Defects in Halide Perovskites: Does It Help to Switch from 3D to 2D?

Two-dimensional (2D) organic–inorganic hybrid iodide perovskites have been put forward in recent years as stable alternatives to their three-dimensional (3D) counterparts. Using first-principles calculations, we demonstrate that equilibrium concentrations of point defects in the 2D perovskites PEA2PbI4, BA2PbI4, and PEA2SnI4 (PEA, phenethylammonium; BA, butylammonium) are much lower than in comparable 3D perovskites. Bonding disruptions by defects are more destructive in 2D than in 3D networks, making defect formation energetically more costly. The stability of 2D Sn iodide perovskites can be further enhanced by alloying with Pb. Should, however, point defects emerge in sizable concentrations as a result of nonequilibrium growth conditions, for instance, then those defects likely hamper the optoelectronic performance of the 2D perovskites, as they introduce deep traps. We suggest that trap levels are responsible for the broad sub-bandgap emission in 2D perovskites observed in experiments.

Hybrid organometal halide perovskites are materializing as candidate semiconductors for new generations of optoelectronic devices such as solar cells and light-emitting diodes. 1,24][5][6] One of the first frequently studied compounds, MAPbI 3 , has favorable optical and charge transport properties, [7][8][9][10] but the MA + (methylammonium) ion is chemically not sufficiently stable, and suffers from degradation reactions. 11,126][17] This tendency can be suppressed to a certain extent by mixing in smaller inorganic cations, such as Cs + , 18,19 but the fundamental issue remains that a stable 3D perovskite lattice requires the sizes of the constituting ions to be of a certain proportion, as expressed by the Goldschmidt tolerance factor, 2,20 and the scale is set by the 3D network of metal halide octahedra in the perovskite.
In recent years, organometal halide perovskites with a Ruddlesden-Popper structure have emerged as alternative materials. 21In these perovskites the metal halide octahedra form a planar 2D network, and these 2D layers are separated by layers of organic cations, where the interlayer interaction is typically Vanderwaals. 22Using organic ions with a quasi-linear structure, such as PEA (phenethylammonium) 23 or BA (butylammonium), 24 the in-plane tolerance factor for a stable crystal structure is easily obeyed, whereas the out-of-plane size of the organic ion becomes relatively unimportant.Although the stability of such 2D perovskites is markedly improved, as compared to their 3D counterparts, presently photoelectric devices based upon 2D perovskites fail to reach the high efficiencies obtained with 3D perovskites. 22In terms of this, defects can play an important role, whereas their concentrations in 2D perovskites and resulting impacts on electronic properties are not yet clear. 22 this paper we explore the defect chemistry and physics of prominent 2D organometal iodide perovskites, PEA 2 PbI 4 , BA 2 PbI 4 , and PEA 2 SnI 4 , and the alloy PEA 2 Sn 0.5 Pb 0.5 I 4 , using first-principles density functional theory (DFT) calculations.The ease with which point defects can be created in a material is an indication for its stability.We therefore focus on the defect formation energy (DFE) as it can be calculated assuming thermodynamic equilibrium conditions.We use the same formalism as applied in our previous work on 3D perovskites. 25A summary of the theory is given in the Supporting Information (SI), Sec.I.
The equilibrium chemical potentials of the different elements are determined by considering the phase diagram of the 2D perovskite, see the SI, Fig. S1.The defect formation energies are calculated using iodine-medium conditions, which are the conditions most typically used.
Even if defects do not occur in large quantities under thermodynamic equilibrium conditions, they may appear more prominently under nonequilibrium growth conditions, or under operating conditions. 26If so, they can seriously affect the electronic properties of the material, as defect states with energy levels inside the semiconductor band gap can act as traps for charge carriers, and as recombination centers for radiationless decay.We explore these energy levels, called charge-state transition levels (CSTLs), associated with the most likely point defects in the 2D materials listed above.
DFT calculations are performed on 2×2×1 supercells of 2D perovskites, with the Vienna Ab Initio Simulation Package (VASP), [27][28][29] employing the SCAN + rVV10 functional 30 for electronic calculations and geometry optimization.The SCAN+rVV10 functional is used aiming at obtaining accurate defective structures and total energies, and therefore the DFEs, which are the main focus of this work.Whereas the DFT band gap error may result in incorrect band edges, the calculated CSTLs are suggested to be correct in a relative sense, as discussed in our previous work Ref. 31.Detailed computational settings and structures are discissed in the SI, Sec.I. We start with point defects in the most popular 2D perovskite, PEA 2 PbI 4 , i.e, the PEA vacancy V PEA , the Pb vacancy V Pb , and the iodine vacancy V I , and the interstitials Pb i and I i .The PEA interstitial is omitted because the structure is too dense for additionally accommodating such an extra large-size organic cation.In addition to these simple point defects, we also study the compound vacancies V PEAI and V PbI 2 , representing missing units of the precursors PEAI and PbI 2 .The layered nature of  temperature of 5.73 ×10 7 cm −3 and 1.09 ×10 7 cm −3 , respectively.Other vacancies, V I + , and the compound vacancies V PEAI 0 and V PbI 2 0 , as well as all interstitial species, have formation energies 1 eV, and are thus unimportant under equilibrium conditions (with the intrinsic Fermi level).A full list of formation energies and concentrations at room temperature of defects is given in Table 1.
