Computational exploration of the diverse functional properties of double antiperovskite materials Na₆AgBiX₂ (X = Cl, Br, I)

Abstract

The ongoing study investigates optoelectronic as well as photovoltaic characteristics of double antiperovskite (DAP) compounds Na₆AgBiX₂ (X = Cl, Br, I) using density functional theory (DFT) with semi-classical Boltzmann transport theory calculations, aiming at potential applications in renewable energy technologies. Optimizations of DAP Na₆AgBiX₂ (X = Cl, Br, I) show lattice constants (a) of ∼11 Å, which are increased from its counter double perovskite. The stability of these compounds was tested using formation energy (E_f), binding energy (E_b), Goldschmidt tolerance factor τ_G, elastically and thermodynamics. Electronic calculations reveal Na₆AgBiCl₂ and Na₆AgBiBr₂ possess direct bandgaps with energies of 1.27 eV and 1.22 eV, respectively, whereas Na₆AgBiI₂ displays an indirect band gap of 1.15 eV. The electron charge density reflects the balance of ionic and covalent interactions in these compounds. According to optical characteristics all substances exhibit considerable absorbing power in the visible region. Photocatalytic studies indicate that Na₆AgBiX₂ (X = Cl, Br, I) show a good response for oxidation. The calculated band-edge positions, evaluated using the electronegativity approach, indicate that the Na₆AgBiX₂ (X = Cl, Br, I) compounds are thermodynamically suitable for photocatalytic water splitting. The greatest value of ZT of 0.79 at 300 K among the compounds indicates that Na₆AgBiBr₂ may be a promising candidate for TE applications, according to its TE characteristics. In terms of solar cell efficiency predicted by SLME, Na₆AgBiBr₂ has a higher efficiency of 8.46% than Na₆AgBiCl₂ (@8.06%) and Na₆AgBiI₂ (@8.12%) for a 0.5 µm thick layer. All these findings underscore the potential of Na₆AgBiBr₂ for advanced renewable energy applications (such as optoelectronics, water splitting, thermoelectrics and photovoltaics).

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1. Introduction

With the continuous growth of the global population and the resulting rise in energy demand, renewable energy sources have emerged as sustainable solutions to energy challenges. Photovoltaic energy among these is particularly promising due to its wide applicability and eco-friendly nature.^1–3 Several technologies have been developed to harness and convert solar energy. During the past several decades, organic–inorganic solar cells have been engineered to produce electricity directly from sunlight. More recently, a new class of perovskite solar cells has emerged, achieving efficiencies of up to 25%, and holds strong potential to replace conventional silicon-based counterparts.^4,5 Their low cost, optimal bandgap, and strong absorption coefficient have accelerated progress in this field, enabling them to outperform previous solar cell technologies restricted by environmental challenges and elevated production costs.^6–8 Researchers have paid particular attention to a perovskite CH₃NH₃PbI₃ called MAPI₃ due to its outstanding energy transformation efficiency.^9,10 However, this material contains lead, which is both toxic and unstable, posing challenges for practical applications.¹¹ As a result, significant research efforts are now focused on developing lead-free perovskite alternatives to enhance the commercial viability of perovskite solar cells. Previous studies indicate that replacing MA cations with Li, Na, K, Rb, or Cs metals can enhance stability. Additionally, silver (Ag) and bismuth (Bi) cations are regarded as excellent alternatives for minimizing toxicity in the optical absorption layer. Another class of materials, having the general formula A₃BX and known as antiperovskites, was discovered in 1915.¹² Recent studies have also investigated the promise these materials hold for energy storage applications. A crucial difference between perovskites and antiperovskites is the distinct arrangement of their cations and anions, which dictates their overall crystal structure.¹³

Other than simple perovskites and antiperovskites researchers widely used double perovskites for solar energy conversion. Double perovskites possess a unique structure that is twice the size of conventional perovskites and have formula A₂BB′X₆, where B and B′ are transition metals, A is a metal and X is a halogen. Recently theoretical and experimental studies on double perovskites Cs₂AgBiCl₆ (2.77 eV), Cs₂AgBiBr₆ (2.2 eV) have investigated their potential applications in optoelectronics more than CH₃NH₃PbCl₃ (3.00 eV), CH₃NH₃PbCl₃ (2.26 eV).^14,15 Researchers have also investigated a variety of halide double perovskites and their corresponding band gap values, such as Cs₂LiAlCl₆ (3.22 eV), Cs₂LiInCl₆ (2.66 eV), and Cs₂NaGaBr₆ (1.76 eV),¹⁶ Rb₂LiTlCl₆ (2.9 eV),¹⁷ Li₂AgBiI₆ (1.26 eV),¹⁸ and X₂AgBiBr₆ (X = Li, Na, K, Rb, Cs with 2.124 eV, 2.222 eV, 2.198 eV, 2.209 eV, 1.902 eV respectively).¹⁹ The prospective applications of double perovskites in optical as well as thermoelectric, photocatalytic and photovoltaic fields have been extensively investigated through theoretical studies.^20–24 In particular, exploring the optical features of materials that combine superior non-toxicity, stability, and favorable optoelectronic characteristics is crucial for the development of high-performance photovoltaic applications, photodetectors, and thermoelectrics.²⁵ Due to their improved solar absorption and adjustable energy gaps, Single perovskites are being replaced by double perovskites in the market for thermoelectric and optical applications. By changing the halogen ions, scientists can modify the energy gaps and perhaps enhance the properties of the material. According to recent studies, double perovskite compounds are well-suited for solar energy systems resulting from their superior visible-light absorption and outstanding quantum efficiency.^26,27

Recently, attention has shifted towards new class of perovskites called double antiperovskites, characterized by a crystal structure that is two times that of conventional antiperovskite and represented by the chemical formula A₆BB′X₂. While antiperovskites typically exhibit wide band gaps due to their strong ionic interactions, studies have shown that double antiperovskites possess comparatively narrower band gaps, making them more favorable for optoelectronic applications. Several theoretical investigations have been carried out on different halide double antiperovskites, including Li₆OSI₂, Li₆NBrBr₂, Li₆NII₂, Li₆NClBr₂, Li₆NBrI₂, Na₆SOCl₂, Na₆SOBr₂, Na₆SOI₂, K₆SOCl₂, K₆SOBr₂, and K₆SOI₂ having band gap ranges from 2.90 eV to 4.5 eV.^28–30 But the most recent research on alkali metal halide double antiperovskites Li₆AgBiCl₂ (1.33 eV), Li₆AgBiBr₂ (1.27 eV), Li₆AgBiI₂ (1.21 eV),³¹ and Cs₆AgBiCl₂ (1.50 eV), Cs₆AgBiBr₂ (1.49 eV) Cs₆AgBiI₂ (1.39 eV)³² revealed their favorable electronic and optical features, and they emerge as strong contenders for optoelectronic and photovoltaic devices. Li₆AgBiX₂ and Cs₆AgBiX₂ (X = Cl, Br, I) are potential materials for energy storage, solid electrolytes, and optoelectronic devices since research indicates that they exhibit thermodynamic, structural, and dynamic stability.^31,32 Unlike Li-based and Cs-based materials, Na-based compounds exhibit enhanced structural stability at ambient conditions. The Na-based systematic trend has not been established in earlier Li- and Cs-based reports. The reduced direct or indirect band gaps demonstrate high UV spectrum absorption as well as the materials’ exhibiting mechanical stability, ductility, and ionic natures. Their thermodynamic properties indicate increased melting temperatures and improved thermal conductivity; several theoretical investigations have been carried out on different halide double antiperovskites, exploring their potential for optical applications which are helpful in the production of batteries and solar cells. There is still much to learn about these materials, though, as not much research has been done on them yet. In order to further this investigation, researchers have explored the attributes of double antiperovskites using computer simulations. Here, first-principles simulations are widely employed to investigate the structural, electronic, elastic, optical, and thermodynamic characteristics of double antiperovskites such as Na₆AgBiX₂ (X = Cl, Br, I).

