Bright triplet excitons in lead halide perovskites

Nanostructured semiconductors emit light from electronic states known as excitons[1]. According to Hund's rules[2], the lowest energy exciton in organic materials should be a poorly emitting triplet state. Analogously, the lowest exciton level in all known inorganic semiconductors is believed to be optically inactive. These 'dark' excitons (into which the system can relax) hinder light-emitting devices based on semiconductor nanostructures. While strategies to diminish their influence have been developed[3-5], no materials have been identified in which the lowest exciton is bright. Here we show that the lowest exciton in quasi-cubic lead halide perovskites is optically active. We first use the effective-mass model and group theory to explore this possibility, which can occur when the strong spin-orbit coupling in the perovskite conduction band is combined with the Rashba effect [6-10]. We then apply our model to CsPbX3 (X=Cl,Br,I) nanocrystals[11], for which we measure size- and composition-dependent fluorescence at the single-nanocrystal level. The bright character of the lowest exciton immediately explains the anomalous photon-emission rates of these materials, which emit 20 and 1,000 times faster[12] than any other semiconductor nanocrystal at room[13-16] and cryogenic[17] temperatures, respectively. The bright exciton is further confirmed by detailed analysis of the fine structure in low-temperature fluorescence spectra. For semiconductor nanocrystals[18], which are already used in lighting[19,20], lasers[21,22], and displays[23], these optically active excitons can lead to materials with brighter emission and enhanced absorption. More generally, our results provide criteria for identifying other semiconductors exhibiting bright excitons with potentially broad implications for optoelectronic devices.

1 An exciton involves an electron in the conduction band Coulombically bound to a hole in the valence band. Its energy depends in part on the spin configuration of these two charge carriers. In organic semiconductors, the lowest energy exciton is a triplet state in which these two carriers have parallel spins. For the electron and hole to recombine and release a photon, one spin must simultaneously flip to satisfy the Pauli exclusion principle. Because this coordinated process is unlikely, triplet excitons are poorly emitting.
In addition to spin, the exciton energy depends on the atomic orbitals that constitute the conduction and valence bands. In many inorganic semiconductors, the orbital motion and spin of the carriers are strongly coupled. Spin is no longer conserved, and the total angular momentum of each carrier (J e and J h ) must be considered. Further, the exchange interaction mixes these so that only the total exciton momentum J=J e +J h is conserved. Due to these and other effects, each exciton state is split into several energy sublevels, known as fine structure.
Studies on various materials have found that the lowest energy sublevel is dark. For example, in CdSe, recombination of the lowest exciton requires a change of two units of angular momentum 17 . Because the photon carries one unit, light cannot be emitted unless another unit is simultaneously dissipated, another unlikely process. Thus, despite the added complexity in inorganic semiconductors, they appear to behave like organic semiconductors, i.e. exhibiting an optically inactive lowest exciton. Indeed, no exceptions are known, leading to the belief that such states must be dark.
We show that this belief is incorrect by examining CsPbX 3 (X=Cl, Br, and I) perovskites.
Their crystals comprise corner-sharing PbX 6 -octahedra with Cs + ions filling the voids between ( Fig. 1a). We first approximate the lattice as cubic and calculate band structures (Methods) for CsPbBr 3 (Fig. 1b), CsPbCl 3 , and CsPbI 3 (Extended Data Fig. 1). The bandgap occurs at the Brillouin zone's R-point, near which the valence and conduction bands are well described within the effective-mass model (see Supplementary Table 1). The top of the valence band arises from a mixture of Pb 6s and Br 4p atomic orbitals, with an overall s symmetry 24-26 .
Thus, including spin, the hole can occupy one of two s-like Bloch states with J h =1/2, i.e. ↑ h = S ↑ or ↓ h = S ↓ , using standard notation 27,28 . The conduction band consists of Pb 6p orbitals, leading to three possible orthogonal spatial components for the Bloch function: X , Y , or Z 24-26 . Because of strong spin-orbit coupling, these are mixed with spin to obtain a doubly degenerate J e =1/2 state for the electron at the bottom of the conduction band: When the momentum of the electron and hole states are then combined, the exciton splits due to electron-hole exchange into a J=0 singlet state, and a threefold degenerate J=1 triplet state, where each Ψ J , J z is labeled with J z , the z-projection of J. The probability of light emission due to electron-hole recombination from these excitons can then be calculated (Supplementary Section 1). We find a probability of zero for Ψ 0,0 and nonzero for Ψ 1, J z =0,±1 , indicating a dark singlet and bright triplet.
These selection rules are confirmed by group theory. At the R-point, the band-edge electron and hole states transform as irreducible representations R 6 − and R 6 + , respectively 3 (superscript denoting parity) 28,29 . Exchange then splits the exciton into a dark singlet ( R 1 − ) and a bright triplet ( R 4 − ). (See Supplementary Section 2 and Supplementary Table 3.) Detailed calculations (Supplementary Section 1) can then reveal the energetic order of these levels. If only short-range exchange is included, the singlet lies below the triplet (Fig.   1c). However, CsPbX 3 perovskites should also exhibit a large Rashba effect 6 . This occurs in semiconductors with strong spin-orbit coupling and an inversion asymmetry. For the closely related hybrid organic-inorganic perovskites, the impact of this effect on photovoltaic and spintronic devices has been heavily discussed [6][7][8][9][10] . Although the cause of the inversion asymmetry (cation positional instabilities 30 or surface effects 10 ) remains unknown, the Rashba effect should alter the fine structure. Indeed, the bright triplet exciton can be lowered below the dark singlet exciton.
To examine this possibility, we studied colloidal nanocrystals of CsPbX 3 (Methods).
Compared to bulk crystals, nanocrystals allow the additional effect of system size to be investigated. Such particles are roughly cube-shaped with edge lengths L=8-15 nm (Fig. 1d).
However, fast decays could also indicate emission from trions (charged excitons). Trions are optically active and suffer from nonradiative Auger recombination. In our singlenanocrystal experiments, any trion contribution is already reduced by spectral filtering (Extended Data Fig. 2). To completely eliminate the trion contribution, the photon stream from individual nanocrystals can be analyzed 31 . By correlating emission intensity with lifetime ( Fig.   2c,d), pure exciton contributions can be extracted. We confirm fast exciton lifetimes (1.2 and 0.4 ns, respectively) for CsPbI 3 and CsPbBr 3 nanocrystals, values consistent with ensemble measurements (Extended Data Fig. 3).
To compare with theory, we calculated radiative lifetimes for perovskite nanocrystals within the effective-mass model. In addition to the wavefunctions in equations (2)- (3), exciton confinement within the nanocrystal must be included via envelope functions for the electron and hole. If CsPbX 3 nanocrystals were spherical, excitonic lifetimes could be calculated with prior methods (Supplementary Section 3). However, for cubes, the electric field of a photon not only changes across the nanocrystal boundary due to dielectric screening (as in spherical nanocrystals), but it also becomes inhomogeneous 32 (Fig. 2e). We included this inhomogeneity, along with the Rashba effect, the orthorhombic lattice distortion in CsPbX 3 nanocrystals 33 , and the possibility that their shape is slightly elongated (which leads to tetragonal or orthorhombic symmetry). Only when the Rashba effect was included could a self-consistent model for CsPbX 3 nanocrystals be obtained, as now described.
The Rashba coefficient was estimated from low-temperature photoluminescence spectra (see below). If the effective Rashba field is parallel to one of the orthorhombic symmetry axes of the nanocrystal (see Supplementary Section 1 for details and other cases), the bright triplet exciton (J=1) is split into three nondegenerate sublevels: which lie below the dark singlet (Fig. 1c). The triplet states represent three linear dipoles polarized along the orthorhombic symmetry axes (x, y, z). Transitions from these three sublevels have the same oscillator strength. Moreover, in cube-shaped nanocrystals, these states still emit as linear dipoles despite the inhomogeneous field (Supplementary Sections 1 and 3).
The triplet exciton radiative lifetime, τ ex , can then be evaluated from 34 : with the angular transition frequency, w, the refractive index of the surrounding medium, n, the free-electron mass, m 0 , the speed of light, c, the Kane energy, E p = 2P 2 / m 0 , and the Kane parameter, P (ref. 27). I ! is an overlap integral that includes the electron and hole envelope functions and the field-averaged transition-dipole moment (Supplementary Section 3).
Figure 2b presents the calculated τ ex for CsPbX 3 nanocrystals (circles). The results can be divided into three regimes, depending on the nanocrystal size 35 . When the nanocrystal is smaller than the exciton Bohr radius a B (strong confinement, orange circles), the predicted radiative lifetime decreases from 2 to 1 ns with increasing emission energy. For large nanocrystals in the opposite limit (green circles), the lifetime should be even shorter as weakly confined excitons exhibit larger oscillator strengths 36 . In this size regime (L~15-25 nm), the calculated lifetimes decrease below 100 ps for CsPbBr 3 and CsPbCl 3 nanocubes.
The lifetime would be decreased further in spheres of the same volume (lower inset, Fig. 2e).

