Lead-free perovskites InSnX3 (X = Cl, Br, I) for solar cell applications: a DFT study on the mechanical, optoelectronic, and thermoelectric properties

This study aims to explore for the first time the mechanical, electronic, optical and thermoelectric properties of cubic lead-free perovskites InSnBr3 and InSnI3 to investigate their potential applications in solar cell devices. Additionally, the previously examined InSnCl3 perovskite is also included. The properties of the perovskites were determined using first-principles calculation based on the well-known Density Functional Theory (DFT) with the Generalized Gradient Approximation (GGA) functional implemented in the Quantum Espresso package. One of the most important findings was that the bandgaps of the compounds decrease and undergo an indirect-to-direct bandgap transition when Cl is replaced by Br and I. This indicates that InSnBr3 and InSnI3 perovskites are more suitable for solar cell applications. The bandgap energies for InSnCl3, InSnBr3, and InSnI3 perovskites are 0.59 eV (R→X), 0.44 eV (R→R), and 0.24 eV (R→R), respectively. The improved band gaps using the HSE06 functional are 2.35 eV, 2.13 eV, and 2.01 eV for the respective perovskites. The materials were found to possess chemical, mechanical, and thermodynamic stability as well as ductile behaviour. Furthermore, the materials exhibit remarkable optical properties, including high absorption coefficients and relatively small reflectivity. The calculated thermoelectric properties indicated high electrical conductivity and reasonable figure of merit values, making them promising candidates for the application in thermoelectric devices.


Introduction
Perovskite solar cells (PSCs) have recently evolved as one of the most promising candidates to replace the conventional silicon-based solar cells, thanks to their high efficiency and cost-effective fabrication process.Currently, Pb-based halide perovskite solar cells exhibit the highest efficiency compared to other types of PSCs on a global scale.However, concerns regarding highly perceived toxicity of Pb in these devices have impeded their large-scale commercialization, hampering their large-scale production [1][2][3][4][5].Consequently, the development of environmentally friendly lead-free PSCs has become crucial to realize their large-scale production.To address this challenge, a number of novel lead-free perovskite materials have been computationally and experimentally explored.
Ideal Pb-free materials as absorbers in PSCs should possess several key characteristics including low toxicity, narrow and direct bandgaps, high optical-absorption coefficients, high mobilities, long charge-carrier lifetimes,

Computational details
In this study, the first-principle calculation of the optoelectronic and elastic properties of InSbX 3 (X = Cl, Br, I) perovskites were evaluated based on the DFT [48] using the Quantum Espresso (QE) package [49].The GGA-PBE functional [50] was applied with the ultrasoft pseudopotential while Broyden Fletcher-Goldfarb-Shanno (BFGS) minimization procedure was used in the structural optimization.In addition, the optical and elastic properties of the perovskites were computed using the thermo_pw program.To obtain more accurate band gap energies, the HSE06 functional was also applied.
The cut-off for the wave function and the charge density energy were chosen to be 50 Ry and 500 Ry, respectively.For the Self-Consistent Field (SCF) calculation, K-points of 6 × 6 × 6 were used whereas for the Non-Self-Consistent Field (NSCF) calculation, k-points of size 12 × 12 × 12 were applied.The convergence threshold for the total energy was set to be 10 −8 Ry.
By using this cubic structure, the total energy of the lead-free perovskites was computed as a function of the lattice parameters.The results are presented in figures 2(a)-(c), for InSnCl 3 , InSnBr 3 and InSnI 3 perovskites, respectively.These figures clearly indicate that the unit cell undergoes expansion when Cl is substituted with Br and I anion, which aligns with the expected behavior.
The Birch-Murnaghan equation of states [51,52] was then used to optimize the structure of the perovskites and the results are depicted in table 1.
It is clear from table 1 that the optimized lattice parameter of InSnCl 3 obtained using the GGA-PBE functional in the present work is slightly lower than the corresponding value reported by Khan et al [41], who used the GGA with Hubbard term (GGA+U).The discrepancy in the lattice parameters of perovskites resulting from the use of different functionals is common, as observed in various other compounds including SrNiO 3 [53] and several Thallium based fluoroperovskites presented in [11].Table 1 also indicates that the replacement of Cl by Br and I anions leads to an increase in the optimized lattice parameters.This can be attributed to the larger ionic radii of the anions, which is generally true for many halide perovskite compounds [54][55][56][57][58][59][60][61][62][63][64][65][66].
The formation energy E f ∆ of the perovskites was also calculated using equation (1) to evaluate their chemical and thermodynamic stability.
Here, N is the number of atoms in the unit cell, E InSnX tot 3

