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        "<!-- image -->\n\nRESEARCH ARTICLE |  NOVEMBER 22 2023 ## Tuning perovskite nanocrystal superlattices for superradiance in the presence of disorder\n\nSpecial Collection: 2023 JCP Emerging Investigators Special Collection\n\nT. P. Tan Nguyen\n\n; Dmitry Baranov\n\n\ue923\n\n<!-- image -->\n\nCheck for updates\n\nJ. Chem. Phys. 159, 204703 (2023)\n\nhttps://doi.org/10.1063/5.0167542\n\n\ue918 CHORUS ## Articles You May Be Interested In\n\nTheory of high-temperature superfluorescence in hybrid perovskite thin films - J. Chem. Phys. (September 2024)\n\nElectromagnetic enhancement spectra of one-dimensional plasmonic hotspots along silver nanowire dimer derived via surface-enhanced fluorescence - J. Chem. Phys. (January 2024)\n\nShape-dependent oxidation rates of nano-structured silver particles - J. Chem. Phys. (September 2024) ## Webinar From Noise to Knowledge\n\n13th Register now May\n\n<!-- image -->\n\nInstruments\n\nUniversit\u00e4t Konstanz\n\n<!-- image -->\n\n\ue929 \ue92d\n\n<!-- image -->\n\n<!-- image -->\n\n<!-- image -->\n\n<!-- image --> ## Tuning perovskite nanocrystal superlattices for superradiance in the presence of disorder\n\n<!-- image -->\n\n<!-- image --> ## AFFILIATIONS - 1 M2I Formation, Sophia Antipolis, Mougins 06250, France\n- 2 Molecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA\n- 3 Division of Chemical Physics, Department of Chemistry, Lund University, P.O. Box, 124, SE-221 00 Lund, Sweden\n\nNote: This paper is part of the 2023 JCP Emerging Investigators Special Collection. - a) Authors to whom correspondence should be addressed: phuctan3108@gmail.com and dmitry.baranov@chemphys.lu.se ## ABSTRACT\n\nThe cooperative emission of interacting nanocrystals is an exciting topic fueled by recent reports of superfluorescence and superradiance in assemblies of perovskite nanocubes. Several studies estimated that coherent coupling is localized to a small fraction of nanocrystals ( 10 -7 -10 -3 ) within the assembly, raising questions about the origins of localization and ways to overcome it. In this work, we examine singleexcitation superradiance by calculating radiative decays and the distribution of superradiant wave function in two-dimensional CsPbBr3 nanocube superlattices. The calculations reveal that the energy disorder caused by size distribution and large interparticle separations reduces radiative coupling and leads to the excitation localization, with the energy disorder being the dominant factor. The single-excitation model clearly predicts that, in the pursuit of cooperative effects, having identical nanocubes in the superlattice is more important than achieving a perfect spatial order. The monolayers of large CsPbBr3 nanocubes (LNC = 10-20 nm) are proposed as model systems for experimental tests of superradiance under conditions of non-negligible size dispersion, while small nanocubes (LNC = 5-10 nm) are preferred for realizing the Dicke state under ideal conditions. - \u00a9 2023 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0167542 ## I. INTRODUCTION\n\nThe 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-assembly 10 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.\n\nconditions. 13,14 Semiconductor nanocrystals, resembling 'artificial atoms,' are interesting as quantum emitters with facile tunability of emission frequency by changing nanocrystals' size or composition. Non
        "Localization is one consequence of disorder that emerges from the recent experiments on perovskite nanocrystal superlattices. For example, Rain\u00f2 et al. proposed the existence of superfluorescent domains within superlattices with an\n\nestimated 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 = 10 -10 6 7 nanocrystals, the yield of cooperativity is low and requires an explanation.\n\nThe CsPbBr3 superlattices with N = 10-10 4 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 &gt; 10 3 ) . 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.\n\nIn 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. ## II. THEORETICAL MODEL\n\nLet us consider the general case of a superlattice consisting of Nx Ny , and Nz nanocrystals along x, y and z axes respectively ( N = NxNyNz ) . Each nanocrystal is assumed to be a cube with an edge length LNC and is covered with a homogeneous ligand shell of thickness Lshell. In the single-excitation regime, the system can be described by using the tight-binding Hamiltonian H rad . 19,21\n\n<!-- formula-not-decoded -->\n\nThe above Hamiltonian is applicable to both 3D and 2D superlattices, the latter by setting Nz = 1.\n\nThe first term in Eq. (1) describes a system of N uncoupled nanocrystals. The corresponding matrix consists of the 3 \u00d7 3 diagonal blocks in the basis of bright triplet states denoted by \u2223 \u03b1 \u27e9 . The complex diagonal term is\n\n<!-- formula-not-decoded -->\n\nwhere h \u03c9 0 is the energy of bright exciton (without fine structure splitting) and \u03b3 r is the radiative recombination rate of an exciton in a single nanocrystal.\n\nThe off-diagonal second term in Eq. (1)\n\n<!-- formula-not-decoded -->\n\noriginates from the radiative coupling between the nanocrystals in the superlattice. 19,21 The real and imaginary parts of J mn are given as\n\n<!-- formula-not-decoded -->\n\n<!-- formula-not-decoded -->\n\nIn the above, j i / y i denotes the spherical Bessel function of the first/second kind of order i . Dmn \u03b1\u03b2 = \u02c6 e \u03b1 \u22c5 \u02c6 e \u03b2 -3 \u02c6 ( e \u03b1 \u22c5 \u02c6 rmn )( \u02c6 e \u03b2 \u22c5 \u02c6 rmn ) , \u02c6 e \u03b1 is the unit vector in the \u03b1 direction, \u02c6 rmn = \u20d7 rmn / rmn and \u20d7 rmn is the position vector connecting the centers of the m th and n th nanocrystals. The resonant wave number k 0 = \u03c9 0 \u221a \u03b5 opt / c in which \u03b5 opt is the optical dielectric constant of the material in the relevant frequency range.\n\nThe eigenvalues and eigenvectors of Hamiltonian H rad can be obtained by diagonalizing of the corresponding 3 N \u00d7 3 N matrix. Th
        "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.\n\nFinally, 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 finitesized monolayers of CsPbBr3 nanocrystals have been made by solvent evaporation 27 and extended 2D monolayers ( \u03a6 \u226b ) 1 of closepacked CsPbBr3 nanocrystals (LNC = 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 ( \u03a6 \u2265 1 ) islands of CsPbBr3 nanocrystals with LNC = 9.7 \u00b1 2.1 nm by electrohydrodynamic inkjet printing. 29 The latest generations of small colloidal CsPbBr3 nanocrystals with LNC = 5 nm ( \u03b4 L = 7.5%) 48 and large ones with LNC = 19.8 \u00b1 1.7 nm 49 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.\n\nFrom a spectroscopic point of view, the 10 -to 10 4 -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-theart commercial streak cameras and could be probed by time-resolved photoluminescence. 1-3,6 ## V. CONCLUSIONS\n\nWe 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\n\nThe reported calculations provided valuable insights into the non-trivial dependence of superradiance on experimentally relevant parameters such as LNC, Lshell, \u03b4 E , and \u03b4 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. ## ACKNOWLEDGMENTS\n\nL.Z.T. was supported by the Molecular Foundry, a DOE Office of Science Us
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