Deep insight into structural and optoelectronic properties of mixed anion perovskites

The optimization could be the very first important step to obtaining a relaxed structure with zero stresses. The relaxed structure could be used as the initial structure for the properties we are finding afterward. During the optimization, the obtained energy and volumes of the structure are fit to the Birch-Murnaghan equation of state as shown in equation (1) to obtain equilibrium cell parameters.

$$\begin{eqnarray}&&E={E}_{0}+\frac{{B}_{0}}{{B}_{0}^{{\prime} }}\left(V-{V}_{0}\right)-\frac{{B}_{0}{V}_{0}}{{B}_{0}^{{\prime} }(1-{B}_{0}^{{\prime} })}\left[{\left(\frac{V}{{V}_{0}}\right)}^{1-{B}_{0}^{{\prime} }}-1\right]\end{eqnarray} \tag{ 1 }$$

where V is the primitive-cell volume, B is the bulk modulus, which provides the behavior of the crystal volume under hydrostatic pressure, [25] and B' is its first pressure derivative. The zero indexes are the values at zero pressure [26]. Figure 1 represents the variation of total energy with volume for Cs₂AgBiX₆ in cubic and tetragonal phases. We found there is a tendency for the tetragonal phase to become more stable when the halogen component is changed from F towards I.

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Lattice parameters of the optimized Cs₂AgBiX₆ structures are calculated using GGA approximation and obtained values are tabulated in table 1. For the sake of completeness, these calculated lattice parameters are tabulated with other earlier reported theoretical and experimental studies. The comparison shows that our results for other known compounds are in good agreement with the reported literature values for cubic-phased Cs₂AgBiX₆ structures. No previous reports are available on lattice parameters for Cs₂AgBiF₆ structures. According to past theoretical studies, the deviation of the value of lattice constant of the cubic phase in bromide and chloride compounds is in the range of 0.1%–0.5%, [15, 27–29] and also experimentally it is between 1%–2% [6, 7] variations can be found. Thus the deviation is less than 5% in both previous experimental and theoretical works, we could rely on the lattice parameters of our relaxed structure and this implies the accuracy of our geometry optimization and the computational simulation method along with the input parameters which are mentioned under the computational methodology. The lattice constant and the unit cell elongation in the c direction for the tetragonal structure of the bromide compound differ from 1.6% to 5% from the experimental findings which were done under the low temperature, 122 K condition of phase transition from cubic to tetragonal [2]. The computed lattice constant for the iodine compound in the cubic phase deviates from 0.5% - 1.4% from other theoretical findings [30, 31]. When compared to the other double perovskites, cubic phases of the Cs₂NaMCl₆ (M = In, Tl, Sb, Bi) exhibited the lattice constants which vary from 0.8%–3.67% which gives some closer values to the chloride-based, Cs₂AgBiCl₆ compound [32]. Another theoretical study was done for the cubic phases of Cs₂NaBX₆ (B = Sb, Bi; X = Cl, Br, I), where the Cs₂NaBiCl₆, and Cs₂NaSbCl₆ lattice constants are different from 0.85% and 2.54% when compared to Cs₂AgBiCl₆. The other compounds' lattice parameters of Cs₂NaBiBr₆ and Cs₂NaSbBr₆ vary from 1% to 2% from Cs₂AgBiBr₆ respectively and finally, the lattice constant of the iodide compound of Cs₂AgBiI₆ differs from 1% to the Cs₂NaSbI₆ [33]. By examining those previous theoretical studies, we can say that changing the B site cation and the X site anion will not affect very much with our computed lattice parameters.

Table 1. Calculated structural parameters for Cs₂AgBiX₆ (where X = F, Cl, Br, and I) along with other experimental (exp.) and theoretical (Theo.) studies.

Compound	Cubic (Fm $\mathop{3}\limits^{̅}$ m)	Primary	Tetragonal (I 4/m)	Primary
	Lattice parameters (Å)	cell volume	lattice parameters	cell volume
Cs2AgBiF6	9.5038 Å	214.60 Å3	a = 6.5267 Å	229.29 Å3
			c = 10.8009 Å
Cs2AgBiCl6	10.9333 Å	326.74 Å3	a = 7.7052 Å	333.68 Å3
	10.965 [27] (Theo.)		c = 11.2410 Å
	10.777 [27] (exp.)
	10.944 [28] (Theo.)
	10.99 [29] (Theo.)
Cs2AgBiBr6	11.4836 Å	378.59 Å3	a = 8.0127 Å	384.96 Å3
	11.462 [15] (Theo.), 11.25 [6] (exp.),11.53 [29] (Theo.), 11.271 [7] (exp.), 11.496 [27] (Theo.),		c = 11.9938 Å
			a = 7.879, c = 11.3236 [2] (exp.)
			a = 7.844, c = 11.425 [15] (Theo.)
Cs2AgBiI6	12.2054 Å	454.56 Å3	a = 8.4932 Å	467.08 Å3
	12.27 [30] (Theo.)		c = 12.9512 Å
	12.03 [31] (Theo.)
	12.28 [29] (Theo.)

The structural parameters of the tetragonal phases of Cs₂AgBiF₆, Cs₂AgBiCl₆, and Cs₂AgBiI₆ are also reported for the first time here. As table 1 shows, both the lattice parameters and the cell volume increase when the halide compound is varied from fluoride to iodide atoms. This can be attributed to the increase in the size of the p orbital of the halogen atom and the decrease in electronegativity when going from F to I. However, the primary cell volume appears larger in the tetragonal phase than in the cubic phase for all these halide compounds. The DFT calculations were done at standard parameters. The specific reasons for this structural difference can vary depending on the materials and conditions, such as external temperature and pressure, but generally, it involves changes in the arrangement of atoms or molecules within the crystal lattice. The elongation in the c-direction in tetragonal phases contributes to a larger overall volume compared to the more symmetrical cubic phases.

Among the halide materials in table 1, Our attention is directed toward the Cs₂AgBiBr₆ material due to its demonstrated favorable absorption properties (refer to the optical studies section for further information). The bromine atoms were replaced gradually with other halide atoms (F, Cl, I) in different concentration levels. The calculated structural parameters for Cs₂AgBiBr_6−x X_x where X = (F, Cl, I) are presented in (table 2). The symmetry of the mixed halides' crystal changes from higher symmetry (cubic phase) to lower symmetry (tetragonal phase) with the increasing amount of the substituted halide atoms of chlorine and iodine (x = 1, 2, 3, 4, 5). This can be attributed to the octahedral (BiBr₆) tilting in Cs₂AgBiBr₆. As the ionic radius of the Cl⁻ ion is smaller than the Br⁻ ion and the electron negativity is higher than for the Br⁻ ion, these factors lead to a decrease in the cell volume of Cs₂AgBiBr_6−xCl_x as x is increased from 1 to 5.

