Numerical simulation of inorganic Cs2AgBiBr6 as a lead-free perovskite using device simulation SCAPS-1D

Double perovskite, Cs2AgBiBr6, is introduced as a lead-free perovskite solar cell. Device modeling of Cs2AgBiBr6 (DP) was accomplished to obtain the optimum parameters using the Solar Cell Capacitance Simulator (SCAPS). Two devices with two different hole transport layers (HTLs) were investigated, including P3HT and Cu2O. For both devices with different HTLs, an optimal thicknesses of 1200 nm and defect densities of 1.0×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.0\times$$\end{document} 1014 cm−3 for DP layer were attained. For both HTLs, conduction band offset, CBO, is − 0.21 eV and valence band offset, VBO, is + 0.16 eV. For shallow acceptor doping concentration of P3HT and Cu2O, the values of 5.0×1019\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.0\times {10}^{19}$$\end{document} and 5.0×1017cm-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.0\times {10}^{17}\,{\mathrm{cm}}^{-3}$$\end{document} were obtained, respectively. As far as the shallow donor density of electron transport layers (ETLs) is concerned, for both cases, the optimum value of 5.0×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.0\times$$\end{document} 1019 cm−3 were achieved. For capture cross section,σn,p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p}$$\end{document}, in absorber layer for both HTLs, the optimal value at σn,pof10-20cm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p} \mathrm{of} {10}^{-20}\, {\mathrm{cm}}^{2}$$\end{document} for Nt,DP(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{t,DP} ($$\end{document}defect density of DP) is1016cm-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{is} {10}^{16} \,{\mathrm{cm}}^{-3}$$\end{document}, at σn,pof10-19cm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p} \mathrm{of} {10}^{-19} \,{\mathrm{cm}}^{2}$$\end{document} for Nt,DPis1015cm-3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{t,DP}\mathrm{ is} {10}^{15} \,{\mathrm{cm}}^{-3},$$\end{document} at σn,pof10-18cm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p} \mathrm{of} {10}^{-18} \,{\mathrm{cm}}^{2}$$\end{document} forNt,DPis1014cm-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{t,DP}\mathrm{ is} {10}^{14} \,{\mathrm{cm}}^{-3}$$\end{document}, at σn,pof10-17cm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p} \mathrm{of} {10}^{-17}\, {\mathrm{cm}}^{2}$$\end{document} forNt,DPis1013cm-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{t,DP}\mathrm{ is} {10}^{13}\, {\mathrm{cm}}^{-3}$$\end{document}, and at σn,pof10-16cm2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sigma }_{n,p} \mathrm{of} {10}^{-16} \,{\mathrm{cm}}^{2}$$\end{document} forNt,DPis1012cm-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N}_{t,DP}\mathrm{ is} {10}^{12} \,{\mathrm{cm}}^{-3}$$\end{document}. For P3HT device, the interface defect density of P3HT/Cs2AgBiBr6 is occurred at 1.0 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 1014 cm−2, and for Cs2AgBiBr6/SnO2 is happened at 1.0 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 109 cm−2. For Cu2O device, the interface defect density of Cu2O/Cs2AgBiBr6 is befallen at 1.0 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 1013 cm−2, and for Cs2AgBiBr6/SnO2 is happened at 1.0 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 1010 cm−2. As for radiative recombination, for P3HT device, the optimal value is happened at 2.3 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 10−13 cm3/s, however, for Cu2O device is occurred at 2.3 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 10−12 cm3/s. Finally, for P3HT device, a maximum power conversion efficiency, PCE, of 11.69% (open-circuit voltage, Voc, of 2.02 V, short-circuit current density, Jsc, of 6.39 mA/cm2, and fill-factor, FF, of 0.90 (90%)) were achieved, and for Cu2O device, a PCE of 11.32% (Voc of 1.97 V, Jsc of 6.39 mA/cm2, and FF of 0.895 (89.5%)) were attained. This is the highest efficiency for Cs2AgBiBr6 double perovskite solar cell which was achieved till now. Finally, our results are providing towards fabricating a lead-free and inorganic solar cell.


