Quantum coupling in colloidal homodimers of epitaxially attached CdSe@CdS dot@platelets probed on single-particle level

Abstract

Coupled quantum-dot (QD) dimers can be viewed as diatomic artificial molecules, thus offering a unique platform to study molecular physics and quantum phenomena. However, controlled synthesis of QD dimers has remained an outstanding experimental challenge, given necessary construction of hundreds of lattice bonds at the neck between two QDs with a unique attachment axis. Here, QD homodimers are synthesized in solution with a high yield by the epitaxial attachment of two monodisperse CdSe@CdS dot@platelet QDs along their common <11$\bar{2}$0> axis. The exact construction of hundreds of lattice bonds in the epitaxial neck are enabled by the large (11$\bar{2}$0) side facets with ideal lattice match between two epitaxial QDs and limited ligands passivation, resulting in controlled spacing between two core QDs (core-spacing). Single-particle spectroscopy reveals that the photoluminescence (PL) of single homodimer with a small core-spacing (7.8 nm) is split into two room-temperature resolvable peaks, consistent with strongly-split bonding and antibonding electron orbitals of “artificial diatomic molecules”. The two peaks of the split PL are orthogonally-polarized, and two types of molecular bi-excitons are observed for the homodimer. These advancements lay a viable foundation for controllable construction of “artificial molecules” in solution.

Introduction

Atom-like electronic states^1,2,3 make semiconductor nanocrystals with strong quantum confinement (quantum dots, QDs) known as “artificial atoms”, which are in rapid development as an integral part of a semiconductor platform for photon manipulation⁴. If two (or a few) QDs are quantum-coupled through single-crystalline neck(s) to form molecule-like electronic states, they should be named as “artificial molecules”^5,6,7. Similar to regular molecules, molecule-like properties of “artificial molecules” would be mostly observed at the band edges as bonding and antibonding frontier orbitals. These engineered molecular architectures serve as the probes for fundamental quantum phenomena and versatile candidates for advancing quantum-enabled devices^8,9,10. However, the molecule-like properties of diatomic artificial molecules (e.g., homodimers) fabricated via molecular-beam epitaxy (MBE) on single-crystalline substrates currently remain resolvable solely under cryogenic conditions^{8,9,11,12,13,14,15}. This is so because the splitting energy between the bonding and antibonding orbitals of an MBE-grown homodimer is only 1–5 meV^5,16,17,18, which is about one order of magnitude smaller than the thermal energy at ambient temperature (≈25 meV). The orbital hybridization of colloidal “artificial molecules” with the controlled coupling strength could be pronounced and resolvable at room temperature. However, colloidal “artificial molecules” are currently blocked by a few synthetic challenges, despite extensive efforts since the 1990s^{6,7,19,20,21,22}.

For MBE-related epitaxial technologies and solution chemistry, challenges for formation of “artificial molecules” with room-temperature detectable molecule-like properties come from different ways. MBE growth of homodimers—the simplest “artificial molecules”—needs to generate a vast amount of identical yet independent symmetry-breaking sites at an atomic level on a single crystalline substrate. As for the construction of complex “artificial molecules” (such as trimers and tetramers), symmetry-breaking sites on the single crystalline substrate must be in a well-organized pattern at an atomic level, which is prohibitive at present. Furthermore, because of the stringent epitaxy conditions through MBE, all existing combinations of single crystalline substrate and epitaxial structure are limited to those with small confinement energy for the atom-like states, and the resulting energy split between the bonding and antibonding orbitals of a homodimer would thus be small and uncontrollable^5,16,17,18.

Since the 1990s, colloidal synthesis of nanocrystal dimers and trimers through attachment in solution has been actively explored through ligands-linking^23,24,25,26 and DNA templating^27,28,29. While these approaches demonstrated a solution to stable chemical connection of a fixed number of nanocrystals, the organic linkers would unlikely establish coherent quantum coupling between two/three inorganic nanocrystals. The challenging issue of inorganic crystalline fusion of two nanocrystals after assembly on silica beads^30,31 was finally solved in the 2010s, which illustrated some interesting optical properties with the crystalline fused homodimers of two CdSe/CdS core/shell QDs^{31,32,33,34,35,36}. This last strategy made a step forward to yield nanocrystal homodimers with solely inorganic neck for each QD dimer, the hetero-facet attachment of the polycrystalline neck, i.e., either non-epitaxially or epitaxially-attached along a random selected axis, remains as next major challenge. With the hetero-facet attached homodimers, extensive optimization of core spacing, neck dimension, and core size^31,32,35 does not offer strong evidences for identifying molecular bonding and antibonding states. For instance, all detectable emissions of dual-peak PL of single CdSe/CdSe dimer were assigned to two uncoupled QDs³³. Time-resolved single-particle spectroscopy measurements confirmed that two emission states were either the single-exciton emissions of two QDs with slightly different emission energy or combination of one neutral exciton and one trion state. Two peaks of such dual-peak (or multi-peak) PL of a dimer would possess either similar polarization angle or randomly-oriented transition dipole, consistent with two quantum-uncoupled QDs^10,33.

Besides attachment, direct epitaxial growth of one quantum system on top of another with barriers between each other has also been explored^{37,38,39,40,41}, which are mostly based on CdS/HgS/CdS³⁷ and CdSe/Cd(Zn)S/CdSe core/well/shell quantum wells³⁸, CdSe@CdS dot@rod QDs^39,40, and type-II CdTe/CdS/CdSe core/shell/shell QDs⁴¹. Albeit grown with epitaxy, two (or multiple) quantum systems in a colloidal nanocrystal did not demonstrate bonding and anti-bonding molecular orbitals, likely because their QD (and/or quantum well) and barrier dimensions were difficult to be optimized into the necessary geometry^19,21,22. As shown by theoretical calculations, quantum coupling is highly sensitive to the size of two quantum systems, the properties of the barrier (or neck), the epitaxial-attachment direction, and the dimensions of the epitaxial neck^{6,19,20,21,22,42}.