Table 2. The equilibrium lattice parameters for Cs₂AgBiBr_6−xF_x, Cs₂AgBiBr_6−xCl_x, and Cs₂AgBiBr_6−xI_x (x = 0, 1, 2, 3, 4, 5, 6).

	Cs2AgBiBr6−xFx		Cs2AgBiBr6−xClx		Cs2AgBiBr6−xIx
	Lattice	Primary	Lattice	Primary	Lattice	Primary
Substitutions	Constants(Å)	Cell volume	Constants(Å)	Cell volume	Constants(Å)	Cell volume
	Space group	(Å)3		(Å)3	Space group	(Å)3
		(PBE)	Space group	(PBE)		(PBE)
x = 0	a = b = c = 11.4836 Å	378.59	a = b = c = 11.4836 Å, 11.4601 Å [17]	378.59	a = b = c = 11.4836 Å	378.59
	Cubic		Cubic		Cubic
	(Fm$\mathop{3}\limits^{̅}$ m)		(Fm$\mathop{3}\limits^{̅}$ m)		(Fm$\mathop{3}\limits^{̅}$ m)
x = 1	a = 8.1390 Å	352.02	a = 8.1255 Å, 8.0986 Å [17]	370.91	a = 8.1835 Å	395.08
	c = 10.6280 Å		c = 11.2358 Å, 11.2104 Å [17].		c = 11.7988 Å
	Tetragonal		Tetragonal		Tetragonal
	(I 4 m m)		(I4/mm)		(I4mm)
x = 2	a = 11.3738 Ǻ	357.63	a = 11.4889 Å, 11.4095 Å [17]	363.24	a = 8.4099 Å	409.01
	b = 7.9349Ǻ		b = 7.9556 Å, 7.9382 Å [17].		b = 11.6165 Å
	c = 7.9250Ǻ		c = 7.9477 Å, 7.9334 Å [17].		c = 8.3731 Å
	Orthorhombic		Orthorhombic		Orthorhombic (Imm2)
	(Imm2)		(Imm2)
x = 3	a = 11.3230 Ǻ	317.02	a = 7.9516 Å	356.07	a = 11.6328 Å	421.78
	b = 9.6876Ǻ		alpha = 60.07°		b = 12.1612Å
	c = 11.5606Ǻ		Trigonal		c = 11.9259 Å
	orthorhombic		(R3m)		Orthorhombic
	(Fmm 2)				(Fmm2)
x = 4	a = 9.9213Ǻ	276.74	a = 10.9787 Å, 10.9418 Å [17].	345.16	a = 12.1987 Å	434.28
	b = 7.5348Ǻ		b = 7.9333 Å, 7.9274 Å [17]		b = 8.4524 Å
	c = 7.4032Ǻ		c = 7.9250 Å, 7.9053 Å [17]		c = 8.4230 Å
	orthorhombic		Orthorhombic		Orthorhombic (Imm2)
	(Imm 2)		(I mm2)
x = 5	a = 7.1208 Å	254.23	a = 7.7566 Å, 7.7527 [17].	336.42	a = 8.6429 Å	445.02
	c = 10.0277 Å		c = 11.1835 Å, 11.1565 Å [17].		c = 11.9150 Å
	Tetragonal		Tetragonal (Imm2)		Tetragonal
	(I 4 m m)				(I4mm)
x = 6	a = b = c = 9.5038 Å	214.60	a = b = c = 10.9333 Å, 10.9354 Å [17].	326.74	a = b = c = 12.2054 Å	454.56
	Cubic		Cubic		Cubic
	(Fm$\mathop{3}\limits^{̅}$ m)		(Fm$\mathop{3}\limits^{̅}$ m)		(Fm$\mathop{3}\limits^{̅}$ m)

The structural parameters and cell volumes of Cs₂AgBiBr_6−xCl_x are almost the same as reported in previous theoretical and experimental studies on mixed halide compounds [17, 19, 27]. Jing Su et al performed a theoretical study from the VASP code based on the Cl atoms mixed Cs₂AgBiBr₆, the deviation (%) of their lattice parameters' values from our computed values is negligible i.e. (x = 1; a = 0.33%, c = 0.22%; x = 2; a = 0.69%, b = 0.22%, c = 0.18% and x = 4; a = 0.33%, b = 0.07%, c = 0.25%; x = 5; a = 0.24%) [17].

In table 2, the opposite structural behavior of the cell volume of Cs₂AgBiBr_6−xCl_x is seen in Cs₂AgBiBr_6−xI_x. As expected, the cell volume increases when more iodine atoms are substituted instead of bromine atoms in the parent compound, Cs₂AgBiBr₆ because the ionic radius of the I⁻ ion is larger than the Br⁻ ion, and the I⁻ ion exhibits lower electron negativity than the Br⁻ ion. In both cases, the cubic phase can be seen in the pure Cs₂AgBiBr₆ but it tends to have a lower symmetry with the added iodine atoms and ended up with the tetragonal structure when x = 5. A similar facial transition effect is seen with chlorine substitutions.

The mixed halides with the fluorine atoms gave the lowest cell volume for all the number of substitutions compared to the chlorine and iodine atoms substitutions because the fluorine atom has the highest electron negativity and the smallest anion radius.

The elastic constant could describe the material's behavior when stress is applied to it. We have computed the single crystal elastic constant using the finite strain technique shown in table 3 for the cubic phase of Cs₂AgBiX₆ (X = F, Cl, Br, I) within the GGA approach. Within the elastic region, stress and strain could be expressed in Voigt notation which satisfies Hooke's law, [22]

$$\begin{eqnarray}&&{\sigma }_{i}=\displaystyle \sum _{j=1}^{6}{{\rm{C}}}_{{ij}}{\varepsilon }_{j}\end{eqnarray} \tag{ 2 }$$

where ${\sigma }_{i}$ and ${\varepsilon }_{j}$ are respectively the stress and strain. Both stress and strain have three tensile and three shear components. i.e. 1 ≤ i,j ≤ 6. ${C}_{{ij}}$ is the second-order elastic stiffness tensor in the 6 × 6 matrix and it could be derived from the first-order derivative of the stress–strain curves as represented in equation (2). The ${C}_{{ij}}$ values depend on the symmetry of the crystal structure. The cubic crystal which has the higher symmetry consists of three independent elastic constants but for lower symmetry like the triclinic structures associated with 21 single elastic constants, when the symmetry reduces, the number of independent elastic constants increases. We calculate elastic constants for the cubic phase (C₁₁, C_12, and C₄₄) using the finite strain method which satisfies the mechanical stability criteria as shown below.

$$\begin{eqnarray}&&{C}_{44}\gt 0{\rm{;}}{C}_{11}\gt {C}_{12}{\rm{;}}{C}_{11}+2{C}_{12}\end{eqnarray} \tag{ 3 }$$

Using the calculated single-crystal elastic constants ${{\rm{C}}}_{{ij}}$ s could predict the elastic moduli of the macroscopic mechanical properties such as bulk modulus, Young's modulus, Poisson ratio, etc. through the Voigt-Reuss-Hill approximations using the following relationships [34, 35].

$$\begin{eqnarray}&&{B}_{v}={B}_{R}={B}_{H}=({C}_{11}+2{C}_{12})/3\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{G}_{V}=({C}_{11}-{C}_{12}+3{C}_{44})/5\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{G}_{R}=5\left({C}_{11}-{C}_{12}\right){C}_{44}/(4{C}_{44}+3\left({C}_{11}-3{C}_{12}\right))\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{G}_{H}=({G}_{V}+{G}_{R})/2\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&E=9{B}_{H}{G}_{H}/(3{B}_{H}+{G}_{H})\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&v=(3{B}_{H}-E)/6{B}_{H}\end{eqnarray} \tag{ 9 }$$

The subscripts R, V, and H represent the Reuss, Voigt, and Hill approximations, respectively.

