Unconventional thermoelectric effect in superconductors that break time-reversal symmetry

We demonstrate that superconductors that break time-reversal symmetry can exhibit thermoelectric properties, which are entirely different from the Ginzburg mechanism. As an example, we show that in the $s+is$ superconducting state there is a reversible contribution to thermally induced supercurrent, whose direction is not invariant under time-reversal operation. Moreover in contrast to Ginzburg mechanism it has a singular behavior near the time-reversal symmetry breaking phase transition. The effect can be used to confirm or rule out the $s+is$ state, which is widely expected to be realized in pnictide compounds ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$ and stoichiometric LiFeAs.

Figure 1

(a) Schematic view of the band structure in hole-doped iron pnictide compound ${\mathrm{Ba}}_{1-x}{\mathrm{K}}_x{\mathrm{Fe}}_2{\mathrm{As}}_2$. It consists of two hole pockets at $\mathrm{\Gamma }$ shown by open circles and two electron pockets at $(0,\pi )$ and $(\pi ,0)$ shown by ellipses. The $s+is$ state is favored by superconducting coupling dominated by the interband repulsion between electron and hole Fermi surfaces, as well as between the two hole pockets. (b), (c) schematically show the two degenerate ground states $s\pm is$. The phases of the order parameter components ${\mathrm{\Delta }}_k=\left|{\mathrm{\Delta }}_k\right|e^{i{\theta }_k}$ are represented by vector diagrams. Circular arrows show the directions of interband Josephson current supported by nontrivial relative phases ${\theta }_{kj}\ne 0,\pi$. (d) Sample whose ends are maintained at different temperatures $T_{1,2}$. The thermophase effect appears due to the temperature-dependent interband phase difference ${\theta }_{kj}={\theta }_{kj}\left(T\right)$. The total thermally induced current $\mathit{j}$ will have opposite directions in $s\pm is$ states. (e) sketches a closed circuit in spatially inhomogeneous bulk superconductor with two branches characterized by different values of the thermophase coefficients ${\gamma }^{(1,2)}\left(T\right)$. The junctions between branches have different temperatures $T_{1,2}$ and the temperature bias generates magnetic flux through the circuit (7).

Figure 2

(a) Phase diagram of the three-band superconductor described by the microscopic model (see text). Solid curves show the critical temperature $T_{{\mathbb{Z}}_2}$ of the BTRS phase transition. (b) Color scale plot of the thermophase coefficient $(T_{{\mathbb{Z}}_2}-T)\gamma \left(T\right)$. The prefactor $(T_{{\mathbb{Z}}_2}-T)$ is added to regularize the square-root singularity of $\gamma \left(T\right)$ at the BTRS phase transition. (c) The temperature dependencies of $\gamma \left(T\right)$ along the lines such as shown by arrow in (b). From right to left $u_{he}/u_{hh}=1.05;1.15;1.25;1.35$. (d) The ratio of magnetic fluxes generated due to the unconventional and conventional thermoelectric effects, ${\mathrm{\Phi }}_T^{BTRS}$ and ${\mathrm{\Phi }}_T^{TRS}$ correspondingly. ${\tilde{S}}_n$ is a dimensionless normal state Seebeck coefficient which can be of the order 1 but typically is much smaller. The other system parameters are explained in text.

Figure 3

Thermally induced magnetic field on a domain wall between $s+is$ and $s-is$ phases. Two faces of the stripe are maintained at different temperatures, thus resulting in a thermal gradient, here $\delta T=0.2T_c$, along the domain wall. The $s+is/s-is$ domain wall generates the magnetic field ${\mathbf{B}}_T$. In the absence of the temperature gradient a domain wall in $s+is$ superconductor does not have any magnetic signature.

Figure 4

A bimetallic ring consisting of two branches in the $s\pm is$ state, but having different chemical composition (see details in Appendix pp2). Both junctions are maintained at different temperatures $T_{1,2}$, so that there is a thermal gradient that varies linearly with the polar angle. This is sketched in (a). The resulting thermally induced magnetic field is shown on (c). In contrast to the usual thermoelectric effect its overall sign depends not only on the temperature bias but also on which of ${\mathbb{Z}}_2$ BTRS ground states (i.e., $s+is$ or $s-is$) is realized. This configuration supports nonzero flux ${\mathrm{\Phi }}_T$ in the inner hole, which is more or less independent of the diameter of the inner hole. The total flux, on the other hand depends on the diameter of the sample (b).