Lead-free halide Rb₂SnCl₆ double perovskite for ultraviolet, water-splitting, and sustainable energy harvesting technologies

Abstract

Beyond traditional ultra-wide bandgap (UWBG) materials such as Ga₂O₃ and diamond, double perovskites (DP) are an emerging family that can exhibit a bandgap (E_g) above 3.4 eV, which is necessary for deep ultra-violet (DUV) applications. In this work, we theoretically studied a lead-free halide Rb₂SnCl₆ DP that displays a large direct E_g of 4.41 eV using a modified Becke–Johnson potential. Further, hydrostatic ([111]) strain modified the E_g from 4.18 (−5%) to 4.71 eV (+5%) and shows a strong optical absorption in the DUV region. The absorption coefficient of the unstrained system peaks at 1.27 × 10⁵ cm⁻¹ for 5.51 eV. Moreover, the elastic aspects indicate a transition from ductile metallic-like to brittle covalent-like character with applied strain. Additionally, photocatalytic water-splitting analysis reveals that it has a redox potential with valence bands (3.57 V vs. NHE) more positive than the O₂/H₂O oxidation potential (1.23 V). Also, their conduction bands (−0.90 vs. NHE) are more negative than the H⁺/H₂ reduction potential (0 V). Also, electron/hole effective mass increases from 0.163 to 0.463/0.359 to 0.808, as the strain varies from −5% to +5%, indicating improved carrier mobility under compressive strain. Concurrently, the static dielectric constant demonstrates changes from 3.41 to 2.255, leading to an increase in the exciton binding energy (E_b) from 0.189 to 0.789 eV. Finally, thermoelectric analysis shows a high figure of merit of 0.71/0.70 at 0%/−5% strain at 1200 K, owing to an enhanced power factor and reduced lattice thermal conductivity. Along with this, the Seebeck coefficient remains positive across all strain levels, indicating p-type conduction, while the electrical conductivity improves significantly under compressive strain. Thus, these results establish the system as a promising UWBG semiconductor for DUV, hydrogen production, and sustainable energy harvesting applications.

We theoretically studied a lead-free halide Rb₂SnCl₆ DP that displays a large direct E_g of 4.41 eV using a modified Becke–Johnson potential.

1. Introduction

The escalating global energy demand along with the depletion of conventional fossil fuels has forced an urgent paradigm shift towards sustainable and clean energy technologies.^1,2 In this regard, ultra-wide bandgap (UWBG) semiconductors (SC), which are characterized by bandgap (E_g) energies surpassing 3.4 eV have emerged as vital candidates for next-generation applications.^3,4 These include power electronics,⁵ deep-ultraviolet (DUV) photodetectors,⁵ and thermoelectric (TE) devices.⁶ This is due to their exceptional ability to withstand high electric fields and operate under extreme thermal conditions.^5,6 Traditional UWBG SC such as AlGaN/AlN heterostructures (E_g = 3.4–6.0 eV),³ β-Ga₂O₃ (E_g = 4.9 eV),³ diamond (E_g = 5.5 eV),³ and cubic boron nitride (E_g = 6.3 eV)³ have remarkable electronic aspects. However, their widespread adoption is constrained by high fabrication costs, limited scalability, and challenges associated with E_g engineering through conventional doping or alloying.^3,7 Consequently, recent years saw an increase in the search for alternative material systems that combine UWBG characteristics with structural flexibility, compositional diversity, and tunable optoelectronic (OE) features.^8,9

In recent decades, the double perovskite (DP) family has emerged as a versatile platform for discovering UWBG SC with tailored functionalities. They offer enhanced chemical flexibility and improved structural stability through ordered cation arrangements at the B-site, which enable precise modulation of their electronic structure.^10,11 For instance, several oxide-based DP have demonstrated UWBG behavior including BaZrO₃ (E_g = 4.90 eV),¹² Ba₂CaTeO₆ (E_g = 5.24 eV),¹² and the triple perovskite Ba₂K₂Te₂O₉ (E_g = 4.65 eV),¹² establishing the DPs for UWBG OE applications. Other than oxides, halide-based DPs (HDPs) have also gained significant attention due to their solution processability, environmental compatibility (lead-free nature), and remarkable E_g tunability.¹³ In particular, vacancy-ordered (VO) HDP with the general formula A₂BX₆ has demonstrated exceptional electronic and optical aspects. For example, Cs₂SnCl₆ exhibits a wide E_g of 4.50 eV, which makes it a promising candidate for DUV photodetectors and OE devices.¹⁴ Moreover, the presence of Sn in the +4 oxidation state within isolated [SnCl₆]²⁻ octahedra separated by alkali metal cations forms an electronically unique VO framework. It minimizes carrier recombination while also maintaining the structural integrity.^15,16