2. Computational details

The structural parameters, including the lattice constants, were optimized using the Wien2k simulation package, which employs the full-potential linearized augmented plane wave (FPLAPW) method within the framework of density functional theory (DFT).³³ The exchange–correlation energy was treated using the Wu–Cohen scheme within the framework of the generalized gradient approximation (GGA).³⁴ To improve the accuracy of exchange–correlation effects, the highly precise Tran–Blaha modified Becke–Johnson (TB-mBJ) potential was employed, as conventional GGA functionals often tend to overestimate or underestimate band gaps. For reliable convergence of the total energy within the muffin-tin spheres, the parameter R_mtK_max was set to 7.³⁵ Here, R_mt represents the lowest muffin-tin radius and K_max the largest k-vector in momentum space. Atomic site relaxation, along with adjustments to the unit cell size and shape, was performed with an energy convergence tolerance of 10⁻⁵ Ry.³⁶ Additionally, WC-GGA was used in conjunction with the modified Becke–Johnson (mBJ) technique to provide precise and reduced band gap estimates for lead-free halide double antiperovskites. The Na₆AgBiX₂ (X = Cl, Br, I) compounds were examined using this method, and the results were consistent with experimental data for Na₆AgBiX₂ (X = Cl and Br). Electronic properties were assessed using a k-mesh of 13 × 13 × 13 and a finer one of 24 × 24 × 24. After the lattice constants and structural parameters were relaxed, the electronic, optical, elastic, and thermodynamic properties were investigated. The thermodynamic parameters were calculated using the quasi-harmonic Debye model and Gibb's code.

3. Results and discussion

3.1 Structural properties

In this study, the crystal structure of the double perovskite Na₂AgBiX₆ (X = Cl, Br, I) is transformed into double antiperovskite Na₆AgBiX₂ (X = Cl, Br, I), as illustrated in Fig. 1. These newly proposed DAP's adopt an FCC structure with space group Fm3m (225).³⁷ Sodium is an alkali metal having a low density, cost, melting point, toxicity, and high conductivity as well as specific heat in this structure. Na is used for Na-ion-battery, Na-ion-capacitor and power electronics applications.³⁸

Fig. 1 Transformation from double perovskite to double anti-perovskites.

The graphs presented in Fig. 2 illustrate the energy vs. volume relationship for three materials: Na₆AgBiCl₂, Na₆AgBiBr₂, and Na₆AgBiI₂. These curves in DFT optimizations, reveal the relationship between the unit cell volume (measured in ų) along the x-axis and energy (in Ryd.) along the y-axis. In DFT studies, energy vs. volume graphs are fundamental for identifying the equilibrium volume and stability of each compound. The minimum point on each curve represents the optimal (or most stable) volume where the energy is lowest, indicating the stability of the compounds and suggesting the favorable structure for each compound. Each colored line on the graph corresponds to some specific material, and the equilibrium lattice parameters (a) in Å are: like Na₆AgBiCl₂ (11.52 Å), Na₆AgBiBr₂ (11.54 Å), and Na₆AgBiI₂ (11.57 Å). By comparing these lines, we observe that the curve for Na₆AgBiI₂ is characterized by the lowest energy, signifying it has the greatest stability of the three materials, as it achieves the lowest energy at its optimal volume. The stability of Na₆AgBiI₂ can be attributed to its chemical bonding and crystal structure, allowing it to achieve a lower energy state. This graph provides insights into the structural properties of these compounds, indicating that Na₆AgBiI₂ is the most stable, followed by Na₆AgBiCl₂ and Na₆AgBiBr₂. For the successful integration of double antiperovskites into devices, a thorough investigation of their structural and thermal stability is essential.³⁹ The Goldschmidt tolerance factor (τ_G) is commonly regarded as a crucial parameter for predicting the structural stability of perovskite materials. In particular, when monovalent and trivalent cations simultaneously occupy the B site of the lattice, this factor plays an important role in evaluating the crystalline stability of the compound.⁴⁰ The ionic radii in Å of Na, Ag, Bi, Cl, Br and I are 1.02, 1.15, 1.03, 1.81, 1.96 and 2.20 respectively. Previous statistical analysis of all existing halide perovskites have shown the tolerance factor (τ_G) ranges from 0.81 to 1.11. If the material's τ_G value does not fall in the given range it will be considered unstable. The calculated values of τ_G given in Table 1 for the encountered compounds satisfy the stability criteria. In addition, thermal stability can also be quantified by calculating the formation energy (E_f) and binding energy (E_b) of the crystal.⁴¹

Fig. 2 Optimization curves of Na₆AgBiX₂ (X = Cl, Br, I).

Table 1 Structural parameters and thermal stability parameters of halide double antiperovskites Na₆AgBiX₂ (X = Cl, Br, I)

Materials	a (Å)	B (GPa)	τG	Ef (eV per atom)	Eb (eV per atom)	Bond distance (Å)
						(Na–X)	(Ag–X)	(Bi–X)
Na6AgBiCl2	11.52	10.73	0.95	−1.65	−2.73	4.0645	4.9780	4.9780
Na6AgBiBr2	11.54	11.72	1.00	−1.50	−2.58	4.0666	4.9805	4.9805
Na6AgBiI2	11.57	11.53	1.08	−1.36	−2.13	4.0666	4.9805	4.9805

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Similarly, by using the chemical potentials obtained by using Guassian code μ_Na = −2.025 eV, μ_Ag = −2.834 eV, μ_Bi = −4.605 eV, and μ_Cl = −9.56 eV, μ_Br = −7.26 eV, and μ_I = −6.74 eV, the formation energy E_f can be calculated. Negative values of the formation energy (E_f) and binding energy (E_b) indicate that the material is thermodynamically stable. Table 1 shows that Na₆AgBiCl₂ has more negative values of E_f and E_b so it is more thermodynamically stable than Na₆AgBiBr₂ and Na₆AgBiI₂. These energy parameter define the molecular structure and bonds within the crystal, providing valuable insights into its stability. Differences in stability and energy may arise due to lattice strain, ionic sizes, and bonding strengths affected by the halide ions (Cl, Br, I) in each material.

3.2 Mechanical properties

The equilibrium structure of a material is intimately related to its elastic qualities, which are defined by how it responds to stress and bounces back. These properties are critical for understanding mechanical stability, stress response, and practical applications. Effective elastic constants, in particular, reveal the ability of a material to endure stresses without losing its structure.⁴² The elastic constants C₁₁, C₁₂, and C₄₄ for cubic crystals must meet stability criteria according to Born–Huang theory.⁴³ These criteria confirm that our proposed materials like Na₆AgBiX₂, (X = Cl, Br, I) maintain elastic stability under various deformation forces.

