Simultaneous atomic-resolution electron ptychography and Z-contrast imaging of light and heavy elements in complex nanostructures

The aberration-corrected scanning transmission electron microscope (STEM) has emerged as a key tool for atomic resolution characterization of materials, allowing the use of imaging modes such as Z-contrast and spectroscopic mapping. The STEM has not been regarded as optimal for the phase-contrast imaging necessary for efficient imaging of light materials. Here, recent developments in fast electron detectors and data processing capability is shown to enable electron ptychography, to extend the capability of the STEM by allowing quantitative phase images to be formed simultaneously with incoherent signals. We demonstrate this capability as a practical tool for imaging complex structures containing light and heavy elements, and use it to solve the structure of a beam-sensitive carbon nanostructure. The contrast of the phase image contrast is maximized through the post-acquisition correction of lens aberrations. The compensation of defocus aberrations is also used for the measurement of three-dimensional sample information through post-acquisition optical sectioning.


Measuring Lens Aberrations through SVD Matrix Inversion.
Because WDD allows residual lens aberrations to be corrected from the 4D data-set, the ability of measuring residual lens aberrations within the 4D data-set becomes particularly beneficial and leads towards aberration free phase imaging. In this section, we introduce an aberration measurement algorithm that measures the aberration coefficients of low and high orders by solving a set of linear equations through a deterministic matrix inversion using singular value decomposition (SVD) 1 . This method makes use of the phase information inside the disc-overlap region, and can be applied to both crystalline and noncrystalline specimens.
Before going through the details of the aberration measurement algorithm, Supplementary Fig.2 demonstrates an example of measuring and correcting lens aberrations using a sample of single layer graphene with an electron probe arising from a poorly aligned aberration corrector. The poor quality of the ADF image and the initial phase reconstruction assuming no aberrations ( Supplementary Fig.2a) means that data like this would usually be discarded. By applying the SVD, aberration coefficients up to 3 rd order were measured (see Supplementary Table 1). The quality of the aberration measurements can be seen from two examples of the overlapping discs showing the good match between the experimental phases and the calculated phases arising from only the measured aberration coefficients (compare Supplementary Fig.2b with 2c, and 2e with 2f). In the Supplementary Fig.2d,g, the maps of difference also show a reasonably flat phase surface, indicating an efficient aberration compensation has been achieved. The phase in the probe-forming aperture in Supplementary Fig.2i and the real space probe in Supplementary Fig.2j calculated using the measured aberration coefficients were directly fed into the WDD to obtain Supplementary Fig.2h an aberration free phase image with significantly improved resolution and quantitative phase values compared to the initial reconstruction in Supplementary Fig.2a.

Mathematical description of the phase in the disc-overlap region
If we assume the weak-phase object approximation then, where ′ ( ) ≪ 1 and is the Dirac delta function. Substituting equation 1 into equation 2 of the main text gives equation 2 represents, for ≠ , the appearance of two disc overlap regions as shown in Fig. 1d,e of the main text. The phase ∠ ( , ) in the two double disc-overlap regions in the detector plane can be described as where ( ) is the aberration function and ∠ is the phase angle operator. The two terms in the brackets describe the two disc-overlap regions formed by -and + , respectively.

Matrix representation of the linear equations
In a ptychographic 4D dataset, for example an experiment dataset with 256x256 probe positions and 264x264 detector pixels, there are a large number of ( , ) observations that are located within the double-overlap regions (orders of magnitude larger than the number of unknown aberration coefficients), leading to a large set of linear equations as in equation 5. This set of linear equations can be written in a matrix format with each term being defined as following, ( 1 ( , ) − 1 ( + 1 , + 1 ) , ⋯ 12 ( , ) − 12 ( + 1 , + 1 ), ⋮ ⋱ ⋮ 1 ( , ) − 1 ( + , + ) , ⋯ 12 ( , ) − 12 ( + , + ), = [ 1 , 12 Again, the above matrix representation describes only one side (− ) of the two double-overlap regions, but the other side (+ ) can be treated in exactly the same way, and the two matrices from the two double-overlap regions can be combined for direct matrix inversion.

