Mapping 2D strain components from STEM moiré fringes

Artificial moirés are created in a STEM by deliberately choosing a low magnification where the scan step is close to the crystalline periodicity (see Figure 1a) [1]. A moiré contrast then results from the interference between the scan and the crystal lattice. The technique has been developed to analyse strain [2], and has been applied to the study of strained‐silicon devices [3]. In reciprocal space, STEM moiré fringes can be understood as the convolution of the lattice created by the scan, characterized by the scan‐step, s in real‐space, or s * in reciprocal space, and the reciprocal lattice of the crystalline lattice, characterized by d * (Figure 1b). Interference between neighbouring periodicities gives rise to moiré fringes of periodicity q M . In general, only a few moiré fringe periodicities will be present in the image as the scan reciprocal lattice is in reality multiplied by the MTF of the probe: the size of the probe limits the possible interference terms (indicated by the yellow area in Figure 1b). The moiré peridocity q M is related vectorially to s * , which is usually 1 or 2 pixel −1 , and the d * (or g‐vector) for a particular set of lattice fringes (Figure 1c). Here we show how the strain information can be extracted using the concept of geometric phase, previously used for the analysis of high‐resolution TEM images [4]. The advantage of this formulation is that the moiré fringes do not need to be aligned exactly with the crystalline lattice, thus freeing up the experimental work. The periodicity of the moiré fringes (Figure 2a) is identified from the power spectrum of the image and the corresponding geometric phase determined (Figure 2b). The strain is in turn calculated from the geometric phase (Figure 2c). It is not necessary to know the exact calibration of the original image providing a reference region of known crystal parameter is present, as the vectorial relationship (Figure 1c) provides a strong constraint. We have also developed a procedure to determine maps of the 2D strain tensor from two differently oriented SMFs [5] (Figure 3). The results from two separate moiré fringe images need to be aligned (Figure 3b) and combined to determine the full 2D in‐plane strain tensor. Whilst the examples here are simulated, we anticipate presenting experimental results from devices and piezo‐electric materials.