The data presented in Table 2 provides comprehensive parameters of Na₆AgBiX₂ (X = Cl, Br, I) compounds which are indicators of a material's stability as well as stiffness. The computed values of C₁₁ for Na₆AgBiCl₂, Na₆AgBiBr₂, and Na₆AgBiI₂ are 27.74 GPa, 27.79 GPa, and 28.72 GPa, respectively, demonstrating a slight increase as the halogens change from Cl to I. This suggests that substitution of larger halogens slightly enhances the longitudinal stiffness. Cl, Br, and I have values of 2.29 GPa, 2.52 GPa, and 3.4 GPa respectively for the coupling between axial strains, denoted by the parameter C₁₂. This pattern indicates increasing resistance to deformation under uniform pressure as the halogen size increases. The shear modulus C₄₄ also follows an increasing trend from Cl (0.89 GPa) to I (2.73 GPa), reflecting enhanced resistance to shear deformation for the heavier halides. Basic elastic properties including the bulk modulus (B), shear modulus (G), and Young's modulus (Y) can be obtained through elastic constants. These calculations are commonly performed using the Voigt–Reuss–Hill approximation method.⁴⁴ These elastic moduli (B, G and Y) are necessary for understanding the responses of materials to stress, including compressibility, deformation, and overall stiffness.⁴⁵ The resistance of materials to uniform compression is measured by B, which consistently displays values of 10.82 GPa for Cl, 10.96 GPa for Br, and 11.91 GPa for I. This modest increase in larger halogens aligns with the observed trends in C₁₁ and C₁₂. The shear moduli, G_V and G_R, representing the Voigt and Reuss bounds, indicate the stiffness in terms of shear deformation. The average shear modulus (G) increases progressively from 3.52 GPa (Cl) to 5.37 GPa (I), consistent with the higher C₄₄ values. The Young's modulus (Y) shows similar behavior, with values ranging from 14.38 GPa for Na₆AgBiCl₂ to 17.06 GPa for Na₆AgBiI₂, indicating a stiffer lattice with heavier halogens.

Table 2 Mechanical properties of Na₆AgBiX₂ (X = Cl, Br, I)

Parameters	Na6AgBiCl2	Na6AgBiBr2	Na6AgBiI2
C11 (GPa)	27.74	27.79	28.72
C12 (GPa)	2.29	2.52	3.4
C44 (GPa)	0.89	1.24	2.73
B (GPa)	10.82	10.96	11.91
GV (GPa)	5.62	5.80	6.76
GR (GPa)	1.42	1.94	3.98
G (GPa)	3.52	3.87	5.37
Y (GPa)	14.38	14.79	17.06
B/G	3.07	2.83	2.22
C′ (GPa)	12.73	12.64	12.81
C″ (GPa)	1.40	1.28	0.37
A^U	14.8	9.95	3.5
AZ	0.07	0.10	0.21
ν/σ	0.28	0.28	0.26
ζ	0.24	0.25	0.27

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The bonding characteristics, elasticity, and plasticity of a material, as well as its tendency to exhibit brittle or ductile behavior, can be inferred from two key metrics: Poisson's (ν) and Pugh's (B/G) ratios. Specifically, a material is deemed ductile if it satisfies the criteria as ν > 0.26 while B/G > 1.75, indicating plastic deformation over brittle fracture.⁴⁶ These requirements are satisfied for Na₆AgBiX₂ (X = Cl, Br, I) compounds, indicating their ductility. A reliable indicator of ductility in materials is Pugh's ratio (B/G). Specifically, a B/G value exceeding 1.75 signifies a propensity for ductile behavior, whereas values below this threshold suggest a tendency towards brittleness. Here, Na₆AgBiCl₂ has the highest B/G ratio of 3.07, followed by Na₆AgBiBr₂ (2.83) and Na₆AgBiI₂ (2.22). This trend suggests a transition towards brittleness as the halogen size increases. Furthermore, the elastic isotropy of Na₆AgBiX₂ compounds was evaluated. A material is considered isotropic if its isotropy index (A_Z) is either 0 or 1.⁴⁷ The analysis indicates that the investigated materials having A_Z values (0.07, 0.10 and 0.21) exhibit isotropic characteristics, signifying that their properties remain consistent and independent of orientation. The anisotropy factor (A^U) is associated with material cracking and decreases from Na₆AgBiCl₂ (14.8) to Na₆AgBiI₂ (3.5).

Further information on the mechanical behavior of the materials can be obtained from Cauchy's pressure (C″ = C₁₂ − C₄₄), which can be either positive or negative as indicated in Table 2. The ductile nature of Na₆AgBiX₂ (X = Cl, Br, I) is further supported by the positive C″ values, which also support their capacity to deform without breaking. The parameters C′ and C″ are additional measures of elastic behavior, where C′ (C₁₁ − C₁₂)/2 indicates shear anisotropy, and C″ provides insights into the deviation from pure shear. Finally, Poisson's ratio (ν) and ζ reflect the material's lateral strain response to axial stress and the overall stability. Theoretically, ν spans a range of −1 to 0.5, with positive values indicating an ability to resist compression and shear deformation, as reflected in its bulk modulus and shear modulus. Typically, materials exhibiting an ionic character display ν values between 0.3 and 0.4, whereas covalent materials tend to show lower values around 0.2. In the case of Na₆AgBiX₂ (X = Cl, Br, I) ν values of 0.28, 0.28, and 0.26, respectively, suggest that instead of having a covalent character, these materials are primarily ionic. Both parameters show slight variations, with ν/σ decreasing slightly from 0.28 to 0.26 and ζ increasing from 0.24 to 0.27 across the series, pointing to consistent elastic behavior with incremental changes. A graphical analysis of the elastic parameters as shown in Fig. 3 and 3-D contour plots of the elastic moduli demonstrated in Fig. 4 are useful for analyzing a material's mechanical properties across various technological and engineering applications. The maximum and minimum values of the elastic moduli are presented in Table 2. In conclusion, the data show that Na₆AgBiX₂ compounds exhibit increasing stiffness and brittleness as halogen changes from Cl to I, with a clear trend in the elastic parameters confirming this transition.

Fig. 3 Graphical comparison of the elastic properties of the halide double antiperovskites of Na₆AgBiX₂ (X = Cl, Br, I).

Fig. 4 Elastic 3D images of Na₆AgBiCl₂ (a)–(c), Na₆AgBiBr₂ (d)–(f), Na₆AgBiI₂ (g)–(i).

3.3 Electron charge density

The charge density maps reveal critical insights into the bonding nature and electron distribution of the encountered materials.⁴⁸ For Na₆AgBiCl₂, the charge density is more concentrated around the Bi and Cl atoms, as evidenced by the tighter and more numerous contour lines, see Fig. 5. This suggests strong ionic interactions between Na, Ag, and Cl. In Na₆AgBiBr₂, the electron density around Br is slightly more diffuse compared to Cl, indicating a reduction in ionic character which can be attributed to the larger ionic radius and lower electronegativity of Br. For Na₆AgBiI₂, the contours are spread further, showing a more delocalized charge density around the I atoms, which aligns with the trend of an increasing covalent character and decreasing electronegativity as we move down the halide group. The Na ions show minimal charge density, confirming their role as cations contributing to overall charge neutrality rather than significant covalent bonding. The Ag atoms display moderate charge density, suggesting their intermediate role in balancing ionic and covalent interactions within the lattice. The Bi atoms exhibit high electron density, which highlights their role as electron acceptors within the lattice.

