First-principles study of lead-free A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) defect -perovskites: structural, electronic, optical, and mechanical properties

Abstract

The development of environmentally benign and stable alternatives to lead-based perovskites remains a key challenge for next-generation photovoltaic materials. In this work, a comprehensive first-principles investigation of lead-free alkali indium halide defect-perovskites A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) is performed to elucidate the influence of cation and anion substitution on their structural, electronic, optical, and mechanical properties. Density functional theory calculations using GGA-PBE and PBEsol functionals confirm that all compositions crystallize in a stable cubic phase with negative formation energies and favorable Goldschmidt tolerance factors, where substitution of the larger Cs⁺ cation enhances lattice stability compared to Rb⁺. Systematic halide substitution from Cl⁻ to Br⁻ to I⁻ induces lattice expansion and a pronounced reduction in the direct band gap at the Γ-point, enabling effective band gap tunability. Electronic structure analysis reveals that the valence-band maximum is dominated by halogen p states, while the conduction band is primarily governed by In-derived states, underscoring the decisive role of anion chemistry in controlling electronic transitions. Optical calculations demonstrate enhanced dielectric response and significantly improved visible-light absorption for iodide-rich compositions, whereas chloride-based compounds retain wide band gaps and stronger structural rigidity. Mechanical and elastic analyses indicate that both cation and anion substitution modulate lattice stiffness, ductility, and hardness, with Cs- and I-based compounds exhibiting increased mechanical softness and improved machinability.

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1. Introduction

The advent of organic–inorganic hybrid halide perovskites, notably CH₃NH₃PbI₃, has catalyzed a revolution in photovoltaics, with certified power conversion efficiencies (PCEs) soaring from 3.8% to over 25% within a decade.^1–4 This remarkable progress is attributed to their exceptional optoelectronic properties, including high absorption coefficients, ambipolar charge transport, long carrier diffusion lengths, and tunable band gaps, all achievable via low-temperature, solution-processable fabrication.^5,6 However, the path to commercialization is obstructed by two formidable challenges: the inherent instability of these materials under environmental stressors like moisture, heat, and light, and the toxicity of water-soluble lead (Pb²⁺), which raises significant environmental and health concerns.^7,8

The quest for lead-free alternatives has led researchers to explore isoelectronic elements. While tin (Sn²⁺) was an initial candidate due to its similar electronic configuration, its propensity to rapidly oxidize from Sn²⁺ to Sn⁴⁺ under ambient conditions results in severe device instability, often making it less stable than its lead counterparts.^9,10 This has shifted focus towards trivalent cations, particularly antimony (Sb³⁺) and bismuth (Bi³⁺), which form stable, non-toxic compounds with the general formulae A₃B₂X₉ and A₃BX₆ (A = MA⁺, Cs⁺, Ag⁺; B = Sb³⁺, Bi³⁺).^11–13 As demonstrated in the provided research, materials like AgBi₂I₇, Cs₃Bi₂I₉, and Ag₃BiI₆ have shown promising air stability and PCEs up to 4.3%.^14–16 However, these materials often suffer from fundamental limitations such as wide, indirect band gaps, high exciton binding energies, low electronic dimensionality, and poor charge carrier mobility, which intrinsically cap their photovoltaic performance.^17,18 Furthermore, studies on Ag₃BiI₆ reveal unique degradation mechanisms driven by concerted ion migration (Ag⁺, Bi³⁺, I⁻) and low decomposition enthalpy, leading to phase segregation and electrode corrosion, underscoring the complex stability challenges even in lead-free systems.¹⁹

In this landscape, indium (In³⁺)-based perovskites with the A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) structure emerge as a compelling yet relatively unexplored class of lead-free candidates. Indium shares a stable trivalent state and possesses an electronic configuration ([Kr] 4d¹⁰ 5s²) that is conducive to forming semiconductors with desirable optoelectronic properties. The A₃InX₆ structure, often crystallizing in a vacancy-ordered double perovskite or related elpasolite structure, offers several potential advantages. The bandgap can be effectively engineered through halide alloying (Cl, Br, I), allowing for optimization of light absorption across the solar spectrum. The all-inorganic composition (using Cs⁺ or Rb⁺) eliminates volatile organic cations, potentially enhancing thermal and structural stability compared to hybrid perovskites. Preliminary theoretical studies suggest that certain vacancy-ordered perovskites can exhibit good defect tolerance, which is crucial for achieving high open-circuit voltages (V_OC).¹⁸ Compared to the fast migration of small Ag⁺ ions observed in rudorffites like Ag₃BiI₆,²⁰ the larger ionic radii of Rb⁺, Cs⁺, and In³⁺ may intrinsically suppress detrimental ion migration, a key factor for operational stability and reduced hysteresis. Despite this promise, the development of A₃InX₆ perovskites for photovoltaics is still in its infancy. A₃InX₆, where A is a Group-1 monovalent cation (Rb⁺ or Cs⁺), and X is a halide anion (Cl⁻, Br⁻, I⁻). Their typically wider band gaps (>2 eV for most compositions) may limit their use as single-junction absorbers but make them excellent candidates for tandem cell top cells or UV-selective photodetectors. The challenges lie in mastering thin-film processing to achieve dense, pinhole-free layers with high crystallinity and in designing optimal device architectures that facilitate efficient charge extraction from these materials.^21–23

Despite the remarkable progress of halide perovskites in optoelectronic research, the presence of toxic lead and the limited long-term stability of many existing compositions continue to impede their sustainable deployment. While several lead-free alternatives based on trivalent cations such as Bi³⁺ and Sb³⁺ have been explored, these materials often suffer from intrinsic drawbacks, including wide or indirect band gaps, reduced charge-transport characteristics, and limited optical absorption in the visible region.²⁴ Consequently, identifying new lead-free perovskite-derived materials with favorable electronic structure, strong light–matter interaction, and robust structural stability remains a critical challenge. In this context, alkali-indium halide defect-perovskites A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) represent a largely unexplored material family with significant potential for optoelectronic applications. The stable trivalent oxidation state of indium, combined with the all-inorganic nature of these compounds, offers prospects for improved chemical and structural stability compared to hybrid perovskites. Moreover, the availability of both A-site cation substitution (Rb⁺/Cs⁺) and halide anion substitution (Cl⁻/Br⁻/I⁻) provides an effective route for systematically tuning lattice geometry, electronic band structure, and optical response.

However, a comprehensive understanding of how cation and anion substitution collectively influence the structural, electronic, optical, and mechanical properties of A₃InX₆ defect-perovskites is still lacking. In particular, detailed insights into band-gap tunability, orbital contributions near the band edges, optical absorption behavior across the infrared to ultraviolet regions, and mechanical robustness are essential to assess the suitability of these materials for future optoelectronic technologies. Motivated by this gap, the present study employs first-principles density functional theory calculations to establish clear structure–property relationships in A₃InX₆ compounds, providing a systematic theoretical foundation to guide future experimental synthesis and materials optimization.

2. Computational method

2.1. Material Studio-based DFT methodology

The physical properties, including structural, electronic, optical, mechanical, elastic anisotropy, charge density distribution, and population characteristics, of A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) halide double perovskites were systematically investigated using first-principles calculations based on Density Functional Theory (DFT), as implemented in the CASTEP package.^25,26 This ab initio approach provides a reliable and predictive framework for exploring the structural stability and physicochemical behavior of crystalline solids. Geometry optimization and property calculations were performed using the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional, while the Generalized Gradient Approximation PBE revised for solids (GGA-PBEsol) was employed to validate the electronic structure and accurately estimate the band gaps.^27–30 Ultrasoft Vanderbilt-type pseudopotentials were adopted for Rb, Cs, In, Cl, Br, and I atoms to efficiently and accurately describe the electron–ion interactions. A plane-wave cutoff energy of 700 eV was selected to ensure total-energy convergence and precise electronic state representation.^31–33 The Brillouin zone was sampled with a Monkhorst–Pack k-point mesh of 8 × 8 × 8, guaranteeing reliable convergence for total energy, stress, and force calculations across reciprocal space.^34,35 X-ray diffraction (XRD) patterns were simulated using the Forcite module to analyze the diffraction behavior of the optimized structures and to verify the consistency of the adopted crystallographic models.³⁶ Structural optimizations were performed under strict convergence criteria, including a maximum energy difference of 1.0 × 10⁻⁶ eV per atom, a maximum force of 0.006 eV Å⁻¹, a maximum displacement of 2.0 × 10⁻⁴ Å, and a maximum stress tolerance of 0.02 GPa.³⁷ The electronic configurations considered in the calculations correspond to the valence states of each element: Rb and Cs with closed-shell p orbitals and a single outer s electron; In with a fully occupied d shell along with s and p valence electrons; and halogen atoms (Cl, Br, and I) with filled s orbitals and partially filled p orbitals. The optical properties, including the complex dielectric function, absorption coefficient, refractive index, reflectivity, and optical conductivity, were derived from the frequency-dependent dielectric response based on interband electronic transitions. The mechanical properties were evaluated through the calculated elastic constants, from which bulk modulus, shear modulus, Young's modulus, Poisson's ratio, and Vickers hardness were derived to assess mechanical stability and ductility. The graphical visualization of the elastic properties was generated using an open-source computational framework based on the ELATE methodology. The elastic tensor components obtained from first-principles calculations were used as input to evaluate the directional dependence of the elastic moduli. The corresponding ELATE-based formulations were implemented within a Python-based environment to compute and visualize the anisotropic elastic behavior. This approach enables the calculation and representation of both three-dimensional and two-dimensional projections of the elastic parameters, providing insight into the anisotropy of the materials. The use of an open and script-based computational workflow ensures transparency and reproducibility of the elastic property analysis presented in this study. The charge density distribution and charge density difference maps were analyzed to elucidate bonding characteristics and charge transfer between constituent atoms. Finally, population analysis, including Mulliken charges and bond populations, was performed to quantify the ionic-covalent nature of chemical bonding in A₃InX₆ double perovskite compounds.

