Comprehensive study of the physical and optoelectronic properties of A₂AgIrF₆ (A = Cs, Rb, and K) double perovskites for energy harvesting applications: a DFT approach

Abstract

In this study, we performed a comprehensive theoretical investigation of lead-free A₂AgIrF₆ (A = Cs, Rb, and K) double perovskites to examine their structural, electronic, optical, mechanical and thermodynamic properties using first-principles calculations. Known parameters, such as the Goldschmidt tolerance factor, octahedral factor and a novel tolerance factor (all indicating that these compounds are expected to form), were applied to validate the stability of the perovskite structure. Additional thermodynamic and dynamic stability checks, such as negative formation energies, positive phonon frequencies, and ab initio molecular dynamics (AIMD) simulations, strongly imply that these compounds can be synthesized experimentally. The band gaps derived from the Perdew–Burke–Ernzerhof Generalized Gradient Approximation (PBE-GGA) method for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆ are 1.07 eV, 1.13 eV and 1.16 eV, respectively. We employed the Tran–Blaha modified Becke–Johnson (TB-mBJ) functional to adjust the band gaps. It yielded electronic band gaps of 1.65 eV, 1.76 eV and 1.83 eV for A₂AgIrF₆ (A = Cs, Rb, and K, respectively). The calculated band gaps are indirect and within the appropriate range for solar cell applications, as evidenced by density of states (DOS) studies. Optical property characterization showed that the solar cells based on all three perovskites could efficiently absorb visible light and achieve a peak absorption coefficient greater than 10⁵ cm⁻¹, designating them as good absorbers. Mechanical behavior is determined through the analysis of elastic constants, and the compounds are found to be stable, ductile and anisotropic. In conclusion, A₂AgIrF₆ perovskites are highly promising, environmentally friendly candidates for use in optoelectronics and solar energy technologies. This theoretical research offers a crucial foundation for future experimental work and practical development.

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1. Introduction

The world is using increasingly more energy, yet most of it comes from fossil fuels, which are running out and are harmful to the environment.^1,2 Solar energy and thermoelectric conversion are used to successfully harness sunlight and thermal energy, respectively, which are essential sources of power.³ The selection of compounds plays a critical role in determining the performance of such photoelectric and thermal applications.⁴

Perovskite solar cells (PSCs) have been recently studied as a promising and low-cost next-generation solar cell technology, which can compete with or even replace silicon-based technologies.⁵ PSCs were initially touted to have a moderate efficiency of up to 3.8% PCE, but research over the last decade has resulted in a record efficiency of 22.7%.^6,7 This development eclipses even the best-performing dye-sensitized solar cells (DSSCs), which took a little over 20 years to achieve an efficiency of 11.9%. The high performance of PSCs can be attributed to the unique features of the perovskite material, such as a superior charge carrier mobility, long diffusion length, tunable direct band gap, high absorption coefficient, low recombination rate and high molar extinction coefficient.⁸ PSCs are third-generation solar cells that are cheap and efficient.^9–11 Furthermore, many studies have been dedicated to the photonic potential of the synthetic monolayer graphene, as other advanced technologies require its thermal stability.^12–14

The chemical formula for perovskite materials is typically ABX₃, where A is a large cation, B is a smaller metal cation, and X represents an anion, typically a halogen or oxygen.¹⁵ In perovskite structures, the A-site cation typically has a charge of +1, +2, or +3. The B-site cation usually has a charge of +2, +3, +4, or +5. The X-site anion generally carries a charge of −1 in halides and −2 in oxides. Lead-free double perovskites, on the other hand, have the structure A₂B′B″X₆, where A is a larger alkali metal cation (e.g., Cs⁺, Rb⁺, and K⁺), B′ is a monovalent cation such as Ag⁺, B″ is a trivalent cation such as Ir³⁺, and X is a halide ion. These halide-based double perovskites have the potential to serve as earth-abundant alternatives for conventional lead-containing perovskites in renewable optoelectronic and photovoltaic systems. In the context of photovoltaics, the stability and optoelectronic properties of this blend are considered attractive.⁶ These materials have long carrier diffusion lengths, excellent thermal stability, and comparatively high absorption coefficients in the UV-vis range; in certain situations, the PCE can reach 25% or higher.^16,17 Lead-free double halide perovskites were identified as some more ecologically friendly options.

Double perovskites (DPs) are considered promising candidates for renewable energy because they hold the key to resolving challenges in global energy demand. Lead-free halide double perovskites have been proposed as promising and environmentally friendly alternatives to lead halide perovskites owing to the toxicity and structural instability of Pb. Pb-free halide double perovskites have recently attracted growing research interest for optoelectronic and photovoltaic applications because of their non-toxicity and improved stability.

Several experimental investigations have revealed the promising characteristics of halide double perovskites for PV applications. For instance, McClure et al.¹⁸ prepared Cs₂AgBiCl₆ and Cs₂AgBiBr₆ materials through solid-state and solution methods and revealed that halide double perovskite semiconductors are environmentally friendly substitutes for lead halide perovskites. Similarly, Wei et al.¹⁹ manifested the lead-free double perovskite Cs₂AgSbBr₆ as a potential photovoltaic absorber, with an indirect optical band gap of 1.64 eV (bulk crystals) and 1.89 eV (thin films), relevant for solar energy conversion. Zhang et al.²⁰ synthesized a new cost-effective double perovskite Cs₂NaBiI₆ using a facile hydrothermal process and found it to exhibit good stability and repeatability for fabricating devices, which satisfies the demands of photovoltaic applications. Mustafa et al.²¹ synthesized Cs₂AgAsX₆ materials with band gaps varying from 1.0 to 1.9 eV, indicating good visible light absorption and possible sunlight conversion.

