Application of Gaussian processes with variable shrinkage method and just-in-time modeling in the semiconductor industry

Virtual metrology of semiconductor PVD process based on combination of tree-based ensemble model

Virtual metrology (VM) is often applied in semiconductor manufacturing when process measurements cannot be obtained without delay or without impacting the cycle time. When the process condition is varying, the modeling results can be inaccurate if a single VM model is adopted. Also, the large number of variables in the process data increases the difficulty of obtaining good predictions. To improve the accuracy of the metrology value prediction, it is necessary to select and apply only the significant variables in modeling. This paper proposes a just-in-time approach for VM modeling, along with a variable shrinkage and selection method that uses Gaussian process regression. The proposed method is validated using semiconductor process data and found to be superior compared to conventional methods.

In semiconductor manufacturing processes, virtual metrology (VM) has been investigated as a promising tool to predict important characteristics of products. Although partial least squares (PLS) is a well-known modeling technique that can cope with collinearity and therefore applied to construction of VM, its prediction performance deteriorates due to changes in process characteristics. In particular, maintenance of equipment strongly affects the process characteristics and the prediction performance. In this paper, VM was developed by using locally weighted PLS (LW-PLS), which is a type of just-in-time modeling technique, and it was used to predict the etching conversion differential of an actual dry etching process. The industrial application results have shown that the developed VM based on LW-PLS is superior to the conventional VM based on the sequential update model and artificial neural network model. In particular, it has been confirmed that the LW-PLS-based VM can keep its high prediction performance even after the maintenance, i.e., replacement of parts.

Plasma etch is a complex semiconductor manufacturing process in which material is removed from the surface of a silicon wafer using a gas in plasma form. As the process etch rate cannot be measured easily during or after processing, virtual metrology is employed to predict the etch rate instantly using ancillary process variables. Virtual metrology is the prediction of metrology variables using other easily accessible variables and mathematical models. This paper investigates the use of Gaussian process regression as a virtual metrology modelling technique for plasma etch data.

Virtual metrology (VM) is the prediction of end-of-batch properties (i.e., metrology data) using process variables and other information available for the process and/or the product (i.e., machine data) without physically conducting property measurement. VM (sometimes augmented with existing metrology) has been utilized in semiconductor process monitoring and control. Besides the economic benefit of replacing or reducing metrology tools, due to the instant availability of high frequency machine data, a good VM can actually provide better process monitoring and control performance compared to the same monitoring and control schemes based on the physical metrology data which often obtained at lower frequencies and usually with delays. In this paper, we propose a statistics pattern analysis (SPA) based VM approach for predicting sheet resistance using optical emission spectroscopy (OES) data. The advantageous properties of the SPA based VM are discussed. And the performance of the SPA based VM is compared with several commonly used VM algorithms in terms of prediction accuracy.

Samsung SDSThe purpose of virtual metrology (VM) in semiconductor manufacturing is to predict every wafer’s metrological values based on its process equipment data without an actual metrology. In this paper, we propose novelty detection-based reliability estimation models for VM in order to support flexible utilization of VM results. Because the proposed model can not only estimate the reliability of VM, but also identify suspicious process variables lowering the reliability, quality control actions can be taken selectively based on the reliance level and its causes. Based on the preliminary experimental results with actual semiconductor manufacturing process data, our models can successfully give a high reliance level to the wafers with small prediction errors and a low reliance level to the wafers with large prediction errors. In addition, our proposed model can give more detailed information by identifying the critical process variables and their relative impacts on the low reliability.

Plasma etch is a key process in modern semiconductor manufacturing facilities as it offers process simplification and yet greater dimensional tolerances compared to wet chemical etch technology. The main challenge of operating plasma etchers is to maintain a consistent etch rate spatially and temporally for a given wafer and for successive wafers processed in the same etch tool. Etch rate measurements require expensive metrology steps and therefore in general only limited sampling is performed. Furthermore, the results of measurements are not accessible in real-time, limiting the options for run-to-run control. This paper investigates a Virtual Metrology (VM) enabled Dynamic Sampling (DS) methodology as an alternative paradigm for balancing the need to reduce costly metrology with the need to measure more frequently and in a timely fashion to enable wafer-to-wafer control. Using a Gaussian Process Regression (GPR) VM model for etch rate estimation of a plasma etch process, the proposed dynamic sampling methodology is demonstrated and evaluated for a number of different predictive dynamic sampling rules.

