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

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 sub-angstrom electron beams, excitation of anti-symmetric 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 inter-column spacing of 1.28(+-.09) Angstroms, over 64% larger than the actual 0.78 Angstrom 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.

💡 Research Summary

The paper addresses the long‑standing challenge of interpreting high‑energy electron propagation in crystals, a problem that is generally treated as a complex multiple‑scattering phenomenon. By focusing on high‑symmetry zone axes, the authors demonstrate that the electron‑channeling problem can be rigorously mapped onto the time evolution of a two‑dimensional (2D) molecular system. Each projected atomic column is approximated by the potential of a screened 2D hydrogenic atom, supporting a set of bound states that act as channeling eigenstates. When two columns lie in close proximity, the bound‑state wavefunctions overlap, forming symmetric (bonding) and antisymmetric (antibonding) combinations that are direct analogues of molecular orbitals.

A key insight is that for sub‑angstrom electron probes—i.e., probe diameters smaller than 1 Å—the excitation probability of the antisymmetric (antibonding) orbital becomes non‑negligible. This excitation dramatically alters the intensity distribution of the transmitted beam, violating the conventional incoherent imaging approximation in which the recorded intensity is simply the sum of contributions from individual columns. Consequently, the familiar “dumbbell” test for resolution, which assumes a one‑to‑one correspondence between projected atomic positions and intensity peaks, can fail dramatically at realistic sample thicknesses.

The authors validate the theory experimentally using high‑angle annular dark‑field scanning transmission electron microscopy (HAADF‑STEM) on silicon crystals oriented along the