calculate moment of inertia from precession period

In these experiments, the finite electron (Bloch-state) lifetime τ was determined to be in the range of τ = 0.12–0.47 ps, depending not only on the selected material but also on its thickness. At these high frequencies, an additional stiffening of the FMR resonance was observed that was quadratic in the probing frequency ω and, consequently, proportional to the moment of inertia.

31 who measured it for Ni 79Fe 21 and Co films near room temperature with ferromagnetic resonance (FMR) in the high-frequency regime (around 200 GHz). Recently, the moment of inertia was experimentally examined by Li et al. Like the Gilbert damping α, the moment of inertia tensor ι has been considered as a material specific parameter in theoretical investigations. This macroscopic model was further analysed numerically in ref. 23 derived a term in the extended LLG equation that addresses “inertial magnetic mass” by including the moment of inertia tensor ι into the equation of motion. From micromagnetic Boltzman theory, Ciornei et al. More recently, nutation was discovered on a single-atom magnetic moment trajectory in a Josephson junction 26, 27, 28. We obtain numerical results that are consistent with experimental values, and elaborate on how inertia can be detected in magnetic materials.Ĭonceptional thoughts related to the velocity of moving a static domain wall with “inertial mass” were already introduced by Döring 24, De Leeuw and Robertson 25. In this work, we provide a theoretical foundation for understanding inertia in magnetisation dynamics, based on electronic structure theory. It may be expected that for magnetisation dynamics that atomic magnetic moments behave in an analogous way on ultrafast time-scales 22, 23 (Fig. In coexistence with damping, this nutation disappears on a short time. If the rotation axis of the gyroscope do not coincide with the angular momentum axis due to a “fast” external force, a superimposed precession around the angular-momentum and the gravity field axis occurs the gyroscope nutates. In its common formulation, the LLG equation does not account for longitudinal fluctuations of magnetic moment 20, quantum mechanical spin currents 21 or, in particular, the influence of magnetic inertia 22, compared to its classical mechanical counterpart of a gyroscope. On one hand, the quantum mechanical equation of motion can be solved 18, 19, but on the other hand one can linger with the classical approach and instead introduce higher order terms in the LLG.

In this time-scale, the applicability of the atomistic LLG equation must be scrutinised in great detail. This limit characterises the blurry boundary where the time scales of electrons and atomic magnetic moments are separable 17 - usually between 10–100 fs. On the theoretical side it has been argued that the classical atomistic Landau-Lifshitz-Gilbert (LLG) equation 11, 12 should be relevant over a time-scale of sub-pico seconds and longer 13 and provides a proper description of magnetic moment switching 14, but is derived within the adiabatic limit 15, 16. Various magnetic excitation methods 8, 9, 10 allow switching of the magnetic moment on sub-ps timescales. For this purpose, “good” candidates are materials exhibiting thermally stable magnetic properties 3, energy efficient magnetisation dynamics 4, 5, as well as fast and stable magnetic switching 6, 7. The research on magnetic materials with particular focus on spintronics or magnonic applications became more and more intensified, over the last decades 1, 2. We propose ways to utilise this in order to tune the inertia experimentally, and to find materials with significant inertia dynamics. The theoretical analysis shows that even though the moment of inertia and damping are produced by the spin-orbit coupling, and the expression for them have common features, they are caused by very different electronic structure mechanisms. Particularly, the method has been applied to bulk itinerant magnets and we show that numerical values are comparable with recent experimental measurements.

We present and elaborate here on a theoretical model for calculating the magnetic moment of inertia based on the torque-torque correlation model. This magnetic counterpart to the well-known inertia of Newtonian mechanics, represents a research field that so far has received only limited attention. Recently the importance of inertia phenomena have been discussed for magnetisation dynamics. This is described by the Landau-Lifshitz-Gilbert equation and the well known damping parameter, which has been shown to be reproduced from quantum mechanical calculations. An essential property of magnetic devices is the relaxation rate in magnetic switching which strongly depends on the energy dissipation.

Calculate moment of inertia from precession period