Design and simulation investigations on charge transport layers-free in lead-free three absorber layer all-perovskite solar cells

Abstract

The multiple absorber layer perovskite solar cells (PSCs) with charge transport layers-free (CTLs-free) have drawn widespread research interest due to their simplified architecture and promising photoelectric characteristics. Under the circumstances, the novel design of CTLs-free inversion PSCs with stable and nontoxic three absorber layers (triple Cs₃Bi₂I₉, single MASnI₃, double Cs₂TiBr₆) as optical-harvester has been numerically simulated by utilizing wxAMPS simulation software and achieved high power conversion efficiency (PCE) of 14.8834%. This is owing to the innovative architecture of PSCs favors efficient transport and extraction of more holes and the slender band gap MASnI₃ extends the absorption spectrum to the near-infrared periphery compared with the two absorber layers architecture of PSCs. Moreover, the performance of the device with p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ architecture is superior to the one with the p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ architecture due to less carrier recombination and higher carrier life time inside the absorber layers. The simulation results reveal that Cs₂TiF₆ double perovskite material stands out as the best alternative. Additionally, an excellent PCE of 21.4530% can be obtained with the thicker MASnI₃ absorber layer thickness (0.4 µm). Lastly, the highest-performance photovoltaic devices (28.6193%) can be created with the optimized perovskite doping density of around E15 cm³ (Cs₃Bi₂I₉), E18 cm³ (MASnI₃), and 1.5E19 cm³ (Cs₂TiBr₆). This work manifests that the proposed CTLs-free PSCs with multi-absorber layers shall be a relevant reference for forward applications in electro-optical and optoelectronic devices.

Graphical abstract

Keywords

Multiple absorber layer PSCs / CTLS-free / Inverted / Narrow-band gap / Simulation

1 Introduction

Organic–inorganic haloid lead (Pb)-based perovskite solar cells (PSCs) have drawn enormous regard in the photovoltaic community during the past decades by reason of their simple preparation processing, high optical absorption coefficient, superior charge transport properties and outstanding power conversion efficiency (PCE) [1–5]. The evolutionary PCE of single-heterojunction PSCs has rocketed from the original 3.8% to above 26% [6–9]. However, the chemical instability and toxic nature of Pb spawns critical blackmail to the environment and obstacles the massive commercialization application [10]. To dispose of the issue, substituting Pb with the non-toxic Sn element in the MAPbI₃ will be a latent light absorption material of the future PSCs. This is owing to the MASnI₃ material having a narrow band gap which brings about the expanded optical absorption up to 1050 nm [11]. However, Sn²⁺ is easily oxidized to Sn⁴⁺, resulting in Sn vacancies and high-concentration p-type doping behavior as well as an increased carrier recombination rate [12–14]. Consequently, the performance of the device will be affected to some extent. To improve the performance of the Sn-based PSCs, many researchers are working on multi-absorber layer heterojunction solar cells that can generate superior performance in a longer region of visible spectrum. Farhadi et al. used a novel construction with a double absorbing layer (MASnI₃/MAPbI₃) PSCs to achieve 30.88% efficiency [15]. Additionally, Abedini-Ahangarkola et al. proposed three active layer structures (MASnI₃/MAPbI₃/FAMASnGeI₃) and achieved 31.44% high efficiency [16]. However, there are still toxic Pb elements in these perovskite layers.

In this work, an inverted p-p-n planar heterojunction structure PSCs with triple Cs₃Bi₂I₉/single MASnI₃/double Cs₂TiBr₆ three absorber layers are proposed. The upper and lower surfaces of the MASnI₃ absorber layer can be effectively covered by the Cs₂TiBr₆ and Cs₃Bi₂I₉ absorber layers, respectively. Additionally, the triple Cs₃Bi₂I₉ and double Cs₂TiBr₆ not only have ideal light absorption properties and non-toxic crises but also have a high level of stability and non-degradability [17–20]. Accordingly, the Cs₃Bi₂I₉ and Cs₂TiBr₆ can may be similar to antioxidant additives that diminish the oxidation of Sn²⁺ in the MASnI₃ absorber layer. In PSCs, the common electron transport layer (ETL) materials are TiO₂, SnO₂ and ZnO. Wherein, the preparation of the TiO₂ layer is usually required at a high post-annealing temperature, which indubitably increases the complicacy of device fabrication [21, 22]. Furthermore, the immediate contact of ZnO, SnO₂ and the perovskite absorber layer will spawn the degradation of perovskite [23, 24]. In the same way, most used hole transport layer (HTL) materials such as Spiro-OMeTAD and PEDOT: PSS require higher preparation costs and are subject to stability limits [25, 26]. Based on this, to avoid the impacts of these detrimental factors on the efficiency of the PSCs and simplify the preparation process, charge transport layers-free (CTLs-free) PSC devices are gradually being investigated. Furthermore, the PSCs with inversion construction (p-i-n) not only provide superior fixity but also have non-existent evident current density–voltage (J–V) hysteresis effect [27, 28], which shall be a novel exploration sphere.

