Types of the jump phenomenon in the angular dependence of the noncollinear exchange bias

Based on the principle of minimal energy and the coherent rotation model, two types of the jump phenomena, complete and incomplete jump phenomenon, are proved to exist in the angular dependence of the exchange bias with noncollinear unidirectional and uniaxial anisotropies. It is found that the transition between complete and incomplete jump phenomena occurs on condition that the exchange-coupling constant exceeds a critical value. Additionally, two different modes of the magnetization rotation, the whole-plane rotation, and the half-plane rotation are present in the magnetization reversal process, and they are dependent on the direction of the external field. Furthermore, the equations of the critical angle, at which orientation the exchange bias field reaches a maximum value and the coercivity disappears, are also derived in this paper. The numerical calculations in this paper are consistent with the relevant experimental observations, indicating that our method to study the angular dependence of the exchange bias as well as the magnetization reversal behaviors is valid. Our discussion about the jump phenomenon, the critical angle, and the modes of the magnetization reversal can explain the observed differences in results between different experiments.