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ABSTRACT Both high resolution and high precision are required to quantitatively determine the atomic structure of complex nanostructured materials. However, for conventional imaging methods

in scanning transmission electron microscopy (STEM), atomic resolution with picometer precision cannot usually be achieved for weakly-scattering samples or radiation-sensitive materials,

such as 2D materials. Here, we demonstrate low-dose, sub-angstrom resolution imaging with picometer precision using mixed-state electron ptychography. We show that correctly accounting for

the partial coherence of the electron beam is a prerequisite for high-quality structural reconstructions due to the intrinsic partial coherence of the electron beam. The mixed-state

reconstruction gains importance especially when simultaneously pursuing high resolution, high precision and large field-of-view imaging. Compared with conventional atomic-resolution STEM

imaging techniques, the mixed-state ptychographic approach simultaneously provides a four-times-faster acquisition, with double the information limit at the same dose, or up to a fifty-fold

reduction in dose at the same resolution. SIMILAR CONTENT BEING VIEWED BY OTHERS SUB-NANOMETER DEPTH RESOLUTION AND SINGLE DOPANT VISUALIZATION ACHIEVED BY TILT-COUPLED MULTISLICE ELECTRON

PTYCHOGRAPHY Article Open access 31 January 2025 INFLUENCE OF PLASMON EXCITATIONS ON ATOMIC-RESOLUTION QUANTITATIVE 4D SCANNING TRANSMISSION ELECTRON MICROSCOPY Article Open access 21

October 2020 LOCAL-ORBITAL PTYCHOGRAPHY FOR ULTRAHIGH-RESOLUTION IMAGING Article 29 January 2024 INTRODUCTION Determining the local atomic arrangement of complex nanostructures can provide

fundamental insights into the properties of materials1,2. Compared to traditional metals and semiconductors, newer materials systems as metal-organic frameworks and organic perovskites are

more radiation sensitive3,4,5, requiring more dose-efficient imaging techniques in order to allow high-resolution imaging with comparable level of detail. Solving the structure of biological

macromolecules or small molecular at atomic level is even more challenging6. The main problem is that, as a consequence of Poisson statistics, the required illumination dose is inversely

proportional to the square of the spatial resolution7, and thus improving spatial resolution means quadratically higher doses. The increased dose may destroy the structure of the sample

before sufficient image signal-to-noise is reached. The widely adopted atomic-resolution imaging methods in scanning transmission electron microscopy (STEM), such as annular dark-field (ADF)

or coherent bright-field (cBF) imaging, are intrinsically dose inefficient as they use only a small fraction of the scattered electrons, being constructed via a simple integration of a

limited portion of phase space. Therefore, conventional STEM imaging methods usually cannot achieve sub-angstrom resolution or even atomic-resolution for electron radiation-sensitive

materials8. Meanwhile, high-precision measurement of local atomic positions is also fundamentally hindered by the poor signal-to-noise ratio of ADF images from electron-radiation sensitive

or weakly scattering samples, such as monolayer 2D materials. Picometer precision via ADF imaging can only be achieved in electron-radiation-robust and strongly scattering bulk

samples9,10,11. Electron ptychography, however, can potentially use the entire diffraction patterns either via a Wigner-distribution deconvolution (WDD)12,13 or iterative algorithms14,15 in

a way that can account for the probe damping effect and extract the electrostatic potential of the sample. Electron ptychography has been demonstrated as a promising phase-contrast imaging

technique with advantages such as high dose efficiency13,16, high resolution5,17,18, and high contrast5,13. In particular, ptychography has now surpassed the resolution of best physical

lenses, reaching deep sub-angstrom resolution5. Simulations have suggested the possibility of extremely low-dose imaging by electron ptychography, in principle beyond that of all other

electron imaging approaches to date including high-resolution TEM imaging widely used in Cryo-EM community—a potential outcome of considerable importance for the study of electron

radiation-sensitive materials including biological macromolecules. Imperfections of the STEM imaging system especially the partial coherence of the illumination probe reduces the image

resolution and contrast of conventional STEM imaging methods19. Partial coherence also limits the performance of ptychography, although in a more indirect manner, impacting the

signal-to-noise ratio of the reconstruction5,20. WDD-based electron ptychographic phase-contrast imaging has been demonstrated to outperform the commonly adopted phase-contrast conventional

high-resolution TEM imaging by retaining resolution beyond the temporal incoherent limit16,20. However, the partial coherence in the probe can be modeled more explicitly in iterative