These findings are in stark contrast with results obtained for 3D perovskites, calculated using the same computational settings. 25In the archetype 3D perovskite MAPbI 2.30× 10 10 2.18×10 −1 1.59×10 −1 2.95×10 −31 a To facilitate comparison to the other compounds, formation energies and defect concentrations in MASnI 3 are calculated under iodine-medium conditions, which are however unrealistic experimentally for this compound. 25fect chemistry of 3D perovskites. 25In MAPbI 3 , besides the vacancy V Pb 2− , the dominant point defects are the interstitials MA i + and I i − .By comparison, interstitials in PEA 2 PbI 4 are unimportant relative to vacancies.
The chemical bonding patterns of such interstitials in 2D perovskites are also quite different from those in 3D.Iodine interstitials in 3D perovskites are inserted between two Pb atoms in the lattice, next to an already present iodine ion, forming a Pb-I 2 -Pb unit with two equivalent Pb-I-Pb bridge bonds. 25,32In contrast, iodine interstitials in PEA 2 PbI 4 prefer to stay in-between two lattice iodines, forming a I 3 trimer structure, Figure 1(i),(j).This difference in bonding pattern is also reflected in the most stable charge state.Whereas the iodine interstitial in 3D perovskites is negatively charged, in PEA 2 PbI 4 it is positively charged.Alternatively, one might call the trimer structure with the lattice iodines a negatively charged The more prominent defects in PEA 2 PbI 4 , the vacancies V Pb 2− and V PEA − , Figure 1(b),(e), and also the less prominent vacancies V I + , V PEAI 0 and V PbI 2 0 , Figure 1(c),(d),(f),(g), have bonding patterns that are qualitatively similar to those in 3D perovskites, and stable charge states that are the same.The DFEs of these defects in PEA 2 PbI 4 are significantly higher, though.Whereas in 3D perovskites the presence of a vacancy can be accommodated to some extent by rearranging the lattice around the vacancy, in a 2D perovskite such a rearrangement is more difficult.[35] The analysis of results obtained for PEA Stabilization of compounds in 2D structures offers an interesting perspective for Sn-based perovskites, where for instance the 3D MASnI 3 perovskite is quite unstable.That is reflected by the ease with which V Sn 2− vacancies are generated spontaneously, making this compound an intrinsically doped degenerate p-type semiconductor. 257][38][39] In fact, calculations indicate that MASnI 3 is thermodynamically stable only under rather extreme iodine-poor conditions. 25gure 2(c) shows the formation energies of defects in PEA 2 SnI 4 , calculated under milder, iodine-medium, conditions.All DFEs are positive, indicating that the material is stable against the spontaneous formation of defects, which is in contrast with MASnI 3 , see SI Table 1.The vacancy V Sn in PEA 2 SnI 4 is the defect with lowest DFE, but the latter is considerably higher than its DFE in MASnI 3 , signaling an increased stability of the material.The vacancy is negatively charged, V Sn − , and, as is common in Sn-based halide perovskites, there is no appreciable concentration of positively charged defects to maintain charge neutrality. 25,40e latter has to be ensured by holes in the valence band, which leads to an intrinsic Fermi level that is only 0.09 eV above the VBM.It results in PEA 2 SnI 4 being an intrinsic p-type semiconductor, albeit not a degenerate one, as is the case for MASnI 3 .
The DFE of the main defect, V Sn − , in PEA 2 SnI 4 is 0.26 eV, which gives an equilibrium concentation (at room temperature) of 3.08 × 10 17 cm −3 .The vacancy V PEA 0 has a DFE of 0.52 eV, and an equilibrium concentration of 3.13 × 10 12 cm −3 , whereas other vacancies and interstitials have a DFE in excess of 0.9 eV, so they do not play a significant role.Compared to PEA 2 PbI 4 , note that the most stable charge states of the prominent defects are +1e higher (V Pb 2− and V PEA − ).This stems from the low lying intrinsic Fermi level in PEA 2 SnI 4 , as compared to that in PEA 2 PbI 4 , consistent with an increased p-type doping, comparing Figure 2(a) and (c).
One can observe that the DFEs of defects in PEA 2 SnI 4 tend to be significantly smaller than the corresponding ones in PEA 2 PbI 4 , see Figure 2(f).The intrinsic defect concentrations in the former are therefore much higher, which is a clear sign of a decrease in stability.
In particular, 2D Sn-based perovskites inherit the problem of a large amount of Sn vacancies from the 3D Sn-based perovskites.To suppress the formation of Sn vacancies, one solution is to mix Sn with Pb. 39,40 We investigate its effect using the perovskite PEA 2 Pb 0.5 Sn 0.5 I 4 .
