Numerical modeling and extensive analysis of an extremely efficient RbGeI₃-based perovskite solar cell by incorporating a variety of ETL and HTL materials to enhance PV performance

Md. Mojahidur Rahman ^a, Md. Hasan Ali *^b, Md. Dulal Haque ^c and Abu Zafor Md. Touhidul Islam *^a
aDepartment of Electrical and Electronic Engineering, University of Rajshahi, Rajshahi-6205, Bangladesh. E-mail: hasan@brur.ac.bd; touhid.eee@ru.ac.bd
^bAdvanced Energy Materials and Solar Cell Research Laboratory, Department of Electrical and Electronic Engineering, Begum Rokeya University, Rangpur, Rangpur-5400, Bangladesh
^cDepartment of Electronics and Communication Engineering, Hajee Mohammad Danesh Science and Technology University, Dinajpur-5200, Bangladesh

First published on 8th August 2024

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1. Introduction

Due to extensive population growth, the technological improvement of civilization, and the progression of developing countries worldwide, the increasing demands for electronic devices as well as electricity are extensively increasing the consumption and utilization of global energy.^1–3 It has been predicted that in 2050, the demand for the consumption of energy will reach a level of about 30 terawatts (TW).^1,4,5 This substantial need for energy is primarily met by the use of fossil fuels, which continue to account for more than 80% of the global energy system.^6–8 Due to the burning of fossil fuels, a tremendous amount of CO₂ has been exhausted and has contaminated the ecosystem, causing climate change, global warming, and the greenhouse effect.^9–13

The current challenge in photovoltaics is to fabricate eco-friendly solar cells with a higher value of efficiency for the conversion of photon energy, as well as longevity, stability, commerciality, and cost-effectiveness.¹⁴ There are several types of solar cells (SCs) that have technologically evolved, including silicon (Si),¹⁵ cadmium telluride (CdTe),¹⁶ antimony selenide (Sb₂Se₃),¹⁷ molybdenum sulfide (MoS₂),¹ copper indium gallium selenide (CIGS),^18,19 copper zinc tin sulfide (CZTS),²⁰ copper iron tin sulfide (CFTS),^21,22 copper indium telluride (CuInTe₂),²³ polymer, inorganic metal chalcogenide, quantum dot (QD), dye-sensitized solar cells (DSSCs), and perovskite-based solar cells.^24–26

Among the different types of PV materials, organic–inorganic (hybrid) halides exhibit various remarkable properties such as a high absorption coefficient, simple synthesis, adjustable band gap, a larger value of diffusion length, efficient solution processing ability, and low cost, and they are desirable materials because of these remarkable properties.^24,25 The first innovation of organic–inorganic halide perovskite-based solar cells (PSCs) was in 2009, with a power conversion efficiency (PCE) of 3.8% but only a few minutes of stability.²⁷ Conversely, the presence of Pb (a hazardous element in nature) in perovskite and concern regarding long-term stability are the biggest impediments to its commercialization.

To overcome these drawbacks and find a replacement for Pb, researchers have explored metal cations such as Sn²⁺ or Ge²⁺, which are divalent and have a +2-level of oxidation and a similar configuration of the outermost shell that is comparable with Pb²⁺.²⁸ The ionic radius of Sn²⁺ and Ge²⁺ is 1.35 Å and 0.73 Å, respectively, and that of Pb²⁺ is 1.49 Å. As a result, when Sn²⁺ and Ge²⁺ are used as divalent cations to substitute for Pb²⁺, the perovskite crystal structure is not affected.^24,29 The oxidization of Sn²⁺ to Sn⁴⁺ occurs more easily because the Ge²⁺ ionic radius is less than that of Sn²⁺ and Pb²⁺. Thus, Ge-based perovskite exhibits greater conductivity than Pb-based and Sn-based perovskite. According to the periodic table, a comparison of the alkali metals Rb and Cs indicates that the reactivity of Rb is lower than that of Cs (i.e., Rb < Cs).²⁴

Furthermore, RbGeI₃ possesses various promising properties such as a lower band gap of 1.31 eV, an electron affinity of 3.9 eV, a higher dielectric permittivity of 23.1 eV, and a higher electron and hole mobility of 28.6 cm² V⁻¹ s⁻¹ and 27.3 cm² V⁻¹ s⁻¹, respectively. Therefore, the choice of RbGeI₃ material as an absorber is much preferable.

