Investigation of the properties of lead-free Cs₂NaBiX₆ (X = I and Br) double perovskites using density functional theory (DFT)

Abstract

In this comprehensive study, we investigate the properties of Cs₂NaBiX₆ (X = I and Br) lead-free double perovskites using density functional theory (DFT). This is the first in-depth DFT-based analysis that delves into the substantial characteristics of these materials, addressing a significant gap in the existing literature. Our analysis focuses on the optical and electronic properties of these materials. Additionally, we examine the impact of strain (tensile and compressive) on their characteristics. The negative values of formation energy and lack of negative frequencies in phonon dispersion band structures provide evidence for the thermodynamic and dynamic stability of considered compounds, respectively. The results reveal that Cs₂NaBiBr₆ is the most thermodynamically and moisture-stable variant, as corroborated by the narrower energy fluctuation range, indicating superior thermal stability. The adsorption energy of water molecule on Cs₂NaBiBr₆ and Cs₂NaBiI₆ are − 0.109 and − 0.114, respectively.Using PBE approximent, bandgap values for Cs₂NaBiBr₆ and Cs₂NaBiI₆, are 2.09 and 3.14 eV, respectively. Furthermore, the DFT analysis suggests that the mechanical moduli increase as the halide changes from iodide to bromide. According to the computed Pugh's ratio, these materials are ductile. These findings highlight the promising potential of lead-free double perovskites Cs₂NaBiX₆ (X = I and Br) for applications in optoelectronic devices.

1 Introduction

The perovskite crystal structure is characterized by the general formula ABX₃, in which A denotes a larger cation, B signifies a smaller cation, and X can be a halogen or an oxygen atom (Qiu et al. 2020; Ramdane et al. 2024). Various oxide perovskites have demonstrated remarkable properties, exemplified by the significant ferroelectric response and the substantial magnetoresistance, which have resulted in numerous practical applications. While perovskite materials are also known as metal oxides or chalcogenides (Bairwa et al. 2023; Bairwa et al. 2024; Pandit et al. 2024a, b), a metal halide perovskite demonstrates exceptional photovoltaic characteristics. Halide perovskites, unlike oxide perovskites, have high ionic conductivity due to their highly ionic crystal structure (Li et al. 2024).

In the past few years, there has been extensive research on perovskite solar cells (Babaei et al. 2022a; Chu et al. 2019). In 2012, Grätzel and colleagues achieved a power conversion efficiency (PCE) of 9.7% by using methylammonium lead iodide (MAPbI₃) as the perovskite absorber material in solar cells (Kim et al. 2012). Currently, the record PCE for perovskite solar cells has reached an impressive 25.5% (Zheng, et al. 2019), which has sparked significant interest and optimism in the field. During this period, various perovskite materials have been investigated and studied for their potential use as absorbers in perovskite solar cell technology (Babaei et al. 2022b; Alidaei et al. 2022; Wu et al. 2023; Zhu et al. 2024a; Zhu et al. 2024b; Wang et al. 2018; Saadatmand et al. 2023a). Researchers have explored alternative perovskite materials to address the stability issues of MAPbI₃, which has a high-power conversion efficiency but is not stable at room temperature and in ambient moisture conditions (Guo et al. 2024). One approach has been to investigate the use of Formamidinium (FA), Cesium (Cs), and Rubidium (Rb) instead of Methylammonium (MA) (Han et al. 2016). Additionally, efforts have been made to find lead-free perovskite materials to mitigate the potential environmental concerns associated with the toxicity of lead. As a result, compounds such as MASnX₃, CsSnX₃, and FASnX₃ have also been extensively studied as alternative perovskite absorbers (Liu et al. 2018; Zhang et al. 2019).

