Crystal thickness and extinction distance determination using energy filtered CBED pattern intensity measurement and dynamical diffraction theory fitting

A new method for measuring thickness and extinction distance of single crystals based on computed adjustment of measured and calculated CBED pattern intensity profiles is presented and discussed. The experimental beam intensity distribution is measured from an energy filtered CBED pattern recorded on a CCD camera. The calculated profile is based on dynamical diffraction theory, and with the two-beam approximation the analytical expression contains only two free parameters: specimen thickness t and extinction distance ξg. Parameter refinement through minimization of the difference between experimental and calculated intensity profiles is carried out using Origin™ 5.0 software from Microcal. The iterative procedure always converges to a unique solution in a few seconds, yielding an accurate value for both thickness and extinction distance. The method is extensively tested on silicon using the (0 0 4) Bragg reflection. On specimens in the usual TEM thickness range, the method gives result similar to the conventional (P.M. Kelly et al., Phys. Stat. Sol. A31 (1975) 771; S.M. Allen, Philos. Mag. A 43 (1981) 325) graphical methods, both based on the measurement of fringe spacing. Moreover, it is shown that the calculation matches perfectly both the positions of the minimums and maximums as well as the amplitude of maximums. For any single intensity profile, specimen thickness and extinction distance can be determined with a precision of about 0.2%. A statistical comparison of our method with the Kelly and Allen techniques, based on more than 50 experiments, shows an improvement in measured extinction distance dispersion. Using 197 keV electrons, and liquid-nitrogen cryo-holder, the new technique yields an experimental value of 161±3 nm for the extinction distance for silicon with the (0 0 4) Bragg reflection. The equivalent tabulated value at 0 K is about 156 nm. Using the Kelly and Allen methods, the extinction distance is found to be 162±6 nm. The improvement in precision is a direct consequence of matching the intensity profile envelope, which contains information on the extinction distance. Also the accuracy of thickness determination is improved and is around 0.5 to 1% for common specimen thickness. The minimum measurable sample thickness is shown to be two to three times thinner than with the Kelly and Allen methods (0.3ξg as opposed to 0.8ξg). With no independent calculation of the extinction distance needed, the method is also applicable on unknown crystals. The method is fast, simple and can be easily automated.