The obtained C_ij values and all the other macroscopic elastic moduli increase from iodide compound to fluoride compound. For Cs₂AgBiX₆ (X = Cl, Br, I), our calculated elastic constant values show good agreement with the past theoretical results [36–38]. This is the first time elastic constant calculations have been carried out for Cs₂AgBiF₆. Anisotropy is defined by A = 2C₄₄/(C₁₁–C₁₂), the material is isotropic if A = 1, otherwise, it will be anisotropic, which is the case for all cubic Cs₂AgBiX₆ structures as illustrated in table 3. The anisotropic nature could be seen along the principle directions in our compounds as C₁₁ ≠ C₁₂ ≠ C₄₄ [38]. The derived bulk and shear moduli could explain the hardness of the material under different strains. Pugh's criterion (B\G) could be used to determine a material's ductile or brittle nature. When B/G > 1.75 material shows the character of ductility when it is less than 1.75, the material is brittle. Our halide compounds show the nature of ductility. The stiffness of the material described by Young's modulus, and the compressibility of the compound increase from the iodide compound to the fluoride compound. The bonding nature of the material could be described from the Poisson ratio (ν). If ν is less than 0.1, the material shows covalent characteristics, and the tabulated ν values are greater than 0.25, thus the materials are ionic as ν > 0.25 means ionic nature [15].

The band gap of the material determines the material's suitability for photovoltaic applications. We employ GGA and the most accurate HSE06 methods to calculate electronic band structures for materials Cs₂AgBiX₆ and the results for the band gaps are shown in table 4. We found that the GGA band gap value underestimates the experimentally verified band gap values reported in the literature. However, HSE06 band gap values are in good agreement with the past theoretical and experimental findings for the cubic phases of Cs₂AgBiX₆ (X = Cl, Br, I). However, there we can see the overestimation of the experimental band gap value because we used the standard 25% Hatree Fock (HF) exchange with the screening parameter of 0.2. These standard values were used for the rest of the electronic property calculation in both pure halides and mixed halides. Deepika et al obtained the band gap value of 3.15 eV for the Cs₂AgBiCl₆, there they implemented the spin–orbit coupling (SOC) along with the HSE06 and obtained the band gap value of 2.60 eV which is closer to the experimental value of 2.77 eV [39]. Also, they observed when increasing the HF exchange term (α) the band gap values are decreasing. McClure et al used the α as 26% and different cutoff energies which vary from our computational method with the incorporation of SOC and they obtained the band gap values of 2.62 eV and 2.06 eV which are close to the experimentally derived values of 2.77 eV and 2.19 eV for Cs₂AgBiCl₆ and Cs₂AgBiBr₆ respectively [7]. As we observed from the past theoretical studies, they employed the SOC along with the exchange–correlation term due to the heavy metal, Bi included in these perovskites [7, 39, 40] which can influence the band degeneracy or the band splitting of Bi-p orbital [17]. Hence this could lead to a decrease in the band gap value than the HSE06 without SOC. Due to the high computational cost and time consumption, we implemented our DFT calculations only with the HSE06 excluding the SOC, hence this can overestimate our bandgap values compared to the experimental findings. The same phenomena influenced the Cs₂AgBiI₆ where we obtain its band gap value of 1.78 eV which is overestimated from the past theoretical studies which are reported as 1.08 eV and 1.32 eV [7, 40]. However, SOC will not be affected greatly for the calculations based on the VASP code as Feng et al discussed that SOC can be addressed by the PAW method which is the accurate pseudopotential approximation [41]. As emphasized earlier, this is the first time the hybrid calculations for the cubic phase of Cs₂AgBiF₆ and GGA and HSE06 calculations on tetragonal structures of Cs₂AgBiX₆ (X = F, Cl, I) are reported. Figure S1 and figure 2 illustrate the electronic band diagrams of the Cubic and Tetragonal phases of Cs₂AgBiX₆ (X = F, Cl, Br, I) respectively. The band structures E(k) were computed on a discrete k mesh along the high-symmetry directions in the BZ.

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It is important to note that the band structures of these double perovskites in both cubic and tetragonal phases show an indirect band gap nature. The computed band gap values are listed in table 4 along with the references. The conduction band minimum (CBM) lies at the L-point whereas the valence band maxima (VBM) lies at the X-point in cubic phase materials (see figure S1). This is in agreement with the work done by Xinbo et al regarding the location of the band edge positions [42].

For the tetragonal phase (figure 2) CBM lies at X and VBM lies at L high symmetric points in the BZ. The band gap decreases from fluoride compound to iodide compound as we change the halogen atom. This could be related to the lattice parameters represented in table 1. The lattice parameters of the Cs₂AgBiX₆ in both cubic and tetragonal phases increase and also the bond length increases when the halogen atom of the material changes from F to I which could influence the reduction of the bandgap values accordingly. Typically, smaller bond lengths mean the electrons are more tightly bound to the atom, and hence it requires more energy to excite the electron from the valence band to the conduction band. Thus fluoride-based compound needs to absorb higher photon energy compared to other compounds as discussed by Rajeev et al [29] Volonakis et al also showed that indirect band gap values increase when moving up the halogen column in the periodic table [8]. The band gap values in the cubic phase are smaller than the tetragonal phases of these relevant double perovskites as seen in table 4. This could be explained by symmetry characterization. The symmetry is reduced from cubic to tetragonal with the influence of structural distortion [43]. This could lead to small band gap values in the cubic phase compared to the tetragonal phase.

We calculated band gap values employing both GGA/HSE methods for a variety of mixed halides of the form Cs₂AgBiBr_6−x X_x (X = F, Cl, I) in table 5. The mixed halide compounds show an indirect band gap nature when the bromine atoms are substituted by chlorine, iodine, and fluorine atoms. The band gaps of Cs₂AgBiBr_6−xCl_x mixed halides varied between the values of pure halide 3.17 eV (Cs₂AgBiCl₆) and 2.51 eV (Cs₂AgBiBr₆) as shown in table 5. The lowest bandgap value was found at 2.35 eV when the one bromine atom was substituted by the chlorine atom. After that band gap increases to 3.01 eV for the substitutions of five bromide atoms by the chloride atoms. Our HSE06 results for Cs₂AgBiBr_6−xCl_x are in agreement with the experimental findings [49]. Matthew et al observed the band gap linearly increases in Cs₂AgBiBr_6−x Cl_x in between the bandgap values of Cs₂AgBiBr₆ (2.19 eV) and Cs₂AgBiCl₆ (2.77 eV) experimentally. However, they observed, the deviation of the law of Vegard when increasing the chlorine concentration to the bromide compound where the chlorine content exceeds 85% [49].

