First-Principles Prediction of a Stable Hexagonal ... - ACS Publications

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First-Principles Prediction of a Stable Hexagonal Phase of CH3NH3PbI3 Arashdeep Singh Thind,† Xing Huang,‡ Jianwei Sun,§ and Rohan Mishra*,†,‡ †

Institute of Materials Science & Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, United States Department of Mechanical Engineering & Materials Science, Washington University in St. Louis, St. Louis, Missouri 63130, United States § Department of Physics, The University of Texas at El Paso, El Paso, Texas 79968, United States ‡

S Supporting Information *

ABSTRACT: Methylammonium lead iodide (CH3NH3PbI3 or MAPbI3) perovskite is a promising new photovoltaic material with high power conversion efficiency. However, its perovskite phase with corner-connected PbI6 octahedra shows poor environmental stability. More recently, MAPbI3 has been shown to be thermodynamically unstable with a positive formation enthalpy. Here, using first-principles density functional theory calculations, we predict a layered hexagonal phase of MAPbI3 consisting of infinite chains of face-shared PbI6 octahedra with P63mc space-group symmetry to be thermodynamically the most stable phase for a wide range of volume and temperature compared to any of the experimentally observed perovskite phases with a different tilt pattern of the corner-connected octahedra. The predicted hexagonal phase is also dynamically stable without any soft phonon modes. The change from corner to face-shared connectivity in the hexagonal phase leads to a predicted band gap of 2.6 eV and a band structure that favors highly anisotropic charge transport.

O

thermodynamic instability combined with the highly toxic nature of lead makes MAPbI3 unattractive for use in commercial photovoltaics.15 Therefore, there is a critical need to overcome the instability issues in MAPbI3 perovskites. In this article, we predict a thermodynamically stable hexagonal polymorph of MAPbI3 with P63mc space-group symmetry based on first-principles calculations of thermodynamic properties. The predicted phase has a layered structure with infinite chains of face-shared PbI6 octahedra. Total energy calculations based on a variety of exchange-correlation functionals, including dispersion-correction and spin−orbit coupling effects, show that the predicted hexagonal phase is more stable compared to the experimentally observed cornerconnected perovskite phases having either orthorhombic, tetragonal, or cubic symmetry, and is, therefore, the ground state. The calculated enthalpy of formation (compared to

rganohalide lead perovskites have shown a remarkable increase in solar power conversion efficiency from 3.8% in 20091 to 22.1% currently.2 Among the different organohalide lead perovskites, methylammonium (MA) lead iodide (CH3NH3PbI3 or MAPbI3) has received the most attention. It exhibits a direct band gap of ∼1.6 eV,3,4 a large absorption coefficient,5 low exciton binding energy,6 long carrier diffusion length,7−9 and high carrier mobility,4,9 which combined with the ease of synthesis makes it an attractive candidate for nextgeneration photovoltaics. However, MAPbI3 is observed to degrade rapidly under ambient atmospheric conditions due to its high sensitivity to moisture, oxygen, and temperature.10,11 Recently, Nagabhushana et. al,11 by means of calorimetric measurements, have shown that MAPbI3 has a positive formation enthalpy, confirming previous predictions based on first-principles density functional theory (DFT) calculations.12,13 As a result, MAPbI3 is thermodynamically metastable and is found to dissociate into binary halide components: CH3NH3I and PbI2, even in the absence of moisture, heat, or light.11,14 This © 2017 American Chemical Society

Received: May 1, 2017 Revised: June 14, 2017 Published: June 15, 2017 6003

DOI: 10.1021/acs.chemmater.7b01781 Chem. Mater. 2017, 29, 6003−6011

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Chemistry of Materials

Figure 1. Crystal structure of (a) cubic (C), (b) tetragonal (T), (c) orthorhombic (O), (d) hexagonal (4H), and (e) (2H)-phases of MAPbI3. The top and bottom panels represent [001] and [100] crystallographic directions, respectively.

