Shell effects in planar electron clusters

We investigate the electronic shell structure of planar metal clusters, having in mind clusters on insulating surfaces with an interface energy such that the cluster covers the surface in a monolayer. In this first survey we concentrate on the shell effects of such a planar electron cloud using the Ultimate Jellium Model where the structural effects of the positive background are completely eliminated. An axially symmetric electron cloud shows shell effects which are, however, somewhat smaller than those of fully free three-dimensional clusters. The free variation of the shape for planar clusters on surfaces, leading to many triaxial clusters, diminishes the shell effects even further, leading to the existence of hybrid-deformed clusters and a lack of energetically favored ''magic'' clusters in an intermediate size range N approximate to 10-30. In contrary to the situation for free clusters the small shell energies have a minor effect on the energetics of the groundstate. As a consequence, electronic shell effects are only one ingredient amongst others to determine the kinetics of cluster growth on (insulating) substrates. With a bold rescaling assumption, we can relate axially symmetric planar clusters to the planar electron cloud in a neutral quantum dot, having the consequence that shell effects persist to play a role in these systems.