Bayesian optimization for stable properties amid processing fluctuations in sputter deposition

TL;DR

This work addresses robust design of Mo thin films deposited by sputtering, aiming to meet target residual stress and sheet resistance while remaining resilient to processing fluctuations. It introduces a unified objective built from four smooth components that encode stress feasibility, low resistance, film density, and stress robustness, and optimizes it with Bayesian optimization using a Gaussian process surrogate and an upper confidence bound acquisition. The approach efficiently guides the deposition parameter search, converging to an optimal condition at $(2\ \text{mTorr}, 620\ \text{W})$ that satisfies all criteria and shows reduced sensitivity to pressure variations, validated by measurements and sensitivity analyses. The method offers a data-efficient route for robust thin-film design with practical implications for device reliability and reproducibility in sputter deposition.

Abstract

We introduce a Bayesian optimization approach to guide the sputter deposition of molybdenum thin films, aiming to achieve desired residual stress and sheet resistance while minimizing susceptibility to stochastic fluctuations during deposition. Thin films are pivotal in numerous technologies, including semiconductors and optical devices, where their properties are critical. Sputter deposition parameters, such as deposition power, vacuum chamber pressure, and working distance, influence physical properties like residual stress and resistance. Excessive stress and high resistance can impair device performance, necessitating the selection of optimal process parameters. Furthermore, these parameters should ensure the consistency and reliability of thin film properties, assisting in the reproducibility of the devices. However, exploring the multidimensional design space for process optimization is expensive. Bayesian optimization is ideal for optimizing inputs/parameters of general black-box functions without reliance on gradient information. We utilize Bayesian optimization to optimize deposition power and pressure using a custom-built objective function incorporating observed stress and resistance data. Additionally, we integrate prior knowledge of stress variation with pressure into the objective function to prioritize films least affected by stochastic variations. Our findings demonstrate that Bayesian optimization effectively explores the design space and identifies optimal parameter combinations meeting desired stress and resistance specifications.

Figures (9)

• Figure 1: Figures \ref{['fig:stress_example']} and \ref{['fig:resistance_example']} illustrate the stress and resistance variations with pressure for a fixed power, respectively. For the given stress and resistance profiles, the behavior of the four design criteria functions $f_1$, $f_2$, $f_3$, and $f_4$ are shown in Figs. \ref{['fig:criteria_1']}, \ref{['fig:criteria_2']}, \ref{['fig:criteria_3']}, \ref{['fig:criteria_4']}, respectively. The pressure values at which these functions reach their maximum satisfy the corresponding design criteria. The pressure that maximizes the unified objective function $f$, shown in \ref{['fig:combined']}, satisfies all design criteria.
• Figure 2: The plots depict the variations of residual stress, shown in \ref{['fig:exp_stress']}, and variations of sheet resistance, shown in \ref{['fig:exp_resist']} with pressure and powers in the dataset before Bayesian optimization. The observations from the plots assisted in constructing the objective functions.
• Figure 3: Vacuum chamber geometry, as shown in \ref{['fig:chamber']}, is used for physical vapor deposition employing a top-down sputter geometry with true planetary sample stage motion (i.e., orbit + spin). Ultra-high-purity argon gas is consistently introduced into the chamber and regulated by a mass flow controller (MFC). \ref{['fig:deposition']} shows the cross-section drawing of key atomistic processes involved in sputtering, transport, and film growth.
• Figure 4: The figures illustrate trends observed during the BayesOpt guided search, including the proposed sputter deposition configuration for the next iteration and the optimal configuration in that iteration. Figs. \ref{['fig:iter_pressure']} and \ref{['fig:iter_power']} show pressure and power trends over iterations, respectively. See \ref{['tab:trends_deposition']} in \ref{['sec:appendix']} for the specific values.
• Figure 5: The figure shows the contour plots of exploitation term, $\mu_\text{pos}$ in \ref{['fig:mu0']}, exploration term $\beta \cdot \sigma_\text{pos}$ in \ref{['fig:sigm0']}, and UCB in \ref{['fig:only_ucb0']} for the first iteration with $\beta=1$. At this iteration, BayesOpt is exploiting the region $\mathcal{X}_1$.