Phonon anomaly near a Fermi surface topological transition

The general behaviour of a T=0K metal phonon spectrum near a Fermi surface (FS) topology transition is studied in the framework of the quasiharmonic adiabatic approximation. The transition parameter is taken as the separation Delta =EF-Ec between the Fermi level and the energy of the electronic Van Hove singularity responsible for the transition. The dominant anomaly of the static electron density response function is first derived within the RPA and substituted in a perturbation formula (derived from the general formulation of Pick, Cohen and Martin (1970)) which relates a phonon energy change to a perturbative variation of the irreducible electronic polarisability. The overall variation Delta omega 2 with Delta has been estimated from a crude model involving a cut-off parameter related to the 'critical states' bandwidth ( Delta omega 2 includes the cut-off independent dominant anomaly Delta omega 2); phonons dips, as observed in Nb or Pb for instance, are expected quite generally when a Van Hove singularity is near EF. These results have been used to study the Delta dependence of a phonon spectrum average of a frequency-dependent function (such as ( omega n)); a C+or- mod Delta mod 3/2 anomaly is found, with different coefficients C+or- when Delta to +or-0. The theory is applied in a semiphenomenological way to the Tl-Bi-Pb alloys phonon softening at the X point.

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