thermodynamically unstable.11 On the other hand, replacing MA cations with the larger formamidinium ions (HC(NH2)2+ or FA) results in a hexagonal structure with P63mc space-group symmetry.4 FAPbI3 has chains of face-shared octahedra stacked along the [001] direction while neighboring octahedra along [100] and [010] directions are disconnected.4 These phase transitions in inorganic perovskites (oxides and halides) with increasing cation size can often be predicted using the Goldschmidt tolerance factor,22 (t = (rA + rB)/√2(rA + rB)) and the octahedral factor (o = rB/rX), where rA, rB, and rX are the radius of A, B, and X, respectively.23,24 For instance, in the case of perovskite oxides, t > 1 often results in the formation of hexagonal phases with face-shared octahedra.17 However, in organohalide perovskites, the effective radius for the noncentrosymmetric organic cation is ambiguous. As a result, t is found to span a wide range depending on the value of rA used. For MAPbI3, suggested values of t range from 0.8325 to 0.95.26 This ambiguity associated with t makes it difficult to predict if MAPbI3 is close to the cubic or to the hexagonal phase boundary. To overcome such ambiguity, we have used first-principles DFT calculations to investigate the energetic stability of experimentally observed O-, T-, and C-phases of MAPbI3 and compare them to two commonly observed hexagonal phases for ABX3 compounds with larger A-cations.17 Figure 1a−e shows the polyhedral representations of all the five phases. The C-, T-, and O-phases (Figure 1a−c) have corner-connected octahedra where every iodine atom is bonded to two lead atoms. The 4Hphase is composed of a combination of corner-connected and face-shared octahedra (Figure 1d), and the 2H-phase (Figure 1e) consists of infinite chains of face-shared octahedra. Both the hexagonal phases belong to the P63mc space group and are referred to as 2H- and 4H-phases as they consist of 2 and 4 formula units (f.u.) of MAPbI3 per unit cell, respectively. The lattice parameters of the optimized structure of different phases obtained using the Perdew−Burke−Ernzerhof exchangecorrelation (XC) functional revised for solids (PBEsol)27 are given in the Supporting Information Table S1 along with their crystallographic information files (CIFs). The theoretical lattice parameters of the O-, T-, and C- phases obtained using the PBEsol functional are within 2.8, 1.4, and 1.5% of their experimental values, respectively.28 Further discussion on the orientation of the MA cations, which are observed to rotate at finite temperatures, but are static in our DFT calculations, is provided in the Supporting Information.

binary halide constituents) for the hexagonal phase is also found to be negative, which indicates that it is thermodynamically stable. We find the hexagonal phase to be stable for a large range of volume and temperature based on free energy calculations obtained using the quasiharmonic approximation. The hexagonal phase is also dynamically stable due to the absence of any phonon modes with imaginary frequencies (or soft modes). The change in octahedral connectivity leads to markedly different electronic properties. The layered hexagonal phase has a larger indirect theoretical band gap of 2.6 eV compared to the perovskite phase. The breaking of octahedral connectivity perpendicular to the chain direction results in relatively flat electronic bands along the corresponding directions in the reciprocal space and dispersed bands along the chain direction, which is expected to lead to anisotropic transport properties.

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RESULTS AND DISCUSSION The ideal perovskite, ABX3, has a cubic structure with the smaller B-site cations present in an octahedral coordination of anions X, while the larger A-site cations occupy cuboctahedral (12-coordinate) cavities within the corner-connected BX6 octahedral network. The perovskite structure is sensitive to the size of the A, B, and X ions and the volume of the unit cell. Reducing the size of the A-site cation or the volume results in cooperative tilts of the BX6 octahedra along different crystallographic directions and is associated with a lowering of the symmetry to tetragonal, orthorhombic, or rhombohedral space groups.16 On the other hand, increasing the A-site cation size leads to a transition from a corner-connected BX6 network to structures where neighboring BX6 octahedra share either faces or a combination of faces and corners.17,18 Experimentally, MAPbI3 has an orthorhombic (O) phase with Pnma space-group symmetry at temperatures below 162.2 K.19,20 The O-phase has out-of-phase (−) octahedral tilts along the a- and c-axes and in-phase (+) rotations along the b-axis, which is expressed by an a−b+a− tilt pattern in Glazer notation.21 On increasing the temperature above 162.2 K19,20 (and consequently the volume), MAPbI3 adopts a tetragonal (T) structure with I4/mcm space group and an a0a0c+ tilt pattern. Further increase of the temperature above 327.4 K19,20 results in a cubic (C) symmetry with Pm3m ̅ space group without any octahedral tilts (a0a0a0). However, recent calorimetric data shows that the room temperature T-phase has a positive formation enthalpy and is, therefore, 6004


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Figure 2. (a) Calculated formation enthalpy of five phases of MAPbI3 using different exchange-correlation functionals with and without including spin−orbit coupling (SOC) effects and van der Waals interactions. (b) Phonon-dispersion spectra of 2H-phase within the harmonic approximation. (c) Free energy (F) of the 2H-phase and C-phase as a function of temperature calculated under the quasiharmonic approximation. (d) DFT total energy vs volume phase diagram for the 2H-phase and the three corner-connected phases of MAPbI3. The percentage volumetric strain is expressed with respect to the ground-state volume of the 2H-phase. The results in (b−d) have been calculated using the PBEsol functional.