In recent years, researchers have studied a new class of perovskites with the chemical formula A₂(B′B″)X₆, known as double perovskites. The formula presented here utilizes A to denote a cation of moderate size, typically Cs⁺ or MA, while B′ and B″ are used to represent either monovalent or trivalent cations, and X can be either oxygen or halogen (Babaei et al. 2022b; Rani et al. 2022; Rani et al. 2024). Recently, there has been a significant surge in interest within the field of photovoltaic research towards halide double perovskites, as they offer the potential to address the instability and toxicity challenges associated with Pb-based perovskites (Xiao et al. 2017). By exploring different combinations of B′ and B″ in A₂B′B″X₆, it is possible to find a viable substitute for Pb-based halide perovskites (Lee et al. 2020; Tang and Tang 2023a; Tang et al. 2022; Yaseen et al. 2022; Qi et al. 2024; Hu et al. 2022). Mc. Clure et al. (McClure et al. 2016) Introduced double perovskites Cs₂AgBiX₆ (X = Cl and Br) with band gaps of 2.77 and 2.19 eV, respectively. However, these compounds had an indirect band gap, which could limit their optical capabilities. Volonakis et al. succeeded in synthesizing Cs₂InAgCl₆ double perovskite with a direct band gap of 3.3 eV (Volonakis et al. 2017).

Recently, there has been research conducted on A₂NaB″X₆ double perovskite materials, as a novel type of double perovskite materials. The double perovskite materials are anticipated to exhibit a narrow and distinct band structure due to the significant difference in electronegativity between Na⁺ and B″(B³⁺) cations. Computational analyses have indicated that the band-edge molecular orbitals of NaB³⁺ double perovskite materials, such as Cs₂NaBiCl₆, and Cs₂NaInBr₆, are primarily composed of [B³⁺X₆]³⁻ orbitals. As a result, NaB³⁺ double perovskite materials possess distinct optical transition characteristics (Lee et al. 2020; Shi and Du 2015). More recently, Ma and colleagues were able to synthesize a new double perovskite, Cs₂NaBiI₆, which has a bandgap of 1.6 eV (Zhang et al. 2018). Theoretical calculations on the structure of Cs₂NaBiI₆ and Cs₂NaBiBr₆ suggest band gaps of 2.23 and 3.07 eV, respectively (Zhao et al. 2018). In addition to having a suitable band gap for optoelectronic applications, the lead-free Cs₂NaBiI₆ is also very stable against moisture and oxygen in the air (Zhang et al. 2018).

The Cs₂NaBiI₆ bandgap is close to the MAPbI₃ bandgap, but on the contrary, it is lead-free and does not have the problems of MAPbI₃ instability. In addition, the mechanical stability and adequate flexibility of perovskites are very important in the design and fabrication of heterostructures of optoelectronics. Elastic coefficients are one of the most important parameters in studying the elastic properties of materials. The elastic coefficient C₁₁ of MAPbX₃ structures (X = Cl, Br, and I) is of the order of 30 GPa (Saadatmand et al. 2023b; Laamari et al. 2019). Thus, MAPbI₃ is softer than MAPbCl₃ and can be used in heterostructures under the influence of inter-layer mismatch.

In the existing research on Cs₂NaBiI₆ double perovskites, the focus has primarily been on their optical properties, while their stability and mechanical characteristics have received relatively less attention. This paper aims to address this gap by examining the thermodynamical, dynamical, moisture, and thermal stability and mechanical properties of Cs₂NaBiI₆ and Cs₂NaBiBr₆ materials, which are promising candidates for optoelectronic applications. We employed Density Functional Theory (DFT) to evaluate the potential of alternative materials as substitutes for lead-based perovskites and to assess their stability in optoelectronic devices. The phonon dispersion and formation energy related to degradation are determined through DFT to analyze dynamic and thermodynamic stability, respectively. We calculate the elastic constants and mechanical parameters to investigate the mechanical stability of the materials. Additionally, we examined the effect of water molecules on the proposed perovskites by determining the adsorption energy of water on the perovskite surface. This research aims to provide a more comprehensive understanding of the properties of stable, lead-free perovskites, which are essential for advancing optoelectronic devices.

2 Computational methods

In this work, the DFT calculations were performed using the Quantum ESPRESSO software (Giannozzi et al. 2009). After converging the input parameters, the cut-off energy and Monkhorst–Pack scheme (Monkhorst and Pack 1976) of k-point sampling is set to 500 eV and 10 × 10 × 10, respectively. The Perdew-Burke-Ernzerhof (PBE) approximation (Perdew et al. 1996) is used for the exchange–correlation term in the DFT calculations. Additionally, the projector-augmented wave (PAW) pseudo-potentials (Blöchl 1994) are employed to accurately calculate the electronic properties (Babaei et al. 2010). The convergence criteria are set to an energy change less than 10^–6 eV and forces less than 10^–5 eV/Å. Crystal elastic constants are estimated with a maximum strain amplitude of 0.003 and steps of 5.The calculations are carried out on the primitive unit cell of Cs₂NaBiX₆ (X = I and Br) structures, each containing 10 atoms (Fig. 1).