In 1989, the thermometric characteristics of RbGeI₃ were explored via XRD and Raman spectroscopy by Theile et al.³⁰ In 2008, long-range-corrected density functional study for dye-sensitized solar cells was performed by Wong et al.³¹ In 2019, different characteristics, including magnetic, electrical, and structural, of RbGeI₃-based (cubic) and rubidium dysprosium oxide (RbDyO₃-based) PSCs were explored by Khursheed et al.³² In 2019, a first-principles study examining various characteristics, such as structural, electrical, and optical, of RbGeI₃-based PSCs was conducted, and several characteristics such as relative permittivity at static (ε₀), relative permittivity at high frequency (ε_∞), hole and electron effective masses energy band gap, lattice parameters, and tolerance factor (0.90) were evaluated via the PBEsol function, as explored by Jong et al.³³ In 2021, Jayan et al. investigated the optoelectronic, thermoelectric, thermodynamic, mechanical, and structural properties of RbGeI₃-based (cubic Pmm space group, three-dimensional structure) PSCs to determine different exchange correlation functions.³⁴ In 2022, Pindolia et al. researched the impact of various types of hole transport layers (HTLs) and electron transport layers (ETLs) on RbGeI₃ as the absorber layer.²⁵ In 2023, a theoretical maximum power conversion efficiency of 23.8% was achieved by Sarkar et al. through RbGeI₃ simulation using the solar cell capacitance simulator-one-dimensional (SCAPS-1D) simulator.²⁴

One of the most popular techniques for ab initio calculations of atomic, molecular, crystallographic, and surface structures and their interactions is density functional theory (DFT). Also, the material's band gap, and electronic, magnetic, and optical properties can be estimated using DFT calculations. In 2024, a DFT study of RbGeI₃ was performed by Qin et al. to discover its excellent mechanical stability and resilience as well as its performance, which was evaluated using SCAPS-1D simulation software.³⁵ The novelty of this research includes the determination of a new structure with all inorganic materials through many trials with various layer configurations for enhancing the power conversion efficiency, and also the use of low-cost, non-toxic, earth-abundant materials for different layer materials.

In this study, RbGeI₃-based solar cells with various HTLs and ETLs were simulated to propose an improved cell structure and discuss the potential of this material. The properties of these materials are derived from previously published articles. The theoretical evolution of 19 different combinations was explored by incorporating ten inorganic HTMs (BaSi₂, AgInTe₂, Cu₂Te, CdSe, SnS, CuO, Sb₂Se₃, MoS₂, CuSCN, and Sb₂S₃), as well as ten inorganic ETMs (In₂S₃, In₃Se₄, CdS, WS₂, SnS₂, IGZO, ZnSe, In₂Se₃, Cd_0.5Zn_0.5S, and TiO₂) with the RbGeI₃-based absorber. After ascertaining the configurations with the greatest potential from 19 structures, we further investigated to optimize several properties such as temperature; work function of the left metal contact; series and shunt resistance; surface recombination velocity of electrons and holes; thickness of the perovskite-based absorber layer, ETM, and HTM; carrier concentration of the perovskite absorber layer, ETM, and HTM; defect density of the perovskite-based absorber layer, ETM, and HTM; and defect density at the interface of the HTL/absorber and absorber/ETL. Additionally, the C–V attribution and Mott–Schottky plot were also investigated. Finally, a comparison with earlier research was performed using the obtained solar cell parameters.

2. Simulation process, material specifications, and device-designing methodology

2.1 Simulation process and material specifications

The numerical simulation method is crucial for quickly comprehending the physical characteristics and functionality of PV devices as well as the behavior of each device parameter without expending a great deal of money and time. There are numerous simulation tools currently available, including PC1D, AFORS-HET, ATLAS, Sentaurus TCAD, AMPS-1D, wxAMPS, and SCAPS-1D, which are frequently utilized to build and assess a photovoltaic device's performance.^36–39 Notably, SCPAS-1D demonstrated significant potential for modeling and simulating various PV device structures with flexible parameters involving spectral response, carrier generation and recombination mechanism, capacitance and voltage relation, capacitance and frequency relation, working temperature, series resistance and shunt resistance, metal contacts, and precisely calculated V_oc, J_sc, fill factor (FF), and PCE through numerical convergence and rapid single and batch calculations with an intuitive user interface.^24,40 Thus, the present investigation was conducted using the SCAPS-1D simulation tool.

Conversely, there are several restrictions on this tool. One of SCAPS-1D's restrictions is that it can only operate in one dimension. Additional drawbacks of the SCAPS-1D simulator are that it has a layer limit of seven, there is a volatile interpretation for a secondary barrier or n–p (rather than p–n) junction, and error in divergence occurs when the number of steps in simulation is limitless.^24,40 SCAPS-1D is a useful tool that was created by the Department of electronics and information systems (EIS) at the University of Gent in Belgium. Alex Niemegeers, Marc Burgelman, Koen Decock, Stefaan Degrave, and Johan Verschraegen are the researchers that have been involved in its development. The application was initially created for the CuInSe₂ and CdTe families of cell architectures.

