Promising Perovskite Solar Cell Candidates: Enhanced Optoelectronic Properties of XSrI3 Perovskite Materials under Hydrostatic Pressure

Density‐functional theory (DFT) has proven to be invaluable for investigating the physical properties of perovskite materials under varying pressure conditions to uncover potential applications in the field of optoelectronics. Herein, lead‐free XSrI3 (X = FA+, MA+, and DMA+ [formamidinium (FA+), methylammonium (MA+), and dimethylammonium (DMA)]) perovskites are designed and utilized using DFT calculations for promising solar cell applications. The application of pressure to these perovskites leads to a reduction in their lattice parameters, thereby enhancing atom interactions within the material. This compression of the crystal lattice also exerts a significant influence on the electronic band structure and the number of available electronic states, providing valuable insights into their semiconducting properties. Moreover, applying pressure results in a narrower bandgap in the perovskite halides, thus broadening the range of light absorption and potentially increasing light‐absorption efficiency. In this work, the feasibility of employing XSrI3 perovskites for enhanced optical performance is highlighted and valuable directions for further exploration in this field are offered. The insights gained from this theoretical study may hold the potential to advance the development of perovskite‐based materials for various optoelectronic applications.


Introduction
Perovskite solar cells (PSCs) have garnered significant attention in the field of solar energy due to their exceptional performance and potential for achieving high power-conversion efficiency (PCE).The reported PCE values for PSCs have rapidly increased from 3.8% to over 26% within just a few years. [1,2]PSCs employ lead halide perovskite materials of the ABX 3 structure as the light-absorbing layer, wherein A = formamidinium (FA þ ), methylammonium (MA þ ), and/or cesium (Cs þ ); B = lead (Pb 2þ ), tin (Sn 2þ ), and/or germanium (Ge 2þ ); and X = iodide (I À ), bromide (Br À ), and/or chloride (Cl À ).Despite the achievement of high PCEs, long-term operational stability remains a significant obstacle to realize commercialization.5] Perovskite materials possess exceptional abilities to accommodate structural distortions and potentially create new compositions through regulating the A, B, and X elements. [6,7]In 2018, Ke et al. reported that hydrogen iodide catalyzed the acidic hydrolysis of dimethylformamide, leading to the formation of dimethylammonium (DMA þ ). [8]Wang et al. demonstrated that DMA þ affects the crystallization kinetics of CsPbI 3 perovskites. [9][12] In contrast, inorganic perovskites containing strontium (Sr 2þ ) as the B-site cation have emerged as promising alternatives to their Pb 2þ -based counterparts due to their superior optical conductivity and absorption properties, making them attractive candidates Density-functional theory (DFT) has proven to be invaluable for investigating the physical properties of perovskite materials under varying pressure conditions to uncover potential applications in the field of optoelectronics.Herein, lead-free XSrI 3 (X = FA þ , MA þ , and DMA þ [formamidinium (FA þ ), methylammonium (MA þ ), and dimethylammonium (DMA)]) perovskites are designed and utilized using DFT calculations for promising solar cell applications.The application of pressure to these perovskites leads to a reduction in their lattice parameters, thereby enhancing atom interactions within the material.This compression of the crystal lattice also exerts a significant influence on the electronic band structure and the number of available electronic states, providing valuable insights into their semiconducting properties.Moreover, applying pressure results in a narrower bandgap in the perovskite halides, thus broadening the range of light absorption and potentially increasing light-absorption efficiency.In this work, the feasibility of employing XSrI 3 perovskites for enhanced optical performance is highlighted and valuable directions for further exploration in this field are offered.The insights gained from this theoretical study may hold the potential to advance the development of perovskite-based materials for various optoelectronic applications.
[15] In addition, the utilization of Sr 2þ -based perovskites can effectively address the environmental concerns associated with Pb 2þ -based materials.Ongoing research is focused on optimizing the synthesis and performance of these materials as well as exploring their potential in various device architectures and applications. [16][19] The application of hydrostatic pressure to halide perovskite materials results in a reduction of lattice parameters and crystal volume, which can impact the arrangement of cations and anions within the perovskite lattice, thus leading to displacements and rotations of octahedral cages.22] By precisely controlling the pressure, researchers can systematically investigate and manipulate diverse phases of halide perovskites with distinct optical and electrical characteristics.For instance, the application of hydrostatic pressure to inorganic halide perovskites, such as CsGeCl 3 , [23] CsGeI 3 , [24] KCaCl 3 , [25] and RbYbF 3 , [26] results in a narrowing of the bandgap and an increase in conductivity.Therefore, by exerting pressure, researchers can potentially tailor the properties of these materials to meet specific device requirements, such as those in solar cells or transistors.It is important to acknowledge that the influence of hydrostatic pressure on perovskite materials can be complex, and further investigation is necessary to fully comprehend the underlying mechanisms and optimize parameters for experimental applications. [27,28]n this study, we designed and investigated the properties of XSrI 3 perovskite materials, with X representing various organic cations, including FA þ , MA þ , and DMA þ .By employing densityfunctional theory (DFT), we explored their structural, electrical, and optical characteristics under different hydrostatic pressures.Our findings revealed that applying hydrostatic pressure led to notable improvements in the dielectric performance, electrical conductivity, and light absorption bandgap of XSrI 3 perovskites.These enhancements in optoelectronic properties make them highly promising for use in PSCs that require efficient light absorption and charge generation.The insights gained from this research contribute to the design and development of highperformance PSCs, bringing us closer to the realization of efficient and sustainable energy-conversion technologies.

