Thermal bias induced charge current in a Josephson junction: From ballistic to disordered

It is known that Josephson junction (JJ) hosting scattering centers with energy-dependent scattering amplitudes, which breaks the $\ensuremath{\omega}\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\omega}$ symmetry (where $\ensuremath{\omega}$ 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 nonzero thermal-bias induced charge current above the gap, when the junction length is of the order of coherence length of the superconductor and the corresponding response confirms to the universal sinusoidal dependence on ${\ensuremath{\phi}}₁₂$, where ${\ensuremath{\phi}}₁₂$ 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-${\ensuremath{\phi}}₁₂$ part of the this response 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 ${\ensuremath{\phi}}₁₂$ as we approach the large disorder limit under disorder averaging.