Understanding the Structural Origin of Chirality in Magic-Size Semiconductor Nanoclusters through Self-Assembly Simulations

TL;DR

The paper addresses the unclear origin of chirality in magic-size semiconductor clusters (MSCs) and proposes a generic self-assembly framework using oscillatory pair potentials (OPPs) to model binary $A$–$B$ interactions. The simulations reproduce a transition from bulk-like zincblende order at large sizes to distorted icosahedral motifs at smaller sizes, with intermediate sizes showing coexisting structures and racemic chirality in 1:1 stoichiometries. A shared distorted $X_{14}Y_{13}$ core is observed in both simulated and several experimentally resolved chiral MSCs, and chirality arises from geometric frustration and symmetry reduction as tetrahedral units assemble within an icosahedral framework, without system-specific parameterization. The findings provide a unifying, parameter-free mechanism for MSC chirality and yield design principles for predicting new cluster geometries, with implications for extending to bulk-like zincblende nanoclusters in the II–VI and III–V families.

Abstract

Semiconductor magic-size clusters (MSCs) are atomically precise nanoparticles exhibiting unique size-dependent properties, but their ultrasmall dimensions hinder structural characterization, limiting our understanding of their formation and stability. A few MSC structures have been fully resolved, revealing either bulk-like zincblende-type structures or a range of non-bulk-like motifs. Here we use a computational model to investigate the relationship between cluster size and atomic structure in zincblende-forming II-VI and III-V semiconductors. Firstly, we find that all non-bulk-like MSCs in these systems exhibit the same distorted icosahedral motif that is intrinsically chiral. Secondly, we reproduce these MSC geometries in small-cluster self-assembly simulations and discover that their chirality emerges from the geometric frustration and symmetry breaking in arranging tetrahedral bonding environments into an icosahedral topology. Overall, this work reproduces experimentally reported motifs without system-specific parameterization, establishes the structural origin of chirality in MSCs, and provides design principles for predicting new cluster geometries.

Figures (5)

• Figure 1: Computational model and structural analysis of self-assembled, stoichiometric bulk and cluster configurations. (a) Isotropic pair potentials for particle interactions in the binary system: $A$–$A$ (blue), $B$–$B$ (red), and $A$–$B$ (black). (b) Unit cell of the zincblende-type structure, with particle types $A$ and $B$ shown in blue and red, respectively. The zincblende-type structure has (c) coordination number (CN) = 4, (d) bond angles 109.47°, and torsion angles (e) ±60° and (f) 180°. Bulk simulations (with $N = 8000$ particles) confirm the formation of a zincblende-type structure: (g) the CN distribution peaks at four, (h) the bond angle distribution is centered around 109.47°, and (i) torsion angles are concentrated at ±60° and 180°. For clusters with $N=100$ to 8000 particles, (j) the fraction of CN = 4 sites and (k) the average bond angle converge toward the ideal zincblende values as cluster size increases. (l) Torsion angle distributions evolve from non-bulk-like to bulk-like patterns (with peaks at ±60° and 180°), which is reflected in (m) the structure metric for each replica (orange dots) shifting from values below 0.35 to above 0.85. (n) Representative structures at different cluster sizes demonstrate that: small systems ($N = 100$) exhibit only non-bulk-like structures, intermediate-size systems ($N = 729$) can form either zincblende-type structure or non-bulk-like structures, and large systems ($N = 8000$) adopt a bulk-like zincblende-type structure. (Particles in zincblende-type structures are colored according to the "Identify diamond structure" order parameter with OVITO.maras_global_2016)
• Figure 2: Torsion angle distributions and representative structures for clusters at 1:1 stoichiometry. (a) Torsion angle distribution for cluster sizes $N = 60$--100, averaged over 30 replicas each. (b) Torsion angle distribution of 30 replicas at $N = 60$, grouped by chirality and normalized by number of replicas (18 replicas in gray, 12 in yellow). Peaks at $-79$° / $+36$° / $+152$° and $-152$° / $-36$° / $+79$° indicate enantiomers of a chiral species. (c) Representative structures at $N = 60$ showing opposite handedness (opaque particles denote configurations of opposite chirality).
• Figure 3: Shared distorted icosahedral motif across experimentally observed chiral MSCs and simulated cluster analogs, and mechanism of chirality emergence. (a) Timeline of the discovery of chiral semiconductor MSCs, showing the core structures of representative compositions. For each cluster type, both chiral enantiomers are displayed side by side. Shown, chronologically from the earliest to most recent reports, are: Zn$_{14}$S$_{13}$khadka_zinc_2012, In$_{37}$P$_{20}$gary_single-crystal_2016, Cd$_{14}$Se$_{13}$bootharaju_structure_2022, Cd$_{26}$Se$_{17}$ma_precision_2023, In$_{26}$P$_{13}$sandeno_ligand_2024, In$_{26}$As$_{18}$sandeno_synthesis_2024, and Cd$_{28}$S$_{17}$xu_chiral_2025. (b--c) Comparison of simulated nanoclusters with experimentally observed chiral MSCs: (b) simulated $B_{37}A_{20}$ and the $B_{37}A_{20}$ core in a $B_{42}A_{26}$ cluster (with shell particles shown translucently), resembling the In$_{37}$P$_{20}$ shown in (a), and (c) simulated $B_{26}A_{13}$ cluster, sharing the same topology as the In$_{26}$P$_{13}$ shown in (a). (d) Shared structural motif across both experimental and simulated chiral clusters: a distorted icosahedral motif composed of 14 $X$ particles and 13 $Y$ particles. Atoms are colored the same as in (a) and are shown in the same chronological order from the earliest to the most recent reports.
• Figure 4: Proposed mechanism for the emergence of chirality in the distorted icosahedral motifs. (a) A regular $X_{4}Y$ tetrahedron can represent the local environment in zincblende- and wurtzite-type structures, but (b) 20 such tetrahedra cannot tile a regular icosahedron without leaving angular gaps of $\sim$2.9°. (c--d) A regular icosahedron can be constructed using 20 slightly distorted tetrahedra, each sharing a common vertex at the icosahedron's center (shown in red). (e--f) Selecting one centroid (cyan) leaves only six of the remaining 19 centroids with which to form ideal tetrahedral bond angles (109.47°). (g) These six centroids can be divided into two sets of three, each forming a tetrahedron with the selected centroid. (h) These two chiral pathways for the cluster construction result in distorted icosahedral motifs with opposite handedness (i.e., related to each other by mirror symmetry).
• Figure 5: Average bond angles (in degrees) (left), average coordination numbers (middle), and frequency of optimal coordination CN $= 4$ (right) across different stoichiometries and system sizes (stoichiometries are varied in increments of $\Delta N = 1$ between compositions 55:55 and 13:55 / 55:13): (a--c) system sizes $N = 26$--110 (10 replica simulations per pixel; black square outlines in each panel highlight the system size range from $N = 60$--100.); (d--f) system sizes $N = 60$--100 (30 replica simulations per pixel; compositions between 50:50 and 30:50 / 50:30).