First-principles investigation of direct band gap double perovskite halides A₂AgIrCl₆ (A = Cs, Rb, K) for enhanced photovoltaic performance

Abstract

This study carefully investigates the structural, electrical, optical, mechanical, and thermodynamic features of A₂AgIrCl₆ compounds (A = Cs, Rb, K) that belong to double perovskite halides (DPH) utilizing density functional theory (DFT). The stability of the predicted compounds in the cubic structure was confirmed through calculations involving the Goldschmidt tolerance factor, octahedral factor, and the new tolerance factor. Analysis of formation enthalpy, binding energy, phonon dispersion relations, and ab initio molecular dynamics (AIMD) results suggests thermodynamic and dynamic stability, indicating possible synthetic viability that should be verified experimentally. To predict the accurate optoelectronic properties, we employed the Tran and Blaha modified Becke-Johnson (TB-mBJ) potential. The electronic band structure study demonstrated that the studied halides exhibit direct band gap semiconductor with band gap values of 1.43 eV, 1.50 eV, and 1.55 eV for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively. The relatively low electron effective masses suggest favorable carrier transport characteristics. In addition, the calculated exciton binding energies and exciton radii indicate a tendency toward efficient generation of free charge carriers. The optical investigation further demonstrated that the A₂AgIrCl₆ compounds exhibit low reflectivity and high absorption coefficients (on the order of 10⁵ cm⁻¹) in the visible region, highlighting their potential for optoelectronic applications. The computed elastic constants fulfill the Born–Huang criteria, confirming mechanical stability, while further analysis indicates ductile and anisotropic behavior. Overall, the calculated results suggest that the A₂AgIrCl₆ compounds exhibit promising optoelectronic descriptors favorable for further experimental and device-oriented evaluation.

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1 Introduction

The limited energy supplies of the Earth are put under stress by the increasing energy consumption caused by population expansion and growing living standards.^1,2 Over the previous 150 years, a notable increase in energy use has outpaced population growth. Fossil fuels, mainly gas and oil, account for over 80% of global use, which has issues with supply and impact on the environment. This clarifies how urgently renewable energy sources like geothermal, wind, and solar power need to be used.^3–5 Conservation and the use of renewable energy are becoming increasingly important to both governments and citizens. Of the possible substitutes, metal halide perovskites have garnered substantial interest and have been thoroughly studied in the past several years.⁶ These materials have drawn much attention over the past ten years due to their unique characteristics, adaptable synthesis methods, and configurable device designs. Perovskites based on lead have become a viable and affordable alternative for high-efficiency solar cells. Meanwhile, their commercialization has been hampered by problems with lead toxicity and chemical instability despite their excellent efficiency.^7–9 Even though a number of lead-free perovskites have been suggested as a solution to the toxicity issue, they still have difficulty reaching comparable levels of efficiency.

Trivalent cations like Bi³⁺ and Sb³⁺ have recently been used to create 2D layered lead-free halide perovskites.¹⁰ Researchers have also incorporated trivalent cations like Bi³⁺ together with monovalent cations such as Ag¹⁺ into the B-sites of halide perovskites, resulting in the formation of B-cation double perovskites, which follow the general formula A₂B′B′′X₆. Among these, Cs₂AgBiBr₆ and Cs₂AgBiCl₆ have emerged as compounds with promising photovoltaic properties and notable stability.^11–13 The experimentally determined optical band gap for Cs₂AgBiCl₆ and Cs₂AgBiBr₆ range from 2.20 eV to 2.77 eV and 1.83 eV to 2.19 eV, respectively.^12–14 In comparison, calculations using density functional theory (DFT) and hybrid functionals yield slightly higher band gap values, between 2.62 eV and 3.00 eV for Cs₂AgBiCl₆ and 2.06 eV to 2.30 eV for Cs₂AgBiBr₆.^12,14 Despite possessing band gap that lie within the visible spectrum, these materials display an indirect band gap nature, which restricts their usefulness for application in thin-film photovoltaic applications. Recent DFT studies have shown that modifying the B-site cations in DPH is an effective strategy for controlling their band gap and improving properties relevant to photovoltaic performance. In particular, recent work¹⁵ on Ag- and Au-containing double perovskites demonstrated that deliberate changes in composition can significantly influence the electronic structure and light absorption behavior, thereby offering valuable understanding of how compositional design governs material properties in these systems. The fabrication of lead-free double perovskite halides with direct band gap has received substantial interest for its potential in optoelectronic and solar applications. Zhang et al.¹⁶ proposed that integrating In¹⁺ or Tl¹⁺ with Bi³⁺ could result in direct band gap, a concept further reinforced by Zhao et al.,¹⁷ who reported a direct band gap of 0.91 eV for Cs₂InBiCl₆. However, compounds like Cs₂InBiX₆ phase stability limits due to the oxidation of In¹⁺ to In³⁺.¹⁸ In a separate study, S. Mahmud et al. conducted a theoretical analysis of the A₂AuScX₆ compounds, finding a band gap range of 1.30 to 1.93 eV, underlining its possibilities for usage in optoelectronic and photovoltaic systems.¹⁹ Transition metals have also been introduced into DPH due to their unique electronic properties, leading to the synthesis and study of various compounds such as Cs₂AgCrX₆,^20,21 Cs₂AgFeCl₆,²² and Cs₂NaVCl₆.²³ Single crystals of Cs₂Ag_xNa_1−xFeCl₆ and Cs₂NaSc_1−xCl₆ : xTb³⁺, have shown promise in photovoltaic and volumetric display technologies.^24–26 Additionally, transition metals from group VIII, including Co, Rh, and Ir, which are commonly found in oxide perovskites, are being studied for their potential in DPHs due to their distinctive optoelectronic properties, which are beneficial for applications in photovoltaic.^27–30 Recent research has renewed interest in Co, Rh, and Ir-based DPHs, with investigations of compounds like Rb₂NaCoF₆,³¹ Cs₂AgRhX₆,^32,33 and Cs₂CuIrF₆.³⁴ Furthermore, a number of recent investigations have utilized integrated first-principles methodologies that simultaneously examine structural, electronic, optical, mechanical, and thermodynamic characteristics to assess the potential of double perovskites in optoelectronic applications.³⁵ These studies offer a unified computational approach for evaluating diverse material properties and serve as an important reference point for the current work. Despite these advances, many lead-free double perovskites still suffer from either indirect band gap or excessively wide band gap (>2.2 eV), limiting their photovoltaic performance.^14,36 More recently, V. Deswal et al. revealed that the DPH combination Cs₂AgInBr₆ had a direct band gap of 1.57 eV and a high predicted power conversion efficiency, highlighting its promise as appropriate B-site cation combinations.³⁷ A density functional theory investigation by Parves et al. (2025)³⁸ explored the DPH X₂NaIrCl₆ (X = Rb, Cs) to evaluate their prospects for optoelectronic and photovoltaic uses. The calculated electronic band structures reveal direct band gap of roughly 1.93 eV for Cs₂NaIrCl₆ and 2.02 eV for Rb₂NaIrCl₆, which lie within a suitable range for solar energy harvesting. In addition, these materials exhibit strong optical absorption, supporting their potential applicability in solar cell devices. Closely related compounds such as M₂KIrCl₆ (M = Cs, Rb)³⁹ have also been examined through first-principles calculations to understand their electronic and optical behavior for photovoltaic applications. The predicted direct band gap is about 1.99 eV for Cs₂KIrCl₆ and 2.10 eV for Rb₂KIrCl₆, suggesting that these materials are capable of efficiently utilizing visible light from the solar spectrum. Furthermore, recent reports have emphasized the importance of correlating optical and mechanical properties with compositional design in double perovskites, demonstrating how subtle changes in cation selection can influence structural stability and optoelectronic response.⁴⁰ These findings emphasize the necessity of systematic investigations to identify compositions that simultaneously satisfy stability and performance criteria.

