Ionization energy as a stability criterion for halide perovskites
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Ionization energy as a stability criterion for halide perovskites
Chao Zheng ∗ and Oleg Rubel Department of Materials Science and Engineering, McMaster University,1280 Main Street West, Hamilton, Ontario L8S 4L8, Canada (Dated: December 15, 2016)Instability of hybrid organic-inorganic halide perovskites hinders their development for photovoltaic appli-cations. First-principle calculations are used for evaluation of a decomposition reaction enthalpy of hybridhalide perovskites, which is linked to experimentally observed degradation of device characteristics. However,simple criteria for predicting stability of halide perovskites are lacking since Goldschmidt’s tolerance and oc-tahedral geometrical factors do not fully capture formability of those perovskites. In this paper, we extend theBorn-Haber cycle to partition the reaction enthalpy of various perovskite structures into lattice, ionization, andmolecularization energy components. The analysis of various contributions to the reaction enthalpy points to anionization energy of a molecule and a cage as an additional criterion for predicting chemical trends in stabilityof hybrid halide perovskites. Prospects of finding new perovskite structures with improved chemical stabilityaimed for photovoltaic applications are discussed.
I. INTRODUCTION
The efficiencies of hybrid organic-inorganic perovskite so-lar cells have already increased to over 20% . Fabrica-tion of hybrid organic perovksite is based on a low temper-ature solution method, thus offering a low-cost alternative tocrystalline thin-film photovoltaic devices. The main obsta-cle hindering the commercialization of hybrid organic per-ovskite solar cells is the instability of the active material. Hy-brid perovskites are prone to a phase separation that takesplace instantly under ambient conditions (moisture, UV ra-diation, atmospheric oxygen, etc.) . The detrimental roleof moisture in creating a degradation pathway for halide per-ovskites was previously discussed from acid-base chemistry ,molecular dynamic simulations , hydrolysis reaction andthermodynamic perspectives. Encapsulation of the per-ovskite cells does not prevent their degradation either. Theactive layer of encapsulated hybrid organic perovskites even-tually decompose after a period of time that ranges from sev-eral days to a month .Intrinsic instability of hybrid halide perovskite structurescan be captured at the level of first-principle calculations by evaluating the enthalpy of the reaction AX + BX → ABX (1)based on the total energy of the solid compounds involved.Here A represents an organic cation, B and X are the metaland halide elements, respectively. The negative reaction en-thalpy ∆ H r indicates stable products. The lower the valueof ∆ H r , the more stable the structure is against decompo-sition. For example, the reaction enthalpy for tetragonalCH NH PbI is within the range of − . . . . . eV per for-mula unit , which renders the structure to be at the bound-ary between weakly stable and unstable agreeing with exper-imental observations . Despite the success of first-principlecalculations in predicting formability of hybrid halide per-ovskite structures, the origin of intrinsic instability and av-enues for its improvement remain unclear.Geometrical factors such as the Goldschmidt’s tolerancefactor and octahedral factor successfully explain formabil- ity of various inorganic perovskite structures . The toler-ance factor t measures compactness of the perovskite struc-ture. The value of the tolerance factor for CH NH PbI is t = 0 . , which is within the range of acceptable values t = 0 . − . . Li et al. pointed out that the tolerancefactor alone does not fully capture formability of perovkitestructures and proposed to add Pauling’s octahedral factor r B /r X ( r B and r X are the ionic radii of cation B and anion X , respectively) as an additional geometrical criterion. In thecase of CH NH PbI the octahedral factor r Pb /r I = 0 . iswithin the allowable range of . − . . This analysissuggests that geometrical factors are not sufficient to explainthe instability of hybrid halide perovskites.Frost et al. attributed the instability of hybrid organichalide perovskites to a relatively low electrostatic