Background: Scatterometry is a fast, indirect, and nondestructive optical method for quality control in the production of lithography masks. To solve the inverse problem in compliance with the upcoming need for improved accuracy, a computationally expensive forward model that maps geometry parameters to diffracted light intensities has to be defined.
Aim: To quantify the uncertainties in the reconstruction of the geometry parameters, a fast-to-evaluate surrogate for the forward model has to be introduced.
Approach: We use a nonintrusive polynomial chaos-based approximation of the forward model, which increases speed and thus enables the exploration of the posterior through direct Bayesian inference. In addition, this surrogate allows for a global sensitivity analysis at no additional computational overhead.
Results: This approach yields information about the complete distribution of the geometry parameters of a silicon line grating, which in return allows for quantifying the reconstruction uncertainties in the form of means, variances, and higher order moments of the parameters.
Conclusions: The use of a polynomial chaos surrogate allows for quantifying both parameter influences and reconstruction uncertainties. This approach is easy to use since no adaptation of the expensive forward model is required.
Recent studies for profile reconstructions of nanostructures produced with self-aligned quadruple patterning (SAQP) indicate the limits for solving the inverse problem with a rigorous simulation. Using Monte Carlo methods for the theoretical investigation of the observed pitchwalk behaviour is not feasible due to the high computational cost of simulating GISAXS measurements by solving Maxwell’s equations with an FEM approach for each proposed structural model. We will show that a surrogate model based on a polynomial chaos expansion is a versatile tool to reduce the computational effort significantly. The expansion provides not only a surrogate for the forward model, but also Sobol indices for a global sensitivity analysis. This enables the study of the sensitivities in GISAXS in detail.
Scatterometry is a fast, indirect and nondestructive optical method for the quality control in the production of lithography masks. Geometry parameters of line gratings are obtained from diffracted light intensities by solving an inverse problem. To comply with the upcoming need for improved accuracy and precision and thus for the reduction of uncertainties, typically computationally expansive forward models have been used. In this paper we use Bayesian inversion to estimate parameters from scatterometry measurements of a silicon line grating and determine the associated uncertainties. Since the direct application of Bayesian inference using MarkovChain Monte Carlo methods to physics-based partial differential equation (PDE) model is not feasible due to high computational costs, we use an approximation of the PDE forward model based on a polynomial chaos expansion. The expansion provides not only a surrogate for the PDE forward model, but also Sobol indices for a global sensitivity analysis. Finally, we compare our results for the global sensitivity analysis with the uncertainties of estimated parameters.
High quality scatterometry standard samples have been developed to improve the tool matching between different scatterometry methods and tools as well as with high resolution microscopic methods such as scanning electron microscopy or atomic force microscopy and to support traceable and absolute scatterometric critical dimension metrology in lithographic nanomanufacturing. First samples based on one dimensional Si or on Si3N4 grating targets have been manufactured and characterized for this purpose. The etched gratings have periods down to 50 nm and contain areas of reduced density to enable AFM measurements for comparison. Each sample contains additionally at least one large area scatterometry target suitable for grazing incidence small angle X-ray scattering. We present the current design and the characterization of structure details and the grating quality based on AFM, optical, EUV and X-Ray scatterometry as well as spectroscopic ellipsometry measurements. The final traceable calibration of these standards is currently performed by applying and combining different scatterometric as well as imaging calibration methods. We present first calibration results and discuss the final design and the aimed specifications of the standard samples to face the tough requirements for future technology nodes in lithography.
