Towards lead-free perovskite photovoltaics and optoelectronics by ab-initio simulations

Lead (Pb) free non-toxic perovskite solar cells have become more important in the commercialization of the photovoltaic devices. In this study the structural, electronic, optical and mechanical properties of Pb-free inorganic metal halide cubic perovskites CsBX3 (B = Sn, Ge; X = I, Br, Cl) for perovskite solar cells are simulated using first-principles Density Functional Theory (DFT). These compounds are semiconductors with direct band gap energy and mechanically stable. Results suggest that the materials have high absorption coefficient, low reflectivity and high optical conductivity with potential application in solar cells and other optoelectronic energy devices. On the basis of the optical properties, one can expect that the Germanium (Ge) would be a better replacement of Pb as Ge containing compounds have higher optical absorption and optical conductivity than that of Pb containing compounds. A combinational analysis of the electronic, optical and mechanical properties of the compounds suggests that CsGeI3 based perovskite is the best Pb-free inorganic metal halide semiconductor for the solar cell application. However, the compound with solid solution of CsGe(I0.7Br0.3)3 is found to be mechanically more ductile than CsGeI3. This study will also guide to obtain Pb-free organic perovskites for optoelectronic devices.

devices like solar cell, diode, laser, etc. However, less study has been carried out on the optical properties of the considered perovskite compounds. In particular the optical properties of CsGeX 3 (X = I, Br, Cl) materials are still unexplored in detail. Also understanding the mechanical properties of these materials including mechanical stability, rigidity and ductility are useful to predict the importance of the materials for industrial applications. To the best of our knowledge, the elastic constants and moduli of the Pb-free CsSnI 3 , CsSnCl 3 , CsGeI 3 and CsGeCl 3 compounds are still not reported. Therefore, we have considered selected metal halide perovskite compounds and investigated their structural, electronic, optical and mechanical properties using first-principles Density Functional Theory (DFT) to obtain suitable Pb-free perovskites for solar cells and other optoelectronic devices.

