Fostering the Dense Packing of Halide Perovskite Quantum Dots through Binary-Disperse Mixing

Due to their versatile applications, perovskite quantum dot (PQD)-based optoelectrical devices have garnered significant research attention. However, the fundamental packing behavior of PQDs in thin films and its impact on the device performance remain relatively unexplored. Drawing inspiration from theoretical models concerning packing density with size mixtures, this study presents an effective strategy, namely, binary-disperse mixing, aimed at enhancing the packing density of PQD films. Comprehensive grazing-incidence small-angle X-ray characterization suggested that the PQD film consists of three phases: two monosize phases and one binary mixing phase. The volume fraction and population of the binary-size phase can be tuned by mixing an appropriate amount of large and small PQDs. Furthermore, we performed multi-length-scale all-atom and coarse-grained molecular dynamics simulations to elucidate the distribution and conformation of organic surface ligands, highlighting their influence on PQD packing. Notably, the mixing of two PQDs of different sizes promotes closer face-to-face contact. The densely packed binary-disperse film exhibited largely suppressed trap-assisted recombination, much longer carrier lifetime, and thereby improved power conversion efficiency. Hence, this study provides fundamental understanding of the packing mechanism of perovskite quantum dots and highlights the significance of packing density for PQD-based solar cells.

*Email: yiwang@cuhk.edu.hk(Y.W.) Theoretical modelling: The gracing-incidence SAXS data processing step contains model building and data fitting process. 1,2 he scattering intensity can be quantified by the following formula: (S1) The first product A is the Rayleigh function, with a constant proportional to particle 〈  ( ,  )〉  volume fraction , where and are the number density and diameter, respectively;  =  3 /6   and is the average form factor which characterizes the intra-particle property of a system.

〈 𝑃 ( 𝑞, 𝑅 )〉
One simplification adopted is that the shape of the PQDs is treated as spheres, 2,3 therefore is given by: 〈  ( ,  )〉 is the structure factor which describes statistically the contribution of inter-particle () correlation to . In this study, a hard sphere structure factor with Percus-Yevick (P-Y) closure () was implemented, for it provides a reasonable approximation of uncharged particles that freely rotate and have a moderate polydispersity.To gain an insight of the derivation of structure factor, it is reasonable to evoke the Ornstein-Zernike (O-Z) equation. 6,7 ourier transforming the O-Z equation leads to the following: where , are the total correlation function and direct correlation function, respectively, in ℎ() () q space.The structure factor therefore becomes:

Defining
as the radial distribution function gives the P-Y approximation the form of: () with the pair potential which takes the following form for hard spheres () In a mono-disperse system, the Fourier transform of combined with (S7) leads to an analytical () where A and B are coefficients related to volume fraction , and is a function of reduced wave    number . 7

𝑦 = 2𝑞𝑅
The single component film with particle radius and number density can be well described  1  1 by the above structure factor.However, if a second component with radius and number density  2 , based on which ( ) and could be defined, was added to form a binary disperse system, the interaction between particles should contain three parts as well as the correlation function, which were solved from P-Y equation by J. L. Lebowitz 8 and N. W.
Ashcroft and his co-workers 9 : Derived from the relation of and , the explicit solutions can be described as:   () () The above structure factor is used here to model the bi-disperse system.

