# Mathematical methods for non-destructive measurements of nano-structures

Categories:
• Metrology for Economy
18.03.2010

The breathtaking development of nanotechnology requires the design and control of components with smaller and smaller details. In particular, the fabrication of up-to-date computer chips and storage devices by lithographic manufacturing processes relies on new accurate measurement techniques suitable to resolve nanoscale features of masks and wafers. In answer to these challenges PTB is developping new methods to determine the critical dimensions of nanostructures, i.e. objects with dimensions smaller than 1/1000 millimeter. These methods require elaborate mathematical modelling and analysis. Scatterometry is a new non-destructive technique for characterizing the dimensions of nanostructures: The sample is illuminated by well prepared laser light and the complicated scattered wave pattern is measured. Instead of a direct image of the probe, an angle-resolved distribution of scattered light is measured. The geometrical dimensions of the probe need to be reconstructed by solving an inverse problem for the electromagnetic wave equations. The integrated development of measurement instruments and algorithm for measurement data processing was achieved in a cooperation of three PTB working groups (Modelling and simulation, Ultra-high resolution microscopy, EUV radiometry) with the Weierstrass Institute for Applied Analysis and Stochastics. It is also part of the project CDuR32 funded by the German Federal Ministry of Education and Research.

Microscopy is the classical method for imaging and measuring small structures. Optical imaging methods are non-destructive and fast. But their spactial resolution is diffraction limited, i.e., they provide details only up to the size of the wavelength and no smaller. To overcome these limits, new non-imaging methods like scatterometry can be applied. An important application of scatterometric metrology is the evaluation of structure dimensions on photo-masks and wafers in lithography, where the geometrical parameters of periodic line space structures are to be determined precisely. In scatterometry, a polarized light ray illuminates the sample under various angles of incidence and with different wavelengths. The light is scattered by the sample, and the outgoing wave is measured in various directions of radiation (cf. Fig. 1). In PTB two different scatterometers were developed, a EUV reflectometer [1], operating in the wavelength range around 13 nm, and DUV scatterometer working at 193 nm [2]. The conversion of measurement data into the desired geometrical parameters depends crucially on a rigorous modeling by Maxwell's equations and on accurate related numerical algorithms [3,4]. The solution of this inverse problem is incomplete without the knowledge of the uncertainties associated with the reconstructed geometrical parameters. Mathematical modeling and elaborate numerical simulations are essential to evaluate the precision and the accuracy of scatterometrical results [4,5] and to define optimal measurement configurations, per example the preferred incident angle or the number of necessary measurement data for an accurate reconstruction [6]. In particular, it was found that geometrical extensions in the range of 50 - 500 nm are determined with relative uncertainties smaller than 2 % for wide range of experimental conditions [4,5] .

Fig. 1: (left) Scheme for scatterometrical suface characterizing. Typical probes are photomasks; results for a profile reconstruction (rigth): Comparison of simulated and measured diffraction orders for a wave length of 13.6 nm, calculated with the reconstructed profile parameters of the measured probe.

[1] C. Laubis, et al. (2006): Characterization of large off-axis EUV mirrors with high accuracy reflectometry at PTB, Proc. SPIE 6151, 61510I.

[2] M. Wurm, B. Bodermann; F. Pilarski (2007): Metrology capabilities and performance of the new DUV scatterometer of the PTB Proc. SPIE 6533, 65330H.

[3] R. Model, A. Rathsfeld, H. Groß, M. Wurm, B. Bodermann (2008): A scatterometry inverse problem in optical mask technology. J. Phys., 135, 012071.

[4] H. Gross, A. Rathsfeld, F. Scholze, M. Bär (2009): Profile reconstruction in EUV scatterometry: Modeling and uncertainty estimates. WIAS Preprint No. 1411(http://www.wias-berlin.de/main/publications/wias-publ/).

[5] H. Groß, A. Rathsfeld, F. Scholze, R. Model, M. Bär (2008): Computational methods estimating uncertainties for profile reconstruction in scatterometry. Proc. SPIE 6995, 6995OT.

[6] H. Groß, A. Rathsfeld (2008): Sensitivity Analysis for Indirect Measurement in Scatterometry and the Reconstruction of Periodic Grating Structures. Waves in Random and Complex Media, 18, 129.

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