Abstract
Organolead halide perovskite solar cells represent a revolutionary shift in solar photovoltaics, introducing relatively soft defect containing semiconductors as materials with excellent charge collection for both electrons and holes. Although they are based on the nominally simple cubic perovskite structure, these compounds are in fact very complex. For example, in (CH_{3}NH_{3})PbI_{3} the dynamics and ensuing structural fluctuations associated with the (CH_{3}NH_{3})^{+} ions and the interplay with the electronic properties are still not fully understood, despite extensive study. Here, using abinitio calculations, we show that at room and higher temperature, the rotation of CH_{3}NH_{3} molecules can be viewed as effectively giving local structures that are cubic and tetragonal like from the point of view of the PbI_{3} framework, though in fact having lower symmetry. Both of these structures are locally polar, with sizable polarization, ~10 μC/cm^{2} due to the dipoles on the organic. They become energetically degenerate in the volume range, V ~ 250 Å^{3}/f.u–265 Å^{3}/f.u. We also find very significant dependence of the band gap on the local structure. This type of transition is analogous to a transition between two ferroelectric structures, where inspite of strong electron phonon coupling, there is strong screening of charged defects which can lead to enhanced mobility and charge collection. The results provide insights into the enhanced light absorption near the band edge and good charge collection in this material.
Introduction
The organicinorganic hybrid perovskites (OIHP) have emerged as an important new class of photovoltaics, exemplified by methylammonium lead iodide CH_{3}NH_{3}PbI_{3} (MAPbI_{3}), Laboratory devices based on these have reported efficiency exceeding 21%^{1} which is comparable to or even higher than the performance of existing solar cell technologies. The pace of research and ensuing progress is also impressive since first report^{2} on perovskite photovoltaic whose progress has outshined those of other solar cell types in photovoltaic research. Many efforts have made to increase the efficiency of OIHP by chemically and structurally adjusting the band gap and other properties^{3,4}. In addition, the specific chemical bonding of divalent group IV elements with halides has been invoked as a possible explanation for the excellent charge collection in these materials. Specifically, the defect tolerance has been associated with high dielectric constants, and therefore defect tolerance^{4}, as discussed in terms of high Born charges on the Pb and halogen atoms in these and related halides^{5,6,7}. There is therefore a strong interplay between the lattice structure and bonding of the PbI_{3} part of the unit cell and the charge collection. This is complicated by the symmetry lowering but presumably electronically inactive organic cation on the perovskite Asite.
In addition to the dielectric properties, important for charge collection in these defected materials, light absorption beginning with optimized band gaps is crucial. The band gap can be tuned in various ways, but it is important that the successful approaches are ones that give not only optimal band gaps, but good absorption near the band edge, without degradation of the charge collection^{7,8,9}. Rashba band splitting due to the spin  orbit coupling has been suggested as a possible cause for the reduced recombination rate^{10,11,12}. For the I4/mcm structure, Zheng et al.^{10} have shown that the spin forbidden transition between conduction band and valence band reduced the recombination rate; a similar phenomenon has also been unveiled by Etienne et al.^{11} who studied the dynamical origin of the Rashba effect. Experimentally, Wang et al., proved that a weak indirect band gap presents in MPbI_{3} as a result of Rashba splitting of the conduction band due to the spinorbit coupling^{12}. However, it is to be emphasized that the Rashba splitting depends on symmetry lowering, closely related to polar order.
