Quantitative Agreement between Electron-Optical Phase Images of ${\mathrm{WSe}}_2$ and Simulations Based on Electrostatic Potentials that Include Bonding Effects

The quantitative analysis of electron-optical phase images recorded using off-axis electron holography often relies on the use of computer simulations of electron propagation through a sample. However, simulations that make use of the independent atom approximation are known to overestimate experimental phase shifts by approximately 10%, as they neglect bonding effects. Here, we compare experimental and simulated phase images for few-layer ${\mathrm{WSe}}_2$. We show that a combination of pseudopotentials and all-electron density functional theory calculations can be used to obtain accurate mean electron phases, as well as improved atomic-resolution spatial distribution of the electron phase. The comparison demonstrates a perfect contrast match between experimental and simulated atomic-resolution phase images for a sample of precisely known thickness. The low computational cost of this approach makes it suitable for the analysis of large electronic systems, including defects, substitutional atoms, and material interfaces.

Figure 1

${\mathrm{WSe}}_2$ lattice geometry. (a) Top view with the red rhombus marking the unit cell used in DFT calculations and the blue rectangle marking the cell used in multislice simulations of electron-optical phase images. (b) Side view of a unit cell, which comprises a stack of two monolayers.

Figure 2

Integrated plane-averaged electrostatic potential per layer (left axis) and mean inner potential (right axis) for few-layer ${\mathrm{WSe}}_2$ calculated using three different methods.

Figure 3

(a) Change in spatial electron density and (b) change in electrostatic potential in a ${\mathrm{WSe}}_2$ monolayer unit cell between the DFT and IDFT methods with $n_0$ and $V_0$ the bulk average electron density and the bulk mean inner potential of ${\mathrm{WSe}}_2$, respectively [29]. Negative values denote lower electron densities and lower electrostatic potentials in the DFT method.

Figure 4

Comparison between simulated and experimental phase images of a ${\mathrm{WSe}}_2$ bilayer structure, shown for a region corresponding to the area marked by a blue rectangle in Fig. 1. Panel (a) shows an averaged experimental phase image of a ${\mathrm{WSe}}_2$ bilayer [11], alongside a calculated phase image obtained with the DFT method. Panel (b) shows differences between phase images obtained with the three different methods and the experimental phase image. Positive values correspond to higher phases in the experimental phase image. The markings on the color bar represent the average difference for each method.