First-Principles Calculations to Investigate the Mechanical Structure and Optical Properties of Lead Halide Perovskite CH3NH3PbI3

We report the study of the mechanical structure and optical properties of lead halide perovskite CH3NH3PbI3 using ab initio methods. +e ground state energy calculations were performed within density functional theory and generalized gradient approximation using the pseudopotential method with plane-wave basis sets.+e norm conserving pseudopotential was used.+e ground state properties of the electronic structure of the perovskite were used and elastic parameters such as bulk modulus B, Young’s modulus E, shear modulus G, and Poisson’s ratio υ were determined and found to be in good agreement with experimental values.+e ratio B/G obtained was found to be greater than 1.75. Poisson’s ratio (υ) was obtained as 0.25 implying that CH3NH3PbI3 is a ductile material. +e absorption coefficient within the energy range of 0 to 6 eV was found to be 5.76×10 cm indicating maximum absorption. +e absorption coefficient compares well with the available experimental and computed values.


Introduction
e desire to achieve high efficiency solar cells has led to research in perovskites that are lead-based, in particular lead halide perovskites. Since lead is not environmentally friendly, there has been a need to have organic lead halide perovskites with less lead, but organic lead-free perovskites are less stable [1]. Saliba et al. [2] reported that lead halide perovskites have become popular as photovoltaic materials due to their high power conversion efficiency (PCE) which stands at over 22%. e optical properties of these new materials are important not only to device design but also because of the insight they provide into less accessible properties such as energy band structures and binding energies, among other properties.
A study by Feng (2014), investigated the mechanical properties of hybrid organic-inorganic CH 3 NH 3 BX 3 (X � Br, I; B � Sn, Pb) perovskites used in photovoltaic cell absorbers. e structure consists of a network of corner-sharing BX 6 octahedra, where the B atom is a metal cation (Sn 2+ or Pb 2+ ) and X is the hybrid anion (Br-, Cl − or I − ). e dependent octahedra give room for a broad readjustment of the B-X-B bond angle, for this case, Pb-I-Pb. e various sets of joint rotations are called tilt transitions [3]. is develops symmetry, which shows different structures at varied temperatures. For instance, the three structural phases in which the compound CH 3 NH 3 PbI 3 exists include pseudocubic at increasing temperatures (above 327 K), tetragonal at room temperature, and orthorhombic at reducing temperatures (below 162.2 K) [3].
According to Zhu et al. 2021, the tetragonal phase changes to a cubic phase at 0.3 GPa. In addition to the 2Hphase, Young's modulus (E) reduces as pressure increases.
is implies that the stiffness reduces with increasing pressure. Nonetheless, in orthorhombic, tetragonal, and cubic phases, as the pressure increases, E first increases and then reduces. In particular, the cubic phase has only three independent elastic constants: C 11 , C 12 , and C 44 which were determined in this work [3,4]. e crystal structures, elastic, and anisotropy properties of the CH 3 NH 3 PbI 3 were studied using the ab initio calculations. e rationale of employing first-principles (ab initio) calculations was premised on the assertion that the mechanical properties of the compounds are relatively difficult to measure through an experimental approach. It was revealed that the absorption performance of perovskite solar cells greatly depends on the crystalline and stress state of the perovskite layer. e study by Bretschneider et al. [5] concurred with previously conducted work, and it indicated that the mechanical properties of perovskite are crucial for practical applications. e study conducted by Sun et al. [6] analyzed factors that influence the mechanical properties of formamidinium lead halides and associated perovskites. e study formulated a systematic way of probing the mechanical properties of hybrid perovskite single crystals under nanoindentation. It was revealed that the shape, size, and hydrogen bonding resulting from the organic cations have a significant influence on their mechanical properties. Similarly, it was found that bonding in the inorganic framework and hydrogen bonding play a vital role in determining elastic stiffness. e perovskite structure with the formula ABX 3 is used in many oxide compounds. In this study, A represents an organic positively charged ion (CH 3 NH 3 + ), X represents n halide specifically Iodide (I − ), and B is a divalent metal ion, in this case Pb 2+ . is structure is a cubic unit cell which contains an A atom (CH 3 NH 3 + ) in the center of the cube, B atoms Pb 2+ at the corners, and X atoms (I − ) at the center of the cell edges. e modelled pseudocubic structure is shown in Figure 1.
e calculated elastic parameters in this work include C 11 , C 12 , C 44 , bulk modulus B, Young's modulus E, shear modulus G, Poisson's ratio υ, and anisotropy which describe the mechanical stability and ductility of CH 3 NH 3 PbI 3 [7,8].
e study also noted that the measured Young's moduli (9.7-12.3 GPa) and hardness (0.36-0.45 GPa) reflected good mechanical flexibility and ductility. e study indicated that the mechanical properties of lead halide and related perovskites are important in device fabrication and performance. e objective of this work is to establish the ideal mechanical and optical properties for photovoltaic applications from the first principle method. e rest of this paper is organized such that Section 2 is computational details, Section 3 is the results and discussions, and Section 4 deals with conclusions.

