Simulation of strain profile

The strain distribution in a 3.5 nm-thick STO thin film pressed with an AFM tip is obtained by solving the elastic equilibrium equation by using Khachaturyan microelasticity theory³⁷ and the Stroh formalism of anisotropic elasticity³⁸. The detailed procedure has been elaborated in previous works³⁹. Here, we discretized three-dimensional space into 64 × 64 × 700 grid points and applied periodic boundary conditions along the x₁ and x₂ axes. The grid spacing was ∆x₁ = ∆x₂ = 1 nm and ∆x₃ = 0.1 nm. Along the x₃ direction, 35 layers were used to mimic the film; the relaxation depth of the substrate featured 640 layers to ensure that the displacement at the bottom of substrate was negligibly small. To estimate surface stress distribution that developed on AFM-tip pressing, we adopted the Hertz contact mechanics of the spherical indenter⁴⁰ with a tip radius of 30 nm and a mechanical force of 1–7 μN. The Young’s moduli and Poisson ratios of the Pt tip and the STO film were E^tip = 168 GPa and υ^tip = 0.38, and E^film = 264 GPa and υ^film = 0.24, adapted from ref. ⁴¹. The electrostrictive and rotostrictive coupling coefficients of STO were adapted from ref. ⁴². See Supplementary Note ² for more details.