Skyrmions under an applied field gradient#
[1]:
import fidimag
from fidimag.micro import Sim
from fidimag.common import CuboidMesh
from fidimag.micro import Zeeman, Demag, DMI, UniformExchange
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
Here we set parameters of the system. We choose a 250 x 80 x 3 nm nanotrack, and apply periodicity in the x-direction to approximate an infinite nanotrack.
[2]:
dx = dy = 2.5
dz = 1.0
Lx = 250.0
Ly = 80.0
Lz = 3.0
nx = int(Lx/dx)
ny = int(Ly/dy)
nz = int(Lz/dz)
mesh = CuboidMesh(nx=nx, ny=ny, nz=nz, dx=dx, dy=dy, dz=dz,
unit_length=1e-9, periodicity=(True, False, False))
We choose material parameters for Fe0.7Co0.3Si as used in the paper “Skyrmions in thin films with easy-plane magnetocrystalline anisotropy” (http://dx.doi.org/10.1063/1.4945262). We set the initial state to be uniform in the +z direction.
[3]:
Ms = 9.5e4 # magnetisation saturation (A/m)
A = 4e-13 # exchange stiffness (J/m)
D = 2.7e-4 # DMI constant (J/m**2)
H = (0, 0, 3.8e5) # external magnetic field (A/m)
K0 = D**2 / A
K1 = -0.16*K0
alpha = 0.5 # Gilbert damping
gamma = 2.211e5 # gyromagnetic ratio (m/As)
sim = Sim(mesh) # create simulation object
# Set parameters.
sim.Ms = Ms
sim.driver.alpha = alpha
sim.driver.gamma = gamma
sim.driver.do_precession = False
# Add energies.
sim.add(UniformExchange(A=A))
sim.add(DMI(D))
sim.add(Zeeman(H))
sim.add(Demag(pbc_2d=True))
# Seed random number generator so that this simulation is reproducible.
np.random.seed(0)
def m_init(pos):
m = np.random.uniform(-1, 1, 3)
return m
sim.set_m(m_init, normalise=True)
We now relax the system to find a metastable state.
[4]:
sim.driver.relax(dt=1e-13, stopping_dmdt=0.1, max_steps=5000, save_m_steps=None, save_vtk_steps=None)
#1 t=1e-13 dt=1e-13 max_dmdt=7.78e+04
#2 t=2e-13 dt=1e-13 max_dmdt=7.99e+04
#3 t=3e-13 dt=1e-13 max_dmdt=8.01e+04
#4 t=4e-13 dt=1e-13 max_dmdt=7.76e+04
#5 t=5e-13 dt=1e-13 max_dmdt=7.97e+04
#6 t=6e-13 dt=1e-13 max_dmdt=7.75e+04
#7 t=7e-13 dt=1e-13 max_dmdt=7.6e+04
#8 t=8e-13 dt=1e-13 max_dmdt=7.94e+04
#9 t=9e-13 dt=1e-13 max_dmdt=8.26e+04
#10 t=1e-12 dt=1e-13 max_dmdt=8.09e+04
#11 t=1.1e-12 dt=1e-13 max_dmdt=7.42e+04
#12 t=1.2e-12 dt=1e-13 max_dmdt=7.66e+04
#13 t=1.3e-12 dt=1e-13 max_dmdt=7.76e+04
#14 t=1.4e-12 dt=1e-13 max_dmdt=7.47e+04
#15 t=1.5e-12 dt=1e-13 max_dmdt=7.08e+04
#16 t=1.6e-12 dt=1e-13 max_dmdt=6.4e+04
#17 t=1.7e-12 dt=1e-13 max_dmdt=6.47e+04
#18 t=1.8e-12 dt=1e-13 max_dmdt=6.4e+04
#19 t=1.9e-12 dt=1e-13 max_dmdt=6.21e+04
#20 t=2e-12 dt=1e-13 max_dmdt=6.02e+04
#21 t=2.1e-12 dt=1e-13 max_dmdt=5.71e+04
#22 t=2.2e-12 dt=1e-13 max_dmdt=5.63e+04
#23 t=2.3e-12 dt=1e-13 max_dmdt=5.47e+04
#24 t=2.4e-12 dt=1e-13 max_dmdt=5.24e+04
#25 t=2.5e-12 dt=1e-13 max_dmdt=5.1e+04
#26 t=2.6e-12 dt=1e-13 max_dmdt=5.1e+04
#27 t=2.7e-12 dt=1e-13 max_dmdt=5.02e+04
#28 t=2.8e-12 dt=1e-13 max_dmdt=4.86e+04
#29 t=2.9e-12 dt=1e-13 max_dmdt=4.64e+04
#30 t=3e-12 dt=1e-13 max_dmdt=4.37e+04
#31 t=3.1e-12 dt=1e-13 max_dmdt=4.24e+04
#32 t=3.2e-12 dt=1e-13 max_dmdt=4.19e+04
#33 t=3.3e-12 dt=1e-13 max_dmdt=4.09e+04
#34 t=3.4e-12 dt=1e-13 max_dmdt=3.95e+04
#35 t=3.5e-12 dt=1e-13 max_dmdt=3.92e+04
#36 t=3.6e-12 dt=1e-13 max_dmdt=3.92e+04
#37 t=3.7e-12 dt=1e-13 max_dmdt=3.95e+04
#38 t=3.8e-12 dt=1e-13 max_dmdt=4.03e+04
#39 t=3.9e-12 dt=1e-13 max_dmdt=4.05e+04
#40 t=4e-12 dt=1e-13 max_dmdt=4.08e+04
#41 t=4.1e-12 dt=1e-13 max_dmdt=4.12e+04
#42 t=4.2e-12 dt=1e-13 max_dmdt=4.13e+04
#43 t=4.3e-12 dt=1e-13 max_dmdt=4.12e+04
#44 t=4.4e-12 dt=1e-13 max_dmdt=4.1e+04