As an example, we study a highly ordered structure of PEA 2 Pb 0.5 Sn 0.5 I 4 , whose construction is discussed in Figure S2  That this is the case is corroborated by optical measurements. 41The DFEs of other defects, such as V PEA − , are comparable to those in pure PEA 2 PbI 4 , or even higher, see Figure 2(f).
In summary, mixing Pb and Sn in the 2D perovskite significantly suppresses the formation of defects compared to the pure Sn-based perovskite, and maintains the defect tolerance of the pure Pb-based perovskite.
Although in equilibrium defect concentrations in the 2D perovskites considered here, with the exception of PEA 2 SnI 4 , are predicted to be small, materials are often grown under highly non-equilibrium conditions, during which a substantial amount of defects can form. 26,42kewise, under solar cell operating conditions, the (quasi) Fermi levels are very different from the intrinsic Fermi level, which may stimulate the formation of certain defects, Figure 2.
These two points motivate investigating whether defects lead to electronic levels inside the band gap of a 2D material that can be harmful to its electronic operation.
These (so-called) charge state transition levels (CSTLs) are shown in Figure 3 for each point defect and compound vacancy in each of the 2D perovskites studied here.Generally, CSTLs are considered to be deep levels when their energy distance from the band edges is much larger than the thermal energy k B T (0.026 eV at the room temperature). 43Deep levels can trap charge carriers and cause significant (nonradiative) recombination, thereby reducing solar cell or light-emitting diode efficiencies.What can be observed in Figure 3 is that most point defects in 2D perovskites lead to deep levels.In fact, only the compound vacancies, V PEAI and V PbI 2 , give shallow (acceptor) levels.This is in remarkable contrast to what is found for 3D perovskites, where most, point or compound, defects produce shallow levels. 25,26This immediately provides a possible explanation of why the efficiency of 2D halide perovskite solar cells is generally smaller than those based on 3D perovskites.
For 3D perovskites it is argued that defects producing mainly shallow levels stems from the fact that defect states in these materials either have valence band or conduction band character, depending on the type of defect, and that the disruption in the chemical bonding pattern caused by the defect, is not so large as to move the defect levels in energy far from the band edges.Figure 3 shows that the CSTLs in 2D perovskites are still either in the top half or in the bottom half of the band gap, depending on the type of defect, but apparently the disruption caused by the defects is now sufficiently large to move defect levels to well inside the band gap.Part of these changes in going from 3D to 2D perovskites can also be explained by a decrease of the dielectric constant, which is also expected to lead to deeper levels.
As an example, consider the organic cation vacancies in 3D perovskites such as MAPbI 3 , FAPbI 3 or MASnI 3 , where V MA − and V FA − only give a very shallow acceptor level just above the VBM. 25 In all 2D compounds studies here, V PEA − or V BA − gives an acceptor level 0.2 eV above the VBM that can act as a trap state, see the first columns in Figure 3(a)-(d).
Likewise, the Pb and Sn vacancies, V Pb 2− , V Sn 2− , which only give very shallow acceptor levels in 3D perovskites, 25 give a couple of deep trap levels in 2D perovskites, as can be observed from the second columns in Figure 3(a)-(d).
An interesting case is the iodine vacancy V I , which in 3D perovskites typically only gives shallow acceptor levels.In 2D perovskites, V I generates different levels according to the position of the vacancy, whether above/below or in a PbI 2 plane, see Figure 1(c,d).Vacancies associated with the in-plane positions generate deep levels that are ∼ 0.65 eV below the CBM, whereas iodine vacancies located outside PbI 2 planes give donor levels that are much closer to the conduction band edge, which is correlated with the larger extent of bonding disruption on the 2D network induced by the former.In experiments, a broad emission of light with frequencies corresponding to the upper half of the band gap is frequently found in photoluminescence spectra of 2D perovskites, where the peak of the emission spectrum is approximately at 0.6 eV below the CBM. 44,45We suggest that this emission may be associated with defect states created by iodine vacancies.The only other defect that gives a level in the upper half of the band gap, the Pb interstitial Pb i , is not likely to occur in appreciable quantities, Figure 2.Moreover, the Sn interstitial, Sn i , only gives a level close to the VBM, and the broad emission of the type discussed above is also observed in 2D Sn-based perovskites.
To conclude, we employ first-principles calculations to study the defect formation energies and charge state transition levels of intrinsic defects in Ruddlesden-Popper hybrid iodide 2D perovskites.We find that the equilibrium concentrations of point defects in the 2D respectively.An interstitial is created by adding to the supercell a cation or an anion in a specific charge state, and then optimize the atomic positions within the supercell.Likewise, a vacancy is created by removing from the supercell a cation or an anion.