Despite these advances, a significant issue persists between theoretically predicted and experimentally measured E_g in DPs: as first-principles calculations even when employing advanced exchange–correlation functionals, frequently underestimate E_g values by 1–2 eV, thereby limiting predictive accuracy and hindering rational device design.^17–19 For example, previous computational studies on Rb₂SnCl₆ (RSC), a VO HDP isostructural to Cs₂SnCl₆,¹⁴ have reported E_g values ranging from 2.38 to 5.25 eV,^20–22 which diverge substantially from the experimental E_g of 4.79 eV.²³ This inconsistency arises because those studies used GGA or G₀W₀@PBE functionals, which are known to underestimate or overestimate E_g. It demands the use of advanced exchange–correlation potentials such as the modified Becke–Johnson (mBJ) or hybrid-functional (HSE0₆) approach to achieve quantitative agreement with experimental observations. The mBJ potential yields results as accurate as HSE06 but at a much lower computational cost. For example, in the isostructural compounds Cs₂SnCl₆ and Cs₂SnBr₆, the mBJ calculated E_g (3.95 eV and 2.449 eV) agrees with HSE06 values (3.83 eV and 2.36 eV), respectively, and also matched experimental data (3.9 eV and 2.7 eV) well.^24–27 Moreover, there are many ways to tune the OE features of DP such as chemical doping,²⁸ alloying,²⁹ dimensional reduction,³⁰ and interface regulation.³¹ One particularly promising approach is also strain engineering,³² a powerful and versatile tool for tailoring the various physical features of the SC without altering their chemical composition. By applying external or inducing internal strain, the lattice constants and bond lengths can be modified, which directly influences the electronic structure of the material, and consequently, the E_g. It also modifies the carrier effective masses (m*), exciton binding energies (E_b), and optical absorption edges.³³ Notably, strain engineering and halogenation have recently been found to significantly improve the photocatalytic and thermoelectric properties of Ag₂Se and Janus Sn₂XY monolayers as well.^34–36

Alongside this, solar-driven water splitting is widely considered a promising route for sustainable hydrogen generation.³⁷ For a SC photocatalyst to drive overall water splitting, its band edges must properly align with the redox potentials of water: the conduction band should be more negative than the H⁺/H₂ reduction level, while the valence band must be more positive than the O₂/H₂O oxidation level. In addition, the material should possess a E_g larger than 1.23 eV, which corresponds to the minimum thermodynamic energy required for water splitting. Such electronic alignment enables photogenerated charge carriers to participate effectively in hydrogen and oxygen evolution reactions.^38,39 Among the isoelectronic Rb₂SnX₆ (X = Cl, Br, I) family, Rb₂SnBr₆ (RSB) and Rb₂SnI₆ (RSI) have an E_g of 2.4 eV and 1.1 eV, respectively.^40,41 RSB is an excellent photocatalyst in the visible region,⁴² while the E_g of RSI is too low for efficient water splitting.⁴³ Although the E_g of RSC is significantly higher than usual, it remains promising for photocatalysis in the UV region. Most established photocatalysts such as TiO₂,^44,45 ZnO,⁴⁶ operate efficiently under UV light, and the quest for ideal photocatalysts in the visible region is still ongoing. Moreover, thermoelectric (TE) materials have attracted significant attention for their ability to directly convert heat into electrical energy through the thermoelectric effect.^47,48 When a temperature gradient is applied across a material, an electrical potential is generated due to the diffusion of charge carriers, a phenomenon known as the Seebeck effect.⁴⁹ The efficiency of TE materials is commonly evaluated using the dimensionless figure of merit ZT = S²σT/κ, where S is the Seebeck coefficient, σ is the electrical conductivity, and κ is the thermal conductivity. Thus, materials with high σ, a large S, and low κ are considered promising for TE energy conversion.⁵⁰

In this work, a comprehensive first-principles investigation of the VO DP RSC under hydrostatic (hydro.) strain is presented. We almost accurately reproduce the experimental E_g of 4.79 eV and systematically examine the effects of ± 5% strain on the structural, mechanical, electronic, optical, photocatalytic, and TE aspects. By establishing a direct correlation between strain engineering and multifunctional property enhancement, this study positions RSC as a compelling candidate for next-generation sustainable energy technologies spanning UV OE, water splitting, and TE energy conversion.

2. Computational and structural details

Spin degenerate density functional theory (DFT) calculations were performed using the full-potential linearized augmented plane wave (FLAPW) method as executed in the WIEN2k code.⁵¹ The generalized gradient approximation (GGA)⁵² in conjunction with the modified Becke–Johnson (mBJ)⁵³ potential is utilized as an exchange correlation potential. The basis set was designed as R_MT × K_max = 7, where R_MT denotes the muffin-tin radius and K_max corresponds to the maximum magnitude of the reciprocal lattice vector. Inside the muffin-tin spheres, the angular momentum expansion was truncated at l_max = 12, while the Fourier series of the charge density in the interstitial region was limited by G_max = 24. During the self-consistent field⁵⁴ procedure, the convergence criteria was set to 10⁻⁵ Ry for total charge/energy (E_t) and 5 mRy per bohr for forces. Brillouin zone integrations were carried out using a Monkhorst–Pack grid of 6 × 6 × 6 k-points. The elastic constants and consequently the mechanical parameters were extracted using ElaTools,⁵⁵ whereas the TE aspects were computed via the BoltzTraP package.⁵⁶

Pb-free VO RSC DP crystallizes in a cubic structure (see Fig. 1) having experimental lattice constants of a = b = c = 10.123 Å with space group Fm3̄m (No. 225).²³ Rb atoms occupy the 8c sites (0.25, 0.25, 0.25), Sn atoms are located at the 4a sites (0, 0, 0), and Cl atoms reside at the 24e sites (0.23924, 0, 0). To investigate the effect of hydro. strain, the experimental lattice parameters vary under a reasonable range of ± 5% as:

1

where a₀/b₀/c₀ are the original (unstrained) experimental lattice parameters and a/b/c are the strained ones. Thus, all three lattice constants vary by the same amount in the hydro. case. The Birch–Murnaghan equation of state was applied to obtain the optimized volume and lattice parameters of the RSC compound:

2

Fig. 1. Crystal structure of the vacancy ordered double perovskite Rb₂SnCl₆.

