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We examine the anisotropic properties of the exchange stiffness constant,$\mathcal{A}$, for rare-earth permanent magnet, Nd$_2$Fe$_{14}$B, by connectinganalyses with two different scales of length, i.e., Monte Carlo (MC) methodwith an atomistic spin model and Landau-Lifshitz-Gilbert (LLG) equation with acontinuous magnetic model. The atomistic MC simulations are performed on thespin model of Nd$_2$Fe$_{14}$B constructed from ab-initio calculations, and theLLG micromagnetics simulations are performed with the parameters obtained bythe MC simulations. We clarify that the amplitude and the thermal property of$\mathcal{A}$ depend on the orientation in the crystal, which are attributed tothe layered structure of Nd atoms and weak exchange couplings between Nd and Featoms. We also confirm that the anisotropy of $\mathcal{A}$ significantlyaffects the threshold field for the magnetization reversal (coercivity) givenby the depinning process.

Anisotropy of exchange stiffness based on atomic-scale magnetic properties in rare-earth permanent magnet Nd$_2$Fe$_{14}$B