  We have investigated the atomic and electronic structure of the
($\sqrt{3}\times \sqrt{3}$)$R30^{\circ}$ SnAu$_2$/Au(111) surface alloy. Low
energy electron diffraction and scanning tunneling microscopy measurements show
that the native herringbone reconstruction of bare Au(111) surface remains
intact after formation of a long range ordered ($\sqrt{3}\times
\sqrt{3}$)$R30^{\circ}$ SnAu$_2$2/Au(111) surface alloy. Angle-resolved
photoemission and two-photon photoemission spectroscopy techniques reveal
Rashba-type spin-split bands in the occupied valence band with comparable
momentum space splitting as observed for the Au(111) surface state, but with a
hole-like parabolic dispersion. Our experimental findings are compared with
density functional theory (DFT) calculation that fully support our experimental
findings. Taking advantage of the good agreement between our DFT calculations
and the experimental results, we are able to extract that the occupied Sn-Au
hybrid band is of (s, d)-orbital character while the unoccupied Sn-Au hybrid
bands are of (p, d)-orbital character. Hence, we can conclude that the
Rashba-type spin splitting of the hole-like Sn-Au hybrid surface state is
caused by the significant mixing of Au d- to Sn s-states in conjunction with
the strong atomic spin-orbit coupling of Au, i.e., of the substrate.