3. Material Studio-based results and discussion

3.1. Optimized structural properties

The metal halide perovskite A₃InX₆ in this research exhibits a crystal structure that possesses the space group Fm3̄m (225). As illustrated in Fig. 1, the sites 8c and 4b, 4a, and 24e are occupied by Cs/Rb, In, and halide (Cl/Br/I) atoms, respectively. This structural assignment is supported by recent theoretical and experimental studies on similar halide perovskites, where compounds with analogous compositions have been reported to stabilize in the cubic Fm3̄m phase. For instance, A₃GaI₆ (A = Cs, K, and Rb) has been demonstrated to adopt a stable cubic structure within the same space group.³⁸ Similarly, recent work such as “Efficient Discovery of Lead-Free A₃BX₃ Halide Perovskites via Machine Learning” also predicts the stability of cubic Fm3̄m phases for related compounds.³⁹ Although alternative structural configurations (e.g., lower-symmetry phases) have been reported in some cases,⁴⁰ the present cubic model is considered a reasonable and physically justified approximation for describing the structural and electronic properties of the studied system.

Fig. 1 The crystal structure of A₃InX₆ (A = Rb, Cs, and X = Cl, Br, I) defect-perovskites materials.

The optimized Cartesian coordinates are given in Tables S1 and S2. Tables 1 and S3 summarize the calculated structural and electronic properties of the investigated halide perovskites using different exchange-correlation functionals, together with previously reported theoretical and experimental data for related compounds reported in the literature.^40–42 As summarized in Table 1, the lattice constant and unit cell volume increase systematically with the halide ionic radius, following the order Cl < Br < I. For instance, the lattice constant expands from 8.026 Å (Rb₃InCl₆) to 9.247 Å (Rb₃InI₆), while the corresponding unit cell volume rises from 365.666 Å3 to 559.137 ų. A similar trend is observed for Cs-based perovskites, where Cs substitution results in larger lattice expansion compared to Rb due to the higher ionic radius of Cs⁺. Although the unit cell volume increases from Cl to Br to I due to the larger ionic radii of the halide ions, the density increases across the series. This is because the significant increase in atomic mass from Cl to Br to I outweighs the effect of lattice expansion. The electronic band gap shows the opposite trend, decreasing from Cl to I (e.g., Rb₃InCl₆: 3.515 eV → Rb₃InI₆: 1.335 eV), which originates from enhanced orbital overlap and narrowing of the conduction-valence band separation in heavier halides.

Table 1 The energy band gap, lattice constants, unit cell volume, and formation enthalpy of A₃InX₆ (A = Rb, Cs, and X = Cl, Br, I) perovskites calculated using the GGA-PBE functional

Compounds	Band gaps (Eg)	Lattice constant a (Å)	Unit cell volume, V (Å^3)	Density (g cm^−3)	Formation energy, ΔEf (eV per atom)
Rb3InCl6	3.515	8.026	365.666	2.651	−3.318
Rb3InBr6	2.440	8.468	429.431	3.289	−2.957
Rb3InI6	1.335	9.247	559.137	3.363	−2.564
Cs3InCl6	3.567	8.297	403.973	2.985	−3.338
Cs3InBr6	2.570	8.706	466.654	3.533	−2.989
Cs3InI6	1.430	9.433	593.578	3.566	−2.610

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Additionally, formation energy (ΔE_f) is a fundamental thermodynamic quantity that measures the energetic stability of a compound relative to its constituent elements in their standard states. It is defined as the difference between the total energy of the compound and the weighted sum of the energies of the isolated elemental phases, normalized per atom.⁴³ A negative ΔE_f indicates that the compound forms exothermically and is thermodynamically stable, while more negative values imply stronger bonding and higher resistance to decomposition.⁴³ The formation enthalpy (ΔE_f) of A₃InX₆ was calculated using the total energy obtained from the optimized crystal structure. The reference energies of the constituent elements were taken from their standard states, where A and In correspond to their bulk phases and the halogen X was evaluated as for the diatomic gas molecules (e.g., , , or ),

ΔH_f = E_tot(A₃InX₆) − 3E(A) − E(In) − 6E(X) (1)

Tables 1 and S1 summarize the calculated formation energies of the alkali-indium halide defect-perovskites Rb₃InX₆ and Cs₃InX₆ (X = Cl, Br, I). All compounds exhibit negative formation energies (from −3.338 to −2.564 eV per atom), confirming their intrinsic thermodynamic stability and feasibility for experimental synthesis. For both Rb- and Cs-based series, ΔE_f becomes less negative when moving from Cl → Br → I. This systematic trend reflects the increasing ionic radius and decreasing electronegativity of the halogen anions, which weakens the In–X bonding strength and slightly reduces lattice stability. Comparing the A-site cations, the Cs₃InX₆ compounds show marginally more negative ΔE_f than their Rb₃InX₆ counterparts for the same halogen, indicating that the larger Cs⁺ ion provides better lattice stabilization in these cubic defect-perovskite frameworks. Overall, the formation-energy analysis in Table 1 demonstrates that all investigated compounds are thermodynamically stable, with chloride-based compositions being the most stable and iodide-based compositions the least stable within each alkali series.

The Goldschmidt tolerance factor (t) is a widely used geometric criterion to evaluate the structural stability and symmetry of perovskite and defect-perovskite materials.^44,45 The tolerance factor of a given compound can be evaluated using the following equation,

In general, a tolerance factor in the range 0.8 ≤ t ≤ 1.0 indicates a stable cubic or near-cubic structure, whereas values significantly lower or higher than this range may lead to octahedral tilting, lattice distortion, or lower-symmetry phases.^46,47

Table 2 lists the calculated tolerance factors for the alkali-indium halide defect-perovskites Rb₃InX₆ and Cs₃InX₆ (X = Cl, Br, I). All compounds exhibit values between 0.924 and 1.000, which fall within the acceptable stability window, confirming their geometric compatibility with the cubic defect-perovskite structure.

Table 2 Tolerance factor variation in A₃InX₆ (A = Rb, Cs, and X = Cl, Br, I) defect-perovskites

Materials	rA	rIn	rX	Tolerance factor (t)
Rb3InCl6	1.72	0.80	1.81	0.956
Rb3InBr6	1.72	0.80	1.96	0.943
Rb3InI6	1.72	0.80	2.20	0.924
Cs3InCl6	1.88	0.80	1.81	1.000
Cs3InBr6	1.88	0.80	1.96	0.984
Cs3InI6	1.88	0.80	2.20	0.961

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For both alkali series, the tolerance factor decreases systematically from Cl → Br → I, reflecting the increase in halide ionic radius that slightly distorts the lattice and reduces structural compactness. Comparing the A-site cations, the Cs-based compounds consistently show higher t values than the corresponding Rb analogues due to the larger ionic radius of Cs⁺. Notably, Cs₃InCl₆ (t = 1.000) represents an ideal geometric match, suggesting a highly stable and nearly perfect cubic framework. Overall, the tolerance-factor analysis in Table 2 indicates that all investigated compounds possess favorable geometric stability, with Cs₃InCl₆ being the most structurally ideal and iodide-based compositions showing slightly increased lattice distortion while remaining within the stable regime.

