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Imaging and thickness measurement of amorphous intergranular films using TEM
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Ultramicroscopy 99 (2004) 103–113
Imaging and thickness measurement of amorphous intergranular films using TEM I. MacLaren* Institut fur . Materialwissenschaft, Technische Universitat . Darmstadt, Petersenstr. 23, D-64287 Darmstadt, Germany Received 26 February 2003; received in revised form 16 September 2003; accepted 17 October 2003
Abstract Fresnel fringe analysis is shown to be unreliable for grain boundaries in yttrium-doped alumina: the determined thicknesses do not agree well with those measured from high resolution transmission electron microscopy (HRTEM), the asymmetry between under- and overfocus is very large, and Fresnel fringes are sometimes shown at boundaries which contain no amorphous film. An alternative approach to the analysis of HRTEM images of grain boundary films is demonstrated: Fourier filtering is used to remove the lattice fringes from the image thereby significantly enhancing the visibility of the intergranular films. The apparent film thickness shows a discrepancy between measurements from the original HRTEM image and the filtered image. It was shown that fringe delocalisation and diffuseness of the amorphous/crystalline interfaces will lead to a significant underestimate of the thickness in unprocessed HRTEM images. In contrast to this, the average thickness can be much more accurately measured from the Fourier-filtered image, provided the boundary is oriented accurately edge-on. r 2003 Elsevier B.V. All rights reserved. PACS: 61.16.B; 61.72.M; 07.05.P Keywords: High resolution transmission electron microscopy (HRTEM); Image processing; Fourier filtering; Fresnel fringe analysis; Amorphous films
1. Introduction 1.1. Imaging and thickness measurement of grain boundary films The presence of thin amorphous films at grain boundaries in ceramics was first demonstrated in Si3N4 by Clarke and Thomas [1] and these have since been found in many different materials, *Corresponding author. E-mail address: [email protected] (I. MacLaren).
especially ceramics such as Si3N4-derivatives, ZnO varistors [2], ZrO2 [3] and Al2O3[3,4]. Three main methods have been developed for the imaging of such films in transmission electron microscopy (TEM). Firstly, a dark field image may be created using the objective aperture to select part of the first diffuse diffraction ring for the amorphous material, the so-called diffuse dark field (DDF) method. If care is taken to ensure that no crystalline reflections come within the aperture then the only regions to show bright contrast are those which are amorphous [5]. Secondly, the
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.10.002
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difference in atomic density or composition at the grain boundary leads to a change in inner potential. When imaged in the TEM this leads to the appearance of Fresnel fringes at the boundary in over- or underfocus conditions. The fringe spacing can be measured for various defocus values, plotted on a graph and extrapolated to Gaussian focus, giving a value for the film thickness [6]. Finally, films can be imaged using high resolution TEM (HRTEM) and the thickness directly measured from the micrograph [1,5,6]. In the latter two cases, the boundary must be oriented edge-on to the electron beam. Even for the dark field imaging method, one finds better contrast when the boundary is approximately edge-on, and this condition must be fulfilled if measurements of film thickness are to be made. In the case of HRTEM there is the additional and more exacting requirement that the specimen must be so oriented that lattice planes in both crystals are also edge on, so as to give a clear high resolution lattice image of both grains. These three methods have been compared by Cinibulk et al. [7]. It was shown that the HRTEM method gave the smallest thickness, that the Fresnel fringe method gave a thickness about 20% larger, and that the diffuse dark field method gave a thickness 50–100% larger than the HRTEM method. It was concluded that the DDF method is good for revealing amorphous films, but unsuited for thickness measurements, the Fresnel method is widely applicable but may overestimate the thickness slightly (although see Ref. [8]), and the HRTEM method is the most accurate for thickness measurements, but is also very exacting and therefore time consuming. Jin et al. [9] showed widespread applicability of the Fresnel fringe method for grain boundaries and noted less of a discrepancy with HRTEM measurements. Nevertheless, the Fresnel fringe spacing does not always obey the proposed functional dependence on defocus W ¼ W0 þ cDf 1=2 ;
ð1Þ
where c ¼ ð3lÞ1=2 :