Due to the experimental fabricating of the distinct levels of multi-layered PSCs is costly and time-consuming, accordingly, the simulation studies are vital for determining reasonable materials and parameters for different functional layers of PSCs [29]. In our work, we simulated and proposed a novel inverted CTLs-free PSCs structure with triple Cs₃Bi₂I₉/single MASnI₃/double Cs₂TiBr₆ three absorber layers through utilizing wxAMPS simulator and contrasted the photoelectric characteristic with two absorber layers (Cs₃Bi₂I₉/Cs₂TiBr₆ and MASnI₃/Cs₂TiBr₆) architecture of PSCs to show the improvement of efficiency in three absorber layer PSCs. We also analyzed and compared of the electrical and photophysical properties of PSCs with different p-p-n and p-n–n heterojunction structures in detail. Moreover, we concentrate on the effects of different double perovskite materials, the thicknesses and doping density of perovskite layers on the photoelectric characteristic of the proposed device at length to assign the optimum material parameters and layered heterojunction architectures. These results can not only pave the path for simplifying the device's structure and fabricating high-performance CTLs-free PSCs with multi-absorber layers but also allows us to better understand the spatial distribution of internal carriers and charge transport mechanism of the different structure devices.

2 Simulation methodology and device configuration

In this work, the numerical simulation of PSCs has been carried out by employing wxAMPS software under the AM 1.5G solar spectrum illumination circumstance and the device operation temperature is taken to be 300 K. The wxAMPS software is available as an updated and optimized version with based on AMPS developed for the numerical simulation of devices by Professor S. Fonash of Pennsylvania State University [30]. The tunnelling effect was contemplated in the wxAMPS mold by incorporating intra-band tunnelling models and trap-assisted tunnelling. The device performance can be analyzed under the simulation program through solving the Poisson equation (Eq. (1)), electrical and hole continuity equations (Eqs. (2) and (3)) [31]. The relational equations are as follows:

\begin{array}{}\text{(1)}& \text{d}(-\epsilon (x)\text{d}\phi /\text{d}x)/\text{d}x=q[p(x)-n(x)+N\begin{array}{c}+\\ \text{d}\end{array}(x)-N\begin{array}{c}-\\ \text{a}\end{array}(x)+p_{\text{t}}(x)-n_{\text{t}}(x)]\mathrm{,}\end{array}

\begin{array}{}\text{(2)}& \text{d}{p}_{\text{n}}/{\text{d}}_{\text{t}}=G_{\text{p}}-(p_{\text{n}}-p_{\text{n}0})/{\tau }_{\text{p}}-p_{\text{n}}{\mu }_{\text{p}}\text{d}\xi /\text{d}x-{\mu }_{\text{p}}\xi \text{d}{p}_{\text{n}}/\text{d}x+D_{\text{p}}{\text{d}}^2{p}_{\text{n}}/\text{d}{x}^2\mathrm{,}\end{array}

\begin{array}{}\text{(3)}& d{n}_{\text{p}}/{\text{d}}_{\text{t}}=G_{\text{n}}-(n_{\text{p}}-n_{\text{p0}})/{\tau }_{\text{n}}-n_{\text{p}}{\mu }_{\text{n}}\text{d}\xi /\text{d}x-{\mu }_{\text{n}}\xi \text{d}{n}_{\text{p}}/\text{d}x+D_{\text{n}}{\text{d}}^2{n}_{\text{p}}/\text{d}{x}^2\mathrm{,}\end{array}

where ε represents the permittivity, φ represents the electrostatic potential, q represents the charge, p, n, p_t, and n_t are defined as the densities of free holes, free electrons, trapped holes and trapped electrons, respectively. N_d and N_a represent the donor and acceptor doping concentration, G represents the generation rate, ξ represents the electric field, and D represents the diffusion coefficient.