6
The measured photoluminescence decays in Fig. 2b  The above calculations depend on knowing the Rashba coefficient a R . This was estimated from photoluminescence spectra of individual nanocrystals, which reveal the fine structure directly. Our nanocrystals exhibit one, two, or three peaks, all with near-linear polarization . This is consistent with the three nondegenerate exciton sublevels in equation (4) under orthorhombic symmetry, which should emit as orthogonal linear dipoles. The Rashba coefficient (a R = 0.38 eV Å) required to fit the observed splittings (~1 meV) is reasonable, lying between values for conventional III-V quantum wells and organic-inorganic perovskites (see Supplementary Section 1.F). We note that for nanocrystals with tetragonal symmetry, Ψ x and Ψ y in equation (4) Fig. 3a-c. Again, good agreement is obtained. Figure 3g presents the experimental statistics for one-, two-, and three-line spectra. One is most common, suggesting that only the lowest sublevel is populated. For the two-and three-line spectra, the measured energy splittings are plotted in Fig. 3h,i. Given three sublevels separated by energies D 1 and D 2 (inset, Fig. 3i), the average splitting Δ is   Traces show "A-type" blinking from the nanocrystals 31 . Such data can be analyzed to separate contributions to Each panel lists the observation direction required relative to the orthorhombic unit-cell axes. g, Experimental statistics for observation of single-peak, two-peak, and three-peak spectra from individual nanocrystals with L=7.5-14 nm (51 spectra with 35 splittings in total). h,i Experimental fine-structure splitting measured for the twopeak and three-peak spectra, respectively. The average splitting in each case is provided.

Synthesis.
The CsPbX 3 (X=Cl, Br, and I) and CsPbBr 2 Cl nanocrystals were synthesized by fast reaction between Cs-oleate and PbX 2 in the presence of OA and OAm (TOP is also added for CsPbCl 3 and CsPbBr 2 Cl nanocrystals). First, the Cs-oleate was prepared by loading Cs 2 CO 3 (0.407 g) into a 50-ml 3-neck flask along with ODE (20 ml) and OA (1.25 ml). The mixture is dried under vacuum for 1 h at 120 °C and then switched to N 2 . Since Cs-oleate precipitates out of ODE at room temperature, it must be pre-heated to 100 °C before injection. The ODE, OA, and OAm were pre-dried before use by degassing under vacuum at 120 °C for 1 h. For the nanocrystal reaction, 0.376 mmol PbX 2 (X=Cl, Br, or I), dried OA (3 ml for PbCl 2 ,1 ml for PbBr 2 , or 1.5 ml for PbI 2 ), dried OAm (3 ml for PbCl 2 , 1 ml for PbBr 2 , or 1.5 ml for PbI 2 ), and dried ODE (5 ml) were combined in a 25-ml 3-neck flask. For CsPbCl 3 , TOP (1 ml) was also added. The mixture was then degassed for 10 min under vacuum at 120 °C, and the flask was filled with N 2 and heated to 200 °C. Cs-oleate (0.8 ml from the stock solution prepared as described above) was swiftly injected when 200 °C was reached. After 10 s the reaction was stopped by cooling the reaction system with a water bath. The solution was centrifuged (4 min, 13750 g) and the supernatant discarded. Hexane (0.3 ml) was added to the precipitate to disperse the nanocrystals and centrifuged again. The obtained precipitate was redispersed in 3 ml toluene and centrifuged (2 min, 2200 g). The supernatant was separated from the precipitate, filtered, and used for our investigations. For CsPbBr 2 Cl, 0.094 mmol PbCl 2 , 0.282 mmol PbBr 2 , dried OA (1.5 ml), dried OAm (1.5 ml), TOP (1 ml), and dried ODE (5 ml) were loaded into a 25-ml 3-neck flask and the same protocol was followed.

Sample preparation.
For single-nanocrystal spectroscopy, the colloidal dispersions from the above syntheses were diluted to nanomolar concentrations in solutions of 3 mass percent polystyrene in toluene.
This dispersion was then spin-casted at 5000 r.p.m. onto intrinsic crystalline Si wafers with a 3-μm-thick thermal-oxide layer. For ensemble measurements, the undiluted nanocrystal dispersions from the previous section were drop-casted on glass substrates.