3.2. X-ray diffraction (XRD) analysis

X-ray diffraction (XRD) analysis was used to examine the structural characteristics of the optimized compounds. The diffraction peak positions were determined according to Bragg's law,⁴⁸ which relates the diffraction angle to the interplanar spacing of the crystal lattice.⁴⁹ Detailed expressions (eqn (S1) and (S2)) used in the calculations are provided in the SI. Fig. 2 presents the calculated X-ray diffraction (XRD) patterns of double perovskite compounds with the general formula A₃InX₆, where A = Rb or Cs and X = I, Br, or Cl. The diffraction intensity is plotted as a function of the Bragg angle (2θ) over a range of approximately 10° to 90°. The cubic Fm3̄m structure (Materials Project mp-1110693) was selected as a high-symmetry reference model to enable consistent comparison across the A₃InX₆ series.⁵⁰ For the Rb-based compounds in Fig. 2(a), the simulated diffraction profiles exhibit multiple reflections corresponding to characteristic crystallographic planes over a wide 2θ range (10°–90°). The main low-angle peaks appear at approximately 12–13° (111), 15–16° (200), 20–22° (220), 24–26° (311), 28–30° (222), 31–33° (400), 36–38° (331), and 39–41° (420). In addition to these major reflections, several higher-angle peaks are observed, including (422), (511), (531), (600), and other higher-index planes, consistent with the cubic symmetry of the adopted model. Among the reflections, the (220) and (222) peaks exhibit relatively higher intensity. A systematic shift of peak positions toward higher 2θ values is observed when moving from Rb₃InI₆ to Rb₃InBr₆ and Rb₃InCl₆, which is consistent with the reduction in lattice spacing associated with the decreasing ionic radius from I⁻ to Br⁻ to Cl⁻. Fig. 2(b) shows the calculated patterns for the Cs-based compounds, where similar reflection sequences are observed across the full 2θ range. The principal peaks appear at approximately 11–12° (111), 14–15° (200), 19–21° (220), 24–26° (311), 27–29° (222), 30–32° (400), 34–36° (331), 36–38° (420), and 40–41° (422). Additional higher-angle reflections such as (511), (600), (620), and related planes are also clearly visible, indicating consistent indexing across the diffraction pattern. Compared with the Rb-based systems, the peaks for Cs compounds are systematically shifted toward lower 2θ values, which is consistent with the larger ionic radius of Cs⁺ and the corresponding expansion of lattice parameters. All major reflections across the full 2θ range can be consistently indexed based on the reflection conditions of the cubic Fm3̄m space group, supporting internal consistency of the adopted structural model. These simulated diffraction profiles are consistent with structural variations differences that arise from both halide substitution and A-site cation size.^51,52

Fig. 2 Calculated X-ray diffraction spectra of (a) Rb₃InX₆ and (b) Cs₃InX₆ of double perovskite materials.

It should be noted that the cubic Fm3m structure employed in this work represents a high-symmetry reference model adopted to enable systematic comparison across the A₃InX₆ compositional series. Similar group materials, such as Q₃GaBr₆ (Q = K and Na), have also been reported to exhibit cubic structures with Fm3̄m symmetry based on simulated XRD analysis.⁴¹ Importantly, previous experimental studies have reported lower-symmetry phases for related compounds, such as the monoclinic C2/c structure for Rb₃InCl₆.⁴⁰ Such variations may arise from differences in synthesis conditions, temperature effects, or structural distortions. In the absence of comprehensive comparative stability analysis across competing phases, the cubic Fm3̄m phase is adopted here as a consistent and physically reasonable reference model for systematic investigation. It is further emphasized that simulated XRD patterns reflect the diffraction response of optimized structural models and cannot independently confirm phase purity or exclude competing polymorphs. In addition, simulated peak intensities depend on normalization and peak-broadening parameters and do not capture experimental effects such as crystallite size, strain, preferred orientation, or defects. A rigorous determination of the ground-state crystal symmetry would require comparative total-energy calculations of candidate structures along with high-resolution experimental diffraction analysis using quantitative refinement methods such as Rietveld fitting, which is beyond the scope of the present study.

3.3. Electronic properties

The electronic band structures of the investigated materials were calculated within the framework of density functional theory (DFT) using the GGA-PBE, GGA-PBEsol, and hybrid HSE06 exchange-correlation functionals. The GGA-PBE functional is widely adopted owing to its computational efficiency, reasonable accuracy, and low computational cost, which makes it suitable for the systematic evaluation of electronic, optical, and other related physical properties using modest computational resources. The GGA-PBEsol functional, a revised version of PBE for solid-state systems, generally provides improved equilibrium structural properties and slightly larger bandgap values.^53,54 For a more reliable description of the electronic structure, particularly the bandgap, the screened hybrid functional HSE06 was additionally employed,⁵⁵ as hybrid functionals are known to reduce the bandgap underestimation typically observed in semi local GGA approaches.^56–58

As shown in Fig. 3(a)–(f), all compounds exhibit a direct bandgap, with both the valence band maximum (VBM) and conduction band minimum (CBM) located at the Γ point of the Brillouin zone (W–L–Γ–X–W–K), which is favorable for optoelectronic and photovoltaic applications. The calculated bandgap values for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆ are 3.515, 2.440, and 1.335 eV using GGA-PBE, and 3.695, 2.595, and 1.429 eV using GGA-PBEsol, respectively, while the HSE06 functional predicts larger bandgaps of 5.004, 3.892, and 2.550 eV for the same compounds. Similarly, Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆ exhibit bandgaps of 3.567, 2.570, and 1.430 eV with GGA-PBE, which increase to 3.801, 2.755, and 1.590 eV when calculated using GGA-PBEsol, while the HSE06 functional further increases the bandgaps to 5.116, 4.038, and 2.556 eV, respectively. For both Rb- and Cs-based compounds, the bandgap decreases systematically from Cl to Br to I, which can be attributed to the increasing halide ionic radius and the enhanced contribution of halogen p states near the valence band edge.^51–53 In all cases, the HSE06 hybrid functional yields significantly larger bandgaps due to the inclusion of a portion of exact exchange, which reduces the typical bandgap underestimation of semi local GGA functionals.

Fig. 3 Electronic band structures of (a) Rb₃InCl₆, (b) Rb₃InBr₆, (c) Rb₃InI₆, (d) Cs₃InCl₆, (e) Cs₃InBr₆, and (f) Cs₃InI₆ calculated using the GGA-PBE and GGA-PBEsol exchange-correlation functionals.

Importantly, Rb₃InI₆ and Cs₃InI₆ possess bandgaps within the optimal photovoltaic bandgap of 1.0–1.80 eV,⁵⁹ combined with a direct bandgap nature. These features indicate that the iodide-based compounds are particularly promising candidates for solar cell applications, whereas the chloride- and bromide-based materials are more suitable for wide-bandgap optoelectronic devices.

The partial density of states (PDOS) and total density of states (TDOS) were analyzed to gain a comprehensive understanding of the orbital-resolved electronic structure and to validate the band-structure results. The TDOS profiles clearly exhibit a finite bandgap for all investigated compounds, with no electronic states crossing the Fermi level (set at 0 eV), confirming their intrinsic semiconducting behavior. The valence band (V_B) edge lies just below the Fermi level, while the conduction band (C_B) edge starts just above it; the energy separation between these two edges defines the bandgap.^60,61 From the PDOS analysis, the upper valence band region is mainly governed by the halogen p orbitals, namely Cl-3p, Br-4p, and I-5p, with small but noticeable contributions from In-5s and In-5p states. This indicates that the valence band electronic states are primarily controlled by the halide sublattice. In contrast, the conduction band is dominated by In-5p states, with additional hybridization from the A-site alkali metal s orbitals (Rb-s or Cs-s). The alkali metal p orbitals contribute negligibly to both the valence and conduction bands, demonstrating that Rb and Cs ions mainly play a structural stabilization role rather than directly influencing the electronic transitions.

The TDOS maximum peak values at the VBM are calculated to be 30.74, 29.178, and 27.822 states per eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, while Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆ exhibit corresponding values of 32.318, 31.59, and 19.676 states per eV, as shown in Fig. 4. On the other hand, the maximum TDOS peaks at the CBM are 18.54, 19.676, and 22.189 states per eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, and 17.786, 18.926, and 24.378 states per eV for Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI_6, as shown in Fig. 4.

Fig. 4 The PDOS and TDOS of A₃InX₆ perovskite materials.

The extracted bandgap values from the TDOS are 3.515 eV, 2.440 eV, and 1.335 eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, and 3.567 eV, 2.570 eV, and 1.430 eV for Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆. These results are fully consistent with the band-structure calculations and show a systematic reduction of the bandgap from Cl → Br → I, originating from the increasing contribution of halogen p states near the valence band edge.⁶² Overall, the combined PDOS and TDOS analyses confirm that the electronic properties of these materials are mainly dictated by In–X orbital interactions, while the alkali metal cations act as charge-balancing and lattice-stabilizing species. The favorable electronic features of the iodide-based compounds further support their suitability for photovoltaic and optoelectronic applications.

