Electronic and optical properties of vacancy ordered double perovskites A2BX6 (A = Rb, Cs; B = Sn, Pd, Pt; and X = Cl, Br, I): a first principles study

The highly successful PBE functional and the modified Becke–Johnson exchange potential were used to calculate the structural, electronic, and optical properties of the vacancy-ordered double perovskites A2BX6 (A = Rb, Cs; B = Sn, Pd, Pt; X = Cl, Br, and I) using the density functional theory, a first principles approach. The convex hull approach was used to check the thermodynamic stability of the compounds. The calculated parameters (lattice constants, band gap, and bond lengths) are in tune with the available experimental and theoretical results. The compounds, Rb2PdBr6 and Cs2PtI6, exhibit band gaps within the optimal range of 0.9–1.6 eV, required for the single-junction photovoltaic applications. The photovoltaic efficiency of the studied materials was assessed using the spectroscopic-limited-maximum-efficiency (SLME) metric as well as the optical properties. The ideal band gap, high dielectric constants, and optimum light absorption of these perovskites make them suitable for high performance single and multi-junction perovskite solar cells.

www.nature.com/scientificreports/ and Cs 2 AgInCl 6 28 . Structure-wise, the A 2 BX 6 perovskites such as Cs 2 SnI 6 or Cs 2 PdBr 6 , can be considered as a derivation from ABX 3 if half of the B cations at the [BX 6 ] cluster center are removed in a checkerboard type pattern 17 . The charge neutrality condition implies that the B-site cation should be a tetravalent, extending the type of cations for substitution into B-site 29 . Generally, it is referred to as antifluorite crystal (K 2 PtCl 6 ) and is described by the [ BX 6 ] 2octahedral cluster bridged by the A-site cations 25 . The A 2 BX 6 structure shows similar features to ABX 3 perovskites and most of them possesses cubic structure.
Recently, the Cs 2 BI 6 with B = Sn and Te have been reported capable of absorption of light in the visible to infrared (IR) region giving new hope for stable materials with a nature friendly operation 25,30,31 . In this framework, Cs 2 SnI 6 with cubic crystal structure containing Sn in its +4 oxidation state is regarded as a potential candidate for applications in perovskite solar cells (PSCs) 30 . Diffuse-reflectance measurements of Cs 2 SnI 6 show an optical band gap of 1.25-1.30 eV in comparison with the thin films band gap of 1.60 eV. The density functional theory results in a direct transition nature with the suggested band gap (0.13 to 1.26 eV) significantly different from the experimental value given above, most likely using functionals with different correlation approximations 29,32 . The compound possesses N-type electrical conductivity, strong absorption power, and moisture stability 30,31,33 . Due to the tetravalent character of Sn, Cs 2 SnI 6 exhibits higher air stability with respect to CsSnI 3 29 . In fact, numerous non-or low-toxic transition metals have stable +4 oxidation state paving the way for finding favorable halide perovskites, for example by replacing the Sn 4+ in Cs 2 SnI 6 by appropriate transition-metal cations 34 . This is confirmed by Sakai et al. who studied Cs 2 PdBr 6 as a novel perovskite for application in PSCs 35 . The optical band gap of Cs 2 PdBr 6 calculated from absorption measurement was 1.60 eV 35 . The effective mass of electrons and holes calculated from first principles technique were reported to be 0.53 m e and 0.85 m e , which indicate an N-type semiconducting behavior for Cs 2 PdBr 6 35 . Ju et al. carried out an integrated experimental and theoretical study of Ti-based vacancy-ordered double perovskites (DPs) A 2 TiX 6 (A = K + , Rb + , Cs + In + ; X = I, Br, or Cl) and Cs 2 TiI x Br 6−x , showing a suitable band gap in the range from 1.38 to 1.78 eV for photovoltaic applications 34 . Zhao et al. studied a new family of vacancy ordered DPs, Cs 2 BX 6 (B = Pd, Sn, Ti, Te; X = Cl, I), claiming the compounds possess diverse electronic structures and optical features 36 . A group of researchers computationally studied compounds of the type A 2 MX 6 (A = K, Rb, and Cs, M = Sn, Pd, Pt, Te, and X = I) by using hybrid functional (HSE06), reporting the variation of band gap and effective mass with the A-site cation changing from K to Rb to Cs 8 .
