X-ray linear dichroic tomography of crystallographic and topological defects

Materials

We purchased V₂O₅ from US Research Nanomaterials and polystyrene latex spheres, 330 nm in diameter, from Thermo Fisher Scientific.

Sample preparation

The examined pillar was extracted from a sintered millimetre-sized pellet. This was prepared from a mixture of nanocrystalline V₂O₅ and polystyrene spheres (85/15 wt%). Mortar and pestle (10 min) was used to homogenize the mixture before the pellet was pressed using a 17 mm die set (3 min, 1.2 t uniaxial load). To increase the V₂O₅ grain size, sinter and create the desired porous structure, we heated the pellet to 590 °C for 5 h (Supplementary Fig. 1 and Supplementary Note 2). The polycrystalline V₂O₅ pillar was prepared by mechanically fracturing the sintered pellet, after which a fracture piece was mounted on an OMNY tomography pin⁵¹ using epoxy resin. The pillar was then pre-shaped using a microlathe⁵², before being reduced in diameter to 6 µm using focused ion beam (FIB) milling. This pillar was then transferred onto a second OMNY pin⁵¹. The tip of the second OMNY pin was sharpened using FIB milling before transferring the pillar. This was necessary to facilitate tomography measurements with a 30° stage tilt²⁶. See Supplementary Fig. 3 for micrographs of the prepared pillar.

General material characterization

Scanning electron microscopy and FIB milling were performed using a Zeiss NVision 40 dual-beam FIB. Powder X-ray diffraction measurements of the sample before and after sintering were acquired using a Cu K-α radiation source with a step size of 0.02° 2θ, (refs. ^53,54) (Supplementary Fig. 2). The sintered sample consists of α-V₂O₅ with a grain size >100 nm.

Origin of linear dichroism in α-V₂O₅

V₂O₅ has a layered orthorhombic crystal structure consisting of distorted [VO₅] pyramids, shown schematically in Fig. 1b. These pyramids tile along the a–b plane and are bound with van der Waals interactions along the c axis. The apical, vanadyl bond of these pyramids, aligned with the crystallographic c axis, is shorter (1.57 Å) compared with the bonds on the base of the pyramid (1.87 Å). This shorter bond breaks the symmetry of an otherwise regular square pyramid⁵⁵. To examine the spatial orientation of the apical bond and, in turn, the orientation of entire grains and deviations within them, the energy of the incident X-rays was set to that of the vanadium K pre-edge peak⁵⁵. This peak arises from the V (1 s) to V (4p-3d) transition, more specifically, to V (3d e_g + 4_p) + O 2p_z mixing states, which become accessible as a result of the deviation of the V coordination from the octahedral symmetry. When the apical bond is parallel to the direction of the electric field of the incident X-rays, the interaction is strong, as the transition V (1 s) to V (4p-3d) is allowed. When the apical bond is instead perpendicular to the incident polarization, the interaction is weaker^29,55. An illustration of the different absorption strengths that result from the relative orientation between the incident X-ray polarization and the apical bond, known as linear dichroism, is shown in Fig. 1b. The X-ray near-edge absorption and phase spectra of V₂O₅ measured using LH and LV polarizations are shown in Supplementary Fig. 4 and a schematic of the layered crystal structure is provided in ref. ²⁹.

In the above-described relationship between the polarization state of the illumination and the examined asymmetry or anisotropy, the linearly polarized light acts as a ‘search light’ for the resonant bond to which the polarization is parallel. This relationship applies, in principle, to all cases of natural linear dichroism^16,42. The connection between the investigated anisotropy orientation and the unit-cell orientation of the material can be obtained through the use of reference samples, as showcased in 2D linear dichroic microscopy applications³⁶, and is already available in the literature for numerous materials. It can also be readily determined with previous knowledge of the material’s crystal structure (or molecular arrangement)⁵⁵.

