Embedding physics domain knowledge into a Bayesian network enables layer-by-layer process innovation for photovoltaics

The study addresses the challenge of optimizing process variables in photovoltaic device fabrication, where traditional black-box optimization (e.g., design of experiments, grid search, Bayesian optimization, particle swarm) can be limited by user-imposed variable ranges and offers little insight into root causes of underperformance. Growth temperature in III–V MOCVD is a key but difficult parameter, influencing growth rate, morphology, dopant incorporation, and defects. The authors propose embedding physics-based relations into a Bayesian network that links process variables (temperature) to material descriptors (e.g., doping, bulk lifetime, surface recombination velocities) and then to device performance, using only low-cost current–voltage measurements under varied illumination (JVi). This approach aims to provide layer-by-layer interpretability, reduce reliance on auxiliary structures and expensive measurements (e.g., SIMS, TR-PL), and discover optimal process windows that would be missed by constant-temperature black-box optimization. A neural-network surrogate device model accelerates inference by over 100x compared to numerical solvers, enabling practical iterative optimization. The work demonstrates this approach on GaAs solar cells, seeking temperature profiles per layer (window, bulk, back surface field) that maximize efficiency.

The paper situates its contribution within prior work on black-box optimization methods for materials and device design and emerging efforts combining physical insights with machine learning. It notes limitations of standard black-box optimizers in constrained variable spaces and lack of causal insight, contrasting with Bayesian inference coupled to physics-based forward models for diagnosing PV device underperformance. Prior literature establishes temperature effects on III–V MOCVD growth (growth rate, morphology, doping, defects) and Arrhenius-type behavior for dopant incorporation. It also highlights known correlations between recombination parameters (bulk lifetime, SRV) and doping. The authors build on these insights to propose a hierarchical Bayesian framework with physics-informed parameterization, augmented by a fast surrogate for numerical device simulation.

- Bayesian network architecture: A two-step hierarchical Bayesian inference links process conditions (growth temperature T) to material descriptors (emitter/base doping NA, NB; bulk lifetime τ; front/rear SRVs at InGaP/GaAs interfaces) and then to device performance (JVi, QE, and efficiency). The conditional structure is P(JV|T)=∫P(NA,NB,τ,FSRV,RSRV|T)·P(JV|NA,NB,τ,FSRV,RSRV,T) d(...). - Physics-informed parameterization: Each material descriptor y(T) is parameterized by a modified Arrhenius form y(T)=T^a·exp(b/T + c), where latent parameters (a,b,c) encode activation energy and pre-exponential dependence; these are inferred with priors reflecting literature ranges and hard constraints (Supplementary Table 2). This reduces dimensionality, mitigates overfitting, and supports interpretability. Small a implies Arrhenius-dominated behavior; large |a| indicates deviations (non-Arrhenius regime). - Device-physics layer and surrogate: A well-calibrated PC1D model provides forward mapping from material descriptors to device JVi, but is computationally heavy. The authors replace it with a deep neural network surrogate comprising: (1) a denoising autoencoder (three convolutional + two dense layers encoder; mirrored decoder with transposed convolutions) trained on 20,000 simulated PC1D JVi curves augmented with Gaussian noise (mean 0, variance 0.2%) to reconstruct noise-free curves; and (2) a regression decoder that predicts JVi from the five material descriptors in the AE latent space. Training uses ADAM (batch size 128, initial learning rate 1e-4), 80/20 train/test split. The surrogate achieves 130x speedup on GPU versus internal PC1D and ~700x if PC1D is called externally. - Bayesian inference: With the surrogate replacing the numerical solver, the posterior over latent process parameters (a,b,c) is sampled using an affine-invariant ensemble MCMC (Goodman–Weare; emcee library). Posterior updates occur with each new experimental JVi dataset, yielding inferred functions y(T) for all material descriptors. - Optimization framing: After learning g(x): T→{NA,NB,τ,FSRV,RSRV}, the objective h(y) (device performance from material descriptors) is evaluated via the surrogate to solve x*=argmax h(g(x)) over a 10°C grid, respecting process constraints. This enables tailoring layer-by-layer temperatures to maximize τ and minimize SRVs where beneficial. - Experimental design: Five MOCVD batches (four cells each) sweep constant growth temperatures from 530–680°C in 20–50°C steps. Devices are 1 cm^2 GaAs cells without ARCs; JVi measured at 0.1–1.1 suns. Bayesian inference yields y(T) trends; then a sixth MOCVD run implements a variable temperature profile per layer suggested by the model. Auxiliary validation includes SIMS (doping) and TR-PL (bulk lifetime) on dedicated structures grown under matched conditions.

