Exploring optoelectronic and photovoltaic properties of Be₃MF₃ (M = P, As, Sb) perovskites via machine learning and numerical simulation

Abstract

This study presents a comprehensive Density Functional Theory (DFT) analysis of the structural, mechanical, electrical, and optical characteristics of cubic Be₃MF₃ (M = P, As, Sb) perovskites, highlighting its potential for optoelectronic applications. The computed elastic constants adhere to the Born stability criterion, affirming mechanical resilience and ductility. Analyses of band structure and density of states reveal that all Be₃MF₃ compounds exhibit indirect band gaps between 2.0 and 3.3 eV, categorizing them as wide-bandgap semiconductors suitable for high-power and UV optoelectronic applications. Optical investigations demonstrate significant absorption, especially in Be₃SbF₃, highlighting its promise as a lead-free absorbing material for solar cell applications. Device-level performance was modeled utilizing SCAPS-1D, incorporating Be₃SbF₃ as the absorber across diverse configurations with four electron transport layers (In₂S₃, TiO₂, Mg:ZnO, IGZO) and six-hole transport layers (Cu₂O, MoO₃, P3HT, NiO, PTAA, V₂O₅). The best configuration, Al/FTO/In₂S₃/Be₃SbF₃/Cu₂O/Ni, attained a maximum power conversion efficiency (PCE) of 18.28%, with open-circuit voltage (V_OC) of 1.79 V, short-circuit current (J_SC) of 11.04 mA/cm², and a fill factor (FF) of 89.6%. Subsequent examinations of defect density, interface states, thickness variation, and temperature impacts clarify charge transport and recombination kinetics. A Random Forest machine learning (ML) model was created to forecast device performance, with high accuracy (R² = 0.987, MSE = 0.00305, MAPE = 0.075) and pinpointing bandgap energy and interface defect density as critical limiting factors. The integrated DFT-SCAPS 1D-ML framework identifies Be₃SbF₃ perovskites as viable, lead-free materials for advanced photovoltaic and optoelectronic technologies.

Introduction

Addressing pollution, climate change, and energy insecurity requires a thorough shift to renewable energy systems, especially solar, wind, and hydropower resources. Developing countries receive sun energy much exceeding the world’s total coal reserves and current global consumption, highlighting solar power’s unparalleled potential^1,2. As a result, photovoltaic (PV) technologies have emerged as a central focus of research and industry advancement. Recent advancements in perovskite-based solar cells have intensified this trend owing to their exceptional efficiency, cost-effectiveness, and adjustable material characteristics^3,4. Perovskite solar cells (PSCs) are chiefly classified into A₃BX₃ and ABX₃ structural kinds, each providing distinct advantages regarding efficiency, stability, and scalability⁵. The ABX₃ structure, consisting of metal cations (A, B) and halogen anions (X), is the most extensively researched and forms the basis of most metal halide perovskites⁶. Despite significant advancements improving PSC performance, cubic inorganic perovskites continue to encounter issues like moisture sensitivity, structural instability, and scalability in manufacturing. Organic perovskites have excellent efficiency but suffer from inadequate stability, whereas lead-based compounds encounter toxicity and deterioration when exposed to humidity, heat, and ultraviolet radiation^7,8. Conversely, materials such as carbon nanotubes (CNTs), graphene, and transition metal dichalcogenides (TMDs) have enhanced stability, flexibility, and conductivity, rendering them optimal for hybrid solar designs^9,10. Carbon nanotubes improve carrier mobility, moisture resistance, and interface quality when combined with perovskites, hence enhancing long-term stability. However, CNT-based devices frequently necessitate intricate surface modifications to attain good band alignment with the absorber layer. Furthermore, bilayer or multi-junction solar cells that incorporate materials with complementary band gaps enhance light absorption and improve power conversion efficiencies; yet, their production necessitates precise lattice and energy-level alignment, frequently resulting in interfacial instability¹¹.

Researchers are investigating lead-free alternatives that utilize non-toxic cations, demonstrating enhanced chemical stability, tunable bandgaps, and extensive optoelectronic applications to surmount these challenges. Parmar et al.¹² investigated RbGeX₃ (X = I, Br, Cl) perovskites by DFT, demonstrating significant structural stability, direct bandgaps, and remarkable optical and thermoelectric properties, with anticipated efficiencies of 15.67%, 22.08%, and 28.93%, respectively. Rehman et al.¹³ examined CsSnX₃ (X = I, Br, Cl) perovskites using DFT-CASTEP (GGA-PBE), validating their narrow bandgaps, elevated absorption coefficients, and malleable characteristics, hence identifying CsSnI₃ as a notably effective contender for solar and LED applications. Zhang et al.¹⁴ examined InXY₃ (X = Ca, Sr, Ba; Y = Cl, Br) perovskites, revealing their mechanical strength, extensive bandgaps, and significant UV absorption coupled with visible–infrared transparency, making them suitable for transparent and UV-protective optoelectronic devices.

Inorganic compounds are pivotal to advancements in solar technology because of their distinctive physical and electrical characteristics. Recent research emphasizes materials characterized by the generic formula A₃BX₃, when A represents a large inorganic cation, B denotes a smaller metal cation, and X signifies a halogen anion¹⁵. Compounds like Mg₃SbF₃, Ba₃PF₃, and Sr₃AsBr₃ exhibit significant thermodynamic stability, ideal bandgaps, elevated absorption coefficients, and anticipated power conversion efficiencies (PCEs) over 25%, making them very interesting candidates for advanced solar cells^16,17,18. Harun-Or-Rashid et al.¹⁹ investigated Pb-free cubic A₃SbI₃ (A = Ba, Sr, Ca) perovskites by first-principles DFT and SCAPS-1D simulations, validating their mechanical and optical stability. Through the optimization of layer thickness, doping, and flaws, Ba₃SbI₃ attained an exceptional power conversion efficiency (PCE) of 30.26%, surpassing previous versions. Hossain et al.²⁰ investigated A₃BF₃ (A = Mg, Ca, Sr, Ba; B = N, Bi) perovskites, identifying cubic symmetry, mechanical stability, and significant visible-light absorption. Mg-based compounds functioned as insulators, but Ca-, Sr-, and Ba-based complexes exhibited semiconducting properties appropriate for solar, thermal, and optoelectronic applications. Ghosh et al.²¹ conducted an in-depth analysis of A₃BX₃ (A = Ca, Sr; B = P, As; X = I, Br) halide perovskites, documenting direct bandgaps ranging from 1.26 to 2.63 eV, substantial absorption, and elevated efficiency, with Sr₃AsI₃ attaining a PCE of 28.16%. Phosphorus (P), Arsenic (As), and Antimony (Sb) are crucial to improving perovskite efficacy. P serves as a bandgap-tuning dopant, enhancing light absorption and charge mobility; As alters the band structure and prolongs carrier lifetimes; and Sb substitutes hazardous lead, thereby enhancing stability and environmental compatibility^22,23,24. Collectively, these components provide efficient, lead-free, and resilient perovskite configurations. Nonetheless, synthesis difficulties remain due to the volatility of halogens particularly fluorine (F) and the necessity for meticulous control of stoichiometry, phase purity, and oxidation states, especially in antimony-based systems²⁵. Notwithstanding these challenges, ongoing enhancement of synthesis procedures and sophisticated modeling approaches, including GW-BSE simulations to evaluate exciton binding energy²⁶, are anticipated to expedite the advancement of these materials for next-generation optoelectronic and solar applications.

This study extensively investigated the structural, mechanical, optical, and electrical properties of Be₃MF₃ (M = P, As, Sb) halide perovskites by first-principles DFT calculations to evaluate their suitability for PV applications. Among the candidates, Be₃SbF₃ demonstrated the most advantageous bandgap, stability, and light absorption properties, establishing it as a formidable prospect for solar energy conversion. The SCAPS-1D simulation tool was utilized to assess the PV performance of Be₃SbF₃-based PSCs. An analysis was conducted on various hole transport layers (HTLs) such as Cu₂O, MoO₃, P3HT, NiO, PTAA, and V₂O₅ and electron transport layers (ETLs) including In₂S₃, TiO₂, Mg:ZnO, and IGZO to determine the best device configuration. The impact of absorber thickness, doping concentration, and defect density on the principal PV parameters was methodically investigated. Furthermore, Nyquist plots, J-V characteristics, and quantum efficiency (QE) investigations were performed to investigate charge transport, recombination, and overall device efficiency. Temperature-dependent performance evaluations validated the material’s thermal stability. A Random Forest (RF) machine learning model was created to forecast the photovoltaic performance of Be₃SbF₃-based solar cells utilizing an extensive dataset of 13,113 data points obtained from device simulations. The model underwent training and evaluation utilizing several interpretative techniques, such as correlation heatmaps, SHAP (Shapley Additive Explanations) importance analysis, parity plots, and confusion matrix assessments, guarantee thorough feature evaluation and predictive reliability. The RF model exhibited significant predictive accuracy, successfully identifying bandgap energy and interface defect density as the primary factors influencing device efficiency. The combined use of DFT-derived material descriptors, SCAPS-1D device simulations, and ML-based predictions creates a cohesive framework for elucidating performance-limiting mechanisms and progressing Be₃SbF₃ perovskite solar cells towards scalable, lead-free, and sustainable photovoltaic applications.