To simplify the architecture of the device and discuss the impact of utilizing multiple absorber layers on the PSC performance, the simulation is depicted on CTLs-free device architecture with an inversion p-p-n planar heterojunction construction comprising the Cs₃Bi₂I₉ absorber layer (Fig. 1a), p-type MASnI₃ absorber layer (Fig. 1b), Cs₂TiBr₆ absorber layer (Fig. 1c), transparent conduction oxide (FTO), and gold (Au) photocathode, as displayed in Fig. 1d. From Fig. 1a − c, the crystal structure of Cs₃Bi₂I₉ shows the hexagonal P6₃/mmc space group where BiI₆ octahedra share faces to form [Bi₂I₉]³⁻ anions [32]. The MASnI₃ is a cubic structure and forms a tetragonal symmetry with the P4mm space group at ambience conditions [33]. Additionally, the double perovskite Cs₂TiBr₆ has a similar cubic structure with MASnI₃. Figure 1e displays the energy level diagram of distinct functional layers in the proposed inverted PSC. The differences in electron affinities and band gaps of three different materials give rise to the band offset of the interface. It can be observed that the negative conduction band (E_c) offset of − 0.77 and − 0.28 eV are formed at the interfaces of Cs₃Bi₂I₉/p-type MASnI₃ and p-type MASnI₃/Cs₂TiBr₆ heterojunction, respectively. Additionally, there is also a negative valence band (E_v) offset of − 0.15 and − 0.42 eV are formed at the interfaces of Cs₃Bi₂I₉/p-type MASnI₃ and p-type MASnI₃/Cs₂TiBr₆ heterojunction, respectively. Therefore, the negative E_c offset promotes electron transmission from the Cs₃Bi₂I₉ layer to the photocathode, and the negative E_v offset can be in favor of hole transmission from the Cs₂TiBr₆ layer to the FTO. The materials parameters used in our simulations were summarized from relational literature as seen in Table 1 [11, 17, 19, 34–36]. The electrons and holes thermal velocity are all 10⁷ cm/s. The materials defect states are neutral Gaussian distribution in each functional layer and the defect energy level is situated in the band gap (E_g) center. Additionally, the holes and electrons capture cross sections are all set to 10⁻¹⁴ cm² in each functional layer.

3 Results and discussion

3.1 Comparison of Cs₃Bi₂I₉/Cs₂TiBr₆, MASnI₃/Cs₂TiBr₆ and Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ active layer PSCs

The J − V curves of the PSCs with triple Cs₃Bi₂I₉/double Cs₂TiBr₆, single MASnI₃/double Cs₂TiBr₆, and triple Cs₃Bi₂I₉/single MASnI₃/double Cs₂TiBr₆ absorber layers are displayed in Fig. 2a. Table 2 displays the minute device performance parameters derived from our simulation. The detections in Fig. 2a and Table 2 manifest that open-circuit voltage (V_oc) increases from 0.9935 V for Cs₃Bi₂I₉/Cs₂TiBr₆ absorber layer PSC to 1.0709 V for MASnI₃/Cs₂TiBr₆ absorber layer PSC. This enhancement in V_oc can be attributed to the smaller band gap of MASnI₃ which involves the band offset at the interface of absorber layers and built-in potential (V_bi). Nevertheless, a slight reduction is perceived in V_oc (1.0569 V) of the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSC may be due to the inferior energy barrier at the interface of Cs₃Bi₂I₉/MASnI₃ absorber layer. Additionally, a remarkable increment of short-circuit current density (J_sc) appears at the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ (19.9235 mA/cm²) and MASnI₃/Cs₂TiBr₆ (14.6741 mA/cm²) absorber layer PSCs concerning the Cs₃Bi₂I₉/Cs₂TiBr₆ absorber layer counterpart (12.4054 mA/cm²). This is owing to the smaller band gap of MASnI₃ and more absorber layers can facilitate a wider range of photon absorption [16]. Eventually, the highest PCE (14.8834%) of the device was obtained by simulating the PSC of the triple Cs₃Bi₂I₉/single MASnI₃/double Cs₂TiBr₆ configuration. This is attributed to the J_sc and fill factor (FF) being substantially elevated.

To further contrast the PSCs' characteristics of these three configurations, Fig. 2b − d show the energy band diagrams of the proposed PSCs configuration with different absorber layers. As can be seen in Fig. 2b − d, the built-in potentials which are defined by the energy level difference of the E_c at the beginning of the absorber layer and the end of the absorber layer, are 1.46, 1.34, and 1.48 V for Cs₃Bi₂I₉/Cs₂TiBr₆, MASnI₃/Cs₂TiBr₆, and Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs, respectively. The larger V_bi can effectively facilitate the separation and migration of carriers in the device, which will be more conducive to higher open circuit voltage [37]. However, this V_bi value is inconsistent with Table 2 which presents that the minimum and maximum V_oc (0.9935 and 1.0709 V) corresponding to the Cs₃Bi₂I₉/Cs₂TiBr₆ and MASnI₃/Cs₂TiBr₆ absorber layer PSCs, respectively. This is attributed to the overlarge band offset at the E_c and E_v interface of Cs₃Bi₂I₉ and Cs₂TiBr₆ which is adverse to the transport of charge carriers in Fig. 2b. Besides, this large band offset perchance brings about the probability of more carrier recombination. Consequently, overlarge band offset and inferior carrier transport can seriously affect the V_oc and PCE of the device even if an ideal value of the V_bi in this device. As can be seen from Fig. 2c and d, two lesser band offsets of 0.32 and 0.52 eV are bespoke at the E_c and E_v interface of MASnI₃ and Cs₂TiBr₆, respectively. The lesser band offset will be favorable for the better separation and transmission of charge carriers from the perovskite layer to the electrode. However, In Fig. 2d, the Cs₃Bi₂I₉-MASnI₃ junction eventuates the energy band upward bent. The upward bending of the energy band not only hinders the transmission of electrons from the Cs₃Bi₂I₉ absorber layer to the MASnI₃ absorber layer but also prompts more recombination of holes and electrons inside the Cs₃Bi₂I₉ absorber layer and near the interface of Cs₃Bi₂I₉/MASnI₃. As a consequence, the reason for the V_oc (1.0569 V) of the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs is marginally lower than that of the MASnI₃/Cs₂TiBr₆ absorber layer PSCs (1.0709 V) can be explained the V_oc of the device is affected by the carrier recombination. Additionally, it is worth mentioning that the three absorber layers of perovskite are more conducive to the transport and extraction of more holes from the valence band to the electrode and the hole quasi-Fermi energy level (F_p) is approach to the valence band, as depicted by the valence band curve in Fig. 2d. There is no doubt that the J_sc of the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs can be prominently augmented. According to the above analysis, the PSCs configuration with three absorber layers shows the best J_sc and PCE.

