Atomic Layer Deposition Optimization via Targeted Adaptive Design.

Authors: marieme  ngom
carlo  graziani
noah  paulson
Venue: NeurIPS_cc_2024_Workshop_BDU
Published: 2024-09-06 13:58:53

Abstract: Atomic Layer Deposition (ALD) is a commonly employed process for producing
conformal nanoscale coatings in the microelectronics and energy materials industries.
ALD processes are composed of cycles of sequential self-limiting chemical
reactions followed by purges with an inert gas to produce atomically thin coatings.
At the end of each cycle, the Growth Per Cycle (GPC) which corresponds to net
mass or thickness change from the previous ALD cycle is determined. Optimizing
ALD processes for stable and uniform GPC for a new combination of reactants
is challenging as the optimal combination of gas timings, temperature, and gas
partial pressures spans a large multidimensional space and in-situ characterization
is typically performed with a limited number of mass sensors. In this work, we use
Targeted Adaptive Design (TAD), a Gaussian Process (GP)-based probabilistic
machine learning framework that aims at efficiently and autonomously locating
control parameters that would yield a desired target within specified tolerance, to
optimize simulated ALD processes.

Main paper: 
 introduction: 
 Atomic Layer Deposition (ALD) [1] is a thin-film deposition technique that employs a cyclical process of alternating precursor exposures and purging steps, enabling self-limiting chemical reactions that deposit one atomic layer per cycle, providing precise control over film thickness and composition. In this work, we consider ALD processes that involve two chemicals, precursor 1 and precursor 2 that reacts cyclically with the surface of a substrate in a sequential and non-overlapping manner. Each cycle is composed of four sequences: a dose period t 1 , during which the substrate is exposed to precursor 1; a purge period t 2 , during which there is no precursor exposure; a dose period t 3 , during which the substrate is exposed to precursor 2; and a final purge period t 4 , during which there is no precursor exposure. At the end of each cycle, the GPC which corresponds to net mass or thickness change from the preceding ALD cycle is determined. In [5] , the authors employed a simple physics model to simulate ALD and proposed three optimization methods-Bayesian Oprimization (BO) [6] with Expected Improvement [4] , Random Optimization (RO) and Expert Systems Optimization (ESO) leveraging expert knowledge-to find timings (t 1 , t 2 , t 3 , t 4 ) that would yield Growth Per Cycle (GPC) saturation in minimum time. In this work, we use Targeted Adaptive Design (TAD) [2] , a Gaussian Process (GP)-based method for efficiently and autonomously locating control parameters that yield a target feature within specified tolerances, to optimize ALD processes. We employ the physics model from [5] as a simulator and compare the performance of TAD to the methods proposed in that paper. 
overview of targeted adaptive design: 
 We start by giving a comprehensive overview of TAD which is a recently proposed algorithm for locating control parameters in a D-dimensional input space X ⊂ R D that would yield a target feature f target in a E-dimensional output space Y ⊂ R E within specified tolerance, while simultaneously learning the unknown mapping f : X → Y . The mapping f is only attainable through noisy measurements g = f (x) + ϵ, where x ≡ {x k ∈ X : k = 1, . . . , N } and ϵ is a zero-mean Gaussian noise vector. TAD is an iterative method with similarities to BO, optimizing an acquisition function at each iteration to propose new candidate solutions.TAD begins by defining three sets which it will update throughout its iterations: a set x 1 of N 1 points in X with associated noisy observations of f g 1 ≡ f (x 1 ) + ϵ 1 , a set x 2 of N 2 points in X associated with latent observations g 2 ≡ f (x 2 ) + ϵ 2 , and a point x ∈ X that corresponds to the initial target candidate solution. Additionally, TAD defines a Targeted Tolerance Region (TTR) E , where f targeti is the ith component of the target vector f targeti and τ i is a tolerance threshold for f targeti . A vector-valued GP prior is assumed on f i.e f ≡ N (µ(.), C(., .)) with µ and C corresponding respectively the mean and covariance vectors of the GP. The GP is trained on the observations (x 1 , g 1 ) and is used to predict the distributions of g 2 |g 1 and of f (x)|g 1 , g 2 . In particular, we haveη i ≡ [f targeti -τ i , f targeti + τ i ], i = 1, . . . ,EQUATIONEQUATIONQ (2|1) ≡ K 22 + Σ 2 -K 21 (K 11 + Σ 1 ) -1 K 12 .(3) andEQUATIONEQUATIONwhereµ 1 ≡ µ(x 1 ), µ 2 ≡ µ(x 2 ), K xx ≡ C(x, x), K x1 = K T 1x ≡ C(x, x 1 ), K x2 = K T 2x ≡ C(x, x 2 ), K 11 ≡ C(x 1 , x 1 ), K 22 ≡ C(x 2 , x 2 ), K 12 = K T 21 ≡ C(x 1 , x 2 ), Σ 1 is the noise covariance matrix associated to ϵ 1 , and Σ 2 is the noise covariance matrix associated to ϵ 2 . The acquisition function is then constructed using these distributions.TAD proposes a new acquisition function that computes the log-predictive probability density (LPPD) of the target design f target at the point x, conditioned on observations (g 1 , g 2 ). By Equation 4and the standard formula for multivariate normal probability density, this isEQUATIONup to a constant. However, in this expression the mean p (f (x)|1+2) is dependent on the latent values g 2 that are yet to be acquired. To obtain the TAD acquisition function, the expectation of the LPPD with respect to the predictive distribution g 2 |g 1 is calculated to obtainEQUATION)The TAD acquisition function L T AD (x, x 1 , g 1 , x 2 ) of Equation 8 has the following closed form:EQUATIONTAD then defines two stopping rules: convergence/success, which occurs when a solution with uncertainties that fit within the TTR is found, and