Direct calculation of the attempt frequency of magnetic nanostructures using the finite element method

Abstract

A detailed knowledge of the thermal stability is of utmost importance for magnetic nanostructures for various applications ranging from hard disc media, MRAM devices to permanent magnets. A well established tool in chemical rate theory for determining the thermal stability is the transition state theory (TST). In the presented work the methods of the TST are implemented in a finite element micromagnetic package (FEMME) which allows for the calculation of the thermal stability of arbitrary shaped magnetic nanostructures. The thermal stability of a minimum energy state M1 is determined by the application of the Arrhenius-Neel law, tau=1/f0 exp (deltaE/kBT).<br />The energy barrier as well as the saddle point configuration S1 that separates M1 and a second stable state M2 is obtained by the climbing image nudged elastic band method. The attempt frequency consists of the dynamical prefactor lambda+ and a statistical factor omega0. The statistical factor omega0 is obtained by the ratio of the determinants of the Hessian matrix of the energy at the minimum and the saddle point.<br />The dynamical prefactor lambda+ is the positive eigenvalue of the derivative of the linearized Landau-Lifshitz Gilbert equation. The implementation of the calculation of lambda+ and omega0 using the finite element method (FEM) bases on the numerical differentiation of the effective field using a nine point stencil method. In order to test the numerical implementation, the analytical result of the attempt frequency of a single magnet moment are compared with the micromagnetic model combining FEM and TST, which shows very good agreement. However, in complex systems the simulations show a strong mesh dependency. Detailed investigations revealed that the higher eigenvalues of the Hessian matrix show strong oscillations, which are regarded as noise. A function fitted to the data, which then substitutes these higher eigenvalues, successfully reduces the mesh dependency.<br />The developed numerical method is applied in order to compare the attempt frequency of conventional hard disc media (single phase media) and state of the art hard disc media (graded media). Comparing these shows the attempt frequency mainly depends on the maximum of the anisotropy constant of the material, rather than on its coercive field.<br />Furthermore an interesting effect in single phase grains could be observed. An external field perpendicular to the easy axis of the grain shows a strongly resonant change of the attempt frequency.