Dynamic approach for micromagnetics close to the Curie temperature

In conventional micromagnetism magnetic domain configurations are calculated based on a continuum theory for the magnetization. This theory assumes that the absolute magnetization value is constant in space and time. Dynamics is usually described with the Landau-Lifshitz-Gilbert (LLG) equation, the stochastic variant of which includes finite temperatures. Using simulation techniques with atomistic resolution we show that this conventional micromagnetic approach fails for higher temperatures since we find two effects which cannot be described in terms of the LLG equation: (i) an enhanced damping when approaching the Curie temperature and, (ii) a magnetization magnitude that is not constant in time. We show, however, that both of these effects are naturally described by the Landau-Lifshitz-Bloch equation which links the LLG equation with the theory of critical phenomena and turns out to be a more realistic equation for magnetization dynamics at elevated temperatures.

Figure 1

(Color online) Relaxation of the magnetization for different temperatures using the atomistic modeling: 9a) normalized perpendicular component (30° excitation); (b) absolute value of the magnetization $m\equiv \mid \mathbf{m}\mid$ (135° excitation).

Figure 2

(Color online) Temperature dependence of longitudinal and transverse relaxation times from the atomistic modeling and the LLB equation, calculated as inverse rates given by Eq. (6).

Figure 3

(Color online) Relaxation of the magnetization for different temperatures as in Fig. 1 but using the macrospin LLB modeling.