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Avoiding critical-point phonon instabilities in two-dimensional materials: The origin of the stripe formation in epitaxial silicene

The origin of the large-scale stripe pattern of epitaxial silicene on the ZrB$_2$(0001) surface observed by scanning tunneling microscope experiments is revealed by first-principles calculations. Without stripes, the ($\sqrt{3}\times\sqrt{3}$)-reconstructed, one-atom-thick Si layer is found to exhibit a "zero-frequency" phonon instability at the $M$ point. In order to avoid a divergent response, the relevant phonon mode triggers the spontaneous formation of a new phase with a particular stripe pattern offering a way to lower both the atomic surface density and the total energy of silicene on the particular substrate. The observed mechanism is a way for the system to handle epitaxial strain and may therefore be more common in two-dimensional epitaxial materials exhibiting a small lattice mismatch with the substrate.