Machine learning aided optoelectric characterization modelling and prediction of the IV parameters of perovskite solar cells with > 90% accuracy

The trained MLLR model is validated using a test dataset obtained from EL spectroscopy. Four devices, P1, P2, P3, and P4, are considered individually and combined as separate test datasets for validation. Firstly, based on the test dataset, the validation of the proposed predictive MLLR model is accessed by comparing IV parameters predicted and actual values when corresponding EL parameters are provided as input. Figure 4 provides predicted vs. actual Values of IV parameters from Day 1 to Day 8 obtained using the proposed FWHM-based predictive MLLR model using the test dataset (combined).

Table S15 shows the prediction accuracy (%) of individual devices (test devices) using the proposed MLLR Model for various IV Parameters. Figure S4 shows the prediction vs. measured values of IV parameters from Day 1 to Day 8 for individual devices P1, P2, P3, and P4 using the FWHM-based predictive MLLR model.

Secondly, the model’s accuracy is computed to estimate its effectiveness in predicting the IV parameters. Figure S5.1 shows the average accuracy of the proposed FWHM-based predictive MLLR Model (when all test datasets are combined) at 25% tolerance. Equation (1) is used to obtain the accuracy of the model. The accuracy of the model for Voc is the highest. This could be because Voc is a simple factor unaffected by as many other factors as Jsc and FF. The accuracy of the model for Jsc is also high. However, the model’s accuracy for Jsc & FF is slightly lower than that for Voc. This is because Jsc & FF are more complex factors that are affected by several other factors, such as temperature, relative humidity, device fabrication, defect densities, and stray light exposure. The model’s accuracy for PV-Eff is lower than Voc, Jsc, and FF models. This is because PV-Eff is a function of Voc, Jsc, and FF, and the model’s accuracy. Overall, the accuracy of the models for Voc and Jsc is high. However, the model’s accuracy for FF is lower, and the model’s accuracy for PV-Eff is the lowest.

At 25% tolerance levels for individual devices, the model’s prediction accuracy is high for Voc, Jsc, and FF and average for PV-Eff. Similarly, when the dataset is combined, the overall prediction accuracy is 100%, 96%, 96%, and 73% for Voc, Jsc, FF, and PV-Eff, respectively. From Table 1, the accuracy is higher for devices P1 and P4 compared to devices P2 and P3. Due to higher non-linearities inside the data, P2 and P3 have poor performance in predicting compared to P1 and P4.

The prediction accuracy results of the proposed MLLR model from predicting PSCs IV parameters from EL experiments are very promising at 25% tolerance. The average prediction accuracy (indicated as “Average” in Table 1) for all devices is 91.5%, which is quite good for this complex prediction task. The combined prediction accuracy (indicated as “Combined” in Table 1) for all devices is also 91.5%, which indicates that the model is able to generalize well to new data and cross-validate the results of the proposed model.

The average prediction accuracy is calculated by averaging the individual prediction accuracies for all devices. The combined prediction accuracy is calculated by combining all devices’ datasets and then predicting the IV parameters. The prediction accuracy for individual devices (P1, P2, P3, and P4) is even higher, with all devices having an accuracy of at least 90%. This suggests that the model is able to learn the complex relationship between IV parameters and can predict the IV parameters for most devices accurately. Overall, the prediction accuracy of the proposed ML model for predicting solar IV parameters from EL experiments of PSCs is quite good. An analysis of how the accuracy changes with different tolerance levels (e.g., 20%, 15%, 10%, and 5%) are provided in Table S19 indicating the scope of improvement in prediction accuracy of the proposed model. However, it is important to note that the prediction accuracy can vary depending on the specific device and the conditions under which it is tested. The model could speed up the development of new PSC materials and devices. However, more research is needed to improve the prediction accuracy for different types of PSCs.

It is important to note that the last two columns of Table 1 show the predicted PV-Eff and calculated PV-Eff values from the proposed MLLR Eq. (1) and conventional Eq. (3), respectively. When considered individually, the predicted efficiency values are almost the same, with a maximum of 10% deviation. When considered together/combined, the prediction accuracies are the same for the average prediction accuracy and combined prediction accuracy. Both indicate that the proposed model estimates the efficiency values accurately irrespective of using the proposed or conventional equation for efficiency. This is another type of cross-validation to the proposed MLLR model.

Where G = input solar irradiance = 100 mW/cm², A = area of the PSC = 0.1cm², V_oc = open- circuit voltage, J_sc = short-circuit current density, and FF represented in percentage (%).