Based on the accurate HSE06 calculations, it is noted that the iodine substitutions result in low band gap values with the lowest band gap of 1.68 eV when the four bromine atoms were substituted by the iodine atoms. The bromide and iodide ions show a good combination of halide compounds which result in band gap values of 2.03 eV, 2.21 eV, and 2.02 eV for the compounds of Cs₂AgBiBr₅I, Cs₂AgBiBr₄I₂, and Cs₂AgBiBr₃I₃ respectively which correspond to the visible light region in the solar spectrum. Experimental and theoretical works were carried out by Hua Wu et al on mixed halides of iodine mixed Cs₂AgBiBr₆. Experimentally, the halide exchange on Cs₂AgBiBr₆ was done using the post-treatment with methylammonium iodide (MAI) salt [50]. When the MAI concentration is 15 mg mL⁻¹ which is correspondence to the substitution of the two bromine atoms in Cs₂AgBiBr₆ with the inclusion of iodine atoms (x = 2) resulted in a band gap value of 2.20 eV which is in good agreement with our work (2.21 eV) under HSE06 approximation [50]. However, their DFT calculations at the GGA level show much deviation between 7% - 18%. This much deviation can occur due to the different material simulational packages along with their own set of input parameters and simulation tools when determining the corresponding band gap values. The Cs₂AgBiF₆ has the highest band gap among the halide compounds. When it comes to the mixed halides with the replacement of bromine atoms by the fluorine atoms, the band gap of the Cs₂AgBiF₆ could be able to reduce with the incorporation of the bromine atoms. The lowest band gaps of 2.55 eV and 2.86 eV were obtained by substituting one and two bromine atoms with fluorine atoms respectively (table 5). The same type of band diagram could be seen for all the mixed halide compositions of Cs₂AgBiBr_6−xF_x , Cs₂AgBiBr_6−xCl_x, and Cs₂AgBiBr_6−xI_x and are shown to possess indirect band gaps. The VBM and CBM are located along the high symmetry points of X and L in the first BZ. Here we present the band diagrams of the Cs₂AgBiBr₅Cl, Cs₂AgBiBr₅I, and Cs₂AgBiBr₅F as shown in figure 3. The rest of the mixed halides' band structures could be seen in figure S2-S4 in the supplementary document. Almost the same electronic features can be seen in the recent theoretical work based on HSE06 for the Cs₂AgBiBr_6−x Cl_x (x = 1, 2, 3, 4, 5) [17]. Jing Su et al showed the bandstructure calculations of mixed halides on Cs₂AgBiBr_6−xCl_x (x = 0, 1, 2, 3, 4, 5, 6) have the indirect band gap nature and the calculated band gap values are in the range of 2.1–2.7 eV using the hybrid functional method, HSE06 [17].

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To have a better understanding of the orbitals of the elements that are responsible for the band edge positions we study their Total Density of States (TDOS) and partial density of states (PDOS) using the HSE06 approximation in both phases. The calculated PDOS depicted in figure S5 and figure 4 near the Fermi level. VBM consists of hybridization of the halide X-p states (Cl-3p, Br-4p, I-5p, and F-2p) and Ag-4 d orbitals with a little admixture of Bi-6s states. The CBM is mainly composed of the antibonding of Bi-6 p orbitals and slightly of X- p. The contribution of the Cs atom for the electron transition is very low near the E_F because there is no obvious DOS of Cs⁺ ions. However, Cs⁺ ion is helpful for the charge neutrality of the structure. The electron transitions could appear as indirect from X-p and Ag-4d antibonding states to the Bi-6p states, in agreement with the previous DFT results [51].

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DOS calculations for mixed halide compounds were carried out employing HSE06. The PDOS of the mixed halide compounds of Cs₂AgBiBr_6−xCl_x, Cs₂AgBiBr_6−xI_x, and Cs₂AgBiBr_6−xF_x are computed near the Fermi level. The same density of states features could be noticed for the varied substitutions of F, Cl and I on the pristine Cs₂AgBiBr₆. We present the PDOS feature of mixed halide compound by considering Cs₂AgBiBr₅Cl, Cs₂AgBiBr₅I, and Cs₂AgBiBr₅F in figure 5 as examples to understand the detail states of the valence band and conduction band. The top of the valence band is mainly composed of Ag - 4d orbitals, Br-4p, and substituted (F / Cl / I) halogen (2p /3p / 5p) orbitals, while the conduction band is mainly contributed by Bi - 6p orbitals.

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The electron/hole motion and the materials' conductivity could be predicted by effective mass calculations [52]. The smaller (bigger) the effective mass (m^*) is the faster (slower) the carriers could migrate. The low effective mass would be a sign of the low resistance and high conductivity of a material predicting it is a good material for practical applications. Here we use the finite difference method for the effective mass calculator (EMC) using PBE and HSE06 functional other than parabolic fitting to investigate the carrier mobilities. Parabolic fitting is not always applicable because it is quite complicated to fit the band to a quadratic polynomial. The computed band structures in figure S1 and figure 2 show the bands have different dispersion characteristics i.e. no proper parabolic bands at all. Therefore we computed our effective masses of electrons and holes using the Finite Difference Method (FDM) where the second and mixed derivates of the equation (11) are evaluated on a three-point stencil (h is step size) by equations (12) and (13) with the order of error estimation O(h² ). This FDM method was used by other researchers and obtained valid effective mass values to compare with the experimental findings [52]. The effective mass(m^*) of a charge carrier is defined as in equation (10).

$$\begin{eqnarray}&&{\left(\frac{1}{{m}^{* }}\right)}_{{ij}}=\frac{1}{{{\rm{ \hbar }}}^{2}}\frac{{\partial }^{2}{E}_{n}(\vec{k})}{\partial {k}_{i}{k}_{j}},i,j=x,y,z\end{eqnarray} \tag{ 10 }$$

where indices i and j denote reciprocal components, k is the wave vector, and ħ is the reduced Planck constant. x , y , and z represent the reciprocal cartesian space (2π/A) and ${{\boldsymbol{E}}}_{{\boldsymbol{n}}}{\boldsymbol{(}}\vec{{\boldsymbol{k}}}{\boldsymbol{)}}$ is the dispersion relation for the n-th band. The explicit form of the symmetric tensor on the right-hand side of equation (10) is given below.