At present, this program is also practicable for amorphous-type silicon (a-Si) cells, crystalline-type silicon (c-Si) SCs, and the GaAs family.^24,41–43 Additionally, SCAPS is a powerful tool for analyzing semiconductor equations (eqn (1)–(3)). By accounting for the boundary conditions, SCAPS-1D can solve Poisson's equation (eqn (1)), as well as the electron and hole continuity equations given in eqn (2) and (3), respectively, at every location within the device.^44–47 SCAPS-1D analyzes the operation of the photovoltaic devices by taking into account the Shockley–Read–Hall (SRH) recombination statistics:⁴²

	(1)

where E(x) denotes electrostatic potential; e denotes the charge of an electron; ε₀ and ε_r denote vacuum and relative permittivity, respectively; n(x) and p(x) denote the concentration of electrons and holes, respectively; N_A and N_D denote the charge density of acceptor and donor, respectively; and ρ_n and ρ_p denote the distribution of electrons and holes, respectively.

	(2)

	(3)

where J_n and J_p denote the electron and hole current density, respectively, and G and R denote the generation rate and recombination rate, respectively.

In semiconductors, carrier transport originates from drift and diffusion, and can be described as eqn (4) and (5).

	(4)

	(5)

where D_n and D_p denote the electron and hole diffusion coefficients, respectively, and μ_n and μ_p denote the electron and hole mobility, respectively. Table 1 shows the values of several specifications for separate layers of the proposed PSC device. Table 2 shows the value of defect density for different interfaces. Table 3 shows the values of various electrical characteristics for the front metal contact, as well as the back metal contact. Table 4 shows the values of various specifications for several HTL materials. Table 5 shows the values of various specifications for several ETL materials.

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2.2 Structure and design process for the device

PV device performance is generally evaluated by four performance specifications: J_sc, V_oc, FF, and PCE. The spectrum of the incident light (the global air mass ‘AM1.5G, 1 Sun’ spectrum at the temperature of 300 K was used in this investigation) and the optical characteristics (recombination, reflection, and absorption) affect the short circuit current density (J_sc). It can be evaluated using the formula in eqn (6):⁵⁹

	(6)

where q denotes the charge of an electron, T(λ) indicates optical transmission, ϕ_i indicates spectral power density, and Δλ_i indicates the interval between the two subsequent values of wavelength.

When there is no net current flowing through the solar cell, open circuit voltage (V_oc) is created and can be calculated using the formula in eqn (7):⁶⁰

	(7)

where n, k, T, J_sc, and J_o denote the diode ideality factor, Boltzmann constant, ambient temperature, short circuit current density, and reverse bias saturation current density, respectively.

The solar cell's maximum power is determined by the fill factor (FF) and is stated in eqn (8):⁶¹

	(8)

where is denoted as normalized V_oc.

The efficiency for the photon's power conversion (PCE) is determined by eqn (9):⁶⁰

	(9)

Herein, P_in denotes the input photon's power from the sun (1000 W m⁻² was used in this investigation).

To finalize the structure of a PV device, a study was performed to obtain additional information regarding the performance of the device with a combination of several HTLs and several ETLs. Unfortunately, charge transport materials (CTMs) consist of organic compounds that are very unstable.^62–65 Additionally, because of their simplest manufacturing technique, inorganic CTMs are inexpensive. Inorganic CTMs exhibit greater thermodynamical and chemical durability than organic CTMs. In comparison with organic CTMs, inorganic CTMs exhibit a wider band gap, greater mobility of the charge carrier, and superior transparency to the different radiations of visible, infrared, and ultraviolet.^66–69

Therefore, in this investigation, to ascertain the primary structure of the PSC device, a PSC device based on inorganic RbGeI₃ was investigated by incorporating ten inorganic HTMs (BaSi₂, AgInTe₂, Cu₂Te, CdSe, SnS, CuO, Sb₂Se₃, MoS₂, CuSCN, Sb₂S₃) as well as ten inorganic ETMs (In₂S₃, In₃Se₄, CdS, WS₂, SnS₂, IGZO, ZnSe, In₂Se₃, Cd_0.5Zn_0.5S, TiO₂). The values of thickness and concentration of the charge carrier of the RbGeI₃-based absorber layer were initially set as 800 nm and 10¹⁶ cm⁻³ respectively, and that of each HTL as 50 nm and 1 × 10¹⁹ cm⁻³, and each ETL as 50 nm and 1 × 10¹⁵ cm⁻³, respectively, while other specifications remained the same as in Tables 1–3. Simulation was performed for different configurations with the combinations of various HTLs and ETLs, and all the primary data on performance parameters (such as V_oc, J_sc, FF, and PCE) were recorded.

