Structural propertiesBy examining the crystal lattice parameter and applying Murnaghan’s equation of state to the volume of the unit cell, the structural characteristics of RPbBr3 (R = Cs, Hg, and Ga) were identified. We performed geometric optimization on our compounds’ intrinsic unit cell architectures, paying close attention to the Pm3m (221) space group with a cubic nature in the CASTEP code20,21. Volume fluctuations may be seen in the equilibrium cell and unit cell. The structural qualities alter when different R atoms are used, as is clear. When the Y element in ABY3 materials (where Y is O, Br, Cl, or F) is changed, the atomic variations of the element have an impact on the structural features. As a result, changing an atom affects its structural and non-structural characteristics24. In this paper, the lattice parameters (a = b = c) were changed by substituting one atom, which led to modifications in the compound volume as a result of the lattice parameter changes.Through the optimized geometry, we determined the lattice parameters and volume, which are given in Table 1. These values, as presented in Table 1, exhibit similarities to previously reported data23,25, highlighting the precision of our first-principles calculations. Each unit cell has five atoms: one atom of Cs(cesium), Ga(gallium), Hg(mercury), one atom of Pb(lead), and three atoms of Br(bromine) as shown in Fig. 1. Figure 2a–c shows a mapped image of the electron density difference (EDD) in the (100) planes. Here, it can be observed that the plane’s Br element is where charge accumulates to the greatest extent, whereas the Pb atom or element is where charge depletes. Stated differently, covalent bonding is indicated by the overlapping of electron clouds between these two components7,12,13. The covalent connection between the Br-Pb atoms is significantly supported by this charge distribution route. It is discovered that the charge distribution around the atoms is almost spherical, indicating ionic bonding and being comparable to other known perovskites. Furthermore, the ionic character of the Br–Pb bond is demonstrated by the MP analysis, which verifies that the bond’s population number is negative and less than zero (− 0.45, − 0.42)7.
Table 1 Compound RPbBr3 (R = Cs, Hg, and Ga) lattice constant a (Å), volume Å3, bandgap BG (eV), density ρ (amu/Å3), formation energy F.E (eV).Fig. 1Crystal structure of perovskite material (a) CsPbBr3, (b) GaPbBr3, and (c) HgPbBr3.Fig. 2Electron density difference (EDD) of compounds (a) HgPbBr3, (b) GaPbBr3, and (c) CsPbBr3.We have not observed any structural distortion or alteration in our compounds. The charge distributions about the planes are unaltered since the relaxation reveals that the cation molecule is not shifting at all. However, over this plane, (100) exhibits a comparable distribution of charges with a minimal cation tilt8.Figures 3a–c show the energy vs volume curve of the substances CsPbBr3, GaPbBr3, and HgPbBr3 as obtained using the CASTEP software. Curves show that as the energy increases the volume of the compounds increases which also increases the compound’s lattice parameters compared with ref.12,13,14. The Fig. 3a–c demonstrates that volume increases continuously as unit cell energy declines. The stable volume of the compounds CsPbBr3, HgPbBr3, and GaPbBr3 are given as 219.94, 200.64, and 208.59 Å3 as given in Fig. 3a–c (see supporting imformation 3). It comes to its lowest value. The comparison of the compoundsCsPbBr3, GaPbBr3, and HgPbBr3 volume (Å)3, bandgap energy, and lattice parameter (Å) is explained in Table 1.Fig. 3Unit cell energy vs volume optimization curve of (a) CsPbBr3, (b) GaPbBr3, and (c) HgPbBr3.The study makes it clear that the values found in this examination are similar to those found in previous research13. Fitting related volume using Birch Murnaghan’s (BM) equation allows energy values of different states to be discussed27. Compound formation energy plays a vital role in explaining phase stability and thermal stability. According to the given formation energy equation:$$\text{F}.