Dynamic dielectric function and phonon self-energy from electrons strongly correlated with acoustic phonons in 2D Dirac crystals

Abstract

The unique structure of two-dimensional (2D) Dirac crystals, with electronic bands linear in the proximity of the Brillouin-zone boundary and the Fermi energy, creates anomalous situations where small Fermi-energy perturbations (e.g., by impurities and quasiparticles) are known to critically affect the electron-related lattice properties of the whole system. The Fermi-surface nesting conditions determining such effects via electron-phonon interaction, require accurate estimates of the crystal’s response function (χ) as a function of the phonon wavevector q for any values of temperature, as well as realistic hypotheses on the nature of the phonons involved. Numerous analytical estimates of χ(q) for 2D Dirac crystals beyond the Thomas-Fermi approximation have been so far carried out only in terms of dielectric response function χ(q, ω), for photon and optical-phonon perturbations, due to relative ease of incorporating a q-independent oscillation frequency (ω) in their calculation. However, models accounting for Dirac-electron interaction with ever-existing acoustic phonons, for which ω does depend on q and is therefore dispersive, are essential to understand many critical crystal properties, including electrical and thermal transport. The lack of such models has often led to assume that the dielectric response function χ(q) in these systems can be understood from free-electron behavior, or statically, and from zero-temperature behavior. Here, we show that, different from free-electron systems, χ(q) calculated from acoustic phonons in 2D Dirac crystals using the Lindhard model, exhibits a cuspidal point at the FSN condition even in the static case and at 0 K. Strong variability of ∂χ/∂q persists also at finite temperatures, while χ(q) may tend to infinity in the dynamic case even where the speed of sound is small, albeit nonnegligible, over the Dirac-electron Fermi velocity. The implications of our findings for electron-acoustic phonon interaction and transport properties such as the phonon linewidth derived from the phonon self energy will also be discussed.