Micromagnetics · Walls and waves

Domain Wall Motion and Spin Waves

Why does a field push a domain wall only so fast, and how does interfacial DMI raise that limit? A field moves a domain wall by making the magnetization inside it precess. Up to a limiting field the moment in the wall tilts to a fixed angle and the wall glides steadily. Past that limit, the Walker field, the moment keeps turning, the wall rocks back and forth and its average speed drops. Interfacial DMI holds the moment across the wall and raises the limit, so the wall can run much faster.

Push a domain wall along a strip of Pt/Co film and watch the moment inside it tilt, or spin once the field passes the Walker limit. Every dot on the speed curve is a separate chain simulation set against the wall model, and an antenna launches spin waves whose spectrum is measured in the same chain.

Wall speed in the chain $77.9\,\mathrm{m\,s^{-1}}$

Wall-model speed $78.8\,\mathrm{m\,s^{-1}}$

Walker field of the wall model $65.5\,\mathrm{mT}$

Wall moment: angle

\Phi

from the Néel direction, or turn period

T

$47.6^{\circ}$

Spin wavelength at the antenna frequency, from the dispersion relation $31.4\,\mathrm{nm}$

Model assumptions

The strip is a one-dimensional chain of 1 nm cells using the Pt/Co/AlOx film of Thiaville and co-workers (2012): saturation magnetization 1100 kA/m, exchange stiffness 16 pJ/m, anisotropy 1.27 MJ per cubic metre. The film-normal demagnetizing field is folded into the anisotropy, and the wall's own stray field is a local hard-axis term with the demagnetizing factor 0.0224 used in that paper's wall model. Long-range dipolar fields, strip edges, temperature and disorder are left out. Damping 0.5 is that paper's value; the spin-wave preset uses a hypothetical 0.1 so that waves travel a few wavelengths. Damping stays at 0.1 or more because, below that, waves radiated by a turning wall reflect inside the short simulated strips and spoil the speed measurement. The strip ends are held in fixed domains and the damping rises to 1 over the last 20 nm so waves leave. The wall-model curve comes from the collective-coordinate model of a rigid wall (its position and moment angle), a separate reduced description.

Yellow is measured in the chain, cream is the wall model or the dispersion relation, and the pink dashes mark the Walker field.

More numbers and the saved record

Wavelength of the antenna wave measured in the chain

\text{antenna off}

Wall width parameter at rest

5.6\,\mathrm{nm}

Exchange length

4.59\,\mathrm{nm}

DMI field of the wall model

127\,\mathrm{mT}

DMI above which the wall at rest is fully Néel

0.121\,\mathrm{mJ\,m^{-2}}

Turn period of the wall moment in the wall model

\text{steady, no turning}

Lowest spin-wave frequency in the up domain

27.5\,\mathrm{GHz}

Chain runs on the speed curve: finished / planned

0\ /\ 20

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

1.11\times10^{-15}

The JSON record holds the parameters, the material constants, the wall position and angle from the chain, every point of the speed curve, the spectrum peaks and the wall-model reference, kept apart.

Start with the essentials

Focus question: Why does a field push a domain wall only so fast, and how does interfacial DMI raise that limit?
One-sentence intuition: Below the Walker field the wall moment settles at a tilt and the wall glides at a speed proportional to the field; above it the moment spins and the average speed falls. DMI makes the Néel wall stiff against that tilt and raises the Walker field.

Core mathematical model

Effective field in the chain, with interfacial DMI

\mathbf{H}_{\mathrm{eff}}=\frac{2A}{\mu_0 M_{\mathrm{s}}}\,\partial_x^{2}\mathbf{m}+\frac{2K_0}{\mu_0 M_{\mathrm{s}}}\,m_z\hat{\mathbf{z}}-\frac{2K}{\mu_0 M_{\mathrm{s}}}\,m_x\hat{\mathbf{x}}+\frac{2D}{\mu_0 M_{\mathrm{s}}}\bigl(-\partial_x m_z,\,0,\,\partial_x m_x\bigr)+H\hat{\mathbf{z}}

Exchange $A$ , effective anisotropy $K_0=K_{\mathrm{u}}-\mu_0M_{\mathrm{s}}^{2}/2$ , the wall's hard-axis term $K$ and the DMI $D$ . The chain uses these terms with 1 nm cells and the Landau-Lifshitz-Gilbert equation.

Wall model: position and moment angle

\dot{\Phi}+\frac{\alpha\,\dot{q}}{\Delta}=\gamma_0 H,\qquad \frac{\dot{q}}{\Delta}-\alpha\,\dot{\Phi}=\gamma_0\bigl(H_{\mathrm{D}}\sin\Phi-H_{K}\sin\Phi\cos\Phi\bigr)

A rigid wall of width $\Delta=\sqrt{A/K_0}$ with centre $q$ and moment angle $\Phi$ . Here $H_{\mathrm{D}}=\pi D/(2\mu_0M_{\mathrm{s}}\Delta)$ and $H_K=2K/(\mu_0M_{\mathrm{s}})$ (Thiaville et al. 2012).