\text{E}= \frac{\left({\text{RPbBr}}_{3}\right)-(\text{E}\left(\text{X}\right)+\text{E}\left(\text{Pb}\right)+3\text{E}(\text{Br}))}{\text{n}}$$Here R represents the Hg, Ga, and Cs bulk energy. According to our results, the formation energy of the compounds CsPbBr3, GaPbBr3, and HgPbBr3 are − 3.46 eV, − 3.14 eV, and − 2.21 eV are negative. More negative values of the compounds are considered to be more stable28.Figures 4a–c show the computed phonon dispersion spectra along the Brillouin zone (BZ) primary symmetry direction29. It is commonly known that the number of atoms in a primitive cell is three times greater than the number of vibrational modes30,31. The stability of the compounds is shown by the phonon dispersion by the distribution of the electrons (see supporting information data 4).Fig. 4Phonon calculations of perovskite materials (a) CsPbBr3, (b) HgPbBr3, and (c) GaPbBr3.The Yellow–Red–Blue spectrum, which highlights the regions from electronic deficiency to enrichment, is used in the EDD plots to depict the location of the distribution of charges as an isosurface produced from the MP analysis of populations (a, b, and c)8.Band structure and DOSThe band structure and density of states for RPbBr3 (R = Cs, Hg, and Ga) are shown in Fig. 5. Insights regarding the conduction band’s energy range and the valence band’s absence of electron availability are provided by the band structure32. The valence VB and conduction band CB are separated by the Fermi energy (EF). The CB is placed above the EF, whereas the valence bands are below it29,33. The band gaps were estimated by subtracting the CB minima (CBM) from the VB maxima (VBM).Fig. 5RPbBr3 (R = Cs, Hg, and Ga) Compound (a, b, and c).band structure and (d) DOS.Depending on the band structure shown in Figs. 5a–c, a semiconductor material’s energy bandgap can either be direct or indirect in nature (see supporting information data 5). When the greatest point of the VB perfectly lines up with the minimum point of the CB, the bandgap is referred to as a direct bandgap (BG).On the other side, an indirect BG exists when the compound VB maximum and CB minimum are not exactly aligned with each other34. The VB maximum and CB minimum of the material under study, RPbBr3 (R = Cs, Hg, and Ga), are unique from one another, pointing to the existence of a direct bandgap in compounds. Figures 6, 7, and 8 show the partial and elemental density of states (DOS) for CsPbBr3, which has direct band gaps of 1.85 eV, 1.56 eV, and 1.71 eV. The major peak in the density of states (DOS) for HgPbBr3, CsPbBr3, and GaPbBr3 is 19.46 eV at − 3.99 eV, 13.56 eV at − 1.59 eV, and 12.21 eV at − 1.16 eV (see supporting imformation data 6, 7, 8).Fig. 6(a) PDOS and (b) Cs-PDOS, (c) Pb-PDOS, and (d) Br-PDOS of CsPbBr3 compound.Fig. 7(a) PDOS and (b) Hg-PDOS, (c) Pb-PDOS, and (d) Br-PDOS of HgPbBr3 compound.Fig. 8(a) PDOS and (b) Ga-PDOS, (c) Pb-PDOS, and (d) Br-PDOS of GaPbBr3 compound.The composition of the compounds HgPbBr3, CsPbBr3, and GaPbBr3 primarily consists of d-states, p-states, and s-states, while the presence of f-states is minimal.At the compound energy Fermi level, there are no prominent peaks in the density of states, suggesting the presence of impurities in the materials35. Despite this, the compounds exhibit small energy bandgap values, indicating that they are not insulators.Therefore, these materials can function as semiconductors, making them suitable for applications in optoelectronics, which is a crucial aspect to consider36. The compound partial DOS and elemental DOS for CsPbBr3 are observed in Figs. 6, 7, and 8. As we observed the partial DOS graph in Figs. 6a, 7a, and 8a, we came to know that the main peak of the p-state for CsPbBr3, HgPbBr3, and GaPbBr3 is 13.49 at − 0.89 eV, 9.41 at − 3.99 eV, and 12.01 at − 1.13 eV, which is the highest peak of the compound. The Figs. 6b, c, and d, Fig. 7b, c, and d, and Fig. 8b, c, and d show the elemental compound PDOS for Cs, Pb, Hg, Ga, and Br, respectively.The main peaks of the compounds are CsPbBr3 Cs: 11.71 at − 7.21 eV, Pb: 3.45 at 3.52 eV, and Br: 13.12 at − 0.89 eV; HgPbBr3 Hg: 19.52 at − 3.99 eV, Br: 11.87 at − 0.81 eV, Pb: 3.50 at 3.72 eV; and GaPbBr3 Ga: 8.18 at 2.65 eV, Pb: 3.38 at 3.61 eV, and Br: 11.78 at − 1.12 eV. The compound population analysis is employed to better comprehend the molecule’s bonding characteristics29. Chemicals are classified as ionic or covalent depending on whether their bonding value is greater than one37. Consequently, we see in Table 2 the Mulliken populations (MP) values for the RPbBr3 (R = Cs, Hg, and Ga) compounds.
