Computational Metrology for Nanostructures: Problems and Solution Methods

In order to achieve effective process control, fast, inexpensive,  nondestructive, and reliable nanometer scale feature measurements      are extremely  useful in nanomanufacturing. Optics-based metrology tools, which have drawn  more and more attention in semiconductor manufacturing because of their  attractive advantages, such as low cost, non-contact, non-destruction, and high  throughput, may be one of the best choices. Overall, optics-based metrology can  be classified into two categories, namely the model-free and the model-based  methods. Model-free metrology, such as the conventional optical microscopy,  seems to be useless for nanoscale dimensional analysis due to the so-called  diffraction limit. In contrast, model-based metrology, such as optical scatterometry,  does not depend on images with well-defined edges and allows for the indirect  determination of the geometrical parameters of the measuring targets from the  measured signatures. Since this model-based metrology relies heavily on forward  optical modeling and inverse parameter extraction, which are both  computationally intensive, we term this metrology as computational metrology  (CM). To the best of our knowledge, it is the first time we give this term  similar to the case of computational lithography (CL), with emphasis on solving  the mathematical problems.

(1) Fundamental concept of  computational metrology

Computational metrology is defined as "a measurement method where a complicated  measurement process is modeled as a forward problem and some measured data are  obtained by a specific instrument under a certain measurement configuration,  and then the measurands are precisely and accurately reconstructed by solving  the corresponding inverse problem." The basic elements of computational  metrology include the measurands, measurement configuration, forward model,  measured data, and solution of measurands. We have investigated and verified some  fundamental problems and solution methods in computational metrology by taking  the Mueller matrix ellipsometry for nanostructure metrology as a case study.

(2) Fast and accurate numerical  methods for forward modeling


The most commonly used method for the scattering field modeling of nanostructures  (especially periodic nanostructures) is rigorous coupled-wave analysis (RCWA).  However, the computational process of RCWA involves intensive calculations involving  eigenvalue problems and dense matrices. Particularly, repeated computations of  the forward model are inevitable when the modeling parameters change, and  thereby leads to a time-consuming process. To realize fast and accurate  forward modeling, we propose to combine the traditional boundary element method  (BEM) with the reduced-basis method (RBM). In this combined method, the  calculation of the boundary integral equation is divided into a  parameter-independent offline process and a parameter-dependent online process.  In addition, the high-dimensional solution space is mapped into a  low-dimensional solution space, which significantly reduces the computational  complexity.

(3) Fast and robust measurand reconstruction  algorithms

Currently, the  traditional method in computational metrology for parameter extraction uses gradient-based  iterative algorithms such as the Gauss-Newton (GN) and Levenberg-Marquardt  (LM), whose core is the iterative minimization of a pre-defined least square  (LSQ) function. The LSQ function implies an assumption of the statistical  property of measurement errors, namely, the assumption of zero mean with normal  distribution. However, the actual statistical property of measurement errors is  far more complex than the simple normal distribution. To realize a more robust  parameter extraction, we propose to add a robust regression procedure at the  end of each iteration of the GN algorithm to obtain a more accurate parameter  departure vector, by which more accurate reconstructed results can be achieved.

(4)  Measurement configuration optimization

In computational metrology, the sensitivity of  the model inputs on the output has a significant impact on the precision of the  reconstructed profile parameters beyond the quality of the measured signatures and  depends on the selected measurement configuration (a  combination of wavelengths, incidence and  azimuthal angles). Thus, we propose to determine an optimal measurement configuration with  the application of global sensitivity analysis (GSA). For each measurement  configuration, we define called the  uncertainty index to evaluate the impact of random noise in measured signatures  on measurement precision by combining the corresponding noise level with the  main effect defined in GSA. Experiments demonstrated that the lower  the uncertainty index, the better the precision of the  profile parameters.

