Slab Thickness Dependence of Rashba Splitting on Au(111) Surface: First-Principles and Model Analyses

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.