Proximity-Induced $\pi$ Josephson Junctions in Topological Insulators and Kramers Pairs of Majorana Fermions

We study two microscopic models of topological insulators in contact with an $s$-wave superconductor. In the first model the superconductor and the topological insulator are tunnel coupled via a layer of randomly distributed scalar and of randomly oriented spin impurities. Here, we demonstrate that spin-flip tunneling dominates over the spin-conserving one. In the second model the tunnel coupling is realized by a spatially nonuniform array of single-level quantum dots with randomly oriented spins. We find that the tunnel region forms a $\pi$ junction where the effective order parameter changes sign. Because of the random spin orientation, effectively both models exhibit time-reversal symmetry. The proposed $\pi$ junctions support topological superconductivity without magnetic fields and can be used to generate and manipulate Kramers pairs of Majorana fermions by gates.

Figure 1

Setups to generate a proximity-induced Josephson $\pi$ junction in TIs. (a) An $s$-wave SC (red) couples to a TI (grey) via an insulator doped with magnetic and scalar impurities [magnetic insulator (MI), blue]. If the spin-flip tunneling rates are larger than the normal tunneling rates, superconducting gaps with opposite sign are induced in the TI samples. (b) Top view: Instead of the MI, the SC is coupled to the TI via a spatially nonuniform array of single-level QDs in the Coulomb blockade regime. The array of QDs is occupied with randomly oriented electron spins.

Figure 2

(a) Setup hosting Kramers pairs of MFs. Two TIs (grey rectangles) are placed on top of an underlying $s$-wave SC (red) such that proximity superconductivity is induced in both TIs. Importantly, the first TI is coupled through a MI (blue) resulting in the $\pi \mathrm{JJ}$. The tunnel barrier (TB) (yellow) between the edges of two TIs extends from $x=0$ to $x=L$. One Kramers pair of MFs ${\gamma }_{1,2}$ [${\gamma }_{3,4}$] (purple) is localized at the $x=0$ [$x=L$] end of the TB and can be manipulated by tuning the length L of the TB. The spectrum of two pairs of TI edge modes is considered in (b) for the first setup and in (c) for the second setup. (b) Edge modes of the same TI are coupled by proximity-induced pairing amplitudes ${\mathrm{\Delta }}_1<0$ and ${\mathrm{\Delta }}_2>0$, respectively The chemical potentials are opposite for the two TIs, ${\mu }_1=-{\mu }_2$. The helicities of the edge states are opposite (indicated by the coloring in red and blue). The tunneling ($t$) couples a right-moving state in the first TI to a left-moving state in the second TI, and vice versa. (c) The two TIs have the same chemical potential, ${\mu }_1={\mu }_2$, and the same helicities. The tunneling ($t$) couples a right-(left-) moving state in the first TI to a right-(left-) moving state in the second TI.