The physical perception about the device performance of the proposed PSCs configuration with different absorber layers is revealed. The calculated electric field distribution, generation rate diagram, recombination rate diagram, and the external quantum efficiency (EQE) curves of PSCs are drawn as given in Fig. 3a − d, respectively. As can be observed in Fig. 3a, the electric field at the interface of p-type MASnI₃/n-type Cs₂TiBr₆ with Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs is slightly lower than MASnI₃/Cs₂TiBr₆ absorber layer PSCs, this meaning that the thickness or doping of the absorber layer is still not adequately designed. Accordingly, more parameters need to be optimized to elevate device performance as will be done hereafter. Additionally, the contrary electric field distribution direction appears at the Cs₃Bi₂I₉/MASnI₃ interface. The contrary electric field behavior undeservedly influences the carrier collection, which heightens the recombination trend near the interface of Cs₃Bi₂I₉/MASnI₃ and results in the V_oc of the three absorber layer PSCs being slightly lower. This result can also be reflected in Fig. 3c. As presented by the generation rate-recombination rate diagrams in Fig. 3b and c, as observed highest carrier generation rate occurs at the surface or interface of the absorber layer in all device configurations (Fig. 3b). This may be interpreted as the presence of heterojunctions at the interface of the absorber layer. Moreover, the overall carrier generation rate of the MASnI₃/Cs₂TiBr₆ and Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs is higher than that of Cs₃Bi₂I₉/Cs₂TiBr₆ absorber layer PSCs. This can be elucidated by the fact that the presence of MASnI₃ with the smallest band gap and three absorber layers can absorb enough ultraviolet–visible light in both device architectures, implying the generation of higher carrier concentration and the J_sc and PCE sensibly improved. As can be observed in Fig. 3c, the low and even recombination rate occurs inside the MASnI₃ and Cs₂TiBr₆ absorber layer. Nevertheless, the recombination losses significantly increase inside and near the top of the Cs₃Bi₂I₉ absorber layer. This is mainly due to the overlarge band offset and the upward bending of the energy band at the interface of Cs₃Bi₂I₉/Cs₂TiBr₆ and Cs₃Bi₂I₉/MASnI₃, respectively, as also illustrated in Fig. 2b and d. This also indicates that there are more defects inside and near the top of the Cs₃Bi₂I₉ absorber layer. Therefore, the material parameters in the Cs₃Bi₂I₉ absorber layer need to be additionally designed and optimized.

As depicted by the EQE response in Fig. 3d, the absorption spectrum of Cs₃Bi₂I₉/Cs₂TiBr₆ absorber layer PSCs is around the wavelength of 770 nm. Nevertheless, in the case of the MASnI₃/Cs₂TiBr₆ and Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer PSCs, the MASnI₃ absorber layer with a low band gap broadens the absorbance spectrum to the nearinfrared wavelength range (around 870 nm), effectively facilitating the augmentation of photo generated electric and J_sc [31]. Additionally, it is worth noting that the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ absorber layer configuration exhibits the highest EQE in the visible light range compared to the Cs₃Bi₂I₉/Cs₂TiBr₆ and MASnI₃/Cs₂TiBr₆ absorber layer configuration. The EQE can be described by

\begin{array}{}\text{(4)}& \text{EQE}=1240\times J_{\text{sc}}/\lambda \times P_{\text{in}}\mathrm{,}\end{array}

where λ and P_in represent the total incident light power and light wavelength, respectively. It can be seen from Eq. (4) that the EQE in the visible light range is directly proportional to J_sc. Therefore, the Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ three absorber layer configuration becomes efficient in terms of photon absorption. The J_sc and PCE of the three absorber layer PSCs can be enhanced remarkably.