Computational Details
The optoelectronic properties of XSrI 3 (X = FA þ , MA þ , and DMA þ ) perovskites were characterized using DFT modeling.The calculations were performed using the Cambridge Serial Total Energy Package.The physical features of XSrI 3 were computed using the generalized gradient approximation (GGA) and the Perdew-Burke-Ernzerhof functional.[31][32][33] Norm-conserving pseudopotentials were used to estimate the optical properties of the perovskite materials. [34,35]The energy cutoff for the calculations was set to 435.4 eV, and the Monkhorst-Pack scheme was used to generate an 8 Â 8 Â 8 k-point grid. [36,37]The maximum stress applied during the calculations was 0.1 GPa, the maximum force convergence criterion was set to 0.05 eV Å À1 , and the maximum displacement criterion was set to 2 Â 10 À3 Å.The convergence tolerance factor for the total energy was set to 2 Â 10 À5 eV atom À1 . [38]The structural optimization of the perovskite structures was carried out using the Broyden-Fletcher-Goldfarb-Shanno method. [37,39]The atomic positions and lattice parameters were relaxed during the optimization process.Following the structural optimization, the optical and electrical characteristics of the materials were calculated. [40] Results and Discussions

Structural Characteristics
Perovskite derivatives of XSrI 3 (X = FA þ , MA þ , and DMA þ ) have been demonstrated to possess three distinct space groups.These space groups are identified by the symbols P 1 (triclinic) for MASrI 3 , P 4 BM (tetragonal) for FASrI 3 , and P 21/c (monoclinic) for DMASrI 3 .The atomic positions and crystal structures are illustrated in Figure 1 and Table 1, respectively.Pugh's ratio, denoted as the B/G ratio, involves the bulk modulus (B) and the shear modulus (G), serves as an indicator of material properties.If this ratio falls below 1.750, the material is characterized as brittle, while values exceeding 1.750 suggest ductility. [41]dditionally, the Poisson's ratio (ν) can be employed to assess a material's brittleness or ductility.When ν surpasses the threshold of 0.26, the material is considered ductile. [42]The significant difference in Young's modulus (Y) suggests these materials will behave quite differently mechanically despite their similar chemical compositions.The FA þ substitution in FASrI 3 appears to dramatically increase stiffness compared to the MA þ and DMA þ variants.Table S1, Supporting Information, presents the values of both Pugh's ratio and Poisson's ratio for MASrI 3 , DMASrI 3 , and FASrI 3 .The observed indicators suggest that the materials MASrI 3 and FASrI 3 exhibit brittleness.