#45 t=4.5e-12 dt=1e-13 max_dmdt=4.06e+04
#46 t=4.6e-12 dt=1e-13 max_dmdt=4e+04
#47 t=4.7e-12 dt=1e-13 max_dmdt=3.93e+04
#48 t=4.8e-12 dt=1e-13 max_dmdt=3.85e+04
#49 t=4.9e-12 dt=1e-13 max_dmdt=3.76e+04
#50 t=5e-12 dt=1e-13 max_dmdt=3.66e+04
#51 t=5.1e-12 dt=1e-13 max_dmdt=3.55e+04
#52 t=5.2e-12 dt=1e-13 max_dmdt=3.43e+04
#53 t=5.3e-12 dt=1e-13 max_dmdt=3.32e+04
#54 t=5.4e-12 dt=1e-13 max_dmdt=3.23e+04
#55 t=5.5e-12 dt=1e-13 max_dmdt=3.19e+04
#56 t=5.6e-12 dt=1e-13 max_dmdt=3.21e+04
#57 t=5.7e-12 dt=1e-13 max_dmdt=3.23e+04
#58 t=5.8e-12 dt=1e-13 max_dmdt=3.26e+04
#59 t=5.9e-12 dt=1e-13 max_dmdt=3.29e+04
#60 t=6e-12 dt=1e-13 max_dmdt=3.31e+04
#61 t=6.1e-12 dt=1e-13 max_dmdt=3.32e+04
#62 t=6.2e-12 dt=1e-13 max_dmdt=3.31e+04
#63 t=6.3e-12 dt=1e-13 max_dmdt=3.3e+04
#64 t=6.4e-12 dt=1e-13 max_dmdt=3.27e+04
#65 t=6.5e-12 dt=1e-13 max_dmdt=3.23e+04
#66 t=6.6e-12 dt=1e-13 max_dmdt=3.18e+04
#67 t=6.7e-12 dt=1e-13 max_dmdt=3.13e+04
#68 t=6.8e-12 dt=1e-13 max_dmdt=3.11e+04
#69 t=6.9e-12 dt=1e-13 max_dmdt=3.09e+04
#70 t=7e-12 dt=1e-13 max_dmdt=3.07e+04
#71 t=7.1e-12 dt=1e-13 max_dmdt=3.04e+04
#72 t=7.2e-12 dt=1e-13 max_dmdt=3.04e+04
#73 t=7.3e-12 dt=1e-13 max_dmdt=3.03e+04
#74 t=7.4e-12 dt=1e-13 max_dmdt=3.01e+04
#75 t=7.5e-12 dt=1e-13 max_dmdt=3.03e+04
#76 t=7.6e-12 dt=1e-13 max_dmdt=3.05e+04
#77 t=7.7e-12 dt=1e-13 max_dmdt=3.05e+04
#78 t=7.8e-12 dt=1e-13 max_dmdt=3.05e+04
#79 t=7.9e-12 dt=1e-13 max_dmdt=3.03e+04
#80 t=8e-12 dt=1e-13 max_dmdt=3.01e+04
#81 t=8.1e-12 dt=1e-13 max_dmdt=2.97e+04
#82 t=8.23e-12 dt=1.27e-13 max_dmdt=2.93e+04
#83 t=8.35e-12 dt=1.27e-13 max_dmdt=2.86e+04
#84 t=8.48e-12 dt=1.27e-13 max_dmdt=2.78e+04
#85 t=8.61e-12 dt=1.27e-13 max_dmdt=2.68e+04
#86 t=8.73e-12 dt=1.27e-13 max_dmdt=2.6e+04
#87 t=8.86e-12 dt=1.27e-13 max_dmdt=2.64e+04
#88 t=8.99e-12 dt=1.27e-13 max_dmdt=2.66e+04
#89 t=9.11e-12 dt=1.27e-13 max_dmdt=2.67e+04
#90 t=9.24e-12 dt=1.27e-13 max_dmdt=2.67e+04
#91 t=9.37e-12 dt=1.27e-13 max_dmdt=2.65e+04
#92 t=9.49e-12 dt=1.27e-13 max_dmdt=2.63e+04
#93 t=9.62e-12 dt=1.27e-13 max_dmdt=2.6e+04
#94 t=9.75e-12 dt=1.27e-13 max_dmdt=2.55e+04
#95 t=9.88e-12 dt=1.27e-13 max_dmdt=2.51e+04
#96 t=1e-11 dt=1.27e-13 max_dmdt=2.45e+04
#97 t=1.01e-11 dt=1.27e-13 max_dmdt=2.44e+04
#98 t=1.03e-11 dt=1.27e-13 max_dmdt=2.43e+04
#99 t=1.04e-11 dt=1.27e-13 max_dmdt=2.4e+04
#100 t=1.05e-11 dt=1.27e-13 max_dmdt=2.37e+04
#101 t=1.06e-11 dt=1.27e-13 max_dmdt=2.33e+04
#102 t=1.08e-11 dt=1.27e-13 max_dmdt=2.29e+04
#103 t=1.09e-11 dt=1.27e-13 max_dmdt=2.24e+04
#104 t=1.1e-11 dt=1.27e-13 max_dmdt=2.18e+04
#105 t=1.11e-11 dt=1.27e-13 max_dmdt=2.12e+04
#106 t=1.13e-11 dt=1.27e-13 max_dmdt=2.06e+04
#107 t=1.14e-11 dt=1.27e-13 max_dmdt=2e+04
#108 t=1.15e-11 dt=1.27e-13 max_dmdt=1.99e+04
#109 t=1.17e-11 dt=1.27e-13 max_dmdt=1.98e+04
#110 t=1.18e-11 dt=1.27e-13 max_dmdt=1.96e+04
#111 t=1.19e-11 dt=1.27e-13 max_dmdt=1.94e+04
#112 t=1.2e-11 dt=1.27e-13 max_dmdt=1.92e+04
#113 t=1.22e-11 dt=1.27e-13 max_dmdt=1.89e+04
#114 t=1.23e-11 dt=1.27e-13 max_dmdt=1.86e+04
#115 t=1.24e-11 dt=1.27e-13 max_dmdt=1.85e+04
#116 t=1.25e-11 dt=1.27e-13 max_dmdt=1.92e+04
#117 t=1.27e-11 dt=1.27e-13 max_dmdt=1.99e+04
#118 t=1.28e-11 dt=1.27e-13 max_dmdt=2.05e+04
#119 t=1.29e-11 dt=1.27e-13 max_dmdt=2.1e+04
#120 t=1.3e-11 dt=1.27e-13 max_dmdt=2.15e+04
#121 t=1.32e-11 dt=1.27e-13 max_dmdt=2.19e+04
#122 t=1.33e-11 dt=1.27e-13 max_dmdt=2.23e+04
#123 t=1.34e-11 dt=1.27e-13 max_dmdt=2.25e+04
#124 t=1.36e-11 dt=1.27e-13 max_dmdt=2.27e+04
#125 t=1.37e-11 dt=1.27e-13 max_dmdt=2.28e+04
#126 t=1.38e-11 dt=1.27e-13 max_dmdt=2.28e+04
