Landau–Lifshitz–Gilbert equation

In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization M in a solid. It is a modification by Gilbert of the original equation of Landau and Lifshitz. The various forms of the equation are commonly used in micromagnetics to model the effects of a magnetic field on ferromagnetic materials. In particular it can be used to model the time domain behavior of magnetic elements due to a magnetic field. An additional term was added to the equation to describe the effect of spin polarized current on magnets.

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 M at each point equals the saturation magnetization M_s (although it can be smaller when averaged over a chunk of volume). The Landau–Lifshitz–Gilbert equation predicts the rotation of the magnetization in response to torques. An earlier, but equivalent, equation (the Landau–Lifshitz equation) was introduced by (Landau Lifshitz):

[math]\displaystyle{ \frac{\mathrm{d}\mathbf{M}}{\mathrm{d} t}= -\gamma \mathbf{M} \times \mathbf{H_\mathrm{eff}} - \lambda \mathbf{M} \times \left(\mathbf{M} \times \mathbf{H_{\mathrm{eff}}}\right) }[/math]

(1)

where γ is the electron gyromagnetic ratio and λ is a phenomenological damping parameter, often replaced by

[math]\displaystyle{ \lambda = \alpha \frac{\gamma}{M_\mathrm{s}}, }[/math]

where α is a dimensionless constant called the damping factor. The effective field H_eff is a combination of the external magnetic field, the demagnetizing field (magnetic field due to the magnetization), and some quantum mechanical effects. To solve this equation, additional equations for the demagnetizing field must be included.

Using the methods of irreversible statistical mechanics, numerous authors have independently obtained the Landau–Lifshitz equation.

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:

[math]\displaystyle{ \frac{\mathrm{d} \mathbf{M}}{\mathrm{d} t}=-\gamma \left(\mathbf{M} \times \mathbf{H}_{\mathrm{eff}} - \eta \mathbf{M}\times\frac{\mathrm{d} \mathbf{M}}{\mathrm{d} t}\right) }[/math]

(2b)

This is the Landau–Lifshitz–Gilbert (LLG) equation, where η is the damping parameter, which is characteristic of the material. It can be transformed into the Landau–Lifshitz equation:

[math]\displaystyle{ \frac{\mathrm{d} \mathbf{M}}{\mathrm{d} t} = -\gamma' \mathbf{M} \times \mathbf{H}_{\mathrm{eff}} - \lambda \mathbf{M} \times (\mathbf{M} \times \mathbf{H}_{\mathrm{eff}}) }[/math]

(2a)

where

[math]\displaystyle{ \gamma' = \frac{\gamma}{1 + \gamma^2\eta^2M_s^2} \qquad \text{and} \qquad\lambda = \frac{\gamma^2\eta}{1 + \gamma^2\eta^2M_s^2}. }[/math]

In this form of the LL equation, the precessional term γ' 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 m = M / M_S:

[math]\displaystyle{ \dot{\mathbf{m}}=-\gamma \mathbf{m}\times \mathbf{H}_{\mathrm{eff}}+\alpha \mathbf{m}\times \dot{\mathbf{m}}+\tau _{\parallel}\frac{\mathbf{m}\times (\mathbf{x}\times \mathbf{m})}{\left|\mathbf{x}\times \mathbf{m}\right|}+\tau _{\perp}\frac{\mathbf{x}\times \mathbf{m}}{\left|\mathbf{x}\times \mathbf{m}\right|} }[/math]

where [math]\displaystyle{ \alpha }[/math] is the dimensionless damping parameter, [math]\displaystyle{ \tau_\perp }[/math] and [math]\displaystyle{ \tau_\parallel }[/math] are driving torques, and x is the unit vector along the polarization of the current.

Further reading

• Gilbert, T.L. (1955). "A Lagrangian formulation of the gyromagnetic equation of the magnetic field". Physical Review 100 (4): 1243. doi:10.1103/PhysRev.100.1235. Bibcode: 1955PhRv..100.1235..  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 Gilbert, T. L. (2004). "A phenomenological theory of damping in ferromagnetic materials". IEEE Trans. Mag. 40 (6): 3443–3449. doi:10.1109/TMAG.2004.836740. Bibcode: 2004ITM....40.3443G. 
• Landau, L.D.; Lifshitz, E.M. (1935). "Theory of the dispersion of magnetic permeability in ferromagnetic bodies". Phys. Z. Sowjetunion 8, 153. 
• Skrotskiĭ, G V (1984). "The Landau–Lifshitz equation revisited". Sov. Phys. Usp. 27 (12): 977–979. doi:10.1070/PU1984v027n12ABEH004101. Bibcode: 1984SvPhU..27..977S. 
• Guo, Boling; Ding, Shijin (2008). Landau–Lifshitz Equations. Frontiers of Research With the Chinese Academy of Sciences. World Scientific Publishing Company. ISBN 978-981-277-875-8. 
• Cimrak, Ivan (2007). "A Survey on the Numerics and Computations for the Landau–Lifshitz Equation of Micromagnetism". Archives of Computational Methods in Engineering 15 (3): 1–37. doi:10.1007/BF03024947. http://www.kst.fri.uniza.sk/~icimrak/publications/surveyLL.pdf. Retrieved 2012-05-30. 
• M, Lakshmanan (2010). "The fascinating world of the Landau–Lifshitz–Gilbert equation: an overview". Phil. Trans. R. Soc. A 369 (1939): 1280–1300. doi:10.1098/rsta.2010.0319. PMID 21320917. Bibcode: 2011RSPTA.369.1280L. 

External links

• Magnetization dynamics applet