Unravelling of theoretical window to fabricate high performance inorganic perovskite solar cells

Perovskite solar cells (PSCs) have celebrated a decade of intense investigation as a promising photovoltaic technology, measuring a power conversion efficiency of >25.2% with the use of lead based light harvester. Recently, inorganic cesium cation based mixed halide perovskite (CsMI 3-x Br x , M=Pb or Sn) as an abosrber in PSCs gave promising results, particularly CsPbI 2 Br demonstrated improved thermal stability and carriers transport properties. However, the performance is far-off from the theoretical limit due to intriguing issues such as high defect density ( N t ), energetic level mismatch. Such barrier can be over come through device modelling and unraveling of the kinetics. Here we have employed computational approach to design and investigate efficient inorganic CsPbI 2 Br based solar cells by elucidating the role of defect density in the perovskite on the performance, by employing different types of hole transport layers via optimizing their valence band offset and barrier height at back contact. By optimizing such parameters for CsPbI 2 Br, the efficiency of 17.71 , 17.44 and 17.54 % using CuSCN, PTAA and Spiro-OMeTAD as HTMs respectively can be reached. Furthermore, lead free CsSnI 3-x Br x (0<x  3) based PSCs were simulated and the effect of band gap variation as a result of Br content was studied on the performance. The influence of defect density of the absorber layer (CsSnIBr 2 ) and at of interfaces was studied and with optimized defect density, CsSnIBr 2 based PSCs gave an efficiency of 20.32 % with V oc of 1.35 V when SnO 2 was used as electron transport layer and Spiro-OMeTAD as HTM. Our simplistic approach suggests way for experimental design protocol to achieve high performance inorganic Pb and Sn based