Results and Discussion
Structural properties. The cubic metal halide perovskites CsBX 3 (B = Pb, Sn, Ge; X = I, Br, Cl) have space group ) fractional coordinates. The calculated equilibrium lattice parameter a and the volume of the optimized cells are in good agreement with the experimental results as well as with the available theoretical results (see Supplementary Table 1). We have also found that the replacement of halogen atom (X) by a lighter and smaller halogen atom reduces the lattice parameter (Fig. 2) and the unit cell volume (Supplementary Figure 1). Similarly, the replacement of Pb by tin (Sn) or germanium (Ge) causes the reduction of lattice parameter of perovskites as shown in Fig. 2. A clean periodic behaviour in term of the change in lattice parameter as well as the unit cell volume is observed with the considered metal halide perovskite materials when changing the size of halides. This is a general features for the most compounds and it is because of the change in atomic size and charge.
The replacement of one type of atoms by a similar atom from a compound can affect the structural parameters of the compound. As a consequence, the material properties like electronic, optical and mechanical properties are Electronic properties. We have calculated the electronic band gap using generalized gradient approximation (GGA) of Perdew-Berke-Ernzerhof (PBE). The calculated electronic band gap values are found to be in good agreement with the available results in the literature using the same theory (see Supplementary  Table 2) while some diversity has been found with other results. Sometimes hybrid potential like HSE (Heyd-Scuseria-Ernzerhof) 13 may give better estimation 14 . But it is not the perfect one as diversity has been found for some compounds (see Supplementary Table 2). Thus, it is still a challenge to find the appropriate potential to predict the theoretical electronic band gap for this system of materials. Similar to the unit cell volume, the electronic band gap is seen to change periodically for changing the halogen atoms as shown in Fig. 3. Here, higher band gap is observed for the compounds containing lighter halogen atoms. However, exception is observed in case of replacement of Pb by Sn and Ge. The Ge containing compounds have higher electronic band gaps than Sn containing compounds as for Ge, the electronic states of conduction band slightly shift towards the high energy and this is very common for this group of compounds 14 .
The study of the electronic band structure (Supplementary Figure 3) suggests that the pattern of the band structure is almost not affected by replacement of a halogen atom with other halogen atoms. Due to this characteristics, metal halide perovskites are suitable to be used in light emitting diodes (LED) as the band gap of these materials can easily be tuned by changing the halogen contents while the other properties remain almost same. Also, from the total and partial densities of states (Supplementary Figure 4) one can expect that the replacement of I by Br and Cl increases the number of electronic states in valence band just below the Fermi level. But the  opposite trend is observed during the replacement of Pb by Sn or Ge. Moreover, the total DOS below the Fermi level mainly comes from the p orbital of halogen atoms. In addition to this the total DOS above the Fermi level is mainly contributed by p orbital of B atoms as from Pb-6p, Sn-5p and Ge-4p.
Optical properties. The detail optical properties including real and imaginary part of dielectric functions, refractive index, absorption spectra, reflectivity and photoconductivity of our considered compounds are investigated up to the photon energy of 30 eV to reveal the response of the materials in solar and high energy radiation (see Supplementary Section "Optical Functions").
The optical absorption coefficient of a material is a measure of the penetration of light at specific wavelength (energy) into the material before it gets absorbed 15 . It also gives information about the optimum solar energy conversion efficiency which is important for the practical application of a material in solar cell. The calculated optical absorption spectra of the considered compounds under the simulated illumination is presented in Fig. 4. In general, there are mainly three light absorption peaks observed for all compounds except those containing Pb for which an additional peak is observed in high energy region. The replacement of Pb by Sn or Ge affects the magnitude of the absorption spectra, but the position of the peaks remain at the same energy. However, the replacement of I by Br and Br by Cl shift the position of the peaks towards the higher energy. Most of the considered compounds show high absorption between the energy ranges from 2-16 eV with first peak at about 4 eV, second broad peak centred at about 10 eV and third peak at about 14 eV. It is also found that the maximum absorption is obtained with Ge based compounds compared to Pb and Sn based counterpart. Therefore, Ge would be a better substitute of Pb in the inorganic perovskites for application in solar cells 10 . The efficiency of the Ge based inorganic perovskite solar cells (CsGeX 3 ) is, however, to be investigated if it can be competitive with the existing organic-inorganic hybrid perovskite solar cells, such as (CH 3 NH 3 )PbX 3 .
Reflectivity is another important optical property for the solar cell and other applications of the perovskite. All the investigated perovskite materials show a low reflectivity in the IR-Visible-UV region of the spectra (see Supplementary Figure 7). This low reflectivity suggests that the materials have high absorptivity and/or transmitivity. However, our study of the dielectric function (see Supplementary Section "Optical Functions") suggests that the materials have high transmitivity at high energy region and low transmittance in low energy region. Therefore, the results of the reflectivity and dielectric function confirm that the perovskite compounds have high absorption at low energy (<10 eV) region of the energy spectrum. On the other hand, the optical conductivity is a good reflection of the photoconductivity 16 . The real part of the optical conductivity is shown in Fig. 5. As can be seen, most of the perovskite compounds have high optical conductivity at low energy region. Similar to the phenomenon observed in absorption, Ge based compounds have higher optical conductivity than other metal cations based materials. Moreover, the magnitude of the spectra is higher for I containing compounds, while the spectra is seen to broaden for Br and Cl. In addition, the higher value of the extinction coefficient (Supplementary Figure 9) in the low energy region of the spectra also reveals that Ge is found to be a better replacement of Pb for solar cells.
Mechanical properties. The elastic constants are essential to predict the mechanical stability of a crystal.
The well-known Born stability criteria 17 for cubic crystal is given by: − > + > > . The three independent elastic constants (C 11 , C 12 and C 44 ) of the cubic perovskite compounds are calculated using the finite strain theory 18 implemented in CASTEP module and presented in Table 1 with available theoretical and experiment results. The calculated elastic constants for CsPbCl 3 are in good agreement with the only available theoretical results reported in literature 19 . Also, the calculated results for CsSnBr 3 and CsGeBr 3 are in good agreement with the available theoretical results reported by Brik 20 . The calculated elastic constants are positive and satisfy the Born stability criteria, which implies that the compounds are mechanically stable.
The results of bulk modulus, shear modulus, Young's modulus, Pugh's ratio and Poisson ratio of the perovskite compounds were calculated using the second order elastic constants and summarized in Table 1. We also compare our results with available theoretical and experimental results previously reported. The results of bulk modulus for CsPbX 3 (X = I, Br, Cl) calculated in this work are found to be in reasonable agreement with previous theoretical calculations [21][22][23] . The only experimental result of elastic moduli for CsPbI 3 reported by Rakita et al. 24 also agrees well with our simulation. In addition, calculated moduli for CsSnBr 3 and CsGeBr 3 match very well with literature 20 . The bulk modulus is an indicator of the stiffness of a volume of material. The calculated values of bulk modulus of the perovskite lie between 6.30 GPa (for CsSnI 3 ) and 27.03 GPa (for CsGeCl 3 ), indicating that these perovskite compounds are flexible. Because of their flexibility and softness, the metal halide perovskites can easily be developed into thin film and make them more useful in applications such as solar cells. By changing the halogen atom from I to Br then to Cl, the value of bulk modulus is seen to increase. Similarly, the replacement of Pb by Sn and then Sn by Ge increases the bulk modulus of the perovskite compounds except CsSnI 3 which shows extremely low value of bulk modulus. Similar trend is observed for shear and Young's modulus with no exception.
The ratio of the bulk modulus to shear modulus (B/G) also known as Pugh's ratio and the Poisson ratio are considered as a measure to predict the failure mode, i.e., ductility and brittleness of a material 25 . The critical value of Pugh's ratio to separate the ductile materials from brittle is found to be 1.75, while the critical value of Poisson ratio for the separation of ductile from brittle is ~0.26 25 . If the value of Pugh's ratio is greater than 1.75 or the value of Poisson ratio is greater than 0.26, then the material is considered as ductile otherwise it is brittle. Also, the higher value of Pugh's ratio and the Poisson ratio indicate more ductility of a material. According to the criteria, CsSnI 3 is brittle, CsGeI 3 is found near brittle-ductile boarder line and the rest of the compounds are    ductile in nature. Figure 6 presents the variation of Pugh's ratio for different compounds. The same characteristics are observed for Poisson ratio (Supplementary Figure 11). The smaller values of Pugh's ratio and the Poisson ratio of CsSnI 3 indicate that the material is very brittle and it would not be convenient to handle for applications in a device. The higher ductility is observed in compounds containing Pb. Among them, CsPbBr 3 shows the highest ductility.