Despite recent progress in Ir-based DPH such as Cs₂NaIrCl₆ and Cs₂KIrCl₆, these systems typically exhibit direct but relatively wide band gap (∼1.9–2.1 eV), which are not optimal for photovoltaic applications. Therefore, identifying alternative B-site cation combinations that can simultaneously preserve direct band gap while reducing their magnitude remains a critical challenge. However, achieving direct band gap within the optimal photovoltaic range (∼1.12–1.77 eV) remains a major unresolved challenge in lead-free DPHs.

In this work, we introduce a distinct compositional strategy by incorporating Ag⁺ alongside Ir³⁺ in A₂AgIrCl₆ (A = Cs, Rb, K) compounds. Unlike previously studied Ir-based systems, the Ag/Ir combination enables strong Ag–Cl–Ir orbital hybridization, which significantly modifies the electronic structure. As a result, the studied compounds exhibit direct band gap in the range of 1.43–1.55 eV, placing them much closer to the optimal range for solar energy conversion.

To the best of our knowledge, Ag–Ir-based double perovskites remain largely unexplored. Therefore, this study presents a comprehensive first-principles investigation of A₂AgIrCl₆ compounds using the FP-LAPW technique based on the DFT framework, implemented via the Wien2k software package, covering their structural stability, electronic, optical, mechanical, and thermodynamic properties. This work thus represents a clear step beyond existing Ir-based double perovskites. The computational models and techniques used are described in Section 2, and the outcomes and their consequences are covered in Section 3. Section 4 provides an overview of our study's results.

2 Computational methods

This work investigated the structural, dynamical, mechanical, thermophysical, electrical, and optical properties of the A₂AgIrCl₆ compounds (A = Cs, Rb, K) using first-principles calculations. The calculations were conducted using DFT framework^41,42 with the FP-LAPW approach, as implemented in the Wien2k software package.^43,44 The exchange-correlation potential for the bulk structure of the double perovskite was implemented utilizing the Perdew–Burke–Ernzerhof (PBE) formulation within the generalized gradient approximation (GGA).⁴⁵ Birch–Murnaghan's equation of state⁴⁶ was used to evaluate the structural parameters. The plane-wave cut-off value R_MT × k_max, was set to 8.0, where R_MT refers to the radius of the smallest atomic muffin-tin sphere, and K_max indicates the maximum wave vector in the plane-wave basis set. Within the muffin-tin sphere, the maximum partial wave expansion was defined by l_max = 10. The muffin-tin radii were set to 2.50 a.u. (A), 2.13 a.u. (Ag), 2.18 a.u. (Ir), and 1.88 a.u. (Cl), with a Fourier expansion cutoff of G_max = 14 (Ry^1/2). A dense 1000 k-point mesh was used for Brillouin-zone sampling, and a core-valence separation energy of 7.0 Ry with a self-consistent convergence criterion of 10⁻⁵ Ry ensured numerical accuracy. The TB-mBJ potential was employed to obtain reliable optoelectronic band gap.⁴⁷ To ensure precise band gap measurement, the spin–orbit coupling (SOC) approach (TB-mBJ + SOC) was implemented. The crystal structure was visualized using VESTA.⁴⁸ Dynamic stability was assessed through phonon calculations using the CASTEP (Cambridge Serial Total Energy Package) package, employing a 2 × 2 × 2 supercell and finite-displacement method to obtain phonon dispersion and thermodynamic properties.⁴⁹

3 Results and discussion

3.1 Structural parameters and stability

The A₂AgIrCl₆ compounds (A = Cs, Rb, K), shown in Fig. 1, crystallize in a face-centered cubic structure belonging to the Fm3̄m (No. 225) space group.⁵⁰ This structure consists of fourteen [IrCl₆] octahedra, thirteen [AgCl₆] octahedra, and eight A-site atoms (A = Cs, Rb, K) positioned within the interstitial spaces of the octahedra, which help maintain crystal stability.⁵¹ Within this particular space group, the crystal structure of A₂AgIrCl₆ compounds places the A⁺¹ cations in the 8c Wyckoff location with fractional coordinates (0.25, 0.25, 0.25). Ag⁺¹ cations are located at the 4a Wyckoff site with coordinates (0.5, 0.5, 0.5), Ir⁺³ cations are at the 4b site at (0.0, 0.0, 0.0), and Cl⁻¹ anions occupy the 24e Wyckoff site at (0.25, 0.0, 0.0). The A₂AgIrCl₆ compounds have undergone optimization using volume optimization, as illustrated in Fig. 2. Table 1 presents the results of the geometry optimization-especially, the unit cell parameter a₀, bulk modulus B₀, its pressure derivative, and the ground state energy E₀. The selection of A-site cation significantly influences the lattice constant of the conventional cell, which reduces sequentially from Cs to Rb to K. There are currently no experimental results available for direct comparison with these findings in the scientific literature.

Fig. 1 The unit cell of Cs₂AgIrCl₆.

Fig. 2 (a–c) The optimized energy versus volume plots fitted using the Birch–Murnaghan equation for A₂AgIrCl₆ compounds where A = Cs, Rb, K.

Table 1 Calculated values of lattice parameter a₀, bulk modulus B₀, it's derivative B^’₀, total energy E_tot, decomposition energy ΔH_D, formation energy E_f and binding energy E_B for A₂AgIrCl₆ compounds A = Cs, Rb, K

DPH	a0 (Å)	B0 (GPa)	B^'0	Etot (Ry)	ΔHD (meV/atom)	Ef (eV/atom)	EB (eV/atom)
Cs2AgIrCl6	10.19	41.53	5.32	−83050.76	73.55	−2.10	−4.29
Rb2AgIrCl6	10.09	44.03	5.60	−63815.54	50.66	−2.06	−4.28
K2AgIrCl6	10.03	44.93	5.47	−54298.10	0.19	−2.03	−4.28

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To comprehensively assess the stability of the A₂AgIrCl₆ compounds, it is necessary to perform calculations for both their dynamic stability, represented by phonon dispersion calculations and their thermodynamic stability. However, since these calculations are computationally demanding and time-consuming, we initially evaluate the stability of the compounds by computing parameters such as final energy, decomposition enthalpy, formation energy, binding energy, Goldschmidt tolerance factor, octahedral factor, and a new tolerance factor. The final energy values per atom for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆ were determined to be −2149.64 eV, −2004.61 eV, and −2023.32 eV, respectively.

To evaluate the thermodynamic stability of A₂AgIrCl₆ compounds, we compute their decomposition energy along different potential pathways. The primary and most significant pathway involves the decomposition of A₂AgIrCl₆ compounds into the respective binary materials. Halide perovskites are generally synthesized via the reverse reactions of their constituent binary materials. Specifically, we determine the decomposition energy, which is defined as follows:

ΔH_D = 2E[ACl] + E[AgCl] + E[IrCl₃] − E[A₂AgIrCl₆] (1)

where, E[ACl], E[AgCl], E[IrCl₃] and E[A₂AgIrCl₆] are the final energies of ACl, AgCl, IrCl₃ and A₂AgIrCl₆, respectively.⁵² The computed ΔH_D values are given in Table 1. These values indicate that all A₂AgIrCl₆ compounds are thermodynamically stable, as reflected by their positive values. However, a clear trend is observed among the compound: K₂AgIrCl₆ exhibits a comparatively lower decomposition energy than Cs₂AgIrCl₆ and Rb₂AgIrCl₆, suggesting reduced thermodynamic stability and a higher tendency toward decomposition under practical conditions. Moreover, to assess the thermodynamical stability of the perovskite materials, formation energy (E_f) and binding energy (E_B) are calculated through the formula mentioned:⁵³In this context, signifies the overall energy of DPH compounds, whereas E_A, E_Ag, E_Ir and E_Cl represent the energies of individual A (A = Cs, Rb, K), Ag, Ir, and Cl atoms, correspondingly. The variable ‘n’ denotes the number of atoms and ‘µ’ symbolizes the free energy of each atom. The computed formation energies (E_f) and binding energies (E_B) for A₂AgIrCl₆ compounds are presented in Table 1. These values exhibit negativity, indicating that these three compounds should be synthesizable, and they comply with thermodynamical stability criteria. Of the three compounds, the E_f and E_B values indicate that K₂AgIrCl₆ compound is thermodynamically the least stable, in agreement with the decomposition energy analysis.

