First principle study of optoelectronic and mechanical properties of lead-free double perovskites Cs2SeX6 (X = Cl, Br, I)

The variant double perovskites are considered novel materials for solar cells and optoelectronic applications. Here, we explored electronic, mechanical, and optical characteristics of Cs2SeX6 (X = Cl, Br, I) by density functional theory (DFT) analysis. The tolerance factor between 0.97–1.0 signifies the structural stability of the investigated compounds while thermodynamic and mechanical stabilities were ensured by positive frequencies of phonon dispersion as well as elastic constants. Moreover, the Poisson and Pugh's ratios are explored for brittle and ductile behavior. The Debye and melting temperatures have also been reported through mechanical analysis. The tuning of the bandgap takes place from 3.10 to 2.64 eV and then 1.15 eV by substitution of Cl with Br and I, respectively. The optical spectra show a shift in absorption region from ultraviolet-to-visible. In addition, the low light reflection and optical energy loss range (0.0–3.0 eV) promises potential of the studied potential for solar cells and optoelectronics uses.


Introduction
To fulfil the stimulating demand for renewable energy in limited resources is very challenging. The conversion of electrical energy from heat or light energy sources is an essential requirement for renewable energy devices. The efficiency of such devices mainly depends on the selection of particular materials. Perovskite materials are intensively studied by researchers because of their potential applications [1][2][3][4][5][6]. From this versatile family of compounds, lead-halide-based solar cells have the highest efficiency reported to date [7]. A large number of configurations have been scrutinized of organicbased perovskites from which the formamidine lead triiodide (FAPbI 3 ) is found outstanding. Its power conversion efficiency (PCE) reaches 25.06% which is most suitable for solar cells [8]. Regardless of this achievement, Pb-based materials are not environmentally friendly. The replacement of Pb-based optoelectronic material with other non-hazardous material is still needed extensive research [9,10]. To fill this gap, a number of researchers have reported Pb-free single and double perovskite halides (DPH's) [11]. In the case of simple perovskites, Parrey et al. [12] performed DFT calculations on RbGeI 3 and studied electro-optical properties. In another study on halides, Zhang et al. [13] investigated the electronic properties for CsGeX 3 (X = Cl, Br and I) and suggested optoelectronic applications of these compounds. Another strong candidate that emerged in the list of photovoltaic applications is vacancy-ordered double perovskite Cs 2 SnI 6 reported recently [14]. The structure of vacancy-ordered double perovskites is A 2 XY 6 , where A is tetravalent and X octavalent cations, while Y is halides (Cl, Br or I). Exploring the hidden attributes of double perovskites useful for optoelectronic applications in industry, Wang et al. [15] have investigated a number of novel DPH's and suggested possible solar cell applications. Among these materials, Sn-and Ge-based DPH's are better alternatives of Pb-free materials in novel optical devices because they belong to the same group of Pb [16]. Experimentally Cs-based DPH's were first prepared by Bagnall et al. [17]. In another study by Morss et al. [18], new Cs-based DPH's were fabricated. Since the last decade, DPH's have earned the name in the list of Pb-free compounds. Recently, Cs 2 AgBiCl 6 and Cs 2 AgBiBr 6 are cubic compounds reported that have shown remarkable optical applications in the visible range [19,20]. Furthermore, recently, the defective double perovskites Cs 2 SeCl 6 and Cs 2 TeCl 6 are elaborated for thermoelectric properties based on classical transport theory by BoltzTraP code. The computed lattice constants (10.18 and 10.51Å), formation energies (−23.56 and −24.13 eV), bandgaps (2.95 and 3.10 eV) and tolerance factor (0.728 and 0.725) are reported for Cs 2 SeCl 6 and Cs 2 TeCl 6 , respectively [21].
The purpose of this study is to explore hidden traits of Cesium-based DPH's for solar cell and thermoelectric applications. In this study, we have particularly focused on Cs 2 SeX 6 (X = Cl, Br and I) DPH's. Based on comprehensive literature survey, the electronic, elastic and optical properties of Cs 2 SeX 6 (X = Cl, Br and I) DPH's are scarcely being reported to date. We have used a stateof-the-art theoretical approach based on DFT to explore the aforementioned properties.

