Suppressed phase separation of mixed-halide perovskites confined in endotaxial matrices

The functionality and performance of a semiconductor is determined by its bandgap. Alloying, as for instance in InxGa1-xN, has been a mainstream strategy for tuning the bandgap. Keeping the semiconductor alloys in the miscibility gap (being homogeneous), however, is non-trivial. This challenge is now being extended to halide perovskites – an emerging class of photovoltaic materials. While the bandgap can be conveniently tuned by mixing different halogen ions, as in CsPb(BrxI1-x)3, the so-called mixed-halide perovskites suffer from severe phase separation under illumination. Here, we discover that such phase separation can be highly suppressed by embedding nanocrystals of mixed-halide perovskites in an endotaxial matrix. The tuned bandgap remains remarkably stable under extremely intensive illumination. The agreement between the experiments and a nucleation model suggests that the size of the nanocrystals and the host-guest interfaces are critical for the photo-stability. The stabilized bandgap will be essential for the development of perovskite-based optoelectronics, such as tandem solar cells and full-color LEDs.


Supplementary Note 1. Theoretical Model Based on Nucleation
A thermodynamic model based on nucleation in phase transformation was developed to understand the mechanisms of the high photo-stability observed. In the model, photoexcited polarons, which were reported previously 1,2 , tend to trigger the phase separation. The cohesive 30 energy at interfaces, an additional factor that we considered in our model given the unique composite material morphology, is not in favor of separating mixed-halide perovskites into iodine-rich and bromine-rich domains. Particularly, when the interface-to-volume ratio is sufficiently large (i.e. the size of embedded perovskite nanocrystals was sufficiently small), the weight of the cohesive energy can dominate the total energy of the system, which eventually 35 eliminates the photo-induced phase separation.

DGdark for System in the Dark (Ground State)
Based on nucleation theory 3 , the change of the free energy DG would be contributed together by the volumetric enthalpy (Dhmix), the volumetric entropy (Dsmix) and all the cohesive energies Σci r 2 Wi. In the dark, DG = DGdark with the initial Br content XBr was expressed as: 40 (1) The coefficient, ci, of the cohesive energies, Wi, accounts for both CsPbI3/CsPbBr3 and CsPb(BrxI1-x)3/Cs4Pb(BrxI1-x)6 interfaces during phase separation with the values obtained from DFT calculations in Supplementary Note 2 (Computational Methods) with W1 = 0.42 eV nm -2 and 45 eV nm -2 . In Fig. 2a, for the CsPb(BrxI1-x)3-dominant thin films, only the interface W1 between CsPbI3 and CsPbBr3 were considered and DW2 (XBr) was treated as zero.
The phase diagram of the alloy could be built by calculating the free energy at 297 K as shown in  Figure 2). When Dgs (DGs per volume) becomes sufficiently large, the strain energy would be released by de-mixing the halogen anions and forming Br-and I-rich domains (i.e., phase separation). The increased strain energy was calculated as following 3 : (2) where a = 6.29-0.46XBr / a I = 6.29 / a Br = 5.83 Å (lattice constant values from PDF 01-076-8588, 60 00-054-0752) is the initial lattice constant for the pseudocubic lattice of CsPb(BrxI1x)3/CsPbI3/CsPbBr3 in dark, μ = 6.858-0.958XBr / μ I = 6.858 / μ Br = 5.9 GPa is the shear modulus of the alloy CsPb(BrxI1-x)3/CsPbI3/CsPbBr3 with the numbers adopted from Cahen et al. 10 and Hantezadeh et al. 11 , and δ is the change (shrink or expand) in lattice constant after photoexcitation.
Therefore, DG = DGlight in the excited state was written as: 65 The change of lattice distortion to trigger the phase separation in pure CsPb(BrxI1-x)3 (113) thin film with r = 35 nm was δ = 0.15 Å (weak illumination condition in the manuscript). This value is in reasonable agreement with previous results calculated based on first principle methods 7 . It is worth noting that the cohesive energy between CsPb(BrxI1-x)3 (113) and Cs4Pb(BrxI1-x)6 (416) was 70 also considered given the spatial confinement from the Cs4Pb(BrxI1-x)6 (416) matrix (Supplementary Figure 3).

Suppression of Phase Separation with Dominant Cohesive Energy
The phase separation under optical illumination could alternatively be suppressed if the domain size of the mixed-halide perovskites is reduced. The stability could be attributed to the 75 dominating role of the cohesive energy Σcir 2 Wi when the surface-to-volume ratio is increased.
When the intensity of polarons is low, the lattice distortion is limited to the localized area. The 95 perturbed region has a size smaller than the critical size for the iodine-rich domains rI * to induce phase transformation. The average lattice distortion δ in a unit volume is small resulting in a small Δgs and negative ΔGlight. As the intensity increases, the distorted areas start to interact each other.
It is then probable for the small nucleates to merge and form larger ones. When the intensity is over the threshold, nucleates with the size exceeding the critical radius of iodine-rich domains rI * 100 are formed. δ is increased leading to larger Δgs and positive ΔGlight, inducing phase separation. δ is positively related to the light intensity, so are Δgs and ΔGlight as described in Supplementary XBr originated from the smaller cohesive energies with more Br content, as described in Supplementary Equation 1 and 3. It's worth noting that, similar intensity dependent phase separation was reported by Kuno's group 5,15 , which is consistent with our model. 110