Fig. 1

Primitive and conventional unit cells of Cs₂NaBiX₆ (X = I and Br) compounds

3 Results and discussion

After performing geometry optimization using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, the lattice vectors of the Cs₂NaBiX₆ (X = I and Br) compounds are reported in Table 1. The obtained lattice vectors are in good agreement with the previously reported results (Yin et al. 2019). The Cs₂NaBiX₆ (X = I and Br) compounds have a cubic structure and belong to the Fm3̅m space group. The lattice parameters decrease when moving from I to Br due to the decrease in the size of the halide ions (Saadatmand et al. 2023c).

Table 1 Crystal data and structure for the proposed materials

3.1 Structural, thermodynamical, moisture, and thermal stability

The stability of the proposed perovskite crystal phase can be assessed using two empirical parameters: the Goldschmidt tolerance factor (t) and the octahedral factor (μ). The tolerance factor is calculated as (Qiu et al. 2020):

$$t=\frac{{r}_{Cs}+{r}_{X}}{\sqrt{2} ( \frac{{r}_{Na}+{r}_{Bi}}{2} +{r}_{X})}$$

(1)

where r_Cs, r_Na, r_Bi, and r_X are the ionic radii of Cs, Na, Bi, and the halide ion (I or Br), respectively. The octahedral factor is calculated as:

$$\upmu =\frac{{r}_{Na}+{r}_{Bi}}{2{r}_{X}}$$

(2)

The formability criterion for perovskites is 0.8 ≤ t ≤ 1.0 and 0.41 ≤ μ ≤ 0.9. For the Cs₂NaBiI₆, the values of t and μ are 0.894 and 0.465, respectively. For the Cs₂NaBiBr₆, the values are 0.909 and 0.522, respectively. So, these materials have structural stability (Babaei et al. 2022b).

The new tolerance factor (τ) is also used to confirm the stability of the perovskites. This factor is calculated as (Babaei et al. 2022b):

$$\tau = \frac{{2r_{X} }}{{r_{Na} + r_{Bi} }} - n_{Cs} \left( {n_{Cs} - \frac{{\frac{{2r_{Cs} }}{{r_{Na} + r_{Bi} }}}}{{\ln \left( {\frac{{2r_{Cs} }}{{r_{Na} + r_{Bi} }}} \right)}}} \right)$$

(3)

where n_Cs is the oxidation state of Cs. The perovskite stability is indicated by τ < 4.18. The τ values for the Cs₂NaBiX₆ (X = I and Br) are 4.17 and 3.93, respectively, which suggests that these compounds are within the perovskite stability range.

Furthermore, to ensure the thermodynamic stability of the double perovskite Cs₂NaBiX₆ (X = I and Br), the formation energy is calculated using DFT. The Cs₂NaBiX₆ (X = I and Br) structures can potentially decompose when exposed to light, as shown by the reaction:

$${\text{Cs}}_{2} {\text{NaBiX}}_{6} \to 2{\text{AX}} + {\text{NsX}} + {\text{BiX}}_{3}$$

(4)

The formation energies (E_f) of the considered structures are calculated using the following equation (Babaei et al. 2022a, b; Zhang et al. 2016):

$${E}_{\text{f }}\left({Cs}_{2}NaBi{X}_{6}\right)={E}_{{Cs}_{2}NaBi{X}_{6}}-{2E}_{\text{CsX}}-{E}_{\text{NaX}}-{E}_{\text{Bi}{X}_{3}}$$

(5)

where ${E}_{{Cs}_{2}AgSb{X}_{6}}$, ${E}_{\text{CsX}}$, ${E}_{\text{NaX}}$, and ${E}_{\text{Bi}{X}_{3}}$ represent the optimized crystal energies of the respective compounds.

A negative formation energy indicates that the structure is thermodynamically stable, and a higher negative value suggests greater stability. The computed formation energies of the double perovskites Cs₂NaBiI₆ and Cs₂NaBiBr₆ are − 0.209 and − 0.33 eV, respectively. Since both perovskites have negative formation energy values, it can be concluded that they are thermodynamically stable. It should be noted that the proposed materials are more stable than MAPbI_{3 (}Saadatmand et al. 2023b).

