Importance of Orbital Interactions in Determining Electronic Band

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Importance of Orbital Interactions in Determining Electronic Band Structures of Organo-Lead Iodide Jongseob Kim, Seung-Cheol Lee, Sung-Hoon Lee, and Ki-Ha Hong J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp5126365 • Publication Date (Web): 13 Feb 2015 Downloaded from http://pubs.acs.org on February 15, 2015

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The Journal of Physical Chemistry

Importance of Orbital Interactions in Determining Electronic Band Structures of Organo-Lead Iodide Jongseob Kim†, Seung-Cheol Lee ¶, Sung-Hoon Lee*,§, and Ki-Ha Hong*,‡

†Samsung Advanced Institute of Technology, Samsung Electronics Co., Ltd. Mt. 14-1, NongseoDong, Giheung-Gu, Yongin-Si, Gyeonggi-Do, 446-712, Korea ¶ Electronic Materials Research Center, Korea Institute of Science and Technology (KIST), Seoul 136-791, Republic of Korea § Center for Artificial Low Dimensional Electronic Systems, Institute for Basic Science, Pohang 790-784, Korea ‡ Department of Materials Science and Engineering, Hanbat National University, 125 Dongseodaero, Yuseong-Gu, Daejeon, 305-719, Korea

Email: [email protected], [email protected]

ACS Paragon Plus Environment

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ABSTRACT

Organic-inorganic perovskites are promising materials for improving the efficiency of solar cells, but there are still uncovered issues on the understanding of their electronic band structures. Using first principles calculations we investigate the electronic band features of organo-lead iodide perovskites and present the efficient model to predict the band gap variation based on the orbital interaction scheme. The orbital interaction between Pb and I atoms can be controlled through the structural modification such as the change in lattice constant and the deviation of I atoms from cubic symmetry sites. The increase of the lattice constant and the positional distortion of I atoms from the cubic symmetry sites lead to the increase of the band gap. With our findings, puzzling band gap variation behaviors in previous experiments and simulations can be understood and we suggest a pathway to precisely control their band gap. Our study can serve as the design rule for band gap engineering for various kinds of organic-inorganic hybrid perovskites.

KEYWORDS Band gap, Density functional theory, Strain, Spin orbit coupling, Perovskite solar cell


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INTRODUCTION Organic-inorganic hybrid perovskites have been one of the most attractive materials in the field of photovoltaic research. Since Kojima et al.1 applied methylammonium (MA, CH3NH3+) lead halides (MALHs) as an absorber for dye sensitized solar cells,2-7 the photon conversion efficiency of solar cells based on organo-lead halide perovskites has quickly risen, and recently a confirmed efficiency of 17.9 % was achieved.8 Zhou et al.9 have reported 19.3% efficiency through interface engineering. For practical applications, however, several issues still remain to be resolved, such as long-term stability, lead-free perovskites, and high efficiency.10-12 To maximally utilize the solar spectrum, it is well known that the optimal band gap of an absorber material in single junction solar cells is around 1.4 eV.13 In the case of methylammonium lead iodide doped with chlorine, the band gap is about 1.55 eV. This implies that photon conversion efficiency could be improved by modulating the electronic band gap of the organo-lead halide perovskites. Attempts have been made to reduce their band gaps to the optimal value by changing organic cation molecules, and several research groups have successfully implemented formamidinium (FA, CH(NH2)2+) within the 3-dimensional perovskite crystal structures. Reported band gaps of formamidinium lead iodide (FAPbI3) are 1.48 eV14 and 1.43 eV.15 In addition to the promising preliminary results16 for FAPbI3, plenty of opportunities remain to apply various different kinds of molecules and atoms to the perovskites. Understanding the physical origin of the band gap modulation of organo-lead halide perovskites is essential to enhancing their solar cell efficiency. Theoretical investigation with first principles calculations is very useful for revealing the key features in the electronic band structures of organo-lead halide perovskites17-23 and can suggest a pathway to control the band gaps. It is well known that electronic structures can be modulated by strain engineering,24-26 and


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strain is inevitable during heteroepitaxial thin film growth. While lattice constants cannot be arbitrarily controlled in the case of experimental studies, first principles calculations can analyze how a lattice constant change affects the band structures without changing the initial lattice symmetry, because simulation studies can be done under controlled conditions designed by the investigators. Although there have been a lot of computational studies on the electronic band structures of organo-lead halide perovskites,17,23,24,27-32 the universal model to explain the nature of their electronic band structures remains lacking so that the developed theory and models cannot be directly applied to the newly synthesized perovskite materials. The band gap narrowing of FAPbI3 is one of most curious problems on the band gap modification of organo-lead halides. A recent investigation by Amat et al.29 discovered that the tilting of Pb-I bonds is the main source of the narrower band gap of FAPbI3. However, it is still not clear why the conduction band shift can be detected only when the spin-orbit coupling is considered and moreover, another theoretical estimation and analysis cannot be avoided for the newly synthesized organicinorganic hybrid perovskites due to the absence of unified model including Pb-I frame change and lattice constant. While prior studies on the effects of lattice constant changes of perovskites focused on the band gap variation and band structure changes,24,31 Pb-I frame changes are not considered so that the influence of cation molecules cannot be analyzed. The unified structural model to understand the band gap of organo-lead halide needs understanding of the bondingantibonding features reflecting both Pb-halide frame distortion and lattice constant changes.

