Cyclic ferroelectric switching and quantized charge transport in ${\mathrm{CuInP}}_2{\mathrm{S}}_6$

The van der Waals layered ferroelectric ${\mathrm{CuInP}}_2{\mathrm{S}}_6$ has been found to exhibit a variety of intriguing properties arising from the fact that the Cu ions are unusually mobile in this system. While the polarization switching mechanism is usually understood to arise from Cu ion motion within the monolayers, a second switching path involving Cu motion across the van der Waals gaps has been suggested. In this work, we perform zero-temperature first-principles calculations on such switching paths, focusing on two types that preserve the periodicity of the primitive unit cell: “cooperative” paths preserving the system's glide mirror symmetry, and “sequential” paths in which the two Cu ions in the unit cell move independently of each other. We find that ${\mathrm{CuInP}}_2{\mathrm{S}}_6$ features a rich and varied energy landscape, and that sequential paths are clearly favored energetically both for cross-gap and through-layer paths. Importantly, these segments can be assembled to comprise a globally insulating cycle with the out-of-plane polarization evolving by a quantum as the Cu ions shift to neighboring layers. In this sense, we argue that ${\mathrm{CuInP}}_2{\mathrm{S}}_6$ embodies the physics of a quantized adiabatic charge pump.

Figure 1

(a) $P^{+}$ polar phase of CIPS in the conventional cell setting. (b) Same, but in the primitive unit-cell setting. The Cu ions reside inside the monolayers and above the monolayer midplanes. The dashed red line in (a) indicates the glide mirror plane of the system. The lattice parameters and cell angles are $a=b=6.11$ Å, $c=13.36$ Å, and $\alpha =\beta =85.{67}^{\circ }$, $\gamma =120.0^{\circ }$.

Figure 2

(a) The representative sequence $P^{+}\to M^{+}\to \overline{M}\to M^{-}\to P^{-}$. The Cu ions in the lower (red) and upper (blue) layers of the unit cell move cooperatively along the $\mathbf{c}$ lattice vector throughout the switching path, as emphasized by the guiding lines. (b) The representative sequence $P^{+}\to S_2^{+}\to R\to S_2^{-}\to P^{-}$. The lower Cu ions move across the vdW gap first, and are subsequently followed by the Cu ions in the upper vdW gap. (c) Sequence of structures from $P^{+}$ to $P^{-}$ passing through $I$ in the middle. The upper Cu ions move through a monolayer first, followed by the bottom Cu ions. The number below a structure indicates its energy relative to $P^{\pm }$.

Figure 3

The full NEB energy profile of the combined sequential switching paths plotted versus the reduced polarization $p_3$. The most significant energy barriers for switching are defined by $S_1^{\pm }$ (cross-gap), and the local maxima on the $P^{\pm }\to I$ paths (through-layer).