The fitting results of this equation are presented in Fig. 2, where the minimum corresponds to the equilibrium cell volume, from which their optimized lattice parameters were obtained. Based on these results, the lattice parameter a₀ for the VO DP RSC was found to be 10.093 Å, which closely aligns with those reported in earlier studies on the same material.^20–22

Fig. 2. Calculated energy vs. volume curve for the vacancy ordered double perovskite Rb₂SnCl₆.

3. Results and discussion

3.1. Formation enthalpy and mechanical stability

First, thermodynamic stability of the RSC is analyzed by computing formation enthalpy (ΔH_f) per atom as:

3

where E_t(Rb₂SnCl₆) is the total energy of the RSC compound and E_t(Rb)/E_t(Sn)/E_t(Cl) are the E_t of Rb/Sn/Cl elements in their respective stable phase. As shown in Fig. 3, ΔH_f remains negative throughout the entire strain range, confirming the thermodynamic stability of the structures. The negative values of ΔH_f further indicates that their formation is exothermic and favor a spontaneous process. Moreover, to assess the dynamical and thermal stability of RSC, we calculated its phonon dispersion relations and performed ab initio molecular dynamics (AIMD) simulations. The calculated phonon spectrum at 0 K (orange curves) exhibits soft modes with imaginary frequencies centered around the Γ-point as shown in Fig. 4(a). This indicates that the highly symmetric cubic phase (Fm3̄m) is dynamically unstable at absolute zero. Conversely, when thermal effects are accounted for at 300 K, the imaginary frequencies are completely eliminated, lifting the acoustic branches entirely above 0.0 THz. This behavior reveals a strong lattice anharmonicity that dynamically stabilizes the RSC at room temperature. This room-temperature stability is further verified by the AIMD simulation presented in Fig. 4(b). The total energy exhibits steady, bounded oscillations over the course of the simulation without any abrupt energy drops or long-term drift. This uniform energy trend along with the fact that the lattice framework remained intact during the entire simulation, confirms that RSC possesses thermal stability at finite temperatures. Next, the mechanical stability of the structure is investigated through the calculation of elastic constants (C_ij), which characterize the crystal's response to different strain modes. For cubic crystals, only three independent C_ij (C₁₁, C₁₂, and C₄₄) exist, which were obtained using the energy-strain method through the ELAST program (also known as Thomas Charpin method) as integrated in the WIEN 2K code. The calculated C_ij satisfy the Born mechanical stability criteria for cubic systems as C₁₁ − C₁₂ > 0, C₁₁ + 2C₁₂ > 0, and C₄₄ > 0,^57–59 confirming that RSC is mechanically stable throughout the studied hydro. strain range as listed in Table 1. Using the C_ij, several mechanical parameters were calculated via the Voigt,⁶⁰ Reuss,⁶¹ Hill,⁶² and Voigt–Reuss–Hill⁶³ averaging schemes, including the bulk modulus (B), shear modulus (G), Young's modulus (Y), Pugh's ratio , Poisson's ratio (ν), linear compressibility (β), longitudinal velocity (v_l), transverse velocity (v_t), average velocity (v_avg.), and anisotropy (A) as:⁶⁴

4

5

6

7

8

9

10

here, B quantifies resistance to volume change, G measures resistance to shear deformation, and Y represents stiffness under uniaxial stress. The variation of these parameters under hydro. strain of ± 5% is summarized in Table 1.

Fig. 3. Variation of formation enthalpy (ΔH_f) per atom of the Rb₂SnCl₆ structure under ±5% hydrostatic ([111]) strain.

Fig. 4. Calculated (a) temperature-dependent phonon dispersion spectrum along high-symmetry paths and (b) total energy fluctuations over the course of the ab initio molecular dynamics simulation of the Rb₂SnCl₆ structure.

Table 1. Calculated numerous elastic parameters in GPa of the Rb₂SnCl₆ structure under 0%, −5%, and +5% hydrostatic ([111]) strains.

