Reversible flexoelectric domain engineering at the nanoscale in van der Waals ferroelectrics

The universal flexoelectric effect in solids provides a mechanical pathway for controlling electric polarization in ultrathin ferroelectrics, eliminating potential material breakdown from a giant electric field at the nanoscale. One challenge of this approach is arbitrary implementation, which is strongly hindered by one-way switching capability. Here, utilizing the innate flexibility of van der Waals materials, we demonstrate that ferroelectric polarization and domain structures can be mechanically, reversibly, and arbitrarily switched in two-dimensional CuInP2S6 via the nano-tip imprinting technique. The bidirectional flexoelectric control is attributed to the extended tip-induced deformation in two-dimensional systems with innate flexibility at the atomic scale. By employing an elastic substrate, artificial ferroelectric nanodomains with lateral sizes as small as ~80 nm are noninvasively generated in an area of 1 μm2, equal to a density of 31.4 Gbit/in2. Our results highlight the potential applications of van der Waals ferroelectrics in data storage and flexoelectronics.

To further understand the microscopic effects of tip imprinting, a simple model was constructed (see Fig. S1a).We used a monolayer of CIPS to simulate the CIPS thin film and capped carbon nanotubes to simulate the probe.The capped carbon nanotube is chemically stable and therefore cannot form bonds with CIPS or cause possible charge transfer, which guarantees that the interaction between the "probe" and CIPS is attributed only to van der Waals (vdW) forces being in accordance with experiments.
In addition, hollow nanotubes can reduce the number of atoms in models, which is beneficial for enhancing computational convergence and the convergence rate.This approach is useful for the calculations of large systems.To simulate the process of tip imprinting, we fix two CIPS unit cells on the right and left sides of our model (see the shadowed sections).Simultaneously, we move the carbon nanotube downward to generate the effects of tip imprinting on the CIPS, where the carbon nanotube is also fixed to optimize the structure of the unfixed part (see the dashed ellipse in Fig. S1a).
Three different distributions of Cu ions are taken into account in the same bending configurations (see Fig. S1b-1d), corresponding to the same downward depth of the carbon nanotube.The total energies of structural relaxation are calculated.The distribution of Cu ions in Fig. S1d has the lowest energy, indicating that the Cu ions just below the probe and on either side of the probe prefer upward polarization, which is in good agreement with our experiments.
To determine the mechanism of anomalous polarization that occurs under tip pressure, it is intriguing and meaningful to study the deformed geometry of CIPS under tip imprinting.The geometric shape of the bending CIPS can be determined by the position of all the In ions.As shown in Fig. S1e, the black dots are the position coordinates (x and z) of the In ions, where x is the coordinate along the x-axis and z is the height.Therefore, we use the position coordinates to fit the function curve and find that they can be perfectly covered by the Gauss-type function.
where  0 and  are constants.The deformation region can be divided into two parts, with the corresponding curvatures being convex and concave.The convex region generates an upward flexoelectric field, while the concave curvature corresponds to a downward flexoelectric field, potentially revealing the opposite polarization configuration.
For a more comprehensive understanding, a detailed analysis of the flexoelectric field distribution is performed.To simplify the calculation, the focus is directed toward the lower segment, which can be represented by an arch model (see schematic in Fig. S3).For thin flakes with a certain thickness d, the upper part experiences compressive strain upon bending, while the lower part undergoes tensile strain.Thus, the strain gradient ∂u/∂z is perpendicular to the surface and can be written as 6 where κ represents the curvature.Equation ( 2) depicts the relationship between the strain gradient and the curvature κ.The curvature is mathematically defined as where () represents the height.This formula solely quantifies the magnitude of the curvature without providing any information about the concavity or convexity of the curvature.Consequently, this approach solely characterizes the magnitude of the strain gradient while neglecting to specify its direction.In flexoelectric physics, we propose the concept of "two-way curvature"   .
where the sign and value of   describe the direction and magnitude of the strain gradient, respectively.Hence, for a known deformation,() ,   can be effectively utilized to quantify the corresponding strain gradient and flexoelectric field, which is essential for advanced studies involving flexoelectric effects.Taking (1) into equation ( 4), the calculated two-way curvature of the Gauss function is written as follows: .

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The corresponding curve is given in Fig. S1f.
Notably, the signs of the two-way curvature exhibit opposition both below and on either side of the probe, implying the existence of opposing flexoelectric fields.This observation offers a credible rationale for the occurrence of two-way polarization induced by tip imprinting in experimental studies.