Theoretical studies have further demonstrated that halide double perovskites are promising materials for photovoltaic and optoelectronic applications. Alotaibi et al.²² reported the mechanical, optical, thermoelectric and thermodynamic properties of Cs₂AgBiX₆ (X = Cl and Br), indicating its feasibility for renewable energy applications. Saeed et al.²³ studied the first-principles calculations of Cs₂AgCrX₆ (with X = Cl, Br, and I) for possible optoelectronic applications. Maryam et al.²⁴ calculated the band structures of Cs₂AgSbX₆ (X = Cl, Br, and I) using density functional theory (DFT) and concluded that it was a promising Pb-free semiconductor for solar cells. Anbarasan et al.²⁵ studied Cs₂AgInX₆ (X = Cl, Br, and I) and identified its applicability as a single-junction solar cell from its corresponding physical and photovoltaic attributes. Nazir et al.²⁶ further investigated the optical, electronic and thermoelectric profiles of Rb₂AgAsX₆ (X = Cl, Br, and I), affirming its use for optoelectronic purposes. Ali et al.²⁷ suggested that new double perovskites K₂AgAsX₆ (X = Cl and Br) can be considered for different energy-related technologies. Similarly, Zanib et al.²⁸ proposed K₂AgAsX₆ (X = Cl and Br) with strong visible light absorption properties as possible solar cell and thermoelectric materials. The structural and optoelectronic properties, and the band gap of Rb₂AgBiI₆ in the visible region of the electromagnetic spectrum, were later established by Bhorde et al.²⁹ In addition, several first-principles studies on lead-free sodium-based double perovskites have demonstrated their structural, optoelectronic, and thermoelectric potential for eco-friendly solar and thermoelectric applications.^30–35

Motivated by the above findings, we employed density functional theory (DFT) to investigate a new series of double perovskites with the general formula A₂AgIrF₆ (A = Cs, Rb, and K). Previous studies have underscored the potential of halide double perovskites in environmentally friendly energy technologies. In this work, we explore the structural, electronic, optical, mechanical, and thermodynamic properties of A₂AgIrF₆ (A = Cs, Rb, and K) compounds using the WIEN2k computational package. As far as we know, no experimental/theoretical work has been reported for these compounds. The calculation results show that A₂AgIrF₆ (A = Cs, Rb, and K) double perovskites are promising candidates for photovoltaic and energy conversion devices owing to their tunable band gap. The results of this work are anticipated to be helpful in the continuous improvement of lead-free double perovskites and for future experimental/theoretical research of these materials in terms of energy applications.

2. Computational methods

Using DFT-based simulations with the Wien2k code, the structural, electronic, thermal, optical, and mechanical properties of the A₂AgIrF₆ DHP compound are analyzed. To solve the Kohn–Sham (KS) equations, the Full-Potential Linearized Augmented Plane Wave (FP-LAPW) method, along with local orbitals, is employed.³⁶ The structural properties are determined using the GGA-PBE exchange-correlation potential.³⁷ Since GGA-PBE underestimates the band gap, the TB-mBJ potential³⁸ is applied after GGA-PBE to correct this discrepancy.³⁹ For the wave functions inside muffin-tin spheres, the basis set comprises spherical harmonics of up to the angular momentum l_max = 10. In the interstitial region, a plane-wave basis set is used, with a cut-off value of R_MTk_max = 7. For Brillouin zone integration for polygons and naked PES, the Monkhorst–Pack k-mesh⁴⁰ consisting of 1000-k points is employed for the calculation of electronic and optical properties. In this work, the stabilization criterion of the energy up to 10⁻⁴ Ry and the charge up to 0.001e is used. During mechanical property calibration, we calculate the elastic constants. The Wien2k code utilizes two packages to analyze mechanical and optical properties: elastic and optic. These analyses provide insights into the mechanical stability and optical traits of DHP based on the electronic structure of the DHP compound, thereby comprehensively establishing its characteristics.

3. Results and discussion

3.1. Structural parameters and stability

The double halide perovskite (DHP) materials, which are of the form A₂AgIrF₆ (A = Cs, Rb, and K), adopt a face-centered cubic structure with space group Fm3̄m (no. 225).⁴¹ It contains [AgF₆] and [IrF₆] octahedra where the A-site cations (Cs, Rb, and K) reside in their interstices to hold the crystal lattice as a whole, as shown in Fig. 1. Specifically, the A-site cations occupy the 8c Wyckoff position (0.25, 0.25, 0.25), Ag is located at the 4a position (0.5, 0.5, 0.5), Ir is at the 4b position (0, 0, 0), and F is at the 24e position (0.25, 0, 0). Geometry optimization was carried out to evaluate the stability and structural properties of the materials. The lattice constants (a₀), bulk moduli (B₀), and total energies (E_tot) of each compound are tabulated in Table 1. The results show a continuous decrease in the lattice constant with increasing A-site cation from Cs to K, in good agreement with the well-established ionic-radii sequence (Cs > Rb > K). Importantly, the existing literature does not contain experimental evidence that directly shows the novelty of these findings.

Fig. 1 Unit cell of Cs₂AgIrF₆.

Table 1 Lattice constant a₀ (Å), bulk modulus B₀ (GPa), pressure derivative , total energy E_tot (Ry), formation energy E_f (eV per atom) and binding energy E_b (eV per atom) for double perovskite (DP) A₂AgIrF₆ (A = Cs, Rb, and K)

DP	a0 (Å)	B0 (GPa)		Etot (Ry)	Ef (eV per atom)	Eb (eV per atom)
Cs2AgIrF6	8.88	68.87	5.32	−78711.02	−3.77	−5.33
Rb2AgIrF6	8.75	72.80	4.44	−59475.81	−3.73	−5.33
K2AgIrF6	8.68	72.22	5.48	−49958.38	−3.69	−5.33

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The thermodynamic stability of A₂AgIrF₆ was confirmed by determining the formation energy (E_f) and binding energy (E_b) using the following equations:⁴²

and

where E_A2AgIrF6 represents the overall energy of the compound in the final state and E_A, E_Ag, E_Ir, and E_F are energies of the respective atoms. The negative energies of E_f and E_b calculated in this work (as presented in Table 1) can further attest to the thermodynamic stability of these compounds, which indicates their potential for synthesis.

Crystallographic stability was further validated by calculating the octahedral factor (u), Goldschmidt tolerance factor (t), and the new tolerance factor (τ) using the following equations:^43–45

and

where R_A, R_B, and R_F are the symbols for the ionic radii of ions A, B, and F, respectively. All the parameters listed in Table 2 have been evaluated and found to lie within the stability range,⁴⁶ which corroborates the crystallographic stability of the compounds.

A set of structural calculations for A₂AgIrF₆ was performed with varying unit cell volume and cubic symmetry to evaluate the stability and structural performance. To characterize the pressure–volume correlation, Birch–Murnaghan equation of state was applied to describe the pressure-volume-total energy relationship. The resulting volume optimization curves illustrated in Fig. 2 indicate the total energy relative to volume. The equilibrium volume and ground state energy of each compound corresponds to the bottom of those curves. The findings, as anticipated, indicate a relationship between the ionic radius of the A-site cation and the equilibrium volume, with a larger volume observed for Cs₂AgIrF₆ and a comparatively smaller volume for K₂AgIrF₆. This work highlights the prospect of the discussed computational method and provides a strong basis for further examinations of the structural and stability properties of such systems.

Fig. 2 Total energy vs. volume curves for the (a) Cs₂AgIrF₆, (b) Rb₂AgIrF₆, and (c) K₂AgIrF₆ compounds fitted using the third-order Birch–Murnaghan equation of state.

