LLG Equation

Intuition

A ferromagnet’s magnetization $\vec{m}$ behaves like a gyroscope: when you push it with a magnetic field, it doesn’t simply align with the field — it precesses around it, just as a spinning top precesses around gravity. Real magnets also lose energy (to the lattice, to electrons, to spin waves), so the precession slowly spirals inward until $\vec{m}$ comes to rest along the effective field. And in modern spintronic devices, an electrical current carrying angular momentum can give the magnet an extra “kick” — a spin-transfer torque — which can sustain oscillations or even switch the magnetization. The Landau–Lifshitz–Gilbert (LLG) equation, in its full Slonczewski form, packages all three effects into a single equation of motion for the unit vector $\vec{m}=\vec{M}/M_s$.

Formal Definition

The full equation of motion, including spin-transfer torque, is the Landau–Lifshitz–Gilbert–Slonczewski (LLGS) equation:

$$\boxed{\frac{\partial \vec{m}}{\partial t}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}+{\alpha }_G\vec{m}\times \frac{\partial \vec{m}}{\partial t}+{\vec{\tau }}^{\text{ST}}}$$

with

• $\vec{m}(\vec{r},t)=\vec{M}/M_s$ the unit magnetization vector ($|\vec{m}|=1$),
• ${\vec{H}}^{\text{eff}}$ the effective field (sum of exchange, anisotropy, demagnetizing, and Zeeman contributions),
• ${\gamma }_0={\mu }_0{\gamma }_G$ where ${\gamma }_G$ is the gyromagnetic ratio — the constant relating angular momentum and magnetic moment,
• ${\alpha }_G\ll 1$ the dimensionless Gilbert damping coefficient,
• ${\vec{\tau }}^{\text{ST}}$ the Slonczewski spin-transfer torque.

The constraint $|\vec{m}|=1$ is built in: each torque term lies in the plane perpendicular to $\vec{m}$, so only the direction of $\vec{m}$ changes, never its magnitude.

Key Results

1. Pure precession — the Landau–Lifshitz (LL) equation

In the absence of damping and external torques:

$$\frac{\partial \vec{m}}{\partial t}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}.$$

This drives the Larmor precession of $\vec{m}$ around ${\vec{H}}^{\text{eff}}$. The minus sign reflects the negative gyromagnetic ratio of the electron: precession is clockwise when viewed along ${\vec{H}}^{\text{eff}}$. Energy is conserved — the cone angle never closes.

2. Adding dissipation — the Gilbert torque

Real magnets dissipate energy through spin–lattice and spin–electron couplings. Gilbert added a viscous torque

$${\vec{\tau }}^G={\alpha }_G\vec{m}\times \frac{\partial \vec{m}}{\partial t},$$

which is perpendicular to ${\partial }_t\vec{m}$ and points inward in the precessional plane. Inserting it gives the LLG equation:

$$\frac{\partial \vec{m}}{\partial t}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}+{\alpha }_G\vec{m}\times \frac{\partial \vec{m}}{\partial t}.$$

The cone angle now closes monotonically: $\vec{m}$ spirals toward ${\vec{H}}^{\text{eff}}$. Typical values of ${\alpha }_G$ range from $\sim 10^{-4}$ (YIG) to $\sim 10^{-2}$ (Co, Fe alloys).

3. Explicit form

The LLG equation is implicit in ${\partial }_t\vec{m}$. Taking the cross product with $\vec{m}$ on the left and using $|\vec{m}|=1$ yields the mathematically equivalent explicit Landau–Lifshitz form:

$$(1+{\alpha }_G^2)\frac{\partial \vec{m}}{\partial t}=-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}-{\gamma }_0{\alpha }_G\vec{m}\times (\vec{m}\times {\vec{H}}^{\text{eff}}).$$

This is the form used by most micromagnetic solvers (e.g. MuMax3, OOMMF).

4. Spin-transfer torque — the LLGS equation

In MTJs and spin-valve nanopillars, a spin-polarized current transfers angular momentum to the local magnetization. The Slonczewski torque has the canonical form

$${\vec{\tau }}^{\text{ST}}={\gamma }_0{a}_J\vec{m}\times (\vec{m}\times \vec{p})+{\gamma }_0{b}_J\vec{m}\times \vec{p},$$

where $\vec{p}$ is the unit polarization vector (set by a reference layer), and $a_J$, $b_J$ encode the damping-like and field-like components, both proportional to the current density $J$. The damping-like term competes directly with ${\alpha }_G$: above a critical current $J_c$ it can sustain auto-oscillations (as in STVOs) or switch the free-layer magnetization (the basis of STT-MRAM).

Summary

Term	Name	Role
$-{\gamma }_0\vec{m}\times {\vec{H}}^{\text{eff}}$	Precessional (LL)	Larmor precession around ${\vec{H}}^{\text{eff}}$
${\alpha }_G\vec{m}\times {\partial }_t\vec{m}$	Gilbert damping	Drives alignment with ${\vec{H}}^{\text{eff}}$
${\vec{\tau }}^{\text{ST}}$	Spin-transfer torque	Current-induced torque (MTJs, spin valves)

Connections

• effective-field — what enters ${\vec{H}}^{\text{eff}}$ and how it’s computed
• larmor-precession — the precessional motion driven by the LL term
• spin-transfer-torque — origin and structure of ${\vec{\tau }}^{\text{ST}}$
• spin-polarized-current — how the polarization $\vec{p}$ arises
• magnetic-tunnel-junction — the device in which LLGS is most often applied
• thiele-equation — collective-coordinate reduction for rigid textures (vortex core)

References

• L. Landau & E. Lifshitz, Phys. Z. Sowjet. 8, 153 (1935).
• T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004) — phenomenological damping.
• J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996) — spin-transfer torque.
• W. F. Brown Jr., Micromagnetics (Interscience, 1963).