Study of the photovoltaic properties of Cs and Cl co-doped FAPbI3 based on first principles

Formamidine lead iodide perovskite (FAPbI3) is often used as a light-absorbing layer in solar cells to alleviate the energy crisis because of its good photovoltaic properties. However, its lack of stability is also an obstacle to the current development. It has been found that doping with different kinds of elements at different sites can enhance its stability and improve the photoelectric conversion efficiency of solar cells. In this study, the geometry, electronic structure, and optical properties of FA1−xCsxPbI3−yCly were calculated using Cs and Cl co-doped with FAPbI3 using first principles. The analysis revealed that the Goldschmidt factors of the doped system were between 0.962 and 0.974, indicating that the systems could maintain a stable perovskite structure and that the doped system had lower energy and a more stable structure. By calculating the energy bands, it was found that the doped ions have a more pronounced effect on the increase in the dispersion at the bottom of the conduction band than on the decrease in the dispersion at the top of the valence band of the system, and the reduction of the effective mass of carriers is more favorable for transport. As for the optical properties, the right amount of doping is favorable for the improvement of light absorption, whereas excess doping shortens the light absorption range and weakens the light absorption effect, in which FA0.875Cs0.125PbI2.958Cl0.125 has the largest light absorption coefficient. It is shown that the photoelectric properties of FAPbI3 can be effectively modulated by the co-doping with Cs and Cl, which can provide a theoretical reference for the precise preparation of more efficient solar cells.


Introduction
The development of economy and technology cannot be supported by energy sources, which are mainly nonrenewable energy sources, such as oil, coal and natural gas. As increased energy are used, there are few reserves of energy left on the Earth, and the environmental problems caused using non-renewable energy sources have also caused concern in many countries, so they hope to vigorously develop renewable and clean energy sources such as solar energy [1][2][3][4][5]. The conversion efficiency of solar cells depends on the photoelectric conversion of their photo absorption layer, and the special properties of perovskite materials, such as direct and tunable band gap, high lattice defect tolerance, high usage efficiency of photogenerated carriers and high light absorption coefficient, are particularly suitable for use as a photo absorption layer in solar cells [6,7]. The conversion efficiency of perovskite solar cells, first prepared in 2009 at 3.8% [8], has recently exceeded 25% [9] after years of unremitting efforts. Although the conversion efficiency has made great progress, perovskite solar cells are prone to react with water vapor in the air, causing structural instability. So, stability is also a difficult point to overcome for perovskite solar cells [10,11].
The crystal structure of perovskite materials as light absorbing layers is ABX 3 type, with methylamine ions (MA + = CH 3  Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. environment like temperature and humidity, and then intrinsic properties such as electron migration and ion migration, which are related to ion size [12,13]. Methylamine ions (MA + ), formamidine ions (FA + ) and chloride ions (Cl − ) are large size ions with soft lattice, so they tend to react to temperature, humidity and light, causing a decrease in stability [14]. The cubic phase band gap of FAPbI 3 is closer to the Shockley-Queisser limit of 1.34 eV than that of MAPbI 3 , which is more suitable for light absorbing materials, while FAPbI 3 is more likely to form the hexagonal phase at room temperature. To improve the performance of solar cells, researchers have tried to dope different sites of perovskite type crystals with different types of ions. MA x FA 1−x PbI 3 was prepared by Pellet et al [15] using a mixture of methylamine and formamidine ions at site A. The hybrid system exhibited better photovoltaic properties than the undoped system. In addition, Eperon et al [16] found that the diffusion length of carriers in the MA x FA 1−x PbI 3 system became longer, revealing the reason for the improved photoelectric conversion efficiency from the internal mechanism. Lee et al [17] use a small amount of Cs ions instead of FA ions, which makes the bond energy of the chemical bonds between ions larger and the internal defects smaller, and the thermal stability of the prepared Cs x FA 1-x PbI 3 system is improved. Hu et al [18] added 10% CsCl to FAPbI 3 was able to suppress the hexagonal phase of FAPbI 3 , making the cubic phase more stable and the system with the lowest density of film defects.
Numerous studies have shown that the simultaneous addition of Cs and Cl to FAPbI 3 leads to the formation of FA 1−x Cs x PbI 3−y Cl y with better optoelectronic properties and stability than FAPbI 3 , however, the components of FA 1−x Cs x PbI 3−y Cl y cannot be precisely detected experimentally. In this paper, we calculate the stability, electronic structure and optical properties of Cs and Cl co-doped FA 1−x Cs x PbI 3−y Cl y by first nature principle in order to find the theoretically best performing perovskite FA 1−x Cs x PbI 3−y Cl y system.