3.2. Phonon stability

Structural stability evaluation is essential for predicting the viability of new compounds. Thus, we systematically studied the vibrational properties of the A₂AgIrF₆ compounds. The phonon dispersion curves (PDC) and PDOS of A₂AgIrF₆ (A = Cs, Rb, and K) were calculated using the finite displacement method (FDM)⁴⁷ with the CASTEP package.⁴⁸ Structural relaxation was performed over a 2 × 2 × 2 supercell, with an atomic displacement of 0.01 Å. The PDCs and PDOS are displayed in Fig. 3, obtained at high-symmetry point (W–L–Γ–X–W–K) parameters, and exhibited no negative phonon frequencies, indicating that all the compounds are stable under natural conditions. Since the unit cell of these compounds contains 40 atoms, the vibrational spectrum includes 120 branches. Three of these are acoustic modes, which describe the collective movements of the crystal lattice, and the remaining 117 are higher-energy optical modes. The data in Fig. 3 do not exhibit such soft modes, supporting the fact that A₂AgIrF₆ structures are dynamically stable and can be synthesized. Phonon DOS analysis provides the vibrational contributions of each atom. Cs₂AgIrF₆ and Rb₂AgIrF₆ show PDC curves in the 0–16 THz range, while K₂AgIrF₆ reaches up to 18 THz. Low-frequency modes are mainly contributed by the A-site cations (Cs, Rb, and K), and atoms with high frequency are dominated by Ir and F atoms.

Fig. 3 PDCs with phonon DOS of the (a) Cs₂AgIrF₆, (b) Rb₂AgIrF₆, and (c) K₂AgIrF₆ compounds.

In addition to vibrational stability, thermal stability is important. The thermal stability was verified from the ab initio molecular dynamics (AIMD) simulations carried out in an NVT ensemble at room temperature (300 K). Fig. 4 shows the evolution of the total energy over time for the three compounds under investigation. In the case of Cs₂AgIrF₆, the plot is shown in Fig. 4(a), and the energy is going down, ranging from −177.63 eV to −174.99 eV for all systems. The average energy (−176.54 eV) and percentage error from the highest to lowest (0.29% and 0.41%) show the strong thermal stability of the compound. The energy response of Rb₂AgIrF₆ is shown in Fig. 4(b), with changes in a narrow range between −196.50 and −194.07 eV, with an average of −195.41 eV. The offsets from the mean yield 0.56% (between the minimum and mean) and 0.68% (between the maximum and mean). These minor changes point to the stability features of the compound regarding its thermal properties. Fig. 4(c) shows that the energy of the third compound K₂AgIrF₆ lies between −178.41 eV and −176.48 eV, averaging at −177.51 eV. The differences from the minima and the maxima to the mean are 0.51% and 0.58%, respectively. The smooth and steady trend observed in this energy profile is a sign of strong dynamic stability. This stability is further supported by enhanced phonon scattering, which helps protect the structure from failing.

Fig. 4 Variation in the energy profiles and temperature as a function of the time step (fs) for the (a) Cs₂AgIrF₆, (b) Rb₂AgIrF₆, and (c) K₂AgIrF₆ compounds.

3.3. Electronic properties

DFT calculations are effective for studying the electronic properties of materials, including band structure, mobility, and density of states. The arrangement of the electronic distribution across an energy level in a material, represented in the band structure, helps in defining the nature of a material as either a conductor, insulator, or semiconductor.⁴⁹ Simplified energy dispersion and density of states are very useful in electronic and optoelectronic applications. One of these parameters is the band gap, which shows the range of energies within the solid and provides details on the electrical characteristics of the material. The band profile is shown in Fig. 5.

Fig. 5 Band structure of the (a) Cs₂AgIrF₆ (GGA PBE), (b) Cs₂AgIrF₆ (TB-mBJ), (c) Rb₂AgIrF₆ (GGA PBE), (d) Rb₂AgIrF₆ (TB-mBJ), (e) K₂AgIrF₆ (GGA PBE), and (f) K₂AgIrF₆ (TB-mBJ) compounds.

The left panel of Fig. 5 illustrates the band structure calculated using the GGA-PBE method, while the right panel displays the band structure calculated with the TB-mBJ method. The Fermi level is set to 0 eV to facilitate the comparison of the bandgaps for both models. The calculated band gap values for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆ are 1.07 (1.65) eV, 1.13 (1.76) eV, and 1.16 (1.83) eV, respectively, as obtained using the GGA-PBE (TB-mBJ) functional and summarized in Table 3. The gradual increase in the band gap from Cs₂AgIrF₆ to Rb₂AgIrF₆ and K₂AgIrF₆ is attributed to the decreasing ionic radius of the A-site cation, which induces lattice contraction and greater distortion of the AgF₆ and IrF₆ octahedra. Such structural modifications weaken the orbital hybridization among the Ag-4d, Ir-5d, and F-2p states, thereby reducing band dispersion and widening the band gap. Hence, replacing Cs with smaller Rb and K ions enhances octahedral distortion and leads to a systematic increase in the electronic band gap.

Table 2 Calculated parameters including Shannon's ionic radii (r), Goldschmidt tolerance factor (t), octahedral factor (u), and new tolerance factor (τ) for the A₂AgIrF₆ (A = Cs, Rb, and K) compounds

DP	Ionic radius of cations (Å)		Ionic radius of anion (Å)	Tolerance factor (t)	Octahedral factor (u)	New tolerance factor (τ)
	rA	(rAg + rIr)/2	rF
Cs2AgIrF6	1.88	0.92	1.33	1.01	0.69	3.32
Rb2AgIrF6	1.72	0.92	1.33	0.96	0.69	3.47
K2AgIrF6	1.64	0.92	1.33	0.93	0.69	3.52

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Table 3 Computed energy band gaps, band gap types, and electron/hole effective mass values for the A₂AgIrF₆ (A = Cs, Rb, and K) DHP compounds

Compounds	Band gap (eV)	Functional	Type	Effective mass of electron,	Effective mass of hole,	Reference
me = mass of electron.
Cs2AgIrF6	1.07	GGA PBE	Indirect	0.46 me	3.64 me	This work
	1.65	TB-mBJ	Indirect	0.29 me	0.99 me
Rb2AgIrF6	1.13	GGA PBE	Indirect	0.47 me	4.98 me	This work
	1.76	TB-mBJ	Indirect	0.28 me	0.94 me
K2AgIrF6	1.16	GGA PBE	Indirect	0.38 me	6.20 me	This work
	1.83	TB-mBJ	Indirect	0.27 me	0.84 me
Cs2AuAsF6	0.518	GGA-PBE	Indirect	—	—	50
	1.144	TB-mBJ	Indirect	0.16 me	0.48 me
Cs2AuSbF6	0.777	GGA-PBE	Indirect	—	—	50
	1.746	TB-mBJ	Indirect	0.25 me	0.72 me
Cs2NaIrCl6	0.92	GGA-PBE	Direct	0.21 me	1.33 me	51
	1.93	TB-mBJ	Direct	0.30 me	1.69 me
Rb2NaIrCl6	0.97	GGA-PBE	Direct	0.20 me	1.10 me	51
	2.02	TB-mBJ	Direct	0.29 me	1.41 me

