Tuning perovskite nanocrystal superlattices for superradiance in the presence of disorder

The optical properties of self-assembled colloidal nanocrystals are of fundamental and practical interest. The reports of superfluorescence^1–6 and superradiance^7–9 in ordered assemblies (superlattices) of perovskite nanocrystals are recent examples of collective optical effects. Combining such phenomena with scalable self-assembly¹⁰ may lead to the low-cost fabrication of miniature coherent light sources for photonics and optical information processing. These systems form a rich playground where the material properties of constituent nanocrystals and the superstructure could be designed to control the cooperative light emission.

The idealized cooperative light emission (e.g., superradiance¹¹ and superfluorescence¹²) relies on identical emitters that are coupled via the electromagnetic field and form a Dicke ladder of macroscopic states. Molecular and atomic gases have been historically used as experimental systems that can approach such ideal conditions.^13,14 Semiconductor nanocrystals, resembling “artificial atoms,” are interesting as quantum emitters with facile tunability of emission frequency by changing nanocrystals’ size or composition. Nonetheless, their assemblies feature numerous inhomogeneities. For example, a realistic batch of colloidal nanocrystals is characterized by a distribution of sizes, shapes, and different surface passivations.¹⁵ A nanocrystal superlattice also contains further imperfections such as vacancies,¹⁶ variation in the superlattice periodicity,¹⁷ and strain.¹⁸ These experimental realities make nanocrystal emitters distinguishable from each other and introduce energy disorder to the system.

The imperfections pose a general question of how the energetic and spatial disorder influence the macroscopic coherence and cooperative light emission. Localization is one consequence of disorder that emerges from the recent experiments on perovskite nanocrystal superlattices. For example, Rainò et al. proposed the existence of superfluorescent domains within superlattices with an estimated average number of coherently coupled CsPbBr₃ nanocrystals N_coupled = 28,¹ Blach et al. estimated N_coupled = 3.⁸ More recently, Adl et al. measured a radiative enhancement factor in the range 1.2–11.0 for an estimated N_coupled = 1000–40 000 CsPbBr₃ nanocrystals in similar superlattices.⁹ Given that a single superlattice typically contains N = 10⁶–10⁷ nanocrystals, the yield of cooperativity is low and requires an explanation.

The CsPbBr₃ superlattices with N = 10–10⁴ nanocrystals have been studied theoretically using a single-excitation model of superradiance.^19,20 These studies focused on the dependence of the superradiant rate enhancement on the total number of nanocrystals, center-to-center distance, aspect ratio (nanocubes vs nanoplatelets), a static energy disorder and the dimension of the superlattice (1D, 2D, 3D). Among the theoretical findings are predictions that arrays of smaller nanocubes are more resistant to thermal decoherence than arrays of larger nanocubes, and the emergence of cooperative robustness against decoherence in large 3D superlattices (N > 10³). Considering the intense research interest in this topic, it is valuable to continue the systematic investigation of nanocrystal superlattice design parameters across different regimes of superradiance, particularly from weak to strong disorder, and in the Dicke limit or away from it.

In this work, we theoretically examine the effects of nanocube size, interparticle separation, energy and spatial disorder on the superradiance and on the spatial localization of the superradiant wave function in a 2D superlattice. We consider the enhancement of superradiance across different regimes of disorder, finding that optimizing the superradiance demands differing strategies of tuning nanocrystal parameters.

In the above, j_i/y_i denotes the spherical Bessel function of the first/second kind of order i. $Dmnαβ=êα⋅êβ−3(êα⋅r̂mn)(êβ⋅r̂mn)$⁠, $êα$ is the unit vector in the α direction, $r̂mn=r⃗mn/rmn$ and $r⃗mn$ is the position vector connecting the centers of the mth and nth nanocrystals. The resonant wave number $k0=ω0εopt/c$ in which ɛ_opt is the optical dielectric constant of the material in the relevant frequency range.