#127 t=1.39e-11 dt=1.27e-13 max_dmdt=2.28e+04
#128 t=1.41e-11 dt=1.27e-13 max_dmdt=2.27e+04
#129 t=1.42e-11 dt=1.27e-13 max_dmdt=2.24e+04
#130 t=1.43e-11 dt=1.27e-13 max_dmdt=2.22e+04
#131 t=1.44e-11 dt=1.27e-13 max_dmdt=2.18e+04
#132 t=1.46e-11 dt=1.27e-13 max_dmdt=2.14e+04
#133 t=1.47e-11 dt=1.27e-13 max_dmdt=2.1e+04
#134 t=1.48e-11 dt=1.27e-13 max_dmdt=2.05e+04
#135 t=1.49e-11 dt=1.27e-13 max_dmdt=1.99e+04
#136 t=1.51e-11 dt=1.27e-13 max_dmdt=1.94e+04
#137 t=1.52e-11 dt=1.27e-13 max_dmdt=1.88e+04
#138 t=1.53e-11 dt=1.27e-13 max_dmdt=1.88e+04
#139 t=1.55e-11 dt=1.27e-13 max_dmdt=1.88e+04
#140 t=1.56e-11 dt=1.92e-13 max_dmdt=1.87e+04
#141 t=1.58e-11 dt=1.92e-13 max_dmdt=1.86e+04
#142 t=1.6e-11 dt=1.92e-13 max_dmdt=1.84e+04
#143 t=1.62e-11 dt=1.92e-13 max_dmdt=1.82e+04
#144 t=1.64e-11 dt=1.92e-13 max_dmdt=1.79e+04
#145 t=1.66e-11 dt=1.92e-13 max_dmdt=1.75e+04
#146 t=1.68e-11 dt=1.92e-13 max_dmdt=1.77e+04
#147 t=1.7e-11 dt=1.92e-13 max_dmdt=1.81e+04
#148 t=1.72e-11 dt=1.92e-13 max_dmdt=1.83e+04
#149 t=1.74e-11 dt=1.92e-13 max_dmdt=1.84e+04
#150 t=1.76e-11 dt=1.92e-13 max_dmdt=1.85e+04
#151 t=1.78e-11 dt=1.92e-13 max_dmdt=1.85e+04
#152 t=1.79e-11 dt=1.92e-13 max_dmdt=1.84e+04
#153 t=1.81e-11 dt=1.92e-13 max_dmdt=1.82e+04
#154 t=1.83e-11 dt=1.92e-13 max_dmdt=1.79e+04
#155 t=1.85e-11 dt=1.92e-13 max_dmdt=1.76e+04
#156 t=1.87e-11 dt=1.92e-13 max_dmdt=1.74e+04
#157 t=1.89e-11 dt=1.92e-13 max_dmdt=1.76e+04
#158 t=1.91e-11 dt=1.92e-13 max_dmdt=1.77e+04
#159 t=1.93e-11 dt=1.92e-13 max_dmdt=1.77e+04
#160 t=1.95e-11 dt=1.92e-13 max_dmdt=1.77e+04
#161 t=1.97e-11 dt=1.92e-13 max_dmdt=1.76e+04
#162 t=1.99e-11 dt=1.92e-13 max_dmdt=1.75e+04
#163 t=2.01e-11 dt=1.92e-13 max_dmdt=1.74e+04
#164 t=2.03e-11 dt=1.92e-13 max_dmdt=1.72e+04
#165 t=2.04e-11 dt=1.92e-13 max_dmdt=1.69e+04
#166 t=2.06e-11 dt=1.92e-13 max_dmdt=1.66e+04
#167 t=2.08e-11 dt=1.92e-13 max_dmdt=1.63e+04
#168 t=2.1e-11 dt=1.92e-13 max_dmdt=1.59e+04
#169 t=2.12e-11 dt=1.92e-13 max_dmdt=1.56e+04
#170 t=2.14e-11 dt=1.92e-13 max_dmdt=1.58e+04
#171 t=2.16e-11 dt=1.92e-13 max_dmdt=1.59e+04
#172 t=2.18e-11 dt=1.92e-13 max_dmdt=1.59e+04
#173 t=2.2e-11 dt=1.92e-13 max_dmdt=1.6e+04
#174 t=2.22e-11 dt=1.92e-13 max_dmdt=1.59e+04
#175 t=2.24e-11 dt=1.92e-13 max_dmdt=1.58e+04
#176 t=2.26e-11 dt=1.92e-13 max_dmdt=1.57e+04
#177 t=2.27e-11 dt=1.92e-13 max_dmdt=1.55e+04
#178 t=2.29e-11 dt=1.92e-13 max_dmdt=1.53e+04
#179 t=2.31e-11 dt=1.92e-13 max_dmdt=1.51e+04
#180 t=2.33e-11 dt=1.92e-13 max_dmdt=1.47e+04
#181 t=2.35e-11 dt=1.92e-13 max_dmdt=1.44e+04
#182 t=2.37e-11 dt=1.92e-13 max_dmdt=1.4e+04
#183 t=2.39e-11 dt=1.92e-13 max_dmdt=1.35e+04
#184 t=2.41e-11 dt=1.92e-13 max_dmdt=1.31e+04
#185 t=2.43e-11 dt=1.92e-13 max_dmdt=1.26e+04
#186 t=2.45e-11 dt=1.92e-13 max_dmdt=1.2e+04
#187 t=2.47e-11 dt=1.92e-13 max_dmdt=1.16e+04
#188 t=2.49e-11 dt=1.92e-13 max_dmdt=1.12e+04
#189 t=2.51e-11 dt=1.92e-13 max_dmdt=1.08e+04
#190 t=2.52e-11 dt=1.92e-13 max_dmdt=1.08e+04
#191 t=2.54e-11 dt=1.92e-13 max_dmdt=1.09e+04
#192 t=2.56e-11 dt=1.92e-13 max_dmdt=1.09e+04
#193 t=2.58e-11 dt=1.92e-13 max_dmdt=1.09e+04
#194 t=2.6e-11 dt=1.92e-13 max_dmdt=1.09e+04
#195 t=2.62e-11 dt=1.92e-13 max_dmdt=1.12e+04
#196 t=2.65e-11 dt=2.88e-13 max_dmdt=1.15e+04
#197 t=2.68e-11 dt=2.88e-13 max_dmdt=1.19e+04
#198 t=2.71e-11 dt=2.88e-13 max_dmdt=1.22e+04
#199 t=2.74e-11 dt=2.88e-13 max_dmdt=1.25e+04
#200 t=2.76e-11 dt=2.88e-13 max_dmdt=1.28e+04
#201 t=2.79e-11 dt=2.88e-13 max_dmdt=1.29e+04
#202 t=2.82e-11 dt=2.88e-13 max_dmdt=1.31e+04
