Large Thermoelectric Effects and Inelastic Scattering in Unconventional Superconductors

The thermoelectric coefficient η(T ) in unconventional superconductors is enhanced below Tc by intermediate strength impurity scattering that is intrinsically particle-hole asymmetric. We compute η(T ) for a strong-coupling d-wave superconductor and investigate the effects of inelastic scattering originating from electron-boson interactions. We show that η(T ) is severely suppressed at temperatures just below Tc by a particle-hole symmetric inelastic scattering rate. At lower temperatures inelastic scattering is frozen out and η(T ) recovers and regains its large amplitude. In the limit T → 0, we have η(T ) ∼ η0T + O[T 3], where the slope η0 contains information about the Drude plasma frequency, the details of impurity scattering, and the change in effective mass by electron-boson interactions. In this limit η(T ) can be used as a probe, complementary to the universal heat and charge conductivities, in investigations of the nature of nodal quasiparticles. Low-temperature transport measurements have provided a wealth of information about nodal quasiparticles in unconventional superconductors [1, 2]. Thermal conductivity is of particular importance, because theory predicts universality in the sense that the low-temperature asymptotic does not depend on the properties of the impurity potential [3]. This prediction has been confirmed experimentally [1]. However, there are some difficulties in analysing the low-temperature thermal conductivity. First, the leading T 2-dependence of the electronic contribution to κ(T )/T is masked by a phonon contribution with the same T 2-dependence [2]. Second, experiments are often done on ultra-clean samples, which means that the T 2 power law of κ(T )/T holds only in a small temperature bracket [4]. In a recent paper we discussed how isotropic elastic scattering by impurities of intermediate strength, i.e described by a phase shift 0 < δ0 < 2 , gives rise to an electron-hole asymmetric scattering time and consequently a large non-universal thermoelectric response [5]. A careful study of the thermoelectric coefficient in the low-temperature regime would reveal information about the bare elastic scattering rate and potential strength. This information is hard to extract from thermal conductivity data. In this report we examine the interplay between elastic and inelastic scattering and show how it affects the thermoelectric coefficient. The thermoelectric coefficient, η(T ), is defined as δ~je = −η(T )∇T = 2N f ∫ d~p f ∫ dε 4π i e~v f δg . (1) The quasiclassical propagator δ ĝK has a closed form in which the self-consistently computed equilibrium Green function ĝ0 and the self-energy Σ̂R serve as input (see Graf et al. [3] and Ref. [5] for details). In the present study we consider a composite self-energy in particlehole space Σ̂R = Σ̂imp + Σ̂ R in, where the impurity selfenergy Σ̂imp is diagonal with Σ R 3,imp and Σ R 0,imp being its particle-hole symmetric and anti-symmetric parts, respectively. The self-energy Σ̂in includes the effects of inelastic electron-boson scattering, but we also assume that this interaction mediates the pairing. Below, Ŵ R is the off-diagonal component, i.e the usual strong-coupling function related to the energy dependent gap as ∆̂R(ε) = Ŵ R(ε)/Z(ε). Contrary to the impurity self-energy the diagonal part of the inelastic self-energy is particle-hole symmetric as Σ0,in = 0. Finally, Z(ε) is the energyrenormalization function defined by the scattering renormalized energy ε̃R = ZR(ε)ε = ε −Σ3,imp −Σ R 3,in. This model is the same as we used in our study of the thermal conductivity [4]. With this input we use Ref. [5] and write down the response function