Rationalizing the light-induced phase separation of mixed halide organic–inorganic perovskites

Mixed halide hybrid perovskites, CH3NH3Pb(I1−xBrx)3, represent good candidates for low-cost, high efficiency photovoltaic, and light-emitting devices. Their band gaps can be tuned from 1.6 to 2.3 eV, by changing the halide anion identity. Unfortunately, mixed halide perovskites undergo phase separation under illumination. This leads to iodide- and bromide-rich domains along with corresponding changes to the material’s optical/electrical response. Here, using combined spectroscopic measurements and theoretical modeling, we quantitatively rationalize all microscopic processes that occur during phase separation. Our model suggests that the driving force behind phase separation is the bandgap reduction of iodide-rich phases. It additionally explains observed non-linear intensity dependencies, as well as self-limited growth of iodide-rich domains. Most importantly, our model reveals that mixed halide perovskites can be stabilized against phase separation by deliberately engineering carrier diffusion lengths and injected carrier densities.


Preliminaries
The total fraction of mixed phase ( mix ), iodide-rich ( I ) and bromide-rich ( Br ) domains in the thin film is given by 1 = mix + I + Br .
Since I = Br (halide segregation simultaneously produces iodide-and bromide-rich domains) one equivalently has 1 = mix + 2 I = mix + ps 6 where ps is fraction of phase-separated domains ( Br + I ). Solving for mix then gives mix = 1 − ps . (1)

Relevant rate processes
The following are relevant kinetic rate processes involving the hybrid perovskite during photoexcitation. The excitation rate ( ) of the hybrid perovskite with incident light is given by = exc ℎ exc (2) where Iexc is the excitation intensity (W cm -2 ), is the hybrid perovskite absorption coefficient (cm -1 ), and ℎ exc is the energy of the incident photon (J photon -1 ). The corresponding photogenerated carrier density is where is the carrier lifetime. The associated emission rate from the mixed halide perovskite is expressed as where is the perovskite's external quantum yield, which for simplicity is assumed to be a constant over all excitation intensities. We take the phase separation rate to be first order for a given excitation intensity G 2 = 2 mix = 2 (1 − ps ) .
In the equation, 2 is a unitless probability associated with inducing phase segregation upon excitation. We treat the Iexc-dependence of k2 below. We treat the phase segregation recovery as first order in domain concentration: where n is the average molar volume of the domains.

Phase segregated fraction, ps
Processes 3 and 4 contribute to changes in the phase segregated fraction, ps . Consequently, the total rate of change of ps can be expressed as Using Supplementary Equations 5 and 6 for r2 and r3 we therefore have ps = 2 (1 − ps ) − 3 ps = 1 − 2 ps where 1 = 2 and 2 = 1 + 3 . The solution to this differential equation with the initial condition ps (0) = 0 is In the limit that photo-induced phase separation occurs on a significantly faster timescale than thermal recovery This is Equation 2 of the main text when 1 ≫ 3 .