The moisture stability of the Cs₂NaBiX₆ (X = I and Br) perovskites is investigated by determining the adsorption energy (E_ads) of these materials concerning water molecules (H₂O) (Saadatmand et al. 2023c).

$${\text{E}}_{\text{ads}}={\text{E}}_{\text{adsorbate}/\text{sub}}-\left[{\text{E}}_{\text{adsorbate}}+{\text{E}}_{\text{sub}}\right]$$

(6)

where E_{adsorbate/sub} is the total energy of the H₂O molecule with the double perovskite. A positive value for E_ads indicates that the double perovskite is hydrophobic, while a negative value shows that the substrate is hydrophilic.

Figure 2 illustrates the method used to calculate the adsorption energy by placing a water molecule on the Cs₂NaBiX₆ (X = I and Br) surfaces. We focused on the (001) crystal surface (Crystal Facet Index) of the Cs₂NaBiX₆ perovskites. The results are presented in Table 2. Based on the calculated adsorption energies, it can be concluded that the Cs₂NaBiI₆ perovskite exhibits lower stability against water compared to the Cs₂NaBiBr₆ perovskite. The results show that the stability of these materials against moisture is better than MAPbI₃.

Fig. 2

Representation of H₂O placed on a Cs₂NaBiI₆ and b Cs₂NaBiBr₆

Table 2 The values of E_f and E_ads for the proposed materials

Using ab initio molecular dynamic (AIMD) simulations, the thermal stability of the Cs₂NaBiX₆ (X = I and Br) is investigated. The simulations are performed for 10 ps at a temperature of 298.15 K, with 10,000 simulation steps taken at intervals of 1 fs (Shokouhi et al. 2024). In the AIMD simulations, we employed the NVT ensemble, which maintains a constant number of particles (N), volume (V), and temperature (T) throughout the simulation. To control the temperature within the NVT ensemble, we utilized a Nosé–Hoover thermostat. This thermostat effectively regulates the temperature of the system by coupling the system to a heat bath, ensuring that the temperature remains stable at 298.15 K during the simulation. The simulation results, as depicted in Fig. 3, illustrate the variations in total energy and temperature over time at 298.15 K. Notably, the ground state energy exhibits very low fluctuations despite significant temperature changes. The range of energy fluctuation for Cs₂NaBiBr₆ is lower than Cs₂NaBiI₆, suggesting greater thermal stability. Importantly, these energy fluctuation values are an order of magnitude smaller than the formation energies of these materials. This indicates that the chemical bonds remain intact, and the lattice parameters remain constant throughout the simulation, suggesting stability at room temperature.

Fig. 3

AIMD simulations of the total energy and temperature for a Cs₂NaBiI₆ and b Cs₂NaBiBr₆

In the following step, it is crucial to assess the dynamic stability of these materials. We utilize a standard approach to analyze the phonon dispersion relations along the high-symmetry directions within the first Brillouin zone, which allows us to evaluate the dynamic stability of the suggested compounds. Figure 4 illustrates the phonon dispersion for the double perovskites Cs₂NaBiX₆ (with X = Cl, Br, I). The lack of negative frequencies indicates the dynamic stability of the materials being examined. Additionally, an analysis of the phonon dispersion curves reveals that as the X ion transitions from Br to I, the frequencies of the phonon modes decrease, suggesting that Cs₂NaBiBr₆ exhibits greater stability compared to Cs₂NaBiI₆.

Fig. 4

Phonon dispersion spectra for perovskite of a Cs₂NaBiI₆, and b Cs₂NaBiBr₆,

3.2 Electronic properties

The study proceeds to investigate the electronic properties of the Cs₂NaBiX₆ (X = I and Br) compounds by calculating their band structures. The band structure for these double perovskite materials are plotted in Fig. 5. The results indicate that the Cs₂NaBiX₆ (X = I and Br) compounds possess indirect band gaps. In the case of Cs₂NaBiBr₆, the maximum of the valence band is located at the W point, while the minimum of the conduction band is at the L point. The calculated band gap of Cs₂NaBiBr₆ is 3.073 eV, which agrees well with previously reported results of 3.07, 3.06, and 3.15 eV (Zhao et al. 2018; Anbarasan et al. 2021; Ouhammou et al. 2024). For the Cs₂NaBiI₆ structure, the maximum of the valence band is at the W point, and the minimum of the conduction band is at the L point. The calculated band gap for this compound is 2.09 eV, which is found to be reasonably consistent with the previously reported results of 2.23, 2.1, and 1.95 eV (Zhao et al. 2018; Anbarasan et al. 2021; Ouhammou et al. 2024). The difference in the calculated band gaps is attributed to the use of different types of pseudo-potentials. While Zhao et al. (Zhao et al. 2018) employed Ultrasoft pseudo-potentials, the current study utilized the more accurate Projector-Augmented Wave pseudo-potentials to enhance the reliability of the electronic property calculations.

