Mathematical modelling and simulation of CH3NH3Pb(I1−xBrx)3-based perovskite solar cells for high efficiency

Mathematical modelling provides a comprehensive understanding of the charge transport process inside solar cells and determines various elements affecting their performance. In the present paper, various proposed CH3NH3Pb(I1−xBrx)3-based perovskite solar cells consisting of Cd1−xZnxS as electron transport layer (ETL) have been mathematically simulated using the SCAPS-1D simulator. Hole transport layers (HTLs) of poly(3-hexylthiophene) (P3HT), Cu2O, CuSbS2, and CuO were each employed separately to form various proposed solar cells. These perovskite solar cells were compared with existing CH3NH3PbI3−xClx-based perovskite solar cells consisting of Cd1−xZnxS as the ETL and CuI as the HTL. Amongst the proposed HTLs, P3HT exhibited the best efficiency of 28.28%. During the simulation, thickness of 700 nm for the absorber layer [CH3NH3Pb(I1−xBrx)3] of the proposed perovskite solar cells was found suitable for effective solar cell design. These mathematically simulated results demonstrate the optimum efficiency of the solar cell, which will aid in the design of high-efficiency solar cells in the near future.


Introduction
The consumption of energy has increased exponentially. Thus, a reliance on conventional energy resources is insufficient to fulfil the expanding energy requirement. Solar cells exhibiting higher efficiency are desirable candidates to meet the essential electrical energy demand by producing maximum electric power output using solar energy. In this regard, perovskite solar cells (PSCs) are one of the most promising candidates because of their exceptional power conversion capabilities in addition to improved light absorption, lower electron-hole binding energy, direct band gap, and costeffective fabrication in ambient conditions. A viable PSC must have appropriate values of fill factor (FF), short-circuit current density (J SC ), open-circuit voltage (V OC ) and power conversion efficiency (PCE) for converting a large amount of solar energy into electrical energy using the photovoltaic effect. Thus, a standard PSC has a versatile configuration consisting of a perovskite layer (active absorber layer), hole transport layer (HTL) and electron transport layer (ETL). Practically, the fabrication of the various layers of a multilayered perovskite solar cell is an expensive and timeconsuming task. Therefore, computational modelling and simulation studies are important initial steps prior to actual fabrication of different cell layers, as these are helpful in identifying appropriate materials required for processing different layers of perovskite solar cells.
An absorber layer of a PSC incorporates a perovskite structure having the general formula CDX 3 , where C denotes an organic cation, i.e., formamidinium (NH=CHNH 3 + ) or methyl ammonium (CH 3 NH 3 + ), D denotes an inorganic cation, i.e., Sn 2+ and Pb 2+ , and X denotes a halide ion, i.e., Cl − , Br − or I − . Methyl ammonium lead iodide (MAPbI 3 ) is the most common perovskite used in PSCs, while MAPbI 3−x Cl x , MAPb(I 1−x Br x ) 3 , MAPbSnI 3 , MAPbI 3 and FASnI 3 are other perovskite materials which are used as absorber layers in perovskite solar cells [1][2][3][4][5]. Various perovskites and 1 3 non-perovskite materials are also examined and numerically simulated for further utilization as active absorber material in multilayer perovskite solar cells. In addition, pure organic perovskite materials have also been explored and found suitable for the absorber layer in perovskite solar cells.
The performance of the perovskite layer can be improved by utilization of an appropriate electron transport material (ETM) which withdraws the photoelectron produced by the absorber material, and the hole transport material (HTM), which promotes the formation of holes. TiO 2 , ZnO, Cd 1−x Zn x S and SnO 2 are the materials used as ETLs [6][7][8][9][10] while CuI, CuO, P3HT, Spiro-OMeTAD, Cu 2 O, CuSCN and CuSbS 2 are the materials used as HTLs [11][12][13]. The stability and efficiency of perovskite solar cells depend primarily on the characteristics and type of these layers. It has been observed that a perovskite solar cell exhibits maximum efficiency when the rutile crystalline phase of TiO 2 is used as ETM. This phase of TiO 2 requires a high annealing temperature for processing, which restricts its use in perovskite solar cells. Similarly, most common HTMs such as PEDOT:PSS and Spiro-OMeTAD are expensive and less stable, which restricts their use in perovskite solar cells. Hence, various research efforts have been carried out for optimization of solar cell performance by using different ETMs and HTMs. A perovskite solar cell exhibits an efficiency of 24.32% while using MASnI 3 , TiO 2 and CuSCN as absorber layer, ETM and HTM respectively. Other perovskite solar cells exhibit efficiency of 24.17%, 24.50% and 25.36% while using MASnI 3 as absorber materials, ZnO as ETM, and Spiro-OMeTAD, PEDOT:PSS and Cu 2 O as HTMs, respectively. However, perovskite solar cells with CuI as HTM, ZnO as ETM and MASnI 3 as perovskite layer exhibit efficiency of 24.82%. The influence of the back contact material on the efficiency of PSC has also been studied. Remarkably, perovskite solar cells that came into existence in 2009 and exhibited efficiency of 3.8% have been improved to exhibit efficiency of 25.6% in 2021 [14][15][16][17]. The efficiencies of perovskite solar cells are still undergoing progressive modification with an unpredictable upper threshold.
In this paper, the CH 3 NH 3 PbI 3−x Cl x -based perovskite solar cell (existing) with Cd 1−x Zn x S and CuI as ETL and HTL respectively, exhibiting efficiency of 25.68%, is compared with various CH 3 NH 3 Pb(I 1−x Br x ) 3 -based perovskite solar cells that have been mathematically simulated using the SCAPS simulator, where HTLs of Cu 2 O, CuSbS 2 , CuO and P3HT [poly(3-hexylthiophene)] (proposed) are used separately using Cd 1−x Zn x S as an ETL. Optical properties including power conversion efficiency, fill factor, short-circuit current density and open-circuit voltage of the proposed device are characterized by varying the thickness, doping concentrations, and interfacial defect density of the absorber layer [CH 3 NH 3 Pb(I 1−x Br x ) 3 ]. Amongst the abovementioned proposed perovskite solar cells, the cell with P3HT [poly(3-hexylthiophene)] as HTL demonstrated better efficiency of 28.28%. Interestingly, P3HT is an organic material and cheaper than the popular organic HTL such as Spiro-OMeTAD.

