Effects of Quantum and Dielectric Confinement on the Emission of Cs-Pb-Br Composites

The halide perovskite CsPbBr 3 belongs to the Cs-Pb-Br material system, which features two additional thermodynamically stable ternary phases, Cs 4 PbBr 6 and CsPb 2 Br 5 . The coexistence of these phases and their reportedly similar photoluminescence have resulted in a debate on the nature of the emission in these systems. Here, we combine optical and microscopic characterization with an effective mass, correlated electron-hole model of excitons in confined systems, to investigate the emission properties of the ternary phases in the Cs-Pb-Br system. We find that all Cs-Pb-Br phases exhibit green emission and the non-perovskite phases exhibit photoluminescence quantum yields orders of magnitude larger than CsPbBr 3 . In particular, we measure blue-and red-shifted emission for the Cs-and Pb-rich phases, respectively, stemming from embedded CsPbBr 3 nanocrystals. Our model reveals that the difference in emission shift is caused by the combined effects of nanocrystal size and different band mismatch. Furthermore, we demonstrate the importance of including the dielectric mismatch in the calculation of the emission energy for Cs-Pb-Br composites. Our results explain the reportedly limited blue shift in CsPbBr 3 @Cs 4 PbBr 6 composites and rationalize some of


Introduction
[17] Thus, understanding the emission properties of Cs-Pb-Br compounds is fundamental for the design of better materials and devices.The Cs-Pb-Br material system includes two additional, non-perovskite-type, ternary phases: The Cs-rich Cs4PbBr6 and the Pb-rich CsPb2Br5, sometimes referred to as the zero-dimensional (0D) and two-dimensional (2D) phases, respectively, owing to their crystal structure (cf. Figure S1).][30][31][32][33][34] This motivated a debate on whether such luminescence is caused by intrinsic factors [35- 43] or by nanocrystals (NCs) of the perovskite-type phase, CsPbBr3 (with a band-gap energy in the green region of the visible spectrum, ~2.4 eV), [7,42,[44][45][46][47][48][49][50][51][52][53][54] embedded in the non-perovskite matrix.Although substantial evidence in favor of the embedded NCs hypothesis have been reported, [49,50,52- 57] open questions remain, notably the exact mechanism that enhances the luminescence and the limited blue shift measured with decreasing NC size. [56] the present work, we combine photoluminescence (PL) and energy-dispersive X-ray (EDX) spectroscopies, as well as cathodoluminescence (CL) hyperspectral imaging and theoretical modeling to investigate the mechanisms behind the green emission in all three ternary phases of the Cs-Pb-Br material system.Our results provide additional evidence for the green emission in Cs4PbBr6 and CsPb2Br5 stemming from embedded CsPbBr3 NCs.Furthermore, we model the exciton emission in CsPbBr3@Cs4PbBr6 and CsPbBr3@CsPb2Br5 composites using an effective mass model, which includes a simplified treatment of electron-hole correlation, as well as the band gap and dielectric mismatch at the interfaces.Comparing this model with the commonly used effective mass model [58,59] suggests that accounting for the effects of finite confinement potentials and the Coulomb interactions with image charges at the interface can result in better estimates of NC sizes.The model explains the limited blue shift in small CsPbBr3 NCs embedded in Cs4PbBr6.Furthermore, differences in the type of confinement of CsPbBr3 NCs in Cs4PbBr6 and CsPb2Br5 explain the stronger emission of the former case and the measured red shift of the latter case.