ptychography algorithms by decomposing the probe wavefunction into a linear combination of pure states, i.e., a mixed quantum state21. As first demonstrated in coherent diffractive imaging22

and X-ray ptychography23, partial coherence of the probe can be well accounted for by introducing such state mixtures into the reconstruction method21,24,25. High coherent field-emission

guns are widely used as the electron sources in modern electron microscopes and so electron ptychography usually assumes only a pure coherent state of the illumination probe5,26,27. This

assumption is often sufficient when an in-focus-illumination probe is adopted5,27 or when a nanometer spatial resolution is targeted26. The effects of incoherence, which are inevitably

present in electron ptychography, on the reconstruction quality remains underexplored. There are only a few proof-of-principle demonstrations of electron ptychography considering the partial

coherence via approaches either Gaussian blind deconvolution18 or modal decomposition28,29, and while these suggest the promise of the approach, to date no sub-angstrom resolution

reconstructions have been achieved, even on instruments capable of sub-angstrom resolution in conventional imaging modes. Here, we demonstrate the capability of fast, sub-angstrom resolution

and picometer-precision imaging with a large field-of-view (FOV) by mixed-state electron ptychography. We find that a complete description of the partial coherence of the electron

wavefunction using a mixed quantum state is required to accomplish reconstructions with a sub-angstrom resolution, high contrast, high precision and a large FOV via defocused probe electron

ptychography. We also demonstrate low-dose atomic-resolution ptychographic imaging using dose levels up to 50 times lower than conventional STEM imaging techniques. RESULTS EXPERIMENTAL

SETUP OF ELECTRON PTYCHOGRAPHY Figure 1a shows a schematic workflow of the defocused electron ptychography. Instead of focusing the electron probe on the sample, the focal plane is set to a

distance away from the sample. Because the illumination area is broadened, a larger scan step can be used given that there are sufficient overlaps between adjacent scan positions required by

ptychographic reconstruction algorithms. The sample was raster scanned along two perpendicular directions as indicated in Fig. 1b. A high-dynamic-range electron microscope pixel-array

detector (EMPAD)30 used for diffraction pattern acquisition enables capture of both the shadow image31 in the center disk formed by the large defocused probe and the much-weaker high

scattering-angle dark-field signal, as illustrated by three simulated diffraction patterns in Fig. 1c. Similar setups with a defocused probe have been adopted previously17,18,26, which has

shown benefits for overcoming the slow readout speed of conventional CCD cameras and limited stability of the imaging systems. However, usually only a few dozen diffraction patterns were

acquired, whereas fast pixel-array detectors like the EMPAD30 enables the acquisition of more than 50,000 diffraction patterns per minute. Conventional iterative ptychographic

algorithms14,15 assume a coherent illumination. The diffraction pattern is considered as the square of the amplitude of a single probe state multiplied by a sample transmission function

within the multiplicative approximation (an extension of the strong phase approximation, allowing for both phase and amplitude variations)12. However, due to experimentally unavoidable

partial coherence of the imaging system, single pure-state coherent probe illumination is never achieved in real experiments. To account for the partial coherence, the illumination is

represented by several mutually incoherent probe modes instead of using only a single coherent probe mode21. Each mode is then propagated independently to the detector as the measured

diffraction is an incoherent superposition of the contributions from all probe modes21,28,32. For practical implementations, we chose the modal decomposition approach21,28 and the probe is

expanded into several eigenmodes of the density matrix formed by a mixed state. The total intensity of all eigenmodes are normalized to the measured intensity of the diffraction patterns. A

flowchart of the algorithms showing the basic principle is given in Supplementary Fig. 1. Details of the reconstruction algorithms are summarized in the Methods section. MIXED-STATE

PTYCHOGRAPHIC IMAGING TO SUB-ANGSTROM RESOLUTION We performed a scanning diffraction experiment on a sample of a monolayer WS2 with bilayer islands of WS2/MoSe2. The MoSe2 and WS2 layers

have the same crystalline orientation, which is verified by the diffractogram of the ADF image, as shown in Supplementary Fig. 2. Owing to the 4% lattice mismatch between WS2 and MoSe2, the

bilayer regions have a complex projected atomic structure with a continuous transition from a full Mo-Se (or W–S) bond length to intermediate misaligned projected distances (a structural

model is given in Supplementary Fig. 2c). The resulting Moiré pattern serves as a good resolution test for our method. Figure 2a shows a mixed-state ptychographic reconstruction of a dataset

with an FOV of 30 × 30 nm2 corresponding to 1500 × 1500 pixels. The selected sample region contains various structural features, such as a monolayer of WS2 and both well-aligned and

misaligned stacking bilayer MoSe2/WS2 regions. An enlarged view of the bilayer region shown in Fig. 2b from the position marked on Fig. 2a shows the sliding structure of bilayer WS2/MoSe2.