The Structure of the Electronic Bands and the Density of the States
Analyzing the band structure and density of states (DOS) is crucial for understanding the electronic behavior of perovskite materials with different compositions.The band structure provides information about the energy levels and electronic states in the material, while the DOS describes the distribution of these states with respect to energy.The band structures of MASrI 3 , FASrI 3 , and DMASrI 3 under pressures from 0 to 40 GPa are illustrated in Figure 2, S1, and S2, Supporting Information, respectively.FASrI 3 and DMASrI 3 are indirect bandgap semiconductors because electrons and holes have different crystal momenta in the conduction and valence bands (CB and VB).MASrI 3 is a direct bandgap semiconductor as the electron and hole momenta align in both bands.The bandgap character is independent of pressure.Table S2, Supporting Information, compares FASrI 3 , DMASrI 3 , and MASrI 3 bandgap values using  local density approximation (LDA) and GGA exchangecorrelation functions under varying hydrostatic pressures.At 0 GPa, GGA predicts higher bandgap values than LDA for MASrI 3 , FASrI 3 , and DMASrI 3 , and as pressure increases from 0 to 40 GPa, the bandgap decreases for all three perovskites with both LDA and GGA.The largest reduction occurs for MASrI 3 which decreases from 4.012 eV at 0 GPa to 1.715 eV at 40 GPa with GGA and from 3.582 to 1.477 eV with LDA, indicating that applied pressure and choice of computational method significantly impact predicted bandgap values, resulting in larger reductions at higher pressures and with LDA compared to GGA.Table 2 presents the bandgap values for pressures ranging from 0 to 40 GPa.When pressure is applied to a material, it can affect the interatomic distances, bond angles, and overall arrangement of atoms in the crystal lattice.These structural changes can have a significant impact on the electronic band structure of the material, including the bandgap.The application of pressure can compress the crystal lattice, leading to a decrease in the bandgap.Applying hydrostatic pressure reduces the bandgap energy required for electrons transitions from the valence to CB in MASrI 3 , DMASrI 3 , and FASrI 3 perovskites.This enables more efficient charge transport.Pressure also expands the valence and CB widths due to intensified atomic interactions.The direct bandgap of MASrI 3 means that the valence band maximum (VBM) and conduction band minimum (CBM) occur at the same momentum.So pressure-induced lattice structure changes directly impact the band energies, greatly reducing the bandgap.The influence of pressure on the bandgaps of DMASrI 3 and FASrI 3 , which possess indirect bandgaps, is comparatively less significant when compared to MASrI 3 with a direct bandgap.In general, hydrostatic pressure notably reduces the bandgap of MASrI 3 more than that of indirect DMASrI 3 and FASrI 3 .It is anticipated that under extremely high pressure, the bandgap will diminish to zero, resulting in metallic behavior of the material.
The partial DOS (PDOS) and total DOS (TDOS) play a crucial role in understanding the distribution and density of energy states in a material, providing valuable insights into its electronic structure and band formation.Electrical conductivity, as a measure of a material's ability to conduct electric current, relies on the availability of charge carriers and their mobility.The number of charge carriers can be estimated by considering the chargecarrier concentration and their effective mass.Higher chargecarrier concentration and mobility contribute to higher electrical conductivity.It should be emphasized that the determination of excited electrons, ground states, and charge carriers involves intricate calculations that consider various material-specific properties and electronic structures.The DOS provides information on the density of available electronic states across different energy levels.
The graph of TDOS versus energy illustrates the distribution of electronic states across the energy spectrum.As shown in Figure 3, with increasing pressure, a reduction in peak intensities was observed in the regions from 0 to À5 eV and from À15 to À17 eV of the TDOS, indicating alterations in the electronic structure of the materials.In particular, our findings demonstrate that application of pressure between 0 and 40 GPa leads to an expansion of the VBs. Figure S3-S5, Supporting Information, depict the PDOS for various elements including hydrogen, carbon, nitrogen, strontium, and chlorine in MASrI 3 , FASrI 3 , and DMASrI 3 .The PDOS provides valuable insights into the contribution of specific electronic states to the overall DOS in a given material, thereby facilitating a more comprehensive understanding of its electronic properties.
By PDOS, one can discern how individual atom's electronic states evolve under pressure, enabling a comprehensive comprehension of how pressure properties of perovskites.Consequently, PDOS enables identification of which atomic orbitals are most susceptible to pressure-induced changes, leading to phenomena such as valence.Correlating PDOS with alterations in the band structure offers valuable insights into the underlying electronic mechanisms governing the response to pressure.Based on the PDOS analysis, it is observed that the VBM is composed of Sr-3d and Sr-5p electronic energy levels, while the contributions to the CBM come from the I-5s-and Sr-3d-like states with high electronic energies.With pressure increasing from zero, the energy levels associated with the peaks in the PDOS at the Fermi level may undergo shifts or reductions to specific energy levels.The compression or shift of energy levels can result in the delocalization of PDOS peaks in both the CB and VB, indicating a higher degree of electron mobility and mixing between different energy states.Analysis of the PDOS reveals that DMASrI 3 has a higher DOS compared to MASrI 3 and FASrI 3 , which suggests a substantial density of electronic states within the energy bands of DMASrI 3 .This can be attributed to the presence of numerous valence electrons in DMASrI 3 , which make significant contributions to the electronic states.

