Ab initio study of electronic, elastic, thermodynamic, photocatalytic properties of double antiperovskite, Cs₆AgBiX₂ (X = Cl, Br, I)

Abstract

In this paper, we use density functional theory (DFT) using full-potential linearized augmented plan wave plus local orbital method (FP-LAPW + lo). The structural, electronic, optical, photocatalytic, mechanical, vibrational, and thermodynamical behaviors of new double antiperovskite (DAP) Cs₆AgBiX₂ (X = Cl, Br, I) were studied. The band structure was calculated with and without spin orbit coupling (SOC). Using the TB-mBJ approach (Hybrid) revealed bandgap values of 1.504 eV, 1.491 eV, and 1.392 eV for Cs₆AgBiCl₂, Cs₆AgBiBr₂, and Cs₆AgBiI₂, respectively. Optical characteristics were studied to ascertain the light absorbing ability of Cs₆AgBiX₂. The elastic and vibrational (phonon) properties demonstrate that Cs₆AgBiCl₂ and Cs₆AgBiBr₂ are stable but Cs₆AgBiI₂ is not. The calculated optimal bandgap and high absorption coefficient of Cs₆AgBiCl₂ and Cs₆AgBiBr₂, suggest their potential for solar cell applications. Moreover, our photocatalytic results suggest that these materials have high oxidizing capacity that can be used to efficiently produce oxygen cheaply using solar water splitting.

---

1 Introduction

Growing industrialization and fossil fuel use has significantly contaminated the environment and there is now a global energy crisis. To address this, researchers are turning to renewable energy sources, with solar energy being a key focus due to its safety and eco-friendliness.^1–3 Perovskite materials are gaining attention for solar energy use because of their great properties like effective light absorption and improving power conversion efficiency. Researchers have been studying these materials extensively.^4–6

Perovskite has the crystal structure CaTiO₃ and the formula ABX₃, it is used in different forms like antiperovskite X₃BA and double perovskite A₂B′B′′X₆. These structures offer unique qualities, like easy synthesis and stability.^7–9 A recent addition is the double antiperovskite X₆B′′B′A₂, showing promise for various applications.¹⁰ Hybrid halide perovskites, like CH₃NH₃PbX₃ are widely studied for efficient solar cells but their sensitivity to moisture and use of toxic lead, pose challenges. Seeking stable, lead-free alternatives has led to the exploration of double perovskite halides, such as Cs₂AgBiX₆. These compounds, like Cs₂AgBiCl₆ and Cs₂AgBiBr₆, are non-toxic and stable, making them promising for solar applications compared to the lead-containing alternatives. The unique structure of halide double perovskites has gained attention as a potential replacement for lead-based perovskite materials.^11–15

van der Waals (vdW) interactions have been shown to have a significant impact on the geometry optimization and electronic properties of inorganic CH₃NH₃(Pb,Sn)(I, Br, Cl)₃ halide perovskites. Studies have focused on the effects of these interactions on the structural and electronic properties of these perovskites. A hydrogen/ionic link between the halogen atoms and an ordinary amine group (NH₃⁺) creates the bonding between organic and inorganic components (van der Waals interaction).^16,17 Wang et al.¹⁸ presented a PD with a hybrid vdW heterostructure made of graphene (Gr)/1D CH₃NH₃PbI₃ and hexagonal boron nitride (h-BN).

However, vdW heterostructures based on double perovskites are still uncommon but are known to have a significant role on the properties of 2D perovskites. Gao et al. constructed different heterostructures made up of CsCl, NaInCl, CsNaInCl, and Cl interfaces of the inorganic double perovskite Cs₂NaInCl₆ and two-dimensional transition metal–sulfur complexes XS2 (X = Cr, Mo, and W).¹⁹ Using DFT calculations, properties of the 2D (C₄H₉NH₃)₂PbI₄/black phosphorus (BP) vdW heterostructure have been investigated.²⁰ Furthermore, density functional theory considering vdW, was used to calculate the electron structure and interface properties of inorganic perovskite (without organic cation), results show weak vdW forces between the interfaces of Cs₂AgBiBr₆ and Ti₃C₂T_x (to maintain its semiconductor characteristics).²¹ There is no study for vdW interactions in 3D inorganic double perovskites or double anti-perovskites as compared to 2D and heterostructures.

Recently, Rani et al. explored some new double antiperovskites Na₆SOCl₂, Na₆SOBr₂, Na₆SOI₂, K₆SOCl₂, K₆SOBr₂ and K₆SOI₂ with bandgaps of 4.34 eV, 3.71 eV, 3.33 eV, 3.99 eV, 3.38 eV and 2.90 eV, respectively, and show that they are good for thermoelectric applications.²² Yu et al. predict that Na₆SOI₂ is specifically appealing for solid sodium-ion battery applications at low temperatures.²³ Mebrouki et al. examined the process by which temperature effects the BaVO₃ elastic constants C₁₁, C₁₂, and C₄₄, less progressively as temperature rises.²⁴ Djebari et al. calculated the octahedral factor, the formation energy, the tolerance factor etc. to confirm the structural stability of A³⁺B⁴⁺(O₂N)⁷⁻ and A²⁺B⁵⁺(O₂N)⁷⁻. Based on the computed bandgaps, photovoltaic and ferroelectric devices could find use for them.²⁵ Gao et al. discovered that the coefficient of thermal expansion is negatively impacted by variations in pressure and temperature.²⁶ Since the double antiperovskite has just recently been proposed, there are not many reports on tests or theoretical calculations, suggesting that this might be a new line of inquiry. Presently, we are examining the features of double antiperovskites Cs₂AgBiX₆ (X = Cl, Br, I). We are examining the expected material’s structural, electrical, elastic and optical properties, focusing on those with a bandgap similar to MAPI.²⁷

2 Computational details

Using the Wien2k code, we ran our calculations using the full-potential linearized augmented plane wave plus local orbitals (FP-LAPW + lo) technique, which is a component of DFT.²⁸ Using the more accurate exchange-correlation potential flavor of the generalized gradient approximation (WC-GGA), structural, electrical, and thermodynamic features are investigated.²⁹ To acquire precise bandgap data, we implemented the modified Becke-Johnson (mBJ) scheme across the WC-GGA.^30,31 In the past, the mBJ plan provided promising band structures and bandgap values for lead halide perovskites and other semiconductors when compared to experimental results.^32–37 The scheme mBJ across WC-GGA gives results close to the experimental results for Cs₆AgBiX₂ (X = Cl, Br and I). The accuracy of our calculations is mainly affected by two factors: one is the total number of k points, so for the Brillouin zone (BZ) integration a k-mesh of 13 × 13 × 13 was used and a finer k-mesh of 24 × 24 × 24 for electronic and optical properties. Another factor is cutoff kinetic energy which determines the number of plane wave expansions inside the muffin-tin radii, here the cutoff energy is set to be −7.0 Ryd, which separates the valence states from the core states of electrons. The plane-wave expansion cutoff was expanded up to G_max = 20 and the angular momentum of the sphere is l_max = 12. All structures were optimized with an energy convergence tolerance of up to 10⁻⁵ Ryd.^38–41