The crystallographic stability of perovskite structures is frequently determined using the Goldschmidt tolerance factor ‘t’ (as specified in eqn (4))⁵⁴ and the octahedral factor ‘u’ (as defined in eqn (5)),⁵⁵ both of which serve as trustworthy indications for predicting structural stability. ‘τ’, a new tolerance factor newly introduced by Bartel and co-workers to eqn (6),⁵⁶ has demonstrated a high level of prediction accuracy. The parameters t, u, and τ are calculated as follows:

In these equations, n_A is the oxidation state of A, with R_A > R_B by definition. R_A, R_B, and R_X are the ionic radii of A, B, and X ions, respectively in an ABX₃ structure. R_B is calculated as the average ionic radius of Ag⁺ and Ir³⁺ for double perovskites. A stable perovskites requires t between 0.81 – 1.11, u between 0.41 – 0.90, and τ ≤ 4.18.^54–56 These values were determined using Shannon's ionic radii⁵⁷ and are listed in Table 2. Although all compounds lie within the acceptable stability range for perovskite structures, K₂AgIrCl₆ lies closer to the lower stability limit and suggesting reduced structural stability compared to the Cs- and Rb-based systems. This observation is consistent with decomposition energy, formation energy and binding energy analysis. This suggests that while K₂AgIrCl₆ remains a viable candidate, its experimental synthesis and long-term stability may require more careful consideration.

Table 2 Shannon's ionic radii (r), Goldschmidt tolerance factor (t), octahedral factor (u), and new tolerance factor (τ) for A₂AgIrCl₆ compounds A = Cs, Rb, K

DPH	Ionic radius of cations (Å)		Ionic radius of Cl ion (Å)	Tolerance factor (t)	Octahedral factor (u)	New tolerance factor (τ)
Cs2AgIrCl6	rCs 1.88	(rAg + rIr)/2 0.92	rCl 1.81	0.96	0.51	3.83
Rb2AgIrCl6	rRb 1.72	(rAg + rIr)/2 0.92	rCl 1.81	0.91	0.51	3.93
K2AgIrCl6	rK 1.64	(rAg + rIr)/2 0.92	rCl 1.81	0.89	0.51	4.04

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3.2 Phonon stability and AIMD simulations

Fig. 3 displays the phonon dispersion curves (PDCs) and phonon density of states (PDOS) for A₂AgIrCl₆ compounds (A = Cs, Rb, K). A dynamics matrix, derived from force constants, illustrates the variation in force experienced by a standard atom due to the orientation of neighboring atoms. The matrices' diagonals are then isolated, representing eigen-values and eigenvectors, indicative of fundamental phonon frequencies and motion. The results show three acoustic and several optical phonon modes, consistent with the phonon dispersion curves. However, upon deploying conventional unit cells, it was found that two modes were degenerate, underscoring the importance of cell structure in determining degeneracy accurately.³³

Fig. 3 (a–c)The phonon dispersion spectra and corresponding phonon DOS for A₂AgIrCl₆ compounds where A = Cs, Rb, K.

The conventional unit cell of A₂AgIrCl₆ compounds comprises forty atoms, resulting in a variety of acoustic and optical phonon modes. Among these, three acoustic modes at the Γ point are characterized by low frequencies ranging from 0 to 1.25 THz, while the remaining optical modes fall within the high-frequency range of 1.25 to 10 THz. Fig. 3 indicates the absence of soft phonon modes in the material, as all phonon modes exhibit positive lattice vibration frequencies. This indicates that the material is dynamically stable within the present computational framework. The phonon stability of A₂AgIrCl₆ compounds is maintained along the pathways W to L, L to Γ, Γ to X, X to W, and W to K, suggests stability across the entire pathway. The PDOS of the compound is presented alongside the dispersion curve. It is shown that lattice vibrations at low frequencies, less than 1.0 THz, correspond to optical modes and are primarily impacted by A-site atoms, namely Cs, Rb, and K. On the other hand, the low and mid-frequency vibrations are mostly controlled by Ag, Ir, and Cl atoms. Higher frequencies primarily affect the vibrational behavior due to the interaction between Ir and Cl ions.

To further assess thermal behavior, AIMD simulations were performed using a 2 × 2 × 2 supercell at 300 K for 10 000 fs within the NVE ensemble.⁵⁸ The variation of total energy with simulation time is illustrated in Fig. 4. For Cs₂AgIrCl₆ shown in Fig. 4a, the total energy oscillates between −149.93 eV and −146.53 eV, giving an average value of −148.56 eV. The corresponding deviations from the mean energy are approximately 0.92% and 1.36%, indicating only minor thermal perturbations. In the case of Rb₂AgIrCl₆ represented in Fig. 4b, the energy values fall within a slightly narrower interval ranging from −149.97 eV to −147.85 eV, with an average of −148.83 eV. The deviations from the mean are relatively small (0.76% and 0.66%), suggesting stable dynamical behavior throughout the simulation period. Similarly, for K₂AgIrCl₆ depicted in Fig. 4c, the total energy varies from −149.59 eV to −147.29 eV, producing an average energy of −148.95 eV, with deviations of 0.42% and 1.11% from the mean value.

Fig. 4 The variation in energy profiles and temperature as a function of time step (fs) for (a) Cs₂AgIrCl₆, (b) Rb₂AgIrCl₆, and (c) K₂AgIrCl₆.

The absence of sudden energy jumps during the entire simulation interval indicates that the crystal frameworks remain preserved at room temperature. Such steady and bounded energy oscillations are characteristic of dynamically stable structures under finite-temperature conditions. Similar stability characteristics have previously been observed in other halide perovskite systems, such as A₂AlAgBr₆ (A = K, Rb, Cs)⁵⁹ and CuMCl₃ (M = Ge, Sn).⁶⁰ We have also extended our AIMD calculations to 30 000 fs at 300 K and 500 K, as shown in Fig. S1 (supplementary part), to further examine the finite-temperature structural stability of the investigated compounds within the simulation timescale. These observations provide additional computational evidence supporting the finite-temperature stability of the A₂AgIrCl₆ compounds within the limits of the present theoretical approach and may be relevant for future experimental and optoelectronic investigations.

3.3 Electronic properties

The features of direct band gap semiconductors make them more favorable for optical applications than those with indirect band gap.⁶¹ The GGA-calculated band gap (E_g) for Cs₂AgIrCl₆ is 0.34 eV, suggesting the potential for electron transitions between the valence band maximum (VBM) and the conduction band minimum (CBM). Substituting lighter alkali atoms (A = Cs, Rb, K) at the A-site in A₂AgIrCl₆ compounds has a minimal impact on E_g, with values ranging from 0.34 to 0.38 (see Table 3). In contrast, the TB-mBJ functionals yield much higher E_g values of 1.43 eV for Cs₂AgIrCl₆, 1.50 eV for Rb₂AgIrCl₆, and 1.55 eV for K₂AgIrCl₆ (see Table 3), respectively. This mismatch shows that compared to the TB-mBJ functional, GGA functionals tend to significantly underestimate E_g while maintaining the direct nature of band gap between VBM and CBM. The GGA functional underestimates the band gap,^62–64 while the TB-mBJ functional provides results that closely match the experimental values documented in the literature.^65–67 Both approximations consistently yield a direct band gap nature at the same high symmetry points. The calculated band gap of A₂AgIrCl₆ compounds exhibit a systematic increase when moving from Cs to Rb to K at the A-site, independent of the exchange–correlation functional used. This trend originates primarily from structural and electronic effects induced by A-site cation substitution, rather than direct electronic participation of the alkali metal states near the band edges. As the ionic radius of the A-site cation decreases (Cs⁺ → Rb⁺ → K⁺), the lattice constant correspondingly contracts, leading to shorter Ag–Cl and Ir–Cl bond lengths and enhanced octahedral tilting. This structural contraction strengthens the orbital overlap between Ir-5d and Cl-3p states, which predominantly define both the VBM and CBM. The increased crystal field splitting and modified p–d hybridization results in a widening of the band gap.

