Geometric effect on Majorana corner modes in mesoscopic superconducting loop systems with Rashba spin-orbit interaction

Based on the Bogoliubov-de Gennes theory, the energy spectra in the gap region are investigated for mesoscopic superconducting asymmetric loops with Rashba spin-orbit interaction. Compared with the symmetric square-loop system, the feature of Majorana zero-energy states bounded in the vicinity of loop corners is sensitive to the geometric effect due to the reconstruction of energy spectra of the condensate. The influences of the rectangular aspect-radio, the off-centered hole, and the surface defect are mainly considered. The evolution of energy levels closest to the Fermi energy as a function of the spin-orbit coupling strength $$V_{\text {Rso}}$$ V Rso as well as the zero-energy local density of states at the gapless points are given. The level crossing does not appear in the weak $$V_{\text {Rso}}$$ V Rso regime for the rectangular loop, and a displacement of the centered hole of a square loop can lead to a deviation from the symmetric location of Majorana corner modes. In particular, the number of zero-energy eigenvalues is disturbed by the surface defects, and novel spatial patterns of Majorana corner states can be obtained. Graphic abstract