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 $\left(A\sim m^2\right)$, comes from the nonlinear spin-wave effects.

Figure 1

(Color online) Scaling behavior of the exchange stiffness (DW Langevin) as obtained from the domain-wall free energy for a generic model with $d/J=0.032$. The solid line is the numerical solution of the CSD method outlined in Sec. 4B. The SW Langevin points are obtained from the SW stiffness approach based on the atomistic LLG-Langevin simulations outlined in Sec. 4A.

Figure 2

(Color online) Scaling behavior of the exchange stiffness as obtained from the DW method applied to the full FePt Hamiltonian. The solid line indicates the numerical solution using the CSD method (see Sec. 4B) for the FePt Hamiltonian without dipole-dipole interaction.

Figure 3

Sketch of the three planes under consideration. The magnetization is in the $x\text{−}z$ plane and tilted by an angle $\psi$ between adjacent planes.

Figure 4

(Color online) Angle dependence of the domain-wall free energy for different temperatures. The data points are the numerical solution of Eq. (13) in connection with Eq. (11) while the solid lines represent the analytical approximation [the second part of Eq. (15)].

Figure 5

(Color online) Magnetization dependence of the domain-wall free energy for different angles $\psi$. The data points are from the numerical solution of Eq. (13) in connection with Eq. (11) while the black line represents the $m_{\psi }^2$ behavior [see Eq. (15)].

Figure 6

(Color online) (a) Power spectrum density as a function of frequency for thermally excited SW in a generic Heisenberg Hamiltonian for a system of $\mathcal{N}=32\times 32\times 32$ moments for various temperatures (from top to bottom) $T=T_C/70$, $T=T_C/4$, $T=T_C/2$, and $T=0.9T_C$, for fixed wave vector $\mathbf{q}=\left[0,0,\pi /\left(4a\right)\right]$, and applied field $H_z=1\text{T}$. (b) Dispersion relations for SWs with wave vector $\mathbf{q}=\left(0,0,q\right)$ computed via the Langevin dynamics simulations (symbols) for the same temperatures (from top to bottom). The lines show the results obtained by the CSDM method, see Sec. 4B.