Micromagnetics · Single-domain switching

Pulse Switching of a Nanomagnet

How can a field pulse shorter than a nanosecond flip a nanomagnet, and why does a longer pulse fail? The pulse does not push the magnetization straight over. It tips the moment slightly out of the film, and the strong field from the film's own magnetic surfaces then spins it around in the plane. If the pulse ends after half a turn, the magnet settles pointing the other way; if it ends after a full turn, the magnet comes back, so a longer pulse can fail.

Hit a 500 nm permalloy bar with a field pulse and follow its magnetization around a sphere. Each cell of the switching map is a full trajectory, so the pulse lengths that flip the bar show up as stripes.

Easy-axis component at the end,

m_x

$-1$

Highest energy, in saddle heights $1.44$

Ringing after the pulse $4.65\,\mathrm{GHz}$

Kittel frequency of the final well $4.66\,\mathrm{GHz}$

One pulse, one trajectory

Kick the nanomagnet

Pulse field

\mu_0 H_{\mathrm{p}}

25\,\mathrm{mT}

Pulse length

\tau_{\mathrm{p}}

140\,\mathrm{ps}

Pulse direction from the easy axis

\phi_{\mathrm{p}}

90^{\circ}

Gilbert damping

\alpha

0.02

Drag the start ring on the sphere, or click a cell of the switching map to load that pulse.

Bias field and start direction

Bias field along the easy axis

\mu_0 H_{\mathrm{b}}

0\,\mathrm{mT}

Start: angle from the film normal

\theta_0

90^{\circ}

Start: in-plane angle from the easy axis

\phi_0

1^{\circ}

Model assumptions

The bar is the 500 nm by 125 nm by 3 nm permalloy element of the muMAG standard problem 4, with saturation magnetization 800 kA/m. Here it moves as one rigid moment (a macrospin), so it has no domains, no edge modes and no spin waves; the real element reverses through nonuniform states. Its only anisotropy is the shape anisotropy of a uniformly magnetized rectangular prism (Aharoni 1998). The Landau-Lifshitz-Gilbert equation is integrated with fourth-order Runge-Kutta steps of 0.25 ps (1 ps for the map). Temperature is left out; a 1 degree starting tilt stands in for thermal disorder. A run counts as switched or returned once the pulse is off and its energy has fallen below the saddle between the two wells.

Outcome: switched to the other well. Each map cell is one full trajectory from the same equation; the white ring marks the current pulse.

More numbers and the saved record

Outcome fixed once the energy fell below the saddle at

0.14\,\mathrm{ns}

Last sign change of the easy-axis component in the window

0.09\,\mathrm{ns}

Switching map: switched trajectories / trajectories run

422\ /\ 1280

Demagnetizing factors along x, y, z

0.0092,\ 0.0382,\ 0.9526

In-plane shape-anisotropy field

29.1\,\mathrm{mT}

Largest change of the unit length within one step, before renormalizing

1.54\times10^{-13}

The JSON record holds the parameters, the model constants, the sampled trajectory with its energy, and every cell of the switching map.

Start with the essentials

Focus question: How can a field pulse shorter than a nanosecond flip a nanomagnet, and why does a longer pulse fail?
One-sentence intuition: A transverse pulse starts a precession around the film normal; the bar switches when the pulse ends after an odd number of half turns and its energy then sits below the saddle.

Core mathematical model

Landau-Lifshitz-Gilbert equation

\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}=-\gamma_0\,\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}+\alpha\,\mathbf{m}\times\frac{\mathrm{d}\mathbf{m}}{\mathrm{d}t}

The first term turns $\mathbf{m}$ around the effective field; the second, with damping $\alpha$ , pulls it toward the field. Both keep $\lvert\mathbf{m}\rvert=1$ . Here $\gamma_0=2.211\times10^{5}\,\mathrm{m\,A^{-1}\,s^{-1}}$ .

