Comparative 1D optoelectrical simulation of the perovskite solar cell

As a promising third-generation solar cell (SC), the perovskite solar cell (PSC) has caused a motivation in scientific research with the major enhancement of the power conversion efficiency (PCE) from 3.8% in 2009¹ to further than 22% in 2017.^2–4 The simpler planar configuration has been often chosen as the PSC structure compared to a porous scaffold pattern.^5,6 The planar PSC commonly contains a heterojunction architecture with three main layers, i.e., electron transport material (ETM), perovskite absorber, and hole transport material (HTM). The commonly used ETM, perovskite, and HTM are TiO₂, CH₃NH₃PbI₃, and Spiro-OMeTAD, respectively. Generally, to fabricate efficient planar PSCs, morphology control of perovskite films, using new materials, optimizing device structure, alternative HTM layer and development of effective fabrication methods are key factors.^7–9 For example about the new HTM layer, inorganic materials are considered as promising materials for the HTM layer because of their favorable characteristics, such as high mobility of hole, low cost of fabrication, better chemical stability, and appropriate valence band position. So, simulation can also play a significant role in choosing an inorganic HTM alternative.^10,11

For optical modeling of the thin-film solar cell, the most valuable and common methods are transfer matrix method (TMM) and finite difference time domain (FDTD).¹² The TMM is one of the most generally used numerical techniques because it is an impressive frequency-domain based on simple matrix operations, which are computationally light. Moreover, the TMM has been proven to be a very effective adaptable tool for simulating the performance of thin-film solar cell structure, geometrical optimization, and contact configuration.^12–16 This method, by considering the coherent-incoherent optical effect, can calculate the reflection and the transmission spectra at each interface as well as the attenuation in each layer.

In relation to the electrical simulation, the software like the Solar Cell Capacitance Simulator (SCAPS-1D)¹⁷ was used. It is designed based on three basic semiconductor equations under steady-state condition (i.e.: the Poisson, the hole continuity, and the electron continuity equation). The SCAPS can describe a solar cell as a series of layers with different optoelectrical properties as well as defect states of each layer and interface. It is well adapted for the analyses of homo and heterojunctions, multi-junctions and Schottky barrier photovoltaic devices, such as PSC.^18–21 The other similar one dimension simulation tools like PC1D, ASA and Amps-1D can be used for thin-film solar cell and were compared with the SCAPS by Burgelman et al.²² 

Beside experimental study, the simulation provides a powerful tool that will lead to a better understanding of physics of the SC and certainly has a major contribution to remove unclear points of the SC.^14,23–25 Despite the significant progress toward high PCE, a more practical simulation study is desirable to achieve an improved understanding of the optical and electrical factors.^11,26–28 There are some 1D simulation studies about planar heterojunction PSC.^11,26–29 According to options of these 1D codes such as SCAPS, most of these studies focus more on various factors such as layer thickness, working temperature, illumination intensity, different solid-state parameters of layers (mobility, band gap, doping type/concentration and…). Therefore, they investigated the influence of these factors on the basic parameters of the SC (FF, V_OC, J_SC, and PCE). This reference provides a comprehensive study in this regard.²⁹ In this respect, proper modeling of the absorption layer is of great importance, especially for the simulation of the EQE spectrum. But, an analytical model is often used for this purpose. Therefore, the main purpose of this study is to determine to what extent the output of the 1D modeling can close to the experimental ones when complementary optical calculations are considered for optical absorption.

On the other hand, this study showed that how the combination of optical and electrical simulations by using simple methods can be considered as a practical tool for investigating the SCs, regardless of another involved effect such as the resistance losses and ionic emigration. The validity of optoelectrical device modeling was investigated by comparing the simulation results with the experimental ones, especially the EQE spectrum and JV curve. Then the difference between the simulated parameters of the PSC and experimental ones were comprehensively investigated.