(5)  Error analysis and uncertainty estimation

In computation metrology, the errors can be  generally categorized into random and systematic errors. The random errors are  mainly induced by random noise in measured data, and can be studied by using probability and statistics. In detail, we  first calculate the Hessian matrix of the measurands, and then obtain the  variances and parameter correlation coefficients from the corresponding  covariance matrix. Finally, the measurement precision and parameter correlation  can be estimated. Systematic errors are induced by both the instrument  and the forward model. First, we do a first order Taylor expansion of the forward  model in the vicinity of the nominal measurement condition and the solution of measurands. Then, the impact of various  systematic errors on the accuracy of final measurement results can be estimated by making some matrix computations.


Model-Based Infrared Reflectrometry for Deep Trench Structures

Fig. 1. Fundamental concept and  basic elements of computational metrology

Model-Based Infrared Reflectrometry for Deep Trench Structures

Fig. 2. The comparison of calculation accuracy and time between the conventional BEM and BEM-RBM combined method

Model-Based Infrared Reflectrometry for Deep Trench Structures

Fig. 3. The comparison of solutions of measurands obtained by the GN-based algorithm (with red circles) and proposed robust algorithm (with blue rectangles). The dash-dotted lines denote the results measured by SEM.

Model-Based Infrared Reflectrometry for Deep Trench Structures

Fig. 4. The achieved optimal azimuthal angles for the measurands - TCD, Hgt, and SWA, are 85°, 0°and 85°, respectively, which are in reasonable agreement with the actually optimal azimuthal angles.

Model-Based Infrared Reflectrometry for Deep Trench Structures

Fig. 5. Error propagation in nanostructure metrology using Mueller matrix ellipsometry

Secleted papers

  1. J. L. Zhu, S. Y. Liu, X.  G. Chen, C. W. Zhang, and H. Jiang, "Robust solution to the inverse problem in  optical scatterometry," Opt. Express 22(18), 22031-22042  (2014). (URL, PDF)

  2.                 Z. Q. Dong, S. Y. Liu, X. G. Chen, and C. W.  Zhang, "Determination of an optimal measurement configuration in optical  scatterometry using global sensitivity analysis," Thin Solid Films 562, 16-23 (2014). (URL, PDF)

  3. X. G. Chen, S. Y. Liu, H. G.  Gu, and C. W. Zhang, "Formulation of error propagation and estimation in  grating reconstruction by a dual-rotating compensator Mueller matrix polarimeter," Thin Solid Films 571, 653-659 (2014). (URL, PDF)

  4.                 S. Y. Liu, "Computational metrology: Problems  and solution methods," J. Mech. Eng. 50(4), 1-10 (2014). (URL, PDF)

  5.                 S. Y. Liu, "Computational metrology for  nanomanufacturing," Presented at the 6th International Symposium on Precision  Mechanical Measurement (ISPMM), Guiyang,  China, August 8-10, 2013, in Proc. SPIE 8916, 891606 (2013). (Invited speech) (URL, PDF)

  6.                 X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang,  "Improved measurement accuracy in optical scatterometry using correction-based  library search," Appl. Opt. 52(27),  6727-6734 (2013). (URL, PDF)

  7.                 X. G. Chen, S. Y. Liu, C. W. Zhang, and H. Jiang,  "Measurement configuration optimization for accurate grating reconstruction by  Mueller matrix polarimetry," J.  Micro/Nanolith. MEMS MOEMS 12(3), 033013 (2013). (URL, PDF)

  8.                 X. G. Chen, S. Y. Liu, C. W. Zhang, and J. L.  Zhu, "Improved measurement accuracy in optical scatterometry using fitting  error interpolation based library search," Measurement 46(8), 2638-2646 (2013). (URL, PDF)

  9. J. L. Zhu, S. Y. Liu, C. W. Zhang, X. G. Chen, and  Z. Q. Dong, "Identification and reconstruction of diffraction structures in  optical scatterometry using support vector machine method," J. Micro/Nanolith. MEMS MOEMS 12(1), 013004  (2013). (URL, PDF)

  10.                 S.  Y. Liu, Y. Ma, X. G. Chen, and C. W. Zhang, "Estimation of the convergence  order of rigorous coupled-wave analysis for binary gratings in optical critical  dimension metrology," Opt. Eng. 51(8), 081504 (2012). (URL, PDF)