Progress in Aberration-Corrected High-Resolution Transmission Electron Microscopy of Crystalline Solids

Transcription

1 Progress in Aberration-Corrected High-Resolution Transmission Electron Microscopy of Crystalline Solids K Tillmann, J Barthel, L Houben, C L Jia, M Lentzen, A Thust and K Urban Institute of Solid State Research and Ernst Ruska-Centre for Microscopy and Spectroscopy with Electrons, Research Centre Jülich, D Jülich, Germany Summary: With impressive improvements in instrumental resolution and a simultaneous minimisation of image delocalisation, high-resolution transmission electron microscopy is presently enjoying increased popularity in the atomic-scale imaging of lattice imperfections in a variety of solids. In the present overview, recent progress in spherical aberration corrected imaging performed in troika with the ultra-precise measurement of residual wave aberrations and the numerical retrieval of the exit plane wavefunction from focal series of micrographs is illustrated by highlighting their combined use for the atomic-scale measurement of common lattice imperfections observed in compound semiconductors and high-temperature superconductors. 1 Introduction High-resolution transmission electron microscopy (HRTEM) is now firmly established as a unique analysis technique for the study of lattice imperfections in crystalline solids. Nearly half a century ever since the first images of dislocations were recorded [1, 2, 3], the technique has enabled the characterisation of lattice imperfections not only in the bulk of single crystals but also at heterointerfaces between common solids of technological relevance. Over and above, the performance of intermediate voltage electron microscopes has undergone major improvements in the decade passed through novel electron optical components of particular sophistication, e. g. spherical aberration corrector elements [4, 5] and monochromators of various design [6, 7, 8]. The use of these hardware components means a tremendous facilitation of solidstate analysis on the atomic scale. Consequently, with the instruments now approaching the half Ångström frontier, HRTEM becomes at long last a versatile tool for truly atomic-scale measurements of heterointerfaces and lattice imperfections in a variety of solids. The sole availability of structural information beyond the Ångström barrier, however, only represents a sine qua non for the imaging of object details of the very same length scale. Additionally, an unaltered transfer of information through the lens system constitutes a mandatory requirement in the direct interpretability of recorded micrographs. Thence, any recorded sub-ångström information needs to be impartially regarded as of equivocal reliability without an optimised tuning of instrumental parameters [9, 10] and, in particular, a precise control of residual wave aberrations. The latter can be minimised to a large extent by hardware prior to experimental analyses [11] and entirely eliminated a posteriori by applying phase retrieval methods making use of focal series [12, 13] or illumination tilt direction series [14] of micrographs as well as by off-axis holography [15, 16]. Against the background that any adequate elimination of wave aberrations entails extremely high demands on hardware correction also coming along with the indispensable necessity to measure associated residual aberrations, a sound strategy is required to overcome the problem of still non-direct image interpretability at ultimate resolution. This is especially true in view of a tran-

2 Fig. 1. Process diagram illustrating the course of successive procedures to minimise residual wave aberrations w ij during setup of a spherical aberration corrected microscope using the ATLAS package (left) followed by the numerical retrieval of the exit-plane wavefunction Ψ(r) from a throughfocus series of micrographs (right) also involving a posteriori correction of residual aberrations w ij by the utilisation of measured phase-plates (bottom). sient behaviour of the corrector as-is state during operation of the electron microscope necessitating a follow-up of the actual instrumental transfer properties. Hence, in the present overview, three co-acting techniques are exemplified by highlighting their combined use for the atomic-scale analysis of lattice imperfections and internal boundaries in crystalline solids. This regards (i) imaging at optimised conditions employing negative spherical aberration, (ii) the ultra-precise measurement of higher-order wave aberrations w ij from amorphous specimen areas under tilted illumination, as well as (iii) the restoration of the exit-plane wavefunction from a focal series of micrographs. The fine-tuning and subsequent numerical elimination of residual aberrations still present in hardware-corrected microscopy is demonstrated not only to end in itself, but in particular to exhaust the instrumental information limit at interpretable contrast features. For illustration purposes, recent progress is reviewed in the atomic-scale measurement of the core structure of partial dislocations and of lattice distortions across stacking faults in compound semiconductors as well as atomic bond reconstructions at tilt boundaries in hightemperature superconductors. 2 Methodical Background and Experimental Modus Operandi A three-step course of action, as illustrated by the process chart displayed in Fig. 1, is suggested to fully make use of the improved information limit offered by field-emission gun instruments equipped with a spherical aberration corrector unit and, especially, to overcome uncertainties in image contrast interpretation due to image delocalisation and residual lens aberrations.

