Origin of versatile polarization state in CuInP2S6

Exotic electric polarization-related phenomena have recently been reported in layered van der Waals material such as ${\mathrm{CuInP}}_{2}{\mathrm{S}}_{6}$ and its derivatives, but the physical origin for such behaviors, especially the formation of quadrupole-well ferroelectric states and the anomalous polarization switching between these states, has not been clearly understood. Here, we provide a simple theoretical explanation using group theory analysis, highlighting the role of local symmetry-determined orbital interactions between copper ions and surrounding ligands. Based on this theory, we predict a surprising effect: carrier doping, which is commonly believed to suppress electric polarization, can lead to enhanced polarization in ${\mathrm{CuInP}}_{2}{\mathrm{S}}_{6}$. Our first-principles simulations confirm this unusual effect and further show that an elevated hole doping level can cause electric polarization to give way to emerging ferromagnetism. Our results not only pave a way to realize ferroelectric metals, but also broaden the scope of magnetoelectric coupling mechanisms and may help enrich the potential applications of layered ferroelectric materials in the future.