  Artificially confined, small quantum systems show a high potential for
employing quantum physics in technology. Ultra-cold atom gases have opened an
exciting laboratory in which to explore many-particle systems that are not
accessible in conventional atomic or solid state physics. It appears promising
that optical trapping of cold bosonic or fermionic atoms will make construction
of devices with unprecedented precision possible in the future, thereby
allowing experimenters to make their samples much more "clean", and hence more
coherent. Trapped atomic quantum gases may thus provide an interesting
alternative to the quantum dot nanostructures produced today. Optical lattices
created by standing laser waves loaded with ultra-cold atoms are an example of
this. They provide a unique experimental setup to study artificial crystal
structures with tunable physical parameters. Here we demonstrate that a
two-dimensional optical lattice loaded with repulsive, contact-interacting
fermions shows a rich and systematic magnetic phase diagram. Trapping a few (N
=< 12) fermions in each of the single-site minima of the optical lattice, we
find that the em shell structure in these quantum wells determines the
magnetism. In a shallow lattice, the tunneling between the lattice sites is
strong, and the lattice is non-magnetic. For deeper lattices, however, the
shell-filling of the single wells with fermionic atoms determines the
magnetism. As a consequence of Hund's first rule, the interaction energy is
lowered by maximizing the number of atoms of the same species. This leads to a
systematic sequence of non-magnetic, ferromagnetic and antiferromagnetic
phases.