Connection to the emission rate
Supplementary Figure 14. Carrier diffusion to iodide-rich domains.
The rate of appearance of the iodide-rich phase (Process 3) is much slower than the rate at which the iodide-rich photoluminescence (Iiodide) changes. Thus, we posit that the largest contribution to Iiodide comes from the migration of carriers generated in the parent mixed halide phase to nearby iodide-rich regions, resulting in subsequent radiative recombination. Note that our DFT calculations below suggest hole diffusion as the dominant process taking place due to the existence of favorable valence band offsets between MAPb(I1-xBrx)3 and iodide-rich regions, but the exact nature of the carriers is immaterial to the basic arguments. The underlying assumption is that the presence of an iodide-rich region within the geometric volume defined by e/h means that charge migration dominates direct photogenerated carrier recombination (Process 2). Plausibility for this can be seen by comparing the time for charge diffusion d to the carrier lifetime () as determined from experimental emission decay measurements. Namely, using the Einstein relation ( d = e/h 2 ) with le/h=100 nm and a diffusion coefficient of D=0.08 cm 2 s -1 1 we obtain d =1.25 ns which is significantly shorter than which is on the order of 62 ns. 1 Consequently, Iiodide is proportional to the Poisson distributed probability (P) of carriers encountering at least one iodide domain upon diffusion. For analytical tractability, this probability can be expressed in terms of the probability of encountering no iodide-rich domain (Pnone) and is = 1 − none (9) where Pnone= − I with nI the average number of iodide domains within the diffusion volume (VD).
We therefore obtain the following expression for iodide ∝ with sat an empirical saturation photoluminescence. In the equation, nI can be expressed in terms of D , the average volume of iodide-rich domains ( ̅ ), and I such that Using the linear approximation of Supplementary Equation 8 then gives from where we obtain the following expression for Iiodine/Isat We use DFT to estimate the ground state formation energy of MAPb(I1-xBrx)3, MAPbBr3, and MAPbI3. The resulting ground state formation difference per formula unit is defined as where , Br and I are the GGA energies per formula units of MAPb(I1-xBrx)3, MAPbBr3, and MAPbI3, respectively. We use the experimentally observed cubic structure for MAPbBr3 (symmetry Pm3 ̅ m 2 ) with a GGA-computed lattice constant of 6.14 Å. We compute the energies of MAPbI3 and the mixed halides starting from the orthorhombic structure (symmetry Pnma (Z=4) 3,4 ). The GGA-computed lattice constants of the iodide orthorhombic phase are 9.25 × 12.89 × 8.62 Å. We choose orthorhombic over tetragonal across the compositional domain for computational convenience, because it avoids orientational ambiguity for MA cations in the tetragonal structure.
To model the mixed halide, we start from four formula unit cells of iodide, containing 12 iodide ions, generate mixed configurations by replacing iodide with bromide, and relax the orthorhombic cell parameters and fractional coordinates. We consider six compositions ranging from x = 1/12 to ½ in 1/12 steps. The configurations are chosen arbitrarily to maximize the separation between the bromide ions and to probe the first-nearest-neighbor interactions that would be expected to dominate any ordering tendencies. A representative structure at x = 0.5 is shown in Supplementary  Figure 4. CONTCARs of all relaxed structures are included as separate files.
Computed EGS values per two formula units, or FU, are shown in Supplementary Figure 5a as solid blue circles. Solid lines are included as a guide to the eye. From the plot it is evident that MAPb(I1-xBrx)3 formation energies are small and positive up to x  1/3, above which formation energies become negative. These results can be compared with the GGA results of Brivio et al. 5 , who employed a more restrictive tetragonal representation of mixed halide structures but considered a greater variety of anion configurations at a given composition. Brivio et al. 5 found formation energies to be similarly positive and of order 0.01 eV formula unit -1 to even higher compositions.
Both the results reported here and those of Brivio et al. 5 are based on modest-sized computational cells that imply some level of ordering and do not consider the consequences of MA dynamics. Consequently, we cannot rule out the existence of much lower energy structures when these assumptions are relaxed. However, there is no evidence for this possibility either in the DFT calculations or experimental observations. Thus, all available DFT data are consistent with ground state EGS formation energies that approach zero in the iodide-rich region (i.e. x less than 0.5).
Next, to account for entropic effects on free energies of formation, we combine Δ GS ( ) with the ideal entropy of mixing on the anion sub-lattice. Resulting free energies of formation per formula unit (Δ GS ( )) are then The results show that MAPbBr3 band offsets are unfavorable for either electron or hole localization within bromide-rich regions of hybrid perovskite films. We consequently focus on band offsets present at the MAPb(I1-xBrx)3 and MAPbI3 interface. We construct an orthorhombic supercell doubled in the long direction, consisting of two layers of mixed perovskite below two layers of MAPbI3. Lateral lattice constants of the supercell match those of MAPbI3. The mixed domain is constructed by replacing every other iodide ion with bromide. After relaxing atomic positions, the layer-by-layer projected density of states is computed and is plotted in Supplementary Figure 5c. In agreement with experimental inferences, we find that MAPb(I1-xBrx)3 and MAPbI3 conduction band edges are essentially isoenergetic and that the valence band maximum of MAPbI3 is 0.2 eV above that of MAPb(I0.5Br0.5)3 (Supplementary Figure 5c). The GGA-computed band gap is also consistent with experiment, likely due to cancellation of relativistic and many-body effects. 8 Because the conduction bands of MAPb(I1-xBrx)3 and MAPbI3 are effectively isoenergetic, their valence band offsets across the compositional range of interest can be captured using known experimental band gap differences. The band gap is a function of the composition x, increasing from 1.57 eV in the iodide form to 2.29 eV in the bromide form. 9 Noh et al. 9 have fit the experimentally-observed band gap (Eg) difference (Δ g ) between MAPb(I1-xBrx)3 ( g ) and MAPbI3 ( I g ) phase as a function of composition according to: Δ g ( ) = g − I g = 0.39 + 0.33 2 eV.
