Channeling of a subangstrom electron beam in a crystal mapped to two-dimensional molecular orbitals

Abstract

The propagation of high-energy electrons in crystals is in general a complicated multiple-scattering problem. However, along high-symmetry zone axes the problem can be mapped to the time evolution of a two-dimensional (2D) molecular system. Each projected atomic column can be approximated by the potential of a 2D screened hydrogenic atom. When two columns are in close proximity, their bound states overlap and form analogs to molecular orbitals. For subangstrom electron beams, excitation of antisymmetric orbitals can result in the failure of the simple incoherent imaging approximation. As a result, the standard resolution test and the one-to-one correspondence of atomic positions of a crystal imaged along a zone axis with closely spaced projected columns (``dumbbells'') can fail dramatically at finite and realistic sample thicknesses. This is demonstrated experimentally in high-angle annular dark-field scanning transmission electron microscope (HAADF STEM) images of [211]-oriented Si showing an apparent intercolumn spacing of 1.28(\ifmmode\pm\else\textpm\fi{}0.09) \AA{}, over 64$%$ larger than the actual 0.78 \AA{} spacing. Furthermore, the apparent spacing can be tuned with sample thickness and probe size to produce a larger, smaller, or even the actual spacing under conditions when the peaks of two adjacent Si columns should not even have been resolved given the electron probe size.