Table 2 Compound RPbBr3 (R = Cs, Hg, and Ga) Mullikan population analysis.Optical propertiesRefractive index, energy loss, conductivity, and non-conductive behavior were among the optical characteristics of RPbBr3 (R = Cs, Hg, and Ga) that were thoroughly examined and characterized. The frequency of electromagnetic waves affects these particular optical properties because they result from the wave-matter interaction between the substance and the waves35. Calculating the dielectric function ε(ω), which may be represented by the following equation, is necessary to evaluate the photo-sensitive behavior ε(ω) = ε1 (ω) + i ε2 (ω).The real (Re) part of the material’s dielectric function (DF), represented as ε1 (ω), and the compound imaginary (Im) part, denoted as ε2 (ω), were employed in the calculation38. The intrinsic properties of the compound, while impart denoted energy losses during the interaction with electromagnetic waves, Additional optical qualities can be found by applying the relevant formulas.These characteristics include the refractive index, denoted by the symbol n (ω), energy loss, denoted by the symbol L (ω), absorption coefficient, denoted by the symbol I (ω), and reflectivity, denoted by the symbol R (ω)39. The reflectivity of CsPbBr3, HgPbBr3, and GaPbBr3 observed in Fig. 9a which varies from a higher value of 0.30 at 14.87 electron volts (eV), 0.30 at 0.0 eV, and 0.32 at 4.81 eV, and reflectivity starts from 0.11, 0.30, and 0.13 at zero eV. To identify how RPbBr3 (R = Cs, Hg, and Ga) reacts to particular light wavelengths, the optical absorption coefficient of the material are determined (see supporting imformation data 9).Fig. 9(a) Reflectivity and (b) absorption, and (c) loss function of compound RPbBr3 (R = Cs, Hg, and Ga).Additionally, this knowledge is necessary for determining the feasibility of solar cell compounds that rely on capturing solar energy on a practical level. CsPbBr3, HgPbBr3, and GaPbBr3 compounds have a maximum absorption peak at 266,893.04 cm-1 at 14.43 eV, 201,188.455 cm−1 at 13.81 eV, and 192,529.7.11 cm-1 at 12.59 eV, as shown in Fig. 9b. It’s interesting to note that compounds CsPbBr3, HgPbBr3, and GaPbBr3 have zero absorption at 2.16 eV, 1.48 eV, and 0.11 eV, as seen in the absorption spectrum. Additionally, peaks of absorption are 10.75 eV, 10.16 eV, and 4.93 eV; the greatest absorption peaks for CsPbBr3, HgPbBr3, and GaPbBr3 are seen in the graph clearly. But Fig. 9c shows the loss Function of compound RPbBr3 (R = Cs, Hg, and Ga) evidently.CsPbBr3, HgPbBr3, and GaPbBr3 display in Fig. 10a a real peak in conductivity at 4.34 fs−1 at 13.80 eV, 3.85 fs−1 at 9.78 eV, and 3.58 fs−1 at 4.25 eV. CsPbBr3, HgPbBr3, and GaPbBr3’s main real conductivity peaks are 4.33 eV, 5.66 eV, and 8.81 eV. In particular, CsPbBr3, HgPbBr3, and GaPbBr3 compounds have an imaginary part shown in Fig. 10b that is almost zero at 0 eV (see supporting imformation data 10). The main highest peaks are shown at 2.90 fs−1 at 14.72 eV, 2.23 fs−1 at 14.09 eV, and 2.21 fs−1 at 12.88 eV for CsPbBr3, HgPbBr3, and GaPbBr3. The conductivity properties of CsPbBr3 play a significant role, particularly in the real part, and notable conductivity occurs at 2.16 eV, 1.48 eV, and 0.11 eV, indicating an important characteristic.Fig. 10Conductivity (a) δ1(ω) and (b) δ2(ω) of compound RPbBr3 (R = Cs, Hg, and Ga).The real part of the compound dielectric constant signifies the ability to interact with a material