3 Results and discussion

3.1 Structural properties

Understanding the different physical characteristics of a substance is crucial and understanding structural characteristics is essential for this. Fig. 1 shows the transformation of double perovskite into a double anti-perovskite crystal structure. The proposal aims to optimize the structural properties of Cs₆AgBiX₂ (X = Cl, Br, I) by reducing the energy to its ground state. The starting point for the optimization procedure is the lattice parameter of experimental Cs₂AgBiCl₆. The volume optimization curve, which represents the relationship between energy and volume, is plotted in Fig. 2 for Cs₆AgBiX₂ (X = Cl, Br, I). To fit the nonlinear plot of energy vs. volume we used the initial lattice parameter values for Cs₂AgBiX₆ (X = Cl and Br) as a reference. The volume optimization was used to calculate structural parameters such as bulk modulus B (GPa) and lattice constant a (Å), for Cs₆AgBiX₂ (X = Cl, Br, I). Table 1 shows the structural characteristics, which were determined by applying Birch–Murnaghan's equation of states.⁴⁴ The optimized lattice constants for all compounds fall within the range 14.14 Å to 14.24 Å, closely aligning with experimentally synthesized double perovskites like Cs₂AgBiCl₆ and Cs₂AgBiBr₆, which have lattice constants of 10.77 Å and 11.27 Å.¹⁴ The optimized bulk modulus (B) ranges from 4.63 GPa to 5.16 GPa as presented in Table 1. Unfortunately, experimental or theoretical data for comparison with these optimized compounds is not available.

Fig. 1 Prospective view of (a) double perovskite and (b) double anti-perovskite crystal structures.

Fig. 2 For cubic double anti-perovskites, the energy vs. volume curve, Cs₆AgBiX₂ (X = Cl, Br, I).

Table 1 Optimized lattice parameters, bulk modulus and bandgap of double antiperovskites Cs₆AgBiX₂ (X = Cl, Br, I)

Materials	a = b = c (Å)	Bulk modulus B (GPa)	GGA Eg (eV)	GGA + SOC Eg (eV)	Hybrid Eg (eV)
Cs6AgBiCl2	14.14	4.63	0.91	0.72	1.50
Cs6AgBiBr2	14.16	5.10	0.84	0.65	1.49
Cs6AgBiI2	14.24	5.16	0.78	0.59	1.39

---

3.2 Electronic band structure

The behavior of a material is profoundly shaped by its band structure, a critical aspect for studying its properties. Analyzing the band structure, including determining its metallic or semiconducting nature, provides valuable insights into the electronic characteristics of the material. This electronic band structure is essential for understanding the optical behaviour and resistivity of solids, which are important for the design of solid-state electronics like solar cells and transistors. The band gap is either direct or indirect, the dispersion of bands informs us of the electronic behavior through which we can specify the physical properties of materials. Consequently, we conducted band structure calculations for Cs₆AgBiX₂ (X = Cl, Br, I) and examined their electronic conduct for further analysis. The initial bandgap calculations for each compound utilized a simple self-consistent field (SCF) approach without considering spin–orbit coupling (SOC). Similarly, this was also done in order to obtain bandgap values that nearly match the experimental bandgap value of MAPI (1.55 eV), producing a straight bandgap as a result at the symmetry point similar to MAPI.⁴² Additionally, mBJ + SOC (Hybrid) was utilized to modify the bandgap values in order to better correlate with the experimental MAPI, as a good solar cell absorber. The bandgap of CH₃NH₃PbCl₃ is 3 eV, while the band gap of CH₃NH₃PbBr₃ is 2.26 eV.⁴³ The experimental (theoretical) bandgap values for Cs₂AgBiCl₆ and Cs₂AgBiBr₆, which were previously reported as 2.77 (2.62) eV and 2.19 (2.06) eV, respectively, were recalculated as a baseline with a similar approach.^14,43 We computed bandgap values of 1.50 eV (Cl), 1.49 eV (Br), and 1.39 eV (I) for Cs₆AgBiX₂ (X = Cl, Br, I) with hybrid, respectively. These are close to MAPI, hence proving that anti-double perovskite could reduce the bandgap values substantially as compared to double perovskites. While with SOC, and without SOC, bandgap results are listed in Table 1 and are explained in Fig. 3 and 4.

Fig. 3 The band structure of Cs₆AgBiX₂ (X = Cl, Br, I) without SOC (black) and with SOC (blue).

Fig. 4 The band structure of Cs₆AgBiX₂ (X = Cl, Br, I) with hybrid.

Our analysis leads to the conclusion that the FP-LAPW + lo technique, without considering mBJ + SOC, tends to substantially underestimate bandgap values in double antiperovskites. Consequently, our objective is to identify a lead-free material with a bandgap value approximately equal to 1.5 eV, serving as a potential replacement for MAPI at room temperature.⁴⁵

The bandgap values for Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂ are 0.91 eV, 0.84 eV and 0.78 eV, in the absence of SOC. The bandgap values of these materials with and without SOC are depicted in Fig. 3a–c. The band structure of Cs₆AgBiCl₂ has a bandgap value of 0.91 eV in the absence of SOC and 0.72 eV in the presence of SOC. The band structure of Cs₆AgBiBr₂ has a bandgap value of 0.84 eV in the absence of SOC and 0.65 eV in the presence of SOC. These two materials have a direct bandgap as shown in Fig. 3a and b while the band structure of Cs₆AgBiI₂ has a bandgap value of 0.78 eV in the absence of SOC and 0.59 eV in the presence of SOC as shown in Fig. 3c.

It is observed that the bandgap values decreased slightly for Cs₆AgBiX₂ (X = Cl, Br, I) after applying SOC. The bandgap values of Cs₆AgBiCl₂ and Cs₆AgBiBr₂ align closely with the experimentally calculated values of CH₃NH₃PbI₃ (1.55 eV). Within the bandgap of 1.39 to 1.50 eV (see Fig. 4), efficient semiconductor solar cells can be developed, enabling the release of electrons without generating a substantial amount of heat. While some recently investigated double antiperovskites have a direct bandgap, their values exceed this optimal range. Others within the mentioned range, although having an indirect bandgap, are not suitable for solar cell applications. Pressure-induced bandgap tuning could potentially bring the bandgap close to 1.5 eV.