The 4H-phase (Figure 1d) forms a complex network of facesharing Pb2I9 bioctahedral units that are corner-connected to six neighboring bioctahedral units. As a result, there are two types of Pb−I bonds in the 4H-phase, ones with the three I atoms at the shared face, which have a bond distance of 3.20 Å, and others with the three I atoms at the shared corners, which have a bond distance of 3.18 Å. The combination of face and corner connectivity also results in two different Pb−I−Pb bond angles. The Pb−I−Pb bond angle involving I atoms at the shared face is 76.3°, while the angle involving I atoms for the cornerconnected octahedra is 148.4°. The difference in energy due to changes in the orientation of the MA cations is discussed in the SI text. The 2H-phase consists of infinite chains of face-shared PbI6 octahedra running along the [001] direction with the chains being disconnected along [100] and [010] directions as shown in Figure 1e. The average Pb−I bond distance increases from 3.17 Å in the O-phase to 3.21 Å in the 2H-phase. Due to the transition from the corner to face connectivity of neighboring PbI6 octahedra, the average Pb−I−Pb bond angle reduces from 156.6° in the O-phase to 75.8° in the 2H-phase. This substantial reduction in bond angle reduces the Pb−Pb distance along the [001] direction of the 2H-phase to 3.93 Å from 6.23 Å in the O-phase. The break in octahedral connectivity along [100] and [010] directions, however, leads to a greater Pb−Pb distance of 8.5 Å in the 2H-phase. The overall volume increases from 238.08 Å3/f.u. in the O-phase to 244.63 Å3/f.u. in the 2H-phase. We find the total energy changes by a maximum of 29 meV/f.u. on changing the orientation of the MA cation as discussed in the SI text. We have calculated the ground-state energy of the five phases of MAPbI 3 using the PBEsol functional. Among the experimentally observed phases, we find the O-phase to be most stable, followed by T-phase and C-phase, which are,

respectively, 8 and 88 meV/f.u. higher in energy. The energetic stability of the three phases is in good agreement with the O → T → C phase transitions observed experimentally with increasing temperature.19,20 However, we find the 2H-phase to be the most stable phase with its ground-state energy being 27 meV/f.u. lower than the O-phase. The 4H-phase competes closely with the O-phase, as its total energy is only 2 meV/f.u. higher than the O-phase. Following recent results from calorimetric studies by Nagabhusana et. al11 that show the room temperature Tphase of MAPbI3 to be thermodynamically metastable having a positive enthalpy of formation of 0.36 eV/f.u., we have also calculated the formation enthalpy of each phase at 0 K to compare their relative thermodynamic stability with respect to their reactants, as shown in the chemical reaction in eq 1. The formation enthalpy ΔHf (CH3NH3PbI3) for each phase is calculated using eq 2 CH3NH3I + PbI 2 → CH3NH3PbI3

(1)

ΔHf (CH3NH3PbI3) = E(CH3NH3PbI3) − E(CH3NH3I) − E(PbI 2)

(2)

where E(CH3NH3PbI3), E(CH3NH3I), and E(PbI2) are the DFT total energy/f.u. of CH3NH3PbI3, and its most stable reactants CH3NH3I and PbI2, respectively. It is important to note here that comparing formation enthalpies of different phases is more conclusive than comparing total energies. The comparison of total energies only gives the order of relative stability irrespective of the formability of a particular phase, whereas the comparison of formation enthalpy provides a complete picture of relative stability as well as the formability of a phase. 6005


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at the local minima of its potential energy landscape and is dynamically stable, lacking any soft modes. To estimate the thermodynamic stability of the structures at elevated temperatures taking vibrational entropy and lattice expansion into consideration, we have calculated the free energy (F) of the C- and 2H-phases within the quasiharmonic approximation (QHA). In QHA, F is described by eq 338,39