3 2.1 Tuning of the Instrument towards Negative Spherical Aberration Corrected Imaging Conditions With the integration of double hexapole correctors, the spherical aberration w 40 (vulgo C 3 ) of the electron microscope becomes a tunable imaging parameter. w 40 can therefore be adjusted to balance phase contrast and residual delocalisation while keeping the point resolution close to the information limit not within reach during operation of traditional uncorrected medium voltage instruments equipped with a field emission gun. Given a sufficiently thin sample with a thickness just smaller than half the extinction distance favourable conditions for negative phase contrast, i. e. for bright-atom contrast, are attained by combining an optimised negative spherical aberration [9] given by: w 40,opt = " 64/ 27 # "3 "4 g max (1) with an overfocus setting w 20 (vulgo Z) of the objective lens: w 20,opt = 16/ 9 " #1 #2 g max (2) minus half the specimen thickness [10] with λ and 1/g max, denoting the electron wavelength and the information limit of the instrument, respectively. When initially neglecting the impact of partially compensated higher-order wave aberrations, a corresponding instrumental set-up yields directly interpretable micrographs accompanied by a residual image delocalisation: R = 16/ 27 g "1 max. (3) With this tuning w 20,opt replaces the Scherzer defocus of traditional HRTEM, and the partially coherent phase contrast transfer function of the instrument is positive up to the information limit and characterised by a broad pass-band. The according negative phase contrast condition not only ensures a substantial contrast improvement but low-nuclear charge elements can also be imaged at bright-atom contrast features in the vicinity of high-nuclear charge atoms, which is in particular viable for the investigation of oxide and nitride materials [17]. Numerical values of the aforementioned optimised imaging parameters are summarised in Table 1 putting to use instrumental parameters of 200 and 300 kv microscopes characterised by information limits in the range between 30 pm and 125 pm, respectively. U [kv] / g max [pm] Δ [nm] w 40,opt [µm] w 20,opt [nm] R [pm] Table 1. Optimised values of spherical aberration w 40,opt and defocus w 20,opt as well as the resulting image delocalisation R according to Eqs. (1) to (3) based on different acceleration voltages U and information limits 1/g max together with the associated defocus spread Δ, respectively. Grey shaded columns apply to parameters of the CM-200-C and Titan instruments used during this study.

4 Fig. 2. (a) Wave aberration coefficients w ij and their sum Σ displayed in phase-plate representation measured during set-up of a Titan microscope. Corresponding numerical data on w ij are summarised in Table 2. Bright (dark) areas indicate positive (negative) phase deviation with grey level jumps occurring in steps of p/2. The white circles denote an information limit of 1/g max = 1/12.5 nm 1 = 80 pm. (b) Time-resolved monitoring of the twofold astigmatism w 22 measured over a period of t ser = 144 s during operation of the instrument at nominally fixed conditions. Astigmatism fluctuations w 22x and w 22y along orthogonal directions x and y are due to instrumental instabilities. The statistical error of a single w 22 measurement is approximately 100 pm with the circle at w 22 = 0.81 nm indicating the p/4 limit not be exceeded to fully exploit the information limit of the microscope. The regression line yields an average variation in the two-fold astigmatism of 6 pm/s. 2.2 High-Precision Measurement of Residual Wave Aberrations for Sub-Ångström HRTEM Quantitative high-resolution electron microscopy requires the measurement and ideally the full elimination of objective lens induced parasitic wave aberrations to high accuracy. This indispensable requirement is because of a decrease of minimum tolerable quantities of higher-order wave aberrations w ij, which have not been considered in traditional high-resolution microscopy so far, with an increase of the instrumental information limit. In this respect the wave aberration function χ(g) describing deviations between ideal and real wavefronts, is the relevant instrumental tuning parameter to be considered. In the isoplanatic approximation [21], χ(g) can be expanded in terms of axial aberration coefficients w ij and their respective azimuths ϕ ij. When transformed to cylindrical coordinates g = (g, ϕ), the aberration function is given by: χ(g,ϕ) = 2" # w ij & i #g i$2, i$ j$0 i+ j % 2 N ( ) i cos[ j (' (' ij )] (4) making the radial (w ij ) and azimuthal (ϕ ij ) dependence of the aberration terms, e. g. defocus w 20, twofold astigmatism w 22, axial coma w 31, and threefold astigmatism w 33 apparent. The other

