Semianalytical treatment of beam broadening in crystals

Abstract

Contrast and specimen resolution in electron micrographs of point defects, voids, aggregates, and precipitates in crystalline material are diminished by multiple scattering of the incident electrons within the crystalline top layer. For the scanning transmission electron microscope (STEM) the current density distribution inside the crystal has been calculated by employing the k · p perturbation expansion used in solid state physics. Neglecting inelastic scattering the solution of the electron wave function within the crystal can be expressed as a sum of Bloch waves. The excited Bloch waves are expanded in a power series of the angle of incidence, and only terms up to the second order inclusively are taken into account. This procedure permits the analytical integration over the illumination angles in STEM or the aperture angles in the case of a fixed-beam electron microscope (FBEM). Considering only the two strongest Bloch waves an approximate formula is obtained for the current density distribution within the crystal which is valid for light atoms and resolution limits larger than twice the lattice constant. In the case of heavy atoms the approximation for the current density is only valid in the region near the atoms. As the incident electrons channel along the atom rows beam broadening within the crystal is largely suppressed up to depths of about 1000 Å. The image contrast of an embedded scatterer resting on an atom row is strongly enhanced, whereas it is diminished when the scatterer is located midway among the atom rows. The proposed method makes possible a fast calculation of the electron wave propagating in crystalline material.