Ptychography, PXCT and phase contrast

Ptychography is a lensless imaging technique in which the phase problem is solved by means of iterative phase-retrieval algorithms²⁷. By applying ptychography to solve the phase problem at different projection angles, its tomographic extension, ptychographic X-ray computed tomography (PXCT)⁵⁶, is able to retrieve the complex-valued transmissivity of the specimen, providing quantitative tomograms of both phase and amplitude contrast³². Both the individual images—or projections—and resulting tomograms obtained using X-ray ptychography are sensitive to changes in the complex-valued refractive index, η. The real part of the refractive index decrement, δ, corresponds to the phase, whereas the imaginary part of the refractive index corresponds to the amplitude, β. The refractive index is fundamentally an expression of the complex atomic scattering factor, f = f₁ + if₂. The refractive index is therefore given by:

with r_e being the classical electron radius and λ the illumination wavelength^57,58. The images and tomograms resulting from measurements performed with incident X-ray energies away from sample-relevant absorption edges can, in the case of tomograms, be converted to quantitative electron-density, n_e, and absorption index, µ, tomograms³². Measurements conducted near sample-relevant absorption edges, that is, examining specific electronic transitions and the associated increase in the photoabsorption cross-section, are subject to anomalous scattering effects^57,58, including dichroism.

The angular dependence of the linear dichroism has previously been used in a microscopy context, in particular in X-ray linear dichroism microscopy with secondary imaging modalities such as photoemission electron microscopy, to provide a 2D spatially resolved microstructural characterization tool^{16,29,30,36,59,60}. The reader is directed to the initial work of Ade and Hsiao¹⁶ and the more recent works of Gilbert et al.^36,61,62,63 and Collins et al.^59,60,64. In the present work, we have developed the capability to map the orientation in 3D by combining X-ray linear dichroism microscopy with PXCT (XL-DOT).

Although XL-DOT can be applied with a range of imaging techniques, such as scanning transmission X-ray microscopy, we have selected X-ray ptychography as the imaging modality, a choice motivated by three factors. (1) PXCT provides quantitative or absolute contrast tomograms, which is ideal for material or component identification and for the detection of marginal signal variations^11,30. (2) As a lensless imaging technique, ptychography excels in terms of signal-to-noise ratio (SNR), spatial resolution and dose efficiency (per resolution element) compared with other methods^65,66,67,68. Given its superior SNR, it is ideal for measuring the relatively weak linear dichroism signal exhibited by V₂O₅ (refs. ^30,61). (3) Ptychography can access the phase component. Phase changes at the vanadium K-edge are twice as large as changes in the absorption, so that the retrieved phase projections have a higher spatial resolution and superior SNR; see Supplementary Figs. 5 and 6 (ref. ⁵⁸). We performed all data analysis on the phase component of the projections and tomograms only.

Ptychographic linear dichroic X-ray tomography

Data acquisition

Experiments were carried out at the coherent small-angle X-ray scattering (cSAXS) beamline of the Swiss Light Source. The photon energy was selected using a double-crystal Si(111) monochromator. The horizontal aperture of slits located 22 m upstream of the sample was set to 20 μm, creating a virtual source point that coherently illuminates a 220-μm-diameter Fresnel zone plate with an outermost zone width of 60 nm and with engineered aberrations designed to improve reconstruction contrast and spatial resolution⁵⁰. Coherent diffraction patterns were acquired using an in-vacuum Eiger 1.5M area detector, with a 75 µm pixel size, placed 5.235 m downstream of the sample inside an evacuated flight tube. Tomography experiments were performed using the positioning instrument described in ref. ⁶⁹.

