  The quantum size effect has a significant impact on electrons, such that it
can even change their topologically protected properties. An example of this
phenomenon is the gap opening in the topologically protected gapless surface
state in finite-thickness topological-insulator films. However, much is not
known about the quantum size effect in topological heterojunctions. In this
study, by calculating the single-particle spectrum of Bi/Bi$_{1-x}$Sb$_{x}$
based on the well-known Liu-Allen model, we found that the strong quantum size
effect in topological heterojunctions yields an unexpected band alignment. The
wave functions permeate each other through the attached materials, and this
occurs even in 80-nm-thick heterojunctions. Furthermore, we theoretically found
that one of the two major spectra obtained from the Bi surface does not
represent the true surface state of Bi. This finding may overturn the previous
understanding of the topological surface state of Bi.
