# Laboratoire de Physique des Solides - UMR 8502

## Ultra-fast magnetization dynamics : Gilbert’s derivation and the generalization to inetial regime

J.-E. Wegrowe

J.-E. Wegrowe*, M. C. Ciornei*, J. M. Rubì +

*Ecole Polytechnique, LSI, France

+ Universitat de Barcelona, Spain.

The well-known Gilbert’s equation describes the dissipative dynamics of the magnetic degrees of freedom coupled to a heat bath. We propose to discuss the derivation of the Gilbert’s equation from both the point of view the Lagrangian approach  and the non-equilibrium thermodynamic (NET) approach.

The derivation based on the Lagrangian approach (used initially by Gilbert in 1955)  is performed in the phase space : the angular momentum (with the kinetic energy) is introduced as starting hypothesis. However, the Gilbert equation is a kinetic equation that does not contain any inertial terms for uniform magnetization. This paradoxical situation appeals at least two comments.

(1) On one hand, a full kinetic derivation of the Gilbert equation (with zero kinetic energy) would solve the paradox. Such a derivation is proposed within the NET approach . This derivation is performed in the configuration space of the magnetic degrees of freedom, and not in the phase space (i.e. no need to introduce the angular momentum).

(2) On the other hand, the Gilbert’s Lagrangian approach can be pushed to its logical consequences, i.e. a generalization of the equation with the introduction of inertial terms proportional to the second derivative of the uniform magnetization . This derivation is obtained straightforwardly by enlarging the configuration space of the magnetization to the phase space, i.e. to the angular momentum. Beyond, the transient nature of this inertial regime is derived in the framework of the NET theory.

 Thomas L. Gilbert, IEEE Trans. Mag. 40 (2004) 3443, Phys. Rev. 100, 1243 (1955) (abstract only), and PhD Dissertation “Formulation, Foundations and Applications of the phenomenological theory of ferromagnetism” Illinois Institute of Technology, 1956.

 J.E. Wegrowe, Phys. Rev B 62, 1067 (2000), J.-E. Wegrowe et al. J. Phys. : Condens. Matter 19, 165213 (2007), Phys. Rev. B 77, 174408 (2008) and Sol. State Com. 150 (2010), 519

 C. Ciornei, M Rubì, and J.-E. Wegrowe, Cond-Mat ArXiv 1008.2177