Multiscale analysis of Rb₂NaInI₆: from electronic structure to device performance for next-generation perovskite solar cells

Abstract

The harmful effects and long-term unpredictability of conventional compounds made from lead have driven a more intense quest for practical, stable, and ecologically acceptable lead-free perovskite components. Among the exciting prospects, the pair of perovskites Rb₂NaInI₆ stands out for its special structural, electrical, and optical properties, therefore offering a possible subsequent-generation light-absorbing substance for photovoltaic energy and optoelectronic use. We study Rb₂NaInI₆ holistically in this work, utilizing a mixed computational method. The density functional theory (DFT) computations using WIEN2k verify their cubic Fm̄3m organization with an optimal lattice consistency of 12.25 Å, guaranteeing strong structural integrity. Whereas its density of states (DOS) assessment shows substantial hybridization concerning In-5s and I-5p orbitals, boosting charge transport, the semiconductor band structure provides a secondary bandgap of 0.68 eV. Strong absorption in the visible to light spectrum, with a significant coefficient of attraction and low reflectance (<30%), makes optical analysis very appropriate for solar power plants and optoelectronic device uses. Furthermore, its dielectric material functions alongside refractive index, pointing to great possible usage in photonic devices and wavelength guides. Using SCAPS-1D, we simulate four straight heterojunction topologies, including various electron transport layers (ETLs), ZnO, WS₂, WO₃, and PCBM, via Pt as the rear contact as well as CBTS as the hole transport layer, thereby exploring its device-level efficiency. Under AM 1.5G enlightenment the FTO/ZnO/Rb₂NaInI₆/CBTS/Pt arrangement obtained the best leads to between these with an open-circuit voltage (V_oc) of 1.39 V, short-circuit current density (J_sc) at 21.39 mA cm⁻², fill factor (FF) of 89.83%, particularly a staggering power conversion efficiency (PCE) of 26.84%. When combined with ideal ETLs, Rb₂NaInI₆ expresses a lucky, lead-free perovskite-absorbing material that opens routes for developing stable, effective, and ecologically conscious solar panels made of perovskite. This work highlights the need to accelerate development and enhance new solar components by merging first-principles modelling with device training exercises.

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1. Introduction

Advancing innovative materials is crucial for progress in research and technology across various sectors, including energy, electronics, healthcare, and the environment. However, traditional experimental methods for discovering and optimizing materials are slow, costly, and labor-intensive—often taking over two decades to move from concept to commercialization.¹ This sluggish tempo engenders a bottleneck in advancement in technology, highlighting the pressing necessity for expedited and more efficient approaches to material discovery. In response to these challenges, computational materials science has emerged as a groundbreaking discipline, enabling researchers to predict material properties and design innovative molecules with unprecedented speed and accuracy. Computational technologies diminish dependence on experimentation and failure methods of testing and substantially decrease scientific and development expenditures. Research by Allison et al. indicates that utilizing computational design can decrease time to market and yield a return on your investment of 300% to 700%.² Density Functional Theory (DFT) has emerged as one of the many potent and adaptable computing approaches. Rooted in the concept of quantum mechanics, DFT provides a first-principles framework for examining the electronic makeup of molecules, atoms, and compressed material systems.^1,3–5 The complicated many-body Schrödinger equation does not need to be solved directly since DFT makes it possible to calculate ground-state parameters based just on electron density. This renders it highly beneficial across several research domains, encompassing molecular morphologies,^6,7 elements and interfaces,⁸ flaws in solids,^9–11 as well as the examination of electrical transport,^12,13 optical characteristics,¹⁴ and contributing behavior.¹⁵ The accuracy of DFT is contingent upon the exchange–correlation functional utilized, which presents difficulties in accurately simulating specific interactions; nonetheless, ongoing advancements and novel approximations have markedly improved its predictive efficacy. Consequently, DFT has become essential in contemporary materials science, connecting theoretical predictions with experimental confirmations. This paper introduces an innovative approach that combines DFT with the solar panel Cell. The term capacitor Simulators (SCAPS-1D) to examine the optoelectronic characteristics and performance of the Rb₂NaInI₆ double perovskite, a potential contender for solar power applications. We seek to deliver an extensive comprehension of the material characteristics and photovoltaic performance of this free from lead double perovskite by integrating atomistic simulations with device-level modeling.^16,17 After identifying viable materials, a crucial stage is the filtration and selection process. Preliminary screening often emphasizes the exclusion of materials containing hazardous substances such as lead (Pb) and mercury (Hg), as well as those that incorporate rare and costly elements like palladium (Pd) and platinum (Pt).^18–20 Advanced Training filtering processes, based on computational projections of properties such as formation electrical power, bandgap, along with thermodynamic rigidity (e.g., hull energy), enhance the refinement of candidate materials, guaranteeing that only the best potential candidates are given preference for exploratory synthesis and testing.^21–23 Perovskites have transformed the domain of photovoltaics in the last decade, mostly due to their beneficial material characteristics and straightforward production. These materials generally possess a crystalline structure defined by the formula ABX₃, in which ‘A’ represents a monovalent cation, ‘B’ denotes a divalent metal cation, and ‘X’ signifies a halide anion.^24–26 They are renowned for their ideal direct bandgaps, robust absorption of visible light, minimal effective carrier masses, and outstanding charge transport properties, including extended diffusion lengths, lifetimes, and elevated mobilities. Furthermore, their inherent defect tolerance and multifunctionality broaden their applicability beyond photovoltaics to encompass photocatalysis, light-emitting diodes (LEDs), unpredictable optical devices, photodetectors, scintillators, memristors, and field-effect transistors.^27–29 Since the initial demonstration of a perovskite solar cell (PSC) in 2009, which exhibited a power conversion efficiency (PCE) of 3.8%, efficiencies have escalated to over 25.8% by 2022. Notwithstanding these notable accomplishments, the actual implementation of PSCs is impeded by two critical concerns: the environmental and health risks linked to lead-based perovskites and their inherent instability under operational settings.^30,31 Strategies to boost stability have involved substituting organic cations with inorganic counterparts such as Cs⁺ and Rb⁺, which fortify lattice connections and improve heat and moisture stability.^32–34 Nonetheless, the issue of lead toxicity persists, prompting significant investigation into lead-free alternatives. Lead-free perovskites, specifically those composed of tin (Sn²⁺) and germanium (Ge²⁺), have surfaced as appealing alternatives owing to their analogous electrical structures and optoelectronic characteristics. Regrettably, these materials exhibit inadequate ambient stability, as Sn²⁺ and Ge²⁺ are susceptible to oxidation into their tetravalent states (Sn⁴⁺ and Ge⁴⁺), leading to fast deterioration of device performance.^35,36 An alternate method is hetero valent replacement, in which two Pb²⁺ ions are substituted by a mixture of monovalent (+1) and trivalent (+3) cations, resulting in lead-free double perovskites (LFDPs) via the following formula: A₂B(I)B(III)X₆.^37–39 These compounds provide enhanced chemical maneuverability and customization of band structure and optical characteristics, expanding the possibilities for material design. The structural paradigm of double perovskites originated in the early 20th century, with K₂NaAlF₆ recognized as one of the initial documented instances.^38–41 Recent studies have concentrated on inorganic LFDPs employing Cs⁺ or Rb⁺ at the A-site, as these cations strengthen the contact between cations and [BX₆]⁻ octahedra, leading to improved structural integrity.^34,42 Sn-based perovskites have commendable photovoltaic properties, featuring band gaps ranging from 1.2 to 1.4 eV—approaching the Shockley–Queisser optimum bandgap of 1.34 eV^43–45 and power conversion efficiencies above 13% in certain studies.^46,47 Oxidation difficulties continue to exist, driving the persistent quest for innovative LFDPs that integrate stability, ideal bandgaps, and enhanced electrical characteristics. Elpasolites, a category of low-dimensional functional materials characterized by the general formula A₂M(I)M(III)X₆, include more than 350 recognized structures.^48,49 Building upon the research of Z. Zhang et al., who achieved a power conversion efficiency of 6.37% with a hydrogenated Rb₂NaInI₆ structure, our investigation advances this field by employing sophisticated computational methods to enhance and model device performance. We utilized an inverted (p-i-n) perovskite solar cell architecture, maintaining ZnO as the electron transport layer (ETL) and Cu₂BaSnS₄ (CBTS) as the hole transport layer (HTL). Simulations were performed under AM 1.5G illumination with SCAPS-1D (version 3.8), enabling the optimization of essential parameters like absorber layer thickness, loading concentration, and defect density. Furthermore, we methodically examined the influence of HTL thickness and doping concentrations on device efficacy. To enhance comprehension of the interfacial characteristics and band alignment, we simulated devices employing four distinct electron transport layers: ZnO, WS₂, WO₃, and [6,6]-phenyl-C₆₁-butyric acid methyl ester (PCBM). ZnO and WS₂ exhibit elevated electron mobility and advantageous energy-level alignment with perovskite absorbers, whilst WO₃ and PCBM provide adjustable electronic characteristics and compatibility with scalable processing methods.^50–54 Our simulations demonstrated remarkable maximum power conversion efficiencies of 26.84% (ZnO), 26.61% (WS₂), 26.73% (WO₃), and 26.26% (PCBM), underscoring the significant potential of hydrogenated Rb₂NaInI₆ in inverted perovskite solar cell topologies. Experimental power conversion efficiencies (PCEs) for analogous materials like Cs₂AgBiBr₆ are modest, varying from 1.66%⁵² to 21.88%⁵⁵ in optimized setups, although theoretical and computational analyses persist in highlighting their promise for high-performance solar applications.^56–58 DFT studies of essential electronic properties—namely band structure, bandgap, density of states (DOS), and electron density mapping—provide critical insights into the applicability of halide perovskites for optoelectronic applications.^59–61 Our research constitutes the inaugural comprehensive endeavor to amalgamate DFT and SCAPS-1D simulations for the assessment of the photoelectric potential of the Rb₂NaInI₆ double perovskite. By integrating atomistic insights with device-level performance modeling, we offer a comprehensive understanding of this material's appropriateness for subsequent-generation, without lead, perovskite solar cells. This study establishes a foundation for subsequent experimental validations and informs the conceptualization of trustworthy, high-efficiency LFDP-based solar energy sources.