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The figures show that the TB-mBJ method outperforms the GGA-PBE by correcting its errors of underestimating band gaps. Band structure calculations were performed along the Brillouin zone path at the high-symmetry points that included W (0.50, 0.25, 0.75), L (0.5, 0.5, 0.5), Γ (0, 0, 0), X (0.5, 0, 0), W (0.50, 0.25, 0.75), and K (0.375, 0.375, 0.75). Both profiles portray a parabolic nature at the conduction band minimum and valence band maximum.

Fig. 5 reveals that the conduction band minimum (CBM) and valence band maximum (VBM) occupy distinct symmetry points, confirming the indirect bandgap nature of all double perovskites (DPs). According to the Shockley–Queisser (S–Q) model, semiconductor materials require an optimal bandgap of ∼1.4 eV to achieve maximum theoretical efficiency in single-junction photovoltaic cells under standard solar illumination.⁵² In addition, the band gap of perovskite within the range of 0.8–2.2 eV makes the perovskite suitable for the photosensitive electrochemical (PEC) process.⁵³ The band gap values calculated for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆ are also within the ideal range of SQ values, and from the aspect of photosensitivity, they can be named as a highly photosensitive material for higher optoelectronics applications.

The effective mass of charge carriers is the effective mass of electrons or holes in materials like double perovskite halides. It elucidates the behavior of charge carriers and how they respond to stimuli, including electric fields. In DHPs, the effective mass depends on the curvature of the band structure of a material and the composition of the frame of the perovskite based on halides and other elements.

For the compounds, A₂AgIrF₆ (A = Cs, Rb, and K), the effective masses of holes and electrons were evaluated at high-symmetry points corresponding to the CBM and VBM. The effective mass m* was determined using the following expression:⁵⁴

where m* refers to the effective mass of the charge carrier and ℏ is less known as the reduced Planck constant. In contrast, ε(k) and d²ε(k)/dk² refer to the energy eigenvalue and the second derivative of ε(k) concerning wave vector k, respectively.

In the case of m*, this equation suggests that the value is directly proportional to the dispersion curve of the electronic band. The calculated effective mass values are provided in Table 3, which are several orders of magnitude lower than those considered for other DHPs.^55,56 These carriers have a lower effective mass, increasing the carrier mobility essential for charge transport. In addition, the carrier mobility is a function of the ratio of hole mass to electron mass . However, the increase in the effective mass decreases the carrier mobility, and the ratio prescribes the ratio of the electron and hole mobility. The calculated band gap energies and carrier effective masses demonstrate excellent agreement with similar reported values compiled in Table 3, supporting the reliability of the present computational approach. Using the GGA–PBE functional, the computed band gaps and charge-carrier effective masses for A₂AgIrF₆ (A = Cs, Rb, and K) closely match the values documented by Yu et al.⁵⁷ This consistency reinforces the reliability, significance, and overall scientific validity of the materials under investigation.

For a clearer understanding of bonding in the compounds, the partial density of states (PDOS) and total density of states (TDOS) are shown in Fig. 6, calculated using the TB-mBJ potential. DOS evaluation shows how individual atoms and orbitals of A₂AgIrF₆ (A = Cs, Rb, and K) provide insights into the interaction among atoms, bonding types, and alterations to the material's physical properties.

Fig. 6 Total and partial density of states of the (a) Cs₂AgIrF₆, (b) Rb₂AgIrF₆, and (c) K₂AgIrF₆ compounds.

For Cs₂AgIrF₆, the states close to E_F consist mainly of Ir 5d and Ag 4d states and F 2p states below E_F. The Cs 5s states contribute little and are primarily situated at higher energy levels. The conduction band is mainly formed by the Ir 5d states, while the F 2p states contribute minimally to the conduction band. Likewise, for Rb₂AgIrF₆ and K₂AgIrF₆, the valence band primarily consists of the Ag 4d and Ir 5d states, while the conduction band primarily comprises the Ir 5d state. The VBM is dominated by the Ag 4d, Ir 5d, and F 2p states in all three compounds. The conduction band mainly originates from the Ir 5d orbitals, demonstrating that the Ir 5d states largely determine the electronic structure of the materials.

3.4. Charge density

As pointed out in the analysis of the A₂AgIrF₆ (A = Cs, Rb, and K) compounds, electronic charge density contour plots provide precise descriptions of bond characteristics shown in Fig. 7. The bonding properties and charge transfer are further enhanced using the TB-mBJ method for charge density derived from a converged wave function.^58,59 The samples of A₂AgIrF₆ contain covalent, ionic, and metallic bonding, which define the distribution of the charge density of the compounds in the crystal lattice. These materials comprise AgF₆/IrF₆ connected by corner-sharing octahedra and A⁺ cations (Cs⁺, Rb⁺, and K⁺) in the interstitial sites. The high electronegativity difference between A⁺ cations and F⁻ anions allows the structural framework to be controlled by the ions' charges; the electrostatic network derived from the ionic bonding proves to be highly stable. Within the AgF₆ and IrF₆ octahedra, the Ag–F and Ir–F bonds exhibit mixed ionic and covalent characteristics. This is attributed to the overlap of Ag/Ir-d-orbitals with F-p-orbitals, leading to enhanced electronic coupling. The high electronegativity of fluorine polarizes the charge density, resulting in localized high-density regions around the F⁻ ions and along the Ag–F and Ir–F bond axes. Furthermore, the positive charge of the Ag⁺ and Ir³⁺ ions leads to semi-delocalized electron contacts, which give rise to the formation of metallic bonds. This delocalization results in intermediate charge density in the Ag–Ir regions, affecting the electrical properties, conductivity, and charge transport. Similar results have been reported in ref. 42, 60 and 61. The intricate balance of charge density, highly localized around F⁻ ions, intermediate along Ag–F and Ir–F bonds and delocalized between Ag and Ir, defines the electronic behavior of A₂AgIrF₆, making these compounds promising candidates for optoelectronic applications.

Fig. 7 Charge density of the (a) Cs₂AgIrF₆, (b) Rb₂AgIrF₆, and (c) K₂AgIrF₆ compounds.

