Lead-free new alkali metals double perovskites A₂InAgF₆ (A = Na, K, and Rb) for optoelectronic applications: a first-principles study

Abstract

The growing demand for sustainable energy and the environmental drawbacks of lead-based perovskites have led to the emergence of lead-free double perovskites as promising alternatives. In this work, first-principles calculations based on density functional theory (DFT) are employed to systematically investigate the structural, mechanical, electronic, optical, and thermodynamic properties of fluoride-based double perovskites A₂InAgF₆ (A = Na, K, and Rb). Structural analysis confirms that all three compounds are stable in the cubic space group, with negative formation energies. Mechanical evaluations demonstrate that all three compounds satisfy the Born-stability criteria and exhibit ductile behavior. Electronic band structure calculations reveal that these materials are direct bandgap semiconductors, with bandgaps increasing from 3.02 eV for Na₂InAgF₆ to 3.88 eV for Rb₂InAgF₆ using the Hybrid-HSE06 functional. Furthermore, the materials exhibit low reflectivity and strong optical absorption in the UV range, alongside stable thermodynamic profiles at high temperature. These findings highlight the potential of A₂InAgF₆ compounds for next-generation optoelectronic, photovoltaic, and thermoelectric applications.

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1 Introduction

The increasing depletion of fossil fuels and the urgent need to reduce carbon emissions have made solar energy a promising renewable solution to the global energy crisis.^1,2 The efficiency of solar cells depends on the optical and electronic properties of the materials used.³ Although silicon-based photovoltaic cells dominate the market due to their reliability and efficiency, their broader use is restricted by high production costs and modest power conversion efficiency (PCE).^4,5 Perovskites have attracted significant attention as promising alternatives for emerging solar cell technologies. Since 2009, perovskite solar cells have achieved PCEs exceeding 25%, comparable to commercial silicon-based cells.⁶ This performance arises from their unique properties, including long charge-carrier diffusion lengths, tunable bandgaps, and low exciton binding energies.^7,8 These features enable applications in solar cells, light emitting diodes, photodetectors, and thermoelectric devices.^8,9 Additionally, they are low cost, lightweight, and amenable to large-scale fabrication.¹⁰ However, conventional Pb-based perovskites suffer from serious drawbacks such as toxicity and poor stability under moisture, heat, and UV exposure,¹¹ motivating the exploration of lead-free alternatives such as double perovskites.¹²

The A₂BB′X₆ double perovskite structure has recently gained attention for sustainable energy applications due to its compositional flexibility and ecofriendly nature. These materials crystallize in the cubic Fm3̄m structure, where the A-site is occupied by alkali metal ions (e.g., K⁺, Rb⁺); the B and B′ sites by mono- and trivalent cations (e.g., Ag⁺, Cu⁺, In³⁺, Bi³⁺), and the X-site by halogens (e.g., F⁻, Cl⁻, Br⁻, I⁻).^13,14 This structural versatility allows tuning of bandgaps, carrier mobility, and optical absorption, making them suitable for photovoltaic and optoelectronic applications. Lead-free double perovskites such as Cs₂AgBiBr₆ and Cs₂AgBiCl₆ show indirect and large bandgaps (>2 eV), limiting efficiency, while Cs₂TiBr₆ exhibits an improved bandgap and a power conversion efficiency (PCE) of 3.3%.^15,16 To overcome such limitations, halide substitution and cation engineering have been widely used to tune material properties. These materials exhibit good stability, long carrier diffusion lengths, and strong optical absorption, making them promising for optoelectronic applications.^17,18

Recent studies further demonstrate the potential of Pb-free double perovskites. For example, A₂YAuI₆ (A = Rb, Cs), studied by Nazir et al.,¹⁹ shows ductile behavior and direct bandgaps (∼1.7 eV) suitable for clean energy. X₂AgBiI₆ (X = K, Rb, Cs) exhibits stability, ideal bandgaps (1.26–1.35 eV), strong absorption, and promising thermoelectric performance.²⁰ Nasir et al.²¹ reported X₂CuAsF₆ (X = Na, K) with bandgaps (0.8 to 1.56 eV) and strong optical response.²¹ A₂AlAuCl₆ (A = Cs, K, Rb) shows ∼2 eV bandgaps with UV-visible absorption and thermoelectric potential.²² A-site substitution significantly influences properties; for example, A₂AgAlI₆ (A = Na, K, and Rb) demonstrates decreasing bandgaps with A-site variation,²³ while Rb₂AlAgCl₆ and Cs₂AlAgCl₆ studied by Lantri et al.²⁴ reveal bandgap tuning and UV responsiveness. Similarly, K₂AlInI₆, Rb₂AlInI₆, & Cs₂AlInI₆ highlight the A site effect on electronic and thermoelectric properties.²⁵ Studies on K₂NaInX₆ (ref. 26) and Rb₂GeSnX₆ reveal halide-dependent tuning of optical and thermoelectric properties.²⁷ Recent first-principles studies on Rb₂NaTlZ₆ (Z = Cl, Br, I) by Shah et al.²⁸ report structural stability, direct bandgaps (3.4–2.1 eV), and thermoelectric performance (ZT up to 0.71). DFT studies on Cs₂NaXCl₆ (X = In, La, Sc, Y) by Shakeel et al.²⁹ show stability, wide bandgaps (4.1–6.4 eV), strong UV absorption, and low thermal conductivity (0.169–0.228 W mK⁻¹). Rb₂CuSbX₆ (X = Cl, Br, I) exhibits pressure-dependent bandgaps (1.08–0.33 eV) and tunable optoelectronic and thermoelectric properties.³⁰ In another study by Shah et al.,³¹ K₂NaTIX₆ (X = Cl, Br, I) exhibits strain-tunable direct bandgaps and improved optoelectronic properties under uniaxial strain (−4% to 4%). Saeed et al.³² reported that X₂BAgCl₆ (X = K, Rb, Cs; B = Sc, Y) exhibits semiconducting behaviour with bandgaps (3.56–4.88 eV), strong UV absorption, and moderate thermoelectric performance (ZT ≈ 0.38–0.42).

Despite these advances, fluoride-based double perovskites remain less explored than iodide, bromide, and chloride systems. In particular, a systematic theoretical study of A-site cation effects on structural stability and optoelectronic properties is still lacking, representing an important scientific gap. In this context, A₂InAgF₆ (A = Na, K, and Rb) provides an ideal platform to investigate the role of A-site cation size and chemistry. These compounds consist of non-toxic elements and are promising for sustainable applications, while Ag⁺ and In³⁺ are expected to enhance structural stability and electronic performance. Therefore, in this work, density functional theory (DFT) is employed to study the structural, electronic, optical, phonon, and thermodynamic properties of A₂InAgF₆ (A = Na, K, and Rb). The objective of this work is to understand the effect of A-site variation and identify the most suitable composition for efficient and stable clean energy applications.