COMPUTATIONAL DETAILS


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Equilibrium lattice structures and electronic band structures were obtained within density functional theory (DFT) using VASP.33,34 Electronic wavefunctions were expanded with plane waves with an energy cutoff of 520 eV. The core-valence interaction was described by the projector-augmented wave (PAW) method.35 We used the PBE-type generalized gradient approximation (GGA)36 to exchange-correlation functional and included 5d electrons in the valence charges for Pb. For all GGA calculations, we also considered spin-orbit coupling (SOC) as previous studies28,37,38 to reveal the band edge change trend. Because GGA calculations including SOC (GGA-SOC) tend to underestimate the band gap, Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional39 is adopted to obtain the band gap close to experiment. HSE calculations present that GGA-SOC is enough to interpret the band gap variation under various circumstances in spite of the underestimated band gaps. Cubic, tetragonal (orthorhombic), and trigonal lattice structures were optimized by including a molecule with an 8 × 8 × 8, 4 × 4 × 4, and 2 × 2 × 2 Γ-centered k-point grid, respectively. For the calculations of cubic phase lattice, cubic symmetry constraint was forced because the cubic phase can only be found at high temperature. The atomic positions were relaxed until residual forces was less than 0.01 eV/Å. Molecular orbital characters were obtained from the projection of each band onto ion-centered spherical harmonics, as implemented in VASP. In order to align eigenvalues between different lattice structures, we corrected calculated eigenvalues by adding the (G=0) components of longrange Coulomb interactions from ions and electrons, which are typically neglected in the conventional formulation and lead to the shift of the vacuum level.40 The corrected eigenvalues are given with respect to the vacuum level and can be compared on an equal footing regardless of different lattice constants.


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RESULTS and DISCUSSION Three kinds of lattice symmetries have been reported for organo-lead halide perovskites, cubic, tetragonal, and trigonal. Figure 1 shows unitcell and superlattice structures obtained by GGASOC calculations for MAPbI3 and FAPbI3. Crystal structures of organo-lead halide perovskites are significantly affected by preparation methods and can be changed by varying the temperature.41 The crystal structure of MAPbI3 is known to be tetragonal (I4cm) at ambient temperature and pseudo-cubic (Pm3ത m) at high temperatures.42 The crystal structure of FAPbI3 is reported to be trigonal14 or tetragonal.15 The lattice constants of the trigonal and tetragonal FAPbI3 are quite similar, with the former being 0.02 Å longer than the latter. The respective band gaps are 1.43 eV and 1.48 eV. Calculated lattice constants of MAPbI3 and FAPbI3 are summarized in Table 1 along with experimental data. Although SOC significantly affects the electronic band structure as shown in previous reports37, it hardly changes the equilibrium lattice constants. In the case of MAPbI3, orthorhombic structures are favored without special treatment of the direction of methylammonium. Aligning the direction of methylammonium stabilizes the pseudo-tetragonal structure (Figure 1b) as reported in prior studies.43 The resulting lattice constants are overestimated by 2-4 % relative to experiments, which is reasonable, considering that GGA calculations typically overestimate lattice constants by ~2%. As it will be discussed later, the overestimation of the lattice constant causes the overestimation of the band gap than experiments and other simulation studies. It is worthy of notice that the sensitiveness to lattice constants would cause different band gap estimation for the MAPbI3 among previous theoretical works43-46 by applied numerical methods, exchange correlations, and convergence criterions.


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The band gaps obtained by GGA without including SOC, shown in Table 1, show good agreement with experiments, but it needs care when interpreting because GGA calculations have an intrinsic limitation of delocalization errors that dictates a significant underestimation of the band gap.47 The accidental agreement for organo-lead halide perovskite is due to neglecting the SOC.37 The proper consideration of SOC results in much reduced band gaps (Table 1) in agreement with the general trend of GGA calculations. Although recent advanced calculations22,48 revealed that experimental band gap of organo-metal halide perovskites can be reproduced by including GW corrections, the orbital character of the band-edge states is not affected by consideration of GW corrections. Thus, GGA-SOC calculation can be used to analyze the band gap modification behavior,49 which can be found in this study also. The GGA-SOC calculated band gaps for various crystal structures, shown in Table 1, present notable points. Firstly, the band gap of tetragonal MAPbI3 is larger than that of trigonal FAPbI3 and the difference is well matched with experiments. The calculated band gap difference between them is 0.14 eV and 0.10 eV with and without considering SOC, respectively, while the experimental value is 0.12 eV. Secondly, the band gap of the cubic phase is lower than that of other crystal structures. The band gaps of cubic MAPbI3 and cubic FAPbI3 are smaller by 0.26 eV and 0.08 eV than those of tetragonal MAPbI3 and trigonal FAPbI3, respectively. Thirdly, there is complex relationship between the band gap and the lattice constant. The calculated band gap of tetragonal MAPbI3 is larger than that of trigonal FAPbI3 and the effective cubic lattice constant of the former is smaller by 0.13 Å than that of the latter. This band gap narrowing for larger lattice constant materials was also observed by Eperon et al.14 However, when we consider only cubic structures, the band gap of cubic MAPbI3 is smaller than that of cubic FAPbI3, even though the former has a smaller lattice constant than the latter. This puzzling dependence of the


