Theory of two-phonon modes in layered charge-density-wave systems

Abstract

A theoretical model with electron-phonon and anharmonic interactions is proposed to explain the two-phonon mode observed in the Raman spectra of layered transition metal dichalcogenides, which exhibit charge density wave (cdw) phase transition. The phonon self-energy, which involves the electron response function and the two-phonon Green’s function, is calculated using the double-time Green’s function formalism. It is shown that in these low-dimensional systems there exists an anharmonicity-mediated two-phonon mode in the phonon spectral function both in the normal and in thecdw phases. In the normal phase since the phonon Raman scattering proceeds through a single optic phonon the calculations are carried out for zero wave vector and hence the contribution of the electron response function to the self-energy vanishes. On the other hand explicit evaluation of the two-phonon Green’s function shows that the frequency of the two-phonon mode is twice that of the Kohn anomaly phonon and decreases with decreasing temperature. The strength of two-phonon peak is found to be comparable to that of the original optic phonon. In thecdw phase the phonon which enters into the Raman scattering is taken to be the one with thecdw wave vectorQ, which when zone-folded becomes equivalent to zero wave vector. The evaluation of the electron response function in this phase generates a phonon corresponding to thecdw-amplitude mode. The two-phonon Green’s function is assumed to be of similar form as in the normal phase. The spectral function evaluated at zero temperature shows a weak two-phonon peak besides the prominentcdw-amplitude mode. Numerical results are presented for the system 2H-NbSe₂ and are found to be in qualitative agreement with the experimental data.