t=2.21e-10 dt=1.81e-12 max_dmdt=216
#515 t=2.22e-10 dt=1.81e-12 max_dmdt=214
#516 t=2.24e-10 dt=1.81e-12 max_dmdt=212
#517 t=2.26e-10 dt=1.81e-12 max_dmdt=209
#518 t=2.28e-10 dt=1.81e-12 max_dmdt=207
#519 t=2.3e-10 dt=1.81e-12 max_dmdt=204
#520 t=2.31e-10 dt=1.81e-12 max_dmdt=201
#521 t=2.33e-10 dt=1.81e-12 max_dmdt=197
#522 t=2.35e-10 dt=1.81e-12 max_dmdt=194
#523 t=2.37e-10 dt=1.81e-12 max_dmdt=191
#524 t=2.39e-10 dt=1.81e-12 max_dmdt=187
#525 t=2.4e-10 dt=1.81e-12 max_dmdt=184
#526 t=2.42e-10 dt=1.81e-12 max_dmdt=180
#527 t=2.44e-10 dt=1.81e-12 max_dmdt=176
#528 t=2.46e-10 dt=1.81e-12 max_dmdt=172
#529 t=2.48e-10 dt=1.81e-12 max_dmdt=168
#530 t=2.49e-10 dt=1.81e-12 max_dmdt=164
#531 t=2.51e-10 dt=1.81e-12 max_dmdt=160
#532 t=2.53e-10 dt=1.81e-12 max_dmdt=156
#533 t=2.55e-10 dt=1.81e-12 max_dmdt=152
#534 t=2.57e-10 dt=1.81e-12 max_dmdt=148
#535 t=2.59e-10 dt=1.81e-12 max_dmdt=144
#536 t=2.6e-10 dt=1.81e-12 max_dmdt=140
#537 t=2.62e-10 dt=1.81e-12 max_dmdt=136
#538 t=2.64e-10 dt=1.81e-12 max_dmdt=132
#539 t=2.67e-10 dt=2.74e-12 max_dmdt=127
#540 t=2.69e-10 dt=2.74e-12 max_dmdt=121
#541 t=2.72e-10 dt=2.74e-12 max_dmdt=115
#542 t=2.75e-10 dt=2.74e-12 max_dmdt=110
#543 t=2.78e-10 dt=2.74e-12 max_dmdt=104
#544 t=2.8e-10 dt=2.74e-12 max_dmdt=98.3
#545 t=2.83e-10 dt=2.74e-12 max_dmdt=92.9
#546 t=2.86e-10 dt=2.74e-12 max_dmdt=87.7
#547 t=2.89e-10 dt=2.74e-12 max_dmdt=82.8
#548 t=2.91e-10 dt=2.74e-12 max_dmdt=78
#549 t=2.94e-10 dt=2.74e-12 max_dmdt=77.7
#550 t=2.97e-10 dt=2.74e-12 max_dmdt=78.4
#551 t=3e-10 dt=2.74e-12 max_dmdt=78.9
#552 t=3.02e-10 dt=2.74e-12 max_dmdt=79.1
#553 t=3.05e-10 dt=2.74e-12 max_dmdt=79.1
#554 t=3.08e-10 dt=2.74e-12 max_dmdt=78.9
#555 t=3.11e-10 dt=2.74e-12 max_dmdt=78.5
#556 t=3.13e-10 dt=2.74e-12 max_dmdt=77.9
#557 t=3.16e-10 dt=2.74e-12 max_dmdt=77.1
#558 t=3.19e-10 dt=2.74e-12 max_dmdt=76.2
#559 t=3.22e-10 dt=2.74e-12 max_dmdt=75.1
#560 t=3.24e-10 dt=2.74e-12 max_dmdt=73.9
#561 t=3.27e-10 dt=2.74e-12 max_dmdt=72.5
#562 t=3.3e-10 dt=2.74e-12 max_dmdt=71.1
#563 t=3.32e-10 dt=2.74e-12 max_dmdt=69.5
#564 t=3.35e-10 dt=2.74e-12 max_dmdt=67.8