Computational methodology

A Density Functional Theory (DFT)-based technique was employed inside the Cambridge Serial Total Energy Package (CASTEP) framework to examine the structural, electrical, and optical properties of Be₃MF₃ (M = P, As, Sb) perovskite material. The Generalized Gradient Approximation (GGA) utilizing the Perdew–Burke–Ernzerhof (PBE) functional was utilized to address the exchange–correlation potential and assess the electronic band structure²⁷. The Vanderbilt-type ultrasoft pseudopotential On–The–Fly–Generated (OTFG) was employed to precisely simulate the interaction between valence electrons and ionic cores, ensuring computational efficiency and accuracy in total energy calculations²⁸. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization technique was employed to optimize the crystal shape, allowing the system to achieve its minimum energy configuration²⁹. The Brillouin zone was sampled with a Monkhorst–Pack k-point grid of 10 × 10 × 10, and a plane-wave cutoff energy of 450 eV was utilized. This selection was established by rigorous convergence testing, which verified that these parameters provide very accurate assessments of the structural, electrical, and optical properties without incurring excessive computational costs. The selected k-point density effectively encapsulates the fundamental electronic interactions stemming from the inherent symmetry of the crystal lattice, guaranteeing numerical accuracy in the calculation of mechanical properties, including the elastic constants. The alignment of these computational settings with previously reported²¹ further validates their reliability and appropriateness for attaining a balanced trade-off between accuracy and computational efficiency. The elastic properties were ascertained utilizing the finite strain–stress methodology, as executed in CASTEP, to compute the elastic stiffness constants (C_ij) under equilibrium conditions³⁰. Geometry optimization adhered to rigorous convergence criteria: a maximum displacement of 5 × 10⁻⁴ Å, a maximum force of 0.01 eV/Å, a maximum stress of 0.05 GPa, and a total energy convergence of 1 × 10⁻⁵ eV per atom. A strain magnitude of 0.003 was utilized in the elastic calculations, guaranteeing computational stability and alignment with previous theoretical studies. These criteria combined guaranteed the dependability and reproducibility of the results obtained. Additionally, to augment the first-principles findings, the SCAPS-1D software was utilized to model charge carrier transit, electrostatic potential distribution, and overall device performance under steady-state conditions^31,32. The model quantitatively resolved Poisson’s and continuity equations for electrons and holes, yielding a comprehensive insight into recombination kinetics, carrier diffusion, and drift phenomena^33,34. This hybrid computational method integrating quantum–mechanical DFT insights with numerical device modeling facilitated a thorough evaluation of the Be₃SbF₃-based PSC, providing essential guidance for enhancing its optoelectronic characteristics and operational efficiency in advanced PV applications.

Result and discussion

DFT result

Structural properties

The beryllium-based single halide perovskite compounds, represented as Be₃MF₃ (where M = P, As, Sb), crystallize in a cubic A₃BX₃-type perovskite structure that corresponds to the Pm-3 m crystallographic space group (No. 221)³⁵. Figure 1a, b illustrates the optimal crystal structure of the Be₃MF₃ systems, showcasing a well-ordered cubic layout including a single formula unit per cell, consisting of seven atoms. The optimized lattice parameters and associated unit cell volumes, derived from GGA-PBE functional, are presented in Table 1. In these structures, the Be atoms predominantly occupy the 3a Wyckoff positions (1/2, 1/2, 0), whereas the M-site elements (P, As, Sb) are located at the 1b Wyckoff locations (1/2, 1/2, 1/2), and the F atoms are situated at the 3c Wyckoff sites (1/2, 0, 0). A comprehensive geometry optimization was conducted, permitting the total relaxation of lattice parameters to achieve the equilibrium configuration. The derived lattice constants and total energy values exhibit remarkable concordance with existing theoretical and experimental findings, hence affirming the computational precision and dependability of the current results. To further validate the geometric stability of the cubic structure, the Goldschmidt tolerance factor (τ_G) was computed using Eq. (1)³⁶.

$$\tau_{G} = \frac{{2 \left( {R_{A} + R_{X} } \right)}}{{\sqrt 2 \left( {R_{B} + 2R_{X} } \right)}}$$

(1)

Fig. 1

Graphic representation of Be₃NX₃ compounds: (a) polyhedral view, (b) ball-and-stick model, for Be₃MF₃.

Table 1 Optimized structural models and associated thermodynamic stability parameters for Be₃MF₃.

where R_A, R_B, and R_X specify the ionic radii of the A-site cation (Be), B-site atom (P, As, or Sb), and X-site anion (F), accordingly. The calculated tolerance factor of τ_G between 0.804 to 1 is comfortably situated within the defined stability range for cubic perovskites (0.71 < τ < 1.00), affirming the maintenance of a stable cubic phase devoid of significant lattice aberrations. This geometric stability is a vital prerequisite for sustaining consistent optoelectronic properties, which are crucial for dependable device performance over prolonged operational durations. The thermodynamic stability of the Be₃MF₃ compounds was assessed by computing the formation energy (E_f) as per Eq. (2)³⁷.

$${\text{E}}_{{\text{f}}} = {\text{ E}}_{{{\text{Be3MF3}}}} - \, \left( {{\text{3E}}_{{{\text{Be}}}} + {\text{ E}}_{{\text{M}}} + {\text{ 3E}}_{{\text{F}}} } \right)$$

(2)

where E_Be3MF3 denotes the cumulative energy of the optimum configuration, and E_Be, E_M and E_F correspond to the aggregate energies of the constituent Be, M, and F atoms, respectively. The negative formation energy estimates (Table 1) indicate that all Be₃MF₃ molecules are energetically favorable and thermodynamically stable, hence affirming their viability for experimental synthesis. The comprehensive evaluation of optimized lattice parameters, tolerance factor, and formation energy confirms the significant structural integrity of the Be₃MF₃ perovskites. This intrinsic stability improves their prospective incorporation into solar, optoelectronic, and advanced energy-harvesting applications, where phase resilience and minimum structural variation are essential for attaining superior device performance and operational longevity.

Electronic band structure and density of states

The electronic band structure and density of states (DOS) are essential instruments for elucidating the electronic characteristics of crystalline solids, since they disclose the distribution of electrons across different energy levels E (in eV). Examining the band structure yields essential insights into a material’s conductive, semiconductive, or insulating properties, which directly influence its electrical, optical, and thermal responses. This comprehension is essential for customizing materials for sophisticated electrical applications, such as semiconductors, field-effect transistors, photodetectors, and PV cells. Figure 2a, c, e illustrates the computed band structures of Be₃MF₃ (M = P, As, Sb) compounds derived from the GGA-PBE functional. The computations were executed along high-symmetry directions (X–R–M–Γ–R) within the Brillouin zone, with the Fermi energy level (E_F) indicated by a red dashed line set at 0 eV. The valence bands are situated beneath the E_F, whilst the conduction bands are positioned above them. The lack of band overlap at E_F signifies that all Be₃MF₃ compounds possess semiconducting properties. The PBE-GGA functional, despite inherent self-interaction inaccuracies, frequently underestimates the band gap owing to its semi-local exchange–correlation characteristics. This constraint is particularly apparent in halide-based perovskites, where robust electron–electron interactions govern the electrical and optical properties⁴¹. Nonetheless, PBE-GGA continues to be highly proficient in capturing qualitative patterns in electrical structure and band dispersion. Advanced methodologies, including hybrid functionals (HSE06) and techniques such as GW or mBJ, enhance bandgap precision by integrating accurate exchange corrections, however they entail significantly more computational expense⁴². PBE-GGA, while producing lower absolute gap values, provides an exceptional balance between computational efficiency and physical reliability, rendering it ideal for extensive and defect-focused simulations. Its strength in consistently recreating electrical and optical trends guarantees dependable insights into structural and compositional changes in intricate perovskite systems.

Fig. 2

The calculated electronic band structure and DOS plots for (a, b) Be₃PF₃, (c, d) Be₃AsF₃, and (e, f) Be₃SbF₃ perovskite.

The investigation of the electronic band structure indicates that the valence band maximum (VBM) and conduction band minimum (CBM) of Be₃MF₃ (M = P, As, Sb) are located at different high-symmetry points (R-G) in the Brillouin zone, hence validating their indirect band gap characteristic. The determined band gaps 3.327 eV for Be₃PF₃, 3.201 eV for Be₃AsF₃, and 2.158 eV for Be₃SbF₃, as shown in Table 1, categorize these compounds as wide-bandgap semiconductors. Indirect band gap materials demonstrate phonon-assisted electronic transitions, leading to less non-radiative recombination and extended carrier lifetimes. These attributes offer improved thermal stability, radiation resistance, and mechanical durability, which are advantageous for high-power, high-temperature, and long-lifespan electronics. Despite their reduced efficacy in light emission, these materials are exceptionally appropriate for nonlinear optical (NLO) and passive photonic applications, where accurate light modulation and elevated damage thresholds are critical^43,44. In optoelectronic device engineering, the indirect characteristics of Be₃MF₃ compounds provide benefits for UV and deep-UV photodetectors, electro-optic modulators, waveguides, thermophotovoltaic converters, and optical limiters, where regulated light absorption and minimal recombination losses are essential. The extensive band gaps indicate robust interatomic bonding, elevated breakdown voltage, and substantial optical phonon energies, resulting in enhanced thermal conductivity and prolonged device lifespan. The intrinsic properties of Be₃MF₃ compounds render them viable candidates for energy-harvesting systems, radiation-resistant photovoltaic cells, and high-performance optoelectronic devices in sophisticated solar and photonic technologies. Among them, Be₃SbF₃ is notable for its optical transitions, substantial mechanical stability, and a moderate band gap, facilitating significant absorption throughout the visible to near-UV spectrum. Despite its band gap being marginally greater than that of optimal single-junction absorbers, Be₃SbF₃ is still highly efficient for tandem solar cell configurations, serving as the wide-bandgap top absorber in conjunction with a low-bandgap bottom cell (such as silicon or tin-based perovskites) to expand the spectral response and improve overall efficiency⁴⁵.