Computational Details
In this work, scalar relativistic electronic-structure calculations on this material were carried out based on densityfunctional theory (DFT) [9], plane waves, and the pseudopotential approach as implemented in the Quantum Espresso computer code [10]. In the calculation of total energy, the exchange-correlation potential is treated with the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA). e lattice constant of CH 3 NH 3 PbI 3 was well optimized [11,12]. e total energy convergence in the iterative solution of the Kohn-Sham equations was fixed at 2 × 10 −8 Rydberg (Ry), and self-consistency was achieved [13,14]. All calculations were carried out under ground state conditions. e energy cut off was obtained as 30 Ry and 4 × 4 × 4 k-point grid or more was found to be sufficient for this material.
Core electrons were replaced by ab initio norm conserving pseudopotentials, generated using the Troullier-Martins scheme [15]. In the Kleinman-Bylander fully nonlocal separable representation [16]. According to this arrangement, s 1 , 2s 2 2p, 2s 2 2p 3 , 5s 2 5p 5 , and 5d 10 6s 2 6p 2 were used as valence electrons for H, C, N, I, and Pb, respectively. e large overlap between the semicore and valence states makes it possible for the semicore 5d electrons of Pb to be treated as valence electrons and distinctly included in the simulations, as explained in [17]. In this study, the electronic density, Hartree, and exchange correlation potentials were calculated in a uniform real-space grid with an equivalent plane-wave cutoff of 55 Ry in the representation of charge density. To integrate the Brillouin zone, we employed a Monkhorst-Pack sampling [18] equivalent to 13 × 13 × 13 in a twelve-atom CH 3 NH 3 PbI 3 unit cell [19]. e ground state energies were determined by running the files in Appendices A and B. e determination of e-cut energy, alat, k-points, and the pseudocubic structure are the preliminary calculations determined for the optical and mechanical properties of CH 3 NH 3 PbI 3 . e approximations used in the calculation include the following: the exchange correlation GGA and the Voigt-Reuss-Hill approximation in calculating bulk, shear, and Young's modulus. Other types of approximations are beyond the objective of this study.

Mechanical Properties of CH 3 NH 3 PbI 3 .
Mechanical properties of solids include elasticity, strength, abrasion, hardness, ductility, brittleness, malleability, and toughness.  Advances in Condensed Matter Physics e elastic constants give the relation between dynamic properties and mechanical properties with respect to forces existing in materials. e elastic constants of CH 3 NH 3 PbI 3 determined in this work are given in Table 1. Table 1 shows the mechanical properties calculation of CH 3 NH 3 PbI 3 using the Quantum Espresso code, and results were compared with other available data. e average of the Voigt-Reuss-Hill approximations was used in calculating bulk (B), shear (G), and Young's modulus (E).
e results show that our calculated elastic constants for CH 3 NH 3 PbI 3 compare well with other work. ese results confirm that pseudocubic CH 3 NH 3 PbI 3 is stable mechanically because it fulfills the Born stability criteria of cubic crystals whereby C 44 > 0, C 11 > C 12 , and C 11 + 2C 12 > 0 [20]. e ratio B/G obtained was found to be greater than 1.75 implying that CH 3 NH 3 PbI 3 is a ductile material. ese results confirm the ductility property of the perovskite [14,[21][22][23].
e computed anisotropic ratio which was obtained as 0.6 indicated that cubic CH 3 NH 3 PbI 3 was elastically anisotropic because of substituting it in the equation A � (2C 44 /C 11 + C 12 ). e value of A was obtained to be smaller than one (A <1).
Poisson's ratio obtained was 0.25 which when rounded off to one decimal place is closely equal to 0.3 [24] which confirms that CH 3 NH 3 PbI 3 is a ductile material.

