If I apply a position dependent temperature gradient to a s wave superconductor, would I get a position dependent superconducting order parameter?

Question

What would happen if a s wave superconductor placed in a position dependent temperature gradient, where $T(x)<T_c$ is always satisfied. Would I get a position dependent superconducting order parameter $\Delta(x)$?

Answer 1

Yes. To see that this must be true, consider a slow gradient from 0K to slightly above $T_c$. One end of the sample is superconducting, and the other in the trivial phase. In fact, the order parameter is generally position dependent, $\Delta \left( \mathbf{r}\right)$ due to actual samples having finite size and disorder.

---

Although that is the full answer to your question, I cannot help myself from diving into a digression about the transport effects from this temperature gradient-induced variation of the order parameter... The subtle thing about the superconducting order parameter is the fact that it's complex-valued, $\Delta \left( \mathbf{r}\right) = |\Delta\left( \mathbf{r}\right)|e^{i\phi\left( \mathbf{r}\right)}$. You might be aware that, in e.g. the Josephson effect, supercurrents can be driven by a phase difference $\Delta \phi$. However, because of time-reversal symmetry, the magnitude of the order parameter does not have a similar effect, see e.g. Galperin et al. (2002).

Now, in bulk superconductors you can have such a phase difference due to a temperature gradient, driving a supercurrent. However, you can't have a stationary electric field in the superconductor, so this supercurrent gets cancelled by an opposite normal current. To get around this, and to see thermoelectric effects in superconductors, people have studied more complicated geometries, such as rings where each half is made from a different superconducting material.