Here, we hypothesize that, with nearly monodisperse core size and shell thickness, epitaxial fusion of two single-crystalline core/shell QDs along a common axis in solution may lead to ideally monodisperse “artificial diatomic molecules”. We understand that this strategy needs to navigate a few additional challenges to realize strong coherent quantum coupling. Firstly, the attachment should be not only epitaxial but also facet-selective, given the quantum coupling is sensitive to epitaxial-attachment direction. Secondly, to form an epitaxial neck with desirable dimensions, the facet-selective attachment requires the precise construction of hundreds of chemical bonds with a fixed lattice axis between two nanocrystals. Thirdly, the epitaxial neck must allow quantum coupling to form a “artificial molecule”, instead of either merging two QDs into a large “artificial atom” or isolating two QDs as two independent quantum systems. Fourthly, optical properties of the initial QDs and resulting epitaxial-attached dimers must be sufficiently defined for identifying the molecular states. Following these guidelines, the faceted wurtzite CdSe@CdS dot@platelet QDs⁴³ with a low dispersity in size are chosen for establishing the solution strategy. In comparison with spheroidal CdSe/CdS core/shell QDs³¹ with a low dispersity in size, two large polar basal facets, i.e., (0001) and (000$\bar{1}$), are bonded with strong ligands and should thus be inert for attachment. Conversely, their nonpolar (11$\bar{2}$0) and/or (10$\bar{1}$0) side facets are relatively large and flat, dynamically passivated with weak ligands⁴⁴, which might thus be selectively accessible for epitaxial attachment with a defined neck. The optical properties of CdSe@CdS dot@platelet QDs are nearly unity PL quantum yield, mono-exponential PL decay dynamics, nearly non-blinking at a single QD level, and a two-dimensional emission dipole (spherically polarized)⁴³, ideal for studying coherent quantum coupling between two nanocrystals.

Results

Facet-selective attachment epitaxially along the <11$\bar{{{{\boldsymbol{2}}}}}$0> axis

The pre-synthesized CdSe@CdS dot@platelet nanocrystals⁴³ (Supplementary Fig. 1) are added into an epitaxial-attachment solution—dodecylamine (DoNH₂, 25 vol.%) and octadecylamine (OdNH₂, 25 vol.%) as dynamic⁴⁵ and entropic⁴⁶ ligands in octadecene (50 vol.%)—at 280 °C (Fig. 1a). After 5-h reaction under the given conditions, the conversion ratio from the nanoplatelet monomers to homodimers for the optimized reactions can reach 50%-60%, and it is ≈30% in average for all reactions studied. Through typical size-selective precipitation⁴⁷, the population of the homodimers can be consistently improved to 80–90% (Supplementary Fig. 2), with about 70–80% recovery ratio of the dimers in the specific fraction. The overall conversion ratio of the nanoplatelets to homodimers through an optimized reaction/separation procedure can be as high as 35–45%.

Fig. 1: Synthesis of (11$\overline{2}$ 0)-attached homodimers and trimers with nearly monodisperse size, neck dimensions, and epitaxial axis.

a Scheme for the synthesis of the (11$\bar{2}$0)-attached homodimers. b, c High-resolution TEM image (b) and selected-area-electron-diffraction pattern (c) of the (11$\bar{2}$0)-attached homodimers with the <0001> zone axis. d, e High-resolution TEM images of an (11$\bar{2}$0)-attached homodimer with the <0001> (d) and <10$\bar{1}$0> (e) zone axes. f–h High-resolution TEM images of individual trimers with various geometries. The bond angles are 180^o, 120^o, and 60^o, respectively. Yellow and white dashed arrows indicate the wurtzite crystal axes. Yellow and white solid lines highlight the various crystal facets. The experiment was repeated more than 5 times. Source data are provided as a Source Data file.

High-resolution transmission electron microscope (TEM) measurements reveal that the epitaxial attachment through the scheme in Fig. 1a is solely along the <11$\bar{2}$0> axis of the wurtzite lattice (Fig. 1b-e), indicating suitable facet-selective and epitaxial attachment. Provided each (11$\bar{2}$0)-attached homodimer with two large basal planes, all homodimers in Fig.1b show clear cross fringes, and selective-area electron diffraction (SAED) of a large number of homodimers deposited on the TEM substrate shows a common <0001> zone axis (Fig. 1c). Along the <0001> zone axis, width of the epitaxial necks (Fig. 1b, d) is approximately 1/3 of that of the nanoplatelet for the specific sample. Occasionally, the homodimers are with their <10$\bar{1}$0> zone axis perpendicular to the TEM substrate, revealing the neck thickness being about the same as that of the QD building blocks (Fig. 1e). Elemental distribution along the <11$\bar{2}$0> axis (Supplementary Fig. 2) is in good accordance with the homodimer configuration.

Formation of trimers (Fig. 1f-h) and tetramers (Supplementary Fig. 2) is not the goal of the current work and thus limited to a small portion under the optimized conditions. These complex structures can be isolated in the first fraction of the size-selective separation. Importantly, these complex nanostructures are also (11$\bar{2}$0)-attached with characteristic bond angles as 180^o, 120^o, and 60^o, suggesting that the solution chemistry for epitaxial attachment is robust and can produce complex “artificial molecules” not easily accessible for the MBE technology.

Epitaxial attachment along the <10$\bar{1}$0> axis

Results above suggest that the epitaxial attachment through the reaction scheme in Fig. 1a is facet-selective solely towards the nonpolar (11$\bar{2}$0) side facets, instead of the nonpolar (10$\bar{1}$0) side facets and the polar basal planes. To understand this selectivity, the CdSe@CdS dot@platelet nanocrystals are facet-reconstructed in a solution to form hexagonal-nanoplatelets with their side facets being solely (10$\bar{1}$0) ones⁴⁴ (Fig. 2b and Supplementary Fig. 3). An optimized epitaxial-attachment scheme (Fig. 2a) only yields 10%-20% of homodimers, yet the homodimers are indeed epitaxially attached along the wurtzite <10$\bar{1}$0> axis (Fig. 2c, d and Supplementary Fig. 3). Different from the (11$\bar{2}$0) attached ones, the neck(s) of a (10$\bar{1}$0)-attached dimer (or a trimer/tetramer) usually consists of only one monolayer of atoms with some point defects (Fig. 2c–g and Supplementary Fig. 3), and the neck thickness terminates at the end of the attached (10$\bar{1}$0) facets (Fig. 2d).

Fig. 2: Synthesis of (10$\overline{1}$0)-attached homodimers and trimers.

a Scheme for synthesis of homodimers epitaxially-attached along the <10$\bar{1}$0> axis. b High-resolution TEM image of a hexagonal-nanoplatelet solely with the (10$\bar{1}$0) side facets. c, d High-resolution TEM images of a (10$\bar{1}$0)-attached homodimer with (c) the <0001> and (d) the <10$\bar{1}$0> zone axes. e–g High-resolution TEM images of individual trimers with various geometries, the bond angles are 180^o, 120^o, and 60^o, respectively. Yellow and white dashed arrows indicate the wurtzite crystal axes. White and orange solid lines highlight the various crystal facets. The experiment was repeated more than 5 times. Source data are provided as a Source Data file.