Results and Discussion
Powder samples of the Cs-Pb-Br ternary phases -CsPbBr3, CsPb2Br5 and Cs4PbBr6were synthesized as described in Section S.1 of the Supporting Information (SI).The three ternary phases exhibit PL emissions in the range from 2.3 to 2.5 eV, as shown in Figure 1.The integrated PL intensities of the CsPb2Br5 and Cs4PbBr6 are about one and three orders of magnitude larger, respectively, than that of the CsPbBr3.Commensurately, the PL quantum yield (PLQY) at 1 sun equivalent illumination is 3 × 10 −3 %, 5%, and 8 × 10 −4 % for CsPb2Br5, Cs4PbBr6, and CsPbBr3, respectively.It is also noteworthy that the PL peak of green luminescence in the non-perovskite-type phases is shifted with respect to that of CsPbBr3.A blue shift of 50 meV and a red shift of 40 meV were measured for the PL peaks of Cs4PbBr6 and CsPb2Br5.The emission shifts are consistent among spectra measured on different sample areas and are significant, considering our measurement resolution (~5 meV, see Section S.2 of the SI) and the thermal energy at room temperature (~25 meV).We investigated the origin of the green emission in the CsPb2Br5 and Cs4PbBr6 samples using scanning electron microscopy (SEM), CL imaging, and EDX spectroscopysee Figures 2 to 4 below.Experimental details are given in Section S.2 of the SI.The CL maps for both Cs4PbBr6 and CsPb2Br5 show that luminescence in the visible spectral range stems from localized emitters, embedded in a solid matrix.Panchromatic (spectral range from 1.95 eV to 2.8 eV) and 500 nm bandpass-filtered maps are identical, confirming that the materials exhibit green emission (peak emission between 2.3 to 2.5 eV, cf. Figure 2 and 3).EDX elemental maps show that for Cs4PbBr6 the emitters exhibit higher Pb and lower Cs and Br counts (see Figure 2c-f ), whereas for CsPb2Br5 the emitters exhibit lower Pb and higher Cs counts, but Br counts remain unchanged (cf. Figure 3c-e).The compositions of the emitting regions relative to the matrix suggest that they are CsPbBr3.Therefore, we conclude that the green emitters are CsPbBr3 NCs embedded in the host Cs4PbBr6 or CsPb2Br5 matrix.In order to investigate the spectral features of the CsPbBr3 NCs, we performed hyperspectral CL mapping, using a low acceleration voltage (3.5 kV) in order to avoid beam damage and reduce the interaction volume.This allows us to improve the spatial resolution of the CL signal around the NCs.The CL map of Cs4PbBr6 acquired at larger magnification shows that luminescence stems from clusters of NCs, rather than individual emitters (see Figure 4a).The NCs exhibit emission maxima in the range from 2.44 to 2.47 eV (spectrum 1 in Figure 4b), which agrees well with the PL measurement (cf. Figure 1).Furthermore, no emission was detected from the matrix material (spectrum 2 in Figure 4b).For the CsPb2Br5 sample, the CL spectra of the emitters peak between 2.35 and 2.37 eV (spectrum 1 in Figure 4e), in good agreement with the PL results (cf. Figure 1).The matrix material exhibits no sharp CL peak.However, we detected a wide band of weak emission centered at ∼1.9 eV (spectrum 2 in Figure 4e) when measuring CL on matrix regions.This observation is consistent with defect emission at the interface, we measured for CsPb2Br5/CsPbBr3 films in previous work (cf.Ref. [60] and its supplemental material).Our CL-EDX correlative characterization clarifies the origin of the green emission in Cs4PbBr6 and CsPb2Br5.In order to understand the recombination mechanism and the nature of emission, we performed intensity-dependent PLQY measurements by exciting only the perovskite phase using a 445 nm laser (see Figure 5).Because of the excitation wavelength used, we can exclude charge transfer from the wide-gap, non-perovskite phases to the narrower perovskite ones.By investigating the slope  of the dependence of the emitted photon flux ( PL =   ) and the slope k of the dependence of the PLQY (PLQY =   ) on the illumination intensity (I) in a log-log plot, it is possible to access the charge recombination mechanisms.[63] In contrast, the values of  = 1.5 and  = 0.5 for CsPb2Br5 indicate a deviation from pure excitonic emission.This suggests that the green emission in this phase is dominated by nonradiative recombination.For the CsPbBr3 phase, we find  = 1.1 and  = 0.1, slightly deviating from a pure excitonic emission, and the absolute PLQY values are lower compared with those of Cs4PbBr6.This behavior suggests a stronger contribution of nonradiative recombination processes occurring in bulk CsPbBr3, as compared with NC domains in Cs4PbBr6, suggesting that the latter is less defective.The contribution of nonradiative recombination processes can also be associated to the low PLQY of out Cs-Pb-Br composites, compared with the previous reports. [31,32,64]he microscopy and spectroscopy results confirm consistently that the non-perovskite-type Cs-Pb-Br samples contain CsPbBr3 NCs embedded in a solid matrix.The confinement of the NCs also explain the blue shift of the PL measured for Cs4PbBr6, with respect to bulk CsPbBr3 emission (Figure 1).The best achievable spatial resolution for our CL experiments is in the range of ~60-80 nm, as estimated from the density of Cs4PbBr6 (4.29 g/cm 3 ) [65,66] and CsPb2Br5 (5.76 g/cm 3 ), [34,67] the electron beam parameters, and Gruen's equation (Eq.S1 in Section S.2 of the SI).This resolution limit, as well as the diffusion of charge carriers before they recombine, limits the accuracy of the estimated NC size directly from CL experiments.Because the magnitude of quantum and dielectric confinement effects is generally related to the size of the NCs, it is interesting to examine whether a theoretical model can be used to provide an explanation for the origin of different blue-and red-shift effects and their dependence on the size of the emitting CsPbBr3 NCs.
An important and commonly used approximation to the emission in such systems, within the effective mass approximation, has been proposed by Brus in the 80s. [58,59,68]This model considers a spherical NC in an infinite potential well, where the Coulomb interaction is strongly screened, such that the exciton (electron-hole) wave function is uncorrelated.However, as stated by Brus in his original paper, [58] this model can be a poor approximation for large band gap materials with moderate NC sizes, because in those systems the Coulomb energy is comparable to the confinement energy and the electron-hole correlation can be important.Considering that CsPbBr3 is indeed a (relatively) large gap material with a (relatively) significant exciton binding energy (35-60 meV), [39,[69][70][71][72] we opt for a simple model, enforcing confinement in one dimension, that includes some form of electron-hole correlation, while remaining mathematically tractable and computationally inexpensive.We show below that this is sufficient for a qualitative explanation of our results.
We apply an effective mass model, proposed by Rajadell et al. for CdSe nanostructures, [73][74][75][76][77] The model explicitly treats the correlated electron-hole pair, including Coulomb interactions and dielectric effects, based on the method of image charges that approximate polarization terms in the exciton Hamiltonian (see Section S.3 in the SI for further details).Within this framework, the Hamiltonian that describes the electron-hole pair is given by: (  ,  ℎ ) =  () (  ) +  (ℎ) where  () (  ) is an effective-mass, single-particle Hamiltonian, that contains image charge effects, which is described in detail in Eq.S2 of the SI.  (  ,  ℎ ) is a generalized electron-hole Coulomb interaction potential, calculated using the method of image charges.This term describes the interaction of a charged particle with interface image charges induced by the other particle and can be written in one dimension as: [77]   (  ,  ℎ ) = ∑ where  ∥, is the in-plane particle position (perpendicular to the confinement direction),   is the particle position in the confined direction,  is the width of the well (NC size), and   = ( where  1 and  2 are the dielectric constant of the NC and host, respectively.We seek a two-particle wave-function solution to this Hamiltonian, namely where the wave function is approximated by [76] Ψ(  ,  ℎ ) =   (  ) ℎ ( ℎ ) where  /ℎ are the single-particle wave functions for electron and hole, and  is a variational parameter.Solving Eqs. ( 1)-( 4) requires electron and hole effective masses as well as dielectric constant values.For the former, we use the values reported by Protesescu et al. [71] for CsPbBr3.For the latter, we used density functional perturbation theory (DFPT) to obtain computed values for all three Cs-Pb-Br phases (see Section S.4 in the SI for details).All parameters used in the model are summarized in Table S1 of the SI.