2 Computational methods

First-principles calculations based on DFT^33,34 were performed using the Cambridge Serial Total Energy Package (CASTEP) module.³⁵ The electronic exchange–correlation interactions were treated using the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE)³⁶ functional for geometry optimization and electronic properties calculation. Additionally, spin–orbit coupling (SOC) was incorporated. For more precise evaluation of the bandgaps and optical properties, the Hybrid-HSE06 functional was employed in the calculations. Ultrasoft pseudopotentials were employed to describe the core-valence electron interactions for all elements. The structural optimization of the crystal was conducted using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization scheme.^37–40 The plane-wave basis set energy cutoff was set to 500 eV, and a Monkhorst–Pack k-point mesh of 4 × 4 × 4 was used to sample the Brillouin zone.^41,42 The convergence thresholds were set to 1.0 × 10⁻⁶ eV per atom for self-consistent field (SCF) cycles, and 1.0 × 10⁻⁵ eV per atom for total energy. Additionally, the maximum force, stress, and atomic displacement tolerances were set to 0.03 eV Å⁻¹, 0.05 GPa, and 0.001 Å, respectively. A k-point mesh of 8 × 8 × 8 was chosen, and the Kramers–Kronig relations were used to derive the frequency-dependent complex dielectric function, which allowed optical constants, including the absorption coefficient, refractive index, and reflectivity, to be calculated.⁴³ Small finite strains were applied to the optimized structure in order to evaluate the second-order elastic constants. The GGA-PBEsol framework, which offers better lattice parameter predictions for solids, was used for these computations. The determined elastic constants were used to investigate mechanical stability and anisotropy. The final crystal structures of A₂InAgF₆ fluoride perovskites are visualized using the VESTA visualization software.⁴⁴ The elastic modulus parameters are produced using three-dimensional rendering by means of the ELATE software.⁴⁵ Phonon dispersion and thermodynamic calculations were performed using a higher cutoff energy of 880 eV and a reduced 2 × 2 ×2 k-point grid to ensure the accuracy of the lattice dynamical properties.⁴¹ The convergence criteria for these calculations were set to 0.5 × 10⁻⁵ eV per atom for SCF energy, 0.1 × 10⁻¹ eV Å⁻¹ for force, 0.2 × 10⁻¹ GPa for stress, and 0.5 × 10⁻³ Å for displacement.

3 Result and discussions

3.1 Structural properties

The cubic space group Fm3̄m (# 225) represents the fundamental crystal symmetry of double perovskites with the general formula A₂B′B″X₆.¹⁵ The fluoride double perovskites A₂InAgF₆ (A = Na, K, and Rb) also crystallize in this highly symmetric cubic structure, which is characteristic of stable double perovskite systems. Each conventional unit cell contains 40 atoms, corresponding to four formula units. The alkali metal cations (Na⁺, K⁺, and Rb⁺) are accommodated within the interstitial sites of the perovskite lattice, where they play a crucial role in stabilizing the overall crystal structure. Fig. 1 presents the schematic crystal structure of the cubic double perovskite A₂InAgF₆. In this structure, the alkali metal cations occupy the 8c Wyckoff positions (0.25, 0.25, 0.25), while the In³⁺ ions are located at the 4a sites (0, 0, 0) and the Ag⁺ ions reside at the 4b sites (0, 0, 0.5). The fluoride anions are positioned at the 24e sites (0, 0, x), where the internal coordinate x shows a slight dependence on the size of the A-site cation.

Fig. 1 The schematic crystal representation of the double perovskite A₂InAgF₆ (A = Na, K, and Rb) structure.

The formation enthalpy values are computed using the following equation:

The formation energy ΔE_f represents the energy change associated with forming the compound A₂InAgF₆ from its constituent elements in their most stable bulk and molecular reference states. In this expression, E_A2InAgF6 denotes the total energy of the optimized compound obtained from DFT calculations. The terms E^bulk_A, E^bulk_In, and E^bulk_Ag correspond to the energies per atom of the alkali metal (A = Na, K, Rb), indium, and silver in their respective bulk crystalline phases. The fluorine contribution is represented by E_F2, which is the total energy of an isolated fluorine molecule, and it is multiplied by 3 because six fluorine atoms in the compound correspond to three F₂ molecules. The factor normalizes the total energy difference per atom, where N is the total number of atoms in the formula unit (10 in this case). A negative value of ΔE_f indicates that the compound is thermodynamically stable with respect to its elemental constituents. The calculated formation energies (ΔE_f) are negative for all three compounds, confirming their stability and spontaneous formation tendency from their constituent elements.⁴⁶ Among the studied systems, Rb₂InAgF₆ exhibits the most negative formation energy (−2.319 eV per atom), indicating the highest structural stability as shown in Fig. 2(a). In comparison, K₂InAgF₆ shows a slightly less negative value (−2.298 eV per atom), while Na₂InAgF₆ has the least negative formation energy (−2.114 eV per atom), suggesting comparatively lower lattice cohesion. The increase in stability with increasing A-site ionic size can be attributed to enhanced lattice strain and reduced geometric compatibility within the perovskite framework. Additionally, we also calculated cohesive and binding energy. The following equations are used to calculate the cohesive and binding energy:

Fig. 2 (a) Formation energy, (b) lattice constant, and volume for A₂InAgF₆ (A = Na, K, and Rb) compounds.

The cohesive energy ΔE_c and binding energy ΔE_b describe the strength of atomic bonding in A₂InAgF₆, but they are defined with slightly different sign conventions. In the cohesive energy expression, E_A2InAgF6 is the total energy of the compound, while E^iso_A, E^iso_In, E^iso_Ag, and E^iso_F represent the energies of isolated (free) atoms of A, In, Ag, and F, respectively. These isolated atomic energies are multiplied by their stoichiometric coefficients (2 for A, 1 for In, 1 for Ag, and 6 for F) to account for all atoms in the formula unit. The cohesive energy equation calculates the energy difference between the compound and its constituent isolated atoms, normalized by the total number of atoms N, and typically yields a negative value, indicating energy release upon formation of the solid from free atoms. In contrast, the binding energy ΔE_b uses the same terms but reverses the subtraction order, effectively measuring the energy required to break the compound into isolated atoms. As a result, binding energy is usually positive for stable systems. Both quantities provide insight into bond strength and structural stability, with larger magnitudes indicating stronger interatomic interactions. The calculated cohesive energies of Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆ are −4.154, −4.231, and −4.226 eV per atom, respectively. This decrease in cohesive energy indicates a weakening of the internal bonding strength, as a lower amount of energy is required to separate the crystal into its constituent isolated atoms. Similarly, the calculated binding energies for Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆ are also found to be 4.154, 4.231, and 4.226 eV per atom, respectively. This well-ordered atomic arrangement gives rise to a highly symmetric cubic lattice and ensures ideal octahedral coordination around the B-site metal ions. The increase in ionic size leads to an expansion of the perovskite lattice, resulting in an increase in the lattice constant from 8.98 Å for Na₂InAgF₆ to 9.05 Å for K₂InAgF₆ and further to 9.12 Å for Rb₂InAgF₆ as shown in Fig. 2(b). The unit cell volume increases from 720.44 ų to 740.15 ų and 760.09 ų, respectively. This behavior is consistent with the fundamental characteristics of perovskite structures, where larger A-site cations enlarge the perovskite cage by increasing the metal–fluorine bond lengths and inter-octahedral spacing.⁴⁹ Laihnuna et al. reported the lattice parameter and formation energy of K₂InAgF₆, while Laihnuna et al. and Ahmad et al. reported comparable structural and energetic parameters for K₂InAgCl₆, K₂InAgBr₆, and K₂InAgI₆; these values are in good agreement with our results, confirming the reliability of the present calculations.^47,48 All the values are given in Table 1.