Overall, successful synthetic schemes for a given type of designated homodimers not only depend on the starting materials (see above) but also the type and concentration of ligands, reaction temperature, reaction time, and concentration of the QDs. Dynamic amine ligands for the (11$\bar{2}$0)-attachment and small cadmium carboxylate (cadmium acetate, Cd(Ac)₂) ligands for the (10$\bar{1}$0)-attachment are necessary to allow formation and sufficient exposure of the designated facets. Excessively high ligand concentrations would provide over-protection for the designated facet(s) and hinder the attachment, thereby reducing the dimer yield. Conversely, overly low ligand concentrations would rapidly destabilize the nanocrystal solution and lead to uncontrolled aggregates with randomly-oriented, polycrystalline, and/or multi-nanocrystal attachments. High temperatures (above 280 °C) often cause inter-particle and intra-particle ripening⁴⁸, and insufficient thermal agitation would not induce precise construction of hundreds of chemical bonds along a designated and common lattice facet between two nanocrystals.

Exclusion of epitaxial attachment of the polar facets including the (0001), ($000\bar{1}$), and the (10$\bar{1}\bar{1}$) facets—the (10$\bar{1}\bar{1}$) ones available for the hexagonal-nanoplatelets as the slope facets connecting the ($000\bar{1}$) basal plane and side facets (see Fig. 2d)—should be a result of their tightly-bonded surface ligands⁴⁴. Such strong ligands would form a closely-packed monolayer of hydrocarbon chains on the given facet and act as a kinetic barrier under the mild epitaxial-attachment conditions. Selection among the two nonpolar side facets for the epitaxial attachment, i.e., the (11$\bar{2}$0) facets over the (10$\bar{1}$0) ones, should be a result of their atomic packing patterns⁴⁹. Supplementary Fig. 4 shows that, ideal for epitaxial attachment, either cation or anion on a (11$\bar{2}$0) facet is always has a single dangling bond. Conversely, a nonpolar (10$\bar{1}$0) facet is terminated with both monodentate and bidentate cations/anions that require a special anion-cation matching between two nanoplatelets for the epitaxial attachment, specifically, “monodentate anions to monodentate cations” and “bidentate anions to bidentate cations” for the construction of hundreds of lattice bonds. This stringent requirement may also cause the appearance of a noticeable amount of point defects in the thin (single monolayer) neck of the (10$\bar{1}$0)-attached dimers, trimers, and tetramers.

In comparison with a relatively small (11$\bar{2}$0) facet of the original nanoplatelet monomers⁴³, the neck dimensions of the homodimer, including its thickness, width, and length (Fig. 1b–d) are large. This suggests that the chemical bonds between two epitaxially-fused QDs should have followed a typical two-step-oriented attachment studied by in-situ TEM^50,51. In the first step, two QDs follow the site-selective coalescence at two relatively small (11$\bar{2}$0) facets to form a facet-selective neck with a small cross-section (Supplementary Fig. 4). In the second step, the surface atoms would move to the low-energy neck area through the known facet-reconstruction process in the given solvent system⁴⁴.

Control on core-spacing of homodimers for quantum coupling

The scheme in Fig. 1a can be readily applied for the synthesis of the (11$\bar{2}$0)-attached homodimers with various core-spacing in high yield (Fig. 3a and Supplementary Fig. 5). Experimentally, the CdS shells with a designated thickness are epitaxially grown onto CdSe core nanocrystals with a given size (≈3 nm)⁴³. The resulting CdSe@CdS dot@platelet nanocrystals with three different CdS shell thicknesses along the lateral direction (i.e., 2.4, 3.3, and 4.2 nm) are isolated, each of which is dissolved in octadecene at room temperature in the reaction flask. Subsequently, each reaction solution is heated up to 280 ^oC under Ar atmosphere, into which the DoNH₂-OdNH₂ entropic and dynamic ligands are introduced for the facet-selective attachment (Fig. 3a). All spectroscopy measurements below are carried out at room temperature. Statistically, the homodimers for the three samples in Fig. 3a are with about the same neck cross section (4.7 nm width and 7.0 nm thickness), which means precise formation of about 200 Cd-S epitaxial bonds at the attached (11$\bar{2}$0) facet for each homodimer. Though the CdSe core size is 3 nm for all three samples, the nearly monodisperse core-spacing dictated by the CdS shell lateral thickness increases from 7. 8, 9.6, to 11.4 nm from top to bottom in Fig. 3a.

Fig. 3: Control on neck dimensions and core-spacing of the (11$\overline{2}$0)-attached homodimers.

a Statistical data and TEM images (inset) of the (11$\bar{2}$0)-attached homodimers with different core spacing. The experiment was repeated more than 5 times, scale bar is 30 nm. Inset: Statistical analysis of the mean and standard deviation for the dimer dimensions (350 dimers). b Normalized ensemble absorption (solid line) and steady-state PL (dotted line) spectra of the monomers (black) and the corresponding homodimers (red) with different core-spacing, the PL peak and PL full-width-at-half-maximum (FWHM) are provided as inset in each plot. Source data are provided as a Source Data file.

Figure 3b demonstrates that, regardless of the differences on the core-spacing, main features of UV-vis spectra of all three samples remain observable after epitaxial attachment, excluding formation of a large “artificial atom” by complete fusion of two QDs. Consistent with theoretical predictions for “artificial diatomic molecules”^{6,19,20,21,22,52}, red-shift and broadening of the band-edge UV-vis absorption peak, red-shift of PL peak, and absorbance increase in the high-energy region relative to the first excitonic absorption peak are all apparent for the homodimers with the smallest core-spacing (top plot in Fig. 3b) and observable to a certain degree for those with the median core-spacing (middle plot in Fig. 3b). Conversely, UV-vis and PL spectra (Fig. 3b, bottom plot) for the homodimers with the largest (11.4 nm) core-spacing are nearly identical to those of the corresponding monomers, indicating two QDs within a homodimer in this sample remain as two independent “artificial atoms” with any possible quantum coupling observable at room temperature. For simplicity, the one with smallest core-spacing shall be denoted as “diatomic molecules” below while those without any evidence of quantum coupling shall be named as “double-QDs”.

Single-particle spectroscopy of <11$\bar{{{{\boldsymbol{2}}}}}$0 > -attached homodimers

TEM-correlated single-particle spectroscopy⁵³ (Supplementary Fig. 6) is adopted here for studying a single diatomic molecule with single QD monomer and a single double-QD dimer as the references. Figure 4a reveals similar PL counts for a single monomer, diatomic molecule (7.8 nm core-spacing), and double-QD (11.4 nm core-spacing) after normalization by the number of QDs in each emitter. The TEM-correlated spectroscopy measurements suggest that this unified intensity level can be applied as a simple indicator to distinguish the residual monomers from the dimers in a given sample. Figure 4a further illustrates that all three PL intensity trajectories are nearly free of dim states (Fig. 4a) and statistically qualified as non-blinking⁵⁴ (Supplementary Fig. 7). In some cases, a single diatomic molecule can exhibit a PL intensity trajectory with a significant number of dim states generated by blinking, which offers a possible way to study optical properties of the dim states (Supplementary Fig. 8). The optical properties described below, including those of single-exciton and bi-exciton, are for the bright state (or the neutral QD) unless otherwise specified.

