Complete and incomplete jump phenomenon in the angular dependence of the noncollinear exchange bias

The angular dependence of the exchange bias (ADEB) has been investigated in detail for ferromagnetic/antiferromagnetic bilayers with noncollinear uniaxial and unidirectional anisotropies. Two different types of the jump phenomenon, complete and incomplete jump phenomena, have been proved to occur at the orientations of the intrinsic easy and hard axes. A special position for the intrinsic easy and hard axes, which makes an angular deviation of 58.2826° from the uniaixal anisotropy axis, has been deduced by analyzing the magnetization reversal processes based on the principle of minimal energy. When the angular deviation of the intrinsic easy or hard axis from the uniaxial anisotropy axis is above the critical value of 58.2826°, the complete jump phenomenon will be shown in the ADEB. On the contrary, once this angular deviation is not more than 58.2826°, the incomplete jump phenomenon occurs, and the critical angle will be observed in the ADEB. The determined formula of the critical angle is also obtained by analyzing the magnetization reversal processes. Additionally, the extreme value problem of the exchange bias is studied in this paper. The coercivity always reaches its maximum value when the external field points along the intrinsic easy axis. The minimal coercivity occurs at the orientation of the critical angle. However, there are two orientations of the applied field to gain the maximal exchange bias field. One is the direction of the intrinsic hard axis when the complete jump phenomenon occurs at this orientation. The other one is the orientation of the critical angle under the condition that the incomplete jump phenomenon takes place at the intrinsic hard axis. The numerical calculations are consistent with the involved experimental observations, indicating that our method is valid to study the ADEB and it is an effective method to achieve the maximal exchange bias field as well as the maximal or minimal coercivity by adjusting the orientation of the external field.