Table 1 Calculated lattice constants a (Å), unit cell volume V (ų), formation energy (ΔE_f), cohesive energy (ΔE_c), and binding energy (ΔE_b)

Compound	Lattice parameter, a (Å)		V (Å^3)	Formation energy, (eV per atom)	Cohesive energy, (eV per atom)	Binding energy, (eV per atom)
	This study	Previous study
Na2InAgF6	8.98	—	720.44	−2.114	−4.154	4.154
K2InAgF6	9.05	9.003 (ref. 47)	740.15	−2.298	−4.231	4.231
Rb2InAgF6	9.12	—	760.09	−2.319	−4.226	4.226
K2InAgCl6 (ref. 47 and 48)	—	10.502, 19.930	—	—	—	—
K2InAgBr6 (ref. 47 and 48)	—	11.0543, 20.9559	—	—	—	—
K2InAgI6 (ref. 47 and 48)	—	11.8904, 22.5263	—	—	—	—

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The structural stability was further assessed using the Goldschmidt tolerance factor (τ_G), and the octahedral factor (µ) defined as:

where r_A, r_B, r_B′ and r_X stand for the elements' respective Shannon ionic radii at locations A, B, B′, and X. Table 2 demonstrates the ionic radius of each atom. For stable cubic perovskites, the tolerance factor typically lies within the range 0.825 < τ_G < 1.059, while the octahedral factor satisfies 0.414 < µ < 0.732.⁵⁰ The calculated values for A₂InAgF₆ (A = Na, K, and Rb), presented in Table 3, fall within these acceptable limits, confirming their structural compatibility with the cubic perovskite framework. The tolerance factors are 0.83 for Na₂InAgF₆, 0.91 for K₂InAgF₆, and 0.94 for Rb₂InAgF_6. The closer proximity of τ_G to unity for K₂InAgF₆ and Rb₂InAgF₆ indicates a more favorable cubic perovskite geometry, whereas the lower value for Na₂InAgF₆ suggests increased octahedral distortion due to size mismatch at the A-site. This confirms that larger alkali ions provide better structural stabilization of the cubic framework.⁵¹

Table 2 Shannon ionic radii of constituent ions in A₂InAgF₆ (A = Na, K, and Rb)

Ion	Ionic radius (Å) [Shannon ionic radii (1976)]
Na^+ (12-coordinated)	1.39
K^+ (12-coordinated)	1.64
Rb^+ (12-coordinated)	1.72
In^3+ (6-coordinated)	0.80
Ag^+ (6-coordinated)	1.15
F^− (6-coordinated)	1.33

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Table 3 Goldschmidt's tolerance factor (τ_G), Bartel's tolerance factor (τ_B) and octahedral factor (µ) for A₂InAgF₆ (A = Na, K, and Rb) compounds

Compound	τG	τB	µ
Na2InAgF6	0.83	4.384	0.73
K2InAgF6	0.91	3.599	0.73
Rb2InAgF6	0.94	3.472	0.73

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Bartel et al.⁵² proposed another tolerance factor to predict the stability of perovskite materials. To further assess structural stability, the data driven tolerance factor proposed by Bartel et al. was used, given by:

where n_A is the oxidation state of the A-site cation. According to this criterion, double perovskites are structurally stable when τ_B < 4.63.⁵² The calculated values (3.599 for K, 3.472 for Rb, and 4.384 for Na) confirm that all three compounds fall within the stability range. However, the relatively higher τ_B value of Na₂InAgF₆ indicates greater lattice strain, while K₂InAgF₆ and Rb₂InAgF₆ exhibit superior geometric compatibility and structural robustness. Therefore, the calculated τ_B values, presented in Table 3, further support the structural stability of the studied compounds.

Overall, the combined analysis of lattice parameters, formation energy, and tolerance factors demonstrates that K₂InAgF₆ and Rb₂InAgF₆ possess highly stable cubic double perovskite structures, whereas Na₂InAgF₆, although stable, shows comparatively higher structural distortion. This systematic dependence of structural stability on the A-site cation size plays a crucial role in tuning the mechanical, vibrational, and optoelectronic properties of fluoride double perovskites.

3.2 Mechanical properties

In density functional theory, the reaction of a material to stress and other conditions is described in terms of its mechanical properties.⁵³ The elastic stiffness constants symbolized by the C_ij matrix in Table 4 describe the mechanical response of Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆ and illustrated in Fig. 3(a). These constants are also significant for their structural stability, anisotropy, and potential applications.

Table 4 Calculated elastic stiffness constants for Na₂InAgF₆, K₂InAgF_6, and Rb₂InAgF₆

Compounds	C11	C12	C44	Cs	Cp	ζ
Na2InAgF6	63.69	26.06	6.22	18.82	19.83	0.55
K2InAgF6	60.26	23.50	7.07	18.38	16.43	0.53
Rb2InAgF6	50.65	23.80	10.72	13.42	13.08	0.60
K2InAgCl6 (ref. 48)	44.42	22.82	7.31	—	15.51	—
K2InAgBr6 (ref. 48)	47.43	14.86	9.81	—	5.05	—
K2InAgI6 (ref. 48)	43.17	8.59	8.24	—	0.36	—

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Fig. 3 The comparison of the A₂InAgF₆ (A = Na, K, and Rb) compounds' (a) elastic constants C_ij, (b) elastic moduli, and (c) Pugh's ratio & Poisson's ratio.

The elastic constant C₁₁ defines how a material resists forces of compression along the principal crystallographic direction. Measurement of elasticity creates higher stiffness along this particular direction.⁵⁴ C₁₂ measurements define the behavior of the material to respond to stress that is equally acting on multiple axes, stress is applied in one direction, and strain in a direction perpendicular [1 1 0] plane.⁵⁵ The material responds to shear strains through the Shear Elastic Constant measurement expressed as C₄₄. The concept of resistance to the change in shape without a change in volume is an important aspect to explore.⁵⁶ The Born stability criteria of the compounds are described through the equations noted below:⁵⁷

C₁₁ > 0, C₄₄ > 0, C₁₁–C₁₂ > 0, C₁₁ + 2C₁₂ > 0 (7)

Among the three compounds, Na₂InAgF₆ has the highest C₁₁ value at 63.69 GPa. This means it resists uniaxial stress better than K₂InAgF₆ (60.26 GPa) and Rb₂InAgF₆ (50.65 GPa). The higher resistance of Na₂InAgF₆ comes from the shorter and stronger In–F bonds, which give improved strength along the axis. For C₁₂, Na₂InAgF₆ has the largest value of 26.06 GPa, while Rb₂InAgF₆ shows 23.80 GPa, and K₂InAgF₆ has 23.50 GPa. Na₂InAgF₆ exhibits more resistance to uniform multidirectional stress compared to the other two compounds. For shear resistance, the C₄₄ values show that Rb₂InAgF₆ (10.72 GPa) has the strongest one. Na₂InAgF₆ and K₂InAgF₆ have the C₄₄ values of 6.22 GPa and 7.07 GPa, respectively. As a conclusion, Rb₂InAgF₆ resists shape changes better than the others. Therefore, Na₂InAgF₆ has the highest resistance to compression, but Rb₂InAgF₆ is stronger against shear deformations. These compounds show moderate stiffness and elastic anisotropy, which makes them suitable for mechanical and optoelectronic applications.⁵⁸ Cauchy pressure (C_p) is an indicator of whether a material is ductile or brittle. Negative values signify brittleness, while positive values suggest ductility.⁵⁹ The parameter known as crystal stiffness (C_s) measures the overall rigidity of the crystal structure.⁶⁰