Scatterometry is a fast indirect optical method for the determination of grating profile parameters of photomasks. Profile parameters are obtained from light diffracted intensities by solving an inverse problem. There are diverse methods to reconstruct profile parameters and to calculate associated uncertainties. To fit the upcoming need for improved accuracy and precision as well as for the reduction of uncertainties different measurements should be combined. Such a combination increases the knowledge about parameters and may yield smaller uncertainties. The Bayesian approach provides an appropriate method to evaluate combined measurements and to obtain the associated uncertainties. However, for computationally expensive problems like scatterometry, the direct application of Bayesian inference is very time consuming. Here, we use an approximation method based on a polynomial chaos expansion. To probe the quality of this approximation, we reconstructed geometry parameters, quantify uncertainties and study the effect of different prior informations onto the obtained grating profile parameters by using simulation data superimposed by noise.
Scatterometry is a common tool for the dimensional characterization of periodic nanostructures. In this paper we compare measurement results of two different scatterometric methods: a goniometric DUV scatterometer and a coherent scanning Fourier scatterometer. We present a comparison between these two methods by analyzing the measurement results on a silicon wafer with 1D gratings having periods between 300 nm and 600 nm. The measurements have been performed with PTB’s goniometric DUV scatterometer and the coherent scanning Fourier scatterometer at TU Delft. Moreover for the parameter reconstruction of the goniometric measurement data, we apply a maximum likelihood estimation, which provides the statistical error model parameters directly from measurement data.
The impact of line-edge (LER) and line-width roughness (LWR) on the measured diffraction patters in extreme
ultraviolet (EUV) scatterometry has been investigated in recent publications. Two-dimensional rigorous numerical
simulations were carried out to model roughness. Simple analytical expressions for the bias in the mean
efficiencies stemming from LER and LWR were obtained. Applying a similar approach for DUV scatterometry
to investigate the impact of line roughness we obtain comparable results.
The precise and accurate determination of critical dimensions of photo masks and their uncertainties is important
in the lithographic process to ensure operational reliability of electronic compounds. Scatterometry is known as
a fast, non-destructive optical method for the indirect determination of geometry parameters. In recent years
novel methods for solving the inverse problem of scatterometry have enabled a more reliable determination of
grating parameters. In this article we present results from maximum likelihood parameter estimations based on
numerically simulated EUV scatterometry data. We approximately determine uncertainties of these parameters
by a Monte Carlo method with a limited amount of samplings and by employing the Fisher information matrix.
Furthermore, we demonstrate that the use of incomplete mathematical models may lead to severe distortions in
the calculations of the uncertainties by the approximate Fisher matrix approach as well as to substantially larger
uncertainties for the Monte Carlo method.
The characterization of nanostructured surfaces by scatterometry is an established method in wafer metrology.
From measured light diffraction patterns, critical dimensions (CD) of surface profiles are determined, i.e., line
widths, heights and other profile properties in the sub-micrometer range. As structures become smaller and
smaller, shorter wavelengths like extreme ultraviolet (EUV) at 13.5 nm ensure a sufficient sensitivity of the
measured light diffraction pattern with regard to the structure details. Obviously, the impact of structure
roughness with amplitudes in the range of a few nanometers can no longer be neglected in the course of the profile
reconstruction. To model line roughness, i.e., line edge (LER) and line width (LWR) roughness, a large number
of finite element (FEM) simulations are performed for domains with large periods, each containing many pairs
of line and space with stochastically chosen widths. These structures are composed of TaN -absorber lines with
an underlying MoSi -multilayer stack representing a typical EUV mask. The resulting mean efficiencies and the
variances of the efficiencies in dependence on different degrees of roughness are calculated. A systematic decrease
of the mean efficiencies for higher diffraction orders along with increasing variances are observed. In particular,
we obtain a simple analytical expression for the bias in the mean efficiencies and the additional uncertainty
contribution stemming from the presence of LER and/or LWR. As a consequence, the bias has to be included
into the model to provide accurate values for the reconstructed critical profile parameters. The sensitivity of the
reconstructed CDs in respect of roughness is demonstrated by using numerous LER/LWR perturbed datasets of
efficiencies as input data for the reconstructions. Finally, the reconstructed critical dimensions are significantly
improved toward the nominal values if the scattering efficiencies are bias-corrected.