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Based on the Green's function method, we investigate the interplay between Majorana zero mode (MZM) and Andreev bound states (ABSs) in a quantum dot molecule side coupled to a topological superconducting nanowire with a pair of MZMs forming a Josephson junction. Since the strong electron–hole asymmetry induced by the nanowire with a topologically non-trivial phase, the MZM suppress the ABSs. The suppression induced by the MZM is robust against the Coulomb repulsion. The interplay between the MZM and the ABSs in Josephson junction presents a feasible experimental means for distinguish between the presence of MZM and ABSs.

Using the time-dependent Schrodinger equation, we present the analytical result of the expectation value of spin injected into a two-dimensional electron gas with respect to an arbitrarily spin-polarized electron state and monitor the spin time-evolution. We demonstrate that the expectation value of spin operator S x is the time-independent, and only the expectation values in the S y – S z plane are time-dependent. A detailed study of spin precession in the spin-valve and spin-transistor geometry is presented, in which the initial spin-polarized electron state point perpendicular and parallel to the current direction, respectively. We put forward the possible reason that the resistance change is independent of gate voltage in the spin-valve geometry. Furthermore, it has been shown that the effective magnetic field generated by the spin–orbit interaction is not same with the truly magnetic field. The main effect of the truly magnetic field is to align the spin along the field direction, but the effective magnetic field generated by the spin–orbit interaction does not.

A study on characteristics of electrons tunneling through semiconductor barrier is evaluated, in which we take into account the effects of Rashba spin–orbit interaction. Our numerical results show that Rashba spin–orbit effect originating from the inversion asymmetry can give rise to the spin polarization. The spin polarization does not increase linearly but shows obvious resonant features as the strength of Rashba spin–orbit coupling increases, and the amplitudes of spin polarization can reach the highest around the first resonant energy level. Furthermore, it is found that electrons with different spin orientations will spend quite different time through the same heterostructures. The difference of the dwell time between spin-up and spin-down electrons arise from the Rashba spin–orbit coupling. And it is also found that the dwell time will reach its maximum at the first resonant energy level. It can be concluded that, in the time domain, the tunneling processes of the spin-up and spin-down electrons can be separated by modulating the strength of Rashba spin–orbit coupling. Study results indicate that Rashba spin–orbit effect can cause a nature spin filter mechanism in the time domain.