Computational details
All calculations in this paper were performed in the VASP (Vienna ab initio Simulatioan Package) software package, where the first principle was based on density functional theory(DFT), and the projective suffix plus plane wave pseudopotential was chosen to describe the interaction between the nucleus and the valence electron, and the generalized gradient approximation Perdew-Burke-Ernzerhof(PBE) generalized gradient approximation to describe the interaction-correlation energy of electron interactions. Since we mainly discuss the impact of Cs and Cl on the performance of the system after the common doping, coupled with the comprehensive consideration of the calculation amount, SOC is not selected in this paper. So, in this paper, the band gap of the system is calculated using the PBE generalized function. The k-point of the integral in the Brillouin zone was 4 × 4 × 4. The 400 eV was chosen as the plane wave truncation energy, and the total energy convergence criterion was 1 × 10 −5 eV atom −1 , and the maximum interaction force between atoms was 2 × 10 −2 eV Å when structural optimization was performed.
The structure of the perovskite crystal itself is one of the main factors affecting its stability, and the Goldschmidt factor t has been used to assess the stability of the ionic radius in relation to the perovskite structure since it was proposed [19]. Regarding the doped system FA 1−x Cs x PbI 3−y Cl y , the weights of the different types of atoms need to be measured, and the formula valid for calculating the Goldschmidt factor t is r eff 1 is the effective radius of the mixture of FA and Cs, r eff 2 is the effective radius of the mixture of I and Cl, r , For the Goldschmidt factor t, the crystals maintain a stable cubic phase structure of perovskite when t is within 0.78 to 1.05. The effective mass of the photogenerated carriers is closely related to the second-order derivative of the energy band edge, and the effective mass of the carriers m * can be calculated by fitting the energy band edge according to the following equation: where E is the dispersion relation function obtained by fitting the energy band structure map and k is the inverse space vector, which is the approximate Planck constant.

Structural stability and adjustment
Experiments by Mark [20] et al showed that FAPbI 3 exhibits a cubic phase with lattice constant a = 6.3620 Å and space group pm3m at 298 K. In this paper, the cubic phase FAPbI 3 model structure is optimized by taking a = 6.3620 Å. The optimized lattice constants are a = 6.392 Å, b = 6.473 Å, c = 6.323 Å, α = β = γ = 90°, it becomes an orthorhombic crystal system after optimization.
When calculating its energy band, the high symmetry K point is chosen according to the orthogonal crystal system. A 2 × 2 × 2 cubic phase FAPbI 3 supercell is shown in figure1(a). The calculated energy band structure is shown in figure1(b), and the calculated forbidden band width Eg = 1.40 eV, which is slightly smaller than the experimental value of 1.48 eV [21]. Figure1(c) shows its total density of states and the density of fractional states. It can be seen that the conduction band bottom is mainly composed of Pb-6p orbitals and some I-5p orbital electrons hybridized, and the valence band top is mainly composed of I-5p orbital electrons.
Charles et al [22] prepared Cs x FA 1−x PbI 3 at 300 K and obtained many stable cubic phases at x < 0.15. Cheng et al [23] prepared FAPbI 3−x Cl x (x = 0 ∼0.4) planar heterojunction photovoltaic cells using a one-step solution method and found that the conversion efficiency did not increase linearly with increasing Cl content, the conversion efficiency was up to 17% at x = 0.2 and decreased with increasing Cl content, and the light absorption pattern also showed that the excess doping of chlorine shortens the light absorption range of the perovskite compound. So, to keep the FA 1−x Cs x PbI 3−y Cl y system in cubic phase at room temperature, the amount of doped Cs should not exceed 15%, i.e., x should be less than 0.15. Because the excess Cl will shorten the light absorption range of the material, so for the proportion of doped Cl is between 0 and 0.4, i.e., y is between 0 and 0.4. Due to the limitation of calculation volume, this paper calculates the substitution of 1 of 8 FA ions by one Cs ion in 2 × 2 × 2 supercell FAPbI 3 (i.e., x = 0.125 < 0.15) and 1 to 9 of 24 I ions by Cl ions (i.e., y = 0.042, 0.083, 0.125, 0.167, 0.208, 0.250, 0.292, 0.333, 0.375) in terms of electronic structure and optical properties.
To investigate the stability of the FA 1−x Cs x PbI 3−y Cl y perovskite system, the Goldschmidt factor t of the system was calculated according to equation (1) and the individual ionic radii (r FA + = 2.53 Å [24], r Cs + = 1.67 Å, r Pb 2 + = 1.2 Å, r I -= 2.20 Å, r Cl -= 1.81 Å), and it was found that all the Goldschmidt factors t were in the range   figure 2(a), indicating that the system can maintain a stable cubic phase structure of perovskite. The variation of energy ΔE before and after Cs + and Cl − doping was further analyzed. As shown by the black line in figure 2

