Majorana corner modes in mesoscopic superconducting square systems with mixed pairing in the presence of spin–orbit interaction

In the framework of the Bogoliubov-de Gennes theory, we investigate the Majorana zero-energy states in mesoscopic superconducting square systems with spin–orbit (SO) interaction. The mixed d-wave and extended s-wave condensates as well as the favored $$s+id$$ s + i d state can be obtained by suitable choice of model parameters. We find that the energy gap is highly sensitive to the introduced SO interaction and the Majorana zero modes can emerge at four outer corners of a perfect square with only the Rashba SO interaction or with combined Rashba and Dresselhaus SO couplings. Furthermore, for a square sample with a centered hole, energy levels can cross the Fermi energy at appropriate Rashba SO-coupling strengths, accompanied with the occurrence of additional four Majorana corner states localized around the inner corners of the loop. The effect of Dresselhaus SO interaction on the number and location of Majorana corner states is also examined, and novel zero-energy modes mainly located at four opposite inner and outer corners along the (anti-)diagonal direction of the square loop can be realized. Graphical abstract