Multi Band Gap Electronic Structure in CH3NH3PbI3

Organo-lead 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 (CH3NH3)PbI3 the dynamics and ensuing structural fluctuations associated with the (CH3NH3)+ ions and the interplay with the electronic properties are still not fully understood, despite extensive study. Here, using ab-initio calculations, we show that at room and higher temperature, the rotation of CH3NH3 molecules can be viewed as effectively giving local structures that are cubic and tetragonal like from the point of view of the PbI3 framework, though in fact having lower symmetry. Both of these structures are locally polar, with sizable polarization, ~10 μC/cm2 due to the dipoles on the organic. They become energetically degenerate in the volume range, V ~ 250 Å3/f.u–265 Å3/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 in-spite 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.

The organic-inorganic 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 A-site.
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 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 co-workers 21 does not abruptly change at the cubic-tetragonal 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 A-site, 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 Pm-3m and tetragonal P4mm phase are more stable. The tetragonal P4mm is obtained from cubic Pm-3m by deforming only one lattice constant along the c-axis, c ≠ a) and the tetragonal I4/mcm is a further deformation from P4mm by the rotation of octahedral PbI 6 anti-phase tilt around the c-axis. Since solar cell is working at room temperature and higher, therefore our research mainly focuses on Pm-3m 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 Pm-3m to tetragonal P4mm and vice versa. This causes a momentum dependent splitting of energy band by Rashba effect due to the spin-orbit 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 energy-volume phase diagram of structures P4mm and Pm-3m (for the PbI 3 part of the cell) is given in Fig. 1 direction then the Pm-3m structure is further stabilized to P4mm structure by reducing the lattice constant from 6.32 Å to 6.30 Å, point 1 t 001 with energy difference E(1 t 001 ) −E(1 c 001 ) = − 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 co-existence.
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 Tran-Blaha Modified Becke-Johnson (TB-MBJ) 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 TB-MBJ 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 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  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 co-exit three phases Pm-3m, 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 c 111 , 1 c 110 ,(a = b = c = 6.32 Å), 1 t 110 (a = b = 6.32 Å c = 6.26 Å) and 1 t 001 (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 Pm-3m 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 spin-orbit coupling. This effect has been mainly studied for the I4/mcm structure and much less studied for cubic Pm-3m and tetragonal P4mm. In principle the spin-orbit coupling causes the energy band splitting in cubic Pm-3m 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 spin-orbit 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 Fig. 3(b-d).
The rotation of MA molecule changes the lattice constants continuously in three directions from a to a-δ and vice versa with δ is the lattice constant difference between cubic Pm-3m 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 non-orientation. 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 quasi-cubic Pm-3m structure or quasi-tetragonal 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 Pm-3m and P4mm structure. The results showed that at V = 250 Å 3 /f.u three structures I4/mcm, P4mm and Pm-3m 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.

Methods
For the calculations of electronic structures and related properties of MAPbI 3 we use the projector augmented wave (PAW) method 27 with the Perdew-Burke-Ernzerhof (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 cut-off 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 Hellman-Feynman forces are less than 0.005 eV/Å. The 6 × 6 × 6 Monkhorst-Pack grid of k-points 29 for Brillouin zone integration was used in calculations for Pm-3m 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 I-5s 2 5p 5 , C-2s 2 2p 2 , N-2s 2 2p 3 and H-1s were considered as valence electrons. The symmetry of Pm-3m 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 Pm-3m or P4mm symmetry. Therefore they are called a pseudo-cubic Pm-3m or a pseudo-tetragonal P4mm.
We applied the Van der Waals correction force (vdW-DF2 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 vdW-D3 correction method by Grimme et al. 32 , which is reported by Thind et al. 33 , underestimate the lattice constant of the cubic structure Pm-3m in comparison to experimental data, see the table S1. Because of this we preferred to use PBEsol method instead of PBE-vdW-D3.
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 Fourier-expanded with G max = 20 Ry. A k-mesh of 10 × 10 × 10 in the full Brillouin zone was used. In addition to the usual valence states, also extra local orbitals for "semi-core" states (Pb-5b, 5d, 6 s, 6p; I-4d, 5 s, 5p; N: 2p, and C:2p) were added and considered as band states.