Micromagnetics · A course with experiments

L4 · Chapter 7

A current pushes a skyrmion at an angle, and edges and weak spots decide where it ends up

A spin-polarized current drags the magnetization along with the electrons. A skyrmion answers like a particle with a sideways, Magnus-like force set by its topological charge, so it moves at an angle to the current: −10.4 degrees for damping 0.3 and a non-adiabatic coefficient of 0.15. In a track it drifts to an edge, the tilted edge moments push it back, and it glides along the edge at $\beta/\alpha$ times the drive. Without the non-adiabatic torque it stops there, and a stronger current forces it through the edge, where it vanishes. A weak spot holds it until the current passes a threshold, between 41.9 and 42.5 m/s on the 2 nm mesh of the Lab.

Sandbox Physics · 2026-10-29

The current as a drift velocity

In the Zhang-Li model a current adds two torques to the LLG equation (Abert 2019, Eqs. 134 and 135). Both act only where the magnetization varies, and both are written with the spin drift velocity $u$ , a velocity proportional to the current density. The first, adiabatic, term carries the texture along with the electrons. The second is $\beta$ times a torque at right angles to it; $\beta$ is called the non-adiabatic coefficient.

Abert notes that the model is local: it ignores the diffusion of spin, and becomes less accurate where the magnetization changes sharply. The skyrmion Lab’s track takes $u$ along $+x$ and up to 200 m/s, and $\beta/\alpha$ from 0 to 2.

\frac{\partial\mathbf m}{\partial t}=-\gamma_0\,\mathbf m\times\mathbf H_{\mathrm{eff}}+\alpha\,\mathbf m\times\frac{\partial\mathbf m}{\partial t}-(\mathbf u\cdot\nabla)\mathbf m+\beta\,\mathbf m\times(\mathbf u\cdot\nabla)\mathbf m

Worked example

The Thiele equation and the skyrmion Hall angle

Treat the skyrmion as rigid, $\mathbf m(\mathbf r-\mathbf R(t))$ , and project the equation onto its two translations. What remains is the Thiele equation for the velocity $\mathbf v=\dot{\mathbf R}$ (Iwasaki, Mochizuki and Nagaosa 2013, Eq. 4). $G=4\pi Q=-4\pi$ is the gyrotropic term from the charge of Chapter 6. $\mathcal D=\int(\partial_x\mathbf m)^2\,\mathrm d^2r$ measures how sharply the texture varies; for the radial profile at $D=3.0\,\mathrm{mJ/m^2}$ it is 16.6. The last term is the force from edges and defects.

Without that force, split the equation into its $x$ and $y$ parts and solve the two linear equations; the result is below. At $u=60\,\mathrm{m/s}$ , $\alpha=0.3$ and $\beta=0.15$ it gives $v_x=55.9$ and $v_y=-10.3\,\mathrm{m/s}$ : the skyrmion moves at −10.4 degrees to the current. This sideways motion is the skyrmion Hall effect.

When $\beta=\alpha$ , $v_y=0$ and $v_x=u$ : the skyrmion moves straight with the electrons. For small $\alpha$ and $\beta$ Iwasaki et al. give the sideways part as $\delta\mathbf v=\hat z\times\mathcal D(\alpha-\beta)\mathbf u/G$ , which is the first-order form of the same result.

The Lab checks this in an unbounded film. With 2 nm cells the skyrmion moves at (54.7, −10.1) m/s; with 1 nm cells at (55.6, −10.2) m/s. The error along the track drops from 2.2 to 0.55 per cent when the cells are halved, as a second-order method should, and the angle matches to 0.03 degrees.

G\,\hat z\times(\mathbf u-\mathbf v)+\mathcal D(\beta\mathbf u-\alpha\mathbf v)-\nabla V=0\;\Longrightarrow\; v_x=u\,\frac{G^2+\alpha\beta\mathcal D^2}{G^2+\alpha^2\mathcal D^2},\quad v_y=u\,\frac{G\mathcal D(\alpha-\beta)}{G^2+\alpha^2\mathcal D^2}

Worked example

An edge pushes back, and the skyrmion glides

In a track the sideways drift carries the skyrmion toward one edge. The edge moments tilt in the plane, as Chapter 6 showed, and Iwasaki et al. explain why that repels a skyrmion: the tilt at the edge points opposite to the tilt at the skyrmion’s perimeter. The skyrmion stops approaching where the edge force balances the gyrotropic term.