Lead Free Perovskites
The summary of the key properties of the metal halide perovskites are presented in Table 2. In comparison, Ge based compounds have a higher optical absorption and optical conductivity than Pb containing compounds, which suggests that Ge would be a better replacement of lead. On the other hand, CsSnI 3 is not suitable to use in any device as it is highly brittle. Furthermore, the Sn based compounds CsSnX 3 (X = I, Br, Cl) have low optical absorption and conductivity than that of Ge based compounds. Considering all these properties, we can conclude that CsGeI 3 would be a better lead-free perovskite material for solar cell application as it has highest absorption and photoconductivity in the solar spectrum. However, the problem with CsGeI 3 is that it is slightly brittle and also having poor film forming ability 10 .
Nevertheless the issue of brittleness should be improved by incorporation of Br with CsGeI 3 in solid solution 26 , and consequently the film forming ability may also be improved.
To find a ductile lead free perovskite, the mechanical properties of solid solutions CsGe(I 1−x Br x ) 3 have been calculated and the variation of the Pugh's ratio for different combinations of I and Br is presented in Fig. 7. According to the results, the maximum ductility is found at x = 0.3 in CsGe(I 1−x Br x ) 3 . Therefore, CsGe(I 0.7 Br 0.3 ) 3 is more ductile than CsGeI 3 also the material is found to be mechanically stable and soft (see Supplementary  Table 3) and can easily be developed into thin film. The electronic band gap of CsGe(I 0.7 Br 0.3 ) 3 is found as 0.83 eV, also the absorption and the optical conductivity is expected between that of CsGeI 3 and CsGeBr 3 as the optical properties is seen to change periodically (Supplementary Figure 10). Therefore CsGe(I 0.7 Br 0.3 ) 3 would be the best Pb-free perovskite for those applications in which ductility is required.

Conclusion
First-principles DFT calculations have been performed to investigate the structural, electronic, optical and mechanical properties of Pb-free inorganic metal halide cubic perovskites CsBX 3 (B = Sn, Ge; X = I, Br, Cl) and compared with Pb based compounds. This study suggests that the electronic band gap value is affected due to the replacement of halogen atom while the other properties remain almost same. This is a key feature of the materials to be used in light emitting diodes (LED) as the band gap of these materials can easily be tuned by changing the halogen contents. The investigated perovskite compounds are mechanically stable and can easily be developed into thin films as the materials have low bulk modulus. According to the optical properties analysis, Ge atom appears superior to Pb for the considered inorganic perovskites, as Ge based compounds have higher optical absorption and optical conductivity than the Pb based compounds. By analysing the combined effects of the electronic, optical as well as mechanical properties of CsGeX 3 (X = I, Br, Cl), it can be concluded that CsGeI 3 is a better candidate as a Pb-free perovskite for solar cell application. However, CsGeI 3 is slightly brittle and may not be superior in applications that require ductility. This issue has been resolved by partially replacing iodine with solid solution of Br atom CsGe(I 0.7 Br 0.3 ) 3 .

Computational Methods
The first-principles density functional theory (DFT) 27,28 calculations have been performed using the plane wave pseudopotential method as implemented in CASTEP 29 module of Materials Studio 8.0. Vanderbilt type ultrasoft pseudopotential 30 was used to treat electron ion interaction for all calculations. The exchange-correlation energy was evaluated by using the generalized gradient approximation (GGA) of Perdew-Berke-Ernzerhof (PBE) 31 . To optimize the crystal structure and find the ground state energy the Broyden-Fletcher-Goldfarb-Shanno (BFGS) 32 minimization technique was used. Elastic constants were calculated by using the finite strain theory 18 which was implemented within CASTEP module.