Introduction
Solar energy is one of the most promising source of energy to counter the adverse effect of climate change on our planet. Emerging photovoltaic technologies can potentially reach to the masses, in particularly perovskite solar cells (PSCs). The hybrid organic inorganic lead (Pb) based perovskite as an light absorber (APbI3) is the most investigated in PSCs due to their excellent electro-optical properties [1][2][3][4][5] and gave record performance. 6,7 However, hybrid organic-inorganic lead halide perovskite suffers from challenges such as, efficiency, 8 thermal instability at high temperature, originating due to volatile nature of organic cations (FA or MA). 9,10 Replacement of the organic cations by inorganic cations such as Cs found to be an effective approach to overcome such bottleneck. 11,12 However, the CsPbI3 is stable only in cubic perovskite (αphase) at high temperature, which can be easily converted to orthorhombic non-perovskite (δ) phase. 13 In this context, CsPbI3-xBrx evolved as a promising material for new architect of PSCs in term of phase stability. Snaith et al reported CsPbI2Br is stable in the cubic phase at room temperature with a band gap of 1.82 -1.92 eV 14 , partial substitution of I to Br into CsPbI3 stabilizes the cubic phase, this led to intense investigation and performance of CsPbI2Br based PSCs up to 16.07 % was reported. 15 Such results were obtained by reducing the defect density in the active layer, growth control of CsPbI2Br through optimized annealing temperature 16 , using effective anti-solvent approach 17 or minimizing the energy level gap at the charge selective layers, 18 similar to hybrid lead halide PSCs. 19 However, the performance remains lower from the theoretical limit of 22.1 % PCE and 1.65 Voc, when 500 nm thick CsPbI2Br was used. 20 To achieve the theoretical efficiency limit, the strategy should be focus on reducing the defect density of the active layer, and it should be less than the current value of 3.64 E15 cm -3 . 21 The energy alignment at the charge selective layers needs further optimization using p-type conducting polymers or inorganic semiconductors to minimized energy loss (Eloss=Eg -eVoc, Eg is the band gap and e is the elementary charge). Arguably, the efficient CsPbI2Br based PSCs should focus on reducing Eloss as much as possible by improving Voc and Jsc, this target can be achieved through energy alignment and high quality CsPbI2Br crystal formation. The most efficient CsPbI2Br based PSCs yielded PCE of 16.79 % with an Voc of 1.32 V, i.e. Eloss = 0.59 eV 22 . The toxicity of the lead (Pb) metal is visualized as a barrier for commercial success of inorganic PSCs. Tin (Sn) as possible metal to replace the problematic Pb is being investigated. 23 Compared to Pb-, Sn-based perovskites showed high optical absorption coefficients, 24,25 narrow optical band gaps and high charge carrier mobilities. 26 In accordance to the perovskite structure, ABX3, the A cation can be (MA, FA and Cs), in case of MA cation an efficiency of 6.4% was achieved by using MASnI3 as light harvester having a band gap of 1.3 eV. 27 Further, using FA cation, slightly larger than MA, Koh et al. prepared FASnI3 with incorporation of 20% SnF2 addition, a PCE of 2.10% was achieved with a band gap of 1.41eV. 28 However, the instability of organic-inorganic halide perovskite is of concern, due to the use of organic cations under outdoor conditions. 29 To resolve this stability issue, Cs as A cation was opted for all inorganic Sn-based perovskites with X = I, Br, which has similar electro-optical properties to FASnI3 and MASnI3. The first report using Sn-based allinorganic PSCs was reported in 2012, where CsSnI3 acted as hole transporter gave a PCE of 0.9 %. 30 However, instability of Sn in the +2 oxidation state, which can be easily converted to the +4 state in the presence of moisture and oxygen 31 was a limiting factor. To resolve the oxidation issues of Sn, SnCl2 and SnF2 as reducing agents was exploited to minimize the extent of p-type self-doping of the perovskite and to reduce the hole carrier density, in order to improve the performance. 32,33 Besides, mixed anion containing CsSnI3-xBrx as light harvester has been studied, and its band gap was tuned from 1.27 eV to 1.74eV by varying the percentage of Br from x = 0-1. 34 Although, these Inorganic Sn-based PSCs shows lower performance compared to their Pb based counterparts, but their low band gap exhibits a high potential to exploit them in perovskite-perovskite tandem solar cells. Further, investigation of macroscopic device model of inorganic Snbased PSCs is still obscure and a lot of issues need to be resolved to improve the Sn-based PSCs, such as prevention of bulk and interface recombination due to tin vacancies, rational designing of charge selective contacts to efficiently extract carriers from perovskite. The issue of high hole carrier density as a result of self p-doping can be controlled by SnF2 addition, which will allow to reduce the defect density from pristine value of 1.1E19 cm -3 to 5.7E17 cm -3 for CsSnI3. This can be retarded further by substitution of I to Br. Taking this into account, we performed theoretical investigation for CsPbI2Br and CsSnI3-xBrx based PSCs. Firstly, we focus on CsPbI2Br by studying the effect of interface engineering at HTL and perovskite interface employing different HTMs such as organic (Spiro-OMeTAD), inorganic (CuSCN) and polymeric (PTAA). Additionally, the influence of defect density of CsPbI2Br absorber layer on performance was studied. The HTM is a channel to transfer the holes from the active layer to back contact, which should be optimized to make better alignment. Most of the studies in PSCs are focused on optimizing the valence and conduction band offset, which is critical for transport and extraction of charges as the height of energy barrier regulates the contact resistance. However, reports showing the effect of energy band alignment between HTM and back contact are in scarce. We reported how the valence band offset can be optimized to boost the performance of PSCs 35 Thus to address the mismatch alignment at perovskite/HTM/back contact, firstly, we have elucidated the impact of valence band maximum (VBM) of HTMs (Spiro-OMeTAD, CuSCN & PTAA) on the performance of CsPbI2Br solar cells and also on the contact between HTM-Back contact. During this process, we discovered that modulating the EV (by varying the electron affinity) of HTM, leads to an arc shape forms at HTM/back contact which changes its curvature with the variation of Ev and that has strong influence on the performance of PSCs. Such optimizations led to an efficiency of 17.73, 17.45 and 17.44% using CuSCN, PTAA and Spiro-OMeTAD as HTMs respectively. Secondly, we ran simulation for lead free Sn-based inorganic PSCs, where the effect of I anion substitution by Br in CsSnI3 absorber was investigated to clarify its impact on the conduction band offset (CBO), i.e., the difference between the conduction band minimum (CBM) energy levels of the ETL and those of the perovskite layer, Ec= Ec_ETL -Ec_absorber . To our understanding, there is no theoretical or experimental studies emphasized on lead free inorganic PSCs. The ''selfdoping'' in Sn-based inorganic PSCs allows lowering the trap density and its impact on solar cell performance was studied.
We studied the influence of defect density variation in absorber layer and its interfaces to elucidate the optimum values that can boost the efficiency and computed 20.35 % PCE with a Voc of 1.35 V.