Table 3 Calculated band gap by different approaches, band gap nature, and calculated electron and hole effective mass values, reduced effective masses, exciton binding energy in meV and exciton radius in nm of the A₂AgIrCl₆ compounds where A = Cs, Rb, K

DPH	Band gap (eV)	Functional	Band gap nature	Effective mass of electron, m^*e	Effective mass of hole, m^*h	Reduced effective mass, µ^*r	Static dielectric constant (εs)	Exciton binding energy (Eb)	Exciton radius (a^*0)	Ref.
Cs2AgIrCl6	0.34	GGA PBE	Direct	0.13 me	1.10 me	—	—	—	—	This
	1.43	TB-mBJ	Direct	0.19 me	1.43 me	0.168 me	4.41	117.53	1.39
	1.42	TB-mBJ + SOC	Direct	—	—	—	—	—	—
Rb2AgIrCl6	0.36	GGA PBE	Direct	0.13 me	1.10 me	—	—	—	—	This
	1.50	TB-mBJ	Direct	0.18 me	1.70 me	0.163 me	4.27	121.63	1.39
	1.53	TB-mBJ + SOC	Direct	—	—	—	—	—	—
K2AgIrCl6	0.38	GGA PBE	Direct	0.13 me	1.64 me	—	—	—	—	This
	1.55	TB-mBJ	Direct	0.18 me	1.14 me	0.155 me	4.20	119.50	1.43
	1.60	TB-mBJ + SOC	Direct	—	—	—	—	—	—
Cs2NaIrCl6	0.92	GGA PBE	Direct	0.21 me	1.33 me	—	—	—	—	38
	1.93	TB-mBJ	Direct	0.30 me	1.69 me	—	—	—	—
Rb2NaIrCl6	0.97	GGA-PBE	Direct	0.20 me	1.10 me	—	—	—	—	38
	2.02	TB-mBJ	Direct	0.29 me	1.41 me	—	—	—	—
Cs2KIrCl6	1.08	GGA-PBE	Direct	0.23 me	1.51 me	—	—	—	—	39
	1.99	TB-mBJ	Direct	0.32 me	1.98 me	—	—	—	—
Rb2KIrCl6	1.12	GGA-PBE	Direct	0.23 me	1.24 me	—	—	—	—	39
	2.10	TB-mBJ	Direct	0.32 me	1.63 me	—	—	—	—
Cs2AgInCl6	3.02	HSE	Direct	0.272 me	0.520 me	0.178 me	2.760	319	0.819	78
Cs2AgIrF6	1.07	GGA PBE	Indirect	0.46 me	3.64 me	—	—	—	—	79
	1.65	TB-mBJ	Indirect	0.29 me	0.99 me	—	—	—	—
Rb2AgIrF6	1.13	GGA PBE	Indirect	0.47 me	4.98 me	—	—	—	—	79
	1.76	TB-mBJ	Indirect	0.28 me	0.94 me	—	—	—	—
K2AgIrF6	1.16	GGA PBE	Indirect	0.38 me	6.20 me	—	—	—	—	79
	1.83	TB-mBJ	Indirect	0.27 me	0.84 me	—	—	—	—

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Fig. 5 illustrates band structure computations for A₂AgIrCl₆ compounds using GGA-PBE (left) and TB-mBJ (right), revealing a significant behavioral similarity. Our discussion exclusively focuses on TB-mBJ for its superior accuracy in energy band gap values over GGA. To further evaluate relativistic effects, spin–orbit coupling (SOC) was included in the TB-mBJ calculations. Although Ir is a heavy element with strong intrinsic relativistic effects, a significant SOC influence might be expected. However, the calculated results show that SOC introduces only relatively minor changes in the electronic structure, mainly in the form of band splitting near high-symmetry points, while the overall band dispersion and band gap remain largely unchanged. This seemingly modest SOC effect can be understood from the nature of the band edge states. Even though Ir 5d orbitals are intrinsically sensitive to SOC, the valence and conduction band edges are not solely governed by Ir states. Instead, they originate from significant hybridization between Ir 5d and Cl 3p orbitals. This hybridization effectively spreads the electronic character over different atomic species, which in turn lessens the direct influence of SOC on the band edge positions. Consequently, SOC primarily perturbs the fine structure of the bands rather than inducing substantial shifts in the band gap. Similar behavior has been reported in related double perovskite systems, where orbital mixing mitigates the expected strong SOC influence of heavy elements.^19,68 Perovskite materials with band gap values between 0.8 and 2.2 eV⁶⁹ are highly fit for a comprehensive range of photovoltaic applications, particularly in photovoltaic conversion processes. The A₂AgIrCl₆ compounds, with their band gap falling within this ideal range, carry significant potential for development as photosensitive materials for future solar cell technologies. Their band gap properties make them promising candidates used for improving the efficiency then effectiveness of photovoltaic conversion in solar energy applications.

Fig. 5 Band structure of (a) Cs₂AgIrCl₆, (b) Rb₂AgIrCl₆, and (c) K₂AgIrCl₆ compounds using GGA-PBE (left panel, green color) and TB-mBJ (right panel, pink color) functionals.

Analyzing the density of states (DOS) and electronic band structure plots for the A₂AgIrCl₆ compounds (see Fig. 5 and 6), our investigation reveals that the VBM predominantly comprises the character of Cl(3P) and Ir(5d). Specifically, the valence band dispersion immediately under the Fermi level is significantly influenced by the Ir 5d and the Cl 3p orbitals. In the case of Cs₂AgIrCl₆, the calculated normalized contribution to the VBM is 65.2% for Ir(5d) and 20.6% for Cl(3p). Similarly, for Rb₂AgIrCl₆ (and K₂AgIrCl₆), the contributions to the VBM for Ir(5d) and Cl(3p) are 65.0 (64.5)% and 19.2 (17.9)%, respectively. Interestingly, the alkali and Ag atoms demonstrate minimal contributions, hovering around 1–2%, to the VBM for A₂AgIrCl₆ compounds. Conversely, our examination of the CBM reveals a predominant derivation from Ir(5d) empty anti-bonding states with significant contributions from Ag, Ir, and Cl states, causing its dispersion well above the Fermi level. The normalized contributions to the CBM for A₂AgIrCl₆ compounds are (57.2, 57.1, 56.8)% for Ir(5d), (27.1, 27.0, 26.8)% for Cl(3p), and (5.8, 5.8, 6.0)% for Ag(4d), respectively. Notably, these contributions exhibit marginal variations upon the substitution of the A-site cation with lighter alkali group elements. Considering the substantial separation between the CBM and VBM, coupled with the VBM's proximity to the Fermi level, these results suggest a valence-band structure dominated by Ir–Cl states, which may favor hole participation in transport.⁷⁰ Very large DOS peaks just below the Fermi level come from the nearly localized Ir(5d) electrons. The electronic properties of A₂AgIrCl₆ DPHs are governed by strong Ir–Cl–Ag hybridization, which controls band gap and reduces effective SOC impact at band edges.

Fig. 6 Density of states of (a) Cs₂AgIrCl₆, (b) Rb₂AgIrCl₆, and (c) K₂AgIrCl₆ compounds.