Fig. 5

Band structure of a Cs₂NaBiI₆ and b Cs₂NaBiBr₆

The electronic structure of the perovskites under consideration can be described using the density of states (DOS) and partial density of states (PDOS). In Fig. 6, the DOS and PDOS for the considered materials are illustrated. It is evident, that substituting I with Br can lead to an increase in the number of electronic states in the valence band. Both the conduction band minimum (CBM) and valence band maximum (VBM) show the contribution of halogen atoms to the total DOS. Consequently, modifying the bandgap of the double perovskites can be achieved by replacing the halogen.

Fig. 6

The calculated total DOS and PDOS for a Cs₂NaBiI₆, and b Cs₂NaBiBr₆

3.3 Optical properties

The study further examines the optical properties of the Cs₂NaBiX₆ (X = I and Br) compounds by calculating their dielectric functions and absorption spectra, as shown in Figs. 7 and 8. The imaginary part of the dielectric function for Cs₂NaBiI₆ exhibits two distinct peaks at energies of 3.58 and 5.4 eV. In contrast, the imaginary part of the dielectric function for Cs₂NaBiBr₆ presents a single peak at 4.6 eV. The real part of the dielectric function at zero energy, known as the static dielectric constant (SDC), is calculated to be 7.1 for Cs₂NaBiI₆ and 4.5 for Cs₂NaBiBr₆.

Fig. 7

The real (ɛ₁) part of the dielectric function of a Cs₂NaBiI₆ and c Cs₂NaBiBr₆, and imaginary (ɛ₂) part of the dielectric function of b Cs₂NaBiI₆ and d Cs₂NaBiBr₆

Fig. 8

a, b Absorption coefficient, c, d loss function, and e, f conductivity of Cs₂NaBiI₆ and Cs₂NaBiBr₆, respectively

The study highlights an interesting aspect related to the absorption coefficient of the Cs₂NaBiX₆ (X = I and Br) compounds, which is in the order of 10⁵ cm⁻¹. The absorption intensity of these double perovskites is lower compared to the well-known perovskite material, MAPbI₃, which has a maximum absorption intensity of 14 × 10⁵ cm⁻¹ (Saadatmand et al. 2023b).

3.4 Strain effects

Strain modifies the lattice parameters of the perovskite structure, leading to changes in bond lengths and angles. This distortion can influence the overlap of electronic wavefunctions, thereby affecting the electronic band structure. Under strain, the potential energy landscape experienced by the electrons changes. Compressive strain can increase the overlap between the valence and conduction bands, potentially decreasing the bandgap, whereas tensile strain may lead to a widening of the bandgap due to reduced overlap. Using density functional theory, the impact of triaxial strains ranging from − 3 to + 3% on the bandgaps and refractive indices of the proposed Cs₂NaBiI₆ and Cs₂NaBiBr₆ perovskites is investigated. The results, as shown in Fig. 9, indicate that by applying strain, the bandgaps of Cs₂NaBiI₆ and Cs₂NaBiBr₆ can be tuned within a range of 0.274 and 0.319 eV, respectively. Furthermore, the refractive indices of Cs₂NaBiI₆ and Cs₂NaBiBr₆ can be tuned within a range of 0.174 and 0.036, respectively. This strain, which can result from shrinkage and expansion due to temperature variations, will alter the properties of these materials. This finding demonstrates the ability to manipulate the bandgap of the double perovskites across a wide range by applying strain (Wu et al. 2023; Tang and Tang 2023b).