3.2 Three absorber layer PSCs with two different p-p-n and p-n–n heterojunctions

In the PSC devices' structure, different heterojunction structures will have specific effects on the spatial distribution and transmission of charge carriers [38]. Figure 4a and b depict the different device architecture diagrams of the proposed three absorber layer PSCs with the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ and p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure, respectively. Combining the PSCs J − V curves with different heterojunction structures in Fig. 4c and the electric field diagrams in Fig. 4d, it can be observed clearly in Fig. 4a that the strongest electric field (E1) in the upward direction is generated at the interface of p-type-Cs₃Bi₂I₉/p-type-MASnI₃, which is owing to the existence of the concentration difference at the interface. Furthermore, an intense electric field (E2) in the downward direction is generated at the interface of p-type-MASnI₃/n-type-Cs₂TiBr₆ due to the presence of the p-n heterojunction. The electrons and holes in the device are separated and transported under the driving of the built-in electric field. The holes in Fig. 4a are transported from the n-type-Cs₂TiBr₆ absorber layer down to the FTO electrode. In virtue of the existence of E1 in the opposite direction of the hole transport, the transport and extraction of the hole are emasculated to a certain extent, thus impacting the J_sc of the devices. Nevertheless, in Fig. 4b, a weak electric field (E3) in the downward direction is generated at the interface of p-type-Cs₃Bi₂I₉/n-type-MASnI₃, and a weaker electric field (E4) in the upward direction is generated at the interface of n-type-MASnI₃/n-type-Cs₂TiBr₆. Since the downward direction of E3 is slightly higher than the upward direction of E4, the remaining electric field in the downward direction will be more favorable for the transport and extraction of the hole compared to Fig. 4a. Therefore, it can be perceived from the performance parameters of devices with different heterojunction structures in Table 3 that the PSCs of p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure has higher J_sc (22.3157 mA/cm²) than the value (19.9235 mA/cm²) from the PSCs of p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterostructure. It is also worth noting that the V_oc and FF of PSCs with p-type-Cs₃Bi₂I₉/ptype-MASnI₃/n-type-Cs₂TiBr₆ heterostructure are higher than those of PSCs with p-type-Cs₃Bi₂I₉/n-type-MASnI₃/ntype-Cs₂TiBr₆ heterostructure, which is perchance attributed to the diminution of series resistance and the strong electric field facilitates the separation of electrons and holes in the PSCs of the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure, effectively downgrading the recombination loss of carriers.

The energy band curves of the proposed PSCs with different heterojunction structures have been shown in Fig. 5a. In contrast with the energy band curve of the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction PSCs, there is no upward bending of the energy band at the interface of the absorber layer of the PSCs with p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure. Consequently, the recombination of the carrier at the interface of the absorber layer with p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction PSCs is lower, resulting in fewer defects. Both electrons and holes will be transported from the position with high energy values to the position with low energy values, which will be more conducive to the transport and extraction of carriers. This also demonstrates again the high J_sc of PSCs with p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure. Figure 5b shows the variation curves of the carrier concentration values of PSCs with different heterojunction structures. We observe that there are hole and electron concentration differences at the interfaces of p-type-Cs₃Bi₂I₉/p-type-MASnI₃ and n-type-MASnI₃/n-type-Cs₂TiBr₆, which is the reason for the presence of positive electric fields at the interface of these two absorber layers. Furthermore, it is worth mentioning that the concentration of the collective hole is higher than the electron concentration within both the p-type-Cs₃Bi₂I₉ and p-type-MASnI₃ absorber layers. However, the overall electron concentration is higher than the hole concentration in the n-type-MASnI₃ and n-type-Cs₂TiBr₆ absorber layers. Correspondingly, the p-type perovskite absorber layer will be more favorable for the transmission of more electrons to the n-type perovskite absorber layer, and the n-type perovskite absorber layer will be more conducive to the transport of more holes to the p-type perovskite absorber layer. Accordingly, more holes and electrons are generated in the p-type and n-type perovskite absorber layers, respectively. As can be seen from the overall carrier concentration values in Fig. 5b, the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction produces higher electron and hole concentrations compared to the p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction. Therefore, the high carrier concentration leads to high V_oc in p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction PSCs. From the recombination rate curves of the PSCs with different heterojunction structures in Fig. 5c, it can be observed that the recombination rate in the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ absorber layer is lower than that in the p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ absorber layer. Additionally, in the carrier life time curves of PSCs with different heterojunction structures shown in Fig. 5d, the carrier life time is only meaningful for minority carriers. The minority carriers are electrons in p-type perovskite semiconductors, and the minority carriers are holes in n-type perovskite semiconductors. Then it is evident that the carrier life time in the p-type MASnI₃ absorber layer is higher than that in the n-type MASnI₃ absorber layer, and the n-type-Cs₂TiBr₆ absorber layer in the p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction PSCs has a higher carrier life time compared to the p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction PSCs. Combined with the above analysis, as can be concluded that PSCs with p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction structure have higher V_oc, FF as well as PCE due to less carrier recombination and higher carrier life time inside the absorber layers. The following are further studies on the PSCs with p-p-n heterojunction structure.

