Landau-Lifschitz-Gilbert equation

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The Landau-Lifschitz-Gilbert equation (with English transcription in German sometimes also called Landau- Lifshitz- Gilbert equation) describes the behavior of the magnetic moments of a ferromagnetic material in an effective magnetic field in solid-state physics . It is named after Lew Dawidowitsch Landau , Evgeni Michailowitsch Lifschitz and Thomas L. Gilbert . It is a common differential equation , from which a complex integro-differential equation arises, however, by taking into account the non-local nature of this effective field with regard to the interaction of the magnetization dipoles . ${\displaystyle {\vec {H}}_{\mathrm {eff} }}$

Landau-Lifschitz equation

The original Landau-Lifschitz equation was established in 1935. it is

${\displaystyle {\frac {\partial {\vec {M}}}{\partial t}}=-\gamma {\vec {M}}\times {\vec {H}}_{\mathrm {eff} }-{\frac {\lambda }{\mathrm {M} _{S}}}{\vec {M}}\times ({\vec {M}}\times {\vec {H}}_{\mathrm {eff} })}$

and describes both the precession of the magnetization and the dissipation that occurs . The amount of remains because it is ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {M}}}$

${\displaystyle {\frac {\partial }{\partial t}}{\vec {M}}^{2}=2{\frac {\partial {\vec {M}}}{\partial t}}\cdot {\vec {M}}=0}$.

This constant amount is called the saturation magnetization , which also appears as a parameter in the Landau-Lifschitz equation. Further parameters of the equation are the gyromagnetic ratio and a phenomenological damping parameter . However, this formula fails in the case of large damping ( ). ${\displaystyle M_{\mathrm {S} }=|{\vec {M}}|}$ ${\displaystyle \gamma }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda \rightarrow \infty }$

In the stationary solution of the system, to which the system strives when it is left to its own devices, magnetization and effective magnetic field are parallel to one another.

Landau-Lifschitz-Gilbert equation

In 1955, Gilbert replaced the damping term and introduced a kind of viscous force. The Landau-Lifschitz-Gilbert equation resulted:

${\displaystyle {\frac {\partial {\vec {M}}}{\partial t}}=-{\frac {\gamma }{1+\alpha ^{2}}}{\vec {M}}\times {\vec {H}}_{\mathrm {eff} }-{\frac {\gamma \alpha }{(1+\alpha ^{2})M_{\mathrm {S} }}}{\vec {M}}\times ({\vec {M}}\times {\vec {H}}_{\mathrm {eff} })\,,}$

which can also be written more easily with the Gilbert damping parameter and the shorthand notation as a unit vector of magnetization in an equivalent form: ${\displaystyle \gamma _{G}={\frac {\gamma }{1+\alpha ^{2}}}}$ ${\displaystyle g=|\gamma _{G}|\alpha }$ ${\displaystyle {\vec {m}}={\frac {\vec {M}}{M_{\mathrm {S} }}}}$

${\displaystyle {\frac {\partial {\vec {m}}}{\partial t}}\,=\,-\gamma _{G}\,{\vec {m}}\times {\vec {H}}_{\mathrm {eff} }\,+\,g\,{\vec {m}}\times {\frac {\partial {\vec {m}}}{\partial t}}\,,}$

It can be shown that the Landau-Lifschitz-Gilbert equation resulting last is identical to the original Landau-Lifschitz equation quoted in the previous subsection, if one identifies with λ ; the decisive difference, however, apart from the greater formal simplicity, is that not and but and are used in regressions . Formal is only replaced by ; the last term contains all damping terms. ${\displaystyle g|\gamma |}$ ${\displaystyle \gamma }$ ${\displaystyle \lambda \,,}$ ${\displaystyle \gamma _{G}}$ ${\displaystyle \alpha }$ ${\displaystyle \,\gamma {\vec {H}}_{\mathrm {eff} }}$ ${\displaystyle \gamma _{G}{\vec {H}}_{\mathrm {eff} }-{g}\,{\frac {\partial {\vec {m}}}{\partial t}}}$

As in the original Landau-Lifschitz equation, the magnetic moment is now asymptotically aligned in the direction of the field, with the damping now also having an effect on the precession frequency, as in mechanics with the damped oscillator . In the case of low damping, the Landau-Lifschitz-Gilbert equation changes into the Landau-Lifschitz equation. ${\displaystyle t\to \infty }$

The "effective field"

Landau and Lifschitz stated in 1935 how the vector depends on all four interactions involved (the “magnetic exchange energy”, the “dipole-dipole energy”, the “anisotropy energy” and the “Zeeman energy”). We cannot go into details here. ${\displaystyle {\vec {H}}_{\mathrm {eff} }}$

Spin waves u. Ä.

A so-called spin wave in a ferromagnetic solid

With the Landau-Lifschitz-Gilbert equations, u. a. Also dynamic conditions (eg. B. spin waves , as in the adjacent image) are treated realistic, wherein all the relevant geometries (for example, thin-layer geometries) and interactions (u. a., the very long-range magnetic dipole-dipole interaction ) may be fully taken into account, if you accept high memory requirements and corresponding computing times in the computer simulations .

The dispersion relations in these systems - these are the relationships between frequency and wavelength of the excitation states - are very complex because of the large number of characteristic lengths of the system and the angles involved .

Web links

• Representation in the dissertation of Massimiliano d'Aquino , 2005

References and footnotes

1. ↑ Landau, Lifschitz, Theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj., Volume 8, 1935, p. 153
2. ^ Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetic field, Physical Review, Volume 100, 1955, p. 1243.
3. ^ TL Gilbert: A Phenomenological Theory of Damping in Ferromagnetic Materials . In: IEEE Transactions on Magnetics . tape 40 , no. 6 , November 2004, ISSN  0018-9464 , p. 3443–3449 , doi : 10.1109 / tmag.2004.836740 (DOI = 10.1109 / tmag.2004.836740 [accessed July 16, 2020]).
4. ↑ J. Miltat, G. Albuquerque, A. Thiaville: An Introduction to Micro Magnetics in the dynamic regime . In: Hillebrands B., Ounadjela K. (ed.): Topics in Applied Physics Vol. 83: Spin Dynamics in Confined Magnetic Structures I . Springer-Verlag, Berlin 2002, ISBN 978-3-540-41191-8 , pp. 1–34 , doi : 10.1007 / 3-540-40907-6_1 ( springer.com [PDF; accessed January 26, 2018]).