Optical characterization.
All optical measurements of single nanocrystals were performed in a self-built microphotoluminescence (μ-PL) setup. The samples were mounted on xyz nano-positioning stages inside an evacuated liquid-helium flow cryostat and cooled down to 5 K. Single nanocrystals were excited by means of a fiber-coupled excitation laser at an energy of 3.06 eV with a repetition rate of 40 MHz and a pulse duration of 50 ps. The excitation beam was sent through a linear polarizer and a short-wavelength-pass filter before being directed toward the sample by a dichroic beam splitter. Typical power densities used to excite single nanocrystals were 2-120 W/cm 2 . For both excitation and detection, a long-working distance 100x microscope objective with numerical aperture of 0.7 was used. The nearly Gaussian excitation spot had a 1/e 2 diameter of 1.4 μm. The emission was filtered using a long-pass filter and dispersed by a 0.75 m monochromator with an 1800 lines/mm grating before detection with a back-illuminated, cooled CCD camera. For polarization-dependent measurements, a liquid crystal retarder was employed to compensate for retardation effects in the setup. For photoluminescence lifetime and time-tagged time-resolved (TTTR3) singlephoton-counting measurements, we filtered the emission with a suitable tunable bandpass filter to either measure only the excitonic photoluminescence decay or to correlate excitonic and trionic emission intensities and decay times with a time-correlated single-photon-counting system with nominal time resolution of 30 ps.
Ensemble measurements were performed in an exchange-gas cryostat at 5 K. Here, the samples were excited with a frequency-doubled Ti:sapphire femtosecond pulsed laser with a repetition rate of 80 MHz at 3.1 eV. The emitted light was dispersed by a 150 lines/mm grating within a 300-mm focal length spectrograph and detected by a streak camera with 2 ps resolution.
Band-structure calculations. Fig. 1 show calculated band structures for CsPbBr 3 , CsPbCl 3 , and CsPbI 3 . We assume that these materials exist in the cubic perovskite structure with a lattice constant of 5.865, 5.610, and 6.238 Å, respectively 39 . The electronic structure of these crystals was determined using the Vienna Ab-initio Simulation Package (VASP) 40-42 with projector-augmented wavefunctions 43 . Our initial calculations used the PBEsol 44,45 generalized gradient approximation, and included spin-orbit coupling. We used an energy cutoff of 400 eV and Γ-centered k-point grid of 6×6×6, which yield 40 k-points in the irreducible Brillouin zone.

Figure 1b and Extended Data
As expected, standard density functional theory (DFT) seriously underestimates the bandgap in these materials. Accordingly, we used a modified version of the Heyd-Scuseria-Ernzerhof "HSE06" hybrid functional 46 , which mixes exact Hartree-Fock exchange with conventional DFT. We initially started with 25% mixing, and planned to adjust the mixing to match the observed bandgap. However, this was not possible, even with 45% Hartree-Fock in the calculation for CsPbBr 3 . This produced a bandgap of 1.4 eV, far smaller than the experimental gap of 2.8 eV. Rather than using even higher mixing, or even a full-scale Hartree-Fock calculation, we instead added a scissors operator to adjust the bandgap to the experimental result. We found that the electron and hole masses were nearly unchanged with Hartree-Fock mixing, leading us to believe that this technique still provides the correct physics.   The R-point of the Brillouin zone is isomorphic to the Γ -point in cubic semiconductors [1]. As a result, the dispersion of electrons and holes at the R-point is described by the familiar 8x8 k · p Hamiltonian matrix that characterizes the band edge of direct-gap cubic semiconductors at the Γ -point. In the perovskites studied here, due to large spin-orbit coupling, a good description of the electron and hole dispersion is obtained by extracting the 4x4 part related to the Γ − 6 and Γ + 6 bands of the conduction and valence bands [2,3]. Using the same standard semiconductor notation [4,5] introduced in the main text, the Bloch wavefunctions of the corresponding band-edge states can be written as: wherep is the momentum operator, P = −i S|p z |Z , p ± = p x ±ip y , p 2 = p 2 x +p 2 y +p 2 z , E c,v are the band-edge energies, and γ e,h are the remote-band contributions to the electron and hole effective masses. Note that in the perovskites considered the band structure is reversed compared to many typical semiconductors in the sense that the valence band (instead of the conduction band) is s -like. The energy gap, E g = E c − E v , is connected with the energy gap E g and the spin-orbit splitting ∆ of the standard 8-band model as E g = E g − |∆| , because ∆ in these perovskites is negative.
The energy spectrum of the carriers is isotropic at the R-point of the Brillouin zone and can be easily found by taking p along the z axis. In this case, the 4×4 matrix is composed of two identical 2 × 2 blocks decoupled from each other, determined as,    The usual procedures lead to the dispersion relation where we have used the Kane energy E p = 2P 2 /m 0 .
These expressions are derived from the parabolic approximation to Eq.(S4) applied for small p and using E p E g . To extract E p , we take the asymptotic limit of Eq. (S4) at large p , such that p 2 E p 6m 0 E 2 g . Assuming that E p is sufficiently large that p 2 m 0 E p / (γ e ± γ h ) , which is satisfied for a very wide range of energies in the conduction and valence bands, we obtain,   Extended Data Fig. 6 shows the slope E p /6m 0 according to Eq.(S6), which is calculated as the energy difference E − E g /2 divided by the corresponding difference in momentum, p = ∆k , where ∆k is the wave number shown in nm −1 . The results of this fitting procedure are summarized in the Table S1 along with the energy gaps E g of the bulk perovskites CsPbX 3 taken from experimental data.
The last parameter needed to analyze the exciton radiative lifetimes is the high-frequency dielectric constant for each material. Using the effective masses summarized in Table S1 we can calculate the high-frequency dielectric constant from the exciton Rydberg when it is known. Taking the measured Rydberg for CsPbCl 3 and CsPbBr 3 , 60 meV and 34 meV, respectively [6], we find in = 4.5 for CsPbCl 3 and in = 4.8 in CsPbBr 3 . The Rydberg has not been measured for CsPbI 3 . Noting that the dielectric constants determined for  Table S1 are used in the calculations of the radiative lifetimes.
C. Exciton fine structure in nanocrystals with cubic lattice structure The total wavefunction of the electron and hole states in nanocrystals can be found using the parabolic-band effective-mass approximation [9,10]. They generally can be written as a product of the Bloch functions defined in Eq.(S1) and envelope functions, i.e. products of the form: Ψ e ⇑,⇓ (r e ) = F e (r e )| ⇑, ⇓ e and Ψ h ↑,↓ (r h ) = F h (r h )| ↑, ↓ h for electron and hole states, respectively, where F e,h are the electron or hole envelope functions.
The total exciton wavefunctions in nanocrystals are the product of the Bloch functions defined in Eqs. (2) and (3) in the main text and the exciton envelope function v(r e , r h ) , which describes spatial motion of the exciton confined in the nanocrystal. The resulting wavefunctions of the exciton Ψ ex J,Jz with momentum J and momentum projection J z have the following form: The electron-hole exchange interaction [1], are the electron and hole Pauli operators, α exc is the exchange constant, and Ω 0 is the volume of the unit cell, conserves the two-particle angular momentum, J = 1 2 (σ e + σ h ) . This exchange interaction splits the fourfold degenerate exciton ground state into an optically passive singlet ( J = 0 ) and a threefold degenerate optically active triplet state ( J = 1 with three momentum projections J z = ±1, 0 ).
The singlet-triplet splitting of the exciton levels can be shown to be equal to 4η , where η = α exc Θ , with Θ = Ω 0 d 3 rv 2 (r, r) . It is known that the splitting is enhanced by spatial confinement [11], which is included via the parameter Θ . In the strong-confinement regime: Θ ∼ Ω 0 /V and is inversely proportional to the nanocrystal volume, V . In the bulk and in the weak-confinement regime: Θ = Ω 0 /πa 3 B . It is easy to demonstrate that the singlet level |Ψ 0,0 is optically passive. This is because the transition-dipole matrix element taken between this state and the vacuum state |0 = δ(r e − r h ) is zero: 0|p|Ψ 0,0 = 0 . In the optical matrix element,p acts only on the conduction-band Bloch functions. Thus, the exciton wavefunction |Ψ 0,0 from Eq.(2) in the main text should be transformed to the electron-electron representation. In this case, using the time-reversal operatorK for transformation of the hole wavefunction to the electron form, one can show that Here we used the following properties of the time-reversal operatorK :K| ↑ = | ↓ and K| ↓ = −| ↑ . Similar calculations show that all three triplet states are optically active.