The eigenvalues and eigenvectors of Hamiltonian H_rad can be obtained by diagonalizing of the corresponding 3 N × 3 N matrix. The superradiant state is defined to be the eigenstate Ψ_SR of H_rad with the smallest (i.e. the most negative) imaginary part Γ_SR, which is defined as the superradiant radiative decay rate. Here we take into account only the radiative coupling and radiative recombination of excitons in the nanocrystals. Other recombination or energy transfer channels, for instance electron-phonon coupling or Auger-type processes, play the role of decoherence sources and are not considered in the current model.

The results presented in this manuscript focus on the described model applied to nanocubes with L_NC ranging from 5.0 to 20.0 nm, which are sizes representative of the CsPbBr₃ nanocubes spanning various confinement regimes and that have been synthesized with high uniformity, high photoluminescence quantum yield, and form close-packed assemblies in previous publications (see Refs. 22 and 23 for an overview).

We begin by examining the effects of the superlattice morphology and the nanocube size on the acceleration of the radiative decay of the superradiant state. Two superlattice morphologies of experimental relevance to CsPbBr₃ nanocubes are considered: a symmetric 2D monolayer N_x × N_y where N_x = N_y, and N_z = 1 [Fig. 1(a)]^27–29 and a symmetric 3D superlattice consisting of N_x × N_y × N_z nanocubes, N_x = N_y = N_z.^1–6,30

For a total number of nanocubes N, the superradiant enhancement is defined as the ratio Γ_SR/γ_r between Γ_SR of the superlattice and the radiative rate of a single nanocrystal γ_r. The calculated Γ_SR/γ_r for 2D and 3D superlattice morphologies, each with L_NC = 5.0, 13.0 and 20.0 nm, are presented in Figs. 1(b)–1(d). The shell length was set to L_shell = 1.5 nm, a value typical of cesium oleate and oleylammonium bromide ligands used in the preparation of superfluorescent assemblies. For most nanocrystals of interest, many-body perturbation theory provides a reasonable description of the dependence of ω₀ and γ_r on L_NC.²⁴ More quantitative calculations of ω₀ and γ_r vs L_NC based on configuration-interaction approach can be found in Ref. 25. For the range of nanocrystal sizes considered here, the nanocrystal energy ℏω₀ varies from 2.32 to 2.52 eV, while the radiative decay times $γr−1$ varies from 0.1 to 0.9 ns, respectively, based on the calculations from these works.^24,25 More details can be found in Table I.

In panels (c) and (d) of Fig. 1, we defined the superlattice size factor Φ = d/λ, where d = N_x(L_NC + 2L_shell) is the length of the superlattice, which is the distance covered by nanocubes along x direction, and $λ=2πc/ω0εopt$ is the resonant wavelength of the bright exciton. In other words, the dimensionless size factor Φ characterizes the relative size of the superlattice to the resonant wavelength. In the calculations, we consider superlattices containing a total number of nanocubes ranging from N = 4 to N = 50² for 2D and N = 25³ for 3D case, which spans the Dicke limit (Φ ≪ 1) to the short wavelength limit (Φ ≫ 1). In the following, we note main observations from the results presented in Fig. 1.

First, similar trends in superradiant enhancement are observed between 2D and 3D superlattices [Figs. 1(c) and 1(d)]. Not only the 2D calculations still capture the key features of the structure-property relation with minuscule computation cost but also they permit a more transparent presentation of the wave functions (Fig. 4) for the discussion of superradiance localization in the later sections. Hence, we will focus on the results for 2D superlattices.

Second, the increase in the radiative enhancement, which grows approximately as ∼N for a small number of nanocubes, shows a sub-linear behavior in N for larger superlattices.¹⁹ This sub-linear behavior can be explained by the fact that the radiative coupling, Eqs. (4) and (5), attenuates for nanocubes that are further apart in a large superlattice. The increase is accompanied by shallow dips at the values indicated by the dashed lines in Figs. 1(c) and 1(d). In Ref. 31, similar dips were seen in dilute superradiant lattices where the inter-particle spacing was comparable to the wavelength, and explained by geometric resonances when the lattice constant approaches certain critical values.