Strain	C 11	C 12	C 44	B	G	Y	ν	β		C p
0%	42.507	26.863	19.361	32.078	14.745	38.32	0.31	0.03117	2.17	7.50
−5%	106.169	27.212	50.784	53.531	46.262	107.63	0.26	0.01869	1.16	−6.03
+5%	38.421	14.632	20.660	22.562	17.154	41.05	0.38	0.04434	1.32	−23.38

The calculated B and G for the unstrained structure are 32.08 and 14.75 GPa, respectively. For −5%/+5% strain, B increases/decreases to 53.53/22.56 GPa which means resistance to volumetric deformation increases/decreases (see Table 1). However, G increases to 46.26/17.15 GPa for −5%/+5% highlighting the enhancement of resistance to shear deformation compared to that of the unstrained case (14.75 Gpa). Also, Y for the unstrained structure is 38.32 GPa and it increases to 107.63/41.05 Gpa for −5%/+5% strains, which means a significant increase in stiffness of the material. Additionally, to determine the ductile, brittle, and bonding nature of the compounds, we computed , ν, and Cauchy's pressure (C_p).⁶⁵ The differentiates brittle for values <1.75 and ductile for >1.75 nature, while ν indicates bonding nature with ν > 0.26 typically reflecting predominantly ionic bonding. For the unstrained system, is 0.32/2.38, indicating ductile metallic-like behavior. However, for −5%/+5% strained ones, ν and are 0.167/1.17 and 0.205/1.36, respectively, highlighting the brittle covalent-like nature. Generally, positive C_p indicates metallic bonding and ductile behavior, whereas a negative value suggests directional covalent bonding and brittleness.^66,67 For the RSC, our calculated C_p is positive for 0% strain, indicating ductile behavior but it is negative for the −5%/+5% strain implying covalent behavior (see Table 1). Our computed elastic moduli are consistent with previously reported studies on the same RSC structure.^20,21

Further, the influence of strain on the bonding nature is discussed by elastic wave velocity calculations (see Fig. 5). These elastic wave velocities such as v_l/v_t/v_avg. (which are computed using eqn (11)–(13)) increases with −5% and +5% strain as compared to the unstrained one. The v_l depends on B, G, and mass density (ρ) whereas v_t only depends on G and ρ. It is found that −5% strain, results in an increase in v_l which is consistent an increase in overall stiffness as reflected by the high value of Y and shows ductile metal-like behavior (MLB). In contrast, 0% and +5% strain states exhibit lower values showing brittle covalent-like behavior (CLB).

11

12

13

Fig. 5. Variation of the elastic wave velocities (longitudinal (v_l), transverse (v_t), and average wave velocities (v_m)) against hydrostatic ([111]) strains in the Rb₂SnCl₆ structure. Shaded regions distinguish the compressive (light blue for −5%), tensile (light coral for +5%), and unstrained system (light grey for 0%).

These mechanical and acoustic trends collectively demonstrate that strain effectively tunes the bonding character and mechanical performance of the RSC. Moreover, the correlation between the key indicators like ν, , and hardness (H) are presented in Fig. 6. The plot describe the strain induced transition where the −5% one clusters in a region of high H but relatively low ν and , which are characteristic of a harder and more brittle solid with an increased covalent-like bonding aspect. Conversely, the +5% strain level exhibits higher ν and intermediate , correlating with lower H and a covalent brittle-like response. The 0% strain state has a large but low H indicating ductile metallic-like regimes. The dashed threshold lines at ν = 0.26 and emphasize the strain driving the material across conventional ductility boundaries, modulating its behavior from ductile metallic-like to brittle covalent-like regions.

Fig. 6. Correlation of Poisson's ratio (ν), Pugh ratio , and hardness (H) under different hydrostatic ([111]) strain levels in the Rb₂SnCl₆ structure. The scatter points are colored by hardness values in GPa and dashed lines at ν = 0.26 and denote boundaries between ductile and brittle behavior, respectively, where labels indicate bonding types and mechanical regions.

Along with this, to analyze the strain-dependent elastic behavior of the RSC material, we calculated its elastic A using by eqn (14)via ElATools,⁵⁵ specifically assessing the 3D anisotropic features under −5%, 0%, and +5% strain levels:.

14

The changes in the 3D anisotropic features in Y, β, G, B, ν, and against strain are shown in Fig. 7(a)–(f), correspondingly. It is found that Fig. 7(a) clearly shows that stiffness increases with strain. While, Fig. 7(b), (c), and (e) show an isotropic pattern. Conversely, Fig. 7(d) and (f) display unusual behavior at −5% despite showing isotropic behavior at 0% and +5%.

Fig. 7. Variation of: (a) Young's modulus, (b) linear compressibility, (c) bulk modulus, (d) shear modulus, (e) Poisson's ratio, and (f) Pugh's ratio, under hydrostatic ([111]) strain of −5%/0%/+5% for the Rb₂SnCl₆ structure.

3.2. Electronic features

To describe the electronic structure of the RSC material, the total density of states (TDOS) and electronic band structure were calculated using GGA-mBJ potential and are presented in Fig. 8(a) and (b), respectively, which clearly exhibits an E_g of 4.416 eV that is close to the experimental value of 4.79 eV,²³ with an accuracy of 92%. Here, it is important to emphasize that previously reported theoretical E_g values are 2.380,²⁰ 3.446,²¹ and 5.25 eV (ref. 22) for the RSC system, which are very far away from the experimental one. However, our computed E_g is in excellent agreement with the experimentally observed value and is more accurate as compared to the above mentioned previous investigations.²³ Also, the band structure describes that it has a direct E_g because the valence band maximum (VBM) and conduction band minimum (CBM) lies at the same symmetry Γ-point (see Fig. 8(b)). Further, the comparison of UWBG RSC with other UWBG SCs is presented in Table 2. This large E_g makes it favorable for UV applications. Moreover, the flatness of the conduction band near the CBM suggests a high m* for electrons , while the dispersive nature of the valence band near the VBM indicates lighter holes . Next, the variation in the E_g under hydro. strain is presented in Fig. 9 where compressive (comp.) strain reduces the E_g from 4.18 to 4.14 eV, reflecting increased orbital overlap and metallic bonding tendencies. On the other hand, tensile (tens.) strain increases the E_g from 4.72 to 4.76 eV, due to reduced overlap and enhanced covalent character. The computed corresponding TDOS and band structure of the RSC structure for various hydro. strains are given in Fig. S1 of the SI. The results indicate that strain strongly affects the electronic structure of the material. With increasing comp. strain magnitude, the conduction band shifts toward the Fermi level (E_F), leading to a reduction in the E_g. Conversely, tens. strain shifts the conduction band away from E_F, resulting in a slight increase in E_g. These changes reflect the strain-induced modification of the electronic states near the band edges.