Effective field of a uniformly magnetized bar

\mathbf{H}_{\mathrm{eff}}=\mathbf{H}_{\mathrm{p}}+\mathbf{H}_{\mathrm{b}}-M_{\mathrm{s}}\left(N_x m_x,\;N_y m_y,\;N_z m_z\right)

For the 500 by 125 by 3 nm prism the demagnetizing factors are $N_x=0.0092$ , $N_y=0.0382$ and $N_z=0.953$ , so $x$ is the easy axis and leaving the film plane costs the most energy.

Damping only removes energy

\frac{\mathrm{d}E}{\mathrm{d}t}=-\frac{\alpha\,\gamma_0\,\mu_0 M_{\mathrm{s}}}{1+\alpha^{2}}\,\bigl\lvert\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}\bigr\rvert^{2}\le 0

With the field held fixed, the energy per volume can only fall. Once the pulse is off and the energy is below the saddle, the moment can no longer reach the other well, so the outcome is decided.

Kittel frequency of small oscillations

f_{\mathrm{K}}=\frac{\gamma_0}{2\pi}\sqrt{\bigl[H_{\mathrm{b}}+(N_y-N_x)M_{\mathrm{s}}\bigr]\bigl[H_{\mathrm{b}}+(N_z-N_x)M_{\mathrm{s}}\bigr]}

Without bias field this gives 4.66 GHz; the ringing measured from the simulated trajectory after a small kick is 4.65 GHz.

Common difficulties

Longer is not stronger

Typical misconceptionIf a pulse flips the magnet, a longer or stronger pulse will flip it too.

Better mental modelPrecessional switching is about timing. The pulse keeps the moment turning; what matters is where the moment is when the pulse ends.

Damping is not the engine here

Typical misconceptionDamping is what drags the moment to the other side.

Better mental modelWith a transverse pulse, precession carries the moment across and damping only settles it. With a field along -x, damping does the work, and the reversal takes several times longer.

What the model leaves out

Typical misconceptionA real 500 nm bar reverses as one rigid moment.

Better mental modelThe bar is the 500 nm by 125 nm by 3 nm permalloy element of the muMAG standard problem 4, with saturation magnetization 800 kA/m. Here it moves as one rigid moment (a macrospin), so it has no domains, no edge modes and no spin waves; the real element reverses through nonuniform states. Its only anisotropy is the shape anisotropy of a uniformly magnetized rectangular prism (Aharoni 1998). The Landau-Lifshitz-Gilbert equation is integrated with fourth-order Runge-Kutta steps of 0.25 ps (1 ps for the map). Temperature is left out; a 1 degree starting tilt stands in for thermal disorder. A run counts as switched or returned once the pulse is off and its energy has fallen below the saddle between the two wells.

Run the experiment

01
Watch one flip
Start from A half-turn pulse flips it and let the run play.
What to observe: The trail lifts slightly off the equator, swings around to -x and spirals in. The energy climbs above the saddle line during the pulse and drops below it once the pulse stops.
02
Make the pulse longer
Drag Pulse length from 140 ps up to 350 ps.
What to observe: The moment makes almost a full turn while the pulse is on and settles back at +x. Same field, same energy kick, opposite result.
03
Read the switching map
Look at the right-hand map, then click cells in different stripes.
What to observe: Red and blue stripes alternate along the pulse length. Each stripe is one more half turn; stronger pulses turn faster, so the stripes bend toward shorter pulses. Below about 20 mT nothing switches.
04
Switch off damping
Choose Without damping it circles forever.
What to observe: After the pulse the energy stays flat above the saddle and the moment keeps circling both wells. Its unit length still holds to better than one part in a trillion per step.
05
Compare with Kittel
Choose A small kick rings at the Kittel frequency, then raise the bias field in the advanced group.
What to observe: The ringing frequency read from the trajectory stays within half a percent of the Kittel formula as the bias changes.

Pulse Switching of a Nanomagnet

Flipping a nanomagnet with a timed kick

Core mathematical model

Landau-Lifshitz-Gilbert equation

Effective field of a uniformly magnetized bar

Damping only removes energy

Kittel frequency of small oscillations

Common difficulties

Longer is not stronger

Damping is not the engine here

What the model leaves out

Run the experiment

Watch one flip

Make the pulse longer

Read the switching map

Switch off damping

Compare with Kittel

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