After validation and optimization of the proposed machine learning model, it needs to be evaluated for its performance. There are different evaluation metrics for several types of problems to evaluate the model objectively. For regression models, the commonly used metrics for regression problems are mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error (MAPE), and R-squared^{53,54,55,56,57,58}. Figure S5.2 shows the model’s MAE, RMSE, and MAPE for respective IV parameters for the combined test dataset. The MAE, RMSE, and MAPE are all measures of the error between the predicted and actual values. The MAE is the average of the absolute errors, the RMSE is the square root of the average of the squared errors, and the MAPE is the average of the absolute errors expressed as a percentage of the actual values. Equations (S2), (S3), and (S4) represent the equations for MAE, RMSE, and MAPE, respectively, that are used to evaluate the proposed MLLR model.

The accuracy of the model closely matches the results obtained from MAPE values. Therefore, MAPE is the best tool to see the effectiveness of the model compared to MAE and RMSE indices. MAPE of 0.04 V, 0.11 mA/cm², 0.12, and 0.18% for Voc, Jsc, FF, and PV-Eff, respectively, indicate that this model predicts the corresponding IV parameters for 95% of the time within 4%, 11%, 12%, and 18% of the actual values. Overall, the MAE, RMSE, and MAPE for the MLLR model are all relatively low, indicating that the model is performing well.

The results of this study demonstrate the potential of using EL spectra to predict IV parameters of PSCs. Future work could expand the dataset to include more PSCs with varied materials and device architectures. This would allow for further validation of the model and improvement of its accuracy. Additionally, investigating the physical mechanisms underlying the relationship between EL spectra and IV parameters could provide a deeper understanding of how PSCs work and could lead to the development of new, more efficient devices. Finally, developing this model using EL spectra to predict IV parameters of PSCs, such as open-circuit voltage, short-circuit current, fill factor, and efficiency, would be a valuable tool for PSC optimization. This work is the first to carry out such a study with a few hundred datasets, which is large in material science40. This could lead to new insights into PSCs’ underlying physics and help improve their performance.

Comparison of MLLR models

The study has successfully completed the electro-optical modeling of PSCs based on EL spectroscopy to estimate the device (PSC) IV parameters (Voc, Jsc, FF, and PV-Eff) using continuous datasets of similar efficiency devices. studies and FWHM-CCT-based hybrid model prediction of IV parameters were conducted. Figure S6 shows the Log-transformed Multiple Linear Regression Equations and plots of the supervised machine learning-based predictive CCT model for estimating IV parameters from CCT values (for forward characteristics). It was found that all of them are CCT-based MLLR Models with an R-square over 0.99. Here, the pattern could be observed from the coefficients of log (CCT/T), t (day of measurement), and the constant terms, which are positive, negative, and negative, respectively. This means that this equation is very good in estimating the degradation characteristics of PSC compared to IV parameters, as the “t” term refers to the same efficiencies. Figure S8 shows the CCT range of values for each day.

Using the test dataset devices P1 to P4, the CCT-based MLLR Model is used to predict the IV parameters. This model predicts Voc and FF values better than PV-Efficiency values and cannot predict Jsc values. Figure S9 shows predicted vs. measured values of IV parameters (a) Voc, (b) Jsc, (c) FF, and (d) PV-Eff from Day 1 to Day 8 obtained using the proposed CCT-based predictive MLLR model using the test dataset. This indicates that the CCT value could be used to understand the device FF and, therefore, the degradation performance on par with the FWHM-based MLLR model. Figure S10 shows the accuracy of the proposed CCT-based predictive MLLR Model (when all test datasets are combined) at 25% tolerance. It can be observed that the accuracy of estimating the IV parameters Voc and FF are almost the same as that of the proposed FWHM-based model and fairly comparable in estimating the PV-efficiency values. In general, Voc and Jsc depend on the device’s inherent properties, whereas FF and PV-Eff are performance properties of the device. Therefore, the FF estimation or FF values are expected to be proportional in nature. If the number of datasets were large enough, then the estimation of PV-Eff and FF would be almost similar for both types of models (FWHM or CCT-based). The lower accuracy of FF and PV-Eff values estimation from the CCT-based model could be due to fewer datasets than the proposed FWHM-based model.