Understanding the photovoltaic properties of solar cells depends on study on the effective masses of electrons and holes as it directly influences carrier mobility, resistivity, and the optical response of free charge carriers.⁷¹ We observed that the VBM is less dispersed compared to the CBM. A comparable flat valence band feature has been observed in bulk Cs₂AgInCl₆, mainly due to the hybridization of Ag 4d and Cl 3p orbitals.⁷² In such cases, it's common for the hole effective mass (m^∗_h) linked with the VBM to be greater than the electron effective mass (m^∗_e) of the CBM. Consequently, the hole mobility is less than electron mobility. This is because, the band curvature is inversely related to the second derivative of energy concerning the wave vector k and is elucidated by the dispersion relationship:⁷³

In this context, the symbols “+” and “–“ represent electrons and holes, respectively. Here, m^∗ refers to the effective mass of either the electron or hole, k stands for the wave vector, E(k) denotes the energy as a function of k, and ℏ represents the reduced Planck constant. The second derivative of energy regarding the wave vector, d²E(k)/dk², is obtained by fitting a parabolic function to the E–k dispersion at the symmetry points. The estimated effective masses of electrons and holes for the investigated compounds are summarized in Table 3. For Cs₂AgIrCl₆, the electron's effective mass is approximately 0.19 m_e and the hole's is 1.43 m_e, with m_e being the free electron mass. For Rb₂AgIrCl₆, these values are 0.18 m_e and 1.70 m_e, respectively; and for K₂AgIrCl₆, they are 0.18 m_e and 1.14 m_e. All these values were computed using the TB-mBJ functionals. It is found that while the effective masses of holes are larger than the free electron mass (m_e), the effective masses of electrons are all smaller. The fact that the hole effective masses (m^∗_h) in Table 3 are significantly greater than the electron effective masses (m^∗_e) indicates a pronounced asymmetry between electron and hole effective masses, which may influence carrier transport behavior.⁷⁴ Additionally, while effective mass and mobility are inversely related, Table 3 shows that both charge carriers have very modest effective masses, indicating increased carrier mobility.⁷⁵ With promising performance across a wide variety of solar applications, our findings demonstrate the great potential of A₂AgIrCl₆ compounds for usage in photovoltaic technology.

The exciton binding energy E_b plays a critical role in solar cell materials. For efficient charge separation, this energy needs to be low. A low E_b means that thermal energy at room temperature can easily separate bound electron–hole pairs (excitons) into free charge carriers. In contrast, a high exciton binding energy keeps electrons and holes paired, reducing carrier mobility, and increasing recombination losses, which negatively impact the solar cell's performance. Excitonic behavior arises from the Coulomb attraction between photo-generated electrons and holes. We have calculated E_b for the A₂AgIrCl₆ compounds using the Wannier–Mott exciton model with the static dielectric constant by the following relation:⁷⁶

where,where, R = 13.6057 eV is the Rydberg energy constant, a₀ = 0.5292 Å is the Bohr radius constant, ε_s is the static dielectronic constant, m_e is the mass of free electron and µ^∗_r is the reduced effective mass of the electron–hole pair. Table 3 lists the resulting excitonic parameters. For Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, we have found exciton binding energies of approximately 117.53 meV, 121.63 meV, and 119.50 meV, and corresponding exciton radii on the order of 1.39 nm, 1.39 nm, and 1.43 nm. These exciton binding energies are somewhat smaller and exciton radii larger when compared with the previous similar studies.^77,78 The comparatively low exciton binding energy and larger exciton radius indicate a weaker electron–hole Coulomb interaction, so that excitons dissociate quickly into free electrons and holes, which is favorable for the solar cell applications. Therefore, the A₂AgIrCl₆ compounds are promising candidates for photovoltaic applications due to their low exciton binding energies and rapid generation of free charge carriers.

3.4 Charge density

To better understand the electronic behavior of the investigated compounds, the cross-sectional charge density distribution of A₂AgIrCl₆ compounds within the (110) crystallographic plane was computed and is shown in Fig. 7. The charge density was calculated using the TB-mBJ approach based on a fully converged wavefunction.^80,81 From the contour plots, one can observe a mixed bonding character consisting of partial ionic and covalent interactions within the crystal. A pronounced charge accumulation is observed along the Ir–Cl bonds, which suggests significant covalent interaction between Ir and Cl atoms. This behavior may be attributed to the hybridization between Ir-5d and Cl-3p orbitals and is consistent with the PDOS results. In contrast, the Ag–Cl bonds exhibit relatively weaker and more symmetric charge distribution, suggesting moderate covalent character with a partial ionic contribution arising from Ag-4d and Cl-3p hybridization. Furthermore, no significant charge overlap is observed directly between Ag and Ir atoms, indicating the absence of direct bonding and suggesting that their interaction is mediated through Cl atoms. On the other hand, the absence of substantial charge accumulation in the regions between the A-site cations (Cs⁺/Rb⁺/K⁺) and Cl⁻ ions, along with the nearly spherical charge distribution around the A-site atoms, is consistent with predominantly ionic A–Cl interactions. Therefore, the bonding in A₂AgIrCl₆ compounds can be qualitatively described as a combination of relatively strong Ir–Cl covalent interactions, weaker Ag–Cl hybridization, and predominantly ionic A–Cl interactions, which collectively govern the electronic structure of the A₂AgIrCl₆ compounds. Across the series from Cs to K, the overall charge distribution pattern remains similar, although a slight increase in charge overlap is observed with decreasing lattice size, which may be associated with enhanced orbital interaction. Similar qualitative trends have been reported in earlier studies.^19,79,82,83 It should be noted that the present bonding interpretation is qualitative in nature and that more quantitative bonding characterization would require additional analyses such as Bader charge analysis, electron localization function (ELF), crystal orbital Hamilton population (COHP), or related approaches.

Fig. 7 Charge density of A₂AgIrCl₆ compounds where A = Cs, Rb, K.

3.5 Optical properties

The optical response of A₂AgIrCl₆ compounds was analyzed to understand light-matter interactions and their relation to the electronic structure. The calculated direct band gaps (1.43–1.55 eV) suggest that these compounds are suitable for visible-light absorption. The frequency dependent complex dielectric function encapsulates the complete material response to electromagnetic radiation-induced perturbations,⁸⁴ expressed by the equation ε(ω) = ε₁(ω) + iε₂(ω).⁸⁵ The real part of dielectric constant ε₁(ω) describes the dispersive response of the material, while the imaginary part of dielectric constant ε₂(ω) represents interband electronic transitions. In Fig. 8a and b, we compared the spectra of ε₁(ω)and ε₂(ω) for A₂AgIrCl₆ compounds within the energy range of 0 eV to 14 eV. The static dielectric function ε₁(0) assumes values of 4.41, 4.27, and 4.20 for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively, mirroring those of Cs₂AgRhCl₆ (A = Li, K, Na, Rb, Cs),³³ Cs₂XRhCl₆ (X = Na, K),⁸⁶ and Cs₂AgBiCl₆.⁸⁷ As the A-site cation changes from Cs⁺ to Rb⁺ to K⁺, the ionic radius decreases, leading to a gradual contraction of the lattice and enhanced distortion of the [AgCl₆] and [IrCl₆] octahedra. This structural contraction reduces the electronic polarizability of the lattice, which directly results in a decrease in the static dielectric constant ε₁(0) from Cs₂AgIrCl₆ to K₂AgIrCl₆. Substitution of heavier alkali metals results in a higher value. Additionally, ε₁(ω) rises with ascending photon energy for each perovskite material, reaching maximum values of 10.84, 10.76, and 10.55 at energies 2.24 eV, 2.33 eV, and 2.38 eV for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively. Following this, there is a rapid decline in ε₁(ω) characterized by oscillations and multiple peaks. In certain intervals, ε₁(ω) records values below zero, indicating an impediment for photons with corresponding energies to penetrate the solid materials. Penn's model delineates an inverse correlation between static polarization and the band gap, articulated as:⁸⁸In this expression, ℏ denotes the reduced Planck constant, and ω_p refers to the plasma frequency.

Fig. 8 (a–i) Optical properties of A₂AgIrCl₆ compounds where A = Cs, Rb, K.