The NH_{3}CH_{3} (MA) molecules have been theoretically shown to play important roles in MAPbI_{3}. Theoretical reports by Ong et al.^{13,14} revealed how the rotation of molecule CH_{3}NH_{3} couples to the PbI_{3} host leading to effective structural phase changes, at least locally. By using the Van der Waals corrected Density Functional Theory, Motta et al.^{15} revealed that the rotation of orientation of CH_{3}NH_{3} molecule in cubic MAPbI_{3} from [111] direction to [110] direction distorts the PbI_{6} octahedral cage and results in indirect band gap. Such an effect has been also investigated by Gao, et al.^{16}. On the opposite, under the hydrostatic pressure, Wang et al.^{12} reported that the Rashba splitting is reduced due to a pressure induced reduction in local electric field around the Pb atom. The role of molecule rotation is excluded in that report. Experimentally at finite temperature, the CH_{3}NH_{3} molecule is reported to orient randomly^{17,18,19}, giving a net overall cubic centrosymmetric state. In the cubic phase, the nuclear magnetic resonance (NMR)^{20} showed that the MA cations reorient themselves with picosecond scale dynamics at high temperature but freeze at low temperatures. Importantly, it has been shown that that there are sizable effects on the optical properties associated with the dynamics of the MA from experiments. In particular, Quarti and coworkers^{21} does not abruptly change at the cubictetragonal phase transition, but gradually changes from 270–400 K, in spite of the phase transition at 327 K^{22}. The temperature dynamical correlations between MA molecules near the phase transition remain to be fully elucidated, but near the transition it is likely that there is substantial local correlation between the MA orientations, and therefore on the time scale of optical and electronic processes it is likely that the behavior is influenced by local structural effects associated with this dynamics. Therefore, the influence of the orientation of molecule to the properties of hybrid perovskite still needs further investigation. In the present work we examine the magnitudes of local structural effects on the electronic and optical properties, using first principles calculations for ordered structures (see Methods).
Considering the PbI_{3} part of the structure (for the discussion, but not in the calculations, which necessarily include the full atomic structure, including the dipoles associated with the organic Asite, we describe the structure of the PbI_{3} with space groups that describe to a close approximation this part of the cell), the tetragonal phase I4mcm is stable only at V < 252 Å/f.u theoretically^{14}, at higher volume the cubic Pm3m and tetragonal P4mm phase are more stable. The tetragonal P4mm is obtained from cubic Pm3m by deforming only one lattice constant along the caxis, c ≠ a) and the tetragonal I4/mcm is a further deformation from P4mm by the rotation of octahedral PbI_{6} antiphase tilt around the caxis. Since solar cell is working at room temperature and higher, therefore our research mainly focuses on Pm3m and P4mm structures. In this report, we study the impact of MA molecules to the efficiency of hybrid perovskite based solar cell. Our results show that the MA units easily rotate in PbI_{6} cuboctahedral, consistent with experiment, and these rotations are coupled significantly to the PbI_{3} lattice inducing structural changes from cubic Pm3m to tetragonal P4mm and vice versa. This causes a momentum dependent splitting of energy band by Rashba effect due to the spinorbit coupling, prevents the electron recombination, and induces a multi band gap electronic structure.
Results and Discussion
To study the impact of molecule rotation on the properties of MAPbI_{3}, we study the influence of volume change to the rotation. We start with the most stable structure of P4mm with a = b = 6.32 Å, c = 6.31 Å^{14} and volume of V = 252 Å^{3}, which is close to experimental data a = b = 6.312 Å, c = 6.316 Å, and V = 251.6 Å^{3} ^{17}. The orientation of MA molecule is studied at three different directions: quasi[111], quasi [001] and quasi[110]. For the P4mm, at each of specific lattice constant a = b and specific orientation of MA molecule the lattice constant c is fully relaxed. Our study shows that the orientation of MA molecule along the quasi[001] direction gives almost the same energy as the quasi[110] direction, see Fig. 1.
The energyvolume phase diagram of structures P4mm and Pm3m (for the PbI_{3} part of the cell) is given in Fig. 1. The result shows that at the same volume there is no preferred direction for MA molecules in tetragonal P4mm. On the other hand, the [110] and [001] directions are preferred directions for MA molecule in Pm3m phase. We notice here that the P4mm structure was reported to be more stable than the Pm3m (MA molecule lies along the [111] direction) with V ~ 250 Å^{3}/f.u–260 Å^{3}/f.u ^{14}. We find that a rotation out of [111] direction of MA molecule further stabilizes the Pm3m structure but the P4mm is still more stable than the Pm3m structure (V < 256 Å^{3}/f.u) with very small energy difference of 1–2 meV. Thus the preference for the MA orientation away from [111] plays an important role in selecting the structure.
