Nanoscale deformation analysis of a crack-tip in silicon by geometric phase analysis and numerical moiré method
Introduction
Material fracture is an important concern for structural and functional materials. Proper investigation of the morphology and deformation fields of a crack-tip in a single crystal is critical to understanding the development of single-crystal failure. Rice [1] predicted a type of patchy plastic deformation around crack-tips in metallic single crystals, and this behavior was confirmed experimentally [2] and investigated numerically [3]. Currently available experimental [4], [5], analytical, and numerical investigations in literature provide some insight into the behavior of ductile single crystals in the presence of a crack or notch [6], [7], but fail to provide results that can be used to predict certain mechanical behaviors at the nanoscale. With a known dopant level and a sharp transition temperature, the single-crystal silicon remains a frequent choice of material for the observation of a single-crystal failure. Micro-crack in silicon could have cleavage that expands alternately along the (1 1 1) and (0 0 2¯) crystal planes, while the principal cleavage plane is in the (1 1 1) crystal plane [8].
The nano-moiré method was used to measure deformation at 0.1 nm [9], [10]. The scanning moiré method [11] was developed and applied to the deformation measurement at the nanoscale. The digital nano-moiré method [12] was proposed and the displacement measurement sensitivity can reach a pitch of the lattice of a single crystal, which is at the sub-nanometer level. Recently, high-resolution transmission electron microscopy (HRTEM) has become a powerful tool for mapping the deformation field at the nanoscale [13]. Geometric phase analysis (GPA) [14] and the numerical moiré (NM) method [15], two recently developed techniques with sub-angstrom sensitivity [16], [17], can detect small displacements of lattice fringes in HRTEM images. The GPA method has been applied to a wide variety of systems, such as Al/Si nanoclusters [18], nanoparticles [19], and dislocations [20]. In this study, a nanoscale experimental study of a micro-crack in silicon is analyzed through a combination of GPA, NM, and TEM.
Section snippets
Geometric phase analysis
An HRTEM image formed at a zone axis of a crystal can be considered as a set of interference fringes corresponding to the atomic planes of the specimen. All the information contained in an HRTEM image can be obtained by analyzing these few components of the image intensity. The displacement and strain can be obtained by the measurement of these atomic plane positions. An image of a perfect crystal can be described as a Fourier serieswhere I(r) is the intensity in the image at
Results and discussion
A TEM bright-field image of the examined micro-crack is depicted in Fig. 1a. The length of the micro-crack is about 1.4 μm. A crack-tip can be seen clearly in the front end of the micro-crack. The TEM image in the boxed area in Fig. 1a is shown in Fig. 1b. The observed orientation is [1 1 1]. The micro-crack is in the direction of [4 1¯ 3¯]. The HRTEM image of the boxed area in Fig. 1b is depicted in Fig. 2a. The observed area is 19.0 nm×19.0 nm. In this image, it can be observed that the crystal
Conclusions
A combination of GPA, NM, and TEM was used to study the deformation field of a crack-tip in a single-crystal silicon. The crystal lattice structure of the crack-tip is shown at an electron microscope image scale of approximately 19.0 nm×19.0 nm. The strain field maps of the crack-tip show that the deformation can only occur at the crack-tip area. The maxima of the strain components εxx, εyy, and εxy found at the crack-tip area can reach 1.47%, 2.91%, and 2.47%, respectively. Linear
Acknowledgements
This work was supported by the National Natural Science Foundation of China no. 10862002.
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