Improved Power Conversion Efficiency with Tunable Electronic Structures of the Cation-Engineered [A_i]PbI₃ Perovskites for Solar Cells: First-Principles Calculations

Abstract

Higher power conversion efficiencies for photovoltaic devices can be achieved through simple and low production cost processing of ${\mathrm{APbI}}_3\left(\mathrm{A}={\mathrm{CH}}_3{\mathrm{NH}}_3,{\mathrm{CHN}}_2{\mathrm{H}}_4,\dots \right)$ perovskites. Due to their limited long-term stability, however, there is an urgent need to find alternative structural combinations for this family of materials. In this study, we propose to investigate the prospects of cation-substitution within the A-site of the ${\mathrm{APbI}}_3$ perovskite by selecting nine substituting organic and inorganic cations to enhance the stability of the material. The tolerance and the octahedral factors are calculated and reported as two of the most critical geometrical features, in order to assess which perovskite compounds can be experimentally designed. Our results showed an improvement in the thermal stability of the organic cation substitutions in contrast to the inorganic cations, with an increase in the power conversion efficiency of the Hydroxyl-ammonium (NH₃OH) substitute to η = 25.84%.

1. Introduction

Perovskite materials with the ABX₃ (A = cations, B = Pb, and X = I) formula have a variety of valuable properties, including both in application-oriented uses and fundamental physics [1,2,3]. One of the main reasons perovskites are so successful is their adaptability—they can be tailored to have a variety of different properties, which allows them to meet specific requirements such as an impressive series of improvements in light to electricity efficiency [4,5,6,7,8]. In just over five years of exhaustive research, the efficiency of (CH₃NH₃)PbI₃ has been around 25% [9,10]. In this family, the perovskite architecture is largely retained, with one ion substituted by an organic ion (usually A or X). This makes the perovskite a hybrid with A and B as the cations, and X as a halogen or oxygen ion—two of the most common anions—while different metal cations can form halide and oxide perovskites [11,12] where the radii of A is larger than that of B. Both these perovskites have crystal structures that are very similar, with BX₆ octahedra located in between the A atoms [13,14]. Studies from first-principles calculations showed that adding substituents such as Br, Cl and F to MAPbI₃ can help facilitate the transfer of charge from the hybrid perovskite to the electrodes [2]. On the other hand, B-site substitutions change the bandgap. In our study, we investigated how the cation substitution at the A-site might affect the organic perovskite’s properties. Our analysis suggests that substituting the A cation may be a way to obtain an optimal band gap (1.1–1.4 eV) and high power conversion efficiency [9,15]. The Goldschmidt tolerance factor (t) has been a key element in the development of perovskites for many years, and it has been expanded to the growing field of organic-inorganic perovskites [4,16,17,18]. The t value should range between 0.8 and 1.11—otherwise, the crystal structure will be distorted, and might be destroyed. A smaller t value will result in a tetragonal or orthotopic shape with low symmetry. The t-value of an ideal cubic structure is from 0.89 to 1.0. The ideal value of the octahedral factor (μ), on the other hand (044 to 0.90), affects the stability of the octahedron, and further influences the stability of the perovskite structure [19,20].

As shown in Figure 1, recent machine learning studies generated about 2352 cations from different protonated amines. Only 742 of these cations showed a tolerance factor (t) of between 0.8 and 1.11, from which about 140 materials are better identified, such as well-characterized hybrid perovskites [13,21].

In our paper, we limited our study to nine cations (CH₄, PH₄, Li₃O, CH₃OH, Li₃S, CH₃NH₃, CH(NH₂)₂, CH₃C(NH₂)₂ and C(NH₂)₃), whose chemical formulas are detailed in Figure 2, and which have ionic radii of 146, 167, 170, 216, 220, 222, 253, 277 and 278 pm, respectively [22,23].

For all substitutions (except for Li₃O and Li₃S), the short circuit current (J_SC), the open-circuit voltage (V_OC) and the power conversion efficiency (PCE) are also calculated. We hypothesized that every photon energy above E is absorbed, always promoting the perovskite to generate electron-hole pairs [20,24,25,26,27,28,29,30,31].

V_OC was calculated, as well as the maximum theoretical limit of the PCE for all configurations, where the efficiency is regarded as a function of the band gap energy (E_g) and the total incident power density (P_S), which can be calculated using solar spectrum data. PCE was obtained by using Jsc, Voc, and the fill factor (FF), as well as P_S [32,33,34,35].

2. Results and Discussion

We used density functional theory (DFT) and ab-initio molecular dynamics (AIMD) to investigate the crystal structure, electronic structure, thermal and dynamic stability, and the power conversion efficiency of the substituted cubic MAPbI₃ (MA = CH₃NH₃⁺).

