Process of Irreversible Nucleation in Multilayer Growth. I. Failure of the Mean-Field Approach

The formation of stable dimers on top of terraces during epitaxial growth
is investigated in detail. In this paper we focus on mean-field theory, the
standard approach to study nucleation. Such theory is shown to be
unsuitable for the present problem, because it is equivalent to considering
adatoms as independent diffusing particles. This leads to an overestimate
of the correct nucleation rate by a factor N, which has a direct physical
meaning: in average, a visited lattice site is visited N times by a
diffusing adatom. The dependence of N on the size of the terrace and on the
strength of step-edge barriers is derived from well known results for
random walks. The spatial distribution of nucleation events is shown to be
different from the mean-field prediction, for the same physical reason. In
the following paper we develop an exact treatment of the problem.