The use of optical proximity correction (OPC) demands increasingly accurate models of the photolithographic process. Model building and analysis techniques in the data science community have seen great strides in the past two decades which make better use of available information. This paper expands upon Bayesian analysis methods for parameter selection in lithographic models by increasing the parameter set and employing posterior predictive checks. Work continues with a Markov chain Monte Carlo (MCMC) search algorithm to generate posterior distributions of parameters. Models now include wafer film stack refractive indices, n and k, as parameters, recognizing the uncertainties associated with these values. Posterior predictive checks are employed as a method to validate parameter vectors discovered by the analysis, akin to cross validation.
The use of optical proximity correction (OPC) demands increasingly accurate models of the photolithographic process. Model building and inference techniques in the data science community have seen great strides in the past two decades which make better use of available information. This paper aims to demonstrate the predictive power of Bayesian inference as a method for parameter selection in lithographic models by quantifying the uncertainty associated with model inputs and wafer data. Specifically, the method combines the model builder's prior information about each modelling assumption with the maximization of each observation's likelihood as a Student's t-distributed random variable. Through the use of a Markov chain Monte Carlo (MCMC) algorithm, a model's parameter space is explored to find the most credible parameter values. During parameter exploration, the parameters' posterior distributions are generated by applying Bayes' rule, using a likelihood function and the a priori knowledge supplied. The MCMC algorithm used, an affine invariant ensemble sampler (AIES), is implemented by initializing many walkers which semiindependently explore the space. The convergence of these walkers to global maxima of the likelihood volume determine the parameter values' highest density intervals (HDI) to reveal champion models. We show that this method of parameter selection provides insights into the data that traditional methods do not and outline continued experiments to vet the method.
The physical process of mask manufacturing produces absorber geometry with significant deviations from the 90-deg corners, which are typically assumed in the mask design. The non-Manhattan mask geometry is an essential contributor to the aerial image and resulting patterning performance through focus. Current state-of-the-art models for corner rounding employ “chopping” a 90-deg mask corner, replacing the corner with a small 45-deg edge. A methodology is presented to approximate the impact of three-dimensional (3-D) EMF effects introduced by corners with rounded edges. The approach is integrated into a full-chip 3-D mask simulation methodology based on the domain decomposition method with edge to edge crosstalk correction.
In the field of model design and selection, there is always a risk that a model is over-fit to the data used to train the model. A model is well suited when it describes the physical system and not the stochastic behavior of the particular data collected. K-fold cross validation is a method to check this potential over-fitting to the data by calibrating with k-number of folds in the data, typically between 4 and 10. Model training is a computationally expensive operation, however, and given a wide choice of candidate models, calibrating each one repeatedly becomes prohibitively time consuming. Akaike information criterion (AIC) is an information-theoretic approach to model selection based on the maximized log-likelihood for a given model that only needs a single calibration per model. It is used in this study to demonstrate model ranking and selection among compact resist modelforms that have various numbers and types of terms to describe photoresist behavior. It is shown that there is a good correspondence of AIC to K-fold cross validation in selecting the best modelform, and it is further shown that over-fitting is, in most cases, not indicated. In modelforms with more than 40 fitting parameters, the size of the calibration data set benefits from additional parameters, statistically validating the model complexity.
The physical process of mask manufacturing produces absorber geometry with significantly less than 90 degree fidelity at corners. The non-Manhattan mask geometry is an essential contributor to the aerial image and resulting patterning performance through focus. Current state of the art models for corner rounding employ “chopping” a 90 degree mask corner, replacing the corner with a small 45 degree edge. In this paper, a methodology is presented to approximate the impact of 3D EMF effects introduced by corners with rounded edges. The approach is integrated into a full chip 3D mask simulation methodology based on the Domain Decomposition Method (DDM) with edge to edge crosstalk correction.
With the introduction of negative tone develop (NTD) resists to production lithography nodes, multiple NTD resist modeling challenges have surpassed the accuracy limits of the existing modeling infrastructure developed for the positive polarity process. We report the evaluation of two NTD resist modeling algorithms. The new modeling terms represent, from the first principles, the NTD resist mechanisms of horizontal shrink and horizontal development bias. Horizontal shrink describes the impact of the physical process of out-gassing on remaining resist edge location. Horizontal development bias accounts for the differential in the peak and minimum development rate with exposure intensity observed in NTD formulations. We review specific patterning characteristics by feature type, modeling accuracy impact presented by these NTD mechanisms, and their description in our compact models (Compact Model 1, CM1). All the new terms complement the accuracy advantage observed with existing CM1 resist modeling infrastructure. The new terms were tested on various NTD layers. The results demonstrate consistent model accuracy improvement for both calibration and verification. Furthermore, typical NTD model fitting challenges, such as large SRAF-induced wafer CD jump, can be overcome by the new NTD terms. Finally, we propose a joint-tuning approach for the calibration of compact models for the NTD resist.
