Finite element calculations of energy barriers in magnetic systems

Abstract

With decreasing size of magnetic nanostructures thermal effects become increasingly important. Prominent examples are magnetization noise in magnetic sensor elements and the thermal stability of magnetic MRAM (Magnetic Random Access Memory) cells or magnetic storage media. In this work we apply both stochastic time integration and path finding techniques in the framework of the finite element method, in order to simulate thermal effects in magnetic nanostructures. Thus it is possible to take into account complex geometries and realistic element shapes. Both methods are complementary. The stochastic time integration is restricted to simulation times of about 10 ns. As a consequence the calculation of barrier crossing by stochastic time integration is limited to small energy barriers. The transition rate for large barriers can be estimated from the barrier height which can be calculated from the minimum energy path. In addition to the energy barrier, the elastic band method provides a global view of the energy landscape such as local minima and saddle points along the path. The magnetization processes as computed from the stochastic time integration method and the minimum energy path are compared for transitions between different ground states in magnetic nano-elements. Studying reversal modes in small softmagnetic elements such as MRAM, complex processes were found even at sizes where homogenous reversal was expected. As a consequence the corresponding energy barrier is lower than when estimated with simpler models. For thin permalloy squares a 8-fold like in plane anisotropy was observed at a certain size, a feature overlooked in previous studies. In granular patterned media the thermal reversal mode and the height of the energy barrier strongly depends on the intergranular exchange strength. Similar, the energy barrier as function of the interlayer exchange is calculated for antiferromagnetically coupled media. For optimum exchange, the energy barrier can be increased by 15% as compared to conventional recording media without an increase of the coercivity. In softmagnetic nanodots a novel reversal mode involving Bloch-point structures was investigated. The observed switching fields agree well with experimental data. For small elements we compared the two methods and obtained good agreement of the predicted results. The calculated saddle points obtained with the nudged elastic band method indeed represent the transition state found with Langevin dynamics.