Third, for a fixed number N of nanocubes, a system with smaller L_NC exhibits stronger enhancement, i.e. larger Γ_SR/γ_r. This is particularly noticeable in the limit of small nanocube size, where the number of nanocubes that can be coupled within a region of size ∼λ scales with their 2D packing density $(LNC+2Lshell)−2$⁠. The optical transition energy ℏω₀ (hence λ) is weakly size-dependent for the range of L_NC studied here. Therefore, the smaller L_NC (or smaller Φ) implies a stronger radiative coupling leading to larger Γ_SR/γ_r.¹⁹ 

Next, we systematically study the effect of shell length L_shell on the superradiant enhancement (Fig. 2). Two sizes of nanocubes L_NC = 5.0 and 20.0 nm, and five different shell lengths L_shell = 1.0, 10.0, 20.0, 40.0, and 80.0 nm were considered. The large L_shell resembles the situation of diluted 2D superlattice where the nanocrystal number density per unit volume is significantly reduced. Generally, as the L_shell increases, the superradiant enhancement is reduced to the point that, at large L_shell values, the superradiance enhancement is virtually independent of L_NC (e.g., L_shell = 40.0 and 80.0 nm, green squares, and orange triangles in Fig. 2, respectively).

As expected, a larger L_shell leads to a larger center-to-center distance between nanocubes. The larger distance weakens the radiative coupling [Eqs. (3)–(5)] and lowers the superradiant enhancement. Intriguingly, the effect of increasing L_shell onto the superradiant wave function is not intuitive.

The color plots in Fig. 4 show the total norm of the superradiant wave functions, obtained by summing over norms of the x, y, and z components of the 3 N complex eigenvector, at each site of the N_x × N_y superlattice. For the case where L_shell = 1.0 nm and L_NC = 5.0 nm, the superradiant wave function spreads out and delocalizes over the entire superlattice, resulting in the largest Γ_SR/γ_r. The biggest nanocubes, L_NC = 20.0 nm, with the same L_shell = 1.0 nm, exhibit the delocalization of Ψ_SR to a lesser extent, resulting in reduced Γ_SR/γ_r (see Fig. 2 for comparison). For a large L_shell = 40.0 nm, the N – particle wave function Ψ_SR fragments into several “pockets” of superradiance [Figs. 4(c) and 4(d)] for both L_NC sizes. The unexpected superradiance localization in a perfect superlattice is attributed to the weakened radiative coupling. To overcome the localization, strategies for enhancing radiative coupling are needed. For example, introducing a cavity may help overcome the delocalization through Purcell enhancement.³² 

In this subsection, we consider the role of static energy disorder stemming from the size dispersion of nanocubes prepared by chemical syntheses. We note that there are other contributions, such as different surface passivation, shape variation, and dynamic disorder, to the energy dispersion, even at low temperatures.¹⁵ Our choice to investigate the dependence of Γ_SR/γ₀ on δ_L and δE is based on the assumption that size dispersion captures the essence of static disorder in colloidal nanocrystals.

It is customary to use batches of colloidal nanocrystals with a narrow size distribution (δ_L of 10% or less) to produce well-ordered superlattices.^33,34 The reported average sizes for CsPbBr₃ nanocubes characterized with electron microscopy yield δL_NC in the range of 0.5–1 nm for L_NC = 5–20 nm.^35,36 This is equivalent to a variation of δ_L in the range from 5 to 10%. Therefore, we study the effect of the static energy disorder on the superradiant enhancement Γ_SR/γ_r for various δ_L as shown in Figs. 3(a)–3(c) and 5.