When  1 >  2 , i.e., the dielectric constant of the NC is larger than that of the host, there is dielectric confinement.The image charge induced at the interface exhibits the same sign as the confined charge and the electric field owing to the confined charge penetrates the matrix region, as shown schematically in Figure 6a.This reduces the effective dielectric constant, with respect to  1 , and enhances the Coulomb interaction between electron and hole. [73,78]The electron-hole Coulomb interaction is further modulated by the overlap between the electron and hole wave functions, which depends on the degree of confinement.Consequently, the exciton binding energy will be a function not only of  1 and  2 , but also of the NC size.
We first use the above model to investigate the effect of a dielectric mismatch for a NC confined by an infinite potential.We calculate the shift of the emission energy, Eem with respect to the bulk band gap of CsPbBr3, Eg, as a function of the NC size , for various values of  2 (see Figure 6b).For small , quantum confinement is dominant and the effect of dielectric mismatch on  em −  g is negligible.As  increases and quantum confinement is reduced, the effect of dielectric mismatch becomes more important, and  em −  g decreases with  2 owing to a stronger dielectric confinement.
Dielectric effects also directly impact the exciton binding energy, Eb, namely the difference between the emission energy and the single-particle bandgap, as shown in Figure 6c,d.Because dielectric confinement reduces the effective dielectric constant and Coulomb screening, it increases  b .Specifically,  b rapidly decreases with  for  2 <  1 .81]  Next, we investigate the effect of a finite confinement potential on the emission energy, by including the band mismatch between NC and matrix material, i.e., Cs4PbBr6 or CsPb2Br5 (see Section S.3 in the SI for details).We use the literature-reported band-gap energies of the Cs-Pb-Br phases, i.e., 2.4 eV for CsPbBr3 (in reasonable agreement with our measurements), [7,[44][45][46][47][48] 4.0 eV for Cs4PbBr6 [28][29][30][31][32] and 3.7 eV for CsPb2Br5. [18,33,34]The band mismatch for CsPbBr3/Cs4PbBr6 results in a type I band alignment, i.e., both hole and electron wave functions are confined, as deduced from both theoretical calculations [82] and (X-ray and UV) photoelectron spectroscopy. [83,84]We model the type I alignment using a fixed band mismatch Δ g 2 (where Δ g =  g Cs 4 PbBr 6 −  g CsPbBr 3 ) for the conduction and valence band mismatch.In the case of CsPbBr3/CsPb2Br5, some theory reports a quasi-type I alignment (only one carrier confined), [85] while experimental characterization assigns a type II band alignment, [86] i.e., the electrons and holes are confined to different regions, which reduces their wave function overlap.We modeled this by setting a small but negative valence band offset ∆ V = −0.1 eV and a corresponding conductions band offset of ∆ C = ∆ g + 0.1 eV (where Δ g =  g CsPb 2 Br 5 −  g CsPbBr 3 ).
The effect of the finite band mismatch is apparent in the calculated shift in emission energy  em −  g , shown in Figure 7, where the finite potential reduces the magnitude of the quantum confinement effect.This effect is stronger in Cs4PbBr6 than it is in CsPb2Br5, partly because of the larger band gap of the former, but mostly due to the more effective quantum confinement of the type I alignment.Generally, for a finite confinement potential the wave function extends into the barrier regions (cf. Figure 6a), making quantum confinement less effective and also reducing the effective dielectric constant of the system, thereby favoring dielectric confinement. [73]At the same time, the decrease in electron-hole overlap reduces the Coulomb interaction, hindering dielectric confinement.This competition results in a complex dependence of the exciton binding energy on the finite potential barriers.In the present calculation, confinement effects are apparent for  < 6.3 nm (< 7 nm) for Cs4PbBr6 (CsPb2Br5), providing a guide to the size regimes for which quantum or dielectric confinement is dominant.While the similarity to the exciton Bohr radius of Bulk CsPbBr3 (~7 nm) [71] is notable, we emphasize that this is a 1D model, which should not be compared fully quantitatively to experiment.We can now qualitatively interpret our experimental data, in light of the computed result from the above model.In the case of Cs4PbBr6, it is clear that the size of the embedded CsPbBr3 NCs lies in the strong confinement regime, i.e., quantum confinement is dominant and results in the abovereported blue shift and PLQY enhancement of the exciton emission.The blue shift in this system is reduced with respect to the conventional estimations using a simple effective mass model, owing to the combined effect of a finite confinement potential and the reduced effective dielectric constant.This rationalizes the limited emission shift achieved experimentally for CsPbBr3@Cs4PbBr6 composites. [32,56]For CsPb2Br5, a plausible explanation for the red shift measured in our PL and CL experiments is that there is a weaker quantum confinement caused by type II band alignment, coupled with larger NC sizes, expected from the similar formation enthalpies of the CsPbBr3 and CsPb2Br5 phases. [26,27,60]For larger NCs, the dielectric mismatch can cause enough of an increase in the exciton binding energy to overcome the effect of the weak quantum confinement, resulting in a net red shift in  em with respect to the bulk  g .This also can explain the large discrepancy between NC sizes estimated with a simple effective mass model and the TEM images, reported for CsPb2Br5.For this case, the error in the estimation of the NC size was found to be as large as 4 nm, which is much larger than the error for Cs4PbBr6 (0.6 nm). [87]ile the simple model provides a plausible qualitative explanation for all emission energy patterns observed in our experiments, we caution that further refinement is needed to gain a complete quantitative understanding of the PL results from the composite Cs-Pb-Br materials.First, the model used in the present work is one-dimensional, whereas the effects of dielectric and quantum confinement could be stronger and have a more complex relationship in a system confined in all three spatial dimensions.For example, the dielectric tensor of CsPb2Br5 is highly anisotropic, i.e., the effect of dielectric mismatch would depend strongly on the confinement direction.Also, in our model we did not consider changes in the electron and hole effective masses, which can also influence the exciton levels. [68,88]Furthermore, considering the exciton fine structure of the CsPbBr3 including effective mass non-parabolicity using an energy-dependent effective mass, as described by Sercel et al., [89,90] will help improving the accuracy of the model, making it more comparable with experimental results.Ghribi et al. recently used similar models, combined with exciton fine structure to study dielectric confinement in CsPbBr3 NCs. [91]Our results on the dielectric tensor, band alignment and finite confinement effect of the NCs in Cs4PbBr6 and Cspb2Br5 could contribute to refining this model, explain experimental results and provide material design guidelines for Cs-Pb-Br composites.Additionally, this work could be extended to study other halide composites, such as Cs4PbCl6 and Cs4PbI6.[94] The change in the halide offers an additional degree of freedom (i.e., composition) for tuning the exciton emission in Cs-Pb-X (X=Br, Cl, I) composites.

