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

Based on first-principles calculations using density functional theory, we study the vibrational properties and thermal expansion of monoatomic 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, a phonon band gap 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 quasiharmonic approximation, we find that all of them show negative thermal expansion at room temperature.