Phonon anharmonicity in multi-layered WS2 explored by first-principles and Raman studies

Explaining Mysterious “Shoulder” Raman Band in TiS2 by Temperature-dependent Anharmonicity and Defects

“…In eqs and , parameters and are the coefficients of third- and fourth-order terms in the Taylor expansion of the interatomic potential with respect to normal phonon coordinates, ,, we denote ω i = ω q , j i , , and the symbol P denotes the principal value. The higher-order derivatives have been calculated using a method based on the frozen-phonon approach. , The method involves calculating phonon dispersion relations ω q , j i for three structures: the equilibrium one (with A 0 ,j = 0) and two distorted structures constructed by slight shifts of atoms toward eigen-displacements corresponding to the desired phonon mode ( 0 , j ) (i.e., with A 0 , j = ±Δ A 0 , j ). We used the values of Δ A 0 , j leading to the spatial shifts of sulfur atoms by 0.10 Å, and we checked this choice to provide converged results.…”

“…Mathematically, we use the following decomposition where the first sum is performed over all Γ-point phonon modes in the nondefected supercell (denoted by l ) and μ l,j are coefficients of the seriesgiven by sums of phonon eigenvectors’ scalar products over all atoms κ (we skip those atoms which are present in the defected supercell and not in the pristine structure). This enabled us (see Section S2.2 in Supporting Information for a further justification) to estimate the intensity of a narrow Raman band due to the j -th phonon mode as follows , where l R denotes Raman-active phonon modes in pristine 1 T -TiS 2 : one out-of-plane A 1g and two degenerate in-plane E g modeswith perfectly synchronized vibrations in each unit cell. In eq , ω j is the mode’s circular frequency, , vectors e i , e s are the polarization vectors of the incident and scattered light, respectively, and ­( l R ) is the Raman tensor of the l R -th phonon mode in the nondefected supercell.…”

“…For any wavevector q, such transformation can be realized by collecting eigenvectors of all q-point phonon modes in the columns of square matrices Ω(q) and Ω ̃(q)�separately for the equilibrium and the distorted systems�which provides the following matrix equation for the desired derivatives 55…”

“…To estimate the Raman intensities corresponding to Γ-point phonon modes in nonstoichiometric TiS 2 supercells (Figure 5), we assumed that the eigen-displacements vector W(κ; 0,j) of an arbitrary j-th Γ-point phonon mode in the defected supercell (defined for each κ-th atom) can be considered equal to its orthogonal projection onto eigenvectors W 0 (κ; 0,l) of the Γpoint phonon modes in a hypothetical nondefected supercell with the same dimensions. 55…”

“…where the first sum is performed over all Γ-point phonon modes in the nondefected supercell (denoted by l) and μ l,j are coefficients of the series�given by sums of phonon eigenvectors' scalar products over all atoms κ (we skip those atoms which are present in the defected supercell and not in the pristine structure). This enabled us (see Section S2.2 in Supporting Information for a further justification) to estimate the intensity of a narrow Raman band due to the j-th phonon mode as follows 55,57 e R e I j n l ( )…”