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Preferential CH3NH3+ Alignment and Octahedral Tilting Affect Charge Localization in Cubic Phase CH3NH3PbI3 Byungkyun Kang and Koushik Biswas* Department of Chemistry and Physics, Arkansas State University, Jonesboro, Arkansas 72467, United States S Supporting Information *

ABSTRACT: Hybrid organic−inorganic perovskites (CH3NH3PbI3) have gained prominence in recent years due to their fascinating electronic properties and potential for commercial application in photovoltaics and optoelectronics. One of their intriguing features is in the structure itself and the role played by the organic cation CH3NH3+ (MA+). In this study, we implement first-principles-based methods to take a static look at this dynamic system, which may shed some light on the preferential orientation of MA+ and its impact. We find there is a lattice energy gain in cubic CH3NH3PbI3, when going from a pristine host supercell with [100] MA+ orientation to a distorted host consisting of preferentially aligned MA+ and tilted PbI6 octahedra. Reoriented MA+ and octahedral tilting are also accompanied by larger number of (N−H3)···I hydrogen bonds. This lattice reconfiguration may support charge localization as evidenced by larger 216-atom supercell calculations. The localization behavior is a consequence of lattice polarization (reoriented MA+ and tilted octahedra) which spreads across multiple unit cells and thus may not be strongly bound small polarons.

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INTRODUCTION The startling photovoltaic performance of organic−inorganic lead halide perovskites (CH3NH3PbI3, hereafter, MAPbI3) has generated a great deal of interest in this general family of materials.1−15 In addition to photovoltaics and optoelectronics, some of these hybrids and their inorganic analogues may also be important in the field of radiation detection16 where the radiation generated carriers are collected under an externally applied bias voltage. Large strides have been made toward understanding their fundamental properties, and there are encouraging signs of further optimization to cater to a diverse set of technologies. Despite the advances, there are a few intriguing features about MAPbI3one being the apparent disconnect between measured moderate-to-low mobility and low effective mass of charge carriers as reported in numerous studies.4,17−20 Also, there is some uncertainty about the role of the organic cations, CH3NH3+ (MA+), and the extent to which they affect bulk electronic properties.21−26 The subject of MA+ orientation has been considered in many studies, and indeed their preferential ordering and rotations have been extensively reported.21−25,27−32 For instance, molecular dynamics (MD) simulations27,28 show free MA rotations in cubic phase of MAPbI3, while quasielastic neutron scattering data seem to demonstrate that they undergo jumplike preferential orientations within the cuboctahedral cages below 370 K.29 Tilting of the PbI6 octahedra is also well-known, especially in the orthorhombic and tetragonal phases, although MD simulations have indicated framework distortions even within the cubic structure.23,27 There is a possible correlation between C−N bond orientation and octahedral tilting which © 2017 American Chemical Society

may be linked with changes in the (N−H3)···I hydrogen bonds.27,28,30,33−35 These studies provide us with a perspective on the complicated interplay between MA+ moieties and the inorganic matrix and how this might impact electronic properties.28,30,31,36,37 The objective of this report is mainly twofold: (i) emphasize the connection between MA + reorientation and unusual octahedral tilting within the cubic structure, where such distortions are generally believed to be less pronounced, and (ii) concerted cationic reorientation and octahedral tilting create the conditions for charge localization. This charge localization otherwise remains absent in cubic MAPbI3. By employing supercells and accurate density functional methods (DFT), we find that MA+ reorientations seemingly trigger those tilts in the PbI6 octahedra. It is also interesting that such framework distortions lead to substantial lattice stabilization compared to an undistorted cubic host. In this context, it is important to note a recent observation regarding the existence of instantaneous symmetry-broken local structure consisting of tilted PbI6 octahedra in cubic MAPbI3 at 350 K.38 The other point of emphasis is that the deviation from cubic toward a distorted framework supports charge localization. A recent work by Neukirch et al.39 used finite cluster calculations to describe the stabilization of strongly bound small polarons caused by cooperative lattice distortion and MA+ rotations. Although our results also indicate similar reorientations and octahedral tilting happening in concert in the cubic Received: February 6, 2017 Revised: March 29, 2017 Published: March 31, 2017 8319

DOI: 10.1021/acs.jpcc.7b01184 J. Phys. Chem. C 2017, 121, 8319−8326

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The Journal of Physical Chemistry C perovskite, we find those distortions spreading over multiple lattice sites through the supercell. This may suggest charge localization at shallow states near the band edge instead of deep, strongly bound polaronic states.