From the above two models, it is clear that Voc estimation is slightly better by the CCT-based MLLR model and Jsc and FF estimation by the FWHM-based MLLR model. PV-Eff estimation is almost the same for both models, but the FWHM-based model outperforms the CCT-based model by 12.6%. Considering the same and the analysis being already explained using the FWHM- based model, a hybrid model is also studied but with the use of estimating the CCT-based model for Voc, PV-Eff estimation, and Jsc and FF estimation through the FWHM-based model. Figure S11 shows the Log-transformed Multiple Linear Regression Equations of the supervised machine learning-based predictive hybrid model for IV parameters from CCT & FWHM values (for forward characteristics). The equations are obtained from FWHM-based and CCT-based models for analysis and are shown in equations S2, S3, S4, and S5. When estimated for their IV parameters from this model, the accuracy of predicting Voc, Jsc, FF, and PV-Eff were 100%, 96.4%, 96.4%, and 64%, respectively. Figure S12 shows the accuracy of the proposed predictive Hybrid MLLR Model (when all test datasets are combined) at 25% tolerance. After understanding the three types of MLLR models using forward IV characteristics datasets, the reverse characteristics datasets were also analyzed using ML regression and corresponding equations were obtained for estimating the IV parameters from the same EL and CCT parameters. Table S15 shows the prediction accuracy of three types of ML regression models for forward and reverse IV characteristics of PSCs in predicting the IV parameters from EL parameters (FWHM and CCT).

Discontinuous datasets analysis

To further explore the validation of this model, discontinuous datasets are considered. A few (4 Nos.) additional fresh devices were prepared on the same day, but their IV and EL measurement sets were taken with a 7-day gap. Further, individual devices were purposefully stored in the glove box or clean room in between continuous measurements to have discontinuous datasets for each of the PSCs. Table S16 shows the EL images corresponding to the measurement day for each PSC. Figures S13 to S28 show the results of various measurements performed on these PSC devices nos. 1, 2, 3, and 4 over 22 days with a 20 mA current injection. Each of the device’s EL images and IV curves for each measurement day is analyzed, and the correlation between PV-Eff and FWHM values is also shown. As defect density increases, the PV-Eff reduces, which is the same for all devices. It can be observed that the relation of FWHM and PV-Eff with time are inversely proportional to each other. Further to FWHM-based analysis, chromaticity coordinates of all the devices at 20 mA current injection in EL with time are analyzed, correlating PV-Eff and CCT values. The PSCs’ regeneration capability (self-healing property) was clearly observed; therefore, the prediction of IV parameters was difficult. It is observed that by applying the FWHM-based and CCT-based MLLR models on the discontinuous datasets for predicting their IV parameters from EL parameters FWHM & wavelength and CCT, respectively, the prediction accuracies are good for both the models, but the FWHM-based model shows better accuracy compared to CCT based model at 25% tolerance level.

EL Intensity, EL emission wavelength, and EL efficiency studies need to be explored further in their relation to material properties of the absorber layer and transport layers with several types of substrates. FWHM from EL spectra is a useful parameter related to the defect density of the PSC, like the PL spectroscopy method. CCT and FWHM values are inclusive parameters that include the variation of IV parameters and could be used to better estimate the PV-Eff of PSC. Future works will concentrate more on similar studies with many datasets by evaluating highly efficient PSCs with different substrates (ITO, flexible, etc.), absorber layers, and environmental conditions.

When such discontinuous datasets are used for estimating the IV parameters using the equations obtained in Eq. (1), a correction factor must be added. Table S17 shows the values of the CFs (Correction Factors) for the respective models of regression equations for estimating IV parameters such as Voc, Jsc, FF, and PV-Eff values. The CFs are used to adjust the measured values of the IV parameters to account for the effects of inherent properties or environmental factors. There is no clear relation between the FWHM-based and CCT-based models. The CFs for the two models are different for all IV parameters. This suggests that the two models are not directly comparable.