3.5. Optical properties

The optical properties of a material are crucial for optoelectronic and photovoltaic devices since they indicate how the material absorbs visible light energy and the internal electric polarization of the available material. In the present work, the dispersive data, such as dielectric function, refractive index n(ω), extinction coefficient K(ω), absorption coefficient α(ω), conductivity, reflectivity R(ω), and loss function L(ω), are computed for these materials. An analysis of the studied photon types shows that ultraviolet photons exhibit higher transmission efficiency through the material compared to infrared or visible photons. Within the visible spectrum, the material demonstrates optical characteristics analogous to metallic behavior, consistent with our initial assumption that all atomic outer shells participate in metallic bond formation. These optical parameters are fully described by the dielectric function ε(ω) = ε₁(ω)+iε₂(ω), where ε₁ and ε₂ denote the real and imaginary components of the dielectric function, respectively. The real part of the susceptibility, ε₁(ω), characterizes the material's response to a field. It can, in turn, be calculated using the Kramers–Kronig relation:⁶²where M stands for a unique integral value attached to the principal component. The absorption of light by a material matches the values of the imaginary dielectric function. Light can transfer electrons from the occupied to unoccupied states of momentum when it travels through any material substance. We can determine ε₂(ω) by the momentum matrix of electronic transitions:⁶³where ‘e’ represents the electric charge, V represents the unit cell volume, ℏ represents the Plank's constant, and |kn|p|kn′| is the momentum transition matrix. Moreover, k represents the wave vector of the conduction band (CB), while kn indicates the valence band (VB) wave vector.

Using the dielectric constants of all the compounds and the equations given below,⁶⁴ we can find the refractive index n(ω), extinction coefficient k(ω), absorption coefficient α(ω), conductivity σ(ω), reflectivity R(ω), and loss function L(ω). These are all important optical constants:

and

The optical properties of all A₂AgIrF₆ (A = Cs, Rb, and K) compounds have been established and described by photon energies in the range of 0 to 5 eV. According to the real and imaginary parts of the presented materials, the dielectric constant is entirely analogous to the nature of the electronic structure within the visible spectrum range. The numerical values of the real part of the dielectric function of the double perovskite are plotted in Fig. 8(a). For ε₁(ω) > 0, photons can pass through the material.^65–67 The materials with higher dielectric constants tend to have lower exciton binding energy, which suits their application in solar cells.⁶⁸ The computed values of the static dielectric function ε₁(0) are 3.77, 3.52, and 3.44 for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆, respectively, which are higher than those of X₂AuYZ₆ (X = Cs, Rb; Z = Cl, Br, I).⁶⁰ The energy of the incident radiation is the main cause of polarization. To obtain desired characteristics in semiconductors, it is important to enhance the polarization values shown in Fig. 8(a), as epitomized by the magnitude of ε₁(ω). In the visible light range, the ε₁(ω) gradually increases with an increase in the incident radiation up to the polarization peak of 10.28, 10.15, and 10.19 at energies of 1.86, 2.00, and 2.05 eV, respectively. The negative values of A₂AgIrF₆ are in the range of 2.14–3.63 eV. Thus, the three materials exhibit metallic characteristics for the energy ranges and can engage in strong reflection that cannot transmit photons. The calculated graphs for the imaginary part (ε₂(ω)) of the dielectric function are presented in Fig. 8(b). ε₂(ω) is linked to the light absorption characteristics of the materials. The absorption begins at 1.55, 1.63, and 1.72 eV for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆, respectively. This energy is similar to the fundamental gap when the system has acquired thermal equilibrium. They rise gradually, and the highest intensities in the visible region are at wavelengths of 11.42, 11.49, and 11.59 for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆, respectively. The rates of the peak values represent the electron transfer from the valence band towards the conduction band. This is mainly attributed to a transfer between the Cs, Rb, and K atoms in A and the Ag, Ir, and F atoms in the valence band and the Ir and F atoms in the conduction band of A₂AgIrF₆.

Fig. 8 Optical properties of the A₂AgIrF₆ (A = Cs, Rb, and K) compounds.

The refractive index n(ω) determines the probabilities of photon absorption and the speed of light in the material. In the case of solid media, another parameter useful for the comprehension of light transmission and interaction is n + ik. It may be helpful for optoelectronics.⁶⁹ As the energy of the incident photon increases, the k(ω) increases and is equal to 0 within the energy band gap. In the region of interband transitions, the k(ω) changes correspondingly. In this study, we only calculated the real part of the complex refractive index. Representative plots of the refractive index values are plotted as per the calculation shown in Fig. 8(c). The specified dependencies n(ω) = ε_r(ω) were positively correlated, confirming that the real part of the dielectric function correlates with the refractive index.⁴⁴

Table 4 presents the zero-frequency optical parameters of A₂AgIrF₆ (A = Cs, Rb, and K). The calculated static refractive indices n(0) range from 1.85 to 1.94, consistent with values typical of semiconducting halide perovskites. The frequency-dependent refractive index n(ω) varies between 2.0 and 4.0, reflecting the degree of electronic polarizability and the strength of light–matter interaction within the compounds.⁷⁰ The peak values of n(ω) are 3.50 at 1.95 eV for Cs₂AgIrF₆, 3.48 at 2.00 eV for Rb₂AgIrF₆, and 3.46 at 2.05 eV for K₂AgIrF₆. These peaks indicate strong optical dispersion associated with interband transitions between the valence and conduction bands. Such refractive index values are appropriate for optoelectronic and photonic materials, as high-quality optical media typically exhibit n(ω) values between 2.5 and 3.5.⁶⁹ The slightly decreasing trend in n(ω) from Cs to K correlates with the increasing optical band gap, consistent with the inverse relationship between the refractive index and band gap energy, as described by the Penn model. Hence, as the band gap widens, the electronic polarizability decreases, leading to a lower refractive index. This behavior suggests that A₂AgIrF₆ (A = Cs, Rb, and K) compounds possess tunable optical responses, making them suitable candidates for photovoltaic, light-emitting, and photonic applications. The extinction coefficient k(ω) consistently mirrors the pattern of ε₂(ω) and α(ω), signifying the identification of electromagnetic wave attenuation in the materials, as depicted in Fig. 8(d). The correlation between k(ω) and α(ω) is expressed as .