D. The order of the singlet and triplet excitons in perovskite nanocrystals
The sign of the exchange-interaction constant α exc affects the level order of the singlet and triplet exciton states. In the absence of spin-orbit coupling, both α exc and η are always positive, resulting in a optically passive spin-triplet exciton ground state. This is the case for organic semiconductors. When strong spin-orbit coupling exists in only one band (for which the corresponding band-edge Bloch functions are described by Eq. 1 of the main text and above), the parameters α exc and η are negative leading to an optically passive singlet exciton ground state. Ignoring the Rashba effect for the moment (see the next subsection, Section S1.E), this optically passive singlet would be the expected exciton ground state for perovskites and perovskite nanocrystals [12][13][14][15][16]. The splitting in this case was intensively analyzed theoretically [17][18][19] in connection with CuCl, for which the conduction and valence band edges have symmetry Γ + 6 and Γ − 6 , respectively. The triplet-singlet splitting, 4η can be expressed in terms of the Bloch functions of the conduction and valence bands [18]: 4η = (2/3)(Θ/Ω 2 0 ) d 3 r 1 d 3 r 2 S * (r 1 )X * (r 2 )V (r 1 − r 2 )S(r 2 )X(r 1 ) , where the Bloch functions are normalized to the unit-cell volume, Ω 0 , the integrals are taken over one unit cell, and V (r 1 −r 2 ) = e 2 /( in |r 1 −r 2 |) is the Coulomb potential between two electrons. This exchange integral is always positive and the optically active triplet always has higher energy.
To quantify the exchange splitting for perovskite nanocrystals (still ignoring the Rashba effect), we conducted first-principles calculations of the band-edge Bloch functions and calculated the exchange constant α exc for all three CsPbX 3 halide perovskites. The results (X=Cl, Br, I) in Table S2. The calculated short-range exchange splitting of the singlettriplet exciton is shown in the third column of the table.
Our experimentally studied CsPbX 3 nanocrystals are known to exhibit an orthorhombic lattice distortion [20]. The reduction of the nanocrystal symmetry generally splits the threefold degenerate triplet states into three exciton sublevels. To find these splittings in CsPbBr 3 nanocrystals, we used G 0 W 0 first-principle calculations (see Methods). Our calculations predict an expected orthorhombic splitting for the triplet with ∆ 1 = 1.9 * 0.004388 = 0.008 meV and ∆ 2 = 3.9 * 0.004388 = 0.017 meV (see inset to Fig. 3i in the main text for definitions of ∆ 1 and ∆ 2 ). These splittings are hundreds of times smaller than the splittings measured experimentally ( ∼ 1 meV) in our perovskite nanocrystals (see Fig. 3). This suggests that the orthorhombic distortion is not responsible for the observed splittings.

