Description of dark-field x-ray microscopy

The geometry and operational principle of dark-field x-ray microscopy (Figure 1) are conceptually similar to that of dark-field transmission electron microscopy (TEM). The diffracted beam passes through an x-ray objective lens, creating a magnified image of a specific region of interest with contrast from local variations in lattice symmetry, orientation, and strain. The sample-to-detector distance is typically 2–6 m, enabling magnification ratios of up to 50 while still maintaining sufficient space around the sample for in situ apparatus. Like the electron microscope, the x-ray microscope can be operated in a variety of modes. Maps of axial strain can be obtained by scanning the objective and detector through the scattering angle 2θ, while the local misorientation can be similarly mapped by scanning the sample through two orthogonal tilt directions (α, β).

Figure 1. Principle of dark-field x-ray microscopy. Monochromatic x-rays illuminate a crystalline element of interest, and the diffracted radiation is imaged by means of an x-ray objective and a 2D detector. The magnification is approximately given by the ratio q′/p′. Scanning the sample tilt (α, β) and 2θ angles facilitates mapping of orientation and strain, respectively, while different projection angles can be obtained by rotating the sample around its scattering vector, Q.^{Reference Simons, King, Ludwig, Detlefs, Pantleon, Schmidt, Stöhr, Snigireva, Snigirev and Poulsen22}

Three-dimensional measurements of axial strain and orientation can be obtained in two ways: First, by using a one-dimensionally focusing condenser to create a narrow line beam^{Reference Stöhr, Wright, Simons, Michael-Lindhard, Hübner, Jensen, Hansen and Poulsen19} that illuminates a “slice” of the material (typically 200 µm × 200 µm × 200 nm), which is then imaged at an oblique angle. A 3D volume is then obtained layer wise by translating the sample through the planar beam in small increments. A second, faster method involves illuminating the entire grain and recording projections from different viewing angles while rotating the sample around the scattering vector, Q.^{Reference Ludwig, Cloetens, Härtwig, Baruchel, Hamelin and Bastie20} Three-dimensional reconstruction can then be accomplished using adapted tomographic algorithms.

A defining feature of the dark-field x-ray microscope is the x-ray objective. Like a visible light microscope, the x-ray objective can be reconfigured to adjust the magnification and numerical aperture, and hence the spatial resolution according to specific experimental requirements. Compound refractive x-ray lenses are typically used,^{Reference Snigirev, Kohn, Snigireva and Lengeler21} however, such optics are chromatic and have a significantly smaller numerical aperture than their visible light counterparts. The implications of this are threefold. First, the angular resolution is on the order of 0.001° to 0.03°, and thus, superior to those of electron microscopes. Second, the objective filters out diffraction signals from other unwanted regions of the sample, potentially allowing the study of a specific element out of 10¹⁰ or more within the illuminated volume. Third, the narrow angular acceptance necessitates prior knowledge of orientation of the region of interest. Such information is readily obtained from fast prescanning techniques, such as 3DXRD or DCT.

The studies described herein were performed ad hoc at beamline ID06 at the European Synchrotron Radiation Source (Grenoble, France), operating in the x-ray energy range of 15–30 keV. The current spatial resolution is ∼100 nm, while the mapping of strain can, in favorable cases, be performed with a resolution of 10^–5. A dedicated x-ray microscope incorporating the dark-field modality is anticipated to be commissioned at this beamline in 2016 with a target resolution of 50 nm.

Processing of plastically deformed metals

During plastic deformation, dislocations proliferate and self-organize to create hierarchical structures comprising grains, cell blocks, and individual subgrains surrounded by dislocation walls. At present, bottom-up modeling approaches are not capable of creating the complex structures observed using TEM. Furthermore, as heterogeneous events such as the nucleation of new grains or cracks tend to govern the structural evolution, state-of-the art models, at best, systematize generic features of the static structure. Hence fundamental questions remain regarding both the deformation process itself, and the nucleation and growth processes during subsequent annealing.

