Informing quantum materials discovery and synthesis using X-ray micro-computed tomography

The presence of inclusions, twinning, and low-angle grain boundaries, demanded to exist by the third law of thermodynamics, drive the behavior of quantum materials. Identification and quantification of these structural complexities often requires destructive techniques. X-ray micro-computed tomography (µCT) uses high-energy X-rays to non-destructively generate 3D representations of a material with micron/nanometer precision, taking advantage of various contrast mechanisms to enable the quantification of the types and number of inhomogeneities. We present case studies of µCT informing materials design of electronic and quantum materials, and the benefits to characterizing inclusions, twinning, and low-angle grain boundaries as well as optimizing crystal growth processes. We discuss recent improvements in µCT instrumentation that enable elemental analysis and orientation to be obtained on crystalline samples. The benefits of µCT as a non-destructive tool to analyze bulk samples should encourage the community to adapt this technology into everyday use for quantum materials discovery.


INTRODUCTION
Materials behavior is critically impacted by extended structure complexity on the micron/nanometer scale. These hierarchical structures can hinder or enhance the desired properties, especially for quantum materials where a balance of energy scales makes the ground states particularly sensitive [1][2][3] . While more focus is directed on the atomic scale and periodicities, bulk features (on the scale of >1 µm) can also mask the intrinsic behavior of a sample. To avoid these is impossible according to the third law of thermodynamics, but, to even controllably tune these intricacies is difficult depending on the synthesis conditions, as seen in materials such as metal alloys, crystal scintillators, and thermoelectrics [4][5][6][7] . This can be troublesome in particular for quantum materials, where inclusions, low-angle grain boundaries, crystal twinning, and other structural imperfections can impact the physical properties of a material, such as in the superconductors YBa 2 Cu 3 O 7-δ (impacting the critical current density) and Sr 2 RuO 4 (impacting the T c ) [8][9][10][11][12] .
Many characterization techniques commonly used probe too small-scales, focus solely on the surface, and/or require destruction of the sample to visualize and quantify such features [13][14][15] . X-ray micro-computed tomography (µCT) is a non-destructive alternative technique that allows one to obtain a 3D representation of the sample with resolution in the µm to nm range. This method uses high-energy X-rays to transmit through the sample, collecting multiple images that are then reconstructed into a 3D object where extended structural features can be visualized through a variety of contrast methods 16 . Since its conception, µCT has been integrated as a routine characterization tool for a variety of fields, including medicine 17 , biology 18 , and geology 19 . While this method of characterization is used extensively in materials science 20 , for the condensed-matter community it has only recently been adopted in the analysis of electronic and quantum materials [21][22][23] . Throughout this paper, we demonstrate examples in which µCT can be an important characterization tool to aid in materials design and show insights that would not be possible without this technique.