Mechanical property prediction plays a crucial role in the steel industry, enabling better materials selection and enhancing production efficiency. In this study, we propose a novel framework that facilitates optimal materials selection for steel’s alloying elements, based on accurate predictions of the Jominy hardness curve. Leveraging Gaussian Process (GP) regression, we provide probabilistic predictions of steel hardenability characteristics from alloy element composition. Taking it a step further, our framework incorporates these accurate predictions into a constrained optimization process, yielding optimal compositions that reduce overall spending while meeting performance specifications. Through data obtained from 1080 steel samples, our GP regression model exhibits high accuracy, achieving an RMSE of 1.37 and showcasing significant improvements in the field. Moreover, our constrained optimization utilizing the GP model and historical market data reveals an average cost reduction of 18% on alloying element expenses, highlighting the tangible cost-saving potential of this approach. By leveraging Gaussian Process (GP) regression, we not only achieve accurate predictions of the Jominy hardness curve based on alloy element composition, but we also introduce a crucial element of uncertainty quantification. This empowers us to place trust in the results of our optimization process, ensuring robust and reliable materials selection. The integration of GP regression and optimization provides a powerful tool for achieving cost-effective materials selection and marks a significant advancement compared to existing studies. This research underscores the promise of machine learning in the steel industry, demonstrating its ability to yield substantial cost savings and enhance decision-making in materials selection.

Soft Sensor for Melt Index Prediction Based on Long Short-Term Memory Network

Earth observation from satellite sensory data pose challenging problems, where machine learning is currently a key player. In recent years, Gaussian Process (GP) regression and other kernel methods have excelled in biophysical parameter estimation tasks from space. GP regression is based on solid Bayesian statistics, and generally yield efficient and accurate parameter estimates. However, GPs are typically used for inverse modeling based on concurrent observations and in situ measurements only. Very often a forward model encoding the well-understood physical relations is available though. In this work, we review three GP models that respect and learn the physics of the underlying processes in the context of inverse modeling. First, we will introduce a Joint GP (JGP) model that combines in situ measurements and simulated data in a single GP model. Second, we present a latent force model (LFM) for GP modeling that encodes ordinary differential equations to blend data-driven modeling and physical models of the system. The LFM performs multi-output regression, adapts to the signal characteristics, is able to cope with missing data in the time series, and provides explicit latent functions that allow system analysis and evaluation. Finally, we present an Automatic Gaussian Process Emulator (AGAPE) that approximates the forward physical model via interpolation, reducing the number of necessary nodes. Empirical evidence of the performance of these models will be presented through illustrative examples of vegetation monitoring and atmospheric modeling.

This paper describes a real-time data collection framework and an adaptive machining learning method for constructing a real-time energy prediction model for a machine tool. To effectively establish the energy consumption pattern of a machine tool over time, the energy prediction model is continuously updated with new measurement data to account for time-varying effects of the machine tool, such as tool wear and machine tool deterioration. In this work, a real-time data collection and processing framework is developed to retrieve raw data from a milling machine tool and its sensors and convert them into relevant input features. The extracted input features are then used to construct the energy prediction model using Gaussian Process (GP) regression. To update the GP regression model with real-time streaming data, we investigate the use of sparse representation of the covariance matrix to reduce the computational and storage demands of the GP regression. We compare computational efficiency of sparse GP to that of full GP regression model and show the effectiveness of the sparse GP regression model for tracking the variation in the energy consumption pattern of the target machine.