3.4. Optical properties

The optical properties of A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) perovskites describe their interaction with electromagnetic radiation across the infrared, visible, and ultraviolet regions and are critical for optoelectronic and photovoltaic applications. Optical analysis provides key information on absorption edges, electronic transition mechanisms, dielectric response, optical conductivity, and refractive behavior. These parameters determine the efficiency of light absorption, charge-carrier generation, and interband transitions.⁶³ Comparing different halide compositions enables the identification of materials with optimal spectral coverage and strong light–matter interaction, supporting their potential use in solar cells, UV-visible photodetectors, LEDs, and laser devices.⁶⁴

3.4.1. Absorption coefficient (α). Fig. 5(a) illustrates the absorption coefficient spectra of A₃InX₆ over the photon-energy range of 0–16 eV, encompassing the infrared (IR), visible, and ultraviolet (UV) regions.⁶⁵ It measures the attenuation of incident radiation per unit path length expressed in the eqn (S3). As shown in Fig. 5(a), absorption begins at 3.515 eV for Rb₃InCl₆, 2.440 eV for Rb₃InBr₆, and 1.335 eV for Rb₃InI₆, while the corresponding values for the Cs-based compounds are 3.567 eV for Cs₃InCl₆, 2.570 eV for Cs₃InBr₆, and 1.430 eV for Cs₃InI₆. This systematic redshift from Cl to I reflects the reduction in bandgap with increasing halide ionic size. In the visible region, Fig. 5(a) shows negligible absorption peaks for Rb₃InCl₆ and Cs₃InCl₆, indicating limited visible-light absorption. In contrast, modest absorption peaks are observed for the bromide-based compounds, reaching 0.07 × 10⁵ cm⁻¹ for Rb₃InBr₆ and 0.0846 × 10⁵ cm⁻¹ for Cs₃InBr₆. The I-based compounds exhibit more pronounced visible-range absorption, with peak values of 0.0938 × 10⁵ cm⁻¹ for Rb₃InI₆ and 0.113 × 10⁵ cm⁻¹ for Cs₃InI₆, confirming their superior visible-light harvesting capability among all studied materials. In the UV region, strong absorption is observed for all compositions, as evident in Fig. 5(a). The maximum absorption coefficients reach 1.76 × 10⁵ cm⁻¹ at 16 eV for Rb₃InCl₆, 1.393 × 10⁵ cm⁻¹ at 7.64 eV for Rb₃InBr₆, and 1.449 × 10⁵ cm⁻¹ at 6.52 eV for Rb₃InI₆. For the Cs-based compounds, even higher maxima are obtained, with 3.035 × 10⁵ cm⁻¹ at 15.07 eV for Cs₃InCl₆, 2.698 × 10⁵ cm⁻¹ at 14.785 eV for Cs₃InBr₆, and 1.335 × 10⁵ cm⁻¹ at 6.257 eV for Cs₃InI₆. These strong UV responses originate from high-energy interband electronic transitions. Overall, the absorption behavior depicted in Fig. 5(a)-characterized by negligible IR absorption, enhanced visible-light absorption for I- and Br-based compounds, and strong UV absorption for all materials-demonstrates the suitability of A₃InX₆ compounds for solar cells, UV-visible photodetectors, and advanced optoelectronic devices.

Fig. 5 Optical properties of (a) absorption coefficient, (b) conductivity, (c) dielectric function, (d) reflectivity, (e) refractive index, (f) loss function of A₃InX₆ defect-perovskite materials.

3.4.2. Conductivity (σ). Optical conductivity describes the electronic response of a material under electromagnetic excitation and is derived from the complex dielectric function using standard relations.^66,67 The detailed expressions used in the calculations are provided in the SI eqn (S4). Fig. 5(b) presents the real part of the conductivity, it is observed that no noticeable response appears in the infrared (IR) region for the Cl- and Br-based compounds, whereas the I-based compounds exhibit finite conductivity contributions. In the visible region, the Cl-based compounds show no observable conductivity peaks, indicating weak optical activity.^68,69 In contrast, the Br-based materials begin to exhibit conductivity at photon energies of 2.44 eV and 2.57 eV for Rb₃InBr₆ and Cs₃InBr₆, respectively, where moderate conductivity values are observed throughout the visible range. Notably, the I-based compounds show pronounced visible-region peaks, with maximum values of 0.4098 at 3.10 eV for Rb₃InI₆ and 0.35 at 3.10 eV for Cs₃InI₆, demonstrating stronger optical conductivity compared to the Cl- and Br-based counterparts. With further increase in photon energy into the UV region, the real part of the conductivity increases significantly for all compounds, accompanied by multiple distinct peaks across the energy spectrum. The maximum UV-region conductivity peaks are observed at 3.08 at 7.91 eV for Rb₃InCl₆, 3.41 at 7.19 eV for Rb₃InBr₆, 3.71 at 6.106 eV for Rb₃InI₆, 5.04 at 13.857 eV for Cs₃InCl₆, 4.31 at 13.95 eV for Cs₃InBr₆, and 3.487 at 5.797 eV for Cs₃InI₆.

The imaginary part of the conductivity is also shown in Fig. 5(b). For all compounds, it initially exhibits negative values at low photon energies. This negative imaginary conductivity originates from a capacitive-like response of bound charge carriers, where the induced current lags behind the applied electromagnetic field due to polarization and carrier inertia effects. As the photon energy increases, the magnitude of the negative conductivity grows across the IR and visible regions and reaches maximum negative peaks in the UV region, with values of −2.165 at 5.905 eV, −2.079 at 5.00 eV, −1.862 at 5.312 eV, −2.1587 at 5.964 eV, −1.87 at 5.70 eV, and −2.103 at 5.045 eV for the A₃InX₆ compounds, respectively. With a further increase in photon energy, the imaginary part decreases in magnitude and crosses zero at 9.375 eV, 7.615 eV, 6.298 eV, 13.367 eV, 7.928 eV, and 6.11 eV, respectively. This zero-crossing behavior signifies a transition in the phase relationship between the induced current and the external field, typically associated with electronic resonance and interband transition thresholds. Beyond these energies, the imaginary part becomes positive, indicating an inductive-like response dominated by high-energy interband transitions, where the current leads the applied field and energy exchange is governed by rapid electronic oscillations.

3.4.3. Dielectric function. The optical dielectric response of the A₃InX₆ compounds is shown in Fig. 5(c). The optical behavior is described by the complex frequency-dependent dielectric function, whose real and imaginary components characterize the polarization and absorption responses of the material, respectively.^65–67 The detailed expressions used for calculating the dielectric function are provided in the SI eqn (S5)–(S7). The real part, ε₁(ω), represents the dispersive response of the material and is associated with electronic polarization and dielectric screening. It reflects how efficiently the material stores electromagnetic energy and influences key optical properties such as refractive index, phase velocity, and reflectivity. Positive values of ε₁(ω) indicate normal dielectric behavior, whereas a decrease or sign change signifies strong interband transitions and plasmon-related effects.

The real part of the dielectric function, ε₁(ω), for the A₃InX₆ defect-perovskite materials is presented in Fig. 5(c). At zero photon energy, ε₁(ω) starts with values of 2.729, 3.154, and 3.750 for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, and 2.805, 3.203, and 3.787 for Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆, reflecting the static dielectric constants of these compounds. As the photon energy increases across the IR and visible regions, the real part of the dielectric function gradually increases, indicating enhanced electronic polarization and dielectric screening. Several pronounced peaks are observed, corresponding to strong interband electronic transitions. The maximum values of ε₁ are 4.015 at 5.75 eV, 4.487 at 4.78 eV, and 5.038 at 3.634 eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, while Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆ reach maxima of 4.123 at 5.63 eV, 4.496 at 4.74 eV, and 4.814 at 3.499 eV, respectively. With further increase in photon energy into the UV region, ε₁(ω) decreases due to reduced polarization response at higher excitation energies. Notably, the Cs-based compounds exhibit zero-crossing behavior at 14.347 eV (Cs₃InCl₆), 14.239 eV (Cs₃InBr₆), and 14.197 eV (Cs₃InI₆), beyond which ε₁(ω) becomes negative, indicating plasma-like behavior and strong reflectivity in the high-energy region.

The imaginary part ε₂(ω), which represents optical absorption due to interband electronic transitions. The onset of ε₂(ω) occurs at photon energies of 3.515 eV, 2.440 eV, and 1.335 eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, while the corresponding thresholds for the Cs-based compounds are 3.567 eV (Cs₃InCl₆), 2.570 eV (Cs₃InBr₆), and 1.430 eV (Cs₃InI₆). These onset energies are closely associated with the fundamental band gaps of the respective materials.

As the photon energy increases, ε₂(ω) increases steadily, indicating enhanced optical absorption due to stronger interband transitions, and reaches pronounced maxima in the UV region. The peak values for the Rb-based compounds are 3.23 at 7.76 eV, 3.96 at 7.04 eV, and 5.06 at 5.91 eV for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively. Similarly, the Cs-based compounds exhibit maximum ε₂(ω) values of 3.09 at 13.41 eV for Cs₃InCl₆, 3.65 at 6.547 eV for Cs₃InBr₆, and 4.99 at 5.69 eV for Cs₃InI₆. The stronger and lower-energy peaks observed for the I-based compounds further confirm their enhanced optical absorption and higher electronic polarizability compared to the Cl- and Br-based systems.