Fig. 8. GGA-mBJ calculated: (a) total density of states (TDOS) and (b) band structure, for the unstrained Rb₂SnCl₆ structure.

Table 2. List of popular ultra-wide bandgap (E_g) semiconductors for comparison with that of the Rb₂SnCl₆ structure.

Material	E g (eV)	Material	E g (eV)	Material	E g (eV)
AlGaN/AlN	3.4 to 6.0 (ref. 22)	β-Ga2O3	4.9 (ref. 22)	Diamond	5.5 (ref. 22)
BaZrO3	4.90 (ref. 3)	Ba2CaTeO6	5.24 (ref. 3)	Ba2K2Te2O9	4.65 (ref. 3)
Cs2SnCl6	4.50 (ref. 14)	Al2O3	>7.0 (ref. 68)	Rb2SnCl6	4.41 (this work)

Fig. 9. Variation of the energy bandgap (E_g) against ±5 hydrostatic ([111]) strain for the Rb₂SnCl₆ structure.

Next, the charge carrier properties of the material are evaluated by determining the around the Brillouin zone center (Γ-point) for 0% and ± 5% strain levels. The m* is a crucial metric for predicting carrier mobility, which is integrally linked to the band curvature at the CBM and VBM. The expression used to determine the m* is given by:⁶⁹

15

where ℏ, E, and k represent the reduced Planck's constant, energy, and wave vector, correspondingly. To implement this computationally, a parabolic function of the form: E = A₀k² + A₁k + A₂ was fitted to the E(k) dispersion relation in the immediate vicinity of the Γ-point. The second derivative, , was extracted from this fit which directly yields the . For practical comparison across different materials, the resulting m* are normalized to the rest mass of a free electron (m₀). This fitting procedure was applied within a constrained k-space region around the band edges to ensure an accurate representation of the curvature. The computed for various strain states are plotted in Fig. 10(a). For the unstrained system, is 0.217/0.563, whereas it increases to 0.463/0.808 for +5% and decreases to 0.163/0.359 for −5% strain. The increase in m* under tens. strain indicates a reduction in the curvature of the conduction and valence bands, suggesting more localized charge carriers and lower carrier mobility. In contrast, the decrease in m* under comp. strain reflects enhanced band dispersion, which facilitates higher carrier mobility. These results indicate that hydro. strain significantly influences the charge transport properties of the material by modifying the band curvature near the band edges. The obtained m* are comparable to those reported for other VO HDP. For example, Cs₂SnCl₆ exhibits an of 0.55 with a corresponding of 2.2,⁷⁰ while Cs₂TiCl₆ shows much larger values .⁷⁰ In comparison, the RSC shows significantly lighter m*, particularly under comp. strain , indicating superior carrier transport properties. Moreover, another crucial parameter controlling OE performance is the E_b that quantifies the strength of electron–hole coulombic coupling and is determined through the expression:⁷¹

16

where ε₁(0) denotes the static dielectric constant (ε(ω)) and m_µ is the reduced mass. The E_b for the unstrained system is 0.333 eV but it significantly increases to 0.789 eV for +5% and reduces to 0.189 eV for −5% strain levels (see Fig. 10(b)). This is because of the ε₁(0) which significantly reduces for +5% strain as compared to −5% strain (discussed later). But overall the E_b for UWBG SC is usually high because of large m* and low ε₁(0). Moreover, the calculated E_b values are comparable to those reported for VO HDP. For instance, Cs₂SnCl₆ exhibits an E_b of 0.73 eV,⁷⁰ while Cs₂TiCl₆ shows a much larger value of 1.73 eV.⁷⁰

Fig. 10. Variation of the effective mass (m*) ratio of holes /electrons in red/blue color; and corresponding exciton binding energy (E_b) in the Rb₂SnCl₆ structure against −5%/0%/+5% hydrostatic ([111]) strains.

