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We consider the bending effect on the formation of Majorana bound states (MBSs) in planar Josephson junctions where the normal stripe is tilted in a V shape. Our results show that the MBSs remain robust for moderate bending angles. Beyond some critical angles, the degradation of MBSs can be revealed by its eigenspectrum as well as the Majorana polarization (MP). Our results show that the parameter space of bending angle for robust MBSs can be significantly enlarged by tuning the superconducting phase difference across the Josephson junction. These findings suggest that the interplay of the junction geometry and the device parameters provides richer degree of freedom in designing topological superconducting devices for future applications. The MP analysis is an indispensable tool for characterizing the Majorana states.

Scanning tunneling microscopy (STM) is an indispensable tool in detecting Majorana bound states (MBSs) in vortices of topological superconductors. By reducing the computational complexity via non-uniform grids, we systematically study the tunnel coupling as well as the temperature dependence of the differential conductance of MBSs in two dimensional devices. Numerical results show that the conductance peak approaches the quantized value 2e 2/h in strong coupling limit at low temperatures which are characteristic features of MBSs. More interestingly, a conductance local minimum in the spatially scanning is observed when the STM tip is placed at the vortex center. The dip structure can be enhanced with increased temperature or enlarged vortex size. We ascribe this observation to the sensitivity of the Andreev reflection processes of carriers at the vortex center where the thermal energy could be comparable to the vanishing pair potential. We also investigate the STM of two-vortex systems where the hybridization of the vortices can lead to oscillatory behavior of the state energy. With small inter-vortex distances, the original MBSs in vortices can merge into topologically trivial states and the conductance peak can be significantly suppressed.