Size oscillations of the work function of a metal film on a dielectric substrate

Analytical expressions for the size dependence of the Fermi electronic energy of ultrathin metal films on a dielectric substrate have been derived in the model of free electrons and finite-depth asymmetric potential well. The work functions have been calculated for Al films on SiO$_2$ and Al$_2$O$_3$. It has been shown that the presence of a dielectric leads to a shift in the work function, while retaining the general character of the size dependences.