lattice en-ergy of their ionic structure as compared to non-halide per-ovskite compounds. For instance, traditional inorganic per-ovskites of the II − IV − VI family, e.g. PbTiO , have the lat-tice energy of − eV. This value is much lower that thelattice energy of − eV for CH NH PbI perovksite, whichbelongs to the I − II − VII family. This argument suggeststhat I − II − VII perovskites have intrinsically lower electro-static energy and thus weaker chemical stability. On the otherhand, the experimental reaction enthalpy for PbTiO is only − . eV , which is orders of magnitude less than its latticeenergy. It is also known that CsPbI perovskite structure isindeed stable up to the temperature of 460 ◦ C , above whichthe material melts without decomposition, despite of its higherlattice energy of − eV. These observations indicate that thelattice energy alone cannot be used as a criterion for stabilityof ionic structures.The Born-Haber cycle is traditionally used for analysis offormation enthalpies. It allows to break the formation energyinto the following components: atomization enthalpy, ioniza-tion enthalpy, and lattice enthalpy . In this paper we extendthe Born-Haber cycle to the analysis of energy components ofthe reaction enthalpies for various perovskite structures usingthe density functional theory (DFT). It will be shown that inI − II − VII organic and inorganic perovskites the lattice en-ergy contribution is largely cancelled by the molecularizationenergy leaving the ionization enthalpy to determine the direc-tion of the reaction. The instability of hybrid organic lead-iodine perovskites can be attributed to the high energy associ-ated with ionization of organic molecules and [PbI ] – . II. BASIC CONCEPTS
The Born-Haber cycle was originally proposed by MaxBorn and Fritz Haber as a way to measure formation ener-gies of ionic structures . The cycle also provides a methodto determine the lattice energy of the structures, which oth-erwise cannot be directly measured experimentally. Here wewill explain the essence of the Born-Haber cycle and its uti-lization for analysis of reaction enthalpy components usingthe CH NH PbI perovksite structure as an example.The formation process of CH NH PbI from solidCH NH I and PbI compounds can be subdivided into sev-eral consecutive steps illustrated in Fig. 1.The initial step—molecularization (similar to the atomiza-tion in the original Born-Haber cycle)—involves breakingthe CH NH I and PbI lattice structures and formation ofCH NH and PbI moleculesCH NH I ( s ) + PbI ( s ) ∆ H mo −−−→ CH NH ( g ) + PbI ( g ) . (2)The rational for using CH NH and PbI molecules as thesmallest units in the Born-Haber cycle is justified by the exis-tence of the corresponding free standing ions , and will bediscussed in section IV.The next step is the ionization of CH NH moleculeCH NH ( g )+ PbI ( g ) ∆ H ion,1 −−−−→ CH NH +3 ( g )+ PbI ( g ) , (3)followed by the ionization of PbI CH NH +3 ( g ) + PbI ( g ) ∆ H ion,2 −−−−→ CH NH +3 ( g ) + PbI − ( g ) . (4)It can be seen from the diagram in Fig. 1 that the formation of[CH NH ] + ion is an endothermic process, whereas the ion-ization of PbI is an exothermic process. The resultant ioniza-tion energy is an additive of two enthalphies ∆ H ion = ∆ H ion,1 + ∆ H ion,2 . (5)Finally, electrically charged [CH NH ] + and [PbI ] – com-plex ions are combined to form CH NH PbI crystallinestructureCH NH +3 ( g ) + PbI − ( g ) ∆ H latt −−−→ CH NH PbI ( s ) . (6)The amount of energy ∆ H latt released in this reaction is calledthe lattice energy of the hybrid organic perovskite structure.This concludes the Born-Haber cycle of CH NH PbI . Thetotal reaction enthalpy is compiled from enthalpies of individ-ual steps of the cycle ∆ H r = ∆ H mo + ∆ H ion + ∆ H latt . (7) III. COMPUTATIONAL DETAILS