3.3 Density of states (DOS)

We examined the density of states (DOS) for Cs₆AgBiX₂ (X = Cl, Br, I) in order to evaluate the individual contributions of Cs, Ag, Bi, Cl, Br and I atoms to the valence band maxima (VBM) and conduction band minima (CBM), as depicted in Fig. 5a–c.

Fig. 5 Density of states (DOS) of (a) Cs₆AgBiCl₂, (b) Cs₆AgBiBr₂ and (c) Cs₆AgBiI₂.

In Fig. 5a and b, it is evident that the VBM is predominantly influenced by the Cs atom in both materials, while the CBM is shared between Cs and Cl/Br atoms. Notably, Br atoms significantly contribute to the bandgap value. Further insights into the orbital contributions of Cl and Br in the VBM and CBM were gained by examining the PDOS, which is also illustrated in Fig. 5 for Cs₆AgBiX₂ (Cl, Br, I).

The range of the valence band is −3 to 0 and the conduction band range is 0 to 3, and the 0 point represents the Fermi level. Fig. 5a reveals that the Cs-s orbital plays a dominant role in both CBM and VBM, while other atoms have a relatively small contribution compared to these two atoms. The p-orbital of Cl also exhibits significance in the CBM, whereas the contributions of d orbitals are minor in VBM. In VBM, the p-orbital of Bi plays a crucial role, with Cl's p-orbital occupying the entire VBM up to 2 eV, however Br's d-orbital does not contribute much to either the CBM or VBM. Also, compared to Ag and Bi, the effect of Cl's p-orbital on VBM is greater. According to Fig. 5b, comparable patterns are seen in the DOS: Br's p-orbital and Cs's d-orbital completely dominate the CBM and Cs's p-orbital completely dominates the VBM. According to Fig. 5c, similar patterns are seen in the DOS: I's p-orbital and Cs's p-orbital completely dominate the CBM and Cs's d-orbital completely dominates the VBM.

3.4 Optical properties

Studying the optical characteristics of materials is essential from the standpoint of the industrial manufacture of optical devices like sensors, lasers, modulators, solar calculators and optical coatings. One promising option for optical materials production is perovskite high-performing optoelectronic devices.^45,46 The optical characteristics such as the complex dielectric function, ε(ω) = ε₁(ω) + iε₂(ω), can be utilized to describe a material's reaction to an external electromagnetic field,⁴⁷ where ε₁(ω) and ε₂(ω) are the real and imaginary parts of ε(ω). In the relation ε₁(ω), is the dielectric constant, which expresses the amount of material polarized by induced electric dipole formation in the presence of an electric field, whereas ε₂(ω) indicates the capacity for electromagnetic waves to transmit and attenuate them. The dielectric characteristics ε(ω) are closely linked to the combined density of states (DOS) and transition momentum matrix components. The real part of the dielectric function, ε₁(ω), can be obtained from ε₂(ω) using Kramers–Kronig relationship.⁴⁸ The wave is diminishing and cannot propagate if the value of ε < 0; if ε₁(ω) > 0 light can propagate through the material.^49,50

The dispersion of the real and imaginary components of the dielectric characteristics can be analyzed to compute the refractive index, optical conductivity, reflection and absorption coefficients among other optical properties.

The optical properties of solids can be explained by the refractive index n(ω) and extinction co-efficients k(ω). In the case of n(ω) < k(ω) the incident rays are reflected and the behavior of materials is dielectric to metallic. It is more advantageous for materials with high dielectric constants to be used in solar cells since they often have low exciton binding energy.⁵¹

Cs₆AgBiX₂ (Cl, Br, I) have ε₁ 9.57, 9.75 and 10.19, respectively, which are greater than Cs₂LiTlBr₆ (1.6), Cs₂NaTlBr₆ (1.7), Cs₂AgCrI₆ (5.7), MAPbI₃ (5.4) and FAPbI₃ (5.7), other potential candidates for solar cell applications.^51–56 It is important to reach maximum polarization characteristics, as indicated by the size of ε₁(ω), in order to obtain optimal semiconductor performance. In the visible light range, ε₁(ω) approaches peak polarization values of 11.96, 8.77 and 8.63 at 1.0 eV, 0.91 eV, and 0.90 eV, respectively; and ε₂(ω) imaginary part shows the light absorption with ε₂(ω) values 8.6, 7.9 and 7.5 at −3.0 eV, −3.8 eV, −4.3 eV. None of these compounds are in the visible region. Computations based on the distribution of the dielectric characteristics with real and imaginary parts are shown in Fig. 6, and provide important new information on the linear optical behavior of these materials. We calculated parameters such as the refractive index (n), extinction coefficient (k), absorption coefficient I(ω), optical conductivity σ(τ), and reflectivity R(ω), in order to analyze the optical properties of Cs₆AgBiX₂ (X = Cl, Br, I), as illustrated in Fig. 7. For Cs₂LiTlBr₆ and Cs₂NaTlBr₆, the static dielectric constants ε(0) are 1.6 eV and 1.7 eV, respectively.⁴⁶ So here our study produced comparable findings. Looking at Fig. 7a, we can see that both materials, Cs₆AgBiCl₂ and Cs₆AgBiBr₂, have a static refractive index (n(ω)) with values of 3.2 and 3.1, respectively. In Cs₂LiTlBr₆ the highest values for n(ω) and k(ω) are 1.8 eV and 2.6 eV, while for Cs₂NaTlBr₆, these values are 2.5 eV for n(ω) and 3.2 eV for k(ω). The optical conductivity (σ) of Cs₆AgBiCl₂ peaks at 650 (Ω cm)⁻¹ at 2.6 eV in Fig. 7b, whereas Cs₆AgBiBr₂ peaks at 500 (Ω cm)⁻¹ at 3 eV. Because of this, Cs₆AgBiCl₂ has exceptionally high optical conductivity in the visible light spectrum (1.77 eV to 3.1 eV) which is important for solar cell applications. The efficiency of solar energy conversion is often described by the absorption coefficient, representing how specific light frequencies penetrate the material before absorption. For Cs₆AgBiCl₂ and Cs₆AgBiBr₂, the absorption coefficient spectrum I(ω) shows peaks of 12 × 10⁴ cm⁻¹ at 2.6 eV (477 nm) and 11 × 10⁴ cm⁻¹ at 3.2 eV, respectively (Fig. 7c). These values are comparable to those of MAPI (methylammonium lead iodide) at 10⁴–10⁵ cm⁻¹. The absorption coefficient increases rapidly as incident photons approach the absorption edge, with Cs₆AgBiCl₂ exhibiting higher absorption in the visible region compared to Cs₆AgBiBr₂. Lastly, in Fig. 7d, the maximum reflectivity R(ω) for Cs₆AgBiCl₂ and Cs₆AgBiBr₂ is 13.5% at approximately 1.6 eV and 9% at 2.2 eV, respectively.

