Symmetries in TEM imaging of crystals with strain

TL;DR

This paper investigates the origins of symmetries in TEM images of strained crystals, distinguishing between imaging process symmetries and those from the crystal's strain profile, supported by mathematical proofs and examples.

Contribution

It provides a mathematical framework to explain symmetries in TEM images based on invariances of the imaging process and strain properties, enhancing understanding of image interpretation.

Findings

Mathematically proven invariance of intensities under specific transformations.

Symmetries explained by combining invariances with strain profile properties.

Application demonstrated on semiconductor nanostructures like quantum wells and dots.

Abstract

TEM images of strained crystals often exhibit symmetries, the source of which is not always clear. To understand these symmetries we distinguish between symmetries that occur from the imaging process itself and symmetries of the inclusion that might affect the image. For the imaging process we prove mathematically that the intensities are invariant under specific transformations. A combination of these invariances with specific properties of the strain profile can then explain symmetries observed in TEM images. We demonstrate our approach to the study of symmetries in TEM images using selected examples in the field of semiconductor nanostructures such as quantum wells and quantum dots.