Fig. 9

Strain effect on bandgaps and refractive indices of a Cs₂NaBiI₆ and b Cs₂NaBiBr₆

3.5 Elastic properties and mechanical parameters

After examining the electronic and optical properties of the Cs₂NaBiX₆ (X = I and Br) compounds through geometry optimization, the study proceeds to investigate their elastic properties and mechanical parameters. To calculate the elastic constants, the unit cells are subjected to three different types of deformation matrices (Jamal et al. 2014):

$$D1 = \left( {\begin{array}{*{20}c} {1 + \varepsilon } & 0 & 0 \\ 0 & {1 + \varepsilon } & 0 \\ 0 & 0 & {1 + \varepsilon } \\ \end{array} } \right), D2 = \left( {\begin{array}{*{20}c} {1 + \varepsilon } & 0 & 0 \\ 0 & {1 - \varepsilon } & 0 \\ 0 & 0 & {\frac{1}{{1 - \varepsilon^{2} }}} \\ \end{array} } \right), D3 = \left( {\begin{array}{*{20}c} 1 & \varepsilon & 0 \\ \varepsilon & 1 & 0 \\ 0 & 0 & {\frac{1}{{1 - \varepsilon^{2} }}} \\ \end{array} } \right)$$

(7)

By applying strain (ɛ) in the range of − 2% to + 2% to the Cs₂NaBiX₆ (X = I and Br) compounds, the energy-strain curves for each of the deformation types are plotted, as shown in Fig. 10. To calculate the elastic coefficients, the study employed the second-order derivative of the energy-strain curve around the equilibrium state. Specifically, for the D2 deformation:

$$\frac{{d^{2} E}}{{d\varepsilon^{2} }} = 2V_{0} \left( {C_{11} - C_{12} } \right)$$

(8)

Fig. 10

Energy-strain curves of Cs₂NaBiX₆ (X = I and Br) compounds under deformations D1, D2 and D3

And D1 deformation:

$$\frac{{d^{2} E}}{{d\varepsilon^{2} }} = 3V_{0} \left( {C_{11} + 2C_{12} } \right)$$

(9)

And also, for D3:

$$\frac{{d^{2} E}}{{d\varepsilon^{2} }} = 4V_{0} \left( {C_{44} } \right)$$

(10)

By calculating the second-order derivative of the energy-strain curves for the different deformation types (Eqs. 8–10), the independent elastic constants of the cubic perovskites (C₁₁, C₁₂, and C₄₄) can be determined. In these equations, E and V₀ represent the energy and volume of the equilibrium state, respectively.

The energy-strain curves for the Cs₂NaBiX₆ (X = I and Br) under the D1, D2, and D3 deformations are shown in Fig. 8. Using Eqs. 8–10, the elastic coefficients of the Cs₂NaBiX₆ (X = I and Br) compounds are calculated and listed in Table 3. The calculations show that the C₁₁ value of Cs₂NaBiBr₆ (C₁₁ = 32.3 GPa) is 3.7 GPa larger than that of Cs₂NaBiI₆ (C₁₁ = 28.6 GPa).

Table 3 Elastic coefficients of Cs₂NaBiX₆ (X = I and Br)

For cubic crystals, the Born stability criteria (Born and Misra 1940), can be written as:\

$$C_{11} > 0,\,C_{11} - C_{12} > 0,\,C_{11} + 2C_{12} > 0,\,C_{44} > 0$$

(11)

The elastic constants of the analyzed compounds are found to be positive and satisfy the Born stability criteria for cubic crystals, thereby signifying the mechanical stability of these materials.

In the following, using the calculated elastic constant, the study proceeds to derive various mechanical parameters for the Cs₂NaBiX₆ (X = I and Br) compounds. The bulk modulus (B) is calculated based on the C₁₁ and C₁₂ constants, as shown in Eq. 12 (Babaei et al. 2022b; Saadatmand et al. 2023b):

$$B = \frac{{C_{11} + 2C_{12} }}{3}$$

(12)

The shear modulus (G) is then computed using the C₁₁, C₁₂, and C₄₄ coefficients, as per Eq. 13 (Liu et al. 2018):

$$G = \frac{{3C_{44} + C_{11} - C_{12} }}{5}$$

(13)

Finally, Young’s modulus (E) and Poisson’s ratio (ν) are derived from the bulk and shear modulus values (Saadatmand et al. 2024; Saadatmand 2024; Shokouhi et al. 2023):

$$E = \frac{9BG}{{3B + G}}$$

(14)

$$\nu = \frac{1}{2} - \frac{E}{6B}$$

(15)