Proposed device simulation
In this paper, design simulation of the mixed-halide perovskite solar cells has been executed on the SCAPS-1D simulator. The device design for the simulation comprises transparent conducting oxide (TCO)/Cd 1−x Zn x S (buffer layer or ETL)/DL I/active layer (CH 3 NH 3 Pb(I 1−x Br x ) 3 )/DL II/HTL/back contact. The proposed cell has been simulated independently using various HTLs including CuI, Cu 2 O, CuSbS 2 , CuO and P3HT. From the various simulations, it was found that solar cell with P3HT as HTL displayed the highest efficiency of 28.28%. A simple configuration with respect to this is revealed in Fig. 1a. SCAPS-1D software used for the simulation was administered by Poisson and continuity equations [18][19][20][21].
where n(x) and p(x) denote the concentration of negative and positive charge carriers, respectively; G n (x) and G p (x) denote the rate of photogeneration; and R n (x) and R p (x) signify the recombination of negative and positive charge carriers, respectively. E(x), D and µ in Eqs. 1 and 2 denote  Table 1.
The defects existing in the device structure highly influence the performance of the cell. The Shockley-Read-Hall (SRH) recombination model applicable for regulating the defect concentration in a solar cell is mathematically expressed as [19,20] where τ n,p is the relaxation time of charge carriers, n and p are the density of the electron and hole, respectively, n i is the intrinsic density, E i is the intrinsic energy level and E t is the energy level of the trap defects.
The thermal velocity for negatively and positively charged carriers in every layer is held steady at 10 7 cm/s.
. N t and σ n,p are the density and capture cross-sectional area pertaining to defects, respectively.
The diffusion length (l) in respect of the mobile charged particles is expressed as [19,20] Here, the diffusion constant is represented by D, which can be found using the relation [19,20] Various parameters can be estimated using the above equations. The calculated values were found to be very close to the simulated results.
The planned design of the solar cell used for the simulation is shown in Fig. 1b. In addition, the simulated energy band diagram and comparison of energy levels of the structure simulated with different HTLs is illustrated in Fig. 2a, b, respectively.

Effect of thickness of mixed-halide perovskite layer
In this paper, mixed-halide perovskite layers with varying thickness from 300 to 1200 nm have been analysed. It has been observed that thin perovskite layers lead to lower current density (J sc ) and lower efficiency (η) values due to low light absorption, which is not advantageous for a solar cell. On the other hand, perovskite layers with greater thickness result in higher recombination due to the formation of a critical path for the movement of electrons and holes. Consequently, it is necessary to select a perovskite layer of optimum thickness to design an efficient solar cell. Experimentally, it has been found that mixed-halide perovskites show optimum efficiency   Table 2. The performance then begins to decline gradually due to an increased rate of recombination. Due to the variation in diffusion length and absorption depth of mobile charged particles, they recombine prior to reaching the interface.
Consequently, there is steady decline in V OC and FF values along with decrease in PCE as shown in Fig. 3.

