Thermoelectric response of Josephson junction: from ballistic to disordered

It is known that Josephson junction (JJ) hosting scattering centers with energy-dependent scattering amplitudes which breaks the $\omega\rightarrow-\omega$ symmetry (where $\omega$ is the excitation energy of electron about the Fermi level) exhibits finite thermoelectric response. In contrast, here we show that even in a ballistic JJ this symmetry is broken and it leads to a non-zero thermoelectric response when the junction length is of the order of coherence length of the superconductor and the corresponding thermoelectric coefficient confirms to the universal sinusoidal dependence on $\phi_{12}$, where $\phi_{12}$ is the superconducting phase bias. In presence of multiple scatterers in the junction region, we have numerically shown that the sign of the even-in-$\phi_{12}$ part of the thermoelectric coefficient fluctuates violently from one disorder configuration to another hence averaging to vanishingly small values while the odd part tends towards the universal sinusoidal dependence on $\phi_{12}$ as we approach the large disorder limit under disorder averaging.