RESULTS AND DISCUSSION
Volumetric measurements and sample porosity Figure 1a shows a general schematic of the key components of a µCT instrument. In this technique, X-rays transmits through the sample from a polychromatic source, usually following a cone or fan beam geometry. The X-rays then hit the scintillation detector, collecting a series of images while the sample rotates on a platform by 180°or 360°(dependent on sample geometry). From this point the collection of images is used to reconstruct the 3D sample, typically done using a filtered back-projection algorithm. There are a variety of collection/reconstruction strategies and algorithms to avoid artifacts that are better addressed in related reviews 19,24 . The reconstruction is a 3D representation, with voxel (3D pixel) by voxel contrast measurement (e.g., Beer's law in the case of X-ray attenuation contrast). This contrast allows for the characterization of a variety of defects that plague electronic and quantum materials, examples of which are shown in Fig. 1b in accordance with the time and length scales they occur. Common inhomogeneities that impact materials' properties, along with their corresponding dimensionalities, are listed in Table 1, adapted from ref. 25 . These are wide ranging in dimensionality and much work has been dedicated to controlling their origin [26][27][28][29][30][31][32][33] .
A simple measurement attained from µCT is sample geometry down to~µm-nm resolution. This is especially helpful for thermal/ electrical transport measurements of geometrically complex samples where an accurate volume or cross-sectional area are still needed. It enables the determination of density, as well as the observation of voids or cracks that can impact the transport properties 34 . Alongside techniques such as resonant ultrasound spectroscopy, it also enables rigid body simulations for determining vibrational response, allowing extraction of phonon properties.
These volumetric measurements can be helpful when one is doing a high throughput search of materials where crystal growth may be too difficult or time-consuming, instead relying on wellcompacted powders using instruments such as a spark plasma sintering system 35 . In this example, being able to calculate the porosity, or % volume of voids in relation to the total volume, would be useful for comparison. Figure 2a shows an example of Cu powder that was cold pressed to better emphasize the size of voids that can occur. As air is less attenuating than the bulk, the voids can be easily visualized and then quantified using µCT. One can quantify the number of detected voids to be 67 and their volume distribution can be visualized in the histogram distribution in Fig. 2b. In this example, most voids are a volume of similar magnitude of each other. While other techniques can be used to quantify pores, µCT gives us a look at the 3D structure, highlighting internal voids that may be more difficult to characterize. This type of measurement expands the characterization methods possible in determining sample quality, bolstering the validity of the results of transport measurements.

Inclusions from flux crystals
The previous example showed an important insight in the application of µCT, that sample components can be distinguished and visualized non-destructively throughout the bulk. This contrast method can occur when the compounds have different attenuation coefficients in the operating X-ray voltage of the instrument. This ability can be put to great use in flux crystal growth. In the flux method, one uses an excess of a compound to dissolve the desired material below its melting point, where upon gradual cooling the flux reaches supersaturation resulting in crystallization of the target sample. This technique is widely used in materials discovery as a low-cost method to grow millimeterscale crystals 36,37 .
For both oxides and metals, one would normally use a flux that is chemically related to the target material, but sometimes an unrelated compound must be used. Poor selection of flux would result in a mixture of phases and poor crystallite size, where µCT could be utilized to help understand the growth formation mechanisms and segregation of phases if combined with a secondary technique such as powder X-ray diffraction to understand phase composition. Even use of a successful flux can still result in inclusions in the sample that can impact materials properties, as well as be indistinguishable without destroying the crystal 38 .
A classic example of this is the material SmB 6 , a well-known Kondo insulator with reported quantum oscillations that supports the suspected topological behavior 39 . The most common method to grow this material is by using an excess of Al flux, which can show up as inclusions 21 . Figure 3a is a graph of the attenuation coefficients of Al and SmB 6 as a function of X-ray voltage showing that provided one is in the right range of energy (in this case the operating range of the instrument), it is easy to distinguish these inclusions from the bulk sample. A comparison of the crystal structure between SmB 6 and Al in Fig. 3b shows that given the similar structure and lattice parameter, the two materials will likely seed the growth of one another. Figure 3c and d demonstrates this, where flux grown SmB 6 crystals with epitaxially grown Al inclusions is visualized using µCT. Measurements of samples while systematically thinning the crystal show a disappearance in the quantum oscillations 40 . Since the presence of Al makes it difficult to elucidate the inherent characteristic behavior of SmB 6 , µCT can be combined with other measurements (such as bulk magnetization) as an effective way to non-destructively screen samples.
Screening samples is helpful, but in this example µCT also gives us information regarding the formation of these extended features. Not only do we see the presence of inclusions, but we also see that they are not uniformly distributed throughout the sample and instead develop a preferred orientation. The absorption data collected lets us quantify the size, distribution, and orientation of these inclusions using software such as Blob3D and Quant3D 41,42 .
One measurement that can be performed is the star volume distribution (SVD), with other examples being the mean intercept length and star length distribution. For a distribution of points in the segmented data, the SVD determines the distance traveled before the interception of a boundary, with the measurement performed over the volume of the specimen 43 . We can use such measurements of the Al to calculate a degree of anisotropy of 0.96 (where a value of 0 is completely isotropic and 1 is completely Fig. 1 µCT setup and the formation of defects. a A representation of the key components in a µCT instrument. The X-ray source shown generates X-rays in a cone-beam geometry as they transmit through the sample and hit the scintillation detector, all while the sample rotates on a stage. b A comparison of the various time and length scales defects can form in electronic and quantum materials. Precipitates 3D Inclusions 3D