2. Materials and methodology

2.1. First-principles computations of Rb₂NaInI₆ absorber utilizing DFT

This work investigates the characteristics of the substances using density functional theory (DFT) and WIEN2k software under the FP-LAPW approach.^62–64 Three primary phases define the computational process: structure optimization and optoelectronic property computation. The Generalized Gradient Approximation (GGA) method—more especially, the Perdew–Burke–Ernzerhof (PBE)⁶⁵ exchange–correlation function—is used for the optimization of structural elements.⁶⁶ Keeping all of the energy regarding the foundational set size (R × K_max = 8) and k-point grid (13 × 13) guarantees convergence. We consider a 10⁻⁶ Ry total energy convergence threshold. Using TB-mBJ potential, which is known for its accuracy in band gap prediction, optoelectronic features are investigated.⁶⁷ The relaxation interval approximated performance (RTA = 10⁻¹⁴ s) helps one to find semi-classical transportation coefficients. Moreover, the thermodynamic durability of the materials⁶⁸ is verified by computing the phonon band arrangement and density of states (DOS) employing the WIEN2k algorithm with the limited displacement approach. Employing the Kramers–Kronig relations, one finds optical parameters, including the absorption coefficients, reflectance, conductivity of electric current, refractive index, energy loss function, and the actual portion of the dielectric property function from the imaginary component of the dielectric property function. The mathematical equations for computing these optical characteristics are presented below; each symbol has a usual interpretation.^69,70

2.2. Numerical simulation employing SCAPS-1D

Comprehending the utility and efficiency of solar energy cells depends on using mathematical models like SCAPS-1D. Through numerical solution of equations such as a single-dimensional Poisson's equation, scientists can obtain an important understanding of the relationship between charge distribution inside a material made of semiconductors and its electrostatic potential. Poisson's equation provides a useful illustration of eqn (1).^71–73 The above equation explains in any particular semiconductor material the link between the charge transportation and the electromagnetic potential. Utilizing numerical solutions, scientists may examine and forecast the effect of different elements on solar cell operation. This analytical method clarifies the main aspects influencing the energy efficiency of sunlight-sensitive cells through significant elements.⁷⁴In this instance, ε₀ signifies the permittivity of free space, ψ denotes the potential of electricity, and ε_r indicates the proportional permittivity. The symbols ND as well as NA refer to the adjustment of charged donors as well as acceptors, while n among p indicate the densities of electrons plus holes. P_n also ρ_p show that the distributions of electrons and holes, respectively, and q represents the electric charge. Considering all processes simultaneously-recombination, generation, drift, and diffusion—the continuity equation emerges as the primary equation of importance. Eqn (2) and (3), which are continuity equations, describe the changes in the counts of electrons and holes, respectively, taking into account the various mechanisms at play.

Under this situation, J_n and J_p correspondingly reflect the current density of holes as well as electrons. R_n and R_p show the relative recombination rates; G_n and G_p show the hole in addition to the electron generation rates. Eqn (4) and (5) help one to determine the current densities of electrons and holes in solar cells utilizing the charged carrier drift-diffusion mathematical equations. The following formulas clarify the way holes as well as electrons migrate across the dielectric material in response to changing electromagnetic fields, concentration shifts, carrier production, and recombination events.

J_n = qμ_nnε + qD_n∂n (4)

J_p = qμ_pnε + qD_p∂p (5)

In this context, D_n and D_p demonstrate these diffusion coefficients, whereas μ_n and μ_p correspondingly reflect the mobility of electrons and holes. Carrier lifetime and mobility, which are represented by the Albert Einstein relation, significantly affect their diffusion coefficient of friction. Furthermore, the absorbance constant of sheets was determined using a modified form of the old sqrt (hν − E_g) model based on the recently updated E_g-sqrt structure. Eqn (6) therefore shows this coupled with using the “Torque laws”, which offer understanding of substance absorption behavior depending upon their band gap energy.

In the provided context, the symbols hν represent the photon energy, and E_g represents the band gap and absorption coefficient of the material. Eqn (7) and (8) demonstrate the relationship between the model constants α₀ and β₀ with the traditional model constants A and B. This relationship allows for a clear understanding of how these parameters are interconnected and how they relate to the material's optical properties.

Fig. 1, therefore, displays our performance in the SCAPS-1D modeling. First, we created a model of our solar cell using the program. This involved organizing all the several layers, including ETLs, and that Rb₂NaInI₆ material we employ as an electromagnetic absorber—the perovskite component. By simulating several varying situations to obtain the J–V curves. We varied the voltage, light intensity (AM 1.5G), and temperature while inputting key material properties like bandgap, electron affinity, and carrier mobility. The simulation results, including J–V curves, capacitance, QE, and Mott–Schottky plots, were matched with experimental data. The software internally solves drift-diffusion, Poisson, and continuity equations to model electron behavior. Finally, we visualized and analyzed the data to assess solar cell performance.

Fig. 1 The operational methodology of SCAPS-1D.

2.3. Structure of the device

Fig. 2a illustrates the stratified architecture of a perovskite solar cell (PSC) including Rb₂NaInI₆ as the light-absorbing layer. The design commences with a fluorine-doped tin oxide (FTO) film at the base, functioning as a translucent conducting electrode that permits sunlight to penetrate the cell. The Electron Transportation Layer (ETL) is positioned beneath the FTO, comprised of one of four substances: ZnO,⁷⁵ WS₂,⁷⁵ WO₃,⁷⁶ or PCBM,⁷⁵ enabling electron transfer from the absorbers to the FTO. The absorbing layer, composed of Rb₂NaInI₆, captures sunlight and produces a pair of electron–holes. Furthermore, the Hole Transport Layer (HTL), consisting of CBTS,⁷⁵ facilitates the transfer of holes to the upper electrode. The rear contact, composed of Platinum (Pt), finalizes the cell by collecting holes and enabling their mobility, thereby generating an ongoing current flow while subjected to sunlight.

Fig. 2 (a) Device architecture based on Rb₂NaInI₆, (b) energy band alignment between FTO/ETLs/Rb₂NaInI₆/CBTS/Pt.

Fig. 2b illustrates that the photoelectric device architecture incorporates Rb₂NaInI₆ as the primary absorber layer, complemented by meticulously positioned holes as well as electron transport layers ETLs along with HTL to facilitate effective separation of charged particles and collection. The heterogeneous structures comprise FTO, an electron transport layer (ETL) such as ZnO, WS₂, WO₃, or intentionally PCBM, Rb₂NaInI₆, CBTS, and additionally Pt, with each layer selected to enhance energy alignment. The ETLs and Rb₂NaInI₆ exhibit compatible band alignments, leading to the quasi-Fermi levels representing electrons (E_fn) along with holes (E_fp) aligning with the conduction band (E_C) and valence band (E_V) edges, for example, facilitating efficient electron and hole transportation to their corresponding electrodes. The HTL CBTS is deliberately designed to promote hole transport to the Pt back contact while reducing recombination losses. The resulting device architecture utilizes work function differences, employing FTO as the front contact (4.0 eV) along with Pt as the rear contact (5.7 eV), thereby inducing electron transportation toward the FTO and hole migration toward the Pt, which improves charge collection. Among the ETLs, WS₂,⁷⁵ WO₃,⁷⁶ or PCBM⁷⁵ exhibit their conduction band edges that are closely aligned with Rb₂NaInI₆, facilitating efficient electron transport and reducing energy loss. Although ZnO⁷⁵ has a lower conduction band minimum, it may create energy barriers that affect carrier dynamics. The HTL CBTS is compatible with Rb₂NaInI₆, facilitating efficient hole extraction. The band alignment underlies the observed solar performance, with ZnO and WO₃ achieving better efficiencies due to advantageous energy level alignment. Table 1 outlines key parameters such as thickness, bandgap, electron affinity, dielectric constant, effective density of states (VB and CB), carrier mobilities, shallow acceptor/donor densities, and defect density (N_t). These values were carefully chosen based on fabrication data from peer-reviewed literature to ensure accurate simulation results.

Table 1 The input settings for FTO, ETLs, HTL, as well as an absorber layer

Parameter	FTO^77	WS2 (ref. 59)	WO3 (ref. 78)	ZnO^59	PCBM^59	CBTS^59	Rb2NaInI6
Thickness (nm)	500	100	50	50	50	100	1000
Bandgap (Eg) (eV)	3.5	1.8	2	3.3	2	1.9	1.702
Electron affinity (χ) (eV)	4	3.95	3.8	4	3.9	3.6	3.779
Dielectric permittivity (εr)	9	13.6	4.8	9	3.9	5.4	5.06
Effective DOS (conduction) (cm^−3)	2.2 × 10^18	1.0 × 10^18	2.2 × 10^21	3.7 × 10^18	2.5 × 10^21	2.2 × 10^18	3.053 × 10^16
Effective DOS (valence) (cm^−3)	1.8 × 10^19	2.4 × 10^19	2.2 × 10^21	1.8 × 10^19	2.5 × 10^21	1.8 × 10^19	9.945 × 10^17
Electron thermal velocity (cm s^−1)	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7
Hole thermal velocity (cm s^−1)	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7	1.0 × 10^7
Electron mobility (cm^2 V^−1 s^−1)	20	100	30	100	0.2	30	155.6
Hole mobility (cm^2 V^−1 s^−1)	10	100	30	25	0.2	10	15.25
Donor density (ND) (cm^−3)	1 × 10^18	1 × 10^18	6.35 × 10^17	1 × 10^18	0	0	0
Acceptor density (NA) (cm^−3)	0	0	0	0	2.93 × 10^17	1 × 10^18	1 × 10^15
Defect density (Nt) (cm^−3)	1 × 10^15	1 × 10^15	1 × 10^15	1 × 10^15	1 × 10^15	1 × 10^15	1 × 10^15

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In the simulations, parameters such as electron and hole thermal velocities (set at 10⁷ cm s⁻¹) were standardized across all layers. Interface defect layers (IDLs) were introduced between the ETL/Rb₂NaInI₆ and Rb₂NaInI₆/HTL interfaces to better reflect real conditions, as detailed in Table 2. Fluorine-doped tin oxide (FTO) served as the transparent, conductive anode, while platinum (Pt) was used as the cathode for its suitable work function. Simulations were conducted under standard AM 1.5G illumination at 1000 W m⁻² and 300 K. Only defect-related recombination was considered, excluding radiative and Auger processes. The voltage range was set from 0 to 1.8 V to capture the full operational window.

