  The search for the Majorana fermions in condensed matter physics has
attracted much attention, partially because they may be used for the
fault-tolerant quantum computation. It has been predicted that the Majorana
zero mode may exist in the vortex core of topological superconductors.
Recently, many iron-based superconductors are claimed to exhibit a
topologically nontrivial surface state, including Fe(Te,Se). Some previous
experiments through scanning tunneling microscopy (STM) have found zero-bias
conductance peaks (ZBCP) within the vortex cores of Fe(Te,Se). However, our
early experimental results have revealed the Caroli-de Gennes-Matricon (CdGM)
discrete quantum levels in about 20% vortices. In many other vortices, we
observed a dominant peak locating near zero bias. Here we show further study on
the vortex core state of many more vortices in FeTe$_{0.55}$Se$_{0.45}$. In
some vortices, if we take a certain criterion of bias voltage window near zero
energy, we indeed see a zero energy mode. Some vortices exhibit zero energy
bound state peaks with relatively symmetric background, which cannot be
interpreted as the CdGM discrete states. The probability for vortices showing
the ZBCP lowers down with the increase of magnetic field. Meanwhile it seems
that the presence and absence of the ZBCP has no clear relationship with the
Te/Se ratio on the surface. Temperature dependence of the spectra reveals that
the ZBCP becomes weakened with increasing temperature and disappears at about 4
K. Our results provide a confirmed supplementary to the early claimed zero
energy modes within the vortex cores of Fe(Te,Se). Detailed characterization of
these zero energy modes versus magnetic field, temperature and spatial
distribution of Te/Se will help to clarify its origin.
