The Majorana zero modes and persistent spin current in mesoscopic $d$-wave-superconducting loops with spin-orbit (SO) interaction are investigated by numerically solving the spin-generalized Bogoliubov–de Gennes equations self-consistently. For some appropriate strength of the SO coupling, Majorana zero-energy states and sharp jumps of the spin-polarized currents can be observed when the highest energy levels cross the Fermi energy in the spectrum, leading to spin currents with opposite chirality flowing near the inner and outer edges of the sample. When the threaded magnetic flux turns on, four flux-dependent patterns of the persistent spin current with step-like features show up, accompanied by Majorana edge modes at flux values where the energy gap closes. Moreover, the Majorana zero mode is highly influenced by the direction of the Zeeman field. A finite in-plane field can lead to the gap opening since the inversion symmetry is broken. Remarkably, multiple Majorana zero-energy states occur in the presence of an out-of-plane field ${\mathbf{h}}_z$, and the number of steps in the spin current evolution can be effectively tuned by the field strength due to the shift of Majorana zero modes. Finally, when the loop sample contains surface indentation defects, zero-energy modes can always show up in the presence of an appropriate ${\mathbf{h}}_z$. Interestingly, multiple Majorana states may be present in the system with a corner defect even if ${\mathbf{h}}_z=0$.

• Received 31 July 2015

DOI:https://doi.org/10.1103/PhysRevB.92.094516