  Annealing has been extensively used to control crystal growth and physical
properties of materials with unfortunately unclear mechanism and quantitative
correlations. Here we present the "annealing temperature - grain size - band
gap" correlation for ZnO nanocrystals with experimental evidence. Findings
revealed that the annealing condition determines the critical size by equating
the thermal and the cohesive energy of the undercoordinated atoms in the
surface skin, which in turn induce local strain and quantum entrapment,
perturbing the Hamiltonian and hence the band gap. The formulation provides a
general guideline for controlling crystal growth and performance of materials,
and makes predictive design and fabrication of functional nanomaterials into
reality.