By considering the coherent-incoherent optical effects, the TMM method was used for calculating the wavelength-dependent total reflection of the front surface of the SC. Moreover, the absorption coefficient, α, which is related to the extinction coefficient, $α=4πkλ$⁠, were calculated for all layers of the PSC. The outputs from this model were saved as input ASCII files so that they could be used in the SCAPS.

The planar PSC is assumed to be composed of different homogeneous and isotropic layers. When considering the structure like this, in terms of the electromagnetic theory of light, several different approaches are possible in order to calculate the transmission-reflection coefficient and parasitic absorption of each layer. One of the more practical approaches for multilayer structures is to use the TMM. The stratified structures with isotropic and homogeneous media and parallel-plane interfaces can be expressed by 2*2 matrices.³⁰ Consider a plane wave incident from the left at a common multilayer structure having m layers between a semi-infinite transparent ambient and a semi-infinite substrate. Each layer (j=1 … m) has a thickness dj and its optical properties are represented by its complex refractive index $nj=ηj+ikj$ or complex dielectric function (⁠$εj=εj′+iεj′′=nj2$⁠) which is a function of wavelength (energy) of the incident light. The TMM code which was developed by D. McGehee was used for the optical simulation¹³ (whenever necessary, the new commands added to the code). Therefore, the model of the planar PSC (Fig. 1) was used to calculate the total front reflection versus the wavelength by TMM. The results of these calculations for the total reflection of each sample as well as the absorption coefficient (⁠$α=4πkλ$⁠) in the form of two-column ASCII files were transferred into the SCAPS code. The refractive indexes of the material that used in this study have been adapted from this Ref. 31.

For the electrical simulation, the previously calculated parameters (include the absorption coefficient files of all layers and total reflection file from the front surface), along with other required parameters, were set into the SCAPS (version 3.3.05 under 1 sun). Baseline input parameters used in the SCAPS are presented in Table S1 (supplementary material). Most data were taken from Refs. 32, 33.

The total trap density 10⁹ cm^-2 in both interfaces and band tail defect 10¹⁴ eV^-1cm^-3 for each layer was introduced to the SCAPS as well as shallow level donor and acceptor density N_D=10% of N_A and N_A=10% of N_D in HTM and ETM, respectively. Fig. 1 shows the PSC structure. The multilayer stack glass (FTO-coated soda-lime glass substrate) was used as the substrate.³¹ 

In the SCAPS, the optical absorption coefficient can be set from either a model or a two-column file format, when it is set from a model, α(λ) is given by the following equation:

If using this SCAPS’ model for the perovskite layer, the absorption parameters A and B is usually set to 10⁵ and 0, respectively. But in this study, the absorption coefficient file method (⁠$α=4πkλ$⁠) was used for simulation.

In relation to modeling defects in the SCAPS, they can be set in both bulk and interface. There are five defect types and distributions available in the software. It considers the defect in bulk and interfaces and provides a variety in types of selection of defect charge. The energy distribution caused by defects can be a single level, uniform distribution, Gauss distribution, and Band tail. The presence of defects and traps in perovskite and details of their impact are so far under theoretical and experimental discussion and investigation.³⁴ However, bulk defects in each solar cell layer, as well as interface defect between active and buffer layers, have been used in this modeling (detailed in supplementary material Table S2 and S3). On the other hand, the weak exponential tail states were used for the E_C and E_V of PSC layers (Table S2). Moreover, two Gaussian distribution defects were also used to simulate deep defect (Table S2). In addition, the interface traps were also introduced for the contact surfaces between the active layer and adjacent buffer layers (Table S3).

The parasitic absorptions of different layers of the PSC and the total reflectance from the front electrode are shown in Fig. 2(a). Moreover, Fig. 2(b) shows the electric field intensity at the different position of the PSC and varying wavelengths.