Introduction
Perovskite solar cells (PSCs) have gained more than 25% efficiency within a very short period (Usiobo et al. 2020). Although PSCs have some advantages such as indicating high charge carrier mobilities, high absorption coefficients, direct and tunable band gaps, long carrier diffusion lengths, and high performance (Zhang and Zhu 2020;Gu et al. 2020;Li and Zhang 2020), they endure from instability issue, mostly owing to perovskite material deterioration upon moisture and light exposure Singh et al. 2020;Lee and Park 2020;Yang et al. 2020a, b). Moreover, the stability of the PSCs are researching (Bouich et al. 2021a, b;Mohseni et al. 2021). Despite some advantages, the presence of lead and the toxicity in these kind of cells is a major blockage to its commercialization (Xu et al. 2020). In order to overcome the existing shortcoming related to the presence of lead in the perovskite solar cell, the main competitor is to replace tin (Xu 2021;Poli et al. 2021;Liu et al. 2020a, b;Jiang et al. 2020;) and germanium (Soto-Montero et al. 2020;Liu et al. 2020a, b) belonging to the same lead group, therefore we expect the exhibition of similar properties. For lead-free perovskite solar cells, since tin has the same diameter as lead, the scientists are encouraged to replace tin with lead to form ASnX 3 (Ogomi et al. 2014;Zuo et al. 2014;Noel et al. 2014;Liao et al. 2016;Xi et al. 2017;Shi et al. 2017). However, Sn 2+ ion is unstable in these materials and easily oxidize to Sn 4+ which results in degradation of photovoltaic performance.
The other way to develop the perovskite family for photovoltaic application is to replace Pb 2+ ion with one monovalent B + and one trivalent B 3+ ion, forming A 2 B + B 3+ X 6 , double perovskite (DP) structure called elpasolite (Zhao et al. 2017). More than 300 diverse elpasolites have been synthesized till now (Giustino and Snaith 2016). Nevertheless, so far only three of them, including Cs 2 AgBiBr 6 (Slavney et al. 2016), Cs 2 AgBiCl 6 (McClure et al. 2016) and (CH 3 NH 3 ) 2 AgBiBr 6 ) have been produced. The bandgap engineering of Cs 2 AgBiBr 6 has been done by inducing defects and alloying of triple metals (Slavney et al. 2017;Du et al. 2017). Among the double perovskite family, Cs 2 AgBiBr 6 is a suitable absorbing layer in solar cell devices, because it has high crystallinity, encouraging photovoltaic properties including long carrier recombination lifetime, good stability against air and moisture, low carrier effective masses and both thermal stability and fabricated in the ambient as compared to the perovskite structure, and long carrier diffusion length of around 700 nm (Yang et al. 2020a, b;Longo et al. 2020;Burwig et al. 2020;Ghasemi et al. 2020;. While the double perovskite solar cell is more stable as compared to perovskite cell, it has much lower inherent efficiency than that of perovskite cell (Wu et al. 2018). For instance, for the device of ITO/SnO 2 /Cs 2 AgBiBr 6 /P 3 HT/Au, the efficiency was attained to be 1.44% experimentally (Wu et al. 2018).
Although P 3 HT (poly(3-hexylthiophene-2,5-diyl)) is a good candidate material for the realization of PSC industrialization, P 3 HT endures from some inherent drawbacks, such as elevated levels of occupied molecular orbital energy (HOMO) and narrow visible light absorption range (< 650 nm), leads to low efficiency and low V oc Chow and Someya 2020). Lately, it has been reported that Cu 2 O (cuprous oxide) has emanated as a strong option for the HTL owing to its very high mobility, optimal direct bandgap and significantly inorganic, consequently it has low degradability characteristic (Mkhalid and Shawky 2021;Islam et al. 2020a, b). Besides, 1.0 gr of P 3 HT is around 547 EURO depending on purity degrees, while 1.0 gr Cu 2 O is around 11.76 EURO depending on purity degrees (www. sigma aldri ch. com/ germa ny. html). As far as price is concerned, it rises to the attention that Cu 2 O is the best example between two HTLs, in consequence of the lower cost. Islam et al. 2020a, b used wxAMPS software to find the efficiency of Cs 2 AgBiBr 6 double perovskite and as they mentioned in the conclusion of their report, they obtained the effiecency of 8.11 certainly. However, for the other double perovskite such as Cs 2 TiBr 6 , the efficiency in the experiment was attained to be 2.15% . After performing optimization, the efficiency is reached to 11.49%, which means that efficiency is improved theoretically by 434.41% (Ahmed et al. 2021).