Conclusions
In summary, we synthesized green-luminescent, ternary phases in the Cs-Pb-Br material system, namely perovskite-type CsPbBr3 and non-perovskite-types Cs4PbBr6 and CsPb2Br5.We investigated their emission properties using PL experiments, which showed that the non-perovskite-type phases exhibit a higher quantum yield, with a blue shift of 50 meV and a red shift of 40 meV (with respect to the PL peak of bulk CsPbBr3) for the Cs4PbBr6 and CsPb2Br5 phases, respectively.Microscopic characterization by means of correlative CL hyperspectral imaging, as well as EDX elemental mapping, showed that the luminescence emission at 2.45 eV and 2.36 eV of Cs4PbBr6 and CsPb2Br5 stems from nanocrystals of CsPbBr3, embedded in the matrix material.We qualitatively explained the experimental results using a one-dimensional effective-mass model that considers electron-hole correlation, as well as band gap and dielectric mismatch at the interface of the nanocrystal and host materials.This model allowed us to rationalize the effect of the quantum and dielectric confinements on the emission shifts.We showed that including dielectric effects is important for estimating the emission energy and particle size.In the light of the correlated model, we conclude that the CsPbBr3@Cs4PbBr6 composite is formed by small, strongly quantum-confined CsPbBr3 nanocrystals in a Cs4PbBr6 matrix, while the CsPbBr3@CsPb2Br5 composite is formed by large and weakly confined CsPbBr3 nanocrystals in a CsPb2Br5 matrix, in which dielectric effects dominate, resulting in a net red shift.quantification.Therefore, Figures 2f and 3 show elemental maps and normalized line-scans, which allow us to extract changes in the relative composition among phases.