3.3. Photocatalytic analysis

The redox ability of the RSC is assessed by determining the energy positions of valence and conduction bands. The positions can be calculated according to the equation as follows using Mulliken electronegativity and the E_g value as:⁷²

17

E_CB = E_VB − E_g 18

where E_VB and E_CB are the VB and CB edge potentials, respectively, χ is the Mulliken electronegativity of the RSC, E_e is the energy of free electrons on the hydrogen scale (4.5 eV), and E_g is the calculated bandgap. The χ of an individual atom is defined as the arithmetic mean of its first ionization energy (I) and electron affinity (A) as:⁷³

19

The total χ (χ_tot.) of the RSC is calculated using the geometric mean of the constituent atoms:

20

The calculated individual Mulliken χ for the Rb, Sn, and Cl atoms are 2.33, 4.23, and 8.29 eV, respectively (see Table 3). So, the computed χ_tot. for RSC is 5.80 eV. Moreover, the large E_g of RSC makes it a UWBG photocatalyst, which are classified as materials with E_g greater than 4 eV such as ZrO₂ (E_g = 5.0 eV),³⁷ Sr₂Ta₂O₇ (E_g = 4.6 eV),³⁷ Zn₂GeO₄ (E_g = 4.6 eV),³⁷ ZnGa₂O₄ (E_g = 4.3 eV),³⁷ NaTaO₃:La (E_g = 4.1 eV),³⁷ Ta₂O₅ (E_g = 4.0 eV).³⁷ The calculated VBM potential of the RSC is 3.57 V, which is more positive than the O₂/H₂O potential (1.23 V). Therefore, the RSC has the ability to oxidize H₂O to produce O₂ or oxidize pollutants. Whereas the conduction band minimum (CBM) potential of RSC is −0.90, which is more negative than the H⁺/H₂ potential (0 V) and therefore it can also reduce H⁺ to H₂. It should be noted that such UWBG SC (≥4 eV) restrict photon absorption primarily to the UV region (∼4–5% of the solar spectrum). Nevertheless, it remains attractive for UV-driven water splitting and as a UV-absorbing top layer in tandem photocatalyst systems paired with visible light absorbers.

Table 3. Calculated ionization energy (I), electron affinity (A), and Mulliken electronegativity (χ) of constituent atoms (Rb, Sn, Cl) in the Rb₂SnCl₆ structure.

Atom	I (eV)	A (eV)	χ (eV)
Rb	4.17713	0.485916	2.331523
Sn	7.34390	1.11207	4.227985
Cl	12.96764	3.612725	8.2901825

Along with this, we present the pH-dependent (0 to 14) band edge alignment of the RSC structure relative to the redox levels of water in Fig. 11, which highlights its thermodynamic viability for water splitting. To study the pH effects, we employed the Nernst equation as:^37,74

E_redox(pH) = E° − 0.059 pH 21

where E_redox(pH) is the redox potential at a given pH, and E° denotes the standard redox potential at pH = 0. As shown in Fig. 11, redox potential shifts upward by increasing pH (approximately 0.059 V per pH unit). As a result, the RSC maintains proper band edge alignment with the H⁺/H₂ (reduction) and O₂/H₂O (oxidation) potentials across pH of 0–14, indicating its thermodynamic suitability for overall water splitting.⁷⁵ The pH-dependent band edge alignment of strained structures (−5% and +5%) is shown in Fig. S2 of the SI. For −5% strain, the CBM and VBM positions are −0.790 and 3.390 V versus NHE, respectively, whereas for +5% strain, the corresponding CBM/VBM values are −1.106 and 3.660 V versus NHE. Under compressive strain, the E_g decreases slightly, shifting the CBM upward (less negative) and the VBM downward (less positive), thereby reducing the water oxidation window while still maintaining suitable band-edge positions for overall water splitting. In contrast, tensile strain increases the E_g, moving the CBM to a more negative potential and the VBM to a more positive potential, thereby enhancing the thermodynamic driving force for both the hydrogen evolution reaction and oxygen evolution reaction. Consequently, the strained structures retain the ability to drive overall water splitting across the entire pH range, although the redox potentials are modulated by the strain-induced changes in the electronic structure. These results indicate that strain engineering provides an effective route to fine-tune the photocatalytic performance of RSC without compromising its thermodynamic viability for overall water splitting. Moreover, our analysis uses bulk band edge positions, while photocatalytic reactions occur at the material surface, where termination, reconstruction or hydroxylation can shift the local band edges.³⁵ Nevertheless, bulk alignment remains an essential first screening for thermodynamic viability.

Fig. 11. Calculated redox potentials versus pH-scale, which varies from 0 to 14 for the Rb₂SnCl₆ structure related to the vacuum scale and normal hydrogen electrode (NHE) potentials for water splitting.

3.4. Optical aspects

Optical features of the UWBG SC are vital to understand for their applications in OE, particularly in UV detectors and photonics devices.⁵ The fundamental optical behavior is captured by the ε(ω) as:

ε(ω) = ε₁(ω) + iε₂(ω) 22

where ε₁(ω) is the real part, which describes the dispersion of the electric field and ε₂(ω) is the imaginary one, which quantifies absorption arising from inter-band electronic transitions. The Kramers–Kronig transformation is used to obtain ε₁(ω), whereas ε₂(ω) has been determined by utilizing the momentum matrix elements, which are represented as:^76,77