Optical Absorption Spectrum
Optical functions play a crucial role in understanding the electrical nature and experimental application of materials, particularly in photovoltaic applications. [43]To observe the improvement of a perovskite candidate for solar cells and other related popular optoelectronic fields, the optical properties of lead-free XSrI 3 perovskite under varying hydrostatic pressures are comprehensively investigated, encompassing absorption spectra, optical photoconductivity, and dielectric functions.The optical absorption coefficient, denoted α(ω), is a measurement that determines the amount of energy that a material absorbs as a percentage per unit of length.As shown in Figure 4, the initiation of the absorption process for all materials does not occur at 0 eV, which can be attributed to the presence of a bandgap under normal pressure conditions.
The optical absorption properties of DMASrI 3 exhibit a higher sensitivity to applied hydrostatic pressure compared to MASrI 3 and FASrI 3 .Alterations in the electronic band structure induced by hydrostatic pressure can lead to either a redshift (a shift toward lower photon energies) or blueshift (a shift toward higher photon energies) of the optical absorption spectrum. [44]A decrease in the fundamental bandgap energy caused by pressure results in a blueshift of the absorption onset, while an increase in the bandgap width leads to a redshift. [45]This change in the spectral position of absorption peaks under pressure can be significant at higher applied pressures up to 40 GPa due to the pronounced influence of pressure on the electronic structure.The larger shift in peak absorption energies for DMASrI 3 suggests a greater pressure coefficient for its bandgap compared to MASrI 3 and FASrI 3 .Under identical pressure conditions, all substances initiate their absorption processes within the visible-to-ultraviolet transition zone (%3-4 eV).
When the pressure is increased to 40 GPa, the onset of absorption shifts to energies between 1.5 and 2.2 eV.The fundamental bandgap (E g ) decreases as the applied pressure increases according to Equation (1): where α is the pressure coefficient and P is the applied pressure.The narrowing of the bandgap is a consequence of the pressureinduced broadening observed in both the VB and CB.There is also an increasing trend in absorption with pressure showing significant peaks at 5-10 eV.In addition, two further significant peaks may be seen at 10-15 and 20-25 eV.The absorption of these compounds mainly peaks in the visible to ultraviolet range, between 5 and 10 eV.They consistently maintain a greater level of steady absorption between 10 and 20 eV.Overall, an increase in pressure leads to an elevation of absorption and a reduction in bandgap, which has significant implications for applications such as solar cells and optoelectronics.