Table 6 shows the data for four performance parameters, and the maximum device performance was obtained when Sb₂S₃ and TiO₂ were individually applied as HTM and ETM. Also, Sb₂S₃ possesses superior optoelectronic properties such as an appropriate band gap (∼1.62 eV), a superior absorption coefficient (∼10⁵ cm⁻¹), ingredient abundance on earth, minimal toxicity, and a low-cost production procedure.⁵² Additionally, the valence band offset (VBO) of Sb₂S₃ and the conduction band offset (CBO) of TiO₂ are much lower in comparison with other HTLs and ETLs.

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After numerous trials, the Al/ITO/TiO₂/RbGeI₃/Sb₂S₃/Pt structure proved to be an effective primary structure. Hence, further investigation, i.e., optimization of several properties such as temperature; work function of the left metal contact; series and shunt resistance; surface recombination velocity of electrons and holes; thickness of the perovskite-based absorber layer, ETM, and HTM; carrier concentration of the perovskite absorber layer, ETM, and HTM; defect density of the perovskite-based absorber layer, ETM, and HTM; and defect density at the interface of HTL and the absorber layer, and also the absorber layer and ETL, were conducted with the structure of the PSC device as Al/ITO/TiO₂/RbGeI₃/Sb₂S₃/Pt.

2.3 Finalized structure of the device

Fig. 1(a) shows the finalized and optimized structure (Al/ITO/TiO₂/RbGeI₃/Sb₂S₃/Pt) of the proposed perovskite-based PV device. It was determined that p-type RbGeI₃ was the most attractive Pb-free, highly stable, efficient, and low-cost perovskite-based absorber material. On the top of the absorber, n-type TiO₂ ETL was used as a heterojunction associated with the RbGeI₃ absorber. It formed an n–p heterojunction interface that enhanced the collection of photogenerated electrons (PGEs) with blocking holes. On top of the TiO₂ layer, highly transparent, conductive, and low-cost n⁺-type ITO was applied as a window layer. At the bottom of the absorber (RbGeI₃), a p⁺-type Sb₂S₃ HTL was used to effectively collect photogenerated holes (PGHs) with blocking electrons that enhanced the overall efficiency of the proposed PSC. Aluminum (Al) was used as the front metal contact material, and platinum (Pt) was used as the back metal contact material. Fig. 1(b) shows the energy band diagram of the suggested device, which shows that there is a low VBO and adequately large CBO at the Sb₂S₃/RbGeI₃ interface, and the band configuration between Sb₂S₃ and RbGeI₃ is excellent for ensuring that hole transmission from the absorber as well as electron opposition are facilitated. Also, because of the adequately large VBO and very low CBO at the RbGeI₃/TiO₂ interface, the band configuration between RbGeI₃ and TiO₂ is excellent for ensuring smooth electron transmission from the absorber and facile hole opposition.

3. Results and discussion

3.1 Consequences of ETL thickness and carrier concentrations on the performance of the device and their optimization

There is considerable impact of the ETL's (TiO₂) carrier concentration as well as thickness on PSC performance, and they can be used to improve it. Therefore, when developing highly efficient PSCs, it is important to carefully select the ETL's characteristics. An optimal ETL for PSCs can assist in decreasing recombination currents and increasing transmittance.⁷⁰ In this investigation, the ETL's carrier concentration and thickness were varied consecutively within the range of 10¹²–10¹⁹ cm⁻³ and 10–100 nm, while other specifications remained the same, as in Tables 1–3.

The variation in the device's performance according to the ETL's thickness is demonstrated in Fig. 2(a), which shows that all the PV performance parameters, i.e., V_oc, J_sc, FF, and PCE, remain unchanged for ‘with HTL’ and ‘without HTL.’ All the PV parameters have a higher value ‘with HTL’ than ‘without HTL.’ Because there is no significant alteration in the PSC performance as the ETL's thickness is varied, 50 nm was selected as the optimal thickness for the ETL (TiO₂) layer to minimize fabrication costs.