Electronic structure calculations have been performedin the framework of DFT and Perdew-Burke-Ernzerhofgeneralized gradient approximation (GGA-PBE) for theexchange-correlation functional. Total energies of all com-pounds were obtained using the Vienna ab initio simula-tion program (VASP) and projector augmented-wave (PAW)potentials .All crystal structures of compounds studied here aretaken at their most stable polymorph at ambient conditions.Among perovskite structures, CH NH PbI adapts a tetrag-onal β -phase at the ambient temperature, CH NH PbBr and CH NH PbCl have a cubic phase , CN H PbI and (CH ) NPbI favor hexagonal structures . CsPbI ,CsPbBr , and CsPbCl prefer an orthorhombic (Pnma) δ -phase . The crystal structure of CH NH I, CH NH Br,and CH NH Cl organic salts correspond to α ′ -tetragonal(P4/nmm) phase at room temperature . Szafra´nskiand Jarek reported the structures of guanidinium iodideCN H I, and the structure of tetramethylammonium iodine(CH ) NI was obtained using (CH ) NAu as a parent struc-ture followed by full relaxation of their structural parameters.Cubic crystal structures of CsI, CsBr, CsCl and NaCl as wellas hexagonal PbI and orthorhombic PbBr were taken fromGraystone and Wyckoff , Gerlach . The crystal structure oforthorhombic PbCl was derived from the structure of PbBr .For reciprocal space integration, × × Monkhorst-Packgrid was used for cubic phases, × × were used for tetrag-onal phases, × × for hexagonal phases and × × fororthorhombic CsPbX phases and × × for ohthorhombicPbBr and PbCl . The cutoff energy for a plane wave expan-sion was set at 400 eV. The lattice constant and atomic posi-tions were optimized such that residual forces acting on atomsdid not exceed 2 meV/ ˚A, and the residual hydrostatic pressurewas less than 50 MPa.Gaseous phases, such as Cs, [CH NH ] + , [PbI ] – , weremodelled as an individual atom/molecule surrounded by 20 ˚Aof vacuum. All calculations related to gaseous phases wereperformed in conjunction with optimization of internal de-grees of freedom. Only Γ -point was used in the Brillouinzone. The ionization energy of positively charged ions wascalculated by subtracting the total energy of cations (e.g. Cs + ,[CH NH ] + , [CN H ] + ) from the energy of neutral atoms ormolecules (e.g. Cs, CH NH , CN H ). Similarly, the electronaffinity of negatively charged ions was modelled by addingone electron to PbCl , PbBr , or PbI molecules to form[PbCl ] – , [PbBr ] – , and [PbI ] – anions. The electron affin-ity of these ions was represented as an energy difference be-tween negatively charged complex ions and neutral species.Monopole, dipole and quadrupole corrections implemented inVASP were used for eliminating leading errors and ac-quiring accurate total energies of all charged ions. VESTA was used to visualize crystal structuresand for computing the Madelung electrostatic energy usingoxidation state as formal charges. In these calculations, theradius of ionic sphere and the reciprocal-space range were setat 1 ˚A and 4 ˚A − , respectively. FIG. 1. Born-Haber cycle of hybrid halide perovskites: Methylammonium (MA) lead iodide obtained with [CH NH ] + and [PbI ] – ions aselementary species. IV. RESULTS AND DISCUSSIONA. Lattice energies of halide perovskites
Calculation of individual energies associated with varioussteps in the Born-Haber cycle requires subdivision of the ionicsolid in question into elementary species. In the case of alkalihalides (such as NaCl, CsCl, etc.), the atomization is an appar-ent choice. Following the same strategy, Cs + , Pb , and I − ions can be used to calculate the lattice energy, which yields ∆ H latt ≃ − eV (Fig. 2).This value agrees well with the Madelung energy of − eV obtained from the point charge model. Gopal no-ticed existence of a trend between the lattice energy ∆ H latt and the melting point T m of alkali halides with the propor-tionality factor of − ∆ H latt /T m ≈ . · − eV/K. Assumingthat the same proportionality holds for perovskite structures,the melting point of I − II − VII perovskites would be near3900 K, which is an order of magnitude greater than the ac-tual values of − K for group-I lead halide perovskites(CsPbI , CsPbBr , and CsPbCl ) .Alternatively, we can separate CsPbI perovskite structureinto two ions Cs + and [PbI ] – . The existence of the corre-sponding free-standing ions was verified experimentally .Using this approach we re-evaluated the lattice energy ofCsPbI as − . eV using the Born-Haber cycle similar tothat shown in Fig. 1. This result translates into a substantiallylower melting point of approximately 750 K, which is remark-ably close to the experimental value of 749 K.Similar calculations of the lattice energy were performedfor other inorganic I − II − VII and II − IV − VI perovksites.Results are summarized in Table I.The plot of the melting point vs the lattice energy of thosecompounds is shown in Fig. 3.From this figure, we can see that the melting point of dif-ferent ionic structures including alkali halides follows a linear trend line. This suggests that formation of A + cations and[BX ] – complex anions is a result from melting of the per-ovskite structures. B. Stability analysis of hybrid organic halide perovksites