3.4.4. Reflectivity. Reflectivity describes the fraction of incident electromagnetic radiation reflected from the material surface and was calculated from the complex dielectric function using standard relations.⁷¹ The detailed expressions used in the calculations are provided in the SI in eqn (S8).⁷²

The presented spectra in Fig. 5(d), the reflectivity values begin around 0.07 to 0.11 at low photon energies (near 0 eV), indicating relatively weak reflection in the infrared region. The highest reflectivity peak is observed for Cs₃InCl₆, reaching above 0.35 at about 15 eV, showing strong reflection in the deep UV region. Cs₃InI₆ also exhibits a significant peak close to 0.27 around 7 eV, while Cs₃InBr₆ and Rb₃InI₆ display intermediate peaks between 0.2 to 0.25 around 6 to 8 eV. In contrast, Rb₃InCl₆ shows the lowest overall reflectivity, with peaks not exceeding 0.18. The closest comparable values appear for Rb₃InBr₆ and Rb₃InI₆ in the visible-UV transition region (3 to 7 eV), suggesting similar photon–electron interaction strengths. These trends imply that Cs-based halide perovskites generally exhibit stronger reflectivity responses than their Rb analogues, especially in the higher-energy UV range. From an application perspective, the high reflectivity of Cs₃InCl₆ at deep UV energies suggests potential as a UV-protective or reflective coating material, while Cs₃InI₆, with its pronounced peak around 7 eV, may serve in UV-visible optoelectronics and detectors. Rb₃InX₆ compounds, with comparatively lower reflectivity across the spectrum, could be more favorable in solar energy harvesting applications where reduced reflection (hence greater absorption) is desired to maximize photon capture. The systematic variation of peaks across halides also demonstrates tunability, making these materials candidates for wavelength-selective mirrors, filters, or optoelectronic interfaces.^73–75 Thus, calculating reflectivity is crucial to evaluate and optimize the suitability of A₃InX₆ perovskites for photovoltaic devices, photodetectors, and coating technologies.

3.4.5. Refractive index. The refractive index were calculated from the complex dielectric function using standard optical relations shown in SI eqn (S9)–(S11).^70–72 The variation of the η(ω) for the A₃InX₆ defect-perovskite compounds is presented in Fig. 5(e). At zero photon energy, the η(ω) starts at 1.675, 1.790, and 1.947 for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, while identical initial values of 1.675, 1.790, and 1.948 are obtained for the corresponding Cs₃InCl₆, Cs₃InBr₆, and Cs₃InI₆ compounds. These values reflect the static refractive indices and increase systematically from Cl to I due to enhanced electronic polarizability. With increasing photon energy, η(ω) increases in the IR and visible regions, reaching maximum values of 2.056 at 6.0 eV for Rb₃InCl₆, 2.150 at 4.89 eV for Rb₃InBr₆, and 2.276 at 3.73 eV for Rb₃InI₆. Beyond these energies, further increase in photon energy leads to a gradual decrease in the refractive index up to 16 eV, indicating reduced optical dispersion at higher energies. A similar trend is observed for the Cs-based compounds, where η(ω) increases with photon energy and attains maximum values of 2.070 at 5.87 eV for Cs₃InCl₆, 2.150 at 4.89 eV for Cs₃InBr₆, and 2.260 at 4.983 eV for Cs₃InI₆. At higher photon energies, the refractive index decreases; however, for the Cs-based materials, η(ω) remains relatively elevated up to approximately 14 eV before showing a more pronounced decline. Overall, the refractive index follows the trend I > Br > Cl, consistent with the dielectric response and optical conductivity results.

As shown in Fig. 5(e), the imaginary part of the refractive index remains nearly zero in the IR region for all A₃InX₆ compounds, indicating negligible absorption at low photon energies. In the visible region, the Cl-based compounds exhibit very weak k(ω) values, confirming their low absorption and wide band-gap nature. In contrast, the Br-based materials show a gradual increase in k(ω), while the I-based compounds display a more pronounced rise, reflecting their enhanced optical absorption due to narrower band gaps. With further increase in photon energy into the UV region, the imaginary part of the refractive index increases significantly for all materials and exhibits multiple distinct peaks. These peaks originate from strong interband transitions involving deeper valence-band states. Among all compounds, the I-based materials show the highest k(ω) values, indicating stronger absorption losses, followed by the Br-based and Cl-based systems. Additionally, the Cs-based compounds generally exhibit slightly higher and broader absorption features than their Rb-based counterparts, suggesting enhanced optical transition strength at higher photon energies. Overall, the variation of the imaginary part of the refractive index confirms that halide substitution plays a dominant role in controlling optical absorption in A₃InX₆ defect-perovskites, with iodine-based compounds being the most optically active across the visible and UV spectral regions.

3.4.6. Loss function. The energy loss function, L(ω), derived from the complex dielectric function, describes the energy dissipation of fast electrons interacting with the material and is commonly used to identify plasmon resonances.⁷⁶ From the plotted spectra in Fig. 5(f), the highest peak is observed for Rb₃InI₆ around ∼12.5 eV, followed closely by Cs₃InI₆ near ∼13.5 eV, indicating strong plasmonic resonance and enhanced electron–photon interactions. The lowest peak response is found for Cs₃InCl₆ (∼9.5 to 10 eV), reflecting weaker plasmon activity due to its wider band gap and lower polarizability. The intermediate responses are seen for Rb₃InCl₆ and Rb₃InBr₆ (∼10 to 12 eV), and Cs₃InBr₆ (∼11 to 12 eV), suggesting moderate plasmonic activity. Based on these peak values, Rb₃InI₆ and Cs₃InI₆ are promising for ultraviolet (UV) plasmonic devices, high-frequency optoelectronics, and radiation detectors; Br-based compounds (Rb₃InBr₆, Cs₃InBr₆) may be suitable for visible to near-UV applications such as light-harvesting and LEDs, while Cl-based perovskites (Rb₃InCl₆, Cs₃InCl₆) are better suited for high-energy UV filters and insulating optical coatings due to their higher stability and weaker loss response. Thus, the loss function analysis highlights how halogen substitution modulates the plasmonic and dielectric properties of these compounds, guiding their application in solar cells, sensors, and high-frequency photonic technologies.

3.5. Elastic and mechanical properties

3.5.1. Elastic properties. The elastic constants provide essential information on the mechanical response, stiffness, and stability of crystalline materials under external stress. Within the linear elastic regime, the stress–strain relationship is governed by Hooke's law,⁷⁷

σ_i = ∑C_ijε_j (3)

where C_ij are the elastic stiffness coefficients. For a cubic crystal system, only three independent elastic constants exist, namely C₁₁, C₁₂, and C₄₄. Here, C₁₁ describes resistance to uniaxial deformation, C₁₂ represents the elastic coupling between longitudinal and transverse strains, and C₄₄ corresponds to resistance against shear deformation.

Fig. 6 illustrates the calculated elastic constants of the A₃InX₆ defect-perovskites. For each investigated compound, the elastic constants follow the relation,

C₁₁ > C₁₂ > C₄₄

indicating that the resistance to longitudinal (axial) deformation is greater than that to shear deformation. This ordering is characteristic of many crystalline solids and reflects the intrinsic mechanical response of the lattice under different modes of strain. It should be noted that this relation describes the relative magnitude of elastic constants within a given compound and is independent of compositional trends.

Fig. 6 Elastic constant of A₃InX₆ defect perovskite materials.

When comparing different halide compositions, the variation of the elastic constants does not follow a strictly monotonic trend. In the Rb-based series, C₁₁ decreases from Rb₃InCl₆ (32.538 GPa) to Rb₃InBr₆ (26.553 GPa) and further to Rb₃InI₆ (16.779 GPa), suggesting a progressive reduction in lattice stiffness with increasing halide ionic radius. A similar decrease is observed for C₁₂ and C₄₄ within this series. In contrast, the Cs-based compounds exhibit non-monotonic behavior. Specifically, C₁₁ increases from Cs₃InCl₆ (31.50 GPa) to Cs₃InBr₆ (40.186 GPa), followed by a significant decrease for Cs₃InI₆ (16.677 GPa). This indicates that substitution of Cl by Br enhances lattice stiffness, possibly due to improved orbital interactions or bonding characteristics, whereas further substitution with the larger I⁻ ion leads to lattice softening due to increased bond length and reduced bond strength. Similar non-monotonic variations are also observed in C₁₂ and C₄₄. These results demonstrate that elastic properties in these systems are governed by a balance between ionic size, bond strength, and structural distortion, and therefore cannot be described by a single systematic trend across all compositions.

The mechanical stability of the cubic A₃InX₆ compounds is evaluated using the Born stability criteria, which for cubic symmetry are given by:^78,79

C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0 and C₁₁ > 0, C₄₄ > 0 (4)

All calculated elastic constants satisfy these conditions, confirming that the investigated compounds are mechanically stable at ambient pressure. Overall, the moderate elastic stiffness and tunable mechanical properties achieved through halide substitution indicate that A₃InX₆ defect-perovskites possess adequate structural stability for optoelectronic and energy-related applications.

