2.3 Linking different timescales

The magnetization dynamics simulations have to be able to deal with different time scales. Experimentally, the magnetization dynamics occur at different time scales starting from femtoseconds (recent pump-probe experiments [Cinchetti 06]), and from fast magnetization dynamics (fast-magnetooptical measurements [Bauer 00]). At seconds-hours range, slow thermal magnetization dynamics are measured in standard magnetometers as magnetic viscosity [Wegrowe 97]. Finally, long-time magnetization stability up to decades is relevant to magnetic recording applications [Weller 99].

From the point of view of simulation, the main parameter is the time-step of the numerical integration of the basic equation of motion (the Landau-Lifshitz-Gilbert equation). In the atomistic modeling, the time step is defined by the exchange interaction and is of the order of $ 10^{-16} s$. Therefore, integrating the equation of motion at atomistic level, one can afford the description of small system up to the time of $ 1 ns$. In the micromagnetic systems the time-step has to be at least two orders of magnitude less than the frequency of precession of the magnetization. The resulting time-step is about $ 10^{-13} s$ and the reachable time simulations of tens of nanoseconds, far from interval of interest in most of the experimental situations. These limits depend on the system size and in the computer technology advances.

The zero-temperature dynamics are deterministic. Different techniques are available to extend the simulation time scale by improving the numerical integration of the dynamical equation [Suess 02] and even trying to solve analytically simple systems under some assumptions (see Section 2.3.2 and [Serpico 04]). However, the temperature effects are very important in dynamics and produce magnetization random walk and eventually thermal decay of initial magnetization stay. The deterministic behavior becomes stochastic at finite temperature and the system will evolve from state to state overcoming energy barriers due to thermal fluctuations. At long time scale in this situation statistical methods have to be used in order to simulate the dynamic of the magnetization. Fig. 2.8 summarizes different methods in the presence of temperature. The stability of the magnetization can be studied with the Kinetic Monte Carlo method, energy barrier calculations and the Arrhenius-Néel law. In between several methods can be used to accelerate the thermal dynamics, as for example, Victora method [Xue 00] or Time Quantified Metropolis Monte Carlo [Chubykalo 03a] reaching ranges of several seconds. Here we will describe several of these methods, which we tried to implement in magnetic systems and check their performance.

Figure 2.8: Numerical methods for the thermal dynamics of the magnetization and their characteristic time scales.