The Moiré-like pattern changes continuously from hexagonal rings in well-aligned stacking regions to stripe features in the misaligned regions (see the structural model in Supplementary Fig.

 2c). Unambiguous sub-Ångstrom resolution can be illustrated in both Fourier space and real space. First, the isotropic information limit shown in the diffractogram of the reconstruction in

Fig. 2d better than 1.4 Å−1 and the diffraction spots circled on Fig. 2d correspond to a real-space distance of 0.69 Å. Second, the line profiles in Fig. 2e from atomic pairs marked on Fig. 

2b demonstrate real-space peak separations down to 0.6 Å. In addition, the Fourier ring correlation (FRC) analysis33, widely adopted for resolution estimation in cryo-electron microscopy and

X-ray ptychography, also demonstrates spatial resolution down to 0.66 Å. This is shown in Supplementary Fig. 3, which is very close to the 0.69 Å resolution determined from the

diffractogram in Fig. 2d. Figure 2c shows a conventional ADF image acquired with the same aperture size and a similar dose using an in-focused probe, limiting the resolution to worse than 1 

Å. Compared with the ADF image in Fig. 2c, the ptychographic reconstruction in Fig. 2b demonstrates a significant and simultaneous improvement in contrast, signal-to-noise, and resolution.

What is more important is that the large FOV and sub-angstrom resolution ptychographic reconstruction are both achieved using a low-dose illumination. Typical doses used for

atomic-resolution (1.5–2 Å) STEM ADF images of monolayer transition-metal dichalcogenides at 80 keV are ~105 e Å−2 34,35. The dose for the dataset used in Fig. 2a is 1.6 × 104 e Å−2. The

ptychographic reconstruction has a 0.69 Å Abbe resolution, which doubles the resolution of ADF images achievable from the same imaging condition, ~1.37 Å (e.g., in Fig. 3). Furthermore,

sub-angstrom resolution was reached for a dataset with doses down to 4.0 × 103 e Å−2 (e.g., in Supplementary Fig. 4)—50 times lower than the dose for ADF images for the same resolution, with

lower dose lattice images discussed below. In addition, ptychography with a large defocused probe allows us to use a large scan step size of about 30% of the probe diameter15,36, because

the real-space pixel size of ptychographic reconstruction is determined by the maximum scattering angle of the diffraction pattern instead of scan step size in the conventional STEM ADF or

BF images. This decouples the scan step size from the real-space sampling required by Nyquist–Shannon sampling theorem37 for a certain resolution. This option enables a fast acquisition

despite the frame-rate of current 2D array detectors being usually two to three orders slower than the acquisition rate of point detectors30,38,39. The dataset used for the ptychographic

reconstruction shown in Supplementary Fig. 4a contains 64 × 64 diffraction patterns and the total acquisition time was 7.6 s. Assuming the same FOV and the typical acquisition conditions for

ADF imaging—a step size of 0.2 Å with 1500 × 1500 pixels, a beam current of 10–40 pA, which requires 64–16 μs per pixel dwell time to achieve 105 e Å−2 dose34,35—the total acquisition time

would be 144–36 s. Therefore, the high-quality ptychographic reconstruction with the same FOV is more than four times faster in acquisition while still using one order of magnitude lower

dose compared with conventional ADF imaging. Further tests of the practical experimental conditions show that much relaxed real-space overlap constraints, defined as (1 _−_ _r_/_D)_36, where

_r_ is the scan step size and _D_ is the diameter of probe, is required by mixed-state ptychography. As shown in Supplementary Fig. 5, mixed-state ptychography with only two probe modes

provides a stable high-quality reconstruction for a much larger scan step size up to 5.08 Å, corresponding to only 72% probe overlap, whereas artifact free reconstruction by single-state

ptychography can only be achieved with a very small step size, 0.85 Å, corresponding to a 95% probe overlap. For larger scan step sizes, shown in Supplementary Fig. 5b, c, the single-state

reconstructions fail to converge to the correct structure and generate artificial periodicities of the sample. At large step sizes, the errors in modeling of the probe partial coherence by

the single-state reconstruction accumulate, leading to a solution that is not able to describe the measured data well and therefore, in combination with the weaker constrains due to larger

scanning step, it leads to nonphysical reconstructions. Even in the case of a large probe overlap where the single-state reconstruction works, 95%, the mixed-state reconstruction has about

two times better resolution and enhanced contrast than that from the single-state (Supplementary Fig. 5). We also find that four eigenmodes are sufficient to capture the incoherence

properties of our probe, and additional modes do not lead to improvement of the reconstruction quality. A comparison of reconstructions with different numbers of modes is presented in