The Total and Partial density of states (TDOS and PDOS) investigations offer significant insights into the electronic configuration and bonding properties of crystalline materials. TDOS provides a comprehensive overview of the total accessible electronic states across various energy levels, whereas PDOS disaggregates these contributions by specific atomic orbitals or elements, thereby clarifying the local electronic environment and the characteristics of orbital hybridization⁴⁶. Collectively, these investigations are essential for elucidating the origins of band structure characteristics and comprehending the chemical bonding mechanisms that dictate material stability and electrical performance. Figure 2b, d, f displays the calculated TDOS and PDOS spectra for Be₃MF₃ (M = P, As, Sb) compounds, plotted against energy with E_F set at 0 eV (denoted by a black dashed line). The valence band, ranging from approximately − 4 to 0 eV, is predominantly influenced by the F-2p orbitals, signifying that fluorine atoms significantly contribute to the characteristics of the upper valence band and the nature of chemical bonding.

Minor hybridization contributions from the P-3 s, As-4p, and Sb-5 s states are found near the valence band edge, indicating slight differences in electronic interactions across the series as the atomic mass of M increases. Conversely, the conduction band mostly derives from the Be-2 s and Be-2p states, supplemented by contributions from the higher-energy orbitals of group-V elements (P, As, Sb). This indicates a robust covalent interaction between Be-F and Be-M, wherein beryllium orbitals substantially affect electronic transitions and play a role in the conduction process. The extent of orbital overlap between beryllium and fluorine states suggests increased charge delocalization, potentially enhancing electron mobility and overall charge transport characteristics. The prevalence of F-2p states at the valence band maximum and Be-2 s states at the CBM confirms the indirect band gap characteristic revealed in the band structure study. The uniform electronic alignment in all three compounds emphasizes their semiconducting properties and shows the influence of fluorine and beryllium orbitals on electronic transitions. Furthermore, the progressive decline of the conduction band edge from Be₃PF₃ to Be₃SbF₃ indicates a trend of band gap shrinking, along with the augmented atomic size and diminished electronegativity of the group-V element. This electrical development is essential for optimizing optical absorption and customizing these materials for PV, photodetector, and light-emitting device applications, where precise band alignment improves energy conversion efficiency.

Optical characteristics

The optical response of a material offers essential insights into its capacity to interact with, absorb, and transmit electromagnetic radiation, which are crucial for assessing its appropriateness for photonic, optoelectronic, and PV applications. Comprehension of optical behavior is essential for materials designed for solar energy harvesting, light-emitting devices, photodetectors, and nonlinear optical systems. These features are predominantly determined by electronic structure and chemical bonding factors, which affect electron transitions between valence and conduction bands when exposed to incident light. The investigation of halide-based perovskites has proliferated in recent years owing to their exceptional light absorption, adjustable band gaps, and elevated carrier mobility. The optical properties of Be₃MF₃ (M = P, As, Sb) were examined by first-principles DFT to evaluate their applicability in advanced energy conversion and optical communication technologies. The analyzed optical functions encompass the absorption coefficient (α(ω)), reflectivity (R(ω)), refractive index (n(ω)), dielectric function (ε(ω)), optical conductivity (σ(ω)), and energy loss function (L(ω)), all assessed up to 30 eV photon energy, as seen in Fig. 3a–f. These features jointly delineate the propagation of electromagnetic waves within the material and the efficiency of their interaction with photons across various frequency spectra.

Fig. 3

The optical characteristics of Be₃MF₃ perovskite compounds encompass (a) absorption spectra, (b) reflectivity, (c) variations in refractive index, (d) the real and imaginary components of the dielectric function, (e) optical conductivity, and (f) electron energy loss function, underscoring their applicability in optoelectronic and PV domains.

The absorption coefficient (α(ω)) indicates the capacity of Be₃MF₃ compounds to absorb photons and generate charge carriers, as illustrated in Fig. 3a, and serves as an essential indicator of their light-harvesting effectiveness. The absorption spectra initiate at specific photon energies that align with their optical band gaps, in accordance with the semiconducting characteristics derived from band structure research. The tight correlation between electronic and optical data validates the trustworthiness and internal consistency of band structure computations⁴⁷. Numerous pronounced absorption peaks are observable over the UV spectrum, indicating effective photon electron interactions at elevated energies. Among the compounds, Be₃PF₃ has the most pronounced absorption peak at approximately 13.62 eV with an intensity of 24.27, while Be₃AsF₃ and Be₃SbF₃ provide peaks around 23.02 and 21.7, respectively. The existence of several maxima and minima throughout the spectrum signifies intricate interband electronic transitions involving Be-F and Be-M orbital hybridizations. Despite the absorption intensity being comparatively low below 2.5 eV because to the indirect bandgap characteristics and fluorine-induced electronic localization, a pronounced rise is noted at elevated photon energies, signifying effective photon–electron coupling in the UV–visible spectrum. The notable optical activity indicates that Be₃MF₃ compounds, especially Be₃SbF₃, hold considerable promise for incorporation into UV photodetectors, radiation-resistant coatings, and tandem solar cell designs⁴⁸. Their elevated absorption coefficients in the ultraviolet spectrum render them advantageous for high-power optoelectronic devices, UV filters, and laser protection applications, where wide bandgap materials with resilient optical responses are essential for performance and longevity.

The reflectivity, R(ω), quantifies the efficiency with which a substance reflects incoming electromagnetic radiation. This characteristic directly affects the optical contrast, thermal regulation, and energy retention of materials employed in photonic systems and solar cells. Figure 3b illustrates the reflectivity spectra of Be₃MF₃ compounds across an extensive photon energy range. At 0 eV, the reflectivity initiates at moderate levels ranging from 0.16 to 0.175, indicating the intrinsic optical density of the compounds. With increasing photon energy, reflectivity progressively ascends in the infrared and visible spectra, reaching peak values of 0.318 for Be₃PF₃, 0.313 for Be₃AsF₃, and 0.332 for Be₃SbF₃. This little difference can be ascribed to the rising atomic mass and the associated alteration in electronic polarizability of the M-site elements (P, As, Sb). At elevated photon energy (exceeding 27 eV), the reflectance of all three substances declines precipitously, nearing zero. The decreasing reflectance signifies a transparent optical window in the UV–visible spectrum, facilitating effective photon transmission through the material⁴⁹. This optical characteristic is essential for anti-reflective coatings, transparent electrodes, and optical encapsulants utilized in PV and photonic devices, as diminished surface reflection improves energy absorption and device performance. Moreover, the prevailing tendency indicates that the Be₃MF₃ perovskites exhibit an intrinsic equilibrium between reflection and absorption a desirable attribute for light-trapping structures in thin-film solar cells.

The refractive index, n(ω), measures the extent of light refraction as it traverses a material and offers insight into optical polarization and dielectric behavior. The real and imaginary components of n(ω) were computed, as depicted in Fig. 3c. The static refractive index values, n(0), vary from 2.3 to 2.5, indicating a modest optical density characteristic of wide band gap semiconductors. In the infrared and visible spectra, the real part of n(ω) progressively ascends, attaining peak values of 2.92 for Be₃PF₃, 2.91 for Be₃AsF₃, and 3.99 for Be₃SbF₃ within the energy range of 3.1–4.8 eV. This increase implies robust light–matter interaction and proposes possible applications in nonlinear optical modulation and light confinement systems. As photon energy increase, the refractive index progressively diminishes, signifying a reduction in transparency and a shift towards elevated optical absorption. The imaginary component of n(ω), denoting the extinction coefficient, stays minimal at elevated energies, indicating diminished photon loss due to scattering or absorption. The near resemblance in the refractive index values of Be₃PF₃ and Be₃AsF₃ suggests analogous bonding properties and electrical behaviors, whereas Be₃SbF₃, with a little elevated peak, may provide improved optical polarization due to its greater atomic size and diminished band gap. The refractive index analysis indicates that Be₃MF₃ compounds exhibit excellent optical transparency and moderate refractive properties, making them appropriate for light-guiding components, photonic waveguides, and electro-optic modulators.

The dielectric function, ε(ω) = ε₁(ω) + iε₂(ω), characterizes a material’s reaction to an external electromagnetic field and offers essential understanding of polarization effects and optical transition processes. The real component, ε₁(ω), signifies the dispersive (polarizable) aspect, whereas the imaginary component, ε₂(ω), pertains to absorption and energy dissipation resulting from interband transitions. The calculated dielectric spectra are displayed in Fig. 3d. At zero photon energy, the actual dielectric constant ε₁(0) ranges from 5.4 to 5.9, signifying substantial electronic polarization⁵⁰. The real component displays peak values between 8.1 and 8.9 at around 4.5 eV, indicating significant dipole excitation within this energy spectrum. The imaginary component, ε₂(ω), has pronounced peaks in the visible spectrum and progressively diminishes to zero beyond 25 eV, indicating negligible energy loss at elevated photon energies. The characteristics indicate that the Be₃MF₃ perovskites exhibit superior dielectric stability and frequency-dependent polarization regulation, essential for electro-optical modulators, optical sensors, and capacitive energy storage systems. The significant correlation between ε₁(ω) and n(ω) profiles further validates the self-consistency of the optical analysis and emphasizes the resilience of the materials’ optical polarization response.