Table 2 Interface characteristics employed for this perovskite solar panel

Interface	ETL/absorber^59	Absorber/HTL^59
Defect type	Neutral	Neutral
Electron cross-section measurement (cm^2)	1 × 10^−17	1 × 10^−18
Acquire cross-sectional data for apertures (cm^2)	1 × 10^−18	1 × 10^−19
Energetic distribution	Single	Single
Defect reference energy level Et	Above the highest EV	Above the highest EV
Characteristic energy (eV)	0.6	0.05
Total density (cm^3)	1 × 10^10	1 × 10^10

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3. Results and discussion

3.1. Structural properties and stability of Rb₂NaInI₆

The structural properties of Rb₂NaInI₆ reveal that it crystallizes in a cubic double-perovskite-type system, adopting the Fm3̄m (space group 225) symmetry.⁷⁶ The optimized atomic arrangement within the unit cell places Rb atoms at the 8c (0.25, 0.25, 0.75) positions, Na at 4b (0.5, 0, 0), In at 4a (0, 0, 0), and I at 24e (0.2415, 0, 0). This well-ordered atomic configuration contributes to the stability and symmetry of the crystal structure. Upon structural optimization, the calculated lattice parameter was found to be a = b = c = 12.19 Å, leading to a unit cell volume of 1810.05 ų. Such a lattice framework indicates a spacious and stable arrangement of cations and anions, reinforcing the perovskite-derived nature of the compound. The three-dimensional visualization of the optimized crystal structure as illustrated in Fig. 3 further confirms the uniformity and connectivity of the octahedral framework, where NaI₆ and InI₆ octahedra share corners, maintaining a stable coordination environment. A crucial aspect of the structural stability of Rb₂NaInI₆ is its formation energy (ΔH_f), which serves as an indicator of thermodynamic feasibility. The negative formation energy of −1.238 eV per atom suggests that this material is energetically stable.^79,80 The formation energy was computed using the following equation:where E_s (Rb), E_s (Na), E_s (In), and E_s (I) represent the individual atomic energies of Rb, Na, In, and I, respectively, while E_tot (Rb₂NaInI₆) denotes the total unit cell energy of the compound, and N signifies the number of atoms in the unit cell.⁸¹

Fig. 3 Crystal structure of Rb₂NaInI₆.

Our computational findings align well with existing literature, reinforcing the validity and robustness of the adopted methodology. Evaluating the dynamic stability of a crystalline material is essential for determining its practical applicability, particularly under time-dependent or external perturbations. To this end, the phonon dispersion curve (PDC) serves as a critical diagnostic tool. In this study, we analyzed the dynamic behavior of Rb₂NaInI₆ by calculating its phonon spectrum along the high-symmetry paths (W–L–Γ–X–W–K) within the Brillouin zone, as depicted in Fig. 4. The absence of imaginary phonon modes across the entire Brillouin zone confirms the dynamical stability of the structure. This result is significant, as imaginary frequencies would indicate structural instabilities and potential lattice distortions. Therefore, the phonon dispersion analysis provides strong evidence that Rb₂NaInI₆ is not only energetically favorable but also dynamically stable, further supporting its potential for reliable real-world applications.

Fig. 4 Phonon dispersion spectrum of Rb₂NaInI₆.

3.2. Band structure of Rb₂NaInI₆ compound

Understanding the electronic properties of a material is crucial in elucidating its bonding nature, photon absorption capabilities, and overall electronic behavior. One of the most significant aspects of these properties is the band structure, which provides insights into the energy levels that govern electrical conductivity and optical transitions. The band structure of Rb₂NaInI₆ was computed along the high-symmetry directions of the first Brillouin zone, as illustrated in Fig. 5. The Fermi level (E_F) is set at 0 eV and is indicated by a red dashed line. In this analysis, it is observed that neither the valence band (VB) nor the conduction band (CB) crosses the Fermi level, confirming the presence of an energy band gap in its electronic structure. The calculated band gap is 0.71 eV, suggesting that Rb₂NaInI₆ exhibits semiconducting behavior. Comparing this band gap with similar double perovskite compounds, it is noted that Rb₂NaInI₆ has a slightly lower band gap than Rb₂CuSbBr₆ (0.72 eV) but is higher than Rb₂CuSbI₆ (0.36 eV).⁸² However, it is smaller compared to Rb₂ScInI₆ (0.90 eV)⁸³ and Rb₂AgGaCl₆ (1.28 eV).⁸⁴ A closer examination of the band structure reveals that the maximum of the valence band (VBM) and the minimum of the conduction band (CBM) are located at different k-points, confirming that Rb₂NaInI₆ is an indirect bandgap semiconductor. This characteristic typically results in lower radiative recombination rates, which could be advantageous or limiting, depending on the targeted application. Additionally, the valence band maximum (VBM) is found to be the closest to the Fermi level, which suggests that Rb₂NaInI₆ primarily exhibits p-type semiconducting behavior.^84,85 This is further supported by the presence of nearly flat valence bands between the F and Z points, which are observed close to the Fermi level. The high dispersion of these bands near E_F indicates a strong potential for high charge carrier mobility, suggesting that Rb₂NaInI₆ could have promising electrical transport properties. Moreover, the flat nature of some valence bands near E_F suggests a higher effective mass of charge carriers, which could influence hole transport dynamics. This balance between charge carrier mobility and effective mass is a key factor in determining the potential electronic and optoelectronic applications of Rb₂NaInI₆.⁸⁶

Fig. 5 Electronic band structure of Rb₂NaInI₆.

3.3. DOS of Rb₂NaInI₆ compound

The total density of states (TDOS) and partial density of states (PDOS) provide critical insights into the electronic structure, chemical bonding, and orbital contributions in Rb₂NaInI₆. The TDOS, depicted in Fig. 6, shows a distribution of electronic states with a notable presence of approximately 12 electrons/eV at the Fermi level (E_F), marked by a vertical dashed line at 0 eV. This line distinguishes the valence band from the conduction band. Although the band structure reveals a finite band gap, the slight non-zero TDOS at E_F may suggest numerical broadening effects or indicate a semi metallic tendency within specific computational limits. In the valence band region (E < 0 eV), the I-5s and I-5p orbitals dominate the DOS, with secondary contributions from Rb-5s, Na-3s, and In-5p orbitals. The strong presence of I-5p and In-5p states reflects their critical role in chemical bonding and hole conduction, with the In–I interaction evident in the hybridized peaks. These features indicate substantial orbital overlap and suggest strong covalent character between Indium and Iodine atoms. In the conduction band (E > 0 eV), the DOS is primarily composed of In-5s, In-5p, and I-5p states, with additional minor contributions from Rb-5p and Na-3p orbitals. The sharp PDOS peaks observed in this region highlight discrete energy states, indicating well-defined conduction channels. The overlapping of I-5p and In-5p states in both valence and conduction bands further emphasizes the degree of orbital hybridization, which plays a key role in enhancing electronic transitions and light absorption efficiency. Furthermore, while the d-orbitals of Indium and Iodine are less prominent, they do appear in the PDOS and contribute subtly to the deeper valence and higher conduction states, implying their involvement in electronic polarization and bonding symmetry adjustments. The overall orbital contributions, as mapped in the PDOS panels, reveal the intricate interplay between s, p, and d states, which collectively define the semiconducting nature and electronic behavior of Rb₂NaInI₆. These findings confirm that Rb₂NaInI₆ exhibits a suitable band gap, strong In–I orbital interactions, and well-structured electronic transitions, supporting its promise for optoelectronic and photovoltaic device applications.⁸⁷

Fig. 6 Total and partial density of states of Rb₂NaInI₆.

3.4. Optical properties of Rb₂NaInI₆ compound

Understanding the optical properties of a material is crucial for evaluating its interaction with electromagnetic waves, which directly influences its suitability for photovoltaic and optoelectronic applications.⁸⁸ The optical properties of Rb₂NaInI₆ are investigated over a broad range of photon energies, considering essential parameters such as absorption coefficient α(ω), optical conductivity σ(ω), reflectivity R(ω), dielectric function ε(ω), refractive index n(ω) and energy loss function L(ω).

3.4.1. Absorption coefficient α(ω). The absorption coefficient is a fundamental optical parameter that quantifies how deeply incident light penetrates a material before being absorbed. A high absorption coefficient is desirable for applications such as solar cells and photodetectors.⁸⁹ The calculated absorption spectrum of Rb₂NaInI₆ shows strong absorption in the visible to ultraviolet (UV) region, making it highly efficient in capturing solar energy. The initial peak in the absorption spectrum is observed at 1.70 eV, aligning with the bandgap value calculated using the TB-mBJ functional. As photon energy increases, the absorption intensity rises markedly, reaching a maximum of approximately 1.5 × 10⁴ cm⁻¹, which signifies strong photon absorption in the ultraviolet region and highlights the material's suitability for optoelectronic applications (Fig. 7).

Fig. 7 Optical properties: (a) absorption, (b) loss function, (c) refractive index, (d) conductivity, (e) dielectric function, (f) reflectivity of Rb₂NaInI₆.