The calculated mechanical parameters for the Cs₂NaBiX₆ (X = I and Br) compounds are listed in Table 4. The results show that the bulk modulus of Cs₂NaBiBr₆ (18.2 GPa) is 2.1 GPa higher than the bulk modulus of Cs₂NaBiI₆ (16.1 GPa). Similarly, Young's modulus of Cs₂NaBiBr₆ (25.1 GPa) is 0.5 GPa greater than that of Cs₂NaBiI₆ (24.6 GPa). To understand the reason behind the higher stiffness of Cs₂NaBiBr₆ compared to Cs₂NaBiI₆, the study investigated the bond dissociation energies (BDE) of the Na–Br and Na–I bonds. The Na–Br BDE (361.1 kJ/mol) is found to be 56.3 kJ/mol higher than the Na-I BDE (306.8 kJ/mol) (Luo 2007). This indicates that the Na–Br bond is stronger than the Na–I bond, which contributes to the enhanced mechanical properties of the Cs₂NaBiBr₆ compound.

Table 4 Bulk modulus (B), Young’s modulus (E), shear modulus (G), B/G ratio, and Poisson’s ratio (ν) of Cs₂NaBiX₆ (X = I and Br) compounds

The ratio of bulk modulus to shear modulus (B/G), known as Pugh's ratio (Babaei et al. 2022b), is listed in Table 4 (Saadatmand et al. 2023b). If the B/G ratio is greater than 1.75, the material is considered ductile, while a B/G ratio less than 1.75 indicates a brittle material. According to this criterion, Cs₂NaBiBr₆ (B/G = 1.84) is classified as a ductile material, while Cs₂NaBiI₆ (B/G = 1.63) is considered brittle.

Then we examine the isotropic nature of the mechanical parameters for the Cs₂NaBiX₆ (X = I and Br) compounds using the ELATE software (Gaillac et al. 2016) to plot the three-dimensional (3D) shapes of the Young’s modulus, linear compressibility, and Poisson's ratio (Fig. 11).

Fig. 11

3D diagram of Young's modulus, linear compressibility, and Poisson's ratio of a Cs₂NaBiI₆, and b Cs₂NaBiBr₆

According to the results, Young’s modulus, linear compressibility, and Poisson’s ratio of the Cs₂NaBiBr₆ structure are completely isotropic, indicating that the material exhibits the same behavior in all directions when subjected to external stresses. In contrast, while Young’s modulus and linear compressibility of the Cs₂NaBiI₆ structure are isotropic, its Poisson's ratio shows a 10% anisotropic behavior.

4 Conclusion

In this comprehensive study, we have conducted a thorough investigation of the Cs₂NaBiX₆ (X = I and Br) double perovskites using density functional theory (DFT). Our analysis has revealed several key findings. Firstly, we have examined the optical and electronic properties of the Cs₂NaBiX₆ (X = I and Br) double perovskites. Additionally, we have investigated the impact of strain (tensile and compressive) on their properties, providing valuable insights into the tunability of these materials. The negative formation energy values and the absence of negative frequencies in the phonon dispersion band structures provide evidence for the thermodynamic and dynamic stability of the compounds examined. By analyzing the electronic structure using the PBE method, the bandgap values for Cs₂NaBiBr₆ and Cs₂NaBiI₆ are found to be 2.1 and 3.07 eV, respectively. This study highlights the potential to modify the electronic bandgap of these materials by substituting the halogen atom, making them particularly suitable for optoelectronic applications. The results of our study indicate that Cs₂NaBiBr₆ emerges as the most thermodynamically and moisture-stable variant, as evidenced by its narrower energy fluctuation range, which suggests superior thermal stability. Furthermore, the DFT analysis has shown that the mechanical moduli of these materials increase as the halide changes from I to Br. The ratio of shear modulus to bulk modulus shows the ductility of considered compounds. Perovskite Cs₂NaBiBr₆ is relatively more flexible than perovskite Cs₂NaBiI₆ because the bond dissociation energy of Na–Br (361.1 kJ/mol) is higher than that of Na-I (306.8 kJ/mol). These findings highlight the promising potential of the Cs₂NaBiX₆ (X = I and Br) double perovskites for applications in optoelectronic devices. The ability to tune their properties through strain and the superior stability of the Cs₂NaBiX₆ (X = I and Br) variant make these materials attractive candidates for further exploration and development in the field of advanced materials and device engineering.