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band gap on crystal structures illustrates the need for a fundamental understanding of what governs the size of the band gap. To untangle the complex relationship between lattice structures and band gaps and clearly understand the electronic band structure of organo-lead halide perovskites, we considered the cubic inorganic frame of PbI3, neglecting the cation molecules, as a minimal structural model. This is justified since the valence band maximum (VBM) and conduction band minimum (CBM) states come from the inorganic frame as shown below. To compensate the absence of cation molecules, an electron is added per cation molecule so that the VBM and CBM states are well defined. Recently, Even et al.50 systematically investigate the calculation method by adding or removing an electron per cation molecule for 2D perovskite compounds. The calculated band structures with and without the cation molecules are almost the same near the band edges regardless of the use of SOC (Figure S1). Because the inorganic frames of the cubic, trigonal, and tetragonal lattices (Figure 1d-1f) can be described by simple deformations from the cubic lattice, such as changing bond lengths or moving iodine atoms (Figure S2), we will discuss the electronic structure of [PbI3]- in the cubic symmetry (Figure 2a) as the basis for understanding the effect of the lattice deformations on the band gap.

The schematic orbital coupling diagram between Pb and I atoms at the R point (k = (0.5, 0.5, 0.5)) in ideal cubic PbI3 (negatively charged), obtained by orbital analysis, is shown Figure 2(c). The energy eigenvalue is obtained by plotting the whole energy eigenspectrum at the R point with GGA and GGA-SOC calculations shown in Figure S3a and S3b, respectively. Our orbital analysis by GGA calculations without SOC shows that the VBM state is nondegenerate but the CBM states are threefold degenerate locating at +1.5 eV above the VBM state. (The spin


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degeneracy is ignored when discussing non-SOC results.) The VBM state is identified as the antibonding hybridization between the Pb 6s and a bonding combination of I 5p orbitals, specifically of I1 5px, I2 5py and I3 5pz orbitals. The bonding counterpart is located at -8.2 eV. Since the I 5p orbitals have a higher energy level than Pb 6s, the VBM state has more contribution from the I 5p orbitals. The threefold degenerate CBM states are the antibonding and bonding coupling pairs between Pb 6px – I1 5s, Pb 6py – I2 5s, and Pb 6pz – I3 5s orbitals, respectively, where each of the Pb 6p orbitals couple with the 5s orbitals of different I atoms, depending on the direction of the orbital lobes. Since Pb 6p orbitals have a much higher energy level than I 5s orbitals, the CBM states have mostly the Pb 6p character, which was not shown in schematic energy level feature in a previous report23. When the spin-orbit coupling is considered, it mostly affects the CBM states. The three 6p orbitals of Pb split into P3/2 and P1/2 states with 4-fold and 2-fold degeneracy (including spin degree of freedom), respectively, with the energy level of the latter being ~1 eV lower. This atomic energy level-splitting is reflected into the CBM states, whose main contribution is from the Pb 6p orbitals. The low-energy P1/2 state defines the CBM levels (Figure S3b), reducing the band gap to ~ 0.4 eV. The VBM state, the antibonding combination of Pb 6s and I 5p orbitals, is largely intact. The SOC also gives level splitting of I 5p orbitals, but they are well below the VBM. Through the orbital coupling diagram, we find that the energy levels of VBM and CBM states are governed by the orbital interactions between Pb 6s and I 5p and between Pb 6p and I 5s, respectively. This implies that the control of orbital interaction between Pb and I could change the band gap of organo-lead iodide perovskites through the structural modification of the inorganic frame. Because the amount of the orbital interactions depends on the molecular


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structure, we investigate the impact of the atomic displacement in [PbI3]- frames and lattice constant changes on the electronic structures to understand the band gap of organo-lead iodide. And we also study the role of the cationic organic molecule on the structural and electronic band gap variation. As models for Pb-I atomic frame distortion, we considered the positional shift of I atoms. The positional deviation of I atoms from their cubic symmetry sites can be divided into two types: One is the shift along the Pb-I-Pb line (blue arrow in Figure 2a) and the other is along the normal direction to the Pb-I-Pb line (red arrow). We first consider the latter case, namely the transverse shift of I atoms and the effects of the [PbI3]- octahedron tilting can be understood by interpreting the transverse shift of I. The change of energy levels at R point by this transverse shift of I is shown in Figure 2b, calculated for a cubic [PbI3]- lattice with SOC and HSE06 correction (HSESOC). The VBM level decreases and the CBM level gets higher with the transvers shift of I1. The energy level lowering of VBM can be ascribed to the anti-bonding nature of VBM between Pb 6s and I 5p because the amount of repulsive interaction becomes weaker as I atoms undergo the transverse shift. This trend is analogous to the finding by Amat et al.29 but the understanding based on the bonding-antibonding of Pb-I frame has not been tried yet.