Theory and Computational details
The simulations were performed using SCAPS 3.3.07 software, 19 based on the Poisson equation (1), and the continuity equations for electrons(2) and holes (3) Here, ε denotes the permittivity, q , the charge of electron, Ψ, the electrostatic potential , n, the free electrons, p, free holes, nt and pt are trapped electrons and holes, ND + and NAare the ionized donor and acceptor-like doping concentrations respectively, Rn and Rp are electrons and holes recombination rate, G is the generation rate, Jn and Jp are, the electron and hole current densities respectively. The symbols presented in the table 1 and 4 can be described as follows: NA and ND denote acceptor and donor densities, εr is relative permittivity, χ is electron affinity, Eg is the band gap energy, Nt is defect density, μn and μp are mobility of electron and hole respectively, and. Computational studies were performed on a planar PSCs with an architecture of FTO/SnO2/interface layer 1(IL1) / inorganic perovskite/interface layer2(IL2)/HTM/Au. The defect type in bulk perovskite is considered neutral with a cross section of electron and hole is 2×10 -14 cm 2 . The electron and hole thermal velocity was 10 7 cm/s. Gold (Au) was used as the back contact while fluorine doped tin oxide (FTO) as the front contact. Before detailing the band alignment at HTM-back contact, it is imperative to understand the condition for the formation of the junction between p-type semiconductor (HTM) and metal for the flow of holes. 36 Fig.1 shows the energy band diagrams for semiconductormetal interfaces, where ϕ is the work function, sp and m refer to p-type semiconductor and metal respectively. EF is the fermi-energy and χ is the electron affinity. If the metal and semiconductor are brought together, two type of contact can be formed depending on the difference of work function of metal and semiconductor: Schottky and ohmic junction as illustrated in Fig. 1a-b. The type of interface is a result of the levelling of work functions to balance the chemical potential. When the work function of p-type semiconducting HTM (ϕsp) is higher than that of the metal, (Fig.1a), an energy barrier is formed for holes and Schottky contact is created. On contrary, if the work function of HTM (ϕsp) is lower than that of the metal (ϕm), an ohmic contact will be formed (Fig.1b). To analyze the effect of these two interfacial contacts on the performance of PSCs, we will vary the valence energy level (Ev_HTM) of Spiro-OMeTAD and PTAA as HTMs (by changing the electron affinity) with respect to the valence energy level of the perovskite absorber (EV_absorber), and rational interfaces at HTM-metal can be observed. The optimized valence band offset (VBO) of each HTM will be the point where the performance of CsPbI2Br PSCs is maximized. The valence band offset (VBO) is the difference between the valence band energy of the perovskite layer and those of HTM, EV = EV_absorber − EV_HTM. The barrier height between a p-type semiconductor and metal can be given by ((ϕBp =Eg-q(Фm-) or ϕBp =(q(Фm-)-Eg)) ; Schottky and ohmic contact respectively.