electric field, storing and transmitting without absorbing energy40. On the other hand, the imaginary component represents the capacity of a material to permanently absorb energy from an electric field that varies over time. For CsPbBr3, HgPbBr3, and GaPbBr3, the maximum main real portion dielectric function, as seen in Fig. 11a, is observed at 6.30 at 3.16 eV, 7.64 at 2.93 eV, and 11.82 at 0.0 eV. In comparison, the imaginary component for CsPbBr3, HgPbBr3, and GaPbBr3 is located at 5.16 at 4.22 eV, 4.94 at 5.33 eV, and 7.09 at 4.04 eV. Figure 11b shows the significant values of ε2(ω) of compound RPbBr3 (R = Cs, Hg, and Ga) (see supporting imformation data 11).Fig. 11Dielectric function (a) ε1(ω) and (b) ε2(ω) of compound RPbBr3 (R = Cs, Hg, and Ga).At 2.48 at 20.30 eV, 1.67 at 15.24 eV, and 1.84 at 20.02 eV, more notable peaks in the loss functions of CsPbBr3, HgPbBr3, and GaPbBr3 are discovered. The compounds have loss function values of 17.90 eV, 17.53 eV, and 14.81 eV. The major main peaks of the refractive index k(ω) for CsPbBr3, HgPbBr3, and GaPbBr3 are found at 2.54 at 3.30 eV, 3.40 at 0.0 eV, and 2.83 at 3.15 eV. Among the examined compounds, CsPbBr3 refractive index n(ω) starts at 2.02, 3.40, and 2.18 at zero eV, and refractive index k(ω) for CsPbBr3, HgPbBr3, and GaPbBr3 values of zero are still 1.90 eV and 1.41 eV, as shown in Fig. 12a and b (see supporting imformation data 12). As a result, HgPbBr3 has greater refractive index (n) values than the other compounds. The matching energy and wavelengths have a significant impact on the transmittance of light. Due to considerable microelectronic changes and little conduction, there is no transmittance observed at higher energies when the wavelength is shorter39. On the other hand, conduction increases with lower wavelengths as a result of fewer microelectronic changes brought on by lower energy.Fig. 12Refractive Index (a) n(ω) and (b) k(ω) of compound RPbBr3 (R = Cs, Hg, and Ga).The compounds under investigation exhibit notable absorption peaks in the energy range, with the greatest peak being noted at 14.43 eV. Because of its optical and electrical properties, this substance may be used in solar cells and light-emitting semiconductor diodes. But HgPbBr3 plays a bigger part between the two because of its smaller band gap. Enhancing electrical contact, reducing density defects during transmission carrier loss, and achieving high power efficiency conversion were the main goals of our research using DFT methodologies41. As a result, the results of the calculations are useful when applied to solar cell and LED applications.Mechanical propertiesThe elastic characteristics of a crystal can be used to predict how it will respond to applied forces. Understanding these characteristics is essential for comprehending a compound’s mechanical behavior. Using the elastic constant values C11, C12, and C44, the mechanical nature and elastic characteristics of the compounds were examined in this work. An essential quality that characterizes a substance’s nature is its strength42. The combination is found to be mechanically stable by examining the correlations C11 – C12 > 0, C11 + 2C12 > 0, and C44 > 0. The adjustable anisotropy factor A determines whether a material is isotropic43. The A value of 1.0 indicates isotropy, while a deviation from 1.0 suggests significant anisotropy. Table 3 shows the detailed value of the elastic coefficient of said compounds.