The variation in the device's performance according to the ETL's carrier concentration is demonstrated in Fig. 2(b). Although there were lower values without HTL than with HTL for all the PV parameters, they remained nearly unchanged, except for FF as the carrier concentration, which was varied. The FF increased very slightly beyond 10¹⁶ cm⁻³. Because this occurred as doping concentrations increased, they also raised the internal electric field of the ETL, thus increasing the transport velocity of the charge carriers and thereby conductivity.⁷¹ The PV parameters have a higher value with HTL than without HTL, and they remained unchanged up to 10¹⁸ cm⁻³. After that, V_oc, J_sc, and PCE slightly decreased, but FF significantly decreased. The reason behind this is that coulomb traps resulted due to high doping concentrations, which in turn decreased the electron mobility.⁷² Therefore, it was determined that the optimal carrier concentration for the ETL (TiO₂) layer was 10¹⁷ cm⁻³.

3.2 Consequences of the absorber's thickness and carrier concentrations on the device's performance and their optimization

The perovskite layer's thickness and carrier concentration are critical elements for improving the PSC's performance, and they influence diffusion length, photocarrier generation, carrier lifetime, and extraction. As the absorber layer thickness increases, the generation and recombination rates of the carriers also increase.⁷³ In this investigation, the absorber layer's thickness and carrier concentration varied with the range (100–1000) nm and (10¹²–10²¹) cm⁻³ respectively, while other specifications remained the same as in Tables 1–3. The alteration in the device's performance according to absorber thickness is demonstrated in Fig. 3(a), where the open circuit voltage (V_oc) significantly increased up to 500 nm for the ‘without HTL’ configuration, and after that, it slightly increased. Although the V_oc for the ‘with HTL’ configuration is higher than that ‘without HTL,’ it slightly diminished with the initial increasing thickness. The reason behind this is that the recombination increases with the increase in absorber thickness, and hence the saturation current also increases.⁷² The current density (J_sc) significantly increased up to 600 nm with the increase in thickness of the absorber layer. The significant increase in photon absorption is the primary cause of the rise of J_sc.⁷² Beyond the 600 nm thickness of the absorber, the current density also increased, but not significantly in comparison. The reasons are that the photon absorption also increases, but not significantly, and the greater absorber layer influences charge carrier diffusion lengths, enhancing photogenerated carrier recombination via SRH recombination.⁷⁴ The fill factor (FF) for ‘without HTL’ significantly increased to 200 nm and that is in the opposite phase for “with HTL”, and after that, the change in FF for both “with HTL” and “without HTL” is slight but in the opposite phase. In the case of “with HTL” structures, the reason for abrupt decreases in FF is that the series resistance (R_s) is growing as the thickness of the cell rises.^75,76 However, the FF is boosted in the ‘with HTL’ configuration rather than ‘without HTL.’ Because the PCE is determined by the parameters V_oc, J_sc, and FF as presented in eqn (9). Although V_oc and FF are slightly decreased for ‘with HTL,’ as a result of a significant increase in J_sc up to 600 nm thicknesses, the PCE significantly intensified up to 600 nm. The PCE (without HTL) increased due to the same reason but is lower than ‘with HTL.’ Therefore, it was determined that 600 nm was the optimal value of the thickness for the absorber (RbGeI₃), and this value was used to reduce fabrication costs.

The device's performance varied according to the absorber's carrier density, as demonstrated in Fig. 3(b). According to Fig. 3(b), the V_oc remained unchanged with an absorber carrier concentration of up to 10¹⁶ cm⁻³ for the ‘with HTL’ and ‘without HTL’ structures, and after that, it significantly increased. The key reason for this is because as the concentration of the carrier increases, the saturation current of the entire cell also increases, which reinforces the increase in V_oc.²¹ Although the V_oc is lower for ‘without HTL’ than ‘with HTL,’ the J_sc remained unchanged with the variation of the carrier concentration for the ‘with HTL” structure. Thus, there is no substantial alteration of photon absorption within the variation of the carrier concentrations, and that is the principal reason for a lack of change in J_sc. The carrier concentration accelerates the recombination process by incorporating traps or recombination centers into the layer. At the front, the collection of the photogenerated carriers is then limited by this recombination process.^1,77 The increase in recombination rate may be the same as the increasing rate of J_sc due to the increase in carrier concentration, and hence, no significant change in J_sc occurred. However, for the ‘without HTL’ structure, the J_sc is the same as that for the ‘with HTL’ structure. Up to 10¹⁶ cm⁻³ and beyond that, it rapidly decreased up to 10¹⁸ cm⁻³, and after that, it gradually decreased. This occurred because the recombination rate may be higher due to the rapid decrease in J_sc. The FF for the ‘with HTL’ configuration remained unchanged up to 10¹⁵ cm⁻³, but after that, it significantly increased up to 10¹⁷ cm⁻³, and beyond that, it slightly increased. The FF for ‘without HTL’ is less than that for the ‘with HTL’ configuration and remained unchanged up to 10¹⁶ cm⁻³; after that, it slightly decreased up to 10¹⁷ cm⁻³, and beyond that, it significantly increased. The fact behind this is that the increase in V_oc is due to an increase in carrier concentration. Because the PCE is determined by the parameters V_oc, J_sc, and FF, as presented in eqn (9), the J_sc remained unchanged for the ‘with HTL’ configuration, but as a result of a major increase in V_oc as well as FF, the PCE then significantly increased up to 10¹⁹ cm⁻³. However, the PCE for ‘without HTL’ significantly decreased due to a considerable diminution of J_sc, although there was a slight increase in V_oc and FF. The optimal value of the carrier concentration for the absorber (RbGeI₃) layer was therefore taken as 10¹⁹ cm⁻³.

