Temperature in micromagnetism: cell size and scaling effects of the stochastic Landau–Lifshitz equation

The movement of the macroscopic magnetic moment in ferromagnetic systems can be described by the Landau–Lifshitz (LL) or Landau–Lifshitz-Gilbert (LLG) equation. These equations are strictly valid only at absolute zero temperature. To include temperature effects a stochastic version of the LL or LLG equation for a spin density of one per unit cell can be used instead. To apply the stochastic LL to micromagnetic simulations, where the spin density per unit cell is generally higher, a conversion regarding simulation cell size and temperature has to be established. Based on energetic considerations, a conversion for ferromagnetic bulk and thin film systems is proposed. The conversion is tested in micromagnetic simulations which are performed with the Object Oriented Micromagnetic Framework (OOMMF). The Curie temperatures of bulk Nickel, Cobalt and Iron systems as well as Nickel thin-film systems with thicknesses between 6.3 mono layer (ML) and 31 ML are determined from micromagnetic simulations. The results show a good agreement with experimentally determined Curie temperatures of bulk and thin film systems when temperature scaling is performed according to the presented model. (paper)