Fig. 2(b) shows that the high field intensity occurs for wavelengths near the infrared region, where the conventional CH₃NH₃PbI₃ has poor absorption.³¹ In this regard, the band gap engineering of the active layer can play an important role in increasing the solar cell’s efficiency. For example, the band gap of formamidinium lead iodide (FAPbI₃) perovskite permits wider absorption of the solar spectrum which results in an efficient SC compared with conventional methylammonium lead iodide (MAPbI₃).² Hence, FAPbI₃-based PSCs achieved greater PCE (∼20%) relative to MAPbI₃-based PCSs(∼15%).² 

Fig. 3(a) and 3(b) show the simulated EQE and JV of 1D models of the planar PSC that its structure was shown in Fig. 1 (The experimental EQE and JV were taken from Ref. 35). For all samples, the same set of parameters and defects-traps are used which they are introduced in Table S1-S3. In the S₀ sample, without using ASCII file format of the absorption coefficient (equation (1) was used and the absorption parameters A and B is set to 10⁵ and 0, respectively) and without total reflection, the EQE curve is more similar to the ideal solar cell’s behavior (the square shape). By considering the absorption coefficient file for S₁ sample as well as front reflection file in the S₂ sample, the EQE curves become more similar to the experimental one (the experimental EQE–JV curve belongs to the laboratory certified CH₃NH₃PbI₃ perovskite SC³⁵).

The zero-edge of the S₀’s EQE corresponds to set band gap (1.5 eV or ∼826 nm) in SCAPS. For S₁ and S₂ samples, the absorption coefficient has zero value after ∼800 nm, which, conforms to the EQE spectra. Fig. 3(b) shows the simulated JV characteristics curves, which by comparison with the experimental one, the results of the S₁ and S₂ samples also seem to be more practical than S₀. Therefore, it can be noted that the S₀ sample (without total reflection and by using the absorption coefficient in the form of formula 1) is simulated more electrically, and its results are similar to other simulations,^18,19 which only used the electrical simulator like SCAPS.

On the other hand, the results of S₀ sample do not seem to be very practical, but, with the step-by-step entrance the optical effects along with the electrical effects, the results in S₁ sample and especially in the S₂ sample will be more similar to the experimental results. Although the simulated curves seem to be practical, there is still no complete match between experiment and simulation. For example, at near UV, there is still no good agreement between the experimental EQE and the simulated curve.

In the following, Fig. 4 (a and b) shows the variation of the PSC device performance versus thickness of perovskite. The absorption coefficient file method (⁠$α=4πkλ$⁠) and the total reflection from the front surface were considered for this simulation (Fig. S1 of supplementary material shows the same results when the total reflection is not considered, but the absorption coefficient file is used).

The simulated solar cell parameters, especially PCE and V_OC, are increasing more or less steeply up to 700 nm and very gradually beyond 800 nm. The results show that ∼800 nm thickness is also satisfactory for proper photovoltaic action. Beyond this thickness photovoltaic performance is growing very slowly due to an increase in series resistance. While, without optical effects, the thickness of about 400 nm seemed to be sufficient.³² Based on relevant reference, the notable laboratory certified values for the conventional CH₃NH₃PbI₃ PSC is PCE=15.0±0.6%, V_OC=1.090 V, J_SC=20.61 mA/cm², FF=66.8%.³⁵ 

At 800 nm thickness of the absorber, the performance of the S₂ sample (Fig. 2) is simulated as PCE=19.68%, V_OC=1.118, J_sc=21.68, and FF=81.18. The V_OC and J_SC are very close to experimental ones but the simulated PCE and FF are higher than experimental ones.

In order to explain the reasons behind the difference between the experimental and the simulated SC parameters, the empirical relation between the V_OC and the FF is:³⁶ 

Where $νOC=qnkTVOC$ is defined as normalized V_OC. According to this equation, the smaller V_{OC (and} V_OC around 1) will result in lesser FF.

Moreover, the V_OC can be expressed as:

Where n is an ideality factor, KT/q is thermal voltage, J_SC is light generated current density, and J₀ is dark saturation current density.