(
)is the total energy of the InSnX 3 (X = Cl, Br, I) perovskites while E In , ( ) E Sn ( )and E X ( )are the energy of In, Sn, and X atoms, respectively.The computed formation energy values are presented in table 1.It is important to note that the negative formation energy values  of the compounds indicate their chemical stability, suggesting that these compounds can be experimentally synthesized under ambient conditions.
To investigate the thermodynamic stability of the studied perovskites, the phonon dispersion curves were calculated and the results are shown in figure 3. It is clear from figure 3 that InSnX 3 (X = Cl, Br, I) perovskites possess positive lattice vibrational frequency for all phonon modes, implying their thermodynamic and dynamical stability.Similar vibrational properties were also shown for various perovskites including lead-free double perovskites Ga 2 PdX 6 (X = Cl, Br, and I) [30].The authors reported that the stability is accompanied by the coincidence of three vibrational modes at Γ direction, which is also true in the current study.

Mechanical and elastic properties
Table 2 summarizes the calculated elastic constants and of mechanical properties of the studied perovskites.In this work, the Born-Huang mechanical stability criteria [67] were used to evaluate the mechanical stability of the perovskites.Based on the calculated data presented in table 2, it can be inferred that InSnCl 3 , InSnBr 3 , and InSnI 3 perovskites are mechanically stable because their elastic constants and the bulk moduli satisfy the Born-Huang criteria: C 11 > 0, C 44 > 0, (C 11− C 12 ) > 0, (C 11 + 2C 12 ) > 0, and C 12 < B < C 11 .The mechanical stability  of InSnCl 3 perovskite was previously predicted by Khan et al [41], where the elastic constants of the compound were found to be in good agreement with results presented in the current study (table 2).
The ductility or brittleness of materials, including perovskites, is an important property for their applications in devices.This behaviour can be determined from the Poisson ratio of the materials, with a value greater than 0.26 indicating ductility while a value lower than 0.26 suggesting the brittleness [68].In the present study, all studied perovskites exhibit the Poisson ratio values greater than 0.26, indicating their ductile nature.Another value, Pugh's ratio, can also be used to evaluate their ductile or brittle behaviour, with a value larger than 1.75 representing ductility and vice versa [69].As seen from table 2, the Pugh's ratio values of InSnCl 3 , InSnBr 3 , and InSnI 3 perovskites are 3.007, 2.913, and 2.577, respectively, further confirming their ductile behaviour.This property can be further confirmed by the Cauchy pressure values of the materials, where positive Cauchy pressure represents ductile behaviour while negative values indicate brittleness [70].It is obvious from table 2 that all the three materials possess positive Cauchy pressure values, further supporting their likelihood of exhibiting ductile behaviour.This significant finding strongly suggests that these materials can be effectively deposited as high quality thin films for a wide range of applications including optoelectronic and solar cells [4,71].
Among the studied perovskites, InSnCl 3 perovskite exhibits the largest bulk modulus (22.749GPa) and shear modulus (7.566 GPa), indicating its supperior resistance to applied pressure and shear forces.However, these mechanical properties slightly decrease when Cl is substituted by Br and I.The reduction in the bulk modulus of the compounds for the anion replacement is closely related to the increase in the lattice parameters of the compounds (table 1).As for the Young modulus, which characterizes the stiffness of materials, InSnCl 3 is also predicted to be the stiffest material in the present study.The calculated elastic and mechanical properties of InSnCl 3 using the GGA-PBE in the present work are in reasonable agreement with those obtained using GGA +U in a previous study [41], implying the reliability of the results of the present study.Consequently, it is anticipated that the corresponding results for the other two isoelectronic compounds, InSnBr 3 and InSnI 3 , are also accurate.