For all nanocube sizes L_NC, the energy disorder clearly diminishes the superradiant enhancement. Generally, in a more polydispersed sample, the larger energy variation among the constituent nanocubes significantly reduces the effect of the coupling defined in Eq. (3) of the total Hamiltonian. Hence, the eigenstates, especially the superradiant solution, in Eq. (1) are pushed towards those of an uncoupled system. For superlattices made of smaller nanocubes, the same size dispersion δ_L results in a larger energy width δE and, thus, more dramatic reduction in radiative enhancement compared to larger nanocubes. Notably, for the case of L_NC = 5.0 nm in Fig. 3(a), a level of size dispersion around 5% (δE = 27.5 meV) is sufficient to quench the cooperative effect. On the other hand, for L_NC = 20.0 nm, a part of the cooperative enhancement at δ_L = 15% (δE = 1.5 meV) is still retained in the superradiant state.

The influence of size dispersion on superradiance can be understood and visualized from the perspective of wave function distribution over the superlattice. Panels (e) and (f) of Fig. 4 show, in the case with energy disorder determined by δ_L = 10%, the total norm of the wave functions Ψ_SR of superlattices made of nanocubes with L_NC = 5.0 nm [panel (e)] and L_NC = 20.0 nm [panel (f)]. We note an important observation from a quick comparison with the case without energy disorder (δ_L = 0% implying δE = 0) in panels (a) and (b) of Fig. 4: the static energy disorder alters the localization property of the N – nanocube wave function from being delocalized over the superlattice to being localized on several nanocubes. The localization of the superradiant wave function gets stronger with the bigger value of energy width δE, making smaller nanocubes more susceptible to the effects of disorder than larger nanocubes.

The effect of energy disorder saturates past a certain point, with disorder greater than δ_L = 10%–15% showing similar emission characteristics in our simulations. Even in this limit, when energy disorder is large enough to cause large reductions in the superradiance rate, emission rates are still expected to increase with the superlattice size factor Φ, as seen in Figs. 3(a)–3(c), albeit at a slower rate than in a perfect superlattice with δ_L = 0. This is surprising because cooperative superradiance is destroyed by localization in this limit [Figs. 4(e) and 4(f)]. Instead, increasing the superlattice size factor in the presence of strong disorder increases the probability of the formation of local regions of strong emission, arising from the concerted local fluctuation of the diagonal energies E_n. That explains the cooperative robustness to disorder reported by Ghonge et al.²⁰ 

Figure 3(d) shows the results of Γ_SR/γ_r for the case of superlattices with δ_r = 0, δE = 0 (filled purple diamonds), with only random displacement δ_r = 10% and no energy disorder δE = 0 (filled green circles), with random displacement and energy disorder, δ_r = δ_L = 5% (empty red triangles) and δ_r = δ_L = 10% (empty green squares). The choice δ_r = δ_L is motivated by the fact that the spatial displacement is due to the size dispersion. Without energy disorder, some slight reduction is observed in the radiative enhancement due to the randomness in nanocube-to-nanocube distance $r̃⃗mn$⁠. Nonetheless, the most significant impact of size dispersion on the superradiant enhancement Γ_SR/γ_r occurs through the energy disorder [Eq. (6)] and not through the imperfect spatial arrangement of nanocubes, especially in superlattices made of nanocubes in strong or intermediate confinement regime.

The above results have uncovered competing effects of the nanocube size L_NC onto superradiant enhancement Γ_SR/γ_r. On the one hand, superradiance from small nanocubes is enhanced due to the higher packing density and shorter center-to-center distances (Figs. 1 and 2). On the other hand, small nanocubes are susceptible to energy disorder and show strongly suppressed superradiance (Fig. 4). For instance, the large value of δE for the case of L_NC = 5.0 nm leads to an almost complete quenching of Γ_SR/γ_r and the localization of the wave function Ψ_SR onto a few nanocube sites (Anderson localization).^8,39–41 The degree of localization is controlled by the energy width δE [Figs. 4(e) and 4(f)] and manifests the order-disorder transition in a nanocrystal superlattice. These results suggest a possible explanation for minuscule values of N_coupled in recent experiments.^7,8