The optical conductivity (σ(ω)), measures a material’s capacity to transfer charge carriers when subjected to optical excitation and is essential for evaluating photoconductive efficiency. Figure 3e illustrates the real and imaginary components of σ(ω) for the Be₃MF₃ series. The real component of σ(ω) displays distinct peaks between 4.7 and 5.3, indicating active interband transitions and effective electron excitation across the optical band gap. The analogous magnitudes among the three chemicals indicate similar electron mobility in optical fields. The imaginary component, linked to phase lag in carrier response, exhibits a gradual decrease over the infrared and visible spectra, signifying diminished reactive losses. The elevated optical conductivity in the visible spectrum indicates robust light-induced carrier excitation and effective current responsiveness when illuminated. Their attributes, along with wide band gap stability, render Be₃MF₃ materials exceptional prospects for photoconductive sensors, transparent conductors, and energy-efficient optoelectronic systems.

The energy loss function (L(ω)) indicates the energy dissipated by a high-velocity electron as it moves through a solid and offers direct insights into plasmon resonances collective oscillations of free electrons. Figure 3f depicts the loss function spectra for the Be₃MF₃ molecules. With reduced photon energies, the loss function diminishes owing to heightened photon absorption within that spectrum. Distinct plasmon resonance peaks arise at elevated energy, corresponding to plasma frequencies where collective electronic oscillations prevail⁵¹.

The peak L(ω) for Be₃PF₃ is around 21.06 eV (≈5.25), whereas Be₃AsF₃ and Be₃SbF₃ display lower peak values of 2.2 and 2.1, respectively. The peaks signify robust electron–photon interaction and moderate energy dissipation, validating the promise of Be₃MF₃ compounds for plasmonic and photothermal applications, where regulated energy loss and resonance characteristics are beneficial. Furthermore, the comparatively low loss function at visible photon energies indicates negligible damping and superior optical quality, rendering these materials suitable for transparent coatings, UV shielding layers, and light management systems in optoelectronic devices.

Mechanical characteristics

The mechanical properties of crystalline materials are essential for comprehending their stability, deformation behavior, and applicability in industrial and optoelectronic fields. Elastic constants (C_ij) are essential markers of interatomic bonding strength, measuring a solid’s response to external stress and strain. This study comprehensively examined the elastic characteristics of Be₃MF₃ (M = P, As, Sb) utilizing the linear finite strain–stress methodology in CASTEP. Figure 4a, b illustrates that the mechanical properties of these cubic systems are determined by three distinct elastic constants C₁₁, C₁₂, and C₄₄ which correspond to resistance against uniaxial, uniform, and shear deformations, respectively. To guarantee mechanical stability, the Born-Huang stability criteria must be fulfilled⁵², articulated as:

$${\text{C}}_{{{11}}} + {\text{2C}}_{{{12}}} > 0\;\;\left( {\text{Spinodal criteria}} \right)$$

$${\text{C}}_{{{11}}} {-}{\text{C}}_{{{12}}} > 0\;\;\left( {\text{Born criteria}} \right)$$

$${\text{C}}_{{{44}}} > 0\;\;\left( {\text{Shear criteria}} \right)$$

Fig. 4

Values of calculated (a) elastic and (b) mechanical constants for Be₃MF₃ (M = P, As, Sb).

Figure 4a demonstrates that all Be₃MF₃ compounds meet the mechanical stability criterion, with C₁₁ significantly above C₁₂ and C₄₄, signifying greater resistance to longitudinal compression compared to shear deformation. Among the series, Be₃PF₃ demonstrates the greatest C₁₁ value, indicating increased bond strength and diminished compressibility, in accordance with its exceptional mechanical stiffness. The Voigt–Reuss–Hill (VRH) averaging approach was utilized to achieve a comprehensive characterization of polycrystalline mechanical behavior⁵³. This approach reconciles the Voigt model (uniform strain) and the Reuss model (uniform stress), yielding a realistic estimation of the effective mechanical moduli. The bulk modulus (B), shear modulus (G), and Young’s modulus (Y) were calculated using Hill’s Eqs. (3–5)^54,55.

$$B = \frac{{B_{v} + B_{R} }}{2}$$

(3)

$$G = \frac{{G_{v} + G_{R} }}{2}$$

(4)

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

(5)

Consequently, Poisson’s ratio (v) was obtained from the subsequent Eq. (6).

$$v = { }\frac{{\left( {3{\text{B}} - 2{\text{G}}} \right)}}{{2\left( {3{\text{B}} + {\text{G}}} \right)}}$$

(6)

The B measures a material’s resistance to volumetric change under hydrostatic pressure, whereas the G assesses its resistance to deformation under applied shear forces. Young’s modulus (Y), a critical indicator of stiffness, denotes the ratio of tensile stress to strain in the elastic range and is inversely related to thermal shock resistance. The computed findings in Fig. 4b, indicate that Be₃PF₃ demonstrates the greatest values of B, G, and Y, succeeded by Be₃AsF₃ and Be₃SbF₃. This signifies that the Be₃PF₃ compound is mechanically the most resilient, exhibiting the highest resistance to deformation and compression. The increased moduli of Be₃PF₃ indicates more robust directional bonding and reduced Be-F and Be-P bond lengths, hence augmenting lattice rigidity. In contrast, when the M-site atom enlarges (from P to when to Sb), the moduli diminish due to bond elongation and diminished charge localization, indicating fewer interatomic interactions and a marginal improvement in flexibility. Elevated values of B, G, and Y indicate that these materials possess structural integrity and mechanical stability, rendering them ideal for high-stress or high-temperature applications, including optoelectronic devices, dielectric substrates, and coating materials in solar energy systems. Pugh’s ratio (B/G) and Poisson’s ratio (ν), which are critical parameters for differentiating ductile from brittle materials. A pivotal Pugh’s ratio threshold of 1.75 distinguishes ductile (B/G > 1.75) from brittle (B/G < 1.75) behavior⁵⁶. The computed B/G ratios for Be₃PF₃ (1.824), Be₃AsF₃ (1.858), and Be₃SbF₃ (1.951) surpass this threshold, so affirming that the complete Be₃MF₃ series demonstrates ductility. Be₃SbF₃ exhibits the highest ratio, indicating superior plastic deformability and enhanced resistance to fracture under mechanical stress. This ductility indicates that these materials can experience moderate elastic–plastic deformation prior to failure, providing robustness against stress building during thin-film development and device assembly. This inherent mechanical compliance is beneficial in the manufacturing of perovskite solar cells, where films frequently undergo stress due to thermal annealing, substrate mismatch, and multi-layer stacking. Ductile absorber layers exhibit reduced vulnerability to microcrack development and interfacial delamination, therefore preserving film continuity, grain integrity, and electrical stability throughout prolonged operation⁵⁷. Moreover, the mechanical flexibility inherent in ductility facilitates possible incorporation into flexible or wearable optoelectronic systems, permitting Be₃MF₃ films to endure bending, rolling, and mechanical vibrations without structural deterioration. Poisson’s ratio (ν) offers additional understanding of the bonding and elastic properties of these materials. Generally, ν > 0.26 signifies ductility and primarily ionic bonding, while ν < 0.26 denotes brittleness and covalent bonding. The calculated Poisson’s ratios for Be₃PF₃, Be₃AsF₃, and Be₃SbF₃ all surpass 0.26, validating their ductile characteristics and affirming a predominantly ionic bonding structure that enhances elastic flexibility. The simultaneous presence of elevated B/G and ν values confirms that the Be₃MF₃ family integrates mechanical strength with processing suitability. This ductility improves compatibility with solution-based and vacuum deposition methods, including spin coating, sputtering, and thermal evaporation, which frequently create localized stress gradients during solvent evaporation and film cooling⁵⁸. A supplementary metric of mechanical significance is the Zener anisotropy index (A), defined in Eq. (7).

$$A = \frac{{2C_{44} }}{{C_{11} - C_{12} }}$$

(7)

The calculated A values for Be₃MF₃ significantly diverge from unity, indicating considerable elastic anisotropy across crystallographic directions. Fig. S1 (Supplementary Information) illustrates that the deformed 2D and 3D elastic contour plots exhibit direction-dependent stiffness resulting from fluctuations in atomic bonding and internal stress distribution within the cubic lattice. Mechanical analysis verifies that all compounds are stable, moderately rigid, and ductile, with Be₃PF₃ demonstrating superior stiffness and Be₃SbF₃ displaying enhanced ductility. The combination of these features, along with robust ionic bonding and anisotropic elasticity, renders Be₃MF₃ materials advantageous for structural, optoelectronic, and thermomechanical applications especially in photovoltaic encapsulation, radiation-resistant optics, and high-frequency device substrates.

SCAPS 1D result

Solar cell performance evolution using Be₃SbF₃ as absorber material

Figure 5a–c depicts the fundamental structure and band alignment of a lead-free PSC, which includes ETL, HTL, and a back contact, incorporating a Be₃SbF₃ absorber layer. The device utilizes an n-i-p configuration, wherein the intrinsic Be₃SbF₃ layer creates an extensive depletion region that improves sensitivity to long-wavelength photons. This facilitates the effective creation and separation of electron–hole pairs both inside and beyond the depletion region, enhancing quantum efficiency and charge collection⁵⁹.