S.3. Effective Mass Exciton Model Including Electron-Hole Correlation
The exciton in a CsPbBr3 NC, confined in a host matrix of Cs4PbBr6 or CsPb2Br5, is modelled using a nanoplatelet effectively confined only in the  direction, with the perpendicular ,  plane modeled as a square with length  = 30 nm, as proposed by Rajadell and Planelles. [7]This approximation allows us to understand how the dielectric confinement depends on the NC size and interpret our experimental results.However, it does not offer quantitative information.The model approximates the effects of the self-interaction and generalized Coulomb interaction, arising from the dielectric mismatch at the interface, using the method of image charges as described by Kumagai. [8]e effective mass, exciton Hamiltonian, and exciton wave function are given in Eqs. ( 1)-( 4) in the main text.Here, we present the single-particle equations for completeness.We start with the single-particle Hamiltonian to describe either the electron or the hole in the system: where the first term is the kinetic energy operator,  pot is the confining potential, which arises from band discontinuity at the interface between the NCs and the host material, and   self is the selfinteraction potential, i.e., the interaction of a charge with the induced charges, caused by the dielectric mismatch at the interface.This self-interaction potential is calculated using the method of image charges in a 1D quantum well: [7,8] where  1 and  2 are the dielectric constants of the confined NC and the host material, respectively,   is the particle coordinate in the confined direction,  is the width of the well (NC size) and . The wave function is found by minimizing the exciton emission energy  em = ⟨Ψ(  ,  ℎ )|(  ,  ℎ )|Ψ(  ,  ℎ )⟩.Details of the calculation, as well as the Mathematica code used for our calculations, can be found in references [7,9].

