Fig. 1

a For the conductive AFM tip-ferroelectric film-conductive substrate, the origin is at the tip–surface contact point; V is the applied voltage; a, the tip apex radius; r, is the lateral distance from the origin; \(r_{0}\), the distance between the intercept of the cone with the surface and the origin; d, the thickness of the film; and \(\theta _0\), the cone half angle. [001] is the c-direction of the film and [100] the a-direction. b Polarization domain expands over time under the electric field applied by the AFM tip. \(r_{i}\) is about a tip diameter away from the origin and where the electric field due to the cone dominates