Phenomenology of the superconductive state of a marginal Fermi liquid

We consider an extension of the marginal-Fermi-liquid model to the s-wave superconducting state by phenomenologically incorporating the superconducting gap into the scattering spectrum. The linear-in-temperature scattering rate due to the high density of low-energy electronic excitations naturally leads to a large pair-breaking rate, which suppresses ${\mathit{T}}_{\mathit{c}}$. Below ${\mathit{T}}_{\mathit{c}}$ the low-energy excitations are self-consistently suppressed due to the opening of a superconducting gap. This leads to a vanishing of both the inelastic-scattering rate and pair breaking below ${\mathit{T}}_{\mathit{c}}$. There are a number of consequences not found in traditional BCS electron-phonon–induced s-wave superconductors. For energies below 3Δ, the quasiparticles become well defined in the superconducting state, while they are marginal (scattering rate proportional to the energy) in the normal state. This produces a two-peaked structure in the one-particle spectra—a sharp feature between Δ and 3Δ (depending on momentum) and a broad hump with an onset at 3Δ (independent of momentum). The transport properties do not obey the usual BCS rules. Transport properties in the q→0 limit and for low frequencies ω≪Δ show peaks below ${\mathit{T}}_{\mathit{c}}$. These are observable in microwave conductivity and in electronic thermal conductivity. Local or momentum-averaged response properties such as the nuclear relaxation rate show no peak, but a sharp drop below ${\mathit{T}}_{\mathit{c}}$. The superconductive gap opens very rapidly below ${\mathit{T}}_{\mathit{c}}$, and the value of 2Δ/${\mathit{T}}_{\mathit{c}}$ can cover a wide range, depending on parameters. The physical origin of these results is discussed, and comparison to experiment is made.