S.4. Calculation of the High-frequency Dielectric Constants
Calculations of high-frequency dielectric constants were performed using the Vienna Ab-initio Simulation Package (VASP), [10] based on the Perdew-Burke-Ernzerhof (PBE) form of the generalized gradient approximation to describe the exchange-correlation interactions [11] and including spin-orbit coupling.Dispersion interactions were included within the Tkatchenko-Scheffler scheme, [12] using an iterative Hirshfeld partitioning of the charge-density. [13,14]Core electrons were described by the projector augmented waves method. [15,16]The orthorhombic structure of CsPbBr3 [3] was used for the electronic structure calculations.A planewave basis set with cutoff of 700 eV was used and -space integration was carried out on a 6 × 6 × 6 grid.For Cs4PbBr6, the atomic structure reported by Velázquez et al. [17] was used as a starting point.A cutoff of 500 eV and a 4 × 4 × 3 -space grid was used.The structure of CsPb2Br5 reported by Cola and Riccardi [4] was used as a starting point; a cutoff of 700 eV and a 3 × 3 × 2 -space grid were used.The geometries were optimized using a conjugated gradient algorithm until the largest force was below 0.001 eV/Å.The high-frequency dielectric constants were calculated using density functional perturbation theory as implemented by Gajdoš et al. [18] in the VASP package.Convergence was tested for all the parameters in the calculations.
Figure S1 summarizes the calculated dielectric tensors of all of the Cs-Pb-Br ternary phases.A particular feature of the dielectric tensors of CsPbBr3 and CsPb2Br5 is their anisotropy.For CsPbBr3, the dielectric constant is larger along the longest lattice vector of the orthorhombic structure ( = 11.75Å).The second largest dielectric constant is along the  = 8.54 Å lattice vector, and the smallest one is along the  = 8.06 Å direction.This suggests that in the CsPbBr3 the polarizability increases as the unit cell is stretched, which results from the tilting of the PbBr6 octahedra.This phenomenon appears to be related to the one described by Kang and Biswas, [19] in which the crossgap hybridization of Pb-Br p-orbitals results in large Born effective charges and dielectric constants.The dielectric constant of Cs4PbBr6 is isotropic, which can be understood when looking at the primitive cell in Figure S1b.The disjoint octahedra not only reduce the band dispersion, but also hinder the lattice polarization, which results in a lower dielectric constant.The orientation of the octahedra relative to one another seems to have little influence on the dielectric response of the material.Finally, CsPb2Br5 exhibits strong polarizability on the  plane, owing to the face-sharing PbBr2 prisms forming planes (Figure S1c).This results in an in-plane dielectric constant that is even larger than that of CsPbBr3, regardless of the larger band-gap energy.The polarizability in the  direction is closer to that of Cs4PbBr6.The discontinuous planes inhibit lattice polarization and reduce band dispersion, which in turn reduces the dielectric constant.In the calculations of the exciton model, we use a dielectric constant of  1 = 4.2 for the confined CsPbBr3 (average of  in all the directions, because the variation is small).The dielectric constant of the host materials is  2 = 3.1 or 3.8 for Cs4PbBr6 and CsPb2Br5, respectively.The lowest  of the dielectric tensor of CsPb2Br5 is used, because the effect of the dielectric mismatch is strongest when  1 >  2 .Naturally, the anisotropy of CsPb2Br5 will reduce the dielectric confinement in two directions.