Figure 2 demonstrates how DCT and dark-field x-ray microscopy can be combined into a unique tool for 3D mapping structures at various length scales in deformed metals. For this study^{Reference Simons, King, Ludwig, Detlefs, Pantleon, Schmidt, Stöhr, Snigireva, Snigirev and Poulsen22} of a needle-shaped Al sample with dimensions of 0.3 mm × 0.6 mm × 8 mm, an x-ray energy of 17 keV was used with image exposure times on the order of one second. Dark-field x-ray microscopy has the potential to directly visualize the formation and dissolution of subgrains, and their interactions with one another and their parent grains.

Figure 2. Multiscale mapping of 10% tensile deformed 1050 (99.5% pure) aluminum. (a) Part of a map of all grains in the specimen. (b) Zooming in on one grain and mapping the intrinsic variation in orientation. A vertical section through the grain is shown. (c) Additional zoom and mapping of selected individual subgrains in 3D. The spatial resolution from left to right is 3.5 µm, 1.5 µm, and 300 nm, and the angular resolution is 0.5°, 0.15°, and 0.03°, respectively. The colors symbolize orientations.^{Reference Simons, King, Ludwig, Detlefs, Pantleon, Schmidt, Stöhr, Snigireva, Snigirev and Poulsen22}

The high-angular resolution of dark-field x-ray microscopy also provides a way to detect fine-scale structures, such as lines of dislocations with low densities. As an example, Figure 3 shows the substructure of a recrystallized Al grain with an angular difference across the observed internal boundaries of the order of 0.05°, corresponding to dislocations separated by more than 0.46 µm.^{Reference Ahl, Simons, Jakobsen, Zhang, Stöhr, Juul Jensen and Poulsen23} The intrusions in the external grain boundary seem to be related to the internal boundaries. Furthermore, unlike conventional techniques such as electron backscatter diffraction, dark-field x-ray microscopy provides a measure of the angular distribution associated with each voxel in the sample (Figure 3c). This is seen as a potential vehicle for improved estimations of local dislocation densities.

Figure 3. Structure of one layer of a recrystallized grain within a ∼500-µm-sized 1050 (99.5% pure) aluminum sample. The distribution of angular spread of the (200) reflection is recorded for each voxel in terms of sample tilt (α, β) (as per Figure 1). (a) Map of the most intense angular setting for each voxel, (b) corresponding map of local misorientations, and (c) map of the full width at half maximum (FWHM) of the mosaic distribution for each voxel.^{Reference Ahl, Simons, Jakobsen, Zhang, Stöhr, Juul Jensen and Poulsen23}

Relatively fast measurements of the dynamics can be conducted by assessing the collective statistics from individual subgrains, such as volume, average orientation, or average strain. During nucleation and growth processes, the evolution of such parameters can be monitored within hundreds or thousands of elements. As an example, Figure 4 shows the subgrain size distribution in an Al sample after deformation at room temperature.^{Reference Ahl, Simons, Jakobsen, Jensen and Poulsen24} As anticipated, the distribution shifts toward larger and fewer grains upon heating.

Figure 4. Histograms of subgrain size (defined as the equivalent spherical radius, R _eq) as a function of a linear increase in temperature during recovery of 1050 (99.5% pure) aluminum. The distribution is prone to experimental error below a radius of 1 µm.^{Reference Ahl, Simons, Jakobsen, Jensen and Poulsen24}

Strain and domain mapping in ferroelectrics

A defining feature of ferroelectric materials is the presence of domains—regions of parallel spontaneous polarization that self-organize across several length scales to minimize the global free energy.^{Reference Damjanovic25} Under external fields, these domains can nucleate, grow, shrink, and annihilate through the collective motion of domain walls, facilitating a diverse range of applications, including memory, microelectromechanical systems, and ultrasound technology. Despite the crystallographic twinning relationship between neighboring domains, ferroelectrics tend to be distorted in the local vicinity of the domain wall, giving rise to complex local phenomena, such as conductivity^{Reference Siedel, Martin, He, Zhan, Chu, Rother, Hawkridge, Maksymovych, Yu, Gajek, Balke, Kalinen, Gemming, Wang, Catalan, Scott, Spaldin, Orenstein and Ramesh26} and polarization rotation.^{Reference Catalan, Lubk, Vlooswijk, Snoeck, Magen, Janssens, Rispens, Rijnders, Blank and Noheda27}

