Parallel mode differential phase contrast in transmission electron microscopy, II: K$_2$CuF$_4$ phase transition

TL;DR

This paper demonstrates the application of parallel mode differential phase contrast in transmission electron microscopy to study phase transitions in a 2D ferromagnetic material, providing insights into critical phenomena and phase transition characteristics.

Contribution

It extends the TEM-DPC technique to experimental cryo-TEM data and compares its effectiveness with the transport of intensity method for phase recovery.

Findings

TEM-DPC outperforms transport of intensity in experimental data

Estimated critical temperature and exponent suggest a Kosterlitz--Thoules transition

Evidence supports the 2-D XY universality class for the phase transition

Abstract

In Part I of this diptych, we outlined the theory and an analysis methodology for quantitative phase recovery from real-space distortions of Fresnel images acquired in the parallel mode of transmission electron microscopy (TEM). In that work, the properties of the method, termed TEM-differential phase contrast (TEM-DPC), were highlighted through the use of simulated data. In this work, we explore the use of the TEM-DPC technique with experimental cryo-TEM images of a thin lamella of a low temperature two-dimensional (2-D) ferromagnetic material, K$_2$CuF$_4$, to perform two tasks. First, using images recorded below the ordering temperature, we compare the TEM-DPC method to the transport of intensity one for phase recovery, and discuss the relative advantages the former has for experimental data. Second, by tracking the induction of the sample as it is driven through a phase transition by heating, we extract estimates for the critical temperature and critical exponent of the order parameter. The value of the latter is consistent with the 2-D XY class, raising the prospect that a Kosterlitz--Thoules transition may have occurred.