To illustrate the point and showcase the immediate utility of the model for interpreting experimental results, we extracted experimental parameters from Refs. 7 and 8 and ran the calculations. In the first case, using L_NC = 9.0 nm, L_shell = 1.5 nm, and an energy width δE = 7.8 meV,⁸ we obtained, from the radiative Hamitonian, the enhancement factors ranging from 4.7 to 6.5 for 3D superlattices with N = 5 × 5 × 5 to 12 × 12 × 12. The calculated enhancement is higher than the reported factor of 2.7, and the difference is reconcilable by a slightly higher δE. In the second case, for 2D superlattices with N = 40 × 40 and L_NC = 3.8 nm, an enhancement factor of 1.9 in the superradiant decay rate was calculated when the ligand length L_shell was changed from 1.5 nm (oleylammonium bromide, experimental δE = 32.3 meV) to 0.35 nm (sodium bromide, δE = 24.2 meV), which is close to the experimental ratio of 1.5 between the fast-components of the decay rates.⁷ 

How to counteract the punishing effect of energy disorder? One strategy is to increase the L_NC   hence reducing the impact of size dispersion δ_L. Figure 5 maps the radiative enhancement Γ_SR/γ_r as a function of δ_L (upper panels) or δE (lower panels) for N_x × N_y = 5 × 5 (Φ ≪ 1, the Dicke regime) and 15 × 15 superlattices. The strategy for maintaining superradiance in the face of large δ_L is to shift to the upper part of the map (albeit with a modest Γ_SR/γ_r = 5–20), in other words, use larger nanocubes (L_NC = 10–20 nm). Under near-perfect conditions, the smaller nanocubes (L_NC = 5–10 nm) are clearly preferred as they deliver the largest radiative enhancement of Γ_SR/γ_r = 20–120.

The employed theoretical model is rooted in molecular aggregates²¹ and it is instructive to compare the described effects of disorder in nanocubes with molecules. Fidder et al. computed the effects of Gaussian distribution of transition energies (δE, diagonal disorder) and inter-molecular distances (off-diagonal disorder) on superradiance for a linear molecular aggregate in Φ ≪ 1 regime.⁴² Since size distribution is not a concern for molecules, the δE was set as a fraction of dipole-dipole interaction energy between neighbours. Both types of disorder were found to result in excitation localization to the ends of the chain. However, the quenching of radiative enhancement was found to be more sensitive to the off-diagonal disorder rather than the diagonal disorder, with one reason being the formation of strongly-interacting dimers. Jang and Silbey found that both types of disorder matter for excitation localization in a model of a molecular ring of light-harvesting complex 2,⁴³ with either one capable of dominating localization depending on their magnitude and correlation. In contrast to detrimental effect of disorder in identical emitters, Moix et al. considered a model of Fenna–Matthews–Olson complex with a built-in energy transfer (a biased system), finding that diagonal disorder enhances delocalization.⁴⁴ The brief comparison with molecular aggregates indicates that the effects of the disorder on radiative enhancement and localization are not axiomatic and depend on the specific emitter configuration and properties of the system. Continuing to adapt various scenarios from molecular aggregates to nanocrystal superlattices offers exciting avenues for future research.

Next, it is important to revisit several fundamental assumptions of the theoretical model. The Hamiltonian defined Eq. (1) is applicable in the ideal case where all dipoles are aligned, i.e. in-phase, and isotropic where the x, y and z components of the optical dipoles have the same oscillator strength. Including the phase decoherence between the optical dipoles or the anisotropy of the optical dipole may be essential to understand more comprehensively the superradiant behaviour.⁴⁵ So far, only the single-excitation regime has been considered.^19–21 At high excitation intensities, the formation of multiple excitons is likely,^46,47 in which the single-excitation Hamiltonian is no longer valid. In this case, a new framework to describe the system beyond single excitation remains to be developed for nanocrystal superlattices.