3.4.1.1 Optical conductivity σ(ω). The optical conductivity represents the ability of free charge carriers to respond to an applied electric field within specific energy ranges.⁹⁰ The optical conductivity spectrum of Rb₂NaInI₆ remains negligible at low photon energies due to its semiconducting nature. The optical conductivity (σ₁) begins to rise noticeably around 6 eV, reaching a distinct peak at approximately 7.5 eV, followed by a steady decline toward near-zero values. This spectral behavior indicates a high degree of photoresponsivity in Rb₂NaInI₆, underscoring its potential for integration into optoelectronic devices, including photodetectors and light-emitting components.
3.4.1.2 Reflectivity R(ω). Reflectivity is an important property that determines how much of the incident light is reflected off the surface of the material.⁹¹ The reflectivity spectrum of Rb₂NaInI₆ starts with a low value of ∼7% at 0 eV, which ensures high light absorption and minimal optical losses—a crucial requirement for solar cell applications. The reflectivity increases with energy, reaching a peak of approximately 60% around 13.6 eV, which is attributed to interbond transitions. However, the overall reflectance remains below 65%, indicating that Rb₂NaInI₆ is an excellent absorber of light.

3.4.2. Dielectric function ε(ω). The dielectric function characterizes the optical response of a material when subjected to an electromagnetic field. It consists of a real part ε₁(ω), which represents the material's ability to store electric energy, and an imaginary part ε₂(ω), which corresponds to energy absorption due to electronic transitions.⁹² The static dielectric constant ε₁(ω) of Rb₂NaInI₆ is relatively high, indicating efficient charge separation, which is beneficial for photovoltaic and optoelectronic applications. The real part ε₁(ω) of the dielectric function exhibits a marked decline beyond 7 eV, suggesting the onset of metallic behavior at elevated photon energies. Meanwhile, the imaginary component ε₂(ω) shows a pronounced peak near 8 eV, indicative of intense interband electronic transitions within the material.
3.4.2.1 Refractive index n(ω). The refractive index characterizes the degree to which light is bent as it passes through a material. For Rb₂NaInI₆, the static refractive index n₁(ω) exhibits fluctuations in the photon energy range of 2 to 8 eV, with a pronounced peak near 7.7 eV, indicating a strong light–matter interaction. The imaginary component n₂(ω) remains relatively high up to approximately 8 eV, supporting the material's potential use in infrared sensors, waveguides, and optical coating applications. Beyond 8 eV, both real and imaginary parts of the refractive index begin to decline, with n₂(ω) approaching unity around 12 eV, which confirms the material's optical transparency in the ultraviolet region.⁹³

3.4.3. Loss function L(ω). The energy loss function describes the loss of energy due to electron interactions in the material.⁹⁴ The plasma frequency ω_p, which represents the energy level at which the material transitions from a dielectric to a metallic state, is identified at approximately 12 eV. This suggests that Rb₂NaInI₆ exhibits excellent stability in the visible and infrared range while showing metallic behavior in the ultraviolet region. The primary loss peak confirms its ability to be used in high-frequency optical applications.⁹⁵

3.5. Evaluation of SCAPS 1D outcomes

3.5.1. Optimization of ETL and HTL material combinations for enhanced photovoltaic performance. In solar cell design, the selection of electron transport layer (ETL) and hole transport layer (HTL) materials plays a critical role in determining the device's overall performance. These transport layers not only facilitate efficient extraction and transport of charge carriers (electrons and holes) but also significantly influence the open-circuit voltage (V_oc), short-circuit current density (J_sc), fill factor (FF), and power conversion efficiency (PCE). An optimal ETL/HTL combination ensures minimal recombination losses, good energy level alignment with the absorber, and high carrier mobility, all of which are essential for achieving superior device efficiency and stability. In this context, Fig. 8 presents a comparative performance analysis of different ETL and HTL materials using Rb₂NaInI₆ as the absorber layer, with a focus on four key parameters: V_oc, J_sc, FF, and PCE. To select the best ETL materials, the HTL CBTS was considered, as it consistently demonstrated strong performance across various ETL combinations. Among the ETLs, ZnO stood out with the highest PCE of 26.84%, coupled with a V_oc of 1.39 V, J_sc of 21.39 mA cm⁻², and FF of 89.83%. These parameters indicate excellent charge extraction capability, minimal energy losses, and strong interfacial contact with CBTS, making ZnO an outstanding candidate. WS₂ also exhibited remarkable performance, achieving a PCE of 26.61%, V_oc of 1.39 V, J_sc of 21.42 mA cm⁻², and FF of 89.03%. The layered structure and good electrical conductivity of WS likely contribute to its efficient charge transport and collection, further validating its suitability. WO₃ emerged as another high-performing ETL, delivering a PCE of 26.73%, V_oc of 1.39 V, J_sc of 21.37 mA cm⁻², and FF of 88.95%. WO₃'s high work function and excellent electron affinity promote efficient electron extraction while suppressing recombination at the interface. Similarly, PCBM demonstrated strong performance with a PCE of 26.26%, V_oc of 1.38 V, J_sc of 21.34 mA cm⁻², and FF of 88.92%. Although slightly lower in FF compared to the top three, PCBM still offers a robust balance of parameters and good compatibility with the CBTS HTL. The consistent performance of these ETL materials in combination with CBTS highlights their strong interfacial alignment, effective charge transport properties, and potential to enhance the overall device efficiency. In summary, the careful evaluation of performance metrics from Fig. 8 justifies the selection of ZnO, WS₂, WO₃, and ZnS as the four best ETL materials when paired with CBTS, underscoring the importance of synergistic ETL/HTL combinations in optimizing perovskite solar cell performance.

Fig. 8 Performance parameters of Variation. i.e. V_oc (V), J_sc (mA cm⁻²), FF (%), PCE (%) of Rb₂NaInI₆ absorber layer.

3.5.2. Interactions of conduction band offset (CBO) as well as valence band offset (VBO). Band offsets, the conduction band offset (CBO) along with the valence band offset (VBO), are essential for comprehending the behavior of charge transporters in thin-film perovskite solar energy cells (PSCs), as emphasized in an overview. The CBO denotes the energy disparity across the conductive band of the electron transport layer (CTL) while participating in the absorber material, whereas the VBO indicates the variation in energy concerning the valence band, or band, in the CTL when the perovskite absorbers, furthermore, ref. 96 Clarifies the energy level requirements for efficient hole extraction and electron blocking in perovskite solar cells. To ensure efficient hole extraction from the perovskite absorber to the hole transport layer (HTL), the highest occupied molecular orbital (HOMO) of the perovskite should indeed be slightly higher (less damaging) than the HOMO of the HTL, as this energy level alignment facilitates the movement of holes from a higher state of energy to a minimal energy state. To inhibit the flow of electrons from the perovskite's LUMO (lower unoccupied molecular orbital) through the hole transport layer (HTL), a substantial energy disparity separating the perovskite's LUMO and the power levels of the HTL is advantageous. This electrical barrier effectively obstructs the movement of electrons to the HTL, thereby enhancing charge separation and minimizing recombination losses. Therefore, carefully adjusting the valence band offset (VBO) and conduction band offset (CBO) at the perovskite/HTL interface is indeed a critical factor in achieving efficient hole extraction and overall high performance in perovskite solar cells, making the optimization of these band offsets through material selection and interface engineering a key focus in the field.⁹⁷ The Conduction Band Offset (CBO) and Valence Band Offset (VBO) can be optimized by using the relations,

CBO = X_absorber − X_ETL (9)

VBO = X_HTL − X_absorber + Eg_HTL − Eg_absorber (10)

To determine the conduction band offset (CBO) and valence band offset (VBO), eqn (8) and (9) can be employed. For instance, when considering WS₂, the CBO at the electron transport layer (ETL)/absorber interface can be calculated using these equations. X_absorber − X_ETL = 3.779–3.95 = −0.171 eV = CBO. The computed negative CBO value illustrates a cliff-like power barrier at the interface of the system. VBO is equal to X_HTL − X_absorber + Eg_HTL − Eg_Absorber = 3.6–3.779 + 1.9–1.702 = 0.019 eV. The absorber/HTL contact at the valence band offset. A positive CBO value indicates the existence of a spike-like energetic barrier. The VBO remained consistent at 0.019 eV across all ETLs. The CBO values varied depending on the ETL: −0.171 eV for WS₂, −0.2998 eV for WO₃, −1.598 eV for ZnO, and −0.298 eV for PCBM. The VBO CBO table is attached in the supporting information.

3.5.3. Band diagram. Fig. 9a–d illustrates the energy band diagrams of the simulated FTO/ETL/Rb₂NaInI₆/CBTS solar cells, highlighting the crucial energy level alignment at the ETL-Rb₂NaInI₆ and Rb₂NaInI₆-CBTS interfaces for efficient charge extraction. For effective electron extraction, the electron affinity of the ETL must be higher than that of the Rb₂NaInI₆ absorber, In each diagram, the quasi-Fermi levels for electrons (F_n) and holes (F_p) are shown relative to the conduction band (E_c) and valence band (E_v) energies, indicating the energy landscape for photogenerated carrier transport. The Rb₂NaInI₆ absorber has a band gap of approximately 1.7–1.8 eV (inferred from the diagrams), while the band gaps of the ZnO, WS₂, WO₃, and PCBM ETLs. The varying band alignments and offsets presented in Fig. 9a–d likely contribute to the differences in V_oc and FF observed for the different ETLs, underscoring the importance of selecting an ETL with optimal energy level matching to both the absorber and the front contact for enhanced charge extraction and overall device efficiency. Further analysis of the specific band offsets and their correlation with device performance metrics is necessary to identify the most promising ETL for this solar cell architecture.

Fig. 9 Band diagram illustrating Rb₂NaInI₆, (a) ZnO, (b) WS₂, (c) WO₃, and (d) PCBM are the first four.