Method for calculations
The structural, electronic, elastic and optical properties of DPH's were analysed by DFT based on the FP-LAPW method [22] performed in WIEN2k code [23]. Additionally, the exchange-correlation potential used in our study is Pardew Burke Ernzerhof Generalized Gradient Approximation (PBE-GGA) [24]. The PBE-GGA approximation solves the properties in the ground state with high precision; however, the calculated bandgap is underestimated. To circumvent this issue, the bandgap is improved by implementing modified Becke and Johnson Potential of Trans and Balaha (TB-mBJ) [25]. It improves the bandgap very accurately and the most versatile potential calculated through a mathematical equation.
where c is convergence factor, ρ σ (r) indicates density of states and t σ (r) denotes its kinetic energy. The charge convergence factor is calculated as where α and β are numeric constant and adjusted in WIEN2k. The number of K-points selected to achieve full convergence was set at 2000. This helped to generate a denser k-mesh of 12 × 12 × 12 k-mesh points. The maximum angular momentum value l max was adjusted at 7. The R MT × K max was set to 8. Where R MT and K max are muffin-tin radii and cut-off wave vectors. Maximum Fourier transformation vector G max was adjusted to 12. The self-consistent field (SCF) convergence was minimized up to 0.00001 Ry. The ground state optimizations of structural reforms were solved using the Murnaghan equation of state. Further, for phonon calculations, the density functional theory perturbation has been incorporated in the phonopy software to extract the phonon dispersion band structures [26].

Results and discussion
In this section, we have reported the structural, electronic, elastic and optical properties of DPH's in detail.
In Section 3.1, structural stability is explained under the Goldschmidt tolerance factor, ground state optimization and enthalpy of formation. Moreover, electronic properties are discussed which includes DOS and bandgap analysis. In Section 3.2, mechanical attributes of the studied compounds is being explained in detail. In the last section, optical properties are discussed to explore possible optical applications.

Structural reforms and electronic attributes
To calculate the DPH's structural stability, ground state optimization of the compound is essential. In our study, we have used the Murnaghan equation of states to achieve equilibrium in structure. The stability in structure is addressed by using tolerance factor T f of cubic crystal systems suggested by Goldschmidt [27].
The T f value around unity portrays cubic structure is stable. In our study, all compounds are stable in the cubic phase. The enthalpy of formation H f of BPH's is also calculated using All calculated values are summarized in Table 1. The negative value of H f (Figure 1) ensures the thermodynamic stability of studied materials. Among the studied compounds, Cs 2 SeCl 6 is more stable due to the large enthalpy of formation [28].   The density functional perturbation theory (DFPT) has been incorporated to confirm the thermodynamic behaviour of the studied compounds. The computed dispersion plots are illustrated in Figure 2(a-c) which shows the studied compounds have positive frequencies. Therefore, the positive frequencies indicate the studied compounds are dynamically stable. Three acoustic modes of phonons dispersion are accurately zero at symmetry direction which is also the confirmation of dynamical stability [29,30].
First, the structural, mechanical and thermodynamic stabilities are calculated according to their formalism as shown in the manuscript. However, we relate structural stability with mechanical and thermodynamic stabilities. The structural stability has been calculated from the tolerance factor which depends upon the atomic radii. Mechanical stability has been depicted from born mechanical stability criteria C 11 < B < C 12 , C 11 -C 12 > 0 and C 44 > 0 [31]. Therefore, the bulk modulus is volumetric deformation after the applied stress and its value related to the lattice constant. The optimized lattice constant for stable structure and relaxed positions adjusts the bulk modulus to keep positive elastic constants for mechanical stability. Secondly, stable and relaxed structures released more energy which is measured in terms of negative formation energy. Therefore, stable structure measures are directly ensured from formation energy. Thirdly, the acoustic phonons have zero frequencies at and L symmetries as depicted in the dispersion plot in Figure  2(a-c) which is confirmation of structural stability from the thermodynamic approach.
The electronic band structure of DPH's is depicted in Figure 3. In this study, spin-polarized TB-mBJ potential is used to investigate the correlation effects. From  Table 1. It is observed that Cs 2 SeI 6 has the smallest bandgap. The nature of the bandgap is further elaborated by the density of states (DOS) plot as shown in Figure 4. The valence band maxima are contributed mainly by the p orbital of the halogen atom in all compounds. While conduction band minima are formed by Se-p orbital. It can be inferred from the partial DOS that inter-band transition will occur only due to the involvement of Se-p to X-p orbitals.