While the DFT formalism to calculate the ground-state energy and electron density is exact, the accuracy of the results depends on the XC functional used, which is an approximate quantity.29 In the case of MAPbI3, the presence of heavy lead and iodine atoms that have large spin−orbit coupling (SOC) effects and the organic MA cation that forms hydrogen bonds with iodine30 makes it even more sensitive to the XC functional. Hence, we have employed a wide range of XC functionals to calculate the formation enthalpy of the different phases of MAPbI3. The functionals used in this study include: local density approximation (LDA),31 generalized gradient approximation (GGA) as implemented by Perdew−Wang (PW91),32 by Perdew−Burke−Ernzerhof (PBE),33 PBE functional revised for solids (PBEsol), and the recently proposed strongly constrained and appropriately normed (SCAN) metaGGA functional, which has been shown to predict accurate geometries and properties especially for materials with strong hydrogen-bonding.34,35 For LDA, PW91, PBE, and PBEsol functionals, geometry optimization was performed both with and without SOC effects. We have performed an additional set of calculations to account for van der Waals (vdW) interactions as implemented in the DFT-D3 correction method (PBE+D3) of Grimme et al.36 The calculated formation enthalpy for the five MAPbI3 phases using different XC functionals is shown in Figure 2a. It is clear that the 2H-phase is the ground state independent of the choice of XC functional and has a negative formation enthalpy that, depending on the functional, ranges between −5 and −90 meV/f.u. The formation enthalpy for the experimentally observed low-temperature O-phase is negative for all the functionals except for PBE and PW91. The room temperature T-phase and the 4H-phase compete closely with O-phase in terms of relative stability, which is dependent on the XC functional. For instance, we find the order of stability to be O (more stable) > 4H > T using the PBEsol functional, whereas, for LDA and PBE+D3, it is O (more stable) > T > 4H. The high temperature, C-phase is found to be thermodynamically unstable at 0 K, as it has a positive formation enthalpy for all XC functionals used. For the remainder of this article, we have used the PBEsol functional (unless specified) based on the good agreement of the lattice parameters for the cornerconnected phases with the experimental data and their ability to reproduce the correct order of phase transitions.1,28 To investigate the dynamical stability of the 2H-phase, we have calculated its phonon dispersion spectra using the frozenphonon approach, which is shown in Figure 2b. The phonon spectra of C- and O-phases are provided in Figure S1a,b, respectively. A structure is dynamically stable if all the force constants or the phonon frequencies are positive. Alternatively, the presence of “soft modes” with imaginary force constants, which is shown in the phonon spectra with negative frequency, suggests the presence of instabilities that lower the space-group symmetry.37 For the 2H-phase, we find phonon modes along Γ−M and Γ−A directions have a negative frequency (energy) of −0.065 THz (−0.007 meV/f.u.) and −0.073 THz (−0.008 meV/f.u.), respectively. However, these negative frequencies do not correspond to soft modes but are due to lack of convergence with respect to the size of the supercell (having 432 atoms) used for the phonon calculations. As shown in Figure S1c, we find the energy of the lattice increases on displacing the atoms away from their equilibrium position along the eigenvector corresponding to the mode having a negative frequency along the Γ−A direction. The 2H-phase is, therefore,

⎧ ⎪ 1 F(T ) = min⎨E(V ) + V ⎪ 2 ⎩

∑ ℏωq,j(V ) q, j

⎫ ⎡ ⎛ −ℏωq, j(V ) ⎞⎤⎪ ⎟⎟⎥⎬ + kBT ∑ In⎢1 − exp⎜⎜ ⎢⎣ ⎝ kBT ⎠⎥⎦⎪ q, j ⎭

(3)