A material's ability to resist changes in shape reflects its mechanical strength, as these characteristics are directly linked to its level of robustness. The Kleinman parameter (ζ) describes how easily the internal coordinates of a crystal can be modified when subjected to an applied strain. The formulas for determining Cauchy pressure, crystal stiffness, and the Kleinman parameter are as follows:

C_p = C₁₂ − C₄₄ (8)

The C_p of Na₂InAgF₆ is 19.83 GPa, which is the highest among the compounds. This shows that it is the most ductile, compared to K₂InAgF₆ (16.43 GPa) and Rb₂InAgF₆ (13.08 GPa). The higher ductility of Na₂InAgF₆ can be explained by the stronger bonding created by the smaller Na⁺ ion, which improves structural integrity and resistance to brittle failure. The crystal stiffness (C_s) values are 18.82 GPa for Na₂InAgF₆, 18.38 GPa for K₂InAgF₆, and 13.42 GPa for Rb₂InAgF₆. Na₂InAgF₆ shows the highest crystal stiffness parameter (C_s), indicating stronger resistance to lattice deformation at the atomic scale. The reason is that the Na–F bonds are shorter and stronger than K–F and Rb–F bonds, as the ionic radius of Na⁺ is smaller, leading to stronger electrostatic attraction. For the Kleinman parameter (ζ), Rb₂InAgF₆ has the highest value at 0.60. Na₂InAgF₆ has the value at 0.55, and K₂InAgF₆ has the lowest value at 0.53. This means Rb₂InAgF₆ is the most flexible among the three compounds. The larger size of Rb⁺ weakens the bonding strength, which makes the lattice more adaptable to internal distortions under stress. Na₂InAgF₆ offers the best balance between ductility and thermal stability. Rb₂InAgF₆ exhibits higher stiffness and lower anisotropy, making each compound suitable for different mechanical applications. However, its lower bulk modulus and melting temperature suggest reduced resistance to volumetric compression and lower thermal stability compared to Na₂InAgF_6. Material's resistance to volumetric deformation under applied pressure is measured by its bulk modulus (B); higher B values indicate more compression resistance. Shear modulus (G) represents the ability to tolerate shape changes and shear deformation, as well as its stiffness against forces that cause angular distortion. Young's modulus (E) measures the stiffness under uniaxial stress. Table 5 shows the evaluated mechanical properties of Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆, and is illustrated in Fig. 3(b). These properties include their strength, elasticity, ductility, and thermal stability. These values were determined using the relationships described below:^61,62

Table 5 Computed mechanical parameters including bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (ν), Pugh's ratio (B/G), anisotropy factor (A), and melting temperature (T_m)

Compounds	B (GPa)	G (GPa)	E (GPa)	ν	B/G	A	Tm (K)
Na2InAgF6	38.60	9.88	27.31	0.38	3.91	0.33	930
K2InAgF6	35.75	10.48	28.65	0.37	3.41	0.38	910
Rb2InAgF6	32.75	11.73	31.44	0.34	2.79	0.80	853
K2InAgCl6 (ref. 48)	30.02	8.55	23.43	0.37	3.51	0.68	—
K2InAgBr6 (ref. 48)	25.72	12.03	31.23	0.29	2.14	0.60	—
K2InAgI6 (ref. 48)	20.12	11.14	28.21	0.27	1.81	0.48	—

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The bulk modulus demonstrates the maximum value in Na₂InAgF₆ (38.60 GPa), which decreases to 35.75 GPa for K₂InAgF₆ and subsequently to the lowest value observed in Rb₂InAgF₆ (32.75 GPa). The highest value of bulk modulus for Na₂InAgF₆ indicates that it is the most incompressible and mechanically stable material, which reflects stronger atomic bonding compared to the other two compounds. The shear modulus shows its highest value in Rb₂InAgF₆ (11.73 GPa), which is greater than K₂InAgF₆ (10.48 GPa) and Na₂InAgF₆ (9.88 GPa). This characteristic suggests that Rb₂InAgF₆ is the most resistant to shear forces and shape distortion, whereas Na₂InAgF₆ is the least resistant compound. Young's modulus, which measures stiffness, is also highest in Rb₂InAgF₆ (31.44 GPa), which shows a decrease in K₂InAgF₆ (28.65 GPa) and Na₂InAgF₆ (27.31 GPa). A higher value of Young's modulus indicates greater stiffness and resistance to elastic deformation. Therefore, K₂InAgF₆ shows moderate flexibility with balanced stiffness and ductility.⁶³ The Poisson ratio, v is the dimensional alteration of a material in directions perpendicular to stress application.⁶⁴ The B/G ratio demonstrates the ductility or brittleness of a material, with greater values of over 1.75 indicating ductility.⁶⁵ Mechanical anisotropy is present to a larger degree in materials exhibiting A factors that are not 1 in all directions.⁶⁶ The material's melting point in Kelvin indicates its thermal stability as given by T_m.⁶⁷ The following formulas are used to calculate these four parameters:

T_m = (554 + 5.911C₁₁) ± 300 K (16)

Material's deformation under stress is quantified by Poisson's ratio (ϑ). Among all three studied compounds, Na₂InAgF₆ has the highest Poisson ratio of 0.38, followed by K₂InAgF₆ at 0.37 and Rb₂InAgF₆ at 0.34. Na₂InAgF₆ has a high v value, indicating a large lateral expansion, which signifies better ductility compared to the rest of the compounds. The ductile versus brittle behaviour of materials is primarily determined by Pugh's ratio (B/G). Na₂InAgF₆ demonstrates the greatest ductility with a B/G value of 3.91, closely followed by K₂InAgF₆ at 3.41, whereas Rb₂InAgF₆ is the least ductile with 2.79. All compounds exhibit ductile behavior, as their B/G ratios exceed the critical value of 1.75 with different values as illustrated in Fig. 3(c).

Mechanical anisotropy is measured by the anisotropy factor (A), where 1 is ideal isotropy. Rb₂InAgF₆ (0.80) is the least anisotropic, making it the most mechanically uniform of the three compounds. Ahmad et al. reported similar elastic and mechanical properties for K₂InAgCl₆, K₂InAgBr₆, and K₂InAgI₆, which are in close agreement with our calculated values, thereby confirming the reliability of the present results.⁴⁸ The thermal stability is measured using the melting temperature (T_m). The highest T_m is exhibited by Na₂InAgF₆ at 930 K, followed by K₂InAgF₆ at 910 K, and Rb₂InAgF₆ at 853 K. The excellent thermal resistance of Na₂InAgF₆ makes it the most appropriate compound for applications at high temperatures, whereas Rb₂InAgF₆ is the least thermally stable. Overall, Na₂InAgF₆ demonstrated the best combination of ductility and thermal stability, while Rb₂InAgF₆ represented the lowest elastic anisotropy, indicating more isotropic mechanical behaviour. The elastic local anisotropy tool, ELATE⁴⁵ was used to create the three-dimensional visualizations of anisotropic Young's modulus (E), shear modulus (G), and Poisson ratio (v) visualization of A₂InAgX₆ (A = Na, K, and Rb) shown in the Fig. 4.