Fig. 6 The dielectric constants for Cs₆AgBiX₂ (X = Cl, Br, I).

Fig. 7 Optical properties such as (a) refractive index, n and extinction coefficient, k, (b) optical conductivity, σ, (c) absorption coefficient, α, (d) reflectivity, R of Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂.

3.5 Photocatalytic properties

Suitable indirect bandgap semiconductors can be used to utilize solar energy to generate hydrogen by dissociating water.^57,58 Therefore, clean renewable energy can be produced through photocatalytic water splitting.^59,60 Water is oxidized by holes and reduced by electrons in the photocatalytic process.⁶¹

The oxidation–reduction potential of O (1.23) eV for this operation needs to be smaller (higher) than the conduction (valence) band photocatalytic water splitting for all materials under consideration (Table 2).

Table 2 Photocatalytic parameters of Cs₆AgBiX₂ (X = Cl, Br, I)

Compounds	χ (eV)	Eg (eV)	ECBM	EVBM
Cs6AgBiCl2	1.17	1.50	−1.06	−2.57
Cs6AgBiBr2	1.16	1.49	−1.10	−2.59
Cs6AgBiI2	1.14	1.39	−1.26	−2.65

---

Mullikan electronegativity is used to explore this:

E_VBM = χ − E_elec + 0.5E_g (1)

E_CBM = E_VBM + E_g (2)

As displayed in Fig. 8, on the hydrogen scale the standard oxidation and reduction potentials for photocatalytic water splitting are (−4.44 eV) and (−5.67 eV), respectively.⁶² The Fermi level is set to (−4.44) eV in order to determine the band edge positions of the VB and CB with respect to standard oxidation.⁶² The CB and VB are set to (0 eV = −4.44 eV) and (1.23 eV = ¬5.67 eV).⁶³ It is clear from the figure that all the materials show good responses for the oxidation of water but fail to produce hydrogen from water.

Fig. 8 Photocatalytic properties of Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂.

3.6 Elastic properties

The effective elastic constants are essential for determining a material's practical applications, providing information about its reactivity to outside influences and structural stability. The elastic constants C₁₁, C₁₂, C₄₄, shear moduli G and Bulk moduli B are the key points of elastic properties. These properties help us to understand the material demand in industrial applications. In the case of Cs₆AgBix₂ (x = Cl, Br, I) the calculated elastic constants are detailed in Table 3 and graphical analysis is given in Fig. 9. The materials Cs₆AgBiCl₂ are Cs₆AgBiBr₂ meet the requirements of Born–Huang stability for cubic crystal stability, that is C₁₁ > 0, C₄₄ > 0, C₁₁ + 2C₁₂ > 0, C₁₁ − C₁₂ > 0 and C₁₂ < B < C₁₁ are satisfied⁶⁴ while Cs₆AgBiI₂ does not meet the criteria. So it is unstable.

Table 3 Elastic properties of Cs₆AgBiX₂ (X = Cl, Br, I)

Constants	Cs6AgBiCl2	Cs6AgBiBr2	Cs6AgBiI2
C11	9.5	10.74	9.15
C12	2.22	2.56	3.17
C44	2.38	1.72	−3.65
B	4.64	5.10	5.16
Gv	2.8	2.67	−0.99
GR	2.7	2.24	−32.67
GH	2.8	2.45	−16.83
Y	7.17	6.86	−3.18
B/G	1.66	2.15	0.30
C′′	−0.16	0.84	6.82
ν	0.24	0.28	0.60
A	0.65	0.42	−1.22
C′	3.64	4.09	2.99
Az	1.73	1.55	0.54
Au	0.22	0.95	−4.84
ξ	0.38	0.39	0.49
T(m)	609.15	616.48	607.08

---

Fig. 9 Graphical view of elastic properties for elastically stable Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂.

The elastic constants can be calculated using the following Voigt–Reuss–Hill approximation where, in the formulas B_V = B_R = C₁₁ + 2C₁₂/3, G_V = C₁₁ − C₁₂ + 3C₄₄ and G_R = 5C₄₄(C₁₁ − C₁₂)/(4C₄₄ + 3C₁₁ − 3C₁₂), subscripts V and R stand for Voigt bound and Reuss bound, respectively.⁶⁵ In Table 3, we can see that the G_v of Cs₆AgBiX₂ (Cl, Br, I) decreases continuously which shows that the resistive behavior of these materials is in the order Cs₆AgBiCl₂ > Cs₆AgBiBr₂ > Cs₆AgBiI₂.

Poisson's ratio (ν) or Pugh's ratio (B/G) can be used to evaluate a material's toughness and brittleness. When ν is less than 0.26 or B/G is less than 1.75, the material is considered ductile.⁶⁶ Here, only one material, Cs₆AgBiCl₂, is ductile while the other two are brittle as their ν and B/G values are greater than 0.26 and 1.75, respectively. Ionic materials typically have a ν range of 0.3–0.4, but the limit for pure ionic materials is 0.5. Covalent materials usually have ν values around 0.2 and for Cs₆AgBiX₂ (X = Cl, Br, I), they are 0.24, 0.28 and 0.59, indicating a greater ionic property than covalent property.⁶⁷

Elastic anisotropy is associated with the development of microcracks during the use of a material, we computed this for Cs₆AgBiX₂ (X = Cl, Br, I) using the anisotropy index A = 2C₄₄/(C₁₁ − C₁₂). A material is isotropic if A is either 0 or 1; otherwise, it exhibits anisotropy.^68,69 The mechanical properties, given in Table 3, were analyzed using standard relations. Cs₆AgBiCl₂ shows greater shear modulus (G_H) and Young's modulus (Y) values, indicating greater stiffness and resistance against plastic deformation compared to Cs₆AgBiBr₂.

Cauchy's pressure (C′′ = C₁₂ − C₄₄) values, whether positive or negative are presented in Table 3, further elucidate the ductile nature of Cs₆AgBiCl₂. The positive values of C′′ for Cs₆AgBiBr₂ and Cs₆AgBiI₂ reinforce the materials' brittleness. The shear constant (G_H), determining a material's dynamic stability, has a positive value for Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂ in Table 3, affirming their mechanical stability.