E. Effect of Rashba terms on the exciton fine structure
We now consider the influence of the Rashba effect on the observed exciton fine structure.
This effect can arise due to inversion-symmetry breaking in CsPbBr 3 , for example due to the instability of Cs + ions in the lattice [21]. Instabilities in the ion positions can result in lattice polarization, which creates Rashba terms in the Hamiltonians describing the electrons and holes.
The Rashba effect for electrons and holes in a nanocrystal made from a cubic crystal lattice can be described as [22]: where α R,c represents either the conduction-or valence-band Rashba coefficient for nanocrystals with cubic lattice structure ( α e R,c and α h R,c , respectively), and σ i are the projections of the Pauli operators for the electron total momentum operator for J = 1/2 and for the hole spin s = 1/2 , ( σ e i and σ h i respectively). In Eq. (S9), n x,y,z are the projections on the cubic axes of a unit vector n defining the direction of the symmetry breaking (see, for example, the inset in Fig. 1c As one can see from Eq. (S10), the Rashba effect for both electrons and holes is fully described with six independent parameters: α z,e;h R,xy , α z,e;h R,yx , α y,e;h R,zx , α y,e;h R,xz , α x,e;h R,yz , and α x,e;h R,zy , respectively, which reflect the material properties and symmetry of the nanocrystal, while again the projections n x , n y , n z of the unit vector n define the Rashba symmetry-breaking direction. For calculations it is convenient to re-write the Rashba Hamiltonian in Eq. (S10) acting on the exciton as a sum of the three termsĤ o R =Ĥ x n x +Ĥ y n y +Ĥ z n z : The wavefunction of the exciton ground state in cube-shaped nanocrystals in the weak confinement regime (which we use to approximate our experimental samples), can be written as: Here ψ 100 (r) is the hydrogen-like function that describes the relative motion of the exciton ground state with r = r h − r e . For the ground state, the wavefunction of the exciton relative motion can be written: where L x , L y , and L z are the edge lengths of the cube-shaped nanocrystal. Finally U J,Jz in Eq.(S12) is the spin part of the exciton function, which for J = 0 and J = 1 (singlet and triplet states, respectively) can be written as: Corrections to the exciton ground state from the Rashba terms in Eqs. (S11) vanish in first-order perturbation theory. In second-order perturbation theory, however, we find corrections that describe coupling among the spin sublevels of the exciton. The resulting coupling matrix contains spin-spin coupling terms and is similar in that respect to an effective exchange Hamiltonian. In second-order perturbation theory this matrix can be written To estimate the matrix in Eq. (S17) we take into account just the first few excited states of the exciton center-of-mass and relative motions. In the later case, the wavefunction can be written as: where ψ 21m (r) is the hydrogen-like wavefunction of the 1P exciton level with angular momentum l = 1 and momentum projections m = 0, ±1 . These wavefunctions can be written as: where Y 1,m are the spherical harmonics with l = 1 [23]. The energy distance for the 1P level is 0.75e 4 µ/ 2 2 in . For the first three excited levels connected with the exciton center-of-mass motion we can write: where the excited wavefunction of the exciton center of mass motion Ψ x,y,z can be written: The energy distances between the ground and excited exciton states are 3 2 π 2 /2ML 2 x , 3 2 π 2 /2ML 2 y , and 3 2 π 2 /2ML 2 z , respectively. Let us now calculate the effective electron-hole spin-coupling Hamiltonian created by the Rashba term. SubstitutingĤ o R into Eq. (S17) we obtain: M where,α z,e;h R,ij = α z,e;h R,ij n z ,α y,e;h R,ij = α y,e;h R,ij n y ,α x,e;h R,ij = α x,e;h R,ij n x , A c R = 128/(27π 2 ) , and A r R = (64/81 √ 3) 2 . The terms proportional to A c R and A r R come from the intermediate states connected with the exciton center-of-mass motion and the relative motion of the electron and hole, respectively. The third term in Eq. (S22) consists of spin-spin coupling terms and has the same form as the effective spin-dependent electron-hole exchange Hamiltonian. Such terms determine the fine structure of the band-edge exciton. One can see that the contributions of the center-of-mass motion and the relative motion of the exciton have different signs and result in a different level order. However, because A c R > A r R it is the center-of-mass motion that determines the level order of the exciton.
The fine structure of the exciton is thus defined by the following matrix: a circularly polarized doublet, with polarization x ± iy . The eigenvalues from Eq. (S25) are reduced to x = y = 0 and z = − d . This level structure we believe was observed recently [16]. The Rashba splitting between the two bright excitons in that case is described . Another analytical expression for the exciton fine structure can be found for the case when n z = 0 and n x , n y = 0 . Diagonalization of the matrix described by Eq. (S23 where . The corresponding eigenstates can be written up to a normalization constant as: In this configuration, in which the Rashba asymmetry direction n contains components along two nanocrystal symmetry axes, that is, n lies in a mirror plane, the dark exciton is activated. It is easy to show that the directions of the dipoles of the former bright states remain orthogonal to each other, despite that their dipole orientations have been changed.
The only non-orthogonal pair of dipoles here correspond to |ψ 1 and |ψ 2 , but they both are polarized along the same z direction.
Going further, we can consider a general orientation of the Rashba asymmetry axis. In that case, we have not yet found a closed-form solution to the expressions above. Nevertheless, it is clear that, for a general asymmetry direction n , the dark state mixes with each of the bright states, creating a higher-order coupling between each state via the "dark" intermediate state. This results in the orthogonality of the dipoles being weakly broken.
Numerical calculations have been performed that confirm this.
The results above can be understood in the context of group theory. If the Rashba asymmetry direction n is parallel to any one symmetry axis of the orthorhombic nanocrystal, the symmetry is reduced to C 2v . In that case, the dark state remains dipole inactive and the three bright states are split into mutually orthogonal, linearly polarized dipoles. But if the Rashba asymmetry also has a component along either of the other two axes, the symmetry is reduced further to C s for which no dark state exists. Finally, for a general Rashba asymmetry direction, with components along all three nanocrystal symmetry axes, the symmetry is reduced to C 1 for which all exciton states are coupled and as a result all dipole components are present for every state. These considerations are further discussed in Section S2.

F. Rashba coefficient in inorganic perovskite nanocrystals
From the results from the previous subsection, Section S1.E, we now estimate the Rashba coefficients necessary to explain our experimental exciton fine-structure splittings, which are ∼ 1 meV. Specifically, we can exploit Eq. (S25). For simplicity, we assume that all Rashba coefficients for the electron and hole are equal to each other, α i,e R,jk = α i,h R,jk = α for any i , j , and k , and that their effective masses are equal: m e = m h = M/2 . The Rashba energy E R = α 2 m e /2 can then be found as This gives E R ≈ 0.92 meV. For comparison, in organic perovskites it was found to be 13 meV [25]. It is more appropriate, however, to compare the Rashba coefficient α R rather than the Rashba energy between different materials, because the Rashba coefficient does not depend on the effective mass. Using m e = 0.13 for CsPbBr 3 from Table S1, we obtain for the traditional definition of the Rashba coefficient α R = α = 0.38 eVÅ. For comparison, the measured value for InSb/InAsSb quantum wells is α R = 0.14 eVÅ[26]; in InGaAs/InP quantum wells, α R = 0.065 eVÅ [27], and in InAs quantum wells, α R = 0.22 eVÅ [28]. In organic-inorganic hybrid perovskites α R is much larger due to their ferroelectricity: α R = 7± 1 eVÅ in ortho-CH 3 NH 3 PbBr 3 ; while α R = 11± 4 eVÅ in cubic-CH 3 NH 3 PbBr 3 [29].
The value of α R we estimate from the experimental data is quite reasonable in comparison with other semiconductors. This leads us to conclude that the Rashba effect indeed is responsible for the observed exciton fine structure in the CsPbX 3 perovskite nanocrystals.

S2. SYMMETRY ANALYSIS OF THE EXCITON FINE-STRUCTURE
Here we consider the point-group symmetry and irreducible representations appropriate for describing the band-edge excitons of quasi-cubic perovskite nanocrystals. Table S3 below shows how degeneracies and selection rules are modified as we descend in symmetry from cubic ( O h ) to tetragonal ( D 4h ) or orthorhombic ( D 2h ) due to lattice or shape distortions.
Note that in the table, the irreducible representation labels are given for the nanocrystal point group rather than the bulk space group. For each of these "parent" point groups, we also show the symmetry breaking effect of a Rashba asymmetry for different orientations n of the asymmetry axis. The groupings for each parent group ( O h , D 4h , and D 2h ) are separated by double vertical lines in Table S3. Optical selection rules for exciton transitions are shown in Table S3 by writing the allowed transition-dipole components for each exciton irreducible representation as x, y, z for linear polarized dipoles or σ ± for circular polarization. In constructing the table, we utilized the irreducible representation labels, and the character and multiplication tables of KDWS [3].
The results summarized in the table show that the cubic perovksite band-edge exciton fine structure consists of a threefold degenerate (triplet) bright state split from a singlet dark state. As the nanocrystal symmetry is reduced by unit-cell or shape distortions the bright triplet is expected to split. For the tetragonal phase D 4h , the triplet splits into a singlet linearly polarized along the axis of symmetry and a doublet circularly polarized perpendicular to the symmetry axis. An orthorhombic distortion of D 2h symmetry will split the bright triplet into three non-degenerate states each linearly polarized along the orthorhombic symmetry axes as follows: . The addition of a Rashba asymmetry further breaks the symmetry of the nanocrystal beyond the shape or lattice distortions just discussed. A Rashba asymmetry directed along the z -axis breaks the symmetry of cubic ( O h ) and tetragonal ( D 4h ) nanocrystals to C 4v , characterized by a dark singlet, a bright doublet and a linearly polarized singlet. The effect of a Rashba asymmetry along the z axis of an orthorhombic nanocrystal takes the symmetry from D 2h to C 2v , maintaining the dark state and three linearly polarized, orthogonal bright Point group O h and its subgroups. Rashba fields are described in the table in terms of an asymmetry direction n = n xx + n yŷ + n zẑ as in Section S1.E. See text for explanation.
excitons. However, if the Rashba asymmetry is directed off the principle axis, the symmetry reduces to C s in the case that the asymmetry is oriented within a mirror plane of the nanocrystal, and to C 1 otherwise. In both cases, the dark exciton state is mixed with the bright excitons. In the case of symmetry C s , the Rashba asymmetry further mixes two of the bright excitons; in the lowest symmetry case, all bright excitons are mixed and the orthogonality of the dipoles is broken, at least in principle. Calculations show, however, that this mixing is a second order effect and the resulting non-orthogonality of the dipoles is expected to be weak.
Note that in Table S3, where relevant, the z-axis is taken as the principle axis unless otherwise specified as a superscript on the group symbol. The x, y axes are then the axes associated with other symmetry elements such as C 2 rotations where they exist. When a Rashba asymmetry is oriented along a particular direction that creates a mirror plane, this mirror plane is given as a superscript on the point-group symbol. For example, C σxz s denotes the C s point group defined with a mirror plane σ xz containing the z and x axes.