Optical Properties of CH 3 NH 3 PbI 3 .
Optical properties of a material include reflectance, conductivity, absorption, refraction, transmittance, dispersion, diffraction, and so on. Solar panels, for example, need to absorb sufficient light energy. erefore, a lower solar reflectance index rating is required of the material for high absorption. e solar transmittance of a surface is the fraction of the sun's radiation that is transmitted through the surface of the solar material. What is utilized by solar materials to produce electricity is the amount of solar radiation absorbed. is therefore prompted the study of absorption other than reflectance and transmittance as the main optical property in this work with respect to its suitability as a photovoltaic material.
e absorption coefficient of CH 3 NH 3 PbI 3 has therefore been computed in this work. e absorption coefficient shows how deep light rays will penetrate into a layer before it is absorbed. Light is rarely absorbed in a material with a low absorption coefficient. For a direct gap semiconductor like CH 3 NH 3 PbI 3 , the absorption coefficient can reach elevated values.
A material that has a high optical absorption is suitable to be used in photovoltaic applications [25]. e high photovoltaic performance of CH 3 NH 3 PbI 3 is associated with optically high absorption characteristics. It is clear from Figure 2 that the material has a high absorption ability. Its absorption coefficient is about 1.4 × 10 6 cm −1 for energy between 0 and 40 eV which contributes to the effective utilization of solar radiation.
is means that, with a high absorption coefficient, CH 3 NH 3 PbI 3 readily absorbs photons, which excite electrons from the valence band into the conduction band. In the solar spectrum, visible light is found within the range of wavelength λ ≈ 380-780 nm which translates to energy of approximately 1.6-3.5 eV. Figure 3 indicates that at about 3.5 eV there is a peak in the absorption of the visible spectrum by CH 3 NH 3 PbI 3 . e peak in Figure 3 gives the absorption coefficient within the given range of energy and it was found to be 5.76 × 10 5 cm −1 indicating maximum absorption.
is value can be compared with the derived absorption coefficient of 1.0978 × 10 5 cm −1 [26] and 8.1448 × 10 4 cm −1 [27] which greatly depends on the range at which the energy or wavelength was measured from. e data plotted in Figure 2 are in Appendix C. e absorption coefficient determines how far into a material light of a particular wavelength or energy can pass through a material before it is absorbed. Absorption of solar energy will occur within the near UV region, visible region, and near infrared region between energy 0.886 eV and 3.98 eV.
is is equivalent to a wavelength of 1400 nm-380 nm. is means that the material has a relatively wide absorption range, as shown by the wavelength. e high absorption coefficient of CH 3 NH 3 PbI 3 indicates that the material readily absorbs photons with energy equivalent to the band gap energy which excites electrons from the valence band into the conduction band. e imaginary part of Figure 4 indicates a maximum absorption of photon energy between 0 and 5.0 eV, which is  Advances in Condensed Matter Physics within the region where solar energy is utilized in a solar cell. CH 3 NH 3 PbI 3 has a band gap of 1.58 eV [28]. is means that only photons of energy equal to 1.58 eV will be absorbed by the material. Photons of higher or lower energy value than the band gap are wasted. Figure 4 shows that the appearance of a sharp peak of the imaginary part of the dielectric constant of the CH 3 NH 3 PbI 3 implies the occurrence of strong absorption in this spectral region. e real part is related to the refractive index, while the imaginary part gives the absorption coefficient. e data plotted in the figure are in Appendix D.
Generally, a material with a high dielectric constant has a relatively less charge carrier recombination rate. As a result, the overall performance of optoelectronic devices is enhanced.

Conclusion
e result from the elastic constants done compares well with previously carried out work. e elastic parameters determined include bulk modulus B, Young's modulus E, shear modulus G, and Poisson's ratio υ. e ratio B/G and Poisson's ratio obtained implied that CH 3 NH 3 PbI 3 is a ductile material. e evaluated elastic parameters enable us to conclude that the investigated compound is mechanically stable. is means that it can be molded into different shapes and sizes; therefore, suitable for the fabrication and modelling of solar cells.
Here, the optical constants which include absorption coefficients and real and imaginary components of the dielectric constant are reported. e value of the absorption coefficient within the spectral range of 0-6 eV within which the visible spectrum exists is 5.76 × 10 5 cm −1 .
ere was, however, inadequate literature from experimental and computational for the absorption coefficient to compare these results with, but from the results obtained, it can be concluded that CH 3 NH 3 PbI 3 has a high absorption coefficient of solar radiation and is highly suitable to be used in the fabrication and modelling of solar cells.