In order to graphically portray the anisotropic and elastic behavior of the materials under study, three-dimensional (3-D) contour plots of the Young's modulus (Y), linear compressibility (β), shear modulus (G), and Poisson's ratio (ν) have been created using the Elate Code.⁷⁰ Fig. 10 and 11, respectively, show the resulting 3-D visual representations for Cs₆AgBiCl₂ and Cs₆AgBiBr₂. Notably, all other constants show large deviations from a fully spherical (3-D) shape, with the exception of linear compressibility, highlighting the anisotropic character of these materials.

Fig. 10 3D spatial dependence of (a) Young's modulus, (b) linear compressibility, (c) shear modulus and (d) Poisson's ratio for Cs₆AgBiCl₂.

Fig. 11 3D spatial dependence of (a) Young's modulus, (b) linear compressibility, (c) shear modulus and (d) Poisson's ratio for Cs₆AgBiBr₂.

Additionally, anisotropic parameters, such as Zener (A_Z) and Universal anisotropic parameters (A_U), have been determined using the second-order elastic constants (SOECs).^71,72 Table 3 presents the computed values for these parameters. In isotropic materials, the Zener elastic parameter, which is mainly related to shear anisotropy in materials, stays constant at 1. But any departure from unity, as we can observe in A_U and A_Z in Table 3, indicates that the material under study exhibits anisotropic behavior. The elastic constants have later been used to calculate the Kleiman parameter (ξ). Table 3 indicates that the materials have a higher resistance against bond angle distortions and bond bending due to the observed low value of ξ.

We determined the melting temperature using the following equation in order to assess their stability at high temperatures,⁷³

T(m) = [553 + 5.911C₁₁] ± 300 (3)

The predicted melting temperatures are determined to be 609.15 ± 300 K for Cs₆AgBiCl₂, 616.48 ± 300 K for Cs₆AgBiBr₂ and 607.08 ± 300 K for Cs₆AgBiI₂. Notably, the high value of the melting temperature suggests that these materials exhibit stability even under extreme temperatures.

3.7 Phonon dispersion and thermal properties

To further confirm the stability, adding to the elastic constant results, we use a 2 × 2 × 2 supercell in the well -established PHONOPY package.^74,75 The calculated phonon band structures of Cs₆AgBiCl₂, Cs₆AgBiBr₂ and Cs₆AgBiI₂ are shown in Fig. 12a–c. Cs₆AgBiCl₂ and Cs₆AgBiBr₂ are dynamically stable which also validates the elastic results, while Cs₆AgBiI₂ has negative phonon frequencies up to −25 cm⁻¹. Hence, Cs₆AgBiI₂ has been shown to be unstable by elastic as well as vibrational analysis.

Fig. 12 Calculated phonon band structures of Cs₆AgBiX₂ (X = Cl, Br, I).

Aside from these elastic constants, Debye temperature (Θ_D) is an important thermal parameter that is closely linked to several physical properties such as a compound’s melting point and specific heat capacity. It is possible to calculate Debye temperature using average sound velocity v_m.⁷⁶

The variables h, k, n, N_A, ρ, and M stand for Planck's constant, Boltzmann constant, atomic number, Avogadro’s number, material density, and molecular mass, respectively. The stronger the connection, the higher the Debye temperature or sound velocity. The average sound velocity in polycrystalline materials can be expressed as follows:⁷⁷where v_t and v_l are the transverse and longitudinal sound velocities, respectively, that may be derived using Navier's equation and elastic constants (B and G).⁷⁸

Table 4 displays the values of v_t, v_l, v_m, and Θ_D that were derived from the aforementioned relations. Cs₆AgBiCl₂ and Cs₆AgBiBr₂ have the highest Debye temperature indicating that these are thermally more conductive. These findings also provide insight into the bond strength hierarchy.

Table 4 Thermal properties of Cs₆AgBiCl₂ and Cs₆AgBiBr₂

Compound	Cs6AgBiCl2	Cs6AgBiBr2
vt (m s^−1)	1006.9	904.04
vl (m s^−1)	1737.84	1670.64
vm (m s^−1)	1086.32	985.35
ΘD (K)	473.6	451.3
ωD (THz)	61.9	58.9

---

4 Conclusion

In this work, structural, electronic, optical, photocatalytic, mechanical, vibrational, and thermodynamical behaviors were studied for new double antiperovskite (DAP) Cs₆AgBiX₂ (X = Cl, Br, I). Band structures were calculated with and without spin orbit coupling (SOC). Using the TB-mBJ approach (hybrid) reveals bandgap values of 1.504 eV, 1.491 eV, and 1.392 eV for Cs₆AgBiCl₂, Cs₆AgBiBr₂, and Cs₆AgBiI₂ respectively. To gain deeper insight into the orbital contributions of chlorine (Cl), bromine (Br), and iodine (I) near the Fermi level, analysis of the total density of states (TDOS) and partial density of states (PDOS) were studied. In addition to electronic properties, the investigation extended to optical properties, including real and fictitious refractive indexes, extinction coefficients, optical conductivities, absorption coefficient, and reflectivity. The elastic and phonon properties demonstrate that Cs₆AgBiCl₂ and Cs₆AgBiBr₂ are stable except Cs₆AgBiI₂. The calculated optimal bandgap (∼1.5 eV) and high absorption coefficient (∼30× 10⁴ cm⁻¹) of Cs₆AgBiCl₂ and Cs₆AgBiBr₂, suggest their potential for solar cell applications. In addition, our photocatalytic results suggest that these Cs₆AgBiX₂ have more oxidizing capacity due to good visible-light absorption capability that can be used for cheap oxygen production via solar water splitting.

Data availability

Data will be made available upon request.

Conflicts of interest

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

Acknowledgements

The authors would like to acknowledge the Researcher's Supporting Project Number (RSP2024R511), King Saud University, Riyadh, Saudi Arabia. The author Y. Saeed would like to thank the Higher Education Commission (HEC) of Pakistan for providing a grant under NRPU-15844.