$$\begin{eqnarray}\frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}^{2}}=\left(\begin{array}{ccc}\frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{x}}^{2}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{x}}\text{d}{\text{k}}_{\text{y}}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{x}}\text{d}{\text{k}}_{\text{z}}}\\ \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{x}}\text{d}{\text{k}}_{\text{y}}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{y}}^{2}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{y}}\text{d}{\text{k}}_{\text{z}}}\\ \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{x}}\text{d}{\text{k}}_{\text{z}}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{y}}\text{d}{\text{k}}_{\text{z}}} & \frac{{\text{d}}^{2}\text{E}}{\text{d}{\text{k}}_{\text{z}}^{2}}\end{array}\right)\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\frac{{\partial }^{2}\text{f}}{\partial {\text{x}}^{2}}{\unicode{x02248}}\frac{1}{12{\text{h}}^{2}}\left(\begin{array}{c}-\left({\text{f}}_{-2}+{\text{f}}_{2}\right)+\\ 16\left({\text{f}}_{-1}+{\text{f}}_{1}\right)+\\ -30\left(\text{f}\right)\end{array}\right)\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&\frac{{\partial }^{2}\text{f}}{\partial \text{x}\partial \text{y}}\text{{x02248}}\frac{1}{600{\text{h}}^{2}}\left(\begin{array}{c}-63\left({\text{f}}_{1,-2}+{\text{f}}_{2,-1}+{\text{f}}_{-\mathrm{2,1}}+{\text{f}}_{-\mathrm{1,2}}\right)+\\ 63\left({\text{f}}_{-1,-2}+{\text{f}}_{-2,-1}+{\text{f}}_{\mathrm{1,2}}+{\text{f}}_{\mathrm{2,1}}\right)+\\ 44\left({\text{f}}_{2,-2}+{\text{f}}_{-\mathrm{2,2}}-{\text{f}}_{-2,-2}-{\text{f}}_{\mathrm{2,2}}\right)+\\ 74\left({\text{f}}_{-1,-1}+{\text{f}}_{\mathrm{1,1}}-{\text{f}}_{1,-1}-{\text{f}}_{-\mathrm{1,1}}\right)\end{array}\right)\end{eqnarray} \tag{ 13 }$$

the effective mass components are the inverse of the eigenvalues of the above equation (11) and the principal directions correspond to the eigenvectors.

Effective masses of electrons and holes of Cs₂AgBiX₆ in both the cubic and tetragonal phases are computed and presented in tables 6 and 7 with the three effective masse components (M1, M2, and M3). A similar method was utilized by Volonikas et al to calculate the effective masses of charge carriers using finite difference calculations for the cubic phases of the compounds Cs₂AgBiX₆ (X = Cl, Br, I) [53]. Our calculated results are in agreement with those results. The effective masses of the cubic phase mentioned in table 6 are derived with the step size of 0.01 (1/Bohr). In table 7, we present the effective masses for the tetragonal phase with the step size of 0.1(1/Bohr). The relevant k-points for the top of the valence band and bottom of the conduction band for both phases are found using the band structure plots of figure S1 and figure 2 where the electrons and holes transitions are taking place. The effective masses we obtain, for the cubic phase, are similar to the values obtained from parabolic fitting from the literature survey [30]. Except for the tetragonal phases of Cs₂AgBiBr₆ other halide compounds for tetragonal phases' effective masses are for the first time calculated values. The effective masses for pure and mixed halide compounds are calculated using GGA and HSE06 approximations and presented in table 6, table 7, table S1, table S2, and table S3. We found that most of the calculations in HSE06 show lesser values for effective masses compared to the values obtained with GGA.

Overall, our effective masses calculated for both phases of Cs₂AgBiX₆ show that they have higher mobility of electrons than the hole mobility and the mobility increases as X (halogen atom) is changed from F to I. The recombination effect will be lower hence the corresponding diffusion length will be higher. The isotropic nature could be seen mostly in the cubic phase than in the tetragonal phase. Among these halide compounds in both phases, the cubic phase of Cs₂AgBiI₆ shows the low effective masses of both electrons and holes making it a promising photovoltaic material. The fluoride compound exhibits larger effective masses of charge carriers as shown in tables 6 and 7 compared to the other effective masses of halide compounds. This may be due to having flat valence bands in reciprocal space (see figure S1 and figure 2). The effective mass of holes is higher than the electrons' due to the band dispersion features in both phases. When considering the other kinds of double perovskites, Shuai et al studied the sodium-based double perovskites, Cs₂NaBX₆ (B = Sb, Bi, X = Cl, Br, I). They too observed the higher values of the holes' effective mass than the electrons' values and also the magnitude of those photocarriers' masses decreased when changing the halogen compound, X-site anion [33]. Wei et al calculated the effective masses using the GGA approximation for the lead-based perovskites which have direct band gaps, CH₃NH₃ PbI₃ for the tetragonal phase. Their obtained values were inferior to our calculated values for the tetragonal phase of the Cs₂AgBiI₆. However, it shows quite closer values to the holes' effective masses as 0.28 and 0.30 me when compared to our values of 0.206 and 0.328 me. [54]. The effective masses of electrons and holes of the Cs₂AgBiBr₆ and Cs₂AgBiCl₆ show lower values than the values obtained by De yuan et al for the Cs₂CuBiX₆ (X = I, Br, Cl) [55]. The effective masses of the hole charge carriers of the tetragonal phase show similar values to the hole effective masses of tetragonal phases of CH₃NH₃ PbX₃ (X = I, Br) [56]. The effective masses of mixed halide compounds are comparable, chlorine substitutions (table S1) show some higher values than the values in the host compound, Cs₂AgBiBr₆. However, the iodine substitutions (table S2) lead to lower values when compared to the parent compound. The fluorine substitutions (table S3) to the pure Cs₂AgBiBr₆ can able to reduce the effective mass of the pure Cs₂AgBiF₆ compound with the mixing of the bromine atoms. There the effective mass was reduced when more bromine atoms were present, comparatively low effective mass was found for the compound of Cs₂AgBiBr₅ F.

When the light irradiates on the surface of the material the phenomena of light reflection and absorption occur. Theoretically, this phenomenon is described by the complex dielectric function, ε(ω) = ε₁(ω) + iε₂(ω) for the optical properties of materials by Kramer–Kronig relationship. Here ε(ω) describes the interaction between photons and electrons. The real part of ε(ω) in the limit of zero energy (or infinite wavelength) is equal to the square of the refractive index. ε₂(ω) is responsible for absorption performance, it reflects the electronic transitions from occupied valence bands to unoccupied conduction bands.

$$\begin{eqnarray}{\epsilon }_{2}(\omega )=\frac{4{\pi }^{2}{e}^{2}}{{m}^{2}{\omega }^{2}}\displaystyle \sum _{r,c}{\int }_{{\rm{BZ}}}\frac{2d{\rm{k}}}{{(2\pi )}^{3}}{\left|⟨{\psi }_{c{\rm{k}}}{\rm{|e}}\cdot {\rm{p|}}{\psi }_{c{\rm{k}}}⟩\right|}^{2}\delta \left[{E}_{c}({\rm{k}})-{E}_{v}({\rm{k}})-{\rm{ \hbar }}\omega \right]\end{eqnarray} \tag{ 14 }$$

Equation (14) implies the sum of all conduction bands and valence bands where their energies are E_c and E_v respectively.

