Electric properties of the twelve-fold vortex structure in hexagonal manganites.

Hexagonal manganites, as a functional ferroelectric (FE) material, receive considerable attention due to their improper ferroelectricity and topological vortex structures. This family exhibits three low-symmetry states accompanied by distinct vortex domain structures. In addition to the FE6and anti-FE (AFE)-31 states accompanied by dual six-fold vortex structures, there is another FE31 state accompanied by a twelve-fold vortex structure. The responses of FE materials to external stimuli, such as external electric fields, are the core ingredients in the physics of FEs and are significant for technological applications. Under external electric fields, the responses of FE materials are determined by special FE domain structures. The electric properties of the FE6and AFE-31 states are very different. However, the electric properties of the FE31 state, which only stabilizes in Ga-substituted In(Mn, Ga)O, are unclear. The present work studies the electric properties of the FE31 state. The electric-field-driven transition of the FE31 state is found to follow two sequences, i.e. (1) twelve-fold31 → nine-fold31 +6→ three-fold6, and (2) twelve-fold31 → six-fold31 → three-fold6. The variation of average polarization withfor the FE31 state with the second transition sequence manifests as an unusual triple-hysteresis loop, different from the usual single-hysteresis loop of FE materials. The results are related to the coexistence of the FE and non-FE domain walls in the FE31 state. Furthermore, it is found that the FE31 state at substitution concentration 0.39 exhibits the highest dielectric response. The results advance our understanding of topological vortex structures in hexagonal manganites.