Fig. 1

Studying diffusion characteristics of oxygen vacancies with KPFM. a Sketch of the sample geometry and KPFM measurement architecture. b Schematic illustration of the contact potential difference (CPD) contrast between the pristine (referred to as 1) and \(V_{\rm{o}}^{ \cdot \cdot }\)-rich (referred to as 2) regions. E _vac denotes the vacuum energy level. c Illustrative KPFM signal from a line scan from position A to B across this \(V_{\rm{o}}^{ \cdot \cdot }\)-rich region. \(\Delta {V_{\rm{s}}}\) and \(\left[ {V_{\rm{o}}^{ \cdot \cdot }} \right]\) denote the net change in the measured KPFM signal and the concentration of \(V_{\rm{o}}^{ \cdot \cdot }\) within the \(V_{\rm{o}}^{ \cdot \cdot }\)-rich region, respectively. d–f Characterisation of diffusion of \(V_{\rm{o}}^{ \cdot \cdot }\) with the KPFM technique. KPFM images around a \(V_{\rm{o}}^{ \cdot \cdot }\)-enriched surface region of a 14-uc-thick d and 120-uc-thick e STO films. The time lag between poling the pristine surface and the time of acquiring an image is indicated on the top-right corner of KPFM image. The time evolution of the degree of equilibrium, S(t) (solid symbols) and fit (solid line) according to Fick’s 2nd law of diffusion f. The scale bar in d, e represents 1 µm