Calculation of Volumetric Gibbs Free Energy of Mixing Δgv for Ground-State CsPb(BrxI1-x)3
We modeled the mixed-halide perovskite CsPb(BrxI1-x)3 at the ground state (in dark) as a statistical ensemble of independent configurations under seven compositions: XBr = 0, 1/6, 1/3, 1/2, 2/3, 5/6, and 1. This was similar to the treatment of a binary alloy system. The mixing enthalpy, 115 Δhi (XBr), of each configuration i with ground-state energy, Ei (XBr), was defined as: (4) where ECsPbI3 and ECsPbBr3 represented the total energies of the pure compounds at the ground state when XBr = 0 and 1, respectively, and both terms were computed in per volume units.
After knowing the mixing enthalpy, Δhi (XBr), and the degeneracy of each configuration i 120 (due to different combinations of the anion positions under a constant XBr), we could estimate the ensemble (or degeneracy)-averaged mixing enthalpy, Δhmix (XBr, T) as functions of the composition XBr and temperature T accordingly to the Boltzmann distribution: The configuration energies Ei (XBr) were each computed within the framework of  Sham density functional theory (DFT) 16 . We considered a cubic supercell with 2 × 1 × 1 expansion of a cubic perovskite building block, which corresponds to six halide anions and 10 atoms in total.
The total number of configurations for this system was 2 6 = 64. We studied the cubic polymorph based on our experimentally confirmed fact that they all exhibited the cubic polymorph ( Fig. 1a-b in main text, Supplementary Figure 1). For such a cubic inorganic perovskite, the three halide 130 sites are almost equivalent (where the symmetry is greatly enlarged due to the presence of the monoatomic Cs + cations), which could reduce the total number of configurations to 21 in total.
We took the symmetry-reduced inequivalent configurations and perform a full structural relaxation for each case.
With Δhmix (XBr, T) calculated, the volumetric Gibbs free energy of mixing per unit volume in the dark, Δgv (XBr, T) could be directly evaluated: (6) 145 where we estimated Δsmix (XBr, T) assuming the ideal mixing behavior among halide anions:

Computation of Cohesive Energy at the Phase-Separated CsPbBr3-CsPbI3 Interface 150
To determine the cohesive energy between pure CsPbBr3 and CsPbI3 W1 (equivalent to interfacial tension), we constructed a supercell (2 × 2 × 2 of CsPbBr3 interfacing with 2 × 2 × 2 of CsPbI3) with an interface between the two pure materials, as shown in Supplementary Figure 5b.
It is important to note that the cohesive energy contains contributions from: (i) the chemical interfacial energy due to chemical potential difference at the interface, and (ii) the strain energy at 155 the interfaces due to slight lattice mismatch between two crystals when they are constrained to have the same lateral dimension in the supercell. The same DFT method and parameters mentioned above were used to relax this supercell, as well as to obtain the optimized configuration and the associated total energy at the ground state, ECsPbI3+CsPbBr3. The cohesive energy, W1, was then calculated as: 160 (8) where ECsPbI3 (2 × 2 × 2) and ECsPbBr3 (2 × 2 × 2) represent the DFT-computed total energies of the individual pure compounds at the ground state, each with a cell height that is one half of the supercell used to compute ECsPbI3+ CsPbBr3, A represents the relaxed interfacial area, and the factor 165 "2" accounts for the two interfaces due to the periodic boundary condition applied in the DFT calculation. With the assumption that the predicted cohesive energy of the ground-state CsPbBr3/CsPbI3 interface does not change after photoexcitation, we obtained a cohesive energy of The CsPbBr3/Cs4PbX6 interface is preferred over the interface of CsPbI3/Cs4PbX6 due to 190 the much smaller cohesive energy between CsPbBr3 and Cs4PbX6. The nucleation of I-rich domains was reported previously in mixed-halide perovskites 1,23 . It is likely that the new I-rich phase starts to nucleate inside each alloy nanocrystal, making the matrix Cs4PbX6 (416) phase interface with a Br-rich CsPbX3 (113) phase in the outer region of the nanoparticle, as illustrated in Supplementary Figure 3. The change of bandgap measured and lattice constant with the 195 composition indicated that 113 and 416 species shared the similar XBr (Fig. 1c&1f, Supplementary   Figure 1). Therefore, we could assume that the interface between CsPb(BrxI1-x)3 and Cs4Pb(BrxI1- where α is the absorption coefficient of mixed-halide perovskite; Iexcitation is the illumination intensity; Eph is the energy of photons. The excess carrier density is calculated as (11) τ is the carrier lifetime, which is estimated as about 20~70 ns from our measurements.
Substitute the numbers with the absorption coefficient 24,25 α~10 5 cm -1 , Iexcitation=0.3 W cm -230 2 , Eph=3.39 eV Substitute τ=20 ns and G from the above calculation, the excess carrier density is calculated as Which should be similar to the conditions used in previous reports 1,2,5,26 . Note that the excess 235 carrier density in CsPbX3/Cs4PbX6 composites might be under estimated in mixed-halide perovskite as Cs4PbX6 phase also have a small amount of absorption with broad excitation peak centered at Eph = 3.39 eV and pass more charge carriers to CsPbX3 nanocrystals.
For the condition of strong illumination (440 W cm -2 ), we used a 405-nm laser source. The numbers for the calculation were then changed to 240

Lifetime (PL lifetime) of Composite Thin Films
The (bottom) form a well-matched interface with two face-sharing PbX6 octahedra.