References

1. T. Wu, M. L. Yao and M. Q. Long, First principle calculations of interface interactions and photoelectric properties of perovskite CsPbX₃ (X = Cl, Br, I) and penta-graphene van der Waals heterostructures, Acta Phys. Sin., 2021, 70, 056301.
2. H. K. Ting, L. Ni, S. B. Ma, Y. Z. Ma, L. X. Xiao and Z. J. Chen, Progress in electron-transport materials in application of perovskite solar cells, Acta Phys. Sin., 2015, 64, 038802.
3. P. Zhao, J. Su, Y. J. Guo, Z. H. Wang, Y. Lin, X. P. Hao, Ouyang and J. J. Chang, Cs₂TiI₆: a potential lead−free all−inorganic perovskite material for ultrahigh−performance photovoltaic cells and alpha-particle detection, Nano Res., 2022, 15, 2697–2705.
4. Y. Guo, Y. B. Xue and L. Q. Xu, Interfacial interactions and enhanced optoelectronic properties of GaN/perovskite heterostructures: insight from first-principles calculations”, J. Mater. Sci., 2021, 56, 11352–11363.
5. G. M. Sui, G. J. Yan, G. Yang, B. Zhang and Y. Q. Feng, Theoretical investigation on structure and optoelectronic performance of two-dimensional fluorbenzidine perovskites, Acta Phys. Sin., 2022, 71, 208801.
6. D. Wang, H. M. Zhu, Z. M. Zhou, Z. W. Wang, S. L. Lv, S. P. Pang and G. L. Cui, Effect of solvent on the perovskite thin film morphology and crystallinity, Acta Phys. Sin., 2015, 64, 038403.
7. Y. B. Fu, H. Wei, L. Wei, H. D. Zhang, X. P. Wang, B. Liu, Y. Y. Zhang, X. S. Lv, J. X. Zhou and H. J. Yu, Origin of the difference in thermal conductivity and anharmonic phonon scattering between LiNbO₃ and LiTaO₃, CrystEngComm, 2021, 23, 8572–8578.
8. Y. N. Xiao, P. F. Sui, Y. C. Zhao, J. Ni, S. Meng and Z. H. Dai, Low lattice thermal conductivity of hydride-based cubic antiperovskites A3HB (A = Li, Na; B = S, Se, Te) with higher−order anharmonicity correction, Int. J. Energy Res., 2022, 46, 13687–13697.
9. Y. Liu, L. H. Ni, G. Xu, C. L. Song, G. R. Han and Y. Zheng, Phase transition in PbTiO3 under pressure studied by the first−principles method, Phys. B, 2008, 403, 3863–3866.
10. D. Han, C. B. Feng, M. H. Du, T. Zhang, S. Z. Wang, G. Tang, T. Bein and H. Ebert, Design of high−performance lead−free quaternary antiperovskites for photovoltaics via ion type inversion and anion ordering, J. Am. Chem. Soc., 2021, 143, 12369–12379.
11. J. Chen, X. Cai, D. Yang, D. Song, J. Wang, J. Jiang, A. Ma, S. Lv, M. Z. Hu and C. Ni, Recent progress in stabilizing hybrid perovskites for solar cell applications, J. Power Sources, 2017, 355, 98–133.
12. M. J. Fang, C. W. Tsao and Y. J. Hsu, Semiconductor nanoheterostructures for photoconversion applications, J. Phys. D: Appl. Phys., 2020, 53, 143001–143026.
13. Y. H. Chiu, T. H. Lai, M. Y. Kuo, P. Y. Hsieh and Y. J. Hsu, Photoelectrochemical cells for solar hydrogen production: Challenges and opportunities, APL Mater., 2019, 7, 080901–080911.
14. Y. Saeed, B. Amin, H. Khalil, F. Rehman, H. Ali, M. I. Khan, A. Mahmood and M. Shafiq, Correction: Cs2NaGaBr6: a new lead-free and direct bandgap halide double perovskite, RSC Adv., 2020, 10, 17444–17451.
15. M. Z. Rehman, S. A. M. Abdelmohsen, E. A. Mahmoud, M. U. Saeed, M. Idress, M. Shafiq, B. Amin and Y. Saeed, First principles study of structural, electronic, elastic and optical properties of Cs₂LiTlBr₆ and Cs₂NaTlBr₆, Mater. Sci. Semicond. Process., 2022, 151, 106993.
16. N. Pandech, T. Kongnok, N. Palakawong, S. Limpijumnong, W. R. Lambrecht and S. Jungthawan, Effects of the van der Waals Interactions on Structural and Electronic Properties of CH₃NH₃(Pb, Sn)(I, Br, Cl)₃ Halide Perovskites, ACS Omega, 2020, 5, 25723–25732.
17. H. Ferjani, Y. B. Smida and Y. Al-Douri, First-Principles Calculations to Investigate the Effect of Van der Waals Interactions on the Crystal and Electronic Structures of Tin-Based 0D Hybrid Perovskites, Inorganics, 2022, 10, 155.
18. G. Wang, B. Han, C. H. Mak, J. Liu, B. Liu, P. Liu, X. Hao, H. Wang, S. Ma, B. Xu and H.-Y. Hsu, Mixed-dimensional van der waals heterostructure for high-performance and air-stable perovskite nanowire photodetectors, ACS Appl. Mater. Interfaces, 2022, 14, 55183–55191.
19. S. Gao, W. Chen, E. Abduryim, C. Chen, C. Dong, Y. Kuai, X. Guan and P. Lu, Double-perovskite van der Waals heterostructure Cs2NaInCl6-XS2 (X=Cr, Mo, W) as great potential material in photovoltaic devices, Surf. Interfaces, 2023, 37, 102734.
20. D. Li, D. Li, A. Yang, H. Zhang, X. Lai and C. Liang, Electronic and Optical Properties of van der Waals Heterostructures Based on Two-Dimensional Perovskite (PEA)₂PbI₄ and Black Phosphorus, ACS Omega, 2021, 6, 20877–20886.
21. L. Yang, P. Hou, B. Wang, C. Dall'Agnese, Y. Dall'Agnese, G. Chen, Y. Gogotsi, X. Meng and X.-F. Wang, Performance improvement of dye-sensitized double perovskite solar cells by adding Ti₃C₂T_x MXene, Chem. Eng. J., 2022, 446, 136963.
22. U. Rani, P. K. Kamlesh, R. Agarwal, J. Kumari and A. S. Verma, Electronic and thermo-physical properties of double antiperovskites X₆SOA₂ (X = Na, K and A = Cl, Br, I): a non−toxic and efficient energy storage materials, Int. J. Quantum Chem., 2021, 121, 26759.
23. Y. Yu, Z. Wang and G. S. Shao, Theoretical design of double anti-perovskite Na₆SOI₂ as a super−fast ion conductor for solid Na⁺ ion batteries, J. Mater. Chem. A, 2018, 6, 19843–19852.