Fig. 4 The physical properties of the elastic parameters of A₂InAgF₆ (A = Na, K, and Rb) (a) Young's modulus, (b) shear modulus, and (c) Poisson's ratio are described in the 3D illustration.

3.3 Electronic properties

The band structure is a crucial element in the evaluation of materials' electronic properties. Band structures, including conductor, semiconductor, and insulator forms, can clarify the electrical properties of a material. The bandgap is the energy variation between the valence band maximum (VBM) and conduction band minimum (CBM) in a material's electronic band structure.⁶⁸ This study analyses the band structures depicted in Fig. 5, calculated using Generalized Gradient Approximation (GGA-PBE) both with and without SOC, as well as with the HSE06 functionals for three halide double perovskites, A₂InAgF₆ (A = Na, K, and Rb). The Band Structure diagrams of A₂InAgF₆ (A = Na, K, and Rb) demonstrate that the VBM and CBM align exactly at the highest symmetry k-point, known as the G point, within the first Brillouin zone. Within GGA-PBE, the calculated direct bandgaps are 0.98 eV, 1.52 eV, and 1.70 eV for Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆, respectively. The impact of relativistic spin orbit coupling (SOC) on the electronic structure is depicted in the top panels of Fig. 5. The inclusion of SOC breaks the degeneracy of several electronic states, with a noticeable effect in the valence band near the M point for A₂InAgF₆ (A = Na, K, and Rb). Furthermore, SOC causes a significant shift in the conduction band, lowering its energy, which leads to a considerable narrowing of the bandgap. As a result, the bandgaps calculated with SOC are consistently smaller than those predicted without SOC. This reduction becomes more pronounced with the substitution of heavier A-site atoms, reflecting the stronger relativistic effects associated with higher atomic numbers. Specifically, the bandgap of Na₂InAgF₆ decreases from 0.98 eV to 0.59 eV with SOC inclusion, while for K₂InAgF₆, it is significantly reduced from 1.52 eV to 1.18 eV, and for Rb₂InAgF₆, it decreases from 1.70 eV to 1.42 eV. The substantial reduction in the bandgap observed in A₂InAgF₆ (A = Na, K, and Rb) compounds underscores the importance of including SOC for accurately describing the electronic properties of halide perovskites containing heavier elements like In and Ag, as SOC can notably modify the band dispersion and reduce the bandgap.

Fig. 5 Calculated band structure of (a) Na₂InAgF₆, (b) K₂InAgF₆, (c) Rb₂InAgF₆ using GGA-PBE with and without spin–orbit coupling (SOC) and calculated band structure using Hybrid-HSE06 functional.

As expected, the HSE06 functional yields larger and more accurate bandgaps of 3.02 eV, 3.63 eV, and 3.88 eV for Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆, respectively as shown in Table 6. The atomic radius increases in the order: Na < K < Rb. Due to differences in atomic shapes and sizes, the bandgap varies, rising from sodium to potassium and rubidium to regulate the total repulsion among the electronic states.⁶⁹ Direct bandgap double halide perovskites provide substantial benefits for optoelectronic applications owing to their effective light absorption and emission characteristics. These attributes render them exceptionally appealing for next-generation solar cells, LEDs, and various photonic gadgets.⁷⁰

Table 6 The computed bandgap and effective mass of the carriers of A₂InAgF₆ (A = Na, K, and Rb) compounds

Compounds	Bandgap (eV)				Effective mass
	GGA-PBE		Previous study (mbj-PBE sol)	HSE06	HSE06
	Without SOC	With SOC
Na2InAgF6	0.98	0.59	—	3.02	0.10	0.82
K2InAgF6	1.52	1.18	—	3.63	0.10	0.86
Rb2InAgF6	1.70	1.42	—	3.88	0.10	0.94
K2InAgCl6 (ref. 48)	—	—	2.48	—	—	—
K2InAgBr6 (ref. 48)	—	—	1.47	—	—	—
K2InAgI6 (ref. 48)	—	—	0.23	—	—	—

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The Density of States (DOS) diagrams offer critical insights into the electronic structures of the A₂InAgF₆ (A = Na, K, and Rb) compounds by depicting both Total Density of States (TDOS) and Partial Density of States (PDOS) across a wide energy spectrum. Fig. 6 illustrates the DOS profiles for the three variants, spanning energies from −10 eV to +10 eV, with the Fermi level (E_f) positioned at 0 eV, separating the occupied and unoccupied states. A detailed observation reveals characteristic features in both the valence and conduction bands, helping determine the bonding nature and electronic conductivity of each composition. In the lower valence region, strong peaks appear. These peaks come from the hybridization between F-2p and Ag-4d orbitals. This shows the presence of strong covalent bonding. The p-states of the A-site cations (Na, K, Rb) appear only weakly. They stay localized in shallow energy ranges. This means they have only a small role in electronic interactions. The conduction band begins just above 0 eV and goes up to +10 eV. The conduction band mainly consists of contributions from In-5s, Ag-5s, and F-2p orbitals. There are clear differences in the start of conduction for the three compounds. From the DOS curves, the bandgaps are estimated. Rb₂InAgF₆ has the widest bandgap of 3.88 eV. This means it has very few electronic states near the Fermi level and is less conductive. K₂InAgF₆ shows a moderate bandgap of 3.63 eV. Na₂InAgF₆ has the smallest bandgap of 3.02 eV. This shows it has the highest density of conduction states near E_F. The bandgap becomes smaller as the A-site cation decreases in size from Rb⁺ to Na⁺. The reduction happens because smaller cations cause more structural distortion and stronger orbital overlap between Ag-F and In-F. This makes electronic excitation easier. Based on these bandgaps, the expected order of conductivity is: Rb₂InAgF₆ < K₂InAgF₆ < Na₂InAgF₆. Na₂InAgF₆ is therefore expected to exhibit the highest intrinsic conductivity. However, the relatively sharp density of states features near the conduction band edge in Na₂InAgF₆ indicate favorable charge transport characteristics, making it a promising candidate for optoelectronic device applications requiring balanced transport properties. On the other hand, Rb₂InAgF₆ has a wide bandgap and low conduction. It may be more useful in applications that need high resistivity or insulating properties. The PDOS confirms the active roles of Ag-d, In-p, and F-p orbitals in shaping the electronic structure of these compounds.

Fig. 6 The Total Density of States (TDOS) and Partial Density of States (PDOS) of the compounds (a) Na₂InAgF₆, (b) K₂InAgF₆, and (c) Rb₂InAgF₆.

The effective mass (m*) of charge carriers in the A₂InAgF₆ (A = Na, K, Rb) series was evaluated to determine the electronic transport properties, as it is inversely proportional to the curvature of the energy bands according to the relation:

where E(k) and ℏ are band energy as a function of the wave vector k and reduced Planck constant, respectively. The HSE06 calculations reveal a consistent electron effective mass of 0.10 across all three compounds, indicating a highly dispersive conduction band minimum (CBM) that favors high n-type mobility. Conversely, the hole effective masses are significantly higher, ranging from 0.82 for Na₂InAgF₆ to 0.94 for Rb₂InAgF₆, suggesting a flatter valence band maximum (VBM) and potentially restricted p-type transport. This pronounced asymmetry between and implies that while these halides are excellent candidates for electron conduction, the relatively heavy holes may limit overall bipolar charge extraction efficiency in optoelectronic applications. Furthermore, the marginal increase in from Na to Rb correlates with the widening of the bandgap from 3.02 eV to 3.88 eV, reflecting the influence of the A-site cation on the electronic structure.