#203 t=2.85e-11 dt=2.88e-13 max_dmdt=1.31e+04
#204 t=2.88e-11 dt=2.88e-13 max_dmdt=1.31e+04
#205 t=2.91e-11 dt=2.88e-13 max_dmdt=1.31e+04
#206 t=2.94e-11 dt=2.88e-13 max_dmdt=1.35e+04
#207 t=2.97e-11 dt=2.88e-13 max_dmdt=1.44e+04
#208 t=2.99e-11 dt=2.88e-13 max_dmdt=1.52e+04
#209 t=3.02e-11 dt=2.88e-13 max_dmdt=1.6e+04
#210 t=3.05e-11 dt=2.88e-13 max_dmdt=1.66e+04
#211 t=3.08e-11 dt=2.88e-13 max_dmdt=1.72e+04
#212 t=3.11e-11 dt=2.88e-13 max_dmdt=1.76e+04
#213 t=3.14e-11 dt=2.88e-13 max_dmdt=1.78e+04
#214 t=3.17e-11 dt=2.88e-13 max_dmdt=1.79e+04
#215 t=3.2e-11 dt=2.88e-13 max_dmdt=1.79e+04
#216 t=3.23e-11 dt=2.88e-13 max_dmdt=1.77e+04
#217 t=3.25e-11 dt=2.88e-13 max_dmdt=1.73e+04
#218 t=3.28e-11 dt=2.88e-13 max_dmdt=1.69e+04
#219 t=3.31e-11 dt=2.88e-13 max_dmdt=1.63e+04
#220 t=3.34e-11 dt=2.88e-13 max_dmdt=1.56e+04
#221 t=3.37e-11 dt=2.88e-13 max_dmdt=1.49e+04
#222 t=3.4e-11 dt=2.88e-13 max_dmdt=1.41e+04
#223 t=3.43e-11 dt=2.88e-13 max_dmdt=1.33e+04
#224 t=3.46e-11 dt=2.88e-13 max_dmdt=1.25e+04
#225 t=3.48e-11 dt=2.88e-13 max_dmdt=1.17e+04
#226 t=3.51e-11 dt=2.88e-13 max_dmdt=1.1e+04
#227 t=3.54e-11 dt=2.88e-13 max_dmdt=1.02e+04
#228 t=3.57e-11 dt=2.88e-13 max_dmdt=9.49e+03
#229 t=3.6e-11 dt=2.88e-13 max_dmdt=8.81e+03
#230 t=3.63e-11 dt=2.88e-13 max_dmdt=8.17e+03
#231 t=3.66e-11 dt=2.88e-13 max_dmdt=8.07e+03
#232 t=3.69e-11 dt=2.88e-13 max_dmdt=8.11e+03
#233 t=3.71e-11 dt=2.88e-13 max_dmdt=8.13e+03
#234 t=3.74e-11 dt=2.88e-13 max_dmdt=8.13e+03
#235 t=3.77e-11 dt=2.88e-13 max_dmdt=8.11e+03
#236 t=3.8e-11 dt=2.88e-13 max_dmdt=8.09e+03
#237 t=3.83e-11 dt=2.88e-13 max_dmdt=8.05e+03
#238 t=3.86e-11 dt=2.88e-13 max_dmdt=7.99e+03
#239 t=3.89e-11 dt=2.88e-13 max_dmdt=7.93e+03
#240 t=3.92e-11 dt=2.88e-13 max_dmdt=7.85e+03
#241 t=3.95e-11 dt=2.88e-13 max_dmdt=7.77e+03
#242 t=3.97e-11 dt=2.88e-13 max_dmdt=7.68e+03
#243 t=4.02e-11 dt=4.37e-13 max_dmdt=7.55e+03
#244 t=4.06e-11 dt=4.37e-13 max_dmdt=7.38e+03
#245 t=4.1e-11 dt=4.37e-13 max_dmdt=7.21e+03
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#247 t=4.19e-11 dt=4.37e-13 max_dmdt=6.83e+03
#248 t=4.24e-11 dt=4.37e-13 max_dmdt=6.63e+03
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#251 t=4.37e-11 dt=4.37e-13 max_dmdt=6.04e+03
#252 t=4.41e-11 dt=4.37e-13 max_dmdt=5.84e+03
#253 t=4.45e-11 dt=4.37e-13 max_dmdt=5.65e+03
#254 t=4.5e-11 dt=4.37e-13 max_dmdt=5.45e+03
#255 t=4.54e-11 dt=4.37e-13 max_dmdt=5.27e+03
#256 t=4.59e-11 dt=4.37e-13 max_dmdt=5.08e+03
#257 t=4.63e-11 dt=4.37e-13 max_dmdt=4.91e+03
#258 t=4.67e-11 dt=4.37e-13 max_dmdt=4.73e+03
#259 t=4.72e-11 dt=4.37e-13 max_dmdt=4.58e+03
#260 t=4.76e-11 dt=4.37e-13 max_dmdt=4.44e+03
#261 t=4.8e-11 dt=4.37e-13 max_dmdt=4.32e+03
#262 t=4.85e-11 dt=4.37e-13 max_dmdt=4.21e+03
#263 t=4.89e-11 dt=4.37e-13 max_dmdt=4.09e+03
#264 t=4.93e-11 dt=4.37e-13 max_dmdt=4.04e+03
#265 t=4.98e-11 dt=4.37e-13 max_dmdt=3.98e+03
#266 t=5.02e-11 dt=4.37e-13 max_dmdt=3.92e+03
#267 t=5.07e-11 dt=4.37e-13 max_dmdt=3.85e+03
#268 t=5.11e-11 dt=4.37e-13 max_dmdt=3.78e+03
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#270 t=5.2e-11 dt=4.37e-13 max_dmdt=3.61e+03
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#275 t=5.55e-11 dt=7.06e-13 max_dmdt=3.35e+03
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#280 t=5.9e-11 dt=7.06e-13 max_dmdt=2.82e+03
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#284 t=6.19e-11 dt=7.06e-13 max_dmdt=2.58e+03
#285 t=6.26e-11 dt=7.06e-13 max_dmdt=2.55e+03
#286 t=6.33e-11 dt=7.06e-13 max_dmdt=2.53e+03
#287 t=6.4e-11 dt=7.06e-13 max_dmdt=2.51e+03