3.5.4. Integrate the effects of absorber and then ETL thickness in photovoltaic cells made from perovskite. The thickness of both the absorber layer and the electron transport layer (ETL) is a key determinant of solar cell performance, and overall energy conversion efficiency. Achieving optimal photovoltaic performance requires precise control over the thickness of these layers. In particular, the ETL thickness must be fine-tuned to avoid excessive light absorption and to enhance photon penetration into the absorber layer.⁹⁸ For high-efficiency solar cells, the entire PV structure must be carefully engineered and optimized.⁹⁷ Fig. 10 illustrates the open-circuit voltage (V_oc) variations of contour maps as a function of absorber and electron transport layer (ETL) thickness for solar cells with different ETLs: (a) FTO/ZnO/Rb₂NaInI₆/CBTS, (b) FTO/WS₂/Rb₂NaInI₆/CBTS, (c) FTO/WO₃/Rb₂NaInI₆/CBTS, and (d) FTO/PCBM/Rb₂NaInI₆/CBTS. For FTO/ZnO/Rb₂NaInI₆/CBTS (Fig. 10a), the V_oc ranges from 1.387 V to 1.405 V, with the highest values observed at a thinner absorber layer, below 0.8 μm. This suggests that reducing layer thickness enhances charge separation and minimizes recombination. A similar trend is observed for FTO/WS₂/Rb₂NaInI₆/CBTS (Fig. 10b), where V_oc values upscale from 1.386 V to 1.404 V, though the slightly narrower range compared to ZnO indicates differences in interfacial properties that could affect charge transport efficiency. In the case of FTO/WO₃/Rb₂NaInI₆/CBTS (Fig. 10c), V_oc values range between 1.379 V and 1.397 V. A minimal absorber thickness (0.8 μm) and thinner ETL (0.05–0.06 μm) needed for achieving highest V_oc, which are slightly lower than those for ZnO and WS₂, possibly due to increased charge recombination or less efficient charge extraction at the interface. FTO/PCBM/Rb₂NaInI₆/CBTS (Fig. 10d) exhibits the lowest V_oc values, ranging from 1.371 V to 1.389 V, indicating that charge transport limitations or higher interfacial recombination might be negatively impacting performance. However, the ETL material selection also plays a critical role in device performance, with FTO/ZnO/Rb₂NaInI₆/CBTS and FTO/WS₂/Rb₂NaInI₆/CBTS demonstrating superior V_oc values compared to FTO/WO₃/Rb₂NaInI₆/CBTS and FTO/PCBM/Rb₂NaInI₆/CBTS. Contour maps show that an absorber thickness of 1.2–1.3 μm yields the highest J_sc, highlighting the need for precise thickness optimization. Overall, selecting a suitable ETL and tuning absorber and HTL thicknesses are key to achieving high-efficiency solar cells.

Fig. 10 Contour modeling of the V_oc (V) with the HTL-like (a) ZnO, (b) WS₂, (c) WO₃, (d) PCBM.

Fig. 11a–d presents contour plots illustrating the short-circuit current density (J_sc) of FTO/ETL/Rb₂NaInI₆/CBTS-based solar cells as a function of both absorber (Rb₂NaInI₆) as well as ETL thicknesses for four different ETLs: ZnO, WS₂, WO₃, and PCBM. Across all cases, J_sc consistently increases with absorber thickness, particularly in the 1.2–1.3 μm range, as thicker absorbers enhance light absorption and electron–hole pair generation. For the FTO/ZnO/Rb₂NaInI₆/CBTS (Fig. 11a) structure, J_sc ranges from ∼20.96 to 21.75 mA cm⁻², with slightly higher values observed at thinner ETLs and thicker absorbers. The FTO/WS₂/Rb₂NaInI₆/CBTS (Fig. 11b) structure depicts the highest performance, achieving J_sc values from ∼21.00 to 21.76 mA cm⁻², indicating that WS₂ gives more efficient electron extraction as well as transport. The FTO/WO₃/Rb₂NaInI₆/CBTS (Fig. 11c) configuration shows a similar trend, with J_sc ranging between ∼20.90 and 21.73 mA cm⁻², showing minimal sensitivity to ETL thickness. Meanwhile, the FTO/PCBM/Rb₂NaInI₆/CBTS (Fig. 11d) device displays a wider but slightly lower J_sc range of ∼20.64 to 21.71 mA cm⁻², with the absorber thickness again playing the dominant role. The Rb₂NaInI₆ absorber thickness is the key factor in maximizing J_sc, with WS₂ proving to be the most effective ETL for enhancing current and overall performance. Contour maps show that an absorber thickness of 1.2–1.3 μm consistently yields the highest J_sc, highlighting the need for precise optimization. Overall, selecting a suitable ETL and fine-tuning absorber and HTL thicknesses are vital for achieving high-efficiency solar cells, with WS₂ emerging as the most promising choose.

Fig. 11 Contour modeling of the J_sc (mA cm⁻²) with the HTL-like (a) ZnO, (b) WS₂, (c) WO₃, (d) PCBM.

The contour plots in Fig. 12a–d illustrate the fill factor (FF) dependency on the thickness of the ETL and absorber layer for FTO/ETL/Rb₂NaInI₆/CBTS solar cells. The FF, which represents the efficiency of charge extraction and recombination losses, exhibits the highest values for thinner absorber layers (0.5–0.6 μm) across all SCs, indicating improved charge transport with reduced recombination. Among the devices, WS₂-based SC (Fig. 12c) demonstrates the best FF performance, peaking at 85.41%, followed by PCBM/Rb₂NaInI₆/CBTS (Fig. 12d) (85.38%), ZnO/Rb₂NaInI₆/CBTS (Fig. 12a) (85.31%), and WO₃/Rb₂NaInI₆/CBTS (Fig. 12c) the lowest (85.08%). ETL thicknesses between 0.05–0.07 μm are optimal, as thicker layers increase resistive losses and hinder charge transport. FF declines when absorber thickness exceeds 0.8 μm, likely due to higher recombination and reduced carrier collection. WS₂ emerges as the most effective ETL for high FF, with PCBM also showing competitive performance.

Fig. 12 Contour modeling of the FF(%) with the HTL-like (a) ZnO, (b) WS₂, (c) WO₃, (d) PCBM.

Fig. 13 shows that the contour plots as a function of Power Conversion Efficiency (PCE) (%) both absorber (Rb₂NaInI₆) and Electron Transport Layer (ETL) thickness for four different solar cell configurations: FTO/ZnO/Rb₂NaInI₆/CBTS, FTO/WS₂/Rb₂NaInI₆/CBTS, FTO/WO₃/Rb₂NaInI₆/CBTS, and FTO/PCBM/Rb₂NaInI₆/CBTS. Across all four architectures, PCE increases noticeably with increasing absorber thickness, particularly in the range of 1.2–1.3 μm, due to enhanced photon absorption and improved generation of charge carriers. The FTO/ZnO/Rb₂NaInI₆/CBTS (Fig. 13a) device exhibits PCE values ranging from 26.49% to 27.10%, peaking with a thinner ZnO layer (0.05–0.07 μm) and a thicker absorber. The FTO/WS₂/Rb₂NaInI₆/CBTS (Fig. 13b) configuration, although not reaching the absolute highest value, achieves a very consistent performance across thickness variations, with PCEs from 26.19% to 26.90%, highlighting WS₂ as an excellent ETL candidate due to its stable and efficient electron transport. The FTO/WO₃/Rb₂NaInI₆/CBTS (Fig. 13c) device reaches up to 27.02% PCE, again performing best with absorber thickness of 1.2–1.3 μm and thin WO₃ layers (0.05–0.07 μm). In contrast, the FTO/PCBM/Rb₂NaInI₆/CBTS (Fig. 13d) structure shows a slightly lower performance, ranging from 25.34% to 26.50%, with its optimum appearing at a somewhat thicker ETL (0.07–0.1 μm) and a comparable absorber thickness, suggesting a more nuanced relationship between layer thickness and efficiency. Overall, this analysis underscores the critical role of optimizing absorber thickness to enhance light absorption and subsequently improve solar cell efficiency, while also indicating that careful selection and thickness control of the ETL are necessary for maximizing charge transport and overall device performance.^99,100

Fig. 13 Contour mapping of PCE (%) when ETL is (a) ZnO, (b) WS₂, (c) WO₃, (d) PCBM.

3.5.5. Combine effect of shallow acceptor density and defect density. In perovskite solar cells, absorber defect density (N_t) refers to the density of traps or imperfections within the perovskite layer that can capture and recombine charge carriers. In contrast, shallow acceptor density (N_a) refers to intentional p-type doping that adds mobile holes to facilitate the modulation of the electrical characteristics of the material that absorbs radiation. When you combine these two factors, the interaction between them strongly influences the device's key performance metrics: open-circuit voltage (V_oc), short-circuit current density (J_sc), fill factor (FF), and overall power conversion efficiency (PCE).^98,99 The combined effect of absorber defect density and shallow acceptor density on photovoltaic (PV) performance is illustrated in Fig. 14–17. In these figures, the defect density varied from 10¹² to 10¹⁷, while the acceptor density exhibited a similar trend.

The four 3D surface plots (Fig. 14a–d) illustrate the combined impact of shallow acceptor density (N_A) and absorber defect density (N_t) on the open-circuit voltage (V_oc) of perovskite solar cells (PSCs) employing different electron transport layers (ETLs): ZnO-based device (Fig. 14a), WS₂-based device (Fig. 14b), WO₃-based device (Fig. 14c), and PCBM-based device (Fig. 14d). Across all ETLs, a clear trend emerges: as the absorber defect density (N_t) increases, V_oc consistently decreases due to enhanced non-radiative recombination, which reduces the quasi-Fermi level splitting and ultimately lowers the open-circuit potential. In contrast, increasing the shallow acceptor density (N_A) generally leads to an increase in V_oc, thanks to the stronger built-in electric field that improves charge separation and reduces recombination. The optimal condition for maximizing V_oc is a low N_t (<10¹³ cm⁻³) combined with intermediate-to-high N_A (10¹⁵–10¹⁶ cm⁻³). At very high N_A levels, V_oc may plate Pt or slightly decline due to adverse effects such as increased Ptger recombination or impaired carrier mobility. Specifically, for the ZnO-based device (Fig. 14a), V_oc ranges from approximately 1.21 V to 1.575 V; for the WS₂-based device (Fig. 14b), it spans about 1.164 V to 1.542 V; for the WO₃-based device (Fig. 14c), it ranges from ∼1.190 V to 1.437 V; and for the PCBM-based device (Fig. 14d), it shows the lowest maximum V_oc, between ∼1.251 V and 1.416 V. This analysis suggests that ZnO and WS₂ are the most favorable ETLs for achieving high open-circuit voltages, especially in scenarios of minimal number of defects and enhanced doping.