To map the local orientation of the apical bond within the examined sample volume in 3D, we exploited its linear dichroism and acquired eight equiangular ptychographic tomograms over 180° at 5.469 keV for different illumination polarizations and sample tilts. Specifically, ptychographic tomograms were acquired with a LH and LV polarization of the incident illumination at 0° stage tilt and at 30° stage tilt (sample in grey and pink in the top two panels on the right of Fig. 1a). Two further tilts were measured, whereby the sample was first rotated by +90° and −90° about the main axis of the pillar, followed by a 30° stage tilt²⁶. The last two tilts are equivalent to tilting towards and away from the beam by 30° (sample in green and blue in the bottom two panels on the right of Fig. 1a). Examination under different sample tilts and X-ray polarizations is required to have sufficient information for the construction of an orientation tomogram representative of the apical bond orientation in 3D^26,47. To change the illumination source native horizontal polarization to vertical, we used a 250-µm-thick diamond crystal phase plate inserted into the illumination path upstream of the zone plate (see Fig. 1a). The phase plate absorbed approximately 65% of the incident photons⁷⁰. The degree of polarization of the X-rays was determined to be approximately 60% using a polarization analyser set-up. The sample tilt was changed using a sample holder insert²⁶. To minimize the acquisition time, we used an adaptive field of view for each group of ptychographic projections. The maximum field of view, horizontal × vertical, was about 24 × 25 μm². The scanning followed a Fermat’s spiral pattern⁷¹. An average step size of 0.8 µm was used for all tomograms. The exposure time per scanning point was 0.1 s. 280 projections were acquired per tomogram.

Finally, using the same acquisition parameters, we acquired an off-resonance ptychographic tomogram of the pillar below the absorption edge at 5.4 keV. This tomogram, being insensitive to any dichroic effects, was used for computing the electron-density tomogram and subsequently used for compositional analysis¹¹. It should be noted that the starting angle and angular spacing of projections was kept constant across all tomograms.

Ptychographic image reconstruction

Ptychographic images (or tomographic projections) were reconstructed using the PtychoShelves package⁷². For each reconstruction, a region of 600 × 600 pixels of the detector was used per scanning point, resulting in an image pixel size of 30.91 nm for the pre-edge and 31.29 nm for the below-edge tomogram. Reconstructions were obtained with 200 iterations of the difference map algorithm⁷³, followed by 300 iterations of maximum likelihood refinement⁷⁴.

Preprocessing of projections

Before any tomogram reconstructions, we: (1) resampled all projections to a pixel size of 30.91 nm using Fourier interpolation; (2) extracted the phase from the reconstructed projections, removed constant and linear phase components and spatially aligned the projections using a tomographic consistency approach³¹; and (3) aligned all projections to a common pillar orientation. As a last step, the different orientations at which projections were measured were characterized by a 3D rotation matrix²⁶, which was input into a specially developed reconstruction code (see the ‘XL-DOT reconstruction’ section below). It should be noted that, owing to the sample tilt and the fixed vertical field of view of the 2D projections, the 3D volume that is commonly sampled in all orientations, and used in the subsequent analysis and visualization, is reduced. (4) Last, to isolate the dichroic component from the isotropic electron-density contribution, the LV projection was subtracted from the LH projection. The resulting set of projections were used in the reconstruction of the XL-DOT dataset, as discussed further below.

Ptychographic tomogram reconstruction

The ptychographic tomogram, acquired with the X-ray energy tuned to below the absorption edge, was reconstructed using a modified filtered back-projection algorithm⁷⁵. This off-resonance phase tomogram was used to derive the electron-density tomogram, which was then used for material component identification^11,32.

XL-DOT reconstruction

A gradient-based iterative reconstruction algorithm was developed to reconstruct the orientation field in 3D. A schematic of the reconstruction process is shown in Supplementary Fig. 7. The process starts with the creation of a 3D starting, random guess of the sample. Using the sample–illumination interaction relationship in equation (2), a set of projections is simulated. These projections are then compared with the measured set of projections and their difference is used to compute a gradient to iteratively correct the initial guess.

