Off-plane polarization ordering in metal chalcogen diphosphates from bulk to monolayer

Vertically (off-plane) ferroelectric ordering in ultrathin films has been pursued for decades. We predict the existence of intrinsic vertical polarization orderings in ultrathin metal chalcogen-diphosphates (MCDs). Taking ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$ as an example, the first-principles calculation and electrostatic-energy model show that, under the open-circuit boundary condition, the ground state of bulk ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$ is ferroelectric (FE) while that of monolayer is antiferroelectric (AFE), and the critical thickness for this FE/AFE transition is around six layers. Interestingly, under the closed-circuit boundary condition, the FE state can hold even for monolayer. Particularly, because of the small energy difference but the large barrier between FE and AFE orderings, the FE state can be stabilized in a free-standing monolayer, giving rise to intrinsic, off-plane two-dimensional ferroelectrics. Applying Monte Carlo simulations, we further calculate the ferroelectric Curie temperature ($T_c$) and electric hysteresis.

Figure 1

(a) Top view of monolayer ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$. (b) Side views of the FE (upper) and AFE (lower) states. (c) Schematic potential of FE monolayer under the $D=0$ (open-circuit) boundary condition. (d) Averaged electric potential of bulk under the $E=0$ (closed-circuit) boundary condition.

Figure 2

Layer-dependent evolution of the energy difference ($E_{\text{FE}}-E_{\text{AFE}}$) of ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$ under the $D=0$ boundary condition. The critical thickness is marked by the cross sign.

Figure 3

(a) Monolayer and bulk sample FE(AFE)-to-paraelectric energy tracks from first-principles NEB calculations. (b) Energy variation with aspect to the applied displacement field $D/{\epsilon }_0$. The energy of the FE state of the freestanding monolayer is set to be zero. (c) Temperature-dependent zero-$E$-field spontaneous polarization $P_s$ in monolayer and bulk ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$.

Figure 4

(a) Electric hysteresis of polarization of monolayer ${\mathrm{CuInP}}_2{\mathrm{Se}}_6$ under 30 K, starting from the AFE ground state, then evolving into and stabilizing at the FE state. (b) Electric hysteresis of FE ${\mathrm{CuInP}}_2{\mathrm{Se}}_6{P}_s$ under different temperatures. (c) Temperature-dependent coercive field with an exponential decay fitting.