Four types of comparative analysis are carried out using discontinuous datasets. Firstly, the dataset is merged into a single dataset, and a combined analysis of EL injection currents at 10 mA and 20 mA (i.e., 50% and 100% of rated current density) was carried out. In this scenario, the prediction accuracies are comparatively low for FWHM and CCT-based MLLR models. Secondly, the dataset containing 20 mA EL injection current (rated short circuit current density) values are alone considered for analysis and found that the prediction accuracy is better than the first type, and the FWHM-based MLLR model showed better rate of prediction than the CCT-based MLLR model. Thirdly, the actual day of measurement is considered as the start of measurement rather than considering the day of preparation as Day 1. In this scenario with modified days of measurement, both models showed better accuracy compared to the past two types of comparison. This indicates that the PSC behavior is not affected much if they are stored immediately after the day of preparation but shows regenerative (self-healing) behaviors in the case of storage between the measurement periods. Lastly, the longest working PSC (D1) is alone considered for estimating IV parameters, i.e., single device estimation. This scenario highlighted that the single device prediction accuracy is higher than the other three types for FWHM and CCT-based MLLR models. This is expected as the dataset reduces the prediction accuracy increases. Voc and Jsc estimation is 100% in the case of the FWHM-based MLLR model and far better than the CCT-based model. If observed closely, in single-device prediction of IV parameters, the prediction of FF and PV-Efficiency is slightly better for the CCT-based MLLR model than the FWHM-based model in the case of single-device prediction. The FWHM-based MLLR model is the best-fitted model to estimate all IV parameters from EL spectroscopy.

Comparison of Voc estimation and Voc loss analysis using thermodynamic model and proposed machine learning approach

The present-day PCE of approximately 25% is undoubtedly impressive for PSCs. However, these devices have yet to realize their full potential in terms of open-circuit voltage (Voc). Despite the theoretical Voc limit of 1.32 V for lead halide perovskites, which typically exhibit a bandgap of 1.6 eV, the actual Voc values achieved under standard AM 1.5 G illumination conditions remain suboptimal. This discrepancy between the theoretical maximum and practical performance highlights a critical area for further research and development in the field of perovskite photovoltaics. One of the purpose of thermodynamic approach is to exceed the Shockley–Queisser limit for PCE and EQE_EL (closer to 1) of a solar cell by optimizing and reducing the non-radiative voltage losses due to defects, traps, geometrical loss (like reflection, waveguiding, etc.). For maximizing the Voc, better EL efficiency is required⁵⁹. Research that simultaneously investigates the performance of devices and their corresponding spectroscopy data is limited, particularly in the context of PSCs and the significance of EL⁶⁰. The analysis of EQE through EL spectroscopy reveals fascinating insights into the fundamental characteristics of light-emitting devices. It is proven that EQE of EL of the solar cells under bias voltage with current injection in dark equal to short circuit current density of the PSC can help estimate the recombination in the devices⁶¹. PSCs limited by radiative recombination can experience a substantial boost in Voc, Jsc, and overall efficiency through optimized internal optics and the strategic utilization of internal luminescence. While internally emitted photons can escape the cell directly, effective confinement using front and back reflectors enables a significant portion to be reabsorbed, a process known as photon recycling. This phenomenon elevates the equilibrium minority carrier concentration, consequently increasing Voc⁶².

Nonradiative losses (NRL) to the Voc is one of the factors limiting the PSC’s PCE. This can be determined from the radiative thermodynamic limit of the photovoltage. The real Voc is the difference between the radiative limit and the total nonradiative losses (\(\:{{V}_{oc},}_{nrad}\)), defined as the sum of the intrinsic (\(\:{V}_{oc}^{intrinsic}\) ) and electrode-induced (\(\:{V}_{oc}^{electrode}\) ) losses⁶³. For high Voc in PSCs, most researchers have linked with Shockley–Queisser limit for a specific bandgap that was determined in different ways from paper to paper. This can be easily replaced by radiative limit of Voc, which can be calculated using the EQE of the PSC. The radiative limit to the Voc can be calculated from \(\:{V}_{oc}=\:{{V}_{oc},}_{rad}-\:{{V}_{oc},}_{nrad}\:\) and \(\:{{V}_{oc},}_{rad}\:=\:{n}_{id}^{rad}*\left(\frac{kT}{q}\right)\:*\:ln\left(\right({EQE}_{EL}*\frac{{J}_{sc}}{{J}_{0},rad})\:+\:1)\)^64,65, where EQE=1 to determine the V_{oc, rad}.