Fig. 5

Designing Be₃SbF₃-based PSCs: (a) device configuration and band alignment with various (b) ETL and (c) HTL materials.

Moreover, Be₃SbF₃ proficiently restricts charge carriers and photons, augmenting photon absorption and establishing ohmic contacts with the doped HTL and ETL layers, so improving device performance. This device utilizes Cu₂O, P3HT, NiO, V₂O₅, MoO₃, and PTAA as HTLs, with Ni functioning as the rear metal contact and Al as the front metal contact. The employed ETLs are In₂S₃, TiO₂, Mg:ZnO, and IGZO, with Be₃SbF₃ serving as the absorber layer. The comprehensive configuration is Al/FTO/ETL/Be₃SbF₃/HTL/Ni. Tables 2 and 3 delineate the simulation input parameters for the absorber layer, ETLs, HTLs, and fluorine-doped tin oxide (FTO), whereas Table 4 presents the values for interfacial defect layers. SCAPS-1D simulations were conducted to assess the efficacy of various PSC topologies under standard operating conditions, which included the AM1.5G solar spectrum, a frequency of 1 MHz, and an ambient temperature of 300 K.

Table 2 Essential input parameters for FTO, ETLs, and Be₃SbF₃ absorber layers in PSCs^51,60,61,62.

Table 3 Key input parameters of diverse HTLs in PSCs^63,64.

Table 4 Investigating primary interface parameters in PSCs.

Optimization of ETL and HTL

The photovoltaic efficacy of Be₃SbF₃-based solar cells is significantly determined by the choice and interaction of appropriate electron transport layers (ETLs) and hole transport layers (HTLs), which directly affect charge extraction, recombination dynamics, and overall device efficiency. This study extensively investigated four ETLs such as TiO₂, In₂S₃, Mg:ZnO, and IGZO alongside six HTLs as Cu₂O, P3HT, NiO, V₂O₅, MoO₃, and PTAA to determine ideal configurations for Be₃SbF₃ absorbers. ETL enables electron flow from the absorber to the front electrode while preventing hole recombination, whereas the HTL effectively extracts holes toward the rear contact, providing balanced charge transport and minimizing recombination losses⁶⁵. Figure 6a–d demonstrates that the PV characteristics: V_OC, J_SC, FF, and PCE exhibit considerable variation depending on the combinations of ETL and HTL. Among the ETLs, TiO₂ attained the highest V_OC (about 1.80 V), whilst Mg:ZnO exhibited the lowest (about 1.61 V). In₂S₃ and IGZO yielded intermediate values, so validating their stable interfacial characteristics. For J_SC, most devices displayed values ranging from 9.65 to 11.04 mA/cm², with IGZO demonstrating a little decreased current density attributable to restricted carrier mobility and interface discrepancies. The FF varied from 86.61% (IGZO) to 91.89% (In₂S₃), with In₂S₃ identified as the most efficient electron transport layer because of its enhanced charge extraction and reduced interface recombination. Table S1 (Supplementary Information) illustrates the four most effective device architectures. The Al/FTO/In₂S₃/Be₃SbF₃/Cu₂O/Ni design achieved the best PCE of 18.28%, surpassing alternative combinations.

Fig. 6

Assessment of charge transport layers on critical performance metrics: (a) V_OC, (b) J_SC, (c) FF, and (d) PCE in Be₃SbF₃ SCs.

The Cu₂O HTL demonstrated exceptional efficacy across all ETLs, owing to its good band alignment, superior p-type conductivity, and chemical stability⁶⁶. In contrast, NiO exhibited comparatively lower power conversion efficiencies because of suboptimal energy-level alignment. The findings underscore the necessity of sustaining optimal HTL thickness and band alignment to reduce recombination and improve carrier transmission. The integration of In₂S₃ as ETL and Cu₂O as HTL establishes an optimal interface design, delivering a synergistic equilibrium of voltage, current, and efficiency. This underscores Be₃SbF₃ promise as a durable lead-free absorber for advanced optoelectronic and PV applications.

Role of back contact work function in determining solar cell characteristics

Back contact is crucial in influencing solar cell efficiency and overall performance. It must minimize resistive losses to ensure efficient current extraction, as inadequate contact elevates series resistance and diminishes power production⁶⁷. Moreover, robust adhesion and chemical stability with the semiconductor are essential for the longevity of the device. Poor design may result in shade and restrict the effective area for energy conversion. An improved back contact enhances charge carrier collection, reduces recombination losses, and consequently improves the performance and dependability of the solar cell. The optimal metal for back contact is primarily determined by its work function (WF), which varies from 4.81 eV to 5.9 eV, as detailed in Table S2. Fig. S2 depicts the PV parameters acquired with diverse back contacts for distinct ETLs. The findings demonstrate that a reduction in WF of the Fe contact corresponds to a decline in the PCE of the SC. An elevation in WFs of metals like Au, Ni, Pd, Pt, and Se demonstrates a consistent positive link with improved PCE. This enhancement is primarily ascribed to the diminution of the Schottky barrier height via improved alignment of the metal work function with the Fermi level of the HTL⁶⁸. Ni is the most appropriate back contact material among these metals, owing to its ideal work function, cost efficiency, plentiful availability, and consistent performance, rendering it a viable option for efficient and scalable device manufacturing.

Energy band diagram (EBD) of Be₃SbF₃

The energy band diagram (EBD) illustrates the conduction and valence band offsets (E_C and E_V) between the Be₃SbF₃ absorber and the HTL, influenced by various ETLs in conjunction with Cu₂O. Optimal energy level alignment at the ETL/Be₃SbF₃ and Be₃SbF₃/HTL interfaces is essential for maximizing device efficiency. Misalignment may result in charge accumulation and interfacial recombination, thereby diminishing performance. For efficient operation, ETL must exhibit a higher electron affinity than Be₃SbF₃ to enable effective electron extraction, whereas HTL should demonstrate a lower ionization energy to promote hole transport⁶⁹. Therefore, optimizing the interfacial electronic properties of both layers is crucial for improving overall PV efficiency. Figure 7a–d illustrates the EBDs of Be₃SbF₃-based solar cell structures that utilize Cu₂O as HTL alongside four different ETLs. The configuration employing In₂S₃ as ETL shows a clear intersection of the Fermi level with the E_C. The ETLs utilized in these devices, ordered by ascending bandgap energy, include In₂S₃ (2.1 eV), TiO₂ (3.2 eV), IGZO (3.05 eV), and Mg:ZnO (3.5 eV). The quasi-Fermi level for holes (F_p) is consistently positioned above EV, whereas the quasi-Fermi level for electrons (F_n) is closely aligned with E_C, demonstrating effective charge carrier separation in all configurations.

Fig. 7

Investigating variations in band structure of SCs with diverse ETLs: (a) In₂S₃, (b) TiO₂, (c) Mg:ZnO, and (d) IGZO.

Role of acceptor density and absorber thickness variations in SC performance

The dynamics of charge carriers, recombination rates, and efficiency in PSCs are significantly influenced by the acceptor density (N_A) and the thickness of the absorber layer. An optimal N_A enhances the built-in electric field, facilitating charge separation and minimizing recombination, while values that are too low or too high may impede carrier collection or elevate trap-assisted losses⁷⁰. Thicker absorber layers enhance light absorption but may elevate recombination and resistance, while thinner layers reduce recombination but limit photocurrent⁷¹. Attaining an appropriate equilibrium of N_A and thickness is essential for maximizing PCE, achieved by interface engineering, doping regulation, and defect passivation to enhance stability and performance. Figure 8a–d illustrates the synergistic impact of N_A and absorber thickness on the PCE of the SCs. This study altered N_A from 10¹³ to 10¹⁹ cm⁻³ and adjusted absorber thickness between 0.3 and 3.0 µm. The findings reveal a notable fluctuation in PCE corresponding to alterations in these two parameters across four distinct SCs including FTO/ETL/Be₃SbF₃/Cu₂O/Ni configurations, utilizing ETLs of In₂S₃, TiO₂, Mg:ZnO, and IGZO. The PCE is generally reduced when the N_A is beneath 10¹⁷ cm⁻³. When the N_A attains 10¹⁷ cm⁻³, the PCE hits its zenith, signifying that elevated doping concentrations result in augmented recombination losses. As N_A surpasses 10¹⁷ cm⁻³, the PCE progressively diminishes. Likewise, for absorber thicknesses between 0.5 and 1.2 µm, the PCE stays elevated but diminishes for thicker absorbers (1.5 to 3.0 µm) due to heightened bulk recombination and series resistance. Among the studied ETLs, the In₂S₃-based design (Fig. 8a) exhibited the highest PCE of around 18.28%, indicating effective charge transport and decreased recombination. The TiO₂-based (Fig. 8b) and Mg:ZnO-based (Fig. 8c)) SCs had PCE values of 18.20% and 18.15%, respectively, indicating a close similarity. The IGZO-based SC (Fig. 8d) exhibited a marginally reduced efficiency of around 17.26%, presumably attributable to fluctuations in carrier extraction characteristics. The results emphasize the importance of absorber thickness, N_A, and the selection of ETL in optimizing PCE. Achieving a balance between low N_A and moderate thickness is essential for minimizing recombination and enhancing charge collection efficiency. Among the assessed ETLs, In₂S₃ exhibited exceptional performance in attaining elevated PCE, establishing it as a viable alternative for the advancement of high-efficiency and robust PSC designs.