5 aberration w ij X k modulus azimuth defocus w 20 C 1 (Z) ± 0.16 nm twofold astigmatism w 22 A ± 0.12 nm ± 1.2 axial coma w 31 3 B ± nm ± 38.1 threefold astigmatism w 33 A ± 5.57 nm 36.4 ± 7.5 spherical aberration w 40 C 3 (C S ) ± 0.60 µm star aberration w 42 4 S ± 0.25 µm ± 1.1 fourfold astigmatism w 44 A ± 0.05 µm 20.6 ± 0.3 fifth-order axial coma w 51 5 B ± µm 0.0 ± 13.7 three-lobe aberration w 53 5 D ± µm 74.0 ± 6.7 fivefold astigmatism w 55 A ± 1.80 µm 60.5 ± 0.6 sixth-order spherical aberration w 60 C ± 0.70 mm sixfold astigmatism w 66 A ± 0.05 mm 39.1 ± 0.1 Table 2. Numerical values of higher order wave aberration coefficients w ij belonging to the phase-plate representations displayed in Fig. 2. Indices i and j specify the spatial frequency order of individual aberrations of the wave aberration function in Taylor expansion and the azimuthal symmetry, respectively. Aberration coefficient denotations X k according to the traditional convention by Hawkes and Kasper [21] are given in the middle column for comparison. During evaluation, sixth-order star aberration w 62 (6 R 5 ) and four-lobe aberration w 64 (6 D 5 ) have been set to zero as their magnitudes are typically found to be negligible and any consideration would only deteriorate the accuracy in the determination of the remaining aberration coefficients. higher-order coefficients w ij are denoted in Table 2 for an expansion of the aberration function to the fifth order. While the introduction of a w 40 imaging corrector raises the instrumental resolution also coming along with virtually directly interpretable micrographs when the electron microscope is operated under optimised conditions, experience with both Jülich based instruments equipped with CEOS double-hexapole w 40 imaging correctors [11], i. e. a CM-200-C prototype as well as a commercial Titan electron microscope, shows that a number of coefficients w ij are not sufficiently stable over a period of a typical TEM session. Some of them, e. g. w 42, hold steady for several weeks, while others run out of the tolerance limits ensuring proper exploitation of the information limit within a few hours, e. g. w 31 and w 33, or even minutes, e. g. w 22. This observation suggests the integration of software solutions diagnosing and rectifying the actual state of the optical transfer system during operation of the electron microscope, which is basically due to two reasons. Firstly, commercially distributed aberration measurement routines [18] are by far not sufficient in precision for a corrector alignment aiming at a target resolution in the sub-ångström regime. Moreover, error limits that are traditionally given only for individual aberrations are no longer a realistic approach in case of a comprehensive ensemble of higher-order aberrations. Secondly, the observation that various aberrations, which may certainly be minimised by hardware, will vary with time, and the marginal constraint that some of them cannot be corrected by hardware at all, already suggest combining any incomplete correction during experiments with an a posteriori software correction via phase retrieval methods. We have, hence, developed new numerical analysis procedures in the framework of the advanced treatment of lens aberrations and stability (ATLAS) software project [19], which base upon the well-known analysis of diffractogram tableaus where the defocus and the two-fold astigmatism

6 induced by intentional beam tilts are observed as a variation of Thon rings in the diffractograms of amorphous specimen areas under investigation [20]. Special emphasis was put on the error analysis, exceeding the so far considered magnitude limits for single aberrations [11] by means of the combination of all actually measured aberrations. With a newly implemented pattern recognition module we succeeded to obtain an accuracy of nearly 100 pm in the determination of defocus and two-fold astigmatism belonging to single diffractograms. This level is by more than one order of magnitude superior to that achieved by the manufacturer-supplied measurement software. The same holds true regarding processing speed and robustness against perturbations originating from residual crystalline signals. By this dramatic gain in accuracy, aberration control is now sufficiently precise for reliable imaging even at an information limit of about 50 picometres. As a further consequence highly precise time-resolved measurements of defocus and two-fold astigmatism, which indicate the stability of the complete optical set-up, are now possible. As an example of use, Fig. 2 (a) displays wave aberrations w ij measured during operation of a Titan electron microscope, as obtained from the analysis of 18 diffractograms taken under tilted illumination with maximum angles of 20 mrad from an amorphous specimen area of a crystalline GaAs sample prepared by argon ion milling employing a final 0.5 kev milling step. When also monitoring individual aberrations, cf. Fig. 2 (b), it is observed that the twofold astigmatism w 22 may run out of it s allowed magnitude after less than 3 minutes without any user interaction with the instrument Numerical Retrieval of the Exit-Plane Wavefunction Beyond imaging at optimised conditions, the numerical retrieval of the exit-plane wavefunction Ψ(r) from a through-focus series of micrographs [13] offers not less than five further improvements. First, Ψ(r) is free from nonlinear imaging artefacts and by the combination of many images taken at different foci, the low-frequency gap in the phase contrast transfer function, i. e. the insufficient contrast transfer of low spatial frequencies caused by employing a rather small w 20 value, is reduced considerably [22]. Second, by extracting information from about N = images, the signal-to-noise ratio can be improved by a factor of N/2 compared to that of a single micrograph taken under w 20,opt conditions. Indeed experimental analyses demonstrate a triplication [23] and even quadruplication [24] of the signal-to-noise ratio dependent on the number of images used during retrieval of the exit-plane wavefunction. Third, even the application of small w 20 and w 40 values, which is a prerequisite to obtain phase contrast, induces a parasitic delocalisation R whereas the numerically retrieved exit-plane wavefunction is ideally free from any delocalisation effects. Fourth, the availability of the complex-valued quantity Ψ(r) allows for the numerical a posterior correction of residual wave aberrations. This aspect is of special practical importance as experience shows, that not all aberrations of the microscope are sufficiently constant over the period of operation but can now be determined with sufficient accuracy before recording individual focal series via the ATLAS package, cf. above. Fifth, since Ψ(r) is complex-valued, we may calculate local diffraction patterns from specimen areas as small as desired. When evaluated during operation of the microscope, the judgement of the symmetry properties of these local diffraction patterns is a most convenient tool for the proper orientation of specimen areas under investigation. Corresponding tuning procedures ensure a proper semiconductor zone axis alignment with accuracy well below 3 mrad [22].