Experimentally, the P4mm and Pm3m structures are stable at room and higher temperature. On the other hand, our result at V < 265 Å^{3}/f.u reveals that energy difference between P4mm and Pm3m phases with different molecule orientations [111], [110], [001] is at maximum of 25 meV or equivalent to ~300 K. Such small energy differences easily allow the rotation of MA molecules in the cuboctahedral PbI_{6} by thermal excitation as is known. It is interesting to explore how the rotation of MA molecules will affect the properties of MAPbI_{3}. Let start from cubic structure Pm3m with a = b = c = 6.32 Å and MA molecule oriented along the [111] direction, point \({1}_{111}^{c}\), see Fig. 1. In principal, from the [111] direction MA molecule can rotate to any directions. To ease the study, we select two most symmetry directions: [110] and [001]. Our study shows that for the Pm3m structure, the rotation of MA molecule from [111] direction to [110] or [001] will further stabilize the Pm3m structure. It is noticed here that both Pm3m structures with MA molecule oriented along [110] and [001] direction give almost the same total energy, E^{C}[110] ~ E^{C}[001], therefore we report only [110] direction. The conclusions for [001] case are the same for [110] direction. The rotation from point \({1}_{111}^{c}\) to point \({1}_{110}^{c}\) results in energy difference E(\({1}_{110}^{c}\)) − E(\({1}_{111}^{c}\)) = −16 meV ~ 186 °K. Our results show that at point \({1}_{110}^{c}\) the MAPbI_{3} structure is further stabilized from cubic Pm3m to tetragonal P4mm structure by changing the lattice cconstant from 6.32 Å to 6.26 Å, point \({1}_{110}^{t}\), see Fig. 1, with E(\({1}_{110}^{t}\)) −E(\({1}_{110}^{c}\)) = −2 meV ~ 23 ^{°}K. If the MA molecule is in [001] direction then the Pm3m structure is further stabilized to P4mm structure by reducing the lattice constant from 6.32 Å to 6.30 Å, point \({1}_{001}^{t}\) with energy difference E(\({1}_{001}^{t}\)) −E(\({1}_{001}^{c}\)) = − 1meV ~ 11.6 ^{°}K. In short, the rotation of MA molecule in different directions results in small energy differences in comparison to thermal energy at room temperature. Therefore MA molecules rotate in the cuboctahedral PbI_{3} under the thermal excitation near room temperature consistent with experiments. As discussed, such rotations are coupled to strain including volume, creating a breathing of the MAPbI_{3}. In general, this type of coupling provides a local strain coupling that favors formatiion of regions or clusters of like orientation, and in the case of phase transitions, favors first order character with coexistence.