ð2Þ
W is the observed fringe spacing, W0 is the film thickness, Df is the defocus, and l the electron wavelength [10]. Jin et al. [8] found that the constant of proportionality, c; was often smaller than expected; however, no explanation was given for this discrepancy. Moreover, an anisotropy is typically observed between the under- and overfocus plots [7,8,11,12]. More detailed studies of Fresnel fringe formation at interfaces including detailed mathematical and computational calculations have been performed by other workers [12–16]. It is clear from these studies that the amorphous-crystalline interface and thus the change in internal potential is rarely atomically sharp, and that grain boundary grooving as a result of preferential etching during specimen preparation is common; both of these effects affect the form of the fringes. Moreover, space charge has also been shown to affect the form of the fringes [15,16]. It is therefore no great surprise if the functional form of the dependence of the fringe spacing on defocus is not always as simple as first derived by Clarke on the assumption of sharp interfaces [6]. What is perhaps more worrying, however, is that the boundary region can have a reduced atomic density even when fully crystalline, resulting in the appearance of Fresnel fringes even in the absence of any amorphous film [3,17,18]. 1.2. Fourier filtering of HRTEM micrographs Fourier filtering has been used for the processing of HRTEM images almost ever since computers became powerful enough to perform fast Fourier transforms in a reasonable time period. For example, the group of van Dyck at the University of Antwerp were among the first to demonstrate the wide variety of possibilities of such an approach in image processing. This included noise removal from HRTEM images, observation of small deviations from periodicity at stacking faults, epitaxial interfaces, and antiphase boundaries, and the imaging of short range order in Au4Cr [19,20]. Since then it has been used for a wide variety of applications. For example, improvement of the visibility of small crystalline clusters or small particles [21–23],
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imaging displacement and strain in HRTEM images [24–26], generation of thickness maps in certain special cases [27], characterisation of antiphase domains and domain boundaries [28] and the localisation of dislocations in HRTEM images [25,29,30]. Whilst the periodic information or small deviations from periodicity at defects, steps etc. have been widely used in Fourier filtering, the nonperiodic information has generally been thrown away. In the present work, use is made of a somewhat neglected suggestion of Coene et al. [19] where Fourier filtering is used to do the opposite and remove the periodic information from HRTEM micrographs in order to allow the better observation of non-periodic information. In that case, it was used for the imaging of Pt particles on a crystalline support. In the current work, a similar procedure is used for the imaging of thin amorphous films at grain boundaries. The results of this approach are compared with those of standard TEM techniques for analysis of such grain boundary films.
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3. Results and analysis 3.1. Fresnel fringe analyses of grain boundaries in Y-doped alumina A grain boundary was tilted to an essentially non-diffracting orientation and a defocal series recorded to produce Fresnel fringe images. Three images from this series are shown in Fig. 1:
2. Experimental procedure An alumina ceramic doped with 500 wt ppm Y2O3 was prepared as described previously [31–33] and was sintered at 1450 C for 96 h and then annealed at 1650 C for 12 h. As a result, abnormal growth of some grains was observed as has been reported in detail elsewhere [33]. TEM specimens were prepared using a standard procedure of slicing, disc cutting, mechanical polishing, dimpling and ion milling. A final treatment of 30 min low energy ion-beam polishing using Ar+ ions accelerated by just 500 V was used to remove surface damage from ion-beam thinning and resulted in a sample having very little amorphous material at the edges of the hole in the specimen. The sample was examined using a JEOL3010 TEM equipped with a high resolution objective lens polepiece (Cs ¼ 0:6 mm) and operated at 297 kV; images were recorded using the CCD camera of the attached Gatan image filter. Processing of these images was performed using the Digital Micrograph software (Gatan Inc., Pleasanton, CA, USA).
Fig. 1. Fresnel images of a grain boundary (an HRTEM image of the same boundary is shown in (Fig. 5): (a) Df E 480 nm; (b) Df E0; (c) Df E þ 480 nm; (d) Intensity profiles across the boundary for each image, averaged 500 pixels along the boundary length.