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#294 t=6.89e-11 dt=7.06e-13 max_dmdt=2.33e+03
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#300 t=7.5e-11 dt=1.07e-12 max_dmdt=2.16e+03
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#304 t=7.93e-11 dt=1.07e-12 max_dmdt=2.09e+03
#305 t=8.04e-11 dt=1.07e-12 max_dmdt=2.07e+03
#306 t=8.14e-11 dt=1.07e-12 max_dmdt=2.05e+03
#307 t=8.25e-11 dt=1.07e-12 max_dmdt=2.02e+03
#308 t=8.36e-11 dt=1.07e-12 max_dmdt=1.99e+03
#309 t=8.47e-11 dt=1.07e-12 max_dmdt=1.96e+03
#310 t=8.57e-11 dt=1.07e-12 max_dmdt=1.93e+03
#311 t=8.68e-11 dt=1.07e-12 max_dmdt=1.9e+03
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#313 t=8.89e-11 dt=1.07e-12 max_dmdt=1.84e+03
#314 t=9e-11 dt=1.07e-12 max_dmdt=1.82e+03
#315 t=9.11e-11 dt=1.07e-12 max_dmdt=1.84e+03
#316 t=9.27e-11 dt=1.61e-12 max_dmdt=1.87e+03
#317 t=9.43e-11 dt=1.61e-12 max_dmdt=1.91e+03
#318 t=9.59e-11 dt=1.61e-12 max_dmdt=1.94e+03
#319 t=9.75e-11 dt=1.61e-12 max_dmdt=1.96e+03
#320 t=9.91e-11 dt=1.61e-12 max_dmdt=1.98e+03
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#330 t=1.15e-10 dt=1.61e-12 max_dmdt=2.5e+03
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#332 t=1.18e-10 dt=1.61e-12 max_dmdt=2.51e+03
#333 t=1.2e-10 dt=1.61e-12 max_dmdt=2.51e+03
#334 t=1.22e-10 dt=1.61e-12 max_dmdt=2.49e+03
#335 t=1.23e-10 dt=1.61e-12 max_dmdt=2.47e+03
#336 t=1.25e-10 dt=1.61e-12 max_dmdt=2.44e+03
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#339 t=1.3e-10 dt=1.61e-12 max_dmdt=2.32e+03
#340 t=1.31e-10 dt=1.61e-12 max_dmdt=2.27e+03
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#343 t=1.36e-10 dt=1.61e-12 max_dmdt=2.11e+03
#344 t=1.38e-10 dt=1.61e-12 max_dmdt=2.05e+03
#345 t=1.39e-10 dt=1.61e-12 max_dmdt=2e+03
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#353 t=1.52e-10 dt=1.61e-12 max_dmdt=1.53e+03
#354 t=1.54e-10 dt=1.61e-12 max_dmdt=1.48e+03
#355 t=1.56e-10 dt=1.61e-12 max_dmdt=1.49e+03
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#358 t=1.6e-10 dt=1.61e-12 max_dmdt=2.13e+03
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#366 t=1.7e-10 dt=1.11e-12 max_dmdt=4.44e+03
#367 t=1.71e-10 dt=1.11e-12 max_dmdt=4.68e+03
#368 t=1.72e-10 dt=8.1e-13 max_dmdt=4.86e+03
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#377 t=1.79e-10 dt=8.1e-13 max_dmdt=4.1e+03
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#380 t=1.82e-10 dt=8.1e-13 max_dmdt=3.79e+03
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#384 t=1.85e-10 dt=8.1e-13 max_dmdt=3.04e+03
#385 t=1.86e-10 dt=8.1e-13 max_dmdt=2.83e+03
#386 t=1.87e-10 dt=8.1e-13 max_dmdt=2.62e+03
#387 t=1.88e-10 dt=8.1e-13 max_dmdt=2.43e+03
#388 t=1.88e-10 dt=8.1e-13 max_dmdt=2.27e+03
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#390 t=1.9e-10 dt=8.1e-13 max_dmdt=2e+03
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#392 t=1.92e-10 dt=8.1e-13 max_dmdt=1.89e+03
#393 t=1.92e-10 dt=8.1e-13 max_dmdt=1.84e+03
#394 t=1.93e-10 dt=8.1e-13 max_dmdt=1.8e+03
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#400 t=1.98e-10 dt=8.1e-13 max_dmdt=1.51e+03
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#402 t=2e-10 dt=8.1e-13 max_dmdt=1.42e+03
#403 t=2.01e-10 dt=8.1e-13 max_dmdt=1.37e+03
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#411 t=2.1e-10 dt=1.31e-12 max_dmdt=947
#412 t=2.12e-10 dt=1.31e-12 max_dmdt=906
#413 t=2.13e-10 dt=1.31e-12 max_dmdt=883
#414 t=2.14e-10 dt=1.31e-12 max_dmdt=860
#415 t=2.16e-10 dt=1.31e-12 max_dmdt=837
#416 t=2.17e-10 dt=1.31e-12 max_dmdt=819