Fig. 8

Exploration of PCE fluctuation in relation to absorber layer thickness and N_A for SCs utilizing various ETLs: (a) In₂S₃, (b) TiO₂, (c) Mg:ZnO, and (d) IGZO.

Bulk and interfacial defect investigation

Organic and inorganic devices are significantly impacted by bulk defects, which are a significant factor contributing to their lower performance. The densities of defect levels are significantly influenced by experimental processing parameters; nonetheless, evaluating device performance at different defect energy levels and concentrations offers a useful comparison framework for optimizing these conditions⁷². Figure 9a–f depicts the fluctuation in solar output parameters (PCE) relative to defect energy and concentration. In the bulk defect study of the absorber layer, the defect energy level for Be₃SbF₃ was modified from − 0.2 to 2.4 eV, and the defect concentration (N_t) was altered between 10¹² and 10¹⁸ cm⁻³.

Fig. 9

Performance Analysis of SCs in various layer configurations with varying defect position and density: (a) Be₃SbF₃, (b) Be₃SbF₃/In₂S₃, (c) Be₃SbF₃/TiO₂, (d) Be₃SbF₃/Mg:ZnO, (e) Be₃SbF₃/IGZO, and (f) Cu₂O/Be₃SbF₃.

The defect energy range (− 0.2 to 2.4 eV) encompasses both shallow and deep-level trap states, facilitating a thorough examination of defect-assisted recombination in Be₃SbF₃. Shallow traps near the band boundaries (about − 0.2 eV) induce minor carrier trapping, whereas deep traps (approximately 2.4 eV) function as non-radiative recombination centers that diminish carrier lifespan and voltage. This range aligns with prior research by Reza et al.⁷³ and others on lead-free perovskites (CaRbCl₃, Ca₃NCl₃), which indicated that deep-level defects (> 2 eV) substantially impair charge collection and solar efficiency⁷⁴. Figure 9a illustrates the associated defect energy levels and N_t for Be₃SbF₃. The data indicates that elevated N_t can significantly impact device performance. Decreasing N_t to 10¹⁴ cm⁻³ yields an enhancement of device performance by as much as 40%. Nonetheless, attaining such a minimal N_t is still impractical within the limitations of conventional thin-film deposition methods. Consequently, N_t of 10¹⁵ cm⁻³ is deemed a more plausible concentration for this investigation.

Interface defects significantly affect device performance. Figure 9b–f illustrates the examination of defect concentrations at the ETL/absorber and Cu₂O/absorber interfaces. The device’s performance is analyzed at different defect concentrations and energy levels. Defect concentrations were changed from 10¹⁰ to 10¹⁶ cm⁻³, whereas defect energy levels spanned from − 0.4 to 2.4 eV. The findings demonstrate that defects at the HTL/absorber interface adversely impact device performance more significantly than those at the ETL/absorber interface. The examination of bulk and interfacial defects elucidates the impact of different defects on overall device performance. In the device bulk layer, defects with energy levels ranging from 0.3 to 2 eV are categorized as deep defects, which adversely impact performance and should be eliminated. Conversely, defects at the absorber/HTL interface, exhibiting energy levels between 0 and 2 eV, are also classified as deep defects. The severity of defects in PV performance is contingent upon the location, concentration, and energy level of each defect, highlighting the necessity of meticulous defect control to enhance device efficiency.

Impact of temperature on PV efficiency

Temperature is crucial in influencing the operational stability and performance of PSCs. Although PSCs provide high efficiency and manufacturing simplicity, their large-scale production encounters obstacles, especially from environmental conditions including UV radiation, temperature variations, wetness, and humidity. Temperature is particularly crucial, as solar cells often operate in direct sunshine, where temperatures can exceed ambient values by as much as 45 °C⁷⁵. An elevation in temperature results in a reduction of the semiconductor bandgap owing to the expansion of interatomic distances inside the material. The working temperature fluctuates for PSCs extends from 275 to 450 K, allowing for the evaluation of temperature fluctuations on cell performance. Figure 10 illustrates that temperature increase consistently diminishes critical metrics, such as V_OC, FF, and PCE, across all cell configurations. Notably, the J_SC exhibits remarkable stability despite rising temperatures. The noted reduction in V_OC is chiefly due to its inverse correlation with the saturation current density (J₀), which demonstrates a significant increase with rising temperature. The temperature-induced increase in J₀ arises from heightened carrier recombination rates in the absorber layer and at the interface regions⁷⁶. The PCE and FF often decrease with elevated temperature, mainly due to the increase in series resistance and augmented non-radiative recombination losses, both of which impede efficient charge transfer and extraction⁷⁷. Conversely, the IGZO-based device exhibits a nearly constant FF, indicating its enhanced thermal and electrical stability compared to alternative ETLs. This stability arises from IGZO enhanced electron mobility, minimal intrinsic defect density, and extensive bandgap (~ 3.05 eV), which together inhibit thermally induced carrier leakage and maintain the built-in electric field at high temperatures⁷⁸.

Fig. 10

Temperature-driven performance variations in PV cells with distinct ETLs.

Moreover, IGZO offers a properly oriented conduction band and a chemically stable interface with the perovskite absorber, mitigating temperature-induced interfacial deterioration. Nonetheless, as temperature escalates, the concomitant increase in series resistance diminishes overall device efficiency and fill factor, resulting in a significant decline in power conversion efficiency. These findings underscore the significance of heat management in sustaining consistent performance of PSCs under practical operating situations.

Nyquist plot analysis

Impedance analysis is an effective method for examining the properties of perovskite materials, especially those containing different halides. This method is recognized as one of the most efficacious techniques for investigating the behavior of perovskite materials with various halide compositions⁷⁹. To enhance comprehension of the effects of different ETLs on critical PV parameters, including V_OC, J_SC, FF, and PCE obtained from the I-V characteristics, complex impedance (Z*) was utilized to investigate the influence of ion migration and charge carrier dynamics within the perovskite solar cell. Figure 11 displays the Nyquist plot of complex impedance, depicting the correlation between the imaginary component (${Z}_{im}$) and the real component (${Z}_{re}$) of impedance for various ETL materials. The X-axis denotes recombination resistance, but the Y-axis illustrates geometrical capacitance, indicating charge carrier concentration at the interface layers. Each curve in the graph approximates a semi-circle within the frequency spectrum of 100 Hz to 1 MHz. The real (${Z}_{re}$) and imaginary (${Z}_{im}$) components of the impedance are graphed against one another, revealing that the impedance spectra for each ETL exhibits a different, approximately semi-circular form. Furthermore, the diameter of the semi-circle differs among various ETL-based structures, signifying variations in electrical properties and potential performance consequences. The diameter of the semi-circle expands with increasing frequencies. The broader semi-circle linked to the TiO₂ ETL-based PSC indicates an elevated impedance of roughly 34,366.681 Ω cm². This indicates increased resistance inside the system, probably attributable to diminished charge transport kinetics or heightened charge transfer resistance at the perovskite-TiO₂ interface. The smaller semi-circle seen in the IGZO ETL-based structure suggests reduced resistance, approximately 32,272.6149 Ω cm², implying enhanced charge transfer, superior interface qualities, or more charge carrier mobility relative to other ETL materials.

Fig. 11

Impedance analysis of Be₃SbF₃ absorber-based solar cells employing various ETL materials (In₂S₃, TiO₂, Mg:ZnO, and IGZO) via Nyquist plot.

At elevated frequencies, the observed resistance indicates the recombination resistance of the material, but the capacitance at these frequencies denotes the geometric capacitance, which relates to charge accumulation at the interfaces. The low-frequency response is intricate, probably reflecting ionic mobility and hysteresis phenomena⁸⁰. As ions collect and the reduction barriers diminish, the impedance response becomes inductive. The Nyquist plot of the solar cell provides a thorough qualitative insight into the device’s capacitance, resistive losses, and the rate of recombination faults present inside the device. This investigation offers critical insights into the electrical characteristics and efficiency of PSCs, enhancing comprehension of the elements affecting their performance.

J-V and QE characteristics

Figure 12a presents the J-V characteristics of four configurations of Be₃SbF₃-based solar cells under applied voltages from 0 to 2.0 V. Initially, all devices demonstrate comparable photocurrent values of approximately 11.03 mA/cm², except for the IGZO-based configuration, which exhibits a significantly lower current density. With an increase in voltage beyond 1.6 V, all structures exhibit a reduction in photocurrent, mainly due to increased charge carrier recombination linked to defect states in the perovskite absorber. The observations indicate that improved crystallinity and reduced defect density in the absorber layer enhance PV performance by facilitating efficient charge separation and transport.

Fig. 12

Analysis of (a) J-V and (b) QE characteristics curve of PSCs.

