As a dye sensitized solar cells (DSCs), perovskite CH3NH3PbI3 (methyl-ammonium lead triiodide) has been widely utilized for solar cell application (E D Indari1 et al., 2017). CH3NH3PbI3 contains octahedral PbI6 units within its perovskite crystal structure with the general formula ABX3, where A is CH3-NH3+, B is Pb2+, and X is I-. CH3NH3PbI3 has several attractive features, such as low cost, tenable band gap (Wan-Jian Yin et al., 2015) and strong optical absorption (Wan-Jian Yin et al., 2014). In 1978, when Dr Klaus Weber, synthesized and characterized CH3NH3PbI3, he found that each Pb2+ cation is coordinated to six I- anions to form PbI6, which are corner connected to each other form a three-dimensional Pb-I framework (D. Weber, 1978). Each CH3NH3+ cation stays at the centre of four PbI6, which interacts with 12 I- anions (Tom Baikie et al., 2013). The symmetry and structure of CH3NH3PbI3 crystals are highly dependent on the temperature according to the report of previous experiments (Tom Baikie et al., 2013). At low temperature, an orthorhombic phase with Pnma space group is found which may transform into a tetragonal structure (space group: I4/m) above 161.4 K. The cubic phase (space group: Pm-3m) is found when the temper-ature is higher than 330.4 K. For improving the symmetry at the higher temperature, the fast-dynamic movement of CH3NH3+ cations within the Pb-I framework is needed. At low temperature, the location of CH3NH3+ cations can only be determined experimentally in case of orthorhombic phase where the lattice constants in orthorhombic crystal vary little (Tom Baikie et al., 2013). In addition, a novel light harvester, which can greatly improve the solar conversion efficiency of dye-sensitized solar cells based on organic/inorganic hybrid CH3NH3PbI3 perovskite. 

The highest reported power conversion efficiency has increased from under 4% in 2009 (Akihiro Ko-jima et al., 2009) to over 22% in recent years (Hanul Min et al., 2021). The ability to tune the band gap and related key properties of CH3NH3PbI3 by doping the ions of various sub-lattices has been demonstrated in huge number of recent studies which broadens the optoelectronics applications (Keith P et al., 2018). 

CH3NH3PbI3 has already been accepted as a potential material for next-generation solar cell. To know the molecular design of organic/inorganic perovskite materials with defined properties, it is obvious to study the structures and chemistries of these prototype light harvesters. Among different perovskite formulations used in solar cells, CH3-NH3PbI3 is the most widely studied (Shenghao Wang et al., 2016). As a result, our investigation is focused on the orthorhombic perovskite CH3NH3-PbI3. By ab initio calculations we have investigated both of structural and electronic properties of organic/ inorganic hybrid perovskite CH3NH3PbI3 crystals. Density functional theory (DFT) is employed as it offers an efficient, yet accurate quantum mechanical method for theorists to optimize structures and determine energies of reactants and products (J. Even et al, 2014). In this letter, we present a theoretical investigation of structural with electronic properties of the orthorhombic phase of CH3NH3PbI3. Different parameters are computed such as the lattice parameters, the interatomic distances, the total energy, density of states with corresponding band structures and so on. 

All these parameters are computed without considering any physicochemical effect. Afterwards, all results are compared with the available experimental data which are the important parameters for enhancing the electron transport and efficiency. In addition to that, it also leads the research in the exploration of material properties of high efficiency photovoltaics, high optical absorption coefficient, excellent charge carrier transportation, material stability and so on. 

Computational Parametrization

Ab initio calculations are performed in the framework of the DFT, based on a plane-wave basis set for the expansion of the single-particle Kohn-Sham wave functions and pseudopotentials. Electron-ion interactions are described using plane wave ultrasoft pseudopotentials with a kinetic energy cut-off of 520.0 eV. The Perdew-Burke-Ernzerhof (PBE) parametrization of the Generalized Gradient Approximation (GGA) (John P. Perdew et al., 1997) is employed for the evaluation of the exchange-correlation energy under electronic and ionic self-consistency, with convergence criteria of 10−4 eV and -10−3 eV·Å−1, respectively. Electron in valence states contained the Pb 6s, 6p, and 5d states, I 5s and 5p states, C 2s and 2p states, N 2s and 2p states, and the H 1s states. All atoms are relaxed after optimized their geometry where cell optimization technique is employed to optimize the lattice constants. For the calculations of structural with electronic properties, we perform Brillouin-zone integrations using Monkhorst-Pack grids of special points with (4×4×4) mesh. We reproduced the experimentally verified simple orthorhombic MAPI supercell that consists of per-ovskite units (a = 8.836Å, b = 12.580Å, c = 8.555Å) at low temperature (Tom Baikie et al., 2013). The orthorhombic supercell of CH3NH3PbI3 includes 48 atoms. DFT methods have been implemented in the Vienna Ab initio Simulation Package (version 5.4.1) (Jürgen Hafner, 2008) for ground state calculations. 