Once it glides steadily along the edge, $v_y=0$ and the edge force points only along $y$ . The $x$ part of the Thiele equation then reads $\mathcal D(\beta u-\alpha v_x)=0$ , so $v_x=(\beta/\alpha)u$ (Iwasaki et al. Eq. 5). The $y$ part fixes how hard the edge must push: $G(u-v_x)$ .

In the skyrmion Lab’s track, with the same drive, the skyrmion settles about 20 nm from the lower edge and glides at 29.0 m/s with 2 nm cells and 29.8 m/s with 1 nm cells, against 30 m/s. Part of the gap is the grid’s slowing of the drift term, seen when $\beta=\alpha$ gives 58.7 and 59.7 m/s instead of 60. Allowing for that factor, the glide speed is 1.1 and 0.2 per cent low.

With no non-adiabatic torque the edge must hold everything

Set $\beta=0$ . Eq. 5 gives $v_x=0$ : the skyrmion drifts to the edge and stops, which Iwasaki et al. call intrinsic pinning. The edge must now push with $|G|u$ , in proportion to the current. The edge barrier is finite, so above some current the skyrmion is forced into the edge and leaves the film.

In the Lab, starting 20 nm from the lower edge, the skyrmion is held at 38.75 m/s but not at 39.375 m/s with 2 nm cells. With 1 nm cells it is held at 35.9 m/s and not at 42.2 m/s; those two runs were placed 8 per cent either side of the 2 nm bracket and only show that the threshold moves by less than that. At 150 m/s the skyrmion is gone in 0.64 ns. Its charge passes through 16 recorded samples between −0.9 and −0.1, at most 0.23 per 10 ps: the charge leaves through the edge continuously.

Iwasaki et al. also find a threshold for pushing a skyrmion out of the end of a magnetic region, about $1.5\times10^{11}\,\mathrm{A/m^2}$ for $\beta/\alpha=1$ in their model. Their lattice model, geometry and material differ from the Lab’s, so read their number as a separate result.

A weak spot and the depinning threshold

A disc of radius 10 nm where the anisotropy is 30 per cent lower attracts the skyrmion, because its wall costs less energy there. At rest, $\mathbf v=0$ , the Thiele equation asks the surroundings for a backward force $\beta\mathcal Du$ and a sideways force $|G|u$ . With $\beta=\alpha=0.3$ the sideways part is 2.5 times larger. Iwasaki et al. write the threshold as $|F_{\mathrm{pin}}|/|G|$ for a free skyrmion and $|F^{\mathrm{pin}}_x|/(\beta\mathcal D)$ in a channel, where an edge supplies the sideways force.

In the skyrmion Lab’s track, with the spot at the centre and $\beta=\alpha$ , the skyrmion is held at 41.875 m/s and breaks free at 42.5 m/s with 2 nm cells. Just below the threshold it rests about 9 nm past the centre of the spot and 7 to 8 nm off the centre line. With 1 nm cells it is held at 38.8 and escapes at 45.6 m/s, so the mesh moves the threshold by less than 8 per cent. There is no independent reference for this number: it is a zero-temperature result for one spot, one start position and the run rule set out in Chapter 8.

What the rigid-particle picture leaves out

The Thiele equation treats the skyrmion as rigid, but near an edge or a weak spot it deforms, which is why the Lab integrates the full LLG equation and keeps the Thiele velocity as a separate reference line. The Lab has no temperature, so its thresholds hold only at zero temperature, and the only current torque is the Zhang-Li torque. The stray field is local, as in Rohart and Thiaville. The track is a 128 nm loop, so the skyrmion meets the weak spot once per lap.

An experiment to think through

Predict, change, explain

With $u=60\,\mathrm{m/s}$ and $\beta/\alpha=0.5$ , predict which edge the skyrmion drifts toward and the speed at which it then glides. Then predict the motion when $\beta=\alpha$ .
In the skyrmion Lab choose “Drift sideways, then glide along the edge”, let it play, then choose “No sideways drift when beta equals alpha”.
Explain why one ratio, $\beta/\alpha$ , sets both whether the skyrmion drifts sideways and how fast it glides once an edge stops the drift.