Understanding the relationship between domain topology, orientation, and stress around domain walls would therefore be a significant advance. Yet, the topologies and dynamics of ferroelectric domains are extremely complex—3D and multiscale—and highly sensitive to internal and external boundary conditions. With dark-field x-ray microscopy, we can directly map the local variations in strain and orientation that occur at domain walls and other critical interfaces. In this example study,^{Reference Simons, Jakobsen, Ahl, Detlefs and Poulsen28} the domain topology and strain distribution were mapped within an isolated region of an orthorhombic KNbO₃ ferroelectric crystal. The microscope imaged the 220 family of reflections at an x-ray energy of 17 keV to give a spatial resolution of 95 nm × 250 nm × 250 nm.

Figure 5 shows a coarse, nonspatially resolved overview of the orientation and strain distributions within the grain obtained by recording the far-field diffraction intensity distribution of the 220 family of reflections (i.e., without the objective in place).^{Reference Simons, Jakobsen, Ahl, Detlefs and Poulsen28} The two intense peaks correspond to two domain variants dominating the sampling volume, however, it is their local structure and the intensity between them that is of interest. The peaks are not ideally Gaussian as expected, rather they are elongated and asymmetric with “wings” of intensity between them. These features are deviations from the presumed orthorhombic crystal symmetry of KNbO₃, and strongly indicate the presence of strain and orientation inhomogeneities associated with the domain topology.

Figure 5. Reciprocal space map around the 220 reflection of a KNbO₃ ferroelectric crystal obtained as a rapid prescanning technique. The peaks correspond to the lattice spacing and orientation of domain variants. Q _x and Q _y are the reciprocal space directions corresponding to α and β. The contours correspond to a normalized log axis, where red is the highest intensity value.^{Reference Simons, Jakobsen, Ahl, Detlefs and Poulsen28}

The topological origin of this intensity distribution can be directly imaged using dark-field x-ray microscopy, as shown in Figure 6.^{Reference Simons, Jakobsen, Ahl, Detlefs and Poulsen28} Both the orientation and strain maps show a topology consistent with contemporary microscopy techniques, containing stripe-shaped domains separated by non-180° ferroelastic domain walls. In particular, the strain map indicates clearly that strain gradients are a pervasive and significant feature in ferroelectric crystals. Zooming in both spatial and angular domains indicates that significant strain and lattice orientation gradients extend up to 2 µm into the material.

Figure 6. False-color maps of (a) orientation and (b) strain in the KNbO₃ crystal. Zoomed in images of the (c) orientation and (d) strain from the centers of the maps show local gradients around stripe-like domains. The orientation and strain maps correspond to the same region of the sample and are shown at the same scale. The local orientation in terms of sample tilt (α, β) (as per Figure 1) in (a) and (c) corresponds to the keys in the bottom left corner of the images. The local lattice strain in (b) and (d) corresponds to the key on the right side of the images.^{Reference Simons, Jakobsen, Ahl, Detlefs and Poulsen28}

These results have significant and far-reaching implications for ferroelectric and other ferroic materials, such as shape memory and superelastic alloys. The cumulative effects of strain and orientation gradients at domain walls imply that the local polarization at such boundaries must be significantly augmented by both direct lattice rotation and flexoelectric effects.^{Reference Zubko, Catalan and Tagantsev29} These observations constitute a significant step forward in the direct characterization of complex multiscale phenomena in these materials, and are directly comparable to similar multiscale models from, for example, phase-field techniques.^{Reference Gu, Li, Morozovska, Wang, Eliseev, Gopolan and Chen30}

Mapping dislocations

Visualizing the 3D dislocation structure within single crystals, grains, or domains is essential to understand micromechanics and is vital for controlling the performance, for example, of photovoltaics and microelectronics. Traditionally, TEM is used to visualize the dislocation structure due to the high-contrast and high-resolution images obtained from the technique, even in 3D.^{Reference Liu, House, Kacher, Tanaka, Higashida and Robertson31} However, the small samples are subject to spurious boundary conditions from the surfaces. X-ray topography, on the other hand, can visualize hundreds of dislocations in centimeter-sized single crystals and can be generalized to 3D reconstructions.^{Reference Hänscke, Helfen, Altapova, Danilewsky and Baumbach32} However, the spatial resolution of the “direct beam” approach used is inherently limited to approximately 10 µm,^{Reference Tanner33} putting a severe upper limit on the detectible dislocation densities for which it is applicable.