Mechanical properties
The three elastic constants for the cubic crystals are prescribed as C 11 , C 12 and C 44 . These constants are calculated using the Charpin method embodied within the WIEN2k code. The stability criteria that cubic compound must satisfy are (C 11 > 0), (C 44 > 0), (C 11 − C 12 > 0), (C 11 + 2C 12 > 0) and (C 11 > B > C 12 ) [31]. From Table  2, all compounds satisfy the stability criteria. Using these  elastic constants, we have calculated bulk modulus (B), Young's modulus (E) and shear modulus (G) by formulas presented in Muhammed et al. [4]. It is found that Cs 2 SeCl 6 has the highest B, G and E values which suggest that it is more rigid than Cs 2 SeBr 6 and Cs 2 SeI 6 . To check whether the compound is ductile or brittle, the Pugh's (B/G) criteria were used. If the value of B/G > 1.75, the studied material is considered ductile in nature; however, brittle otherwise. In our study, all compounds are found ductile. Additionally, its nature was also confirmed by Poisson's ratio "υ" for which υ > 0. 26 suggests the materials are ductile in nature. Moving further, the Kleinman parameter (ξ ) [32] is used to estimate the bond strength whether it is bending or stretching. For ξ near 0, it suggests bond bending while for ξ around unity implies bond stretching. It is observed that Cs 2 SeI 6 has the highest bond-stretching nature. To check the covalent and ionic nature of the bonds present in the unit cell Cauchy parameter C is computed as C = C 12 − C 44 . The C < 0 portrays the dominancy of majority bonds as covalent. While for C > 0 ionic-bonding dominancy is expected. From analysis, all studied compounds exhibit positive value which reflects the dominancy of ionic bonding. It is found that Cs 2 SeCl 6 has strong ionic nature than Cs 2 SeBr 6 and Cs 2 SeI 6 . We have also checked the Debye temperature to estimate the influence of lattice stability in bonds among atoms by relation [33]. where h and k are Plank's and Boltzmann's constant, ρ density of alloy, M molecular weight, N A Avogadro number and n number of atoms. Also, V is total velocity computed by relation, where the longitudinal mode of velocity and transverse velocity is given by Ionic compounds exhibit high Debye and melting temperature which is affirmed by results and mentioned in Table 2. Melting temperature T m [34] is estimated by relation Thermal conductivity is used to understand the material response under influence of external heat. The minimum thermal conductivity is estimated by utilizing the Cahill criteria [35,36].
where n o is no of atoms per unit volume. The highest value of K min is observed in the case of Cs 2 SeCl 6 which suggests the material applications in high-temperature sensors. The hardness parameter H a [37][38][39][40] is used to explain the property of a material to resist being dented under stress. It is calculated using the following relation, All computed values are enlisted in Table 2 which suggests that Cs 2 SeCl 6 has the highest ability to resist the applied stress.