MD Simulations:
All-atom (AA) molecular dynamics (MD) simulations were first employed to investigate ligand conformation and distribution over the PQD surface in octane.Same as in the experiments, OAm+/OA-were modeled as the long-chain ligands and FA + /OAc -as the short-chain ligands.Four cubic PQD models were constructed, which consisted of 4.5, 6.5, 8.5 and 10.5 unit cells, respectively, where each cubic unit cell had a dimension of 0.62 nm.Given the stability of lead halide octahedral within various perovskite-related materials, 10,11 CsI-rich (100) crystallographic planes were chosen to represent the PQD surface, consistent with previous modeling studies.The surfaces of PQDs were found to be passivated by ammonium in previously proposed models, which explained their high tolerance towards degradation.Following these studies, in our PQD model, 12,13 50% of surface Cs + atoms were replaced by pre-inserted OAm + .
A given PQD was then placed in a simulation box with at least 4 nm between its surface and the box boundaries.The anionic ligands OA -and the solvent octane molecules were randomly inserted into the box.To mimic experimental conditions, the molar ratio of OA -:octane was set to 1:16.
Free cationic OAm + ligands were then inserted into the system to ensure that the final simulation system (Figure S3) was charge-neutral.The simulation systems of PQDs with short-chain ligands were constructed using the equilibrated PQD-long-ligands structures as templates.The long-chain ligands attached to the PQD surface were replaced by the short-chain ligands, mimicking the ligand exchange effect in the corresponding experiments.
The CHARMM36 14 and CHARMM General Force Field (CGenFF) 15,16 were adopted in all AA simulations.The CHARMM-GUI 17 and CGenFF program 18,19 were used to model ligands without existing parameters.For the FA + molecule that received a high penalty score from the CGenFF program, additional optimization was carried out using the VMD Force Field Toolkit Plugin (FFTK) 20 and Gaussian 21 .Nonbonded parameters for Cs + and I -ions were adopted from Joung et al. 22 and Pb 2+ parameter was adopted from Li et al. 23 After energy minimization all systems were equilibrated for 1 ns in the NVT ensemble, followed by 20 ns equilibration in the NPT ensemble.
Simulated annealing was then performed in which the systems were heated up from 300 to 500K at the rate of 50 K/ns; then the temperature was kept at 500K for 1 ns and reduced to 300 K at the same rate of 50 K/ns.After another 20 ns equilibration, the resulting structures were simulated for 100 ns in the NPT ensemble as the production run.For each PQD model, four replica production runs were performed.
In all AA simulations, van der Waals interactions were smoothly switched off from 0.8 nm to 0.9 nm, while electrostatic interactions were calculated using the particle mesh Ewald (PME) method with a cutoff of 0.9 nm.The temperature of all systems was maintained at 300 K by velocity rescaling with a stochastic term 24 and Berendsen coupling was used to maintain the pressure at 1 bar.All bonds with hydrogen atoms were constrained using the LINCS algorithm.All simulations were performed with the GROMACS software package 25 and visualized using the Visual molecular dynamics (VMD) program 26 .The 3D electrostatic potentials shown in Figure S5 were evaluated using the VMD PMEPot Plugin 27 .
To construct a model mimicking a PQD after spin coating, when most of the solvent and free ligands were cast off, only ligands within 2 nm of a PQD were retained from the aforementioned simulations.More specifically, two such PQD models each with a size of 6.5-unit cells were retained, neutralized and then simulated for 100 ns in the NVT ensemble, during which they were only allowed to move in the surface-to-surface orientation shown in Figure 3c.Finally, the surface ligand density on a PQD was defined as the number of ligand heads divided by the PQD surface area.Measurement of the normalized mass density of ligands was performed by first aligning the PQD to the center of the simulation box.The ligand densities were then scanned and averaged in the ±x, ±y, and ±z directions over the duration of the four replicas of production runs.
To further investigate the interactions between multiple PQDs, we next conducted coarse-grained (CG) simulations using the MARTINI3 force field.Large PQDs (10.5-unit cell) and small PQDs (6.5-unit cell) with long-chain ligands were constructed based on their equilibrated AA models without the octane solvent or ligands beyond 2 nm of a PQD surface.Altogether three CG systems were constructed: all-large (40 large PQDs), all-small (70 small PQDs), and 1:1-mix (25 large and 25 small PQDs).The CG models of these PQDs were inserted in a vacuum box of 50*50*60 nm 3 and neutralized by additional long-chain ligands.A baseboard (carbon atoms, 10 atoms/nm 2 ) was placed in the -plane at z = 0, while a flat bottom potential was placed for z > 55 nm to prevent , the particles from crossing the top periodic boundary.CG parameters of OAm + and OA -were obtained from the DOPC molecule in MARTINI3 while nonbonded parameters of PQDs were retained from their all-atom models.All systems were equilibrated for 100 ps in an NVT ensemble after energy minimization and then simulated in the NVT ensemble for 40ns as the production run.
Each system at a given large:small PQD number ratio (1:0, 1:1, and 0:1 for the all-large, 1:1-mix and all-small systems, respectively) was simulated for N = 40 replicas, resulting in a total CG simulation time of ~5 μs.
Based on the CG simulations we measured the probability distribution function P(r) of the minimum distance dm between any two PQDs (excluding ligands): where is the number of PQD pairs with their minimum distance in the range   =   , [,  +0.2) from the i-th simulation replica, and is the total number of PQD pairs from the i-th  , simulation replica.Next, for each Cs atom (denoted as atom A) on the surface of a given PQD, we measured the frequency with which it came into contact ( ≤ 4 nm, see Figure S9) with a Cs atom   (denoted as atom B) of a second PQD.We further distinguished the location of atom B based on whether it was on the edge, corner, face (anywhere on a PQD surface that is not the edge or corner), or any of the above three locations of a PQD.The resulting contact maps are thus labelled as 'edge', 'corner', 'face' or 'any', respectively.For a given atom A at , its contact frequency (,,) with atom B of a second PQD is therefore: where is the number of PQDs with a given size type (st) (large/small), N = 40 is the number of   simulation replicas and is the minimum distance between atom A at and atom B   (,,) (edge/corner/face/any) that belongs to a second PQD.More specifically, we set the center of the first PQD to the origin (0,0,0) and place its edges parallel to the x, y, and z axes.Denoting the edge length of the PQD as 2L, we obtain the coordinates of atoms on its surface as , ( ± , ,) (, ± ,