In this work, the Cs 2 AgBiBr 6 double perovskite is offered as a lead-free perovskite as an active layer in solar cell. Two devices with two different HTLs including P 3 HT and Cu 2 O are comperehensively surveyed in the simulation using SCAPS software and optimizing widely most of the parameters to get the highest efficiency.

Methodology
The Solar Cell Capacitance Simulator (SCAPS) is implemented in this research. The ver. 3.3.07 of this package established by University of Gent is used in our calculations (Burgelman et al. 2000). The SCAPS software is one dimensional simulation program that computes energy bands, concentrations, J-V characteristics, ac characteristics (C (Capacitance) and G (Conductance) as functions of V and f) and spectral response (QE) using the three basic semiconductor equations comprise of the continuity equations for hole and electron and Poisson's equation, as shown in Eqs. (1)-(3) under the steady-state conditions, where G n , G p indicates the generation rate of electrons and holes, n and p are electron and hole mobilities, is the electric field, D is diffusion coefficient, is the life time of electrons and holes, is permittivity, is electrostatic potential, q is electron charge, p t (x) and n t (x) are the concentrations of trapped holes and electrons, N D and N A are shallow donor and acceptor concentrations. Figure 1 shows the schematic diagram of multilayer used in the double perovskite solar cell. The device contains of indium tin oxide (ITO)/electron transport layer (SnO 2 )/ absorber layer (Cs 2 AgBiBr 6 )/hole transport layer (P 3 HT)/Au (Wu et al. 2018). We used the following arrangement to perform the simulation. (2) dp n dt = G p − p n − p n 0 p + p n p d dx + p dp n dx

Device structure and simulation parameters
In Table 1, the simulation parameters of all layers such as thickness, band gap energy E g , electron affinity χ, relative permittivity ε r, , effective density of states of conduction band N C , effective density of states of valence band N V , mobility of electron μ n , mobility of hole μ p , shallow donor density N D, shallow acceptor density N A and defect density N t are presented.
Some other parameters and settings are given in the following. The thermal velocity of electron and hole is 1.0 × 10 7 cm/s. The N ts suggests the defect density at the interface. The N ts quantities at SnO 2 /Cs 2 AgBiBr 6 and Cs 2 AgBiBr 6 /P 3 HT interfaces are fixed to 2.3 × 10 10 cm −2 to achieve the recombination velocities of 0.74 cm/s at both interfaces. The defect energy level is selected at the center of band gap and defect type is neutral. The energetic distribution is Gaussian and its characteristic energy is 0.1 eV. R s is the series resistance with the unit of Ω cm 2 . The N t of the Cs 2 AgBiBr 6 , shown in Table 1, is fixed to 3.678 × 10 16 cm −3 , as stated by the following formula: Fig. 1 Device multilayer structure, ITO/SnO 2 /Cs 2 AgBiBr 6 / P 3 HT/Au, used in solar cells where ε 0 is the vacuum permittivity and ε is the dielectric permittivity of Cs 2 AgBiBr 6 film. L is the thickness of the Cs 2 AgBiBr 6 film and q is the elementary charge. The V TFL (the trap-filled limit voltage) of the Cs 2 AgBiBr 6 is 1.29 V, according to the Supporting Information of reference (Wu et al. 2018). For this compound, the carrier lifetime is 843.6 ns, L n of 5.1 μm and L p of 1 μm (Wu et al. 2018). Moreover, the optical reflection of front contact, see Fig. 1, is assumed to be 40%. The obtained capture cross section of electron and hole is 3.22 × 10 −18 cm 2 , according to the following formula: where n,p is capture cross section of electron and hole, N t is defect density of Cs 2 AgBiBR 6 layer, V th is thermal velocity of electron and hole and n,p is the carrier lifetime of electron and hole. Right and left contact work function are 4.8 eV (ITO) and 5.1 eV (Au), respectively. Pre-factor A for Cs 2 AgBiBr 6 is set to 2.62 × 10 4 cm −1 eV −1/2 to acquire absorption coefficient, , as computed by  Table 1.