Optical properties
The optical behaviour of materials is strictly dependent on the interaction among light energy and matter, the transition rate of electrons from the valenceto-conduction bands, and the recombination rate. In the case of wide bandgap semiconductors, inter-band transitions are taken into consideration and intra-band transitions are ignored because excitation and recombination within the band are not possible [41]. We have analysed the optical spectra of Cs 2 SeX 6 (X = Cl, Br, I) through graphical representation in Figures 5  and 6. The dielectric response has been explained by complex dielectric constant ε(ω) = Re ε(ω) + Im ε(ω), where Re ε(ω) is the real part of dielectric constant which enlighten the polarization and dispersion of light energy from lattice while its imaginary part Im ε(ω) describe the absorption of light energy when the frequency of light is greater than the threshold limit. The dependence of Re ε(ω) and Im ε(ω) is explained by the Kramer-Krong relation [42].
The real dielectric constant Re ε(ω) elaborates the dispersion and polarization of light on interaction with the material of a slightly changing refractive index. The frequency of light depends upon phase velocity which reacts to maximum dispersion and polarized light at plasma resonance of lattice waves. The reported values of Re ε(ω) have been depicted in Figure 5(a). The zero-energy value static Re ε(0) of Cs 2 SeCl 6 is 3.59 which increases to 4.13 and 6.17 by the replacement of Cl with Br and I because of decreasing bandgap values and large dispersion from Cl to I. The resonance frequencies occur at 3.3 eV (7.9), 2.80 eV (8.93) and 2.0 eV (11.7) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively. These are the peak values at which dispersion of light is maximum. Above the resonance frequency, the Re ε(ω) decreases to minimum at 3.69 eV (−0.22), 3.24 eV (1.12) and 2.97 eV (−4.84) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively. The two negative peaks for Cs 2 SeCl 6 and Cs 2 SeI 6 show the metallic behaviour for the light which reflects the light falling on the material in this region.  Moreover, bandgap E g and Re ε(0) are according to Penn's model relation ε(0) ≈ 1 + (hω p /E g ) 2 [43].
The imaginary dielectric constant Im ε(ω) which measures the absorption of light energy is presented in Figure 5(b). The threshold values of absorption are noted as 3.1, 2.66 and 1.15 eV for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively [44]. The maximum absorption peaks occur at 3. 54 eV (7.4), 3.09 eV (7.6) and 2.4 eV (15) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively which shows the absorption intensity of light increases from Cl to I and ultraviolet region is shifted to visible. Furthermore, absorption bandwidths 0.62 eV (3.2-3.8 eV), 0.75 eV (2.65-3.4 eV) and 1.36 eV (1.78-3.14 eV) increase from Cl to I which depict the Cs 2 SeI 6 is optimal for absorption of light. The energy regions instigate the ultraviolet region of the spectrum for Cs 2 SeCl 6 , ultraviolet violet edge to the visible region for Cs 2 SeBr 6, and the visible region of Cs 2 SeI 6 . Therefore, being a large absorption bandwidth in the visible region of light the Cs 2 SeI 6 is the novel replacement of organic-based perovskites for solar cell applications. The ultraviolet region absorption of Cs 2 SeCl 6 also makes them equally important for sterilizing surgical equipment and other optoelectronic devices.
Absorption coefficients α (ω) also explain the decline of light into the material similar to the imaginary dielectric constant, shown in Figure 5(c). The critical values of absorption coefficient 3.15, 2.66 and 1.16 eV for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively. The peak values at 3.6 eV (5.8), 3.15 eV (5.9) and 2.61 eV (6.8) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 , respectively, show the maximum absorption similar like imaginary dielectric constant. In absorption coefficient, the shifting of peaks towards higher energy region and decreasing of intensities as compared to imaginary dielectric constant is the consequence conversion of dielectric constants into absorption coefficient which depend on frequency. The formula of absorption coefficient contains both real and imaginary parts of dielectric constants.
The refractive index n (ω) elucidates the dispersion of light and transparent behaviour of the material. The pattern of n (ω) and Re ε(ω) are similar and they are related to each other through relation n 2 − k 2 = Reε(ω). The calculated values of n (ω) are presented in Figure 5(d), which also explains the zero-frequency values n (0) and Re ε(0) that satisfied the relation n 2 0 = Re E (0) as shown in Table 3. The peak values of refractive index are reported at 3.62 eV (1.6), 3.14 eV (1.61) and 2.56 eV (2.57) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 halides, respectively. For the visible region, the refractive index lies between 2 and 3 [32] which confirms Cs 2 SeI 6 is more suitable for visible region operation and its reducing intensity for Cs 2 SeBr 6 and Cs 2 SeCl 6 show the shifting of absorption from the visible region to the ultraviolet region. Table 3. The calculated bandgap E g (eV), static dielectric constant ε 1 (0), static refractive index n (0) and reflectivity R (0) of Cs 2 SeX 6 (X = Cl, Br, I). The morphology of the materials depends upon the reflection of light which provides information about the roughness of the material surface. The calculated values of reflectivity have been presented in Figure 6(a). The reflection increases from zero-frequency value and reaches to maximum at 3.62 eV (0.31), 3.11 eV (0.32) and 3.09 eV (0.62) for Cs 2 SeCl 6 , Cs 2 SeBr 6 and Cs 2 SeI 6 . Therefore, the reflection in the visible region is less as compared to the ultraviolet region. The optical loss of light energy in the form of scattering, heating, etc. has been reported in Figure 6(b). The optical loss of energy up to 3.0 eV is very negligible. Therefore, the optical characteristic analysis confirms the maximum absorption, less reflectivity and optical loss in the visible region increase the importance of studied materials for solar cells and other optoelectronic applications.

Conclusion
In the present article, the electronic, mechanical and optical properties are investigated thoroughly to understand the potential of the studied materials for solar cells and optoelectronics. The value of tolerance factor in the range (0.97-1.0) shows the structural stability while negative formation energy and positive frequencies of phonon dispersion certify thermodynamically stability. The studied materials are also mechanically stable and have ductile nature along with a high melting point. The Cs 2 SeCl 6 has the highest absorption in the ultraviolet region and shifted to the visible region by substituting Cl with Br and I. The variant perovskites Cs 2 SeI 6 is new potential material for visible light solar cells. Additionally, other parameters such as reflection of light and secondly the optical loss possess minimum value in the visible region. Therefore, the highest visible region absorption along with minimum loss of energy makes them interesting compounds for practical applications for solar cells and in optoelectronics.