Now we will utilize the Born-Haber cycle in order toevaluate components of the reaction enthalpy of hybridhalide perovskits. The lattice energies of CH NH PbCl ,CH NH PbBr and β -CH NH PbI perovskites are listed inTable I. All three compounds have similar values of the latticeenergies ( ∼
10% max-min difference). However, their stabil-ity characteristics are quite different. Buin et al. demon-strated that under ambient conditions CH NH PbCl andCH NH PbBr do not undergo a phase separation, unlike β -CH NH PbI . Both CH NH PbCl and CH NH PbBr re-main stable up to the temperature of approximately 520 K,above which they decompose . Lattice energies of thecorresponding inorganic perovskites (CsPbI , CsPbBr andCsPbCl ) are very similar to their organic counterparts. Infact, these inorganic perovskites are chemically stable underthe ambient environment. Remarkably, the lattice energy of β -CH NH Pbl and δ -CsPbCl structures are identical, in spiteof the distinct stability characteristics. Therefore, we can con-clude that the lattice energy cannot be used as a criteria topredict the chemical stability of compounds.The analysis of various contributions to the reaction en-thalpies of hybrid halide perovskites (Table I) shows thatthe molecularization and lattice energies largely cancel eachother. The ionization energy is the remaining contribution tothe reaction enthalpy in Eq. (7) that ultimately controls thebalance of the reaction. The lower ∆ H ion is, the more stablethe compound.Let us examine the chemical trends in ionization energy ofvarious perovskites. The total ionization energy (Eq. 5) com- FIG. 2. Born-Haber cycle of inorganic halide perovskites: Caesium lead iodide obtained with Cs + , Pb , and I − ions as elementary species.TABLE I. Components (eV) of the reaction enthalpies extracted from Born-Haber cycle as well as the melting temperature and stability againstspontaneous decomposition for halide perovskites and other ionic structures.Compounds ∆ H mo ∆ H latt ∆ H ion ∆ H r T m (K) Stability δ -CsPbCl − − − Y δ -CsPbBr − − − Y δ -CsPbI − − − YCH NH PbCl − − − · · · YCH NH PbBr − − − · · · Y β -CH NH PbI − · · · NCN H PbI − − − · · · Y(CH ) NPbI − − − · · · YCsCl 2.59 − − YNaCl 3.01 − − Y prises of two components: the ionization energy for the cation(Cs + or [CH NH ] + ) and that for the complex ion ([PbI ] – ,[PbBr ] – , or [PbCl ] – ). Caesium has a lower ionization en-ergy than CH NH (Table II), which explains trends in thehigher chemical stability of Cs-based perovskites as comparedto their CH NH -based counterparts.Switching halides in the complex ions from PbI to PbCl lowers their electron affinity (Table II) and, thus, leads tothe lower total ionization energy. This explains increase ofthe chemical stability when changing the inorganic cage fromPbI to PbBr and PbCl .In order to achieve a chemically stable hybrid halide per-ovskite structures, the necessary requirements are favourablegeometrical factors (t-factor and octahedral factor) in conjunc-tion with the low ionization energy ( ∆ H ion . eV). Twostrategies can be used to achieve this goal: (i) find a cationwith the low ionization energy or (ii) select an inorganic cagewith the low electron affinity. The second avenue is not verypromissing, since the band gap of PbBr - and PbCl -based hy-brid perovskites (2.3 eV and 2.9 eV , respectively) is out- side of the favourable range for single-junction solar cells.Since caesium has the lowest ionization energy in the pe-riodic table, it is a challenging task to find molecules withsmaller or similar ionization energy. Among the variety of or-ganic cations listed in the Table II, [CN H ] + and [(CH ) N] + have the ionization energies lower than that for [CH NH ] + cation making them favourable candidates for perovskiteswith improved stability. However, the size of CN H and(CH ) N molecules is significantly greater than CH NH ,which raises the tolerance factor above the upper formabilitylimit of 0.95 (Table III).From two structures, CN H PbI and (CH ) NPbI showsfavourable