Supplementary Fig. 6. Therefore, under practical conditions of partially coherent illumination, mixed-state ptychography significantly relaxes the requirement of the real-space constraint,

i.e., scan step, and makes large FOV atomic-resolution imaging feasible. LOW-DOSE ATOMIC-RESOLUTION IMAGING Low-dose electron ptychography was first suggested via the Wigner-distribution

deconvolution (WDD) approach13, but only 104 e Å−2 or higher dose for atomic-resolution imaging has been realized experimentally by this approach16. In principle, the low-dose performance

should be comparable for WDD and iterative ptychography algorithms5, but the high sampling density required and currently slow detector readouts make working at low dose more challenging for

WDD40. For iterative electron ptychography, the dose-limit has been explored previously via simulations5,18,41,42 and some experimental attempts40. Simulations usually show one to two

orders lower dose than what has been realized by experiments5. This discrepancy may be caused by experimental limitations, such as sample stability, limited beam stability, and partial

coherence, which were neglected in simulations. Recent work using a highly coherent defocused probe and single-state electron ptychography has demonstrated a lattice resolution image from a

monolayer MoS2 using low-dose illumination40. However, only resolution worse than 1 Å was achieved and partial coherence of the illumination was not considered in the ptychographic

reconstruction. To explore the impact of partial coherence on the practical limit of the low-dose imaging, we show the dose dependence of ptychographic reconstructions using experimental

datasets and find benefits for both resolution and required dose after accounting for partial coherence. Figure 4 shows a series of reconstructions using datasets acquired from monolayer WS2

by varying the illumination dose via changing the beam current. For a dose higher than 3300 e Å−2 (Fig. 4a, b), both the W and S sublattice can be resolved unambiguously and the isotropic

information transfer is higher than 1.1 Å−1, corresponding to a real-space resolution better than 0.9 Å as shown in Fig. 4f, g. For doses of 1500 and 790 e Å−2 shown in Fig. 4c, d, the

reconstruction quality is slightly reduced but both sublattices are still resolvable and the information transfer is higher than 0.73 Å−1, corresponding to a real-space resolution of 1.37 Å.

Even for a dose of 375 e Å−2, shown in Fig. 4e, the lattice can still be recognized, and the information transfer is up to ~0.63 Å−1, corresponding to a real-space resolution of 1.59 Å. As

mentioned above, the dose commonly used for atomic-resolution (~1.5–2.0 Å) ADF images is about 105 e Å−2 34,35. Therefore, the minimum dose in the ptychographic reconstruction with

atomic-resolution demonstrated here is more than two orders smaller. We also compared the reconstructions from mixed-state and single-state ptychography in the low-dose conditions. As

demonstrated in Fig. 4a–e, k–o, both the resolution and contrast of the reconstructions from mixed-state ptychography are enhanced compared with single-state ptychography using the same

dose. The mixed-state ptychographic reconstruction has a better resolution at half the dose than that for single-state reconstruction; see for example, Figs. 4m and 4d. Similarly, the

mixed-state reconstructions in Fig. 4 show better quality and higher information limit compared with single-state reconstructions at similar doses reported recently40. A summary of the

dose-resolution dependence including the results in Fig. 4 and previous reported ptychographic results5,40,41 is given in Fig. 3. Some general trends and scaling behaviors can be noted, even

though the samples and experimental conditions for all the data are not identical, which may slightly affect the absolute resolution value at a certain dose. In the low dose-limit where

performance is limited by counting statistics, the resolution is well-aligned along a family of dose-limit lines, \(k/\sqrt N\). _N_ is the dose, and _k_ is a constant that depends on the

imaging method and the scattering power of the diagnosed samples43. Compared with conventional ADF images, for a certain information limit, the dose required for the mixed-state

ptychographic reconstructions is 10–50 times lower. At higher doses, the resolution is limited by the largest angle of information transfer. Ptychography is limited by the cut-off angle of

the diffraction patterns used in the reconstruction and can demonstrate a much better resolution limit than ADF imaging, which is limited by the probe-forming aperture. The negligible

readout noise (1/40 e−) and single-electron sensitivity of detector used30 is critical for approaching the limits of the low-dose ptychographic reconstruction. As shown in Supplementary Fig.