3.5.2. Mechanical properties. The mechanical properties of crystalline materials describe their ability to withstand external forces without failure and are essential for evaluating structural integrity, durability, and practical applicability. These properties govern how a material responds to compression, tension, and shear, and they play a critical role in determining mechanical reliability in device-level and engineering applications.⁸⁰

The bulk modulus (B) quantifies the resistance of a material to uniform volume compression and serves as a direct measure of incompressibility. Materials with higher bulk modulus values exhibit stronger interatomic bonding and greater resistance to externally applied pressure. Conventionally, materials with B > 40 GPa are considered mechanically hard, whereas those with B < 40 GPa are classified as relatively flexible or soft. In the present work, the bulk modulus is evaluated using the Voigt–Reuss–Hill (VRH) approximation, where the Hill average provides a reliable estimate of the polycrystalline bulk modulus:⁸¹

with the Voigt and Reuss bounds for cubic symmetry given by:

The calculated bulk modulus values are presented in Fig. 7(a). The Rb-based compounds exhibit bulk moduli of 14.027 GPa (Rb₃InCl₆), 11.79 GPa (Rb₃InBr₆), and 8.672 GPa (Rb₃InI₆), showing a systematic decrease from Cl to I, which reflects the weakening of interatomic interactions with increasing halide ionic radius. Similarly, the Cs-based materials show bulk moduli of 13.753 GPa (Cs₃InCl₆), 25.495 GPa (Cs₃InBr₆), and 8.221 GPa (Cs₃InI₆). Among all compounds, Cs₃InBr₆ exhibits the highest bulk modulus, indicating comparatively stronger resistance to compression. Overall, all investigated A₃InX₆ compounds possess bulk modulus values well below 40 GPa, confirming their mechanically soft and flexible nature, which is advantageous for applications requiring strain tolerance and mechanical adaptability, such as flexible optoelectronic and energy-conversion devices.

Fig. 7 Mechanical properties in the (a) bulk modulus, (b) shear modulus, (c) Young's modulus, (d) Poisson ratio, (e) Pugh's ratio, and (f) hardness of the A₃InX₆ defect-perovskite materials.

The shear modulus (G) describes a material's resistance to shape change under applied shear stress and is an important indicator of rigidity against transverse deformation. A higher shear modulus signifies stronger resistance to distortion and improved mechanical rigidity. The shear modulus is given by:⁸²

The calculated shear modulus values are presented in Fig. 7(b). The Rb-based compounds exhibit shear moduli of 5.242 GPa (Rb₃InCl₆), 4.305 GPa (Rb₃InBr₆), and 4.123 GPa (Rb₃InI₆), indicating a gradual reduction in rigidity with halide substitution from Cl to I. Similarly, the Cs-based materials show shear modulus values of 4.322 GPa (Cs₃InCl₆), 4.785 GPa (Cs₃InBr₆), and 3.201 GPa (Cs₃InI₆). Among all compounds, Rb₃InCl₆ and Cs₃InBr₆ exhibit comparatively higher shear resistance, while the I-based materials display lower values, confirming their mechanically softer nature.

The Young's modulus (Y) characterizes the tensile stiffness of a material and defines the extent of elastic deformation under uniaxial tensile or compressive loading. A higher value of Y indicates greater stiffness and stronger resistance to elongation, whereas lower values correspond to mechanically softer and more flexible materials.⁸³

The calculated Y values are displayed in Fig. 7(c). The Rb-based compounds exhibit Y of 13.985 GPa (Rb₃InCl₆), 11.515 GPa (Rb₃InBr₆), and 10.677 GPa (Rb₃InI₆), showing a decreasing trend from Cl to I due to progressive weakening of interatomic bonding. Similarly, the Cs-based materials show Young's moduli of 11.737 GPa (Cs₃InCl₆), 13.511 GPa (Cs₃InBr₆), and 8.501 GPa (Cs₃InI₆). Among all compounds, Rb₃InCl₆ and Cs₃InBr₆ exhibit the highest tensile stiffness, while Cs₃InI₆ shows the lowest Young's modulus, indicating enhanced mechanical flexibility.

The ductile or brittle nature of the A₃InX₆ defect-perovskite compounds is assessed using both Poisson's ratio (ν) and Pugh's ratio (B/G), as shown in Fig. 7(d) and (e). According to established criteria, materials with ν > 0.26 and B/G > 1.75 are classified as ductile, whereas lower values indicate brittle behavior.⁸⁴

As presented in Fig. 7(d), Poisson's ratio values for the Rb-based compounds are 0.333 (Rb₃InCl₆), 0.337 (Rb₃InBr₆), and 0.294 (Rb₃InI₆), while the Cs-based materials exhibit values of 0.357 (Cs₃InCl₆), 0.411 (Cs₃InBr₆), and 0.327 (Cs₃InI₆). All values exceed the critical limit of 0.26, indicating a clear ductile mechanical response.

Consistently, the Pugh's ratio values shown in Fig. 7(e) further confirm the ductile nature of these materials. The Rb-based compounds exhibit B/G ratios of 2.676, 2.739, and 2.103 for Rb₃InCl₆, Rb₃InBr₆, and Rb₃InI₆, respectively, while the Cs-based compounds show significantly higher values of 3.182 (Cs₃InCl₆), 5.328 (Cs₃InBr₆), and 2.568 (Cs₃InI₆). All values are well above the ductility threshold of 1.75. Overall, the combined analysis of Poisson's ratio and Pugh's ratio demonstrates that all A₃InX₆ compounds are mechanically ductile, with Cs-based systems-particularly Cs₃InBr₆-showing superior plastic deformability. This intrinsic ductility enhances mechanical reliability and supports the suitability of these materials for flexible and strain-tolerant optoelectronic and energy-related applications.

Hardness is an important mechanical parameter that reflects a material's resistance to permanent (plastic) deformation, scratching, and wear. The Vickers hardness (H_V) is estimated using Tian's empirical model. It is expressed as:⁸⁵

The calculated hardness values are presented in Fig. 7(f). The Rb-based compounds exhibit hardness values of 1.639 GPa (Rb₃InCl₆), 1.380 GPa (Rb₃InBr₆), and 1.179 GPa (Rb₃InI₆), showing a systematic decrease from Cl to I. This trend is attributed to the progressive weakening of interatomic bonding and increased lattice softness with heavier halide substitution. Similarly, the Cs-based compounds show hardness values of 1.374 GPa (Cs₃InCl₆), 0.636 GPa (Cs₃InBr₆), and 0.940 GPa (Cs₃InI₆). Among all materials, Rb₃InCl₆ exhibits the highest hardness, while Cs₃InBr₆ shows the lowest value, indicating enhanced mechanical softness.

The machinability index (µ_M) is an important mechanical parameter that evaluates the ease with which a material can be machined, cut, or shaped without inducing excessive damage. It is defined as,⁸⁶

A higher machinability index indicates better machinability, implying lower resistance to shear deformation and improved workability. The calculated machinability indices are presented in Fig. 8(a). The Rb-based compounds exhibit µ_M values of 5.968 (Rb₃InCl₆), 5.885 (Rb₃InBr₆), and 2.740 (Rb₃InI₆), showing a notable reduction for the I-based compound, which reflects increased resistance to shear deformation. In contrast, the Cs-based materials display higher machinability indices of 8.896 (Cs₃InCl₆), 10.100 (Cs₃InBr₆), and 4.218 (Cs₃InI₆). Among all investigated compounds, Cs₃InBr₆ exhibits the highest machinability index, indicating superior machinability and ease of fabrication.

Fig. 8 Mechanical parameters of (a) machinability index, (b) elastic Debye temperature, (c) average sound velocity of A₃InX₆ defect perovskite materials.

The Debye temperature (Θ_D) is an important thermophysical parameter that provides insight into lattice vibrations, bonding strength, and thermal behavior of crystalline materials. It is closely related to the average sound velocity and elastic properties, and higher Debye temperatures generally indicate stronger interatomic bonding, higher phonon frequencies, and improved lattice rigidity. In this work, the elastic Debye temperature is evaluated from the elastic moduli through the average sound velocity, reflecting the vibrational contribution of the crystal lattice. The calculated Debye temperatures are presented in Fig. 8(b). The Rb-based compounds exhibit Debye temperatures of 129.303 K (Rb₃InCl₆), 100.558 K (Rb₃InBr₆), and 95.357 K (Rb₃InI₆), showing a clear decreasing trend from Cl to I. This reduction is attributed to the increasing atomic mass of the halide ions and the corresponding softening of lattice vibrations. Similarly, the Cs-based compounds show Debye temperatures of 102.082 K (Cs₃InCl₆), 103.757 K (Cs₃InBr₆), and 78.372 K (Cs₃InI₆). Among all investigated materials, Rb₃InCl₆ exhibits the highest Debye temperature, indicating comparatively stronger bonding and higher lattice vibrational frequencies, whereas Cs₃InI₆ shows the lowest value, reflecting enhanced lattice softness and weaker bonding interactions.