Electronic properties
The energy band diagrams and the total density of states of the studied perovskites have been calculated and the results are plotted and shown in figures 4(a)-(c) for InSnCl 3 , InSnBr 3 , and InSnI 3 , respectively.The numerical values of the band gap energies of the perovskites are presented in table 3.
As can be seen from table 3, the energy gaps of InSnCl 3 perovskite reported in this study are in excellent agreement with those reported previously in [41], providing validation to the findings.The energy gaps for InSnBr 3 and InSnI 3 are presented and discussed for the first time in the present work.It is found that InSnCl 3 possesses an indirect band gap of 0.59 eV (R→X).Interestingly, when Cl is replaced by Br and I, the energy gap decreases and becomes direct, i.e. 0.44 eV (R→R) and 0.24 eV (R→R) for InSnBr 3 and InSnI 3 perovskites, respectively.This is also confirmed by the results of our calculation using the HSE06 functional, with band gap energies of the compounds found to be 2.35 eV, 2.13 eV, and 2.01 eV, respectively.This phenomenon is clearly demonstrated in figures 4(a)-(c).Due to their direct and narrow band behaviour, InSnBr 3 and InSnI 3 are more favourable for photovoltaic applications compared to InSnCl 3 perovskite.
Overall, perovskite materials with a direct bandgap exhibit faster charge carrier recombination and better radiative efficiency compared to those with indirect bandgap [72].Wang et al [72] demonstrated that methylammonium lead iodide, which originally exhibited an indirect bandgap, can be transformed into a material with a more direct bandgap by applying external hydrostatic pressure.This transformation further enhances the material's efficiency for the application in optoelectronic devices.Inducing indirect-to-direct bandgap transition in perovskites can also be done using various approaches such as mechanical strains [73], order to disorder transformation [74], and mixing cations [75].Furthermore, Wu and co-workers [76] have recently reported a thermally induced indirect-to-direct bandgap transition which is associated with defect formation in CsPb 2 Br 5 perovskite.They observed a significant change in the optical properties of the perovskite as a result of this indirect-direct transition.
The partial Density of States (PDOS) of the perovskites were also computed and the results are displayed in figure 5. Overall, the three compounds have similar general features, as expected.Near the top of the valence bands, the 5 s orbital of the Sn atom was more dominant than the other orbitals while near the bottom of the conduction band, In-5p state was the most dominant state, followed by the Sn-5p state and finally by the halide orbitals.The (100) and (110) electron density of the materials are visualized in figures 6(a)-(c) for the respective compounds.It is apparent from the figures that electronic distribution around In and X (X = Cl, Br, I) are almost spherical, strongly implying that In-X bonds are mostly ionic.Conversely, there is a noticeable distortion in the electron distribution between Sn and X atoms, which suggests the presence of a covalent bond between Sn and X atoms.These bonding characters have also been reported in a wide range of perovskite materials [54].
The covalent bonding character between Sn-X atoms (right part of figure 6) is believed to make considerable contribution to the narrow band gap energies of the compounds.This is in contrast to ionic crystals such as BaLiX 3 (X = F, Cl, Br, I) which have relatively wider electronic band gaps [58].Figures 6(a)-(c) (right) also illustrate that the covalent character of the Sn-X bonds is quite strong as seen by some accumulation of electrons between Sn and X atoms and by nearly elliptical shapes of electronic distribution between them.This can also be confirmed by using equation (2) [58,77].

= --
- Here, IC stands for the ionic character while X 1 and X 2 are the electronegativity of atom 1 and atom 2, respectively.Using this equation, the ionic character percentages of Sn-Cl, Sn-Br, and Sn-I are 22.89%, 18.94%, and 12.63%, respectively.This clearly demonstrates the decrease in the ionic character bond when Cl is substituted with Br and I, which also indicates an increase in the covalent character for the anion replacement.This is also partly due to the increase in the lattice parameters of the compounds, i.e., the increase in the bond length, making the valence electrons become less bound to their parent atoms in the compounds.As a result, less energy will be required to facilitate the free movement of these electrons as conduction electrons within the compounds.This leads to a decrease in the band gap energy of the compounds [58].