Results and discussion
Figure 2a presents the current density-voltage (J-V) curves computed for ITO/SnO 2 / Cs 2 AgBiBr 6 /P 3 HT/Au cell by using the factors given in Tables 1 and 2 and comparing with the experiment (Wu et al. 2018). As Fig. 2b shows, the computed external quantum . 2 a Comparison of our simulation and experimental results of current density-voltage curves. b Calculated external quantum efficiency of the double perovskite solar cell, for ITO/SnO 2 /Cs 2 AgBiBr 6 /P 3 HT/Au cell (Wu et al. 2018) efficiency, EQE, is commenced from zero at 330 nm and achieved quickly to the apex of 23.60% at 340 nm and then decreased to zero at 610 nm. The simulation and experimental results are compared in Table 3.
In the next step, we are employing the other HTL such as Cu 2 O. According to the data given in Tables 1 and 2, the current density-voltage curve and external quantum efficiency with Cu 2 O layer are plotted in Fig. 3a, b, respectively. For ITO/SnO 2 /Cs 2 AgBiBr 6 /Cu 2 O/ Au cell, the performance parameters are V oc of 1.17 V, J sc of 3.42 mA/cm 2 , FF of 0.78 and PCE of 3.13%. As shown in Fig. 3b, the computed external quantum efficiency is started at 330 nm and reached swiftly to the apex of 44.12% at 340 nm and then decreased to zero at 610 nm.  The performance parameters for two different HTLs are shown in Table 4, before optimizing parameters.
To get the required efficiency, we used the obtained optimal value of each quantity. Nevertheless, in this research, the improvements made in the efficiency have been explicated in the following sub-sections for ITO/SnO 2 /Cs 2 AgBiBr 6 /P 3 HT/Au cell.

Effect of the double perovskite (absorber layer) thicknesses
The effect of absorber layer thickness on the parameters of the double perovskite solar cell was calculated. The thickness of the double perovskite layer was altered from 100 to 2500 nm, the attained parameters as a function of thickness are revealed in Fig. 4. Figure 4a exhibits the current density-voltage curves for different thicknesses of absorber layer changing from 100 to 2500 nm. As a consequence of increasing the thickness of the absorbing layer, photons with longer wavelengths were also absorbed. This effect is associated with the enhancement of the collection of photogenerated carriers. According to Eq. (6), V oc is function of both J sc and the saturation current density of the device, J 0 , which is also influenced by the thickness of the absorber layer: Figure 4b represents the performance parameters of V oc , J sc , FF and efficiency of the device as a function of thicknesses of absorber layer altering from 100 to 2500 nm. According to Fig. 4b, the efficiency was started from 1.05% at 100 nm and then amended to the amount of 3.65% at 1200 nm and decreased to 2.92% at 2500 nm, see Supporting Information of Dang et al. (2020), Longo et al. (2020), Keshavarz et al. (2020) and . The FF, J sc , and V oc are also presented in Fig. 4b. Figure 4c exhibits the external quantum efficiency of Cs 2 AgBiBr 6 device for different thicknesses altering in the same range. Therefore, the optimized parameter for this section is 1200 nm. The V oc of 1.08 V, J sc of 6.33 mA/cm 2 , FF of 0.53 and efficiency of 3.65% are attained.