- Physics–process relations: Inferred log(Zn emitter doping), log(Si base doping), and log(FSRV) vary approximately linearly with −1/T, consistent with Arrhenius behavior (a≈0). Bulk lifetime τ and RSRV show nonlinear, non-Arrhenius dependence (significant a values). - Latent parameter means (a,b,c): Zn doping (0.0018, −0.1494, −0.1948), Si doping (0.0016, 0.1551, 0.2970), bulk lifetime (−4.5973, 2.7984, 2.3687), front SRV (0.0015, −0.1440, −0.1892), rear SRV (2.1194, −1.1119, −0.7300). - Distinct optimal temperatures per recombination channel: τ peaks near ~620°C; FSRV and RSRV exhibit opposite trends with temperature. This indicates different optimal growth temperatures for back contact, bulk, and front contact layers rather than a single constant temperature. - Layer-by-layer temperature profile (10°C resolution): Buffer 580°C; InGaP BSF 580°C; GaAs bulk 620°C; InGaP window 650°C; GaAs contact 650°C. - Device performance gains: Conventional grid search over constant temperatures yields a 1.4% relative efficiency improvement over baseline (630°C). The Bayesian network–informed variable temperature profile achieves a 6.5% relative efficiency increase over baseline in the sixth MOCVD run, surpassing constant-temperature optimization. - Extracted material parameters (Bayes Net vs best baseline): FSRV 1.2×10^3 cm/s vs 4.1×10^4 cm/s; RSRV 5.4×10^4 cm/s vs 6.1×10^4 cm/s; τ 29 ns vs 26 ns. EQE indicates improved short-wavelength response (<820 nm), consistent with reduced front and bulk recombination. Both Jsc and Voc contribute to efficiency gains. - Surrogate efficiency: The neural surrogate is ~130× faster than internal PC1D and ~700× faster than external calls, enabling tractable Bayesian inference over large parameter spaces. - Context: Devices lack ARCs; best cells are estimated to reach 24–25% with optimal ARC. The approach identifies root causes of underperformance and optimal process windows with only five initial MOCVD sweeps plus one targeted run and no secondary measurements needed for optimization (auxiliary samples used only for validation).

The hierarchical Bayesian framework effectively addresses the process-optimization challenge by inferring causal links from JVi data to layer-specific material properties and back to process temperatures. By embedding physics (modified Arrhenius relations) and using a fast, noise-robust surrogate model, the method identifies that different recombination channels (bulk, front, rear) favor different growth temperatures, a key insight inaccessible to constant-temperature black-box optimization. Implementing the inferred layer-by-layer profile reduces SRVs (particularly at the front interface) and increases bulk lifetime, translating into simultaneous Jsc and Voc improvements and a 6.5% relative efficiency gain. The approach reduces experimental burden (few growth runs, no extensive auxiliary characterization for optimization), enhances interpretability, and provides a principled path to root-cause diagnosis. Given its modularity, the framework can generalize to other PV materials and multilayer energy devices, substituting other physics-based or black-box mappings where appropriate.

The study introduces a physics-informed Bayesian network that couples process variables to material descriptors and device performance via a fast neural surrogate, enabling layer-by-layer process optimization for GaAs solar cells. In six MOCVD experiments, it delivers a 6.5% relative efficiency improvement over a 630°C baseline, outperforming conventional grid search in constant-temperature space. The method provides interpretable, layer-resolved insights (Arrhenius vs non-Arrhenius behavior across properties) and requires only routine JVi data. Future directions include applying the framework to other PV technologies and multilayer systems, integrating more comprehensive physics or alternative surrogates, and extending the first-layer mapping with regularized black-box regressors when physics is unknown while preserving interpretability and robustness.

- Dependence on prior parameterization: Accuracy of first-layer mappings relies on appropriate physics-informed forms (modified Arrhenius) and chosen prior ranges; when physics is unclear, replacing with black-box regression reduces interpretability and is sensitive to hyperparameters, especially under data scarcity. - Surrogate modeling and calibration: Training the surrogate requires a calibrated numerical device model (PC1D) and domain expertise; model mismatch and noise can affect inference despite denoising. - Validation samples: Although optimization did not require secondary measurements, the study used several auxiliary SIMS and TR-PL samples for validation; broader generalization may require similar calibrations. - Process constraints: Hardware-imposed temperature resolution (10°C) and avoidance of extreme conditions limit the explored space; interactions with other process variables (e.g., pressure, flow rates) were not optimized and may be tool-specific. - Generalizability: The demonstrated trends and optimal profiles are specific to the reactor, materials stack, and process window studied; transfer to other tools/systems may require re-training and re-parameterization.