Introduction

Lead halide perovskites have recently emerged as one of the most promising materials for a broad range of applications ranging from photovoltaics to optoelectronics1,2,3,4 and potential applications as quantum light sources5,6,7,8,9,10. While intense experimental and theoretical efforts have been made to explore their unique optical and electronic properties, the physics of band-edge charge complexes (e.g. excitons, trions, biexcitons), whose recombination is at the origin of the luminescence, remains unclear.

For instance, the ordering of bright and dark sublevels in the exciton fine structure is a subject of ongoing debate, though essential for the development of quantum light sources11,12,13 and for spin-based technologies14,15,16. The band-edge exciton of perovskites is expected to split by electron-hole exchange interaction into a dark ground singlet state (with zero angular momentum) and a bright triplet (with angular momentum unity)17,18,19,20. However, since cesium lead halide perovskite nanocrystals (NCs) exhibit strong photoluminescence (PL) with triplet-line emission at cryogenic temperatures19,21,22,23,24, it has been proposed that a Rashba effect reverses the fine structure level ordering and places the dark singlet above the bright triplet24,25,26,27, making perovskites an exception among all bulk semiconductors and all existing semiconductor heterostructures. Studies on Mn-doped and undoped ensembles of CsPbCl3 NCs28 have suggested that the dark state is positioned below the bright triplet and that a slow bright–dark state relaxation at cryogenic temperatures gives rise to almost exclusively bright-state emission. Then, low-temperature single-NC magneto-optical spectroscopy29 has provided direct signatures of the dark singlet exciton emission, showing that the singlet is positioned few meV below the bright triplet in hybrid organic–inorganic formamidinium lead bromide (FAPbBr3)30 and inorganic cesium lead iodide (CsPbI3)9 NCs, in agreement with theoretical estimations of confined exchange interaction.

Similarly, no general picture is available for many-body Coulomb interactions in perovskites, besides investigations of the size dependence of the biexciton and trion binding energies in single FAPbBr3 NCs31. The biexciton binding energy has important consequences in the development of perovskite-based light-emitting technologies, since a large shift of the biexciton recombination lines with respect to those of the band-edge exciton would reduce the emission color purity under high-flux excitation. Its sign also fixes the transition that may sustain population inversion and NC gain for lasing applications32. For CsPbBr3 perovskites, contradictory results have been reported on the amplitude and sign of the biexciton energy shift in the range −100 meV to 100 meV, using time- and spectrally- resolved PL33,34,35, transient absorption36,37,38,39 or nonlinear40 spectroscopic methods on NC ensembles. Heralded single-NC spectroscopy on CsPbBr3 and CsPbI3 perovskites lead to estimations of the biexciton binding energy at room-temperature with inherently low spectral precision and even uncertainties on the sign of the interaction41.

Here, we use low-temperature single-NC magneto-optical spectroscopy to reveal the entire band-edge exciton fine structure and charge-complex binding energies of nearly bulk CsPbBr3 perovskite NCs, the reference materials for emerging applications and theoretical modeling. We demonstrate that the dark singlet exciton lays at least 3.6 meV below the bright triplet, at variance with the models based on the Rashba effect24,25,26. This settles the debate on the bright-dark exciton level ordering in halide perovskite nanostructures. Moreover, the spectral fingerprint of biexciton recombination is also evidenced in these materials and provides precise measurements of the biexciton binding energy. Importantly, we show evidence for universal scaling laws that relate the exciton fine structure splitting, the trion and biexciton binding energies to the band-edge exciton energy in lead-halide perovskite nanostructures. For that purpose, we combine the present results on CsPbBr3 single NCs with spectroscopic data on various perovskite NCs previously investigated9,21,22,24,30,31,42. Additional measurements on single CsPb(ClxBr1-x)3 NCs further extend the validity of these dependences over a broader emission spectral range. These scaling laws solely based on quantum confinement effects and dimensionless energies offer a general predictive picture for the interaction energies within charge complexes photo-generated in lead halide perovskites, regardless of their chemical composition.