Fig. 14 Three-dimensional surface plot of both open-circuit voltage (V_oc) being a function of shallow acceptor density (N_A) in addition to absorber defect density (N_D) for perovskite solar cells (PSCs) utilizing (a) ZnO, (b) WS₂, (c) WO₃, and (d) PCBM electron transport layers (ETLs).

Fig. 15a–d illustrates how shallow acceptor density (N_A) and absorber defect density (N_t) influence the short-circuit current density (J_sc) of PSCs across four configurations. Overall, J_sc shows a strong negative dependence on absorber defect density (N_t), as higher defect densities introduce recombination centers that reduce the collection of photogenerated carriers. Unlike V_oc, J_sc is less sensitive to variations in shallow acceptor density (N_A), though moderate improvements are sometimes observed at intermediate to high N_A, likely due to better charge extraction. However, excessively high doping levels may slightly decrease J_sc, potentially becPtse of enhanced carrier scattering or mobility limitations. For the ZnO-based device (Fig. 15a), J_sc ranges from approximately 19 to 20 A m⁻², with similar values seen for the WS₂-based device (Fig. 15b), WO₃-based device (Fig. 15c), and PCBM-based device (Fig. 15d). Across all ETLs, minimizing absorber defect density emerges as the most critical factor in maximizing J_sc, underscoring the importance of material quality and defect passivation strategies in high-performance PSC designs.

Fig. 15 Three-dimensional surface plot of both short-circuits voltage (J_sc) being a function of shallow acceptor density (N_A) in addition to absorber defect density (N_D) for perovskite solar cells (PSCs) utilizing (a) ZnO, (b) WS₂, (c) WO₃, and (d) PCBM electron transport layers (ETLs).

The combined effect of absorber defect density (N_t) and shallow acceptor density (N_A) on the fill factor (FF) of perovskite photovoltaic cells, shown in Fig. 16a–d, demonstrates that FF is moderately sensitive to both parameters, though less dramatically than V_oc. The analysis of the fill factor (FF) plots reveals that the ZnO-based device exhibits the highest FF values (∼90–91%) when the shallow acceptor density (N_A) is in between 10¹⁶ and 10¹⁷ cm⁻³, provided the absorber defect density (N_t) remains below 10¹⁵ cm⁻³. This indicates that the ZnO ETL can maintain excellent carrier extraction and minimize recombination even at relatively high doping levels, as long as defect densities are kept low. In contrast, WS₂ (Fig. 16b), WO₃- (Fig. 16c), and PCBM-based (Fig. 16d) devices show initial FF values above ∼91%, which decline to ∼67–73% at the highest N_t and N_A regions. This decline can be attributed to enhanced recombination losses and poor charge extraction when the absorber layer has excessive defects or over-doping, both of which disrupt the optimal electric field and series resistance balance. Notably, the WS₂ and PCBM devices seem to retain slightly higher FF across a broader defect and doping range compared to ZnO and WO₃, suggesting their better resilience against performance degradation. Overall, these plots indicate that while FF is fairly robust under low to moderate absorber imperfections, it degrades significantly when both N_t and N_A exceed critical thresholds, aligning with known recombination and resistive loss mechanisms.

Fig. 16 Three-dimensional surface plot of both fill factor (FF) being a function of shallow acceptor density (N_A) in addition to absorber defect density (N_D) for perovskite solar cells (PSCs) utilizing (a) ZnO, (b) WS₂, (c) WO₃, and (d) PCBM electron transport layers (ETLs).

The 3D contour plots in Fig. 17a–d show how the power conversion efficiency (PCE) of perovskite solar cells varies with absorber shallow acceptor density as well as absorber defect density for four different electron transport layers (ETLs): ZnO, WS₂, WO₃, and PCBM. Across all four cases, a clear and consistent trend emerges—as the absorber defect density increases (moving upward in the plots), the PCE decreases sharply, underscoring the critical importance of minimizing defects in the absorber layer to maintain high photovoltaic performance. For ZnO (Fig. 17a), the highest PCE (∼29–30%) is observed in a broad region where the shallow acceptor density ranges from approximately 10¹⁵ to 10¹⁶ cm⁻³ under low defect densities; however, efficiency deteriorates rapidly as defects rise. WS₂ (Fig. 17b) exhibits the most extensive high-efficiency region (reaching ∼30–31%), again centered around a shallow acceptor density of 10¹⁵–10¹⁶ cm⁻³, suggesting that WS₂ offers slightly better tolerance and potential compared to ZnO. WO₃ (Fig. 17c) shows a narrower optimal region, with maximum PCE (∼27–28%) shifting slightly towards higher shallow acceptor densities (around 10¹⁶ cm⁻³), while PCBM (Fig. 17d) demonstrates the smallest high-efficiency window and the lowest maximum PCE (∼26–27%), indicating it is less effective in this material system.

Fig. 17 Three-dimensional surface plot of both power conversion efficiency being a function of shallow acceptor density (N_A) in addition to absorber defect density (N_D) for perovskite solar cells (PSCs) utilizing (a) ZnO, (b) WS₂, (c) WO₃, and (d) PCBM electron transport layers (ETLs).

The results highlight the importance of maintaining low absorber defect density to achieve high power conversion efficiency. Optimizing shallow acceptor density is also critical, with the optimal range varying slightly based on the chosen ETL. Among the materials studied, WS₂ emerged as the most effective ETL, followed by ZnO, while WO₃ and PCBM were less favorable. These insights provide valuable guidance for material selection and device design, emphasizing the strong correlation between absorber quality and ETL compatibility in enhancing perovskite solar cell efficiency.

3.5.6. Effectiveness of HTL alongside absorber thickness. Fig. 18 presents an in-depth analysis of the impact of Rb₂NaInI₆ and CBTS thickness variations on the photovoltaic performance of a solar cell employing different Electron Transport Layers (ETLs): ZnO, WS₂, WO₃, and PCBM. The study evaluates key parameters, including Power Conversion Efficiency (PCE), Fill Factor (FF), Short-circuit Current Density (J_sc), and Open-circuit Voltage (V_oc), providing valuable insights into absorber and buffer layer optimization for enhanced device performance. Here, the absorber thickness varied from 0.7 to 1.3 μm, while the CBTS thickness varied from 0.1 to 0.6 μm. In Fig. 18a, V_oc decreases with increasing Rb₂NaInI₆ absorber thickness, falling from approximately 1.413 V at 0.7 μm to around 1.37 V at 1.2 μm. Among the ETLs, ZnO and WS₂-based devices exhibit better voltage retention, with V_oc stabilizing near 1.39 V, while PCBM-based cells show the lowest V_oc, indicating greater recombination losses at higher thicknesses due to elongated charge transport paths alternatively, J_sc steadily improves as the absorber layer becomes thicker, increasing from about 20.8 mA cm⁻² to 21.6 mA cm⁻² across all configurations. This enhancement reflects more effective light harvesting and photocarrier generation with thicker active layers. In terms of fill factor (FF), the values remain relatively stable, fluctuating between 89.0% and 90.5% for ZnO, WS₂, and WO₃-based devices, which indicates minimal impact of absorber thickness on charge extraction efficiency for these ETLs. However, the PCBM-based device displays a slight FF degradation, decreasing from approximately 89.5% to 88%, likely due to increased internal resistive losses. Lastly, the power conversion efficiency (PCE) shows a noticeable enhancement as the Rb₂NaInI₆ thickness increases from 0.7 μm to 1.2 μm, where WS₂-based cells reach up to ∼27.0%. At the same time, ZnO-based configurations maintain slightly better overall efficiency, reinforcing ZnO's consistent top performance. Meanwhile, PCBM-based cells exhibit the lowest PCE, affirming their weaker overall performance. The initial increase in PCE is due to enhanced absorption of sunlight and carrier production; however, the enhancement rate diminishes at greater thicknesses, indicating a saturation threshold beyond which further width results in decreasing returns.

Fig. 18 The influence of modifications in (a) Rb₂NaInI₆ and (b) CBTS thickness demonstrates the effects on V_oc (V), J_sc (mA cm⁻²), FF (%), and PCE (%).

In Fig. 18b, V_oc remains nearly constant across all CBTS thicknesses (0.1 μm to 0.6 μm), with ZnO and WS₂-based devices maintaining values around 1.398 V, followed by WO₃ at approximately 1.39 V, while PCBM-based cells continue to exhibit the lowest V_oc near 1.385 V, confirming the stability of voltage across thicknesses but highlighting PCBM's relatively weaker performance due to its higher recombination tendency. In contrast, J_sc consistently improves with increasing CBTS thickness, rising from ∼21.35 mA cm⁻² at 0.1 μm to ∼21.45 mA cm⁻² at 0.6 μm for WS₂, which outperforms the others, while ZnO and WO₃ show moderate improvements and PCBM lags slightly behind, reinforcing the significance of thicker absorbers in enhancing photocurrent through improved light absorption. Regarding FF, all devices maintain high fill factors ranging from approximately 89.8% to 90.1%, with only slight fluctuations as the CBTS thickness increases, indicating efficient charge transport and minimal resistive losses across all ETLs. Lastly, the PCE increases modestly with CBTS thickness, where ZnO-based devices show the highest performance peaking at ∼27.0%, followed by WS₂ (∼26.8%), WO₃ (∼26.6%), and PCBM (∼26.3%). This gradual improvement in efficiency stems from enhanced J_sc while V_oc and FF remain stable, demonstrating that optimizing CBTS thickness primarily benefits current generation without compromising charge extraction or voltage stability.

