Landau–Lifshitz–Gilbert equation

The LLG equation describes a more general scenario of magnetization dynamics beyond the simple Larmor precession.

In particular, the effective field driving the precessional motion of M is not restricted to real magnetic fields; it incorporates a wide range of mechanisms including magnetic anisotropy, exchange interaction, and so on.

Recent developments generalizes the LLG equation to include the influence of spin-polarized currents in the form of spin-transfer torque.

An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by Landau & Lifshitz (1935):[1] where γ is the electron gyromagnetic ratio and λ is a phenomenological damping parameter, often replaced by where α is a dimensionless constant called the damping factor.

In 1955 Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization: This is the Landau–Lifshitz–Gilbert (LLG) equation, where η is the damping parameter, which is characteristic of the material.

The terms of the Landau–Lifshitz–Gilbert equation: precession (red) and damping (blue). The trajectory of the magnetization (dotted spiral) is drawn under the simplifying assumption that the effective field H eff is constant.