Using Four-Dimensional Scanning Transmission Electron Microscopy to Determine the Strain Profiles and Thermal Expansion Coefficient of Polycrystalline Epitaxial WSe₂

Thermal transport properties can be greatly impacted by the presence of grain boundaries and interfaces in two-dimensional (2D) materials. Previous techniques for determining the thermal expansion coefficient of 2D materials lacked the necessary spatial resolution in temperature measurements. We present a nanoscale approach to overcome this obstacle using Four-Dimensional Scanning Transmission Electron Microscopy (4D-STEM) and analysis by local statistical quantification of crystallographic parameters to determine the thermal expansion coefficient for epitaxial WSe₂ (2D material). From this approach, we determined the localized structural parameters as a function of temperature with high precision by analyzing the histogram of peak positions in a virtual peak profile. Experimentally, WSe₂ was heated through a temperature range (18^oC to 564^oC) using 4D-STEM to observe temperature-induced structural changes to WSe₂ which can be performed without additional alterations or destruction to the sample. From this analysis methodology, we determined lattice parameters with sub-picometer accuracy, microstrain, and coherent crystalline domains, (related to the sample thickness). The robustness of combining 4D-STEM measurements with quantitative structural analysis for 2D materials is illustrated by the direct determination of thermal expansion coefficients for epitaxial WSe₂.