3.3 Consequences of the HTL's thickness and carrier concentrations on device performance and their optimization

The HTL's (Sb₂S₃) thickness and the concentration of the carrier have the ability to greatly influence the PSC's performance, and can improve it. The HTL increased the SC effectiveness due to minimization of the surface recombination velocity (SRV) by producing a strong field near the rear electrode.¹ To examine the consequences of concurrent changes in thickness and concentration of the carrier of the HTL (Sb₂S₃) on the PSC device performance, a contour map was constructed with simultaneous variation of thickness and density of the carrier of the HTL (Sb₂S₃) within the range of 10–200 nm and 10¹⁴–10²¹ cm⁻³, respectively, while other specifications remained the same as in Tables 1–3. Fig. 4 illustrates the influence of simultaneous variation of thickness and concentration of the carrier of the HTL on the parameters that determine the performance of the PV devices.

According to Fig. 4(a), for the range of the variation of carrier density from 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³, the V_oc increased very negligibly within the increase of thickness for the range of 10 nm to 60 nm, and there was no alteration in the V_oc after 60 nm. Beyond 8 × 10²⁰ cm⁻³, the change in thickness did not considerably impact the V_oc of the proposed PSC device. Also, for a particular thickness, the V_oc was nearly unchanged for the range of 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³, and the V_oc slightly increased beyond 8 × 10²⁰ cm⁻³. The reason for this negligible increase in V_oc is that the Sb₂S₃ HTL produced an advanced built-in potential at the Sb₂S₃/RbGeI₃ interface,^56,78 as well as an increase in J_sc.⁷⁹

Fig. 4(b) shows a very negligible increase in J_sc within the increase in thickness from 10 nm to 200 nm for the range of carrier concentrations of 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³, and the J_sc slightly increased beyond 8 × 10²⁰ cm⁻³ within the increase in thickness. Also, for the range of thickness of 10–100 nm, J_sc was almost unchanged for the range of 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³, and for further than 8 × 10²⁰ cm⁻³, the J_sc slightly increased for the range of thickness of 100–200 nm. Photon absorption at longer wavelength spectra results in a slight increase in electron–hole pair generation through the tail-states-assisted (TSA) two-step photon upconversion process, which is the main reason for the slight increase in J_sc.^16,19,56

According to Fig. 4(c), for the range of 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³ for carrier concentration, the FF increased very negligibly within the increase in thickness for the range of 10 nm to 30 nm, and afterwards, the FF remained nearly unaffected. Beyond 8 × 10²⁰ cm⁻³, the variation in thickness had no impact on the FF of the proposed PSC device, and this occurred because V_oc has no impact on the thickness variation. Additionally, for a particular thickness, the FF was nearly unchanged from 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³. Away from 8 × 10²⁰ cm⁻³, the FF slightly increased because higher carrier concentrations reduced the series resistance of the cell, hence grading the FF of the cell.⁸⁰

From Fig. 4(d), it is clear that the PCE increased very negligibly for the range of carrier concentrations from 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³ within the increase in thickness for the range of 10 nm to 60 nm, and afterward, the PCE remained nearly unaffected. Beyond 8 × 10²⁰ cm⁻³, the variation in thickness had no impact on the conversion efficiency of the photon's power in the proposed PSC device. Additionally, for a particular thickness, the PCE was nearly unchanged for the range of 10¹⁴ cm⁻³ to 8 × 10²⁰ cm⁻³, and outside of 8 × 10²⁰ cm⁻³, the PCE slightly increased. This occurred because of all the combined effects of V_oc, J_sc, and FF.