Optical properties
In this section, the computed optical properties of the InSnX 3 (X = Cl, Br, I) perovskites within photon energy range [0,15] eV are presented and discussed.
e w e w e w = + 3 1 2

( ) ( ) ( ) ( )
Where ε 1 (ω) and ε 2 (ω) are the real and imaginary parts of the dielectric function.ε 1 (ω) reflects the degree of polarization of these materials under the external electric field, while the ε 2 (ω) shows the ability of these materials to absorb the light.These tensors can be calculated using the following equations Based on the dielectric constants e w 1 ( )and e w , 2 ( ) the optical parameters including the refractive index w n , ( ) the reflectivity w R ( )and the absorption coefficient a w ( ) were then calculated using the following equations: h w e w e w e w = + + show the computed values of the real part of the dielectric function e w , 1 ( ) the imaginary part of dielectric function e w , 2 ( ) the reflectivity w R , ( ) and the refractive index w n , ( ) of the InSnX 3 (X = Cl, Br, I) perovskites for photon energy between 0 and 15 eV.As discussed in the electronic properties section, the optical properties of the compounds experience reasonable changes due to replacement of the halide anions.
In figure 7(a), it is evident that the static dielectric constant e 0 1 ( ) rises as the Cl anion is substituted by Br and I anions, with e 0 1 ( ) values for InSnCl 3 , InSnBr 3 , and InSnI 3 being 5.03, 5.91, and 7.25, respectively.This observed trend aligns well with the Penn's model [80], which suggests that the static dielectric functions are inversely proportional to the electronic band gap energy of the compounds.Moreover, it is generally true that materials with larger static dielectric constants tend to exhibit better optoelectronic properties [81,82].Based on these values, it can be inferred that InSnI 3 is expected to exhibit the highest optoelectronic performance, followed by InSnBr 3 , and InSnCl 3 .
Figure 7(b) depicts the computed imaginary part of the dielectric function e w 2 ( ) for the energy interval [0,15] eV, which represents the absorption behaviour of InSnX 3 (X = Cl, Br, I) perovskites.As expected, the absorption threshold of the materials is in excellent agreement with their computed electronic band gap energies.InSnI 3 exhibits the smallest light absorption threshold, followed by InSnBr 3 and InSnCl 3 , which is consistent with previous findings that InSnI 3 has the smallest electronic bandgap, followed by InSnBr 3 and InSnCl 3 perovskites.It can also be seen from the figure that InSnI 3 shows the strongest absorption in visible and lower ultraviolet regions, making it more promising as absorbers for solar cells compared to its counterparts.The highest peaks of e w 2 ( ) for InSnI 3 , InSnBr 3 , and InSnCl 3 are 9.47 at 2.86 eV, 7.49 at 4.18 eV, 6.90 at 4.84 eV, for the respective materials.The reflectivity w R( ) of the compounds is shown in figure 7(c), where all materials have relatively low reflectivity for the entire energy range considered.The maximum reflectivity values for the compounds are approximately 35%, with the exact values predicted to be 34.71% at 5.72 eV for InSnCl 3 , 35.01% at 4.84 eV for InSnBr 3 , 35.38% at 2.86 eV for InSnI 3 .These relatively low reflectivity values imply minimal light loss during the light absorption, highlighting the materials' effectiveness as absorbers.The refractive index w n( ) of the perovskites has also been computed and plotted in figure 7(d).n(ω) gradually decreases, suggesting that the interaction of incident photons with the material is diminished.The computed static refractive index values n 0 ( ) for InSnCl 3 , InSnBr 3 , and InSnI 3 are 2.24, 2.43, and 2.69, respectively.
The absorption coefficient a w ( )is a significant optical property used to assess the performance of materials in optoelectronic applications.Figure 8 presents the calculated optical properties of the studied materials.The figure demonstrates that InSnI 3 is predicted to possess stronger light absorption compared to InSnBr 3 and InSnCl 3 in the energy range suitable for solar cell applications (from 0 eV to 4 eV), which is consistent with the previous discussions related to the computed imaginary part of the dielectric functions (figure 7(b)).Overall, the three perovskites possess strong absorption behaviour throughout the entire energy range, with absorption coefficients on the order of 10 5 cm −1 .