It is interesting to further explore the effect of such rotation on the electronic properties of MAPbI_{3}. To investigate this effect we use the TranBlaha Modified BeckeJohnson (TBMBJ) potential in general potential linearized augmented planewave (LAPW) method as implemented in the WIEN2k code^{23,24} to improve the value of band gap, which is under estimated by using the PBE^{25} or PBEsol^{26} GGA calculations. The TBMBJ has been proved to give good band gap in comparison to GW method for s and p –electron systems, which is applicable to MAPbI_{3} ^{23}. Our results, see Fig. 1, show that the [110] direction is the most stable direction for MA molecule in both Pm3m and P4mm structure. The rotation of MA molecule causes a large difference in band gap and in general MAPbI_{3} gets (i) highest band gap with the tetragonal P4mm symmetry and MA molecule in [110] direction and (ii) lowest band gap with cubic Pm3m symmetry and MA molecule in [111]. Figure 1 shows that at V < 265 Å^{3}/f.u the maximum energy difference between phases is about 22 meV~255 ^{0}K which means that MA molecules are easy to rotate by thermal excitation at room temperature and higher. Such rotation not only changes the structure of MAPbI_{3} but also the energy band gap such as, at V = 252 Å^{3}/f.u the rotation of MA molecule from [110] direction to [001] direction does not change much the energy band gap of MAPbI_{3}, E_{gap} ~ 1.22 eV, while the rotation from [110] direction to [111] direction reduces the energy band gap to E_{gap} = 1.02 eV, 0.2 eV smaller. At higher volume the rotation of MA molecule results in larger band gap difference, larger shrinkage and expansion of volume. For example at V = 267 Å^{3}/f.u the energy band gap of MAPbI_{3} is 1.2 eV in Pm3m with molecule oriented in [111] direction. The rotation of molecule from [111] (E_{gap} ([111]) = 1.02 eV) to [110] results in band gap of 1.32 eV. When MA molecule is in [110] direction, a further deformation from Pm3m structure to P4mm by reducing the lattice constant c results in volume reducing from V~ 267 Å^{3}/f.u to V~257 Å^{3}/f.u and the band gap gets value of 1.4 eV, see Fig. 2.
At lower volume V < 250 Å^{3}/f.u the rotation of MA molecules in cuboctahedral PbI_{3} will transform the MAPbI_{3} from tetragonal P4mm to tetragonal I4/mcm^{14}. Our calculations show that the transition results in big jump of energy band gap, such as at V = 252 Å^{3}/f.u the P4mm has band gap of 1.22 eV while the I4/mcm at V = 251.02 Å^{3}/f.u has energy band gap of 1.55 eV which is in agreement with experimental report^{1}. Therefore at V ~ 251–252 Å^{3}/f.u, the structure is sensitive with structural change and may coexit three phases Pm3m, P4mm and I4/mcm due to the rotation of molecules under thermal excitation. At lower volume V < 250 Å^{3}/f.u the MAPbI_{3} is stabilized by I4/mcm structure due to larger energy difference between I4/mcm and P4mm phase and low thermal excitation energy.
The rotation of MA molecule varies the energy band gap of MAPbI_{3} with an average amount of 0.2 eV down to the redshifted spectra, which help to enhance the efficiency of solar absorber. It is more insightful to know how such rotation will affect the band structure of MAPbI_{3}. The band structures of MAPbI_{3} at points \({1}_{111}^{c}\), \({1}_{110}^{c}\),(a = b = c = 6.32 Å), \({1}_{110}^{t}\) (a = b = 6.32 Å c = 6.26 Å) and \({1}_{001}^{t}\) (a = b = 6.32 Å, c = 6.30 Å) are given in Fig. 3. The results show that for cubic structure when the MA molecule is in the [111] direction then the energy band gap is direct band gap at A[½ ½ ½]. The rotation of MA molecule out of the [111] direction results in indirect bandgaps and higher energy band gap for both Pm3m and P4mm structure, see Fig. 3. The rotation of molecules therefore not only changes the crystal structure, tunes the band gap of MAPbI_{3} but also tunes the nature of the band gap from direct to indirect and vice versa.
