Cooper-pair splitters as circuit elements for realizing topological superconductors

Abstract

Advances in materials and fabrication of superconducting devices allows the exploration of novel quantum effects in synthetic superconducting systems beyond conventional Josephson junction arrays. As an example, we introduce a new circuit element, the Y-splitter, a superconducting loop with three leads and three Josephson junctions, smaller or comparable in size to the superconducting coherence length of the material. By tuning magnetic flux through an array of Y-splitters, Cooper-pair transport can be made to interfere destructively, while spatially separated split Cooper pairs propagate coherently. We consider an array of Y-splitters connected in a two-dimensional star [Archimedean (3,$12^2$)] geometry, deformable into the kagome lattice, and find a rich phase diagram that includes topological superconducting phases with Chern numbers $\pm 2$. Experimental realization appears feasible.

Figures (17)

• Figure 1: The Y-splitter, the superconducting network component introduced in this work, in two sizes, small (a,c) and large (b,d) compared to the coherence length, $\xi$, and two Josephson coupling strengths, strong (a,b) and weak (c,d). Blue circle represents the coherence length, red segments the insulating barrier thickness. Dominant contribution of Cooper pair transport from the left to the upper arm shown as orange trajectories. The flux $\phi$ through the triangular loop controls interference of single electrons and Cooper pairs. Applying 1/2 (mod 1) flux quanta through the triangular loop leads to destructive interference of Cooper pairs but not single electrons.
• Figure 2: (a) Network of Y-splitters arranged into an Archimedean (3,$12^2$) or star lattice. By shrinking legs that connect a pair of Y-splitters (once instance is highlighted in green) to a point, we arrive at the kagome lattice shown in (b), with corresponding highlighted green region. (b) Magnetic flux $\phi$ is staggered such that the net magnetic field is zero (mod 2$\pi$). Josephson junctions are located along segments of each triangle, represented in red. The vectors $\mathbf s_i$ (depicted in blue) locate the three nearest left-pointing triangles relative to a lattice site $\mathbf{r}$ at the center of right-pointing triangles.
• Figure 3: Phase diagram of the effective model. The diagram shows the chirality $\chi$ and Chern numbers $\cal{C}$ as functions of the flux $\phi$ and ratio $\Gamma/\Delta$. The chirality takes specific values $\chi_{23}=0,\pm 1$ per Eq. \ref{['eq:chirality']}, denoted as $\chi=0, >0, <0$ with different colors. The limit $\Gamma\ll \Delta$ is the "classical" Josephson regime, which is studied in more detail in Appendix \ref{['classical']}. Regions with non-zero Chern numbers (hashed) only appear in the regime in which Cooper-pair splitting is active.
• Figure 4: Bands in the flux configuration $\phi=3\pi/2$ for (a) $\Delta=0$, where each one of the three bands is double (spin) degenerate; (b) $0<\Delta< \sqrt{3}\, \Gamma/2$, where the degeneracy in each one of the bands is split by $\pm \Delta$ as in Eq. \ref{['eq:spectrum-phi-equal-pi-half']}; and (c) $\Delta>\sqrt{3}\Gamma/2$, where the bands are separated into two sets of spectrum of $h_{\mathbf k}$. The light blue and pink colored bands in (b) and (c) indicate anti-symmetric and symmetric spin states, which are non- degenerate when $\Delta\neq 0$.
• Figure 5: Model of the wire network that accounts for the finite extent of the wires. The model can be understood as a network of X-shaped crossings comprising two intersecting wires at the sites of the kagome lattice. Intra-molecule tight-binding hopping have amplitude $t$, and inter-molecule hopping have amplitude $\gamma$. Intra-molecule superconductivity is accounted for within the BdG formalism. We label the three X-shaped crossings 1, 2, and 3, according to our site labeling convention in Fig. \ref{['fig:kagome_lattice']}(b)