Effect of defect densities, N t , of Cs 2 AgBiBr 6
For supplementary enhancement in performance, defect density is another serious parameter which should be disputed. Defect density in Cs 2 AgBiBr 6 are mostly point defects including Vacancy, Interstitial, and Antisite. We do not specifically report each of these defects (Islam et al. 2020a, b). The performance of double perovskite solar cell is noticeably reformed by the morphology and quality of absorber layer film. As the quality of film is not ideal, the defect density improved and the recombination rate of the carriers enlarged in the absorbing layer, thus it disturbs the performance of the solar cell Li et al. 2019;Tan et al. 2020).  Figure 5a demonstrates the current density-voltage curves for numerous defect density (N t ) of Cs 2 AgBiBr 6 , changing from 10 12 to 10 17 cm −3 . Figure 5b indicates the performance parameters of V oc , J sc , FF and efficiency against defect density (N t ) of device varying from 10 12 to 10 17 cm −3 . As shown in Fig. 5b, the efficiency of device was initiated from 4.06% at 10 12 , 10 13 and 10 14 cm −3 and reduced to 4.04% at 10 15 cm −3 , 3.93% at 10 16 cm −3 , and 3.27% at 10 17 cm −3 . The FF, J sc and V oc are also plotted in Fig. 5b. As Fig. 5b suggests, the optimized value of these parameters can be acquired at defect density of 10 14 cm −3 . It is appeared that the high crystallinity of the double perovskite result in the minimum of defect density thereby dwindling charge recombination and thus higher efficiency are gained Li et al. 2019;Tan et al. 2020). The V oc of 1.09 V, J sc of 6.35 mA/cm 2 , the FF of 0.58 and efficiency of 4.06% are gained. Fig. 4 a Current density-voltage curves for different thicknesses of Cs 2 AgBiBr 6 . b The performance parameters of V oc , J sc , FF and efficiency of the device as a function of Cs 2 AgBiBr 6 thicknesses. c The external quantum efficiency for various thicknesses of Cs 2 AgBiBr 6 Page 9 of 22 319

Effect of the valence band offset at the HTL side
The serious factor between Cs 2 AgBiBr 6 and P 3 HT layers is band offset which controls the carrier recombination at the interface and it measures V oc . By changing the band gaps of P 3 HT (1.5-3.2 eV), the band offset can be attuned, according to Table 1 and the following  formula: where DP and HTL are the affinity, E g,DP and E g,HTL are the band gap of Cs 2 AgBiBr 6 and P 3 HT, respectively. VBO is enhanced from − 1.54 to + 0.16 eV. Figure 6a exhibits the (7) VBO = HTL + E g,HTL − ( DP + E g,DP )

Fig. 5 a
The current density-voltage curves for various defect density (N t ) of device of Cs 2 AgBiBr 6 changing from 10 12 to 10 17 cm −3 . b The performance parameters of V oc , J sc , FF, and efficiency against defect density (N t ) of device varying from 10 12 to 10 17 cm −3 Fig. 6 a The current density-voltage curves for VBO of P 3 HT/Cs 2 AgBiBr 6 changing from − 1.54 to + 0.16 eV. b The energy band diagrams for VBO of P 3 HT/Cs 2 AgBiBr 6 varying from − 1.54 to + 0.16 eV current density-voltage curves for various band gaps of P 3 HT changing from 1.5 to 3.2 eV. Figure 6b displays the energy band diagrams for the band gaps of P 3 HT varying from 1.5 to 3.2 eV. As indicated by Fig. 6, the optimal value for VBO is happened at + 0.16 eV (the band gap of P 3 HT is 3.2 eV). Since VBO is positive, the spike is established. Meanwhile, the formed spike is lower than 0.2 eV, the hole can move to P 3 HT (Minemoto and Murata 2015). The V oc of 1.87 V, J sc of 6.38 mA/cm 2 , FF of 0.83 and efficiency of 10.0% are obtained.

