given as $V_{\rm Au,exp} = 22\pm1$ V. For small Au clusters on TiO$_2$ substrates a sharp increase of $V_0$ for clusters smaller than 4 nm was observed with values of up to $V_{\rm Au,cluster} \sim 50$ V for clusters smaller than 2 nm.[@IAO+03] We confirm these findings in Sec. \[sectionFilmChar\].

For the a-C substrate films the situation is less consistent. We observe phase shifts of the substrate film with respect to the vacuum of $\sim 1.9 - 2.8$ radians. Associating these phase shifts with the numbers of the manufacturer for the film thickness of 9.5 to 14 nm we obtain a potential of $V_{\rm subs} = 27.4$ V.

On the other hand, the experimental value for a-C films of the same density as ours has been given by Harscher and Lichte[@HL98] as $V_{\rm HL} = 10.7$ V. This number compares satisfactorily with the values for graphite summarized by Sánchez and Ochando[@SO85] of $V_{\rm graph,theo}\sim 12 - 15$ V (theory) and $V_{\rm graph,exp} \sim 11 - 13$ V (experiment) since the ratio of the potentials $V_{\rm HL} / V_{\rm graph} \approx \rho_{\rm
  subs} / \rho_{\rm graph}$ corresponds roughly to that of the densities, i.e., the inner potential scales roughly with the 