#417 t=2.18e-10 dt=1.31e-12 max_dmdt=821
#418 t=2.2e-10 dt=1.31e-12 max_dmdt=823
#419 t=2.21e-10 dt=1.31e-12 max_dmdt=825
#420 t=2.22e-10 dt=1.31e-12 max_dmdt=826
#421 t=2.24e-10 dt=1.31e-12 max_dmdt=828
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#423 t=2.27e-10 dt=1.97e-12 max_dmdt=831
#424 t=2.29e-10 dt=1.97e-12 max_dmdt=833
#425 t=2.31e-10 dt=1.97e-12 max_dmdt=834
#426 t=2.33e-10 dt=1.97e-12 max_dmdt=835
#427 t=2.35e-10 dt=1.97e-12 max_dmdt=835
#428 t=2.37e-10 dt=1.97e-12 max_dmdt=835
#429 t=2.39e-10 dt=1.97e-12 max_dmdt=834
#430 t=2.41e-10 dt=1.97e-12 max_dmdt=834
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#432 t=2.45e-10 dt=1.97e-12 max_dmdt=830
#433 t=2.47e-10 dt=1.97e-12 max_dmdt=828
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#435 t=2.51e-10 dt=2.97e-12 max_dmdt=836
#436 t=2.54e-10 dt=2.97e-12 max_dmdt=845
#437 t=2.57e-10 dt=2.97e-12 max_dmdt=853
#438 t=2.6e-10 dt=2.97e-12 max_dmdt=859
#439 t=2.63e-10 dt=2.97e-12 max_dmdt=864
#440 t=2.66e-10 dt=2.97e-12 max_dmdt=869
#441 t=2.69e-10 dt=2.97e-12 max_dmdt=872
#442 t=2.72e-10 dt=2.97e-12 max_dmdt=873
#443 t=2.75e-10 dt=2.97e-12 max_dmdt=874
#444 t=2.78e-10 dt=2.97e-12 max_dmdt=872
#445 t=2.81e-10 dt=2.97e-12 max_dmdt=870
#446 t=2.84e-10 dt=2.97e-12 max_dmdt=866
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#448 t=2.9e-10 dt=2.97e-12 max_dmdt=854
#449 t=2.93e-10 dt=2.97e-12 max_dmdt=846
#450 t=2.96e-10 dt=2.97e-12 max_dmdt=836
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#452 t=3.05e-10 dt=4.47e-12 max_dmdt=815
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#456 t=3.23e-10 dt=4.47e-12 max_dmdt=800
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#462 t=3.5e-10 dt=4.47e-12 max_dmdt=699
#463 t=3.54e-10 dt=4.47e-12 max_dmdt=676
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#465 t=3.63e-10 dt=4.47e-12 max_dmdt=656
#466 t=3.68e-10 dt=4.47e-12 max_dmdt=649
#467 t=3.72e-10 dt=4.47e-12 max_dmdt=640
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#469 t=3.81e-10 dt=4.47e-12 max_dmdt=619
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#471 t=3.9e-10 dt=4.47e-12 max_dmdt=593
#472 t=3.94e-10 dt=4.47e-12 max_dmdt=579
#473 t=3.99e-10 dt=4.47e-12 max_dmdt=564
#474 t=4.03e-10 dt=4.47e-12 max_dmdt=548
#475 t=4.08e-10 dt=4.47e-12 max_dmdt=532
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#477 t=4.17e-10 dt=4.47e-12 max_dmdt=511
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#479 t=4.26e-10 dt=4.47e-12 max_dmdt=512
#480 t=4.3e-10 dt=4.47e-12 max_dmdt=512
#481 t=4.35e-10 dt=4.47e-12 max_dmdt=511
#482 t=4.39e-10 dt=4.47e-12 max_dmdt=509
#483 t=4.43e-10 dt=4.47e-12 max_dmdt=508
#484 t=4.48e-10 dt=4.47e-12 max_dmdt=506
#485 t=4.52e-10 dt=4.47e-12 max_dmdt=503
#486 t=4.57e-10 dt=4.47e-12 max_dmdt=500
#487 t=4.61e-10 dt=4.47e-12 max_dmdt=497
#488 t=4.68e-10 dt=6.76e-12 max_dmdt=493
#489 t=4.75e-10 dt=6.76e-12 max_dmdt=487
#490 t=4.82e-10 dt=6.76e-12 max_dmdt=480
#491 t=4.88e-10 dt=6.76e-12 max_dmdt=473
#492 t=4.95e-10 dt=6.76e-12 max_dmdt=466
#493 t=5.02e-10 dt=6.76e-12 max_dmdt=458
#494 t=5.09e-10 dt=6.76e-12 max_dmdt=450
#495 t=5.15e-10 dt=6.76e-12 max_dmdt=441
#496 t=5.22e-10 dt=6.76e-12 max_dmdt=432
#497 t=5.29e-10 dt=6.76e-12 max_dmdt=423
#498 t=5.36e-10 dt=6.76e-12 max_dmdt=414
#499 t=5.42e-10 dt=6.76e-12 max_dmdt=404
#500 t=5.49e-10 dt=6.76e-12 max_dmdt=408
#501 t=5.56e-10 dt=6.76e-12 max_dmdt=412
#502 t=5.63e-10 dt=6.76e-12 max_dmdt=416
#503 t=5.69e-10 dt=6.76e-12 max_dmdt=419
#504 t=5.76e-10 dt=6.76e-12 max_dmdt=421
