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

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