Effect of the conduction band offset at the ETL side
The earnest factor between SnO 2 and Cs 2 AgBiBr 6 layers is band offset which controls the carrier recombination at the interface and it evaluates V oc . By altering the affinity of SnO 2 in the range of 4.0 to 4.5 eV, the band offset can be adjusted. According to Table 1 and the  following formula: where DP and ETL are the affinity of Cs 2 AgBiBr 6 and SnO 2 , respectively. The obtained CBO is changed from + 0.19 to − 0.31 eV. Figure 7a exhibits the current density-voltage curves for the CBO of Cs 2 AgBiBr 6 /SnO 2 changing from + 0.19 to − 0.31 eV, also Fig. 7b displays the energy band diagrams. As indicated by these figures, the optimal value for CBO is occurred at − 0.21 eV (affinity of SnO 2 is 4.4 eV). The CBO is negative which means that the cliff is formed, the electron can simply drift to SnO 2 (Minemoto and Murata 2015). The V oc of 1.87 V, J sc of 6.39 mA/cm 2 , FF of 0.88 and efficiency of 10.57% are achieved. Figure 8a displays the current density-voltage curves for various doping concentration of P 3 HT changing from 5× 10 12 to 5× 10 21 cm −3 . According to this figure, the larger Fig. 7 a The current density-voltage curves for CBO of Cs 2 AgBiBr 6 /SnO 2 changing from + 0.19 to − 0.31 eV. b The energy band diagrams for CBO of Cs 2 AgBiBr 6 /SnO 2 changing from + 0.19 to − 0.31 eV concentrations caused higher efficiency due to better charge transport and charge extraction at the P 3 HT/absorber interface. Figure 8b denotes the effect of N A on the performance parameters. It determines that PCE is low at low level of N A which is owing to high series resistance in line with the former studies (Daraie and Fattah 2020;Jamal et al. 2020;Wang et al. 2020). Thus, according to Fig. 8b, PCE is maximized at N A = 5× 10 19 cm −3 and lager values. The V oc of 1.87 V, J sc of 6.39 mA/cm 2 , FF of 0.88, and efficiency of 10.59% are found.

Effect of doping concentration (N D ) of SnO 2
Figure 9a displays the current density-voltage curves with varying doping concentration of SnO 2 changing from 5× 10 12 to 5× 10 21 cm −3 . According to this figure, the greater concentrations produce the better charge transport and charge extraction at the SnO 2 /absorber interface. Figure 9b shows the effect of N D on the performance parameters. Again it defines that PCE is low at low level of N D which is owing to high series resistance in agreement with aforementioned studies (Daraie and Fattah 2020;Jamal et al. 2020;Wang et al. 2020). Thus, according to Fig. 9b, PCE is maximized at 10.59% for N D = 5× 10 19 cm −3 , 10.55% for N D = 5× 10 20 cm −3 , and 10.48% for N D = 5× 10 21 cm −3 . As shown in Fig. 9b, the efficiency of 10.59%, V oc of 1.87 V, J sc of 6.39 mA/cm 2 and FF of 0.88 are obtained. Figure 10 demonstrates the current density-voltage curves for N t,DP (defect density of absorber layer) and different capture cross section ( n,p ) varying from 10 −20 to 10 −12 cm 2 (Shikoh et al. 2020;Bruzzi et al. 2020;Nithya and Sudheer 2020). After optimizing, we achieved the optimum values of n,p and N t,DP (Jamal et al. 2019;Rai et al. 2020;Teimouri et al. 2020). Fig. 8 a The current density-voltage curves, and b the performance parameters of V oc , J sc , FF and efficiency for various doping concentration, N A , of P 3 HT changing from 1× 10 12 to 1× 10 21 cm −3 Finally, here we report the highest PCE. It should be noted that we only draw the PCE curve. We avoid to draw V oc , J sc , and FF curves. The highest PCE is achieved at optimum values of at n,p of10 −20 cm 2 for N t,DP of10 16 cm −3 , at n,p of10 −19 cm 2 for N t,DP of10 15 cm −3 , at n,p of10 −18 cm 2 for N t,DP of10 14 cm −3 , at n,p of10 −17 cm 2 for N t,DP of10 13 cm −3 , and at n,p of10 −16 cm 2 for N t,DP of10 12 cm −3 . According to Fig. 11, the highest efficiency of 10.60%, V oc of 1.87 V, J sc of 6.39 mA/cm 2 , and FF of 0.88 are obtained.