The Electron Localization Function (ELF) analysis quantifies the charge distribution within a given region of space. The ELF analysis shown in Fig. 7 along the (100) and (200) crystallographic planes provide valuable insights into the physical properties and bonding characteristics of A₂InAgF₆ (A = Na, K, and Rb) double perovskites. The charge density map on the right side of Fig. 7 uses a color scale where red areas represent higher electron density, reaching up to 0.4713 e Å⁻³, and blue areas indicate lower electron density approaching 0 e Å⁻³. The presence of overlapping elliptical contours between In and F atoms on the (100) plane confirms covalent bonding. In contrast, spherical contours around Ag and F atoms on the (200) plane, with no atomic overlap, suggest that ionic bonding predominates.

Fig. 7 Electron localization function (ELF) analysis along the (100) and (200) crystallographic plane of (a) Na₂InAgF₆, (b) K₂InAgF₆, and (c) Rb₂InAgF₆.

3.4 Optical properties

The optical characteristics prescribe how the materials interact with electromagnetic (EM) radiation, which includes absorption, reflection, transmission, and optical conductivity. An overall knowledge of these properties has been vital in the design and optimization of materials in the optoelectronic structures, such as solar cells, light emitting diodes (LEDs), and photodetectors.⁷¹ The optical parameters of A₂InAgF₆ (A = Na, K, and Rb) have been determined at photon energies of 30 eV, revealing the regulation of the material's optical characteristics.

A convergence test with respect to the k-point mesh was performed to ensure the accuracy and reliability of the optical spectra. The optical properties of Na₂InAgF₆ were calculated using k-point meshes of 4 × 4 × 4, 6 × 6 × 6, 8 × 8 × 8, and 10 × 10 × 10. The results showed that the optical spectra for the 8 × 8 × 8 and 10 × 10 × 10 k-point meshes were nearly identical, as shown in Fig. 8, indicating that increasing the k-point mesh beyond 8 × 8 × 8 does not significantly affect the calculated optical properties. That's why this k-point mesh was subsequently used for the calculation of the optical properties for all other A₂InAgF₆ (A = Na, K, and Rb) compounds.

Fig. 8 The convergence test of k-point mesh for the calculation of (a) real part, (b) imaginary part of the dielectric function, (c) absorption, and (d) reflectivity of Na₂InAgF₆ calculated using k-point meshes of 4 × 4 × 4, 6 × 6 × 6, 8 × 8 × 8, and 10 × 10 × 10.

3.4. 1 Dielectric functions. The dielectric function is an indispensable quantity for analyzing the optical properties, which are related to absorption characteristics, as indicated by the following equation for materials:

ε(ω) = ε₁(ω) + iε₂(ω) (18)

Here, ε₁(ω) represents the dielectric constant (real part), while ε₂(ω) denotes the imaginary part. The dielectric function physically represents the frequency-dependent relative permittivity of the material. The material's ability to polarize and reflect incident light is governed by ε₁(ω), while optical absorption is primarily described by ε₁(ω), which also influences device performance. The capacity of a material to dissipate energy is dictated by its ε₂(ω), which also confirms its appropriateness for a certain application within a defined energy spectrum.⁷² The real dielectric function ε₁(ω) and imaginary dielectric function ε₂(ω) defined as:Here, P denotes the momentum matrix element between the electronic states of bands α and β under the crystal momentum k.

The comparatively larger ε₁(0) values are associated with the materials' narrower bandgaps, which are useful for minimizing carrier recombination rates and improving optoelectronic device efficacy.⁷³ Fig. 9(a) indicates the dielectric function of A₂InAgF₆ (A = Na, K, and Rb) compounds. The real component of the static dielectric function ε₁(0), for the evaluated double perovskites, is summarized as follows: ε₁(0) = 1.55 for Na₂InAgF₆, ε₁(0) = 1.64 for K₂InAgF₆, and ε₁(0) = 1.74 for Rb₂InAgF₆. As the photon energy increases, the real part ε₁(ω) initially rises to a maximum in the low-energy region (∼5–8 eV) and then gradually decreases, eventually attaining negative values at higher energies. This behavior indicates a transition from dielectric to metallic-like response due to strong electronic excitations. Among the studied compounds, Rb₂InAgF₆ exhibits the most intense peak, indicating enhanced optical polarization, while K₂InAgF₆ and Na₂InAgF₆ show comparatively lower and broader peaks, reflecting reduced polarization strength.

Fig. 9 The visualization of the A₂InAgF₆ (A = Na, K, and Rb) compounds' (a) real part, (b) imaginary part of the dielectric function, (c) reflectivity, and (d) loss function.

The imaginary part of the dielectric function, ε₂(ω), provides important information about interband electronic transitions between the valence and conduction bands. From Fig. 9(b), ε₂(ω), remains nearly zero up to the bandgap energy for all compounds, indicating the absence of optical transitions in this region. As the photon energy exceeds the bandgap, ε₂(ω), increases sharply and exhibits multiple pronounced peaks, corresponding to strong optical transitions. Rb₂InAgF₆ shows the highest peak intensity (∼2.67 near ∼20.6 eV), followed by K₂InAgF₆ and Na₂InAgF₆. These variations arise from differences in their electronic band structures and transition probabilities. Overall, the dielectric response confirms strong optical activity in the UV region, highlighting their potential for optoelectronic applications.

3.4. 2 Reflectivity. Reflectivity in density functional theory (DFT) refers to the portion of incident light reflected by a substance's surface, based upon the material's electronic structure. This property can be determined using the following equation:

Fig. 9(c) presents an analysis of the reflectance spectra for A₂InAgF₆ (A = Na, K, and Rb) double halide perovskite compounds. The obtained static coefficients of reflection R (0) are 0.011, 0.015, and 0.018 for A₂InAgF₆ (A = Na, K, and Rb) compounds, respectively. With increasing photon energy, R(ω) shows peaks due to interband transitions. Rb₂InAgF₆ has the highest reflectivity (∼0.212 at ∼20.6 eV), K₂InAgF₆ shows moderate peaks (∼0.101 at ∼28.9 eV), while Na₂InAgF₆ remains lower (0.049). Low visible-region reflectivity and moderate UV values suggest efficient absorption, making these materials suitable for optoelectronic applications.

3.4. 3 Loss function. The loss function represents fundamental insights into plasmonic events and dielectric responses by specifying the mechanisms via which electrons dissipate energy during interactions with external electromagnetic fields.⁷⁴ The computation is done according to the equation as shown below:

Fig. 9(d) shows the energy loss function L(ω), representing electron energy dissipation. All compounds exhibit low loss at low photon energies and prominent peaks at higher energies. The main peaks appear around ∼24.7 eV for Rb₂InAgF₆, ∼24.5 eV for Na₂InAgF₆, and ∼29 eV for K₂InAgF₆, indicating plasma resonance. Rb₂InAgF₆ shows the highest intensity (∼2.07), followed by K₂InAgF₆ (∼1.86), while Na₂InAgF₆ has lower values (∼0.85). These peaks correlate with optical conductivity and reflect collective electronic excitations. The low loss in the visible region and strong high-energy peaks suggest reduced energy dissipation, highlighting their potential for optoelectronic and photovoltaic applications.

