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Recent discoveries have spurred the theoretical prediction and experimentalrealization of novel materials that have topological properties arising fromband inversion. Such topological insulators are insulating in the bulk but haveconductive surface or edge states. Topological materials show various unusualphysical properties and are surmised to enable the creation of exoticMajorana-fermion quasiparticles. How the signatures of topological behaviorevolve when the system size is reduced is interesting from both a fundamentaland an application-oriented point of view, as such understanding may form thebasis for tailoring systems to be in specific topological phases. This workconsiders the specific case of quantum-well confinement definingtwo-dimensional layers. Based on the effective-Hamiltonian description of bulktopological insulators, and using a harmonic-oscillator potential as an examplefor a softer-than-hard-wall confinement, we have studied the interplay of bandinversion and size quantization. Our model system provides a useful platformfor systematic study of the transition between the normal and topologicalphases, including the development of band inversion and the formation ofmassless-Dirac-fermion surface states. The effects of bare size quantization,two-dimensional-subband mixing, and electron-hole asymmetry are disentangledand their respective physical consequences elucidated.

Manipulating topological-insulator properties using quantum confinement