4.3 The Nudged Elastic Band Method

In order to find possible paths of a micromagnetic system through its energy landscape to a local minimum of the total energy, we have implemented the static energy minimization method (cf. Sec. 4.1) and the time integration of the Landau-Lifshitz-Gilbert equation (cf. Sec. 4.2). We applied it to nucleation (cf. Sec. 8) and domain wall pinning problems (cf. Sec. 7) as well as investigations of dynamic magnetization reversal processes (cf. Sec. 9, Sec. 10). However, the investigation of thermal stability, which is an important topic especially in the area of magnetic storage devices, requires the calculation of transition rates between stable equilibrium states of the system. The transition rate between two stable equilibria is determined by the lowest energy barrier (saddle point), which separates them. Henkelman and Jónsson proposed the nudged elastic band method to calculate these minimum energy paths [55]. This method has been successfully applied to complex micromagnetic systems [56,57] and it is especially suitable for parallelization.

The path is represented by a sequence of ``images'' (magnetization distributions), which connects the two given stable equilibrium states $\boldsymbol{M}_i$ and $\boldsymbol{M}_f$. These equilibria may be obtained using the static energy minimization method, for example. The initial path is given by the initial magnetization distribution $\boldsymbol{M}_i$ and the final magnetization distribution $\boldsymbol{M}_f$ and a number of images $\boldsymbol{M}_k$ in between, which can be obtained by simple linear interpolation. Then, an optimization algorithm is applied, which moves the ``elastic band of images'' through the energy landscape towards the optimal path, which is defined by

$$(\nabla E(\boldsymbol{M}_k)\cdot \boldsymbol{t}) \boldsymbol{t}= \nabla E(\boldsymbol{M}_k) \quad,$$ (4.17)

where $\nabla E(\boldsymbol{M}_k)$ denotes the gradient of the total energy at image $\boldsymbol{M}_k$ and $t$ is the unit tangent vector along the path. In order to avoid kinks in the path the tangent is calculated using forward, backward, or second order central differences. Eq. (4.17) requires the component of the gradient parallel to the tangent to be equal to the gradient. In other words, the optimal path is characterized by fact that the gradient of the total energy is parallel to the tangent for any image $\boldsymbol{M}_k$.

Starting from the initial path an iterative optimization scheme is applied, which moves the images $\boldsymbol{M}_k$ in a direction $\boldsymbol{D}(\boldsymbol{M}_k)$, which is given by

$$\boldsymbol{D}(\boldsymbol{M}_k)= -\biggl( \nabla E(\bold... ...mbol{M}_k)\cdot \boldsymbol{t}) \boldsymbol{t} \biggr) \quad.$$ (4.18)

The interpretation of this expression is obvious: The images are moved along the negative gradient of the total energy perpendicular to the tangent. The negative gradient determines the direction towards lower energy while the distance between the images is preserved by taking only the component perpendicular to the tangent. Thus, an ordinary differential equation can be formulated

$$\frac{\partial \boldsymbol{M}_k}{\partial t}=\boldsymbol{D}(\boldsymbol{M}_k) \quad,$$ (4.19)

where $t$ denotes some artificial time parameter, which is integrated using any ODE solver.

After the discussion of the static energy minimization method and the time integration the Landau-Lifshitz-Gilbert equation, the implementation of the nudged elastic band method has become very simple, because we can reuse parts of both of them. The static energy minimization provides us with the gradient of the total energy and for the integration of Eq. (4.19), we can use the same methods explained in Sec. 4.2 by just replacing the right hand side of Eq. (4.7) with Eq. (4.18).

The parallelization of this method can be done by distributing the images across the processors. Thus, every processors needs the full set of matrices (but only one copy independent of the number of images), which are required for the calculation of the local fields (and gradients). However, there is no need to partition the finite element mesh any more, because every processor has to do the full calculation for the images. This has the advantage, that no communication is required during the calculation of the gradient of the total energy. Only for the calculation of the tangents the magnetization of some images has to be copied to ``neighboring'' processors.