convergence/failure, which occurs when the search space is exhausted without finding such a solution (based on information theory). TAD iterates until one of these stopping conditions is met or the computational budget is reached. TAD thus offers an efficient search (with the x 2 points) at the cost of more function evaluations. A detailed discussion of the properties of this acquisition function, including its incorporation of the exploration/exploitation trade-off, as well as numerical simulations and comparisons with existing methods, can be found in [2] .3 Optimization of Atomic Layer Deposition (ALD):The authors in [5] Prior to running the BO algorithm, an initial set of 10 observations is sampled using a Latinhypercube design of experiments (LH-DOE) [3] to provide a representative starting point for the GP surrogate model. The objective function for BO/EI and RO is defined as a combination C total of four cost functions C 1 , C 2 , C 3 and C 4 . C 1 focuses on minimizing the sum of the dose and purge timings relative to the sum of the maximum allowed time for each sequence. C 2 aims at reducing the divergence of the average GPC over multiple cycles from a reference GPC of a stable ALD process.C 3 aims at controlling the difference in GPC between an ALD process and one resulting from perturbed timings while C 4 does the same n comparison to the initial GPC. The three different methods were run over 40 iterations for 10 different initializations of the physics model. We reproduce their experiments for the Al 2 O 3 system at 200 o C and a noise value of 0.1. Figure 1a shows the mean and 95% confidence intervals of the normalized GPC ĥ across these runs with respect to the number of ALD cycles (each iteration consists of 21 ALD cycles). BO/EI and RO produced timings that would yield desirable GPC but also explored values that led to excessive GPC during optimization. ESO, on the other hand, while reliable, suffered from overly conservative precursor dose times. Note that the variability during the first 210 ALD cycles for BO/EI is due to the LH-DOE sampling. We now add TAD to the optimization strategies. We define the unknown function f to be f (t 1 , t 2 , t 3 , t 4 ) = (C total , ĥ), with C total and ĥ defined above, resulting in a 2-dimensional output space. We then set f target = (0.4, 1.) where 0.4 is a conservative value derived from the ESO and ĥtarget = 1 ensures the normalized GPC remains stable around 1 during the optimization. We initialize the candidate solution to (t 1 , t 2 , t 3 , t 4 ) = (1, 1, 1, 1) and x 2 is initialized as in [2] . Finally, we set the tolerance for the TTR to be 10% across both output space dimensions. Since the GPC is dependent from previous ALD cycles, one needs to be conservative about the number N 2 of x 2 points to be acquired at each TAD iteration. Furthermore, the number of ALD cycles per TAD iteration is N 2 times more the number of ALD cycles per BO/EI iteration. To avoid having a prohibitive number of ALD cycles, we set N 2 = 1. We also run TAD for 10 different initializations of the physics model and an observation noise level of 0.1. The initial (x 1 , g 1 ) observations for the initial GP fitting are also constructed using the same LH-DOE procedure as before. If TAD reaches convergence/success, only the successful candidate solution is acquired after the last iteration. For each initialization, TAD successfully found a solution within the prescribed tolerance in 14 to 21 iterations (i.e. between 4 to 11 iterations after the 10 initial LH-DOE steps) corresponding to 357 to 651 total ALD cycles, as illustrated in Figure 1 . Furthermore, the normalized GPC remained stable across the 10 different runs. Note that Figure 1a shows TAD results for the smallest number of iterations across the 10 different initializations while Figure 1b shows TAD normalized GPC values for each of the runs over each corresponding number of iterations.For illustration purposes, we show uptake curves which are commonly used graphical represen- tations of the growth or deposition rate of material as a function of the precursor exposure time or dose. Uptake curves are obtained by varying one timing and fixing all the others at values resulting in saturation. Figure 2 shows uptake curves from one of the TAD runs (solid lines) superposed with reference uptakes curves for a noise level of 0.1. TAD saturated slightly above the reference horizontal line y = 1 with saturation speed similar to the reference growth.04 Discussion and future work:We applied TAD, a recently developed batch iterative algorithm for locating control parameters that yield a target design, to the optimization of ALD. We compared TAD to existing optimization methods for ALD, including Bayesian optimization with expected improvement and have shown the effectiveness of our method in terms of the number of ALD cycles required for convergence and the stability of the normalized growth per cycle during optimization. In order to avoid a large number of ALD cycles per TAD iterations, we used a conservative number of points to be acquired at each TAD iteration, and aim to relax this constraint in the future by modifying the objective function and refining the function sampler require fewer ALD cycles per evaluation. Furthermore, we plan to analyze the effect of lower noise levels on TAD. In fact, in the zero-noise limit, the covariance matrices Q (f (x)|1+2) can become ill-posed especially in the presence of redundant x 2 points. The authors in [2] proved that matrix has a finite limit when the noise goes to zero and it would be interesting to diagnose how TAD is affected in this case and compare with the other three methods presented here.Workshop on Bayesian Decision-making and Uncertainty, 38th Conference on Neural Information Processing Systems (NeurIPS 2024). 
 
