We numerically investigate the spin-orbit (SO) coupling effect on the magnetic flux evolution of energy and supercurrent in mesoscopic $d$-wave superconducting loops by solving the spin-generalized Bogoliubov–de Gennes equations self-consistently. It is found that the energy spectrum splits when the SO interaction is involved and the Majorana zero mode can be realized in the [100] edges of square systems for an appropriate SO coupling strength. Superconducting phase transitions appear when the energy gap closes, accompanied by energy jumps between different energy parabolas in the ground state, which provides a possible mechanism to support fractional flux periodicity of supercurrent. Moreover, in the case of rectangular loops with SO coupling, the jumps of the ground-state energy gradually disappear by increasing the ratio of length to height of the sample, and a paramagnetic response with opposite direction of the screening current around zero flux value can occur in such systems.

• Received 27 May 2014
• Revised 8 July 2014

DOI:https://doi.org/10.1103/PhysRevB.90.014522