Andrea Schirru | University of Pavia (original) (raw)
Papers by Andrea Schirru
Abstract In semiconductor manufacturing, the state of the art for wafer quality control is based ... more Abstract In semiconductor manufacturing, the state of the art for wafer quality control is based on product monitoring and feedback control loops; the related metrology operations, that usually involve scanning electron microscopes, are particularly cost-intensive and ...
Increasingly semiconductor manufacturers are exploring opportunities for virtual metrology (VM) e... more Increasingly semiconductor manufacturers are exploring opportunities for virtual metrology (VM) enabled process monitoring and control as a means of reducing non-value added metrology and achieving ever more demanding wafer fabrication tolerances. However, developing robust, reliable and interpretable VM models can be very challenging due to the highly correlated input space often associated with the underpinning data sets. A particularly pertinent example is etch rate prediction of plasma etch processes from multi-channel optical emission spectroscopy data. This paper proposes a novel input-clustering based forward stepwise regression methodology for VM model building in such highly correlated input spaces. Max Separation Clustering (MSC) is employed as a pre-processing step to identify a reduced set of well-conditioned, representative variables that can then be used as inputs to state-of-the-art model building techniques such as Forward Selection Regression (FSR), Ridge regression, LASSO and Forward Selection Ridge Regression (FSRR). The methodology is validated on a benchmark semiconductor plasma etch dataset and the results obtained are compared with those achieved when the state-of-art approaches are applied directly to the data without the MSC pre-processing step. Significant performance improvements are observed when MSC is combined with FSR (13%) and FSRR (8.5%), but not with Ridge Regression (-1%) or LASSO (-32%). The optimal VM results are obtained using the MSC-FSR and MSC-FSRR generated models.
In a Bayesian learning setting, the posterior distribution of a predictive model arises from a tr... more In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.
Abstract Epitaxy is a process strongly dependent on wafer temperature. Unfortunately, the perform... more Abstract Epitaxy is a process strongly dependent on wafer temperature. Unfortunately, the performance of the pyrometers in charge of sensing wafer temperature deteriorate with the usage. This represents the major maintenance issue for epitaxy process engineers who have to frequently calibrate pyrometers emissivity coefficient. At the present state the change of the emissivity coefficient is heuristically based on fab tradition and process engineers experience. We present a statistical tool to map the relationship between change in the ...
Ion Implantation is one of the most sensitive processes in Semiconductor Manufacturing. It consis... more Ion Implantation is one of the most sensitive processes in Semiconductor Manufacturing. It consists in impacting accelerated ions with a material substrate and is performed by an Implanter tool. The major maintenance issue of such tool concerns the breaking of the tungsten filament contained within the ion source of the tool. This kind of fault can happen on a weekly basis, and the associated maintenance operations can last up to 3 hours. It is important to optimize the maintenance activities by synchronizing the Filament change operations with other minor maintenance interventions. In this paper, a Predictive Maintenance (PdM) system is proposed to tackle such issue; the filament lifetime is estimated on a statistical basis exploiting the knowledge of physical variables acting on the process. Given the high-dimensionality of the data, the statistical modeling has been based on Regularization Methods: Lasso, Ridge Regression and Elastic Nets. The predictive performances of the aforementioned regularization methods and of the proposed PdM module have been tested on actual productive semiconductor data.
In semiconductor manufacturing, the purpose of chamber matching is the alignment of process and y... more In semiconductor manufacturing, the purpose of chamber matching is the alignment of process and yield results of distinct chambers performing in parallel the same process step on different silicon wafers. In this paper, multilevel linear models and statistical process control techniques are jointly employed to define control charts for monitoring chamber matching accuracy and preemptively report chamber misalignments. Specifically, multilevel versions of the classic T 2 Control Chart, MEWMA Control Chart and Self-Starting Control Chart are defined and tested against experimental and simulated data.
Predictive Maintenance methods are aimed to obtain reliable estimates of the remaining life cycle... more Predictive Maintenance methods are aimed to obtain reliable estimates of the remaining life cycle of an equipment from time series of suitable process parameters, named “health factors”, typically exhibiting a monotone evolution associated with the equipment wear. The present study was motivated by the predictive maintenance of a dry etching equipment within a semiconductor manufacturing process. The optimal prediction of the health factor, represented by the cooling helium flow, must cope with noisy measurements of the health factor (possibly masking its monotonicity) and non uniform sampling times. The problem is formulated as a stochastic filtering problem in which a stochastic process has to be optimally predicted based on noisy and irregularly sampled observations. In particular, a hidden Gamma process model is proposed in order to capture all the features of the health factor, namely its nonnegativity and nonnegativity of its derivative. Since this filtering problem is not amenable to a closed form solution, a numerical Monte Carlo approach based on particle filtering is developed. Additionally, an adaptive parameter identification procedure is proposed to achieve the best trade off between promptness and noise insensitivity.
The present paper is motivated by the application of Predictive Maintenance (PM) techniques in th... more The present paper is motivated by the application of Predictive Maintenance (PM) techniques in the semiconductor manufacturing environment: such techniques are able, using process data, to make reliable predictions of residual equipment lifetime. The employment of PM yields positive fallouts on the productive process in form of unscheduled downtime reduction, increased spare parts availability and improved overall production quality. One of the main challenges in PM modeling regards the data-driven assessment of relevant process variables when insufficient expert knowledge is available. In this paper, survival models theory is employed jointly with ℓ1 penalization techniques: this allows to obtain sparse models able to select the meaningful process variables and simultaneously predict the remaining lifetime of an equipment. Additionally, frailty modeling techniques are employed to concurrently handle several productive equipments of the same type, exploiting their similarities to increase prediction accuracy. The proposed methodology is validated, illustrating promising results, by means of a semiconductor manufacturing dataset.
