Imaging screw dislocations at atomic resolution by aberration-corrected electron optical sectioning

Screw dislocations play an important role in materials' mechanical, electrical and optical properties. However, imaging the atomic displacements in screw dislocations remains challenging. Although advanced electron microscopy techniques have allowed atomic-scale characterization of edge dislocations from the conventional end-on view, for screw dislocations, the atoms are predominantly displaced parallel to the dislocation line, and therefore the screw displacements are parallel to the electron beam and become invisible when viewed end-on. Here we show that screw displacements can be imaged directly with the dislocation lying in a plane transverse to the electron beam by optical sectioning using annular dark field imaging in a scanning transmission electron microscope. Applying this technique to a mixed [a+c] dislocation in GaN allows direct imaging of a screw dissociation with a 1.65-nm dissociation distance, thereby demonstrating a new method for characterizing dislocation core structures.

E dge dislocations are characterized by columns of atoms in tension or in compression in a plane normal to the dislocation, above and below the slip plane. Such dislocations are conventionally viewed end-on at high resolution in transmission electron microscopes because the tensile and compressive displacements are clearly visible [1][2][3][4][5] . Screw dislocations are characterized by equivalent atoms in columns parallel to the screw dislocations being displaced along the screw axis to form a helicoid, which cannot be observed end-on because the displacements are parallel to the viewing direction. Such screw dislocations do however show small rotational displacements in thin foils due to surface stress relaxations known as the Eshelby twist 6 , and these can affect both the contrast and peak positions in high-resolution images [7][8][9][10] . Here we show that the helicoidal displacements around a screw can be imaged directly with the dislocation lying in a plane transverse to the electron beam by optical sectioning using annular dark field (ADF) imaging in a scanning transmission electron microscope (STEM). This novel technique is illustrated by application to the screw component [c] in the dissociation of a mixed [a þ c] dislocation in GaN, which has only been observed end-on previously 11 .
Because of the relatively large lattice mismatch with the substrate, GaN films grown parallel to (0001) generally contain threading dislocations, and during growth the dislocations tend to align themselves close to the [0001] direction. Three types of dislocations are found: edge, screw and mixed dislocations with Burgers vectors [a], [c] and [a þ c], respectively. The [a þ c] mixed dislocations are frequently found to be dissociated 4,5 , and a recent study in ref. 11 showed that the edge component dissociates by climb according to the reaction [a] ¼ ½[a] þ ½[a] to form a fault lying in the 2 1 10 f g plane (a-plane). Since the ½[a] shift would give rise to Ga-Ga and N-N bonding and therefore a very high-energy interface, an additional ½[c] shift along [0001] was proposed so that correct Ga-N pairings across the interface are possible. This implies that the screw component is also dissociated according to , and therefore the total dissociation reaction proposed is [a þ c] ¼ ½[a þ c] þ ½[a þ c], and the two partials are bounding a fault with a structure similar to the one proposed by Drum in ref. 12, with the dissociation distance of B1.8 nm measured on the electron micrographs 5,11 .
The development of aberration correctors in (S)TEM (transmission electron microscopy) has led to a dramatic improvement in spatial resolution in both STEM 13,14 and conventional TEM 15 ; however, there is also a considerable reduction in the microscope depth of field, which in a state of the art microscope may be just a few nanometres. This creates an opportunity to focus at specific depths within the sample and to extract information from the layers at those depths-a process referred to as 'optical sectioning'. Achieving the full-depth resolution available for a general sample requires a confocal configuration; however, at atomic resolution a conventional STEM configuration can be used 16,17 .