The conduction band increase by transverse shift of I can be understood by noting that in nonSOC calculations the CBM states are three fold degenerate (neglecting spin degeneracy) with major orbital characters of Pb 6px, 6py, and 6pz orbitals, respectively. Upon the shift of the I1 atom along the y direction, the coupling between Pb 6py and I1 5px changes from a nonbonding to an antibonding character, which is depicted in Figure 3a (before shift) and 3b (after shift). This raises the level of the CBM state of Pb 6py character. Charge density profile for the antibonding


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feature of Figure 3b is represented in Figure 3c. Similarly the CBM state of the Pb 6px character gets higher due to the coupling with the I1 5py orbital (Figure 3d). But, the CBM state of the Pb 6pz character does not experience such a change and keeps the nonbonding character. Thus, upon the transverse shift, the three-fold degeneracy of the conduction bands is broken as two of them get higher while one is almost intact (see Figure S4a). The last intact one defines the CBM edge and thus the CBM level becomes immune to the shift within the non-SOC calculations. When ௠ୀ±ଵ/ଶ

SOC is included, the CBM is defined by two Pb P1/2 states where ܲଵ/ଶ

= ∓|‫ ;ݖ‬±ۧ ±

√2|‫ ݔ‬± ݅‫ ;ݕ‬∓ۧ with the first and second arguments in the bracket representing the spatial and spin states, respectively. Because all the Pb 6px, 6py, and 6pz orbitals are locked to intermix by the spin-orbit interaction in the 6P1/2 states, both of the CBM states are affected upon the transverse shift, leading to the increase of the CBM level (see Figure S4b) and the band gap as shown in Figure 2b. Although the HSE-SOC predicted eigenvalues in Figure 2b tends to well match with real band gap of organo-lead iodide, it is well known that HSE correction is computationally expensive. Comparison of CBM and VBM produced by HSE-SOC and GGASOC (see Figure S5) shows that GGA-SOC is enough to trace the trend of the band edge changes.

The longitudinal shift of the I1 atom (blue arrow in Figure 2a) makes the bond lengths of Pb(left)-I1 and Pb(right)-I1 bonds asymmetric. While this asymmetric bond length hardly affects the VBM level, it makes the CBM level higher, leading to an increase of the band gap (Figure 4a). The dependence of the VBM and CBM levels can be understood through the orbital interactions as follows. By the longitudinal shift, the VBM states, antibonding states of Pb 6s and I 5p orbitals, experience both energetically favorable and unfavorable effects, i.e., a favorable one due to the increase of the Pb(left)-I1 bond and an unfavorable one with the decrease of the


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Pb(right)-I1 bond. On the other hand, for the CBM states, the shift of I1 along the x direction breaks the balance between the bonding and antibonding characters and brings a stronger antibonding coupling (Figure 4b). This causes the energy level of the CBM state of the Pb 6px character to elevate (see Figure S7). The bonding counterpart of the Pb 6p – I1 5p coupling is contained in I 5p states that are deep in the valence band and gets lower due to the longitudinal shift. Similarly, that of the Pb 6py character undergoes a net antibonding coupling with the I1 5py orbital (Figure 4c), though the coupling strength is weaker. The inclusion of SOC which mixes Pb 6p orbitals tightly leads to an overall substantial increase of the two degenerate CBM states as shown in Figure 4a.

Besides the shift of I from cubic symmetry, the band gap of organo-lead halide perovskites can be significantly affected by the change of lattice constant. The lattice size (or spacing) effect is analyzed with the ideal cubic structure of [PbI3]-. When 1-, 2-, and 3-D lattice strains are applied on the cubic lattice by changing the lattice constant, the relative positions of I are kept fixed. The calculated band structures are shown in Figure 5, Figure S8a, and Figure S8b. The energy levels of both CBM and VBM decrease with 1-D tensile strain, which can be easily deduced from the antibonding character of both the CBM and VBM states discussed already. 2-D and 3-D lattice strains show the same trend and only the magnitude of the change increases (see Figure S8a and Figure S8b). Since the antibonding coupling of the CBM state is rather weaker than those of the VBM state, the CBM state decreases less sensitively than the VBM state and thus the band gap increases with tensile strain or longer lattice constants. This well explains why the band gap of cubic FAPbI3 is larger than those of cubic MAPbI3, considering that the lattice constant of cubic FAPbI3 is larger.


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Combining the effects of the I shift from cubic symmetry with the contribution of lattice constant changes makes it possible to interpret above mentioned complex band gap variation behaviors for experiments and simulations. For MAPbI3, the incorporation of dipolar molecules such as MA induces the positional distortion of I atoms from the cubic symmetry sites even for the cubic perovskites. I atoms in cubic MAPbI3 show both a transverse shift of 2.0% and a longitudinal shift of 1.3% (with respect to the lattice constant) from its cubic symmetry sites (Figure S9). While the Pb 6px and I2 5py orbitals are nonbinding in the ideal cubic structure of PbI3 (see Figure 6a), they becomes antibonding in cubic MAPbI3, as shown in Figure 6b. Interestingly the longitudinal shift of I1 enhances the antibonding coupling between the Pb 6px and I2 5py orbitals by asymmetrizing the sizes of the positive and negative lobes of the Pb 6px orbital, indicating that the transverse and longitudinal shifts of I atoms work collaboratively. We also consider the effect of cation molecules on the band gap through their effect on the structural modification of the inorganic frame. The molecular incorporation changes the dependence of the band gap on the lattice constant. Under hydrostatic strain, the band gap of cubic MAPbI3 increases with tensile strain and decreases with compressive strain (Figure 7a), which is consistent with the result in cubic PbI3 (Figure 5). However, the CBM energy level shows a different dependence on strain. Under the tensile strain, the CBM level for MAPbI3 gets higher (Figure 7b), while it gets lower for PbI3 (Figure 5). This difference can be attributed to the enhanced deviation of iodine atoms from the cubic symmetry sites, to maintain the optimal distance between the MA molecules and the halide anion atoms (Figure 7c). As shown above, the deviation of I atoms from the cubic symmetry elevates the CBM energy level. This results in a much steeper increase of the band gap for tensile strained MAPbI3 (Figure 7a).