3.5.7. Dominance of series (R_s) and shunt (R_sh) resistance. Fig. 19 examines the impact of series resistance (R_s) and shunt resistance (R_sh) on the photovoltaic performance of a solar cell utilizing different ETLs (ZnO, WS₂, WO₃, and PCBM). The series resistance (R_s) significantly impacts the efficiency of photovoltaic cells. In perovskite solar cells, series resistance arises from the resistance of various layers, including the ETL, HTL, absorber, and both the front and back electrodes.¹⁰¹ In Fig. 19a, as R_s increases from 1 Ω cm² to 6 Ω cm², V_oc exhibits a slight reduction, with ZnO and WS₂ maintaining the highest values (∼1.395 V initially, dropping by ∼0.7%), while PCBM shows the lowest V_oc (∼1.385 V, decreasing by ∼0.8%). J_sc remains nearly constant across all R_s values, suggesting minimal impact on charge generation. An increase in R_s significantly reduced both FF and PCE, similar to trends in inorganic and organic solar cells.¹⁰² FF declines significantly, with an approximate 6% drop for ZnO and WS₂, while PCBM experiences a steeper decline (∼8%). This reduction in FF indicates that an increasing R_s obstructs efficient charge extraction, raising resistive losses. Consequently, PCE follows a similar decreasing trend, with ZnO and WS₂ showing a total drop of ∼2%, whereas PCBM undergoes a more significant decrease (∼3%), reinforcing that excessive series resistance hinders the overall power conversion process. The reduction of FF and PCE with increasing R_s is linked to the rise in power loss.⁵⁹ Selecting the minimum R_s helps reduce its impact on PCE and FF, ultimately enhancing solar cell efficiency.

Fig. 19 Effectiveness variations in (a) Shunt resistance (R_s) and (b) series resistance (R_sh) both on V_oc (V), J_sc (mA cm⁻²), FF (%), and PCE (%).

Shunt resistance (R_sh) plays an important role in determining the efficiency of photovoltaic cells. Increased R_sh often leads to power losses, typically due to issues during manufacturing. Power losses in solar cells are caused by low shunt resistance, as it offers an alternative current path for the light-induced current.¹⁰³ In Fig. 19b, as R_sh grows from 10 Ω cm² to 10⁶ Ω cm², V_oc initially rises sharply from ∼0.3 V to ∼1.39 V before stabilizing, indicating a significant reduction in leakage currents. J_sc exhibits a moderate increase, with ZnO and WS₂ reaching around 21.36 mA cm⁻², while PCBM remains slightly lower (∼21.30 mA cm⁻²). FF improves substantially, rising by ∼10% for all ETLs as R_sh increases, highlighting the suppression of unwanted recombination pathways. As a result, PCE experiences a sharp enhancement, increasing from ∼5% to ∼25% before stabilizing, emphasizing the crucial role of high R_sh in achieving efficient solar cell operation. Among the ETLs, WS₂ consistently performs the best, maintaining superior values in V_oc, J_sc, FF, and PCE, while PCBM demonstrates the weakest performance across all parameters. These findings confirm that minimizing R_s while maximizing R_sh is critical for enhancing device efficiency, with WS₂ emerging as the most promising ETL for stable as well as high-performance solar cell applications. The Shockley equation, outlined in eqn (10) and (11), describes the expected current–voltage (I–V) behaviors of a solar cell under optimal one-sun illumination.¹⁰⁴

In this case: q_e = elementary charge, J_PH = photocurrent density, J₀ = reverse saturation current, n = ideality factor, T_e = 298 K, k = Boltzmann constant. To achieve optimal device performance, it is crucial to avoid localized shunting and current leakage.¹⁰⁵

3.5.8. Guide on temperature effect on PSCs. As the operating temperature increases from 300 K to 425 K, all four photovoltaic performance parameters—V_oc, J_sc, FF, and PCE—undergo significant changes, influenced by the choice of electron transport layer (ETL). The impact of temperature on these SCs is shown in Fig. 20a–d. This range was selected to simulate real-world operational conditions, as solar cells often experience elevated temperatures under sunlight. The observed behavior is attributed to temperature-induced changes in material properties – specifically, the reduction in bandgap and increased carrier recombination at higher temperatures, which can affect key performance metrics such as V_oc and FF. The mechanism is primarily related to enhanced phonon interactions, increased intrinsic carrier concentration, and possible interface degradation at elevated temperatures, which collectively influence the device efficiency. This minor improvement in J_sc can be attributed to enhanced carrier mobility and reduced recombination resistance at elevated temperatures, though the gain is insufficient to offset the losses in other parameters. The fill factor (FF), which reflects the quality of charge extraction and internal resistive losses, steadily decreases for all devices as temperature rises. ZnO-based solar cells, which begin with a high FF of approximately 90%, drop significantly to around 83%, indicating increased series resistance and reduced diode ideality at higher thermal loads. WS₂, WO₃, and PCBM-based devices also experience FF degradation, though the decline is relatively less severe in PCBM (from ∼88% to ∼85%), implying a slightly better thermal resilience in charge transport mechanisms. Consequently, the power conversion efficiency (PCE) drops across all ETLs, driven by the combined effect of reduced V_oc and FF. ZnO-based solar cells, which initially show the highest PCE of nearly 26% at 300 K, decline to around 22% at 425 K. Similarly, PCBM-based devices drop from ∼25.2% to ∼21%. Despite ZnO's initial superior performance, it is more susceptible to thermal degradation, especially in FF and V_oc, whereas PCBM, though starting with slightly lower values, shows relatively stable thermal behavior. This highlights the importance of selecting ETL materials with better thermal stability to ensure reliable solar cell performance under varying environmental conditions.

Fig. 20 The temperature variation from 300 K toward 425 K is represented via (a) V_oc (V), (b) J_sc (mA cm⁻²), (c) FF (%), and (d) PCE (%).

3.5.9. Control of capacitance and Mott–Schottky (1/C²). Fig. 21 provides an in-depth analysis of the electrostatic and photophysical parameters for the optimized Rb₂NaInI₆-based solar cells using various ETLs—ZnO, WS₂, WO₃, and PCBM. The Mott–Schottky (M−S) method, based on capacitance–voltage (C–V) analysis, enables the determination of the built-in voltage (V_bi) and charge carrier density (N_d). This approach is widely utilized in standard semiconductor devices, particularly those with p–n junctions and semiconductor/metal interfaces containing space charge regions and fixed depletion layers. The junction capacitance for each unit area (C) is calculated using eqn (12).

Fig. 21 The improved device's (a) capacitance, (b) Mott–Schottky (1/C²), (c) generation rate, as well as (d) recombination rate.

In this context, ε₀ is the vacuum permittivity, ε_r is the dielectric constant of the donor material, q represents the electronic charge, and V indicates the applied voltage.⁷⁷ In Fig. 21a, the capacitance–voltage (C–V) characteristics reveal an increasing trend with forward bias for all devices, indicating enhanced charge accumulation at higher voltages. Among the ETLs, PCBM-based devices exhibit the highest capacitance, peaking at approximately 10.8 nF cm⁻² at 0.8 V, while ZnO-based devices show the lowest capacitance, around 8.5 nF cm⁻². This suggests a broader depletion width in ZnO, which is beneficial for efficient charge separation and suppression of interface recombination. Fig. 21b displays the Mott–Schottky (1/C² – V) characteristics. All devices exhibit a typical linear inverse square capacitance profile with increasing voltage. The ZnO-based cell demonstrates the steepest slope, indicating a lower carrier concentration and wider depletion region, which is aligned with its enhanced V_oc performance. In contrast, PCBM presents a flatter slope, implying higher doping concentration and possibly inferior junction quality.

3.5.10 Dominance generation and recombination rate. Fig. 21c shows the generation and recombination rates of these four PSCs. As part of the carrier production process, an electron moves from the valence band to the conduction band, leaving behind a hole in the valence band and forming an electron–hole pair.⁵⁹ The maximum generation rates for both designs occur at approximately 1.1–1.2 mm, as depicted in Fig. 21c. To calculate G(x), the electron–hole pair production, SCAPS-1D is employed along with the incoming photon flux, Nphot (l, x), as given in eqn (14).

G(l, x) = a(l, x). Nphot(l, x) (14)

In contrast to the generation process, recombination involves the interaction and separation of electrons and holes within the conduction band. In a solar cell, the recombination rate is influenced by factors such as the density and lifetime of the charge carriers.¹⁰⁶ A decrease in electron–hole recombination is observed due to the presence of defect states in the absorber layer. The formation of energy states within the solar cell impacts the distribution of electron–hole recombination. The recombination rate distribution becomes irregular due to the presence of defects and grain boundaries.¹⁰⁷ The ZnO-based device exhibits the highest generation rate, peaking at approximately 6.9 × 10²¹ cm⁻³ s⁻¹, followed by WS₂ and PCBM. This superior performance of ZnO highlights its strong ability to support efficient photogeneration under optimized conditions. Fig. 21d depicts the recombination rate across the device thickness. Here, ZnO and WS₂ again outperform PCBM and WO₃, with ZnO-based devices showing a minimal recombination peak of approximately 2.1 × 10¹⁹ cm⁻³ s⁻¹ near the rear interface, while PCBM-based devices exhibit a significantly higher peak of about 9.5 × 10¹⁹ cm⁻³ s⁻¹, indicating greater carrier losses. This stark contrast underscores ZnO's superior passivation capabilities and reduced defect density at interfaces. Overall, ZnO emerges as the most favorable ETL, offering the best balance of low capacitance (indicating wide depletion width), high carrier generation, and minimal recombination. These electrostatic and optical advantages explain the consistently superior device metrics (V_oc, J_sc, FF, and PCE) observed in earlier figures at optimized absorber and HTL thicknesses.