Fig. 4: Single-particle spectroscopy of single (11$\overline{2}$0)-attached homodimer and monomer.

a PL intensity trajectories of single monomer (gray), homodimer with 7.8 nm core-spacing (diatomic molecule, red), and homodimer with 11.4 nm core-spacing (double-QD, black) with background noises (soft gray in each plot). The binning time was 30 ms. b PL spectra with proper fittings of single monomer (gray), diatomic molecule (red), and double-QD (black). Inset in top plot, the polar graph of PL intensity of single monomer. Inset in middle plot, TEM image of a homodimer obtained after the single-particle spectroscopy measurements with a scale bar as 10 nm, the experiment was repeated 3 times. c Intensity ratios between PL₁ and PL₂ of single diatomic molecule vs. energy difference between PL₁ and PL₂ (ΔE) with a Boltzmann distribution fitting (black line). Inset: Energy difference (ΔE) of the mean and standard deviation for the homodimers (52 dimers), and Boltzmann distribution equation. d PL decay dynamics of a single diatomic molecule (red) and corresponding single monomer (black). Inset, the average lifetime and the PL QY. e A schematic illustration of the band-edge energy levels for diatomic molecule. f Rotation polarization angle dependent PL intensity (open circle) of PL₁ (blue) and PL₂ (green) for a single diatomic molecule, each of which is fitted with a sine function (solid lines). Inset: the polar graph of PL₁ and PL₂ intensity, with the PL₂ intensity multiplied by 2 for better visibility. All optical measurements were performed at room temperature. Source data are provided as a Source Data file.

While the PL spectrum of single monomer shows a single Lorentzian peak (Fig. 4b, top), single homodimer with the smallest core-spacing (7.4 nm) possesses dual-peak PL (Fig. 4b, middle), which is distinctly divergent from a single yet broad PL peak exhibited by hetero-facet attached homodimers of the spheroidal CdSe/CdS core/shell QDs^31,32. Conversely, the PL spectrum of a double-QD (core-spacing 11.4 nm) is nearly identical in its contour and peak width with the corresponding monomer (Fig. 4b bottom). Within statistical error (Supplementary Fig. 9), all double-QD spectra can be well-fitted with a single Lorentzian peak, similar to that of the single monomer, consistent with two independent QDs in a double-QD. Given the original monomers of two reactions with similar ensemble-PL peak width (Fig. 3b top and bottom), results of double-QD measurements—each with a single Lorentzian peak—suggest that the commonly-observed dual-peak PL of a single diatomic molecule should not be a result of emissions from its two independent QDs in a homodimer.

Within the statistical error, each homodimer with the smallest core-spacing (7.8 nm) judged by the TEM-correlated spectroscopy and/or the unified-intensity indicator (50-60 in total for a specific sample) shows the dual-peak PL, each of which can be deconvoluted into two peaks, i.e., PL₁ and PL₂ in Fig. 4b (middle). The splitting energy between PL₁ and PL₂ (∆E) of 50-60 diatomic molecules of the specific sample is $36.4\pm 11.0$ meV (Supplementary Fig. 10). The intensity ratio between PL₁ and PL₂ of the diatomic molecules follows a Boltzmann distribution⁵⁵ (Fig.4c), in accordance with single exciton emission from two split electron levels that are accessible through thermo-equilibrium^{56,57,58,59,60}. Statistical results reveal that, relative to the corresponding monomer’s PL peak at 2.014 ± 0.020 eV, PL₁ and PL₂ exhibit blue-shift to 1.999 ± 0.018 eV and red-shift to 2.036 ± 0.021 eV, respectively (Supplementary Fig. 10). These results are in agreement with the electron bonding and antibonding energy level splitting caused by quantum coupling between two QDs in a homodimer.

The PL decay lifetime of single diatomic molecule remains nearly constant across different PL intensity fractions and at different PL emission wavelengths (Supplementary Fig. 11), supporting dual-peak PL originating from two split electron levels in thermo-equilibrium^{55,56,57,58,59,60}. Transient PL measurements (Fig. 4d) reveal a double-channel PL decay dynamics for a single diatomic molecule with a much-increased average lifetime (40 ns, ensemble PL quantum yield as 68%) relative to the corresponding monomer (25 ns, ensemble PL quantum yield as 97%), implying the unlikelihood of emissions from two independent QDs in a dimer. Conversely, the PL decay dynamics of a single double-QD—determined as a dimer emitter by the two indicators—and its corresponding monomer are nearly identical (Supplementary Fig. 12), in accordance with two independent QDs in a double-QD. Presumably, the relatively low PL quantum yield of a typical sample of diatomic molecules could be due to the subpopulation of trimers (8%), given a purified trimer sample always with a very low PL quantum yield ( <10%). In addition, possible defects in the epitaxially-attached neck area of diatomic molecules could cause delayed photoluminescence with a somewhat reduced PL quantum yield and quasi-double channel PL decay⁴⁹.

Results in Fig. 3b and Fig. 4a–c all suggest that quantum coupling of the band-edge electron atomic orbitals (1S(e)) likely exists between two QDs in a homodimer with a small (7.8 nm) core-spacing (see Fig. 3a), which would form the band-edge bonding and antibonding electron orbitals (or molecular states)^{5,6,19,20,21,22,42,52} (Fig. 4e). Because of the deep valence band of the epitaxial CdS neck and large hole effective mass of CdSe, two band-edge hole levels (1S_3/2(h)) in the two cores would remain as two localized nonbonding orbitals^{6,19,20,21,22,52} (Fig. 4e). The average splitting energy ∆E ( ≈35 meV) between PL₁ and PL₂ is in the same magnitude of the bonding-antibonding splitting energy predicted for diatomic molecules with similar dimensions in ref. ¹⁹, though the exact value of energy splitting between bonding and antibonding electron orbitals is under debate⁴². It should be noted that the homodimers discussed in this subsection are all epitaxially-attached along the <11$\bar{2}$0> axis that is different from those often considered in the literature^{6,19,20,21,22}. In addition, all calculations in the literature were based on spherical CdSe/CdS core/shell QDs, the coupling energy of which may differ from that for homodimers of CdSe@CdS dot@nanoplatelet QDs. Presumably, largely different spatial distributions of the photo-generated electron in either bonding or antibonding molecular orbital and hole in a localized orbital of either core reduces their wavefunction overlapping roughly by a half, leading to a much elongated single-exciton radiative decay lifetime.

