Spin Transport Properties Of Superconducting Junctions In Arbitrary Magnetic Configurations Based On Narrow Two-dimensional Topological Insulators

In recent years, the spin-polarized transport properties of superconducting junctions based on topological insulators (TIs) have attracted much attention. TIs have a bulk band gap and metallic surface states due to the time-reversal symmetry. When superconductor (SC) and ferromagnetic insulator (FI) touch on the surface of a TI, the proximity effect leads to superconductivity and ferromagnetism on the surface, which can change the electronic transport properties. Particularly, for a narrow two-dimensional (2D) TI, the coupling between the states of the two edges emerges,thus could feature exotic transport phenomena. Applying the method based on an extended Bogoliubov-de Gennes (BdG) equation, we study the spin-polarized transport properties in the ferromagnetic superconducting junction composed of FI and SC on a narrow 2D TI in arbitrary magnetic configurations.Firstly, we investigate spin-polarized transport features in the FI/SC/FI topological superconducting junction on a narrow 2D TI in arbitrary magnetic configurations, which can be used to create spin entangled electron pairs. It is shown that due to the noncollinear magnetizations inducing spin mixing, there exists not only normal nonlocal Andreev reflection (AR) but also novel nonlocal AR. For the former,the incident electron and the nonlocal Andreev-reflected holes come from the different spin subbands,forming spin-singlet entangled electron pairs. For the latter, the incident electron and the nonlocal Andreev-reflected holes belong to the same spin subband, leading spin-triplet entangled electron pairs. All tunneling processes are suppressed for incident spin-down electrons during the low energy scope. For incident spin-up electrons, at some energy values, with angle a between the directions of the two magnetic exchange fields increased, the probability of novel nonlocal AR firstly increases, then decreases, exhibiting a peak at ? = ?/2. However, for the probability of normal nonlocal AR, the situation is just contrary. And there is a valley at ? = ?/2.For any arbitrary magnetic configurations, the sum of probabilities of normal and novel nonlocal ARs can be approximate to 100%, indicating pure spin entangled electron pairs with high efficiency of producing rate. Moreover, we can get various ratio of pure spin-singlet and spin-triplet entangled electron pairs by manipulating angle a. Particularly, in magnetic configuration with ? =?/2, the probabilities of two kinds of nonlocal ARs at some energies can be as high as 50%. Besides, only during the low energy scope, has the angle a an obvious influence on the differential conductance and noise power, which also reflects the properties of tunneling processes.Secondly, we study spin-polarized transport properties in the FI/FI/SC topological superconducting junction on a narrow 2D TI in arbitrary magnetic configurations. The results show that tunneling processes can emerge similarly only for incident spin-up electrons during low energy scope. There exists the novel local AR besides the normal local AR similarly due to the spin mixing induced by nonlinear magnetizations. The incident electrons and the two kinds of local Andreev-reflected holes come from different and the same spin subbands, so the spin-singlet and spin-triplet pairing states are respectively formed. Interestingly, the properties of probabilities of novel local AR are similar to that of novel nonlocal AR in the FI/SC/FI junction mentioned above, exhibiting the peak at ? = ?/2. Particularly, in magnetic configuration with ? = ?/2, the probabilities of novel local AR at E = ?and normal local AR at E = 0.4? can approximate to 100%. This means that the junction can be used to create pure spin-singlet and spin-triplet pairing states with high efficiency. Similarly, the influence of angle a on differential conductances and noise power also manifests the properties of the tunneling processes.Experimentally, the devices corresponding to the two kinds of heterojunctions mentioned above may be realized with current technologies, one for producing the spin-singlet and -triplet entangled electron pairs and the other for creating spin-singlet and -triplet spin currents. These devices have import practical use in topological superconducting spintronics and quantum information.