The significance of ε₂(ω)lies in its pivotal role in determining the peak optical absorption and monitoring interband electronic transitions within materials. Due to inherent limitations in DFT, there are slight deviations in the electronic transition points between the valence and conduction bands, as evidenced in Fig. 8b. The maximum value of ε₂(ω), often referred to as the first absorption peak (FAP), is indicative of significant electronic transitions near the Fermi energy. In the A₂AgIrCl₆ compounds, the FAP appears at photon energies of approximately 2.38 eV, 2.46 eV, and 2.52 eV, respectively, highlighting minor variations in electronic structure influenced by the choice of A-site cation. These are situated at energies higher than those observed in the ε₁(ω) curves (refer to the above). The primary focus for these compounds is on the highest FAP values. These values fall within the visible range, indicating an effective absorption capacity for visible light. In comparison to the DOS, the first absorption spectrum (FAS) primarily stems from the Ir-d and Cl-p orbitals at the VBM and the CBM points.

The refractive factor n(ω), extinction factor k(ω), absorption coefficient α(ω), conductivity σ(ω), reflectivity R(ω), and loss factor L(ω), which are essential optical constants, are derived from the dielectric properties of all compounds using prescribed equations given below:^89,90

The complex refractive index (n + ik) is an essential characteristic of solid materials, offering information about how light travels through them and their possible use in optoelectronic technologies.⁹¹ Using eqn (12) and (13), we determined the real components of the complex refractive index. The value of n(ω) varies depending on the material, while semiconductors typically have a small k(ω).⁹² The outcomes of the refractive index calculation are presented in Fig. 8c. The real part of dielectric function shows a behavior akin to that of the refractive index since n²(ω) = ε₁(ω).⁵⁵ The static refractive index values for A₂AgIrCl₆ compounds at zero frequency are 2.09, 2.06, and 2.04, respectively. Consequently, the higher dielectric screening in Cs₂AgIrCl₆ leads to a larger refractive index, while reduced screening in K₂AgIrCl₆ yields a smaller n(0). The peak values of n(ω) at energies of 2.27, 2.35, and 2.41 eV for A₂AgIrCl₆ compounds are 3.44, 3.43, and 3.40. The measured refractive indices fall within the typical range (2.5 to 3.5)⁹¹ reported for DPHs, indicating consistent optical behavior. The extinction coefficient k(ω) exhibits a consistent resemblance to the patterns of ε₂(ω) and α(ω), indicating the detection of electromagnetic wave loss in materials, as illustrated in Fig. 8d. The relationship between k(ω) and α(ω) is as follows: .

Device performance is strongly influenced by the material's light absorption capability.⁹³ The absorption coefficient, α(ω), in semiconductors plays a crucial role in explaining how materials absorb photons and produce electron–hole pairs, which is fundamental to the functionality of photovoltaic devices like solar cells. In the present case, the absorption coefficient α(ω) exhibits a pattern similar to that of ε₂(ω), as evidenced by their coefficients at different energies and wavelengths in Fig. 8e. For A₂AgIrCl₆ compounds, the absorption edge aligns well with the electronic band gap, confirming the direct-gap nature of these materials. The curve displays initial peaks at 2.46, 2.57, and 2.62 eV, respectively, all within the visible range. This emphasizes their suitability for visible-light-driven applications such as solar cells and photodetectors.⁹⁴ The appearance of many peaks within the higher energy range (6.0 to 13.5 eV), each with differing intensities, shows a diverse of transitions occurring between occupied and unoccupied electronic states. The first peak of the absorption coefficient (α_max(ω)) is observed approximately 4.9 × 10⁵, 5.1 × 10⁵, and 5.2 × 10⁵ cm⁻¹ at photon energies of 2.49, 2.57, and 2.62 eV for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively. The absorption coefficients of our investigated compounds surpass those of others, ranging from 6.12 to 6.58 × 10⁴ cm⁻¹ for A₂CuSbX₆ (A = Cs, Rb, K; X = Cl, Br, I),⁹⁵ 3 to 6.5 × 10⁴ cm⁻¹ for Cs₂CuBiX₆ (X = Cl, Br, I),⁹⁶ 2.3 to 3.4 × 10⁵ cm⁻¹ for Cs₂AgBiX₆ (X = Cl, Br, I).⁹⁷ They are consistent to those of Cs₂XRhCl₆ (ref. 86) and Cs₂AgRhCl₆ (ref. 33) (around 10⁵), as observed in the highest peaks of DPH compounds within the visible range (400–700 nm). As shown in Fig. 8f, the absorption peaks for the A₂AgIrCl₆ compounds are observed at 498 nm, 482 nm, and 477 nm, respectively. These peaks all fall within the visible light spectrum of 380–780 nm, indicating that these DPH have strong absorption coefficients for visible light. The optical band gap in the studied DPH is determined through the established Tauc plot derived from absorption coefficient spectra.⁹⁸ Precisely determining the optical band gap is crucial for predicting semiconductor properties. However, improper application of the Tauc plot may lead to misinformation, especially in estimating the band gap.⁹⁹ The absorption behavior of a material can be analyzed using the Tauc equation, (αhν)^1/η = A(hν − E_g), where, α for absorption coefficient, h for Planck's constant, ν for photon frequency, A is a constant specific to the material, and E_g is the optical band gap energy.⁹⁸ The exponent η is determined by the nature of the band gap, generally set to 1/2 for direct band gap semiconductors and 2 for indirect ones. Based on the electronic band structure, A₂AgIrCl₆ compounds exhibits direct band gap; therefore, η = 1/2 was used in constructing the Tauc Plots. The linear fitting was performed in the near-edge region of the absorption spectra, where (αhν)² shows approximately linear behavior, while avoiding both the low-energy tail and higher-energy non-linear regions. As shown in Fig. 8f, the estimated optical band gap for A₂AgIrCl₆ compounds is calculated as 1.40, 1.43, and 1.48 eV, respectively, using the TB-mBJ functional. These values are consistent with those obtained from the electronic structure calculations, reinforcing the reliability of the results.

Optical conductivity, described by eqn (15), governs electron conduction in materials when exposed to specific photon frequencies. Optoelectronic devices require the optical conductivity, linked to interband electron movement, to be between 1.4 eV and 4.0 eV in the visible spectrum.³⁶ Alkali metal substitution has minimal impact on conductivity. The primary conductivity peaks for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆ occur at 2.38 eV (3534 Ω⁻¹ cm⁻¹), 2.46 eV (3691 Ω⁻¹ cm⁻¹), and 2.51 eV (3699 Ω⁻¹ cm⁻¹), respectively, with the highest peak observed in the range of 5200 to 5500 Ω⁻¹ cm⁻¹ from 11 to 13 eV, as illustrated in Fig. 8g. These peaks, predominantly influenced by Ir-5d states, play a crucial role in the material's overall electronic structure. These findings affirm the high optical conductivity of these materials in the visible region.

The reflectance spectra of A₂AgIrCl₆ compounds are analyzed in Fig. 8h. Calculated static reflection coefficients R(0) for A₂AgIrCl₆ compounds are 0.125, 0.121, and 0.118. Minimal reflectivity R(ω) suggests efficient light capture within the energy range from zero to the band gap. R(0) increases with the substitution of heavier alkali atoms, and peak positions around 2.5 eV for A₂AgIrCl₆ compounds correspond to a maximum reflectance of about 38%. Fig. 8i displays the energy loss function, representing the electrons' energy loss as they move. Notable peaks at approximately 3.52 eV (Cs₂AgIrCl₆), 3.68 eV (Rb₂AgIrCl₆), and 3.76 eV (K₂AgIrCl₆) indicate electron dispersion around these energies, correlating with optical conductivity. Materials with low reflectivity and loss function values are valuable for potential use in solar cell absorbing layers. The computed values of real part of dielectric constant ε₁(0), refractive index n(0), reflectivity R(0), first peak of the absorption coefficient α_max(ω), and optical band gap using Tauc plot for A₂AgIrCl₆ compounds are given in Table 4. Overall, the calculated optical properties indicate that A₂AgIrCl₆ compounds exhibit favorable light absorption and electronic transitions for optoelectronic applications.