Results

Tailoring nearly-bulk CsPbBr3 NCs

Magnetic brightening of the dark exciton state is challenging in CsPbBr3 perovskite NCs. Indeed, the PL intensity of the dark state is inversely proportional to the square of the bright-dark energy splitting, while the bulk electron-hole exchange interaction in this material is particularly strong among lead-halide perovskites9. Moreover, confinement effects in NCs enhance the exciton fine structure splittings, which makes magnetic brightening of the dark singlet even more difficult. The CsPbBr3 NCs specifically prepared for this study are cuboids with sides that largely exceed the bulk exciton Bohr radius \({a}_{{{{{{\rm{B}}}}}}}\) ~3.1 nm in order to minimize quantum confinement effects on the dark-bright exciton splittings. These large NCs are synthesized under ambient conditions with the ligand-assisted reprecipitation approach, using a novel ligand molecule, oleylguanidinium bromide (OGB), to obtain dispersible and large CsPbBr3 perovskite NCs (See Methods, Supplementary Figs. 1 and 2 for details of the NC synthesis and characterization). The 3D size and shape of these OGB-capped CsPbBr3 NCs with orthorhombic crystal phase are characterized using 45°-tilted scanning electron microscopy (Fig. 1a). With average base lengths 195 ± 18 nm, 41 ± 8.5 nm and a thickness of ~40 nm, these NCs are nearly bulk-like particles with reduced quantum confinement. The ensemble absorption spectrum presents a threshold at ~520 nm (inset Fig. 1b), which roughly corresponds to the room-temperature bulk electronic bandgap. Under light excitation at 405 nm, these NCs exhibit a bright green PL centered at 2.36 eV, with a high color purity (full width at half-maximum, FWHM, of 77 meV) and a PL quantum yield of 32 % (see inset Fig. 1b).

Fig. 1: Characterization of the large OGB-capped CsPbBr3 NCs.
figure 1

a 45°-Scanning electron microscopy image of the OGB-capped CsPbBr3 NCs. Their room-temperature absorption and emission spectrum is shown in the inset of b. b Representative PL spectrum of a single NC at 3.5 K under laser excitation with wavelength 488 nm and intensity 50 W cm−2 (integration time 10 s). It displays the bright exciton triplet ZPLs, the trion recombination line as well as longitudinal optical (LO) phonon replicas.

Full size image

Band-edge exciton fine structure

At low temperature (\(3.5\,\pm 0.3\) K) and under continuous-wave laser excitation with a wavelength of 488 nm (2.54 eV), the PL spectra of single NCs are characterized by a triplet of resolution-limited zero-phonon lines (ZPLs) corresponding to the recombination of the bright triplet exciton, followed by optical phonon replica, as exemplified on Fig. 1b. The photon energy of these recombination lines is distributed in the range 2.3-2.35 eV, i.e. across the low-temperature electronic bandgap\(\,{E}_{{{{{{\rm{gap}}}}}}}\) = 2.342 eV of bulk CsPbBr343. The spread of emission energies among the nearly-bulk CsPbBr3 NCs may be due to stochastic deviations of cation positions or torsions of the halogen octahedra, which translate into a complex potential landscape felt by the exciton. This is a consequence of the very strong lattice anharmonicity or intrinsic polymorphous nature of the perovskite structure44,45,46. In order to get an overview on the general consequences of these fluctuations, a theoretical framework is proposed in Supplementary Note 1 to show the smearing of the electronic density of states, which leads to the broadening of the energy distribution of the emission lines.

