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Phonon spectrum and bonding properties of Bi$_{2}$Se$_{3}$: Role of strong spin-orbit interaction

Phonon dispersions of one typical three-dimensional topological insulator Bi$_{2}$Se$_{3}$ have been studied within density functional theory. The soft modes of two acoustic branches along the $Z$$-$$F$ and $Γ$$-$$F$ directions within the pure local density approximation will transit to show imaginary frequency oscillating after including the spin-orbit interaction (SOI). Similar phenomenon has also been observed for Bi$_{2}$Te$_{3}$. Besides, we have found that the weak van der Waals forces between two Se1 layers in Bi$_{2}$Se$_{3}$ are strengthened by turning on the SOI.