  We report a post-growth aging mechanism of Bi$_2$Te$_3$(111) films with
scanning tunneling microscopy in combination with density functional theory
calculation. It is found that a monolayered structure with a squared lattice
symmetry gradually aggregates from surface steps. Theoretical calculations
indicate that the van der Waals (vdW) gap not only acts as a natural reservoir
for self-intercalated Bi and Te atoms, but also provides them easy diffusion
pathways. Once hopping out of the gap, these defective atoms prefer to develop
into a two dimensional BiTe superstructure on the Bi$_2$Te$_3$(111) surface
driven by positive energy gain. Considering the common nature of weakly bonding
between vdW layers, we expect such unusual diffusion and aggregation of the
intercalated atoms may be of general importance for most kinds of vdW layered
materials.