All triplets observed in this study are non-degenerate, which is consistent with triplet splittings mainly set by the NC anisotropic crystal phase for such large crystals21,47,48. Indeed, the triplet splittings between extreme lines range from 0.5 to 2.3 meV (Supplementary Fig. 3), with an average of ~1.2 meV that matches the theoretical estimations for nearly-bulk CsPbBr3 NCs with an orthorhombic crystal structure48. One can also notice on Supplementary Fig. 3 that the triplet splittings have a slight correlation with the NC emission energies, which points to residual effects of exciton quantum confinement. The observation of triplet emission in perovskite NCs indicates that acoustic-phonon-assisted relaxation between triplet sublevels is much longer than the exciton recombination lifetime49. This is due to the strong lattice anharmonicity in these soft materials, where the acoustic vibrational density of states is that of a glassy state.50

For most of the NCs, the emission switches over time between the neutral exciton triplet and trion emission characterized by a single, red-shifted ZPL that invariably splits into two Zeeman components under the application of an external magnetic field, as shown in Fig. 2a. The \(g\)-factor associated with the trion doublet is narrowly distributed around 2.4 (Supplementary Fig. 4). This value is in broad agreement with the sum of electron and hole \(g\)-factors measured on CsPbBr3 crystals with time-resolved Kerr rotation51. It is also very similar to the trion \(g\)-factors measured for ~10 nm sized CsPbBr3 NCs21, FAPbBr3 NCs30 and CsPbI3 NCs9. Overall, these results point to a nearly isotropic character of the magnetic response of charged perovskite NCs, regardless of their size and chemical composition. The red shift of the trion line from the central triplet line provides a measure of the trion binding energy and spans from 3 to 18 meV among the studied NCs (Fig. 2b). A correlation emerges between the trion binding energy and the exciton recombination energy (Fig. 2a). This correlation becomes obvious when adding to this set of data the trion binding energy measured on smaller CsPbBr3 NCs having sizes 14.5 nm, 9 nm and 5.5 nm (Fig. 2b). Their corresponding low-temperature emission energies are respectively 2.355 eV, ~2.40 eV21 and 2.434 eV, as a signature of increasing charge-carrier confinement.

Fig. 2: Band-edge exciton fine structure and trion binding energy in CsPbBr3 NCs.
figure 2

a The upper panel shows the spectral trail of a single NC at 7 T, showing switches between the entire exciton fine structure manifold (triplet + singlet) and the trion Zeeman doublet. The singlet and triplet lines clearly undergo identical variations of their PL intensities. The intensity variations of the doublet and the exciton manifold are anti-correlated, showing evidence that they belong to different charge complexes of the same NC. The lower panel displays the spectrum integrated over the spectral trail. The inset indicates the band-edge exciton fine-structure sublevels, where X, Y, Z arbitrarily refer to linear and orthogonal emission dipoles associated with the NC axes. b Distribution of trion binding energies in CsPbBr3 single NCs as a function of the exciton recombination energy taken at the central triplet ZPL. The black dots correspond to nearly-bulk NCs, the blue triangle to 9 nm sized NCs21. The blue star and the blue square respectively correspond to 5.5 nm sized and 14.5 nm sized NCs (Supplementary Fig. 5). c Dark-bright exciton splittings of nearly-bulk CsPbBr3 NCs as a function of the exciton recombination energy.

Full size image

Bright triplet—dark singlet level ordering

The application of a magnetic field also reveals the entire band-edge exciton fine structure of the nearly-bulk CsPbBr3 NCs, as shown in Fig. 2a. While the triplet further splits under increasing fields, a red-shifted line indeed develops in the PL spectrum as a hallmark of magnetic brightening of the ground dark singlet exciton state (See also Supplementary Fig. 6). The ZPL of the dark state at fields limited to 7 T is often very weak and its observation requires PL spectra with a remarkable quality, which requires excellent photostability and long acquisitions times. The dark singlet lays 3.6 meV to 5 meV below the middle bright triplet sublevel in these CsPbBr3 NCs (Fig. 2c). These results invalidate recent models of exciton fine structure, which predict for weakly confined CsPbBr3 NCs a bright-dark exciton level inversion caused by the Rashba effect24,25,26. The strong triplet luminescence of lead halide bulk-like perovskites NCs at liquid helium temperatures is actually a signature of reduced phonon-assisted relaxation to the dark ground exciton, even if the dark-bright splittings coincides with the LO-phonon energies. Triplet-to-singlet relaxation with a LO-phonon is indeed inhibited due to the fact that phonons do not carry angular momentum52,53. The ground singlet exciton can nevertheless be populated by relaxation from high-energy continuum states excited at 2.54 eV, with important consequences on the statistics of photons emitted by individual bulk-like NCs. As a matter of fact, strong photon bunching is evidenced in the low-temperature PL intensity autocorrelation function of these NCs (Supplementary Fig. 7), while a flat correlation is expected for bulk-like particles with Poissonian photon statistics. Magnetic coupling of dark and bright states leads to shortening of the long component of the PL decay and to weakening of the photon bunching, as shown in Supplementary Fig. 7 and modeled in Supplementary Fig. 8. These observations indicate that the population of the long-lived ground exciton state favors the formation of biexcitons in perovskites9.