The interaction between the electric field of the incident linearly polarized X-rays, \(\overrightarrow{E}\), and the orientation of the apical vanadyl bond, \(\overrightarrow{a}\), can be described as:

Here f is the total scattering factor, which contains the isotropic charge contribution, f₀, and the linear dichroism contribution, \({(\overrightarrow{E}\cdot \overrightarrow{a})}^{2}\), with a pre-factor f_lin that depends on the electronic transition under resonance. Keeping with the experimental geometry (Fig. 1a); using X-rays with a LH polarization parallel to the x axis and denoting an arbitrary polarization angle as φ, in which φ = 0° is LH polarization and φ = 90° is LV polarization, the tomographic rotation and tilting of the sample can be quantitatively represented by the 3D rotation matrix R. In transmission, the measured projection can then be described by the integral given in equation (3). Index summation notation is used to give the rotation of the relevant components of the orientation, a_j. The integration is evaluated along the X-ray propagation direction, the z axis.

Knowing the form of the interaction, the reconstruction algorithm was formulated by generating a guess structure, from which projections were simulated at the same orientations that the sample was measured. These simulated projections, \(\hat{P}\), were then compared with the corresponding measured projections, P. Their square difference was used to define an error metric, ϵ, quantifying how well the guess could reproduce the measured projections, given by

in which m represents the projection index. The error metric was reduced using gradient descent, therefore improving the ability of the guess structure to represent the internal c-axis orientation of the measured sample. By differentiating the error metric in equation (4) with respect to each component, we obtain the following analytical expression for calculating the gradient:

The gradient was evaluated and applied to the guess structure at every iteration. During the reconstruction process, the magnitude of the linear dichroic contrast, corresponding to f_lin, was not constrained and was therefore also optimized during gradient descent. As a result, it is not necessary to predetermine the f_lin value. As the iterative gradient descent reconstruction is prone to converging at local minima, 40 individual reconstructions were performed using different random, non-zero initial conditions. The individual reconstructions are combined by averaging all components to obtain a final reconstruction. The difference in the angular orientations between the individual reconstructions and the final, averaged reconstruction was used to evaluate the standard deviation of the orientation, which is an estimate of the uncertainty in orientation.

Notably, using equation (3), it can be shown that LV polarization (φ = 90°) projection measurements evaluate to

Because there are no vector rotations in this expression, it is equivalent to examining a scalar consisting of two components: the isotropic charge background, f₀, and the (out-of-plane) \({a}_{y}^{2}\) component. This can be reconstructed with conventional tomography and gives contrast between grains that are in-plane (x–y plane) and out-of-plane oriented. This contrast was used for further validation of the final reconstruction, as shown in Supplementary Fig. 12.

Multiaxis tomography

To obtain a first estimation of how many sample tilts and linear polarization states are necessary for a robust XL-DOT reconstruction, we performed a series of numerical simulations and tomographic reconstructions with fewer sample tilt axes (Supplementary Fig. 14). Preliminary reconstructions can be obtained with as little as two tilt axes using LV and LH polarizations only. Both our simulations (not shown) and recent literature^30,47 indicate further that the numerous tilt axes can be replaced by measurements with extra X-ray polarizations⁷⁶. Similar results can also be achieved using laminography^46,48. This offers a route to fewer or even single tilt-axis measurements.

Dose estimation

The total deposited dose over the duration of the experiment and the entire volume of the V₂O₅ pillar was approximately 10⁹ Gy. This estimate is based on the mass density of the sample and the average flux density per projection⁷⁷. No actions were taken to limit the dose, as V₂O₅ is not known to degrade under the present experimental conditions^11,29. For radiation-sensitive materials, preventative measures can be used to mitigate or account for potential radiation damage⁷⁸. Dose-limiting options include scanning and projection sparse acquisition schemes^11,79 that reduce the total deposited dose, changes to the ptychography acquisition such as using an out-of-focus acquisition with micrometre-sized scanning probes which lead to a reduction of both the total and peak dose per area, as well as the implementation of cryogenic and inert atmosphere measurement conditions^80,81.