Defect formation energy
The defect formation energy ∆H f is calculated from the expression where E tot D q and E tot (bulk) are the DFT total energies of the defective and pristine supercells, respectively, and n i and µ i are the number of atoms and chemical potential of atomic species i added to (n i > 0) or removed from (n i < 0) the pristine supercell in order to create the defect.
Creating a charge q requires taking electrons from or adding them to a reservoir at a fixed Fermi level.The latter is calculated as E F + E VBM , with 0 ≤ E F ≤ E g , the band gap, and E VBM the energy of the valence band maximum.As it is difficult to determine the latter from a calculation on a defective cell, one establishes E VBM in the pristine cell, shifted by ∆V , which is calculated by lining up the core level on an atom in the pristine and the neutral defective cell that is far from the defect.S11,S12 We neglect the vibrational contributions to the DFEs, and the effect of thermal expansion on the DFEs, as these are typically small in the present compounds.The intrinsic Fermi level can be determined by the charge neutrality condition, which expresses the fact that, if no charges are injected in a material, it has to be charge neutral p − n + D q q c(D q ) = 0, (S2) where p and n are the intrinsic charge densities of holes and electrons of the semiconductor material, c(D q ) is the concentration of defect D q , and the sum is over all types of charged defects.The concentrations can be calculated from Boltzmann statistics where c 0 (D q ) is the density of possible sites for the defect (defined by the number of possible sites for the defect D q in the unit volume), T is the temperature, k B is the Boltzmann constant, and ∆H f (D q ) follows from Equation S1.Obviously, p, n, and c(D q ) are functions of E F , so the charge neutrality condition, Equation S2, serves to determine the intrinsic position of the Fermi level E (i) Since the E (i) F is relatively close to the valence band maximum, the intrinsic hole density p is large and important for maintaining the charge neutrality together with the charged defects, while the intrinsic electron density n is negligibly small.The hole concentration p can be calculated from the density of states near the VBM:  The main interaction of ions in 2D perovskites is within the inorganic layer.Therefore, Pb and Sn are mixed in the same layer.Whereas the stacking pattern of adjacent layers has two possibilities, as shown in Figure S2, the second configuration is 0.03 eV per unitcell less stable than the first one.Therefore, we choose the one in Figure S2(a) as the model for studying defects.Another possible configuration can be one pure Sn layer stacked with one pure Pb layer, but apparently the local interaction is expected to be similar to the pure Snor Pb-based perovskites.