Internal voids 3D
A list of some common inhomogeneities that can impact a material's properties, adapted from ref. 25 .
anisotropic). The ratio of the largest eigenvalue over the smallest is another measure of the degree of anisotropy (DA) and is 220 (20) for Fig. 3c. We can then further generate a 3D rose diagram, a projection of the calculated SVD measurements, to visualize the directionality of these features relative to the crystal itself, as seen in Fig. 3e. The diagram shows that the inclusions preferentially grow along the length of the crystal.
As the same material can form different crystal habits, µCT as a tool affords us the opportunity to compare the inclusions between different crystal shapes of SmB 6 , previously never investigated in the community. The more cuboid-shaped crystals, as rendered in Fig. 3d, show sheets of aluminum form perpendicular to the main crystal axis. The eigenvalue DA determined through Quant3D is. 2.75 (9). A visualization of the Al using a 3D rose diagram of the   6 and Al, demonstrating that given the similar lattice parameter, both systems mutually support the growth of one another. c A 3D rendering of a crystal of SmB 6 showing the presence of Al inclusions (colored yellow) that are epitaxially oriented. d A rendering of a different crystal of SmB 6 with Al inclusions showing up as epitaxially oriented layered sheets. e A 3D rose diagram generated from the star volume distribution (SVD) of the inclusions of (c). The data are oriented along the length of the crystal, showing a preferred growth direction following the crystal habit. The image is colored according to distance from origin. f A 3D rose diagram generated from the SVD of the inclusions of (d), showing the impact the opposing layers have on the overall anisotropy. The image is colored according to the distance from the origin. Fig. 3f shows a different form of anisotropy compared to the SmB 6 rods. In this case, two competing sheet-like layers are developed in opposing crystalline directions. The data shows that in the discovery of quantum materials, extended features do matter and techniques such as µCT should be implemented to carefully examine the distribution of such inhomogeneities. Further work can be performed using this method to analyze inclusions of other materials exhibiting quantum oscillations to correlate the orientation with the quantum oscillation frequency 44 .