3.4. 4 Refractive index. The index refraction of a substance is a significant characteristic for assessing photon absorption during the chemical degradation process in mixtures. A high refractive index is correlated with a denser medium, as documented in an earlier study.⁷⁵ Complex refractive index (n + ik) is an important characteristic of a solid material, giving information about the velocity with which light travels, and has prospective uses in optoelectronics.⁷⁶ The calculations follow the equation stated below:

Applying eqn (17), the actual components of the complex refractive index were obtained. The value of n(ω) fluctuates based on the material, although semiconductors often have a low k(ω). The refractive index n(ω) of A₂InAgF₆ (A = Na, K, Rb) shown in Fig. 10(a) follows a trend similar to the real part of the dielectric function, ε₁(ω). At zero photon energy, the static refractive indices are about 1.25, 1.28, and 1.32 for Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆, respectively. With increasing photon energy, n(ω) shows small peaks in the low-energy region (∼6–10 eV), reaching ∼1.4–1.45, and then gradually decreases, becoming less than unity at higher energies due to reduced polarization and strong dispersion. This behavior indicates normal dispersion at low energies and anomalous dispersion at higher energies. The moderate refractive index values suggest good transparency and confirm the dielectric nature of these compounds, making them suitable for optoelectronic applications.

Fig. 10 The visualization of the A₂InAgF₆ (A = Na, K, and Rb) compounds' (a) refractive index, and (b) extinction coefficient.

3.4. 5 Extinction coefficient. The extinction coefficient, k(ω), can be defined to measure the loss of electromagnetic radiation through material expression. As can be seen in Fig. 10(b), the calculated k(ω) of A₂InAgF₆ (A = Na, K, and Rb) has a trend that is similar to that of the imaginary component of the dielectric function, ε₂(ω). This correspondence spells the direct correlation between optical absorption with k(ω), seeing that ε₂(ω) is the capacity of the material to be able to absorb photon energy.

A₂InAgF₆ exhibits the highest peak (∼1.1 at ∼20–21 eV), followed by K₂InAgF₆ (∼0.6 at ∼25–26 eV), and Na₂InAgF₆ shows lower values (∼0.4). Multiple peaks in the 18–26 eV range indicate various interband transitions in Fig. 10(b). Overall, strong UV response and low visible absorption highlight their suitability for ultraviolet optoelectronic applications.

3.4. 6 Conductivity. Optical conductivity σ(ω) is a crucial parameter that indicates the response of charge carriers at optical frequencies. The optical conductivity of a material is intrinsically linked to its electronic band structure, as it originates from interband electronic transitions. Optical conductivity arising from interband electronic transitions within the visible to ultraviolet energy range is essential for optoelectronic applications.⁷⁷ Substitution of alkali metals exerts only a minor influence on the optical conductivity within the visible spectrum. The conductivity was obtained using the following equation:

Fig. 11(a) and (b) show the real and imaginary parts of the optical conductivity, σ₁(ω) and σ₂(ω), of A₂InAgF₆ (A = Na, K, and Rb). The real part σ₁(ω) indicates charge transport due to photon absorption. All compounds exhibit negligible conductivity at low energies (visible region), confirming insulating behavior, while conductivity increases sharply in the UV region due to enhanced interband transitions. Rb₂InAgF₆ shows the highest peak (∼6.66 (fs⁻¹) at ∼20.6 eV), followed by K₂InAgF₆ (∼3.63 (fs⁻¹) at ∼25.5 eV), whereas Na₂InAgF₆ exhibits lower (∼1.7 (fs⁻¹) at ∼12.7 eV) and broader peaks. The imaginary part σ₂ (ω) reflects energy storage and dissipation. It remains negative at low energies, indicating weak optical response, and becomes positive at higher energies with multiple oscillations, especially in the 18–28 eV range. Rb₂InAgF₆ again shows the highest intensity (∼2.5–2.7 (fs⁻¹)), followed by K₂InAgF₆ and Na₂InAgF₆. Overall, the absence of conductivity in the visible region and strong response in the UV region confirm their wide bandgap nature and suitability for ultraviolet optoelectronic applications.

Fig. 11 The visualization of the A₂InAgF₆ (A = Na, K, and Rb) compounds' (a) real part, (b) imaginary part of conductivity, and (c) absorption.

3.4. 7 Absorption coefficient. The effectiveness of optoelectronic and photovoltaic devices is significantly determined by their capacity to absorb sunlight, as the creation and transportation of charge carriers are directly correlated with the absorption coefficient.⁷⁸ The absorption coefficient α(ω) is essential for describing the mechanisms by which materials absorb photons and generate pairs of electrons and holes, which is vital for the operation of photovoltaic devices such as solar cells. An increased absorption coefficient results in enhanced light absorption and a more significant decrease in the intensity of light passing. This eventually boosts the device's efficiency. This is calculated by the equation:

The absorption coefficient α(ω) of A₂InAgF₆ (A = Na, K, Rb) in Fig. 11(c) shows clear dependence on the A-site cation, with slight shifts in the absorption edge indicating variation in optical bandgaps. Na₂InAgF₆ exhibits the earliest onset, followed by K₂InAgF₆ and Rb₂InAgF₆. All compounds display weak absorption at low energies and strong absorption in the ultraviolet (UV) region. Rb₂InAgF₆ shows the highest peak (∼3.61 × 10⁵ cm⁻¹ at ∼21–22 eV), followed by K₂InAgF₆ (∼2.45 × 10⁵ cm⁻¹ at ∼26 eV), while Na₂InAgF₆ has the lowest intensity (∼1.1 × 10⁵ cm⁻¹). Multiple peaks in the 18–26 eV range indicate various interband electronic transitions, arising from differences in their band structures. Negligible absorption in the visible region suggests transparency to visible light. The overall behavior of α(ω) follows ε₂(ω), with minor deviations due to the independent-particle approximation. The strong UV absorption highlights their potential for ultraviolet optoelectronic applications.

A comparable analysis of dielectric response, absorption, and optical behavior has been reported by Shah et al.⁷⁹ for halide double perovskites, highlighting the strong dependence of optical properties on compositional variation. The A₂InAgF₆ (A = Na, K, and Rb) compounds exhibit a systematic enhancement in optical properties with increasing A-site cation size, where Rb₂InAgF₆ demonstrates the strongest dielectric response, absorption, and optical conductivity, indicating superior ultraviolet optoelectronic performance. Substituting the halide with hydrogen may enable these compounds to function as hydrogen storage materials.⁸⁰ Beside double perovskite⁸⁰ A₂B₇- and AB₂-type materials also used for hydrogen storage (Table 7).^81,82

Table 7 Key optical properties of A₂InAgF₆ (A = Na, K, and Rb)

Optical property	Na2InAgF6	K2InAgF6	Rb2InAgF6
Static dielectric constant, ε1(0)	1.55	1.64	1.74
Maximum imaginary dielectric function, ε2(ω)	∼2.1	∼2.4	∼2.67
Static reflectivity, R(0)	0.011	0.015	0.018
Maximum reflectivity, R(ω)	0.049	0.101	0.212
Static refractive index, n(0)	1.25	1.28	1.32
Maximum extinction coefficient, k(ω)	0.4	0.6	1.1
Maximum absorption coefficient, α (cm^−1)	1.1 × 10^5	2.45 × 10^5	3.61 × 10^5
Maximum optical conductivity, σ1(ω)	1.7	3.63	6.66