Fig. 5 Electron charge densities of Na₆AgBiX₂ (X = Cl, Br, I).

The electron charge density reflects the balance of ionic and covalent interactions in these compounds. The ionic character decreases as the halide changes from Cl to I owing to the increasing radius and polarizability of halide ions. The covalent character increases for the heavier halides, as seen in the smoother and more diffuse charge density contours. A high electron density is localized around the Ag and Bi atoms, as shown by the bright contours, indicating valence electrons involved in bonding. Na and Cl exhibit moderate electron densities, suggesting ionic bonding with weaker covalent interactions. The tighter contour lines near Ag and Bi reflect stronger electron localization, while interstitial regions show sparse electron density. The electron density distribution helps in predicting the chemical reactivity and bonding nature, which allows the exploration of potential applications of materials in optoelectronics or thermoelectric devices.⁴⁹

3.4 Electronic band structure

Band structure helps distinguish between metallic and semiconducting behavior and identifies whether the band gap is direct or indirect.⁵⁰ Furthermore, electrical conductivity can be interpreted from the band structure, which also provides a foundation for designing new materials for optoelectronic applications.⁵¹ The WC-GGA exchange–correlation potential was used to compute the electronic band structures for the enlisted compounds. Under various conditions, the band gap values for Na₆AgBiX₂ (X = Cl, Br, I) were computed: excluding spin–orbit coupling (SOC), with SOC, and a hybrid method (SOC + TB-mBJ). Table 3 lists all these results. In the band structure graphs shown in Fig. 6, the red lines represent SOC, blue lines represent without SOC, and magenta lines represent SOC + mBJ calculations. The y-axis represents the energy in electron volts (eV), while the x-axis denotes the momentum along high-symmetry points in the Brillouin zone for all three compounds. The gap between them shows the semiconductor band gap, which is direct or indirect depending on whether CBM and VBM occur at the same or different k-points.⁵² The compounds Na₆AgBiCl₂ and Na₆AgBiBr₂ show a direct band gap, whereas Na₆AgBiI₂ has an indirect band gap.

Table 3 Band gaps in eV and effective masses of electrons and holes of halide double antiperovskites Na₆AgBiX₂ (X = Cl, Br, I)

Materials	E^wosocg	E^wsocg	E^hybridg			Band's nature
Na6AgBiCl2	0.99	0.76	1.27	0.32	2.12	Direct
Na6AgBiBr2	0.98	0.75	1.22	0.31	1.42	Direct
Na6AgBiI2	0.90	0.69	1.15	0.11	0.85	Indirect

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Fig. 6 Band structures of halide double antiperovskites Na₆AgBiX₂ ((a)–(c) X = Cl, Br, I).

The electronic band structures without SOC (blue lines) show the band gaps: approximately 0.99 eV for Na₆AgBiCl₂, 0.98 eV for Na₆AgBiBr₂, and 0.90 eV for Na₆AgBiI₂. Introducing SOC (red lines) reduces the band gaps slightly due to band splitting: around 0.76 eV for Na₆AgBiCl₂, 0.75 eV for Na₆AgBiBr₂, and 0.69 eV for Na₆AgBiI₂. SOC is particularly influential in Na₆AgBiI₂, where the heavier iodine atom intensifies relativistic effects. With SOC + mBJ (magenta lines), the band gaps are slightly increased and more accurate: 1.27 eV for Na₆AgBiCl₂, 1.22 eV for Na₆AgBiBr₂, and 1.15 eV for Na₆AgBiI₂. This makes SOC + mBJ the preferred method for predicting realistic semiconductor band gaps, and suggesting potential applications in optoelectronics.

The effective mass of charge carriers is a key factor governing the carrier mobility in optoelectronic materials. In this work, the effective masses were estimated by applying the Gaussian fitting technique to the valence band maximum (VBM) and conduction band minimum (CBM) of the band structure; the effective masses of electrons and holes were calculated using mathematical relations.⁵³ A higher carrier mobility is correlated with a lower effective mass. In comparison to Na₆AgBiCl₂ and Na₆AgBiBr₂, the data presented in Table 3 indicate that Na₆AgBiI₂ has the lowest effective mass for electrons (0.11) and holes (0.85). This property increases the power conversion efficiency of Na₆AgBiI₂ for photovoltaic applications and makes it extremely efficient for light absorption.

3.5 Density of states (DOS)

Study of the DOS enables a comprehensive understanding of electronic transitions across energy bands and provides crucial information about atomic state hybridization.^54,55

For Na₆AgBiX₂, the DOS were computed using a WC-GGA technique with the overlapping TB-mBJ method. This highlighted the contributions of orbitals and elements (Na, Ag, Bi, Cl/Br/I) to the DOS. Both TDOS (total DOS) and PDOS (partial DOS) exhibit a clear bandgap, with the conduction band from 0 eV to 3.0 eV and valence band from −3.0 eV to 0 eV. With the vertical axis representing states per eV, the TDOS shows that occupied states are on the left and unoccupied states are on the right in Fig. 7. For Na₆AgBiX₂ (X = Cl, Br, I), on the left the TDOS shows multiple distinct peaks in various energy regions. For instance, in Na₆AgBiX₂ (X = Cl, Br, I), the major contribution from Na is shown by the biggest peak in the valence band, which lies between −2.5 eV and −3 eV. The conduction band's smallest peak, which spans from 1.3 eV to 1.8 eV, is mostly caused by (Bi, Cl), (Bi, Br), and (Bi, I), as seen in Fig. 7(a)–(c). Ag, Cl, Br, and I are the primary contributors in all three compounds at 0 eV which is the Fermi level, with Bi making a minor contribution. The PDOS for Na₆AgBiX₂ (X = Cl, Br, I) is displayed in Fig. 8(a)–(c). These figures show that the Na-p, Ag-p, Bi-s, Cl-s, Br-s, and I-s orbitals are the main contributors in the valence band. The contributions of Bi-p, Cl-s/p/d, Br-s/p/d, and I-s/p/d orbitals to the conduction band are relatively small. Comparing the DOS of Na₆AgBiI₂ to that of the other halides, Na₆AgBiCl₂ and Na₆AgBiBr₂, reveals more noticeable peaks close to the Fermi level. This results in broader electronic states because of I's bigger atomic size and lower electronegativity. While the d orbitals of Ag contribute at deeper energy levels, the p orbitals of I and Bi are important close to E_F.

Fig. 7 (a)–(c) Total DOS of Na₆AgBiX₂ (X = Cl, Br, I).

Fig. 8 (a)–(c) Partial DOS of Na₆AgBiX₂ (X = Cl, Br, I).