7 3 Experimental Details Experimental analyses were performed using two different FEI instruments equipped with imaging correctors, namely a CM-200 prototype operated at 200 kv and a commercial Titan instrument operated at 300 kv. Both systems were equipped with CEOS double hexapole w 40 correctors which allow for correcting axial aberrations up to the third order and for partially compensating for fourth and fifth order aberrations. Instrumental parameters of these microscopes are listed in the grey shaded columns of Table 1. High-resolution micrographs discussed below were recorded at optimised conditions with a slightly negative value for w 40. Focal series of N = images were recorded at a sampling rates well below the Nyquist frequency with regard to half of the instrumental information limit 1 / (2 g max ) for sampling the full instrument potential, knowing that the modulation transfer function of many CCD cameras exhibit poor transfer at higher spatial frequencies. The focal range of each series included the focus setting with w 20,opt for optimised phase contrast. From these series, the exit-plane wavefunction Ψ(r) was retrieved for the frequency band between 1 nm -1 and respective g max values applying a doughnut-shaped restoration filter. 4 Simulation Study: Impact of Image Delocalisation and Residual Wave Aberrations on Dumbbell Lengths Measured from HRTEM Micrographs of ZnO (1120) In recent years, the measurement of in-plane distances between adjacent contrast dots associated with atomic column positions, has become a widespread technique for mapping lattice distortions in the vicinity of defects and across heterointerfaces. Underlying numerical analysis algorithms meet uncertainties in the measurement of contrast dot positions given by 2 σ confidence intervals as far as to 4 pm [23]. Against the background we are currently witnessing the transition to sub- Ångstöm microscopy, a point of principle emerges regarding whether micrographs recorded with aberration corrected instruments substantially allow such small tolerance limits to be measured facing system-inherent residual image delocalisation and parasitic higher-order wave aberrations. For exemplification purposes Fig. 3 displays a series of calculated w 20,opt images Ι(r) and the phase Φ(r) of the exit-plane wavefunction of ZnO viewed along the [112 0] zone axis together with associated intensity line profiles taken along the direction of the zinc oxygen dumbbell assuming different instrumental information limits in the range between 100 and 30 picometres. Line profiles show a decreased peak-to-background ratio and an increased asymmetry of individual peaks with decreasing information limit. This behaviour, which is more pronounced for the lighter oxygen columns, is basically due to an increase of image delocalisation with decreasing 1 / g max values as expressed by Eq. (3). From these line profiles, dumbbell lengths d have been measured by an algorithm basically employing dedicated image intensity thresholding operations and a centre-of-mass analysis inside individual bright contrast dots supported by adequate refining procedures as specified in [25]. Systematic deviations Δd with regard to the actual dumbbell length of d ZnO = pm have been added to Fig. 3. As can be seen from these numerical data, Δd values measured from w 20,opt images Ι(r) clearly exceed 4 pm as long as the information limit is below 80 picometres. To come to a more general description, Fig. 3 (g) displays calculated peak distances d between two columns of equal atomic species in dependence on the instrumental information limit g max. Both parameters, d and g max, are plotted normalised to the actual column distance d real. As can be seen from the figure, d / d real values trick to believe in an superficial resolution characterised by d / d real < 1 at g max d real values in the range between 0.7 and Contrastingly, d/d real values are well above 1.03 at g max d real in the range between 0.85 and 1.30.

8 Fig. 3. Calculated optimum focus Ι(r) and Φ(r) images of ZnO [112 0] and intensity line profiles extracted along the dumbbell direction based on a specimen thickness t = 3 nm and a 300 kv instrument characterised by an information limit of (a) 100 pm, (b-d) 80 pm, (e) 50 pm, and (f) 30 pm. Images displayed in (c) consider residual lens aberrations amounting to w 22 = 2 nm (148 ), w 33 = 50 nm (343 ), and w 31 = 20 nm (339 ) with the magnitudes representing allowed aberration coefficients to fulfill the p/4 limit according to [11] and the values in parantheses indicating randomly generated azimuths with the [0001] direction, respectively. Δd values specify the measurement error of the dumbbell length d with regard to the actual distance of pm. (g) Normalised peak distance d/ d real between two columns of equal atomic species in dependence on the normalised instrumental information limit g max d real.