2.1. Optimization

We optimized the structures after substituting the cations while observing the distortions in the nascent crystallites relative to the pristine crystal of MAPbI₃. We noticed clear distortions in some structures, especially those with larger cations, or inorganic (Li₃S, Li₃O) size cations, while the other structures did not change significantly. Figure 3a,b shows the engineered structures before and after optimization for all the studied systems.

Table 1. The theoretical lattice constants a, b, c, α, β, γ, and space groups of A_iPbI₃ with the different cations.

	a (Å)	b (Å)	c (Å)	α (°)	β (°)	γ (°)	Space Group
NH4PbI3	6.3079	6.2407	6.3912	90.0000	90.0000	90.0000	P m
PH4PbI3	6.5455	6.5483	6.5574	90.0000	88.7342	90.0000	P m
Lis3OPbI3	6.8934	6.9410	6.9572	90.0000	90.0000	90.0000	P m m 2
HAPbI3	6.3388	6.3572	6.3517	90.0000	84.496	90.0000	P m
Li3SPbI3	6.6913	6.6140	6.5984	89.0179	90.4295	90.7352	P1
MAPbI3	6.4271	6.4006	6.4547	90.0000	90.1803	90.0000	P m
FAPbI3	6.4918	6.5457	6.3295	90.0000	90.0000	90.0000	P m m 2
AcePbI3	7.6505	7.0395	6.2866	90.2916	90.0465	90.2715	P1
GuPbI3	7.2249	6.5031	6.4847	90.5220	89.8453	89.5245	P1

gives an exhaustive list of all calculated parameters [14,36]. This is attributed to many reasons, such as the difference in electrical polarization between the anion and the cation, as well as the size of the cation in relation to a unit cell.

2.2. Tolerance Factor and Octahedral Factor and Parameters Optimization

In our study, we substituted the MA cation with a number of different-sized organic cations and their constituting atoms within the cubic phase, and for a first examination of the stability of the crystal lattice with each cation, we calculated the tolerance factor (t) and the octahedral stability factor (μ), and matched them with the permissible ranges, which were arranged in an ascending order of size, as shown in

Table 2. Estimated radii (R_A), band gap (E_g), formation energy (E_f), polyhedral factor (μ) and tolerance factor (t) of (A_i)PbI₃ Perovskites with GGA-PBE (vdW) by theoretical calculations.

	RA (pm)	Eg (eV)	Ef (eV)	μ	t
NH4PbI3	146	1.7611	−0.32564	0.468182	0.801242
PH4PbI3	167	1.8411	−0.34885	0.468182	0.847215
[Li3O] PbI3	170	0.5932	−0.34170	0.468182	0.853782
HAPbI3	216	1.5348	−0.35955	0.468182	0.954485
[Li3S] PbI3	220	0.4146	−0.34944	0.468182	0.963241
MAPbI3	222	1.5971	−0.36116	0.468182	0.967620
FAPbI3	253	1.4815	−0.36469	0.468182	1.035485
AcePbI3	277	1.9645	−0.37708	0.468182	1.088025
GuPbI3	278	2.1517	−0.37980	0.468182	1.090214

. The results indicated that every one of these falls inside the stable range [37].

2.3. Stability

Formation Energy:

We can clearly see that this perovskite can be produced in a laboratory based on the calculated formation energy values. Additionally, we also noticed that the formation energy rises as the cation size increases in all structures (see Table 1). The reason may be attributed to the poor thermal stability. The formation energies were calculated by Equation (1)

$${\mathrm{E}}_{\mathrm{F}}^{\mathrm{Ai}}=\frac{{\mathrm{E}}_{\mathrm{Total}}^{{\mathrm{A}}_{\mathrm{i}}{\mathrm{BX}}_3}-{\mathrm{yE}}_{\mathrm{Total}}^{{\mathrm{BX}}_3}-{\mathrm{xE}}_{\mathrm{Total}}^{{\mathrm{A}}_{\mathrm{i}}}}{\mathrm{x}+\mathrm{y}}$$

(1)

where A_i is the different cation, and x and y are the numbers of A_i and BX₃ atoms in a unit cell, respectively.

2.4. Dynamical Stability

The total phonon density of states (Ph-DOS) is calculated at the equilibrium volumes for different cations of A_i, and displayed in Figure 4. For all these cations, a small imaginary frequency was observed for the CH₄, PH₄, CH₃OH, CH₃NH₃, CH(NH₂)₂, CH₃C(NH₂)₂, and C(NH₂)₃, indicating that these structures are stable or at least relatively stable at ambient conditions. Other than this, a clear imaginary frequency was observed in the alkali (Li₃O, and Li₃S) structures, indicating that these structures are unstable at ambient conditions.