Biexciton binding energy

Our spectroscopic investigations of single perovskite NCs also reveal the spectral fingerprint of the biexciton-to-exciton transitions. Indeed, raising the excitation intensity leads to the onset of a spectral triplet (XX) that is red-shifted with respect to the exciton (X) bright triplet (Fig. 3a), and whose temporal variations in PL intensity are correlated with those of the X triplet (Supplementary Fig. 9). Moreover, the ratio of the integrated intensities of the XX and X triplets is proportional to the intensity of the X triplet, as a distinctive signature of XX biexciton emission22,54 (Supplementary Fig. 10). Additionally, the XX triplet components XXi perfectly match the X triplet components Xi (i = 1,2,3) under mirror symmetry and are attributed to the three transitions that link the ground zero-angular-momentum biexciton state to the bright exciton triplet sublevels (Fig. 3a). Further evidence of the correspondence between the XXi and Xi lines comes from their polarizations, as shown on Fig. 4b–e for another perovskite NC. For this NC, the transition dipoles associated to X1 and X3 (as well as XX1 and XX3), have nearly linear and orthogonal polarizations in the focal plane, while X2 (as well as XX2) with weak intensity and polarization character is assigned to a transition dipole oriented along the optical axis. This demonstrates that XXi and Xi transitions involving the same bright state have identical polarizations, as expected from symmetry considerations (Supplementary Note 2).

Fig. 3: Biexciton-to-exciton transitions and biexciton binding energy in perovskite NCs.
figure 3

a Low-temperature PL spectrum of a single CsPbBr3 NC at high excitation intensity (50 kW cm−2) and 7 T. The lines named X1, X2, X3 are assigned to the exciton triplet recombination ZPLs. The red-shifted lines named XX1, XX2, XX3 are assigned to the biexciton-to-exciton transitions and perfectly map onto the exciton triplet ZPLs under mirror image (left inset), allowing a straightforward correspondence of the transition lines (right inset). The spectral trail of this NC over time is shown in Supplementary Fig. 9. b, c Polarized PL spectra zoomed on the exciton triplet ZPLs (b) and the biexciton recombination ZPLs (c) for a single CsPb(ClxBr1-x)3 single NC with x~0.3 and size ~30 nm, for various analyzer angles ranging from 0° to 180°. The whole series of spectra is displayed in Supplementary Fig. 12. The evolutions of the exciton and biexciton ZPL-intensities as a function of the polarizer angle are displayed in the polar plots d and e, respectively. These polar plots are remarkably similar, demonstrating that X1 and XX1, X2 and XX2, X3 and XX3 have identical transition dipole orientations nearly aligned along the orthogonal unit vectors \({{{{{{\bf{e}}}}}}}_{{{{{{\rm{Y}}}}}}}\), \({{{{{{\bf{e}}}}}}}_{{{{{{\rm{Z}}}}}}}\), \({{{{{{\bf{e}}}}}}}_{{{{{{\rm{X}}}}}}}\), respectively.