The dark-field microscope provides an image of the regions in the sample that are both within the direct-space field of view and give rise to diffraction in the range of angles accepted by the objective. The angular resolution is sufficiently good in that the microscope can be set to focus on parts of the tails of the scattering, thus excluding the response from the main peak. The “weak beam” scattering defined in this way will be approximately kinematical in nature. Furthermore, by acquiring images as a function of distance to the main peak in reciprocal space, one can map strain components in 3D. This is illustrated in Figure 7 for the case of a diamond crystal.^{Reference Jakobsen, Simons, Ahl, Detlefs, Härtwig and Poulsen34} As expected, upon crossing a dislocation line, the strain is inverted, as seen by the change from red to blue color in Figure 7. The sharpness of the dislocation topology in the high-strain regions near the core implies that the detectable dislocation density is dictated by the spatial resolution of the microscope. These results point to a route for experimentally addressing dislocation dynamics.

Figure 7. False-color image of three dislocations within a diamond crystal. Two images are superposed, the red and blue ones representing an offset in the axial strain of +3 × 10^–4 and –3 × 10^–4, respectively. The sample was fully illuminated and, as such, the image is a projection.^{Reference Jakobsen, Simons, Ahl, Detlefs, Härtwig and Poulsen34}

We emphasize that only three of the nine components of the displacement gradient tensor can be measured from the type of mapping previously described, as we only probe one diffraction vector, Q. To map all components, it is required to repeat the data acquisition for at least two additional reflections. (Likewise, to map full orientations and not just pole figures, two reflections are required.) To facilitate this, the microscope at the ID06 beamline can operate with the objective and far-field detector both in the horizontal and vertical planes.

Outlook

Dark-field x-ray microscopy is a promising technique for comprehensive multiscale studies of self-organization, domain evolution, transport, and the role of defects within crystalline materials. In addition, we can highlight two further potential applications:

• • In operando studies of devices–Spatially localized phenomena govern the performance and lifetime of many devices or components. Dark-field x-ray microscopy would enable one to directly record movies of the structural changes taking place at embedded interfaces in, for example, electrochemical cells or microelectromechanical devices.

• • Nucleation–Nucleation processes (e.g., phases, grains, and domains) determine transformation rates, yet mechanisms are often not known, and quantitative data for simulation models is sparse. With this technique, we can map sufficiently large specimens to identify such events, and zoom in on individual nuclei to retrieve crucial local parameters, such as wetting angles and orientation relationships.

Like electron microscopes, this approach can easily incorporate additional modalities, such as bright-field microscopy,^{Reference Mathiesen, Arnberg, Li, Meier, Schaffer, Snigireva, Snigirev and Dahle35} high-resolution x-ray microscopy,^{Reference Bosak, Snigireva, Napolskii and Snigirev36} and small-angle x-ray scattering microscopy,^{Reference van Schooneveld, Hilhorst, Petukhov, Tyliszczak, Wang, Weckhuysen, de Groot and de Smit37} as well as the use of coherent x-ray nanobeams for ultimate resolutions.^{Reference Robinson and Harder38–Reference Godard, Carbone, Allain, Mastropietro, Chen, Capello, Diaz, Metzger, Stangl and Chamard40}

With continual improvements in hard x-ray optics, such as multilayer Laue lenses, improved spatial resolution is also expected in the coming years.^{Reference Morgan, Prasciolu, Andrejczuk, Krzywinski, Meents, Pennicard, Graafsma, Barty, Bean, Barthelmess, Oberthuer, Yefanov, Aquila, Chapman and Bajt41} While temporal resolution (and thus the suitability for in situ measurements of structural dynamics) is presently limited by the photon flux of the x-ray source, upcoming “ultimate storage rings”^{Reference Eriksson, van der Veen and Quitmann42} in fourth-generation synchrotrons promise to improve this by several orders of magnitude.