Dirac Strings in Kagome ASI

Here we demonstrate how flatspin was to produce Dirac Strings following the experimental setup given by [Mengotti et al., 2011].

“Real-space observation of emergent magnetic monopoles and associated Dirac strings in artificial kagome spin ice” Nature Phys 7, 68–74 (2011)

The flatspin results are discussed in more detail in our paper [Jensen et al., 2022].

Creating the Dataset

We set our parameters to closely match the experimental setup in [Mengotti et al., 2011].

Below is the flatspin-run command used to generate the dataset for this section, followed by an explaination of the parameters used.

flatspin-run -m KagomeSpinIceRotated -e Triangle -p phase=0 -p phi=-3.6 -p "size=(29,29)" -p temperature=300 -p H=0.1 -p periods=1  -p sw_beta=2.5 -p sw_b=0.212 -p sw_gamma=3 -p sw_c=1 -p hc=0.216 -p alpha=0.00103 -p use_opencl=1 -p neighbor_distance=10 -p disorder=0.05  -p "m_therm=0.05*1.29344e-15" -p timesteps=2000 -p spp=2000 -o diracStringsPublish2

-m KagomeSpinIceRotated and -p size=(29,29) define the geometry and the number of magnets similar to the experimental setup.

-e Triangle, -p phi=-3.6 and -p phase=0 uses a Triangle encoder to set up a reversal field at an angle of -3.6 degrees. As H=0.1, the field starts at -0.1 T ramps up linearly to 0.1 T then back down to -0.1 T

A temperature of 300 K is used to simulate room temperature.

Using a magnetization saturation (\(M_S\)) of 860 kA/m, and the volume of the magnets given in the experimental setup (~1.5e-21 m^3), the m_therm parameter is taken to be 5% of volume * msat.

The value of alpha is calculated from \(\alpha = \frac{\mu_0 M}{4\pi a^3}\) (see Theory), with \(M\) = 860e3 * 1.504e-21 Am^2 and \(a\) = 500 nm

Micromagnetic simulations of magnets with this msat and the given dimensions (470 nm * 160 nm * 20 nm) were used to obtain the switching parameters sw_beta=2.5, sw_b=0.212, sw_gamma=3, sw_c=1 and hc=0.216.

5% disorder was used to account for variations in the magnets used in the experiments.

Calculate \(H_c\)

First we analyze the dataset created by the above flatspin-run command to calculate the switching field of the full lattice, \(H_c\).

from flatspin import data

ds = data.Dataset.read("/data/flatspin/diracStringsPublish2")

# for now we're only interested in the fist half of the run
t = slice(None, 1000)

# using grid_size=(1,1) returns the average magnetization over the whole lattice
mag = data.load_output(ds, "mag", grid_size=(1, 1), t=t)

# get timestep where array switches (the time where the absolute magnetization in the x-direction is minimized)
t_min = np.argmin(abs(mag[:, 0]))
print(f"Timestep where array switches, t_min = {t_min}")

# load the data for the external field and use t_min to get find HC
h_ext = data.load_output(ds, "h_ext", t=t)
Hc = h_ext[t_min][0]
print(f"H_c = {Hc}")

Timestep where array switches, t_min = 738

H_c = 0.04750607227318573

Find the field values of interest

In [Mengotti et al., 2011] they show the state of ASI at field values: [0.8HC, 0.85HC, 0.92HC, 0.95HC, 0.99HC, 1.06HC]. To allow us to compare the results of flatspin to the experimental setup, we will find the timesteps in our dataset where the field is closest to these values.

# calculate the fields of interest in terms of our HC
foi = [0.8, 0.85, 0.92, 0.99, 1.06]
foiHC = Hc * np.array(foi)
print(f"foiHC = {foiHC}")

# find the nearest times by minimizing the absolute difference between the field and the HC
nearest_time = [np.argmin(np.abs(field - h_ext[:, 0])) for field in foiHC]
print(f"nearest_time = {nearest_time}")
print(f"nearest fields = {[str(round(h_ext[t, 0]/Hc,2))+'HC' for t in nearest_time]}")

foiHC = [0.03800486 0.04038016 0.04370559 0.04703101 0.05035644]
nearest_time = [690, 702, 719, 736, 752]
nearest fields = ['0.8HC', '0.85HC', '0.92HC', '0.99HC', '1.06HC']

Below we animate the state of the ASI, as it evolves from the first to the last field value of interest (0.8HC to 1.06HC).

from IPython.display import HTML
from matplotlib.animation import FuncAnimation
from flatspin.plotting import plot_vectors

def animate_dirac_strings(ds, times):
    fig, ax = plt.subplots(figsize=(7.2, 7.2), facecolor=(0.4, 0.4, 0.4))
    fig.subplots_adjust(left=0, right=1, bottom=0, top=0.95, wspace=0, hspace=0)
    ax.set_axis_off()

    _, UV = data.read_vectors(ds.tablefiles(), "mag", times)
    positions, _ = data.read_geometry(ds.tablefile('geometry'))

    def animate(i):
        plot_vectors(positions, UV[i], arrows=True, replace=True, ax=ax, cmap="peem180")
        ax.set_title(f"{round(h_ext[times[i],0]/Hc,2)}$H_c$", fontsize=20, color="white")


    anim = FuncAnimation(
        fig, animate, init_func=lambda: None,
        frames=len(times), interval=100, blit=False
    )
    plt.close() # Only show the animation
    return HTML(anim.to_jshtml(fps=8))
#animate_dirac_strings(ds, times=nearest_time)
#animate_dirac_strings(ds, times=list(range(710,750,1)))
animate_dirac_strings(ds, times=list(range(nearest_time[0], nearest_time[-1]+1)))

Once Loop Reflect

Hysteresis

Now we plot the hysterisis of our dataset, as well as a sketch of the hysteresis shown in [Mengotti et al., 2011].

mag = data.load_output(ds, "mag", grid_size=(1, 1))  # now we want full run length
h_ext = data.load_output(ds, "h_ext")
plt.figure(figsize=(8, 5))
plt.plot(h_ext[:, 0] / Hc, mag[:, 0], label="flatspin")

meng = np.loadtxt("scaledMeng.txt").round(2)  # our rough estimate of the Mengotti et al. dataset inferred from their graph
plt.plot(meng[:, 0], meng[:, 1], label="Mengotti")

#make text marker at the fields of interest
rom_num = ["I", "II", "III", "IV", "V"]
offsetx = [-.1,0,.1,-.3,-.3]
offsety = [-.15,-.15,-.15,0,0]
for i, field in enumerate(foi):
    plt.plot(field, mag[nearest_time[i], 0], ".", label=f"{rom_num[i]}", color="black")
    plt.text(field+offsetx[i], mag[nearest_time[i], 0] + offsety[i], rom_num[i], color="black")
    
plt.xlabel("$H/H_c$")
plt.ylabel("$M/M_S$")
plt.ylim(-1.2, 1.2)
plt.legend(["flatspin", "Mengotti et al."]);