Figure 1 :
Figure 1: (a) Top and side views of a 2 × 2 × 2 PEA 2 PbI 4 supercell; optimized structures of vacancies (b-g) and interstitials (h-j) in their most stable charge states in PEA 2 PbI 4 .The positions of the defects are marked in red.The labels (in-plane) and (out-of-plane) refer to positions of iodine vacancies and interstitials either within an PbI 2 plane or above/below it.

Figure 2 :
Figure 2: (a-d) Defect formation energies in PEA 2 PbI 4 , BA 2 PbI 4 , PEA 2 SnI 4 , and PEA 2 Pb 0.5 Sn 0.5 I 4 as function of the position of the Fermi level; the intrinsic Fermi level is indicated by the black dashed line.Comparison of DFEs at the intrinsic Fermi level to MAPbI 3 (e), and among 2D perovskites with different metal cations (f); the DFEs of Sn i are 2.2 eV and 3.1 eV, for PEA 2 SnI 4 and PEA 2 Pb 0.5 Sn 0.5 I 4 , respectively.

2 2
Figure 2(a)-(d).The defect in PEA 2 Pb 0.5 Sn 0.5 I 4 that is easiest to form is the vacancy V Pb 2− with a DFE of 0.65 eV, whereas the DFE of V Sn 2− is 0.19 eV higher.Although this particular difference may be the result of the particular ordered structure we have chosen to represent PEA 2 Pb 0.5 Sn 0.5 I 4 , we would argue that DFEs of metal vacancies in this compound are larger than in pure PEA 2 SnI 4 , provided Sn and Pb metal atoms are well mixed on an atomic scale.

Figure 3 :
Figure 3: Charge state transition levels (CSTLs) of defects in PEA 2 PbI 4 (a), BA 2 PbI 4 (b), PEA 2 SnI 4 (c), and PEA 2 Pb 0.5 Sn 0.5 I 4 (d); the most important ones are indicated by colored lines, representing a change of a single unit ±e starting from one stable charge state of a defect; the bottom and top gray areas represent the valence and conduction bands (calculated with SCAN+rVV10), aligned at the VBM.In (d), M represents metal cations, Pb and Sn.

Figure S1 :
Figure S1: Calculated stability diagrams of PEA 2 PbI 4 and PEA 2 SnI 4 .µ I = µ I 2 ,molecule /2 corresponds to ∆µ I = 0 in the figures, and µ M = µ M,bulk defines ∆µ M = 0 for M = Pb, Sn.The points A and B define iodine-rich and iodine-poor conditions, respectively.Iodinemedium conditions (point C) are defined as halfway between points A and B.

.
The outer bounds define I-poor (or Pb-rich) or I-rich conditions, respectively, with µ Pb,bulk and µ I 2 ,molecule the DFT total energies of bulk Pb metal, and an I 2 molecule.I-poor and I-rich conditions are indicated by points B and A, respectively, in Figure S1(a).The I-medium condition (point C) is defined as the halfway between points A and B, which is the focus of this work.The stability diagram of BA 2 PbI 4 is similar to PEA 2 PbI 4 .For PEA 2 SnI 4 , the allowed interval for µ X has to be narrowed down to prevent the formation of other phases, such as SnI 4 , S14-S16 see Figure S1(b).For PEA 2 Sn 0.5 Pb 0.5 I 4 , we take the point C in the stability diagram of PEA 2 SnI 4 to get the µ I and µ Sn .With the µ I , the µ Pb is then determined in Figure S1(a).

)
In Equation S4, N 2D V is the 2D effective density of states of the valence band edge, which can be modeled assuming parabolic bands, using the effective hole masses m * h at the VBM.The m * h for PEA 2 PbI 4 , BA 2 PbI 4 and PEA 2 SnI 4 , taken from Ref. S17, are 0.25m 0 , 0.39m 0 , and 0.15m 0 , respectively, where m 0 is the mass of the electron.For PEA 2 Sn 0.5 Pb 0.5 I 4 the S2 Mixed Pb-Sn perovskite

Figure S2 :
Figure S2: Optimized structures of two possible configurations of PEA 2 Pb 0.5 Sn 0.5 I 4 with well mixed Pb and Sn.

