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 m1.66 for systems on a simple cubic lattice and as m1.76 for an FePt Hamiltonian parametrized through ab initio calculations. The additional reduction in the scaling exponent, as compared to the mean-field theory (A~m²), comes from the nonlinear spin-wave effects.