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The physical mechanisms discussed so far readily explain the puzzling dependence of the band gap on the lattice constants. The larger band gap of cubic FAPbI3 compared to cubic MAPbI3 can be attributed to the 1.1% larger lattice constant of the former. Similarly, the larger band gap of trigonal FAPbI3 over cubic FAPbI3 can be attributed to the 1.8% larger lateral lattice constant of the former. On the other hand, in spite of a smaller lattice constant, tetragonal MAPbI3 has a larger band gap than trigonal FAPbI3 because of the large transverse distortion of I atoms involved in the octahedral rotation in the tetragonal lattice. Moreover the developed model can be applied to various types of hybrid perovskites and mixed halides such as MAPbBr3 and MAPbI3-xClx. While the comparable band gap of MAPbI3-xClx to MAPbI3 was already shown by prior studies,43,51 the physical origin has not been clarified. According to the recent theoretical studies, the inorganic lattice structures are significantly affected by the polarity of cation molecules19 and the band gap can be affected by the dynamic motion of cation molecules52,53. Thus, the combined study of the dynamic behaviors of cation molecules with the present inorganic structural model will be followed.

CONCLUSION In summary, we investigate the nature of electronic band structures of organo-lead iodide perovskites with density functional theory calculations including spin-orbit coupling to devise a pathway for engineering their band gap. Based on the molecular orbital analysis and electronic band features of lead iodide perovskites, we reveal that the factor governing their band gap engineering is the orbital interaction between Pb and I through the structural modification in inorganic Pb-I frame, i.e. lattice constant, and the positional distortion of iodine from cubic


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symmetry sites. The antibonding characters of both conduction band and valence band edges and the p-orbital hybridization in Pb atoms imposed by spin orbit coupling are responsible for the unique band gap dependence on lattice deformation. Our results suggest that reducing lattice constants while keeping the cubic symmetry of the PbI3 inorganic framework is essential for the development of lead iodide perovskites having a lower band gap. This might be achieved by incorporating less-polar molecules as the cation molecules. Although this study focuses on organo-lead iodide perovskites, our understanding can be applied to other perovskites. The intense competition to develop more efficient photovoltaic cells using organo-metal halide perovskites today drives the synthesis of various kinds of alloys, and trials involving substitutions of elements. We expect that our strategy for tuning their band gap could provide useful guidance for the development and analysis of perovskite materials.

ASSOCIATED CONTENT Validation of minimal structural model through the band structure diagram. Lattice structures of organo-lead halide perovskites. Energy eigenspectrum calculated by GGA and GGA-SOC. GGA and GGA-SOC results on the effects of the shift of I. The CBM and VBM changes by HSE-SOC and GGA-SOC. Band structure changes induced by 2-D and 3-D lattice strain. Energy eigenspectrum of cubic PbI3 obtained by GGA calculations with SOC (GGA-SOC). Longitudinal and transverse shift of I atoms in the cubic of MAPbI3. This material is available free of charge via the Internet at http://pubs.acs.org

AUTHOR INFORMATION


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Corresponding Author *E-mail: [email protected] *E-mail: [email protected] Funding Sources The author(s) declare no competing financial interests

ACKNOWLEDGMENT This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A1011302) and by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2014-C2-016).


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FIGURE CAPTIONS

Figure 1. Crystal structures of organo-lead iodide perovskites. (a,d) Cubic MAPbI3 (methylammonium

lead

iodide).

(b,e)

Tetragonal

MAPbI3.

(c,f)

Trigonal

FAPbI3

(formamidinium lead iodide). For each case, unit cell and supercell are shown. H/C/N/I/Pb atoms are represented with white/light gray/light blue/violet/thick gray balls, respectively.


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Figure 2. (a) Iodine numbering and directions of iodine shift. I1/I2/I3 atoms are I atoms in Pb-IPb chains along the x/y/z direction. Blue and red arrows correspond to the longitudinal and transverse shift of I1, respectively. (b) Effects of the transverse shift of I atoms on the CBM and VBM in cubic [PbI3]-. The energy level of the valence band maximum of 0 % shifted system is set to 0. The eigenvalues are obtained by HSE-SOC calculation. (c) Schematic diagram for the coupling between Pb and I atoms in ideal cubic [PbI3]- and the effect of transverse shift of I in the cubic cell.


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Figure 3. Schematic orbital features at the R point of (a) non-bonding between Pb 6py and I1 5px when I1 was in the cubic symmetry site and (b) anti-bonding between Pb 6py and I1 5px induced by the transverse shift I1. Charge density distributions of the split CBM states of the (c) Pb 6py and (d) 6px characters when I1 is transversely shifted toward y-direction obtained through GGA.