 7a–d, a single diffraction pattern with dose smaller than 3300 e Å−2 contains only a few electrons per pixel in the center disk and a very large fraction of pixels with fewer than single

electron in dark-field region. Despite the very low dose, which provides only ~500 electrons per scan position for a dose of 790 e Å−2, the averaged diffraction patterns shown in

Supplementary Fig. 7e–h still retain sharp edges of the diffraction disks. This demonstrates that the interference between different diffraction paths has been well encoded in these

sparse-signal diffraction patterns. The lattice fringes in Fig. 4d reveal that the sample phase information contained in the low-dose diffraction patterns have been retrieved via electron

ptychography. HIGH-PRECISION ATOMIC POSITION MEASUREMENTS One of the advantages of ptychography is the ability to correct for positioning errors44,45,46 by maximization of mutual consistency

between the adjacent regions. These methods can serve as an independent estimation of the position measurements, which can in principle correct for sample drifts and improve the precision

of atomic position measurement. We have selected an area of monolayer WS2 (4.0 × 5.2 nm2) shown in Fig. 5a. The contrast from light sulfur elements in the ptychographic reconstruction is

very strong, because the contrast of the phase image is roughly linearly proportional to _Z_, the atomic number47, whereas the contrast in ADF is close to a _Z_2 dependence8,48. Therefore,

atomic positions of sulfur atoms can be readily obtained with high precision even in the vicinity of heavy W atoms. In conventional ADF images, the sulfur atoms are shadowed by the strong

scattering from W atoms, which hinders the estimation of the sulfur positions. The recorded EMPAD dataset shows sample drift during the data acquisition. However, the relative probe position

errors can be very well estimated via the gradient-based probe position refinement algorithm49. At first, the global geometry errors, such as scan step size scaling, axis rotation or

skewness, are fitted and corrected via an affine transformation between the nominal probe positions and estimated probe positions. Owing to the raster position scanning, a constant velocity

drift of the sample will result in a skewness of up to a few degrees for long acquisition datasets. When the geometry model is refined, the residual position errors are estimated. Figure 5b,

c shows the residual position errors along the horizontal (fast scan) and vertical (slow scan) direction, respectively. The standard deviation (st.d.) of the residual errors along both

directions shown in Fig. 5b, c is ~0.16 Å with a maximum error of ~0.6 Å. The atomic column positions were estimated using a 2D Gaussian function fit of the reconstructed phase. The

positions were used to calculate the nearest neighbor W–W and S–S atomic distances. The distributions of the S–S and W–W distances are plotted in Fig. 5d, e. The st.d. of the S–S distance is

5.8 pm and the st.d. of W–W distance is 5.2 pm, which is a precision measurement of the atomic positions as the monolayer WS2 region shown in Fig. 5a has negligible structural distortions.

The intensity variations in Fig. 5a are from residual polymer residue, and likely degrade the precision of the bond-length measurements. Therefore, the reported precision of the ptychography

method should be viewed as an upper bound. For transition-metal dichalcogenides imaged by 80 keV electrons, the dose must be kept below ~106 e Å−2 to avoid significant structural damage50.

An even lower dose beneath 104 e Å−2 is required to avoid the formation of sulfur point vacancies or structural alterations around deficient regions51. Therefore, it is important to see the

performance of ptychography for estimating atomic positions at low dose. Figure 5f shows the dose-dependent precision estimation of W–W and S–S distances. Precisions of about 10 pm for both

W–W and S–S distances can still be achieved using doses as low as 104 e Å−2. However, from a conventional ADF image, the precision of W–W atomic distance with a single fast scan (6 μs per

pixel chosen to match dose and pixel sampling) is only 13.6 pm as shown in Supplementary Fig. 8, which is about twice as large as that from ptychography with a similar dose. Using a more

stable imaging system, multiple scans and drift correction algorithms, precision for both ADF imaging and ptychography can be further improved10,11. As detector speeds increase, the multiple

scan strategy becomes more practical for ptychography. Denoising and deconvolution algorithms can also help with peak location41,52. However, sulfur atoms in the ADF image are not visible

due to their much-weaker scattering, shown in Supplementary Fig. 8a. Although sulfur atoms can be seen by using a lower collection angle for ADF35, the precision of S atomic positions

determined from low-dose ADF image is much worse than that from W. Therefore, the low-dose and high contrast capabilities of ptychography provide a picometer-precision technique for atomic