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(\omega \right)=1+\frac{2}{\pi }P{\int }_{0}^{{\rm{\infty }}}\frac{\omega ^{\prime} {\varepsilon }_{2}(\omega ^{\prime} )}{{\omega ^{\prime} }^{2}-{\omega }^{2}}d\omega ^{\prime} \end{eqnarray} \tag{ 15 }$$

The ε₁(ω) see equation (15) gives information on polarizability, From ε₂(ω), the real part ε₁(ω) can be derived by using the Kramers-Kronig relation. Using the dielectric constant we could compute the frequency-dependent linear optical properties such as refractive index n(ω), absorption coefficient α(ω), extinction coefficient K(ω), reflectivity R(ω), and electron energy-loss spectra, L(ω) [57, 58].

We investigate the optical properties including dielectric function and optical absorption for the considered double perovskites using HSE06 functional for both phases. The calculated optical features are in the range of energy 0 eV to 6 eV.

The absorption coefficient α(ω) describes the attenuation of the photon energy of the material and it plays a major role in identifying the suitability of the material for optoelectronic applications. The absorbance plots in figure 6 were calculated to the corresponding double perovskites in both the cubic phase and tetragonal phase and they did not show any major difference between those phases. The redshift can be seen in figure 6 when changing the halogen atom (X) from F to I. In figure 6, absorption spectra: there we can find the optical band gap. The absorption spectrum edge corresponds to the direct optical transitions from the top of the valence band to the bottom of the conduction band. We could find this direct optical transition of the Cs₂AgBiX₆ using the absorbance plot with the help of Origin software see table S4. The optical band gaps (table S4) are similar in both phases except for the little variations of magnitude. The absorption intensity is quite lower in low symmetry structure as shown in figure 6 other than that in general both phases show the same optical characteristics. The same feature was obtained for Cs₂AgBiBr₆ in cubic and tetragonal phases by Tahar et al [15].

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The two phases of Cs₂AgBiBr₆ and Cs₂AgBiI₆ cover the visible and ultraviolet regions. Their optical band gap values (table S4) could be comparable to the reported values of CH₃NH₃PbBr₃ and CH₃NH₃PbCl₃ at 2.26 eV and 3.00 eV respectively [7]. These two halide compounds show the attenuation of solar irradiation high compared to the fluoride and chloride compositions. Cs₂AgBiF₆ and Cs₂AgBiCl₆ show the absorbed radiation falls to the ultraviolet region. The Cs₂AgBiX₆ in both phases achieves a high absorption coefficient of over 10⁵ cm⁻¹ in the regions of both ultraviolet and visible regions.

The experimental studies revealed that Cs₂AgBiBr₆, extends the absorption to the ultraviolet region as well the absorption begins around 400 nm wavelength which belongs to the ultraviolet region, [47] in agreement with our theoretical observations for bromide compound, Cs₂AgBiBr₆. Cs₂AgBiBr₆ of two phases shows high absorption among these halide components. The substitutions for the cubic phase of Cs₂AgBiBr₆ are done as it shows the highest absorption intensity among these halide compounds (see figure 7 where the absorption coefficient enlarged between the 4.0 eV—5 eV.)

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The similar features of the absorption spectra for the substitutions of the cubic phase of Cs₂AgBiBr₆ with F, Cl, and I atoms can be seen in figure S6, figure S7, and figure S8 respectively. These substitutions consist of the indirect band gap nature. Figure S6 and figure S7 show that the blue shift of the absorption curve occurs when increasing the substitutions of the fluorine and chlorine atoms. Figure S8, the redshift occurs when increasing the iodine atoms in the bromide site. The absorption intensity of these mixed halides is situated in between the pure bromide and iodide compounds and it changes corresponding to the mixed halide ratios i.e. Br/F, Br/Cl, and Br/I.

The calculated real and imaginary parts of dielectric functions of Cs₂AgBiX₆ in both phases show similar features. Concerning the real part (ε₁) of the dielectric function for both the cubic(tetragonal) phases which is shown in figure S9. The ε₁ starts from 2.48 (2.38), 3.5 (3.41), 4.1 (4.05), and 5.37 (4.98) for Cs₂AgBiX₆ (X = F, Cl, Br, I). These values are the static dielectric function ε₁ (ω→0). At these starting points, materials have the ability to interact with an electric field without absorbing the photon energy [57]. Afterward, the dielectric function increases with the photon energy and attains the maximum values. The maximum ε₁(ω) values for Cs₂AgBiF₆ and Cs₂AgBiI₆ are at 2.47 eV (2.68 eV) and 5.13 eV (4.95 eV) were obtained for cubic(tetragonal) phases respectively. Cs₂AgBiCl₆ and Cs₂AgBiBr₆ are having two peaks of the ε₁(ω). Cs₂AgBiCl_6, the first and second peaks occur at 3.88 eV (3.92 eV) and 4.48 eV (4.55 eV) for cubic (tetragonal) phases respectively. For the Cs₂AgBiBr₆ first peak occurs at 3.24 eV (3.31 eV) and the second peak at 3.82 eV (3.88 eV) corresponds to the cubic (tetragonal) phases. This implies there is a shift could occur from higher energy to lower energy from F to I. Tripathy et al observed that similar peak shifting occurs to lower energy in imaginary dielectric function by substituting larger halide ions in Cs₂AgBiCl₆, Cs₂AgBiBr₆, and Cs₂AgBiI₆ [30]. Typically these starting values are the dielectric constant where ε₁(0). After reaching the imaginary function to these maxims it starts to decrease by increasing the photon energy. There are no negative ε₁ values that occur in our computed dielectric function as shown in figure S9 which means it shows the prominent optoelectronic properties in both phases.

The Real part of the dielectric function (ε₁) is plotted as shown in figure S10 for the mixed halide compounds. When increasing the substitutions of the fluorine and chlorine atoms to the pure Cs₂AgBiBr₆, the graphs figure S10 (a), and figure S10 (b) show the blue shift of the ε₁ values. For the Cs₂AgBiBr_6−xI_x in figure S10 (c), we could see the redshift of the real part of the dielectric function when increasing the iodine atoms. There are no negative values ε₁ values in mixed halide components which indicates that the interior of these materials always exhibits prominent dielectric behavior. De-Yuan et al discussed the same observation for the Cs₂CuBiX₆ [55]. The computed static dielectric function for mixed halide compounds is given in table S5.