The high DOS regions of all three investigated compounds suggest enhanced electrical conductivity, whereas low DOS regions represent energy gaps or reduced electron availability. The PDOS plots for the materials Na₆AgBiX₂ (X = Cl, Br, I) provide detailed insights into the contributions of specific orbitals to the total DOS and their role in the Fermi energy (E_F). For Na₆AgBiCl₂, the p orbitals of Bi and Cl/Br/I show significant contributions near E_F. These orbitals dominate the conduction and valence bands, indicating their critical role in electronic transitions. The overall higher PDOS contributions from the p orbitals imply they are a key factor affecting the electronic behavior of this material. The larger atomic size and lower electronegativity of I compared to Cl and Br result in broader and more prominent p orbital contributions near the Fermi level. The PDOS peaks near E_F indicate the material's potential for high electrical conductivity and improved optical properties. These graphs highlight the available electron energy states, essential for understanding electrical, optical, and thermal behaviors. The unique combination of properties exhibited by double antiperovskites makes them attractive materials for use in solar energy conversion and optoelectronic devices.^56,57

3.6 Optical properties

Analyzing a material's interaction with light and bandgap dynamics requires an understanding of its optical properties, which is critical for creating devices like sensors, solar cells, and lasers.⁵⁸ Comprising the real part ε₁(ω) and the imaginary part ε₂(ω), the complex dielectric function provides crucial insight into the interaction between light and solid-state materials. Specifically, ε₁(ω) describes the polarization response of the material, whereas ε₂(ω) characterizes its interaction with and absorption of electromagnetic waves. Understanding the correlations between a few other variables is essential for assessing the interaction between materials and light in addition to the complex dielectric function.⁵⁹ To analyze the various optical characteristics of a material, the dielectric function serves as a key quantity.⁶⁰

The static values of ε₁(ω) for Na₆AgBiX₂ (X = Cl, Br, I) are 6.2, 7.5, and 9.2 at 0 eV (Table 4), rising to 12, 15, and 18 in the infrared spectrum at 1.7 eV. These values are significantly higher than those of Li₂LiTlBr₆ (1.6), Li₂NaTlBr₆ (1.7), Li₂AgCrI₆ (5.7), MAPbI₃ (5.4), and FAPbI₃ (5.7), making them promising materials for solar cell applications. Incident radiation primarily contributes to polarization, and enhanced polarization properties, as reflected by ε₁(ω), are necessary to maximize the performance of semiconductors. Similarly, at 1.7 eV in the visible range, the imaginary component, ε₂(ω), which represents light absorption, exhibits values of 10, 13, and 16. The computed real and imaginary dielectric properties provide valuable information on these materials' linear optical properties, as demonstrated in Fig. 9.

Table 4 Optical properties of halide double antiperovskites Na₆AgBiX₂ (X = Cl, Br, I)

Materials	ε1(0)	n(0)	R(0)
Na6AgBiCl2	6.2	2.5	0.19
Na6AgBiBr2	7.5	2.7	0.22
Na6AgBiI2	9.2	3.0	0.26

---

Fig. 9 (a) ε₁(ω) and (b) ε₂(ω) are the real and imaginary components of the dielectric function of the double antiperovskite Na₆AgBiX₂ (X = Cl, Br, I), respectively.

The refractive index n(ω) measures how much the light's speed is reduced within a material and is crucial for light-bending applications. In Fig. 10(a), it is observed that static index of refraction n(0) at 0 eV is 2.5 for Na₆AgBiCl₂, 2.7 for Na₆AgBiBr₂, and 3 for Na₆AgBiI₂. Na₆AgBiI₂ has the highest refractive index, indicating a stronger interaction with light and potential advantages in waveguiding. Na₆AgBiCl₂ and Na₆AgBiBr₂ show lower refractive indices, suggesting more transparency in specific energy ranges.

Fig. 10 (a) Refractive index (n), (b) optical conductivity σ, (c) extinction coefficient (k), (d) absorption coefficient I(ω), and (e) reflectivity R(ω) of Na₆AgBiX₂ (X = Cl, Br, I).

Optical conductivity σ serves as a key parameter describing the interaction of a material's electrons with electromagnetic fields. The optical conductivity rises with energy, with peaks aligning with ranges where electron transitions are prominent. In Fig. 10(b), Na₆AgBiCl₂, Na₆AgBiBr₂, and Na₆AgBiI₂ show optical conductivities σ(ω) which first reach their maximum peak at 1.7 eV, with values of 2200 (Ω cm)⁻¹, 3000 (Ω cm)⁻¹, and 4000 (Ω cm)⁻¹, respectively. High optical conductivity in the visible spectrum renders Na₆AgBiI₂ a potential material for efficient solar cell devices. Na₆AgBiI₂ exhibits high optical conductivity, indicating a denser electronic state conducive to transitions at these energies, followed by Na₆AgBiCl₂ and Na₆AgBiBr₂.

The extinction coefficient k(ω) defines how absorption contributes to the decrease in light intensity within the material. In Fig. 10(c), Na₆AgBiX₂, (X = Cl, Br, I) exhibit a first highest peak of extinction coefficient k(ω) of 1.7, 2.1, and 2.7 respectively, at 2 eV. The extinction coefficient trends are similar to those of the absorption coefficient, with Na₆AgBiI₂ showing the highest values, implying greater light attenuation. Na₆AgBiCl₂ and Na₆AgBiBr₂ show relatively lower extinction values, reflecting their lower light attenuation characteristics.

The absorption coefficient (α) characterizes the fraction of incident light absorbed per unit length during its propagation through the material. Observations show that the absorption coefficient generally increases with energy, peaking before gradually decreasing, which indicates an absorption edge, marking the onset of high absorption.⁶¹ α(ω) has a crucial role in evaluating solar energy harvesting efficiency by quantifying the fraction of incident light captured within the material rather than lost through reflection or transmission. As shown in Fig. 10(d), the absorption coefficient α(ω) rises sharply near the absorption edge, with Na₆AgBiX₂ (X = Cl, Br, I) exhibiting strong absorption approximately ranges from 30 to 58 cm⁻¹ at 2 eV in the visible range. Among the materials, Na₆AgBiI₂ exhibits the highest peak absorption, suggesting an enhanced light-absorbing capability, likely due to the iodine ions, which tend to increase optical absorption. Na₆AgBiCl₂ and Na₆AgBiBr₂ show lower absorption peaks, consistent with the lower absorptive effects of chlorine and bromine atoms.

Reflectivity represents the ratio of incident light that bounces off the surface of a material. Higher reflectivity can reduce effectiveness in applications that require absorption. Fig. 10(e) presents the static reflectivity R(ω), recorded as 0.19 for Na₆AgBiCl₂, 0.22 for Na₆AgBiBr₂, and 0.26 for Na₆AgBiI₂. The highest reflectivity peak values are 0.35 and 0.41 at 1.7 eV for Na₆AgBiCl₂ and for Na₆AgBiBr₂, and 0.55 at 2.2 eV for Na₆AgBiI₂. The trends indicate that reflectivity increases with energy, peaking at higher energy values. Na₆AgBiI₂ shows the highest reflectivity, while Na₆AgBiCl₂ and Na₆AgBiBr₂ have relatively lower reflectivity values. The quantities R(ω) and α(ω), representing reflectivity and absorption respectively, can be derived.³²

For Na₆AgBiX₂ (X = Cl, Br, I), the halogen identity significantly impacts its optical response. Na₆AgBiI₂ exhibits the strongest optical response, with high absorption and conductivity but increased reflectivity, making it ideal for applications requiring these traits. Na₆AgBiCl₂ and Na₆AgBiBr₂ offer moderate optical responses, suitable for applications needing balanced properties and lower reflectivity. The contrasting behaviors of the refractive index n(ω) and absorption coefficient I(ω) further emphasize the suitability of these double antiperovskites for high-performance optoelectronic devices.⁶²

