J.-E. Wegrowe

The study of the conservations laws of the magnetization and spins in the context of spin-transfer effects led us the reconsider the equation of the dynamics of the magnetization by adding an inertial term to the well-known Ladau-Lifshitz-Gilbert (LLG) equation [1]. In the Gilbert form, the new equation reads :

The effect of the inertial term is to add nutation oscillations to the well-known precession of a magnetic dipole. A new resonance peak is expected to occure at high frequency in the context of ferromagnetic resonance experiments, if a oscillating field is superimposed perpendicular to a static field [2].

Fig. 1. Resonance peak due to inertial nutation of the magnetization at about 10^15 Hz. The usual ferromagnetic resonance is located here at about 10^11 Hz (from reference [2]).

The LLG equation generalized to inertial effects can be straightforwardly  derived in the Lagrangian formalism, generalizing Gilbert’s approach [3]. This derivation is instructive from the pedagogical point of view because it is a generalization of the model of Ampere’s molecular currents to more realistic atomic model (see Fig. 2).

In this context, the analogy between the dynamic of the magnetization and the dynamics of a spinning top is complete.

Fig 2. (a) Illustration of Ampère's model of molecular current. The electron of mass m, charge q and angular velocity Ω3 is confined in a circular loop of radius r. The angular momentum due to the rotating mass is L 3= I3 Ω3 = mvr $ and the magnetic moment due to the rotating charge is Ms=qvr/2. (b) Generalization of Ampère's molecular current: The components of the angular momentum are now defined by the three principal moments of inertia I1=I2 and I3.

The experimental study of this effect is ongoing.

[1] Magnetization dynamics in the inertial regime: Nutation predicted at short time scales, M.-C. Ciornei, J. M. Rubi, and J.-E. Wegrowe, Phys. Rev. B 83, 020410(R) (2011).
[2] Beyond Ferromagnetic Resonance: the Inertial Regime of the Magnetization,
E. Olive, Y. Lansac and J.-E. Wegrowe, Appl. Phys. Lett. 100 (2012).
[3] Magnetization Dynamics, Gyromagnetic Relation, and Inertial Effects,
J.-E. Wegrowe, M.-C. Ciornei, Am. J. Phys. 80 (6) (2012)