The nature of the band gap (indirect vs. direct) comes from the momentum dependent splitting of energy bands by Rashba effect due to the spinorbit coupling. This effect has been mainly studied for the I4/mcm structure and much less studied for cubic Pm3m and tetragonal P4mm. In principle the spinorbit coupling causes the energy band splitting in cubic Pm3m and tetragonal P4mm symmetry but there is no momentum dependent splitting of the energy band, the Rashba effect. The momentum dependent splitting of energy bands that occurs in these structures (Fig. 3) is due to the rotation of MA molecules in the PbI_{3} cuboctahedral. This rotation lowers the symmetry. However, the fast rotation of MA molecules in PbI_{3} cuboctahedral at room and higher temperature in a time average would make the MA molecule behave as a point like particle (Fig. S1, supporting information). The effect of spinorbit coupling on the momentum dependent splitting of energy band due to the rotation of MA molecules in PbI_{3} cuboctahedral is reported in supporting information. The study reveals a strong dependence of Rashba interaction coefficient on the volume and the orientation of MA molecule in PbI_{3} cuboctahedral_{.}
Since the rotational dynamics of the MA molecules may be correlated through strain coupling, [11] it may be useful to consider a new band structure, which is a combination of band structures of MAPbI_{3} with MA molecules in different directions. There are four structures in our cases: Pm3m structure with MA molecule in [111] and [110]/[001] direction, P4mm structure with MA molecule in [110] and [001] direction, see Fig. 4. This is germane to the case where the correlation length for MA rotation is significantly longer than the unit cell dimension, so that local band structures are important. The new energy band structure has features of good solar absorber materials with (1) multi energy bandgaps and (2) indirect energy bandgaps. The most extension of energy band gap to the lower value comes from the rotation of MA molecule to the [111] direction. The multi energy band gap structure (1) allows more photons with different wavelengths are absorbed. The indirect energy band gap (2) prevents the recombination of electrons in the conduction band with holes in the valence band. This mechanism enhances the electron density in the conduction band of MAPbI_{3} and their lifetime.
The rotation of MA molecule in the cuboctaheral PbI_{3} causes an electronic polarization. The momentum dependent splitting of energy band only appears in case of anisotropy. To investigate this effect, we calculate the volume evolution of electronic polarization when the MA molecules are in different orientations. The results are shown in Fig. 5. The result clearly shows that for cubic structure when the MA molecule is in [111] direction then the polarizations in three different directions are almost the same. Such effect results in a very weak energy band splitting as shown in Fig. 3a. When MA molecule is in [110] or [001] direction, the electronic polarizations in different directions are very much different, see Fig. 5(b–d), causing a strong energy band splitting as shown in Fig. 3(b–d).
The rotation of MA molecule changes the lattice constants continuously in three directions from a to aδ and vice versa with δ is the lattice constant difference between cubic Pm3m and tetragonal P4mm when MA molecule is in the [110] direction. Since we are studying the MAPbI_{3} by using density functional theory i.e., one unit cell with periodic conditions, therefore this phenomenon is only valid when all molecules rotate at the same direction. In general, without any external constrain, at the same time all MA molecules in MAPbI_{3} rotate nonorientation. Such free rotation induces a continuous change of PbI_{3} cuboctahedral, shrink and expand in all directions. Therefore the lattice constant may be effectively considered as the quasicubic Pm3m structure or quasitetragonal P4mm structure with lattice constants a = b~c.
Conclusions
In summary, we have studied the impact of correlated orientations of MA molecules on the evolution of crystal structures, energy band structures and energy band splitting of MAPbI_{3} focusing on Pm3m and P4mm structure. The results showed that at V = 250 Å^{3}/f.u three structures I4/mcm, P4mm and Pm3m coexist. The energy band gap of I4/mcm structure is 1.55 eV which is in perfect agreement with experimental report but it is about 1.2 eV for Pm3m and P4mm structure when MA molecule is in the [001] and [110] direction. When MA molecule is in the [111] direction, the MAPbI_{3} is stabilized with Pm3m structure with energy band gap of 1.02 eV. At higher volume the P4mm and Pm3m is more stable than I4/mcm. Although the MAPbI_{3} is stable with Pm3m structure and MA molecule is in [110] direction, the rotations of MA molecules induce the change of local lattice constants. Therefore MAPbI_{3} can exist in both Pm3m and P4mm structures. The volume evolution of band structure shows an increasing trend of energy band gap with volume. The rotation of MA molecule defines the direct and indirect nature of energy band gap with an extension 0.2 eV of energy band gap down to redshifted region when MA molecule is in [111] direction. A new band structure has been proposed with new features (1) multi energy bandgaps and (2) indirect energy bandgaps. Study on the spinorbit induced band splitting effect shows strong dependence of Rashba interaction coefficient on the volume and the orientation of MA molecule in cuboctahedral PbI_{3}.
