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The atomic structure of the $\sqrt{3} \times \sqrt{3}$ phase of silicene on Ag(111)

The growth of the $\sqrt{3} \times \sqrt{3}$ reconstructed silicene on Ag substrate has been frequently observed in experiments while its atomic structure and formation mechanism is poorly understood. Here by first-principles calculations we show that $\sqrt{3} \times \sqrt{3}$ reconstructed silicene is constituted by dumbbell units of Si atoms arranged in a honeycomb pattern. Our model shows excellent agreement with the experimentally reported lattice constant and STM image. We propose a new mechanism for explaining the spontaneous and consequential formation of $\sqrt{3} \times \sqrt{3}$ structures from $3 \times 3$ structures on Ag substrate. We show that the $\sqrt{3} \times \sqrt{3}$ reconstruction is mainly determined by the interaction between Si atoms and have weak influence from Ag substrate. The proposed mechanism opens the path to understanding of multilayer silicon.