reaction enthalpies of − . eV and − . eV, re-spectively(Table I). A large size of the organic molecule hin-ders formability of CN H PbI and (CH ) NPbI perovksitestructures. They both prefer hexagonal structures at ambi-ent temperature . Marco et al. successfully synthesizedand characterized CN H PbI perovskite solar cells. It wasfound that CN H PbI solar cell is also unstable under theambient environment, which is evident from degradation of FIG. 3. Correlation between the lattice energy and melting tempera-ture of ionic compounds. The line is a guide to the eye.TABLE II. Ionization energies (eV) of atoms and molecules.Ions ∆ H ion,1/2 [(CH ) N] + + H ] + NH ] + NH ] + ] + + PH ] + SH ] + ] + PH ] + ] + ] – − ] – − ] – − the power conversion efficiency over time. Interestingly, therate of the efficiency decay is slower for CN H PbI as com-pared to CH NH PbI . Szafra´nski found that CN H PbI crystals transform from orange-reddish phase to yellow phaseafter several hours at ambient pressure and temperature. Thiscolor changing demonstrates that the bandgap increases dur-ing phase transformation. From the reaction enthalpy ofCN H PbI (Table I), we conclude that the drop of power con-version efficiency of CN H PbI photovoltaic device is due tothe phase transformation, and the CN H PbI structure won’t TABLE III. Size of organic cations, the tolerance factor, volume ofthe unit cell and the band gap of selected perovskites.Perovskite Cation radius Tolerance Volume Bandgap(pm) factor ( ˚A /f.u.) (eV) β -CH NH PbI H PbI ) NPbI go through phase separation over time.The ionization energies of onium ions in Table II corre-late with the proton affinity of the corresponding molecules .Molecules with the low ionization energy exhibit strong pro-ton affinity and vice versa . For instance, the proton affinityof PH is 785 kJ/mol, which is much lower than 901 kJ/molfor CH NH . It turns out that methylamine has one of thestrongest proton affinity among organic compounds. Therevery few organic molecules (including (CH ) NH studiedhere) with stronger proton affinity than CH NH , but none ofthem have a size compatible with the PbI cage. V. CONCLUSIONS
The Goldschmidt’s tolerance and octahedral geometricalfactors do not fully capture prerequisites for formability ofhybrid halide perovskites. Here we used DFT calculations inconjunction with a Born-Haber cycle to evaluate contributionsof the lattice, ionization and molecularization energies to thedecomposition reaction enthalpy of hybrid halide perovskites.It was previously assumed that the instability of halide per-ovskite is due to a lower lattice energy of their ionic struc-ture. We observe a correlation between the lattice energiesand melting temperatures, but not with reaction enthalpiesthat are ultimately linked to the chemical instability of theperovskites. Analysis of Born-Haber cycle components sug-gests that the reaction enthalpy of hybrid halide perovskites isgoverned by the sum of ionization energies of a cation, e.g.,[CH NH ] + , and an anion, e.g., [PbI ] – . The lower total ion-ization energy, the more stable is the structure, provided thegeometrical conditions are fulfilled (the tolerance and octa-hedral factors). This explains chemical trends in stability ofhybrid and inorganic halide perovskites. For instance, the rel-atively high stability of CH NH PbCl is attributed to a lowerionization energy of [PbCl ] – complex ion, whereas the sta-bility of CsPbI is due to the lower ionization energy of Cs + .The ionization energy of organic cations correlates with theirproton affinity. In the search for hybrid perovskite with im-proved chemical stability and the band gap suitable for pho-tovoltaic applications, several cations were investigated. Thepromising candidates are [CN H ] + and [(CH ) N] + with theionization energies even lower than Cs + . The correspondingCN H PbI and (CH ) NPbI structures have the decomposi-tion reaction enthalpy approximately 0.3 eV more favourablethan CH NH PbI . However, these ions has a prohibitivelylarge size that translates into a large band gap. It is the factthat CH NH has the highest proton affinity among moleculesof comparable size. It makes challenging to find a cation suit-able for PbI cage as a stable activate layer for photovoltaics. ACKNOWLEDGMENTS
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