2.5. Thermal Stability

Furthermore, we simulated the chemical reaction using an ab-initio molecular dynamics simulation that was based on the force calculations from density functional theory. The simulations were run on models with 3 × 3 × 3 supercells, for a total of 25 ps per case (see Figure 5). The smaller the organic component of a system, the more it is over-correlated with other parts of the system. This suggests that the smaller system is more easily moved around by electrostatic forces. A large-scale system mitigates this finite-size artifact, and produces a more reliable explanation of the anisotropic rotational behavior of cations. To compare the dynamics performance of the perovskites with the cations, the AIMD simulations are performed for the cubic NH₄PbI₃, PH₄PbI₃, Li₃OPbI₃, NH₃OPbI₃, Li₃SPbI₃, MAPbI₃, AMPbI₃, AcPbI₃ and GuPbI₃ perovskites with a 3 × 3 × 3 supercell. Figure 6 shows the AIMD simulation results for cubic Li₃OPbI₃ and Li₃SPbI₃ perovskites after 25 ps.

According to the AIMD simulation of the A_iPbI₃ perovskite depicted in Figure 6, the A_iPbI₃ perovskites exhibit no evident structural change after a 500 ps AIMD simulation at 300 K.

The A_i⁺ ions in the A_iPbI₃ perovskites do not congregate, the PbI₃ frame does not crumble (see Figure 5), and the temperature and energy curves indicate balance with a little shock (see Figure 6).

Furthermore, given the strong interaction between Li₃O⁺ and Li₃S⁺ ions in super-alkali perovskites, as well as the large-size cations, investigating the dynamics performance of perovskites based on various cations (A_i) is critical for identifying perovskites with stable dynamics performance [38]. This may be due to the hydrogen bonding between the cation and anions (PbI₃ frame) and the difficulty of movement of the cation inside the cavity of the structure, due to its large size, and might also be attributed to the electrostatic forces resulting from the transfer of charges between positive cations A+ and negative anions PbI6- (see Figure S3).

2.6. Electronic

In photovoltaic applications, determining the absorber layer material with appropriate electronic features plays an important role in improving device performance. The size of a band gap is important in estimating the number of photons that are absorbed, and the rate of photogenerated charge carriers with minimal optical losses. To generate and transport free charge carriers efficiently, low exciton binding energy and low effective masses of holes and electrons are necessary. Moreover, DFT studies reveal that, apart from LiO₃, LiS₃ and Gu, the electrical structures of the examined substituted perovskites were like those of MAPbI₃, although their bandgaps marginally decreased or increased compared to those of MAPbI₃. Our findings imply that adjusting the band gap and enhancing thermal and dynamical stability may both be accomplished by substitution in halide perovskite. To help us better understand the electronic structure of the optimized A_iPbI₃ perovskite structures, we calculated their band structures using the GGA-PBE method for A_i = NH₄, PH₄, Li₃O, NH₃O, Li₃S, CH₃NH₃, CH(NH₂)₂, Ace, and Gu along the high symmetry points of the Brillouin zone, which has taken three space groups—cubic, triclinic and orthorhombic—as shown in Figure 7a. Here, the Fermi level is set to zero. The maximum energy of valence band (VBM) and the minimum energy of conduction band (CBM) for all computed band structures are clearly shown in the figure, to be located along points R, Q and E of the Brillouin zone for (PH₄, Li₃O, CH(NH₂)₂), (Li₃S, Ace, Gu-NH₄) and (NH₃O, MA), respectively. Since these band gaps represent the direct nature of the band gaps, these materials can be called as having good optical absorption properties (Figure 7b,c). The calculated band gaps for NH₄PbI₃, PH₄PbI₃, Li₃OPbI₃, NH₃OPbI₃, Li₃SPbI₃, CH₃NH₃PbI₃, CH(NH₂)₂PbI₃, AcPbI₃ and GuPbI₃ are 1.7611, 1.6101, 0.5932, 1.5348, 0.4146, 1.5971, 1.4815, 1.9645 and 2.1517 eV, as given in

Table 3. Calculated open-circuit voltages (V_OC), short circuit currents (J_SC) and power conversion efficiencies (η) for the hypothetical perovskites A_iPbI₃ and MAPbI₃.

	JCS	FF	VOC	η (%)	Eg
NH4PbI3	24.84	66.4391	1.02	21.39543	1.7611
PH4PbI3	24.45	62.3652	1.05	20.34961	1.8410
HAPbI3	26.94	68.6404	1.10	25.84328	1.5348
MAPbI3	26.04	68.2843	1.09	24.63394	1.5971
FAPbI3	27.38	69.4876	1.006	24.32676	1.4815
AcePbI3	23.75	63.1115	1.01	19.24151	1.9645
GuPbI3	20.97	61.6615	1.005	16.51673	2.1517

respectively, which are in fair agreement with the experimental results and other theoretical calculations.