S3. CALCULATION OF EXCITON AND TRION RADIATIVE LIFETIMES
A. Radiative lifetime of excitons in cube-shaped nanocrystals The probability of optical excitation of or recombination from any exciton state |Ψ ex is proportional to the square of the matrix element of the operator ep between that state and the vacuum state, where e is the polarization vector of the emitted or absorbed light, andp is the momentum operator. In cube-shaped nanocrystals, the calculation of these matrix elements is complicated by the fact that the electric field of a photon inside the nanocrystal not only changes its value from outside due to dielectric screening, as in spherical nanocrystals, but it also becomes inhomogeneous.
As we discussed in Section S1.E, the triplet state in perovskite nanocrystals with orthorhombic symmetry is always split into three orthogonal dipoles with the same oscillator transition strength. Therefore, let us consider the optical transition to the triplet exciton state with J z = 0 , which has a linear z dipole. The square of this matrix element can be written: where I = d 3 re z z (r)v(r, r)/E z ∞ , and e z z (r) is the z component of the electric field of the light in the nanocrystal created by a photon with an electric field E z ∞ , which is linearly polarized along z and defined at infinite distance from the nanocrystal. The inhomogeneous distributions of the electric field in the cube-shaped nanocrystals created by the external homogeneous electric field is calculated in the next Section S3.B for various ratios of dielectric constants and are shown in Fig. 2e of the main text. Due to the even parity of the electron and hole envelope functions participating in the band-edge optical transitions This analysis shows that despite the inhomogeneous distribution of the photon electric field in the cube-shaped nanocrystals the linearly polarized dipoles of each sublevel of the triplet only emit or absorb photons having a nonzero electric-field projection on the respective dipole orientation. In short, the linearly polarized dipoles emit linearly polarized light.
Substituting the matrix elements from Eq. (S29) into the expression for the radiative decay rate from ref. 30 we find the radiative lifetime of the triplet exciton τ ex : where ω is the transition (angular) frequency, n is the refractive index of the surrounding media, m 0 is the free-electron mass, c is the speed of light in vacuum, and E p = 2P 2 /m 0 is the Kane energy. The calculated radiative lifetimes in perovskite nanocrystals can be directly compared with experimental results at low temperature because the experimental data are not obscured by contributions of a low-energy dark exciton. The largest uncertainty in the radiative lifetime defined by Eq. (S30) is connected to the uncertainty in the high-frequency dielectric constant, in , for the perovskites, which together with out = n 2 determines the depolarization of the photon electric field in the nanocrystals.
Using Eq. (S30) we have calculated the radiative lifetimes in CsPbX 3 (X=I, Br, Cl) nanocrystals and their optical-transition energies. These are plotted in Fig. 2b in the main text. In these calculations, we used the energy-band parameters and dielectric constants from Table S1 and a refractive index of n = 1.6 for the surrounding medium, which yields out = n 2 . Calculations were conducted for the three nanocrystal size regimes: (i) strong spatial confinement when the nanocrystal size L is smaller than the exciton Bohr radius a B , (ii) weak spatial confinement when L a B , and (iii) intermediate confinement when L ∼ a B . The discussion of these three cases is presented further in Section S3.C.

