Electron time correlation microscopy and the stability of the numerical evaluation

Abstract text (incl. figure legends and references)

The high spatial and temporal resolution of electron microscopy opens new ways of characterizing matter. Electron Correlation Microscopy (ECM) is a comparably new method analyzing time-intensity correlations in thermodynamic equilibrium [1,2]. For amorphous materials, transmission electron microscopy (TEM) dark-field (DF) images show speckle contrast arising by local structural arrangements. ECM measures basically a speckle lifetime. In the present work, the ECM data evaluation and the resulting accuracy are discussed. Measuring ECM out of thermodynamic equilibrium at room temperature (RT) was shown recently for a metallic glass [3,4]. ECM measurements out of thermodynamic equilibrium are here denoted Electron time Correlation Microscopy (EtCM). This brings additional evaluation criteria, visualized by calculating two-time correlation functions.

A schematic of EtCM is shown in Figure 1. An amorphous sample is imaged in TEM-DF mode to detect structural arrangements fulfilling the chosen DF condition leading to speckle contrast. Typically about 50 nm in thickness samples and a beam current density of about 105 e-/nm²s at exposure times of seconds are applied. From the image time series, a correlation function (g2(t)) is calculated, which is fitted to a stretched exponential function ( A*[1-exp(-t/τ)β], see Figure 1 top-right). A β=1 leads overall to a single exponential function, β<1 to a stretched, and β>1 to a compressed exponential function. The β therefore reflects the energy landscape of the underlying processes. Depending on the material, the correlation time at RT can be extremely large (> 10 k seconds). Consequently, the corresponding data quality is limited due to the limited experimental accessible time. One approach is to extrapolate τ and β to infinitely long measurement times [3] by using e.g. a*[1-exp(-x/b)], where a is the value of τ or β for time to infinity, b is a parameter reflecting how fast the data analysis comes to a numerical steady state and x denotes the experimentally determined τ or β for different, finite times. It should be noted, that τ and β also depend on the DF condition as the beam tilt selects the scattering vector magnitude.

In Figure 2, an example of β and τ as a function of the evaluated stack length is shown. Evaluating short measurement time leads to numerically unstable values. Evaluating longer times leads to more stable values, underestimating the real values. The dependence of β and τ on the evaluated time are different. However, the data analysis using different fractions of a DF-time series is a robust way to gain insights into the stability of the fitting parameters β and τ. Moreover, by extrapolating β and τ to timeàinfinity allows to estimate the real values (at a given beam current), or at least to estimate a proper measurement time for an EtCM experiment. Based on literature and numerical tests, the τ and β values are reviewed and discussed.

[1] L. He at al., Microsc. and Microanal. 21 (2015) 1026-1033.

[2] P. Zhang et al., Nature Communications (2018) 9:1129.

[3] K. Spangenberg et al., Adv. Funct. Mater. 31 (2021) 2103742.

[4] M. Stringe et at., J. Alloys Compd. 915 (2022) 165386.

Fig. 1: Schematic EtCM measurement: (left) data acquisition and (right) data analysis including a two-time correlation plot to confirm a quasi-equilibrium condition.

Fig. 2: Streching exponent β and correlation time τ obtained by fitting a (fraction of a) dataset.