Using the Penn model, The dielectric constant value (ε₁(0)) is inversely proportional to the band gap of the material (E_g ) as shown in the below equation (16) [59];

$$\begin{eqnarray}&&{\varepsilon }_{1}\left(0\right)=1+{\left(\frac{{\rm{ \hbar }}\omega }{{E}_{g}}\right)}^{2}\end{eqnarray} \tag{ 16 }$$

From equation (16), the higher band gap values correspond to the lower ε₁(0) values in comparison with the band gap values shown in tables 4 and 5.

Imaginary part (ε₂(ω)) describes the interband transitions i.e. electron transitions from occupied states to the unoccupied states, [58] where the electrons transitions could take place from occupied X - p states and Ag - 4d orbitals to the unoccupied Bi-6p states which could be verified with the density of states plots. ε₂(ω), which is mentioned in equation (14) could be derived from the band structure and the dipole matrix elements [60]. This equation could be related to the absorption coefficient at a given frequency ω [60]. This could be the reason figure 6, figure S6, figure S7, and figure S8 appear with the same optical features as figure S11 and figure S12 i.e. The imaginary part of the dielectric function and absorption spectra show the same optical behavior in our pure and mixed halide compounds.

Overall, the results suggest that the material Cs₂AgBiI₆ and Cs₂AgBiBr₆ have a higher dielectric constant (figure S9) in low energy region (from 2 to 4 eV) following broad optical spectra which covers most of the region in the solar spectrum including visible and ultraviolet regions compared to other compounds. Therefore, the high value of the dielectric function and absorption intensity in the solar spectrum implicated these two compounds could be promising candidates for photovoltaic applications. For the mixed halides, the substitution of fluorine and chlorine atoms increases in Cs₂AgBiBr₆, figures S12 (a) and S12 (b) demonstrate a clear blue shift in the ε₂ values. On the other hand, in the instances of Cs₂AgBiBr_6−xI_x, illustrated in figure S12 (c) there is a discernible redshift observed in the imaginary part of the dielectric function as the number of iodine atoms increases.

The Refractive index is also an important physical parameter related to microscopic atomic interactions. This could measure the transparency of the incident photon [61]. The obtained static refractive index n(0) is related to the static real part of the dielectric function as shown in below equation (17) [62].

$$\begin{eqnarray}&&n\left(0\right)=\sqrt{{\varepsilon }_{1}\left(0\right)}\end{eqnarray} \tag{ 17 }$$

n(0) values are listed in table S6 for the Cs₂AgBiX₆ in both phases, the plot in figure S13 (a). They are in good agreement with past theoretical works [57, 58, 62]. The n(0) value increase when changing the halogen (X) compound from F to I. The computed n(0) values for the mixed halide compounds are listed in table S7. Fluorine and chlorine atoms substituted compounds show the n(0) values are decreasing with the increase of fluorine and chlorine atoms in figure S16 (a) and figure S14 (a). However n(0) values of figure S15 (a) increase when the iodine atoms are substituting to the pristine, Cs₂AgBiBr₆ compound.

The scattering and reflection of light from materials are related to the R(ω). The computed R(0) values are listed for pure and mixed halide compounds see table S6 and table S7. Reflectivity spectra for the lead-free double perovskite, Cs₂AgBiX₆ for both phases are shown in figure S13 (b). The reflectivity R(0) of the mixed halide compounds decreases from the 9% and 11.3% reflectivity to 5.17% and 9.25% when 'F' and 'Cl' atoms substitute in place of 'Br' [figure S16 (b) and figure S14 (b)]. However, reflectivity increases up to 15.42% when the iodine substitutions take place [figure S15 (b)].

The extinction coefficient (K(ω)) implies the ability to absorb the photon energy by the material. The higher values of extinction coefficient could be seen in the lower energy range compared to the higher energy range in pure halide compounds see figure S13 (c). For the mixed halide compounds when the fluorine and chlorine atom substitutions are increasing the spectra shift to a higher energy range [figure S14 (c) and figure S16 (c)], nevertheless, the iodine substitutions shift the spectra to a lower energy range [figure S15 (c)]. In these lower regions, the material could absorb more photon energy which is described by their absorption spectra, thus absorption coefficient and extinction coefficient are related by the following equation (18) [63].

$$\begin{eqnarray}&&\alpha \left(\omega \right)=\sqrt{2}\omega K(\omega )\end{eqnarray} \tag{ 18 }$$

Higher values of extinction coefficient relate to higher absorption.

The energy loss function is an important factor in photovoltaic applications. It describes the energy loss of the moving charge carriers inside the material. Here we computed the energy loss function for pure and mixed halide components in figure S13 (d), figure S14 (d), figure S15 (d), and figure S16 (d). The peaks of energy loss functions are related to the corresponding abrupt reduction values of R(ω).

The transition dipole moment (TDM) is typically expressed as the probability of electron transition between states in the valence band and the conduction band [22]. The TDM calculations provide information regarding the height of the intensity of absorption spectra instead of the transitions of electrons between two states which mainly depends on the selection rules. We performed the optoelectronic properties on the cubic and tetragonal phases of Cs₂AgBiX₆ perovskite and present the HSE06 calculated band structures and corresponding TDM diagrams as shown in figures 8 and 9 below. In this configuration, there are a total of 76 valence electrons. Following that, we proceed to calculate the square of the transition dipole moment originating from the highest valence band (indexed at 38) to the lowest conduction band (indexed at 39). This computation yields the square modulus of the electronic wave functions at the edges of these bands. As we can see the resultant absorption spectra show the different peaks are visible in the absorption region that arises as a result of transitions of electrons between VBs and CBs. More transitions occur along the high symmetry k-points; G to L, G to K, and G to X for both phases. The intensity of the height increases when it moves from F to I because the band gap decreases as more transitions could happen from the valence band to the conduction band. The transition dipole moment occurs from Ag - d (VBM) to Ag - s (CBM) states and also more electron transitions are strong and allowed at the Gamma point for all the compounds including the mixed halides. This results in strong intra-atomic transitions. Cs₂AgBiI₆ in both phases shows a higher intensity of the TDM compared to other double perovskite materials as it shows the lowest band gap values compared to other materials.

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Compared to cubic phase compounds, the tetragonal phase compounds of Cs₂AgBiF₆, Cs₂AgBiCl₆, and Cs₂AgBiBr₆ show a higher value of transition probabilities than the cubic phase for these compounds. However, the cubic phase of Cs₂AgBiI₆ shows more electron transitions among all the halide compounds of both phases as it shows the lowest band gap value of 1.78 eV (table 4).