#505 t=5.83e-10 dt=6.76e-12 max_dmdt=422
#506 t=5.9e-10 dt=6.76e-12 max_dmdt=423
#507 t=5.96e-10 dt=6.76e-12 max_dmdt=423
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#913 t=1.4e-08 dt=2.1e-10 max_dmdt=5.44
#914 t=1.42e-08 dt=2.1e-10 max_dmdt=5.37
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#928 t=1.71e-08 dt=2.1e-10 max_dmdt=4.93
#929 t=1.74e-08 dt=3.19e-10 max_dmdt=4.94
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#1077 t=3.34e-08 dt=1.13e-10 max_dmdt=9.72
#1078 t=3.35e-08 dt=1.13e-10 max_dmdt=9.36
#1079 t=3.37e-08 dt=1.13e-10 max_dmdt=9.01
#1080 t=3.38e-08 dt=1.13e-10 max_dmdt=8.68
#1081 t=3.39e-08 dt=1.13e-10 max_dmdt=8.41
#1082 t=3.4e-08 dt=1.13e-10 max_dmdt=8.16
#1083 t=3.41e-08 dt=1.13e-10 max_dmdt=7.93
#1084 t=3.42e-08 dt=1.13e-10 max_dmdt=7.7
#1085 t=3.43e-08 dt=1.13e-10 max_dmdt=7.48
#1086 t=3.44e-08 dt=1.13e-10 max_dmdt=7.27
#1087 t=3.46e-08 dt=1.13e-10 max_dmdt=7.07
#1088 t=3.47e-08 dt=1.13e-10 max_dmdt=6.87
#1089 t=3.48e-08 dt=1.13e-10 max_dmdt=6.69
#1090 t=3.49e-08 dt=1.13e-10 max_dmdt=6.5
#1091 t=3.5e-08 dt=1.13e-10 max_dmdt=6.33
#1092 t=3.51e-08 dt=1.13e-10 max_dmdt=6.16
#1093 t=3.52e-08 dt=1.13e-10 max_dmdt=5.99
#1094 t=3.54e-08 dt=1.13e-10 max_dmdt=5.83
#1095 t=3.55e-08 dt=1.13e-10 max_dmdt=5.68
#1096 t=3.56e-08 dt=1.13e-10 max_dmdt=5.53
#1097 t=3.57e-08 dt=1.13e-10 max_dmdt=5.39
#1098 t=3.58e-08 dt=1.13e-10 max_dmdt=5.25
#1099 t=3.59e-08 dt=1.13e-10 max_dmdt=5.12
#1100 t=3.6e-08 dt=1.13e-10 max_dmdt=4.99
#1101 t=3.61e-08 dt=1.13e-10 max_dmdt=4.86
#1102 t=3.63e-08 dt=1.13e-10 max_dmdt=4.74
#1103 t=3.64e-08 dt=1.13e-10 max_dmdt=4.62
#1104 t=3.65e-08 dt=1.13e-10 max_dmdt=4.51
#1105 t=3.66e-08 dt=1.13e-10 max_dmdt=4.4
#1106 t=3.67e-08 dt=1.13e-10 max_dmdt=4.29
#1107 t=3.68e-08 dt=1.13e-10 max_dmdt=4.19
#1108 t=3.69e-08 dt=1.13e-10 max_dmdt=4.09
#1109 t=3.7e-08 dt=1.13e-10 max_dmdt=3.99
#1110 t=3.72e-08 dt=1.13e-10 max_dmdt=3.9
#1111 t=3.73e-08 dt=1.13e-10 max_dmdt=3.81
#1112 t=3.74e-08 dt=1.71e-10 max_dmdt=3.7
#1113 t=3.76e-08 dt=1.71e-10 max_dmdt=3.57
#1114 t=3.78e-08 dt=1.71e-10 max_dmdt=3.45
#1115 t=3.8e-08 dt=1.71e-10 max_dmdt=3.33
#1116 t=3.81e-08 dt=1.71e-10 max_dmdt=3.22
#1117 t=3.83e-08 dt=1.71e-10 max_dmdt=3.11
#1118 t=3.85e-08 dt=1.71e-10 max_dmdt=3.01
#1119 t=3.87e-08 dt=2.71e-10 max_dmdt=2.89
#1120 t=3.9e-08 dt=2.71e-10 max_dmdt=2.74
#1121 t=3.93e-08 dt=2.71e-10 max_dmdt=2.6
#1122 t=3.96e-08 dt=2.71e-10 max_dmdt=2.47
#1123 t=3.98e-08 dt=2.71e-10 max_dmdt=2.35
#1124 t=4.01e-08 dt=2.71e-10 max_dmdt=2.24
#1125 t=4.04e-08 dt=2.71e-10 max_dmdt=2.13
#1126 t=4.06e-08 dt=2.71e-10 max_dmdt=2.03
#1127 t=4.09e-08 dt=2.71e-10 max_dmdt=1.94
#1128 t=4.12e-08 dt=2.71e-10 max_dmdt=1.85
#1129 t=4.15e-08 dt=2.71e-10 max_dmdt=1.76
#1130 t=4.17e-08 dt=2.71e-10 max_dmdt=1.68
#1131 t=4.21e-08 dt=4.18e-10 max_dmdt=1.58
#1132 t=4.26e-08 dt=4.18e-10 max_dmdt=1.47
#1133 t=4.3e-08 dt=4.18e-10 max_dmdt=1.37
#1134 t=4.34e-08 dt=4.18e-10 max_dmdt=1.28
#1135 t=4.38e-08 dt=4.18e-10 max_dmdt=1.2
#1136 t=4.42e-08 dt=4.18e-10 max_dmdt=1.12
#1137 t=4.46e-08 dt=4.18e-10 max_dmdt=1.04
#1138 t=4.51e-08 dt=4.18e-10 max_dmdt=0.973
#1139 t=4.55e-08 dt=4.18e-10 max_dmdt=0.909
#1140 t=4.59e-08 dt=4.18e-10 max_dmdt=0.849
#1141 t=4.63e-08 dt=4.18e-10 max_dmdt=0.794
#1142 t=4.67e-08 dt=4.18e-10 max_dmdt=0.742
#1143 t=4.72e-08 dt=4.18e-10 max_dmdt=0.694
#1144 t=4.76e-08 dt=4.18e-10 max_dmdt=0.649
#1145 t=4.8e-08 dt=4.18e-10 max_dmdt=0.607
#1146 t=4.84e-08 dt=4.18e-10 max_dmdt=0.568
#1147 t=4.88e-08 dt=4.18e-10 max_dmdt=0.532