9 This behaviour demonstrates that an excess resolution of about 30 per-cent will be necessary to measure column distances correctly from w 20,opt images with a precision in the order of three per-cent. A further increase in excess resolution will be necessary when focusing on column distances between comparatively heavy and light atomic species where the weak signal associated with the latter accounts for additional systematic errors in the determination of column distances. Systematic errors in d values measured from the optimum focus images Ι(r) further increase significantly when also considering the impact of non-fully compensated higher-order wave aberrations which, when assuming reasonable quantities of two- and three-fold astigmatism and axial coma, result in an approximate reduplication of Δd values as can be seen from comparison of Figs. 3 (b) and (c). As illustrated before, a highly efficient strategy to overcome the problems coming along with residual image delocalisation and wave aberrations will be, not to focus on single image Ι(r) but rather on the retrieved phase Φ(r) of the exit-plane wavefunction displayed in Fig. 3 (d) which already ensures Δd = 0.3 pm at information limit of 1 / g max = 80 pm. 5 Experimental Results: Materials Science Applications of Use The analysis and optimisation techniques described in section 2 have been applied to a fairly wide range of defect structure problems arising in solid-state research. In the following we highlight their combined use by discussing three materials science applications related to specific cases. 5.1 Frank Partial Dislocation Cores in Chromium Implanted GaN (1120) As an introductory example of use, we report on the structure of partial dislocation cores introduced by chromium implantation into GaN layers [26] which were grown by metal organic chemical vapour deposition on Al 2 O 3 (0001) substrates under silicon doping to a concentration of cm -3. Chromium ions were implanted with an energy of 200 kv at a dose of cm -2 and a temperature of 350 C to avoid amorphisation. A high density of basal plane stacking faults was observed besides spherical chromium rich precipitates in the surface near regions after implantation and rapid thermal annealing at 700 C for 5 min in N 2 atmosphere [27]. A by-product of the implantation and annealing process is the creation of extrinsic and intrinsic basal plane stacking faults, which are formed as a result of the precipitation of excess interstitials. The majority of the basal plane stacking faults are extrinsically bound by Frank partial dislocations with Burgers vectors of type b = c/2 [0001]. Figs. 4 (a) and (b) display an optimum focus micrograph Ι(r) together with the associated phase image Φ(r) numerically retrieved from a through-focus series of micrographs, respectively, of a Frank partial dislocation core at the terminating zone of a dislocation loop of several 10 nm in size. Gallium and nitrogen columns at a dumbbell length of 114 pm, considerably smaller than the information limit 1 / g max = 125 pm of the employed CM-200-C instrument, are not fully resolved. Nonetheless, the N polarity of the sample is clearly visible from the directly interpretable bright contrast in the phase image. The tetrahedral coordination across the faulted layer stacking is, however, confirmed in the left parts of both images, indicating that no foreign chromium-gallium alloy phase is connected with the planar defects. Although the optimum focus micrograph Ι(r) and the phase image Φ(r) appear quite similar at first glance, the improved signal-to-noise ratio in the phase image enhances the visibility of the nitrogen positions and the dumbbell orientation down to the core of the dislocation. The phase image Φ(r) strengthens the presence of further nitrogen atoms within the core surrounded by the cage of the five marked gallium columns. The faint phase shift in Φ(r), in which non-linear image components and image delocalisation are eliminated, indicates a nitrogen-filled core in favour of a chromium- or gallium-rich core. Based on these observations, a tentative 5/7 ring configuration for the core is presented in Fig. 4 (d). The nitrogen atoms shared by the 5 and 7 membered rings, respectively, are coordinated

10 Fig. 4. Frank partial dislocation with a projected Burgers vector b = c/2 [0001] viewed along the [112 0] direction. (a) Optimum focus micrograph Ι(r) slightly distorted by parasitic wave aberrations amounting to w 22 = 2.7 nm (110 ), w 31 = 240 nm (320 ), and w 33 = 50 nm (80 ) with the values in parantheses indicating respective azimuth angles inclined with the [111 0] direction. (b) Phase image Φ(r) retrieved from the focal series of images together with (c) the associated numerical phase-plate used for the correction of residual wave aberrations. (d) Stick-and-ball model of the 5/7 ring configuration for the core of the Frank partial dislocation [26]. threefold with gallium and nitrogen. Due to the wrong bond and the presence of unpaired electrons, this structure is not expected to be the most stable configuration. Since the phase image Φ(r) is retrieved from a through-focus series taken over a period of a few ten seconds, the image conceivably reflects a transient state of the core that gathers further interstitials during observation. The w 20,opt image may therefore contain complementary but not necessarily identical information when compared to Φ(r). Indeed, in the present case the w 20,opt micrograph, characterised by an inferior noise level compared to Φ(r), also supplies an alternative structure which contains two wrong Ga-Ga bonds connecting threefold-coordinated gallium atoms at positions 1-5 and 2-3, respectively. 5.2 Lattice Distortions in the Vicinity of Extrinsic Stacking Fault Ribbons in GaAs (110) As a further materials science application, we report on locally inhomogeneous distortions of atomic dumbbells across extrinsic stacking faults in a GaAs capping layer grown by molecular