Dielectric Function
The dielectric properties of perovskite materials play a critical role in determining their suitability for use in optoelectronic devices. [46]A comprehensive analysis of a material's optoelectronic properties under light exposure necessitates the incorporation of both the real and imaginary components of its dielectric function.The most valuable features, such as absorption coefficient and refractive index, are determined by the dispersion of these components. [47]The dielectric function ε ω ð Þ can be represented as a linear expression of the real and imaginary components ε 1 ω ð Þ and ε 2 ω ð Þ, as shown in Equation (2): The imaginary part of the dielectric function, ε 2 ω ð Þ, connects the band structure and DOSs for a particular material, while the real part, ε 1 ω ð Þ, determines the polarizability.The Kramers-Kronig relations [48] expressed as a quantum integral are often used to calculate the real component of a dielectric function from its imaginary part, as depicted in Equation ( 3) and ( 4): where P represents the principal value, while light frequency is denoted by ω.Ψ c k stands for the wave functions of the CB, while Ψ v k represents those of the VB at the specific k point.Moreover, e signifies the charge of electrons, Ω corresponds to the volume of the unit cell, and U is the vector aligned with the polarization of the incoming electric field.The delta function ensures the conservation of both energy and momentum when transitioning between occupied and unoccupied electronic states (E).Additionally, E c k and E v k are used to denote the electron energy within the CB and VB, respectively, at a given k-vector. [49]igure 5A,C,E depicts the real part, ε 1 (ω), of the dielectric function, and it can be observed that the static values of ε 1 (ω) are significant for all materials at 40 GPa.From zero photon energy, ε 1 at 40 GPa increases to a peak at 3.76 eV for MASrI 3 , 3.65 eV for FASrI 3 , and 3.19 eV for DMASrI 3 .Beyond the peak, there is a steep decrease for all materials at 0 and 40 GPa pressure.
Furthermore, the tendency of larger ε 1 ω ð Þ and ε 2 ω ð Þ at low photon energy and lower ε 1 ω ð Þ and ε 2 ω ð Þ at high photon energy emphasizes the potential of XSrI 3 in microelectronics and integrated circuits. [50]The imaginary part of the dielectric function, ε 2 (ω), is defined as the absorption coefficient of a material, which arises from transitions between the VB and the CB.Simulations of ε 2 (ω) spectra for halide materials MASrI 3 , FASrI 3 , and DMASrI 3 at 0 and 40 GPa are presented in Figure 5B,D,F.The primary peaks observed at zero pressure are located at 6.57, 5.70, and 6.97 eV for MASrI 3 , FASrI 3 , and DMASrI 3 , respectively.Under applied hydrostatic pressure, these prominent peaks increase in intensity while shifting toward higher photon energies.Moreover, the high dielectric constants of all three result in strong IR-visible absorption properties, indicating their potential for use in device applications.In summary, the ε 2 (ω) values for all the materials are sufficiently large to enable their use in optoelectronic and photovoltaic applications.This study suggests that MASrI 3 , FASrI 3 , and DMASrI 3 materials are poised to become promising candidates for optoelectronic and solar cell applications.

Loss Function
The energy loss function (L(ω)) of XSrI 3 is depicted in Figure 6A-C, representing the energy dissipated by an electron during a rigid collision while traversing through the substance.The plasma frequency, denoted as ω p , corresponds to the resonant frequency of the maximum L(ω) associated with plasma resonance.At 0 GPa pressure, the highest peak values for MASrI 3 , FASrI 3 , and DMASrI 3 were 2.124, 1.601, and 1.099 located at 17.27, 23.03, and 15.57eV, respectively.As the pressure increases from 0 to 40 GPa, the peak energy moves higher, indicating movement of the perovskite's plasma frequency.It can be shown that the ultraviolet region has the most energy loss since the photon energy there is greater than the bandgap energy. [51]The energy loss function describes the energy loss of electrons or photons traveling through a material due to interactions with its electrons.It depends on the complex dielectric function ε(ω) of the material, which itself is related to the electronic band structure.The expression for the energy loss function, denoted as L(ω), as shown in Equation ( 5): [52,53] L ω Physically, Equation (5) represents the efficiency with which the material can convert the energy of an incoming electron or photon into excitations of its own electrons.A higher L(ω) at a given frequency indicates more efficient energy loss, absorption, and excitation of electrons in the material.
However, the semiconducting nature and greater effective mass of conduction electrons may result in relatively low values of ω p and L(ω).Nonetheless, the overall optical behavior of these materials predicts high absorption and minimal energy loss, rendering them promising compounds for cutting-edge optoelectronic applications and devices.

