Weak localization corrections to the thermal conductivity in s-wave superconductors

Abstract

We study the thermal conductivity in disordered s-wave superconductors. Expanding on previous works for normal metals, we develop a formalism that tackles particle diffusion as well as the weak localization (WL) and weak antilocalization (WAL) effects focusing on the two-dimensional case. Using a Green's functions diagrammatic technique, which takes into account the superconducting nature of the system by working in Nambu space, we identify the system's low-energy modes, the diffuson and the cooperon. The timescales that characterize the diffusive regime are energy dependent; this is in contrast to the normal state, where the relevant timescale is the impurity scattering time τe, independent of energy. The energy dependence introduces a novel energy scale ɛ*, which for dirty superconductors (τeΔ≪1, with Δ the gap) is given by ɛ*=Δ/τe. From the diffusive behavior of the low-energy modes, we obtain the WL correction to the thermal conductivity. We give explicit expressions in two dimensions. We determine the regimes in which the correction depends manifestly on ɛ* and propose an optimal regime to verify our results in an experiment. In particular, we find a parametrically large reduction of the weak localization correction in a dirty superconductor, in comparison with its value in the normal state, when the temperature is lowered by 10% below the transition temperature.

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