Albeit measured at cryogenic temperatures⁶¹, two transition dipoles involving a localized hole state and either bonding or antibonding molecular electron state of a diatomic molecule grown by MBE have been confirmed to be perpendicular to each other. With its <0001> axis perpendicular to the substrate, the PL intensity of a single monomer remains unchanged upon rotating the polarization angle of detection (inset, Fig. 4b top), consistent with a two-dimensional dipole on a plane parallel to the basal planes of a CdSe@CdS dot@nanoplatelet⁴³. On the contrary, either PL₁ or PL₂ of a diatomic molecule is linearly polarized (see raw data in Supplementary Fig. 13) at room temperature, with their transition dipoles perpendicular to each other (Fig. 4f). The orthogonally-polarized dual-peak PL is distinctively different from a pair of emission peaks resolved at cryogenic temperatures of a single hetero-facet attached homodimer of two spheroidal CdSe/CdS core/shell QDs³³. Although the mechanism of orthogonally-polarized dual-peak PL is unclear, it at least suggests that the two PL peaks are very unlikely from emissions of two uncoupled QDs in a dimer, provided the distinctive polarization of a monomer (inset, Fig. 4b top).

Except correlation with the molecular bonding and anti-bonding orbitals, the dual-peak PL of single diatomic molecule could also be caused by three other mechanisms, namely, size variations of the two QDs in single diatomic molecule, Förster resonance energy transfer arising from size dispersity²⁴, and charged-exciton states (i.e., trion)⁶². Figure 1 shows that the lateral dimensions of CdSe@CdS dot@nanoplatelets are somewhat broad, which is known to affect optical uniformity less than their better-controlled thickness⁴³. In addition, the single Lorentzian peak emission of single double-QD (Supplementary Fig. 9), the polarization measurements in Fig. 4b (inset, top plot) and Fig. 4f, the much increased PL decay lifetime (Fig. 4d), and the Boltzmann distribution of peak areas of the two peaks based 50–60 dimers (Fig. 4c) all suggest that the dual-peak PL would be unlikely from two independent QDs in a dimer. Statistical results for ≈50 single QD and ≈50 early-termed single diatomic molecules are summarized in Supplementary Fig. 10. The Statistic results illustrate that the average energy position of the PL peak of single QD is between those of the two peaks of dual-peak PL, and the distribution profile of the high-energy (or low-energy) peak for dual-peak PL extends significantly beyond the low-energy (or high-energy) end of single QD, respectively. These results are against either simple size variation or Förster resonance energy transfer arising from size dispersity of two QDs in a homodimer with the smallest core-spacing in Fig. 3a.

Results in Supplementary Fig. 8 and Fig. 4c exclude the possibility of the dim-state emission (i.e., trion emission of singly-charged nanocrystal) as the high-energy PL₂ in Fig. 4b (middle). Instead of a blue-shifted peak, the trion-state spectrum of single diatomic molecule shows a similar dual-peak PL with a minute red-shift of both peaks (Supplementary Fig. 8). In comparison with the bright-state emission, the PL₂/PL₁ intensity ratio of the dim-state emission of single diatomic molecule increases slightly due to increased thermal population of the antibonding orbital^{55,56,57,58,59,60}. PL decay dynamics of the trion emission can be well fitted with mono-exponential function with a very short lifetime ( ≈ 6.6 ns, Supplementary Fig. 8). Such a short lifetime is expected for the trion state from a QD with strong nonradiative Auger recombination⁶³. As mentioned above, the dual-peak PL in Fig. 4b (middle) shows no sign of contribution from this short-lifetime channel in the entire intensity range (Supplementary Fig. 11), disapproving significant contribution from trion-state emission. In addition, it is difficult to imagine the intensity ratio between the dim- and bright-state emissions follows the Boltzmann distribution observed in Fig. 4c.

Biexciton properties of the <11$\bar{2}$0 > -attached homodimers

“Artificial molecules”, including diatomic ones, are expected to demonstrate the special molecular biexciton properties^36,63,64. For single QD monomer, the second-order photon intensity correlation (g⁽²⁾) measurement (Fig. 5a, top) reveals a typical g⁽²⁾ contrast—equivalent to the biexciton emission quantum yield relative to that of single-exciton for single quantum emitter⁶⁵—as ≈0.05 (see statistics in Supplementary Fig. 14). Figure 5a (middle) demonstrates that, for single homodimer with the smallest core-spacing in Fig. 3a—early-termed as diatomic molecule, a modest increase of the typical g⁽²⁾ contrast ( ≈0.25), with an average of 0.26 (see statistics in Supplementary Fig. 14). This mild increase of biexciton quantum yield is expected for a diatomic molecule (see below).

Fig. 5: Single-particle and ensemble measurements of biexciton properties.

a Second-order photon correlation curves with fittings (black solid lines) of single monomer (gray), homodimer with 7.8 nm core-spacing (diatomic molecule, red), homodimer with 11.4 nm core-spacing (double-QD, black). The g⁽²⁾ contrast (g₀²) is calculated as a ratio between the g⁽²⁾ values at t = 0 ns versus t = 1000 ns. b Transient PL spectra of the diatomic molecules under different excitation power measured by the micro-liquid film approach. Inset: Difference in transient PL spectra between <N > = 1.0 and 0.1 (orange), and between <N > = 3.0 and 1.0 (blue). c Pump-power dependence of PL intensity for total PL (red), Auger-free emission (straight dotted line), single-exciton (gray), and two bi-exciton channels (blue and orange) of the diatomic molecules. Dashed black lines are the fits to the ‘PL saturation’ model. d Pump-power dependence of normalized PL spectra for the diatomic molecules measured by the micro-liquid film approach. e–g The configurations of the band-edge electron (green and gray) and hole (red) wavefunctions of single-exciton and two biexciton states for a diatomic molecule. Source data are provided as a Source Data file.

Contrary to early-termed diatomic molecule, the g⁽²⁾ contrast surges to a value as high as ≈0.83 for an atypical double-QD (Fig. 5a, bottom) and an average of ≈0.6 without counting the atypical ones (Supplementary Fig. 15), excluding dominant contributions from biexciton emission in a single quantum emitter. Theoretically, within the strongly quantum confined size regime (i.e., with strong nonradiative Auger recombination), the g⁽²⁾ contrast of two independent QDs in a double-QD could be calculated as g⁽²⁾ = (1 + QY_XX/QY_X)/2 ≈ [1 + QY_XX (1+ k_NR/k_r)]/2 under weak excitation³⁴. Here, quantum yields of biexciton and single-exciton by exciting either QD in a double-QD dimer are respectively denoted as QY_XX and QY_X, and k_NR and k_r represent the rate constants of nonradiative and radiative decay of its single exciton, respectively. With the values of QY_XX and QY_X, the calculated g⁽²⁾ contrast is close to the average g⁽²⁾ contrast of the double-QD. The atypical g⁽²⁾ could be a result of occasional excessively low single-exciton QY_X due to accelerated nonradiative decay by formation of a double-QD.

