  We have used ab initio density functional theory to compute the magnetic
ground states of the surface alloy systems FeAu$_2$/Ru(0001) and
MnAu$_2$/Ru(0001). For both systems, we find that the lowest energy magnetic
configuration corresponds to a left-rotating spin spiral, in which the sense of
rotation is determined by the Dzyaloshinskii-Moriya interaction. These spirals
are lower in energy than the ferromagnetic configuration by 3-4 meV per nm$^2$.
We also find that FeAu$_2$/Ru(0001) has a significantly high magnetic
anisotropy energy, of the order 1 meV per Fe atom. By comparing with the
corresponding freestanding alloy monolayers, we find that the presence of the
Ru substrate plays a significant role in determining the magnetic properties of
the surface alloy systems.