position determination of single atoms including light sulfur atoms in 2D materials. DISCUSSION We have demonstrated that mixed-state electron ptychography provides simultaneously improved

imaging capabilities, including high resolution, large FOV, low dose, high contrast/signal-to-noise ratio, and high precision. A mixed-state model is beneficial for ptychographic

reconstructions using the data from current electron microscopes, as it is able to account for many of the different factors that result in decoherence-like effects in measured diffraction

patterns20,21,42,53, such as the finite electron source size, chromatic aberration, sample vibration, and fast instabilities of the image-forming system. With further improved coherence of

the electron source, such as a cold field-emission gun, a larger scan step sizes with a reduced overlap ratio could be achieved54. Loss of speckle contrast in diffraction, which has similar

effects as partial spatial coherence of the probe, can be introduced due to a finite detector pixel size and limit the largest applicable probe size20,55. Therefore, with further improvement

of the source coherence and the area of the detectors, increased scan step sizes can be utilized, which further enlarges the FOV given the same number of scanning positions. Furthermore,

requirements of the illumination stability in ptychography can be relaxed by use of multiple scans with shared object information56 or other probe relaxation extension49,57. These approaches

can largely overcome the long-term stability limitations of the current scanning systems, which makes even micron length scale with sub-angstrom resolution imaging feasible. Ptychography

requires a forward model for the interaction of the beam with the sample. One limitation of the current mixed-state ptychographic imaging is that it can only be applied in relatively thin

samples because it uses a generalized strong phase approximation that neglects the effects of beam propagation. The generalized phase grating approximation for the interaction of the

incident probe with a projected object function can be written as, \(\psi _{{\mathrm{exit}}}({\mathbf{r}}_i,{\mathbf{r}}) = \psi _{{\mathrm{in}}}\left( {{\mathbf{r}} - {\mathbf{r}}_i}

\right)O\left( {\mathbf{r}} \right)\), where _ψ_exit (R_i_, R) is the electron wavefunction passing through the sample, _ψ_in (R − R_i_) is the incident electron wavefunction centered at

position \({\mathbf{r}}_i\), with \({\mathbf{r}}_i\) and \({\mathbf{r}}\) being 2D coordinates. The complex object function, _O_ (R) is a generalized strong phase object, \(O({\mathbf{r}}) =

A\left( {\mathbf{r}} \right){\mathrm{exp}}\left( {i\sigma V\left( {\mathbf{r}} \right)} \right)\), where _A_ (R) is the amplitude, _σ_ is the interaction constant depending on the electron

energy and _V_ (R) is the projected electrostatic potential of the sample. The amplitude term is included to allow for a weak absorption effect, e.g., scattering outside the detector and

should be close to unity if the sample is thin58. Failures of the model for practical samples might be suspected if the amplitude either deviates by more than 10% from unity or resembles the

phase term such that phase-amplitude mixing has likely occurred59. If the probe shape changes significantly during propagation within the sample, then the probe–sample interaction cannot be

well described in a single plane and a full multi-slice calculation may need to be considered. Both probe free-space propagation and scattering by the sample can change the probe shape. The

thickness limit _T_ due to the propagation effect can be expressed as60, \(T = 1.3\lambda /\theta _{{\mathrm{max}}}^2\), where _θ_max is the maximum scattering angle of the diffraction

pattern and \(\lambda\) is the wavelength of electrons. For a typical scattering angle targeting a resolution better than 0.5 Å, \(\theta _{{\mathrm{max}}} = 20\) mrad at 300 keV, the

thickness limit is ~6.4 nm, which is within the achievable thickness for many samples. For heavy elements, a single atom can induce a large phase shift and a strong amplitude damping to the

electron wavefunction, and the probe shape can be changed significantly by only a few atoms. Therefore, a much more rigorous thickness limit must be adapted for samples containing high

atomic number elements61. Recent attempts to solve the multiple scattering problem in thick samples include multi-slice ptychography60,62,63 and scattering matrix phase retrieval64. Although

the robustness and convergence must be further improved to achieve practical applications in thick samples63,65, mixed-state ptychography could be extended to include multiple scattering49.