Figure 1 .
Figure 1.PL spectra of the three Cs-Pb-Br ternary phases.All the phases exhibit green luminescence between 2.3 and 2.5 eV.Spectra are plotted in semi-logarithmic scale to show the differences of the PL intensity.Insets show photographs of the powder samples under ambient light (left) and 205 nm LED illumination (right).

Figure 2 .
Figure 2. (a) SEM image of Cs4PbBr6 crystal and (b) a corresponding CL intensity map, filtered at 500 nm, showing the localized emitters embedded in the crystal.(c-e) EDX elemental distribution maps and (f) normalized line-scansalong the yellow arrow in (e)of a magnified region, showing clear enrichment of Pb as well as depletion of Cs and Br, correlated with the yellowcircled position of highly luminescent clusters.

Figure 3 .
Figure 3. (a) SEM image of a CsPb2Br5 crystal and (b) a corresponding CL intensity map, filtered at 500 nm.EDX elemental distribution maps of (c) Br, (d) Cs, and (e) Pb show a clear enrichment

Figure 4 .
Figure 4. (a) CL intensity map, filtered at 500 nm, on a flat surface of a Cs4PbBr6 crystal.(b) Corresponding CL spectra (hyperspectral measurement without filter) of selected areas.(c) SEM image of the surface of CsPb2Br5.(d) Corresponding CL map, filtered at 550 nm, and (e) CL spectra of selected areas.Spectrum 2 is magnified x5 for visibility.

Figure 5 .
Figure 5. (a) Photoluminescence emitted photon flux and (b) PLQY as a function of the excitation intensity.The data are plotted in a logarithmic scale and linearly fit.The slope of the PL () and PLQY () curves reveals the type of dominant recombination, see text for details.

Figure 6 .
Figure 6.Effect of confinement and dielectric mismatch in a simple 1D model of a CsPbBr3 NC.(a) Top: schematic representation of a single-particle wave function, e/h, confined in an infinite (dashed) and finite (solid) potential.Bottom: schematic representation of the effect of dielectric mismatch on the electrostatic potential , , for  1 >  2 (solid) and  1 <  2 (dashed).(b) Calculated shift of the emission energy with respect to the bulk band gap energy, Eem -Eg, as a function of the NC size for different values of  2 .(c) exciton binding energy, Eb, as a function of  2 for various NC sizes.(d)  b as a function of  for various  2 .

Figure 7 .
Figure 7. Emission energy shift,  em −  g , of a CsPbBr3 NC embedded in (a) Cs4PbBr6, or (b) CsPb2Br5, as a function of the NC size.Dielectric mismatch is modeled with  1 = 4.2 for CsPbBr3,  2 = 3.1 for Cs4PbBr6, and  2 = 3.8 for CsPb2Br5.Results for infinite (dashed curves) and finite (solid curves) confinement potentials, with type I and type II alignments for Cs4PbBr6 and CsPb2Br5, respectively, are compared.

Figure S2 .
Figure S2.Schematic representations of (a) CsPbBr3, (b) Cs4PbBr6, and (c) CsPb2Br5.The diagonal elements of the dielectric tensor are color-coded to the lattice vector.The geometry of the crystal determines the magnitude of the dielectric constant (see text for details).

Table S1 .
Electron and hole effective masses  /ℎ * (Ref.[20]),dielectric constants , band-gap energies  g and the band mismatch parameter Δ g /2, used in the model of the finite potential barriers