Singular value decomposition
Solving the unknown aberration coefficients can therefore be performed by solving the set of linear equations as follows through direct matrix inversion. As discussed above, because the number of observations M in a hugely redundant 4D dataset is much larger than the number of unknown parameters N, the matrix ̃ is over-determined. Inversion of matrix ̃ can be solved using a method called singular value decomposition. A detailed description of SVD will not be shown in this text, and the reader is referred to the literature for example in 1 for more details.

Matrix preparation and Selection of
Even though the ∠ ( , ) in the double-overlap regions from the entire range of < 2 can be used in the linear equations, practically it's sufficient to use only a number of values whose ∠ ( , ) have the best signal-to-noise ratio, which corresponding to those with the largest modulus of ( , ) summed over . Therefore, the selection of can be fully automated by ranking the modulus of ( , ) summed over . An important requirement for the selection of is to make sure vectors of different directions are selected so that astigmatism aberrations can be effectively measured.

Iterative algorithm for solving aberrations from low to high orders
Solving the set of linear equations involves a step to unwrap the phase surface of ∠ ( , ) if the values go beyond 2π. Ideally if the unwrapping is perfect, all of the values can be used for matrix inversion, and all of the low and high aberration coefficients can be solved instantly using SVD without the use of an iterative approach. However, when working with low dose datasets with a low signal-to-noise ratio, unwrapping the phase becomes non-trivial. This problem can be solved by adopting an iterative approach to measure the low order aberrations and compensate the phase due to low order aberrations first, which iteratively helps to "flatten" the phase surface and reduce the errors induced by phase unwrapping, thus leading to a more precise measurement of higher order aberrations iteratively. The iteration procedure can be described as following: Step 1: Prepare matrix ̃ using equation 6, from k number of spatial frequencies from to and the scattering vector values located inside the disc overlap regions under each spatial frequency.
Step 2: Start with iteration number = 1. Initialize −1 = following equation 7 and set all the aberration coefficients inside 0 to zero. Initialize −1 = 0 as described in equation 8.
Step 3: ∆ is defined as ̃• ∆ = −1 . Calculate ∆ through the SVD matrix inversion 1 . Retain only 1 st order aberrations values in ∆ , and setting the higher orders to zero.
Step 4: Update = −1 + • ∆ , where is the updating step parameter whose maximum value is 100%. In this experiment, we set = 0.5 to apply 50% correction in each iteration.
Step 5: Calculate the overlapping discs ( ) * ( + ) using the aberrations . Step 8: Iterate Step 3 to Step 7, and gradually apply correction to include higher order aberrations in step 3 if ∆ is smaller than threshold and a convergence is reached.

13
C NMR spectra were recorded at room temperature on the following spectrometers: 100 MHz: Bruker DQX400, 125 MHz: Bruker DRX500. Unless reported otherwise, spectra were recorded in CDCl3 solution. Chemical shifts are reported in δ units relative to CDCl3 (δC = 77.23 central line of triplet).

