Singularities in the Polarizability of Superconductors at Nesting Vectors

We calculate the electronic polarizability in the superconducting state at nesting vectors of the Fermi surface. A pole appears in the polarizability at frequencies ω near the superconducting gap 2∆ for arbitrary small couplings in two dimensions. This leads to the appearance of a sharp peak at ω just below 2∆ in the lattice vibrations at the nesting vectors. The results are compared with the recent inelastic neutron scattering measurements in compounds, ANi2B2C, A = Y or Lu. 74.20.-z,74.25.kc Typeset using REVTEX 1 Strong momentum dependence of the electronic polarizability leads to characteristic features in the dispersion relation of phonons in simple metals–the Kohn anomalies. [1] Transition metals and compounds are particularly beset with wiggles and dips in their phonon dispersions. Strong electron-phonon interactions and local field effects are responsible together with nesting features in the band structure for the anomalies and for the occurrence of charge density wave transitions [2]. It is well known that the leading order polarizability Π0 ( ~ Q, ω), Fig. 1 (a), of an electron gas in d-dimensions for ~ Q = 2~kF [3] in the normal state is ReΠ0 (2 ~kF , ω) ≃ N(0)[1 + (N(0)ω)ln( EF ω )]. (1) In s-wave superconductor with gap 2∆, the polarizability for ω >> 2∆ is similar to that of the normal state except for ~q << ξ, where ξ is the superconducting coherence length. When a Fermi-surface is nested at wave-vectors ~ Q0, so that ImΠ n 0 ( ~ Q0, ω) has a peak for ~ Q ≃ ~ Q0, a diminution in the superconducting state is again expected for ω → 0. We show the existence of singularities in the polarizability at ω ≃ 2∆ in s-wave superconductors near the nesting vectors which are stronger than those of the normal state at ω → 0. These singularities strongly affect the phonon spectrum of superconductors in some cases specified later. We explain the remarkable difference between the phonon spectrum of Y Ni2B2C in the superconducting state and that of the normal state recently observed by Kawano et al [4] and in LuNi2B2C by Stassis et al [5]. In the normal phase of (Y, Lu)Ni2B2C the transverse acoustic dispersion curve in the ΓZ direction has a strong temperature dependent dip around ~ Q = (0.5, 0, 8). Below the superconducting transition temperature the spectral weight splits into two parts, a very sharp resolution limited line at an energy ≃ 2∆ and a broad part around the phonon peak of the normal state with the total spectral weight conserved. Weight is transferred from the sharp peak to the broad peak at low temperatures by a magnetic field. As a model, we consider a simple ellipsoidal Fermi-surface, ǫ~k = ∑d i ~k2 i 2mi , and calculate Π(~q, ω) in the superconducting state for ~q near ~ Q0, the spanning vector across the flattest 2 part of the Fermi-surface. Results remain qualitatively the same for any particular Fermisurface in the same direction. We write the model Hamiltonian H = Hph +Hel +Hint, (2)