Stress/Strain Analysis using CBED
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CBED patterns present the deficiency lines in the zero-order disc, which result from excitation of reflections in higher order Laue zones (HOLZ) and are called HOLZ lines. The positions of the HOLZ lines are related to the symmetry and lattice parameters of the crystal. They are changed in the direction of the applied stress if the material is stressed and those changes induce a change in the Bragg condition and thus a shift of the HOLZ lines. By analyzing their positions in combination with a reference CBED patern, a quantitative analysis of the three-dimensional (3-D) strain state of the crystal can be performed. However, it is important to note that HOLZ reflections are sensitive to the changes of lattice parameters associated with the direction perpendicular to the electron incidence, but they are insensitive to the changes of lattice parameters associated with the electron incidence.

As an example, Figure 3871a shows a typical HOLZ pattern from Si (silicon) with [230] electron incidence at an acceleration voltage of 200 kV. Si [230] HOLZ patterns are often used in the strain measurements. The dark lines are the HOLZ lines corresponding to the HOLZ reflection of each indicated index. The intersections, formed by the HOLZ lines, crossing at acute angles (e.g. indicated by the black arrow heads) would improve the measurement accuracy since these positions are sensitive to the variations in HOLZ line positions and thus to the variations in the strain components. The intersections marked by the white circles were used in strain measurement by Toda et al. [1] From the changes in the distance of HOLZ line intersections, the lattice strain was measured.

Si [230] HOLZ pattern

Figure 3871a. Si [230] HOLZ pattern obtained at an accelerating voltage of 200 kV. [1]

CBED technique probably provides the highest measurement accuracy in low strain region among all TEM-based strain measurement methods. However, the TEM samples need to be tilted far away from low index zone axis which results in multiple layers (e.g. in semiconducting device) overlapping, and HOLZ line splitting due to large strain gradient affects the measurement accuracy.

A CBED method of strain measurement is to use the kinematical approximation to simulate HOLZ patterns, and then the strain is obtained by matching the simulated and experimental HOLZ lines based on χ2 procedure. In this method, the dynamic effect of electrons is treated by a constant shift in the momentum dispersion surface of the incident electrons. Note that this approximation is only valid in some off zone axis orientations where the dispersion surface is relatively flat. The basic procedure of this strain-evaluation method is:
        i) Determine the effective acceleration voltage and the effect of the dispersion surface [2].
        ii) Determine the camera length using an HOLZ pattern from an unstrained specimen.
        iii) Based on i) and ii), measure the strain components (e.g. the normal strain in the x-direction εxx, the normal strain in the y-direction εyy, and share strain εxy) using HOLZ patterns from strained specimen.

Note that strain analysis can also be done with CBED patterns in other crystalline orientations such as <340> from crystalline Si [3].





[1] Akio Toda, Nobuyuki Ikarashi, Haruhiko Ono, Local lattice strain measurements in semiconductor devices by using convergent-beam electron diffraction, Journal of Crystal Growth 210 (2000) 341-345.
[2] J. M. Zuo, Ultramicroscopy 41 (1992) 211.
[3] M.J. Hÿtch, F. Houdellier, A. Claverie, and L. Clément, Comparison of CBED and Dark-field Holography for Strain Mapping in Nanostructures and Devices, 2009. ESSDERC '09. Proceedings of the European Solid State Device Research Conference, (2009) 307 - 310.




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