The GGA-PBE calculation is performed for bulk of CH3NH3PbI3 without incorporating the spin-orbit coupling (SOC) effect to the DFT methods. As it is not possible to compute the contributions to the energy, forces, and stresses for an infinite number of cell replica, a tolerance parameter defining the number of cells to be considered is used in the DFT. Here, the Gaussian smearing is used and the stopping criterion on the residual of the forces on each atom is set to 10−7 eV·Å−1, for the geometry optimizations. 

Structural properties

 Table 1: Computed lattice parameters included with unit cell volume are calculated by Perdew-Burke-Ernzerhof (PBE) parametrization, compared with experimental data (Tom Baikie et al., 2013).

To evaluate the performance of various functional, the of organic/inorganic hybrid perovskite CH3NH3 PbI3 crystal structures are firstly optimized where lattice parameters must be released numerically. This is done by allowing the atoms to move in a way that minimizes the forces present in the cell. The structure relaxation allows the determination of the atomic positions at the equilibrium.

 Fig. 1: The orthorhombic atomic structures of perovskite CH3NH3PbI3 crystals without (left) or with (right) showing the [PbI6] octahedra. Key: grey- Pb, purple- I, light blue- N, brown- C, and pink- H. Green solid line indicates the unit cell area.

The initial lattice constants and the system of atomic coordinates in orthorhombic CH3NH3PbI3 crystals for theoretical optimizations are from the X-ray diffraction (XRD) experiments at 100K (Tom Baikie et al., 2013).The optimized lattice constants included with volumes using the various functional are listed into Table 1 with the available experimental values (Tom Baikie et al., 2013; Mostari et al., 2020).

Our results are in good agreement with experimental results. The relative error with respect to the experimental value for lattice constants does not exceed 5% where the total volume of unit cell becomes 7 % larger than the experimental results (Tom Baikie et al., 2013). The optimized orthorhombic atomic structure, CH3NH3PbI3 crystals using the PBE/GGA functional is shown in Fig. 1.   

 Table 2: Displacement parameters for Pb1, I1 and I2 atoms of orthorhombic CH3NH3PbI3.

The reduced coordinates of each atom are also modi-fied due to the relaxation. The orthorhombic symmetry is still conserved. In relaxed system, the total energy is found to be equal to -203.8332 eV. In ideal crystallographic description, I- anions show the transverse displacement from the mid-point of the Pb-Pb distance within [PbI6] octahedral structures in XRD refinement experiments (Fig. 1). By using PBE functional relevant with experimental data, Table 2 lists the theoretical displacement parameters: Pb1, I1 and I2 atoms, marked in Fig. 1. The atomic fractional co-ordinates between optimized and ideal crystallographic structures in displacement parameters are the main differences of them. Without the displacement of I1 atoms along a direction and I2 atoms along b direction, the overall theoretical displacement parameters show well agreement with the experimental values. The I2 atom shift better along both a, b direc-tions but I2 atoms displacement in c directions are smaller than those of the I1 atoms.

Table 3: The optimized interatomic distances and internal bond angles among organic and inorganic atoms of CH3NH3BbI3, verification with the experimental data.

The optimized Pb-I bond length calculated with PBE is 3.26Å while the C-H, N-H and C-N bond lengths are 1.09, 1.04, and 1.49Å, respectively which are satisfied with the experimental values of 3.18, 1.1, 1.0, and 1.57Å (Tom Baikie et al., 2013). The apical (along the b axis) and equatorial (along the c axis) Pb-I-Pb angles are respectively equal to 159.88 o and 152.78 o and compare well with the standard values of 161.9 o and 150.7 o (Tom Baikie et al., 2013). All the results concerning the interatomic distances and the internal bond angles are listed in below Table 3. The Pb-I1-Pb angle is about 159o in our PBE results, which is 7o larger than the Pb-I2-Pb angle. These structural differences mark that I1 and I2 atoms are inequivalent atoms. These parameters are all within 5% from the corresponding experimental values (Tom Baikie et al., 2013).

Table 4: The lattice parameters variations with k-point summations and cut off energies.

To investigate the dependence of k-point summation and cut-off energies, we have optimized the structure with various k-points grid and cut-off energies for obtaining corresponding lattice parameter variations. We found the maximum lattice constant difference is less than 0.25% for cut off energy variation in x direction and the angular change of them are not exceed 0.05% for both of k-point summations and cut off energy variations in x direction. The standard values of the cut-off energy for calculating other compounds from some literatures such as 400 eV/ 500 eV. In our system includes Pb ions which carry d electrons that have high localizations requiring higher cut-off energy, because the cut-off energy is related to minimum wavelength of electrons in/near the ionic core. Moreover, we noticed the dynamic movement of organic part, CH3NH3+ cations with in the Pb-I framework is negligible and the internal structural change is not dominant. By increasing the k-point summations and cut off energy, a bit of dependence is observed, thus, chosen a set of minimum k-points summation, 4 x 4 x 4 and cut off energy, 520 eV is sufficiently enough for structural properties analysis if the cell size become in large. The data for optimized structure with k-point grid and cut off energy variations are shown in Table 4.