Full size image
Fig. 4: Scaling laws of charge-complex energies in lead-halide perovskite single NCs.
figure 4

Evolutions of the single-NC trion binding energies in \({R}_{{{{{{\rm{y}}}}}}}\) units (a), biexciton binding energies in \({R}_{{{{{{\rm{y}}}}}}}\) units (b) and dark-bright exciton energy splittings in \({{R}_{{{{{{\rm{y}}}}}}}}^{2}/{E}_{{{{{{\rm{gap}}}}}}}\) units (c) as a function of the exciton recombination energy expressed in \({R}_{{{{{{\rm{y}}}}}}}\) units after bandgap subtraction. The splittings and binding energies are measured with respect to the central line of the exciton triplet. Open blue triangles: CsPbBr3 NCs; Open black squares: CsPb(ClxBr1-x)3 NCs; Green triangles: FAPbBr3 NCs30; Red circles: CsPbI3 NCs9. Green squares: FAPbBr3 NCs (data extracted from ref. 31). Plain blue triangle in (a): average data for 9 nm sized CsPbBr3 NCs21. Blue star in (a): CsPbBr3 NC with 5.5 nm size. Blue diamond in (a): CsPbBr3 NC with 14.5 nm size. Plain black square in (a): CsPbClBr2 NCs from ref. 24. Black diamond in (a): CsPb(ClBr)3 NCs from ref. 42. Red disks in (a) and (b): CsPbI3 NCs22. For these perovskites, the low-temperature bandgaps and bulk exciton binding energies taken for the normalization are deduced from refs. 43,62,63 (See Supplementary Table 1): \({E}_{{{{{{\rm{gap}}}}}}}\) = 2.342 eV and \({R}_{{{{{{\rm{y}}}}}}}\) = 33 meV for CsPbBr3, \({E}_{{{{{{\rm{gap}}}}}}}\) = 1.723 eV and \({R}_{{{{{{\rm{y}}}}}}}\) = 15 meV for CsPbI3, \({E}_{{{{{{\rm{gap}}}}}}}\) = 2.233 eV and \({R}_{{{{{{\rm{y}}}}}}}\) = 24 meV for FAPbBr3, \({E}_{{{{{{\rm{gap}}}}}}}\) = 2.513 eV and \({R}_{{{{{{\rm{y}}}}}}}\) = 42 meV for CsPb(ClxBr1-x)3 with x = 0.24. This value of x was adjusted using empirical interpolations of the bandgaps of bulk CsPbCl3 and CsPbBr3 materials composing the alloy. It matches reasonably well the estimation x~0.3 after the growth procedure (see Methods). The trion binding energies in a experience a larger spread than the biexciton ones in b, as a result of spectral jumps accompanying the neutral-to-charged switches of the NCs, as illustrated in Supplementary Fig. 9.

Full size image

Importantly, the red shift of the XX triplet with respect to the X triplet indicates that the biexciton interaction is attractive. In the large CsPbBr3 particles, the biexciton binding energy ranges from 17 to 25 meV with a trend of correlation with the corresponding exciton recombination energy (Supplementary Fig. 11), as in the case of trion emission. These measurements provide the bulk CsPbBr3 biexciton binding energy ~17 meV, a key parameter for the development of refined models of many-body Coulomb interactions in perovskites55. This value is in good agreement with that extrapolated from two-dimensional electronic spectroscopic measurements performed on small CsPbBr3 NC ensembles40.

In order to strengthen the generality of these findings, we have also conducted magneto-optical spectroscopic investigations of mixed-halide CsPb(ClxBr1-x)3 single NCs with an average size ~30 nm (See Methods). These perovskites follow the same behaviors as the other compounds in terms of exciton sublevel ordering and signs of the interactions within trion and biexciton charge complexes. Their exciton dark-bright splittings are found in the range 4-6 meV, while their trion and biexciton binding energies are respectively of the order of 5 meV and 24 meV (Supplementary Fig. 13).

