Micromagnetics · Chiral textures

Skyrmion Motion in a Track

Why does a current push a skyrmion sideways, and when does a weak spot or an edge stop it? A current moves a skyrmion, but because its magnetization wraps the sphere once, the push also sends it sideways: the skyrmion Hall effect. In a track it drifts toward one edge until the tilted edge magnetization pushes back, then glides along that edge. A weak spot holds it below a threshold current, and a strong current forces it out through the edge.

Drive a magnetic skyrmion along a Pt/Co track with a spin-polarized current. Watch it drift toward an edge and glide, get caught by a weak spot or leave through the edge, and compare two ways of counting its topological charge as it goes.

Skyrmion velocity in the simulation, along and across the track $v_x = 53.3,\ v_y = -9.11\,\mathrm{m\,s^{-1}}$

Thiele velocity of a free skyrmion $v_x = 55.9,\ v_y = -10.3\,\mathrm{m\,s^{-1}}$

Topological charge

Q

from lattice solid angles $-1.000$

Topological charge

Q

from finite differences $-0.947$

Core radius, where

m_z=0

$8.07\,\mathrm{nm}$

One track, one skyrmion

Drive the skyrmion

Spin drift velocity of the current

u

60\,\mathrm{m\,s^{-1}}

Non-adiabatic torque over damping

\beta/\alpha

0.5

Weak spot: anisotropy lowered by

\delta K/K

0\,\%

Interfacial DMI

D

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

Drag the skyrmion to place it, and drag the weak spot once it is on. The edge switch turns the track into an unbounded film.

Damping and start position

Gilbert damping

\alpha

0.3

Start position along the track

x_0

24\,\mathrm{nm}

Start position across the track

y_0

30\,\mathrm{nm}

Model assumptions

The track is a two-dimensional grid of 2 nm cells, 128 nm long and 60 nm wide; its two ends are joined, so the skyrmion loops around. The film is the Pt/Co/AlOx film of Rohart and Thiaville (2013): exchange stiffness 16 pJ/m, effective anisotropy 510 kJ per cubic metre, saturation magnetization 1100 kA/m, with the stray field folded into the effective anisotropy as in that paper. Temperature, disorder and the long-range stray field are left out. The current acts through the Zhang-Li spin-transfer torque, with a spin drift velocity proportional to the current density; the damping 0.3 and the ratio of the non-adiabatic torque to the damping are assumed values. The free edges obey the DMI boundary condition, which tilts the edge magnetization. The weak spot is a 10 nm disk with lowered anisotropy. On 2 nm cells the free skyrmion is 2 percent smaller and moves about 3 percent slower than the continuum limit; below 2.8 mJ per square metre it collapses on the grid although the continuum keeps it, so the DMI range stops there. The Thiele velocity is a rigid-texture reduction computed from the continuum profile, a separate reduced description.

Yellow and green count the same charge two ways. Cream dashes are the Thiele velocity of a free skyrmion, pink the glide speed along an edge.

More numbers and the saved record

Glide speed against an edge, from the Thiele equation

30\,\mathrm{m\,s^{-1}}

Direction of motion in the simulation, from the current

-9.7^{\circ}

Hall angle of the free Thiele velocity

-10.4^{\circ}

Core radius of the isolated skyrmion in an unbounded continuum film

9.4\,\mathrm{nm}

Dissipation integral of the continuum profile

16.6

In-plane tilt of the first cell at the lower edge, in the simulation

0.447

Continuum edge tilt at that cell centre

0.449

Lattice charge of the painted and relaxed start texture

-1.000

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

2.41\times10^{-12}

Cell size and time step

a = 2\,\mathrm{nm},\ \Delta t = 0.2\,\mathrm{ps}

The JSON record holds the parameters, the material constants, every 10 ps sample of both charges, the centre and the radius, the final texture, and the continuum, Thiele and edge references, kept apart.

Physics tutorial

Where does a current send a skyrmion?

