Due to the antiferromgnetism in the room temperature multiferroic $\mathrm{BiFe}{\mathrm{O}}_3$, the electric field manipulation of exchange bias has been considered to be an efficient way to control the pinned magnetization of ferromagnetic layer. However, the irreversibility strongly impedes the practical application and the underlying mechanism needs clarification. Bilayers of Co with $\mathrm{BiFe}{\mathrm{O}}_3$ of various orientations, including (001), (110), and (111) in pseudo-cubic index, are fabricated. Exchange bias is observed in all the samples, originating from the interfacial spins of perfect crystal and surface defects in $\mathrm{BiFe}{\mathrm{O}}_3$ layer, respectively. The first reversal of out-of-plane component of polarization leads to the irreversible suppression of exchange bias, while the influence of in-plane component is little. Only the exchange bias from perfect crystalline $\mathrm{BiFe}{\mathrm{O}}_3$ is switchable, while that from surface defects remains unaffected. This can be understood by the in-plane component of cycloidal spins in $\mathrm{BiFe}{\mathrm{O}}_3$, anchored by the out-of-plane component of polarization, whose reversal leads to the rotation of cycloidal spins and break of unidirectional exchange coupling. This work provides a clue to the understanding of multiferroic exchange bias and the design of novel energy-efficient magnetoelectric devices.

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• Antiferromagnetism
• Exchange bias
• Ferroelectricity