Bulk mediated surface diffusion: finite bulk case

Within the framework of a Master Equation scheme, we address the dynamics of adsorbed molecules (a fundamental issue in surface physics) and study the diffusion of particles in a finite cubic lattice whose boundaries are at the z = 1 and the z = L planes where $L = 2; 3; 4;\ldots$, while the x and y directions are unbounded. As we are interested in the effective diffusion process at the interface z = 1, we calculate analytically the conditional probability for finding the particle on the z = 1 plane as well as the surface dispersion as a function of time and compare these results with Monte Carlo simulations finding an excellent agreement. These results show that: there exists an optimal number of layers that maximizes $\langle r^2(t)\rangle$ on the interface; for a small number the layers the long-time effective diffusivity on the interface is normal, crossing over abruptly towards a subdiffusive behavior as the number of layers increases. 02.50.Ey Stochastic processes 05.10.Ln Monte Carlo methods 46.65.+g Random phenomena and medi