3.3 Three absorber layer PSCs with different double perovskite materials

The double perovskite materials of the proposed three absorber layer PSCs play a crucial role in transporting electrons from the perovskite layer to the metal electrode [39]. Since Cs₃Bi₂I₉ and MASnI₃ are already optimal triple perovskite absorber layer materials and single perovskite absorber layer materials in our simulated three absorber layer PSCs. Accordingly, to simplify the content of our manuscript, the impact of utilizing various double perovskite materials (Cs₂TiBr₆, Cs₂TiF₆, Cs₂TiCl₆, Cs₂TiI₆) on the photoelectric performance of the proposed three absorber layer PSC is merely investigated in this section. The details of double perovskite materials parameters used in our simulations were summarized from cognate literature as displayed in Table 4 [20, 36, 39, 40]. The J − V curves and performance parameters of three absorber layer PSCs with various double perovskite materials are presented in Fig. 6a and Table 5, respectively. The detections in Fig. 6a and Table 5 display the highest J_sc in the three absorber layer PSCs with double Cs₂TiBr₆ materials. Combining the carrier life time curves and energy band alignment of PSCs with various double perovskite materials in Fig. 6b and c, respectively. It can be interpreted clearly that the highest carrier (hole) life time occurs in the Cs₂TiBr₆ absorber layer and more holes can be transported and extracted into the conductive substrate. Nevertheless, the maximum V_oc and FF occur in the PSCs with double Cs₂TiF₆ materials, which is attributed to the ideal band alignment at the interface of p-type-MASnI₃/n-type-Cs₂TiF₆ and lower carrier recombination loss in the Cs₂TiF₆ and MASnI₃ absorber layers from Fig. 6d. To sum up, the maximum V_oc and FF give rise to the highest PCE of 16.1736% when the Cs₂TiF₆ is regarded as the double perovskite material of the three absorber layer PSCs.

3.4 Effect of Cs₃Bi₂I₉/MASnI₃/Cs₂TiF₆ absorber layer thicknesses on the performance of the three absorber layer PSCs

The thicknesses and photoelectric parameters of the absorber layer have become decisive factors affecting the performance of the PSCs [41]. We have devised seven combinations of the Cs₃Bi₂I₉/MASnI₃/Cs₂TiF₆ absorber layer thicknesses (noted from A to G) while maintaining the overall thickness of the three absorber layers invariant at 0.6 µm. Figure 7a and b and Table 6 display the performance parameters of the device with different Cs₃Bi₂I₉/MASnI₃/Cs₂TiF₆ absorber layer thicknesses. The excessive absorber layer thickness may eventuate an evident increment in the carrier recombination and series resistance, whereas the thin absorber layer thickness perchance induces less shunt resistance [42]. Among the simulated results with various three absorber layer thicknesses. We can observe case C (0.1/0.4/0.1 µm) in which the thickest MASnI₃ absorber layer (400 nm) bespeaks the highest PCE of 21.4530% corresponds to an evident enhancement compared to the PCE of case A (0.2/0.2/0.2 µm (16.1736%)). On the one hand, this enhancement in PCE is due to the increasing thickness of the MASnI₃ absorber layer, which has the narrowest band gap (1.41 eV) and can hasten a wider range of photon absorption, thus more photogenerated carriers are generated. On the other hand, the Cs₃Bi₂I₉ absorber layer can be comprehensively covered with the increasing of MASnI₃ thickness, effectively eschewing the direct contact between the Cs₂TiF₆ absorber layer and the Cs₃Bi₂I₉ absorber layer. Consequently, not only the series resistance is possibly reduced, but also the leakage is effectively avoided. However, when the thickness of MASnI₃ absorber layer is less than 0.4 µm, the V_oc and J_sc shall gradually dilute with the increasing of the Cs₃Bi₂I₉ or Cs₂TiF₆ absorber layer thickness. This is attributed to the excessive Cs₃Bi₂I₉ or Cs₂TiF₆ absorber layer thickness with a wide band gap eventuating more defects and carrier recombination losses inside and at the interface of the absorber layer, and the carriers need to be diffused over a longer distance before reaching the corresponding electrodes.