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METHODS Geometry optimization and electronic structure calculations of the cubic phase MAPbI3 are based on the projector augmented wave method40,41 as implemented in the Vienna Ab-initio Simulation Package (VASP).42,43 Experimental lattice constants (a = b = 6.3115 Å, c = 6.3162 Å) have been used throughout,44 while relaxing all free internal atomic coordinates. The lattice parameters are usually more precisely obtained from experiment than DFT-based methods. A comparison between experiment and fully relaxed cell volumes is presented later in this article. The results are based on hybrid PBE0 functional which mixes a fraction (α = 0.25) of exact Fock exchange.45 All presented results also include spin−orbit interaction, and it is referred to in the text as PBE0-SOC. The internal atomic coordinates are relaxed to obtain the optimized geometry of the perfect and distorted hosts using a planewave energy cutoff of 300 eV and force convergence criteria of 0.05 eV/Å. The reciprocal cells of the 96-atom calculation are sampled using a Γ-centered 2 × 2 × 2 k-point mesh. Larger 216-atom supercells are sampled at two k-points, Γ and R. The energy vs volume calculations of 96-atom supercells are based on PBE functional,46 using force convergence criteria of 0.01 eV/Å. The lattice parameters and volume obtained from fully relaxed PBE0-SOC calculations confirm that the ground state structure, in addition to electronic properties, is well described by this hybrid functional (see Supporting Information). Electronic and structural properties of MAPbI3 unit cell were also obtained by including van der Waals interactions (vdW) at the PBE-SOC level via the TS scheme.47 Although there are no significant differences in the electronic structure, vdW indeed improves lattice parameters in comparison to PBE-SOC calculation.48

Figure 1. Calculated band structure (solid lines) of cubic MAPbI3 using PBE0-SOC. The energy of the valence band maximum is set to zero. The dashed (gray) lines show corresponding bands obtained using PBE0. The solid lines are colored according to their primary orbital character. The inset is a magnified view of the conduction and valence band-edge near R-point.

Recent quasiparticle calculations within the GW approximation also yielded similar band characteristics.53,56 The band structure presented in Figure 1 shows the already well-documented features in the electronic properties of MAPbI3. It can be used to highlight a few distinctions with conventional semiconductors. Unlike cation s states that are commonly found in compound semiconductors, here the spin− orbit split Pb 6p orbitals, which are usually more delocalized than s, make up the conduction band minimum (CBM). They contribute to the low effective mass of bands. Similarly, I 5p that primarily makes up the valence band maximum (VBM) is also hybridized with antibonding Pb 6s, increasing the dispersion and lowering effective hole mass. Another prominent feature is the appearance of Pb 6p near VBM, which is reminiscent of cross-gap hybridization found in ferroelectric oxides and some Tl and Pb halides.57−60 It also undergoes spin−orbit splitting. The MA states are localized deep inside the valence band (∼8 eV below VBM, not shown) and does not seem to directly affect band-edge properties that are controlled by Pb and I valence orbitals. Preferential CH3NH3+ Alignment. One question that continues to intrigue researchers is whether the charge carriers are really free in MAPbI3. Another question revolves around the true role of MA in MAPbI3. Are they behaving only as scaffolds within the perovskite crystal, or do they have a connection with the electronic performance of this material? These have been contested issues for some time.4,17−26 An overall [100] MA+ orientation has been speculated to be energetically stable with other possible orientations along [110] and [111] separated by small energy barriers of around 15−20 meV.29,36,61,62 These numbers are estimated from first-principles calculations, and experimental inputs seem to agree that barrier heights are too low to accurately resolve any specific differences in MA orientation. Notwithstanding this averaged [100] orientation, recent literature also provides clues that other MA orientations may be possible due to small barrier heights. Room temperature vibrational spectroscopy, neutron scattering measurements, and ab initio molecular dynamics all point toward preferential alignment or “jump-like” reorientation of MA+.27−29,62−64 This serves as our motivation to study the effects of MA reorientation in the cubic structure. We proceed by constructing two 96-atom host supercells, with differing MA alignments. The 2 × 2 × 2 supercell (96-atom) contains eight unit cells with the MA+ centered in cages surrounded by eight octahedra. The perfect host consists of all MA+ pointing along [100],