Table 4 Computed zero-frequency parameters for the A₂AgIrF₆ (A = Cs, Rb, and K) DHP compounds: real part of the dielectric constant ε₁(0), static refractive index n(0), reflectivity R(0) and optical band gap using a Tauc plot

DHP	ε1(0)	n(0)	R(0)	Optical band gap (eV)
Cs2AgIrF6	3.77	1.94	0.10	1.52
Rb2AgIrF6	3.52	1.88	0.09	1.61
K2AgIrF6	3.44	1.85	0.09	1.75

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Optoelectronic and photovoltaic devices are related to light absorption because the generation and carrier transport processes are proportional to the absorption coefficient α(ω).⁷¹ Semiconductor absorption coefficients follow those of photons, which determine the basic properties of solar cells. These molecules absorb light more efficiently, leading to better device performance. As shown in Fig. 8(e), the A₂AgIrF₆ (A = Cs, Rb, and K) absorption spectra display strong transitions in the visible region. The characteristic peaks are at 2.96, 3.00 and 3.03 eV for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆, respectively. The positions of these peaks are located in the visible range, confirming potential applications for photovoltaic and photodetector devices based on these double halide perovskites. The static peak values are listed in Table 4. The absorption coefficients of the three highest peaks are estimated to be approximately 5.1 × 10⁵ cm⁻¹, 5.4 × 10⁵ cm⁻¹, and 5.5 × 10⁵ cm⁻¹ for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆, respectively. The studied compounds show higher absorption coefficients than other halide perovskites, including Cs₂AuMF₆ (M = As and Sb),⁵⁰ X₂AuYZ₆ (X = Cs and Rb; Z = Cl, Br, and I),⁷² A₂AuScX₆ (A = Cs and Rb; X = Cl, Br, and I),⁴² and Cs₂AgBiX₆ (X = Cl, Br, and I).⁷³ The strong absorption coefficients (∼10⁵ cm⁻¹) and suitable optical band gaps (1.52–1.75 eV) indicate that A₂AgIrF₆ (A = Cs, Rb, and K) compounds can serve as efficient absorbing layers in optoelectronic and photovoltaic devices. Materials with absorption coefficients above 10⁴ cm⁻¹ are generally capable of absorbing most incident visible light within a few hundred nanometers, which reduces material thickness requirements for solar absorbers.⁷⁴

Fig. 8(f) shows that the A₂AgIrF₆ (A = Cs, Rb, and K) compounds exhibit peak absorption at 532, 503, and 482 nm, respectively. These wavelengths exist in the visible light region (400–700 nm), so the compounds can strongly absorb visible light. The optical band gap of the present double halide perovskites was calculated using Tauc plots based on their absorption coefficient spectra.⁷⁵ An accurate value of the bandgap is important for developing reliable information about semiconductor properties, and applying the Tauc plot in the wrong way can cause an incorrect estimation of band gap values.⁷⁶ The absorption behavior follows the Tauc relation, (αhν)^1/η = A(hν − E_g), where α is the absorption coefficient, h is Planck's constant, ν is the photon frequency, A is a material-specific constant, E_g is the optical band gap, and η depends on the band gap type (1/2 for direct and 2 for indirect).⁷⁵ Using the TB-mBJ functional, the optical band gaps of A₂AgIrF₆ (A = Cs, Rb, and K) are calculated to be 1.52 eV, 1.61 eV, and 1.75 eV, respectively (Fig. 8(f)), consistent with the electrical band gaps reported in Fig. 5 and Table 3. Furthermore, the calculated optical band gaps are smaller than those reported for related optical materials, such as BaVO(IO₃)₅, BaNbO(IO₃)₅, and BaTaO(IO₃)₅ (2.9–4.2 eV).⁷⁷ The comparatively narrower band gaps of A₂AgIrF₆ suggest enhanced absorption in the visible region of the solar spectrum. Consequently, these compounds may be more suitable for visible-light-driven optoelectronic and photovoltaic applications.

Optical conductivity determines the electron transport in a compound under illumination by specific photon energies. For optoelectronic applications, optical conductivity arising from interband electron transitions should fall within the visible range of 1.4–4.0 eV.⁷⁰ Substituting different alkali metals has little effect on conductivity. The main conductivity peaks for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆ are observed at 2.00 eV (3073 Ω⁻¹ cm⁻¹), 2.11 eV (3348 Ω⁻¹ cm⁻¹), and 2.19 eV (3416 Ω⁻¹ cm⁻¹), respectively (Fig. 8(g)). These peaks, mainly contributed by Ir 5d states, are key to the electronic structure and confirm the high optical conductivity of the compounds within the visible spectrum.

The reflectivity spectra R(ω) of the A₂AgIrF₆ (A = Cs, Rb, and K) double halide perovskite compounds are presented in Fig. 8(h), and these systems exhibit different reflectivity behaviors over the whole photon energy range. The static reflectivity values for these materials are as follows: 10% for Cs₂AgIrF₆, 9% for Rb₂AgIrF₆, and 9% for K₂AgIrF₆. The low R(ω) values in the energy range of 0 to the band gap confirm its light trapping ability; thus, they are suitable for optoelectronics and photovoltaic applications. The reflectivity rises steeply towards the absorption edge region, and the peak is observed at a higher energy region, which may be attributed to the interband transitions. This behavior is quite typical for materials with pronounced photon-matter coupling. The low reflectivity in the visible spectrum also supports the applicability of these compounds in light-harvesting technologies, as the compounds limit reflection while enhancing absorption capability. The energy loss function, shown in Fig. 8(i), illustrates the energy lost by electrons as they propagate through the material. Prominent peaks are observed at approximately 3.25 eV for Cs₂AgIrF₆, 3.52 eV for Rb₂AgIr₆, and 3.66 eV for K₂AgIrF₆, indicating significant electron scattering at these energies, which aligns with their optical conductivity behavior. Compounds with low reflectivity and reduced energy loss are ideal candidates for solar cell absorber layers. Owing to their adjustable band gaps, high dielectric strength, excellent absorbance, and efficient photoconductivity, such compounds are highly promising for optoelectronic and solar cell device applications.

3.6. Mechanical properties

The mechanical properties of a crystal are principally governed by its elastic constants, which determine the reaction of the crystal to loads. It is important to acquire such information in studies concerned with the mechanical behavior of materials. In this work, the mechanical properties and the marks related to the elasticity use deformability coefficients, C₁₁, C₁₂ and C₄₄, for different compounds. The mechanical stability of the perovskite compounds was evaluated using the following criteria: C₁₁ – C₁₂ > 0, C₄₄ > 0, and C₁₁ + 2C₁₂ > 0.⁷⁸ Strength, another first-order texture property that provides some insight into the nature of a material, was also considered. The individual elastic coefficients of the studied compounds are reported in Table 5.