The average sound velocity (V_m) is a key elastic parameter that reflects the propagation of acoustic phonons in a crystal lattice and is directly related to the elastic moduli and atomic density of the material. It plays an important role in determining thermophysical properties such as the Debye temperature and lattice thermal conductivity. Generally, higher sound velocity indicates stronger interatomic bonding and greater lattice stiffness. The calculated V_m for the A₃InX₆ defect-perovskites are presented in Fig. 8(b). The Rb-based compounds exhibit sound velocities of 1443.661 m s⁻¹ (Rb₃InCl₆), 1183.347 m s⁻¹ (Rb₃InBr₆), and 1226.575 m s⁻¹ (Rb₃InI₆). Among these, Rb₃InCl₆ shows the highest sound velocity, consistent with its higher elastic stiffness and Debye temperature. For the Cs-based compounds, the average sound velocities are 1178.226 m s⁻¹ (Cs₃InCl₆), 1256.532 m s⁻¹ (Cs₃InBr₆), and 1028.383 m s⁻¹ (Cs₃InI₆). The lower V_m observed for Cs₃InI₆ reflects its softer lattice and weaker bonding interactions. Overall, a general decrease in V_m is observed with halide substitution from Cl to I, which is attributed to the increased atomic mass and reduced force constants.

These results are in good agreement with the calculated Debye temperatures, confirming that the vibrational and elastic properties of A₃InX₆ defect-perovskites are strongly influenced by halide composition. The relatively low sound velocities indicate soft lattice dynamics, which may be beneficial for applications requiring mechanical flexibility and could also impact thermal transport behavior in energy and optoelectronic devices.

3.5.3. Anisotropy properties. Elastic anisotropy reflects the directional dependence of mechanical properties and plays a critical role in understanding deformation behavior, crack propagation, and mechanical reliability of crystalline materials. To achieve a comprehensive evaluation of anisotropic behavior, both the Zener anisotropy factor (A) and the universal elastic anisotropy index (A^U) are employed, as they provide complementary measures of elastic anisotropy. The Zener anisotropy factor, defined for cubic symmetry as⁸⁷quantifies shear anisotropy. For an elastically isotropic material, A = 1, whereas deviations from unity indicate anisotropic mechanical behavior.⁸⁷ As shown in Fig. 9(a), the Rb-based compounds exhibit Zener factors of 0.339 (Rb₃InCl₆), 0.362 (Rb₃InBr₆), and 1.041 (Rb₃InI₆), indicating pronounced anisotropy for the Cl- and Br-based compounds and nearly isotropic behavior for Rb₃InI₆. Similarly, the Cs-based materials show values of 0.232 (Cs₃InCl₆), 0.458 (Cs₃InBr₆), and 0.615 (Cs₃InI₆), confirming anisotropic elastic behavior across the Cs series, with reduced anisotropy for the I-based compound. The universal elastic anisotropy index (A^U), defined asprovides a global measure of elastic anisotropy, where A^U = 0 corresponds to perfect isotropy and increasing values indicate stronger directional dependence. The calculated anisotropy indices are shown in Fig. 9(b). The Rb-based compounds exhibit A^U values of 4.888 (Rb₃InCl₆), 4.447 (Rb₃InBr₆), and 0.529 (Rb₃InI₆), indicating moderate to low elastic anisotropy, with Rb₃InI₆ approaching near-isotropic behavior. In contrast, the Cs-based materials show values of 8.067 (Cs₃InCl₆), 3.112 (Cs₃InBr₆), and 1.872 (Cs₃InI₆), revealing generally higher anisotropy, particularly for Cs₃InCl₆, which exhibits the most pronounced directional dependence among all compounds. Overall, both the Zener anisotropy factor and the universal elastic anisotropy index consistently demonstrate that elastic anisotropy decreases with halide substitution from Cl to I in the A₃InX₆ defect-perovskites. The near-isotropic elastic response observed for Rb₃InI₆, along with reduced anisotropy in other I-based compounds, is advantageous for minimizing stress concentration and enhancing mechanical reliability, which are essential for optoelectronic and energy-related applications.

Fig. 9 Mechanical parameters of (a) Zener isotropic factor (A), (b) anisotropy (A^U), (c) equivalent Zener anisotropy measure (A^eq), and (d) anisotropy in shear (A^G) of the A₃InX₆ defect perovskite materials.

To further quantify elastic anisotropy in a unified and comparable manner, the equivalent Zener anisotropic measure (A^eq) is employed. This parameter transforms the universal elastic anisotropy index into an equivalent Zener-type measure, allowing direct comparison of anisotropy strength on a common scale. It is defined as:⁸⁸

The calculated A^eq values are presented in Fig. 9(c). The Rb-based compounds exhibit equivalent Zener measures of 7.09 (Rb₃InCl₆), 6.568 (Rb₃InBr₆), and 1.631 (Rb₃InI₆), indicating strong anisotropy for the Cl- and Br-based compounds and significantly reduced anisotropy for Rb₃InI₆, which approaches near-isotropic behavior. In contrast, the Cs-based materials show A^eq values of 11.115 (Cs₃InCl₆), 4.722 (Cs₃InBr₆), and 3.359 (Cs₃InI₆). Among all investigated compounds, Cs₃InCl₆ exhibits the highest A^eq value, confirming the strongest elastic anisotropy and pronounced directional dependence of mechanical property.

Elastic anisotropy in shear deformation is quantified using the shear anisotropy index (A^G), which evaluates the directional dependence of shear response in polycrystalline materials. It is defined as:⁸⁹

A value of A^G = 0 corresponds to isotropic shear behavior, while increasing values indicate stronger anisotropy in shear response. The calculated shear anisotropy indices are presented in Fig. 9(d). The Rb-based compounds exhibit A^G values of 0.328 (Rb₃InCl₆), 0.308 (Rb₃InBr₆), and 0.050 (Rb₃InI₆), indicating moderate shear anisotropy for the Cl- and Br-based materials and nearly isotropic shear behavior for Rb₃InI₆. In contrast, the Cs-based compounds show A^G values of 0.446 (Cs₃InCl₆), 0.237 (Cs₃InBr₆), and 0.158 (Cs₃InI₆), revealing comparatively stronger shear anisotropy, particularly for Cs₃InCl₆, which exhibits the highest directional dependence in shear deformation. Overall, the A^eq and A^G follows the same halide-dependent trend observed in other anisotropy measures, decreasing from Cl to I substitution. The low A^eq and A^G values of the I-based compounds-especially Rb₃InI₆-indicate nearly uniform shear resistance along different crystallographic directions, which is beneficial for reducing stress concentration and improving mechanical reliability in device applications.

Anisotropy refers to the directional dependence of physical properties of a material, meaning that its mechanical, electronic, or optical responses vary when measured along different crystallographic directions.⁹⁰ In perovskites such as A₃InX₆ defect perovskite materials, anisotropy is particularly important because it provides insight into the stability, mechanical flexibility, and suitability of these compounds for optoelectronic applications. The elate tensor analysis was investigated with the aid of Google Colab using Python 3.0.⁹¹ Fig. S1–S3 show 2D and 3D representations of Young's modulus, shear modulus, and Poisson's ratio, clearly reflecting the degree of anisotropy.

Fig. 10 Charge density mapping of A₃InX₆ (A = Rb, Cs, and X = Cl, Br, I) perovskite.

Fig. S1 compares two Rb-based compounds with different halides. In both, the Young's-modulus maps show the classic four-lobed, cross-shaped anisotropy, but the chloride has longer, sharper lobes and a larger 3D surface; it is stiffer and more anisotropic than the bromide. The shear-modulus plots are nearly perfect circles with gently oblate 3D shells for both materials, indicating an almost orientation-independent resistance to shear and only minor differences between Cl and Br. The Poisson's-ratio maps form four-petal flowers; the bromide exhibits slightly broader petals (greater directional variation), while the chloride is more uniform. Overall, replacing Cl by Br in the Rb compound softens the lattice and slightly increases ν anisotropy while leaving shear behavior essentially isotropic, useful guidance when balancing stiffness against compliance in Rb-based devices.

Fig. S2 places the softest and the stiffest ends of the series side-by-side. Rb₃InI₆ has the smallest, roundest Young's-modulus surface, signaling a compliant, weakly anisotropic lattice, whereas Cs₃InCl₆ shows the largest and most faceted E-surface, marking it as the stiffest member. Shear-modulus plots for both remain close to isotropic, with only subtle differences in shape and magnitude. In the Poisson's-ratio maps, Rb₃InI₆ presents wider petals (larger directional ν), indicating greater lateral strain for a given axial load; Cs₃InCl₆ is comparatively restrained. The juxtaposition demonstrates how composition can strongly tune mechanical response: moving from Rb–I to Cs–Cl dramatically increases stiffness while maintaining nearly uniform shear behavior critical for choosing between strain-accommodating layers (iodide) and mechanically robust layers (chloride).