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RESULTS AND DISCUSSION Electronic Structure. Before discussing MA+ reorientations, let us first briefly rehash this material’s ground state electronic structure. Recent computational studies have made significant contributions toward understanding the structural and electronic properties of this material. The importance of spin−orbit coupling on the band-edge properties is now wellestablished.36,37,49,50 The PBE0 functional (see Methods section for details) predicts a large gap of 2.68 eV near Rpoint in the Brillouin zone of cubic MAPbI3. Figure 1 shows the calculated band structure of cubic MAPbI3 and highlights the differences induced by spin−orbit coupling (PBE0-SOC functional). Spin−orbit splitting of Pb 6p conduction states drastically reduces the direct gap to about 1.45 eV and shifts it slightly off the R-point. Notice that the emergence of k-space splitting near the band edge, described earlier as Rashba− Dresselhaus splitting,51−55 is a consequence of inversion asymmetry of the structure. It should be clarified that the band structure offers only a snapshot view of a static structure, which may alter due to the rotational dynamics of MA+ and its interaction with PbI6 octahedra. There are also noticeable differences in the band curvature, especially near the conduction band edge, which reinforces the importance of spin−orbit effect causing reduced effective mass of free carriers. 8320


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The Journal of Physical Chemistry C

Figure 2. (a) Optimized structure of 96-atom perfect host having [100] MA+ orientation. (b) Dimensions of the polyhedral cage that holds MA+ in the perfect host (m = n = 6.30 Å) indicating minimum octahedral tilt. (c) (N−H3)···I hydrogen bonds in 96-atom perfect host. (d) Optimized structure of 96-atom distorted host with [110] and [101] MA+ orientation. (e) Octahedral tilt and polyhedral cage dimensions (m′ = 5.30 Å and n′ = 7.31 Å) signify distortion. (f) (N−H3)···I bonds in 96-atom distorted host. (g) Distribution of hydrogen bonds in 96-atom perfect and distorted hosts. (h, i) Optimized structure of 216-atom perfect and mixed host, respectively.

of about 15°, which is the averaged I−Pb−Pb−I dihedral angles along the c-axis. As shown in Figure 2, the dimensions of the cage holding the reoriented MA evolve, allowing the coordinated tilting of the octahedra in the distorted host. This type of out-of-phase octahedral rotation is observed in the tetragonal phase of MAPbI3. The existence of significant hydrogen bonding at room temperature in all three halides, i.e., MAPb(I/Br/Cl)3, has been suggested in a recent experimental study.33,66 In the present case, going from the perfect to distorted cubic supercell, we find a similar scenario with increased hydrogen bonding. Figure 2g shows an increased number of (N−H3)···I bonds in the distorted host, within a 2.5−3.5 Å range and bond angles greater than 110°.67 We also find that the Pb−I and I−I separations are generally extended in each of the distorted octahedra. On average, the Pb−I bonds are elongated by about 0.04 Å in the distorted host compared to that in the perfect host (Table 1). Longer cation−anion and anion−anion distance causes weak hybridization between the orbitals, resulting in narrower valence bands. We must note that similar band gap modulation caused by octahedral tilts and changes in ionicity/covalency of Pb−I bonds has been discussed in previous studies.27,33 However, what is interesting about the distorted structure is the observed anisotropies in the MA cavity brought about by the reoriented organic cations and octahedral tilting (see Supporting Information Figure S1). Coincidentally, it looks similar to the elongation and shortening of MA cavities described recently

toward a cube face. The other supercell, which we refer to as the distorted host, has four MA+ ions initially oriented along [110] and the remaining four along the [101] direction. After geometry optimization, we find significant differences between these two hosts caused by MA+ realignments (Figure 2 and Table 1). The energy of the distorted host is lower than perfect Table 1. Comparison of Electronic and Structural Properties of 96-Atom Perfect and Distorted Host (PBE0-SOC Functional) electronic properties (eV)

