Fig. 2: Lattice instabilities in monolayer 1T-VS₂ from first principles.

a Acoustic phonon dispersion from DFPT. LA, TA, and ZA stand for dominant longitudinal, transverse, and out-of-plane atomic displacements. The insets show selected displacement patterns corresponding to indicated modes. b Total energy from DFT as a function of the displacement amplitude for atomic displacements toward relaxed crystal structures and their projections onto soft phonon modes. c–e Relaxed crystal structures on \(9\times \sqrt{3}{{{{{{{\rm{R}}}}}}}}3{0}^{\circ }\), \(7\times \sqrt{3}{{{{{{{\rm{R}}}}}}}}3{0}^{\circ }\), and 4 × 4 unit cells from DFT. Vanadium and sulfur atoms are represented by black and yellow dots, their undistorted positions by gray shadows. Arrows represent the projections of the atomic displacements onto soft phonon modes. (Only arrows longer than 2% of the lattice constant are shown.) The contributions of different phonon modes are quantified in the figure titles. The displacements in c, d are drawn to scale, those in e have been magnified by a factor of three for better visibility. The primitive cell of the structure in c, which is in agreement with the results of ref. ²¹, is outlined in yellow. Dashed lines and crosses mark reflection planes and inversion centers.