Subtlety of modes trapped by vortices in a topological superconducting heterostructure

In a topological superconducting heterostructure comprising an $s$-wave superconductor and a semiconductor with Rashba spin-orbit coupling, several distinct modes emerge when an external magnetic field is applied: Majorana zero-energy modes, trivial Caroli--de Gennes--Matricon modes, and edge modes. We find three subtle properties of the modes under the fine tuning of the external magnetic field and Rashba spin-orbit coupling. (1) The spatial configuration of Majorana zero-energy modes undergoes a dramatic change when flipping the direction of magnetic field. (2) In the topological nontrivial regime, the Caroli--de Gennes--Matricon modes and edge modes exhibit spatial oscillatory behavior across the entire sample scale and approach zero energy as the strength of the Rashba spin-orbit coupling decreases. (3) In the topological trivial regime, another type of zero-energy mode may also manifest at the vortex center. The key distinction from the Majorana zero-energy mode is its lack of spatial coherence and no coexistence of edge modes. These properties highlight the modes' sensitivity to the tuning parameters and impose constraints on the experimental determination of Majorana zero-energy modes.