Discussion

We now aim at giving a general and predictive picture for the exciton dark-bright energy splittings as well as the trion and biexciton binding energies in lead halide perovskite NCs from various compositions, on the basis of single-NC spectroscopic data gathered in this study and from previous reports9,21,22,24,30,31,42. As compared to II-VI and III-V semiconductor, perovskites exhibit a simpler band structure (a single valence band, and a single conduction band due to the giant spin-orbit coupling). Moreover, for each perovskite compound, the electron and the hole have nearly identical and effective masses \({m}^{*}\)18,24 and confinement potentials. The Hamiltonian \({{{{{\mathcal{H}}}}}}\) describing the kinetic and Coulomb energies of the charge carriers in a NC, where the confinement potential with infinitely high barriers is taken to zero, can be written

$${{{{{\mathcal{H}}}}}} =\mathop{\sum}\limits_{{{{{{\rm{A}}}}}}}\frac{-{{{\hslash }}}^{2}}{2{m}^{*}{a}_{{{{{{\rm{B}}}}}}}^{2}}\frac{1}{{\left({L}_{{{{{{\rm{eff}}}}}}}/{a}_{{{{{{\rm{B}}}}}}}\right)}^{2}}{\hat{{{{{{\boldsymbol{\nabla }}}}}}}}_{{{{{{\rm{A}}}}}}}^{2}\,+\,\mathop{\sum}\limits_{{{{{{\rm{A}}}}}} < {{{{{\rm{B}}}}}}}\,\frac{{e}^{2}}{4{{{{{\rm{\pi }}}}}}{{{{{{\rm{\varepsilon }}}}}}}_{0}\varepsilon {a}_{{{{{{\rm{B}}}}}}}}\frac{{c}_{{{{{{\rm{A}}}}}}}{c}_{{{{{{\rm{B}}}}}}}}{\left({L}_{{{{{{\rm{eff}}}}}}}/{a}_{{{{{{\rm{B}}}}}}}\right)}\,\frac{1}{\left|{\hat{{{{{{\bf{r}}}}}}}}_{{{{{{\rm{A}}}}}}}-{\hat{{{{{{\bf{r}}}}}}}}_{{{{{{\rm{B}}}}}}}\right|} \\ ={R}_{{{{{{\rm{y}}}}}}}\,\left(\mathop{\sum}\limits_{{{{{{\rm{A}}}}}}}\frac{-1}{{2\left({L}_{{{{{{\rm{eff}}}}}}}/{a}_{{{{{{\rm{B}}}}}}}\right)}^{2}}{\hat{{{{{{\boldsymbol{\nabla }}}}}}}}_{{{{{{\rm{A}}}}}}}^{2}+\,\frac{2}{\left({L}_{{{{{{\rm{eff}}}}}}}/{a}_{{{{{{\rm{B}}}}}}}\right)}\,\mathop{\sum}\limits_{{{{{{\rm{A}}}}}} < {{{{{\rm{B}}}}}}}{c}_{{{{{{\rm{A}}}}}}}{c}_{{{{{{\rm{B}}}}}}}\frac{1}{\left|{\hat{{{{{{\bf{r}}}}}}}}_{{{{{{\rm{A}}}}}}}-{\hat{{{{{{\bf{r}}}}}}}}_{{{{{{\rm{B}}}}}}}\right|}\right)$$
(1)

where \({L}_{{{{{{\rm{eff}}}}}}}\) is the effective NC confinement size, \({\hat{{{{{{\boldsymbol{\nabla }}}}}}}}_{{{{{{\rm{i}}}}}}}\) and \({\hat{{{{{{\bf{r}}}}}}}}_{{{{{{\rm{i}}}}}}}\) the dimensionless (normalized to \({L}_{{{{{{\rm{eff}}}}}}}\)) Nabla-operators and coordinates of the charge carriers i (i = A, B…), \({c}_{{{{{{\rm{i}}}}}}}\) the signs of their charges having absolute value \(e\), and \({R}_{{{{{{\rm{y}}}}}}}={\hslash }^{2}/\left({m}^{*}{a}_{{{{{{\rm{B}}}}}}}^{2}\right)\) the bulk exciton Rydberg energy, \(\varepsilon\) being the perovskite dielectric constant. Such normalization in Rydberg’s units makes interaction energies independent of the material and shows that trion and biexciton binding energies are essentially set by the quantum confinement regime, i.e. the dimensionless ratio \({L}_{{{{{{\rm{eff}}}}}}}/{a}_{{{{{{\rm{B}}}}}}}\). This simple statement explains the universal laws relating in Rydberg’s units the experimental trion and biexciton binding energies to the exciton recombination energy \({E}_{{{{{{\rm{X}}}}}}}\), once the bandgap of the material is subtracted (Fig. 4a, b). Indeed, using these scales, the experimental points strikingly spread along a trend line from the bulk limit (abscissa approaching −1) to the strong confinement regime (abscissa of several units), regardless of the perovskite compositions.

