Fuzzy Band Structure of Quantum Dots by Bloch State Expansion of Single-Electron Eigenstates

Fundamental and application-oriented QD research, in particular in view of light emission, has greatly benefited from the theoretical or computational analysis of QDs. Effective mass and theory, for example, describe quantized states starting from the bulk band structure, where well-established semiconductor characteristics, such as effective masses and band offsets, are considered known parameters. The resulting single-electron states are, by design, linear combinations of bulk Bloch states, and can be combined to describe excited states as interacting electron-hole pairs, in good agreement with experimental findings. Not unlike atomistic tight-binding or pseudopotential methods, however, effective-mass theory cannot be used to relax structure, analyze surface-related properties or predict band offsets within a heteronanostructure.<br/>Opposite from solid-state physics methods, density-functional theory (DFT) provides molecular orbitals calculated from an atomistic model structure that includes, by default, the QD surface. Using DFT, stable QD structures can be identified through energy minimization, different surface terminations can be implemented, and single-electron states are not predesigned as consisting of bulk Bloch states. As a result, DFT has been successfully used to identify trap states, determine ligand binding energies, or calibrate molecular dynamics simulations, even if the method has limitations to describe the energetics and spin properties of excited states. However, incorporating the surface in the analysis often leads to highest occupied and lowest unoccupied orbitals that seem dominated by surface contributions, while the delocalized states derived from the bulk Bloch states can be difficult, if not impossible, to identify. Bridging this gap between DFT and solid-state physics methods would create an entirely new perspective for the computational study of QDs, with ramifications well beyond the particular case of semiconductor nanocrystals.<br/>In this work, we show that projecting the single-electron eigenstates as provided by DFT on Bloch states provides a unique tool to classify orbitals as surface-induced or Bloch-state derived. After outlining the concepts, we show that through a so-called Bloch-state expansion (BSE), the QD eigenstates transform into a fuzzy band structure that overlaps with the semiconductor bulk bands calculated at the same level of theory. Using a wide range of QDs, we demonstrate that for states around the band-edges, BSE enables surface-localized and delocalized states to be distinguished, and the dominant symmetry of delocalized states to be identified. Next, we implement BSE to highlight the crucial role of surface reconstructions for eliminating mid-gap surface-induced states and obtain QDs featuring delocalized highest occupied and lowest unoccupied orbitals. Finally, we extend BSE to core/shell QDs, a step that creates the first pathway to predict band-offsets between two semiconductors within a heteronanostructure. These examples underscore that connecting DFT results with a solid-state physics approach through BSE, creates a powerful framework to assess the outcome of DFT calculations, and use the ab-initio character of DFT to predict QD properties that are otherwise taken as adjustable parameters.