Origin of acoustic-optical phonon coupling in Cu₂SnSe₃. DFT-calculated phonon dispersions for (a) pristine, (b) Cd-doped, and (c) Ag-alloyed Cu₂SnSe₃. (d) Comparison of phonon dispersions along ΓX direction for pristine (left), Cd-doped (middle) and Ag-alloyed (right) Cu₂SnSe₃. The gradually softening transverse acoustic branches (TA and TA′) and longitudinal acoustic branch (LA) along with significant acoustic-optical coupling (red circles) are shown in the figure. (e) Comparison of total scattering rate within low-frequency phonons ranges for pristine (left), Cd-doped (middle) and Ag-alloyed (right) Cu₂SnSe₃. (f) Comparison of Grüneisen parameter for pristine (left), Cd-doped (middle) and Ag-alloyed (right) Cu₂SnSe₃, indicating the increased anharmonic lattice vibration from acoustic phonons. Energy profile of (g) Ag atom substituting Cu at different positions. The x = 0 suggests that the atom occupies the center of the Se tetrahedron; the calculated result shows that the off-centered Ag atoms displace 0.016 Å away from the center of the Se tetrahedron. (h) The crystal orbital Hamiltonian populations (COHP) are classified by different chemical bonds for Cu‒Se and Ag‒Se. (i) The schematic atomic orbital hybridization for Cu‒Se and Ag‒Se bonding, where the atomic orbital energy of Cu, Ag, and Se are obtained from NIST database LDA calculations [69]. The corresponding energy of bonding states is deduced from the atomic orbital energies.