BackgroundA skyrmion is a small whirl in a perpendicular magnetic film: the magnetization points down in its core, up far away, and turns through every direction in between. Interfacial DMI in films such as Pt/Co makes such whirls metastable and sets their handedness.

Why it mattersSkyrmions are proposed as bits that a current moves along a track. Two things get in the way: they do not move straight along the current, and defects or edges can hold or destroy them.

Start with the essentials

Focus question: Why does a current push a skyrmion sideways, and when does a weak spot or an edge stop it?
One-sentence intuition: Because the texture wraps the sphere once, the current exerts a sideways, gyrotropic push as well as a forward one. Only when the non-adiabatic torque matches the damping does the skyrmion follow the current. An edge stops the sideways drift, after which the skyrmion glides at $(\beta/\alpha)\,u$ .

Core mathematical model

Landau-Lifshitz-Gilbert equation with the Zhang-Li torque

\dot{\mathbf{m}}=-\gamma_0\,\mathbf{m}\times\mathbf{H}_{\mathrm{eff}}+\alpha\,\mathbf{m}\times\dot{\mathbf{m}}-(\mathbf{u}\cdot\nabla)\mathbf{m}+\beta\,\mathbf{m}\times(\mathbf{u}\cdot\nabla)\mathbf{m}

The current enters through the spin drift velocity $\mathbf{u}$ , proportional to the current density, and the non-adiabatic ratio $\beta$ . The effective field holds exchange, interfacial DMI and the perpendicular anisotropy.

Topological charge, continuum and lattice

Q=\frac{1}{4\pi}\int\mathbf{m}\cdot\bigl(\partial_x\mathbf{m}\times\partial_y\mathbf{m}\bigr)\,\mathrm{d}x\,\mathrm{d}y,\qquad \tan\frac{q_{ijk}}{2}=\frac{\mathbf{m}_i\cdot(\mathbf{m}_j\times\mathbf{m}_k)}{1+\mathbf{m}_i\cdot\mathbf{m}_j+\mathbf{m}_i\cdot\mathbf{m}_k+\mathbf{m}_j\cdot\mathbf{m}_k}

The Lab evaluates the integral with finite differences and, separately, sums the signed solid angles $q_{ijk}$ of the triangles between neighbouring cells (Kim and Mulkers 2020). For this core-down skyrmion $Q=-1$ .

Rigid-texture (Thiele) velocity of a free skyrmion

v_x=u\,\frac{G^{2}+\alpha\beta\mathcal{D}^{2}}{G^{2}+\alpha^{2}\mathcal{D}^{2}},\qquad v_y=u\,\frac{G\mathcal{D}\,(\alpha-\beta)}{G^{2}+\alpha^{2}\mathcal{D}^{2}},\qquad G=4\pi Q

Projecting the equation on a texture that moves without changing shape gives these velocities, with $\mathcal{D}=\int(\partial_x\mathbf{m})^{2}\,\mathrm{d}x\,\mathrm{d}y$ from the continuum profile. The sideways part vanishes only when $\beta=\alpha$ .

Gliding along an edge

v_y=0\quad\Rightarrow\quad v_x=\frac{\beta}{\alpha}\,u

When an edge force stops the sideways motion, the component of the Thiele equation along the track leaves this speed, whatever the size of the edge force (Iwasaki, Mochizuki and Nagaosa 2013).

Edge condition with interfacial DMI

\frac{\partial\mathbf{m}}{\partial n}=\frac{1}{\xi}\,(\hat{\mathbf{z}}\times\hat{\mathbf{n}})\times\mathbf{m},\qquad \xi=\frac{2A}{D},\qquad \sin\theta_{\mathrm{edge}}=\frac{\Delta}{\xi}

The magnetization tilts at a free edge, with $\Delta=\sqrt{A/K}$ (Rohart and Thiaville 2013). At 3 mJ per square metre the tilt is 0.525 at the edge and 0.449 at the centre of the first 2 nm cell; the grid gives 0.447.

Common difficulties

The sideways drift is not a force from the edge

Typical misconceptionThe skyrmion moves sideways because something pulls it toward the edge.

