Excitonic condensation state in the assistance of the adiabatic and anti-adiabatic phonons.

The excitonic condensate in semimetal and semiconducting materials with the assistance of the phonons in both the adiabatic and anti-adiabatic regimes has been investigated. In the framework of the random phase approximation, we analyse the static excitonic susceptibility function once the electrons and holes are described in the extended Falicov-Kimball model involving the electron-phonon interaction. The excitonic condensate transition point is detected as a divergence of the susceptibility function. The complex phase diagrams exhibit the significant impact of the adiabatic phonons on the stability of the excitonic condensation state. In contrast, phonons in the anti-adiabatic regime play a less role in the formation and the condensation of the excitons. Depending on the temperature or the external pressure, the excitonic condensate stability and its Bardeen-Cooper-Schrieffer-Bose-Einstein condensation crossover in the systems are also discussed in detail.