SVD in
While SmB 6 serves as a shining example of the power of µCT, this technique can be used beyond detecting defects to also help optimize crystal growth parameters. This is demonstrated in the case of rutile GeO 2 (r-GeO 2 ), an ultra-wide band gap semiconductor with predicted ambipolar doping. Much work has been invested in the development of controlled defects in bulk and thin film samples to invoke these aforementioned properties for electronic devices [45][46][47] . Here, we discuss the improvement in crystal quality of flux grown single crystals of r-GeO 2 . In flux growths, the cooling rate can have a large impact on not only the size, but also the quality of the crystals obtained. In Fig. 4a, we see a single crystal of r-GeO 2 grown using flux method with a cooling rate of 3°C/h. Laue diffraction shown in Fig. 4b gives clear, defined spots along one of the facets.
Further inspection using X-ray µCT in Fig. 4c shows voids present in the sample. We calculate the degree of anisotropy to be 0.79, with a 3D rose diagram generated in Fig. 4d. The eigenvalue DA determined through Quant3D is 7.6(7). In this instance, the axes were aligned to the [011] direction, showing a preferred orientation of the voids parallel to the crystal facet. A similar crystal of r-GeO 2 cooled at a rate of 0.5°C/h through the same temperature range shows no sign of voids above the µm scale. The improved quality of this crystal is seen in the optical microscope image of Fig. 4e, and the volumetric rendering shown in Fig. 4f. This example follows the common logic that a slower growth leads to better quality crystals, although this need not always be the case. Through this comparison, we show how µCT can be implemented in flux crystal growth to screen crystals for inclusions as well as help optimize the growth parameters. We also see that dependent on the crystal habits, one can quantify features accurately along crystallographic orientations. Extended defects in floating zone samples Flux growth has been the primary form of synthesis discussed so far, but X-ray µCT can assist other synthesis approaches and crystal growth methods. Techniques involving directional solidification, such as Bridgman and floating zone, can also be affected by inclusions as well as voids and air pockets that incorporate into the grown crystal [48][49][50] . These defects impact crystal quality but can be prevented by improvements in sample preparation or changes in the growth parameters. An example of this can be considered in the floating zone technique. Internal voids can form in this technique due to air pockets in poorly densified rods as well as the generation of gas as a side product when the compound melts 51 . A less obvious method of void formation is due to the gas environment itself. While applied pressure of different gas species can be useful in floating zone growths to reduce volatility or stabilize certain oxidation states, it can be detrimental to growth stability as well 52,53 .
We demonstrate the applicability of µCT in analyzing these internal voids in the floating zone growth of YbMgGaO 4 . This material, whose crystal structure is shown in Fig. 5a, is a quantum spin liquid candidate due to the frustrated triangular lattice Yb 3+ ions occupy 54,55 . µCT data collected on a section of a sample grown in PARADIM's 300 bar optical high-pressure floating zone is visualized in Fig. 5b to highlight the presence of internal voids. In this case, a larger applied pressure of 20 bar of O 2 gas led to void formation compared to an applied pressure of 5 bar done in previous papers, likely due to the dissolution of the gas at higher pressures. Figure 5c shows an optical microscope image of the melt as support for both the presence of detectable voids as well as their directionality. To further understand the anisotropic nature of these features, the relative orientations of the voids according to their volume are shown using a contour plot in Fig. 5d. The larger voids tend to cluster around certain areas of orientation, showing a weak anisotropy present as a function of volume, supported by the calculated degree of anisotropy being 0.39. This intuitively makes sense, as the larger size pockets are present in the growth for a longer time and hence have more directionality applied to them because of the growth method itself. Examining the distance between the largest pockets show angles of~30°and 120°, supporting the proposed trigonal structure of the material.

Recent developments in instrumentation
All experiments shown so far were accomplished using a typical lab-bench-sized µCT instrument, showcasing the power of this technique in an accessible research environment. Further advances in instrumentation open up possibilities in using µCT to inform materials synthesis. The onset of accessible hyperspectral detectors has given µCT the ability to perform 3D bulk elemental analysis on the scale of µm 56 . This capability could easily be applied to materials synthesis involving complex phase diagrams. In these cases, distinguishing between the different components may not be naively easy without the use of hyperspectral detectors. Where the hyperspectral detector allows for elemental analysis, Diffraction µ-Computed Tomography (DCT) allows for orientation analysis of 3D volumes of both polycrystalline and single crystalline samples 57 . This has already had some use in ceramics to track grain growth while sintering and has been shown to inform crystal morphology in materials discovery 58,59 . Being able to follow grain selection in floating zone growths using DCT could be used alongside current methods to allow for a better optimization of growth conditions 60 .
An example of such visualization is shown in Fig. 6a, where we have performed DCT on a floating zone-grown crystal of rutile TiO 2 . Since rutile and other similar materials are desired for their large birefringence, low-angle grain boundaries can be harmful to their optical properties 61,62 . The grains are clearly highlighted in the diffraction data, as this was a piece from early in the growth where multiple domains are still competing. An interesting  Fig. 6b, where a Sobel filter was applied to emphasize edges. Without any knowledge of the diffraction data, one can still visualize some grains due to inherent gaps created between the domains. This can lead to normal µCT being a useful screener prior to DCT in identifying potential boundaries present. We can use the combined data to determine domains for sample extraction, as well as to understand more about growth mechanisms occurring, in particular for cases where competing polytypes and intergrowths could be occurring 63,64 .
Techniques such as in situ XRD available at synchrotron facilities already allow for insights into sample synthesis routes 65,66 . Current µCT instruments allow for the application of heating, cooling, and force to examine changes over time. Applying in situ µCT in a crystal growth process would take us one step further and allow one to have a pseudo-real-time response to conditions to avoid defects and inclusions. A traditional floating zone setup that incorporates translating/rotating samples could be adapted for such a purpose, where once steady state has been achieved, a crystal growth is occurring at a (normal) slow rate and rotating for each collection of images. While data sets of µCT are normally gigabytes to terabytes in size, to process and describe this amount of data is an example of "big" data challenges the community needs to address in the future of guided materials discovery.
Throughout this review, we have shown the capabilities of µCT in aiding materials design. The data collected through this technique can aid other characterization methods through the calculation of volumes and quantification of pore sizes for transport measurements. The detection of inclusions can help non-destructively screen samples for quality as well as give insight into defect formation/orientation. Sample synthesis can be improved upon through the analysis of extended features, such as voids that are sometimes hidden in other techniques. Finally, we showed how advances in µCT instrumentation can expand upon the current capabilities by allowing for elemental and orientation analysis as well as in situ measurements. Overall µCT is an effective non-destructive characterization tool that can be widely implemented throughout the materials community to assist in the synthesis of electronic and quantum materials.