Table 5 Mechanical properties of the A₂AgIrF₆ (A = Cs, Rb, and K) compounds

Parameters	Cs2AgIrF6	Rb2AgIrF6	K2AgIrF6
C11 (GPa)	127.79	162.82	162.67
C12 (Gpa)	35.70	31.17	28.21
C44 (Gpa)	16.22	15.42	11.14
B (Gpa)	66.39	75.05	73.03
G (Gpa)	25.02	28.90	25.15
Y (Gpa)	66.69	76.84	67.67
B/G	2.65	2.60	2.90
ν	0.33	0.33	0.35
CP	19.48	15.75	17.07
A	0.35	0.23	0.17

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Specific mechanical properties of A₂AgIrF₆ (A = Cs, Rb, and K) were predicted, which include bulk modulus (B), Pugh's ratio (B/G), Young's modulus (Y), shear modulus (G), and Poisson's ratio (ν). The bulk modulus (B) is the measure of the stiffness of a material with respect to changes in its volume when pressure is applied. Shear modulus (G) refers to the ratio of the applied shear stress to the resulting shear strain. From the ratio of stress to strain at uniaxial deformation, Young's modulus (Y) was deduced as an indicator of the elasticity of the material. The formulas used for these calculations are as follows:^79–81

and

In A₂AgIrF₆, the C₁₁ (mechanical linear stiffness) is larger than C₁₂ and C₄₄, which suggests that the materials are more durable when pressure is applied. The compounds, therefore, stand in the order of Rb₂AgIrF₆, which has the highest C₁₁ (162.82 GPa) and, thus, the best B of 75.05 GPa because the higher C₁₁ means a higher resistance to volumetric compression. Shear modulus (G) is also closely related to the anti-deformation of the shape, and it is, therefore, profound in Rb₂AgIrF₆ with 28.90 GPa, meaning that a greater force is needed to afford the compound this shape than Cs₂AgIrF₆ or K₂AgIrF₆.

The tensile elastic modulus Y, which predicts the stiffness of a material under tension or compression, is higher in Rb₂AgIrF₆, with values of up to 76.84 GPa. From the results above, it can be inferred that the B/G ratio is greater than 1.75 for all the compounds under study, indicating their ductility.⁸² However, for the three compounds, K₂AgIrF₆ has the highest B/G ratio (2.90) and is, therefore, more ductile than the other two compounds. High ductility and room temperature stability make these compounds useful in flexible applications, such as the thin film absorber layers of foldable electronics and biomedical devices.^83,84 Poisson's ratio (ν) is the other mechanical property of concern, which represents the lateral contraction per unit length when the material is subjected to axial loads, which is a measure of flexibility.⁸⁵ The lattice flexible character of the compounds can be estimated using the value of ν, which is higher than 0.26 for all target compounds, whereas K₂AgIrF₆ has the highest value of ν = 0.35. Thus, their capability of stress engineering is demonstrated, and the band gap can be tuned by controlled stress. Non-zero positive values of reported Cauchy pressure (CP) values for the compounds suggest the existence of ionic bonding between the atoms.⁸⁶ Moreover, the values of elastic anisotropy coefficients (A), which are determined from the relationship⁸⁷ are not equal to 1, which confirms the anisotropic nature. Of all the compounds, Cs₂AgIrF₆ has the highest A value of 0.35, indicating that it is more directional. These outcomes hint at the anisotropic character of the researched compounds and their high potential for use in modern technologies as materials with desired mechanical and structural properties. The anisotropic mechanical response of the A₂AgIrF₆ lead-free double perovskites was examined using elastic constants obtained from first-principles calculations, analyzed through the ELATE elastic tensor visualization tool.⁸⁸ Three-dimensional orientation-dependent plots of Young's modulus, shear modulus, and Poisson's ratio are illustrated in Fig. 9(a–c). In such a diagram, the degree of anisotropy is shown with different colored contours, and its sharpness is in proportion to the amount of curvature. A larger curvature of the contours inward to the sphere center is associated with a stronger anisotropy, and a nearly circular outline corresponds to weaker anisotropy.⁸⁹ As calculated from the spatial dependence maps, shear modulus and Poisson's ratio exhibit a strong directionality, consistent with the computed anisotropy factor that confirms the studied compounds to be anisotropic (Table 6).

Fig. 9 3-D representations of the (a) Young's modulus, (b) shear modulus, and (c) Poisson's ratio for the A₂AgIrF₆ (A = Cs, Rb, and K) compounds.

Fig. 10 Thermodynamic properties of the A₂AgIrF₆ (A = Cs, Rb, and K) double perovskite compounds.

Table 6 Minimal and maximal values of the elastic modulus and elastic anisotropy of the A₂AgIrF₆ (A = Cs, Rb, and K) compounds

HDPs	Young's modulus (GPa)		Linear compressibility (TPa^−1)		Shear modulus (GPa)		Poisson's ratio
	Ymin	Ymax	βmin	βmax	Gmin	Gmax	νmin	νmax
Cs2AgIrF6	44.99	112.20	5.02	5.02	16.22	46.05	0.10	0.63
Rb2AgIrF6	43.29	152.80	4.44	4.44	15.42	65.82	0.05	0.71
K2AgIrF6	31.80	154.33	4.56	4.56	11.14	67.23	0.04	0.78
Elastic anisotropy Ax
	AY		Aβ		AG		Aν
Cs2AgIrF6	2.49		1.00		2.84		6.13
Rb2AgIrF6	3.53		1.00		4.27		12.81
K2AgIrF6	4.85		1.00		6.03		20.55

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3.7. Thermodynamic properties

The Debye temperature (θ_D) is a fundamental thermodynamic parameter that governs the thermal properties and heat capacity of solids across different temperature ranges. It serves as an indicator of how material properties evolve with temperature variation. Several theoretical approaches have been formulated to evaluate θ_D, most of which incorporate the mean sound velocity (v_m) as a critical factor in describing lattice vibrations and thermal response. The Debye temperature can be determined using the following equation:⁹⁰where ℏ is the normalized Planck's constant, k_B is Boltzmann's constant, n represents the number of atoms, V is the unit-cell volume, and v_m denotes the average acoustic velocity. The latter is derived from the following relation:⁹⁰where v_t and v_l denote the transverse and longitudinal sound velocities, respectively. The shear modulus (G), bulk modulus (B), and density (ρ) are used to obtain these additional parameters as follows:⁹⁰

and

In the presented data, θ_D increases progressively from 253.34 K (Cs₂AgIrF₆) to 291.25 K (Rb₂AgIrF₆) and 296.09 K (K₂AgIrF₆), which correlates with the increase in sound velocities (v_l, v_t, and v_m) and the decrease in density. The Debye temperature increases by ∼15–17% as Cs is replaced by Rb and K. This rise reflects the influence of the smaller ionic radii of Rb⁺ and K⁺, which cause lattice contraction and stronger interatomic bonding. A denser bond network enhances the average sound velocity (v_m) and, therefore, increases θ_D. The increase in θ_D signifies that the lattice of K₂AgIrF₆ is stiffer and vibrates at higher frequencies compared to Cs₂AgIrF₆. This improvement in vibrational energy correlates with better thermal stability, stronger bonding strength, and higher thermal conductivity potential.