#565 t=3.38e-10 dt=2.74e-12 max_dmdt=66.1
#566 t=3.41e-10 dt=2.74e-12 max_dmdt=64.2
#567 t=3.43e-10 dt=2.74e-12 max_dmdt=62.3
#568 t=3.46e-10 dt=2.74e-12 max_dmdt=60.4
#569 t=3.49e-10 dt=2.74e-12 max_dmdt=58.4
#570 t=3.52e-10 dt=2.74e-12 max_dmdt=56.4
#571 t=3.54e-10 dt=2.74e-12 max_dmdt=54.3
#572 t=3.57e-10 dt=2.74e-12 max_dmdt=52.2
#573 t=3.6e-10 dt=2.74e-12 max_dmdt=50.1
#574 t=3.63e-10 dt=2.74e-12 max_dmdt=48
#575 t=3.65e-10 dt=2.74e-12 max_dmdt=45.9
#576 t=3.68e-10 dt=2.74e-12 max_dmdt=43.8
#577 t=3.71e-10 dt=2.74e-12 max_dmdt=41.7
#578 t=3.74e-10 dt=2.74e-12 max_dmdt=39.7
#579 t=3.76e-10 dt=2.74e-12 max_dmdt=37.7
#580 t=3.79e-10 dt=2.74e-12 max_dmdt=35.8
#581 t=3.82e-10 dt=2.74e-12 max_dmdt=34
#582 t=3.85e-10 dt=2.74e-12 max_dmdt=32.2
#583 t=3.87e-10 dt=2.74e-12 max_dmdt=31.5
#584 t=3.9e-10 dt=2.74e-12 max_dmdt=31.8
#585 t=3.93e-10 dt=2.74e-12 max_dmdt=32
#586 t=3.96e-10 dt=2.74e-12 max_dmdt=32.1
#587 t=3.98e-10 dt=2.74e-12 max_dmdt=32.1
#588 t=4.01e-10 dt=2.74e-12 max_dmdt=32.1
#589 t=4.04e-10 dt=2.74e-12 max_dmdt=32
#590 t=4.06e-10 dt=2.74e-12 max_dmdt=31.8
#591 t=4.09e-10 dt=2.74e-12 max_dmdt=31.5
#592 t=4.12e-10 dt=2.74e-12 max_dmdt=31.2
#593 t=4.15e-10 dt=2.74e-12 max_dmdt=30.8
#594 t=4.17e-10 dt=2.74e-12 max_dmdt=30.3
#595 t=4.2e-10 dt=2.74e-12 max_dmdt=29.8
#596 t=4.23e-10 dt=2.74e-12 max_dmdt=29.3
#597 t=4.26e-10 dt=2.74e-12 max_dmdt=28.7
#598 t=4.28e-10 dt=2.74e-12 max_dmdt=28.1
#599 t=4.31e-10 dt=2.74e-12 max_dmdt=27.4
#600 t=4.34e-10 dt=2.74e-12 max_dmdt=26.7
#601 t=4.37e-10 dt=2.74e-12 max_dmdt=26
#602 t=4.39e-10 dt=2.74e-12 max_dmdt=25.2
#603 t=4.42e-10 dt=2.74e-12 max_dmdt=24.4
#604 t=4.45e-10 dt=2.74e-12 max_dmdt=23.6
#605 t=4.48e-10 dt=2.74e-12 max_dmdt=22.8
#606 t=4.5e-10 dt=2.74e-12 max_dmdt=21.9
#607 t=4.53e-10 dt=2.74e-12 max_dmdt=21.1
#608 t=4.57e-10 dt=4.11e-12 max_dmdt=20
#609 t=4.61e-10 dt=4.11e-12 max_dmdt=18.8
#610 t=4.65e-10 dt=4.11e-12 max_dmdt=17.5
#611 t=4.69e-10 dt=4.11e-12 max_dmdt=16.3
#612 t=4.74e-10 dt=4.11e-12 max_dmdt=15.1
#613 t=4.78e-10 dt=4.11e-12 max_dmdt=14
#614 t=4.82e-10 dt=4.11e-12 max_dmdt=13
#615 t=4.86e-10 dt=4.11e-12 max_dmdt=12.9