Figure 12b illustrates the quantum efficiency (QE) spectra of Be₃SbF₃-based solar cells employing different ETLs across the 300–620 nm wavelength range. The In₂S₃-based device initiates with a reduced quantum efficiency of approximately 68%, whereas other ETL designs commence at approximately 89.9%, signifying enhanced carrier collection near the absorption edge. The QE increases significantly between 350–530 nm, exceeding 95%, indicating robust optical absorption and effective charge production. This plateau indicates superior interfacial charge transport and negligible recombination losses at optimal photon energy. Nonetheless, QE diminishes beyond 590 nm owing to non-radiative recombination and free-carrier absorption in the doped front and ETLs phenomena characteristic of wide-bandgap perovskite and oxide-based devices⁸¹. The spectral variations among ETLs result from disparities in band alignment, carrier mobility, and interface defect density, which significantly affect charge extraction performance. The significant quantum efficiency in the visible spectrum validates the robust optoelectronic coupling and effective photo response of Be₃SbF₃, underscoring its potential as a lead-free perovskite absorber for high-performance photovoltaic applications.

An evaluation and comparison of SCAPS-1D findings with prior studies

This study obtained remarkable performance metrics for the Al/FTO/ETL/Be₃SbF₃/HTL/Ni SC configuration. The study concentrated on integrating diverse ETLs, including In₂S₃, TiO₂, Mg:ZnO, and IGZO, into the device architecture, with Cu₂O functioning as the HTL. The systematic optimization of essential parameters, including layer thickness, doping, defect densities, temperature, J-V, and QE properties, was conducted. After optimizing these variables, the proposed structure exhibited exceptional PV performance, attaining an efficiency of 18.28%, a V_OC of 1.7991 V, a J_SC of 11.0423 mA/cm², and an FF of 92%. Although achieving this high PV performance may provide difficulties for experimental researchers, this study offers a framework and guidance for future inquiries. The efficacy of SCs is predominantly determined by structural design, material quality, and production methodologies. Table 5 compares the best configurations documented in prior studies with the PV characteristics of the four device types introduced in this research.

Table 5 A comparative assessment of PV performance in Be₃SbF₃-based PSCs: insights from previous research (T = Theoretical, E = Experimental, and T.W. = This Work).

Machine learning analysis

Machine learning (ML) has significantly transformed PV research, facilitating swift, data-driven progress in PSCs. Conventional experimental and computational approaches tend to be time-consuming and resource-demanding, while machine learning techniques effectively analyze extensive datasets to identify patterns, forecast performance parameters, and inform material and structural optimization. This capability is essential for PSCs, as efficiency relies on the complex interactions among composition, processing conditions, and device architecture. Machine learning models are essential for predicting PV parameters. Researchers optimize electron and hole transport layers, absorber materials, and interface properties by integrating machine learning with simulation tools such as SCAPS-1D to reduce losses. Research conducted by Shrivastav et al.⁹⁷, Khan et al.⁹⁸, and Li et al.⁹⁹ illustrates the efficacy of machine learning in enhancing lead-free and heterojunction PSCs, resulting in elevated predictive accuracy and increased efficiency. The Random Forest (RF) algorithm is notable among machine learning methods for its robustness, interpretability, and capacity to manage nonlinear, multivariate data. RF models demonstrate superior capability in identifying key parameters that influence PSC performance, frequently surpassing linear regression, support vector machines (SVM), and k-nearest neighbors (KNN) models. Advanced techniques, including Gradient Boosting, XGBoost, Artificial Neural Networks, and Deep Learning, enhance the capabilities of machine learning in high-throughput screening and autonomous material discovery¹⁰⁰. The incorporation of machine learning, particularly random forests, into polymer science and engineering research signifies a transition towards intelligent and automated material optimization. The integration of statistical learning, physical modeling, and experimental insights facilitates the accelerated design of efficient, stable, and sustainable perovskite materials, thereby advancing the development of next-generation lead-free solar technologies.

Comprehending the RF algorithm

This study utilized RF technique as the primary machine learning model due to its remarkable proficiency in managing nonlinear interactions, high-dimensional feature spaces, and intricate correlations among PV parameters. The RF approach functions by creating a collection of many decision trees, each trained on randomly chosen subsets of data and input attributes. The ultimate forecast is derived by averaging the outputs of various trees, which successfully reduces overfitting while preserving excellent predictive accuracy. This ensemble-based methodology offers a dependable equilibrium among robustness, precision, and interpretability, rendering it particularly appropriates for evaluating the interaction of material, structural, and operational elements that collectively affect solar cell efficiency^101,102. The SCAPS-1D simulator was employed to methodically alter essential device parameters, such as absorber layer thickness, electron affinity, defect and doping density, interface defect, bandgap, and operational temperature, to create a comprehensive dataset. This simulation campaign produced 13,113 data points, with 80% (10,490 samples) designated for model training and 20% (2623 samples) for testing and validation. The predictive performance of the trained RF model was assessed using the coefficient of determination (R²) and mean absolute error (MAE) metrics, both indicating the model’s robust capacity to reliably predict PV performance based on input data. The RF algorithm’s primary advantage is its capacity to quantify feature importance, therefore pinpointing the most significant elements affecting PSC efficiency. This feature analysis allows researchers to prioritize elements for further optimization. The hyperparameter tweaking process was conducted with precision to guarantee optimal model performance. The model utilized 100 decision trees, attaining an equilibrium between computational efficiency and precision. The maximum tree depth was limited to avoid overfitting, and a minimum of two samples per leaf node was mandated to guarantee significant splits. Furthermore, bootstrap sampling and feature subsampling methods were employed to incorporate randomness, improve model stability, and avert the dominance of any feature in the learning process.

Figure 13 depicts the significance of essential physical parameters affecting the performance. Each bar illustrates the contribution of a particular variable to the overall PCE and electrical attributes. Bandgap energy is the most crucial element, highlighting its pivotal function in regulating light absorption, carrier production, and voltage output. An adjusted bandgap guarantees a balance between extensive spectrum absorption and reduced thermalization losses, hence directly improving J_SC and V_OC¹⁰³. The interface defect density significantly affects the system, suggesting that non-radiative recombination at the absorber/transport layer junctions is a major loss mechanism. Mitigating interface defects via surface passivation or lattice matching can significantly improve charge extraction efficiency. Parameters like absorber thickness and acceptor density exhibit mild effects, as they affect optical absorption depth and the intensity of the internal electric field. In contrast, electron affinity, temperature, and bulk defect density exert relatively negligible effects under typical conditions. The research indicates that optimizing the bandgap and reducing interface defects are the most effective methods for enhancing perovskite solar cell performance.

Fig. 13

Identifying critical parameters for the evaluation of ML models in Al/FTO/In₂S₃/Be₃SbF₃/Cu₂O/Ni SCs.

An exhaustive evaluation of the model’s performance

The performance of the model was meticulously assessed using a range of statistical indicators, as detailed in Table 6, to guarantee the accuracy and generalizability of the RF predictions. The R² values indicated a significant correlation between predicted and experimental results, attaining 0.9894 for V_OC, 0.9881 for PCE, 0.9892 for J_SC, and 0.9814 for FF. The results demonstrate the model’s capacity to accurately represent complex physical dependencies and nonlinear connections among PV parameters. Moreover, the reduced values recorded for Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE) corroborate the model’s stability and minimum divergence from experimental standards. These indications collectively yield an overall prediction accuracy of 98.7%, highlighting the robustness and trustworthiness of the RF method in correlating material attributes with device performance results. The feature selection process was crucial in improving the model’s interpretability and computational efficiency by methodically eliminating redundant or weakly correlated variables. This enhancement maintained the model’s predictive ability while expediting training convergence and augmenting generalization to novel data. Fig. S3a–d illustrates a nearly perfect correlation between the projected and actual data points along the ideal y = x line, affirming the model’s robust consistency and minimal prediction bias. These findings confirm the RF approach as a robust and interpretable machine learning framework for quantitative prediction, sensitivity analysis, and design optimization of PSCs. Its capacity to amalgamate several performance indicators while preserving precision and stability establishes RF as a highly dependable method for directing the future development of high-efficiency, lead-free perovskite devices.

Table 6 Evaluating forecast accuracy for key PV parameters through various types of measures.

Analysis of correlation heatmap

Figure 14 displays a correlation heatmap that illustrates the linear correlations among several design parameters and the PCE of the PSC. Correlation coefficients between − 1 and + 1 signify the strength and direction of relationships, with positive values denoting direct associations and negative values representing inverse trends. The analysis indicates that absorber thickness demonstrates a modest positive association with PCE (r = 0.25), implying that increased absorber thickness somewhat improves device performance. In contrast, bandgap energy exhibits a significant negative correlation (r = − 0.65), indicating that bigger bandgaps diminish efficiency. Parameters like acceptor density (r = − 0.16) and defect density (r = − 0.11) exhibit weak negative associations, suggesting that excessive doping or structural imperfections marginally impair performance. Likewise, interface imperfections exhibit a moderate negative correlation (r = − 0.36), substantiating their adverse impact on charge transport and overall efficiency.

Fig. 14

Utilizing a correlation heatmap to illustrate the interactions between factors in the proposed SC configuration.

Additional parameters such as electron affinity, dielectric permittivity, electron mobility, hole mobility, and temperature demonstrate weak or negligible correlations, indicating minimal direct impact on PCE. The heatmap indicates that bandgap energy and absorber thickness are the primary factors affecting device efficiency. The relatively weak correlations among most variables highlight their relative independence, hence validating the application of nonlinear machine learning models such as Random Forest to elucidate the intricate interdependencies influencing solar cell performance.

SHAP-based importance analysis for comprehending feature contributions

Figure 15 presents the SHAP (Shapley Additive Explanations) summary graphic, which delineates the contribution of each design parameter to PCE as predicted by the Random Forest model. SHAP values on the x-axis measure the impact of each feature positive values denote an enhancement in PCE, whilst negative values suggest a decline in performance. The research finds bandgap energy and interface defect density as the most significant detrimental factors, demonstrating that elevated values of both parameters substantially decrease PCE.

