Spherical topological insulator nanoparticles: Quantum size effects and optical transitions

We have investigated 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 versatile continuum-model description of the TI bulk band structure and the assumption of 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. As expected from spherical symmetry, quantized levels in TI nanoparticles can be labeled by quantum numbers j and m = - j, - j + 1, ..., j for total angular momentum and its projection on an arbitrary axis. The fact that TIs are narrow-gap materials, where the charge-carrier dynamics is described by a type of two-flavor Dirac model, requires j to assume half-integer values and also causes a doubling of energy-level degeneracy where two different classes of states are distinguished by being 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. All optical transitions satisfy the usual selection rule Delta m = 0, +/- 1. We treat intra- and interband optical transitions on the same footing and establish ways for observing unusual quantum-size effects in TI nanoparticles, including oscillatory dependencies of the band gap and of transition amplitudes on the nanoparticle radius. Our theory also provides a unified perspective on multiband models for charge carriers in semiconductors and Dirac fermions from elementary-particle physics.