The micro-liquid film approach is applied for studying excitation power-dependence of ensemble PL (Methods section) to avoid the influence of photo-ionization of the QDs⁶³. Results (Supplementary Fig. 16) confirm that the QD monomers in a micro-liquid film show a biexciton PL decay channel ( ≈0.9 ns) with ≈5% PL quantum yield, consistent with the biexciton properties of CdSe/CdS core/shell nanocrystals with a similar core size and shell thickness^63,64. Similar ensemble measurements using the micro-liquid film approach reveal distinctive features for the early-termed diatomic molecules in their steady-state and transient PL spectra of biexcitons. Figure 5b, c demonstrates that a long-lived biexciton (lifetime ≈4.8 ns) with relatively high quantum yield ( ≈ 26%) becomes visible by just increasing the average photon absorption per pulse (<N>) to 1.0. Further increase of <N> to ≈3.0 makes another biexciton decay channel (lifetime ≈1.4 ns, quantum yield ≈5%) detectable (see detail in Supplementary Fig. 17). It is interesting to notice that, though one of the biexciton channels possesses high PL quantum yield (≈26%) and should contribute significantly to the total emission at relatively high excitation power, the normalized steady-state PL spectra for the entire series (Fig. 5d and Supplementary Fig. 18) barely vary from the single-exciton spectrum.

All results about biexcitons for the homodimers with the smallest core-spacing can be interpreted using the band-edge energy diagram of diatomic molecule in Fig. 4e, which leads to a simplified scheme for its band-edge single-exciton in Fig. 5e, i.e., with the electron in either bonding or antibonding state and the hole in one of two uncoupled core states. In this molecular diagram, there would be two types of distinguishable biexcitons in a diatomic molecule. If two holes are separately localized on two epitaxially-attached core QDs in a diatomic molecule (Fig. 5f), there would be only one effective Auger nonradiative recombination channel. Specifically, one spatially-delocalized electron in either bonding or antibonding orbital and one localized hole recombine and pass the energy to the other spatially-delocalized electron. Given the bonding and antibonding electron states being delocalized in both QDs in a homodimer, this would result in suppression of the Auger nonradiative recombination in comparison with that in the corresponding monomer. As a result, if the homodimers with the smallest core-spacing in Fig. 3a would be diatomic molecules, a long biexciton decay lifetime (4.8 ns) and an increased relative quantum yield (26%) for the molecular biexciton state would be observed in comparison with the corresponding monomers. Different from this electron-delocalized and hole-segregated molecular biexciton (sBX), the two holes of the second molecular biexciton state (cBX) are colocalized in one of the two epitaxially-attached core QDs in the diatomic molecule (Fig. 5g), activating the other Auger nonradiative decay involving two holes. Similar to single (3 nm) CdSe monomer⁶³, the two holes colocalized in the same CdSe core QD result in highly efficient Auger nonradiative recombination. This would make the cBX behave almost the same as the biexciton of the corresponding monomer within the experimental accuracy, namely, a low biexciton quantum yield (5%) with a short lifetime (1.4 ns). The biexciton decay lifetime of the cBX (1.4 ns) is slightly longer than that of the corresponding monomer (0.9 ns), which should be a result of the relaxed Auger nonradiative decay involving two electrons of the cBX. As a result, the g⁽²⁾ contrast of a single diatomic molecule would be dominated by the long-lifetime biexciton sBX (Fig. 5a, middle).

According to Fig. 5b, c, contributions to the total PL from the molecular biexciton cBX are negligible ( <5%) within the excitation power range. The wavefunction configuration of the long-lifetime molecular biexciton (sBX, Fig. 5f) is almost the same as that of either single-exciton (Fig. 5e) or the corresponding trion, implying the same dual-peak contour with slightly different PL₂/PL₁ intensity ratio for these three states. Thus, the different PL₂/PL₁ intensity ratio and subtle peak shift are irresolvable for the ensemble steady-state PL spectra recorded under various excitation intensity (Fig. 5d). This conclusion is also consistent with theoretic calculations³⁶, which predicted that biexciton binding energies of the sBX and cBX biexciton states of homodimers with similar dimensions should be ≈0 meV and ≈35 meV, respectively.

Discussion

Different from ligands-linking^23,24,25,26, DNA-templating^27,28,29, and solid state-fusion on silica beads^30,31, facet-selective epitaxial attachment of colloidal nanocrystals in solution results in monodisperse epitaxially-attached homodimers with a fixed attachment direction, i.e., preferentially the <11$\bar{2}$0> axis of the wurtzite structure of CdSe@CdS dot@nanoplatelets nanocrystals. The facet-selective attachment strategy is also extendable to epitaxial-attachment of complex structures (such as trimers and tetramers) with a single epitaxial lattice axis. Several design rules are identified for the precise construction of hundreds of lattice bonds at a designated facet between any two epitaxially-attached nanocrystals, including atomic building blocks with sufficiently large and well-defined facets, a self-matched atomic packing pattern on the designated family of facets, and dynamic yet entropic ligands on the selected facet(s) under synthetic conditions.

With a single epitaxial-attachment axis and well-controlled neck dimensions, the distinct optical properties expected for molecular bonding and antibonding states of “artificial molecules” are reproducibly identified, suggesting the monodisperse homodimers with the smallest core-spacing (i.e., 7.8 nm) should be artificial diatomic molecules. Overall, this work lays the foundation for colloidal synthesis of a long-expected class of quantum materials, i.e., “artificial molecules” monodispersed in their size, neck dimensions, crystal orientation, and molecular quantum states, which holds promise in numerous optoelectronic devices¹⁰, quantum technologies^8,9, and photocatalysis⁶⁶. Meanwhile, the homogeneous or heterogeneous QDs artificial molecules (diatomic, triatomic or polyatomic) can offer a suitable platform to study the molecular physics, including the atomic orbital hybridization and interatomic bond formation.

Methods

Materials

Cadmium oxide (CdO, 99.99%), cadmium acetate dihydrate (Cd(Ac)₂·2H₂O, 98.0%), selenium power (100 mesh, 99.99%), 1-octadecene (ODE, 90.0%), trioctylphosphine oxide (TOPO, 90.0%), octylamine (OcNH₂), 99.0%), oleylamine (OlNH₂, 70.0%), dodecylamine (DoNH₂, 98.0%), octanoic acid (HOc, 98%), oleic acid (HOl, 90%) and poly(methyl methacrylate) (PMMA, average Mw ≈ 120,000 g mol^-1 by gel permeation chromatography) were purchased from Sigma-Aldrich Corporation. Trioctylphosphine (TOP, 90%), sulfur powder (sublimed, −100 mesh particle size, 99.5%) and octadecylamine (OdNH₂, 97.0%) were purchased from Alfa-Aesar. Tributylphosphine (TBP, 95%) was purchased from Acros. All organic solvents were purchased from Sinopharm Chemical Reagents. All the chemicals were used as received without further purification.

