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, . . . , 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 ∆m = 0, ±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.

Optical response of a topological-insulator–quantum-dot hybrid interacting with a probe electric field

“…While it is difficult to perform this kind of experiments with topological insulators nowadays, we envision that a similar experimental setup could be engineered to test our results with a spherical topological insulator nanoparticles (as those predicted in Ref. [71]) and CdSe quantum dots. It is worth mentioning that our results can also be tested with linear magnetoelectrics such as Cr 2 O 3 [72] or in some multiferroics [73].…”

Surface‐Mediated Interconnections of Nanoparticles in Cellulosic Fibrous Materials toward 3D Sensors

“…While the quantum size effect is known for atomic layer‐by‐layer metal thin‐film growth, [ 24 ] we ruled out its operation to account for the electrical oscillation phenomenon because of the wet state in our sintering process. Interestingly, for NPs and assemblies in solutions, the electrical oscillation is reminiscent of quantized capacitive charging (QCC).…”

Tailoring topological states of core–shell nanoparticles

“…In recent years, nanoparticles based on TIs have been thoroughly investigated for their interest in photonics, optically controlled quantum memory and quantum computing [8][9][10][11][12][13][14]. Surface states * Corresponding author show little sensitivity to disorder, which is beneficial for optical applications at the nanoscale [10].…”