3.5.11. Power of JV characteristics and QE curve. Fig. 22a and b presents the J–V characteristics and Quantum Efficiency (QE) spectra of FTO/ETL/Rb₂NaInI₆/CBTS-based solar cells using four different Electron Transport Layers (ETLs): ZnO, WS₂, WO₃, and PCBM. The J–V curves show that all configurations—FTO/ZnO/Rb₂NaInI₆/CBTS, FTO/WS₂/Rb₂NaInI₆/CBTS, FTO/WO₃/Rb₂NaInI₆/CBTS, and FTO/PCBM/Rb₂NaInI₆/CBTS—exhibit similar short-circuit current densities (J_sc) in the range of approximately 21–22 mA cm⁻², indicating efficient light harvesting and comparable photogenerated charge carrier collection. However, there is a clear variation in open-circuit voltage (V_oc), with PCBM yielding the maximum V_oc (∼1.40 V), followed up WO₃ (∼1.39 V), ZnO (∼1.38 V), and WS₂ (∼1.38 V), demonstrating that the ETL significantly influences the built-in electric field and recombination losses. The fill factor (FF) appears similar across the devices based on the curve shape, suggesting comparable series and shunt resistance characteristics. While PCE is not plotted directly, it is inferred that FTO/PCBM/Rb₂NaInI₆/CBTS and FTO/WO₃/Rb₂NaInI₆/CBTS structures would exhibit higher efficiencies due to their superior V_oc. The QE spectra further reinforce these observations, showing nearly identical spectral responses across all devices with high quantum efficiencies (∼100%) in the visible range (400–700 nm) and a sharp cutoff near 720 nm, indicating the absorber's bandgap limit. These similar QE responses confirm that the absorber, Rb₂NaInI₆, governs the light absorption and charge generation, while the ETL predominantly impacts charge transport and extraction, particularly influencing V_oc. Overall, the findings underscore that while J_sc and QE remain largely unaffected by the ETL type, PCBM and WO₃ offer superior V_oc performance, making them the most favorable ETL candidates in this device configuration for enhanced power conversion efficiency.

Fig. 22 The (a) J–V characteristics as well as (b) QE curve of the twin PSCs.

3.5.12. Nyquist plot analysis. Fig. 23 presents the Nyquist plot derived from electrochemical impedance spectroscopy (EIS) to compare the interfacial characteristics of various electron transport layer (ETL) materials-ZnO, WS₂, WO₃, and PCBM-used in Rb₂NaInI₆-based perovskite solar cells. The plot displays the imaginary component of impedance (−Z_im) versus the real component (Z_re), allowing a clear visualization of charge transport resistance and recombination behavior. Among the materials studied, ZnO and WS₂ exhibit the largest semicircular arcs, with ZnO reaching a peak-Z_im of approximately 240 000 Ω cm², indicating higher interfacial resistance and lower charge recombination rates. This behavior is attributed to their wide bandgaps, favorable conduction band alignment, and superior electron mobility, which collectively enhance their charge-blocking capability and minimize leakage current.^108,109 On the other hand, WO₃ and PCBM show significantly smaller arc diameters, peaking below 100 000 Ω cm², which suggests faster charge transfer kinetics but possibly at the expense of increased recombination losses.¹¹⁰ The pronounced impedance of ZnO and WS₂ is indicative of their strong recombination suppression abilities and higher series resistance, which may translate into higher open-circuit voltage (V_oc) and device stability.¹⁰⁸ Although PCBM remains a widely adopted ETL for its excellent electronic properties and compatibility, its lower impedance values may correspond to limited recombination control in this specific device architecture.¹¹⁰

Fig. 23 Nyquist plot of the diverse HTL materials derived from the construction using the absorber Rb₂NaInI₆.

3.6. Comparative evaluation with previous works

Table 3 compares previously reported photovoltaic device structures and the optimized configurations proposed in this work. The comparative performance evaluation of the proposed device architectures based on the Rb₂NaInI₆ absorber layer demonstrates a remarkable advancement over previously reported structures. As shown in the table, prior studies employing various absorbers such as Cs₂BiAgI₆, MAPbI₃, Cs₂CuBiBr₆, and Cs₂AgBiBr₆ paired with different ETL/HTL combinations exhibited power conversion efficiencies (PCE) ranging from 16.23% to 23.05%. Among these, the structure ITO/TiO₂/CCSCNCs/Cu₂O/Pt reported the highest efficiency (23.05%) with a short-circuit current density (J_sc) of 25.83 mA cm⁻² and a fill factor (FF) of 82.44%. In contrast, the present work introduces novel FTO-based configurations with Rb₂NaInI₆ as the absorber and CBTS/Pt as the back contact, integrated with varying ETL materials (PCBM, WS₂, WO₃, and ZnO). These optimized designs achieved open-circuit voltages (V_oc) up to 1.39 V, J_sc values up to 21.39 mA cm⁻², and FF exceeding 89%, resulting in an impressive peak PCE of 26.84%. Notably, the configuration FTO/ZnO/Rb₂NaInI₆/CBTS/Pt delivered the best performance (V_oc = 1.39 V, J_sc = 21.39 mA cm⁻², FF = 89.83%, PCE = 26.84%), significantly outperforming all previously reported structures. This enhancement is attributed to the superior band alignment, minimal recombination losses, and improved charge extraction at the interfaces facilitated by the optimized Rb₂NaInI₆ absorber and ZnO ETL. Thus, this work highlights the potential of lead-free, double perovskite-based solar cells in achieving high efficiency while maintaining environmental stability and reduced toxicity, thereby paving the way for scalable and sustainable photovoltaic technologies.

Table 3 Comparative evaluation of the performance of diverse photovoltaic cell architectures with varied absorber and ETL/HTL the layout

Structure	Voc (V)	Jsc (mA cm^−2)	FF (%)	PCE (%)	Ref.
FTO/TiO2/Cs2AgBiBr6/spiro-OMeTAD/Au	1.511	3.89	51.76	3.04	52
ITO/SnO2/Cs2AgBiBr6/spiro-OMeTAD/Au	0.92	11.4	60.93	6.37	54
Glass/FTO/TiO2/Cs2AgBiBr6/Cu2O/Au	1.5	11.45	42.1	7.25	56
FTO/TiO2/Cs2AgSbBr6/spiro-OMeTAD/Ag	0.94	22.49	50.2	10.69	111
FTO/Ws2/Cs2CuBiBr6/spiro-OMeTAD/Ag	0.60	34.59	67.36	14.08	111
Ohmic contact/spiro-OMeTAD/Cs2BiAgI6/TiO2/SnO2: F(ZnO2)/ohmic contact	1.18	16.2	80.20	15.90	112
FTO/PCBM/Cs2BiAgI6/PTAA/Pt	1.08	19.94	74.87	16.23	113
ITO/WO3/MAPbI3/Mg-CuCrO2/Pt	0.89	23.24	83.95	17.53	114
ITO/WS2/Cs2CuBiBr6/CBTS/Ni	0.71	35.63	77.57	19.70	115
ITO/TiO2/Cs2BiAgI6/CBTS/Pt	1.08	23.80	83.61	21.55	116
FTO/TiO2/CCSCNCs/Cu2O/Pt	1.30	21.35	83.02	23.07	117
FTO/AZnO/Cs2AgBiBr6/CNTS/Pt	1.37	21.38	79.93	23.50	118
ITO/WS2/Cs2KIrCl6/V2O5	1.52	14.809	85.12	19.18	119
ITO/ZnO/Cs2KIrCl6/V2O5	1.54	15.479	89.32	21.30	119
FTO/PCBM/Rb2NaInI6/CBTS/Pt	1.38	21.34	88.92	26.26	This work
FTO/WS2/Rb2NaInI6/CBTS/Pt	1.39	21.42	89.03	26.61	This work
FTO/WO3/Rb2NaInI6/CBTS/Pt	1.39	21.37	88.95	26.73	This work
FTO/ZnO/Rb2NaInI6/CBTS/Pt	1.39	21.39	89.83	26.84	This work

---

4. Conclusion

In this study, a comprehensive investigation into the potential of the lead-free double perovskite Rb₂NaInI₆ was carried out using a dual approach involving first-principles Density Functional Theory (DFT) calculations and one-dimensional solar cell simulation via SCAPS-1D. The DFT analysis confirmed that Rb₂NaInI₆ crystallizes in a stable cubic Fm̄3m phase, exhibiting a moderate bandgap of 0.73 eV (via GGA functional) and 1.70 eV (via TB-mBj functional), which, along with strong In-5s and I-5p hybridization in the density of states, ensures favorable charge transport characteristics. The material also demonstrated high absorption coefficients in the visible to ultraviolet (UV) range-peaking around 12 eV-low reflectivity (<30%), and a high dielectric constant, all of which suggest its promise for efficient light harvesting and minimal optical loss in photovoltaic and optoelectronic applications. On the simulation side, SCAPS-1D modeling complemented the theoretical predictions by revealing a wider effective indirect bandgap of 1.702 eV, optimal for solar absorption. Charge distribution analysis highlighted sodium atoms as dominant carriers, with strong covalent In–I bonds and ionic Rb–I/Na–I interactions contributing to the overall material stability. Among 36 device configurations studied using six different ETL and HTL combinations, CBTS emerged as the best-performing HTL when coupled with ZnO as the ETL. The structure FTO/ZnO/Rb₂NaInI₆/CBTS/Pt achieved a peak PCE of 26.84%, slightly surpassing the WO₃-based counterpart. It was also observed that increasing the absorber thickness significantly enhanced performance, whereas ETL thickness had a negligible impact. The simulation further validated device robustness through Nyquist plots, J–V, and QE curves under various operating conditions, including thermal variation and defect-induced recombination. These results affirm the viability of Rb₂NaInI₆ as a lead-free and environmentally benign material and underscore its capability to deliver high photovoltaic efficiency in tandem solar architectures. This combined theoretical–simulation approach confirms Rb₂NaInI₆ as a strong candidate for high-efficiency, environmentally friendly, lead-free perovskite solar cells, offering a promising pathway toward sustainable and scalable photovoltaic technologies.

Author contributions

Akram Hossan Mahedi: investigation, methodology, data curation, conceptualization, writing original manuscript; Salah Uddin: investigation, methodology, data curation, review-editing; Mohammad Yasin Hayat Khan: formal analysis, software, methodology, review-editing; Md. Tarekuzzaman: formal analysis, software, conceptualization, review-editing; O. Alsalmi: formal analysis, methodology, data curation, review-editing; Md. Rasheduzzaman: formal analysis, validation, review-editing; S. M. G Mostafa: formal analysis, data curation, review-editing, M. Moazzam Hossen: formal analysis, data curation, review-editing, Md. Zahid Hasan: formal analysis, validation, supervision, review-editing.

Conflicts of interest

There is no conflict to declare.

Data availability

Data will be made available on request.

Acknowledgements

The authors extend their appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4300099GSSR11.

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