Sb₂S₃ with a small thickness cannot offer a contact channel of low resistance, whereas increased layer thickness prevents carriers from transferring to an electrode. Hence, extending the length of diffusion and allowing recombination to occur via a trade-off may be considered to determine the optimal value of the HTL's thickness, which was determined to be 50 nm. Also, the reasons behind the increase in overall performance are that increased carrier concentration moves the Fermi level towards the valence band (VB), which accelerates the hole build-up at the anode via creation of an ohmic contact with the rare metal contact,⁸¹ and also due to increased carrier concentration. The HTL's internal electric field also increases, which implies a higher transport velocity for the charge carriers, and consequently, the conductivity.⁷² The HTL's carrier concentration should be greater than the absorber layer's carrier concentration, and therefore, the optimal HTL carrier concentration was chosen as 10²¹ cm⁻³.

3.4 Consequences of the bulk defect density of different layers on device performance and their optimization

3.5 Consequences of defect densities at different interfaces on the performance of the device and their optimization

3.6 Consequences of different resistances on device performance and their optimization

The electrical resistances in the PSCs are introduced by the metal contacts used as front and back contacts in the perovskite absorbers, HTLs, and ETLs. The series (R_s) and shunt (R_sh) resistances play a key role in influencing the PV characteristics of SCs. Therefore, it is obligatory to understand the impacts of R_s and R_sh to improve the cell function. The Shockley equations determine the J–V properties of SCs when considering ideal one-sun illumination conditions, as in eqn (11) and (12).^86,87

	(11)

where q, A, k, T, J_sc, J_ph, J_o, V, R_s, and R_sh denote the elementary charge, ideality factor of the heterojunction, Boltzmann constant, ambient temperature, short circuit current density, photo-irradiated constant current density, reverse bias saturation current density, output voltage, and series and shunt resistances, respectively.

Because of an open-circuit condition, i.e., J_sc ≈ 0 mA cm⁻², V_oc is denoted as follows:

	(12)

3.7 Consequences of surface recombination velocity (SRV) on device performance and its optimization

The consequences of SRV at the back metal contact on the performance-determining parameters of the planned PSC device were thoroughly investigated, as seen in Fig. 8.

3.8 C–V attribution and Mott–Schottky plot analysis of the device

3.9 Consequences of the back metal contact work function on device performance and its optimization

The back metal contact's work function exerts a crucial influence on the PSC's stability and performance. To gather holes from the external circuit on the HTM layer, an electrode of metal is situated. Making contact with an ohmic type is a prerequisite for the suitable assemblage of the charge of the majority carrier, i.e., a hole through the contact of the back metal. In this investigation, to study the consequence of the work function of the back metal on the performance-determining parameters of PV devices, ten back metal work functions were used (5.00 eV (C, Co, Ge, Ir), 5.04 eV (Ni), 5.12 eV (Pt), 5.22 eV (Ni, Pd, Pt, W), 5.31 eV (Au), 5.35 eV (Ni), 5.37 eV (Au), 5.42 eV (Ir), 5.47 eV (Au), and 5.60 eV (Pd)),^48,49,94 while other specifications remained the same, as in Tables 1–3. Fig. 11(a) illustrates the influence of alteration of the back metal work function on the performance-determining parameters of the PV devices. For the ‘without HTL’ structure, all the performance-determining parameters of the PV device, i.e., V_oc, J_sc, FF, and PCE, remained almost unchanged until 5.22 eV. After that, they substantially increased to 5.42 eV, and beyond that, they again remained unchanged. The greater the metal's work function for the back contact, the lower the barrier height, and therefore, from the absorber, the transmission of holes becomes smoother. Consequently, cell efficiency has been substantially increased.⁵² The figure also shows that there is no consequence of the back metal work function on the device's performance for the ‘with HTL’ structure. Hence, a suitable back metal, Pt (5.22 eV), was used in this investigation.

3.10 Consequences of temperature on device performance

Temperature crucially impacts the PSC's stability and performance. In this investigation, real environmental conditions were used to study the impact of temperature on the performance-determining parameters of the PV devices and the stability of the proposed PSC device. The temperature was varied within the range of 280–500 K, while other specifications remained the same, as in Tables 1–3. The influence on the performance value due to the alteration of the temperature is demonstrated in Fig. 11(b). According to the figure, for ‘with HTL’ and ‘without HTL,’ except for J_sc, all the performance parameters of the PV device, i.e., V_oc, FF, and PCE, significantly decreased from the initiation of the temperature variation. More electron–hole pairs were formed with the increase in temperature, which implies increasing recombination rates between energy bands. The reverse saturation current was increased by the carrier's internal recombination rate, which lowered the V_oc,⁹⁵ and it can be realized by eqn (7). In addition, raising the temperature increased interface defects, which may be responsible for lowering V_oc. When the temperature was increased, the FF and PCE values decreased, which was probably due to a decrease in the shunt resistance.⁹⁶ However, all the PV performance parameters ‘with HTL’ have a higher value as compared to ‘without HTL.’ Fig. 11(b) recommends that to chose at room temperature (300 K) of the proposed PSC device with HTL.