Enhancing non-radiative recombination within the cell leads to a rise in saturation current and a reduction in the observed V_oc. Consequently, we can identify and measure the impact of non-radiative recombination losses using \(\:{{V}_{oc},}_{nrad}\:=\:{n}_{id}^{rad}*\left(\frac{kT}{q}\right)\:*\:ln\left(\right({EQE}_{EL})\)⁶⁶, where V_oc ^rad is the radiative V_oc; k is Boltzmann’s constant; T is the temperature in Kelvin (T = 293 K in this case); q is the elementary charge; J_sc under 1 Sun illumination with AM1.5 spectrum; J_{o, rad} is the radiative saturation current density, and most importantly, ideality factor most commonly used is replaced with ectypal factor (an alternate yet proven) for PSCs i.e. \(\:{n}_{id}^{rad}=2.6\:\)for considering a multi-diode model of PSC (\(\:{n}_{id}^{rad}\) = 1 for single diode model for ideal solar cells and mostly applicable to silicon crystalline types). Most common ideality factor used by many researchers so far for PSCs is either 1 or 2 depending on the type of recombination and diode model used as per classical theory^67,68,69. Many research works on PSCs in calculating ideality factor shows that the voltage drop across each layer of PSCs is significant by including the ion migration effects within the absorber layer⁷⁰. Non-ideal voltage blocking effects in ETL or HTL could be one the reasons for ideality factors going beyond 2⁷¹. This is also supported by performing IMVS, OCVD, impedance and IMPS studies showing that PSCs have this ectypal factor can range from 2.6 to 5.2 for same set of manufactured batches^72,73. The measurement of ideality factor or ectypal factor for PSCs is necessary to understand the type of recombination and their effects on efficiency. More research is required into the underlying recombination processes in working devices. This study has assumed the ideality factor to be 2.6 considering the literature data for PSCs⁷⁴. Applying the many-diode model allows for accurately determining \(\:{{V}_{oc},}_{rad}\) from these spectral data and justifies that neglecting the defect response is a valid method to straightforwardly determine \(\:{{V}_{oc},}_{rad}\). The mathematical expressions of related parameters like Jsc, J_0,rad, and Black body spectral photon flux (φbb) are provided in^75,76.

For PSCs, if all non-radiative recombination losses are reduced, the V_oc could be increased about 150mV high. Despite good optoelectronic properties, increasing the crystallinity and reducing the non-radiative recombination with help of voltage-dependent EL of PSC could help understand the effect of interlayers, contact selectivity, and defects in these devices. This relationship manifests in a voltage-dependent behavior that significantly impacts photon emission. Having higher V_oc or reaching near to Shockley-queisser limits by reducing defects and non-radiative recombination for PSC can lead to better efficiency⁷⁷. At voltages below the EL intensity peak, electrons struggle to acquire sufficient energy for photon release, relying on environmental energy absorption. Conversely, when voltages surpass this peak, an extreme quasi-Fermi level separation occurs, resulting in electrons possessing excess energy beyond what’s emitted during recombination⁷⁸. Despite the potential for radiative recombination above the peak EL energy, energy loss during down-conversion is unavoidable. These observations underscore the critical importance of precise voltage regulation in minimizing unnecessary energy dissipation, paving the way for more efficient optoelectronic devices⁷⁹.

Table S20 to S23 (in Supplementary Information) contains mainly the Voc measured from 1 sun irradiance (using IV characterization tool) i.e. the actual Voc, predicted Voc by well-know and existing thermodynamics model, and predicted Voc of the proposed FWHM-based MLLR model, along with other parameters such as Voc_rad, Voc_nrad, EQE_EL, EG_EL, Jo_rad, for corresponding PSC devices. The comprehensive Voc loss and prediction evaluation of three predictive models – FWHM-based, CCT-based, and thermodynamic – across four PSC devices (P1-P4) revealed valuable insights into their performance and applicability. All models demonstrated commendable predictive capabilities, as evidenced by their low mean absolute error (MAE) and root mean square error (RMSE) values. Figures S29, S30, S31, S32, and S33 provides the information on comparative statistical metrics, correlation analysis, error analysis, residual analysis, and time-series analysis for test PSC devices (used for predicting from the proposed models) P1 to P4. However, the FWHM and CCT-based models consistently outperformed the thermodynamic model in terms of accuracy and consistency. These models exhibited lower average MAE (0.0346 and 0.0340, respectively) and RMSE (0.0394 and 0.0391) compared to the thermodynamic model (MAE: 0.0598, RMSE: 0.0670). Moreover, the FWHM and CCT-based models demonstrated remarkable adaptability across different devices, suggesting their robustness for general use in various PSC configurations. While the thermodynamic model exhibited lower overall performance, it may still provide valuable insights into specific physical aspects of PSCs, potentially making it useful for particular applications. In conclusion, the FWHM and CCT-based models emerged as the preferred choices for precise Voc predictions in PSCs due to their superior performance and adaptability .