Fig. 15

Thorough examination of variable influences on PCE using SHAP values, with rankings determined by their significance in the rf regression model.

These data align with the correlation heatmap, validating their significant detrimental effect on charge transport and recombination. Defect density exerts a slight negative impact, underscoring the necessity of reducing structural flaws. Conversely, absorber thickness exhibits a significant positive SHAP effect, indicating that an increase in thickness enhances light absorption and device efficiency. Electron mobility has a minor positive contribution, whereas temperature, electron affinity, acceptor density, hole mobility, and dielectric permittivity show minimal impact, with SHAP values around zero. The SHAP analysis identifies bandgap energy and interface defect density as primary factors limiting efficiency, whereas absorber thickness and carrier mobility serve as secondary enhancers. These insights offer essential recommendations for enhancing PSC architecture and attaining superior performance.

Evaluating model performance through parity plot and confusion matrix

The predictive efficacy of RF model was assessed by regression and classification studies, yielding a thorough comprehension of its correctness in modeling PSCs. Figure 16a illustrates that the scatter plot predicted against actual values for both the training and testing datasets demonstrates a robust correlation along the diagonal line, signifying exceptional concordance with experimental data. The substantial overlap between training (blue) and testing (magenta) points validates the model’s superior prediction accuracy and strong generalization to novel data, effectively reducing overfitting. The results underscore the RF model’s capability in identifying nonlinear relationships among essential PV parameters, including bandgap, defect density, and carrier mobility.

Fig. 16

Performance evaluation of the proposed solar cell design as demonstrated by (a) parity plot and (b) confusion matrix.

Figure 16b presents a confusion matrix that further corroborates the model’s classification efficacy in differentiating between high- and low-efficiency PSCs. The matrix demonstrates flawless classification accuracy, comprising 97 true negatives (Low PCE) and 103 genuine positives (High PCE), with no erroneous predictions. This illustrates the model’s dependability in accurately categorizing device efficiency levels without error. The regression and classification outcomes collectively validate that the RF model can properly forecast PV performance and effectively distinguish across device quality categories. This dual predictive capability enhances data-driven decision-making in material selection, structural design, and process optimization. The results highlight the revolutionary impact of machine learning in PV research, facilitating the methodical and intelligent design of next-generation PSCs with improved efficiency and stability.

Feasibility of device fabrication

To actualize the computationally predicted performance, the fabrication feasibility of Be₃SbF₃-based perovskite solar cells (PSCs) necessitates a meticulously organized experimental framework that includes material synthesis, device assembly, stability assessment, advanced characterization, and strategies for efficiency enhancement. This methodical methodology not only corroborates theoretical conclusions but also establishes a solid basis for enhancing performance in practical scenarios.

• Synthesis and preparation of materials

• 1. Synthesis of Be₃SbF₃: High-purity Be₃SbF₃ can be produced using solid-state reaction, chemical vapor deposition (CVD), or solution-based processes, contingent upon the required film thickness and scalability. Maintaining stoichiometric accuracy and phase consistency is essential, as these elements directly affect charge transfer and optical absorption efficiency.

• 2. Structural and morphological characterization: X-ray diffraction (XRD) will ascertain crystal structure and phase purity, while scanning electron microscopy (SEM) and transmission electron microscopy (TEM) will offer comprehensive insights into surface morphology, grain connectivity, and defect density.

• 3. Compositional engineering and doping: Strategic doping and alloying techniques can be employed to precisely adjust electrical conductivity and enhance stability. Inductively coupled plasma mass spectrometry (ICP-MS) and energy-dispersive X-ray spectroscopy (EDS) guarantee compositional uniformity and identify trace contaminants.

• Fabrication of devices

• 1. Solar cell construction: Layer deposition can be accomplished using spin coating, thermal evaporation, or sputtering, resulting in a planar heterojunction device structure, exemplified by Al/FTO/In₂S₃/Be₃SbF₃/Cu₂O/Ni. To avert deterioration from oxygen and moisture, manufacture must take place within an inert environment glovebox.

• 2. Quality control: The uniformity of layer production and the accuracy of thickness control can be assessed by atomic force microscopy (AFM) and profilometry. Elevated interfacial uniformity guarantees efficient charge extraction and reduces recombination losses.

• Optimization of efficiency using optical engineering

Considering that Be₃SbF₃ possesses a moderate indirect bandgap, the use of anti-reflective coatings (ARCs) can markedly improve light absorption, especially in the ultraviolet (UV) and visible spectrum regions where reflectivity losses are substantial. Materials like MgF₂, SiO₂, Si₃N₄, or TiO₂ nanolayers can serve as anti-reflective coatings to diminish surface reflection, enhance photon absorption, and elevate overall PCE¹⁰⁴. Additionally, textured transparent conducting oxides (TCOs) or graded refractive index coatings may be investigated to prolong optical path lengths and improve light trapping within the absorber layer. The integration of ARCs is particularly advantageous for low-absorption materials, as it efficiently mitigates the constraints of optical thickness without elevating recombination concerns.

• Assessment of stability

• 1. Environmental stability: Devices must undergo humidity and oxygen exposure assessments, along with regular J-V measurements to evaluate degradation kinetics. Accelerated aging experiments under constant illumination and elevated temperature circumstances can replicate long-term operating stability.

• 2. Thermal and mechanical stability: Thermal cycling and mechanical bending assessments are crucial for evaluating the durability of flexible or temperature-sensitive setups. These results confirm that Be₃SbF₃ preserves both structural integrity and efficiency under varying operational settings.

• Comparative and benchmarking evaluation

• 1. Reference cells: Performance benchmarks should be established by fabricating reference solar cells utilizing silicon and lead-based perovskites under identical conditions. A comparative assessment of J_SC, V_OC, FF, and PCE yields essential insights into the competitiveness of Be₃SbF₃ absorbers.

• 2. Simulation correlation: Experimental results must be juxtaposed with SCAPS-1D simulation data, emphasizing critical parameters such carrier lifespan, recombination rate, and defect density. Discrepancies between experimental and simulated data will inform iterative enhancements in both modeling and manufacturing processes.

• Sophisticated characterization methods

• 1. Interface and defect analysis: Interfaces and defect states can be examined by X-ray photoelectron spectroscopy (XPS) and Fourier-transform infrared spectroscopy (FTIR) to evaluate chemical bonding and energy-level alignment. Deep-level transient spectroscopy (DLTS) can provide additional quantification of trap states and charge carrier dynamics.

• 2. Microscopic and electrical mapping: Techniques such as Kelvin probe force microscopy (KPFM) and conductive atomic force microscopy (c-AFM) provide the imaging of localized potential changes and charge transport dynamics at the nanoscale, providing enhanced understanding of interfacial quality and electronic uniformity.

• Scalability and market potential

Commercializing Be₃SbF₃-based solar cells necessitates addressing problems pertaining to stability, scalable production, interface optimization, and cost efficiency. Utilizing encapsulation methods and moisture barriers can enhance environmental resistance. Concurrently, solution-processable large-area deposition techniques, like slot-die coating and blade coating, might enhance industrial scalability. Moreover, including ARCs in large-scale production could significantly enhance module efficiency without modifying absorber chemistry. Alongside economical synthesis and robust interfacial engineering, these advancements establish Be₃SbF₃ as a feasible lead-free option for next-generation, high-performance optoelectronic and photovoltaic devices. This comprehensive method connects theoretical modeling with practical verification, thereby progressing the development of commercially viable Be₃SbF₃ perovskite solar systems.

Conclusion

This study conducts a thorough examination of the structural, optoelectronic, and PV characteristics of Be₃MF₃ (M = P, As, Sb) perovskites utilizing an integrated methodology that merges DFT with SCAPS-1D simulations. First-principles of DFT computations were utilized to investigate the structural stability, electronic band structure, charge transport, and optical characteristics of these materials. All compounds demonstrated a persistent cubic phase, in accordance with theoretical predictions, and an indirect semiconducting band structure. Optical study demonstrated a wide absorption spectrum and a tight band gap, designating Be₃SbF₃ as a viable solar absorber with significant light-harvesting capability. Photovoltaic simulations utilizing Be₃SbF₃ as the absorber across four device designs revealed that the Al/FTO/In₂S₃/Be₃SbF₃/Cu₂O/Ni configuration attained exceptional performance, with V_OC of 1.79 V, J_SC of 11.04 mA/cm², FF of 89.6%, and PCE of 18.28%. The improved performance is due to advantageous band alignment, effective charge extraction, and minimal interfacial recombination. The influence of absorber and ETL thickness, defect density, acceptor density, and temperature was examined, indicating best efficiency at an absorber thickness of 1 μm, although increased temperature diminished V_OC, FF, and PCE. Further assessments of QE and J-V characteristics enhanced understanding of device functionality. The RF machine learning model was employed to enhance performance utilizing SCAPS-1D data. connection heatmap and SHAP studies revealed that bandgap energy and interface defect density are significant negative determinants of PCE, but absorber thickness exhibited a positive connection. The parity plot demonstrated exceptional prediction accuracy (R² = 0.987), while the confusion matrix attained classification accuracy. These findings collectively establish a data-driven paradigm for the design and optimization of Be₃MF₃-based PSCs, providing vital insights for next-generation, high-efficiency PV systems.