Spherical topological-insulator nanoparticles: Quantum size effects and optical transitions

We investigate the interplay between band inversion and size quantization in spherically shaped nanoparticles made from topological-insulator (TI) materials. A general theoretical framework is developed based on a continuum-model description of the TI bulk band structure subjected to a hard-wall mass confinement. Analytical results are obtained for the wave functions of single-electron energy eigenstates and the matrix elements for optical transitions between them. Quantized levels in TI nanoparticles can be labeled by angular momentum quantum numbers j and m = −j, −j+1, ... , j. Additionally TIs possess a doubling of energy-level degeneracy due to different parity eigenstates with eigenvalues (−1)^j±1/2. The existence of energy eigenstates having the same j but opposite parity enables optical transitions where j is conserved, in addition to those adhering to the familiar selection rule where j changes by ±1. We treat intra- and inter-band optical transitions on the same footing and establish ways for observing unusual quantum-size effects in TI nanoparticles. Our theory also provides a unified perspective on multi-band models for charge carriers in semiconductors and Dirac fermions from elementary-particle physics.