Comparative study of phonon spectrum and thermal expansion of graphene, silicene, germanene and blue phosphorene

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.