### Abstract:

Topological insulators, a new class of materials discovered recently, have very
unique topological properties. While the surface states are gapless and are
topologically protected by the time reversal symmetry, the bulk of topological
insulator shows a full gap. We study the properties of reflected and transmitted
electron waves on the surface of topological insulator nanofilm with a defect. We
show that an electron incident on a defect produces modes, which are localized at the
defect. Such modes result in an enhancement of electron density at the defect. If an
electron with a given energy E is incident on a region of the TI nanofilm which does
not support electron propagating waves at energy E, then there is a strong
enhancement of local electron density near the boundary of such region. We calculate
the reflectance and transmittance of an electron from such a defect for different
parameters of the topological insulator nanofilm and different parameters of the
defect. The amplitude of the electron wave strongly depends on the position of
electron energy. If electron energy is in the energy gap of defect, then the reflectance
is 1. This result corresponds to total internal reflection. If the electron energy is not
in the energy gap of defect, then the reflectance is small and the electron is almost
totally transmitted. The critical points dividing these two boundaries are defined by
the condition that the electron energy is at the band edge of energy dispersion. The
dependence of the electron wave on the parameters of the system, such as thickness
of the nanofilm or electron energy, is dominated near the critical points.