where E(V) is the total energy at 0 K of the optimized structure at different volumes V, which has been varied from −5.4% to 6.4% (235.8 to 265.26 Å3/f.u.) and −4.9% to 4.6% (232.63 to 255.76 Å3/f.u.) around the ground-state volume for the C- and 2H-phases, respectively, ℏ is the reduced Planck constant, q is the Brillouin zone wave vector and j is the band index of the phonon dispersions, ωq,j is the phonon frequency at q and j, and kB is the Boltzmann constant. The second term on the right-hand side of the equation is the energy contribution from the zero-point vibrations, and the third term is the vibrational free energy at any finite temperature T and volume V. The free energy is then calculated at each temperature (T) by minimizing the sum of all terms on the right-hand side of the eq 3 with respect to volume (V). The free energy of the 2Hand C-phases as a function of temperature as calculated under QHA is shown in Figure 2c. Even on including the vibrational free energy and the zero point energy terms, it is clear that the 2H-phase remains more stable by 80 meV/f.u. than the C-phase at 300 K. We find the difference in energy between the two phases decreases with increasing temperature until a transition from the 2H-phase to the C-phase occurs at 750 K (shown in Figure S1b). However, MAPbI3 is likely to decompose into its reactants CH3NH3I and PbI2 at such high temperatures given the positive enthalpy of formation of the C-phase.11 The presence of the organic cations that are experimentally observed to rotate and change their orientation can also be expected to increase the configuration entropy with increasing temperatures.40 We have sampled several static orientations of the MA cations in the 2H-phase and the O-phase. We find the maximum energy difference between different orientations of the MA cations in the two phases to be similar (29 and 32 meV/f.u., respectively). However, even in the O-phase at up to 100 K, the MA cations are experimentally observed to be highly anisotropic as they are preferentially oriented along a lowenergy configuration.41 Hence, we have compared the lowestenergy static orientation of the MA cations in the two phases, and we find the 2H-phase to be most stable irrespective of the exchange-correlation functional used. In corner-connected perovskites, increasing volume (which could be the result of increasing temperature) is known to result in phase transitions. Calculating the free energy phase diagram for all the phases of MAPbI3 using QHA is, however, computationally restrictive. Instead, to study the relative stability of the different phases of MAPbI3 with changing volume, we have calculated their total energy at 0 K under the application of an isotropic stress that was varied from −0.75 GPa (tensile) to +2.5 GPa (compressive). Figure 2d shows a plot of energy/f.u. of the three corner-connected phases and 6006


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Figure 3. Calculated band structures with and without SOC effects (left), DOS without SOC effects (center), and with SOC effects (right) for (a) 2H-phase and (b) O-phase of MAPbI3.

the 2H-phase as a function of unit cell volume, where the energy has been normalized to the optimized 2H-phase at zero stress. The change in volume due to the application of stress leads to phase transitions among the corner-connected phases, which are mediated through a change in the octahedral tilt pattern. Specifically, on increasing the volume (tensile stress), we find the structure transitions from O-phase to T-phase at 247.06 Å3/f.u. and from T-phase to C-phase at 276.95 Å3/f.u. Our results on the order of stability of corner-connected phases with volume agree well with the literature.42 However, we find the 2H-phase to be most stable in the overall volume range. To get insights about the electronic properties of the 2Hphase of MAPbI3 that we predict to be stable, we have calculated its band structure and density of states (DOS) without and with SOC effects using the PBEsol functional and the hybrid Heyd−Scuseria−Ernzerhof (HSE06) functional.43,44 Figure 3a,b shows the band structure and DOS of the 2H-phase and O-phase calculated using PBEsol, respectively. The band structure of the 2H-phase using the HSE06 functional is shown in Figure S3a. The band structure and DOS for the C-phase, Tphase, and 4H-phase are also provided in Figure S3b−d, respectively. Table 1 summarizes the trend in the band gap, its

nature (direct or indirect), and the effect of SOC for all the phases using PBEsol. The calculated band gaps without SOC as listed in Table 1 are in good agreement with experimental band gap of 1.55 eV1,3 and are within 0.15 eV of previous theoretical results for the C-, T-, and O-phases.30,45 The agreement of the PBEsol band gap without SOC effects with the experimental band gap is due to error-cancellation.9,46,47 Typically, the band gap is underestimated using LDA/GGA functionals. However, due to the heavy Pb and I atoms, SOC effects decrease the band gap substantially. Indeed, on including SOC, the PBEsol band gap decreases by 0.87, 0.56, and 0.56 eV for the C-, T-, and O-phases, respectively. The corner-connected phases of MAPbI3 show a direct band gap and have highly dispersive bands near the valence and conduction band edges, in agreement with previous studies.9,48,49 Large band dispersion leads to high electron- and hole-mobility, and consequently to large diffusion length for the charge carriers.50 The effective mass of the holes in the O-phase changes from 0.25m0 to 0.18m0 on including SOC along the Γ−Z direction, whereas the effective mass of the electrons changes from 0.92m0 to 0.15m0 on including SOC along the Γ−Z direction, where m0 is the rest mass of the electron. The calculated effective masses for the O-phase agree well with previously reported values.9 However, on moving from the corner-connected phases to the face-shared 2H-phase, we observe an increase in band gap to 2.62 eV using PBEsol without including SOC effects. The inclusion of SOC brings down the band gap to 2.18 eV. Analysis of the band structure for the 2H-phase reveals relatively flat bands along all the paths in the Brillouin zone, except for Γ−M and Γ−K directions, as shown in Figure 3a. These directions in the reciprocal space correspond to the direction parallel to the c-axis of the hexagonal structure along which the neighboring octahedra have face connectivity. Other directions along which the octahedra are disconnected show relatively flat bands. Consequently, the effective mass of electrons and holes in the 2H-phase increases compared to the corner-connected phases. The holes in 2H-phase have an

