High-Throughput Screening for Band gap Engineering by Sublattice Mixing of Cs$_2$AgBiCl$_6$ from First-Principles

The lead-free double perovskite material (viz. Cs$_2$AgBiCl$_6$) has emerged as an efficient and environmentally friendly alternative to lead halide perovskites. To make Cs$_2$AgBiCl$_6$ optically active in the visible region of solar spectrum, band gap engineering approach has been undertaken. Using Cs$_2$AgBiCl$_6$ as a host, band gap and optical properties of Cs$_2$AgBiCl$_6$ have been modulated by alloying with M(I), M(II), and M(III) cations at Ag-/Bi-sites. Here, we have employed density functional theory (DFT) with suitable exchange-correlation functionals in light of spin-orbit coupling (SOC) to determine the stability, band gap and optical properties of different compositions, that are obtained on Ag-Cl and Bi-Cl sublattices mixing. On analyzing the 64 combinations within Cs$_2$AgBiCl$_6$, we have identified 19 promising configurations having band gap sensitive to solar cell applications. The most suitable configurations with Ge(II) and Sn(II) substitutions have spectroscopic limited maximum efficiency (SLME) of 32.08% and 30.91%, respectively, which are apt for solar cell absorber.

degradation of photovoltaic performance.
In this Letter, we present an intensive theoretical study on band gap transmutation of Cs 2 AgBiCl 6 by means of sublattice mixing. The sublattice mixing is done by substituting M(I) at Ag-sites, M(II) at Ag-and Bi-sites simultaneously, and M(III) at Bi-sites in various concentrations for enhancing the optical properties of Cs 2 AgBiCl 6 . A high-throughput screening is performed by carrying out the hierarchical computations employing state-ofthe-art first-principles based methodologies under the framework of density functional theory (DFT). We start doing a lot of pre-screening of a large number of configurations with DFT using generalized gradient approximation based exchange-correlation ( xc ) functional (PBE 29 ) and following that the promising candidate structures are further analyzed using hybrid DFT with HSE06. 30 The latter xc functional helps for more accurate understanding of the excited state properties. Note that in all the above calculations (viz. PBE or HSE06), the effect of spin-orbit coupling (SOC) is always taken into consideration. This is a crucial step to determine the accurate band gap and band-edge positions of these systems due to presence of heavy metal atoms. We have started with 64 sets of different combinations of metals M(I), M(II), and M(III) respectively. Firstly, the structural stability is predicted using the Goldschmidt's tolerance factor and octahedral factor. It is worth noting that structural stability alone is not sufficient for the formation of perovskites. Hence, to validate the material's thermodynamic stability, the enthalpy of decomposition (∆H D ) is calculated.
Then, from ∆H D and band gap range (which expands the spectral response in visible region), the promising stable double perovskite configurations are identified. Following identification of such potential candidate structures, detailed electronic structure is carried out alongside of computing optical properties. The real and imaginary parts of the dielectric function are analyzed to understand the effect of sublattice mixing in Cs 2 AgBiCl 6 for transmutation of band gap. Subsequently, we determine spectroscopic limited maximum efficiency (SLME) of all the stable configurations that possess direct band gap, to determine efficient solar cell absorber.
Initially, the benchmarking of xc functional has been performed by calculating the band gap of pristine system viz. Cs 2 AgBiCl 6 . The band gap of Cs 2 AgBiCl 6 with PBE xc functional is 2.06 eV, which is not in agreement with the experimental value of 2.77 eV. 31 As this system contains heavy metal atom (viz. Bi), the inclusion of SOC becomes important. 6 However, incorporation of SOC with PBE underestimates the band gap (1.68 eV) further due to splitting of the conduction band minimum (CBm). The latter gets shifted to lower energy toward Fermi-level, whereas the valence band maximum (VBM) remains unaffected (see Figure S1a in Supplementary Information (SI)). On the other hand, to include the selfinteraction error of a many-electron system in the expression of xc functional, advanced hybrid xc functional viz. HSE06 becomes essential. It gives a band gap of 3.15 eV (without SOC, HSE06 only) and 2.60 eV (with SOC, HSE06+SOC) respectively with default (0.25) Hartree-Fock exchange fraction (α) (see SI Figure S2). On increasing α to 0.30 and 0.35, we have obtained band gaps of 2.79 and 2.99 eV, respectively using HSE06+SOC (see Figure S1 and S2 in SI). Note that using HSE06+SOC w.r.t the experimental value, though the band gap with α = 0.30 is more accurate than that of default value (i.e. α = 0.25), we still have proceeded with default one for further calculations. This is due to the fact that on alloying with various metals, the value of α can vary from one system to other, and determining the same without experimental results is next to impossible for new configurations. Therefore, we expect, atleast the default α should give a correct trend qualitatively even if the actual numbers may differ marginally as in the case of pristine Cs 2 AgBiCl 6 . (Here, red, blue and green color circular dots correspond to M(I) (e.g., substitution of 25% Au at Ag-site ((25%)Au Ag )), M(II) (e.g., substitution of Sn at Ag-and Bi-site simultaneously (Sn Ag,Bi )) and M(III) (e.g., substitution of 25% Sb at Bi-site ((25%)Sb Bi )), respectively.) We have started with 64 configurations of double perovskites obtained on mixing the Ag-Cl and Bi-Cl sublattices with M(I) (viz. Au, Cu, In, K, Na, and Ti), M(II) (viz. Cd, Co, Cu, Ge, Mn, Mo, Ni, Sn, V, Zn, and Rh), and M(III) (viz. Cr, Ga, In, Tl, Sb, and Y). Here, we have varied the concentration of the alloying atom viz. 25%, 50%, 75% and 100%. Two fundamental factors need to be satisfied for the stability of double perovskites to exist in high symmetry cubic structure, viz. Goldschmidt's tolerance factor (t) 32 and octahedral factor (µ). 33 For structural stability, t should lie between 0.8 and 1.0, and µ should be greater than 0.41. 34 To calculate these two fundamental factors for various double perovskite configurations, we have employed a strategy. 35 All the selected double perovskite configurations satisfy these stability criteria (see Table S1 in SI). After structural stability, In Equation 1 and 3, x can have value 1, 2, 3 or 4. Based on the band gap and ∆H D (calculated using PBE+SOC), a pre-screening process has been employed to find the suitable configurations. In Figure 6(a), the promising configurations lie within the shaded region for which band gap is varying from 0.0 to 1.5 eV (for further details see Table S2, S3 and S4 in SI). For all the prescreened configurations of double perovskite, ∆H D is negative, indicating that all the considered systems are stable and these will not decompose into respective binary components. Subsequently, we have calculated ∆H D and band gap using HSE06+SOC xc functional for aforementioned selected configurations (see Figure 6(b)).
