  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.