Thermoelectric properties
In addition to the mechanical and optoelectronic properties of the perovskites, it is crucial to examine their thermoelectric properties to assess their potential applications as thermoelectric materials.Here, the thermoelectric properties of InSnX 3 (X = Cl, Br, I) perovskites were investigated for the first time.These properties encompass the Seebeck coefficient S (figure 9 Figure 9(a) shows that the temperature dependence of the Seebeck coefficients for the three perovskites, revealing an increase in the values with increasing temperature.Notably, the Seebeck coefficients of InSnBr 3 are slightly higher compared to those of InSnCl 3 and InSnI 3 .The highest values of the Seebeck coefficient were found to be at 800 K for the perovskites: 152 μV/K for InSnBr 3 , 150 μV/K for InSnCl 3 , and 141 μV/K for InSnI 3 perovskites.It is worth mentioning that the positive values of Seebeck coefficients indicate the likelihood of the materials being p-type semiconductors.
In figure 9(c), it is evident that the materials have high electrical conductivity and the electrical conductivity of all perovskites increases with increasing temperature.As anticipated, this trend resembles the general pattern observed for the electronic thermal conductivity (figure 9(b)).The electrical conductivity remains relatively constant within the temperature range of 50 K to 200 K, followed by a slight increase up to the room temperature (300 K).Subsequently, there is a dramatic and continuous increase until it reaches its maximum values at 800 K.As seen from figure 9(c), InSnI 3 is projected to have the highest electrical conductivity across the entire temperature range, followed by InSnBr 3 and InSnCl 3 perovskites.The figure of merit ZT is an important property of materials used to evaluate their thermoelectric performance [84], with higher ZT values represent better thermoelectric properties [85].From figure 9(d), it is obvious that the ZT values increase as a function of temperature and therefore it can also be inferred that the materials are expected to have better thermoelectric properties at higher temperature.InSnBr 3 demonstrates the highest ZT values, which is followed by InSnCl 3 and InSnI 3 compounds.At 800 K, the ZT values for the respective compounds are 0.54, 0.51, and 0.49.

Conclusion
The structural, mechanical, optoelectronic, and thermoelectric properties of lead-free perovskites InSnX 3 (X = Cl, Br, I) have been explored through a first-principles calculation based on DFT.The results revealed that the materials are chemically, mechanically and thermodynamically stable as well as ductile in nature.Notably, the perovskite materials possess narrow bandgaps and a transition from indirect to direct bandgap was observed when Cl is replaced by Br and I.Moreover, the perovskites exhibit outstanding optical properties, making them suitable for solar cell applications.Finally, the study on their thermoelectric properties indicated that the materials have high electrical conductivity and reasonable figure of merit.

Figure 1 .
Figure 1.The unit cell structure of the cubic InSnCl 3 perovskite.The exact structure is possessed by InSnBr 3 and InSnI 3 with Cl substituted by Br and I, respectively.

Figure 2 .
Figure 2. The total energy as a function of unit cell volume of the cubic InSnCl 3 (a), InSnBr 3 (b), and InSnI 3 (c) perovskites.
(a)), the thermal conductivity per relaxation time (κ e /г) (figure 9(b)), the electrical conductivity per relaxation time (σ/г) (figure 9(c)), and the figure of merit (ZT) (figure 9(d)).The investigation was carried out within the temperature range of [50, 800] K to capture the thermoelectric properties across a wide range of temperature spectrum.

Table 2 .
[41]elastic and mechanical properties of the cubic InSnX 3 (X = Cl, Br, I) perovskites.The results for InSnCl 3 are also compared with those reported in[41].