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480 nm, Gaussian focus, and +480 nm, together with an intensity profile across the boundary for each image (averaged in each case from 500 pixels along the length of the boundary). As is normal, a bright central fringe surrounded by two dark fringes is seen in underfocus, and a dark central fringe surrounded by two bright fringes is seen in overfocus. At Gaussian focus a dark line was also seen at the boundary. This may be attributed to increased scattering due to the segregation of relatively heavy Y ions at the boundary, as is normally observed in these materials [31,33]. Fringe spacings were measured from the intensity profile for each image and plotted against defocus resulting in Fig. 2. This graph shows the commonly observed asymmetry between under- and over-focus [7,8,11,12]. The overfocus- and underfocus curves can both be fitted using a line of the form given in Eq. (1). In the case of the underfocus series, W0 ¼ 0:35 nm and c ¼ 0:053 nm0.5. For the overfocus series, W0 ¼ 0:5 nm and c ¼ 0:071 nm0.5. For comparison according to Eq. (2), c should equal 0.78 nm0.5. Taking the two series together, however, one would expect a film thickness under 0.5 nm from these results. The same film was examined by HRTEM and Fourier filtering as . . . . . . .
Fig. 2. Plot of Fresnel fringe spacings against defocus for the boundary shown in Figs. 1 and 5. Curves which fit the fringe spacings have been drawn on, the parameters for which are in the text. In dashed lines are curves drawn according to Eq. (1) using c as defined in Eq. (2) and W0 ¼ 0:7 nm.
Fig. 3. High resolution images of a grain boundary showing Fresnel fringes in under- and overfocus conditions, but containing no detectable amorphous film: (a) Df EF48 nm; (b) Approximately Gaussian focus; (c) Df E þ 64 nm.
described later and a film thickness of E0.75 nm was determined. It may be noted that the FWHM of the dark line at the boundary at zero defocus (Figs. 1(b) and (d)) was measured as 0.71 nm, in much better agreement with the HRTEM and Fourier filtering measurement. Fig. 3 shows HREM images of a grain boundary recorded at defocus values of (a) 48 nm, (b) E0 nm, and (c) +64 nm. None of these images shows any sign of an amorphous film at the grain boundary. Nevertheless, Fresnel fringes are apparent in Figs. 3(a) and (c). In Fig. 3a, a bright fringe is found at the centre surrounded by two dark fringes. In Fig. 3(c), a dark fringe is at the boundary centre, with bright fringes on either side. Both appear just as would be expected in the presence of an amorphous film. The bright central fringe in underfocus and dark central fringe in overfocus would seem rather to indicate a reduced GB inner potential [16]. This is perhaps surprising in view of the fact that all grain boundaries in such an yttrium-doped alumina have yttrium segregated to the boundary at levels roughly 3–9 cat nm2 (E0.25–0.75 monolayer) [31,33]. This not-insignificant concentration of high-Z yttrium segregated at the boundary could perhaps be expected to increase the inner potential of the grain boundary region. A recently published
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theoretical study investigated the effect of doping on grain boundary structure in alumina using a variety of different dopant ions including Sc, Y and La [34]. Both in special (prismatic S3) and more general (S13) boundaries it was shown that the dopants led to a significant GB expansion. In the case of Y at similar doping levels to those seen here, this was found to result in an expansion perpendicular to the boundary of about 6.5%. This increase in GB volume and consequent reduction in density probably more than offsets the increased potential of the large-Z yttrium atoms. Thus, grain boundary segregation induces an increase in grain boundary volume and therefore a reduction in density and thus inner potential as previously concluded by Ruhle . et al. [3]. As a further example, a plot of Fresnel Fringe spacings is shown in Fig. 4 for another boundary between small grains where no amorphous film would be expected [33] (although in this case this could not be confirmed by HRTEM as the material was too thick in this place). In this case there is a huge anisotropy between the under- and overfocus results. For the underfocus series, W0 ¼ 0:1 nm whereas for the overfocus series, W0 ¼ 0:75 nm. Meanwhile, the FWHM of the dark line at Gaussian focus was 0.72 nm. It is difficult in such a case with such huge underfocus–
. . . . . .