 Appendix:  



Reviews: review: title: title: Review of Atomic Layer Deposition Optimization via Targeted Adaptive Design
main_review: main_review: This paper applies targeted adaptive design (TAD) to a material science problem. TAD is a recent method that aims to find a "configuration" $\mathbf{x}$ such that a function $f(\cdot)$ evaluates close to a "target region." To do so, TAD uses a certain expected log predictive density as an acquisition function. In the proposed application, TAD is applied to an atomic layer deposition problem and compared to Bayesian optimization, a random search, and an "expert systems optimization" appearing in the literature.

## Strengths
- I'm not an expert in TAD, but this seems like a reasonable application domain which TAD is designed for.
- While I'm skeptical of the loss function used for Bayesian optimization, the performance of TAD compared to the expert systems optimization method is promising.
## Weaknesses
- The paper can be somewhat difficult to read. The introduction to TAD is brief and lacks motivation, and some concepts are not defined (e.g., what "normalized GPC" is normalized w.r.t.).
- I'm a bit confused by the comparisons to Bayesian optimization. Results on the cost function $C_{\text{total}}$ are not shown, and instead, all comparisons are on normalized GPC, where Bayesian optimization is comparable to random sampling. Since the optimization problem is ostensibly smooth and low-dimensional, this seems suggestive of a poorly chosen loss function rather than a poor showing from Bayesian optimization.
  
  To this end, it seems more natural and fair to compare to Bayesian optimization with a loss function incorporating normalized GPC, since TAD is allowed to do this. For example, $C(x) = C_\text{total}(x) - \lambda (\hat{h}(x) - 1.0)^2$ for some constant $\lambda$. This also sidesteps the issue of defining a reasonable value of $C_\text{total}$ to imitate, like is necessary in TAD.