Table 4 Computed values of real part of dielectric constant ε₁(0), refractive index n(0), reflectivity R(0), first peak of the absorption coefficient α_max(ω), and optical band gap using Tauc plot for A₂AgIrCl₆ compounds where A = Cs, Rb, K

DPH	ε1(0)	n(0)	R(0)	αmax(ω) × 10^5 cm^−1 at eV	Optical band gap (eV)	Ref.
Cs2AgIrCl6	4.41	2.10	0.125	4.90 at 2.49	1.40	This
Rb2AgIrCl6	4.27	2.07	0.121	5.10 at 2.57	1.43	This
K2AgIrCl6	4.20	2.05	0.118	5.20 at 2.62	1.48	This
Cs2KIrCl6	2.77	1.66	0.06	3.50 at 3.00	—	39
Rb2KIrCl6	2.68	1.64	0.06	3.80 at 3.20	—	39
Cs2NaIrCl6	2.78	1.67	0.06	3.90 at 4.00	—	38
Rb2NaIrCl6	2.65	1.63	0.06	3.50 at 4.20	—	38
Cs2AgIrF6	3.77	1.94	0.10	5.10 at 2.96	1.52	79
Rb2AgIrF6	3.52	1.88	0.09	5.40 at 3.00	1.61	79
K2AgIrF6	3.44	1.85	0.09	5.50 at 3.03	1.75	79
Cs2AuYCl6	2.44	1.55	0.11	0.23	—	82
Rb2AuYCl6	2.36	1.52	0.11	0.35	—	82

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3.6 Mechanical properties

Though the study of opto-electronic properties of the titled compounds is our prime motivation, we have also studied the mechanical properties with the intention of checking the mechanical stability and their ductility (ductility is always desired for fabricating any devices), which is then further extended up to elastic moduli as a routine check. Studying important characteristics such bulk modulus, shear modulus, Young's modulus, Poisson's ratio, and Pugh's ratio help one to have a thorough knowledge of a material's mechanical performance. These characteristics offer vital information about the material's stiffness, brittleness or ductility, and toughness characteristics—all of which are crucial for determining if it is appropriate for a certain technical application. These properties collectively dictate how the material reacts to strain, significantly influencing its overall mechanical performance. The change in the A site from Cs to Rb to K in A₂AgIrCl₆ (A = Cs, Rb, K) compounds induces a shift in the lattice parameter, significantly impacting the material's elastic constants, as outlined in Table 5. It is imperative to assess mechanical stability before delving into the analysis of these characteristics. Typically, C_ij coefficients are used to provide the criteria for Born stability, which determines the mechanical stability of a lattice.^100,101 These coefficients play an important role in determining the structural integrity of a material under varied mechanical stresses. C₁₁, C₁₂, and C₄₄ are the three particular elastic constants (C_ij) of the cubic double perovskite structure. Determining the lattice's mechanical stability requires knowledge of these constants. For a cubic crystal, Born stability conditions are C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0, and C₄₄ > 0. These conditions correspond to the Born criterion, the spinodal criterion, and the shear criterion, respectively. More precisely, to guarantee the material's mechanical stability, the spinodal criteria, which is connected to the bulk modulus, needs to be positive.¹⁰² Table 5 shows that A₂AgIrCl₆ compounds meets all Born, spinodal, and shear criteria, demonstrating mechanical stability. Compared to other considered perovskites, the Rb₂AgIrCl₆ double perovskite exhibits the highest elastic C_ij value. By using the Cauchy pressure, which is Cʺ = C₁₂ − C₄₄, one can determine if a material is brittle or ductile. A negative Cʺ value indicates brittleness, whereas a positive number indicates ductile behavior. All of the compounds under investigation have positive Cauchy pressure values, as shown in Table 5, which validates their ductile nature. Of them, K₂AgIrCl₆ has the highest Cʺ value, indicating that it is the most ductile, while Cs₂AgIrCl₆ has the lowest, showing that it is somewhat less ductile. Additionally, a distinct pattern is seen in the A₂AgIrCl₆ compounds, where the Cauchy pressure falls as the A-site cation's ionic radius increases. This inverse association suggests that the compound's decreased ductility is a result of higher A-site cations.

Table 5 Elastic constants C_ij (GPa), mechanical stability criteria, Cauchy pressure, and calculated mechanical parameters-bulk modulus (B), Shear modulus (G), Young modulus (Y), Poisson's ratio (ν), Pugh's ratio (B/G), and anisotropy coefficient A_Z-for A₂AgIrCl₆ compounds where A = Cs, Rb, K

Parameters		Cs2AgIrCl6	Rb2AgIrCl6	K2AgIrCl6
Born stability	C11 (GPa)	90.86	99.71	97.74
	C12 (GPa)	19.55	19.59	18.15
	C44 (GPa)	18.18	15.62	13.37
	C11–C12 (GPa)	71.31	80.12	79.59
	C11 + 2C12 (GPa)	129.96	138.89	134.04
Cauchy pressure, CP (GPa)		1.37	3.97	4.78
Bulk modulus, B (GPa)		43.32	46.30	44.68
Shear modulus, G (GPa)		23.97	23.03	21.07
Young modulus, Y (GPa)		60.70	59.26	54.63
Poisson's ratio, ν		0.27	0.29	0.30
Pugh's ratio, B/G		1.81	2.01	2.12
Zener anisotropy index, AZ		0.51	0.39	0.34

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The Shear modulus (G), Bulk modulus (B), B/G, Young modulus (Y), and Poisson's ratio (ν) are calculated based on elastic constants using the Voigt–Reuss–Hill approximation¹⁰³ (refer to Table 5). The values assigned to Y, namely 60.70, 59.26, and 54.63 GPa for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively, serve as indicators of the respective materials' rigidity. Notably, the higher Y value for Cs₂AgIrCl₆ implies a greater stiffness when compared to Rb₂AgIrCl₆, and K₂AgIrCl₆. The observation that Rb₂AgIrCl₆ exhibits a larger B value than other two compounds further reinforce its superior resistance to deformation. Moving on to brittleness and ductility classification via the parameter ν, the Poisson's ratio with a threshold value of 0.26,¹⁰⁴ Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆ are categorized as ductile materials with ν values of 0.27, 0.29, and 0.30, respectively. The B/G ratio, another crucial factor indicative of ductility or brittleness,¹⁰⁵ supports the notion of ductility for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, with computed values of 1.81, 2.01, and 2.12, respectively. These values align with the calculated ν values, collectively suggesting a susceptibility to heat shocks. The positive Cauchy pressure values, specifically 1.37 for Cs₂AgIrCl₆, 3.97 for Rb₂AgIrCl₆, and 4.78 GPa for K₂AgIrCl₆, further confirm the ductile nature of the studied compounds, providing additional validation for their resistance to external stresses. An important metric for material characterization is the Zener anisotropy index (A_Z), determined by the relation A_Z = 2C₄₄/(C₁₁ − C₁₂).¹⁰⁶ The resulting A_Z values of 0.51 for Cs₂AgIrCl₆, 0.39 for Rb₂AgIrCl₆, and 0.34 for K₂AgIrCl₆ unequivocally classify all the studied materials as anisotropic in nature. This aligns with findings from other perovskite materials, such as Cs₂CuIrF₆,³⁴ A₂CuSbX₆ (A = Cs, Rb, K; X = Cl, Br, I),⁹⁵ Cs₂AgBiX₆ (X = Cl, Br, I)⁹⁷ which were identified as mechanically stable, hard, incompressible, and anisotropic.

Additionally, the corresponding anisotropic elastic parameters for each phase may be used to determine the characteristics of homogeneous isotropic and anisotropic polycrystals, such as B, G, Y, and ν (see Table 6). The ELATE software is utilized to calculate these attributes.¹⁰⁷ Additional proof of anisotropic characteristics can be seen in the 3D contour plots for A₂AgIrCl₆ compounds in Fig. 9. Anisotropy is shown in Y when A is greater than one, meaning that none of the materials under study are spherical. A symmetric spherical shape exhibiting isotropy in linear compressibility (β) is generated when A = 1. Based on the β values, it appears that the materials being studied behave in an isotropic manner. Moreover, anisotropy is shown in the ranges of maximum and least deformation for G and ν under applied stresses. Cs₂AgIrCl₆ < Rb₂AgIrCl₆ < K₂AgIrCl₆ is the order of anisotropy in elastic moduli.