19
F NMR spectra were recorded at room temperature a 376 MHz on a Bruker AVIII HD 400. Unless reported otherwise, spectra were recorded in CDCl3 solution. Chemical shifts are reported in δ units HRTEM: SWNT samples and hybrids were dispersed in EtOH [10 μg/mL], sonicated for 30 to 60 min, placed dropwise (4 x 6 μL) onto a Lacey carbon copper support grid and dried by evaporation. The JEOL JEM-3000F with a field-emission gun operated at 300 kV. The microscope was equipped with a 1k Gatan 794 MultiScan camera. The data were analyzed with DigitalMicrograph. EDX: An Oxford Instruments EDS detector equipped with an ultra-thin polymer window and controlled by Oxford Instruments Isis software INCA was used for the acquisition on the JEOL JEM-3000F. XPS: Data were collected under high vacuum (approx. 10 9 torr) on a VG Escalab XPS Spectrometer by Ashley Shepherd in the Surface Analysis Facility, Chemistry Research Laboratory. Analysis was undertaken with CasaXPS software. TGA: SWNT samples were measured in synthetic air flow (20 mL/min) using the model STA 449 F1 Jupiter® from Netzsch by Magdalena Kierkowicz, Institut de Ciència de Materials de Barcelona (ICMAB). An initially isothermal run profile was maintained for 15 min and then increased at 10 °C/min to 900 °C. HRMS: Data was determined under conditions of ES + on a Micromass Q-Tof micro (resolution = 4 x 10 3 D) using H3PO4-clusters as a lock-mass in positive ion mode. HPLC: was performed on a Dionex Ultimate 3000 system. Gradients were established for a Phenomenex Jupiter 4u Proteo 90A (250 x 4.6 mm) and Synergi 4u Hydro-RP 80A (100 x 4.6 mm, 4 micron and 100 x 21.2 mm, 4 micron). The solvent system was mixed appropriately from H2O (MilliQ purity) and MeCN, 0.1% formic acid (FA) each. MALDI-ToF spectra were recorded on a Waters® Micromass® MALDI micro MX TM Mass Spectrometer in the positive reflectron mode. The data were analysed with MassLynx V4.1. Infrared spectra for small organic molecules were recorded on a Bruker Vector 22 IR spectrometer with the sample being pressed into KBr pellets. The IR of CNT samples were recorded using attenuated total reflection Fourier-transformed infrared (ATR-FTIR) on a Varian FTS-7000 Spectrometer and a DuraSamplIR II diamond crystal by Dr. Robert Jacobs in the Surface Analysis Facility, Chemistry Research Laboratory. Raman spectra were recorded on a Jobin Yvon spectrometer (Horiba) equipped with a microscope, through a 50-fold magnification objective (Olympus Co.), by combining three sets of spectra. A 20 mW He-Ne laser (632 nm) was used. The 1800 L/mm grating provides a resolution starting from 1.0 cm -1 at 100 cm -1 up to 0.5 cm -1 at 3000 cm -1 . The abscissa was calibrated with a silicon standard. Raman spectra were directly taken from solid samples. Reagents and Solvents: All reagents and solvents were obtained from Sigma-Aldrich, Alfa Aesar, Carbosynth, GLS Biochemicals, Fluka, and Frontier Scientific and were used directly as supplied, unless otherwise reported. Anhydrous solvents were bought from Sigma-Aldrich. All non-aqueous reactions were performed in oven-or flame-dried apparatus under argon or nitrogen atmosphere using anhydrous solvents. Carbon nanotubes were supplied by Thomas Swan (produced using iron as a catalyst via arc-discharge method 3,4 ) and steam-purified by Dr. Gerard Tobias, ICMAB 5,6 . Reaction Techniques: Reactions were monitored by thin layer chromatography on pre-coated aluminium-backed plates (Merck Kieselgel 60 with fluorescent indicator UV254). Spots were visualised by quenching of UV fluorescence and/or by staining with potassium permanganate, iodine, ninhydrin, p-anisaldehyde or vanillin. Flash column chromatography was performed according to the method described by Still, Kahn and Mitra 7 with silica gel 60 (0.040-0.063 mm) (Geduran® Si 60) applying head pressure by means of manual flushing.