Table 1. Calculated Band Gaps (eV) for Different Phases of MAPbI3 Using PBEsol Functional without and with Spin− Orbit Coupling Effects band gap (eV) structure

without SOC

with SOC

type

cubic (C) tetragonal (T) orthorhombic (O) hexagonal (4H)

1.44 1.45 1.61 2.33 2.47 2.62 2.97

0.57 0.89 1.05 1.69 1.79 2.18 2.45

direct direct direct indirect direct indirect direct

hexagonal (2H)

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Chemistry of Materials effective mass of 1.33m0 along the Γ−K direction (both, with and without SOC), while the effective mass of the electrons along the Γ−K direction is 0.51m0 without SOC, which increases to 0.86m0 after including SOC effects. The reduction in the dispersion of the bands in the 2H-phase is due to a decrease in the overlap of the Pb and I states. It is evident from the DOS plot for the 2H-phase (Figure 3a) that the valence band is predominantly composed of I 5p states with a smaller contribution from Pb 6s states. The conduction band is dominated by Pb 6p states along with a smaller contribution from I 5p states. The organic MA cations do not affect the band edges directly as the states from C, N, and H are either 1.5 eV above the conduction band minima (CBM) or 0.8 eV below the valence band maxima (VBM). However, as mentioned earlier in the discussion of the tolerance factor, the A-cations control the Pb−I bond length, Pb−I−Pb bond angle, and the connectivity of the octahedra, which subsequently affect the band structure. On moving from 2H to O to T to C-phase, there is a continuous increase in the Pb−I−Pb bond angle from 75.78° to 156.63° to 161.31° to 168.9°, respectively, which leads to increasing overlap of Pb and I states. This enhanced overlap in turn increases the hybridization between Pb and I states and results in a greater dispersion of the conduction and valence bands, which eventually reduces the band gap. Furthermore, within the 2H-phase, the face connectivity of the PbI6 octahedra along the chain length facilitates better overlap of Pb and I states and leads to a large dispersion, whereas the separation of the chains along the remaining directions results in flat bands along corresponding directions in the reciprocal space. Similar features have also been reported recently in Bi-based ternary iodides with directional facesharing of BiI6 octahedra.51 A detailed discussion of the optical properties and the Born-effective charges in 2H-MAPbI3 is provided in the Supporting Information. Overall, our band structure calculations highlight toward highly anisotropic charge transport with low effective mass within the face-shared octahedral chains and high effective mass of carriers between different chains in 2H-MAPbI3. As noted earlier, the band gap of the corner-connected phases calculated using PBE or PBEsol (without SOC) shows the best agreement with the experimental band gap due to a cancellation of errors. If the same argument holds true for the hexagonal phases, we predict the 2H-phase to have an indirect band gap of 2.62 eV and a direct band gap of 2.97 eV. We have also calculated the band gap using the HSE06 hybrid functional, which performs better in predicting the band gap of many semiconductors.52 We obtain an indirect band gap of 3.81 and 3.12 eV, using the HSE06 functional without and with including SOC effects, respectively.