METHODS
Crystals of SmB 6 were grown following a similar procedure of Phelan et al. 21 . Crystals of r-GeO 2 were grown following a similar procedure of Niedermeier et al. 46,67 . Initial powder and floating zone rods of YbMgGaO 4 were prepared following a similar procedure to Li et al. 54 . Crystals were grown using the 300 bar high-pressure floating zone furnace (SciDre GmbH) at the Platform   The point-spread function radius for each CT dataset is shown to give an estimation of the length of distinguishable features.
for the Accelerated Realization, Analysis and Discovery of Interface Materials (PARADIM) user facility at Johns Hopkins University. A 5 kW Xenon arc lamp was utilized as a heating source at a pressure of 20 bar with an oxygen flow rate of 0.1 L/min. Feed and seed rods were rotated together at 20 rpm. The feed rod and the seed rod were advanced at rates of 10 to 12 mm/h and 10 mm/h, respectively. Crystals of r-TiO 2 were grown following the procedure of Wright et al. with no sample rotation applied for either the seed or feed rod 60 . Standard µCT data were collected using a Bruker SkyScan 1172 with an operating range of 20-100 kV for the X-ray source and 0.9 µm maximum pixel resolution using an Al+Cu filter. Specifically, the point-spread function for each measurement is listed in Table 2 following the procedure of Ketcham et al. 68 . Reconstructions were done using NRecon with ring artifact removal and beam hardening correction algorithms applied. The reconstructed data were visualized and analyzed using Dragonfly software, Version 2021.1 for Windows (Object Research Systems (ORS) Inc, Montreal, Canada, 2020; software available at http:// www.theobjects.com/dragonfly). Measurements used for the 3D rose diagrams were generated using Quant3D (https:// www.ctlab.geo.utexas.edu/software/quant3d/). Diffraction computed tomography data were collected using a Zeiss Versa 620 with a voxel size of 2.5 µm. DCT projections were reconstructed using the commercial software GrainMapper3D TM developed by Xnovo Technology ApS. Back-scattered X-ray Laue diffraction was collected with a tungsten source operating at 10 kV and 10 mA with a Multiwire Laboratories MWL 110 real-time back-reflection Laue camera. The tool MuCalc (https://www.ctlab.geo.utexas.edu/ software/mucalctool/) was used to plot attenuation versus X-ray energy, drawing data from the NIST XCOM database 69 . Crystal structures were visualized using Vesta software 70 .

DATA AVAILABILITY
Raw data from the facility related to these experiments are available at https:// doi.org/10.34863/sd37-3694.