Results and discussion
3.1. All inorganic CsPbI2Br based PSCs

Performance of CsPbI2Br based PSCs using different HTMs
Photon absorption leads to the creation of an electron-hole pair in a PSC, the electrons travels to the front side through electron selective layers, while hole selective layers conducts the transfer of holes to the back side. To make this process efficient, the energy levels of the interface layers should be in the equilibrium conditions to maximize charge transfer. For efficient transfer of holes, rational HTM with a suitable energetic level as of CsPbI2Br absorber layer), and the metallic cathode (Au) as the back contact is a prerequisite. We investigated the effect of tuning of the valence band by studying series of HTMs, Spiro-OMeTAD, PTAA and CuSCN, in a structure FTO/SnO2/IDL1/CsPbI2Br/IDL2/HTMs/Au, and the preliminary results are shown in the Table3 .  Table 1, b) corresponding energy band diagrams of the CsPbI2Br based PSCs using HTMs illustrating energy barrier at HTM and gold (back contact), and c) corresponding energy band diagrams for the device structure used in the simulation. employing the parameters (Table 1) and for Spiro-OMeTAD from Table 2. It can be deducted that the HTMs represents similar Jsc of ~15.67 mA/cm 2 which does not influence the external quantum efficiency (EQE) of PSCs, while the fill factor (FF) significantly varies by changing the HTM and reaches to 76.66, 79.01 and 81.94 % for Spiro, PTAA and CuSCN respectively, suggesting favorable interfacial contact at CsPbI2Br/HTM (Fig.2b) 2b). Our simulated PV results of inorganic PSCs (Table 2) are based on CsPbI2Br with the same configuration that we have used. We noted that the valence band maximum (Ev) value of -5.30 eV for the CuSCN can deliver the highest PCE of 14.90% in contrast to PTAA (-5.25 eV) of 14.28%, while Spiro-OMeTAD (-5.45 eV) gave slightly lower PCE of 14.09%.
The value of the defect density of CsPbI2Br is equal to 3.64E15 cm -3 which is the finest value 21 while the work function of back contact is -5.1 eV (Au). The difference in the performance between HTMs, can be related firstly; the effect of VBO and the secondly to the barrier height (ϕBp). We can deduct from Table 3 that the efficient PSCs must have a balanced values between the VBO and ϕBp to facilitate the extraction of the holes from the perovskite layer and also to make an efficient transfer of the holes from the HTM to the load.   Fig.1a).

Influence of CsPbI2Br defect density (NT) on performance of PSCs
The performance of the architect simulated here, is in accordance to the recent reports 37,42 , but is still far from the theoretical values. Arguably, the optimization of the selective layers will be appropriate, if the active layer is also of high quality (low defect density). The defect density influence the performance of PSCs, thus, we will elucidate the effect of this parameter on the performance of PSCs with different HTMs (Fig.3), using the SRH recombination model as follows 8 : Here, σn and σp are capture cross-section for electrons and holes, is thermal velocity, Nt is defect density, n and p are the concentration of electron and hole, is the intrinsic density , is the intrinsic energy level , is the energy level of the trap defect. Based, on Shockley-Read-Hall recombination formula is directly proportional to the defect density (Nt) of the absorber layer. 49,50 Fig.3 depicts the efficiency and the Voc variation as a function of defect density in CsPbI2Br with different HTMs. Defect density of the absorber layer directly affects the performance of PSCs, which is a limiting factor to achieve the Schockley-Queisser limit. 51,52 It can be deducted from Fig.3b that by decreasing the Nt of absorber layer up to a certain value (Nt=1E13 cm -3 ), increases the PCE for all the HTMs studied. The lower defect density resulted into the increased lifetime due to increment in diffusion length (a measure of absorber quality) which reduces the rate of recombination in absorber. 3 By comparing the PCE value (Table 3) for high defect density of absorber (3.64E15 cm -3 ) and the optimized defect density (Nt=1E13 cm -3 ) increment in PCE was noted to 16.16, 17.44 and 17.71 % for Spiro-OMeTAD, PTAA and CuSCN respectively (Fig.3a). Similar trend was observed for Voc increase with decreasing Nt until reaching the optimized value of 1E13 cm -3 (Fig.3b). However, the improvement in Voc was higher in case of CuSCN and PTAA and gave the value of 1.43 and 1.42 V respectively, compared to Spiro-OMeTAD of 1.28 V. In case of Spiro-OMeTAD (Table  3), the VBO is less but the barrier height is large in contrast to other HTMs. Even, by considering the low defect density of the absorber layer (Nt=1E13 cm -3 , for high quality layers) and low value of VBO, will not allow to reach the theoretical value of Voc in case of Spiro-OMeTAD. However, if the barrier height is low with large VBO, the performance can be near to the Shockley limit as shown for CuSCN and PTAA.
To validate these assumptions, Table 5, depicts the effect of VBO and barrier height. Valence band energy level of Spiro-OMeTAD and PTAA is optimized by varying their electron affinity and keeping it closer, to the VBO and ϕBp values of CuSCN, which is experimentally achievable (Table 5).  When the Ev of the HTM decreases with respect to the Ev of the absorber (-5.66eV), the PCE and the Voc increases. Upon acquiring the value of VBM closer to the work function of back contact (ϕBp =-5.1 eV), the PCE and the Voc saturates, but when the electron affinity is higher than 2.25 eV and 2.3 eV for Spiro and PTAA and corresponds to the Ev= -5.25 eV in both cases (Fig. 4a & c) respectively, the performance dropped suggesting the optimized value of Ev is -5.25 eV when gold is used as back contact. In the context of interface engineering in optoelectronic devices, we can explain these results in two folds, when the VBM of HTMs affects directly and/or at the same time the VBO and the barrier height (ϕ ) at HTM-Back contact. It implies that an optimized value of VBO and ϕ should be establish for each HTM   To further elucidate this, we analyzed the band diagram of devices with different value of Ev for each HTM (Fig.4b,d) By focusing on the area between the HTM and the back contact with the change of the Ev of HTMs, the performance increases with increasing the VBO value until -0.41 eV. This corresponds to the electron affinity of 2.25 and 2.3 eV for Spiro and PTAA respectively and, beyond this optimized values of VBO (corresponds to Ev_HTM of -5.25eV), the drop in performance was noted due to the formation of Schottky barrier ϕBp for holes. The value of ϕBp was calculated using the equation described in the section 2 and increases with increasing the Ev_HTM as illustrated in the inset of ( Fig.4.b,d) and Table 5. If the Schottky barrier is greater than -0.15 eV, it will hinder the transfer of the holes to the back contact, which decrease the performance of CsPbI2Br based PSCs. Table 4the Schottky contact is preferable with -0.15 eV to the barrier height.