Impact of total defect density (defect concentration) in mixed-halide perovskite
Defect density is a significant factor that plays a very crucial role in determining the performance of perovskite solar cells. The photo-generated charge carriers produced in the perovskite layer are separated under the effect of the electric field and migrate towards the corresponding ETL and HTL. The quality of the mixed-halide perovskite  layer directly influences the device performance which can be disrupted through substantial amount of defects in the layer. These defects result in early recombination of holes and electrons, therefore restricting their active involvement. Further, Gaussian defect states have a greater influence on hole and electron trapping than that of the Urbach tail which affects cell performance [27]. The variations of device parameters by varying total defect density is represented through graphs in Fig. 4. From the graphs, it is seen that there is a decline in cell parameters-defect density curves in the range of 10 8 -10 15 cm −3 . The device is extremely efficacious at the lowest defect density value of 10 8 cm −3 which indicates the suitability of the perovskite layer with minimum defects [22,[28][29][30]. Nonetheless, the synthesis of such a fine, smooth and immaculate surface is a Herculean task. The existing solar device with CH 3 NH 3 PbI 3−x Cl x as the perovskite layer exhibits optimized device parameter values (power conversion efficiency = 25.68%, fill factor = 84.11%, short-circuit current = 25.33 mA/cm 2 ) with a defect density of 10 13 cm −3 [21]. A comparison of existing cells with the proposed device with defect density of 10 12 cm −3 in the mixedhalide perovskite, i.e. CH 3 NH 3 Pb(I 1−x Br x ) 3 , including the optimized values of various parameters achieved using different HTLs, is tabulated in Table 3.

Effect on recombination rate due to defect density of the CH 3 NH 3 Pb(I 1−x Br x ) 3 layer
The influence of the defect density of a perovskite layer on cell operation can be easily obtained using the Shockley-Read-Hall (SRH) recombination model by ascertaining Fig. 4 Graphical representation of solar cell parameters with respect to total defect density of the active layer: a efficiency vs total defect density, b open-circuit voltage vs total defect density, c current density vs total defect density and d fill factor vs total defect density ▸ 1 3 the influence of defect density on the recombination rate [18][19][20][21]. A graph of rate of recombination and depth from surface at varying defect densities is illustrated in Fig. 6. An increase in recombination rate can be observed by increasing the defect density. Consequently, the output of the solar cell parameters is reduced with an increase in defect density, as shown in Fig. 4. From Eq. 4, it is seen that relaxation time of mobile charged particles is inversely proportional to defect density. Also, from Eq. 3, it is visible that recombination rate is inversely proportional to relaxation time of mobile charged particles. The abovementioned equations mathematically confirm that with increase in defect density, relaxation time of mobile charged particles reduces and finally recombination rate increases.
The relaxation time of charge carriers is also useful in conveying diffusion length (l). From Eq. 5, it is confirmed that the higher charge carrier mobility results in larger diffusion length. Therefore, a low recombination rate and large diffusion length results in higher efficiency of 28.28%. Energy diagrams of the proposed solar cell are illustrated in Fig. 2a, b.

Conclusion
In summary, using the SCAPS-1D simulator, the optimization of a perovskite solar cell with a perovskite layer of CH 3 NH 3 Pb(I 1−x Br x ) 3 has been exhaustively examined. The effect of different HTLs on the solar cell performance has been analysed to obtain maximum efficiency. From the simulation, it was found that a perovskite solar cell comprising Cd 1−x Zn x S as the ETL and P3HT as the HTL exhibits higher efficiency than the other combinations. During simulation, the effect of various properties including doping concentration, thickness, and total defect density on the perovskite layer of CH 3 NH 3 Pb(I 1−x Br x ) 3 using different HTLs has been investigated to enhance the cell performance. Interestingly, it was observed that differences in HTL thickness had a negligible effect on solar cell performance. Further, doping in the perovskite layer was found to have an immense influence on cell performance. The proposed perovskite solar cell with an active layer of CH 3 NH 3 Pb(I 1−x Br x ) 3 and P3HT as HTL has displayed an optimized efficiency of 28.28%, which is much higher than the maximum efficiency (25.68%) of existing solar cells, having an active layer of CH 3 NH 3 PbI 3−x Cl x, and CuI as HTL. This prototype of solar cell may potentially assist researchers in subsequent investigation towards practical realization of mixed-halide perovskite solar cells.