The dark-bright exciton energy splitting \({\varDelta }_{{{{{{\rm{BD}}}}}}}\) measured on single NCs of CsPbBr3, CsPb(ClxBr1-x)3, CsPbI39 and FAPbBr330 also displays a universal scaling law as a function of the exciton quantum confinement. This splitting mainly set by the long-range contribution of the exchange interaction is directly related to the energy difference between longitudinal and transverse excitons in bulk semiconductors, which is proportional to \({{R}_{{{{{{\rm{y}}}}}}}}^{2}/{E}_{{{{{{\rm{gap}}}}}}}\) (Supplementary Note 1). Adding the short-range contribution20, we show in the Supplementary Table 2 that the total exchange interaction expected for bulk perovskites with various compositions converge to nearly equal values, once normalized with \({{R}_{{{{{{\rm{y}}}}}}}}^{2}/{E}_{{{{{{\rm{gap}}}}}}}\). Moreover, the short-range and long-range contributions of the exchange interaction are predicted to have the same NC size-dependence in the strongly confined regime56. We have thus plotted the normalized splittings \({\varDelta }_{{{{{{\rm{BD}}}}}}}/\left(\frac{{{R}_{{{{{{\rm{y}}}}}}}}^{2}}{{E}_{{{{{{\rm{gap}}}}}}}}\right)\) of all these NCs as a function of the corresponding exciton recombination energies in Rydberg’s units, after bandgap subtraction (Fig. 4c). This figure shows evidence that the data points line up on a same curve, regardless of the cation and halide contained in the perovskite.

Overall, the dimensionless experimental dark-bright exciton splittings as well as trion and biexciton binding energies are evidenced to display a universal behavior mainly set by the quantum confinement of the charge carriers in perovskite nanostructures, without the need for prior characterization of the NC size or composition. Beyond settling the debate on the bright and dark exciton sublevel ordering and its main origin in lead halide perovskites, the laws unraveled by single-NC spectroscopy offer a general predictive picture for the interaction energies within charge complexes photo-generated in these materials. They also pave the way to the development of accurate models of charge complex interactions in lead halide perovskites, which share common physics and form a unique class of semiconductors.

Methods

Synthesis of N-(octadec-9-en-1-yl)guanidinium hydrobromide (oleylguanidinium bromide, OGB) ligands

Oleylamine (1 eq) was added to S-ethylisothiourea hydrobromide (1.2 eq) suspended in tetrahydrofurane (THF). The resulting mixture was stirred at room temperature overnight. Then THF was evaporated under reduced pressure and the resulting residue was recrystallized with diethylether (Et2O). The obtained precipitate was washed several times with Et2O and dried under reduced pressure overnight at 50 °C.

N-(octadec-9-en-1-yl)guanidinium hydrobromide (3, OGB). Yield 60%; white solid. 1H NMR (400 MHz, DMSO) δ, ppm ( J, Hz): 7.48(t, J = 5.6 Hz, 1H), 7.20 (s, 2H), 6.86 (s, 1H), 5.34 (dt, J = 14.7, 4.3 Hz, 1H), 3.09 (q, J = 6.6 Hz, 1H), 1.98 (q, J = 6.1 Hz, 1H), 1.93 (t, J = 5.8 Hz, 1H), 1.44 (q, J = 7.0 Hz, 2H), 1.25 (d, J = 9.5 Hz, 18H), 0.86 (s, 1H), 0.86 (d, J = 13.5 Hz, 1H). 13C NMR (101 MHz, DMSO) δ, ppm:157.19; 130.54; 130.51; 130.11; 130.07; 41.20; 32.41; 31.75; 29.60; 29.56; 29.52; 29.47; 29.44; 29.32; 29.29; 29.18; 29.16; 29.08; 29.06; 28.94; 28.91; 27.08; 27.04; 26.52; 22.56; 14.41.