The influence of HTM's valence band on the performance of PSCs
illustrates the performance of CsPbI2Br PSCs with the optimized parameters. Nt =1E13 cm -3 for absorber layer, VBO=-0.41 eV and (ϕ )=-0.15 eV.

Bromide substitution on the performance of CsSnI3-xBrx
It is reported that the addition of SnF2 (20 mol%) allows to decrease the defect density in CsSnI3 from 1.1E19 cm -3 to 5E17 cm -3 , by influencing the morphology of CsSnI3. 19 Similarly, in of CsSnI3-xBrx band gap on the PCE and Voc of PSCs. Present work, we have focused on the CsSnI3 where Iodine was partially replaced by bromide anion, to tune the band gap. The band gap variation was studied on the performance of the CsSnI3-xBrx based PSCs. Fig. 5(b) shows the simulated trend of PCE and Voc as a function of band gap for the planar device structure FTO/SnO2/CsSnI3-xBrx/Spiro-OMeTAD/Au and using the parameter given in Error! Reference source not found.
Corresponding to Br/ (I + Br ratio, (x) ranges from 0-3, the band gap was varied from 1.27 −1.75 eV as shown in Fig.5a. Similar trend was noted for CH3NH3PbI3 and CsSnI3, 34,53 experimentally validating our investigation. Fig.5b illustrates that the open circuit voltage (Voc) escalates with the increase in band gap due to increased Br fraction, while PCE starts dropping beyond x=2.  The improvement of Voc is a result of the band gap increment due to high Br content, however, it reduces the absorption in the visible region. To emphasize here, SnO2 was employed as ETL, as the selective layer plays crucial role to reduce the recombination rate particularly the mismatch in energy level alignments at ETL/perovskite interface may increases the recombination. This could be explained by the conduction band offset (CBO), i.e. the difference between the conduction band minimum(CBM) of ETL and those of perovskite layer (±∆ ), 54 the (+) sign represents the spike structure and the (-) sign is the cliff structure. 35 The spike structure is shown to be favorable for improving the performance of PSCs, which can build a potential barrier at ETL/perovskite to reduce the recombination rate 55 . From our theoretical and experimental observation, with the increase in band gap of perovskites, only Ec is assumed to shift upwards thus, the CBO become smaller with optimum value of +0.09 eV (spike structure) for CsSnIBr2 and gave the best performance Fig.5a. The PCE decreases dramatically after x=2 and the Voc can be described by the following equation. 56,57 Voc is open circuit voltage, EA is activation energy, n is diode ideality factor, K is Boltzmann constant, T is temperature, J00 is current prefactor and Jsc is short circuit current density. For T = 0 K, Voc= EA/q. The CBM of bulk CsSnI3 compared to the CBM of ETL is much deeper, thus a large energy barrier with ∆ = +0.47eV affects the performance of PSC. By adding Br to the CsSnI3 leads to the increase in the band gap by shifting the conduction, band upwards. This enhances the energetic level with ETL and the ∆ become small enough to facilitate the transfer of electrons toward the front side of load with the increase of activation energy as well as Voc. The performance of our first model using CsSnIBr2 is still low and matches with the experimental finding in studies, which can be ascribed due to the low defect density in the absorber and at the selective layers interfaces.