Table 3 Compound RPbBr3 (R = Cs, Hg, and Ga) elastic coefficients (Cij) at ambient pressure and anisotropic factor A.For RPbBr3 (R = Cs, Hg, and Ga), we have performed calculations to determine the various mechanical properties of the material. These include the bulk modulus (B), shear modulus (G), Young’s modulus (E), Pugh’s ratios (B/G), and Poisson’s ratios (v). The bulk modulus characterizes the material’s resistance to volume decrease under pressure44. The shear modulus represents the relationship between shear stress and shear strain. Young’s modulus is a measure of the material’s elasticity, obtained by calculating the stress-to-strain ratio during uniaxial deformation45. These mechanical parameters can be determined using the following equations:$$\text{B}= \frac{{C}_{11}+2{C}_{12}}{3}$$$$\text{G}= \frac{3{C}_{44}+{C}_{11}-{C}_{12}}{5}$$$$\text{Y}= \frac{9\text{BG}}{3B+G}$$$$\text{v}= \frac{3\text{B}-\text{E }}{2(3B+G)}$$$$\text{A}= \frac{2{C}_{44}}{3{C}_{11}-{C}_{12}}$$The anisotropic factor A values are 2.135, 3.651, and 10.602 of CsPbBr3, HgPbBr3, andGaPbBr3and these results are better because of large values as compared to the given references12,23,26. The bulk modulus can be used to gauge a material’s hardness. The discovered compound RPbBr3 (R = Cs, Hg, and Ga) has a greater B value, indicating that it is a harder material. Its high Young’s modulus and shear modulus values attest to RPbBr3’s (R = Cs, Hg, and Ga) hardness. To establish whether a compound behaves in a ductile or brittle manner, utilize Poisson’s ratio (v)46. If the value is greater than 0.26, a material is categorized as ductile, and in other cases, it is a brittle material. The Poisson’s ratio (v) values for the compounds CsPbBr3, HgPbBr3, and GaPbBr3 under study show that they are ductile as observed in Table 4.
Table 4 Compound RPbBr3 (R = Cs, Hg, and Ga) modulus (GPa), Poison’s ratio (v), Vickers’s hardness Hv, Pugh’s ratio B/G, and lame lambda λ (GPa).The RPbBr3 (R = Cs, Hg, and Ga) B/G ratio is used to calculate a material’s malleability or brittleness37. Table 5 depicts that RPbBr3 (R = Cs, Hg, and Ga) has fantastic Elastic Debye temperature (K), Compressibility β (1/GPa), Averaged sound velocity vm (m/s), and mechanical stability values, respectively. The material is categorized as ductile if the B/G ratio exceeds 1.75, while a ratio below 1.75 indicates a brittle nature. According to the B/G ratios, RPbBr3 (R = Cs, Hg, and Ga) exhibits a ductile nature22,38. The mechanical stability µM (= G/C44) defines the plastic strain and lubricating properties of the compounds. Hence, it is concluded that GaPbBr3 has the highest lubricating properties as compared to other compounds.
Table 5 Compound RPbBr3 (R = Cs, Hg, and Ga) Elastic Debye temperature (K), Compressibility β (1/GPa), Averaged sound velocity vm (m/s), and mechanical stability.
Computational investigation on physical properties of lead based perovskite RPbBr3 (R = Cs, Hg, and Ga) materials for photovoltaic applications