To conduct further research aimed at the use of various metals substitution for photovoltaic and optoelectronic applications, we make use of the density functional theory to explore new variants in A 2 BX 6 family with possible A = Rb, Cs; B = Sn, Pd, or Pt; and X = Cl, Br, or I. We begin from the structural properties and then examine the electronic structure as well as optical spectra of these compounds. In addition, we also report on the thermodynamic stability of these compounds by calculating their formation energies.

Computational methods
Structures and other physical properties of existing as well as hypothetical compounds can be approximated with considerable success using the density functional theory. To manipulate the vacancy-ordered DPs, we used wien2k 37 code based on the density-functional theory (DFT) 38 by employing the Full Potential Linearized Augmented Plane Wave (FP-LAPW) method with Perdew, Burke, and Ernzerhof functional (PBE) 39 , modified Becke Johnson (mBJ) semi-local exchange potential 40 , and hybrid functional HSE06 41 . The band structure and the density of states were further examined taking into consideration the spin-orbit coupling (SOC) 42 interaction. In HSE06 calculations, we used 25% of the exact Hartree-Fock exchange fraction together with 75% exchange and 100% correlation energies of PBE functional. Both the mBJ and HSE06 have been shown to produce more accurate band gap as compared to standard LDA/GGA functionals 43,44 . Further details about the computational technique can be found in the supplementary information.

Results and discussion
Structural properties. Our calculations show that the vacancy-ordered DPs A 2 BX 6 (A = Rb and Cs; B = Sn, Pd, and Pt; and X = Cl, Br, and I) have a face centered cubic structure with space group Fm3m (No. 225). The atomic positions and geometric configuration of A 2 BX 6 is illustrated in Fig. 1 6 ] octahedra in A 2 BX 6 structure is isolated from the others forming a 12-fold coordination environment of discrete X anions.  Table 1. The lattice constants are found in good agreement with experiments showing an increasing trend with changing Cl to Br and then to I in agreement with their changing geometry (Fig. S2, in Supplementary Information). The calculated bond lengths obtained after energy minimization for A 2 BX 6 compounds are presented in Supplementary Information Table S1. The phase stability was also assessed using the tolerance factor 't' proposed by Goldschmidt 45 and was found well within the proposed range (0.8 ≤ t ≤ 1.11) 46 for stable 3D cubic halide perovskites, as shown in Table 1.
The thermodynamic stability of the compounds was explored via the commonly used convex hull approach (details are given in the Supplementary Information). The full list of the competing phases as well as the calculated formation energies for the set of the compounds are given in Table S2. The accessible range of their chemical potentials in a two dimensional plane is shown in Figs  Band structure and density of states. Generally, the band gap of inorganic Pb-free perovskites solar material ought to be within the optimal range: 0.9 to 1.6 eV (efficiency > 25%) 55 . The band gap calculated with various exchange-correlation functionals is given in  (Fig. 3). The valence band maximum (VBM) and the conduction band minimum (CBM) are localized at the Г (0.0, 0.0, 0.0) symmetry point. The band gap calculated using mBJ are 2.42 eV (Rb 2 SnBr 6 ) and 0.84 eV (Rb 2 SnI 6 ) slightly larger than that calculated with PBE-GGA. When we include the spin-orbit coupling (SOC) effect, the computed band gap for Rb 2 SnI 6 reduces by 0.16 eV bringing it to 0.68 eV. The mBJ calculated band gap