The effect of defect density of absorber layer on the performance of cells
To explain the low performance of CsSnI3-xBrx based PSCs as mentioned in above section, we have selected CsSnIBr2 as absorber layer, which has favorable energetic level alignment with SnO2 as ETL. We assume that the defects in inorganic Snbased perovskite will have significant effect on the performance of inorganic lead free PSCs, and for this we investigated the effect of defect density(Nt) . Fig.6a-c represents the influence of defect density in the CsSnIBr2 absorber layer on the photovoltaic properties From Fig. 6 a & b we noted that up to a certain value of Nt=1E15 cm -3 , the Jsc decreases, which is also reflected in the EQE curve (Fig. 6b). However, beyond this value of Nt, the Jsc becomes unaffected and remains constant. Further, Nt=1E18 to 1E15 cm -3 , J-V characteristics shows the curvature and double diode like behavior up to 1E15cm -3 along with the decreased value of FF. The value of FF and PCE increases as Nt decreases and remains unchanged when the values of Nt become <1E12 cm -3 and leads to high fill factor and PCE (Fig. 6c). High crystallinity, corresponds to optimal quality of perovskite to enhance the performance of Sn-based inorganic PSCs. [58][59][60][61] The improvement in film quality as a result of low value of Nt can be seen in EQE curve Fig.6c, the absorption of light in a thin film, can also be affected by the defect density into the absorber layer. We noted that EQE increases rapidly with decrease of the perovskite defect density and acquire saturation when Nt < 1E12cm -3 . For our further calculations, we took Nt =1E12 cm -3 as an optimum value of defect density for CsSnIBr2 absorber.

Influence of interface defect density at CsSnIBr2/HTM
The quality and nature of interface at absorber/HTM is paramount to fabricate efficient PSCs, here we have varied the Nt from 10 17 cm -3 to 10 12 cm -3 at CsSnIBr2/HTM interface as shown in Fig.7.a-c. We noted abrupt drop in the performance when Nt >10 15 cm -3 and remains nearly unchanged if Nt<10 15 cm -3 , this enhancement was obtained due to the reduction in defect density at CsSnIBr2/HTM interface. The value of contact resistance is higher, when high defect density Nt10 15 cm -3 is present which also led to lower fill factor (Fig.7b). However, the Jsc change is insignificant to defect density variation that can be manifested from the EQE graph Fig.7c. Thus, we have selected 10 15 cm -3 as an optimum value for CsSnIBr2/HTM interface which could delivers a PCE =18.17 %, Voc = 1.29 V, Jsc =20.29 mA/cm 2 and FF = 69.29 %.

Influence of interface defect density at ETM/CsSnIBr2
We continue our simulation studies with the optimization of ETM/CsSnIBr2 interface, which can further contribute to improve the performance of PSCs Fig.7d-f By decreasing the value of Nt, the performance increases linearly, and achieve an optimum value of Nt, beyond this value it saturates. From Fig. 7b, we can set Nt =10 14 cm -3 as an optimal value for ETM/CsSnIBr2 interface, and using this value, a PCE of 20.35 %, with a Voc of 1.35V, Jsc = 20.30 mA/cm 2 and FF = 74.17 % can be derived. It can be concluded that the defect density at ETM/CsSnIBr2 has a strong effect on the performance, as solar irradiation passes through this interface first, thus at this interface photo-generated carrier concentration is higher than of CsSnIBr2/HTM interface. The high defects density will create more recombination sites and traps thus a higher recombination rate. The investigation of the EQE (Fig.7f) as a function of defect density of ETM/CsSnIBr2 interface further support our hypothesis, decrease of Nt leads to enhanced light absorption in the region of 300-700 nm, contributing to the improved EQE.

