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 0K 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 A for an undulation length of 81 A. 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.