Fig. 1: Crystal structure, bonding, and rattling vibrations.

a Chain-like crystal structure of TlInTe₂, where In³⁺ cations (blue sphere) are tetrahedrally coordinated by Te²⁻ anions (dark gray sphere), forming InTe₄ tetrahedron. These InTe₄ tetrahedra share their edges, and extend along the c-axis, giving rise to a chain-like structure. Tl¹⁺ cations (orange sphere) occupy the spaces between the chains and stabilize the structure through electron transfer from Tl¹⁺ to the [InTe₂]¹⁻ sublattices. The conventional unit cell is outlined by the solid black line. b A top-down view of the crystal structure, where it can be seen that Tl atom occupies the empty space inside the square anti-prismatic (outlined by dashed blue line) environment formed by eight Te atoms. Visualization of the c longitudinal (R_LO) and d transverse (R_TO) optical rattling (R) phonon modes at the X point in the Brillouin zone, where the atomic displacements are indicated by the red arrows.