Results
Image simulations of the screw displacements. Figure 1 shows that the shearing of atomic planes around a screw dislocation can be imaged in a transverse view of the dislocation line using optical sectioning. A model of the c-type screw dislocation along [0001] in GaN is used, and atomic positions in single layers parallel to 2 1 10 ð Þ planes normal to the electron beam are shown at various depths relative to the dislocation line. The atomic positions are calculated for an infinite crystal assuming isotropic elasticity (see Supplementary Note 1), an assumption based on the fact that GaN is isotropic on the basal plane 18 . In the TEM thin foil, the surface relaxation due to image dislocations was considered, and the displacements due to such image dislocations were found to be negligible compared with the screw shear of up to ½[c] (see Supplementary Fig. 1 and Supplementary Note 2). Far from the dislocation core, the screw displacements vary slowly across the field of view as expected from the lower shear strain that exists further from the dislocation core (Fig. 1a,c,g,i). Close to the core (Fig. 1e), screw displacements with a rapidly varying shear of the (0002) planes are observed giving an apparent displacement of ½[c] across the dislocation core (as expected for a total Burgers vector of [c]).
In the simulated ADF images (100 keV beam energy, probe convergence semi-angle 30 mrad), the peaks associated with the atomic columns become double peaks as the focal plane approaches the dislocation depth, and the relative intensities of these peaks change with defocus ( Fig. 1d,f,h). (For a smaller convergence semi-angle, the depth-dependent information cannot be observed, see Supplementary Fig. 2 and Supplementary Note 3.) The evolution of double peak intensities with changing defocus can be explained in terms of the elongated nature of the illuminating probe along the electron propagation direction. When the screw dislocation lies transverse to the electron beam, the atomic columns follow an arctangent shape (Fig. 2). As the probe is scanned at various depths, the strength of electron scattering will depend on the number of atoms included in the most intense part of the illuminating probe. Because of the elongated probe, the section of an atomic column that is aligned with the beam will have a stronger projected intensity than the more sheared sections. Figure 2 explains why a column appears as a double peak with comparable intensities when focused close to the core, and why the relative intensity of double peaks is reversed when the focus is changed from above to below the core. One of the peaks of the doublet arises from the column just above the screw dislocation, the other from the column just below. The shear across the screw, and therefore the magnitude of the Burgers vector, can be determined from the positions of, say, the line of strong peaks across the screw (or equally from the weak peaks). The shear measured using the stronger peaks appears reversed when the focal plane changes from above to below the core allowing us to determine the sign of the screw Burgers vector. An approximately 2-nm pre-focusing effect due to electron channelling is taken into account 19 in order to determine the correct depths of focal series images.