or
, where .Due to symmetry, the computed contact frequency at ) (,, ± ) - ≤ , ≤  (,,) can be mapped onto where .The averaged is then mapped onto a by square with , while missing data at Cs atoms substituted by ligands was obtained via 2D interpolation.Finally, the results are smoothed by 2D interpolation and normalized to yield the contact maps shown in Figure S10.
Finally, with defined in the previous equation, the contact ratio of a given PQD surface was  , obtained as: , where missing data due to ligand substitution were obtained via linear interpolation.For each PQD, the maximum contact ratio among its six surfaces was then computed and shown in Figure 3f.

Ligand density from NMR and ICP-OES measurement:
The ligand density is estimated as the number of ligands attached per area on a PQD surface: where and are the molar concentration of ligands and PQDs, respectively; and is the      average edge length of PQDs.
The molar concentration of ligands was determined by NMR measurement.Specifically, a certain amount of dried QD with a mass of was dispersed in volume of deuterated chloroform.By  1  1 setting pure oleyl species as the standard sample in NMR test, the was acquired via the   digital ERETIC 28 .

The
was measured and calculated via the ICP-OES technique. 29The same amount of dried   QDs was dissolved in thick nitric acid (69% in vv) and diluted to 1%.The prepared solution was then to perform the ICP-OES test which give the concentration of Pb 2+ ion in the solution.Under the assumption that all the Pb 2+ were sourced from the PQDs, the quantity of CsPbI 3 was  3 thus gained.With the lattice constant of PQDs given, the amount of QDs can therefore be  obtained: Combining , , and formula S23 and S24, the ligand density can be calculated.

𝑐 𝑙𝑖𝑔𝑎𝑛𝑑 𝑐 𝐶𝑠𝑃𝑏𝐼3
Softness: For a core-shell structure particle, its softness is given by the ratio of the length of the ligand and the diameter or edge length of the particle. 30Specifically, if a particle with a core diameter of and covered with a ligand whose length is , the softness for the particle is To analyze ligand conformation and distribution, the normalized mass densities of OAm + and OA - were measured from all-atom MD simulations as a function of their distance to a PQD surface.As shown in Figure S19, the normalized mass densities reach the bulk values at 1.9, 1.9, 1.9, 2.0 nm for PQDs of 4.5, 6.5, 8.5, and 10.5 unit cells, respectively.Therefore, the thickness of the longchain ligand layer was taken to be approximately 2 nm regardless of the PQD size.Based on further simulations mimicking the experimental conditions after spin coating, the average distance between two PQDs with attached long-chain ligands was found to be 2.1 nm (Figure 3c), reflecting ligand interpenetration between neighboring PQDs.Based on this average surface-to-surface distance between two PQDs, an average ligand length of 1.05 nm was estimated for the PQD film.
The softness values for QD@170 and QD@120 were therefore calculated to be 0.167 and 0.242, respectively.The total volume fractions of the core-shell PQDs thus reached 55.1% and 68.4%, respectively.It is worth mentioning that, after considering ligands in the calculation of volume fraction, the packing densities of the pure QD@170 and pure QD@120 are comparable to the limit of random packing 31 .

