Ultrasound attenuation peaks due to order parameter collective modes in impure superconductors with strong electron-hole asymmetry

Abstract

First we develop the general theory of order parameter collective modes and their sound attenuation in dirty superconductors with even-parity pairing. For the BCS state the theory yields a peak in the attenuation of longitudinal ultrasound whose height is proportional to ω_o[ηJ(s)]² and whose positiont _p ≡1 -T _p /T _c and width are proportional to ω_o/Γ = 2sql(s=v _s /v _F ). Here, η is the particle-hole asymmetry parameter,J(s) a given function, ω_o the sound frequency, and Γ=1/2τ ⁻¹ _N the normal state scattering rate. For η ∼T _c /T _o ∼ 0.1,s=2/3, and ω_o/Γ ∼ql « 1, the peak agrees qualitatively with the peaks found earlier for the polar and axial states. However, at low temperatures the attenuation goes like T⁴ instead ofT ³. We conclude that the observed longitudinal ultrasound peaks in the heavy-fermion superconductors UBe₁₃ and UPt₃ can be explained equally well by any anisotropic state having nodes in the gap, in particular, by a conventional state whose gap has the full symmetry of the Fermi surface. The nature of the state is reflected mainly in the dependence of the peak on the direction of sound propagation.