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

## TL;DR

This paper develops a theoretical framework to understand quantum size effects and optical transitions in spherical topological-insulator nanoparticles, revealing unique energy level degeneracies and selection rules.

## Contribution

It introduces a continuum-model approach for TI nanoparticles, deriving analytical wave functions and optical transition matrix elements, highlighting novel quantum-size effects and parity-based transitions.

## Key findings

- Energy levels labeled by half-integer j with parity distinctions.
- Optical transitions include both standard and parity-enabled processes.
- Band gap and transition amplitudes oscillate with nanoparticle radius.

## Abstract

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 $m=-j, -j+1, \dots, j$ 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 $(-1)^{j\mp 1/2}$. 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 $\pm 1$. All optical transitions satisfy the usual selection rule $\Delta m = 0, \pm 1$. We treat intra- and inter-band optical transitions on the same footing and establish ways for observing unusual quantum-size effects in TI nanoparticles, including oscillatory dependences of the band gap and of transition amplitudes on the nanoparticle radius. Our theory also provides a unified perspective on multi-band models for charge carriers in semiconductors and Dirac fermions from elementary-particle physics.