#1148 t=4.92e-08 dt=4.18e-10 max_dmdt=0.497
#1149 t=4.97e-08 dt=4.18e-10 max_dmdt=0.465
#1150 t=5.03e-08 dt=6.44e-10 max_dmdt=0.428
#1151 t=5.09e-08 dt=6.44e-10 max_dmdt=0.386
#1152 t=5.16e-08 dt=6.44e-10 max_dmdt=0.349
#1153 t=5.22e-08 dt=6.44e-10 max_dmdt=0.315
#1154 t=5.29e-08 dt=6.44e-10 max_dmdt=0.284
#1155 t=5.35e-08 dt=6.44e-10 max_dmdt=0.256
#1156 t=5.42e-08 dt=6.44e-10 max_dmdt=0.231
#1157 t=5.48e-08 dt=6.44e-10 max_dmdt=0.209
#1158 t=5.55e-08 dt=6.44e-10 max_dmdt=0.188
#1159 t=5.61e-08 dt=6.44e-10 max_dmdt=0.17
#1160 t=5.67e-08 dt=6.44e-10 max_dmdt=0.153
#1161 t=5.74e-08 dt=6.44e-10 max_dmdt=0.138
#1162 t=5.8e-08 dt=6.44e-10 max_dmdt=0.125
#1163 t=5.87e-08 dt=6.44e-10 max_dmdt=0.112
#1164 t=5.93e-08 dt=6.44e-10 max_dmdt=0.101
#1165 t=6.03e-08 dt=9.85e-10 max_dmdt=0.0889
[5]:
def plot_field(field, sim, fieldname, layer=0):
nx = sim.mesh.nx
ny = sim.mesh.ny
n_layer = sim.mesh.nx * sim.mesh.ny
m = field.reshape(-1, 3)
fx = m[:, 0][layer*n_layer:(layer+1)*n_layer]
fy = m[:, 1][layer*n_layer:(layer+1)*n_layer]
fz = m[:, 2][layer*n_layer:(layer+1)*n_layer]
fx.shape = (ny, nx)
fy.shape = (ny, nx)
fz.shape = (ny, nx)
extent = [0, nx*sim.mesh.dx, 0, ny*sim.mesh.dy]
plt.figure(figsize=(20, 10))
plt.subplot(1, 3, 1)
plt.imshow(fx, extent=extent, cmap='RdBu', origin='lower')
plt.title('{}_x'.format(fieldname))
plt.xlabel('x (nm)')
plt.ylabel('y (nm)')
plt.subplot(1, 3, 2)
plt.imshow(fy, extent=extent, cmap='RdBu', origin='lower')
plt.xlabel('x (nm)')
plt.ylabel('y (nm)')
plt.title('{}_y'.format(fieldname))
plt.subplot(1, 3, 3)
plt.imshow(fz, extent=extent, cmap='RdBu', origin='lower')
plt.xlabel('x (nm)')
plt.ylabel('y (nm)')
plt.title('{}_z'.format(fieldname))
plot_field(sim.spin, sim, 'm')
We now apply a field gradient which acts to move the Skyrmions. Below we construct a function to impose this gradient, and just plot the resulting Zeeman field.
[6]:
H0 = 0.3 / (4*np.pi*1e-7) # Apply a field of 0.4T
def H_Gradient(pos):
x, y, z = pos
return (0, 0, H0 * y/Ly)
zee = Zeeman(H_Gradient)
sim.add(zee)
plot_field(zee.field, sim, 'HZeeman')
We now evolve in time, showing snapshots of the magnetisation every 1ns.
[7]:
sim.driver.reset_integrator()
for i, t in enumerate(np.linspace(0, 20e-9, 21)):
print('i = {}, t = {}ns'.format(i, t))
sim.driver.run_until(t)
plot_field(sim.spin, sim, 'm')
i = 0, t = 0.0ns
i = 1, t = 1e-09ns
i = 2, t = 2e-09ns
i = 3, t = 3.0000000000000004e-09ns
i = 4, t = 4e-09ns
i = 5, t = 5e-09ns
i = 6, t = 6.000000000000001e-09ns
i = 7, t = 7.000000000000001e-09ns
i = 8, t = 8e-09ns
i = 9, t = 9.000000000000001e-09ns
i = 10, t = 1e-08ns
i = 11, t = 1.1000000000000001e-08ns
i = 12, t = 1.2000000000000002e-08ns
i = 13, t = 1.3e-08ns
i = 14, t = 1.4000000000000001e-08ns
i = 15, t = 1.5000000000000002e-08ns
i = 16, t = 1.6e-08ns
i = 17, t = 1.7e-08ns
i = 18, t = 1.8000000000000002e-08ns
i = 19, t = 1.9e-08ns
i = 20, t = 2e-08ns
/tmp/ipykernel_767011/123838774.py:15: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`). Consider using `matplotlib.pyplot.close()`.
plt.figure(figsize=(20, 10))
We can see that the core of the Skyrmions in the high-field region is smaller than than in the low field region, and there is some motion away from this area boundary.
We can also plot particular fields:
[8]:
# Plot the Effective Field
plot_field(sim.driver.field, sim, 'H')
[9]:
plot_field(sim.get_interaction('UniformExchange').field, sim, 'HExchange')
plot_field(sim.get_interaction('DMI').field, sim, 'HDMI')
plot_field(sim.get_interaction('Demag').field, sim, 'HDemag')