#616 t=4.9e-10 dt=4.11e-12 max_dmdt=13.1
#617 t=4.94e-10 dt=4.11e-12 max_dmdt=13.1
#618 t=4.98e-10 dt=4.11e-12 max_dmdt=13.2
#619 t=5.02e-10 dt=4.11e-12 max_dmdt=13.1
#620 t=5.06e-10 dt=4.11e-12 max_dmdt=13
#621 t=5.11e-10 dt=4.11e-12 max_dmdt=12.8
#622 t=5.15e-10 dt=4.11e-12 max_dmdt=12.5
#623 t=5.19e-10 dt=4.11e-12 max_dmdt=12.2
#624 t=5.23e-10 dt=4.11e-12 max_dmdt=11.8
#625 t=5.27e-10 dt=4.11e-12 max_dmdt=11.4
#626 t=5.31e-10 dt=4.11e-12 max_dmdt=11
#627 t=5.35e-10 dt=4.11e-12 max_dmdt=10.6
#628 t=5.39e-10 dt=4.11e-12 max_dmdt=10.1
#629 t=5.44e-10 dt=4.11e-12 max_dmdt=9.56
#630 t=5.48e-10 dt=4.11e-12 max_dmdt=9.03
#631 t=5.52e-10 dt=4.11e-12 max_dmdt=8.5
#632 t=5.56e-10 dt=4.11e-12 max_dmdt=7.96
#633 t=5.6e-10 dt=4.11e-12 max_dmdt=7.43
#634 t=5.64e-10 dt=4.11e-12 max_dmdt=6.91
#635 t=5.68e-10 dt=4.11e-12 max_dmdt=6.42
#636 t=5.72e-10 dt=4.11e-12 max_dmdt=5.95
#637 t=5.76e-10 dt=4.11e-12 max_dmdt=5.52
#638 t=5.81e-10 dt=4.11e-12 max_dmdt=5.5
#639 t=5.85e-10 dt=4.11e-12 max_dmdt=5.56
#640 t=5.89e-10 dt=4.11e-12 max_dmdt=5.59
#641 t=5.93e-10 dt=4.11e-12 max_dmdt=5.59
#642 t=5.97e-10 dt=4.11e-12 max_dmdt=5.55
#643 t=6.01e-10 dt=4.11e-12 max_dmdt=5.49
#644 t=6.05e-10 dt=4.11e-12 max_dmdt=5.41
#645 t=6.09e-10 dt=4.11e-12 max_dmdt=5.3
#646 t=6.13e-10 dt=4.11e-12 max_dmdt=5.17
#647 t=6.18e-10 dt=4.11e-12 max_dmdt=5.02
#648 t=6.22e-10 dt=4.11e-12 max_dmdt=4.85
#649 t=6.26e-10 dt=4.11e-12 max_dmdt=4.67
#650 t=6.3e-10 dt=4.11e-12 max_dmdt=4.48
#651 t=6.34e-10 dt=4.11e-12 max_dmdt=4.28
#652 t=6.38e-10 dt=4.11e-12 max_dmdt=4.07
#653 t=6.42e-10 dt=4.11e-12 max_dmdt=3.85
#654 t=6.46e-10 dt=4.11e-12 max_dmdt=3.63
#655 t=6.5e-10 dt=4.11e-12 max_dmdt=3.4
#656 t=6.55e-10 dt=4.11e-12 max_dmdt=3.18
#657 t=6.59e-10 dt=4.11e-12 max_dmdt=2.97
#658 t=6.63e-10 dt=4.11e-12 max_dmdt=2.76
#659 t=6.67e-10 dt=4.11e-12 max_dmdt=2.56
#660 t=6.71e-10 dt=4.11e-12 max_dmdt=2.38
#661 t=6.75e-10 dt=4.11e-12 max_dmdt=2.26
#662 t=6.79e-10 dt=4.11e-12 max_dmdt=2.31
#663 t=6.83e-10 dt=4.11e-12 max_dmdt=2.33
#664 t=6.89e-10 dt=6.18e-12 max_dmdt=2.34
#665 t=6.96e-10 dt=6.18e-12 max_dmdt=2.33
#666 t=7.02e-10 dt=6.18e-12 max_dmdt=2.28
