Physics-informed acquisition weighting for stoichiometry-constrained Bayesian optimization of oxide thin-film growth

Technical Summary: Physics-informed Acquisition Weighting for Stoichiometry-Constrained Bayesian Optimization of Oxide Thin-Film Growth

Problem Statement
The epitaxial growth of complex oxides, such as LaAlO₃ (LAO), faces a critical challenge in achieving precise stoichiometric control. Even minor deviations from ideal cation ratios can induce point defects, dislocations, secondary phases, and structural distortions, severely degrading electronic and optical properties. Traditional optimization relies on empirical tuning of coupled parameters (flux ratios, temperature, pressure), a process that is time-consuming, operator-dependent, and often unreliable. While Bayesian optimization (BO) has emerged as a data-driven approach to navigate multidimensional growth spaces, standard BO algorithms often propose off-stoichiometric conditions that lead to experimental failure or poor film quality. There is a need for an optimization framework that integrates physical prior knowledge to guide the search toward chemically balanced regimes without sacrificing the algorithm's ability to explore.

Methodology
The authors propose a Physics-informed Bayesian Optimization (PIBO) framework that incorporates crystal-growth physics directly into the acquisition function via a multiplicative weighting scheme.

• Core Mechanism: The method modifies the standard Expected Improvement (EI) acquisition function by multiplying it with a weight function, $w(\delta)$, where $\delta$ represents the nominal supply ratio of the constituent elements (e.g., La/Al).
• Weighting Function Design: The weight function is designed to encode thermodynamic priors regarding phase stability. It features a "flat window" of half-width $\tau$ around the ideal stoichiometric center $c$ (where $w=1$), ensuring that proposals within this physically plausible region are not penalized. Outside this window, the weight decays according to a Gaussian function controlled by a width parameter $\sigma$.
• Exploration Control: A floor factor $\epsilon \in [0,1]$ is introduced to ensure a non-zero probability of exploring off-stoichiometric regions. This prevents the algorithm from becoming overly rigid if the physical model is imperfect or if experimental noise shifts the true optimum.
• Flexibility: The formulation allows for asymmetric weighting to account for volatility differences between elements (e.g., compensating for the desorption of volatile species like RuO₂ in SrRuO₃) and can be extended to include temperature and oxygen partial pressure constraints based on thermodynamic data.
• Implementation: The study applies this method to the Molecular Beam Epitaxy (MBE) growth of LAO films. The search space includes the La/Al flux ratio ($\delta$), growth temperature, and ozone-nozzle-to-substrate distance. The objective is to minimize $\Delta c$, the difference between the film's c-axis lattice constant and the bulk LAO value, which serves as a proxy for stoichiometry.

Key Contributions

1. Minimal Workflow Modification: The approach requires only a concise modification to standard BO pipelines (multiplying the acquisition function by a weight), making it readily applicable to existing experimental workflows without altering the underlying surrogate model (Gaussian Process) or optimization logic.
2. Stoichiometry-Constrained Search: Unlike previous PIBO methods that embed physics into the surrogate model, this method explicitly steers the acquisition function. It successfully constrains the search toward stoichiometric regions while retaining sufficient flexibility to explore neighboring conditions.
3. Robustness to Model Inaccuracy: By maintaining a non-zero exploration probability outside the ideal window (via $\epsilon$), the method avoids getting trapped in local minima caused by inaccuracies in the prior physical model or experimental noise.

Results
The PIBO framework was validated through the closed-loop optimization of LAO film growth:

• Rapid Convergence: Within only 15 growth runs, the lattice constant of the grown LAO film converged to the bulk value ($\Delta c = 0$ Å).
• Optimal Conditions: The algorithm identified an optimum slightly beyond the strict stoichiometric window ($\delta = 0.88$), suggesting that the weighting scheme successfully navigated small calibration offsets or non-unity sticking coefficients that a naive range restriction would have missed.
• Comparison with Standard BO: In contrast, a standard "bare" EI approach frequently proposed off-stoichiometric conditions (e.g., $\delta = 0.75$) that resulted in the failure of phase formation (disappearance of LAO diffraction peaks).
• Film Quality: The optimized films exhibited a lattice constant identical to bulk LAO and showed a more than tenfold increase in X-ray diffraction (XRD) peak intensity compared to unoptimized films. Scanning transmission electron microscopy (STEM) and atomic force microscopy (AFM) confirmed the single-crystalline nature, atomically smooth surfaces, and step-and-terrace structures indicative of high-quality layer-by-layer growth.

Significance
The paper claims that this PIBO framework offers a general and practical route to AI-driven materials synthesis. By incorporating physics knowledge solely through the weighting function, the method effectively biases exploration toward physically reasonable regimes while preserving the robustness required for autonomous synthesis systems. The results demonstrate that integrating prior crystal-growth physics into the acquisition function can significantly accelerate the optimization of complex oxide thin-film growth, enhancing film quality and reducing the number of experimental runs required to achieve ideal stoichiometry. The authors emphasize that this approach is extensible to other material systems and autonomous synthesis platforms, providing a balanced trade-off between constraint and exploration.