STEM optical sectioning of GaN dislocations at the atomic scale.
The optical sectioning technique is now applied to understand the mixed dislocations in GaN. An [a þ c] dislocation lying approximately along [0001] in the plane of the sample parallel to 2 1 10 ð Þ was identified by the weak-beam technique; however, no dissociation was apparent in this experiment. A focal series of high-resolution STEM ADF images was acquired. The 30-mrad probe convergence aperture gives a depth of field of 7.28 nm. Focused close to the dislocation core (Fig. 3), the (0002) planes both above and below the dislocation core, sheared in opposite directions, are simultaneously resolved, with the higher-intensity planes (indicated by solid curves) corresponding to atoms located closer to focus and the low-intensity planes (dashed curves) being further away from focus and on the opposite side of the core. The rapid shear occurs along two distinct bands suggesting that the screw dislocation is dissociated into two partials. The magnitude of the shear across each partial is 1 4 [c], and therefore the Burgers vector of each screw partial is ½[c], giving a total screw Burgers vector of [c] as expected for GaN. The screw shear is further confirmed by quantifying peak positions from the ADF focal series Image simulations of dissociated dislocation models. To measure the dissociation distance accurately, structure models of the dissociated dislocation were built and multislice frozen phonon image simulations were carried out. The atomic positions in the dissociated ½[a þ c] þ ½[a þ c] dislocation models were determined using an isotropic elastic strain field in an infinite crystal, assuming that the cores of the screw and edge partials are at the same positions. The positions of the cores of the edge dislocation partials were determined by considering several candidate positions 20 (Supplementary Fig. 4). By placing the cores at the midpoint between neighbouring columns (Fig. 4a), the resultant dissociated dislocation structures (Fig. 4b)  Quantification of the screw dissociation distance. The key difference between the images of the dissociated screw (Fig. 4c) and undissociated screw (Fig. 4d) is the rate at which the screw displacements change across the dislocations. The screw displacements in the fault region between two dissociated screw partials change more slowly than corresponding displacements in an undissociated screw. Instead of measuring the screw displacements directly, the structure of the GaN crystal allows us to adopt the following more sensitive method, which is also robust to scan distortions. As explained further in the Supplementary Note 5, the images in the fault region consist of pairs of closely spaced  Fig. 5a) whose relative positions correlate with the amount of screw displacement. The screw displacement can be quantitatively analysed by measuring the change in the orientation of the line joining the column pair across the dislocation (Fig. 5a). Details of the angle measurement using a Radon transform are included in the Supplementary Fig. 6 and Supplementary Note 5. This approach provides a method for measuring the screw displacements that does not rely on correlating displacements over large distances in the image, which would be much more sensitive to scan distortions than the very local information used here.
Measurements on simulated images clearly show a more rapid change in the angles in the undissociated dislocation than in the dissociated dislocation, with the most significant difference found in the fault region between the two dissociated screw partials (Fig. 5b). Moreover, the rate of change in screw displacements across the dislocations depends on the dissociation distance, with the rate increasing with decreasing dissociation distance, and to a lesser extent on the probe defocus. By fitting the angle plots with four-parameter sigmoid functions, the rate can be represented by the slope-controlling parameter of the sigmoid functions (see Supplementary Note 5). As shown in Fig. 5c, by comparing the fitted slope-controlling parameter from both experiment and simulated images of the undissociated and dissociated dislocations with different dissociation distances, the experimental measurements (taking into account a s.d. coming from measurements of several lattice planes) agree with a 1.65-nm dissociation distance   under various focusing conditions. It is clear from Fig. 5b that for this dissociation distance the angles measured from the experimental image match those from the simulated images over the whole range plotted. Figure 5c also shows that the significant difference in the fitted slope-controlling parameters between the undissociated and dissociated dislocations suggests that the sensitivity of the method should be sufficient to detect a screw dissociation with a dissociation distance as small as 1.1 nm in GaN. The sensitivity demonstrated here is sufficient to distinguish changes in dissociation distance of 0.55 nm. The structure shown in Fig. 2 in the paper described in ref. 11 corresponds to a dissociation distance of 1.65 nm as shown in Fig. 4 22 , or the use of forbidden reflections 23,24 . Moreover, the measured dissociation distance of 1.65 nm is at the borderline of what might be observable by the conventional weakbeam technique. Weak-beam requires an objective aperture about an order of magnitude smaller than the one used here, and therefore the diffraction-limited spatial resolution of weak-beam imaging is close to the dissociation distance measured here. Furthermore, both we and previous authors 4 have attempted weak-beam imaging on this type of dislocation and have failed to detect this dissociation. For these reasons, and because our analysis suggests that a dissociation distance as small as 1.1 nm could be detected for GaN (see Fig. 5), we believe that this approach offers a high-resolution alternative to weak-beam imaging for the screw components of a dislocation. The method used here to measure the dissociation distance quantitatively is specific to the structure of GaN, and other structures may require other approaches to measure atomic displacements in STEM images, including, for example, careful correction of scan distortion. There will be a dislocation depth beyond which clear images cannot be formed because of probe scattering by the crystal above the dislocation, but Fig. 5 shows that our method is robust at least to a dislocation depth of 10 nm in a foil of 15-nm thickness (see Supplementary Fig. 7 and Supplementary Note 5). Investigations of the limits of this approach require further work; however, we note that improvements in electron optics will continue to offer higher numerical apertures and smaller depths of field leading to expected improvements in the sensitivity of the optical sectioning approach.

Methods
Specimen preparation. The sample studied consists of a thin GaN foil parallel to (0001). A 1-mm-thick GaN film was grown at 1,000°C on a sapphire substrate by metalorganic vapour phase epitaxy. The TEM specimen was prepared by mechanical grinding and polishing, followed by ion milling using a Gatan Precision Ion Polishing System.
Scanning transmission electron microscopy. The high-angle annular dark field STEM images were obtained using the Nion UltraSTEM100 electron microscope at SuperSTEM, Daresbury, UK. This microscope is equipped with a field emission gun operated at 100 kV, and a Nion fifth-order spherical aberration (Cs) corrector. A probe convergence semi-angle of 30 mrad was used to achieve a 7.28-nm depth of field (see Supplementary Fig. 8 and Supplementary Note 3). The images were captured by an ADF detector with a collection angle ranging from 90 to 200 mrad.
Multislice frozen phonon image simulations. To simulate the STEM ADF images, we have used the multislice frozen phonon approach implemented in the QSTEM software using the same imaging parameters (accelerating voltage and convergence angle) as those used in the experimental work 25 . Thermal diffuse scattering is introduced using the frozen phonon method and up to 30 configurations of thermal displacements of atoms are averaged in the simulated images.