23

where, P is the integral prime integer and

24

V represents a measure of cell size, p̂ denotes the momentum operator, ℏ is a modified version of Planck's constant, while φ_c and φ_v are the wave functions associated with the CB and VB, respectively. The evaluation of the ε₁(ω) and ε₂(ω) in relation to photon energy (ℏω) for the unstrained and ± 5% hydro. strained structures are illustrated in Fig. 12(a) and (b), respectively. The ε₁(0) for the unstrained system is 2.53, which is related to the electronic polarizability of the material, this increases to 2.84 for −5% strain, however, it decreases to 2.25 for the +5% strain state. The ε₁(ω) decreases from 3.41 at −5% to 2.67 at 0% and further decreases to 2.26 at +5% (see Fig. 12(a)). As we mentioned, ε₁(0) has an inverse squared relation with E_b, thus −5% has a low value of E_b (0.189 eV) as compared to +5% (0.789 eV). On the other hand, ε₂(ω) drops from 0.80 to 0.75 to 0.62, indicating reduced polarization and weaker interband transitions with tens. strain (see Fig. 12(b)). Next, other key optical parameters can be derived form ε₁(ω)/ε₂(ω) such as absorption coefficient (α(ω)), refractive index (n), extinction coefficient (k), and reflectivity (R) using ε(ω). The α(ω) is calculated as:⁷⁶

25

Fig. 12. Variation of optical responses with photon energy of Rb₂SnCl₆ under −5%/0%/+5% hydrostatic ([111]) for (a) real dielectric function (ε₁(ω)), (b) imaginary dielectric function (ε₂(ω)), and (c) absorption coefficient (α(ω)).

The α(ω) peak (1.27 × 10⁵ cm⁻¹) for the unstrained system is slightly lower under −5% strain (1.20 × 10⁵ cm⁻¹) and further decreases for +5% strain to 1.14 × 10⁵ cm⁻¹ at 5.51 eV (see Fig. 12(c)). Other optical parameters at this high photon energy are also summarized in Fig. 13. The material's optical E_g is perfectly aligned with the experimental one, while showing the first peak at 5.51 eV (see Fig. 12(c)). Thus, to analyze the material's optical behavior at a high absorption photon energy, we selected this point for further discussion. The n declines from 1.86 to 1.65 to 1.52 from −5% to +5% but the k from 0% is 0.228. It deceases to 2.15 for −5% and to 0.20 for +5%. The R follows the trend being maximum for comp. strain and reduces to tens. strain (from 9.55% to 4.82%) illustrating enhanced transparency under tens. strain. Thus, Fig. 13 highlights the tunability of optical parameters in response to mechanical strain, which is crucial for tailoring materials for photonic and UV applications.

Fig. 13. Numerous optical parameters at a photon energy of 5.51 eV for the Rb₂SnCl₆ structure under −5%/0%/+5% hydrostatic ([111]) strains.

3.5. Thermoelectric features

The global demand for sustainable energy has accelerated the search for efficient TE materials that are capable of directly converting waste heat into electricity. Nearly 60% of wasted thermal energy is instantly transformed by these materials into a cost-effective energy source without harming the environment.^47,48 They have diverse applications which include solid-state refrigeration,⁷⁸ advanced heating and cooling systems,⁷⁹ solar thermal energy conversion,⁸⁰ thermal management of laser diodes,⁸¹ and energy harvesting.⁸² Consequently, implementing BoltzTrap⁵⁶ theoretical algorithm as integrated in the Wien 2K code, we calculated several TE parameters in a relaxation period of τ = 10⁻¹⁴ over a temperature (temp.) range of 0–1200 K for the structures. Materials having high σ, S and low κ are desired for TE applications. The potential of a substance to transfer charge carriers is defined by its σ, which is defined as:⁸³

26

where e, ρ(ε), and τ(ε) represent electron charge, number of DOS, and relaxing period, f(ε) denotes the Fermi distribution parameter, and v²(ε) is the carrier velocity. To gain better understanding, we analyzed the temp.-induced fluctuations of in DP RSC as illustrated in Fig. 14(a) for 0% and ±5% hydro. strains. It increases with temp. due to the fall of electrical resistance, which is a crucial characteristic of a SC. The magnitude of begins at 50 K with a value of 1.20, 1.99, and 0.77 × 10¹⁸ (Ω ms)⁻¹ for −5, 0%, and +5 hydro. strains, respectively (see Fig. 14(a)). It increases steadily with temp., reaching maximum values of 22.59, 18.50, and 8.29 × 10¹⁸ (Ω ms)⁻¹ at 1200 K for the respective strains. Next, the ability of TE material to transform temp. fluctuation in electric voltage via S has been proven and this effect is illustrated by the expression , wherein the ΔV corresponds to a change in voltage and ΔT denotes the temp. fluctuation. Moreover the relation between m* and S is given as:⁴⁹

27

Fig. 14. Numerous thermoelectric features against temperature under −5%/0%/+5% hydrostatic ([111]) strains for the Rb₂SnCl₆ structure.

This expression illustrates the direct dependence of S on the m* and inversely to n (carrier concentration), which demonstrate that S decreases with a rise in n and increases with higher values of m*. For −5%/0% states, the S starts at 50 K from 15.61/18.28 × 10⁻⁵ and initially increases then decreases to 18.017/16.14 × 10⁻⁵V/T as shown in Fig. 14(b). On the other hand, for +5% it reduces to 11.90 from 23.14 × 10⁻⁵V/T at 1200 K. But overall it remains positive across the studied range (50–1200 K) confirming p-type conduction (see Fig. 14(b)). Another crucial measure to evaluate a materials capabilities of heat transformation into electricity, is power factor (PF), which is expressed as . It increases with temp. as illustrated in Fig. 14(c) starting from 0.29, 0.40, and 0.41 × 10¹⁰ W mK⁻² s⁻¹ for −5%, 0% and, +5% strains, respectively, which further increases to 7.32, 4.83, and 1.29 × 10¹¹ W mK⁻² s⁻¹ at 1200 K for respective strains.

