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We study two microscopic models of topological insulators in contact with an$s$-wave superconductor. In the first model the superconductor and thetopological insulator are tunnel coupled via a layer of scalar and of randomlyoriented spin impurities. Here, we require that spin-flip tunneling dominatesover spin-conserving one. In the second model the tunnel coupling is realizedby an array of single-level quantum dots with randomly oriented spins. It isshown that the tunnel region forms a $\pi$-junction where the effective orderparameter changes sign. Interestingly, due to the random spin orientation theeffective descriptions of both models exhibit time-reversal symmetry. We thendiscuss how the proposed $\pi$-junctions support topological superconductivitywithout magnetic fields and can be used to generate and manipulate Kramerspairs of Majorana fermions by gates.

Proximity-induced Josephson $\pi$-Junctions in Topological Insulators