perfect host distorted host

average bond length (Å)

relative energy

energy gap (at R)

valence bandwidth

I−I

Pb−I

N− Pb

+0.48

1.62

4.26

4.48

3.17

4.75

0

1.83

3.99

4.52

3.21

4.72

host by about 60 meV per formula unit (+0.48 eV for the perfect supercell in Table 1). The fundamental gap of the distorted host at R-point increased by about 0.21 eV while its valence bandwidth decreased almost 0.3 eV. These changes lend credence to preferential MA orientations and its impact on the dynamic band gap behavior of the material.27,28,65 The structure of the distorted host is also considerably altered. The PbI6 octahedra undergo out-of-phase rotations having tilt angles 8321


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[100] alignment. In this way, the mixed host may be in approximate agreement with neutron diffraction experiments and molecular dynamics simulations which have shown a preference toward cube face MA alignment and small energy barriers (in the order of thermal energy) between other orientations.29,36,61,62 Once again, after geometry optimization we find a similar trend where the total energy of the mixed 216atom supercell is lowered by about 0.15 eV, compared to the perfect structure. We refer to it as the lattice stabilization energy, ELS = 0.15 eV, which is discussed later (see Summary and Conclusions). There is an associated 0.07 eV increase in its band gap. The average Pb−I distances are also extended in the mixed structure by ∼0.01 Å. The octahedral tilt is most pronounced where the MAs are reoriented, although we find minor tilts spreading across the entire mixed supercell (Figure 2i). We estimate an energy barrier of about 0.22 eV between the 216-atom perfect supercell with [100] MA+ alignment and the aforementioned mixed host (see Supporting Information Figure S3 for details). It amounts to ∼12 meV/MA+ (there are 18 MAs in the 216-atom cell). The distinctions between the perfect and distorted framework structures discussed above provide an impetus to study their excited state behavior and any possible differences in charge localization. In order to disentangle the effect of MA+ orientation, we start with three relaxed host structures: (i) a perfect cubic 216-atom supercell with [100] MA orientation, (ii) a mixed 216-atom host having partially reoriented MAs and tilted octahedra as described in the previous paragraph, and (iii) a 216-atom cell having tilted octahedra as in (ii), but all MA maintaining [100] alignment. The excited structures are then simulated by adding an excess electron or a hole to the respective supercell. Structure (i) yields a completely delocalized solution where the additional electron or hole is spread throughout the supercell. Structure (iii) relaxes back toward a perfect host with almost no distortion of the initially tilted octahedra. Its excess charge is also delocalized, similar to structure (i). A hint of charge localization is observable only in structure (ii), which is what we expect based upon the hypothesis that hopping rotational dynamics of MA at higher temperatures together with octahedral tilting create a local lowsymmetry structure which supports this phenomenon. Figure 4 shows the fractional orbital character of the lowest occupied Pb 6p states near conduction band edge of perfect and mixed host calculations containing an excess electron. While the extra electron is dispersed uniformly through all Pb2+ residing in the front and back layer of the perfect supercell (see Figure 4a), a charge separation is recognizable in the mixed host where the Pb2+ ions in the back layer holds additional charge than front layer. A more detailed analysis of extra charge distribution in front and back layers is given in Supporting Information Figure S4. The binding energy of this charge localization in structure (ii) is obtained from total energy difference between the completely delocalized and polaronic solution:

within the context of a local instantaneous structure in cubic MAPbI3 at 350 K.38 We will consider the issue of local lowsymmetry structure and its possible implication on charge localization in MAPbI3. In order to verify that the above results are not an artifact of finite size of the supercell, which has been kept fixed based upon experimental lattice constants, we performed total energy calculations at different volumes. Figure 3 shows calculated