2.7. Optic

The optical properties of the solar cell, including the absorption edge, the intensity of the absorption at various wavelengths and the reflection coefficient, are crucial to determine its external and internal quantum efficiencies (see Equation (S5) in the Supplementary Materials). These optical properties are the result of light waves interacting with its valence electrons. The optical response of a material when it is interacted with an electromagnetic radiation, such as light waves, is given by the complex dielectric functions ε₁(ω) and ε₂(ω). For energies greater than E_g, our predicted absorption (α) compares favorably to a recent absorption measurement, providing important validation of our technique. According to the experiment (see Figure 8), the calculated α is 10⁴ cm⁻¹ at the start, and increases linearly (on a logarithmic scale) with energy. The visible slope variation around α = 2.5–3 eV replicates a comparable characteristic in the experimental spectrum [39]. The absorption curves of A_iPbI₃ and GaAs are identical throughout a large energy range, except for the energy onset corresponding to the different E_g of the studied materials.

2.8. Power Conversion Efficiency

Despite the cation substitution in the identical structures, these power conversion efficiencies were observed to be substantially closer to those of single-junction solar cells (shown in Table 3), which were estimated using Equation (S5) [40]. For each simulation, we considered a power input of P_S = 100 mW/cm², which is the power under the maximum AM1.5 solar illumination spectrum with T_Cell = 300 k and T_S = 6000 k.

The electron-hole pairs are easily separated and moved in the CBM and VBM states, which suggests that the perovskite-based solar cells studied so far have the potential to have high PCE, which may give them an edge over other cells in terms of power output. Figure 9a shows the PCE of solar cells based on these perovskites as a function of the band gaps. The cubic HAPbI₃ perovskite-based solar cells with a PCE of 25.84% are the best option, followed by the MAPbI₃ perovskites with a PCE of 24.63%. Experimental reports have shown that the efficiency of solar cells tends to be around 25.2% [10], while the maximum efficiency conversion is observed for a 1.4–1.6 eV bandgap. Moreover, the cubic GuPbI₃ perovskite with a direct maximum band gap of 2.15 eV also shows a PCE of 16.52%. The PCE results of these perovskites indicate that they are a good choice for single-junction solar cells. We have calculated PCE for our intrinsic perovskite solar cell with E_g of perovskite structures shown in Table 3. The efficiency of these structures, according to the Shockley–Queisser model, is represented in Figure 9b. Based on these results, the considered perovskites can reach an efficiency limit of 25.84% in ideal conditions, with T_C = 300 k and T_S = 6000 k for the AM1.5 spectrum.

Figure 9b shows the variation of the PV parameters as a function of the absorber’s thickness. Our results depicted from the simulations show that at a thickness of 60 nm, the efficiency rose as high as 24.63%, while FF = 68.28%, J_SC = 26.04 mA/cm² and V_OC = 1.09 V. The defects were considered in the simulated absorber layer. We found that the efficiency increases up to 25.84% as the thickness increases from 10 nm to 90 nm, thus representing the increased electron-hole pair generation rate in the absorber MAPbI₃ [41], while at higher thickness, the recombination of charges starts to occur earlier in the metal, before they reach the contact points—this results in a saturation of charges at thicker metal layers [42]. It was found that the amount of VOC decreased as the thickness of the absorber material, MAPbI₃, was increased. This was related to the short-circuit current (J_SC) and dark saturation current (J₀), and at higher thickness the dark saturation current rises, which causes the recombination probability of the charge carriers to increase. The Shockley-Queisser Equation (S10) clearly describes the relationship between V_OC, J_SC and J₀ [43]. Figure 10 illustrates the J-V curves of all structures, and the structure parameters are shown in Table 3.

3. Conclusions

In conclusion, we found that there is a high probability of the formation of 3D perovskite structures by calculating tolerance factors and octahedral factors for 486 permutations of ABX₃ organic-inorganic hybrid compounds reported in the literature. Nine different organic monoammonium cations were studied in combination with Pb metal ions and I halide. We used CH₃NH₃⁺ as a reference to estimate the steric bulk of molecular cations. We have found that the structures are dynamically stable and thermally efficient, and as a consequence, the adjustment in A+ revealed a significantly increased PCE of 25.84% in Hydroxylammonium (NH₃OH), compared to 24.63% in the MA of the control device, as well as an obvious improvement in its operational and thermal stabilities. There are many previously unknown compounds that could form stable perovskite phases at ambient conditions. This study is expected to lead to increased research efforts in designing perovskite structures with specific optoelectronic properties, as well as their great structural flexibility. This, combined with their promising physical properties, makes perovskites a powerful chemical structure for ferroelectrics and future photovoltaics technologies.