Mark-Alexander Henn

A maximum likelihood approach to the inverse problem of scatterometry

Scatterometry is frequently used as a non-imaging indirect optical method to reconstruct the critical dimensions (CD) of periodic nanostructures. A particular promising direction is EUV scatterometry with wavelengths in the range of 13 - 14 nm. The conventional approach to determine CDs is the minimization of a least squares function (LSQ). In this paper, we introduce an alternative method based on the maximum likelihood estimation (MLE) that determines the statistical error model parameters directly from measurement data. By using simulation data, we show that the MLE method is able to correct the systematic errors present in LSQ results and improves the accuracy of scatterometry. In a second step, the MLE approach is applied to measurement data from both extreme ultraviolet (EUV) and deep ultraviolet (DUV) scatterometry. Using MLE removes the systematic disagreement of EUV with other methods such as scanning electron microscopy and gives consistent results for DUV.

Modeling of line roughness and its impact on the diffraction intensities and the reconstructed critical dimensions in scatterometry

We investigate the impact of line-edge and line-width roughness (LER, LWR) on the measured diffraction intensities in angular resolved extreme ultraviolet (EUV) scatterometry for a periodic line-space structure designed for EUV lithography. LER and LWR with typical amplitudes of a few nanometers were previously neglected in the course of the profile reconstruction. The two-dimensional (2D) rigorous numerical simulations of the diffraction process for periodic structures are carried out with the finite element method providing a numerical solution of the 2D Helmholtz equation. To model roughness, multiple calculations are performed for domains with large periods, containing many pairs of line and space with stochastically chosen line and space widths. A systematic decrease of the mean efficiencies for higher diffraction orders along with increasing variances is observed and established for different degrees of roughness. In particular, we obtain simple analytical expressions for the bias in the mean efficiencies and the additional uncertainty contribution stemming from the presence of LER and/or LWR. As a consequence this bias can easily be included into the reconstruction model to provide accurate values for the evaluated profile parameters. We resolve the sensitivity of the reconstruction from this bias by using simulated data with LER/LWR perturbed efficiencies for multiple reconstructions. If the scattering efficiencies are bias-corrected, significant improvements are found in the reconstructed bottom and top widths toward the nominal values.

First steps towards a scatterometry reference standard

Supported by the European Commission and EURAMET, a consortium of 10 participants from national metrology institutes, universities and companies has recently started a joint research project with the aim of overcoming current challenges in optical scatterometry for traceable linewidth metrology and to establish scatterometry as a traceable and absolute metrological method for dimensional measurements. This requires a thorough investigation of the influence of all significant sample, tool and data analysis parameters, which affect the scatterometric measurement results. For this purpose and to improve the tool matching between scatterometers, CD-SEMs and CD-AFMs, experimental and modelling methods will be enhanced. The different scatterometry methods will be compared with each other and with specially adapted atomic force microscopy (AFM) and scanning electron microscopy (SEM) measurement systems. Additionally novel methods for sophisticated data analysis will be developed and investigated to reach significant reductions of the measurement uncertainties in critical dimension (CD) metrology. To transfer traceability to industrial applications of scatterometry an important step and one final goal of this project is the realisation of different waferbased reference standard materials for calibration of scatterometers. The approaches to reach these goals and first design considerations and preliminary specification of the scatterometry standards are presented and discussed.

Improved grating reconstruction by determination of line roughness in extreme ultraviolet scatterometry

The accurate determination of critical dimensions and roughness is necessary to ensure the quality of photoresist masks that are crucial for the operational reliability of electronic components. Scatterometry provides a fast indirect optical nondestructive method for the determination of profile parameters that are obtained from scattered light intensities using inverse methods. We illustrate the effect of line roughness on the reconstruction of grating parameters employing a maximum likelihood scheme. Neglecting line roughness introduces a strong bias in the parameter estimations. Therefore, such roughness has to be included in the mathematical model of the measurement in order to obtain accurate reconstruction results. In addition, the method allows to determine line roughness from scatterometry. The approach is demonstrated for simulated scattering intensities as well as for experimental data of extreme ultraviolet light scatterometry measurements. The results obtained from the experimental data are in agreement with independent atomic force microscopy measurements.