In semiconductor manufacturing, metrology operations are expensive and time-consuming, for this reason only a certain sample of wafers is measured. With the need of highly reliable processes, the semiconductor industry is interested in developing methodologies covering the gap of missing metrology information. Virtual Metrology (VM) turns out to be a promising method; it aims to predict wafer and/or site fine metrology results in real time and free of costs. Using virtual measurements as the input of a sampling decision system (SDS), an optimal strategy for measuring productive wafers can be suggested. Since sampling decisions strongly depend on the accuracy of the VM system, it is a key requirement to monitor the reliability of the obtained predictions. In this paper, we present approaches for dynamically assessing VM reliability using real metrology data. A Bayesian dynamical linear model (BDLM) handles increasing VM model uncertainty over time. Model parameters are updated whenever new real measurements become available. VM prediction quality is monitored applying a probability integral transform (PIT) and scoring rules for predictive probability distributions. Inferring equipment health factors (EHF), unreliable predictions can be detected before being delivered to the SDS. Based on the likelihood of the predicted measurements, VM trust factors are introduced. A Bayesian model for the prediction precision matrix allows updating the virtual measurements' uncertainty whenever real measurements are available. Taking account of the proposed methods, one is led to an improved accuracy of the SDS.

BackgroundThe analysis of gene expression from time series underpins many biological studies. Two basic forms of analysis recur for data of this type: removing inactive (quiet) genes from the study and determining which genes are differentially expressed. Often these analysis stages are applied disregarding the fact that the data is drawn from a time series. In this paper we propose a simple model for accounting for the underlying temporal nature of the data based on a Gaussian process.ResultsWe review Gaussian process (GP) regression for estimating the continuous trajectories underlying in gene expression time-series. We present a simple approach which can be used to filter quiet genes, or for the case of time series in the form of expression ratios, quantify differential expression. We assess via ROC curves the rankings produced by our regression framework and compare them to a recently proposed hierarchical Bayesian model for the analysis of gene expression time-series (BATS). We compare on both simulated and experimental data showing that the proposed approach considerably outperforms the current state of the art.ConclusionsGaussian processes offer an attractive trade-off between efficiency and usability for the analysis of microarray time series. The Gaussian process framework offers a natural way of handling biological replicates and missing values and provides confidence intervals along the estimated curves of gene expression. Therefore, we believe Gaussian processes should be a standard tool in the analysis of gene expression time series.

Polydimethylsiloxane (PDMS) silicone membranes, such as Silastic®, have been used widely in place of mammalian tissue in the determination of percutaneous absorption. While many experiments have shown correlations between the permeability across both membranes, Moss et al. demonstrated in a systematic study that PDMS membranes tend to exhibit greater permeability than mammalian skin, and that the relationship between permeability across both membranes was not found when the lipophilicity of the penetrant was greater than 3. Further, it was shown previously that when five commonly used physicochemical descriptors were applied to human, pig and rodent membranes they cannot represent the main characteristics of the PDMS dataset when using Gaussian Process (GP) regression to predict skin permeability. However, the previous study in which this process was modelled employed a small dataset (n=19, as part of the wider aims of that work to investigate the effect of dataset construction on model quality).

Kepler-78b is a transiting planet that is 1.2 times the radius of Earth and orbits a young, active K dwarf every 8 hours. The mass of Kepler-78b has been independently reported by two teams based on radial velocity measurements using the HIRES and HARPS-N spectrographs. Due to the active nature of the host star, a stellar activity model is required to distinguish and isolate the planetary signal in radial velocity data. Whereas previous studies tested parametric stellar activity models, we modeled this system using nonparametric Gaussian process (GP) regression. We produced a GP regression of relevant Kepler photometry. We then use the posterior parameter distribution for our photometric fit as a prior for our simultaneous GP + Keplerian orbit models of the radial velocity datasets. We tested three simple kernel functions for our GP regressions. Based on a Bayesian likelihood analysis, we selected a quasi-periodic kernel model with GP hyperparameters coupled between the two RV datasets, giving a Doppler amplitude of 1.86 $\pm$ 0.25 m s$^{-1}$ and supporting our belief that the correlated noise we are modeling is astrophysical. The corresponding mass of 1.87 $^{+0.27}_{-0.26}$ M$_{\oplus}$ is consistent with that measured in previous studies, and more robust due to our nonparametric signal estimation. Based on our mass and the radius measurement from transit photometry, Kepler-78b has a bulk density of 6.0$^{+1.9}_{-1.4}$ g cm$^{-3}$. We estimate that Kepler-78b is 32$\pm$26% iron using a two-component rock-iron model. This is consistent with an Earth-like composition, with uncertainty spanning Moon-like to Mercury-like compositions.