Effect of radiative recombination of Cs 2 AgBiBr 6
In Cs 2 AgBiBr 6 , the Shockley-Reed-Hall (SRH) and radiative recombination are the most prominent recombinations (Islam et al. 2020a, b). However, there is Auger recombination in Cs 2 AgBiBr 6 when the double perovskite material is highly doped (Jani et al. Fig. 9 a The current density-voltage curves, and b the performance parameters of V oc , J sc , FF and efficiency for different doping concentration, N D , of SnO 2 changing from 1× 10 12 to 1× 10 21 cm −3 Fig. 10 The current density-voltage curves for a n,p = 10 −20 cm 2 ,b n,p = 10 −19 cm 2 , c n,p = 10 −18 cm 2 , d n,p = 10 −17 cm 2 , e n,p = 10 −16 cm 2 , f n,p = 10 −15 cm 2 , g n,p = 10 −14 cm 2 , h n,p = 10 −13 cm 2 , and i n,p = 10 −12 cm 2 2020). Figure 13a represents the current density-voltage curves and Fig. 13b shows the PCE against radiative recombination changing from 2.3 × 10 −8 to 2.3 × 10 −14 cm 3 /s. According to Fig. 13b, the amount of radiative recombination is maximized at 2.3 × 10 −13 cm 3 /s, with efficiency of 11.69%, V oc of 2.03 V, J sc of 6.39 mA/cm 2 , and FF of 0.90. The V oc , J sc and FF curves are not shown. In addition to current density-voltage, we only draw the PCE curve.
As Fig. 14a shows, after performing all these optimizations for device of ITO/SnO 2 / Cs 2 AgBiBr 6 /P 3 HT/Au, the performance parameters of V oc of 2.03 V, J sc of 6.39 mA/cm 2 , FF of 0.90, and PCE of 11.69% are attained. For this device, the external quantum efficiency is depicted in Fig. 14b. As shown in this figure, the external quantum efficiency is commenced at the wavelength of 330 nm and improved rapidly to 58.87% at 340 nm and then reduced to zero at 610 nm.
In this step, we examined the other HTL, named Cu 2 O. Figure 15a exhibits the current density-voltage curve and Fig. 15b illustrates the external quantum efficiency, for ITO/SnO 2 /Cs 2 AgBiBr 6 /Cu 2 O/Au cell. After optimizing the mentioned parameters, the performance parameters are obtained as V oc of 1.97 V, J sc of 6.39 mA/cm 2 , FF of 0.895 (89.5%) and efficiency of 11.32%. As specified by Fig. 15b, the external quantum  . 11 The PCE of solar cell against defect density of double perovskite for a n,p = 10 −20 cm 2 , b n,p = 10 −19 cm 2 , c n,p = 10 −18 cm 2 , d n,p = 10 −17 cm 2 , e n,p = 10 −16 cm 2 , f n,p = 10 −15 cm 2 , g n,p = 10 −14 cm 2 , h n,p = 10 −13 cm 2 , and i n,p = 10 −12 cm 2 Fig. 11 (continued) Fig. 12 a PCE against interface defect density of P 3 HT/Cs 2 AgBiBr 6 changing from 10 12 to 10 18 cm −2 . b PCE against interface defect density of Cs 2 AgBiBr 6 /SnO 2 varying from 10 7 to 10 14 cm −2 efficiency is started at the wavelength of 300 nm and enlarged rapidly to the apex of 59.34% at 340 nm and then decreased to zero at 610 nm.
Double perovskites are a new class of material with good inherent stability. After performing all optimization, for both P 3 HT and Cu 2 O, the PCE of 11.69% and 11.32% are achieved, respectively. It is required to mention that, the PCE of 7.92% were reported in the simulation studies of Cs 2 AgBiBr 6 , employing different ETL and HTL layers (Savory et al. 2016).