Hybrid Synthesis Overview
The synthesis of a hybrid, tethered system as envisaged in Fig.2 of the main manuscript was attempted as detailed below. Carbon nanotubes (CNT) were oxidized to introduce carboxylic acid groups for functionalization following literature procedures [8][9][10][11] . For clarity, samples are referred to number 1, 2 and 3 through the main text and supplementary information, and their structures are shown in Fig.2 of the main manuscript, Supplementary Scheme S1 and Supplementary Scheme S2. The analysis of the resulting complex, product carbon nanomaterial was undertaken by means of infrared (IR, Supplementary Fig. 7) and Raman spectroscopy (see Supplementary Fig. 4, 5 and 10, respectively), high resolution transmission electron microscopy (HRTEM as shown in Supplementary  Fig. 6 and 11), energy dispersive x-ray spectroscopy (EDX), X-ray fluorescence spectroscopy (XRF), Xray photoelectron spectroscopy (XPS, Supplementary Fig. 9) and thermogravimetric analysis (TGA, Supplementary Fig. 8).
Characterizations by these conventional techniques suggested fC60-functionalized carbon nanotube hybrids, as intended. These constructs were subjected to scanning transmission electron microscopy (STEM) with implemented ptychography to resolve the structure making use of the 'iodine-tagged' fullerene. In both representative samples 2 and 3, 'peapod' structures were observed, indicating the presence of incomplete covalent functionalization. We speculatively conclude that the CNT-peptide-C60 product mixtures comprise a mixture of covalently-attached fullerenes ( Supplementary Fig.3A), potential ionic interactions [12][13][14] of CNT carboxylic acids with amine and guanidine groups in the peptide linker ( Supplementary Fig.3B), and van der Waals-forced 'peapod'-type interactions ( Supplementary Fig.3C).

Synthesis of CNT-COOH (1)
The suspension of CNTs
After stirring for 14.5 h DIPEA (78.9 μL, 0.45 mmol, 6 eq.) was added. The reaction mixture was stirred for one hour until tlc indicated complete conversion of starting material and the solvent removed under reduced pressure. The crude peptide was first purified by trituration with cold Et2O/hexane 1:1 (2 x 30 mL), centrifugation (10 min, 7197 G) and decantation to give the crude mixture (87.7 mg, 97%) which was further purified by HPLC to give peptide 9 as a white solid (40.7 mg, 45%).

The bounds of the 3D contrast transfer function.
To explain why WDD reconstruction offers a true optical sectioning effect rather than a Fresnel propagated version of the exit wave, we start with equation 2 of the main text. Consider a thin sample that is located a distance, z, from the focal plane of the microscope along the electron beam direction. To make this effective defocus explicit in the mathematics, we write the aperture function, A, as the product of an amplitude term, H, that has a value of unity inside the aperture and zero outside, and a phase term reflecting the usual defocus aberration term, or equivalently the Fresnel propagation term, Substitution into equation 2 of the main text gives, after some simplification, The phase variation in the overlap region due to aberrations can be seen clearly in Supplementary  Fig.2. In the single side-band ptychography method used previously 18 , one side of the disc doubleoverlap region is simply integrated in to get the overall amplitude and phase of the Fourier component of the reconstructed phase image at the spatial frequency, . If the phase is not constant, for example if the object is out of focus, then the integration over the phase variation across the overlap region will result in a lower overall magnitude of that Fourier component and a reduced contribution to the reconstruction. Objects in focus, where there will be no phase variation, will contribute more strongly.
In the WDD approach, the deconvolution step uses a kernel that corresponds to a disc overlap region, and an intentional defocus term can be used in the kernel resulting in a phase variation across the disc overlap region. Because the deconvolution step aims to reduce a disc overlap feature in ( , ) to a single point after deconvolution, it can be regarded as having an implicit summation over the disc overlap region. If the defocus set in the kernel matches that in the experimental data, the phase variation will be corrected and the sum over the disc overlap region will give a strong contribution. If there is a mismatch between the experimental and deconvolution defocus, there will be a resultant phase variation and the strength of the contribution will be reduced.
Under the WPO approximation, the 2D transfer function with zero aberrations is given by the integration over the disc-overlap region 19 . To extend this approach to 3D in the case of WDD, the strength of the transfer for a spatial frequency, , for an object a distance, z, from the defocus used for the deconvolution, can be computed by integrating over the convolution kernel in equation 13,