3.7 Thermodynamic properties

Understanding thermodynamic properties plays a vital role in boosting the effectiveness of the solar energy production in photovoltaic devices, ultimately leading to improved solar energy harvesting capabilities.⁶³ Simulations were performed on Na₆AgBiX₂ (X = Cl, Br, I) under elevated temperatures and fixed pressure in order to examine these characteristics. To optimize with relevance to energy transformation technologies, the Quasi-harmonic Debye approximation and ground-state structural parameters were used to identify a few thermodynamic properties. Important variables that affect solar cell performance include total efficiency of energy conversion, the open-circuit voltage, and the short-circuit current. Furthermore, the design and optimization of materials for optoelectronic and photovoltaic applications heavily relies on thermodynamic parameters.⁶⁴ Materials with high thermodynamic stability must be used since solar cells must endure harsh environmental conditions. We can forecast the behavior of the suggested materials under different operating situations by looking at the thermodynamic data shown in Table 5.

Table 5 Thermodynamic parameters of halide double antiperovskites Na₆AgBiX₂ (X = Cl, Br, I)

Parameters	Na6AgBiCl2	Na6AgBiBr2	Na6AgBiI2
M (Kg)	3484.9 × 10^−27	4082.5 × 10^−27	4706.7 × 10^−27
V (Å)	1529	1537	1548
ρ (Kg m^−3)	2280	2660	3030
v1 (m s^−1)	1240	1280	1330
vt (m s^−1)	2610	2560	2510
Vm (m s^−1)	1395	1444	1487
ΘD (K)	123	128	130
Tm (K)	717	717	723
ωD (THz)	16	17	17
HV (GPa)	0.60	0.73	1.22

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In Na₆AgBiX₂ (X = Cl, Br, I), physical characteristics like density (ρ), transverse velocity, volume (V), and molecular mass (M) all systematically increase as the halogen element's atomic number rises. This pattern provides insightful information on the real-world uses of these materials. Moreover, as it represents the strength of atomic connections and affects characteristics like elasticity, heat capacity, and melting temperatures, the thermal properties of solids can be effectively described using the Debye temperature (Θ_D), which is a fundamental quantity.⁶⁵ Stronger atomic interactions are indicated by a larger Θ_D value, which raises melting points and improves heat conductivity. Table 5 displays the computed Θ_D values for the double perovskite materials. The calculations for transverse sound velocity (v_t) and longitudinal sound velocity v_l were performed by using standard relations.³¹ High temperatures are frequently applied to components used in the construction of solar cells, particularly when processes like the creation of metal contacts or crystalline silicon are being carried out. In these processes, a material's melting point is essential. Low melting point materials run the risk of melting, cracking, or deforming, which can impair the dependability and the solar cell's performance. However, materials with unreasonably high melting points make manufacturing more difficult and expensive. With certain formulae connecting C₁₁ to melting temperatures, an important factor in determining melting temperatures is the elastic constant C₁₁. This connection offers important information on how materials behave at high temperatures during the manufacturing of solar cells.⁶⁶

The suggested compounds' melting temperatures have been calculated and are shown in Table 5. In particular, the melting temperature of Na₆AgBiI₂ (723 K) is greater than that of Na₆AgBiCl₂ (717 K) and Na₆AgBiBr₂ (717 K), suggesting a stronger atomic link between iodine atoms than between chlorine and bromine atoms. The maximum vibrational frequency that phonons in a solid can achieve during heat transmission is known as the Debye frequency (ω_D). Certain equations can be used to calculate a material's hardness (H_V) and Debye frequency,⁶⁷ providing important information about its thermal resistance and mechanical characteristics. Hardness (H_V) also reflects differences in directional bonding characteristics and compound formation. Table 5 shows that the atomic number of the halogen element in the specified compounds increases the Debye temperature, Debye frequency, melting temperature, and hardness of these materials.⁶⁸ A key thermodynamic characteristic is heat capacity, which measures how much heat a substance can absorb without experiencing appreciable temperature changes. As the temperature increases from 0 K to 200 K, C_V increases from 0 J kg⁻¹ K⁻¹ to 250 J kg⁻¹ K⁻¹, as shown in Fig. 11. The heat capacities stay constant up to 1000 K after 200 K, suggesting thermal stability and modest heat absorption. Interestingly, at 1000 K, the Na₆AgBiX₂ show a high heat capacity of roughly 250 J kg⁻¹ K⁻¹, indicating their capacity to absorb and hold heat. Their thermal stability is improved by this characteristic, which increases their resistance to temperature changes. The most thermodynamically stable materials are Na₆AgBiX₂ because of their high heat capacity, which is essential for preserving structural integrity and avoiding thermal deterioration in high-temperature situations (Fig. 12 and 13).

Fig. 11 Variation in heat capacity (C_V) of Na₆AgBiX₂ (X = Cl, Br, I).

The stability of Na₆AgBiX₂ (X = Cl, Br, I) is affected by temperature-dependent thermodynamic characteristics. Enthalpy (H) is a measure of a system's total heat content. A reaction's endothermic (ΔH > 0) or exothermic (ΔH < 0) status is indicated by changes in enthalpy (ΔH).

A system's disorder or randomness is measured by its entropy (S). As particles acquire more kinetic energy and are able to occupy more states, it rises with temperature. The spontaneity of a reaction or process is determined by its Gibbs free energy (G), which is determined by the formula G = H − TS, where T is the temperature. The conditions for spontaneity are:

• G < 0: Reaction is spontaneous in the forward direction.

• G > 0: Reaction is non-spontaneous in the forward direction.

• G = 0: Reaction is at equilibrium.

While enthalpy marginally increases with temperature, suggesting endothermic activity, Gibbs free energy drops, making reactions more spontaneous (Fig. 12). At increasing temperatures, the T·S term rises, indicating more chaos. The highest enthalpy and lowest free energy are displayed by Na₆AgBiCl₂, indicating higher stability. These insights support stability analysis, synthesis, and possible thermoelectric or catalytic applications. Na₆AgBiI₂, in conclusion, is the most promising material for applications that require high thermal stability because of its better heat capacity, which enables it to withstand thermal stress more successfully than Na₆AgBiCl₂ and Na₆AgBiBr₂.

Fig. 12 The thermodynamic properties versus temperature such as T*entropy, enthalpy and free energy of Na₆AgBiX₂ (X = Cl, Br, I).