The Fluorine substitutions mixed halides (figure S17) represent the more electron transitions that take place on Cs₂AgBiBr₅F, Cs₂AgBiBr₄F_2, and Cs₂AgBiBr₃F₃. The pure Cs₂AgBiF₆ has low electrons' transitions for both phases but with the substitution of the bromine atoms with the fluorine atoms, we were able to increase the height of TDM resulting in more transitions taking place from VB to CB. When it comes to the chlorine atom substitutions, the lowest band gaps were obtained for the compounds of Cs₂AgBiBr₅Cl and Cs₂AgBiBr₄Cl₂ (table 5) showing that more transitions occurred in our TDM figures (figure S18). These two compounds carry the same height of intensity of the absorption spectrum as well for the TDM. All the iodine-substituted compounds (figure S19) gave more transition probabilities compared to the fluorine and chlorine substitutions. Among them, Cs₂AgBiBr₂I₄ and Cs₂AgBiBrI₅ show the high intensity of TDM because their computed band gaps were lower compared to the other iodine substitutions.

The phonon dispersion relations are defined as the k dependency of the frequencies for all branches and selected directions in the crystal. The phonon dispersion relations are drawn along the crystal high-symmetry points of the Brillouin zone. Phonon calculations are done for all polytypes to understand the dynamical stability. The dynamical matrices are calculated from the force constants, and also the phonon dispersion curves, at the theoretical equilibrium volume were computed (see figures 10 and 11). We found that two of the tetragonal structures for Cs₂AgBiBr₆ and Cs₂AgBiCl₆ are dynamically stable. While all the cubic structures show negative frequencies and are dynamically unstable. Such instability causes the entire lattice to undergo a structural change. From figures 10 and 11, it is clear that a phonon band gap between the acoustic and optical modes is found in all compounds (cubic Cs₂AgBiI₆) except in the cubic and tetragonal phases. This separation affects the phonon scattering processes and the lattice thermal conductivity.

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Thermal properties of the solids can be derived from lattice vibrations in terms of phonons. Here we examined the thermodynamical behavior of the Cs₂AgBiX₆ (X = F, Cl, Br, and I) in both phases. For these double perovskites, we computed the thermodynamic properties such as vibrational energy (kJ/mol), free energy (kJ/mol), heat capacity (J/K/mol), and entropy (J/K/mol). The temperature range was set for these compounds from 0 to 1000 K to compute the above-mentioned thermal properties. Figures 12 and 13 show that the vibrational energy increases up to 200 kJ mol⁻¹ where the temperature meets 1000 K and the free energy decreases exponentially which satisfies Debye's law [64]. Entropy increases in both phases when the temperature is increasing which proves the third law of thermodynamics and also it reaches the absolute zero value at very low temperatures [64]. A recent theoretical study evaluated the thermodynamic properties such as free energy, enthalpy, and entropy on the cubic phase of Cs₂AgBiX₆ (X = Cl, Br, and I) using the phonon vibrational modes. They observed when raising the temperature, those thermodynamic properties approach zero at extremely low temperatures which corresponds to our findings [65]. At the constant volume, the heat capacity was computed as C_V for both phases. All the compounds exhibit high C_V which is around 200 J K⁻¹ mol⁻¹ which does not change with the rising of the temperature.

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The lead-free halide components could be of interest in the field of photocatalytic water splitting. For the first time, Wang et al used Cs₂AgBiBr₆ to produce H₂ under visible light irradiation [66]. The band edge positions of the semiconductor play a vital role. The optimal water splitting materials are selected where the conduction band edge (CBM) of the semiconductor must be more negative than the H₂O/H₂ level of water (0 eV) and its valence band edge (VBM) must be more positive than the H₂O/O₂ level of water (1.23 eV) [67]. Thus, we determine the CBM and VBM of halides and mixed halide compounds relative to the normal hydrogen electrode potential (NHE). The resulting VBM and CBM are present in figures 14 and 15 where the VBMs and CBMs correspond to their eigenvalues obtained from the DOS calculations under HSE06 approximation (table S8-S11). The visible region between 2 to 2.75 eV was considered [68]. In figure 14, Under UV irradiation, Cs₂AgBiF₆ shows the ability to lead the redox process as the CBM had more negative values (−0.317 eV and −0.707 eV) than the reduction level of the water (0 eV) and more positive values (3.303 eV and 2.341 eV) than the oxidation level of the water (1.23 eV) for cubic and tetragonal phases respectively. The tetragonal fluoride compound shows strong reduction power whereas the cubic phase shows strong oxidation power. Cs₂AgBiCl₆ shows the oxidation ability under UV irradiation as its VBM levels are more positive values (1.935 eV and 1.917 eV) than the oxidation level of water in cubic and tetragonal phases respectively. It also has a good reduction ability the CBM level is 0.103 eV than the reduction level of water. We could expect higher reduction power in the tetragonal phase of the Fluoride compound and higher oxidation power in the cubic phase of Cs₂AgBiF₆ among the halide compounds of both phases. Significantly under visible light irradiation the Cs₂AgBiBr₆ compound shows photocatalytic ability in both oxidation and reduction for both phases. The tetragonal phase of Cs₂AgBiBr₆ shows more reduction ability than the cubic phase. Under visible light irradiation, both the phases of Cs₂AgBiI₆ show the ability of oxidation than the Bromide compound. We could expect more oxidation power in the iodide compounds under visible light irradiation and also the best visible-light photocatalytic material will be the tetragonal phase of Cs₂AgBiBr₆ among our halide compounds.

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We did the substitutions for the cubic phase of the Cs₂AgBiBr₆ as it shows the direct transitions of the electrons from the valence band to the conduction band with the 2.64 eV energy (table S4) emphasizing the good optical absorption among the halide compounds under the visible light irradiation and also shows the highest absorption coefficient among the halide compounds. We have shown the band edge positions of these mixed halides in figure 15. The gradual substitutions of the fluorine atoms to bromine atoms in Cs₂AgBiBr₆ show the photocatalytic ability under UV irradiation will be better for Cs₂AgBiFBr₅. The chloride substitutions show the visible photo-catalytic ability for the compounds of Cs₂AgBi Br₅Cl and Cs₂AgBiBr₅Cl₂. The VBM of Cs₂AgBiClBr₅ is the same as the Cs₂AgBiBr₆, therefore Cs₂AgBiBr₅Cl and Cs₂AgBiBr₆ should have the same oxidation power. Among the chlorine substitutions, Cs₂AgBiBr₂Cl₄ shows a stronger oxidation power than the pure Cs₂AgBiBr₆ compound under visible light irradiation. Under UV light irradiation the chloride-substituted compound, Cs₂AgBiBrCl₅ can be a better candidate for oxidation than the pure bromide compound. Iodine substitutions show the ability of oxidation under visible light irradiation except for the Cs₂AgBiI₄Br₂ and Cs₂AgBiBrI₅ compounds that have better oxidation ability in the infrared region. Cs₂AgBiBr₃I₃, Cs₂AgBiBr₄I₂, Cs₂AgBiBr₂I_4, and Cs₂AgBiBrI₅ show better oxidation ability than the Cs₂AgBiBr_6. However, the Cs₂AgBiIBr₅ shows a lower ability of oxidation power than the pure bromide compound. Overall our results show that the fluorine-substituted Cs₂AgBiBr₆ perovskites will be good photocatalytic materials under both the UV and visible regions.