24. M. Mebrouki, T. Ouahrani and Y. O. Ciftci, Unraveling thermal and dynamical properties of the cubic BaVO₃ perovskite from first−principles calculation, Int. J. Thermophys., 2016, 37, 71–80.
25. K. Djebari, A. Dahani, M. Djermouni, K. Dine, A. Cherif, O. Arbouche, A. Zaoui and S. Kacimi, Spontaneous polarization study in A³⁺B⁴⁺(O₂N)⁷− and A²⁺B⁵⁺(O₂N)⁷− perovskite-type oxynitrides: a first principles investigation”, Appl. Phys. A: Mater. Sci. Process., 2022, 128, 398–410.
26. Q. Gao, R. Y. Ma, L. Li, H. H. Xie, J. B. Deng and X. R. Hu, First-principle study on compensated half metallic double perovskite compound Ba₂PrVO₆, J. Supercond. Novel Magn., 2017, 30, 545–554.
27. H. X. Gao, De-Y. Hu, T.-Yu Tang, Qi-Qi Liang, Qi Dai, S.-S. Wu and Y.-L. Tang, First-principles study on the structural, electronic, elastic, optical and thermodynamic properties of double antiperovskites X₆BiSbN₂ (X = Mg, Ca, Sr), J. Phys. Chem. Solids, 2004, 187, 111859.
28. P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka and J. Luitz, WIEN2k; an Augmented Plane Wave Plus Local Orbital Program for Calculating Crystal Properties, Vienna University of Technology, Austria, 2001.
29. Z. Wu and R. E. Cohen, More accurate generalized gradient approximation for solids, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 235116.
30. J. P. Perdew, K. Burke and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett., 1996, 77, 3865.
31. E. Engel and S. H. Vosko, Exact exchange-only potentials and the virial relation as microscopic criteria for generalized gradient approximations, Phys. Rev. B: Condens. Matter, 1993, 47, 13164.
32. Z. Ali, I. Ahmed, I. Khan and B. Amin, Electronic structure of cubic perovskite SnTaO3, Intermetallics, 2012, 31, 287.
33. G. Murtaza, I. Ahmed, B. Amin and M. Zahid, Investigation of structural and optoelectronic properties of BaThO3, Opt. Mater., 2011, 33, 553.
34. H. Jiang, Band gaps from the Tran-Blaha modified Becke-Johnson approach: A systematic investigation, J. Chem. Phys., 2013, 138, 134115.
35. J. A. Camargo-Martinez and R. Baquero, Performance of the modified Becke-Johnson potential for semiconductors, Phys. Rev. B: Condens. Matter Mater. Phys., 2012, 86, 195106.
36. R. A. Jishi, O. B. Ta and A. A. Sharif, Modeling of lead halide perovskites for photovoltaic applications, J. Phys. Chem. C, 2014, 118, 28344–28349.
37. R. A. Jishi, Modified Becke-Johnson exchange potential: improved modelling of lead halides for solar cell applications, AIMS Mater. Sci., 2016, 3, 149–159.
38. M. Z. ur Rehman, S. A. Abdelmohsen, E. A. Mahmoud, M. U. Saeed, M. Idress, M. Shafiq, B. Amin and Y. Saeed, First principles study of structural, electronic, elastic and optical properties of Cs₂LiTlBr₆ and Cs₂NaTlBr₀, Mater. Sci. Semicond. Process., 2022, 151, 106993.
39. Y. Saeed, B. Amin, H. Khalil, F. Rehman, H. Ali, M. I. Khan, A. Mahmood and M. Shafiq, Cs₂NaGaBr₆ : a new lead-free and direct band gap halide double perovskite, RSC Adv., 2020, 10, 17444–17451.
40. U. Khan, M. U. Saeed, H. O. Elansary, I. M. Moussa and Y. Saeed, A DFT study of bandgap tuning in chloro-fluoro silicene, RSC Adv., 2024, 14, 4844–4852.
41. M. Usman Saeed, T. Usman, S. Pervaiz, Z. Ali, Y. Mohammed Alanazi, A. U. R. Bacha and Y. Saeed, DFT Analysis of Alkali Scandium Sulfides AScS 2 (A = K, Rb): Unveiling Structural, Electronic, and Optical Properties for Enhanced Photocatalysis”, ACS Appl. Opt. Mater., 2024, 2, 1955–1964.
42. C. Y. Wang, S. R. Li, S. F. Wang, P. X. Zhao, R. S. Zhuo and B. Y. Yu, Effect of biaxial Pre-proof strain on monolayer MoS₂ and its vacancy defect system: a first-principles study, J. Solid State Chem., 2023, 317, 123711.
43. J. Liang, C. Wang, Y. Wang, Z. Xu, Z. Lu, Y. Ma, H. Zhu, Y. Hu, C. Xiao, X. Yi, G. Zhu, H. Lv, L. Ma, T. Chen, Z. Tie, Z. Jin and J. Liu, All-Inorganic Perovskite Solar Cells, J. Am. Chem. Soc., 2017, 139, 2852.
44. E. T. McClure, M. R. Ball, W. Windl and P. M. Woodward, Cs₂AgBiX₆ (X= Br, Cl): new visible light absorbing, lead-free halide perovskite semiconductors, Chem. Mater., 2016, 28, 1348.
45. J. Liang, C. Wang, Y. Wang, Z. Xu, Z. Lu, Y. Ma, H. Zhu, Y. Hu, C. Xiao, X. Yi, G. Zhu, H. Lv, L. Ma, T. Chen, Z. Tie, Z. Jin and J. Liu, All-inorganic perovskite solar cells, J. Am. Chem. Soc., 2016, 138, 15829–15832.
46. Z. Abbas, N. Jabeen, A. Hussain, F. Kabir, T. Alshahrani, H. H. Raza, S. Muhammad, S. Azam and I. Gorczyca, Effect of Nb, Ta and V replacements on electronic, optical and elastic properties of NbCu₃Se₄: a GGA+U study”, J. Solid State Chem., 2021, 301, 122338.
47. P. Mondal, N. A. Shahed, S. Khanom, M. K. Hossain and F. Ahmed, Effect of oxygen defects in tuning the half-metallic and optical nature of Sr2CoXO6 (X = Mo and W): a first-principles study, Mater. Chem. Phys., 2022, 277, 125429.
48. L. D. Whalley, J. M. Frost, Y. K. Jung and A. Walsh, Perspective: Theory and simulation of hybrid halide perovskites, J. Chem. Phys., 2017, 146, 220901.
49. D. Y. Hu, X. H. Zhao, T. Y. Tang, L. Li, L. K. Gao and Y. L. Tang, First-principles calculations to investigate structural, elastic, electronic and optical properties of lead-free perovskite derivatives Cs2SeX6 (X = Cl, Br, I), Opt. Mater., 2021, 119, 111316.
50. L. Y. Gong, P. Zhang, Q. Chen, Z. H. Lou, J. Xu and F. Gao, First principles study of structure and property of Nb5+ doped SrTiO3, Acta Phys. Sin., 2021, 70, 227101.
