Undulated silicene and germanene freestanding layers: why not?

Silicene and germanene freestanding layers are usually described as a honeycomb lattice formed by two hexagonal sub-lattices presenting a height difference, namely the layer buckling. In this work, first-principles calculations show that silicene and germanene can be rippled at 0 K with various wavelengths, without any compressive strain of the layer. For germanene, the height difference between two Ge atoms from the same sub-lattice can be as high as 4.7 [Formula: see text] for an undulation length of 81 [Formula: see text]. The deformations are related to slight (lower than 1.7°) bond angle modifications, and the energy cost is remarkably low, lying between 0.1 and 0.8 meV per atom. These undulations modify the electronic structure, opening a gap of 15 meV.