3.8. Photocatalytic properties

Semiconductors with optimal bandgaps are capable of efficiently utilizing solar energy to achieve hydrogen production by means of photocatalytic water splitting and offering a clean renewable energy. The photocatalytic process occurs when a photocatalyst interacts with water, where incident light possessing energy at or above the band gap promotes electron excitation, producing electron–hole pairs.⁶⁹ However, current photocatalysts are limited by high charge carrier recombination rates and insufficient solar sensitivity. Thus, designing advanced photocatalysts with suitable bandgaps, appropriate redox potentials, and reduced electron–hole recombination is essential.⁷⁰ These electron–hole pairs play a crucial role in proton reduction, resulting in H₂ production and in the oxidation of H₂O to yield O₂. A minimum 1.23 eV bandgap is required to initiate the water-splitting reaction. However, the photocatalytic activity of semiconductor materials is significantly influenced by the edge potentials of the valence band (VB) and conduction band (CB).⁷¹ To assess the photocatalytic water-splitting potential of Na₆AgBiX₂, we calculate band-edge potentials based on Mulliken's electronegativity. These potentials are derived from Mulliken's electronegativity (χ), the standard hydrogen electrode potential (E_e = 4.5 eV), the bandgap (E_g), and the energies of the valence band maximum (E_VBM) and conduction band minimum (E_CBM). Table 6 summarizes the edges at pH = 0. The CB and VB edge potentials at this pH are −4.44 eV and −5.67 eV, or 1.23 eV and 0 eV, respectively. Furthermore, the electronegativity of an atom can be quantitatively estimated using a simple formula, which calculates it as the average of the atom's first ionization energy (E_ion) and electron affinity (E), as expressed in ref. 72 and 73.

Table 6 Photocatalytic parameters of Na₆AgBiX₂ (X = Cl, Br, I) in eV

Compounds	χ (eV)	Eg (eV)	ECB (eV)	EVB (eV)
Na6AgBiCl2	1.33	1.27	−1.26	−2.54
Na6AgBiBr2	1.32	1.22	−1.35	−2.57
Na6AgBiI2	1.30	1.15	−1.48	−2.63

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In this context, χ represents the absolute electronegativity of the constituent atoms in the compounds, namely Na, Ag, Bi, and the halogen atoms (Cl, Br, I). Furthermore, E_g denotes the electronic bandgap energy of the Na₆AgBiX₂ (X = Cl, Br, I) compounds. The electron energy level is conventionally referenced as 4.5 eV with respect to the normal hydrogen electrode (NHE).

The rate of electron–hole recombination is an important aspect influencing the photocatalytic performance of a material. The materials Na₆AgBiCl₂, Na₆AgBiBr₂, and Na₆AgBiI₂ have been studied for their photocatalytic properties, which are highly influenced by their band gap alignments and electronic properties. The photocatalytic effectiveness of these materials is largely determined by the positions of their conduction band minimum (CBM) and valence band maximum (VBM) relative to standard redox levels, such as the hydrogen evolution (H⁺/H₂) and oxygen evolution (H₂O/O₂) potentials. These properties impact the materials' abilities to generate charge carriers (electrons and holes) when exposed to light, which is crucial for driving photocatalytic reactions such as water splitting. Among these materials, Na₆AgBiBr₂ has a band alignment that is particularly favorable for photocatalytic reactions (Fig. 13). Its CBM and VBM positions are ideally situated to facilitate the necessary electron transfer processes for water splitting. The slightly higher VBM and lower CBM values of Na₆AgBiBr₂ compared to Na₆AgBiCl₂ and Na₆AgBiI₂ suggest an optimal band gap for visible light absorption, balancing photon energy absorption with efficient charge separation. Furthermore, the electronic structure of Na₆AgBiBr₂ suggests that it has moderate permeability and effective charge carrier mobility, which are critical for photocatalytic performance. Effective permeability in this context refers to the material's ability to allow generated electrons and holes to move freely within the lattice, minimizing recombination losses. This property enhances its overall photocatalytic efficiency as it improves the likelihood that charge carriers will reach the surface and participate in chemical reactions, rather than recombining internally.

Fig. 13 Photocatalytic properties at pH = 0 of Na₆AgBiX₂ (X = Cl, Br, I).

In conclusion, on the band alignment basis Na₆AgBiBr₂ emerges as the most promising photocatalyst among the three materials. Its ideal band edge positions support both the water reduction and oxidation processes required in photocatalysis, while its balanced charge transport properties enhance the separation and mobility of photogenerated carriers. This combination makes Na₆AgBiBr₂ highly effective for visible-light-driven photocatalytic applications compared to Na₆AgBiCl₂ and Na₆AgBiI₂. In photocatalytic reactions, electrons facilitate the reduction of water, whereas holes enable its oxidation. Notably, the investigated compounds Na₆AgBiX₂ (X = Cl, Br, I) exhibit exceptional photocatalytic performance, rendering them promising candidates for efficient solar-driven oxygen production in industrial applications.

3.9 Thermoelectric properties

Atomic vibrations and electron motion are indicated by thermoelectric properties, which show that thermal and electrical conductivities increase with temperature.^74,75 Fig. 14(a) illustrates how the Seebeck coefficient S (µV K⁻¹) varies with temperature. Temperature causes S to decrease in all compounds, and out of all of them, Na₆AgBiCl₂ has the smallest value, at 165.6 µV K⁻¹ at 300 K. The overall trend indicates that as the temperature rises, the positive S value decreases, and at 900 K Na₆AgBiCl₂ has the lowest S among the compounds.

Fig. 14 Thermoelectric properties with temperature variation from 100 K to 900 K for Na₆AgBiX₂ (X = Cl, Br, I).

The temperature change of electrical conductivity, σ/τ (Ω⁻¹ m⁻¹ s⁻¹), is displayed in Fig. 14(b). At 300 K (room temperature), the greatest σ/τ value was calculated to be at 6.96 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹ for Na₆AgBiCl₂, and 4.54 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹ is the lowest recorded value of Na₆AgBiI₂. At 600 K, Na₆AgBiCl₂ has the highest σ/τ of 8.41 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹. It is true that σ/τ increases with temperature for all compounds. However, at 900 K, the σ/τ of all compounds are around 7.6 × 10¹⁸ Ω⁻¹ m⁻¹ s⁻¹.

The temperature versus thermal conductivity (electronic portion) κ_e/τ (W m⁻¹ K⁻¹ s⁻¹) is shown in Fig. 14(c). Mathematically, σ and κ_e/τ are linked through Wiedemann–Franz Law κ_elec = LσT,⁷⁶ where L denotes the Lorenz number. The lowest estimated κ_e/τ at 300 K is of Na₆AgBiI₂ at 0.65 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹, whereas the highest estimated value at the same temperature is of Na₆AgBiCl₂ at 0.78 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹. The trend becomes opposite at high temperatures of 900 K, where Na₆AgBiI₂ has reached value of 1.11 × 10¹⁴ W m⁻¹ K⁻¹ s⁻¹.

In Fig. 14(d), the variation of the electronic power factor, PF (W m⁻¹ K⁻² s⁻¹), with temperature is shown. The PF is determined using the expression PF = S²σ/τ. At 300 K, the highest PF value of 1.95 × 10¹¹ W m⁻¹ K⁻² s⁻¹ is exhibited for Na₆AgBiBr₂ while Na₆AgBiI₂ shows the lowest value of 1.53 × 10¹¹ W m⁻¹ K⁻² s⁻¹. The maximum PF for Na₆AgBiCl₂ is 1.92 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 300 K, while it is 2.01 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 350 K for Na₆AgBiBr₂ and 1.80 × 10¹¹ W m⁻¹ K⁻² s⁻¹ at 450 K for Na₆AgBiI₂.