Influence of defect density on diffusion length of CsSnIBr2 absorber
The absorber quality can be determined by the diffusion length for the electron and hole in an absorber layer and can be calculated by using the equation 7-9.
= √ τ (7) , , =  , Here L is the diffusion length, n,p, the charge carrier life time, σn and σp are capture cross-section for electrons and holes, is thermal velocity, and D is the diffusion coefficient. and µ represent Boltzmann constant and charge carrier mobility, and represent the magnitude of charge and temperature in Kelvin. The equations (7,8 and 9) highlight the direct relation between the defect density (Nt) and lifetime n,p, of free charges in bulk perovskite, which control the performance of PSCs. Fig.8 shows the diffusion length (L) of charge carriers as a function of defect density in CsSnIBr2 this provide ideas about the range of the diffusion length required the interfaces. As discussed in previous section (3.2.), PSCs based on the n-ip configuration, FTO/SnO2/CsSnIBr2/Spiro-OMeTAD/Au was optimized through studying the impact of defect density in CsSnIBr2 and at its interfaces. By using the optimum values of defect densities for CsSnIBr2 (10 12 cm -3 ), CsSnIBr2/HTM (10 15 cm -3 ) and ETM/CsSnIBr2(10 14 cm -3 ) respectively, we can estimate a value of 278 m as the diffusion length for the charge carriers in absorber layer, and 27.84 m, 88.06 m in case of CsSnIBr2/HTM and ETM/CsSnIBr2 interface layers respectively. Further using these optimum values of Nt, a theoretical value of a PCE =20.32 %, Voc =1.35 V, Jsc = 20.30mA/cm 2 and FF 74.17 % can be achieved (Table 7).  Table 6 summarizes the experimental studies reported and compared to our first model and the optimized model. The optimized model shows promising results in the field of leadfree inorganic PSCs and is in close vicinity of the Shockley-Queisser-limit 43 . Adopting new experimental protocol for syntheses of tin based materials in Pyrex-tube at high temperature under vacuum can avoid the oxidation and improve the quality (crystallinity) of absorber to harvest more light with low defect density. 62 Further, the passivation mechanism could be a helpful strategies to minimize the trap density at the interfaces of selective layers.

Conclusions
To summarize, using one dimensional computational approach, device structure of FTO/SnO2/CsPbI2Br/HTMs/Au was designed by selecting different organic, polymeric and inorganic types hole transport materials such as Spiro-OMeTAD, PTAA and CuSCN, and the defect density of absorber layer was investigated suggesting significant influence on the performance of PSCs. Tuning the valence band maximum of Spiro-OMeTAD and PTAA demonstrated competitive performance as compared to CuSCN. An optimized value of VBO (-0.41 eV) and the barrier height between the HTM and back contact (-0.15 eV) with schottky contact was found to be beneficial to achieve high performance. Lead free inorganic PSCs based on CsSnI3-xBrx was also investigated. The addition of Br to substitute I in CsSnI3 allows to improve energy level alignment with the ETL. The quality of absorber layer and its interface is paramount to design efficient inorganic Pb-Free PSCs that can be controlled by optimizing defect density. We put forward the design of high performance Pb-Free PSCs having the structure of FTO/SnO2/CsSnIBr2/Spiro-OMeTAD/Au, with Nt=10 12 cm -3 for absorber layer, and Nt=10 15 cm -3 in case of CsSnIBr2/HTM and Nt=10 14 cm -3 for ETM/CsSnIBr2 yielding a PCE of 20.32%. Our computational approach can pave the way to boost the performance of inorganic Pb and Pb-Free PSCs by taking into account developed parameters.

Conflicts of interest
There are no conflicts to declare.