The doctoral dissertations of the former Helsinki University of Technology (TKK) and Aalto University Schools of Technology (CHEM, ELEC, ENG, SCI) published in electronic format are available in the electronic publications archive of Aalto University - Aaltodoc.
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Novel Genetic Fitting Algorithms and Statistical Error Analysis Methods for X-Ray Reflectivity Analysis

Jouni Tiilikainen

Dissertation for the degree of Doctor of Science in Technology to be presented with due permission of the Faculty of Electronics, Communications and Automation for public examination and debate in Auditorium AS1 at Helsinki University of Technology (Espoo, Finland) on the 3rd of October, 2008, at 12 noon.

Overview in PDF format (ISBN 978-951-22-9541-8)   [2361 KB]
Dissertation is also available in print (ISBN 978-951-22-9540-1)

Abstract

This thesis deals with x-ray reflectivity (XRR) analysis. XRR is a very accurate technique for the metrology of thin films but the analysis of measurements has been difficult thus limiting every day material research. In this thesis, novel genetic algorithms (GAs) for XRR curve fitting and statistical error analysis methods are developed. The XRR analysis utilizes very accurate Parratt's formalism combined with Nevot–Croce interface roughness. The analysis concentrates on the atomic layer deposited materials by using models mimicking their properties. The properties of GAs are studied using aluminium oxide/zinc oxide nanolaminate models. Models of aluminium oxide layers on silicon substrate are used in the case of the error analysis.

The demonstrated novel GAs are utilizing the rotation of coordinates during the crossover phase to reduce interparameter dependencies. The new basis is formed from the eigenvectors of Hessian and statistical covariance matrices. The crossover is performed in the rotated coordinates and the new combinations are transformed back to the original coordinates. It is shown that the coordinate rotation improves the convergence properties of GAs in complex XRR curve fitting problems and a statistical approach is more powerful than the Hessian matrix method. Furthermore, a GA using independent component analysis gives additional robustness to the curve fitting by utilizing a nonorthogonal linear transformation technique.

The interdependency of XRR parameters is studied using fitness landscapes. The fitness landscape analysis utilizes subspace projection of the original parameter space where the projection is done using an experimental model. The work reveals that the error in the determined mass density can compensate the error in surface roughness thus diminishing the accuracy of both of these parameters. This result is also verified later with other methods.

The effect of Poisson noise on the accuracy of XRR analysis is studied statistically. Thickness determination accuracy of an aluminium oxide layer is ±0.09 nm with 99% confidence in the studied case which represents the lower limit for the error. Here the analysis assumed a perfect fit to the measurement. The upper error is achieved by taking into account a nonideal fit by separating the effect of noise from the fitness value. In a case of the studied measurement, the determined thickness error is ±0.12 nm with 99% confidence.

This thesis consists of an overview and of the following 5 publications:

  1. J. Tiilikainen, J.-M. Tilli, V. Bosund, M. Mattila, T. Hakkarainen, V.-M. Airaksinen and H. Lipsanen, Nonlinear fitness–space–structure adaptation and principal component analysis in genetic algorithms: an application to x-ray reflectivity analysis, Journal of Physics D: Applied Physics 40 (2007) 215-218. © 2007 Institute of Physics Publishing. By permission.
  2. J. Tiilikainen, V. Bosund, J.-M. Tilli, J. Sormunen, M. Mattila, T. Hakkarainen and H. Lipsanen, Genetic algorithm using independent component analysis in x-ray reflectivity curve fitting of periodic layer structures, Journal of Physics D: Applied Physics 40 (2007) 6000-6004. © 2007 Institute of Physics Publishing. By permission.
  3. J. Tiilikainen, V. Bosund, M. Mattila, T. Hakkarainen, J. Sormunen and H. Lipsanen, Fitness function and nonunique solutions in x-ray reflectivity curve fitting: crosserror between surface roughness and mass density, Journal of Physics D: Applied Physics 40 (2007) 4259-4263. © 2007 Institute of Physics Publishing. By permission.
  4. J. Tiilikainen, J.-M. Tilli, V. Bosund, M. Mattila, T. Hakkarainen, J. Sormunen and H. Lipsanen, Accuracy in x-ray reflectivity analysis, Journal of Physics D: Applied Physics 40 (2007) 7497-7501. © 2007 Institute of Physics Publishing. By permission.
  5. J. Tiilikainen, M. Mattila, T. Hakkarainen and H. Lipsanen, Novel method for error limit determination in x-ray reflectivity analysis, Journal of Physics D: Applied Physics 41 (2008) 115302. © 2008 Institute of Physics Publishing. By permission.

Keywords: atomic layer deposition, genetic algorithm, fitting, error analysis, x-ray reflectivity

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© 2008 Helsinki University of Technology

Last update 2011-05-26