Enhanced Measurements of Displacements and Strains in Quasiperiodic Nanostructures
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Enhanced Measurements of Displacements and Strains in Quasiperiodic Nanostructures Maciej Wielgus1,2, Daniel Koguciuk3, Zofia Sunderland1, Krzysztof Patorski1 and Anna Piotrowska2 1 Institute of Micromechanics and Photonics, Warsaw University of Technology, ul. Św. A. Boboli 8, 02-525 Warszawa, Poland 2 Institute of Electron Technology, Al. Lotników 32/46, 02-668 Warszawa, Poland 3 Faculty of Mechatronics, Warsaw University of Technology, ul. Św. A. Boboli 8, 02-525 Warszawa, Poland ABSTRACT Displacements and strains can be calculated from the microscopic image of a quasiperiodic structure by the analysis of its spectral content, consisting of a discrete set of peaks (Bragg spots in the case of the crystal structure). Typically one would choose certain peak and evaluate the displacements by investigating its neighborhood. However, there is a large amount of redundancy in such an image, as similar measurement can be performed by choosing a different Bragg spot. We demonstrate an approach which in a systematic manner employs information from multiple Bragg spots for the displacement evaluation. This has a positive influence on the quality and robustness of the measurements. INTRODUCTION Measurements of displacements and strains are of great importance for the characterization of the electronic materials, as they may inform about quality of the sample and its mechanical or electrical properties. Displacements can be measured by analyzing the spectral content of the image of the sample crystalline structure (obtained with, e.g., transmission electron microscope), strains can be calculated knowing the displacements. If the displacements of periodic crystal lattice are considered, a discrete set of peaks is present in the Fourier space (Bragg spots). Displacements (or, more precisely, local departures from the average reciprocal lattice) are coded by values of the spectral components in the close neighborhood of each peak and can be decoded with signal processing techniques such as the geometric phase method [1], typically using a single peak. However, depending on their order, peaks possess different properties, as signal to noise ratio is diminishing with the peak order. On the other hand, peaks of the higher order may code more delicate variations of phase, allowing for more sensitive measurements of displacements and strains [2]. This is because of the so-called phase multiplication phenomenon, well known in optical interferometry [3], but similarly present in measurements based on the electron microscopy [4] or the scanning probe microscopy. In this paper we follow a simple idea to utilize multiple Bragg spots for the displacements calculation and develop a system that performs it in a systematic manner with the algorithm which we refer to as Multiple Phase Fourier (MPF) approach. ALGORITHM DESCRIPTION The Multiple Phase Fourier algorithm can be outlined in the following steps:
1. 2. 3. 4. 5.
Localizing the Bragg spots in the Fourier spectrum, Subpixel refining of the Bragg spots maxima location, Phase decoding fr
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