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Using first-principles calculations we model the out-of-plane switching oflocal dipoles in CuInP$_2$S$_6$ (CIPS) that are largely induced by Cuoff-centering. Previously, a coherent switching of polarization via aquadruple-well potential was proposed for these materials. In the super-cellswe considered, we find multiple structures with similar energies but withdifferent local polar order. Our results suggest that the individual dipolesare weakly coupled in-plane and under an electric field at very lowtemperatures these dipoles in CIPS should undergo incoherent disorderedswitching. The barrier for switching is determined by the single Cu-ionswitching barrier. This in turn suggests a scale-free polarization with aswitching barrier of $\sim$ 203.6-258.0 meV, a factor of five smaller than thatof HfO$_2$ (1380 meV) a prototypical scale-free ferroelectric. The mechanism ofpolarization switching in CIPS is mediated by the switching of each weaklyinteracting dipole rather than the macroscopic polarization itself aspreviously hypothesized. These findings reconcile prior observations of aquadruple well with sloping hysteresis loops, large ionic conductivity even at250~K well below the Curie temperature (315~K), and a significant wake-upeffects where the macroscopic polarization is slow to order and set-in under anapplied electric field. We also find that computed piezoelectric response andthe polarization show a linear dependence on the local dipolar order. This isconsistent with having scale-free polarization and other polarization-dependentproperties and opens doors for engineering tunable metastability by-design inCIPS (and related family of materials) for neuromorphic applications.

Scale-free switching of polarization in the layered ferroelectric material CuInP$_2$S$_6$