Using Aberrations to Improve Dose-Efficient Tilt-corrected 4D-STEM Imaging

TL;DR

This work shows that aberration-corrected bright-field imaging (acBF) in 4D-STEM can maximize dose-efficient information transfer by correcting per-off-axis CTFs before summation, effectively turning a tilt-series into a continuous, interpretable focal-like dataset. By incorporating both symmetric (tcBF) and antisymmetric (tcDPC) scattering contributions, acBF delivers a continuously nonzero contrast transfer up to the information limit, even in the presence of non-defocus aberrations such as spherical aberration and astigmatism. The authors derive and validate the theory with simulations (single atoms, MOFs) and experimental data (twisted WSe$_2$ bilayers), demonstrating that acBF outperforms tcBF and tcDPC and approaches the performance of STEM phase-plate-based methods without requiring additional hardware. The approach offers a robust, non-iterative path to high-resolution, dose-efficient phase-contrast imaging in challenging aberration regimes, with clear guidance on limitations and practical shift-estimation strategies.

Abstract

Tilt-corrected imaging methods in four-dimensional scanning transmission electron microscopy (4D-STEM) have recently emerged as a new class of direct ptychography methods that are especially useful at low dose. The operation of tilt correction unfolds the contrast transfer functions (CTF) of the virtual bright-field images and retains coherence by correcting defocus-induced spatial shifts. By performing summation or subtraction of the tilt-corrected images, the real or imaginary parts of the complex phase-contrast transfer functions are recovered, producing a tilt-corrected bright field image (tcBF) or a differential phase contrast image (tcDPC). However, the CTF can be strongly damped by the introduction of higher-order aberrations than defocus. In this paper, we show how aberration-corrected bright-field imaging (acBF), which combines tcBF and tcDPC, enables continuously-nonzero contrast transfer within the information limit, even in the presence of higher-order aberrations. At Scherzer defocus in a spherically-aberration-limited system, the resultant phase shift from the probe-forming lens acts as a phase plate, removing oscillations from the acBF CTF. We demonstrate acBF on both simulated and experimental data, showing it produces superior performance to tcBF or DPC methods alone, and discuss its limitations.

Figures (13)

• Figure 1: (a) Phase contrast transfer function (PCTF) and (b) Detective quantum efficiency (DQE) of tcBF, tcDPC and acBF at 300 kV, 30 mrad convergence semi-angle. The tcDPC curve in (a) shows the imaginary part of its PCTF. The tcDPC envelope is equal to the PCTF of in-focus tcDPC. acBF fills in the difference between tcBF and tcDPC, which yields constantly non-zero contrast transfer.
• Figure 2: Phase contrast transfer functions (PCTF) at 300 kV, 30 mrad convergence semi-angle in the case of (a,b) $C_s=50 \ \mu$m with Scherzer defocus and (c,d) two-fold astigmatism $C_{12}$ of 10 nm defocus of 10 nm. (b) and (d) are line profiles of the 2D PCTFs in (a) and (c), respectively. In the case of spherical aberration, the PCTF of tcBF is significantly damped due the summation over images with different effective defocus, while acBF corrects for this variation and yields a smooth PCTF. In the case of equal defocus and 2-fold astigmatism, along one direction the defocus effectively doubles, while along the other direction the defocus is zero. tcBF has no contrast along the zero-defocus axis, while acBF recovers the antisymmetric scattering only. Astigmatism of the probe leads to irrecoverable information loss in the dataset.
• Figure 3: Imaging of a simulated single C atom at infinite dose, 300 keV and 30 mrad convergence semi-angle with $C_s = 50\ \mu$m and Scherzer defocus (121 Å) using different methods: (a) axial BF, (b) tcBF, (c) tcBF with post-summation CTF correction (sign flipping), (d) acBF. (e–h) Corresponding fast Fourier transform (FFT) power spectra of (a–d). The dashed circles on the FFTs indicate the spatial frequencies corresponding to 1$\alpha$ and 2$\alpha$, and line traces through the center of each plot are overlaid. The tcBF images both show worse resolution and more substantial tails than the axial image due to the summation over images with different defocus and astigmatism. acBF has a sharp response and no negative tails which would cause contrast delocalization, and smooth transfer of information at all frequencies up to 2$\alpha$.
• Figure 4: Imaging of a simulated 3-nm-thick MOF NU-1000 sample at 200 e-/$\text{Å}^2$, 300 kV, 10 mrad semiconvergence angle, 1 Å scan step size, $C_{s}$ = 2.7 mm, and Scherzer defocus (892 Å) to show the impact of spherical aberration using (a) axial BF, (b) tcBF with post-summation CTF correction (sign flipping), (c) acBF. TcBF and acBF have been upsampled by a factor of 4. (d–f) Corresponding fast Fourier transform (FFT) power spectra of (a–c). The dashed circles on the FFTs indicate the spatial frequencies corresponding to 1$\alpha$ and 2$\alpha$. The atomic structure schematic is displayed on the left with color denoting Zr (gold/green), O (red), C (brown) and H (pink). Within each Zr cluster, the Zr pairs with strong and weak projected electrostatic potential intensity are labeled in gold and green, respectively. The structure is barely resolvable with tcBF, while acBF resolves the structure clearly and locates the Zr atom pairs in the nodes.
• Figure 5: Reconstructed images of an experimental 4D-STEM dataset (adapted from zhang2025atom) of a twisted tungsten diselenide (WSe$_2$) bilayer acquired at 80 kV and 25 mrad semi-convergence angle with an aberration-corrected instrument, using: (a) axial BF, (b) tcBF, (c) tcBF with post-summation CTF correction (sign flipping), (d) acBF, along with their respective Fourier transforms (f-h). TcBF and acBF have been upsampled by a factor of 2. The iterative, multislice ptychography reconstruction performed by zhang2025atom is shown in (e) with its FFT in (j) for comparison to indicate the true atomic positions. The yellow arrow highlights a region where two W atoms closely spaced in projection are resolved by acBF but not tcBF. The dashed circles on the FFTs indicate the spatial frequencies corresponding to 1$\alpha$ and 2$\alpha$, and the tilt corrected images show an information limit of better than 1 Å.