As shown in Fig. 4, If the thickness of Cs 2 AgBiBr 6 is chosen 600 nm (Islam et al. 2020a, b), after doing the same procedure as mentioned in this manuscript, PCE of 8.47%, V oc of 1.97 V, J sc of 4.73 mA/cm 2 , and FF of 0.908 were obtained, which means that the efficiency of device is increased theoretically by 488.19% compared to the experiment. And for the other double perovskite, Cs 2 TiBr 6 , the thickness of this double perovskite is obtained as 800 nm (Jani et al. 2020). We are performing the same procedure in this thickness, PCE of 9.95%, V oc of 2.01 V, J sc of 5.47 mA/cm 2 , and FF of 0.90 Fig. 13 a The current density-voltage curves for radiative recombination of Cs 2 AgBiBr 6 , b PCE against radiative recombination of Cs 2 AgBiBr 6 changing from 1.0 × 10 −8 to 1.0 × 10 −14 cm 3 /s Fig. 14 a The current density-voltage curve, and b the calculated external quantum efficiency for ITO/ SnO 2 /Cs 2 AgBiBr 6 /P 3 HT/Au cell, after doing all optimization using SCAPS were attained, which means that PCE is enhanced theoretically by 590.97% against the experiment.

Conclusion
In conclusion, due to the existing of lead in the structure of perovskite solar cell as well as the inherent low efficiency of the double perovskite solar cell, the current study aims to overcome the mentioned shortcomings by comparing two HTLs using SCAPS software. Two devices with two different HTLs were examined, P 3 HT and Cu 2 O. For both devices, an optimal thickness of 1200 nm and defect density of 1.0× 10 14 cm −3 for Cs 2 AgBiBr 6 layer were attained. For both devices, the highest efficiency is obtained as CBO of − 0.21 eV and VBO of + 0.16 eV. For shallow acceptor doping concentration of HTLs (P 3 HT and Cu 2 O), the values of 5 × 10 19 and 5 × 10 17 cm −3 were obtained, respectively. As far as the shallow donor density of ETLs is concerned, for both cases, we achieved the optimal value of 5 × 10 19 cm −3 . The optimal values of capture cross section ( n,p ) in absorber layer for both HTLs are achieved to be n,p of10 −20 cm 2 for N t,DP of10 16 cm −3 , n,p of10 −19 cm 2 for N t,DP of10 15 cm −3 , n,p of10 −18 cm 2 for N t,DP of10 14 cm −3 , n,p of10 −17 cm 2 for N t,DP of10 13 cm −3 , and n,p of10 −16 cm 2 for N t,DP of10 12 cm −3 . For P 3 HT device, the optimum value of the interface defect density of P 3 HT/Cs 2 AgBiBr 6 is occurred at 1.0 × 10 14 cm −2 , and for Cs 2 AgBiBr 6 /SnO 2 is happened at 1.0 × 10 9 cm −2 . For Cu 2 O device, the optimal value of the interface defect density of Cu 2 O/Cs 2 AgBiBr 6 is befallen at 1.0 × 10 13 cm −2 , and for Cs 2 AgBiBr 6 /SnO 2 is taken place at 1.0 × 10 10 cm −2 . About radiative recombination, for P 3 HT device, the optimal value is happened at 2.3 × 10 −13 cm 3 /s, however, for Cu 2 O device is occurred at 2.3 × 10 −12 cm 3 /s. Finally, for P 3 HT device, PCE of 11.69% (V oc of 2.02 V, J sc of 6.39 mA/cm 2 , and FF of 0.90 (90%)) were accomplished, and for Cu 2 O device, a PCE of 11.32% (V oc of 1.97 V, J sc of 6.39 mA/cm 2 , and FF of 0.895 (89.5%)) were attained. To sum up, the PCE for P 3 HT device has been enhanced theoretically by 711.80%, and for Cu 2 O device has been enlarged by 686.11% compared with the 1.44% efficiency Fig. 15 a The current density-voltage curve, and b the computed external quantum efficiency for ITO/ SnO 2 /Cs 2 AgBiBr 6 /Cu 2 O/Au cell, after doing all optimization using SCAPS reported in the experiment. The future research could be undertaken in these kind of double perovskite materials, due to having vigorous stability against decomposition and demonstrates flexible tunability band gaps of optoelectronic properties in the range of infrared to ultraviolet. Overall, the elicited results suggest that Cs 2 AgBiBr 6 and some other double perovskites can play a momentous role as an absorbing layer towards the highly efficient lead-free and inorganic PSC technology, in designing eco-friendly PSC for future technologies.