Synthesis of (11$\bar{{{{\boldsymbol{2}}}}}$0) attached homodimers with different core spacing

The CdSe@CdS dot@platelet nanocrystals were used as starting materials to prepare homodimers. The CdSe@CdS dot@platelet nanocrystals with different lateral dimensions and the same thickness were synthesized and purified according to ref. ⁴³. OdNH₂ (25 vol.%) and DoNH₂ (25 vol.%) were dissolved in 3.0 mL of ODE and degassed under argon atmosphere at 160 °C for 10 min and heated to 280 °C, the purified pre-prepared nanoplatelets in ODE (1.0 mL) were injected into the flask. The mixture was stirred at 280 °C for 5 h under an argon atmosphere, and then the reaction mixture was allowed to cool down to room temperature. The resulting homodimers were precipitated with methanol and dispersed in 2.0 mL toluene.

Isolation and purification of (11$\bar{2}$0) attached homodimers

To obtain the purified homodimers, the gradient precipitation method was used to separate the as-synthesized homodimers. The ethyl acetate and acetone (2.0 mL, volume ratio, 1:1) with the addition of free ligands (OcNH₂ (50 µL), OlNH₂ (50 µL) and TOP (100 µL)) as a precipitation reagent for gradient centrifugation. Consequently, by dropwise adding the precipitation reagent into the solution of as-synthesized homodimers and the precipitation for many times, and many cycles of size selective precipitation, yielding a sample with 80–90% nanoplatelets in the form of homodimers reproducibly.

Surface treatment of (11$\bar{{{{\boldsymbol{2}}}}}$0) attached homodimers and the corresponding monomers

The surface treatment of the isolated monomers and homodimers according our previous work with a slight modification^43,67. OlNH₂ (1.0 mL) was dissolved in 1.0 mL of ODE and degassed under argon atmosphere at 160 °C for 10 min. After cooling down the mixture to 110 °C, the isolated monomers or homodimers in ODE (1.0 mL) and 1.0 mL S-ODE (0.1 mol L^-1, dissolving sulfur powder in ODE) were successively injected into the flask. The mixture was stirred at 110 °C for 20 min, and then, the nanocrystals were precipitated by methanol. This sulfur treatment process was repeated until no fluorescence of the nanocrystals was detected, indicating that the nanocrystals were completely treated with sulfur. After completion of the sulfur treatment, 0.2 mL TBP was added into the reaction mixture at 220 °C. After 3 min, the reaction mixture was allowed to cool down to room temperature, and Cd(Ac)₂ (0.2 mL, 0.1 mol L^-1 in OcNH₂) was added dropwise into the solution. Subsequently, the sample was irradiated under a 395 nm LED until the surface traps of the nanocrystals were fully removed.

Synthesis of (10$\bar{{{{\bf{1}}}}}$0) attached homodimers

The CdSe@CdS dot@platelet nanocrystals were used as starting materials to prepare the homodimers with (10$\bar{1}$0) facets attachment. Cd(Ac)₂·2H₂O (0.08 mmol) was added to 0.4 mmol of DoNH₂ and 0.4 mmol of OdNH₂, the mixture was stirred at 100 °C for 15 min, until a colorless solution was obtained. In a three-neck flask, 2.0 mL of ODE was degassed under argon atmosphere at 120 °C for 10 min and heated to 220 °C, the above colorless solution of Cd(Ac)₂ in fatty amines, and the purified pre-prepared nanoplatelets in ODE (1.0 mL) were successively injected into the flask. The mixture was stirred at 220 °C for 1 h under argon atmosphere, and then the reaction mixture was allowed to cool down to room temperature. The resulting (10$\bar{1}$0) attached homodimers were precipitated with methanol and dispersed in 2.0 mL toluene.

Optical measurements and structural characterization for ensemble nanocrystals

For all measurements in solution, toluene was chosen to be the solvent. Absorption spectra were measured using an Agilent Technologies Cary 4000 UV-visible spectrophotometer. Steady-state and transient PL spectra were performed on an Edinburgh Instruments FLS920 spectrometer. The transient PL spectra were acquired on a time correlated single-photon counting (TCSPC) module of the same spectrometer with a 405 nm ps-pulsed laser at a repetition frequency of 0.2 MHz. The absolute PL quantum yield was measured using a calibrated Ocean Optics FOIS-1 integrating sphere coupled with a QE65000 spectrometer. All optical measurements were performed at room temperature. As described in the main text, the biexciton properties of monomers and homodimers can be accurately measured with micro-liquid film approach. The micro-liquid film of monomers and homodimers are made by sandwiching ≈10 μm thick samples solution between two pieces of cover glass. Under a low repetition frequency (1 MHz), the nanocrystals in the micro-liquid freely diffusing in and out the beam area would avoid interference of potentially charged nanocrystals during the saturation measurements. Furthermore, all micro-liquid measurements were performed with a low repetition frequency (1 MHz) to avoid artifacts in the decay traces recorded by TCSPC. An oil-immersion objective with numerical aperture of 1.49 is used to focus the excitation light to the center of the samples micro-liquid film and collect the emission signals for both steady-state and transient PL measurements. Samples were prepared by drop-casting dilute toluene dispersions onto copper grids (400-mesh) coated with a pure carbon support film. Transmission electron microscopy (TEM) images were acquired on a Hitachi 7700 transmission electron microscope with an acceleration voltage of 100 kV. High-resolution TEM images and selective-area electron diffraction (SAED) patterns were collected by a JEM 2100 F transmission electron microscope operated at 200 kV. High-angle annular dark field (HAADF) image and energy dispersive spectroscopy (EDS) line of the homodimer were obtained using the JEOL JEM-F200 high-resolution transmission electron microscope operated at 200 kV. The dimensions of the samples were quantified via the Nano Measurer.

Optical measurements for a single nanocrystal

Diluted samples solution in toluene with 2.5 wt.% PMMA was spin-casted on a clean glass coverslip for single particle measurements. Spectroscopic properties of single nanocrystal and micro-liquid films were measured using a modified inverted epifluorescence microscope (Olympus IX 83) with a 60× oil immersion objective (NA = 1.49) and suitable spectral filters under ambient conditions. The 405 nm continuous-wave laser (PicoQuant) was used as an excitation light source, the emission signal of a single nanocrystal can be imaged and recorded by an EMCCD camera (Andor iXon Ultra 897). The steady-state PL spectra were recorded using an Andor Kymera 193i spectrometer, the integration time for each PL spectrum was 2 s. The polarized emitted light of single nanocrystal can also be guided to the spectrometer after passing through a motorized λ/2 waveplate and an additional polarizer. For time-resolved PL measurements, the 405 nm laser was switched to a pulsed mode with a 1 MHz repetition frequency. The emission signals from a single nanocrystal were collected and recorded with an avalanche photodiode (PicoQuant, τ-SPAD). The PL decay traces of single nanocrystal and micro-liquid films were measured using a time-correlated single-photon counting system in the time-tagged time-resolved mode. With different excitation power densities for liquid films, a series of PL decay traces were obtained, and thus the biexciton lifetime and QY could be calculated. Second-order photon correlation measurement was performed with a Hanbury Brown and Twiss setup with two avalanche photodiodes. For single nanocrystal measurements, the average photon number (〈N〉) absorbed was kept below 0.1 to ensure single-exciton emission unless otherwise mentioned. The visualization and fitting of all spectroscopic data were performed using Microsoft Excel or Origin.