In Figure 6 Note that on increasing the concentration of the external element, if the band gap is increased (or decreased), it increases (or decreases) consistently on further increasing the concentration. 37 However, in some cases, we have noticed an irregular change in band gap on varying the concentration of the alloying atoms. For instance, on increasing the percentage of Sb at Bi-sites, band gap decreases up to 75% substitution, and thereafter, an increment in band gap has been observed on 100% substitution. Similar kind of change in band gap has also been observed on complete substitution of other elements at Ag or Bi site (see Table S2, S3 and S4 in SI for details). To understand this change in the band gap on 100% substitution of Sb, we have plotted the band structures of Sb-alloyed system with different concentrations of Sb (see Figure 5). We can clearly see that on 100% substitution of Sb at Bi-sites, the lowest energy level in the conduction band corresponding to Bi-orbitals (that is  Figure 5(a-c)) disappears. Consequently, there is an increment in the band gap (see Figure 5(d)). In addition, from Figure 5, we can see that SOC effect is attributed to the fundamental mismatch of Ag-and Bi-orbitals, which is also mentioned by Savory et al. 38 Similar SOC effect can be observed from band structures for Au substitution at Ag-sites as well (see Figure S3 in SI). Hence, in some cases, on complete substitution either at Ag-or Bi-sites, there is an inconsistent change in the band gap (i.e., on complete substitution of Ag with Na, K etc., similar inconsistency has been observed). For more details also see Figure   S4 and S5.
Next, we have determined the optical properties of the 19 potential candidate structures + Im(ε) has been calculated using HSE06+SOC xc functional as shown in Figure 3 (the results of optical properties without SOC are shown in Figure S6 and S7 in SI for bechmark purpose). Therefore, using Re(ε) and Im(ε) of dielectric function, various optical properties, e.g., refractive index (η), extinction coefficient (κ), and absorption coefficient (α) can be computed using following expressions: Here, in Equation 6, ω and c correspond to angular frequency and speed of light, respectively.
In Figure 3  constant (ω = 0) increases, with increase in the concentration of Au and decreases, with increase in the concentration of Cu (see Figure 3(b)). For high degree of charge screening, which can prohibit radiative electron-hole recombination, a large value of Re(ω) at ω = 0 is indispensable. 39 Hence, the solar cell absorber, which exhibits large Re(ω) at ω = 0 is more efficient. In view of this, Au substitution is more beneficial than Cu for replacing Ag-sites.
In Figure 3(c), we can clearly see that peaks corresponding to Au substitution are red revealed that partial Sn substitution acts as a promising candidate to enhance the optical properties of Cs 2 AgBiCl 6 without degrading the stability. In addition, it also exhibits direct band gap property (see Figure S4 in SI). Thus Sn Ag,Bi acts as a rational candidate for solar cell application.
Lastly, to design highly efficient solar cell absorber, spectroscopic limited maximum efficiency (SLME) 34 has been calculated. The SLME is based on the improved Shockley-Queisser model. It depends on the absorption coefficient, thickness of the film absorber and nature of the band gap of the material. For an efficient solar cell, Yu et al. 40 have proposed the concept of SLME based on Fermi Golden rule. According to Fermi Golden rule, the optical absorption is directly proportional to where ν|Ĥ|c is the transition matrix. ω is the photonic energy required for transition from states in valence band (ν) to the states in conduction band (c). In Equation 7, integration goes over the whole reciprocal space. We have calculated the SLME of those alloyed systems, which possess direct band gap, as a function of thickness of the absorber layer (see Figure 7). Here, in Table 3, we have shown SLME of few double perovskites at 5 µm and compared our results with other efficient hybrid perovskites that are reported recently. From  tively. These numbers are very much encouraging from application perspective in solar cells.
In addition, they are more stable, while in contact with air and moisture as compared to IO hybrid perovskites. 31,42 In summary, we have presented a thorough study of alloying double perovskite Cs 2

Supporting Information Available
The following files are available free of charge. Details regarding band gap using different functionals for various conformers, optical properties using HSE06, and partial density of states (pDOS) have been given in the supporting information file.        In Figure 9, through partial density of states (pDOS), we have presented a clear picture of different orbitals' contribution in VBM and CBm. We have noticed that molecular orbitals that contribute to VBM and CBm consist of hybridization of different atomic orbitals. Hence, complete substitution of Bi with Sb, eliminates the contribution of Bi s-orbital in CBm. As a consequence, the band gap increases. Similarly, in some cases, for 100% removal of Ag and Bi, a sudden change in band gap has been observed, which is ascribed to the elimination of non-degenerate energy levels.

Supplemental
Optical properties using HSE06 xc functional