seems to be significantly underestimated as compared with a UV-visible experiment (1.32 eV) 56 and a calculated (HSE06) value for cubic (1.02 eV) 8 and tetragonal (1.13 eV) 56 phase. However, our calculated HSE06 values are close to the experiment as well as other theoretical works (Table 2). For these compounds, the valence band is mainly derived from Sn-5s and X-5p antibonding orbitals whereas the conduction band is derived entirely from Sn-5p and Rb-3d anti-bonding orbitals as shown in Fig. 4. For the Pd-based compounds, Rb 2 PdCl 6 , Rb 2 PdBr 6 , and Rb 2 PdI 6 , the computed band structures are shown in Fig. S4.    Figure S5 shows the band structure of Cs 2 PtCl 6 , Cs 2 PtBr 6 , and Cs 2 PtI 6 . The band gap seems to increase in the order Cs 2 PtCl 6 > Cs 2 PtBr 6 > Cs 2 PtI 6 , consistent with the trend observed in CH 3 NH 3 PbX 3 (X = Cl, Br, I) 57 . Only the Cs 2 PtCl 6 perovskite possesses a direct band gap (Fig. S5)  For Cs 2 PtI 6 , our calculations show an indirect band gap of 1.22 and 1.11 eV (mBJ and mBJ-SOC) at Г − X symmetry lines. The mBJ based band gap is in a better agreement with an earlier report (E g = 1.34 eV) 8 determined using HSE-SOC functional, justifying using the mBJ approximation in our calculations. The lower part of the valence band is predominantly formed by Cs-p (~ -7 eV) whereas the VBM is mainly due to Cl/Br/I-p states (Fig. S6). The conduction bands are composed of the Pt-d and halogens-p states. Interestingly, we found that among the eight semiconductors, Rb 2 PdBr 6 (1.31 eV) and Cs 2 PtI 6 (1.22 eV) have favorable band gap in the optimal range of 0.9-1.6 eV, suggesting their possible use in single-junctions PSCs. These findings are also supported by our calculated favorable effective mass for electrons and holes for Rb 2 PdBr 6 and Cs 2 PtI 6 as shown in Table S3. We used single parabolic band approximation to calculate the carrier effective mass for these compounds. Despite the favorable band gap and the effective mass, the Pt and Pd based compounds show multiple minima (at X and K/U symmetry points) in the conduction band. The energy difference between the two minima is ~ 0.05 eV which gives rise to the valley degeneracy responsible for the high figure of merit 58 . This difference can be eliminated by strain engineering. It is reported in literature 58-60 that the band convergence improves the thermoelectric performance (TEP) of the system, thus we can also predict that with applying the suitable strain in Pt and Pd based compound, their TEP can be improved. However, calculating TEP of the present class of materials is beyond the scope of the present investigation. Hopefully, our observation may motivate the scientific community to consider working on TEP of Pt and Pd based compounds.
Optical properties. The computed real ε 1 (ω) and imaginary ε 2 (ω) part of the dielectric function are depicted in Fig. 5 and Supplementary Table S4. The static dielectric constant ε 0 of Rb 2 PdI 6 was found to be 6.76, which is larger than that of the Pb-based CH 3 NH 3 PbI 3 perovskite (~ 5.2) 61 . A large value of the static dielectric constant is essential for efficient light absorption. It can promote low level of charge defects and prohibit radiative electron-hole recombination rate. Further, the real part, ε 1 (ω) , increases to a maximum (9.9 for Rb 2 PdI 6 at 1.15 eV) and then decreases asymptotically to negative values making re-inversions to secondary maxima and minima. From Fig. 5a-c, it is obvious that the curves are redshifted towards the visible region by changing the halide ions (Cl → Br → I) across the selected compounds, causing an increase in ε 1 (ω) and shifting of peaks towards the low energies.
