Multiscale modeling of magnetic materials: Temperature dependence of the exchange stiffness

For finite-temperature micromagnetic simulations the knowledge of the temperature dependence of the exchange stiffness plays a central role. We use two approaches for the calculation of the thermodynamic exchange parameter from spin models: (i) based on the domain-wall energy and (ii) based on the spin-wave dispersion. The corresponding analytical and numerical approaches are introduced and compared. A general theory for the temperature dependence and scaling of the exchange stiffness is developed using the classical spectral density method. The low-temperature exchange stiffness $A$ is found to scale with magnetization as ${m}^{1.66}$ for systems on a simple cubic lattice and as ${m}^{1.76}$ for an FePt Hamiltonian parametrized through ab initio calculations. The additional reduction in the scaling exponent, as compared to the mean-field theory $(A\ensuremath{\sim}{m}^{2})$, comes from the nonlinear spin-wave effects.