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

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.

arXiv comment13 pages, 4 figures, RevTex4.2, v2: extended discussion of optical transitions, including new Figs. 3 & 4, to appear in PRB

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• Finite-size effects in cylindrical topological insulators
• Optical intersubband properties of a core-shell semiconductor-topological insulator quantum dot described by $\theta$-Electrodynamics
• Field induced charge symmetry in topological insulator Bi${}_2$Te${}_3$ revealed by nuclear magnetic resonance
• Effects of first- and second-order topological phases on equilibrium crystal shapes
• Hard wall edge confinement in 2D topological insulators and the energy of the Dirac Point
• Equatorial magnetoplasma waves
• Tailoring topological states of core-shell nanoparticles