Quantum Dots as Chemical Building Blocks: Elementary Theoretical Considerations

Abstract

The link between chemical orbitals described by local degrees of freedom and band theory, which is defined in momentum space, was proposed by Zak several decades ago for spinless systems with and without time-reversal in his theory of "elementary" band representations. In a recent paper [Bradlyn et al., Nature 547, 298-305 (2017)], we introduced the generalization of this theory to the experimentally relevant situation of spin-orbit coupled systems with time-reversal symmetry and proved that all bands that do not transform as band representations are topological. Here, we give the full details of this construction. We prove that elementary band representations are either connected as bands in the Brillouin zone and are described by localized Wannier orbitals respecting the symmetries of the lattice (including time-reversal when applicable), or, if disconnected, describe topological insulators. We then show how to generate a band representation from a particular Wyckoff position and determine which Wyckoff positions generate elementary band representations for all space groups. This theory applies to spinful and spinless systems, in all dimensions, with and without time reversal. We introduce a homotopic notion of equivalence and show that it results in a finer classification of topological phases than approaches based only on the symmetry of wavefunctions at special points in the Brillouin zone. Utilizing a mapping of the band connectivity into a graph theory problem, we show in companion papers which Wyckoff positions can generate disconnected elementary band representations, furnishing a natural avenue for a systematic materials search.

Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing

“…It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands. 5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics, 6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays. 7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array.…”

Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing

“…It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands. 5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics, 6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays. 7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array.…”

On the application potential of gold nanoparticles in nanoelectronics and biomedicine

“…Furthermore, low-temperature STS investigations gave clear evidence of discrete energy levels (Zhang et al 2003). Generally, the level spacing, as well as the position of the Coulomb gap in small metal clusters, is also influenced by the size, shape and the surrounding medium, including the ligand shell and the interparticle distances (Remacle & Levine 2001).…”

Effects of Disorder on Electronic Properties of Nanocrystal Assemblies

“…32 Remacle et al studied the effects of charging energy and disorder on charge transport in metal NC solids. 33 Artemyev et al. studied the existence of delocalized electron states in CdSe NC ensembles with different sizes and packing ratios.…”

Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing

“…It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands. 5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics, 6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays. 7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array.…”

On the application potential of gold nanoparticles in nanoelectronics and biomedicine

“…Furthermore, low-temperature STS investigations gave clear evidence of discrete energy levels (Zhang et al 2003). Generally, the level spacing, as well as the position of the Coulomb gap in small metal clusters, is also influenced by the size, shape and the surrounding medium, including the ligand shell and the interparticle distances (Remacle & Levine 2001).…”

Effects of Disorder on Electronic Properties of Nanocrystal Assemblies

“…32 Remacle et al studied the effects of charging energy and disorder on charge transport in metal NC solids. 33 Artemyev et al. studied the existence of delocalized electron states in CdSe NC ensembles with different sizes and packing ratios.…”

Measuring Ligand-Dependent Transport in Nanopatterned PbS Colloidal Quantum Dot Arrays Using Charge Sensing

“…It has recently been shown that, in complement to the tunability achieved by a modification of the dot size, the electronic properties of coupled QD arrays could also be tuned with the surface ligands. 5 There has also been considerable interest in using dots as building blocks for artificial solids to gain insight into correlated electron physics, 6 and interesting theoretical predictions exist for the observation of charge ordering and other novel aspects of charge transport in dot arrays. 7−9 These effects are difficult to observe experimentally because of the finite size distribution of the dots and packing disorder in the array.…”

On the application potential of gold nanoparticles in nanoelectronics and biomedicine

“…Furthermore, low-temperature STS investigations gave clear evidence of discrete energy levels (Zhang et al 2003). Generally, the level spacing, as well as the position of the Coulomb gap in small metal clusters, is also influenced by the size, shape and the surrounding medium, including the ligand shell and the interparticle distances (Remacle & Levine 2001).…”

Effects of Disorder on Electronic Properties of Nanocrystal Assemblies

“…32 Remacle et al studied the effects of charging energy and disorder on charge transport in metal NC solids. 33 Artemyev et al. studied the existence of delocalized electron states in CdSe NC ensembles with different sizes and packing ratios.…”