Drive a skyrmion along a track

The macrospin Lab treats the strip of muMAG Standard Problem 4 as one moment. The wall and skyrmion Labs use the Pt/Co film of Thiaville et al. with local stray-field terms. All three run at zero temperature.

Try explaining it yourself

1. Double the drive to $u=120\,\mathrm{m/s}$ with $\alpha=0.3$ and $\beta=0.15$ . What is the free Thiele velocity, and does the Hall angle change?

Read an explained answer

The velocity is linear in $u$ : $(111.8,\,-20.6)\,\mathrm{m/s}$ . The angle stays at −10.4 degrees.

2. Set $\beta=2\alpha=0.6$ at $u=60\,\mathrm{m/s}$ . Which way does the skyrmion drift?

Read an explained answer

$v_x=68.2$ , $v_y=+20.6\,\mathrm{m/s}$ . The sign of $\alpha-\beta$ has flipped, so it drifts toward the other edge and moves faster than $u$ along the track.

3. Keep $\beta/\alpha=0.5$ but lower the damping to $\alpha=0.1$ . What happens to the Hall angle?

Read an explained answer

With $\beta=0.05$ , $v=(59.5,\,-3.90)\,\mathrm{m/s}$ , an angle of −3.75 degrees. The gyrotropic term dominates more when damping is small, so the skyrmion follows the current more closely.

Sources to read alongside

Abert, Micromagnetics and spintronics: models and numerical methods, European Physical Journal B 92, 120 (2019), arXiv:1810.12365v2
Read: the exchange energy (Eq. 10), the demagnetizing field (Sec. 2.3), uniaxial anisotropy (Sec. 2.4), interfacial DMI (Eq. 33) and its boundary term (Sec. 3.5), the Landau-Lifshitz-Gilbert equation in Gilbert and explicit form (Eqs. 117 and 118), the conservation of the magnetization length and the energy dissipation (Eqs. 120 to 125), the Zhang-Li torque (Eq. 134) and the finite-difference method (Sec. 6.1). Other sections were not read.
Rohart & Thiaville, Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya interaction, Physical Review B 88, 184422 (2013), arXiv:1310.0666
Read in full: the Pt/Co/AlOx parameters and the local treatment of the stray field in ultrathin films, the free-edge boundary condition and edge tilt (Eqs. 3 to 8, with the full-dipolar check of Fig. 1), the isolated-skyrmion equation (Eqs. 15 and 16) and the approximate radius (Eq. 18).
Iwasaki, Mochizuki & Nagaosa, Current-induced skyrmion dynamics in constricted geometries, Nature Nanotechnology 8, 742 (2013), arXiv:1310.1655
arXiv version read in full: the Thiele equation for a skyrmion driven by spin-transfer torque (Eq. 4), the edge-glide speed (Eqs. 5 and 6), the pinning thresholds in a free plane and in a channel, edge repulsion and annihilation at the edge (Fig. 5). The simulations are on a lattice spin model, not the Pt/Co film of this course. The journal version was not opened.

The current as a drift velocity

The Thiele equation and the skyrmion Hall angle

An edge pushes back, and the skyrmion glides

With no non-adiabatic torque the edge must hold everything

A weak spot and the depinning threshold

What the rigid-particle picture leaves out

Predict, change, explain

Try explaining it yourself

1. Double the drive to u=120 m/su=120\,\mathrm{m/s}u=120m/s with α=0.3\alpha=0.3α=0.3 and β=0.15\beta=0.15β=0.15. What is the free Thiele velocity, and does the Hall angle change?

2. Set β=2α=0.6\beta=2\alpha=0.6β=2α=0.6 at u=60 m/su=60\,\mathrm{m/s}u=60m/s. Which way does the skyrmion drift?

3. Keep β/α=0.5\beta/\alpha=0.5β/α=0.5 but lower the damping to α=0.1\alpha=0.1α=0.1. What happens to the Hall angle?

Sources to read alongside

1. Double the drive to $u=120\,\mathrm{m/s}$ with $\alpha=0.3$ and $\beta=0.15$ . What is the free Thiele velocity, and does the Hall angle change?

2. Set $\beta=2\alpha=0.6$ at $u=60\,\mathrm{m/s}$ . Which way does the skyrmion drift?

3. Keep $\beta/\alpha=0.5$ but lower the damping to $\alpha=0.1$ . What happens to the Hall angle?