This study quantifies the impact of systematic mask errors on OPC model accuracy and proposes a methodology to reconcile the largest errors via calibration to the mask error signature in wafer data. First, we examine through simulation, the impact of uncertainties in the representation of photomask properties including CD bias, corner rounding, refractive index, thickness, and sidewall angle. The factors that are most critical to be accurately represented in the model are cataloged. CD bias values are based on state of the art mask manufacturing data while other variable values are speculated, highlighting the need for improved metrology and communication between mask and OPC model experts. It is shown that the wafer simulations are highly dependent upon the 1D/2D representation of the mask, in addition to the mask sidewall for 3D mask models. In addition, this paper demonstrates substantial accuracy improvements in the 3D mask model using physical perturbations of the input mask geometry when using Domain Decomposition Method (DDM) techniques. Results from four test cases demonstrate that small, direct modifications in the input mask stack slope and edge location can result in model calibration and verification accuracy benefit of up to 30%. We highlight the benefits of a more accurate description of the 3D EMF near field with crosstalk in model calibration and impact as a function of mask dimensions. The result is a useful technique to align DDM mask model accuracy with physical mask dimensions and scattering via model calibration.
This paper extends the state of the art by demonstrating performance improvements in the Domain
Decomposition Method (DDM) from a physical perturbation of the input mask geometry. Results from four
testcases demonstrate that small, direct modifications in the input mask stack slope and edge location can result in
model calibration and verification accuracy benefit of up to 30%. All final mask optimization results from this
approach are shown to be valid within measurement accuracy of the dimensions expected from manufacture. We
highlight the benefits of a more accurate description of the 3D EMF near field with crosstalk in model calibration
and impact as a function of mask dimensions. The result is a useful technique to align DDM mask model accuracy
with physical mask dimensions and scattering via model calibration.
This paper extends the state of the art by describing the practical material’s challenges, as well as approaches to minimize their impact in the manufacture of contact/via layers using a grapho-epitaxy directed self assembly (DSA) process. Three full designs have been analyzed from the point of view of layout constructs. A construct is an atomic and repetitive section of the layout which can be analyzed in isolation. Results indicate that DSA’s main benefit is its ability to be resilient to the shape of the guiding pattern across process window. The results suggest that directed self assembly can still be guaranteed even with high distortion of the guiding patterns when the guiding patterns have been designed properly for the target process. Focusing on a 14nm process based on 193i lithography, we present evidence of the need of DSA compliance methods and mask synthesis tools which consider pattern dependencies of adjacent structures a few microns away. Finally, an outlook as to the guidelines and challenges to DSA copolymer mixtures and process are discussed highlighting the benefits of mixtures of homo polymer and diblock copolymer to reduce the number of defects of arbitrarily placed hole configurations.
The Domain Decomposition Method (DDM) for approximating the impact of 3DEMF effects was introduced nearly ten years ago as an approach to deliver good accuracy for rapid simulation of full-chip applications. This approximation, which treats mask edges as independent from one another, provided improved model accuracy over the traditional Kirchhoff thin mask model for the case of alternating aperture phase shift masks which featured severe mask topography. This aggressive PSM technology was not widely deployed in manufacturing, and with the advent of thinner absorbing layers, the impact of mask topography has been relatively well contained through the 32 nm technology node, where Kirchhoff mask models have proved effective. At 20 nm and below, however, the thin mask approximation leads to larger errors, and the DDM model is seen to be effective in providing a more accurate representation of the aerial image. The original DDM model assumes normal incidence, and a subsequent version incorporates signals from oblique angles. As mask dimensions become smaller, the assumption of non-interacting mask edges breaks down, and a further refinement of the model is required to account for edge to edge cross talk. In this study, we evaluate the progression of improvements in modeling mask 3DEMF effects by comparing to rigorous simulation results. It is shown that edge to edge interactions can be accurately accounted for in the modified DDM library. A methodology is presented for the generation of an accurate 3DEMF model library which can be used in full chip OPC correction.