In Fig. S3, the A-site cation is fixed (Cs) and only the halide changes. Young's-modulus maps show Cs₃InBr₆ with a larger, sharper four-lobed pattern than Cs₃InI₆, so the bromide is stiffer and more anisotropic, while the iodide is softer and more rounded. The shear-modulus plots again appear as near-circles with slightly oblate 3D shapes for both compounds, confirming that shear resistance is mostly orientation-independent and only marginally affected by the halide. Poisson's-ratio maps reveal stronger angular variation for the iodide (broader petals) than for the bromide, but ν remains positive throughout, consistent with elastic stability. Collectively, this figure isolates the halide effect in Cs-based materials: Br → I decrease stiffness and increases ν anisotropy, a practical lever for tuning crack resistance and stress accommodation in Cs-containing perovskite layers. Collectively, the figures demonstrate that these perovskites are elastically stable, with shear behavior that is nearly orientation-independent, and that their axial stiffness and lateral-strain coupling can be tuned primarily through the halide harder, more anisotropic for Cl; softer, more compliant for I. This is mechanically significant for crystal growth and device engineering because it identifies orientations least likely to crack, guides texture control in thin films, and clarifies how composition can be used to balance robustness (chlorides) against compliance and stress accommodation (iodides).

3.6. Charge density

Fig. 10 illustrates the charge density mapping of A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) halide double perovskites, highlighting the electronic distribution and bonding features.⁶⁷ The Rb-based compounds are presented in the top panels, while the Cs-based analogues are shown in the bottom panels. The color scale represents electron density, where red indicates regions of high electron concentration and blue corresponds to low electron density, with values ranging approximately from −1.95 to 10.86 e Å⁻³. A pronounced charge localization is observed around the In and halogen (Cl, Br, I) atoms, with maximum density values approaching ∼10.8 e Å⁻³, reflecting their dominant role in bonding interactions. In contrast, the Rb and Cs atoms exhibit significantly lower electron density, indicating their largely ionic character and weaker participation in covalent bonding.

A systematic variation is observed across the halogen series: as X changes from Cl → Br → I, the electron density distribution becomes more spatially extended, indicating increased charge delocalization and a gradual shift in bonding character. Similarly, substitution of Rb with the larger Cs cation leads to a broader distribution of charge around the A-site, suggesting modifications in the local electrostatic environment and possible changes in orbital interactions with neighboring halogen atoms. These observations provide qualitative insight into the bonding nature and orbital hybridization within the A₃InX₆ lattice.

However, it should be noted that charge density analysis primarily reflects electronic distribution and bonding characteristics, and does not directly determine carrier effective mass or transport properties. Further quantitative analysis based on band structure calculations would be required to establish such relationships.⁹²

3.7. Population analysis

Mulliken and Hirshfeld charge population analysis tabulated in Table S4 provides valuable insights into the electronic distribution and bonding character of A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) defect perovskites materials. Mulliken analysis partitions electron density according to atomic orbital contributions and is therefore used here only to examine qualitative trends in charge transfer and bonding. However, it is well known that Mulliken population analysis has inherent limitations, particularly its sensitivity to the choice of basis set and its limited reliability in describing absolute atomic charges in extended solid-state systems. Therefore, in the present work, Mulliken analysis is employed strictly for qualitative interpretation of charge redistribution and bonding tendencies, and not for quantitative evaluation of absolute atomic charges. To complement this, Hirshfeld charge analysis is also employed to provide a comparatively more balanced description of electron-density distribution, although it is likewise considered only for qualitative interpretation.^93,94 Both analyses consistently indicate that the A-site cations (Rb, Cs) carry positive charge, reflecting their electropositive nature and supporting an overall ionic contribution to bonding within the lattice.

Hirshfeld charges follow the same trend but with smaller magnitudes, typically 0.14 to 0.42 e (Table S4), reflecting the method's more delocalized description. Indium atoms generally acquire positive Mulliken charges, e.g., 0.74 e in Cs₃InCl₆, while Hirshfeld charges are lower, 0.42 e, highlighting their intermediate role between the electropositive A-site and electronegative halides. Conversely, halogen atoms (Cl, Br, I) display negative charges across all compounds, with the strongest Mulliken localization at −0.49 e (Cl in Cs₃InCl₆) and the weakest at −0.00 e (Br in Cs₃InBr₆), while Hirshfeld charges range from about −0.24 to −0.21 e, as shown in Table S4. A systematic trend is observed: moving from Cl → Br → I, the magnitude of negative halogen charge decreases, reflecting reduced electronegativity and increasing covalency. Likewise, substitution of Rb with Cs enhances charge transfer, as evidenced by larger positive Mulliken charges on Cs compared to Rb, consistent with the stronger interaction of Cs with halogens. Overall, Mulliken results tend to emphasize ionic contributions, whereas Hirshfeld analysis suggests a comparatively more delocalized electron distribution. These observations provide a qualitative indication of bonding characteristics, with relatively more ionic behavior in Rb–Cl systems and a tendency toward increased covalent contribution in Cs–I systems. It is important to emphasize that the present population analysis is intended only to describe general bonding trends. No direct conclusions regarding band-edge properties, carrier transport behavior, or device performance are drawn from these results. A more quantitative assessment of such properties would require advanced charge-partitioning methods (e.g., DDEC or Bader analysis) or explicit calculations of effective mass and carrier mobility, which are beyond the scope of the present work.

4. Limitations

Although this work offers a comprehensive theoretical investigation of the structural, electronic, optical, and mechanical properties of A₃InX₆ defect-perovskites, several limitations should be acknowledged. First, all results are derived from density functional theory calculations using GGA-based exchange-correlation functionals, which are known to underestimate band-gap values; therefore, the absolute band gaps reported here should be interpreted qualitatively, while the observed trends with cation and anion substitution remain reliable. Second, the present study focuses exclusively on ideal, defect-free bulk crystals at zero temperature and ambient pressure. Effects arising from intrinsic point defects, grain boundaries, surface states, and finite-temperature lattice vibrations-which can significantly influence real material behavior-are not explicitly considered. Additionally, excitonic effects and many-body interactions are not included in the optical calculations, which may affect quantitative predictions of absorption and dielectric response. Finally, this work does not address thin-film growth, interfacial phenomena, charge-carrier transport, or device-level performance, which are essential aspects for practical optoelectronic applications. Consequently, while the present results provide valuable fundamental insights and identify promising compositions, experimental validation and more advanced theoretical treatments are required to fully assess the applicability of A₃InX₆ defect-perovskites in real devices.

5. Conclusion

In this work, first-principles density functional theory calculations were performed to investigate the structural, electronic, optical, and mechanical properties of lead-free alkali-indium halide defect perovskites A₃InX₆ (A = Rb, Cs; X = Cl, Br, I) for potential photovoltaic and optoelectronic applications. All compounds are found to be structurally stable within the adopted cubic framework, satisfying the Born mechanical stability criteria and exhibiting negative formation energies, indicating favorable thermodynamic feasibility. The calculated Goldschmidt tolerance factors confirm good geometric compatibility, with Cs-based systems showing slightly improved lattice matching due to the larger ionic radius. Electronic structure analysis reveals direct band gaps at the Γ-point, with systematic band-gap narrowing from Cl to I driven by increased halogen p-orbital contributions near the valence-band maximum. Iodide-based compositions demonstrate band gaps suitable for solar absorption, while chloride analogues remain promising for ultraviolet optoelectronic devices. Optical results indicate enhanced absorption coefficients, dielectric response, and optical conductivity in iodide-rich systems, confirming superior visible-light harvesting capability and tunable optoelectronic performance through anion engineering. Mechanical analysis further suggests that all materials are ductile and relatively soft, with Cs- and I-containing compounds exhibiting greater deformability, advantageous for flexible devices. The predicted properties should therefore be interpreted as representative of an idealized cubic structural model that provides a consistent theoretical reference for future experimental validation, while acknowledging that alternative polymorphs may exist under different synthesis conditions.

Ethical statement

All procedures performed followed ethical standards.

Author contributions

Imtiaz Ahamed Apon, Md. Sakib Hasan, Rifat Rafiu, Md. Azizur Rahman: formal analysis, data curation, visualization, conceptualization, methodology, investigation, software, writing, review & editing – original draft. Mohamed Benghanem, Amnah Mohammed Alsuhaibani, Moamen S. Refat, Noureddine Elboughdiri: software, validation, visualization, resources, writing – review & editing.

Conflicts of interest

Authors declare that they have no competing interests.

Data availability

The datasets collected and analyzed during the current study are available upon request from the corresponding author. The corresponding author had full access to all data in the study and took responsibility for the integrity and accuracy of the data analysis.

Supplementary information (SI): additional data, figures, and equation details supporting this study. See DOI: https://doi.org/10.1039/d6ra00473c.

Acknowledgements

The researchers wish to extend their sincere gratitude to the Deanship of Scientific Research at the Islamic University of Madinah (KSA) for the support provided to the Post-Publishing Program. Princess Nourah bint Abdulrahman University researchers supporting project number (PNURSP2026R65), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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