---
title: "Domain Wall Motion and Spin Waves Interactive Lab | Sandbox Physics"
description: "Why does a field push a domain wall only so fast, and how does interfacial DMI raise that limit?"
canonical_url: "https://sandboxphysics.com/labs/domain-wall-spin-waves/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/domain-wall-spin-waves/"
last_modified: "2026-10-30"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

[Course: how magnetization finds its own texture](https://sandboxphysics.com/stories/micromagnetics/)

Physics tutorial

## How fast can a field push a domain wall?

**Background** In a film magnetized perpendicular to its surface, regions pointing up and down are separated by domain walls a few nanometres wide. Racetrack memories and wall-based logic move such walls along narrow tracks.

**Why it matters** A field along the up direction favours the up domain, so the wall should move to enlarge it. The surprise is that the wall moves only by turning the moment inside it, and that turning sets a speed limit.

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 misconception** The field exerts a force on the wall the way gravity pulls a ball downhill.

**Better mental model** The 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 misconception** A wall with DMI moves faster at every field.

**Better mental model** Below 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 misconception** A real Pt/Co wire behaves like this chain.

**Better mental model** 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.

### Run the experiment

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

Physical conclusion

A domain wall moves only by turning its own moment; the largest steady turn sets the Walker field, and interfacial DMI raises it.

## Related experiments and articles

Micromagnetics · Reading path

### Micromagnetics: how magnetization finds its own texture

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Exchange smooths the magnetization and sets how fine the mesh must be**](https://sandboxphysics.com/stories/micromagnetics/exchange-and-the-mesh/)
2. [Lab **Skyrmion Motion in a Track** Drive a skyrmion along a track](https://sandboxphysics.com/labs/skyrmion-track-pinning/)
3. [Article **A field drives a domain wall only so fast**](https://sandboxphysics.com/stories/micromagnetics/domain-walls-and-walker-breakdown/)
4. [Article **How a small tilt travels through a magnet as a spin wave**](https://sandboxphysics.com/stories/micromagnetics/spin-waves-and-dispersion/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)