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The zigzag graphene nanoribbon (ZR) is characterized by the distinct pseudoparity combined with valley-selection rule, which could feature exotic transport phenomena, especially in ZR-based superconducting spintronic devices. However, the ZR with superconductivity induced by proximity of a bulk superconductor (SC) on it still keeps original band properties. Herein, we present a superconducting heterostructure with an SC directly coupling to two ZRs, which is characteristic of pseudoparity-mixing, resulting in pseudoparity nonconservation elastic cotunneling (EC) and crossed Andreev reflection (CAR) processes. It is shown that the mixing leads to the switch effect of the EC and CAR processes manipulated by the SC length, particularlly the full spin polarization. In the context of only one magnetized ZR lead, a novel bipolar spin diode behavior on a scale of small SC length and unipolar spin entanglement pairing at some large SC lengths, are both exhibited on a large scale of forward and/or reverse bias voltages. More importantly, the spin-diode can be combined with the quantum spin Hall (QSH) insulator to provide smoking gun evidence for the helical spin texture of the (QSH) insulator, which is still lacking.

We present an anomalous Josephson effect in a TI-based Josephson junction with a ferromagnetic insulator (FI) trilayer which has noncoplanar magnetizations. It is shown that there exist equal spin-singlet and -triplet correlations due to the magnetism-tuning chiral Dirac energy band structure combined with the spin precession and selective equal spin Andreev reflection by chiral Majorana modes (CMMs). The consequent anomalous Josephson supercurrent is exhibited, in which a 0-π or similar 0 - π state transition through phase shift is induced only by exchange field strengths of the first FI region, while the ϕ0 supercurrent and the maximum one gradually drop with the increase of exchange field strengths of the second and third FI regions without relative state transitions. The much different features are found by varying the lengths of trilayer. The Andreev bound states without hybridization for the CMM administrate these features, which could be used to probe and confirm the zero energy CMM. In addition, the corresponding free energies are presented and discussed.

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