Quantitative measurement of mean inner potential and specimen thickness from high-resolution off-axis electron holograms of ultra-thin layered WSe₂

The phase and amplitude of the electron wavefunction that has passed through ultra-thin flakes of WSe₂ is measured from high-resolution off-axis electron holograms. Both the experimental measurements and corresponding computer simulations are used to show that, as a result of dynamical diffraction, the spatially averaged phase does not increase linearly with specimen thickness close to an [001] zone axis orientation even when the specimen has a thickness of only a few layers. It is then not possible to infer the local specimen thickness of the WSe₂ from either the phase or the amplitude alone. Instead, we show that the combined analysis of phase and amplitude from experimental measurements and simulations allows an accurate determination of the local specimen thickness. The relationship between phase and projected potential is shown to be approximately linear for extremely thin specimens that are tilted by several degrees in certain directions from the [001] zone axis. A knowledge of the specimen thickness then allows the electrostatic potential to be determined from the measured phase. By using this combined approach, we determine a value for the mean inner potential of WSe₂ of 18.9±0.8V, which is 12% lower than the value calculated from neutral atom scattering factors.