Methods
For the calculations of electronic structures and related properties of MAPbI_{3} we use the projector augmented wave (PAW) method^{27} with the PerdewBurkeErnzerhof (PBE)^{25} and the PBE revised for solids (PBEsol)^{26} generalized gradient approximation (GGA) exchange correlation potentials as implemented in the VASP code^{28}. The cutoff energy for the plane wave expansion of the wave functions is 500 eV, and all atoms in the unit cell are fully relaxed till the HellmanFeynman forces are less than 0.005 eV/Å. The 6 × 6 × 6 MonkhorstPack grid of kpoints^{29} for Brillouin zone integration was used in calculations for Pm3m and P4mm structures. The semicore states of the Pb atoms are treated as valence electrons; i.e., 14 valence electrons for Pb (5d^{10}6s^{2} 6p^{2}). The I5s^{2}5p^{5}, C2s^{2}2p^{2}, N2s^{2}2p^{3} and H1s were considered as valence electrons.
The symmetry of Pm3m and P4mm is built based on the PbI_{3} frame without the presence of molecule CH_{3}NH_{3}. After the frame is built then the molecule CH_{3}NH_{3} is added to the center of the PbI_{3} frame at different orientations [001], [110] and [111]. At each of specific orientation the MAPbI_{3} crystal structure is fully relaxed without any constrain on the symmetry. The obtained structures are therefore at P1 symmetry in general due to the presence of CH_{3}NH_{3} molecule and they are very close to the Pm3m or P4mm symmetry. Therefore they are called a pseudocubic Pm3m or a pseudotetragonal P4mm.
We applied the Van der Waals correction force (vdWDF2 or D2)^{30}, which is proved to be the best in comparison the lattice constant of CH_{3}NH_{3}PbI_{3} with experimental data^{31}, in our calculation and find that the PBEsol and PBE + vdWDF2 give the same results as reported by Menendex, ref.^{31}. On the other hand, our results based on the vdWD3 correction method by Grimme et al.^{32}, which is reported by Thind et al.^{33}, underestimate the lattice constant of the cubic structure Pm3m in comparison to experimental data, see the table S1. Because of this we preferred to use PBEsol method instead of PBEvdWD3.
To calculate the energy band structure of MAPbI_{3} we use the WIEN2k software package^{23}. This program allows to compute the electronic structure of MAPbI3 within DFT utilizing the full potential (linear) augmented plane wave + local orbitals (APW + lo) method and applying the MBJ method^{24}. The atomic sphere radii of MAPbI_{3} are chosen as 2.5 a.u for Pb and I; 1.28 a.u for N, 1.34 a.u for C and 0.69 a.u for H. Inside the atomic spheres, the partial waves were expanded up to l_{max} = 10 and the number of plane waves was limited by a cut off K_{max} = 4.64 (a.u^{−1}). The charge density was Fourierexpanded with G_{max} = 20 Ry. A kmesh of 10 × 10 × 10 in the full Brillouin zone was used. In addition to the usual valence states, also extra local orbitals for “semicore” states (Pb 5b, 5d, 6 s, 6p; I4d, 5 s, 5p; N: 2p, and C:2p) were added and considered as band states.
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Acknowledgements
This work was supported by Institute of High Performance Computing, Agency of Science, Technology, And Research (A*STAR). Work at Nanyang Technological University (NTU) was supported by Singapore Ministry of Education through AcRF Tier1 grant MOE2017T1002142.
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Theoretical calculations were carried out by K.P.O., S.W., Z.F. and M.B.S. All authors contributed to interpretation of the results. The manuscript was prepared by K.P.O., S.W., D.J.S., Z.F., T.H.N., M.S. and C.D.
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Ong, K.P., Wu, S., Nguyen, T.H. et al. Multi Band Gap Electronic Structure in CH_{3}NH_{3}PbI_{3}. Sci Rep 9, 2144 (2019). https://doi.org/10.1038/s41598018380232
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