Data processing speed is another limiting factor for applications of ptychography. However, with graphics processing unit (GPU) acceleration, the reconstruction of the large FOV image shown

in Fig. 2a only takes less than one hour on a typical GPU card. The processing time largely scales linearly with number of diffraction patterns, therefore, fewer patterns with the defocused

probe setup can significantly accelerate the reconstruction. The flexibility and robustness of mixed-state electron ptychography enable many potential applications. Large FOV high

resolution imaging is critical for uncovering both the overall morphology and the local atomic arrangement in complex nanostructures66. High contrast and high precision coupled with low-dose

imaging can be used to measure the atomic scale dynamics of light elements, such as Li, O, or S in lithium battery materials. Fast single-pass acquisition with scan drift correction opens

the door to in situ phase-contrast STEM imaging of dynamical processes during heating, cooling, or even chemical reaction. Use of a large-illuminated-area probe enables a larger scan step

size and fewer diffraction patterns for a given FOV, which reduces the computational effort during data reconstruction and analysis and accelerates data processing of ptychographic

reconstruction. Rapid data processing is critical for live imaging and 3D structural reconstruction, such as ptychographic tomography67,68. Our demonstration of low-dose imaging at

atomic-resolution is within the allowable dose for many beam-sensitive materials69. Further dose reduction could be potentially realized by improvements of the reconstruction algorithm and

experimental setup such as the averaging of multiple images from structurally identical particles that is commonly used in single-particle Cryo-EM. The high dose efficiency of mixed-state

ptychography may also be helpful for reconstructing biological molecules using cryo-electron microscopy, potentially reducing the number of particles needed in an averaging class to achieve

a desired resolution41. With the anticipated next generation pixel-array detectors that will be even larger and faster, the probe size can be further increased and thus a larger scan step

size can be chosen, which will further enlarge the FOV and increase the acquisition speed. Furthermore, mixed-state electron ptychography, besides the retrieval of the probe mixture, can

also be adapted to retrieve mixed quantum states within a sample21,32, which could expand the reach of quantum-state tomography by using a matter wave. METHODS EXPERIMENTAL METHOD The

scanning diffraction experiments were carried out using an electron microscope pixel-array detector (EMPAD)30 installed on a probe aberration corrected Thermo Scientific™ Titan Themis

electron microscope. The EMPAD has 128 × 128 pixels, a readout speed of 0.86 ms per frame, and 1,000,000:1 electron linear response. All the datasets were acquired using a probe at 80 keV

beam energy, 21.4 mrad probe-forming semi-angle, and ~55 nm defocus value. The exposure time was 1 ms per frame. The beam current varied from 0.09 to 14.3 pA via defocusing a monochromator.

The coherence of the electron probe increases slightly when the beam current reduces but the change is not significant (<3%) as the beam current used (0.1–14 pA) is always much lower than

the coherent current of the source (~50 pA). The large 30 × 30 nm2 FOV image in Fig. 2 is reconstructed from a dataset with a scan step size of 2.36 Å containing 128 × 128 diffraction

patterns, which were selected in an interval of two from a larger dataset with 256 × 256 diffraction patterns. A reconstruction of a four-times down-sampled dataset with a 4.72 Å scan step

size is shown in Supplementary Fig. 4. The imaged sample contains MoSe2 islands on a large area of monolayer WS2. The relative orientation of MoSe2 and WS2 is identical, as we have verified

using the Fourier transform of the ADF image shown in Supplementary Fig. 2. MIXED-STATE PTYCHOGRAPHY We adapted the generalized maximum-likelihood ptychography method49,70 initially

developed for X-ray ptychography. This method solves the phase retrieval problem by preconditioned gradient descent-based optimization. Optimization of amplitude likelihood function49,70

provides more robust convergence than direction optimization of Poisson likelihood. Multiple optimization directions, such as probe and object updates, probe position displacements, or

wavefront variation can be carried out jointly in a consistent way. In combination with a neural-networks-inspired mini-batch optimization scheme49, our approach enables a compromise between

convergence speed and noise robustness, and thus it significantly improves the usability of ptychography for low-dose imaging. The mixed-state description is implemented using the modal

decomposition approach21. Minor time variations of the illumination probe due to the instability of the electron optics or small sample height variation are accounted for by using an

illumination wavefront correction method49, which is a computationally faster and more memory-efficient approximation of the orthogonal probe relaxation (OPR) approach57. The OPR method

describes small variations of the illumination probe by a linear decomposition into a set of several orthogonal mutually coherent modes. The workflow of the algorithm is schematically shown

in Supplementary Fig. 1 and more details are described as a ptychography toolkit in ref. 71. We have observed that the reconstruction quality of the mixed-state ptychography algorithm is not

sensitive to the initial guess of the illumination probe and the number of the mixed modes for datasets with a sufficient probe overlap, although a good initial guess may accelerate the

convergence. On the other hand, the single probe mode ptychography requires a good initial probe guess to provide stable convergence for our experimental datasets. This seems

counterintuitive but it is not surprising. Because mixed-state ptychography can account for the nonnegligible partial coherence of the probe and provide a more accurate reciprocal model,

whereas single-mode ptychography is not sufficient for modeling the probe incoherence and its convergence can become instable if the initial probe deviates from the real probe significantly.

