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Ferroelectric control of two-dimensional magnetism is promising infabricating electronic devices with high speed and low energy consumption. Thenewly discovered layered MnBi$_2$Te$_4$(Bi$_2$Te$_3$)$_n$ and their Sbcounterparts exhibit A-type antiferromagnetism with intriguing topologicalproperties. Here, we propose to obtain tunable magnetic multistates in theirthin films by ferroelectrically manipulating the interlayer magnetic couplings(IMCs) based on the Heisenberg model and first-principles calculations. Ourstrategy relies on that interfacing the thin films with appropriateferroelectric materials can switch on/off an interlayer hopping channel betweenMn-$e_g$ orbitals as the polarizations reversed, thus resulting in a switchableinterlayer antiferromagnetism-to-ferromagnetism transition. On the other hand,the interface effect leads to asymmetric energy barrier heights for the twopolarization states. These properties allow us to build ferroelectricallyswitchable triple and quadruple magnetic states with multiple Chern numbers inthin films. Our study reveals that ferroelectrically switchable magnetic andtopological multistates in MnBi$_2$Te$_4$ family can be obtained by rationaldesign for multifunctional electronic devices, which can also be applied toother two-dimensional magnetic materials.

Ferroelectrically switchable magnetic multistates in MnBi$_2$Te$_4$(Bi$_2$Te$_3$)$_n$ and MnSb$_2$Te$_4$(Sb$_2$Te$_3$)$_n$ (n = 0, 1) thin films