Better mental modelThe drift appears in an unbounded film too. It comes from the gyrotropic term of the Thiele equation, which turns any push into motion partly at right angles to it. The edge is what stops the drift.

A non-integer charge is not always an error

Typical misconceptionA computed topological charge that is not an integer means the simulation is wrong.

Better mental modelWith free edges the magnetization along the boundary is tilted, so the texture does not cover the sphere a whole number of times and the charge may take any value. Without edges the lattice definition is an integer; the finite-difference one still misses by a few percent on 2 nm cells because of its truncation error.

What the model leaves out

Typical misconceptionA real Pt/Co racetrack behaves exactly like this grid.

Better mental modelThe track is a two-dimensional grid of 2 nm cells, 128 nm long and 60 nm wide; its two ends are joined, so the skyrmion loops around. The film is the Pt/Co/AlOx film of Rohart and Thiaville (2013): exchange stiffness 16 pJ/m, effective anisotropy 510 kJ per cubic metre, saturation magnetization 1100 kA/m, with the stray field folded into the effective anisotropy as in that paper. Temperature, disorder and the long-range stray field are left out. The current acts through the Zhang-Li spin-transfer torque, with a spin drift velocity proportional to the current density; the damping 0.3 and the ratio of the non-adiabatic torque to the damping are assumed values. The free edges obey the DMI boundary condition, which tilts the edge magnetization. The weak spot is a 10 nm disk with lowered anisotropy. On 2 nm cells the free skyrmion is 2 percent smaller and moves about 3 percent slower than the continuum limit; below 2.8 mJ per square metre it collapses on the grid although the continuum keeps it, so the DMI range stops there. The Thiele velocity is a rigid-texture reduction computed from the continuum profile, a separate reduced description.

Run the experiment

01
Watch the sideways drift
Start from Drift sideways, then glide along the edge and let it play.
What to observe: The skyrmion leaves at about 53 m/s along the track and 9 m/s toward the lower edge, close to the free Thiele velocity of 55.9 and 10.3 m/s. About 20 nm from the edge it stops drifting and glides at 29 m/s, near $(\beta/\alpha)\,u=30\,\mathrm{m/s}$ .
02
Match beta to alpha
Choose No sideways drift when beta equals alpha.
What to observe: The skyrmion runs straight along the centre line at 58 m/s. In the continuum it would move exactly at u, 60 m/s; the 2 nm grid slows it by about 3 percent.
03
Pin it, then tear it free
Choose A weak spot holds the skyrmion, then A stronger current tears it free.
What to observe: At 20 m/s the skyrmion spirals into the weak spot and stops just beside its centre. At 80 m/s it is pulled in and deflected as it crosses, then breaks away and continues around the loop.
04
Push it out through the edge
Choose Pushed out through the edge and watch the two charge curves.
What to observe: With $\beta=0$ and $u=150\,\mathrm{m/s}$ the edge cannot hold it. Within a nanosecond the skyrmion runs into the edge and the charge falls smoothly from minus one to zero in both definitions. The change happens through the boundary, so it is not a numerical error.
05
Remove the edges
Choose No edges: the Hall angle stays.
What to observe: In the periodic film the skyrmion keeps moving at minus 10.4 degrees to the current, as the Thiele velocity predicts, and the lattice charge stays at minus one to every printed digit. The finite-difference charge reads about minus 0.956 on the same texture.

Skyrmion Motion in a Track

Where does a current send a skyrmion?

Core mathematical model

Landau-Lifshitz-Gilbert equation with the Zhang-Li torque

Topological charge, continuum and lattice

Rigid-texture (Thiele) velocity of a free skyrmion

Gliding along an edge

Edge condition with interfacial DMI

Common difficulties

The sideways drift is not a force from the edge

A non-integer charge is not always an error

What the model leaves out

Run the experiment

Watch the sideways drift

Match beta to alpha

Pin it, then tear it free

Push it out through the edge

Remove the edges

Related experiments and articles

Micromagnetics: how magnetization finds its own texture