Thermal transportation characteristics are important, particularly κ_e and κ_L, where κ_e relates to carrier transport and κ_L defines phonon-triggered heat effects. According to Wiedemann–Franz law, a relation between κ_e and σ is as follows:^84,85

κ_e = LσT 28

where L stands for the Lorentz factor and κ_e directly relates to σ. The calculated values of for the unstrained and ± 5% are illustrated in Fig. 14(d) and a similar pattern is observed as σ because both are directly correlated. It increases to a value of 12.40, 8.11, and 2.30 × 10¹⁴ W mK⁻² s⁻¹ for −5%, 0%, and +5% hydro. strains, respectively. Since, the BoltzTraP method is limited to evaluating only the electronic component of κ, the κ_L was separately calculated using the Slack's equation as:⁸⁶

29

here γ/M_avg./V/T are the Grüneisen parameter/average molar mass per atom/volume per atom/temp. and A is a constant that depends on γ, which further relies on ν, expressed as:⁸⁷

30

31

Along with this, Θ_D serves to assess the vibrational features and is given as:⁸⁸

32

where h, k_B, ρ, n, N_A, M, and v_avg. are the Planck's constant, Boltzmann constant, mass density, number of atoms in the unit cell, Avogadro's constant, molecular mass, and average sound velocity, respectively. The calculated κ_L decreases with increasing temp. to a value of 0.15, 0.024, and 0.017 W K⁻¹ for −5%, 0%, and +5% strains, respectively (see Fig. 14(e)). Finally, the corresponding ZT is computed as:⁵⁰

33

Materials possessing higher PF and lower κ = κ_e + κ_L demonstrate higher ZT which ultimately enhances the material power conversion efficiency. Fig. 14(f) shows that ZT increases to 0.71 for the unstrained ones and for −5%/+5% strain its value is 0.70/0.61 at 1200 K. For comparison, the average ZT values of the Cs₂BI₆ (B = Pt, Pd, Te, Sn) are 0.88, 0.85, 0.95, and 0.78, respectively,⁸⁹ while Cs₂SnBr₆ achieves ZT = 0.55 at 450K.⁹⁰ Likewise, ZT values reported for Cs₂MCl₆ (M = Se, Sn, Te, Ti) approach 1.⁹¹ Moreover, recent theoretical work on Pt-doped Cs₂SnCl₆ achieved a ZT of 2.27 through bandgap engineering.⁹² This suggests that the ZT of RSC could potentially be further enhanced through similar substitutional doping strategies.

4. Conclusion

In summary, a comprehensive first-principles investigation of the hydrostatic ([111]) strain influence on the ultra-wide bandgap (UWBG) double perovskite Rb₂SnCl₆ for deep-ultraviolet (UV) and optoelectronic (OE) applications are conducted. Our system reveals an energy gap (E_g) of 4.41 eV, which is in excellent agreement with the experimental value of 4.79 eV, having an accuracy of 92%, and confirms its UWBG nature. We can tune its electronic properties as E_g varies from 4.18 (−5%) to 4.71 eV (5%) with charge carrier properties like effective mass and binding energy. Moreover, the strain effectively tunes the mechanical behavior from ductile metallic-like (0%) to brittle covalent-like (±5%). It shows maximum absorption in the DUV region for the unstrained (1.27 × 10⁵ cm⁻¹) and strained systems (1.20 × 10⁵ cm⁻¹/1.14 × 10⁵ cm⁻¹ for −5%/+5%), making it a promising candidate for DUV applications. Additionally, electron/hole effective mass increases from 0.163 to 0.463/0.359 to 0.808, as the strain varies from −5% to +5%, indicating improved carrier mobility under compressive strain. Also, its redox ability for water splitting shows that it has a redox potential with valence bands of 3.57, greater than O₂/H₂O oxidation potential (1.23 V), while the conduction band is −0.90 which is more negative than the H⁺/H₂ reduction potential (0 V). Notably, the thermoelectric (TE) analysis revealed a high figure of merit of 0.71 for the unstrained structure, while with −5% compressive strain maintains a high value of 0.70 at 1200 K. These findings establish Rb₂SnCl₆ as a versatile UWBG semiconductor, suitable for the DUV, UV-driven water-splitting, and sustainable energy-harvesting applications. Its tunable electronic, mechanical, and TE properties under strain make it a benchmark for exploring new UWBG materials beyond conventional compounds.

Author contributions

Muhammad Bin Javed: writing – original draft, investigations, formal analysis, data curation. Huda A. Alburaih: validation, formal analysis, and visualization. Arslan Zulfiqar: writing, investigations, and formal analysis. S. Nazir: writing – review and editing, validation, supervision, project administration, conceptualization.

Conflicts of interest

The authors declare no competing interests.

Data availability

The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.

Supplementary information (SI): calculated numerous physical properties of the material. See DOI: https://doi.org/10.1039/d6ra02163h.