#667 t=7.08e-10 dt=6.18e-12 max_dmdt=2.22
#668 t=7.14e-10 dt=6.18e-12 max_dmdt=2.13
#669 t=7.2e-10 dt=6.18e-12 max_dmdt=2.02
#670 t=7.27e-10 dt=6.18e-12 max_dmdt=1.91
#671 t=7.33e-10 dt=6.18e-12 max_dmdt=1.78
#672 t=7.39e-10 dt=6.18e-12 max_dmdt=1.64
#673 t=7.45e-10 dt=6.18e-12 max_dmdt=1.5
#674 t=7.51e-10 dt=6.18e-12 max_dmdt=1.35
#675 t=7.57e-10 dt=6.18e-12 max_dmdt=1.22
#676 t=7.64e-10 dt=6.18e-12 max_dmdt=1.09
#677 t=7.7e-10 dt=6.18e-12 max_dmdt=0.979
#678 t=7.76e-10 dt=6.18e-12 max_dmdt=1
#679 t=7.82e-10 dt=6.18e-12 max_dmdt=1.01
#680 t=7.88e-10 dt=6.18e-12 max_dmdt=1.01
#681 t=7.95e-10 dt=6.18e-12 max_dmdt=0.994
#682 t=8.01e-10 dt=6.18e-12 max_dmdt=0.968
#683 t=8.07e-10 dt=6.18e-12 max_dmdt=0.933
#684 t=8.13e-10 dt=6.18e-12 max_dmdt=0.891
#685 t=8.19e-10 dt=6.18e-12 max_dmdt=0.843
#686 t=8.25e-10 dt=6.18e-12 max_dmdt=0.789
#687 t=8.32e-10 dt=6.18e-12 max_dmdt=0.732
#688 t=8.38e-10 dt=6.18e-12 max_dmdt=0.672
#689 t=8.44e-10 dt=6.18e-12 max_dmdt=0.612
#690 t=8.5e-10 dt=6.18e-12 max_dmdt=0.552
#691 t=8.56e-10 dt=6.18e-12 max_dmdt=0.496
#692 t=8.63e-10 dt=6.18e-12 max_dmdt=0.448
#693 t=8.69e-10 dt=6.18e-12 max_dmdt=0.42
#694 t=8.75e-10 dt=6.18e-12 max_dmdt=0.429
#695 t=8.81e-10 dt=6.18e-12 max_dmdt=0.432
#696 t=8.87e-10 dt=6.18e-12 max_dmdt=0.43
#697 t=8.93e-10 dt=6.18e-12 max_dmdt=0.422
#698 t=9e-10 dt=6.18e-12 max_dmdt=0.41
#699 t=9.06e-10 dt=6.18e-12 max_dmdt=0.394
#700 t=9.12e-10 dt=6.18e-12 max_dmdt=0.375
#701 t=9.18e-10 dt=6.18e-12 max_dmdt=0.353
#702 t=9.28e-10 dt=9.45e-12 max_dmdt=0.323
#703 t=9.37e-10 dt=9.45e-12 max_dmdt=0.283
#704 t=9.47e-10 dt=9.45e-12 max_dmdt=0.243
#705 t=9.56e-10 dt=9.45e-12 max_dmdt=0.21
#706 t=9.65e-10 dt=9.45e-12 max_dmdt=0.184
#707 t=9.75e-10 dt=9.45e-12 max_dmdt=0.189
#708 t=9.84e-10 dt=9.45e-12 max_dmdt=0.188
#709 t=9.94e-10 dt=9.45e-12 max_dmdt=0.181
#710 t=1e-09 dt=9.45e-12 max_dmdt=0.171
#711 t=1.01e-09 dt=9.45e-12 max_dmdt=0.158
#712 t=1.02e-09 dt=9.45e-12 max_dmdt=0.142
#713 t=1.03e-09 dt=9.45e-12 max_dmdt=0.125
#714 t=1.04e-09 dt=9.45e-12 max_dmdt=0.109
#715 t=1.05e-09 dt=9.45e-12 max_dmdt=0.0946