Reflectivity
The surface response of the MASrI 3 , FASrI 3 , and DMASrI 3 materials is influenced by the reflection of incident light.Figure 7A-C provides the reflectivity or reflectance of these materials as a function of energy. [54]Under 0 GPa hydrostatic pressure, MASrI 3 exhibits a peak reflectance of 0.24 at 7.03 eV, FASrI 3 displays a peak reflectance of 0.23 at 5.67 eV, and DMASrI 3 demonstrates a peak reflectance of 0.124 at 7.17 eV.Under hydrostatic pressure of 40 GPa, the peak reflectance values of MASrI 3 , FASrI 3 , and DMASrI 3 undergo changes.Specifically, MASrI 3 exhibits a peak reflectance of 0.36 at 7.98 eV, FASrI 3 displays a peak reflectance of 0.34 at 7.58 eV, and DMASrI 3 shows a peak reflectance of 0.36 at 8.65 eV.These variations indicate that the surfaces of the materials reflect incoming light at different angles and energies under increased pressure, which can be attributed to modifications in their crystal lattice, electronic bandgap, or electronic states.

Refractive Index
The extinction coefficient k ω ð Þ denotes a material capacity to absorb a specific wavelength of incident light, while the refractive index n ω ð Þ indicates the degree of transparency of the material toward incoming radiation.The following dielectric function relationships are utilized to calculate n ω ð Þ and k ω ð Þ: [55] The real component of the complex refractive index determines the phase velocity of an electromagnetic wave propagating through a material, while the imaginary component represents its attenuation or absorption.The real part of the refractive index (n ω ð Þ) characterizes the extent to which light is decelerated or refracted as it traverses a material, and determines the phase velocity, i.e., the speed at which wave phases propagate through said material.A higher refractive index indicates a slower phase velocity, resulting in greater refraction or bending of light when transitioning between media.In contrast, the imaginary part of the refractive index (k ω ð Þ) represents the extinction coefficient.It quantifies the absorption or attenuation of the electromagnetic wave as it travels through the material.If the extinction coefficient is zero, it indicates that the material does not absorb or attenuate the incident light at that specific wavelength.Figure S6A-C, Supporting Information, illustrates the relationship between the complex refractive index and incident light energy under hydrostatic pressures.In FASnI 3 , the maximum peak exhibits a refractive index (n ω ð Þ) of 2.51 at 5.05 eV.For MASrI ð Þ) values are 1.90 at 8.46 eV for DMASrI 3 , 1.90 at 7.86 eV for MASrI 3 , and 1.77 at 7.52 eV for FASrI 3 .At high energies, all the light will be absorbed, resulting in no transmittance.In contrast, at low energies, there are no relevant electronic transitions, leading to strong transmission.

Conclusions
In summary, we have predicted the properties of lead-free perovskite XSrI 3 (X = MA, FA, and DMA) under various hydrostatic pressures.As the hydrostatic pressure increases, the bandgap of the perovskites decreases.At 40 GPa, MASrI 3 , FASrI 3 , and DMASrI 3 exhibit optimum bandgaps of 1.715, 1.594, and 2.178 eV, respectively.Under hydrostatic pressure, the perovskites demonstrate improved dielectric performance, enhanced electrical conductivity, and stronger light absorption capabilities compared to when they are not under pressure.This suggests that applying hydrostatic pressure can enhance the optoelectronic performance of these materials and yield a more desirable energy bandgap, making the perovskites more promising for PSCs.This study not only contributes to the understanding of the effects of hydrostatic pressure on perovskite materials but also provides valuable theoretical guidance for the development and manufacture of PSCs.Hydrostatic pressure induces alterations in the optical properties of materials through its influence on the electronic band structure, which plays a pivotal role in determining how materials interact with light.From an electronic physics perspective, this phenomenon is characterized by shifts in energy levels, changes in electron wave functions, and variations in the likelihood of electronic transitions within the material.By leveraging the advantages of hydrostatic pressure, researchers can further optimize the optoelectronic performance of perovskites and potentially enhance the efficiency and stability of PSCs.