B. Calculation of the interior electric field in cube-shaped nanocrystals
We consider the inhomogeneous electric field inside a cube-shaped nanocrystal, modeled as a dielectric cube. The field is induced by an arbitrarily oriented electric field that is homogeneous at a large distance from the cube. Such a field can always be decomposed into three components created independently by the three projections of this remote electric field along the cube axes. In Fig. 2e in the main text, we show the distribution of the normalized z component of the electric-field magnitude, E z z /E z ∞ , in the cross-section passing through the middle of the cube, created by a homogeneous external electric field, E z ∞ z , calculated for several ratios of internal in to external out dielectric constants. We describe here the approach followed to compute it.
The generalized Gauss's law states that the total electric flux through any closed surface in space of any shape drawn in an electric field is proportional to the total electric charge enclosed by the surface. The differential expression of this law, obtained via the divergence theorem, represents the local conservation of charge in the form of the well known partial differential equation (PDE) where and ρ v represent the electric permittivity (dielectric constant) of the medium and the electric charge density, respectively. In the context of electrostatics, the electric field is computed as the negative gradient of the electric potential scalar field φ , which, together with the charge conservation Eq. (S31), yields which is Poisson's equation for the electric potential. This PDE is discretized and solved numerically using the finite element method (FEM). It is applied for a computational domain that involves a rectangular electrode capacitor configuration where the upper electrode is set to φ = 1 V and the bottom one is grounded at φ = 0 V .
The proper distance between the capacitor plates in relation to the size of the embedded dielectric cube was determined by performing successive FEM analyses at various distances such that the far-field difference among all solutions remained less that 1% for various values of the dielectric constants. The distribution of both the electric potential and the electric-field magnitude along the z -axis line that coincides with the intersection of the zx and zy planes as it extends between the two electrodes of the capacitor assembly is shown in Extended Data Fig. 7 for the final selected configuration.
Contour plots of the normalized electric-field magnitude, E z z /E z ∞ , as a function of the ratio in / out are displayed in Extended Data Fig. 8. This figure shows that as the in / out ratio increases, the overall magnitude of E z z /E z ∞ decreases. It should also be noted that the perturbations of the contours near the corners are artifacts of the interpolation resolution utilized by the software employed to construct them.
A contour plot of the normalized electric-field component E z z /E z ∞ on the xz mid-plane for in / out of 6 is shown in Fig. 2e in the main text. By symmetry this is also valid for the yz mid-plane. Contours for the normalized electric-field component E z x /E z ∞ for the xz mid-plane are shown in Extended Data Fig. 9 for the case of in / out = 9 . Note that the E z y /E z ∞ distribution on the yz mid-plane is identical to Extended Data Fig. 9.
In the strong-confinement regime, when the exciton Bohr radius a B is larger than the nanocrystal size L , then v(r, r) = |Φ c gr (r)| 2 . The ground-state wavefunctions of electrons or holes can be written as Φ c gr (x, y, z) = (2/L) 3/2 cos(πx/L) cos(πy/L) cos(πz/L) , where L is the cube edge length. Introducing dimensionless variablesx = x/L ,ỹ = y/L , and z = z/L , we can rewrite I in dimensionless form: This expression was used in Eq. (S30) to calculate the radiative lifetimes of CsPbX 3 (X=I, Br, and Cl) nanocrystals with L = 6 and 7 nm, L = 5 and 6 nm, and L = 4 and 5 nm, respectively. The energy of the optical transition in this limit is well described by where E g is the bulk energy gap of the perovskite and µ = (1/m e + 1/m h ) −1 is the reduced effective mass of the exciton. We used the energy-band parameters of the perovskites from Table S1.
In the weak-confinement regime, when L a B , the exciton wavefunction can be written as a product of the relative motion of the electron and hole φ(r e − r h ) and the exciton center-of-mass motion confined in the nanocrystal, Φ c gr (R) , where R = (m e r e + m h r h )/M and M = m e + m h is the exciton effective mass [9]: v(r e , r h ) = φ(r e − r h )Φ c gr (R) . In the weak-confinement regime it is more convenient to directly calculate I 2 : where V = L 3 is the volume of the cube-shaped nanocrystal. One can see that under weak confinement, the ratio V/a 3 B in I 2 dramatically shortens the radiative decay times of the exciton in Eq.(S30) due to its giant oscillator transition strength [31]. The energy of the optical transitions in the weak-confinement regime is described as is the exciton binding energy. Figure 2b in the main text shows the results of the calculations of the exciton radiative lifetime in perovskite nanocrystals with L = 17 to 25 nm. One can see that the lifetime is strongly reduced, becoming shorter than 100 ps in CsPbBr 3 and CsPbCl 3 nanocrystals.
Further, one can see in Fig. 2b that the experimentally measured decay times are in between the lifetimes predicted for strong ( L a B ) and weak ( L a B ) confinement. This is because the correlation of the electron and hole motion in nanocrystals, which shortens the radiative decay time, can already be seen in nanocrystals with intermediate size L ≥ a B .
To demonstrate this, we studied the energy of the confined excitons in nanocrystals with L ≥ a B using a one-parameter ansatz function: where β is a variational parameter and C is a normalization constant determined by the condition d 3 r e d 3 r h v 2 (r e , r h ) = 1 . Using the results of these calculations (see the next Section, Section S3.D) we show in Fig. 2b in the main text the exciton radiative lifetimes of CsPbX 3 (X=I, Br, and Cl) nanocrystals with L = 6 to 16 nm, L = 5 to 16 nm, and L = 4 to 16 nm, respectively.

D. Variational calculation for the intermediate-confinement regime
In the variational approach we calculate the expectation value of the two-particle Hamiltonian in a cube with edge length L and minimize this with respect to the variational parameter β . The Hamiltonian iŝ Introducing the dimensionless variablesr e = r e /L ,r h = r h /L , and the dimensionless parameter b = βL , the expectation value v|Ĥ|v reduces to the calculation of three dimensionless integrals. The first integral describes the average kinetic energy: whereΦ c gr (x,ỹ,z) = cos(πx) cos(πỹ) cos(πz) . The second integral describes the average Coulomb interaction: Finally, the third integral determines the normalization constant C : The normalization constant is connected with this integral as C = L −3 / I 3 (b) .
Using the integrals defined in Eqs. (S38), (S39), and (S40), we can rewrite the expectation value of the Hamiltonian as, where a e is the electron Bohr radius: a e = in 2 /(m e e 2 ) . We find the dependence of all three integrals on b using Monte-Carlo integration and determine the value of b that minimizes the energy for a range of the ratios L/a e . The results of these calculations are shown in Extended Data Fig. 10a.
Now we can calculate I , which is defined as Using the ansatz function definition in Eq.(S36) and the relation of the normalization constant C with I 3 (b) , we can directly connect I inter in the intermediate-confinement regime with the corresponding result in the strong-confinement limit, I strong , defined in Eq.(S34): The radiative lifetime is proportional to I 2 . To describe the dependence of the radiative lifetime on the nanocrystal size, L , we plot the dependence |I inter /I strong | 2 as a function L/a e in Extended Data Fig. 10b.

E. Radiative lifetime in spherical nanocrystals
It is interesting to compare the radiative lifetimes obtained for spherical and cube-shaped nanocrystals. In spherical nanocrystals, the electric field of the photon is homogeneous and the ratio e(r e )/E ∞ at each point r of the nanocrystal is equal to the depolarization factor D = 3 out /(2 out + in ) . Substituting this ratio into Eq. (S30) we find the radiative lifetime of the triplet exciton τ ex in spherical nanocrystals: where K = | d 3 rv(r, r)| 2 is the overlap integral squared.
The radiative lifetime defined in Eq. (S44) depends strongly on the nanocrystal radius a via the size dependence of the overlap integral K . In small nanocrystals that are in the strong-confinement regime ( a < a B ), the photoluminescence is determined by the optical transitions between the ground quantum confinement levels of the electrons and holes [9].
In this case, the exciton wavefunction v(r, r) = Φ 2 gr (r) is the product of the two identical wavefunctions Φ s gr (r) = 1/2πa sin(πr/a)/r for the electron and hole ground states resulting in K = 1 independent of size. The radiative lifetime in Eq.(S44) also weakly depends on nanocrystal size via the size dependence of the transition frequency, 1/τ ex ∝ ω .
In the weak-confinement regime, when the nanocrystal radius a a B , shortening of the radiative lifetime is expected at low temperatures due to the giant oscillator transition strength of the exciton localized in the nanocrystal [31]. As discussed in Section S3.C, the exciton wavefunction can then be written as a product of the relative motion of the electron and hole φ(r e − r h ) and the exciton center-of-mass motion confined in the nanocrystal.
For spherical nanocrystals, the latter is Φ s gr (R) yielding: v(r e , r h ) = φ(r e − r h )Φ s gr (R) . Substituting this wavefunction into the overlap integral gives [9]: K = (8/π 2 )(a/a B ) 3 . The resulting dramatic increase of K is due to the correlation of the electron and hole motion, increasing the oscillator transition strength and shortening of the radiative lifetime. This shortening can already be observed in nanocrystals with radius a ≥ a B .
The ratio of the exciton radiative lifetime in spherical and cube-shaped nanocrystals that have the same volume is equal to I 2 /KD 2 . The result of this comparison is shown in the inset of Fig. 2e in the main text for nanocrystals in the strong-and the weak-confinement regimes. One can see that the exciton radiative lifetime in spherical nanocrystals is always shorter than in cube-shaped nanocrystals of the same volume. This is because the electric field of the photon in spherical nanocrystals is larger than the average field in cube-shaped nanocrystals.