Steady motion below the Walker field

v=\frac{\gamma_0\,\Delta(\Phi)}{\alpha}\,H,\qquad \Delta(\Phi)=\sqrt{\frac{A}{K_0+K\cos^{2}\Phi}}

With a constant moment angle the speed grows linearly with field. The wall narrows a little as its moment turns toward the Néel direction (Mougin et al. 2007). At 40 mT the rigid wall model gives 78.8 m/s, this width-corrected formula 78.3 m/s and the chain 77.9 m/s.

Spin waves in the up domain

f=\frac{\gamma_0}{2\pi}\sqrt{\bigl(H+H_{K_0}+\lambda k^{2}\bigr)\bigl(H+H_{K_0}+H_{K}+\lambda k^{2}\bigr)},\qquad \lambda=\frac{2A}{\mu_0M_{\mathrm{s}}}

Linearizing the same equation about $\mathbf{m}=\hat{\mathbf{z}}$ gives a gap near 27 GHz at 40 mT. On the 1 nm chain $k^{2}$ becomes $(4/a^{2})\sin^{2}(ka/2)$ ; DMI drops out for waves along the strip in a perpendicular domain.

Common difficulties

The field does not push the wall directly

Typical misconceptionThe field exerts a force on the wall the way gravity pulls a ball downhill.

Better mental modelThe field makes every moment precess. Inside the wall, that precession turns the moment out of its rest angle, and the restoring torque from the wall's own energy is what rotates the moments in front of it. When the field is too large for any steady tilt, the moment spins and the wall stalls.

DMI does not make the wall more mobile

Typical misconceptionA wall with DMI moves faster at every field.

Better mental modelBelow the Walker field the speed is the same with and without DMI. DMI raises the Walker field, so the steady regime, and the top speed, extend to much larger fields.

What the model leaves out

Typical misconceptionA real Pt/Co wire behaves like this chain.

Better mental modelThe strip is a one-dimensional chain of 1 nm cells using the Pt/Co/AlOx film of Thiaville and co-workers (2012): saturation magnetization 1100 kA/m, exchange stiffness 16 pJ/m, anisotropy 1.27 MJ per cubic metre. The film-normal demagnetizing field is folded into the anisotropy, and the wall's own stray field is a local hard-axis term with the demagnetizing factor 0.0224 used in that paper's wall model. Long-range dipolar fields, strip edges, temperature and disorder are left out. Damping 0.5 is that paper's value; the spin-wave preset uses a hypothetical 0.1 so that waves travel a few wavelengths. Damping stays at 0.1 or more because, below that, waves radiated by a turning wall reflect inside the short simulated strips and spoil the speed measurement. The strip ends are held in fixed domains and the damping rises to 1 over the last 20 nm so waves leave. The wall-model curve comes from the collective-coordinate model of a rigid wall (its position and moment angle), a separate reduced description.

Run the experiment

01
Watch the Néel wall glide
Start from A Néel wall glides at 40 mT and let it play.
What to observe: The up domain grows and the wall crosses the strip at about 78 m/s. In the compass its moment tilts from the Néel direction to about 48 degrees and then stays there.
02
Remove the DMI
Choose Without DMI, steady only below 8 mT, then raise the field to 20 mT.
What to observe: At 5 mT the Bloch wall glides at about 10 m/s. Above the 7.7 mT Walker field the moment spins, turning once every 2.4 ns at 20 mT, and the wall advances in jerks.
03
Read the speed curve
Watch the yellow dots fill the speed panel, then compare them with the cream wall-model curve.
What to observe: Each dot is a separate chain run. Speed rises linearly up to the Walker field near 65 mT and then drops sharply. The chain and the wall model agree within about 2 percent.
04
Reverse the DMI
Choose Reversed DMI flips the wall moment.
What to observe: The speed is unchanged, but the wall moment now starts along the opposite Néel direction. The sign of D sets the handedness of the wall.
05
Launch spin waves
Choose An antenna launches spin waves, then change the antenna frequency in the advanced group.
What to observe: Ripples run from the antenna along the front edge of the strip and fade within about 100 nm, drained by the damping. From 40 to 150 GHz the wavelength measured in the chain agrees with the damped dispersion relation to better than 0.1 percent, and the antenna marker sits on the measured spectrum.

Domain Wall Motion and Spin Waves

How fast can a field push a domain wall?

Core mathematical model

Effective field in the chain, with interfacial DMI

Wall model: position and moment angle

Steady motion below the Walker field

Spin waves in the up domain

Common difficulties

The field does not push the wall directly

DMI does not make the wall more mobile

What the model leaves out

Run the experiment

Watch the Néel wall glide

Remove the DMI

Read the speed curve

Reverse the DMI

Launch spin waves

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