Thermal conductivity in solid materials typically shows a decreasing trend with increasing temperature. A widely used approach for evaluating the temperature dependence of thermal conductivity in solids, and thus understanding their heat transport behavior, is Slack's equation.^77,78 The lattice thermal conductivity κ_L (W m⁻¹ K⁻¹) is depicted using Slack's equations in Fig. 14(f). The observed decrease in κ_L associated with these compounds under rising temperatures results from a number of contributions from phonon scattering, thermal expansion, and diminished interatomic bond strength. Since Na₆AgBiCl₂ exhibits the highest κ_L of 1.55 W m⁻¹ K⁻¹ at 300 K and the lowest at 1.05 W m⁻¹ K⁻¹ for Na₆AgBiBr₂ (see Fig. 14f), which is further decreased at higher temperature.

At increasing temperatures, phonons, or lattice vibrations, become more energetic and frequency-dependent. As a result, there is more phonon scattering, which reduces thermal conductivity and stops thermal energy from flowing smoothly.^79,80 Additionally, the majority of materials expand when heated, altering the atom-to-atom distances that impact phonon dispersion and, consequently, thermal conductivity. Moreover, the material's interatomic bonds may be weakened by high temperatures, which would decrease phonon group velocities and, consequently, thermal conductivity. Recognizing these contributing factors facilitates a comprehensive explanation of the decline in thermal conductivity with increasing temperature.

The thermoelectric figure of merit (ZT) serves as a key indicator of a material's efficiency. ZT is inversely proportional to (κ_e + κ_L) and directly proportional to the PF; ZT = S²σT/(κ_e + κ_L). The ZT at 300 K with the average κ over the electronic and lattice contributions is 0.74, 0.79 and 0.71 for Na₆AgBiCl₂, Na₆AgBiBr₂, and Na₆AgBiI₂, respectively, as Fig. 14(e) illustrates. In contrast, Na₆AgBiBr₂ has the lowest κ_L of any material, which results in the highest ZT value of 0.84 at 900 K. Hence, Na₆AgBiBr₂ will be a promising candidate for thermoelectric applications at room temperature as well as at 900 K (Table 7).

Table 7 Thermoelectric properties, S (µV K⁻¹), σ/τ (10¹⁸ Ω⁻¹ m⁻¹ s⁻¹), κ_e/τ (10¹⁴ W m⁻¹ K⁻¹ s⁻¹), κ_L (W m⁻¹ K⁻¹), PF (10¹⁰ W m⁻¹ K⁻² s⁻¹) and ZT (with calculated κ_L) at 300 K for Na₆AgBiX₂ (X = Cl, Br, I)

Compounds	S	σ/τ	κe/τ	κL	PF	ZT@κL
Na6AgBiCl2	165.9	6.97	0.78	1.55	1.92	0.74
Na6AgBiBr2	199.8	4.88	0.74	1.05	1.95	0.79
Na6AgBiI2	183.8	4.54	0.65	1.25	1.53	0.71

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3.10 SLME analysis

Photovoltaic performance of Na₆AgBiX₂ (X = Cl, Br, I) was analyzed through the spectroscopic limited maximum efficiency (SLME) method. The power conversion efficiency of solar cells is evaluated within the SLME model using the extended Shockley–Queisser formalism.⁸¹ The SLME calculation relies on parameters such as the material's fundamental and direct band gaps (E_g), the standard AM 1.5G solar spectrum, and absorption spectra obtained from DFT.^82,83 In Fig. 15, the SLME values (%) of Na₆AgBiX₂ (X = Cl, Br, I) are displayed for different thicknesses and temperatures. Among the studied parameters, device temperature, band gap, and thickness exert the most significant influence on SLME. Fig. 15 reveals that SLME grows steadily with thickness before reaching a saturation point. In comparison to Na₆AgBiCl₂ and Na₆AgBiI₂ SLME (@8.06% and @8.12% for 0.5 µm thick layer), Na₆AgBiBr₂ has a higher SLME efficiency (8.46@0.5 µm thick layer) according to standard SLME analysis. Which may further increase above 30% with the increase in thickness of slab of Na₆AgBiBr₂, positioning it as a strong candidate for solar cell applications. However, recently double perovskites A₂Au^IAu^IIII₆ (A = Rb, Cs), whose efficiency predicted using SLME over 29% with a film thickness of 0.5 µm.^20,21

Fig. 15 SLME efficient (η) vs. thickness (µm) of Na₆AgBiX₂ (X = Cl, Br, I).

4. Conclusion

In this work, the physical properties of Na₆AgBiX₂ (X = Cl, Br, I) were systematically examined through advanced computational methods, specifically density functional theory (DFT) combined with the full-potential linearized augmented plane wave plus local orbitals (FP-LAPW + lo) technique. The structural and thermal stability assessments based on tolerance factor (τ_G), formation energy (E_f) and binding energy (E_b) calculations show all materials are stable. Calculated charge density, explain trend from Cl to I highlights a transition from predominantly ionic to predominantly covalent bonding. The calculated electronic band structures, revealing direct band gaps for Na₆AgBiCl₂ and Na₆AgBiBr₂, and an indirect band gap for Na₆AgBiI₂. The calculated band gaps (hybrid) range from 1.15 eV (Na₆AgBiI₂) to 1.27 eV (Na₆AgBiCl₂). The effective masses of electrons and holes in Na₆AgBiX₂ (X = Cl, Br, I) materials are computed by employing the Gaussian fitting technique. The TDOS and PDOS analysis revealed that Bi, Cl, Br and I p-orbitals dominate electronic transitions near the Fermi level, while Ag d-orbitals contribute at deeper levels. Optical properties showed halogen-dependent variations, with Na₆AgBiI₂ displaying the highest refractive index and optical absorption, making it suitable for solar cells and waveguiding. In contrast, Na₆AgBiCl₂ and Na₆AgBiBr₂ demonstrated balanced optical characteristics, ideal for moderate-absorption applications. All Na₆AgBiX₂ are considered elastically and thermodynamically more stable. Additionally, photocatalytic studies indicated that Na₆AgBiX₂ have optimal band-edge alignment, efficient charge carrier separation, and superior water-splitting capability, framing it as the most an attractive option for solar oxygen generation. Based on the electronegativity-derived band-edge alignment, the Na₆AgBiX₂ (X = Cl, Br, I) series exhibits band positions that are thermodynamically suitable for photocatalytic water splitting, suggesting potential for future photocatalytic applications. TE properties reveal that Na₆AgBiBr₂ emerges as a strong candidate for TE applications because of its superior ZT value of 0.79 at 300 K and 0.84 at 900 K among other compounds. The solar cell efficiency predicted by SLME, Na₆AgBiBr₂ have high efficiency of 8.46% than Na₆AgBiCl₂ (@8.06%) and Na₆AgBiI₂ (@8.12%) at for 0.5 µm thick layer which is further increase to over 30% with increase in thickness. All the reported results underscore the feasibility of Na₆AgBiBr₂ for advanced uses in eco-friendly energy systems (such as optoelectronics, water splitting, Thermoelectric and photovoltaics).

Conflicts of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

Data will be made available on request.

Funding

This research was funded by Ongoing Research Funding program – Research Chairs (ORF-RC-2025-5529), King Saud University, Riyadh, Saudi Arabia.

Acknowledgements

The authors extend their appreciation to Ongoing Research Funding program – Research Chairs (ORF-RC-2025-5529), King Saud University, Riyadh, Saudi Arabia. The author Y. Saeed would like to thank Higher Education Commission (HEC) of Pakistan for providing grant under NRPU-15844.

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