Table 1 :
25I 4 ,25Figure 2(e).Moreover, interstitials generally play a significant role in the Formation energies ∆H f (eV) and concentrations c (cm −3 ) at T = 300 K of charged defects in 3D and 2D halide perovskites calculated at the intrinsic Fermi level, and iodinemedium conditions, using the SCAN+rVV10 functional.Formation energies and concentrations of the dominant defects in each perovskite are underlined.For those defects which are stable at a different charge state at the intrinsic Fermi level, the corresponding stable charge state is specified in the brackets.Pb 0.5 Sn 0.5 I 4 1.16×10 7 71.44× 10 13 6.10×10 11 1.37 × 10 14 6.36×10 10 PEA 2 PbI 4 5.37 × 10 7 1.09 × 10 7 3.27×10

Table S1 :
perovskites PEA 2 PbI 4 , BA 2 PbI 4 , and PEA 2 SnI 4 are much lower than in comparable 3D perovskites, indicating improved material stability of 2D perovskites.The stability of 2D Sn iodide perovskites can be further enhanced by alloying with Pb.Moreover, unlike the prominence of interstitials in 3D perovskites, 2D perovskites are dominated by vacancies.The difficulty in forming defects in 2D perovskites is attributed to two factors.One is that the bonding disruptions by defects are more detrimental in 2D than in 3D networks.Another is that the dielectric constants are smaller for 2D perovskites.These factors also cause the formation of deep defect levels in the band gap of 2D perovskites.Consequently, should point defects emerge in sizable concentrations, then those defects can hamper the optoelectronic performance of the 2D perovskites.Finally, we suggest that the trap levels of iodine vacancies are responsible for the broad sub-bandgap emission in 2D perovskites observed in experiments.(9)Haruyama,J.;Sodeyama,K.;Han, L.; Tateyama, Y. Surface Properties of CH 3 NH 3 PbI 3 for Perovskite Solar Cells.Acc.Chem.Res.2016,49,554-561.(10)Holzhey,P.;Yadav,P.;Turren-Cruz, S. H.; Ummadisingu, A.; Grätzel, M.; Hagfeldt, A.; Saliba, M. A chain is as strong as its weakest link -Stability study of MAPbI 3 under light and temperature.Mater.Today 2019, 29, 10-19.(11)Juarez-Perez,E.J.; Hawash, Z.; Raga, S. R.; Ono, L. K.; Qi, Y. Thermal degradation of CH 3 NH 3 PbI 3 perovskite into NH 3 and CH 3 I gases observed by coupled thermogravimetry-mass spectrometry analysis.Energy Environ.Sci.2016, 9, 3406-3410.S5 meta-generalized gradient approximation (meta-GGA) functional with the long-range van der Waals interactions from the revised Vydrova-van Voorhis non-local correlation functional (rVV10).S6 It has emerged as a reliable functional for calculating defect properties of metal halide perovskites in our previous work, Ref. S7.The spin-orbit coupling (SOC) is omitted, as it has little effect on the formation energies of defects.S7Our calculations use a plane wave kinetic energy cutoff of 450 eV, with a maximum cutoff energy of the plane-wave-basis set for all elements being 280 eV (see TableS1for the convergence test), and a Γ-point only k-point mesh.The energy and force convergence criteria are set to 10 −4 eV and 0.02 eV/ Å, respectively.Spin-polarization is included in all calculations.Convergence test of the cutoff energy of the plane-wave-basis set using the PEA 2 PbI 4 unitcell.For each cutoff energy, the lattice volume and ionic positions are fully relaxed.PbI 4 , BA 2 PbI 4 and PEA 2 SnI 4 are taken from the experimentally determined lattices from Refs.S8-S10, respectively.PEA 2 Sn 0.5 Pb 0.5 I 4 is constructed by substituting Pb with Sn in PEA 2 PbI 4 , see section S2 for the detailed discussion about the substitution strategy.Then the pristine structures are reoptimized using the SCAN+rVV10 functional, including reoptimizing the volume of the unit cell.
S7,S13Taking PEA 2 PbI 4 for example, the chemical potentials µ i of atomic species i are calculated by assuming that the perovskite is stable, so using 2µ PEA + µ Pb + 4µ X = µ PEA 2 PbI 4 as a constraint, where for µ PEA 2 PbI 4 we use the DFT total energy per formula unit of the PEA 2 PbI 4 perovskite.Furthermore, we assume that the perovskite is in equilibrium with the PbI 2 phase, so µ Pb + 2µ I = µ PbI 2 , with µ PbI 2 the DFT total energy per formula unit of PbI 2 .All µ i can now be expressed in terms of a single parameter, µ I , which is constrained by µ PbI 2 − µ Pb,bulk ≤ 2µ I ≤ µ I 2 ,molecule