Landau–Lifshitz–Gilbert equation

Template:Short description In physics, the Landau–Lifshitz–Gilbert equation (usually abbreviated as LLG equation), named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the dynamics (typically the precessional motion) of magnetization Template:Math in a solid. It is a modified version by Gilbert of the original equation of Landau and Lifshitz. The LLG equation is similar to the Bloch equation, but they differ in the form of the damping term. 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 Template:Math is not restricted to real magnetic fields; it incorporates a wide range of mechanisms including magnetic anisotropy, exchange interaction, and so on.

The various forms of the LLG equation are commonly used in micromagnetics to model the effects of a magnetic field and other magnetic interactions on ferromagnetic materials. It provides a practical way to model the time-domain behavior of magnetic elements. Recent developments generalizes the LLG equation to include the influence of spin-polarized currents in the form of spin-transfer torque.

Landau–Lifshitz equation

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.

In a ferromagnet, the magnitude of the magnetization Template:Math at each spacetime point is approximated by the saturation magnetization Template:Math (although it can be smaller when averaged over a chunk of volume). The LLG equation describes the rotation of the magnetization in response to the effective field Template:Math and accounts for not only a real magnetic field but also internal magnetic interactions such as exchange and anisotropy. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by Template:Harvtxt: Template:NumBlk where Template:Math is the electron gyromagnetic ratio and Template:Math is a phenomenological damping parameter, often replaced by

$\lambda =\alpha \frac{\gamma }{M_{\mathrm{s}}},$

where Template:Math is a dimensionless constant called the damping factor. The effective field Template:Math is a combination of the external magnetic field, the demagnetizing field, and various internal magnetic interactions involving quantum mechanical effects, which is typically defined as the functional derivative of the magnetic free energy with respect to the local magnetization Template:Math. To solve this equation, additional conditions for the demagnetizing field must be included to accommodate the geometry of the material.

Landau–Lifshitz–Gilbert equation

In 1955 Gilbert replaced the damping term in the Landau–Lifshitz (LL) equation by one that depends on the time derivative of the magnetization: Template:NumBlk This is the Landau–Lifshitz–Gilbert (LLG) equation, where Template:Math is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation: Template:NumBlk where

${\gamma }^{\prime }=\frac{\gamma }{1+{\gamma }^2{\eta }^2{M}_s^2}\text{and}\lambda =\frac{{\gamma }^2\eta }{1+{\gamma }^2{\eta }^2{M}_s^2}.$

In this form of the LL equation, the precessional term Template:Math depends on the damping term. This better represents the behavior of real ferromagnets when the damping is large.

Landau–Lifshitz–Gilbert–Slonczewski equation

In 1996 John Slonczewski expanded the model to account for the spin-transfer torque, i.e. the torque induced upon the magnetization by spin-polarized current flowing through the ferromagnet. This is commonly written in terms of the unit moment defined by Template:Math:

$\dot{𝐦}=-\gamma 𝐦\times {𝐇}_{\mathrm{e}\mathrm{f}\mathrm{f}}+\alpha 𝐦\times \dot{𝐦}+{\tau }_{\parallel }\frac{𝐦\times (𝐱\times 𝐦)}{|𝐱\times 𝐦|}+{\tau }_{\perp }\frac{𝐱\times 𝐦}{|𝐱\times 𝐦|}$

where $\alpha$ is the dimensionless damping parameter, ${\tau }_{\perp }$ and ${\tau }_{\parallel }$ are driving torques, and Template:Math is the unit vector along the polarization of the current.

Further reading

• Template:Cite journal This is only an abstract; the full report is "Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956", but was never published. A description of the work is given in Template:Cite journal
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External links

• Magnetization dynamics applet