Figure 3. Calculated energy vs volume of perfect and distorted host supercells (96-atom) obtained using PBE functional and its fit to the Birch−Murnaghan (BM) equation of state. The zero of energy is set at the minimum of perfect host. The volumes obtained from fully relaxed PBE0-SOC calculation of perfect and distorted structures are shown as large solid circles. Details of this result are in the Supporting Information. Dashed vertical line shows the volume according to experimental lattice parameters.44

energy vs volume of perfect and distorted host supercells, fitted to the Birch−Murnaghan (BM) equation of state.68 The results are based on PBE calculation without including vdW interaction. The fitted minima of each host are larger than the experimental volume, which is expected of a PBE functional. Total energy of distorted host is consistently lower than the corresponding perfect supercell. The estimated bulk moduli are 10.0 and 11.2 GPa for perfect and distorted structures, respectively. These are comparable to experimental and computed values reported in the literature.38,48,69 A full relaxation including the volume and shape of the two supercells at the PBE0-SOC level yields parameters close to experimental values,44,70,71 while the same trend in total energy is maintained between the two structures. The obtained cell parameters of this calculation are given in Supporting Information Figure S2. We may therefore infer that symmetry lowering reorientations of the MA+ and corresponding octahedral tilts in the distorted host structure may be possible via lattice stabilization (energy gain), aided by stronger and more numerous (N−H3)···I bonds. Charge Localization. Following on with the above idea of perfect vs distorted host structures and associated lattice stabilization, we proceed with a larger 216-atom supercell. Working under the presumption that these static calculations only provide an instantaneous view of a complicated coupling between the organic and inorganic components, we propose a mixed host structure where not all MAs are preferentially aligned (Figure 2i). It may be worthwhile to investigate this type of structure, as opposed to a completely polar or nonpolar MA+ orientation. Here, part of the MAs are realigned to [110] and [101] like the 96-atom distorted host discussed above, and the remaining about 80% of the organic cations retain their

ΔE = E(delocalized e− or h+) − E(polaronic e− or h+) (1) −

+

where E(delocalized e or h ) refers to the perfect supercell having a delocalized electron (e−) or hole (h+) at CBM or VBM. Similarly, E(polaronic e− or h+) refers to that of a partially localized e− or h+ below CBM or above VBM. The mixed host structure (ii) which consists of partially reoriented MA and tilted octahedra has considerable binding energy as given in Table 2. It also signals that the localized charge is more 8322


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possibility of such a local symmetry-broken, instantaneous structure.38 We have used 216-atom mixed host as a prototype to investigate polaronic behavior of charge carriers. An excess electron or hole in such a supercell has significant binding energy with respect to the pristine, cubic host (Table 2). However, these may not be strongly bound, small polarons as prescribed by Neukirch et al.39 The binding energies reported in ref 39 are considerably larger, on the order of 1 eV, which suggests deep states inside the band gap of MAPbI3. Our DFT supercell calculations could not localize such small polarons. It is worthwhile to note that self-trapped holes or Vk centers72 are commonly found in insulating alkali- or alkaline-earth halides with calculated binding energies up to several tenths of an eV.73−78 In MAPbI3, we find a delocalized hole near VBM is energetically favored over a Vk by about 0.1 eV. Similarly, small electron polarons that are stable in cubic structures of Cs2LiYCl6 or Cs2HfCl676,77 are unstable in MAPbI3. In three dimensions, short-range coupling between charge carrier and acoustic phonons generally requires significant ionic distortion that will initiate polaron self-trapping. The perturbation induced by reorientations of the organic cation and dispersive conduction and valence bands of MAPbI3 may not support such deep polaronic states inside the host gap. In general, the polaron binding energy (ΔE) consists of a competition between energy cost due to lattice strain and electronic energy gain caused by localizing a carrier at a deep gap state. When the electronic energy gain trounces the cost of lattice strain, a small polaron is favored against a free-carrier-like state. In case of cubic MAPbI3 we find an interesting departure from this consensus, where lattice distortion caused by reoriented MA+ and accompanying octahedral tilt contribute toward binding a polaron-like state. It may be understood from a schematic one-dimensional configuration coordinate (cc) diagram shown in Figure 5 which depicts the binding of an

Figure 4. (a) Schematic view of the front (even-numbered ions) and back (odd-numbered ions) Pb layers in a 216-atom supercell. (b) Fractional orbital character of the lowest occupied Pb 6p states in 216atom perfect and mixed host calculations containing an excess electron.

dispersed throughout the back layer in the mixed host, which is a signature of large polaron behavior. Structure (iii) has vanishingly small ΔE.