One of the big challenges in fabricating solar cells is to identify materials that can withstand extreme heat during manufacturing without breaking down or without driving costs too high. Low-melting-point materials run the risk of melting or cracking, forming defects, while very high-melting-point materials are expensive and hard to work with.⁹¹ The melting behavior is commonly described using the elastic constant (C₁₁), which gives the measure of a material's resistance to thermal deformation. The melting temperature (T_m) of the A₂AgIrF₆ materials was calculated using the empirical formula reported by Fine et al.:⁹²

T_m (K) = 553 + (5.911)C₁₁. (23)

The obtained melting temperatures suggest that the investigated double perovskites are thermodynamically stable and can be synthesized under ambient conditions. Among them, K₂AgIrF₆ displays a greater melting point than that of Cs₂AgIrF₆ and Rb₂AgIrF₆. A summary of the thermodynamic parameters derived from the elastic constants is presented in Table 7.

Table 7 Calculated crystal density ρ (gm cm⁻³), longitudinal, transverse, and average sound velocities (v_l, v_t and v_m) (m s⁻¹), Debye temperature Θ_D(K), and melting temperature T_m(K) for the A₂AgIrF₆ (A = Cs, Rb, and K) compounds

Compounds	ρ	vl	vt	vm	ΘD	Tm
Cs2AgIrF6	6.45	3932.91	1969.72	2209.44	253.34	1308.25
Rb2AgIrF6	5.80	4425.28	2232.25	2502.86	291.25	1514.40
K2AgIrF6	4.99	4639.54	2253.84	2532.51	296.09	1515.28

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The stability of the sample was investigated by phonon mode analysis at various temperatures, enabling the determination of fundamental thermodynamic parameters, such as enthalpy, free energy, and entropy. These quantities are directly related to phonon vibration frequencies, expressed as follows:⁴⁷

and

where g(ω) denotes the phonon density of states, K_B is the Boltzmann constant, E_tot is the minimum total energy, and ℏ represents the normalized Planck constant. The temperature-dependent variations of the enthalpy, free energy, entropy, heat capacity, and Debye temperature for A₂AgIrF₆ (A = Cs, Rb, and K) are illustrated in Fig. 10. The enthalpy of the system exhibits a linear increase with temperature, whereas the free energy decreases following an exponential decay trend, as illustrated in Fig. 10(a). Entropy, which determines the degree of disorder in a material, rises exponentially with an increase in temperature. This behavior results from the enhanced thermal motion of atoms at higher temperatures, which increases molecular disorder.⁹³ Heat capacity, reflecting the material's ability to store thermal energy through contributions from both lattice vibrations and electronic motion, shows a sharp increase between 0 and 200 K, consistent with the Debye model⁹³ (Fig. 10(b)). At up to 500 K, most low-energy vibrational modes become thermally active. Beyond 500 K, the heat capacity reaches a plateau, with further temperature increases producing negligible change, in accordance with the Dulong–Petit law.⁹⁴ The temperature-dependent behavior of the Debye temperature is presented in Fig. 10(c). A clear compositional trend (K → Rb → Cs) is observed in the thermodynamic properties shown in Fig. 10(c). This behavior originates from the systematic variation in the A-site ionic radius (K⁺ < Rb⁺ < Cs⁺), which affects lattice volume, bonding strength, and phonon dynamics. At 0 K, the Debye temperatures for Cs₂AgIrF₆, Rb₂AgIrF₆, and K₂AgIrF₆ are approximately 200.99 K, 266.36 K, and 269.74 K, respectively. This trend agrees with Anderson's formula, since smaller A-site cations reduce the lattice volume and increase the sound velocity, leading to higher Debye temperatures. This is consistent with Anderson's formula of Debye temperature by elastic constants. Under constant pressure, the Debye temperature increases with temperature, reflecting the thermal stiffening of lattice vibrations.

4. Conclusions

In this study, the structural, electronic, optical, mechanical, and thermodynamic properties of lead-free double perovskites A₂AgIrF₆ (A = Cs, Rb, and K) were systematically investigated using density functional theory (DFT). The calculated tolerance factors, formation energies and phonon dispersion curves demonstrate that these compounds are thermodynamically and dynamically stable based on AIMD simulations, further confirming their dynamic stability at room temperature. Electronic structure studies show that the indirect band gaps lie in the range of 1.65–1.83 eV (TB-mBJ), which are well within the optimum photovoltaic window. The small band gap, coupled with low carrier effective masses, are suggestive of efficient charge transport and weak recombination losses; the latter is critical in designing high-performance optoelectronic devices. The optical calculations demonstrate large absorption coefficients (>10⁵ cm⁻¹) in the visible region, large dielectric constants, as well as the low reflectivity (<10%) of these materials, indicating their significant light-harvesting capabilities. The B/G ratios and Poisson's ratio values are all high, which is consistent with the ductile properties and anisotropic mechanical behavior that demonstrate structural flexibility for applications in thin-film or flexible electronics. We also measured the Debye and melting temperatures, which verify high thermal stability and ensure their long-term durability in working status. In summary, A₂AgIrF₆ (A = Cs, Rb, and K) double perovskites appear as earth-abundant, mechanically stable and opto-electronically effective materials. Their stability, visible-light absorption ability and ductility mean that the materials could be useful in future solar cells and energy-harvesting devices that adopt a more environmentally friendly approach than toxic lead-based perovskites for next-generation green-energy technologies. This study will guide the development of lead-free perovskite materials, which can be employed in highly efficient and environmentally safe energy technologies.

Author contributions

M. A. Rayhan: writing – original draft, methodology, conceptualization, formal analysis, data calculations, validation. M. M. Hossain: writing – review & editing, validation. M. M. Uddin: writing – review & editing, validation. M. A. Ali: conceptualization, formal analysis, validation, writing – review & editing, supervision, software.

Conflicts of interest

We declare that we have no competing interests.

Data availability

The relevant data from this study are available from the corresponding author upon reasonable request.

Acknowledgements

This work was carried out with the aid of a grant (grant number 21-378 RG/PHYS/AS_G-FR3240319526) from UNESCO-TWAS and the Swedish International Development Co-operation Agency (SIDA). The views expressed herein do not necessarily represent those of UNESCO-TWAS, SIDA or its Board of Governors.

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