  We use first-principles calculations based on density functional theory to
investigate the magnetic exchange interaction of Fe clusters on Rh(111) and
Ru(0001). We consider dimers, trimers, tetramers, and pentamers of different
shape in fcc and hcp stacking as well as infinite atomic and biatomic chains.
From the dimer calculations we extract the exchange interaction as a function
of adatom distance by mapping total energies to a Heisenberg model. The
nearest-neighbor (NN) exchange constant is about one order of magnitude smaller
than reported for other substrates due to the strong hybridization between the
Fe atoms and the partly filled $4d$-band of the surface. We also find a
transition from a ferromagnetic NN exchange interaction for Fe dimers on
Rh(111) to an antiferromagnetic one on Ru(0001). The distance-dependent
exchange coupling displays a RKKY-like oscillatory behavior which is nearly
inverted for Fe dimers on the Rh(111) surface compared to those on Ru(0001).
Unexpectedly, for Fe clusters beyond dimers, a complex trend of the magnetic
ground state is observed which alternates between ferro- and antiferromagnetic
configurations depending on cluster size and shape. In view of the exchange
constants obtained for dimers, it is surprising that on both surfaces small
compact clusters are ferromagnetic while open structures such as linear trimers
or tetramers become antiferromagnetic. We demonstrate that both vertical and
lateral structural relaxations of the clusters are crucial in order to
understand this unexpected trend of magnetic order and connected to the
competition of direct ferromagnetic exchange among Fe atoms in the cluster and
the hybridization with the substrate.