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Figure 4. (a) Effects of the longitudinal shift of I atoms on the CBM and VBM in cubic [PbI3]-. The energy level of the valence band maximum of 0 % shifted system is set to 0. The eigenvalues are obtained by HSE-SOC calculations. Inset shows the change of the VBM and CBM levels produced by the shift. Charge density distributions of the split CBM states of the 6px (b) and 6py (c) characters calculated by GGA. The position of eigenvalues of (b) and (c) is found in the eigenspectrum plot in the Figure S6.


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Figure 5. Effects of lattice strain on the band structure of ideal cubic PbI3 obtained by GGA with considering SOC (GGA-SOC). Atomic configuration in the inset represents the unit cell transformation under 1-D tensile strain along the x-axis. Red/green/blue dots represent the eigenvalues when no strain/2% tensile strain/2% compressive strain is applied to the cubic lattice. Only eigenvalues near R point and Fermi level are plotted.


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Figure 6. Effects of cation molecules on the electronic structure in cubic MAPbI3. Charge density distributions of the CBM state representing the Pb 6px character in (a) cubic PbI3 (without the molecule) and (b) cubic MAPbI3. In (b), I2 experiences a notable transverse shift, leading to the antibonding coupling of I2 5py with Pb 6px.


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Figure 7. Effects of hydrostatic strain on the electronic and atomic structures of cubic MAPbI3 calculated by GGA-SOC. (a) Band gap variation as a function of the lattice constant. (b) CBM and VBM energy levels as a function of the lattice constant. The valence band maximum of the unstrained lattice is set to 0. (c) Transverse and longitudinal shifts of I atoms as a function of the lattice constant. The relative lattice constant, the ratio of strained lattice constant to unstrained lattice constant, represents the measure of the 3-dimensional hydrostatic strain.


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Table 1. Band gaps and lattice constants of MAPbI3 and FAPbI3. MA and FA stand for methylammonium and formamidinium, respectively. The lattice constants are obtained with GGA-SOC calculations. a Experimental band gap for MAPbI3 from Ref. 1. b,c Experimental band gap for FAPbI3 from Ref. 15 and Ref. 14, respectively MAPbI3 Phase GGA Band Gap (eV) GGA-SOC Band Gap (eV) HSE-SOC Band Gap (eV)

FAPbI3

Orthorho Exp.a Tetragonal Cubic Trigonal Cubic mbic Tetragonal 1.82

1.85

1.60

0.74

0.75

0.49

1.90

1.55a

1.75

1.71

0.61

0.53

8.86 a

9.37

6.51

Exp.b Trigonal 1.43b ~1.48c

1.53 6.44

8.99b

a (Å)

9.25

9.10

b (Å)

8.63

9.02

8.86 a

9.37

8.99b

c (Å)

12.89

12.77

12.66 a

11.01

11.01b


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REFERENCES (1)

(2)

(3) (4)

(5) (6)

(7)

(8) (9)

(10) (11) (12) (13)

(14)

(15)

(16)

Kojima, A.; Teshima, K.; Shirai, Y.; Miyasaka, T. Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells. J. Am. Chem. Soc. 2009, 131, 6050– 6051. Lee, M. M.; Teuscher, J.; Miyasaka, T.; Murakami, T. N.; Snaith, H. J. Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites. Science 2012, 338, 643–647. Liu, M.; Johnston, M. B.; Snaith, H. J. Efficient Planar Heterojunction Perovskite Solar Cells by Vapour Deposition. Nature 2014, 501, 395–398. Kim, H.-S.; Lee, C.-R.; Im, J.-H.; Lee, K.-B.; Moehl, T.; Marchioro, A.; Moon, S.-J.; Humphry-Baker, R.; Yum, J.-H.; Moser, J. E.; et al. Lead Iodide Perovskite Sensitized All-Solid-State Submicron Thin Film Mesoscopic Solar Cell with Efficiency Exceeding 9%. Sci. Rep. 2012, 2. Eperon, G. E.; Burlakov, V. M.; Goriely, A.; Snaith, H. J. Neutral Color Semitransparent Microstructured Perovskite Solar Cells. ACS Nano 2014, 8, 591–598. You, J.; Hong, Z.; Yang, Y. M.; Chen, Q.; Cai, M.; Song, T.-B.; Chen, C.-C.; Lu, S.; Liu, Y.; Zhou, H.; et al. Low-Temperature Solution-Processed Perovskite Solar Cells with High Efficiency and Flexibility. ACS Nano 2014, 8, 1674–1680. Li, Z.; Kulkarni, S. A.; Boix, P. P.; Shi, E.; Cao, A.; Fu, K.; Batabyal, S. K.; Zhang, J.; Xiong, Q.; Wong, L. H.; et al. Laminated Carbon Nanotube Networks for Metal Electrode-Free Efficient Perovskite Solar Cells. ACS Nano 2014, 8, 6797–6804. Green, M. A.; Ho-Baillie, A.; Snaith, H. J. The Emergence of Perovskite Solar Cells. Nature Photon. 2014, 8, 506–514. Zhou, H.; Chen, Q.; Li, G.; Luo, S.; Song, T. B.; Duan, H. S.; Hong, Z.; You, J.; Liu, Y.; Yang, Y. Interface Engineering of Highly Efficient Perovskite Solar Cells. Science 2014, 345, 542–546. Park, N.-G. Organometal Perovskite Light Absorbers Toward a 20% Efficiency LowCost Solid-State Mesoscopic Solar Cell. J. Phys. Chem. Lett. 2013, 4, 2423–2429. Snaith, H. J. Perovskites: the Emergence of a New Era for Low-Cost, High-Efficiency Solar Cells. J. Phys. Chem. Lett. 2013, 4, 3623–3630. Gao, P.; Grätzel, M.; Nazeeruddin, M. K. Organohalide Lead Perovskites for Photovoltaic Applications. Energy Environ. Sci. 2014. Boix, P. P.; Nonomura, K.; Mathews, N.; Mhaisalkar, S. G. Current Progress and Future Perspectives for Organic/Inorganic Perovskite Solar Cells. Mater. Today 2014, 17, 16– 23. Eperon, G. E.; Stranks, S. D.; Menelaou, C.; Johnston, M. B.; Herz, L. M.; Snaith, H. J. Formamidinium Lead Trihalide: a Broadly Tunable Perovskite for Efficient Planar Heterojunction Solar Cells. Energy Environ. Sci. 2014, 7, 982–988. Pang, S.; Hu, H.; Zhang, J.; Lv, S.; Yu, Y.; Wei, F.; Qin, T.; Xu, H.; Liu, Z.; Cui, G. NH2CH═NH2PbI3: an Alternative Organolead Iodide Perovskite Sensitizer for Mesoscopic Solar Cells. Chem. Mater. 2014, 26, 1485–1491. Lee, J.-W.; Seol, D.-J.; Cho, A.-N.; Park, N.-G. High-Efficiency Perovskite Solar Cells


