| Zugriffsnummer | 29367 |
| Dokumenttyp | Dissertation |
| Peer Review | unbekannt |
| Sprache | Englisch |
| Titel | Statistical approaches to the inverse problem of scatterometry |
| Autor(in); Institution |
Henn, Mark-Alexander; 8.4, Mathematische Modellierung und Datenanalyse, PTB-Berlin
|
| Quelle/Jahr | (2013), XXI, 90 S. |
| Dissertationsvermerk | Dissertation, Technische Universität Berlin, 2013 |
| Verlag | Berlin: |
| Freie Schlagworte | Scatterometry ; Metrology ; Inverse Problems |
| Zusammenfassung | In the present work statistical approaches to the inverse problem of scatterometry are discussed. Scatterometry is the dimensional characterization of periodic nanostructures as they are used in the manufacturing of lithographic masks. In contrast to direct imaging methods, such as electron microscopy, scatterometry is a non-imaging indirect measuring method. The critical dimensions (CDs) such as line widths and heights of the surface profile are determined from the measured light diffraction pattern. The classical way to solve the inverse problem is the least squares (LSQ) approach. Starting with a model function that depends on the parameters to be reconstructed, the norm of the difference between the measured and simulated data is minimized. The right choice of weights that account for the variances in the measurement data plays a crucial role here, as an inappropriate choice of weights may cause an unsatisfying reconstruction and furthermore an overestimation of the associated uncertainties of the reconstructed parameters. Therefore the maximum likelihood estimation (MLE) is introduced as a method to solve the inverse problem of scatterometry. By doing this, it is possible to determine the variances of the measurement data in addition to the determination of the critical dimensions. In the case of a simplified model function, in which significant effects are not considered, MLE yields estimates for the variances of the measurement data that are way too large. Thus two types of systematic errors and the effect they have on the measured diffraction pattern are investigated. Namely errors stemming from line roughness and variations of the absorbing structure beneath the periodic line structure are discussed. It is shown how the estimated variances for the measurement data reduce if the systematic errors are included into the model function. Furthermore this procedure yields estimates for the critical dimensions that are consistent with result from alternative measurement method. In the last part an example for a Bayesian approach to solve the inverse problem of scatterometry is given. In contrast to LSQ and MLE, the solution to the inverse problem in Bayesian terms is not a single estimate for the parameters of interest but rather their corresponding probability distribution. An advantage of the Bayesian approach is that information about the critical dimensions obtained by alternative methods can be incorporated as prior knowledge. It is demonstrated that several measurement methods can be combined. As a result the uncertainties for the critical dimensions can be drastically reduced. |