Improved reconstruction of critical dimensions in extreme ultraviolet scatterometry by modeling systematic errors

Scatterometry is a non-imaging indirect optical method that is frequently used to reconstruct the critical dimensions (CD) of periodic nanostructures, e.g. structured wafer surfaces in semiconductor chip production. To solve the inverse problem, we apply a maximum likelihood estimation, introduced in Henn et al (2012 Opt. Express 20 12771–86). Along with the CD values, further relevant quantities like noise parameters of the measured diffraction intensities and the strength of line roughness can be estimated from the measured scattering efficiencies. We investigate three different models for extreme ultraviolet (EUV) scatterometry at an EUV photo mask with increasing complexity by successively including two major sources of systematic errors, namely line roughness and deviations in the multilayer substrate of the EUV mask. Applying the different models to reconstruct the CDs from both simulation and measurement data, we demonstrate the improvements of the reconstruction in terms of simulated and real measurement data. The inclusion of systematic errors in the maximum likelihood approach to the inverse problem leads to a significant reduction of the variances in the estimated CDs implying reduced measurement uncertainty for scatterometry.

On numerical reconstructions of lithographic masks in DUV scatterometry

The solution of the inverse problem in scatterometry employing deep ultraviolet light (DUV) is discussed, i.e. we consider the determination of periodic surface structures from light diffraction patterns. With decreasing dimensions of the structures on photo lithography masks and wafers, increasing demands on the required metrology techniques arise. Scatterometry as a non-imaging indirect optical method is applied to periodic line structures in order to determine the sidewall angles, heights, and critical dimensions (CD), i.e., the top and bottom widths. The latter quantities are typically in the range of tens of nanometers. All these angles, heights, and CDs are the fundamental figures in order to evaluate the quality of the manufacturing process. To measure those quantities a DUV scatterometer is used, which typically operates at a wavelength of 193 nm. The diffraction of light by periodic 2D structures can be simulated using the finite element method for the Helmholtz equation. The corresponding inverse problem seeks to reconstruct the grating geometry from measured diffraction patterns. Fixing the class of gratings and the set of measurements, this inverse problem reduces to a finite dimensional nonlinear operator equation. Reformulating the problem as an optimization problem, a vast number of numerical schemes can be applied. Our tool is a sequential quadratic programing (SQP) variant of the Gauss-Newton iteration. In a first step, in which we use a simulated data set, we investigate how accurate the geometrical parameters of an EUV mask can be reconstructed, using light in the DUV range. We then determine the expected uncertainties of geometric parameters by reconstructing from simulated input data perturbed by noise representing the estimated uncertainties of input data. In the last step, we use the measurement data obtained from the new DUV scatterometer at PTB to determine the geometrical parameters of a typical EUV mask with our reconstruction algorithm. The results are compared to the outcome of investigations with two alternative methods namely EUV scatterometry and SEM measurements.

Comparison of CD measurements of an EUV photomask by EUV scatterometry and CD-AFM

EUV scatterometry is a potential high-throughput measurement method for the characterization of EUV photomask structures. We present a comparison of angle resolved extreme ultraviolet (EUV) scatterometry and critical dimension atomic force microscope (CD-AFM) as a reference metrology for measurements of geometrical parameters like line width (CD), height and sidewall angle of EUV photomask structures. The structures investigated are dense and semidense bright and dark lines with different nominal CDs between 140 nm and 540 nm. The results show excellent linearity of the critical dimension measured with both methods within a range of only 1.8 nm and an offset of the absolute values below 3 nm. A maximum likelihood estimation (MLE) method is used to reconstruct the shape parameters and to estimate their uncertainties from the measured scattering efficiencies. The newly developed CD-AFM at PTB allows versatile measurements of parameters such as height, CD, sidewall angle, line edge/width roughness, corner rounding, and pitch. It applies flared tips to probe steep and even undercut sidewalls and employs a new vector approaching probing (VAP) strategy which enables very low tip wear and high measurement flexibility. Its traceability is ensured by a set of calibrated step-height and reference CD standards.

The effect of line roughness on DUV scatterometry

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.

Improved geometry reconstruction and uncertainty evaluation for extreme ultraviolet (EUV) scatterometry based on maximum likelyhood estimation

The task of solving the inverse problem of scatterometry is considered. As a non-imaging indirect optical metrology method the goal of scatterometry is, e.g., to reconstruct the absorber line profiles of lithography masks, i.e., profile parameters such as line width, line height, and side-wall angle (SWA), from the measured diffracted light pattern and to estimate their associated uncertainties. The impact of an appropriate choice of the statistical model for the input data on the reconstructed profile parameters is demonstrated for EUV masks, where light with wavelengths of about 13.5 nm is applied. The maximum likelihood method is proposed to determine more reliable estimations of all model parameters, including the sought profile dimensions. Finally, this alternative approach is applied to EUV measurement data and the results are compared to those obtained by a conventional analysis.

Measurement comparison of goniometric scatterometry and coherent Fourier scatterometry

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.