Table 6 Minimal and maximal values of elastic modulus and elastic anisotropy of A₂AgIrCl₆ compounds where A = Cs, Rb, K

DPH	Young's modulus (GPa)		Linear compressibility (TPa^−1)		Shear modulus (GPa)		Poisson's ratio
	Ymin	Ymax	βmin	βmax	Gmin	Gmax	νmin	νmax
Cs2AgIrCl6	47.85	83.94	7.69	7.69	18.18	35.65	0.11	0.47
Rb2AgIrCl6	41.86	84.81	7.64	7.64	15.62	36.06	0.10	0.53
K2AgIrCl6	36.47	92.05	7.46	7.46	13.37	39.79	0.07	0.61

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Elastic anisotropy Ax
	AY	Aβ	AG	Aν
Cs2AgIrCl6	1.75	1.00	1.96	4.19
Rb2AgIrCl6	2.03	1.00	2.31	5.37
K2AgIrCl6	2.52	1.00	2.98	8.30

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Fig. 9 Three-dimensional directional profiles of (a) Young's modulus, (b) linear compressibility, (c) shear modulus, and (d) Poisson's ratio for the Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆ compounds.

3.7 Thermal properties

One important thermodynamic parameter that is necessary to comprehend the characteristics and heat capacity of solids at various temperatures is the Debye temperature (θ_D). It provides information on how material properties change as a result of temperature changes. To compute θ_D, a number of theoretical models have been established; these models usually use formulas that take into account the average sound velocity (v_m).¹⁰⁸ This parameter plays a crucial role in studying and predicting the thermal behavior and stability of materials.In the equation mentioned, ℏ denotes the normalized Planck's constant, k_B signifies Boltzmann's constant, n stands for the number of atoms, V indicates the volume, and v_m refers to the average velocity of sound. Additionally, v_m can be determined using the following formula:¹⁰⁸In the above expression, v_t represents the transverse sound velocity, while v_l denotes the longitudinal sound velocity. Both v_t and v_l can be computed using the given equation:¹⁰⁸

The Debye temperatures have been found to be 246.24 K, 256.63 K, and 263.82 K for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively.

Materials used in the manufacturing of solar cells are heated to high temperatures for procedures like the formation of metal contact or the development of crystalline silicon. A material with a low melting point runs the danger of deforming, cracking, or even melting in these circumstances, creating flaws that impair the finished product's functionality. However, if the melting point of the material is too high, the fabrication process becomes more difficult and costly. Since it offers a helpful indicator of the material's resistance to deformation under heat, the elastic constant C₁₁ is frequently employed to calculate a material's melting temperature. The following formula can be used to determine the melting temperature:¹⁰⁹

T_m = (553 + 5.91C₁₁)K (22)

The estimated melting temperatures indicate that these compounds can be synthesized under ambient conditions. Of the three, Rb₂AgIrCl₆ has a higher melting temperature than both Cs₂AgIrCl₆ and K₂AgIrCl₆. The thermodynamic properties calculated from the elastic constants are listed in Table 7.

Table 7 Calculated density (ρ), sound velocities—longitudinal (v_l), transverse (v_t), and average (v_m) along with Debye temperature (θ_D) and melting temperature (T_m) for A₂AgIrCl₆ compounds where A = Cs, Rb, K

Parameters	Cs2AgIrCl6	Rb2AgIrCl6	K2AgIrCl6
ρ × 10^3(Kg m^−3)	4.89	4.42	3.89
vl (m s^−1)	3930.01	4164.74	4327.21
vt (m s^−1)	2217.58	2277.51	2328.53
vm (m s^−1)	2466.74	2539.68	2599.67
θD (K)	246.24	256.63	263.82
Tm (K)	1089.97	1142.31	1130.65

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We assessed the stability of the sample by analyzing its phonon modes at various temperatures, which allowed us to calculate key thermodynamic values like enthalpy, free energy, and entropy. These values are influenced by the frequency of phonon vibrations, which are derived by:¹¹⁰

In this context, g(ω) represents the phonon density of states, k_B stands for the Boltzmann's constant, E_tot denotes the minimum total energy, and ℏ refers to the reduced Planck constant. Fig. 10 shows how the predicted thermodynamic parameters of the A₂AgIrCl₆ compounds with temperature owing to phonon states. These variables include enthalpy, free energy, entropy, heat capacity, and Debye temperature. The data shows that while free energy falls with temperature, enthalpy, entropy, and heat capacity all rise.

Fig. 10 (a–c)Thermodynamical properties of A₂AgIrCl₆ compounds where A = Cs, Rb, K.

Additionally, we note a more significant enhancement in the heat capacity of the A₂AgIrCl₆ compounds compared to changes in enthalpy and entropy, suggesting its heightened sensitivity to temperature variations. According to the data in Fig. 10a, the enthalpy, entropy, and free energy values at room temperature are determined as 0.56 (0.56, 0.47) eV, 1.33 (1.36, 1.13) eV, and 0.77 (0.80, 0.66) eV, respectively, for Cs₂AgIrCl₆ (Rb₂AgIrCl₆, K₂AgIrCl₆). The computed enthalpy values at 1000 K are 2.35 eV, 2.34 eV, and 2.07 eV for Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, respectively. The heat capacity gradually rises with temperature and eventually stabilizes at 59.45 (58.85, 53.03) cal/cell.K for Cs₂AgIrCl₆ (Rb₂AgIrCl₆, K₂AgIrCl₆) shown in Fig. 10b, a phenomenon known as the Dulong-Petit limit.¹¹¹ The variation of Debye temperature as a function of temperature is shown in Fig. 10c. For the compounds Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆, the estimated Debye temperature at 0 K is about 116.96 K, 85.87 K, and 122.04 K, respectively. Under constant pressure, the Debye temperature rises with temperature. These thermodynamic characteristics provide insights for comparing with experimental results to predict phase stability.

4 Conclusions

This study systematically addresses three new A₂AgIrCl₆ compounds (A = Cs, Rb, K) and analyzes their structural, electrical, optical, mechanical, and thermophysical characteristics using first-principles calculations. Stability analyses suggest that these compounds are structurally, thermodynamically, and dynamically stable within the present theoretical framework. The computed electronic structures indicate that the direct band gap of Cs₂AgIrCl₆, Rb₂AgIrCl₆, and K₂AgIrCl₆ are 1.43 eV, 1.50 eV, and 1.55 eV, respectively, which fall within a favorable range for optoelectronic applications. The relatively low value of electron effective masses and exciton binding energies suggest beneficial charge transport and separation characteristics. In addition, the study of the optical properties indicates strong visible light absorption characteristics (up to 10⁵ cm⁻¹) and very low reflectivity, which suggests their promise for use as light-harvesting materials. Mechanical parameters such as Pugh's ratio, Cauchy pressure, and Poisson's ratio indicate ductile and anisotropic characteristics, reflecting structural robustness. The calculated mechanical and thermophysical descriptors suggest potentially favorable stability within the limits of the present theoretical approach. Among the investigated compounds, K₂AgIrCl₆ appears comparatively less robust than the Cs- and Rb-based analogues based on the calculated stability-related parameters. Collectively, this study provides theoretical evidence that A₂AgIrCl₆ compounds are promising lead-free double perovskites with favorable optoelectronic characteristics. These findings offer useful guidance for future experimental studies and device-oriented investigations.

Ethical statement

This work did not require ethical approval from a human subject or animal welfare committee.

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

Data used in this study are available from the corresponding author upon appropriate request.

Supplementary information (SI) is available. See DOI: https://doi.org/10.1039/d6ra02981g.

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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