25


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(17)

(18) (19)

(20) (21)

(22)

(23) (24) (25) (26) (27)

(28)

(29)

(30)

(31) (32)

(33) (34)

Page 26 of 28

Based on the Black Polymorph of HC(NH2)2PbI3. Adv. Mater. 2014, 26, 4991–4998. Giorgi, G.; Fujisawa, J.-I.; Segawa, H.; Yamashita, K. Small Photocarrier Effective Masses Featuring Ambipolar Transport in Methylammonium Lead Iodide Perovskite: a Density Functional Analysis. J. Phys. Chem. Lett. 2013, 4, 4213–4216. De Angelis, F. Modeling Materials and Processes in Hybrid/Organic Photovoltaics: From Dye-Sensitized to Perovskite Solar Cells. Acc. Chem. Res. 2014, 11, 3349-3360. Frost, J. M.; Butler, K. T.; Brivio, F.; Hendon, C. H.; van Schilfgaarde, M.; Walsh, A. Atomistic Origins of High-Performance in Hybrid Halide Perovskite Solar Cells. Nano Lett. 2014, 14, 2584–2590. Kim, J.; Lee, S.-H.; Lee, J. H.; Hong, K.-H. The Role of Intrinsic Defects in Methylammonium Lead Iodide Perovskite. J. Phys. Chem. Lett. 2014, 5, 1312–1317. Chang, Y. H.; Park, C. H.; Matsuishi, K. First-Principles Study of the Structural and the Electronic Properties of the Lead-Halide-Based Inorganic-Organic Perovskites (CH3NH3)PbX3 and CsPbX3 (X = Cl, Br, I) J. Korean Phys. Soc. 2004, 44, 889–893. Brivio, F.; Butler, K. T.; Walsh, A.; van Schilfgaarde, M. Relativistic Quasiparticle SelfConsistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers. Phys. Rev. B 2014, 89, 155204. Umebayashi, T.; Asai, K.; Kondo, T.; Nakao, A. Electronic Structures of Lead Iodide Based Low-Dimensional Crystals. Phys. Rev. B 2003, 67, 155405. Feng, J. Mechanical Properties of Hybrid Organic-Inorganic CH3NH3BX3 (B = Sn, Pb; X = Br, I) Perovskites for Solar Cell Absorbers. APL Mater. 2014, 2, 081801. Hong, K.-H.; Kim, J.; Lee, S.-H.; Shin, J. K. Strain-Driven Electronic Band Structure Modulation of Si Nanowires. Nano Lett. 2008, 8, 1335–1340. Kim, J.; Lee, J. H.; Hong, K.-H. A Pathway to Type-I Band Alignment in Ge/Si Core– Shell Nanowires. J. Phys. Chem. Lett. 2013, 4, 121–126. Wang, Y.; Gould, T.; Dobson, J. F.; Zhang, H.; Yang, H.; Yao, X.; Zhao, H. Density Functional Theory Analysis of Structural and Electronic Properties of Orthorhombic Perovskite CH3NH3PbI3. Phys. Chem. Chem. Phys. 2013, 16, 1424–1429. Giorgi, G.; Fujisawa, J.-I.; Segawa, H.; Yamashita, K. Cation Role in Structural and Electronic Properties of 3D Organic–Inorganic Halide Perovskites: a DFT Analysis. J. Phys. Chem. C 2014, 118, 12176–12183. Amat, A.; Mosconi, E.; Ronca, E.; Quarti, C.; Umari, P.; Nazeeruddin, M. K.; Grätzel, M.; De Angelis, F. Cation-Induced Band-Gap Tuning in Organohalide Perovskites: Interplay of Spin–Orbit Coupling and Octahedra Tilting. Nano Lett. 2014, 14, 3608– 3616. Lang, L.; Yang, J.-H.; Liu, H.-R.; Xiang, H. J.; Gong, X. G. First-Principles Study on the Electronic and Optical Properties of Cubic ABX3 Halide Perovskites. Phys. Lett. A 2014, 378, 290–293. Jin, H.; Im, J.; Freeman, A. J. Topological Insulator Phase in Halide Perovskite Structures. Phys. Rev. B 2012, 86, 121102. Kim, M.; Im, J.; Freeman, A. J.; Ihm, J.; Jin, H. Switchable S = 1/2 and J = 1/2 Rashba Bands in Ferroelectric Halide Perovskites. Proc. Natl. Acad. Sci. U.S.A. 2014, 111, 6900–6904. Kresse, G.; Hafner, J. Ab Initio Molecular Dynamics for Liquid Metals. Phys. Rev. B 1993, 47, 558. Kresse, G.; Furthmüller, J. Efficiency of Ab-Initio Total Energy Calculations for Metals


