Phonon properties, thermal expansion, and thermomechanics of silicene and germanene

We report a hierarchical first-principles investigation on the entangled effects of lattice dimensionality and bond characteristics in the lattice dynamics of silicene and germanene. It is found that bond bending (stretching) negatively (positively) contributes to Gr\"uneisen constant $\ensuremath{\gamma}$, which results in the negative acoustic (positive optical) $\ensuremath{\gamma}$. The layer thickening (bond weakening) caused by chemical functionalization tends to increase (decrease) the acoustic (optical) $\ensuremath{\gamma}$, due to the increased (decreased) bond-stretching effect. The excitation of the negative-$\ensuremath{\gamma}$ modes results in negative thermal expansion, while mode excitation and thermal expansion compete with each other in thermomechanics. The sensitive structural and electronic responses of silicene and germanene to functionalization help us to derive a generic physical picture for two-dimensional lattice dynamics.