We study quasiparticle states on a surface of a topological insulator (TI)
with proximity-induced superconductivity under an external magnetic field. An
applied magnetic field creates two Majorana bound states: a vortex Majorana
state localized inside a vortex core and an exterior Majorana state localized
along a circle centered at the vortex core. We calculate the spin-resolved
local density of states (LDOS) and demonstrate that the shrinking of the radius
of the exterior Majorana state, predicted in Ref. [R. S. Akzyanov et al., Phys.
Rev. B 94, 125428 (2016)], under a strong magnetic field can be seen in LDOS
without smeared out by non-zero-energy states. The spin-resolved LDOS further
reveals that the spin of the exterior Majorana state is strongly polarized.
Accordingly, the induced odd-frequency spin-triplet pairs are found to be
spin-polarized as well. In order to detect the exterior Majorana states,
however, the Fermi energy should be closed to the Dirac point to avoid
contributions from continuum levels. We also study a different two-dimensional
topological-superconducting system where a two-dimensional electron gas with
the spin-orbit coupling is sandwiched between an s-wave superconductor and a
ferromagnetic insulator. We show that the radius of an exterior Majorana state
can be tuned by an applied magnetic field. However, on the contrary to the
results at a TI surface, neither the exterior Majorana state nor the induced
odd-frequency spin-triplet pairs are spin-polarized. We conclude that the
spin-polarization of the Majorana state is attributed to the spin-polarized
Landau level which is characteristic for systems with the Dirac-like
dispersion.Comment: 16 pages, 13 figure