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3.5 Phonon dispersion and thermodynamic properties

The dynamic stability of these double perovskites is analyzed by their phonon dispersion spectra. The calculated phonon spectra of A₂InAgF₆ (A = Na, K, and Rb) along the high-symmetry paths (X–R–M–G–R) in the Brillouin zone are shown in Fig. S1. The presence of imaginary frequencies in a material's phonon dispersion is a key indicator of dynamic instability at 0 K. Conversely, a lattice is considered dynamically stable if all its phonon frequencies are real and positive. As there are imaginary frequencies of Na₂InAgF₆, K₂InAgF₆ and Rb₂InAgF₆ phonon spectra, it shows the dynamic instability of them leading to significant structural changes at certain temperatures. Importantly, theoretical dynamical instabilities at 0 K do not necessarily rule out experimental feasibility. For example, the well-known perovskite BaTiO₃ exhibits imaginary phonon modes in its cubic phase at 0 K, yet it becomes dynamically stable at finite temperatures due to anharmonic lattice vibrations and thermal effects.⁸³ Similarly, compounds such as AlH₃,⁸⁴ superconducting Y₂C₃,⁸⁵ and high-pressure H₃S⁸⁶ have been predicted to display dynamical instabilities in theoretical studies, but have nonetheless been successfully synthesized and experimentally characterized. In the present work, all A₂InAgF₆ (A = Na, K, and Rb) compounds exhibit negative formation enthalpies and satisfy the Born mechanical stability criteria, indicating their intrinsic thermodynamic and mechanical stability. Therefore, the observed imaginary phonon modes may suggest a tendency toward lower-symmetry structures, which could be stabilized under finite-temperature conditions.

Thermodynamic properties are the measurable parameters of a system that describe its state and the surrounding environment. The thermodynamic properties of Na₂InAgF₆, K₂InAgF₆, and Rb₂InAgF₆ compounds are systematically analyzed to comprehend their vibrational and thermal behaviour. These properties, including enthalpy, entropy, free energy, heat capacity, and Debye temperature, are estimated using density functional perturbation theory across the temperature range of 0–1000 K.

The parameters are determined from the phonon density of states g(ω) according to the following relations:

where k_B is the Boltzmann constant. As shown in Fig. 12(a), the enthalpy increases monotonically with temperature for all studied compounds. This behaviour indicates the accumulation of vibrational energy with increasing thermal energy. Among the materials, Rb₂InAgF₆ shows the highest enthalpy at elevated temperatures, which may be attributed to its heavier cationic mass and greater unit-cell energy storage capability. As depicted in Fig. 12(a), the entropy increases with temperature, which is consistent with the increasing vibrational disorder and population of phonon modes at higher temperatures. Among the examined systems, Rb₂InAgF₆ shows a slightly higher entropy, reflecting enhanced vibrational degrees of freedom compared to Na₂InAgF₆ and K₂InAgF₆. The free energy, as illustrated in Fig. 12(b), exhibits a decreasing trend with temperature. This decrease results from the growing entropy contribution to the Helmholtz free energy, which dominates at higher temperatures and reduces the total free energy. The negative values throughout the temperature range confirm the thermodynamic stability of these compounds. Fig. 12(c) displays the heat capacity behaviour as a function of temperature. A rapid rise is observed initially, levelling off at higher temperatures, consistent with the Dulong–Petit law for classical solids. The slightly lower saturation heat capacity of Rb₂InAgF₆ may be associated with its distinctive lattice stiffness and phonon cut-off frequencies. In line with the third law of thermodynamics, all the calculated thermodynamic functions approach zero as the temperature approaches absolute zero, reflecting the disappearance of lattice vibrations in a perfectly ordered state. The Debye temperature has been measured to see if the compounds were good enough to be used in industrial equipment, as illustrated in Fig. 12(d). The Debye temperature (θ_D) is the highest temperature at which a crystal can vibrate in its normal way. It is a key factor in connecting a solid's elastic and thermodynamic properties, like its melting point, specific heat, and vibrational entropy. It was calculated by utilizing an equation that has already been published to find the average sound velocity by adding up the elastic wave velocities in the crystal.⁸⁷The symbols h and k, respectively, are used to denote Planck's and Boltzmann's constants. In addition, the symbols N_A, n, M, and ρ represent the numerical values of Avogadro's number, the number of atoms present in the unit cell, the molecular mass, and the density, respectively. When calculating the velocity of the sound (v_m), it is possible to utilize both the transverse (v_t) and longitudinal (v_l) sound velocities.

Fig. 12 The visualization of the A₂InAgF₆ (A = Na, K, and Rb) compounds' (a) enthalpy and entropy, (b) free energy, (c) heat capacity, and (d) Debye temperature.

Debye temperature increases in a linear manner as temperature increases. At elevated temperatures, it is evident that Na₂InAgF₆ exhibits the highest Debye temperature among the three compounds.

4 Conclusion

This research provides a comprehensive theoretical investigation into the influence of A-site cation substitution on the physical properties of fluoride-based double perovskites A₂InAgF₆ (A = Na, K, and Rb). All three compounds are structurally stable, with the Rb-based variant exhibiting the highest stability based on its highly negative formation energy. The lattice parameters and unit cell volumes expand predictably with the increasing ionic size of the A-site cation. The compounds are identified as semiconductors with direct bandgaps ranging from 3.02 to 3.88. The bandgap is tunable through A-site cation engineering, where smaller cations (Na) result in smaller gaps due to structural distortion and stronger orbital overlap. The electron localization function analysis indicated a mixed covalent-ionic bonding nature. The materials are mechanically stable and inherently ductile, standing out for their high resistance to compression and superior thermal stability, making them suitable for high-temperature applications. The optical conductivity and absorption spectra exhibit notable features extending beyond the visible region, indicating promising potential for applications in ultraviolet (UV) technologies. The negative Helmholtz free energy values across a wide temperature range (0–1000 K) confirm that these compositions remain thermodynamically stable under varying thermal conditions, indicating a promising candidate for high temperature applications. Overall, Rb₂InAgF₆ is identified as a promising candidate for charge transport devices due to its pronounced density of states features, while the entire family offers a non-toxic, efficient, and robust platform for the development of future clean energy technologies.

Ethical statement

This article does not contain any studies with human participants performed by any of the authors.

Author contributions

Sharmin Islam: investigation, methodology, data curation, writing – original draft. Shatha A. Aldaghfag, Ali El-Rayyes, Abdullah Al Mahmud, Samah Saidi, Omar Alsalmi, Mohd Taukeer Khan: formal analysis, reviewing and editing. Md Saiduzzaman: formal analysis, conceptualization, supervision, writing – original draft, reviewing and editing.

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.

Supplementary information (SI): Fig. S1: phonon dispersion spectra of A₂InAgF₆ (A = Na, K, and Rb) compounds. See DOI: https://doi.org/10.1039/d6ra01318j.

Acknowledgements

The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R81), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. The authors extend their appreciation to Northern Border University, Saudi Arabia, for supporting this work through project number (NBU-CRP-2026-2985). This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2026/R/1447). The researchers wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah (KSA) for the support provided to the Post-Publishing Program.

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