Δ g increases monotonically with x and reaches 0.28 eV at x = 0.5, for instance. This monotonic trend is consistent with the preferential localization of photogenerated holes in iodide-rich regions of hybrid perovskite film upon photoexcitation. 10 Optical excitation Next, we consider how this band gap difference alters the thermodynamic preference for phase separation when a MAPb(I1-xBrx)3 sample is under illumination. We estimate this effect by evaluating the free energy of complete phase separation in the presence of a photo-generated hole: MAPb(Br I 1− ) 3 * → MAPbBr 3 + (1 − )MAPbI 3 * .
Here n represents some number of formula units. We write the free energy of this process as where mix * , FBr, and I * are free energies of photoexcited MAPb(I1-xBrx)3, MAPbBr3, and MAPbI3. This is Equation 1 of the main text. Supplementary Figure 5a shows the resulting F * computed for n = 2 formula units at 300 K. The excitation energy difference is greater than the mixing free energy and is substantially greater than the DFT-computed mixing energy. Thus, optical excitation carries sufficient energy to drive multiple formula units of the mixed phase into bromide-and iodide-rich phases, supporting the kinetic model. Supplementary Equation 16 is written for separation into pure domains. More precisely, the separated domain compositions will be a compromise between how the free energies and the band gap vary with composition, how these are influenced by interfacial energies, and will even be influenced by the dynamics of anion mobility. Taking decomposition into pure phases yields the maximum Δ GS and thus the free energy cost of phase separation. Separation into iodide-and bromide-enriched phases will yield correspondingly smaller Δ GS values. Supplementary Figure  5a shows Δ * to be near zero in the domain x less than 0.1 with a predominantly iodide-rich phase composition of MAPb(I0.9Br0.1)3. The estimate is consistent with the results of McGehee et al. 11 who have previously suggested that iodide-rich phases following prolonged photoexcitation adopt a composition with x approximately 0. The non-linear Iexc-dependence of kforward,em (Figure 1d, main text) indicates that it is a nonlinear function of photogenerated carrier density (). This observed dependence is readily rationalized using a simple model where the probability of creating a phase-separated domain within the geometric volume defined by the carrier diffusion length (le/h) is the same irrespective of the number of carriers in that volume. To this end, we assume that kforward,em is proportional to the Poisson-distributed probability that at least one carrier exists within VD. For analytical tractability, this probability is expressed in terms of the probability for having no carrier within VD (Pnone) so that forward,em ∝ 1 − none .
Next, since the average number of carriers within D is D we express Pnone as none = − D so that where sat is an empirical saturation rate constant at large excitation intensities and  is an empirical fitting parameter. This is Equation 4 of the main text. Note that Supplementary Equation 19 can be used as a check for the model's self-consistency. Namely, if Supplementary Equation 18 is fit to the experimental data in Figure 1d  =10 ns and = 10 5 cm -1 , we find an associated diffusion length of = 320 nm, which agrees well with known literature values of le/h~100 nm. 12 At sufficiently low Iexc-values, the phase segregation recovery rate exceeds phase separation. Consequently, Supplementary Equation 19 allows us to estimate an excitation intensity threshold (Ith) below which phase segregation is unfavored. We assume this to occur when kforward,em is an order of magnitude smaller than kreverse,em. Using kreverse,em=6.8 × 10 −3 s -1 , sat = 0.7 s -1 and = 0.29 cm 2 mW -1 , we estimate an intensity threshold of Ith ~ 20 W cm -2 , which agrees well with the experimentally-determined threshold value of Ith approximatively 40 W cm -2 .
Mixed halide perovskites were synthesized using a previously reported procedure 13 with slight modifications. Cs-oleate was synthesized by combining Cs2CO3 (0.814g, Sigma-Aldrich 99.9%), 2.5 mL of oleic acid (Sigma-Aldrich, 90%), and 40 mL of octadecene (Sigma-Aldrich, 90%) in a 100 mL 3-neck flask. The reaction mixture was then degassed at 120 ○ C for 1 hr. Following degassing, the vessel was backfilled with N2 and was heated to 150 ○ C. The resulting solution was used without any further purification.
Mixed halide perovskite nanocrystals were synthesized by adding the desired molar ratio of PbBr2 (Sigma-Aldrich, 99.999%) to PbI2 (Sigma-Aldrich, 99.999%) to 5 mL of octadecene in a 10 mL 3-neck flask. The mixture was degassed under vacuum for 1 hr at 120 ○ C and was then backfilled with N2. Then, 1 mL of oleylamine (Acros, 80-90%) and 1 mL of oleic acid were injected into the mixture. After the lead salts had dissolved completely, the reaction mixture was