The structural formula of OGB is displayed in Supplementary Fig. 1.

Ligand-assisted-reprecipitation synthesis of OGB-capped CsPbBr3 nanocrystals

The hybrid precursor solution was prepared by mixing solutions of 90 µL of PbBr2 (0.67 M in DMF), 465 µL of CsBr (0.043 M in DMF-DMSO in ratio 1:1) and 300 µL of OGB (1 M in DMF), where DMF denotes dimethyl formamide and DMSO dimethyl sulfoxide. 75 µL of the hybrid precursor solution was injected into a 4 mL vial containing 2.5 mL of vigorously stirred mesitylene to initiate fast nucleation of the CsPbBr3 NRs. The green solution with bright green luminance could be observed in 15-20 s. For purification of the CsPbBr3 NCs, 0.25 mL of ethyl acetate was added to 0.5 mL of the crude solution, followed by centrifugation for 4 min at 4250 \(\times\) g. The supernatant was discarded and the precipitate was dispersed in 0.4 mL of toluene.

The synthesis and characterization of the OGB-capped CsPbBr3 NCs are depicted on Supplementary Fig. 2.

The solution of CsPbBr3 NCs was then diluted 50 times into a solution of polystyrene (1 wt%) in toluene, and 25 µL of the resulting solution was spin-coated on a clean 1 cm x 1 cm sapphire substrate with a rotation speed of 2000 rpm for 60 s, for low-temperature magneto-optical spectroscopic measurements.

Synthesis of CsPb(ClxBr1-x)3 NCs with x~0.3

CsPb(Br/Cl)3 NCs with size ~30 nm were prepared by partial anion-exchange of CsPbBr3 NCs57,58.

First, ∼30 nm-sized CsPbBr3 NCs were synthesized with the following modifications with respect to Ref. 59. The injection of Cs-oleate and further annealing were conducted at 225 °C. Such large CsPbBr3 NCs were colloidally stabilized by lecithin as a capping ligand. While the reaction mixture was cooling down, at 50 °C 2 ml of lecithin in toluene (0.01 M) were added and the solution was stirred for 30 min. The crude solution was centrifuged for 5 min at 1600 \(\times\) g. The resulting supernatant was discarded and the precipitate was dispersed in a lecithin/toluene mixture (0,01 M, 2 mL) and kept undisturbed for 1 h. Then, the solution was centrifuged a second time for 4 min at 1600 \(\times\) g. The precipitate was redispersed in toluene/1,2-dichlorobenzene (v/v 0.6:1).

The anion-exchange reaction was conducted at room temperature. As a source of chloride ions the mixture of didodecyldimethylammonium chloride (DDAC)/PbCl2 in toluene was used. The solution of DDAC/PbCl2 in toluene was prepared by mixing 42 mg DDAC and 13.8 mg PbCl2 with 7.5 mL toluene under moderate heating. 200 µL of CsPbBr3 NC colloid prepared as described above were mixed with 6.5 µL of the DDAC/PbCl2 mixture.

An estimation x ~ 0.3 for the relative amount of chloride atoms in CsPb(ClxBr1-x)3 NCs is derived from the empirical Vegard’s law60,61, using the room temperature bandgap of this alloy\(\,{E}_{{{{{{\rm{gap}}}}}}}\left({{{{{\rm{x}}}}}}\right)\) = 2.57 eV measured from the absorption spectrum.

A TEM image of these NCs is presented in Supplementary Fig. 13.