FOURIER RING CORRELATION For Fourier ring correlation (FRC)33,72, we used two phase images reconstructed from two separate datasets from the same scan region, which serves as two

independent measurements. Practically, two datasets were selected from one single dataset in every two diffractions at each dimension but with different starting points. After ptychographic

reconstruction, a global linear phase term due to the inherent ambiguities of ptychography73 is removed by fitting as a 2D linear function. Two phase images are aligned using the sub-pixel

precision cross-correlation algorithm74. Before FRC analysis, the edges of the phase images were Apodized to avoid the artifacts introduced from the boundary discontinuities33. The

resolution is estimated by using the 1-bit threshold33. DATA AVAILABILITY A small (200 Mb) data subset is available from PARADIM, a National Science Foundation Materials Innovation Platform

[https://doi.org/10.34863/G4WA-0J57]75. Full datasets are available from the corresponding author ([email protected]) on request. CODE AVAILABILITY The codes developed at Cornell

University is published on GitHub, muller-group-cornell [https://github.com/muller-group-cornell]. The ptychography reconstruction toolkit, PtychoShelves developed at Paul Scherrer Institut,

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NSF Materials Innovation Platform_, https://doi.org/10.34863/G4WA-0J57 (2020). Download references ACKNOWLEDGEMENTS Z.C. and D.A.M. are supported by the PARADIM Materials Innovation

Platform program in-house program by NSF Grant DMR-1539918. Y.H. is supported by the NSF MRSEC program (DMR-1429155). This work made use of the Cornell Center for Materials Research facility

supported by NSF grant DMR-1719875. We thank Tsai Esther Hsiao Rho and Manuel Guizar-Sicairos for useful discussions. AUTHOR INFORMATION Author notes * Michal Odstrcil Present address: Carl

Zeiss SMT, Carl-Zeiss-Straße 22, 73447, Oberkochen, Germany * Yimo Han Present address: Department of Molecular Biology, Princeton University, Princeton, NJ, 08544, USA * Lain-Jong Li

Present address: Department of Electronic Engineering and Green Technology Research Center, Chang-Gung University, Taoyuan 333, Taiwan AUTHORS AND AFFILIATIONS * School of Applied and

Engineering Physics, Cornell University, Ithaca, NY, 14853, USA Zhen Chen, Yimo Han & David A. Muller * Paul Scherrer Institut, 5232, Villigen PSI, Switzerland Michal Odstrcil * Advanced

Photon Source, Argonne National Laboratory, Lemont, IL, 60439, USA Yi Jiang * Physical Science and Engineering, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi

Arabia Ming-Hui Chiu & Lain-Jong Li * Kavli Institute at Cornell for Nanoscale Science, Ithaca, NY, 14853, USA David A. Muller Authors * Zhen Chen View author publications You can also

search for this author inPubMed Google Scholar * Michal Odstrcil View author publications You can also search for this author inPubMed Google Scholar * Yi Jiang View author publications You

can also search for this author inPubMed Google Scholar * Yimo Han View author publications You can also search for this author inPubMed Google Scholar * Ming-Hui Chiu View author

publications You can also search for this author inPubMed Google Scholar * Lain-Jong Li View author publications You can also search for this author inPubMed Google Scholar * David A. Muller

View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS Experiments and data analysis were performed by Z.C. under the supervision of D.A.M. The

main ptychographic algorithms were implemented by M.O. for X-ray ptychography with the adaption to electron ptychography by Z.C. and Y.J. Sample preparation was done by Y.H. from thin films

synthesized by M.C. and L.J. Z.C. wrote the manuscript with revisions from M.O. and D.A.M. All authors discussed the results and implications throughout the investigation, and all authors

have given approval to the final version of the manuscript. CORRESPONDING AUTHOR Correspondence to David A. Muller. ETHICS DECLARATIONS COMPETING INTERESTS Cornell University has licensed

the EMPAD hardware to Thermo Scientific. ADDITIONAL INFORMATION PEER REVIEW INFORMATION _Nature Communications_ thanks Christian Dwyer, Jianwei Miao and the other, anonymous, reviewer(s) for

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