3.11 Optimized features of the proposed PSC device

4. Conclusion

In this investigation, various HTMs and ETMs were incorporated into the Pb-free perovskite-based (RbGeI₃) absorber to discover the primary structure of the proposed SC device with higher performance. To evaluate the PV performance of the proposed PSC device for different structures, the SCAPS-1D simulator was utilized. After many trials, Al/ITO/TiO₂/RbGeI₃/Sb₂S₃/Pt proved to be an effective primary structure. After obtaining the finalized structure, different parameters such as temperature; work function of the left metal contact; series and shunt resistance; surface recombination velocity of holes and electrons; perovskite layer, ETM, and HTM thickness; perovskite layer, ETM, and HTM carrier concentration; defect density of the perovskite layer, ETM, and HTM; and HTL/absorber and absorber/ETL interface defect density were varied so that the proposed device would be more practical. For the ‘without HTL’ and ‘with HTL’ configurations, all the influencing factors were optimized. The optimized thickness of the ETL, absorber, and HTL for the ‘without HTL’ and ‘with HTL’ configurations was found to be 50 nm, 600 nm, and 50 nm, respectively. Also, the optimized carrier concentrations for the ETL, absorber, and HTL for ‘without HTL’ and ‘with HTL’ were found to be 10¹⁷ cm⁻³, 10¹⁹ cm⁻³, and 10²¹ cm⁻³, respectively. Moreover, the temperature impact also ensured that the suggested device was unaffected by the typical operating temperature range of 300 K to 321 K. The optimized structure ‘without HTL’ showed optimal values for V_oc, J_sc, FF, and PCE of 0.923 V, 24.97 mA cm⁻², 84.26%, and 19.42%, respectively, whereas the optimized structure ‘with HTL’ showed optimal values for V_oc, J_sc, FF, and PCE of 1.067 V, 33.15 mA cm⁻², 85.82%, and 30.35%, respectively. Comparative performances with other structures were also studied, and the investigated new structure seems to be important in the long run for experimental designs.

Author contributions

Md. Mojahidur Rahman – conceived, designed, and performed the simulation process; analyzed and interpreted the data; prepared figures; and wrote the manuscript. Md. Hasan Ali – conceived, designed, analyzed, and interpreted the data; prepared figures; wrote, reviewed, and edited the manuscript. Md. Dulal Haque – conceived, designed, and analyzed the data; and reviewed the manuscript. Abu Zafor Md. Touhidul Islam – conceived, designed, and interpreted the data; and reviewed the manuscript.

List of abbreviations

Abbreviations Elaborations

PV	Photovoltaic
TW	Terawatts
CTL	Charge transport layer
HTL	Hole transport layer
ETL	Electron transport layer
FF	Fill factor
PCE	Power conversion efficiency
QE	Quantum efficiency
PGC	Photo generated carrier
PGH	Photo generated hole
PGE	Photo generated electron
PSC	Perovskite solar cell
CB	Conduction band
VB	Valence band
CTM	Charge transport material
HTM	Hole transport material
ETM	Electron transport material
TSA	Tail-states-assisted

List of symbols

Symbols Meaning

L	Thickness
E g	Band gap
χ	Electron affinity
ε r	Dielectric permittivity (relative)
N C	CB effective density of states
N V	VB effective density of states
v T,n	Electron thermal velocity
v T,p	Hole thermal velocity
μ n	Electron mobility
μ p	Hole mobility
N D	Shallow uniform donor density
N A	Shallow uniform acceptor density
σ n	Capture cross section (electrons)
σ p	Capture cross section (holes)
N t	Total defect density (bulk)
n t	Total defect density (interface)
R s	Series resistance
R sh	Shunt resistance
Φ	Work function
ε 0	Dielectric permittivity (vacuum)
ρ p	Hole distribution
ρ n	Electron distribution
J p	Hole current density
J n	Electron current density
J sc	Short circuit current density
J o	Reverse bias saturation current density
q	Charge of an electron
n	Diode ideality factor
k	Boltzmann constant
T	Ambient temperature
V oc	Open circuit voltage
η	Power conversion efficiency

Data availability

The data that has been used is confidential. The data will be available upon reasonable request to the corresponding author(s).

Conflicts of interest

There are no conflicts of interest among the authors.

Acknowledgements

This research has not been funded by any research organization or university. The authors express their heartfelt appreciation to Dr Marc Burgelman and his colleagues at the EIS Department at the University of Gent in Belgium for providing us with the opportunity to conduct research using the SCAPS-1D software.

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