Negative-$U$ superconductivity on the surface of topological insulators

We study the effects of a finite density of negative-$U$ centers (NUCs) on the surface of a three-dimensional topological insulator. The surface Dirac fermions mediate a power-law interaction among the local Cooper pairs at the NUCs, and the interaction remains long-ranged for weak disorder. Superconductivity can be generated in the presence of a random distribution of NUCs. The NUCs play dual roles as both pair creators and pair breakers, and the competition of the two effects results in a nonmonotonic dependence of the mean-field superconducting transition temperature on the density of NUCs. Global phase coherence is established through coupling the locally superconducting puddles via Josephson coupling. Rare fluctuations play important roles, and a globally superconducting phase can only be achieved at large enough concentration of NUCs. The $p\text{−}\mathrm{wave}$ component of the superconducting order parameter gives rise to frustration among the superconducting grains, which is captured by a Potts-XY type model. New phases with chiral order, glass order, and, possibly, topological order can then emerge in the system of superconducting grains.

Figure 1

Superconducting transition temperature as a function of the concentration of negative-$U$ centers for Dirac fermions with large Fermi surface. Here, $\pi J^{2}N\left(0\right)/8=5,15,25$ (from left to right). We have used $1/2{\tau }_e=\pi x{J}^{2}N\left(0\right)/8$ for $x<1/2$ and $1/2{\tau }_e=\pi (1-x)J^{2}N\left(0\right)/8$ for $x>1/2$.

Figure 2

The same plot as Fig. 1 for Dirac fermions with $\mu =0$ and $J^2/\left(8\pi v_F^2\right)=1.5,3,6$ (from left to right).

Figure 3

Schematic phase diagram for rare fluctuation generated superconductivity. The dashed line represents a crossover from a high temperature (semi)-metallic phase to a phase with locally superconducting puddles. The solid line represents a phase transition into the globally superconducting phase, in which global phase coherence is established among the local superconducting puddles via Josephson coupling.