The lack of experimental reports on the observation of a 2Hphase of MAPbI3 in contrast to its predicted theoretical stability, while puzzling, highlights the need for a better understanding of various basic properties of MAPbI3, such as the role of anharmonic interactions and factors affecting its growth process including kinetic factors, role of solvents, surface energy, nonstoichiometry, and defects. The use of quasiharmonic approximation in our study does not capture any anharmonic effects. The organic MA cations are observed to rotate and undergo polar fluctuations at finite temperatures.40 Their rotational motion is also observed to be significantly different between the different corner-connected phases.53 First-principles molecular dynamics simulations have also shown that the dipole interaction between the neighboring rotating organic MA cations leads to the formation of domains with similar orientation.54−56 The strong hydrogen bonding between H atoms in the MA cation and the halides also results in large anharmonic fluctuations of the halides in response to the dynamical reorientation of the MA cations.57 More recently, anharmonic effects have also been reported in an all-inorganic perovskite CsPbBr3.58 While, we have shown that the energy difference between the static configuration of MA cations with various orientations is similar in the corner-connected phase and the 2H-phase, the effect of their dynamic disorder on the relative stability of the two phases remains to be understood. As we have discussed earlier, the A-site cation plays an important role in stabilizing a postperovskite structure with face-shared connectivity of the octahedra.18 In this regard, it is instructive to note that structures having face-shared or edgeshared connectivity of the PbI6 octahedra have been experimentally reported in ternary lead iodides wherein the A-site cation is either larger (FA) or even smaller (Cs) in size compared to the MA cation. In FAPbI3, the 2H-phase is observed to be stable below 403 K.4 This is in good agreement with our calculated DFT total energy of the 2H-phase being lower by 46 meV/f.u. than the high-temperature trigonal phase that has corner-connected octahedra. Likewise, in the case of CsPbI3, a phase transformation is observed from a nonperovskite δ-phase having edge connectivity of the PbI6 octahedra to the corner-connected perovskite α-phase on heating to temperatures above 573 K.4 Once again, we find that the DFT total energy calculated using the PBEsol XC functional for the edge-shared δ-phase is 211 meV/f.u. lower than the corner-connected α-phase which is comparable to a previously reported value of 170 meV/f.u.59 obtained using the PBE functional. The ability of the DFT calculations to reproduce the experimentally observed phases of CsPbI3 and FAPbI3, and also the observed trend in transition from O-phase to T-phase to Cphase among the corner-connected phases of MAPbI3 with increasing volume, is encouraging. This suggests that the experimentally observed stability of the metastable perovskite phase of MAPbI3 over the 2H-phase could possibly be due to factors beyond the free energy of the bulk. For instance, in several inorganic compounds, the solvent is known to play an important role in stabilizing a metastable phase over the ground state by changing the surface thermodynamics.60,61 The crystallization of CaCO3 in seawater is observed to result in the metastable polymorph Aragonite rather than the stable phase Calcite,60 wherein the presence of impurity Mg2+ ions in seawater is known to promote the nucleation thermodynamics and kinetics of Aragonite over Calcite. Indeed, solvent effects have been observed to change the phase transition temper-

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CONCLUSIONS In conclusion, we have carried out an extensive first-principles analysis of the thermodynamic stability of different phases of MAPbI3 and predict a previously unreported hexagonal 2Hphase that consists of infinite chains of face-shared PbI6 octahedra to be the ground state. Unlike the experimentally observed perovskite phases of MAPbI3 that have recently been reported to have a positive formation enthalpy and as a result disintegrate into its binary constituents PbI2 and MAI,11−13 the 2H-phase has a negative formation enthalpy that varies between −5 and −90 meV/f.u. depending on the choice of XC functional used. It is, therefore, expected to be thermodynamically stable. 6008


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eV. Additionally, to calculate the force-constant matrices, a 2 × 2 × 2 supercell expansion was performed for the C-phase and O-phase, while a supercell expansion of 3 × 3 × 2 was carried out for the 2H-phase. The optical properties were calculated using the PBEsol functional. Additionally, for accurate frequency dependent optical properties, the parameter NBANDS was quadrupled with respect to the default values for all phases to accommodate sufficient empty conduction bands. For the HSE06 calculations, the fraction of Hartree−Fock exchange (α) was fixed at 0.25.

