Mechanical-force-induced non-local collective ferroelastic switching in epitaxial lead-titanate thin films

Ferroelastic switching in ferroelectric/multiferroic oxides plays a crucial role in determining their dielectric, piezoelectric, and magnetoelectric properties. In thin films of these materials, however, substrate clamping is generally thought to limit the electric-field- or mechanical-force-driven responses to the local scale. Here, we report mechanical-force-induced large-area, non-local, collective ferroelastic domain switching in PbTiO3 epitaxial thin films by tuning the misfit-strain to be near a phase boundary wherein c/a and a1/a2 nanodomains coexist. Phenomenological models suggest that the collective, c-a-c-a ferroelastic switching arises from the small potential barrier between the degenerate domain structures, and the large anisotropy of a and c domains, which collectively generates much larger response and large-area domain propagation. Large-area, non-local response under small stimuli, unlike traditional local response to external field, provides an opportunity of unique response to local stimuli, which has potential for use in high-sensitivity pressure sensors and switches.


Supplementary Note 1. Theoretical calculations of domain coexistence
The Landau-Ginsburg-Devonshire-type phenomenological theory with the free energy expanded up to the sixth orders in polarization is employed to study the coexistence of different domain configurations in a thin film subjected to epitaxial strain. We assume that the misfit strain is epitaxially uniform and transversely isotropic. The renormalized thermodynamic potential after the Legendre transformation of the Gibbs free energy can be expressed with respect to the primary order parameters of polarization i P and internal mechanical stresses i  in the film as 1 2 2 2 4 4 4 2 2 2 2 2 2 2 2 2 1 1 2 3 11 1 2 3 12 1 2 2 3 3 1 123 1 2 3 Since the mechanical stress i  is related to the elastic strains including the total strain and the phase transition strain, F also takes into account the coupling between the polarization and elastic strain through the total elastic strain. Parameters for our calculations are taken from Refs 1,2 .
For pure c domains, the spontaneous polarizations can be calculated as 3 : To examine the possible existence of each stable or metastable states, Hessian's functions and canonical distribution within the framework of statistical mechanics are adopted to calculate the corresponding distribution probability. For each statistically equivalent ensembles i, the distribution probability of the ensemble being in the energy level Gi can be written as: is the energy for the i th lowtemperature phase with i F the system free energy densities and Vi the corresponding volume.

Supplementary Note 2. Structural characterization using X-ray diffraction
Wide-angle -2 X-ray diffraction pattern shows the epitaxial growth of the heterostructures (Supplementary Figure 2). Besides the diffraction peaks from SmScO3 substrates, only 00l or l00 peaks from PbTiO3 films are observed, suggesting epitaxial growth with the coexistence of c and a domains. Note that because the lattice of (Ba0.5Sr0.5)RuO3 electrode is very close to the SmScO3 substrate, its diffraction peak overlaps with the substrate peak.

Supplementary Note 5. Illustration of "collective ferroelastic switching"
In our experiment, the applied stress (force) is completed in, for example, a 2 × 2 array of points (the tip radius is ~25nm) at the corners of a 1 × 1 µm area within a 2 × 2 µm scanned area ( Supplementary Figures 6a,b). A dramatic change in the domain structures, even well away (microns) from the tip contact area, has occurred after application of a stress (Supplementary Figure 6). That is, the change can extend across nearly the entire 2 × 2 µm scanned area (4×10 6 nm 2 area) when the force is applied only in a small fraction of that area (e.g., the tip-sample contact area has a radius of ~110 nm under force of 600 nN on four points, that is just ~3.8% of the entire switched area; Supplementary Figure 6d). As a result, we consider it as and refer to it as a nonlocal response. To exclude the possible emergence of domains induced by the long-range stress field we further verify the stress state and the stressed area in the film under the scanning-probe tip using finite-element methods. To simplify our calculation, the elastic field is simulated by using isotropic materials constants. Here, the Young's module is 1.3×10 11 Pa, and Poisson's ratio is 0.3. The size of the thin plate is set as 2 × 2 × 0.1 µm. The tip force is simulated by using a pressure applied on a circle area with a radius of 25 nm on the four corners of 1 × 1 μm on the film surface. We assume that the substrate is rigid enough that all the energy applied by the tip is absorbed by the film and transforms to elastic changes, thus the substrate on the boundary can be fixed with zero displacements. As shown in the commonly used von Mises stress contour map (Supplementary Figure 7a), and elastic stresses ( Supplementary Figures 7b-d),

Supplementary
only local area surrounding the tip is affected by the tip force. Based on this simulation result, we conclude that it is not possible for such tip-induced stress field to propagate widely (across the entire area over which switching is observed) in a normal elastic material, and enable the domain mergence in a large distance. We should note that the value colored in blue is on the order of 10 3 , and can be ignored as noise compared with the force on the order of 10 8 . We can further calculate the induced flexoelectric field in the film of the steady state as following:

Supplementary
where,

Supplementary Note 7. Energy barrier from Landau phenomenological theory
The Landau free energy densities of the a1/a2 and c/a domain structures at high temperature of 500K and 300K are shown in (Supplementary Figure 13). We note that the Curie temperature for the films under strain of 0.46% is ~891 K (618℃). At high temperature, the a1/a2 domain structure dominates, and is kept during the cooling by the misfit strain until external perturbation is given. Since a1/a2 and c/a domains have equal free energy at 300K, c/a domains are favorable even without other external field, thus the a1/a2 domains under applied stress with a much higher energy potential can be abruptly switched into c/a domains with a collective, non-local behavior.

Supplementary Note 9. Thickness effects -phase-field simulations of domain structures
In the current phase-field simulation, we chose a simulation size of 128Δx ×128Δx × 32Δx (Δx = 1 nm) with periodic boundary conditions to simulate the non-local domain switching and compare with experiments. We believe this is a balance between computation accuracy and efficiency. Admittedly, our simulated domain size and film thickness are smaller than the actual samples. In order to explore if there was a thickness effect, we performed additional phasefield simulations for 20 nm, 30 nm, 50 nm, and 70 nm thick PbTiO3 thin films under 0.5% and 1.0% tensile strains (Supplementary Figure 15). Despite different thickness of the film, the results show a similar trend. Under misfit strain of 0.5%, large area domain switching is observable; while under misfit strain of 1.0%, the initial domain structure is stable. It is also seen that the depth of the switched c+ and c-domains is around 30 nm. Therefore, we believe