Fractal nature of metallic and insulating domain configurations in nearly grain-boundary-free VO$_{2}$/TiO$_{2}$ thin films

We investigated evolution of the surface work function ($W_{S})$ maps of epitaxial 15-nm-thick VO$_{2}$/TiO$_{2}$ thin films using Kelvin probe force microscopy (KPFM) measurements while the film undergoes the metal-insulator transition (MIT). The metallic and insulating domains coexist in the VO$_{2}$ thin films during the transition, since the MIT is the first-order phase transition. Nearly grain-boundary-free samples allowed observation of metallic and insulating domains with distinct $W_{S}$ values, throughout the transition. Each domain allowed us to obtain real space domain maps with nanoscopic spatial resolution. The two-dimensional percolation model well explained the relationship between the metallic domain fraction and the measured the resistivity. The domain maps also suggested that the percolation clusters formed a fractal surface.