Spatial resolution

Spatial resolution estimates of projections and tomograms were obtained using Fourier ring correlation and Fourier shell correlation, respectively⁸².

To evaluate the spatial resolution of the acquired projections, we acquired projections under identical conditions, that is, at the same rotation angle, calculated the correlation between these two images in the Fourier domain and estimated the spatial resolution based on the intersection with a one-bit threshold (see Supplementary Fig. 6). This gives spatial resolutions close to the pixel limit of 30.91 nm and 31.29 nm for the on-resonance (5.469 keV) and off-resonance (5.4 keV) measured projections, respectively.

To evaluate the spatial resolution of the electron-density tomogram acquired below the absorption edge, we halved the entire dataset and reconstructed two independent tomograms (Supplementary Fig. 10). This gives a 3D spatial resolution of 44 nm.

To evaluate the spatial resolution of the orientation vector field, the corresponding dataset was similarly split in half and two tomograms of the orientation vector field were calculated. Using Fourier shell correlation, we calculated spatial resolution estimates for each of the orientation scalar components (LD_x, LD_y, LD_z), as shown in Supplementary Fig. 8, providing a lower bound for their spatial resolutions of 84 nm, 45 nm and 89 nm, respectively. Also, we measured edge profiles across sharp features such as 90° grain boundaries, which revealed a maximum edge sharpness of 40 nm, with an average edge sharpness of 73 nm, which we take as the spatial resolution of the orientation tomogram.

Measurement error estimation

To estimate the voxel-level electron-density uncertainty, we calculated the standard deviation (σ) of the electron density in a region of air surrounding the imaged pillar. The average electron density in air and uncertainty was calculated as 0.004 ± 0.007 Å⁻³.

To estimate the uncertainty in the detected linear dichroism, that is, spatial variations in the pre-edge peak intensity, we independently reconstructed the LV and LH phase tomograms with the sample at a fixed sample tilt and then subtracted them from each other. We then isolated a region of air and calculated the standard deviation in the phase shift associated with the voxels in this region. This standard deviation of the phase associated with the air region corresponds to the uncertainty of the dichroic signal. On the basis of this procedure, the uncertainty of the dichroic signal is found to be 1.3 × 10⁻⁴ rad, which corresponds to a refractive index decrement, δ, error of 1.9 × 10⁻⁷.

To estimate the error in the determined orientation, we isolated an elongated grain with a volume of 0.85 µm³ and long-edge length of 3.2 µm that showed the least variance in electron density and V₂O₅ orientation, that is, which is assumed to be single crystal, and calculated the standard deviation (σ) in orientation to be ±10° for azimuth (x–y-plane angles) and ±8° for elevation (out-of-plane angles) (Supplementary Fig. 11).

The critical concentration for element detection can be estimated to correspond to a dichroic magnitude (difference between tomograms taken with different polarizations) of at least twice the reconstruction error. The dichroic contrast of the V₂O₅ is 1.8 × 10⁻³ and the noise in the reconstruction is an order of magnitude weaker at 1.3 × 10⁻⁴. As a result, in V₂O₅, our dichroic contrast is 12 times the error. We can estimate that, if all other parameters are held constant, the concentration of V can be decreased by a factor of 6 and still be measurable.

Present XL-DOT acquisition time and future prospects

The total acquisition time for the XL-DOT dataset used in this work was around 85 h, including sample tilting, changing the polarization and alignment and dead-time overheads. The pure measurement time, however, was only about 24 h. This discrepancy is largely because of the lack of automation. There exist several opportunities to reduce the acquisition time as follows:

1. 1.

   Reduce oversampling: reconstructions using 50% of the tomograms provide similar results (Supplementary Fig. 14).

2. 2.

   Automation and imaging geometry: the measurement of intermediate linear X-ray polarization angles^30,47,70,76 and/or use of the laminography geometry^46,48 will eliminate most of the present acquisition overheads.

