We study local oxidation induced by dynamic atomic force microscopy (AFM), commonly called tapping mode AFM. This minimizes the field-induced forces, which cause the tip to blunt, and enables us to use very fine tips. We are able to fabricate Ti-TiOx line grids with 18 nm period and well-defined isolating barriers as small as 15 nm. These junctions show a non-linear current-voltage characteristic and an exponential dependence of the conductance on the oxide width, indicating tunnelling as the dominant conduction mechanism. From the conductance-barrier width dependence we derive a barrier height of Phi = 178 meV. Numerical calculations of the lateral field distribution for different tip geometries allow us to design the optimum tip for the most localized electric field. The electron beam deposition technique makes it possible to produce tips of the desired geometry.