51. E. Oyeniyi, Electronic and optical properties of Mg3XN (X = P, As, Sb, Bi) antiperovskites: the GW/BSE approach”, Solid State Commun., 2022, 355, 114927.
52. W. F. Goh and W. E. Pickett, Topological and thermoelectric properties of double antiperovskite pnictides, J. Phys.: Condens. Matter, 2020, 32, 345502.
53. C. Y. Wang, S. R. Li, S. F. Wang, P. X. Zhao, R. S. Zhuo and B. Y. Yu, Effect of biaxial[110] strain on monolayer MoS2 and its vacancy defect system: a first-principles study”, J. Solid State Chem., 2023, 317, 123711.
54. Z. Abbas, N. Jabeen, A. Hussain, F. Kabir, T. Alshahrani, H. H. Raza, S. Muhammad, S. Azam and I. Gorczyca, Effect of Nb, Ta and V replacements on electronic, optical and elastic properties of NbCu₃Se₄: a GGA+U study”, J. Solid State Chem., 2021, 301, 122338.
55. P. Mondal, N. A. Shahed, S. Khanom, M. K. Hossain and F. Ahmed, Effect of oxygen defects in tuning the half-metallic and optical nature of Sr2CoXO6 (X = Mo and W): a first−principles study, Mater. Chem. Phys., 2022, 277, 125429.
56. S. Berri, Theoretical analysis of the structural, electronic and optical properties of tetragonal Sr2GaSbO6, Chin. J. Phys., 2017, 55, 2476–2483.
57. K. Maeda and K. Domen, Photocatalytic water splitting: recent progress and future challenges, J. Phys. Chem. Lett., 2010, 1, 2655–2661.
58. R. M. Navarro Yerga, M. C. Alvarez Galvan, F. Del Valle, J. A. Villoria de la Mano and J. L. Fierro, Water splitting on semiconductor catalysts under visible-light irradiation, Chem. Sustain. Ener. Mater., 2009, 2, 471–485.
59. F. E. Osterloh, Inorganic materials as catalysts for photochemical splitting of water, Chem. Mater., 2008, 20, 35–54.
60. X. Hu, G. Li and J. C. Yu, Design, fabrication and modification of nanostructured semiconductor materials for environmental and energy applications, Langmuir, 2010, 26, 3031–3039.
61. A. Kudo, Photocatalysis and solar hydrogen production, Pure Appl. Chem., 2007, 79, 1917–1927.
62. J. Liu, X. Fu, S. Chen and Y. Zhu, Electronic structure and optical properties of Ag₃PO₄ photocatalyst calculated by hybrid density functional method, Appl. Phys. Lett., 2011, 99, 191903.
63. M. Saeed, I. U. Haq, A. S. Saleemi, S. U. Rehman, B. U. Haq, A. R. Chaudhry and I. Khan, First-principles prediction of the ground-state crystal structure of double-perovskite halides Cs₂AgCrX₆ (X = Cl, Br, and I), J. Phys. Chem. Solids, 2022, 160, 110302.
64. J. Li and H. Chen, First-principles investigation of the mechanical and thermodynamic properties of the antiperovskite centrosymmetric platinum-based superconductors APt3P (A = Ca, Sr, La and Ce), Phys. B, 2020, 577, 411791.
65. M. Y. Rudysh, M. Piasecki, G. L. Myronchuk, P. A. Shchepanskyi, V. Y. Stadnyk, O. R. Onufriv and M. G. Brika, AgGaTe₂ The thermoelectric and solar cell material: structure, electronic, optical, elastic and vibrational features, Infrared Phys. Technol., 2020, 111, 103476.
66. B. T. Wang, W. X. Zhang and W. D. Li, Mechanics, Lattice dynamics, and chemical bonding in ZrB₂ and ZrB₂ from first-principles calculations, Sci. Adv. Mater., 2013, 5, 1916–1921.
67. X. H. Zhao, Y. L. Tang, T. Y. Tang, X. F. Diao, L. K. Gao, Q. Xie, B. Shi, L. Yuan and L. M. Lu, Study on mechanical, electronic and optical properties of Pb-free double halide perovskites In2TiX6 (X = Cl, Br, I) for solar cells based on first-principles, Mater. Today Commun., 2021, 26, 102180.
68. L. K. Gao, X. H. Zhao, X. F. Diao, T. Y. Tang and Y. L. Tang, First-principles study of photoelectric properties of CsSnBr3 under hydrostatic pressure, Acta Phys. Sin., 2021, 70, 158801.
69. J. Li and H. Chen, First-principles investigation of the mechanical and thermodynamic properties of the antiperovskite centrosymmetric platinum-based superconductors APt3P (A = Ca, Sr, La and Ce), Phys. B, 2020, 577, 411791.
70. R. Gaillac, P. Pullumbi and F. X. Coudert, ELATE: An open-source online application for analysis and visualization of elastic tensors, J. Phys.: Condens. Matter, 2016, 28, 275201.
71. S. I. Ranganathan and M. Ostoja-Starzewski, Universal elastic anisotropy index, Phys. Rev. Lett., 2008, 101, 055504.
72. S. F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag., Ser. 5, 1954, 45, 823.
73. M. Y. Sofi and D. C. Gupta, Scrutinized the inherent spin half-metallicity and thermoelectric response of ff-electron-based RbMO3 (M = Np, Pu) perovskites: A computational assessment, Sci. Rep., 2022, 12, 19476.
74. A. Togo, L. Chaput, T. Tadano and I. Tanaka, Implementation strategies in phonopy and phono3py, J. Phys.: Condens. Matter, 2023, 35, 353001.
75. A. Togo, First-principles Phonon Calculations with Phonopy and Phono3py, J. Phys. Soc. Jpn., 2023, 92, 012001.
76. Q. Mahmood, M. Yaseen, M. Hassan, M. S. Rashid, I. Tlili and A. Laref, The first-principle study of mechanical, optoelectronic and thermoelectric properties of CsGeBr3 and CsSnBr3 perovskites, Mater. Res. Express, 2019, 6, 045901.
77. S. Kuma and M. M. Woldemariam, Structural, electronic, lattice dynamic, and elastic properties of SnTiO₃ and PbTiO₃ using density functional theory, Adv. Condens. Matter Phys., 2019, 2019, 1–12.
78. Z. Sun, S. Li, R. Ahuja and J. M. Schneider, Calculated elastic properties of M₂AlC (M = Ti, V, Cr, Nb and Ta), Solid State Commun., 2004, 129, 589–592.

---