Electronic properties 3.2.1. Band structure
The PBE generalized FA 1−x Cs x PbI 3−y Cl y system was used for energy band calculation, and it was found that when x = 0.125, y = 0.042, 0.125, 0.292, 0.333, the FA 1−x Cs x PbI 3−y Cl y systems were the direct band gap, and the other doped ones were the indirect band gap. The indirect band gaps are not suitable to be used as light absorbing materials, which will increase the difficulty of interband electron leap, making the photoelectric conversion efficiency is greatly reduced. Therefore, we only discuss the electronic structure of the doped system with direct band gap. The forbidden band width, the average effective masses of electrons, and holes of the FA 1−x Cs x PbI 3−y Cl y system when x = 0.125, y = 0.042, 0.125, 0.292, 0.333 were further calculated immediately. As can be seen from table 1, the co-doped system has a larger forbidden band width compared to the pure FAPbI 3 system. From the experiments [25], it is known that the forbidden band widths of the doped systems become larger due to the doping of Cs ions. The average effective mass of electrons is reduced for all doped systems, which indicates that the doping of Cs ions and Cl ions increases the dispersion at the bottom of the conduction band, which facilitates the interband electron leap as well as the transport. While the average effective mass of holes in the doped system increases, indicating that the dispersion at the top of the valence band is reduced. The co-doping of Cs ions and Cl ions has a greater effect on the electron effective mass of the system, which leads to an increase in the hole effective mass but has a more obvious effect on the reduction of the electron effective mass. Overall, the co-doping of Cs ions and Cl ions is beneficial to the carrier leap and transport, which promotes the conversion efficiency of solar cells. When x = 0.125 and y = 0.042, the average effective masses of electrons and holes of the doped system are the smallest, 1.56m 0 and 0.85m 0 .
The energy band diagrams in figure 3 show that the bottom of the conduction band and the top of the valence band are located at the same high symmetry point Γ, and the forbidden band widths are 1.73 eV, 1.77 eV, 1.97 eV, and 2.00 eV, respectively, which are all direct band gap semiconductors, indicating that the interband electron leap is favorable. First-principles calculations indicate that the conduction bands are mainly composed of Pb-6p, C-2p, N-2p, Cs-5p and I-5p orbital electrons hybridized, and the valence bands are mainly composed of I-5p, N-2p, Pb-6p and Cl-3p orbital electrons hybridized. Comparing the energy band diagrams before and after doping, it is still found that the dispersion at the bottom of the conduction band increases in the doped system compared with the undoped system, indicating that the effective mass of electrons in the energy band decreases, while the dispersion at the top of the valence band decreases in the doped system compared with the undoped system, indicating that the effective mass of holes in the energy band increases.

Density of state
To further analyze the electron occupation of molecular orbitals, the total and fractional densities of states of the corresponding systems are given in figure 4. As known from figure 4, the conduction band bottom of the undoped system consists of Pb-6p orbital and I-5p orbital electron hybridization, and the valence band top consists of I-5p orbital electrons. After doping with Cs + and Cl − , the Cl-3p orbital electrons interact with I-5p, Pb-6p and Cs-5p orbital electrons, causing the density of states peak of I-5p electrons at the top of the valence band to move toward the low energy region and the density of states peak of Pb-6p electrons at the bottom of the conduction band to move toward the high energy region, so that the forbidden band width becomes larger.

Optical properties
The absorption coefficient is the characteristic that best reflects the optical properties of the material. The larger the absorption coefficient of the material used as the light absorbing layer of the solar cell, the more beneficial it is to improve the photoelectric conversion efficiency of the solar cell. As shown in figure 5,

Conclusion
In this paper, the geometry, electronic structure, and optical properties of Cs + and Cl − co-doped FAPbI 3 to form FA 1−x Cs x PbI 3−y Cl y were investigated by first principal calculations, and firstly, all the doped systems were found to maintain a stable perovskite structure by the judgment of Goldschmidt factor, and further by analyzing the energy of the FAPbI 3 system before and after doping. The contrast shows that Cs + and Cl − doping makes the structure of perovskite system more stable, among which FA 0.875 Cs 0.125 PbI 2.958 Cl 0.042 is the most stable. The study of the energy band reveals that the forbidden band width of the system becomes larger as the amount of Cs + and Cl − doping increases. The increasing effect of dispersion at the bottom of the conduction band is stronger than the decreasing effect of dispersion at the top of the valence band, which is favorable to carrier transport. The analysis of absorption spectra reveals that the light absorption coefficient of the system has a significant redshift with a certain amount of Cl − doping, and the excess shortens the light absorption range of the material, and the light absorption of the doped system is best when x = 0.125 and y = 0.125. These findings help to provide theoretical guidance for the experimental preparation of efficient and stable solar cells.