Abstract In semiconductor manufacturing, the state of the art for wafer quality control is based ... more Abstract In semiconductor manufacturing, the state of the art for wafer quality control is based on product monitoring and feedback control loops; the related metrology operations, that usually involve scanning electron microscopes, are particularly cost-intensive and ...
Increasingly semiconductor manufacturers are exploring opportunities for virtual metrology (VM) e... more Increasingly semiconductor manufacturers are exploring opportunities for virtual metrology (VM) enabled process monitoring and control as a means of reducing non-value added metrology and achieving ever more demanding wafer fabrication tolerances. However, developing robust, reliable and interpretable VM models can be very challenging due to the highly correlated input space often associated with the underpinning data sets. A particularly pertinent example is etch rate prediction of plasma etch processes from multi-channel optical emission spectroscopy data. This paper proposes a novel input-clustering based forward stepwise regression methodology for VM model building in such highly correlated input spaces. Max Separation Clustering (MSC) is employed as a pre-processing step to identify a reduced set of well-conditioned, representative variables that can then be used as inputs to state-of-the-art model building techniques such as Forward Selection Regression (FSR), Ridge regression, LASSO and Forward Selection Ridge Regression (FSRR). The methodology is validated on a benchmark semiconductor plasma etch dataset and the results obtained are compared with those achieved when the state-of-art approaches are applied directly to the data without the MSC pre-processing step. Significant performance improvements are observed when MSC is combined with FSR (13%) and FSRR (8.5%), but not with Ridge Regression (-1%) or LASSO (-32%). The optimal VM results are obtained using the MSC-FSR and MSC-FSRR generated models.
In a Bayesian learning setting, the posterior distribution of a predictive model arises from a tr... more In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.
Abstract Epitaxy is a process strongly dependent on wafer temperature. Unfortunately, the perform... more Abstract Epitaxy is a process strongly dependent on wafer temperature. Unfortunately, the performance of the pyrometers in charge of sensing wafer temperature deteriorate with the usage. This represents the major maintenance issue for epitaxy process engineers who have to frequently calibrate pyrometers emissivity coefficient. At the present state the change of the emissivity coefficient is heuristically based on fab tradition and process engineers experience. We present a statistical tool to map the relationship between change in the ...
Ion Implantation is one of the most sensitive processes in Semiconductor Manufacturing. It consis... more Ion Implantation is one of the most sensitive processes in Semiconductor Manufacturing. It consists in impacting accelerated ions with a material substrate and is performed by an Implanter tool. The major maintenance issue of such tool concerns the breaking of the tungsten filament contained within the ion source of the tool. This kind of fault can happen on a weekly basis, and the associated maintenance operations can last up to 3 hours. It is important to optimize the maintenance activities by synchronizing the Filament change operations with other minor maintenance interventions. In this paper, a Predictive Maintenance (PdM) system is proposed to tackle such issue; the filament lifetime is estimated on a statistical basis exploiting the knowledge of physical variables acting on the process. Given the high-dimensionality of the data, the statistical modeling has been based on Regularization Methods: Lasso, Ridge Regression and Elastic Nets. The predictive performances of the aforementioned regularization methods and of the proposed PdM module have been tested on actual productive semiconductor data.
In semiconductor manufacturing, the purpose of chamber matching is the alignment of process and y... more In semiconductor manufacturing, the purpose of chamber matching is the alignment of process and yield results of distinct chambers performing in parallel the same process step on different silicon wafers. In this paper, multilevel linear models and statistical process control techniques are jointly employed to define control charts for monitoring chamber matching accuracy and preemptively report chamber misalignments. Specifically, multilevel versions of the classic T 2 Control Chart, MEWMA Control Chart and Self-Starting Control Chart are defined and tested against experimental and simulated data.
Predictive Maintenance methods are aimed to obtain reliable estimates of the remaining life cycle... more Predictive Maintenance methods are aimed to obtain reliable estimates of the remaining life cycle of an equipment from time series of suitable process parameters, named “health factors”, typically exhibiting a monotone evolution associated with the equipment wear. The present study was motivated by the predictive maintenance of a dry etching equipment within a semiconductor manufacturing process. The optimal prediction of the health factor, represented by the cooling helium flow, must cope with noisy measurements of the health factor (possibly masking its monotonicity) and non uniform sampling times. The problem is formulated as a stochastic filtering problem in which a stochastic process has to be optimally predicted based on noisy and irregularly sampled observations. In particular, a hidden Gamma process model is proposed in order to capture all the features of the health factor, namely its nonnegativity and nonnegativity of its derivative. Since this filtering problem is not amenable to a closed form solution, a numerical Monte Carlo approach based on particle filtering is developed. Additionally, an adaptive parameter identification procedure is proposed to achieve the best trade off between promptness and noise insensitivity.
The present paper is motivated by the application of Predictive Maintenance (PM) techniques in th... more The present paper is motivated by the application of Predictive Maintenance (PM) techniques in the semiconductor manufacturing environment: such techniques are able, using process data, to make reliable predictions of residual equipment lifetime. The employment of PM yields positive fallouts on the productive process in form of unscheduled downtime reduction, increased spare parts availability and improved overall production quality. One of the main challenges in PM modeling regards the data-driven assessment of relevant process variables when insufficient expert knowledge is available. In this paper, survival models theory is employed jointly with ℓ1 penalization techniques: this allows to obtain sparse models able to select the meaningful process variables and simultaneously predict the remaining lifetime of an equipment. Additionally, frailty modeling techniques are employed to concurrently handle several productive equipments of the same type, exploiting their similarities to increase prediction accuracy. The proposed methodology is validated, illustrating promising results, by means of a semiconductor manufacturing dataset.