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 paper, by calculating the single-particle spectrum of $\mathrm{Bi}/{\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_x$ 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.

• Received 17 August 2022
• Revised 15 November 2022
• Accepted 22 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.245303