atures of CsPbI3, FAPbI3, and MAPbI3.4 While cooling down the single crystals of FAPbI3, the transition from α-phase to δphase is observed in a liquid interface at ∼360 K, while dry crystals show a transition from α-phase to β-phase at ∼200 K.4 In this case, δ-phase of FAPbI3 exhibits similar octahedra connectivity as the predicted 2H-phase for MAPbI3, while the α- and β-phases show corner-connected PbI6 octahedra. Recently, the stability of MAPbI3 nanoparticles obtained using colloidal synthesis routes has been demonstrated to be critically sensitive to the solvent used.62 The presence of moisture and air could also be playing a role in stabilizing the metastable perovskite phase. For instance, the metastable αphase of CsPbI3 is found to be stable for weeks at room temperature when its exposure to ambient air is avoided during synthesis.63 Our results, therefore, suggest that a better understanding of the crystallization thermodynamics and kinetics is required to reveal the stability of the thermodynamically unstable perovskite phase of MAPbI3 over the stable 2Hphase. Overall, the predicted stability of the 2H-phase combined with the lack of its experimental observation poses a challenge to the community; a better understanding of the factors that lead to this dichotomy will provide crucial insights to stability issues in these perovskites. Organohalide lead perovskites have attracted attention not only for photovoltaic applications but also as cheap and high-efficiency semiconductors with applications in lasers, light-emitting diodes, and thermoelectric devices.64 Low-dimensional halide perovskites with layered structures and anisotropic charge transport are being explored as materials for high-efficiency optoelectronic devices.65,66 The highly anisotropic charge transport predicted here from the band structure of 2H-MAPbI3 along with its predicted thermodynamic stability makes it particularly exciting for such applications.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.7b01781. Structural information on the three experimental phases and two hexagonal phases; description of the orientation of organic cations for all the phases; phonon band structure for C- and O-phases, eigenmode analysis for the 2H-phase, Helmholtz free energy for the C-, O-, and 2Hphases within the harmonic approximation; electronic band structure and density of states for C-, T-, and 2Hphases; comparison of optical properties of C-, T-, O-, and 4H-phases (PDF) Crystallographic information files for C-, T-, O-, 2H-, and 4H-phases (ZIP)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Xing Huang: 0000-0003-4615-2758 Rohan Mishra: 0000-0003-1261-0087 Notes

The authors declare no competing financial interest.

COMPUTATIONAL DETAILS

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DFT calculations were performed using the Vienna Ab-initio Simulation Package (VASP).67 For the formation enthalpy calculations, we employed a wide range of XC functionals, including the local density approximation (LDA),31 generalized gradient approximation (GGA) as implemented in the Perdew−Wang (PW91) functional,32 the Perdew−Burke−Ernzerhof (PBE) functional,33 and the strongly constrained and appropriately normed (SCAN) metaGGA functional.34 We also used the DFT-D3 correction method (vdW-D3) of Grimme et al.36 for the PBE functional. The core and valence electrons were modeled using the projector-augmented-wave (PAW) method.68 We used a plane-wave basis set with a cutoff of 500 eV and performed relaxation until the Hellmann−Feynman forces on the atoms were less than 0.001 eV/Å. To obtain accurate total energies, we repeated the relaxation calculations for the PBEsol functional using a cutoff energy of 700 eV. For the ground state and isotropic stress calculations, a 12-atom unit cell was used for the Cphase, a 48-atom supercell with √2 × √2 × 2, a periodicity of the 12atom cubic cell is used for the T- and O-phases, a 48-atom unit cell was used for the 4H-phase, and a 24-atom unit cell was used for the 2Hphase. A Monkhorst−Pack69 k-points mesh was used for sampling the Brillouin zone where the number of k-points (nk) is changed to keep (nk × a), with a being the lattice constant equal to ∼50 and ∼130 for structural relaxations and electronic calculations, respectively. For the 4H and 2H-phases, we used a Γ-centered k-points mesh. For the relaxation and electronic calculations including SOC effects, we used a less dense k-points mesh (nk × a ≈ 50). The phonon calculations were performed using the frozen-phonon approach, and the dispersion spectra were calculated using the phonopy-package.70,71 For accurate phonon calculations, a higher cutoff energy of 700 eV for the planewave basis set was used with a tighter electronic convergence of 10−8

ACKNOWLEDGMENTS This work was supported by a start-up funding from Washington University and by a Ralph E. Powe Junior Faculty Enhancement Award from Oak Ridge Associated Universities to R.M. This work used computational resources of the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by the National Science Foundation grant number ACI-1053575, and the Washington University Center for High Performance Computing, which was partially funded by NIH grants 1S10RR022984-01A1 and 1S10OD018091-01. Finally, we would like to thank the anonymous reviewer for the constructive criticism.

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REFERENCES

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