Fig. 2: Total density of states (TDOS) for orthorhombic CH3NH3PbI3 crystals and its bandgap energy using PBE/GGA functional.

Electronic properties

The crucial factors of organic/inorganic hybrid CH3 NH3PbI3 materials are the electronic structures where the sunlight absorption used as light harvesters in DSCs (Julian Burschka et al., 2013).The bandgap energy is estimated 1.80 eV calculated from total density of states (TDOS) using the PBE/GGA functional (see Fig. 2) where the experimental lattice constants were used (Tom Baikie et al., 2013). The structural properties have the great impact on band-gap energies and highlight the significance of using approximations which accurately reproduce the correct geometry (Ivo Borriello et al., 2008). By employing optB86b+vdWDF functional (J. Even et al., 2014), the theoretical bandgap energy is 1.74 eV which is 0.06 eV smaller than PBE functional. This difference can be focused on the various theoretical lattice constants taken by various functional.

Fig. 3: Scheme of the Brillouin Zone for an ortho-rhombic lattice. Where Γ (0.0,0,0); Z (0,0,1/2); T (0.0,1/2,1/2); Y (0.0, 1/2,0); S (1/2, 1/2,0); X (1/2,0.0,0); U (1/2,0.0, 1/2); R (1/2, 1/2, 1/2) are respective coordinates.

At low temperature (Ishihara T, 1994)] in optical absorption experiments, the electronic bandgap is 1.68 - 1.72 eV in case of orthorhombic CH3NH3PbI3 crystals. Although, the theoretical predictions at the non-local and sami-local level of bandgap energies of hybrid CH3 NH3PbI3 are almost close to the experimental values, but in some case, the bandgap energies deviation of solid-state semiconductors is about 30% (Haijun Yu et al., 2012) by using DFT functional. This is why good band gaps are not normally to be expected from functional of the types tested here. To know the band structure of CH3NH3PbI3 is interesting as it gives information about the electronic transition. Representing the 3D grid band structure of the Brillouin zone is impossible as it requires a four-dimensional plot. A more convenient way to represent it consists in defining paths along high symmetric directions. The acquirement of a band structure is a two-step process. Firstly, the density must be obtained thanks to a self-consistent calculation.

Secondly, the density used when solving the Kohn-Sham equation for many different k points, along the defined paths in Brillouin zone. It ensures that the potential will not vary during the scan of different k points lines. The Brillouin zone of the orthorhombic cell is represented as in Fig. 3. Lets consider the following directions of high symmetry: Γ→ Z→T →Y→S →X→U→R.

Fig. 4: Band structure of the orthorhombic cell for specific symmetries directions in the Brillouin zone: Γ-Z-T -Y-S -X-U-R. Dash line represented the fermi energy level. A band gap of 1.80eV is found.

The electronic band structure (depicted at the converged parameters) is represented in Fig. 4. Upper lines than fermi energy level are conduction bands while lower ones are valence bands. The orthorhombic phase of CH3NH3PbI3 shows a direct band gap occurring at the point (0, 0, and 0) which is the highest symmetrical point of the Brillouin zone. The estimated band gap is almost equal to ⇠1.80 eV. In general, when using LDA or GGA calculation methods, the band gap energy is often underestimated which is famous DFT band-gap problem. This effect probably comes from the fact that the spin-orbit coupling is not considered. Thus, the priorities of SOC in the DFT calculation has been pointed out (Jacky Even et al., 2013). The harmony with the experimental result degrades seriously if this interaction is incorporated into the DFT calculation with-in PBE or the local-density approximation and the Eg reduces significantly down to 0.5 eV (Giacomo Giorgi et al. 2013).

When the SOC effect is considered using the GW approximation (Paolo Umari et al., 2014) or a hybrid functional (E. Menéndez-Proupin et al., 2014), the experimental Eg of 1.6 eV can be reproduced. How-ever, the band structures obtained from these sophisticated calculations are essentially like that deduced from the simple PBE calculation (Wan-Jian Yin et al., 2015). Therefore, the PBE calculation is performed without incorporating the SOC effect.

In summary, first-principle DFT calculations were performed to study structural with electronic properties of orthorhombic perovskite CH3NH3PbI3 materials. Our results concluded that the optimized lat-tice constants using PBE/GGA does not exceed 5%, compared to the experimental value and the displacement parameters included interatomic distances with internal bond angles satisfy the experimental results. The chosen set of minimum k-points summation, 4 x 4 x 4 and cut off energy, 520 eV is sufficiently enough for structural properties analysis if the cell size becomes in large. The electronic properties analysis concluded that orthorhombic CH3 NH3 PbI3 exhibits as a direct band gap crystal with mini-mum band gap is about 1.80 eV at the gamma symmetric point.

 Im grateful to all the dear Professors for providing their information regarding this research.

 The author of this manuscript declares agreement with the statements and has no conflicts of interest.