Thermopower and thermophase in a $d$-wave superconductor

TL;DR

This paper investigates thermoelectric effects in a $d$-wave superconductor, revealing how impurity scattering and surface states influence thermopower and thermophase, with potential for experimental thermovoltage measurements.

Contribution

It provides a self-consistent quasiclassical analysis of thermoelectric response in $d$-wave superconductors, highlighting the impact of impurity scattering phase shift and surface states on thermoelectric effects.

Findings

Thermopower of several microvolts per kelvin in superconductor structures.

Surface-bound zero-energy Andreev states reduce thermoelectric effects.

Thermovoltage measurements could probe thermoelectricity in unconventional superconductors.

Abstract

In an unconventional superconductor, the interplay of scattering off impurities and Andreev processes may lead to different scattering times for electronlike and holelike quasiparticles. Such electron-hole asymmetry appears when the impurity scattering phase shift is intermediate between the Born and unitary limits and leads to an expectation for large thermoelectric effects. Here, we examine the thermoelectric response of a $d$-wave superconductor connected to normal-metal reservoirs under a temperature bias using a fully self-consistent quasiclassical theory. The thermoelectrically induced quasiparticle current is cancelled by superflow in an open circuit setup, but at the cost of a charge imbalance induced at the contacts and extending across the structure. We investigate the resulting thermopower and thermophase and their dependencies on scattering phase shift, mean free path, and interface transparency. For crystal-axis orientations such that surface-bound zero-energy Andreev states are formed, the thermoelectric effect is reduced as a result of locally reduced electron-hole asymmetry. For a semiballistic superconductor with good contacts, we find thermopowers of order several ${\mu}V/K$, suggesting a thermovoltage measurement as a promising path to investigate thermoelectricity in unconventional superconductors.