Thiele Equation

Intuition

The full LLGS equation is a continuum equation of motion for $\vec{m}(\vec{r},t)$ — millions of degrees of freedom in a typical simulation. But many magnetic textures (vortices, skyrmions, domain walls) are rigid objects: they translate and rotate without changing their internal shape. For such textures it is enough to track a single collective coordinate — the position $\vec{X}(t)$ of the texture’s center — and the entire LLGS dynamics collapses to a Newton-like equation on $\vec{X}$. For a magnetic vortex core in a thin ferromagnetic disk this reduction is the Thiele equation, and the vortex core moves like a charged particle in a magnetic field living in the plane.

Formal Definition

For a rigid magnetization texture $\vec{m}(\vec{r},t)={\vec{m}}_0(\vec{r}-\vec{X}(t))$, inserting the ansatz into the LLGS equation and integrating over space yields the Thiele equation:

$$\boxed{\overline{\stackrel{ˉ}{M}}\cdot \ddot{\vec{X}}+\overline{\stackrel{ˉ}{G}}\cdot \dot{\vec{X}}=-{\nabla }_{\vec{X}}W-\overline{\stackrel{ˉ}{D}}\cdot \dot{\vec{X}}+{\vec{F}}_{\text{ext}}}$$

where, evaluated for the rigid profile ${\vec{m}}_0$,

• $\overline{\stackrel{ˉ}{M}}$ — mass tensor (usually negligible for thin films),
• $\overline{\stackrel{ˉ}{G}}$ — gyrotropic tensor, producing a Magnus-like transverse force,
• $-{\nabla }_{\vec{X}}W$ — restoring force from the confinement energy $W(\vec{X})$,
• $\overline{\stackrel{ˉ}{D}}$ — damping tensor, the collective image of Gilbert dissipation,
• ${\vec{F}}_{\text{ext}}$ — external forces, notably the spin-transfer torque in current-driven devices.

Key Results

1. Thin-disk reduction (the form used in practice)

For a thin circular free layer of thickness $L$ hosting a magnetic vortex, the planar symmetry collapses the tensors to scalars / a single vector:

$$\vec{G}\times \dot{\vec{X}}=-k\vec{X}-D\dot{\vec{X}}+{\vec{F}}_{\text{ext}},\vec{G}=G{\hat{e}}_z,$$

so that the gyrotropic term becomes

$$\overline{\stackrel{ˉ}{G}}\cdot \dot{\vec{X}}=G({\hat{e}}_z\times \dot{\vec{X}}).$$

The constants have transparent micromagnetic meanings:

• Gyrovector magnitude:

$$G=-\frac{2\pi pL{M}_s}{{\gamma }_0},$$

where $p=\pm 1$ is the vortex-core polarity (out-of-plane direction of $\vec{m}$ at the core). The sign of $G$ determines the sense of gyration.

• Confinement stiffness: $k$ comes from the quadratic expansion of the magnetostatic + exchange energy in $\vec{X}$ near the center: $W(\vec{X})\approx \frac{1}{2}k|\vec{X}{|}^2$.
• Damping coefficient: $D\propto {\alpha }_G$ — the collective image of Gilbert damping in the rigid-profile reduction.

2. Free gyrotropic motion

With ${\vec{F}}_{\text{ext}}=0$ and $D=0$, the Thiele equation reduces to

$$G({\hat{e}}_z\times \dot{\vec{X}})=-k\vec{X},$$

whose solution is circular gyration of the core around the disk center at the gyrotropic frequency

$${\omega }_G=\frac{k}{|G|}.$$

This is the eigenmode that dominates the low-frequency response of vortex disks — the working point of STVOs.

3. Force balance under spin-transfer torque

Adding a current-induced force ${\vec{F}}_{\text{STT}}$ from the Slonczewski torque gives the steady-state balance

$$\vec{G}\times \dot{\vec{X}}-k\vec{X}={\vec{F}}_{\text{STT}}-D\dot{\vec{X}}.$$

Above a critical current, ${\vec{F}}_{\text{STT}}$ overcomes damping and the core enters sustained auto-oscillation on a stable limit cycle — the operating regime of a vortex-based nano-oscillator.

Summary

Term	Meaning	Role
$\overline{\stackrel{ˉ}{M}}\cdot \ddot{\vec{X}}$	Inertial	Usually negligible in thin films
$\overline{\stackrel{ˉ}{G}}\cdot \dot{\vec{X}}$	Gyrotropic	Magnus-like transverse force; sign set by core polarity $p$
$-{\nabla }_{\vec{X}}W$	Restoring	Confinement of the texture
$\overline{\stackrel{ˉ}{D}}\cdot \dot{\vec{X}}$	Damping	Gilbert dissipation, collective form
${\vec{F}}_{\text{ext}}$	Driving	Spin-transfer torque, applied fields

The Thiele equation turns a continuum LLGS problem into a 2D Newton equation for the vortex core, exposing the gyrotropic eigenmode and the spin-transfer-driven limit cycles that power STVOs.

Connections

• llg-equation — the full continuum equation from which Thiele is derived
• spin-transfer-torque — origin of ${\vec{F}}_{\text{ext}}$ in spintronic devices
• magnetic-vortex — the topological texture whose center is $\vec{X}$ (stub)
• gyrovector — geometric origin of $\vec{G}$ (stub)
• spin-torque-vortex-oscillator — device exploiting the gyrotropic mode (stub)

References

• A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973) — the original derivation.
• D. L. Huber, Phys. Rev. B 26, 3758 (1982).
• K. Yu. Guslienko, J. Nanosci. Nanotechnol. 8, 2745 (2008) — vortex Thiele dynamics.
• B. A. Ivanov & C. E. Zaspel, Phys. Rev. Lett. 99, 247208 (2007).