11 Fig. 5. Inhomogeneous distortion of atomic dumbbells across an extrinsic stacking fault in GaAs (110). (a) Retrieved phase image Φ(r) with atomic column positions superimposed and dumbbell distortions indicated exemplarily in dependence on specific positions along the [111] direction. (b) Average projected bond length d and (c) misorientation angle δ of the dumbbells along the [111] direction. The lattice planes belonging to the double stacking fault ribbon are indicated in lighter grey colour. (d) Numerical phase-plate used for the correction of residual wave aberrations during evaluation of the phase image [24]. epitaxy on top of a plastically relaxed In 0.3 Ga 0.7 As layer. Measurements on actual dumbbell lengths, amounting to 141 pm in case of unstrained material when viewed along the [110] direction, have been performed on the phase image Φ(r) displayed in Fig. 5 (a) which was retrieved from a focal series of N = 30 micrographs taken with a CM-200-C instrument. The centre of the phase image is 5.2 nm left of a 90 partial dislocation core terminating the faulted double ribbon [24]. In order to avoid systematic errors due to dumbbell distances close to the information limit of the instrument, actual lengths and orientations of individual dumbbells have been measured and normalised to mean values associated with the lower (planes no. 0-3) and upper (planes no ) areas of the Φ(r) image. For the reduction of the measurement error 15 data points have been averaged along the [1 1 2]direction and median values are plotted in Figs. 5 (b) and (c) in dependence on specific (111) lattice plane positions. As can be seen from both plots, the dumbbells on either side of the double ribbon rearrange roughly antisymmetric with regard to the faulted (111) planes. The dumbbells of the bottommost lattice plane of the upper domain (plane no. 8) are compressed to a length of d = 133 pm ± 4 pm and turn towards the double ribbon with a misorientation angle d 2.6 ± 1.3 compared to the reference lattice planes. In contrast, the uppermost dumbbells of the lower crystal area (plane

12 no. 5) are stretched to a length of d = 150 pm ± 4 pm and bend away from the double ribbon at a misorientation angle of d 4.9 ± 1.5. In order to gauge whether the observed antisymmetric distortions represent a genuine structural property or not, potential implications of scattering and imaging artefacts need to be ruled out. Since a hypothetic global misalignment of the sample cannot give rise to any local torsion and dilatation of projected atomic dumbbells we may disenfranchise from this explanation. An argument of the same kind holds true of the potential impact of not fully compensated aberrations as they would take effect on the entire image and, hence, would distort all atomic dumbbells of the same orientation equally. Strictly speaking any potential impact of both, on-axial and off-axial aberrations would need to be evaluated separately. Beyond electron-optical reasoning, a supposed strictly antisymmetric lattice distortion caused by a dedicated combination of higher-order aberrations with the faulted ribbon incidentally acting as the symmetry plane, may be ruled out from a probability point of view because of an almost immense number of possible combinations which will not give raise to the observed distortion behaviour. The observed alteration of atomic dumbbell lengths in the vicinity of stacking fault ribbons of about 10 pm is in fair agreement with recent measurements focusing on intrinsic stacking faults ribbons in heavily beryllium doped GaAs [28]. This analysis revealed an average expansion of dumbbell lengths up to 158 pm in the faulted lattice plane which was explained by the segregation of beryllium dopant atoms and the subsequent formation of antisite defects inside of the plane of the faulted ribbon. Additionally, lattice displacements around the central stacking fault of Z- shaped dipoles connecting two stair rod dislocations in indium doped GaAs revealed the very same antisymmetric distortion of dumbbell related contrast dots as were observed in the present analysis. Lateral displacements of gallium and arsenic atom pair related contrast dots along the [1 1 2] direction were found to be as much as 20 % to 50 % of the dumbbell length [29]. As the specimen investigated in the present study was undoped we may, in the first instance, rule out the aforementioned explanations but only speculate about indium diffusion from the underlying In 0.3 Ga 0.7 As layer to the GaAs layer along the faulted ribbon at most. As long as an inhomogeneous incorporation of indium atoms is not considered this approach, thus, cannot explain the observed lattice distortions. Admittedly, the antisymmetric distortions become perspicuous when considering the elastic distortions associated with the 90 partial dislocations which basically follow the measured characteristics of the dumbbell's measured expansion and torsion but, alas, yield only alterations of dumbbells lengths smaller than ± 3 pm and misorientation angles smaller than ± 0.4 for the image area under investigation [30]. Hence, a more likely explanation would be that the terminating partial dislocation biases the rearrangement of atomic columns in the vicinity of the faulted ribbons and that next but one neighbour interactions between different atomic species will indeed play an important role during this process. 5.3 Structural Reconstruction at 90 Tilt Boundaries in YBa 2 Cu 3 O 7-δ (100) The quantitative analysis of atomic column positions of light and heavy elements is in particular valuable for the examination of oxide materials, since changes in bondlenghts between cations and oxygen can have a considerable effect on the electronic properties especially in the vicinity of defects. A practical example is the meticulous analysis of the structural reconstruction of a tilt grain boundary in YBa 2 Cu 3 O 7-δ, which gave evidence for local doping and disorder affecting the superconductive property in a few atomic layers around the grain boundary. The prerequisites for the above mentioned displacement analysis with picometre accuracy are the accurate correction of aberrations, a good signal-to-noise ratio, the extraction and exploitation of the full signal besides a quantification of the residual noise as a means to quantify the statistical measurement error, and a refinement by comparison with image simulation in order to avoid systematic errors related e.g. to the finite frequency transfer of the microscope already discussed in section 4. The three co-acting strategies of employing a negative spherical aberration set-up, a pre-

