Autocorrected off-axis holography of two-dimensional materials

The reduced dimensionality in two-dimensional materials leads to a wealth of unusual properties, which are currently explored for both fundamental and applied sciences. In order to study the crystal structure, edge states, the formation of defects and grain boundaries, or the impact of adsorbates, high-resolution microscopy techniques are indispensable. Here we report on the development of an electron holography (EH) transmission electron microscopy (TEM) technique, which facilitates high spatial resolution by an automatic correction of geometric aberrations. Distinguished features of EH beyond conventional TEM imaging are gap-free spatial information signal transfer and higher dose efficiency for certain spatial frequency bands as well as direct access to the projected electrostatic potential of the two-dimensional material. We demonstrate these features with the example of $h$-BN, for which we measure the electrostatic potential as a function of layer number down to the monolayer limit and obtain evidence for a systematic increase of the potential at the zig-zag edges.

Figure 1

(a) Off-axis electron holography (EH) setup. (b) Signal transfer for imaging a weak phase object (WPO) and signal-to-noise (SNR) transfer functions for HRTEM in negative $C_{\mathrm{s}}$ conditions ($-C_{\mathrm{s}}$) and EH [see Eq. (9)]. In case of HRTEM, only the signal transfer from object phase to image amplitude described by the phase contrast transfer function (PCTF) contributes to the image. In case of EH, also the signal transfer from object phase to image phase described by the amplitude contrast transfer function (ACTF) contributes to the image. Additionally, the corresponding transferred bands for HRTEM in $-C_{\mathrm{s}}$ conditions with ± 2-nm defocus variation are also plotted. A defocus drift of about 2 nm is commonly observed after about 5 min in aberration-corrected TEM instruments [37].

Figure 2

Aberration correction of the $h$-BN image wave reconstructed by off-axis electron holography. (a), (b) As-reconstructed wave image in amplitude and phase. The symmetric aberration phase plate (c) as determined numerically from Eq. (11) displays all spatial frequencies, including those where noise is dominating because no object information is present. (d) The color-coded representation of panel (c) is overlayed with the Fourier spectrum of the image wave (a), (b) to highlight the meaningful part of the numerical phase plate. (e), (f) Amplitude and phase corrected for symmetric aberrations using the continuous phase plate (g) fitted from panel (c). (h), (i) Amplitude and phase corrected also for antisymmetric aberrations (coma and threefold astigmatism) obtained from Eq. (12) using the phase plate (j). The crystal structure of a $h$-BN monolayer is indicated in panel (i).

Figure 3

Denoising of the aberration-corrected $h$-BN phase image reconstructed by off-axis electron holography. (a) Original autocorrected data and corresponding standard deviations of phase ${\sigma }_{\phi }$ and intensity ${\sigma }_{\mathrm{I}}$ in vacuum. (b) PCA-denoised data after truncation to the first 11 principal components. The selection criterion is the last kink in the scree plot (e). Both the difference (d) and the histogram (c) reveal no noticeable deviation from Gaussian noise between the original and denoised image except at some structurally fluctuating edges.

Figure 4

Average potential analysis of $h$-BN. The overview image in panel (a) contains several flakes of different layer numbers, shaped by electron-beam irradiation. The potential data histogram in panel (b) reveals the presence of several monolayers, separated by well-defined potential offsets $\mathrm{\Delta }V$ (c). The linescans in panel (d) provide local measurements for the latter and reveal the presence of potential bumps at edges and steps.

Figure 5

(a) High-resolution potential analysis. Insets: Zig-zag steps comprising two atomic layers (b), (c), a monolayer region (d), and a BN void defect (e). The monolayer potential is depicted together with the DFT result in panel (f).

Figure 6

Normalized holographic centerband intensity from the same hologram as used for high-resolution potential analysis (Fig. 5).