26

Page 27 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(35) (36) (37)

(38) (39) (40) (41)

(42)

(43)

(44) (45)

(46)

(47) (48) (49)

(50) (51)

(52)

and Semiconductors Using a Plane-Wave Basis Set. Comp. Mater. Sci. 1996, 6, 15–50. Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953–17979. Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868. Even, J.; Pedesseau, L.; Jancu, J.-M.; Katan, C. Importance of Spin–Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications. J. Phys. Chem. Lett. 2013, 4, 2999–3005. Amato, M.; Ossicini, S.; Rurali, R. Band-Offset Driven Efficiency of the Doping of SiGe Core−Shell Nanowires. Nano Lett. 2011, 11, 594–598. Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207–8215. Ihm, J.; Zunger, A.; Cohen, M. L. Momentum-space formalism for the total energy of solids. J. Phys. C: Solid State Phys. 1979, 12 4409-4422. Stoumpos, C. C.; Malliakas, C. D.; Kanatzidis, M. G. Semiconducting Tin and Lead Iodide Perovskites with Organic Cations: Phase Transitions, High Mobilities, and NearInfrared Photoluminescent Properties. Inorg. Chem. 2013, 52, 9019–9038. Baikie, T.; Fang, Y.; Kadro, J. M.; Schreyer, M.; Wei, F.; Mhaisalkar, S. G.; Graetzel, M.; White, T. J. Synthesis and Crystal Chemistry of the Hybrid Perovskite (CH3NH3)PbI3 for Solid-State Sensitised Solar Cell Applications. J. Mater. Chem. A 2013, 1, 5628. Mosconi, E.; Amat, A.; Nazeeruddin, M. K.; Grätzel, M.; De Angelis, F. First-Principles Modeling of Mixed Halide Organometal Perovskites for Photovoltaic Applications. J. Phys. Chem. C 2013, 117, 13902–13913. Yin, W.-J.; Yang, J.-H.; Kang, J.; Yan, Y.; Wei, S.-H. Halide Perovskite Materials for Solar Cells: a Theoretical Review. J. Mater. Chem. A 2015, doi:10.1039/c4ta05033a. Egger, D. A.; Kronik, L. Role of Dispersive Interactions in Determining Structural Properties of Organic–Inorganic Halide Perovskites: Insights From First-Principles Calculations. J. Phys. Chem. Lett. 2014, 5, 2728–2733. Feng, J.; Xiao, B. Crystal Structures, Optical Properties, and Effective Mass Tensors of CH3NH3PbX3(X = I and Br) Phases Predicted From HSE06. J. Phys. Chem. Lett. 2014, 5, 1278–1282. Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Challenges for Density Functional Theory. Chem. Rev. 2011, 112, 289–320. Umari, P.; Mosconi, E.; De Angelis, F. Relativistic GW Calculations on CH3NH3PbI3 and CH3NH3SnI3 Perovskites for Solar Cell Applications. Sci. Rep. 2014, 4, 4467. Grote, C.; Ehrlich, B.; Berger, R. F. Tuning the Near-Gap Electronic Structure of TinHalide and Lead-Halide Perovskites via Changes in Atomic Layering. Phys. Rev. B 2014, 90, 205202. Even, J.; Pedesseau, L.; Katan, C. Understanding Quantum Confinement of Charge Carriers in Layered 2D Hybrid Perovskites. ChemPhysChem 2014, 15, 3733–3741. Colella, S.; Mosconi, E.; Fedeli, P.; Listorti, A.; Gazza, F.; Orlandi, F.; Ferro, P.; Besagni, T.; Rizzo, A.; Calestani, G.; et al. MAPbI3-xClx Mixed Halide Perovskite for Hybrid Solar Cells: the Role of Chloride as Dopant on the Transport and Structural Properties. Chem. Mater. 2013, 25, 4613–4618. Quarti, C.; Mosconi, E.; De Angelis, F. Interplay of Orientational Order and Electronic Structure in Methylammonium Lead Iodide: Implications for Solar Cell Operation.


27


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(53)

Page 28 of 28

Chem. Mater. 2014, 26, 6557–6569. Carignano, M. A.; Kachmar, A.; Hutter, J. Thermal Effects on CH3NH3PbI3 Perovskite From Ab-Initio Molecular Dynamics Simulations. 2014, http://arxiv.org/abs/1409.6842.

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Importance of Orbital Interactions in Determining Electronic Band

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