While J_SC normally has a slight variation, the key effect on the study of V_OC variations is J₀, since this may change by orders of magnitude. J₀ is directly related to recombination, and thus, inversely related to the quality of the SC.³⁷ Therefore, since the experimental SC is influenced by the manufacturing steps, the lower quality of the experimental cell (parasitic resistive losses) is expected to lead to an increase in J₀, and thus a decline in V_OC and therefore a decrease in FF.

Equation (3) also reveals the importance of the ideality factor. The ideality factor is a quantity on the junction quality and the kind of recombination in an SC. For the ordinary recombination mechanisms (direct recombination); the n-factor has a value unity. However, some recombination mechanisms (trap-assisted recombination) particularly if they are large, may introduce high n-value in the experimental cell that not only reduces the FF but also offers low open-circuit voltages.^36,38

Graphically, the FF is a degree of the "squareness" and is also the area of the largest rectangle which will fit in the JV curve. As can be seen from Fig. 3(b), the experimental JV curve with smaller V_OC and J_SC as well as a more rounded portion will result in lower FF. In fact; some properties such as resistance (shunt and series) and ionic migration are not introduced in this simulation. The shunt resistance is typically due to processing defects, rather than weak solar cell design and the series resistance is mainly due to contact resistance due to the different layer of the SC. The main impact of resistances, especially series resistance, is to decrease the fill factor. Therefore, the simulated SC parameters, in particular, FF, are larger than laboratory values. So, according to the relation between FF and PCE (⁠$η=VocIscFFPin$⁠), decrease in FF, result in lower efficiency.

Finally, looking at Fig. 3(a) again, the quantum efficiency can be observed as the collection probability caused by the generation profile of a single wavelength, integrated over the device thickness and normalized to the incident number of photons. For a typical SC, while quantum efficiency ideally has the rectangular shape like the sample S₀, the quantum efficiency is decreased because of recombination effects. The difference between the ideal curve and the experimental one in the three areas of UV, IR, and visible take place because of different reasons. Front surface passivation affects carriers generated near the surface. Since blue light is absorbed very near to the surface high front surface recombination will disturb the "blue" section of the quantum efficiency. Moreover, the red response usually reduced due to rear surface recombination, decreased absorption at long wavelengths, and short diffusion length. But, the visible region is absorbed into the bulk of SC and low diffusion length and reflectance decrease the quantum efficiency in the visible portion of the spectrum. As can be seen in Fig. 3(a), unlike the electrical simulation (S₀), the results of this optoelectrical simulation are different from ideal and are closer to the experimental one. In the case of the PSC, maybe a little reflection and enough diffusion length (exceeding one micrometer¹) have led to the ideal value (∼100%) in both experimental and simulated EQE. However, there is a discrepancy between the results in the range of 350 to 550 nm. This discrepancy may be due to the disregard of factors like the resistance losses and ionic emigration.

In summary, an optoelectrical model of the planar heterojunction PSC has developed. By considering the coherent and incoherent optical effects, the TMM has used for optical analysis. Along with the optical parameters such as total reflection and wavelength-dependent absorption coefficient, all the capacities of SCAPS electrical simulator, such as absorption coefficient, optical reflection, defects, and interface trap have been used in this optoelectrical simulation. It has been observed that the solar cell parameters such as JV and EQE curves, which have been obtained by this optoelectrical model, are very practical and similar to the experimental results. This simulation revealed that due to high field intensity at larger wavelengths, broader absorption could lead to a significant improvement to the SC efficiency. It has been observed, by considering the optoelectrical effects, the suitable thickness of the active layer should be extended to ∼ 800 nm. The comparative values of V_OC, J_SC, FF, and PCE of the optoelectrical model with laboratory certified experimental one, reveal that achieving proper values of SC parameter, especially FF and PCE, requires more precision in SC engineering. It has been found that unsuitable values of shunt and series resistances of the SC, can absolutely degrade the FF and PCE values. Lastly, this 1D model cannot provide a very close similarity between the simulation and experimental results due to the lack of parameters such as the resistance losses and ionic emigration.

The supplementary material is available. It mainly contains data used in electrical stimulation by SCAPS.