13 Fig. 6. (a) Optimum focus micrograph I(r) and (b) phase image Φ(r) retrieved from a focal series of 20 micrographs of a 90 [100] tilt grain boundary in YBa 2 Cu 3 O 7-δ. viewed along the [100] zone axis. Arrows indicate the grain boundary plane. The framed area in (b) highlights a single repeat cell in the periodic arrangement of the grain boundary. (c) Numerical phase-plate used for the a posteriori correction of residual wave aberrations in addition to the defocus and a small negative spherical aberration: w 22 = 9 nm (165 ) and w 31 = 140 nm (10 ) with the azimuth information referring to the image x-axes, respectively. Phase angles are displayed in modulo π/2 representation. cise measurement and control of wave aberrations and the restoration of the exit plane wavefunction taken together with the simulation of wavefunction data most charmingly meets these prerequisites. Details of the procedure are given in [23]. Figs. 6 (a) and (b) display images of the grain boundary viewed along the [100] direction close to optimum defocus w 20,opt and the corresponding phase Φ(r) of the exit plane wave function retrieved from 20 images in the focal series. The symmetry relation between the two domains was exploited for the numerical correction of the parasitic coma w 31 in addition to the considerable twofold astigmatism. The resulting numerical phase-plate for the tuning of the relevant parasitic aberrations up to w 40 is shown in Fig. 6 (c). The notable number of π/2 phase wraps within the information limit g max = 8 nm -1 of the microscope emphasises that direct interpretability in the optimum focus micrograph I(r) is not given and that a posteriori correction of aberrations is required in order to exploit the full information provided by the experimental data. Atomic column positions, the scattering intensity and their statistical uncertainties were calculated by peak regression in the Φ(r) image, owing to the linear relationship between the projected potential and the phase of the exit plane wavefunction for a weak phase object. Picometre accurate quantitative data for single atom column displacements could be derived. The accordingly meas-

14 Fig. 7. Atom column displacements in the retrieved structure of the 90 [100] tilt grain boundary in YBa 2 Cu 3 O 7- δ. The structure model displays columns in the repeat cell marked in Fig. 6 (b). Displacements with respect to the position in the bulk structure are indicated by arrows and given in picometres together with the 2 σ confidence intervals for the statistical measurement error. Significant displacements are present in the grain boundary plane and the two neighbouring planes. Evident is the shift of the O1 atom towards the Cu1 atom in the grain boundary plane. ured column displacements with respect to the periodic structure in the two domains are displayed in Fig. 7 for a single repeat cell of the periodic grain boundary structure. 2 σ confidence intervals of 4 pm were achieved for columns of the cation sub-lattice. The weaker scattering signal on the oxygen positions is responsible for the larger 2 σ confidence intervals, up to more than 10 pm. In order to avoid systematic errors, a refinement by comparison with simulated exit plane wavefunction data was conducted when the measured column distances approached the information limit of the microscope. Despite the nearly vanishing over-all lattice mismatch between the domains, column displacements occur due to the mismatch between the size of the smaller central perovskite-like block and the b-axis in the unit cell structure of YBa 2 Cu 3 O 7-δ. Displacements are restricted to the grain boundary plane and directly neighbouring planes. Elsewhere, the bondlengths are in excellent agreement with neutron scattering data for orthorhombic YBa 2 Cu 3 O 7-δ. In particular, changes in the bondlengths between the Cu1 atom in the basal plane, the O1 atom in the BaO plane and the Cu2 atom in the superconducting CuO 2 plane (see Fig. 7) are important since the shift of the oxygen atom O1 towards the Cu1 atom signifies a local doping effect and goes along with largely increased static or dynamic disorder on the Cu1 site in the boundary [23]. Both factors will change the superconductivity locally in the grain boundary as well as in the neighbouring planes. 6 Present Achievements and Challenges for the Future The examples discussed in this overview demonstrate that the resolution power of neoteric highresolution instruments equipped with an imaging w 40 corrector may very well be exploited for the atomic-scale analysis of lattice imperfections and internal boundaries in a variety of crystalline solids. Very significant advantages regarding the direct interpretation of retrieved phase images Φ(r) and the enhancement of the signal-to-noise ratio arise from the combination of (i) negative spherical aberration corrected imaging, (ii) the ultra-precise measurement of residual higherorder wave aberrations w ij together with (iii) the numerical retrieval of the exit-plane wavefunc-