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SUMMARY AND CONCLUSIONS The above results provide a static picture of a dynamic system where the organic MAs undergo thermally activated rotations. However, the distorted and mixed host supercells employed in this study captures the essence of selective or preferential orientations of the organic rotors and its consequences. Preferential MA orientations has already been discussed in the literature, and there is experimental evidence about these reorientations occurring within picosecond time scale.29,63 In our static calculation such reorientations and strong octahedral tilting, which are the hallmarks of low temperature phases, is associated with stabilization of the lattice. Both 96-atom distorted and 216-atom mixed hosts show a resulting energy gain. These structures may be key to charge separation and localization in MAPbI3. One recent work indicates the

Figure 5. Schematic 1-D configuration coordinate diagram showing lattice energy gain (ELS) going from perfect to mixed host and further electronic energy gain (EEL) after capturing a charge carrier.

electron polaron. The perfect host geometry corresponds to Q = 0, and the energy gain involving the transition to a mixed 216-atom host occurs at ΔQ1 = 23.06 amu1/2 Å (Table 2). Here, ΔQ1 is a measure of the lattice distortion going from perfect to mixed host and defined as (ΔQ1)2 = ∑α,iMαΔRαi2, mixed ΔRαi = Rmixed − Rperf and αi αi where Mα is atomic mass and Rαi

Table 2. Polaron Binding Energy ΔE (≈ ELS + EEL), Obtained Using Eq 1; Contributions to ΔE from Lattice Stabilization (ELS) and Electronic Energy Gain (EEL) and Corresponding Displacements in the 1-D Configuration Coordinate −

localized e localized h+

ΔE (meV)

ELS (meV)

EEL (meV)

ΔQtotal (amu1/2 Å)

ΔQ1 (amu1/2 Å)

ΔQ2 (amu1/2 Å)

180 185

150 150

30 35

24.92 26.38

23.06 23.06

1.86 3.32

8323



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Rperf αi refer to relaxed coordinates i of atom index α in the mixed and perfect host, respectively.79 We refer to the resulting energy gain as lattice stabilization energy (ELS) caused by preferential MA orientation, tilted octahedra, and stronger (N−H3)···I bonds in the mixed host. In the presence of an excited charge carrier there is an additional displacement ΔQ2, accompanied by a smaller electronic energy gain (EEL) involving its localization at a shallow state near the band edge. This scenario is depicted in the second cc diagram with ΔQ2 = 1.86 amu1/2 Å (Table 2). Note that ΔQ1 and ΔQ2 are two separate displacements on the cc diagram, and ΔQtotal = ΔQ1 + ΔQ2 refers to the total lattice distortion corresponding to a localized charge in a mixed host versus a free carrier in a perfect host. The total energy gain during this process is the polaron binding energy, ΔE (≈ ELS + EEL), defined earlier in eq 1. The calculated values of ΔE are given in Table 2. Thus, the possibility of preferential MA orientation sets up the condition for capturing a charge carrier in a polaron-like state involving a lattice energy gain of ELS, followed by an additional electronic energy gain of EEL. We emphasize that the values of ELS and EEL will depend on the size of local disorder containing reoriented MA and tilted octahedra. Our reported binding energies are based within the limitations of a 216-atom mixed supercell, containing about 20% reoriented MA. However, it seems that preferential MA alignment and concerted octahedral tilting spreading across multiple unit cells likely cause charge localization in shallow states near band edge instead of strongly bound small polarons.

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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.jpcc.7b01184. Optimized perfect and distorted host structures, fully relaxed parameters of 96-atom supercells, estimate of energy barrier between 216-atom perfect and mixed host, and additional information on 216-atom mixed host with an excess electron (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; phone (870) 972-2427. ORCID

Koushik Biswas: 0000-0002-7875-9821 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the U.S. Department of Homeland Security under Grant Award 2014DN-077-ARI075-04. The support does not constitute an expressed or implied endorsement on the part of the Government. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract DE-AC02-05CH11231. Computational resource at Arkansas State is partially funded by NSF Grant ECCS-1348341. The authors acknowledge helpful discussions with Audrius Alkauskas regarding carrier capture in semiconductors. 8324


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