The imaginary parts ε 2 (ω) of the dielectric function in Fig. 5d-f exhibits the optical transitions between VBM and CBM at the threshold energy 2.41 (Rb 2 SnBr 6 ) and 0.86 eV (Rb 2 SnI 6 ), which are direct as evident from the band structure plot (Fig. 3). For Rb 2 PdCl 6 , Rb 2 PdBr 6 , and Rb 2 PdI 6 , the threshold is 2.12, 1.33, and 0.49 eV,   Table S4. It is obvious from Fig. 5d-f and Table S4 that the computed intensities of ε 2 (ω) for the iodide-based compounds are higher than those of the Br-and Cl-containing compounds. The main reason is that the band gap of the former phases is smaller than the latter structures. Hence, according to the Fermi golden rule, the transition probability is reduced in the Br-and Cl-based compounds. Figure 6a shows the calculated absorption coefficients α(ω) with the absorption edges at 0.5-2.9 eV for different compounds in reasonable agreement with the corresponding band gap (mBJ). Several absorption peaks with increasing trend can be attributed to the electronic transitions from bonding states to the anti-bonding states. In the visible energy range, the maximum absorption (5.71 × 10 5 cm −1 ) corresponds to Rb 2 PdBr 6 perovskite. Therefore, Pd can be considered a suitable alternative to Pb in inorganic perovskites solar cells. Within this group (Rb 2 PdX 6 ), the maxima of the absorption peaks have a significant downward trend in unison with photon energy as a function of increasing halogen radii. The rest of the compounds have peaks in the ultraviolet region  www.nature.com/scientificreports/ suitable for optical devices working in this range. The optical conductivity as well as reflectivity spectra have also been calculated, their full details is provided in the Supplementary Information (Figs. S7, S8). A successful criteria for assessing the photovoltaic efficiency of a solar absorber is the spectroscopic-limitedmaximum-efficiency (SLME) 62 . It takes into account the band gap, the optical absorption spectrum, the recombination mechanism, and the fundamental transition. The computed SLME for our compounds are shown in Fig. 6b. The SLME of Rb 2 PdBr 6 , Rb 2 PdI 6 , and Cs 2 PtI 6 are higher than the rest of the compounds which can be attributed to their favorable band gap and optimum light absorption.

Conclusions
We have performed first-principles calculations employing the mBJ potential to investigate the electronic structure as well as the optical properties of defect perovskites A 2 BX 6 (A = Rb and Cs; B = Sn, Pd, and Pt; and X = Cl, Br, and I). The structural analysis shows a monotonic increase of the lattice constant and volume by changing the halide ion from Cl to Br and then to I which results in a gradual increase in the B-X bond length. The calculated enthalpies of formation for the investigated A 2 BX 6 family are found to be negative, except for Rb 2 SnBr 6 , Rb 2 SnI 6 , and Rb 2 PdI 6 , and also the calculated tolerance factor of all the compounds ranges from 0.96 to 1.07, lying within the specified range for stability of the cubic halide perovskites. The results show that all the compounds possess optimum electronic and optical properties to be used as visible-light absorbing materials for PV applications. We also applied different exchange-correlation functionals, namely PBE-GGA, mBJ, mBJ-SOC, and HSE06 to get a close approximation of the true band gap of the A 2 BX 6 family. The HSE06 functional gives the nearest approximation to the available experimental results. The band gap varies most likely due to the electronegativity or size difference of B-and X-site atoms. Among the entire group of the compounds studied, the ideal band gap was obtained only for Rb 2 PdBr 6 (1.31 eV) and Cs 2 PtI 6 (1.22 eV) in the optimal range of 0.9-1.6 eV. This indicates that both the compounds are potential candidates for single-junction solar cell application in the future. We also calculated different optical properties, more specifically, the complex dielectric function, absorption coefficient, and SLME which support the use of these materials in various optoelectronic applications. The compounds, Rb 2 PdBr 6 , Rb 2 PdI 6 , and Cs 2 PtI 6 , possess suitable band gap and relatively high optical absorption as compared to other members of the A 2 BX 6 family. We hope that our results will motivate further research in this direction in order to use lead-free perovskite variants in efficient photovoltaic or other optoelectronic devices.