15 tion. When performed in troika, all three techniques allow for the elimination of artificial contrast features caused by non-fully compensated wave aberrations w ij in a most straightforward manner. Seen from a different perspective, there has been recently considerable debate concerning the benefit of aberration correction in HRTEM beyond a beneficial minimisation of image delocalisation as well as an increase of the information limit. Truly, it would makes sense from an ergonomic viewpoint to record directly interpretable high-resolution micrographs to outflank the present imperative to apply numerical post-processing operations, e. g. the retrieval of the exit-plane wavefunction. Corresponding technical solutions, however realised in detail, will at least need to ensure (i) an improved stability of stages allowing an increase of the exposure time to several seconds to measure up the signal-to-noise ratio with retrieved phase images, (ii) sufficient contrast transfer at low spatial frequencies or, as the case may be, an additionally impressed phase shift of about p/2 between diffracted and transmitted beams thus enhancing contrast, (iii) an opportunity to sufficiently minimise residual wave aberrations and, especially, to maintain long-time stability during operation of the instrument against outside influences and goniometer tilt and shift operations on a sustained basis as well as (iv) a sufficient minimisation of off-axial aberrations at increased resolution in order to prevent contrast variations to be associated with identical objects residing at different positions with regard to the microscope s principal axis. Acknowledgements The authors are most grateful to Y Divin, A Förster, V Guzenko, and U Poppe for providing the samples investigated in this study. Also the technical assistance of D Meertens and W Sybertz at certain stages of TEM specimen preparation is most appreciated. References 1. Hirsch P B, Horne W H and Whelan M J 1956 Phil. Mag. 1, Menter J W 1956, Proc. Roy. Soc. A 236, Bollmann W 1956 Phys. Rev. 103, Rose H 1990 Optik 85, Haider M, Rose H, Uhlemann S, Schwan E, Kabius B and Urban K 1998 Ultramicroscopy 75, Tiemeijer P C 1999 Ultramicroscopy 78, Kahl F and Rose H 2000 Proc. EUREM-2000, Vol. 3, eds P Schauer, I Müllerová and L Frank (Brno: Czek Microscopy Society) pp 8. Su D S, Zandbergen H W, Tiemeijer P C 2003 Micron 34, Lentzen M, Jahnen B, Jia C L, Thust A, Tillmann K and Urban K 2002 Ultramicroscopy 92, Lentzen M 2006 Microsc. Microanal. 12, Uhlemann S and Haider M 1998 Ultramicroscopy 72, Coene W M J, Janssen G, Op de Beeck M and van Dyck D 1992 Phys. Rev. Lett. 69, Thust A, Coene W M J, Op de Beeck M and van Dyck D 1996 Ultramicroscopy 64, Kirkland A I, Saxton O W, Chau K L, Tsuno K and Kawasaki M 1995 Ultramicroscopy Lichte H 1986 Ultramicroscopy 20, Lehmann M and Lichte H 2002 Microsc. Microanal 8, Jia C L, Lentzen M and Urban K 2003 Science 299, Hartel P, Müller H, Uhlemann S and Haider M 2004 Proc. EMC-2004, eds N Schryvers and J P Timmermanns (Antwerp: Belgian Society for Microscopy), pp IM01.P Barthel J 2007 PhD Thesis RWTH Aachen University

16 20. Zemlin F, Weiss K, Schiske P, Kunath W and Herrmann K H 1978 Ultramicroscopy 3, Hawkes P and Kapser E 1989 Principles of Electron Optics (London: Academic Press) 22. Tillmann K, Thust A and Urban K 2004 Microsc. Microanal. 10, Houben L, Thust A and Urban K 2006 Ultramicroscopy 106, Tillmann K, Houben L and Thust A 2006 Phil. Mag. 86, Kilaas R, Paciornik S, Schwartz A J and Tanner L E 1994 Journal of Computer- Assisted Microscopy 6, Tillmann K, Houben L, Thust A and Urban K 2006 J. Mater. Sci 41, Guzenko V A,Thillosen N, Dahmen A, Calarco R, Schäpers T, Houben L, Schineller B, Heuken M and Kaluza A 2004 J. appl. Phys. 96, Kisielowski C, Freitag B, Xu X, Beckmann S P and Chrzan D C 2006 Phil. Mag. 86, Lim S H, Shindo D, Yonenaga I, Brown P D and Humphreys C J 1998 Phys. Rev. Lett Hirth J P and Lothe J 1968 Theory of Dislocations (New York: McGraw-Hill)