F. Trion radiative lifetime and polarization
The electron spin is not conserved during optical transitions in perovskite nanocrystals.
As a result the trion optical transition rate is given by summing over the two possible radiative transitions the trion can undergo. The rate is the same for both positive and negative trions. Using the notation introduced in the main text, we can write for the positive trion in the strong-confinement regime: The three matrix elements in Eq.(S45) describe transitions that are accomplished by emission of photons with three different orthogonal polarizations. All these matrix elements are equal to each other, and as a result, the trion photoluminescence is not polarized. This can be confirmed by symmetry analysis. For nanocrystals of cubic symmetry (point group O h ), electrons and holes have symmetry Γ ∓ 6 , respectively, and are two-fold degenerate. Therefore, a positive/negative trion has symmetry Γ ∓ 6 and is also two-fold degenerate, since Γ + 6 × Γ − 6 × Γ ± 6 = Γ ∓ 6 . Moreover, optical decay from a trion state to a single-carrier state is allowed for all polarizations: The x, y, and z components of the dipole operator all transform as Γ − 4 , and Γ ∓ 6 × Γ − 4 contains Γ ± 6 . Furthermore, the matrix elements are equal for the x, y, and z components by symmetry.
Expressing the matrix elements via I , we obtain for the trion lifetime 1 τ trion = 2ωnE p 3 · 137m 0 c 2 I str 2 . (S46) Comparison of this expression with the exciton radiative lifetime in the strong-confinement regime from Eq. (S30) shows that trion lifetime is shorter: τ trion = (2/3)τ ex .
In Fig. 2b in the main text we show the experimental decay times measured in perovskite nanocrystals via single-nanocrystal experiments. The photoluminescence traces shown in faster than that of the exciton. The ∼ 2.5 -fold drop in photoluminescence intensity sug-gests however that for these nanocrystals, non-radiative Auger recombination significantly contributes to the decay, further shortening the trion lifetime.
Another indicator to distinguish between exciton transitions and those from trions is their polarization dependence. According to Fig. 3a-c, we observe that excitonic transitions exhibit typical dipolar emission with a high degree of linear polarization. In Extended Data Fig. 2a we plot the spectrum of a single CsPbBr 2 Cl nanocrystal that exhibits two emission peaks at 2.5158 and 2.5175 eV and an additional trion-emission peak that is red-shifted by approximately 16 meV. For measuring the polarization, we analyzed the emitted intensity of both exciton peaks and the trion peak as function of the linear polarizer angle in front of the spectrograph, which can be seen in Extended Data Fig. 2b. Both exciton peaks, depicted in blue and red, show again typical dipolar emission along different axes of polarization, which are indicated by the blue and red straight lines. The trion peak is essentially unpolarized, in agreement with theory.

S4. RELATIVE INTENSITIES OF PHOTOLUMINESCENCE FROM THREE ORTHOGONAL DIPOLES AND THEIR POLARIZATION PROPERTIES
The relative intensity of the photoluminescence created by three orthogonal emitting dipoles polarized along the x , y , and z axes and its polarization properties depend on the observation direction. The probability of emitting light for each of the dipoles is proportional to ∝ |e · x| 2 , ∝ |e · y| 2 , and ∝ |e · z| 2 , where the polarization unit vector e is perpendicular to the light-propagation direction. To calculate these dependences we denote the lightpropagation direction by the unit vector u with components given by u x = sin θ cos φ , u y = sin θ sin φ , where θ, φ are the standard polar angles. We then form a light polarization unit vector e in the plane perpendicular to u by e = (0, −u z , u y )/ u 2 z + u 2 y .
Finally, we gradually rotate the vector e around u and calculate the scalar products of the form ∝ |e · x| 2 , ∝ |e · y| 2 , ∝ |e · z| 2 at each angle. Each one of these scalar products represents the probability that the corresponding dipole will emit light in the direction u with linear polarization e .
The rotation of e around u is performed using the following transformation where Cs = cos α , Si = sin α , and α is a rotation angle. In fact, our calculations directly simulate the measurements performed by placing a linear polarizer perpendicular to a certain direction with respect to the emitting dipoles, and recording the intensity of the transmitted light as a function of the polarizer angle for each of the dipoles.
To obtain the total photoluminescence intensity emitted in the direction u for each of the lines, we integrate |e·x| 2 , |e·y| 2 , and |e·z| 2 over all polar angles α . In Extended Data Fig.   11 we provide several examples of the angular dependence of the emission probability in the plane perpendicular to the light-propagation direction and the relative photoluminescence intensity which can be observed from three orthogonal emitting dipoles, for four different directions. The calculations were conducted for: (i) a high temperature, T , that results in equal occupation of all three exciton levels (Extended Data Fig. 11a-d) and (ii) a temperature T that provides thermal energy that is comparable to the fine-structure splitting, kT = ∆ 1 = ∆ 2 (Extended Data Fig. 11e-h).
One can see in Extended Data Fig. 11a-d that two dipoles contribute for any observation direction. The one photoluminescence line can be observed only if the upper exciton sublevels are unoccupied (compare photoluminescence spectra in panels a and b with the ones in panels e and f). The relative photoluminescence intensities of two lines whose polarizations are perpendicular to each other allows us to measure the relative population of the exciton spin sublevels and therefore the effective temperature (compare panels b and f). One can also see that at high temperature when all exciton sublevels are populated the detected photoluminescence intensity of the upper energy line can be larger than that of the lower energy line (compare panels d and h).
Extended Data Fig. 11 shows photoluminescence intensity peaks and their polarization calculated for cube-shaped nanocrystals. In perovskite nanostructures with orthorhombic symmetry the triplet exciton state is always split into three orthogonal dipoles. As a result, the polarization curves should look very similar to the curves shown in the insets of Extended Data Fig. 11. The intensity of photoluminescence emitted by each of these dipoles can be very sensitive to the nanocrystal shape, due to the different screening of the photon electric field by the different facets of nanocrystals with non-cube shapes. The fluctuation of the nanocrystal shape could also affect the radiative decay time of the nanocrystals.
Varying the observation directions we can describe the photoluminescence polarization curves measured in individual-nanocrystal experiments (compare insets in Fig. 3a-c and