Figure S17
The EIS measurement of fabricated PQD device.However, the summation of all the R's gives the DC resistance, R DC = R 1 +R 2 +R 3 .The R 1 +R 2 can provide the information about transport resistance 33,34 in the selective contacts while the summation of R 1 +R 2 +R 3 can be considered as the recombination resistance 35,36 .(b) Plotted the transport resistance and recombination resistance of the devices fabricated with pure QD@170 and binary film QD@170+120-0.36.

Figure
Figure S2 GISAXS 2D patterns and linecut curves.(a) GISAXS 2D patterns of pure QD films and blend film with various mix number ratios.(b) to (h) are the linecut fitting of pure QD films and blend films with various number ratios.The samples are labelled with the film components and distinguished by the suffixes that correspond to the concentration of QD@120 in the film.

Figure S3
Figure S3 Simulation snapshot of representative initial and final structures of a single PQD in octane.The PQD (edge length = 6.5-unit cell) is attached with long-chain ligands.The PQD is represented by beads: Cs + (cyan), Pb 2+ (yellow) and I -(pink), while OA -(red) and OAm + (blue) are shown in thin lines, with the pre-inserted OAm + and any ligand whose head group is within 1 nm of the PQD at t=100 ns represented by thick sticks.The octane solvent is represented by transparent surfaces.

Figure
Figure S4 (a) Simulation snapshot of a single PQD with short-chain ligands in octane.OAc-and

Figure S5
Figure S5 3D electrostatic potentials of a single 6.5-unit sized PQD with pre-inserted OAM + computed using the PMEpot plugin 26 of VMD [1.9.4].The sliver, orange, and blue transparent surfaces represent increasingly strong (positive) electrostatic potential.The PQD is represented by beads (Cs + : cyan, Pb 2+ : yellow and I -: pink) and for clarity, only the nitrogen atoms of pre-inserted OAm + are shown (dark blue).

Figure S6 .
Figure S6.Interpolation method to determine the concentration of Pb in QD solutions via ICP-OES.

Figure S7 .
Figure S7.The concentration of attached ligands determined by ERETIC technique via NMR spectra.

Figure S8 .
Figure S8.Illustration of softness.The red cube cores represent the QDs while the transparent shells represent the surface ligand as they are assumed to be "transparent" under X-ray due to the low electron density.

Figure S9 .
Figure S9.Probability distributions of the minimum distance between any two PQDs in coarsegrained MD simulations.

Figure S10 .
Figure S10.The contact maps of all-large (left column) and all-small (right column) PQD systems.

Figure S14
Figure S14 Band structure of assembled QD device

Figure S 18 .
Figure S 18. (a)The equivalent circuit used to fit the EIS curves.32Rs is the series resistance originated by wires, connections and FTO substrate.C G represents the geometric capacitance of the device at high frequency.The C IF , C LF represents the capacitance at intermediate and low frequency, respectively.It is hard to direct interpret the physical meaning of R 1 , R 2 and R 3 .However, the summation of all the R's gives the DC resistance, R DC = R 1 +R 2 +R 3 .The R 1 +R 2 can provide the information about transport resistance33,34 in the selective contacts while the summation of R 1 +R 2 +R 3 can be considered as the recombination resistance35,36 .(b) Plotted the transport resistance and recombination resistance of the devices fabricated with pure QD@170 and binary film QD@170+120-0.36.

Figure S 19
Figure S 19 Normalized mass density of head atoms of ligands within a given distance of a PQD surface obtained from all-atom MD simulations of various sized PQDs with long-chain ligands in octane.

Table S1
Fitted volume fraction  of mono-sized QD films

Table S3
Fitted volume fraction  1 of mono-sized QD films and total volume fraction  2 under the consideration of particle softness.The ligands involved in calculation for softness is OA and OAm with the length of around 0.9 nm.Table S4Fitted volume fraction  of binary sized QD film before and after ligand exchanged.TableS5Averaged photovoltaic parameters of QD devices with champion data included in the bracket.TableS6TRPL fitted parameters of binary-sized QD film with varied concentration of QD@120. 1  2 1 +  2  2