Cooper-Pair Injection into Quantum Spin Hall Insulators

We theoretically study tunneling of Cooper pairs from a superconductor spanning a two-dimensional topological insulator strip into its helical edge states. The coherent low-energy electron-pair tunneling sets off positive current cross correlations along the edges, which reflect an interplay of two quantum-entanglement processes. Most importantly, superconducting spin pairing dictates a Cooper pair partitioning into the helical edge liquids, which transport electrons in opposite directions for opposite spin orientations. At the same time, Luttinger-liquid correlations fractionalize electrons injected at a given edge into counterpropagating charge pulses carrying definite fractions of the elementary electron charge.

Figure 1

Schematic of the model. (a) Same-edge (${\overline{H}}^{\prime }$) and cross-edge (${\check{H}}^{\prime }$) singlet electron-pair injection processes from the superconductor into the upper and lower QSHI edges are shown. (b) A single right-moving electron injected into an edge LL “pumps” a charge of $e(1-g)/2$ to the left. The net fractionalized charges propagating to the right (left) are thus, respectively, given by $e_{\pm }=e(1\pm g)/2$ [18]. (c) A schematic bending of the edges shows a close analogy of our model to the proposed entangled CP injection into carbon nanotubes [11]. The essential differences stem from the LL correlations along the edges vs those at the carbon nanotubes’ ends and the natural spin filtering provided by the edge-state helicity in the QSHI.