Correlation of optical and TEM measurements

The optical measurements of a single homodimer are confirmed by the correlating fluorescence microscopy and TEM. Diluted monodisperse SiO₂ microspheres (size: 1μm) in toluene were spin-casted on copper grids (200-mesh) coated with pure formvar supporting film as makers (large particles in optical and TEM images in the first row in Supplementary Fig. 6a to correlate spectroscopic and TEM measurements for a single homodimer. The pure formvar film was used as the optical and TEM substrate, because this film is transparent in TEM and shows a low fluorescence background. A diluted solution of the homodimers in toluene with 2.5 wt.% PMMA was spin-casted on the pure formvar supporting film, then the spectroscopic properties of a single homodimer were measured using a modified inverted epifluorescence microscope. After all relevant spectroscopic data were collected, the samples were examined by the Hitachi 7700 transmission electron microscope with an acceleration voltage of 100 kV. TEM was performed after the fluorescence measurements to avoid sample contamination and degradation.

Model of emission saturation for micro liquid film approach

Calculation of biexciton QY from the PL saturation curve is according to the published methods⁶⁴. Here, we consider that single monomer constituent of the homodimers is independent for the light absorption process, the photon number absorbed per monomer per excitation pulse is considered as a Poisson distribution:

$$P\left(m, < N > \right)=\frac{{ < N > }^{m}}{m!}{{{{\rm{e}}}}}^{- < N > },\,m=0,\,1,\,2,\,3$$

(1)

where $< N >$ is the average number of photons absorbed per monomer per pulse.

At a given excitation, the total PL intensity measured from monomer can be written as

$$I={\Sigma }_{m=1}^{\infty }P\left(m, < N > \right){\Sigma }_{m=1}^{N}{{{{\rm{QY}}}}}_{m{{{\rm{X}}}}}$$

(2)

where ${{{{\rm{QY}}}}}_{m{{{\rm{X}}}}}$ is the quantum yield of the m exciton state.

Only single exciton emission of monomer is observed at long delay time (e.g., 20–30 ns), when multi-exciton recombination completes. As excitation power increases, the single exciton emission intensity saturates of monomers following the function

$${I}_{{{{\rm{X}}}}}={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-P\left(0, < N > \right)\right)={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-{{{{\rm{e}}}}}^{- < N > }\right)={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-{{{{\rm{e}}}}}^{-C\mu }\right)$$

(3)

where P (m, <N>) is a Poisson distribution function; ${{{{\rm{QY}}}}}_{{{{\rm{X}}}}}$ is the single exciton quantum yield; C is the scaling factor depending on the absorption cross section; μ is the excitation power. Considering ${{{{\rm{QY}}}}}_{{{{\rm{X}}}}}$ is unity for high quality monomer sample, C and thus the x axis <N> can be determined by fitting the single exciton PL saturation curve with the function above.

The total emission intensity saturation curve is essential a function of ${{{{\rm{QY}}}}}_{{{{\rm{BX}}}}}$(${{{{\rm{QY}}}}}_{2{{{\rm{X}}}}}$) and <N > . Fitting the total emission saturation curve in Supplementary Fig. 16 with the model above, the biexciton quantum yield for monomers was determined as ${{{{\rm{QY}}}}}_{{{{\rm{BX}}}}}\,$= 0.05. Due to the lower quantum yield of multiple excitons, only PL emission of single exciton and biexciton is considered here.

For single exciton emission of homodimers, when single monomer constituent of the homodimers absorbs photons (one or more), the hole localized on either of the two attached monomers, and the electron is thermally distributed in the two spatially-delocalized molecular orbitals. The probability can be calculated via the opposite of this event (both of monomer constituent do not absorb photons). As excitation power increases, the single exciton emission intensity saturates of homodimers following the function

$${I}_{{{{\rm{X}}}}}={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-{P\left(0, < N > \right)}^{2}\right)={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-{{{{\rm{e}}}}}^{-2 < N > }\right)={{{{\rm{QY}}}}}_{{{{\rm{X}}}}}\left(1-{{{{\rm{e}}}}}^{-2C\mu }\right)$$

(4)

where ${{{{\rm{QY}}}}}_{{{{\rm{X}}}}}$ is the single exciton quantum yield; C is the scaling factor depending on the absorption cross section of monomers; μ is the excitation power. Considering ${{{{\rm{QY}}}}}_{{{{\rm{X}}}}}$ is unity for high quality homodimer sample, C and thus the x axis <N> can be determined by fitting the single exciton PL saturation curve with the function above (Fig. 5c).

For biexciton emission of homodimers, two holes separately localized on two epitaxially-attached monomers (Fig. 5f), the electrons are delocalized in the whole homodimers, which is called electron-delocalized and hole-segregated biexciton (sBX). Each monomer must absorb a photon or more. The corresponding emission intensity ${I}_{s{{{\rm{BX}}}}}$ can be written as follows:

$${I}_{s{{{\rm{BX}}}}}={{{{\rm{QY}}}}}_{s{{{\rm{BX}}}}}{(1-P(0, < N > ))}^{2}={{{{\rm{QY}}}}}_{s{{{\rm{BX}}}}}\left(1-2{{{{\rm{e}}}}}^{- < N > }+{{{{\rm{e}}}}}^{-2 < N > }\right)$$

(5)

When the two holes of the second biexciton state are colocalized in one of two epitaxially-attached monomers (Fig. 5g), the electrons delocalized in the whole homodimers. The electron-delocalized and hole-colocalized biexciton (cBX) involves at least two photons absorbed by one monomer. The opposite event is that both two monomers absorb fewer than two photons. The corresponding emission intensity ${I}_{c{{{\rm{BX}}}}}$ can be written as follows:

$${I}_{c{{{\rm{BX}}}}} ={{{{\rm{QY}}}}}_{c{{{\rm{BX}}}}}\left(1-{\left(P\left(0, < N > \right)+P\left(1, < N > \right)\right)}^{2}\right)\,\, \\ ={{{{\rm{QY}}}}}_{c{{{\rm{BX}}}}}\left(1-{\left({{{{\rm{e}}}}}^{- < N > }+ < N > {{{{\rm{e}}}}}^{- < N > }\right)}^{2}\right)$$

(6)

Fitting the saturation curve of sBX and cBX in Fig. 5c with the model above, the quantum yield of sBX and cBX for homodimers are 0.26 and 0.05, respectively.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.