  We study the dependence of the spin splitting on the number $N$ of atomic
layers, using first-principles calculation for Au(111) surface. When the slab
of the atomic layers is sufficiently thick, the lower split state has a minimum
away from $\bar{\Gamma}$, which is known as the Rashba effect. As the number of
layers decreases, the minimum approaches $\bar{\Gamma}$, and it is located at
$\bar{\Gamma}$ for $N \leq 14$. This crossover is analyzed in detail using two
models: a tight-binding model and a bilayer nearly-free-electron model. It is
demonstrated that the features of surface band dispersion are clearly
understood as the result of the competition between the interference of the
surface states on both sides and the spin-orbit interaction.