3.5 Effect of Cs₃Bi₂I₉ acceptor doping concentration on the performance of the three absorber layer PSCs

The doping concentration of the absorber layer has become a crucial factor for regulating the photoelectronic properties of PSCs, which directly affects the generation and transport of photogenerated carriers, hence involving the performance of the device [43, 44]. From a practical point of view, we took Cs₃Bi₂I₉ acceptor doping concentration between E10 cm⁻³ and E15 cm⁻³ to maintain the rationality of device parameters. The impacts of Cs₃Bi₂I₉ acceptor doping concentration on J − V curves and energy band distributions are plotted in Fig. 8a and b, respectively. The performance parameters of three absorber layer PSCs with various Cs₃Bi₂I₉ acceptor doping concentrations are displayed in Table 7. Figure 8a and Table 7 display that the J_sc and V_oc improve smoothly with the Cs₃Bi₂I₉ acceptor doping concentrations increases. The bending of the energy band increases in the Cs₃Bi₂I₉ absorber layer giving rise to the enhancement of V_bi, as seen in Fig. 8b, and the larger V_bi can effectively facilitate the separation and migration of carriers toward corresponding electrodes, thus making for a higher V_oc. Additionally, this result can be interpreted from the electric field diagrams in Fig. 8c and d. It can be noticed definitely that the strongest electric field is generated at the interface of MASnI₃/Cs₂TiF₆ when the Cs₃Bi₂I₉ acceptor doping concentration is increased to E15 cm⁻³, which facilitates the transport and extraction of carriers. The J_sc and FF are all improved marginally with the increased doping concentration of the Cs₃Bi₂I₉ absorber layer, which can be elucidated by

\begin{array}{}\text{(5)}& N_{\text{A}}=2\epsilon {\epsilon }_0/(q{W}^2)\mathrm{.}\end{array}

In Eq. (5), W and N_A represent depletion region width and acceptor doping concentration, respectively. With increased Cs₃Bi₂I₉ acceptor doping concentration, the depletion layer width narrows as Cs₃Bi₂I₉ acceptor doping concentration increases, lowering the impediment to majority carrier movement and thus improving the FF. In the case of J_sc, the collection of photogenerated carriers may be diminished with the increases in neutral region width, making for higher bulk recombination. In that way, the J_sc may be affected somewhat. Nevertheless, the J_sc shows a slow growth inclination. This is attributed to the increased Cs₃Bi₂I₉ doping concentration resulting in the energy band bending of the Cs₃Bi₂I₉ absorber layer in Fig. 8b, thus facilitating that the carriers can be efficiently transported. Additionally, it can be also seen from the hole concentration and electron concentration curves in Fig. 8e and f that both hole concentration and electron concentration are all significantly increased in the Cs₃Bi₂I₉ absorber layer and Cs₂TiF₆ absorber layer, respectively, with the increasing of Cs₃Bi₂I₉ acceptor doping concentrations. Then more electrons and holes can be efficiently transported and collected to the respective electrodes. Consequently, it can be concluded that when the doping concentration is extracted E15 cm⁻³ in the Cs₃Bi₂I₉ absorber layer shall yield 23.7982% high-performance photoelectric devices.

3.6 Effect of MASnI₃ acceptor doping concentration on the performance of the three absorber layer PSCs

Figure 9a and b depict the alterations in J − V curves and energy band distributions of the three absorber layer PSCs as a function of different acceptor doping concentrations (E14 cm⁻³ − E19 cm⁻³) in the MASnI₃ absorber layer, respectively. The relevant performance parameters of PSCs are plotted in Table 8. As can be observed in Fig. 9a and Table 8, the V_oc and FF improve continuously with increased doping concentrations of the MASnI₃ absorber layer. This can be also interpreted by Eq. (5), the width of the depletion layer in the MASnI₃/Cs₂TiF₆ heterojunction narrows as MASnI₃ acceptor doping concentration increases, lowering the impediment to majority carrier movement and thus the existence of majority carriers (holes) in the p-type MASnI₃ absorber layer heightening the FF. From energy band curves in Fig. 9b, when the doping concentration exceeds E16 cm⁻³ in the MASnI₃ absorber layer, two lesser band offsets in the conduction band and valence band at the interface of Cs₃Bi₂I₉/MASnI₃ can be conducive to the improvement of V_oc. Additionally, the stronger electric field at the interface of Cs₃Bi₂I₉/MASnI₃ and MASnI₃/Cs₂TiF₆ when the doping concentration of MASnI₃ absorber layer exceeds E17 cm⁻³, as displayed in Fig. 9c, further facilitating the separation and transport of carriers, and thereby enhancing the V_oc of the device significantly. Nevertheless, the J_sc diminishes gradually as the doping concentration of the MASnI₃ absorber layer increases, this is due to the collection of photogenerated carriers being impeded as the increases in neutral region width under the high doping concentration, inducing a gradual increase in bulk recombination. As can be seen obviously from the current density curves of the carriers in Fig. 9d, the hole current density is also diminished gradually as the MASnI₃ absorber layer doping concentration increases, which directly affects the J_sc of the device. Therefore, the excessively high doping concentration of the MASnI₃ absorber layer should not be selected. Overall, based on the above analysis, the high PCE of 25.0713% can be successfully obtained when dominating the MASnI₃ absorber layer doping concentration at about E18 cm⁻³.

