Influence of three-dimensional dynamics on the training effect in ferromagnet-antiferromagnet bilayers

Training effect in exchange-bias systems consists of a variation in coercivity and symmetry between the first reversal after field cooling and the following loops. It has been shown, in the frame of a two-dimensional coherent-rotation approach, that training might be explained in terms of an initial noncollinear arrangement of the antiferromagnetic spins after field cooling, which relaxes to a collinear arrangement during the first reversal [A. Hoffmann, Phys. Rev. Lett. 93, 097203 (2004)]. In this paper, we extend the model to three dimensions, by numerically solving the Landau-Lifshitz-Gilbert equation describing the precession motion of magnetic moments. We are thus able to discuss the validity of Hoffmann's model within a three-dimensional approach, with parameter values similar to those in the original publication, and to enlighten the role of out-of-plane anisotropies and Gilbert damping in determining the occurrence of training. Moreover, when realistic values are considered for the magnetocrystalline anisotropy of the system, we find that no training is reproduced within our extended model, suggesting that symmetry-driven irreversibilities might not be as relevant as previously believed for training effect.