  Using scanning tunnel microscopy (STM), we investigate the superconductivity
and vortex properties in topological insulator Bi$_{2}$Te$_{3}$ thin films
grown on the iron-based superconductor FeTe$_{0.55}$Se$_{0.45}$. The
proximity-induced superconductivity weakens in the Bi$_{2}$Te$_{3}$ film when
the thickness of the film increases. Unlike the elongated shape of vortex cores
observed in the Bi$_{2}$Te$_{3}$ film with 2-quintuple-layer (QL) thickness,
the isolated vortex cores exhibit a star shape with six rays in the 1-QL film,
and the rays are along the crystalline axes of the film. This is consistent
with the sixfold rotational symmetry of the film lattice, and the
proximity-induced superconductivity is still topologically trivial in the 1-QL
film. At a high magnetic field, when the direction between the two nearest
neighbored vortices deviates from that of any crystalline axes, two cores
connect each other by a pair of adjacent rays, forming a new type of electronic
structure of vortex cores. On the 3-QL film, the vortex cores elongate along
one of the crystalline axes of the Bi$_{2}$Te$_{3}$ film, similar to the
results obtained on 2-QL films. The elongated vortex cores indicate a twofold
symmetry of the superconducting gap induced by topological superconductivity
with odd parity. This observation confirms possible topological
superconductivity in heterostructures with a thickness of more than 2 QLs. Our
results provide rich information for the vortex cores and vortex-bound states
on the heterostructures consisting of the topological insulator and the
iron-based superconductor.