Finally, it is valuable to put the results in the context of the available superlattices and CsPbBr₃ nanocubes, and discuss the feasibility of future experiments. Current literature suggests a few viable methods for producing 2D superlattices. For example, irregular finite-sized monolayers of CsPbBr₃ nanocrystals have been made by solvent evaporation²⁷ and extended 2D monolayers (Φ ≫ 1) of close-packed CsPbBr₃ nanocrystals (L_NC = 10–15 nm) were achieved by solvent vapor annealing.²⁸ In a demonstration of control over lateral dimensions of 2D superlattices, Cohen et al. fabricated sub-micron (Φ ≥ 1) islands of CsPbBr₃ nanocrystals with L_NC = 9.7 ± 2.1 nm by electrohydrodynamic inkjet printing.²⁹ The latest generations of small colloidal CsPbBr₃ nanocrystals with L_NC = 5 nm (δ_L = 7.5%)⁴⁸ and large ones with L_NC = 19.8 ± 1.7 nm⁴⁹ have been reported as high-quality single photon emitters and are feasible building blocks for superradiant assemblies. However, the reported size distributions need to be substantially reduced to avoid localization, as suggested by our model.

From a spectroscopic point of view, the 10- to 10⁴-fold acceleration of radiative decay suggests superradiance timescales in the range from tens of femtoseconds (fs) to tens of picoseconds (ps) (using 200–400 ps radiative lifetimes of CsPbBr₃ nanocubes near 4 K^50–53). In the lower limit (i.e. tens of fs), the timescale of superradiance becomes comparable to that of the exciton formation.⁵⁴ In the intermediate range (1–2 ps), superradiance is comparable to the carrier cooling (0.3–0.7 ps in 8 nm CsPbBr₃ nanocubes at room temperature)⁵⁵ and the polaron formation (0.8 ps in a single crystal CsPbBr₃ at room temperature).⁵⁶ Given the typical duration of ultrashort laser pulses (10–100 fs), detecting superradiance requires femtosecond spectroscopies such as transient absorption or multidimensional spectroscopies. On the higher end of the range (tens of ps), the superradiance falls within the time resolution of state-of-the-art commercial streak cameras and could be probed by time-resolved photoluminescence.^1–3,6

We have presented a systematic theoretical investigation of the superradiant Hamiltonian for 2D nanocrystal superlattices and argued that they are feasible testbeds of collective phenomena. Although the focus of the study is on the lead halide perovskite nanocubes, the model can be easily adapted to study nanoscale emitters of other materials. The results presented here are in agreement with the earlier works of Mattiotti et al. and Ghonge et al., who studied superradiance and its suppression by energy disorder and thermal decoherence using the same model.^19,20

The reported calculations provided valuable insights into the non-trivial dependence of superradiance on experimentally relevant parameters such as L_NC, L_shell, δ_E, and δ_r, their role and interplay in excitation localization. The results clearly show that, in pursuing cooperative effects, having as identical nanocubes as possible in the superlattice is more important than achieving a perfect spatial order. And if perfect nanocubes are not possible, larger imperfect nanocubes are preferred over smaller ones. These predictions await experimental verifications and we are optimistic about the possibility of such tests given the interest in these materials and superradiance phenomena.

L.Z.T. was supported by the Molecular Foundry, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

D.B. was funded by the European Union (ERC Starting Grant PROMETHEUS, project No. 101039683). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

The authors have no conflicts to disclose.

T. P. Tan Nguyen: Conceptualization (equal); Data curation (lead); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (lead); Validation (equal); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Liang Z. Tan: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Dmitry Baranov: Conceptualization (equal); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (equal); Methodology (supporting); Project administration (lead); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.