There are also some small typographical or editorial issues. Here are a few suggestions I would encourage the authors to incorporate:
- All acronyms should be defined in the text. For example, GPC is only defined in the abstract, and LPPDD is not defined at all.
- I think $\mathcal{L}(\cdot)$ is chosen because the acquisition function is a log-likelihood, but it is a somewhat unfortunate notation to maximize a quantity called $\mathcal{L}(\cdot)$ in an optimization paper. I'd encourage the authors to find an alternative notation, or at least highlight that $\mathcal{L}(\cdot)$ is maximized.
- Vector/matrix notation should be consistent, e.g. $x$ is inconsistently bolded throughout the text, and $\mathbf{p}$ and $\mathbf{Q}$ are inconsistently bolded between Eq. (1) and Eq. (4).
- "tp" should probably be "to" in the first paragraph of Section 3, and $Q^{(f(x) | 1 + 2)}$ is missing the closing parenthesis in Section 4.
- The title on OpenReview has a typo ("layed" instead of "layer").
- Figure 1 is not readable without colors (e.g., when printed in grayscale). It would be great if the figure is modified with symbols to be more readable in grayscale.

## References
[1] Graziani, C., & Ngom, M. (2024). Targeted Adaptive Design. _arXiv preprint arXiv:2205.14208_.
title: title: Optimize ALD via GP-based TAD
main_review: main_review: This paper applies a Gaussian Process (GP)-based probabilistic machine learning framework Targeted Adaptive Design (TAD) for optimizing the Atomic Layer Deposition (ALD). The optimization of ALD is challenging due to the multidimensional parameter space involved. The paper compares TAD with other methods such as Bayesian Optimization with Expected Improvement (BO/EI), Random Optimization (RO), and Expert Systems Optimization (ESO), showing that TAD-based ALD optimization achieves stable, optimized outcomes with fewer cycles.

Quality: 
The quality of the paper is fair. It is technically sound and all claims are supported by the section 2 Overview of Targeted Adaptive Design (which I haven't reviewed in depth) are adopted from the original Targeted Adaptive Design paper. The final results on the ALD optimization is promising compared with other methods presented in this paper.

Clarity:
The paper is well written and easy to read. Despite some minor spelling error in the Conclusion, such as Q(f (x)|1+2 . Also, the link for citation [2] appears to be wrong.

Originality: 
The paper is an application of using an existing algorithm TAD to optimize the ALD problem. The authors use this well-established GP-based techniques to solve a challenging problem that spans multidimensional space. In this regard, I do not think the method itself is particularly novel; However, the applied work resulting promising outcomes compared to other methods in the experiment section.

Significance:
Despite the potential practical significance of the work, I think the theoretical contributions are comparably small. 

Pros:
- Apply the Targeted Adaptive Design (TAD) for optimizing ALD is an interesting approach.
- This work is focus on the practical usage by simulating and testing the current algorithm on optimizing ALD.

Weakness:
- It would be informative to see the optimization performance under different noisy level in the experiment section instead of focusing on the single set noisy level 0.1
- Add the statistical analysis. the authors are comparing the performance of TAD against other methods (e.g., BO/EI, RO, ESO) and have multiple experimental runs, statistical tests can be used to determine whether the observed differences in results are statistically significant or just due to random chance.
title: title: Accept – Optimizing Atomic Layer Deposition Using Targeted Adaptive Design
main_review: main_review: The paper presents a well-formulated optimization strategy for ALD processes using Targeted Adaptive Design (TAD). The proposed method improves upon existing optimization techniques, including Bayesian Optimization and Expert Systems Optimization, by offering a more efficient way to determine optimal control parameters for ALD cycles. The use of a Gaussian Process (GP)-based probabilistic framework enhances the quality of predictions, especially in the presence of noise, making it highly suitable for real-world applications.

In terms of quality, the methodology is sound and clearly articulated, with a solid theoretical foundation in probabilistic machine learning. The clarity of the paper is commendable, particularly in the explanation of TAD and its comparison to other optimization techniques. The visual aids, including graphs, effectively support the results.

The originality is notable, as TAD offers an innovative approach to optimizing ALD cycles, reducing both the number of cycles required for convergence and the variability in growth per cycle. The paper also makes significant contributions to the field of machine learning in industrial applications.

Cons: The method's limitations in lower-noise environments are acknowledged but could benefit from deeper exploration. Additionally, the scalability of TAD for more complex ALD systems needs further investigation.