3. 3.

   The increase in coherent flux expected from fourth-generation synchrotron light sources promises to reduce scan times for radiation-hard materials⁸³.

4. 4.

   Further innovations such as multibeam ptychography and sparse tomography offer routes to even faster data acquisition^11,84, providing acquisition times compatible with operando measurements^48,85.

Data analysis

Analysis of the dichroic tomogram was performed using in-house-developed MATLAB routines, ParaView and Avizo. To account for the damage caused during the FIB milling step of the sample preparation, we defined a mask that excluded the outermost 90 nm of the sample cylinder from orientation and electron-density volume analysis (Supplementary Fig. 9).

Component identification and isolation

Materials were identified by comparing the tabulated electron densities of the known sample and reference components, listed in Supplementary Table 1, with the PXCT-measured electron densities. Shown in Supplementary Fig. 9 is a volume rendering and a horizontal cut slice through the electron-density tomogram with the corresponding electron-density histogram. The V₂O₅ volume was isolated using threshold segmentation with a lower bound of 0.74 Å⁻³ and an upper bound of 0.90 Å⁻³.

Analysis of topological defects

The topological charge can be determined by considering the winding number associated with a given topological defect. The winding number corresponds to how the crystallographic orientation changes when moving around a circle enclosing the defect in a clockwise manner. For the comet (trefoil) defect, the c axis rotates clockwise (anticlockwise) by +180° (−180°) for one complete revolution. As the crystallographic orientation has completed half a revolution of a full circle (360°), the topological numbers ±1/2 are assigned to them.

Microstructural analysis of V₂O₅ domains

To isolate the V₂O₅ grains and facilitate a correlation between orientation and electron density, we applied the above-defined threshold mask (electron densities between 0.74 Å⁻³ and 0.90 Å⁻³) to the orientation tomogram. To identify and characterize individual V₂O₅ grains, we downsampled the masked XL-DOT reconstruction by a factor of three (transforming a group of 3 × 3 voxels into 1 voxel with an average intensity value of the same size), thus reducing the sensitivity to intragranular variations. Segmentation was then performed by separating regions along high-angle grain boundaries, showing a c-axis orientation difference larger than 10°. Following segmentation, we then calculated the volume of these grains, their mean diameter and their sphericity⁸⁶. Shown in Supplementary Fig. 13 are the corresponding distributions and correlations of the segmented grains.

Sample diameter and photon energy resolution considerations

As linear dichroic phenomena occur near absorption edges or resonant X-ray energies, the X-ray penetration depth at these energies determines the sample diameter that can be investigated with XL-DOT. For most materials, it is the penetration depth at the X-ray energy of the examined chemical element that sets an upper limit on the sample diameter. Taking pure transition metals as an example, this imposes a typical upper limit to the sample size of around 10 µm. Transition-metal-rich functional materials such as catalyst bodies, cathode materials, ferroelectrics, biominerals and concrete, which are also of interest for XL-DOT measurements, exhibit a substantially larger upper sample size limit owing to their internal porosity or composite nature. For instance, a 100 µm-thick V₂O₅ sample transmits around 10% of the incident beam in the pre-edge region (https://henke.lbl.gov/). 3D or nanotomography measurements of such sample diameters are increasingly typical for operando measurements^48,87,88,89.

Although XL-DOT measurements should ideally be performed at the X-ray energy of an absorption edge at which linear dichroic contrast is strongest to maximize contrast in the projections, the range of energies at the absorption edge at which dichroism can be measured can be large. For instance, the full width at half maximum of the near-edge peak in our V₂O₅ spectra used for XL-DOT is approximately 3 eV, which means that even an X-ray energy resolution of up to 3 eV would be sufficient for XL-DOT measurements, albeit at a decreased SNR. There is therefore a degree of flexibility in terms of the required energy position and resolution for XL-DOT measurements.