3.7 Effect of Cs₂TiF₆ donor doping concentration on the performance of the three absorber layer PSCs

The donor doping concentration is also a momentous parameter, which directly impacts the photoelectric performance of the device. Therefore, an exploration of the impact of the Cs₂TiF₆ donor doping concentration on the performance of the three absorber layer PSCs was conducted. Figure 10a and Table 9 present the J–V curves and performance parameters of the three absorber layer PSCs as a function of the donor doping concentration of the Cs₂TiF₆ absorber layer variations from E19 cm⁻³ to 3.5E19 cm⁻³. The V_oc, FF, and PCE improve with the increases of Cs₂TiF₆ doping concentration from E19 cm⁻³ to 1.5E19 cm⁻³ and then decline marginally as the Cs₂TiF₆ absorber layer doping concentration increases from 1.5E19 cm⁻³ to 3.5E19 cm⁻³. However, the J_sc changes barely with the increasing doping concentration of the Cs₂TiF₆ absorber layer. The PCE reaches the maximum when the Cs₂TiF₆ absorber layer doping concentration is 1.5E19 cm⁻³. From the carrier concentration curves (Fig. 10b) and electric field distribution (Fig. 10c), respectively, it can be visibly seen that the electron concentration inside the Cs₂TiF₆ absorber layer and the electric field strength at the interface of MASnI₃/Cs₂TiF₆ all increase gradually with the increase of Cs₂TiF₆ doping concentration. Additionally, there are fewer carriers' recombination when the doping concentration of the Cs₂TiF₆ absorber layer surpasses E19 cm⁻³ (as shown in Fig. 10d). Accordingly, fewer carriers will be trapped by the defect, and more carriers will be efficiently separated and transported to the corresponding electrode. In summary analysis, it can be determined explicitly that a moderate donor doping concentration (1.5E19 cm⁻³) in the Cs₂TiF₆ absorber layer will yield 28.6193% optimalperformance PSCs devices.

4 Conclusions

A numerical simulation investigation on the novel design of CTLs-free inverted PSCs with triple Cs₃Bi₂I₉/single MASnI₃/double Cs₂TiBr₆ absorber layers has been unfolded in this work to pick the optimal conditions by employing wxAMPS software. The simulation demonstrates that the three absorber layer architecture of PSCs is more conducive to the transport and extraction of more holes from the valence band to the electrode and has better photoelectric characteristics than that of the two absorber layers architecture of PSCs, and the higher PCE (14.8834%) of the simulated device with Cs₃Bi₂I₉/MASnI₃/Cs₂TiBr₆ heterojunction has been obtained. Meanwhile, the simulation results illuminate that the electrical and photophysical properties of the device with p-type-Cs₃Bi₂I₉/p-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction architecture (p-p-n) is superior to the one with p-type-Cs₃Bi₂I₉/n-type-MASnI₃/n-type-Cs₂TiBr₆ heterojunction architecture (p-n–n) due to less carrier recombination and higher carrier life time inside the absorber layers. The oxidation of Sn²⁺ in MASnI₃ materials will result in p-type doping of the MASnI₃ layer. Therefore, we also believe that the p-type MASnI₃ is generally caused by the quantitative oxidation of Sn²⁺ may have a positive impact on devices of Sn-based perovskites under our device structure (p-p-n). Additionally, after optimizing the parameters of each absorber layer, we obtained the larger V_oc and FF give rise to the better PCE of 16.1736% when the Cs₂TiF₆ is regarded as the double perovskite material of the three absorber layer PSCs. Varying the thickness of each absorber layer, it was revealed that the thicknesses of absorber layers have a remarkable influence on the J_sc of the device, and the thicker MASnI₃ absorber layer can obtain an excellent PCE of 21.4530%. Lastly, the highest-performance (28.6193%) photoelectric devices can be created with the optimized doping density of around E15 cm³, E18 cm³, and 1.5E19 cm³ in the Cs₃Bi₂I₉ absorber layer, MASnI₃ absorber layer, and Cs₂TiBr₆ absorber layer, respectively. This work not only simplifies the fabrication process of the devices but also allows us to better understand the spatial distribution of internal carriers and charge transport mechanism of the different structure devices. What's more, this work will also tender theoretical guidance for the large-scale fabrication of high-performance CTLs-free inverted PSCs with multi-absorber layers.

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