  Based on first-principles calculation using density functional theory, we
study the vibrational properties and thermal expansion of mono-atomic
two-dimensional honeycomb lattices: graphene, silicene, germanene and blue
phosphorene. We focus on the similarities and differences of their properties,
and try to understand them from their lattice structures. We illustrate that,
from graphene to blue phosphorene, phonon bandgap develops due to large
buckling-induced mixing of the in-plane and out-of-plane phonon modes. This
mixing also influences their thermal properties. Using quasi-harmonic
approximation, we find that all of them show negative thermal expansion at room
temperature.