#716 t=1.06e-09 dt=9.45e-12 max_dmdt=0.0805
#717 t=1.07e-09 dt=9.45e-12 max_dmdt=0.0812
#718 t=1.08e-09 dt=9.45e-12 max_dmdt=0.0814
#719 t=1.09e-09 dt=9.45e-12 max_dmdt=0.0792
#720 t=1.1e-09 dt=9.45e-12 max_dmdt=0.0752
#721 t=1.11e-09 dt=9.45e-12 max_dmdt=0.0697
#722 t=1.12e-09 dt=9.45e-12 max_dmdt=0.0633
#723 t=1.13e-09 dt=9.45e-12 max_dmdt=0.0564
#724 t=1.14e-09 dt=9.45e-12 max_dmdt=0.0494
#725 t=1.14e-09 dt=9.45e-12 max_dmdt=0.0425
#726 t=1.15e-09 dt=9.45e-12 max_dmdt=0.0357
#727 t=1.16e-09 dt=9.45e-12 max_dmdt=0.0356
#728 t=1.18e-09 dt=1.42e-11 max_dmdt=0.0355
#729 t=1.19e-09 dt=1.42e-11 max_dmdt=0.0338
#730 t=1.21e-09 dt=1.42e-11 max_dmdt=0.0305
#731 t=1.22e-09 dt=1.42e-11 max_dmdt=0.0262
#732 t=1.23e-09 dt=1.42e-11 max_dmdt=0.0214
#733 t=1.25e-09 dt=1.42e-11 max_dmdt=0.0166
#734 t=1.26e-09 dt=1.42e-11 max_dmdt=0.0155
#735 t=1.28e-09 dt=1.42e-11 max_dmdt=0.0156
#736 t=1.29e-09 dt=1.42e-11 max_dmdt=0.0147
#737 t=1.31e-09 dt=1.42e-11 max_dmdt=0.0131
#738 t=1.32e-09 dt=1.42e-11 max_dmdt=0.0111
#739 t=1.33e-09 dt=1.42e-11 max_dmdt=0.00887
Plot the magnetisation components of the relaxed state.
[8]:
#NBVAL_IGNORE_OUTPUT
plot_magnetisation([['m0.npy', 'mx'], ['m0.npy', 'my'], ['m0.npy', 'mz']])
The DW is at the maximum value of \(|m_z|\) or \(|m_y|\). Consequently, the domain wall position is:
[9]:
m0_z = np.load('m0.npy').reshape(-1, 3)[:, 2]
x = np.arange(len(m0_z))
index_max = np.argmax(np.abs(m0_z))
print('Maximum |m_z| at x = %s' % x[index_max])
Maximum |m_z| at x = 124
Using the obtained domain wall equilibrium state, we now simulate its motion in presence of a spin-polarised current.
[10]:
#NBVAL_IGNORE_OUTPUT
excite_system(mesh, 1.5e-9, 151)
We plot once again to compare the initial state with the ones after a SP current was applied.
[11]:
#NBVAL_IGNORE_OUTPUT
# We can plot the m_x component for a number snapshots
# to observe the DW motion
# We will plot the 100th and 150th files (we can also compute
# until the system reaches ~5 ns to improve the effect)
plot_magnetisation([['m0.npy', 'mx'],
['dyn_npys/m_100.npy', 'mx'],
['dyn_npys/m_150.npy', 'mx']])
plt.xlim([0, 200])
[11]:
(0, 200)