Fig. 4. Plot of Fresnel fringe spacings against defocus for another boundary not expected to have any amorphous film, showing significant anisotropy between under- and overfocus.
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overfocus anisotropy, to reach a definite conclusion about whether an amorphous film is present, and if so, how thick it is. Nevertheless, this illustrates very well the difficulties in performing reliable Fresnel fringe analysis on this material. Thus, it can be concluded that Fresnel fringe analysis of amorphous film thickness at grain boundaries in yttria-doped alumina is rather unreliable. Widths of grain boundary films measured from Fresnel fringes are significantly less than those measured by HRTEM. This could be a consequence of diffuse crystalline-amorphous interfaces or of compositional gradients across the film itself. In particular, Fresnel fringes are found at segregated grain boundaries which do not possess disordered cores, probably due to grain boundary expansion as a result of the Y-doping. Thus, the use of Fresnel fringe measurements cannot be considered a trustworthy technique for either the detection of amorphous films or the measurement of their thickness in alumina, in agreement with earlier studies [18]. 3.2. Fourier filtering of HRTEM images to reveal amorphous material The principle of the method here is very simple. Instead of using Fourier filtering to enhance periodic information, it is used to remove it [19]. This significantly enhances the visibility of all nonperiodic information, in particular that from amorphous phases, although surface relief is also more clearly revealed. Fig. 5(a) shows an HRTEM image of a grain boundary at which a thin disordered film is apparent. A fast Fourier transform (FFT) of this image is shown in Fig. 5(b). The spots belonging to crystalline reflections were then blanked out using circular masks, resulting in Fig. 5(c). This was then inverse Fourier transformed to produce the image shown in Fig. 5(d). This last image shows no lattice fringes, except as image artefacts at the edges, and shows clearly the amorphous core of the grain boundary. A certain amount of nonperiodic information in the form of surface relief over the whole picture area is also visible. In such a case where the amorphous film is very thin
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Fig. 5. Fourier filtering of an HRTEM image of a grain boundary with a thin amorphous film: (a) original HRTEM image; (b) Fourier transform; (c) Fourier transform with crystalline reflections masked out; (d) reverse filtered image, now showing the amorphous film at the boundary as well as other non-periodic information.
Fig. 6. HRTEM images of a second grain boundary: (a) original HRTEM image; (b) Fourier filtered version showing only the grain boundary film and other non-periodic features.
(o1 nm), the Fourier filtered image shows the film much more clearly than the original HRTEM image. Fig. 6(a) shows a HRTEM image of another grain boundary, this time cut so that only the
boundary is shown (a square image is preferable for the FFT). Fig. 6(b) shows the Fourier-filtered (FF) image, also cut to just show the boundary region. This shows clearly the improvement in visibility that can be achieved using this method.
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For small values of y these can be approximated to w1 ¼ w ty and w2 ¼ w þ ty:
Fig. 7. Schematic diagram showing the effect of tilt of the boundary away from the edge-on orientation on the apparent width in the HRTEM image, w1 ; and the apparent width after Fourier filtering, w2 :
Thus, for y ¼ 0:5 ; t ¼ 20 nm, ty =0.17 nm. For w ¼ 1 nm, this would give w1 ¼ 0:83 nm and w2 ¼ 1:17 nm. It is therefore clear that even very small tilts can give rise to significant discrepancies between the thicknesses estimated from each image. In this case, the true film thickness would then be the mean of the two measurements. A further complication can arise because of delocalisation of the lattice fringe images [5,7,35] causing the lattice fringes to extend over the intergranular film reducing its apparent width. The delocalisation, DR; of a set of lattice fringes with the spatial frequency, u, is given by the equation: DR ¼ Cs l3 u3 þ Df lu;
3.3. Thickness measurement of amorphous grain boundary films It may be noted, however, that the apparent thickness of the amorphous film is consistently larger in the FF-HRTEM image than in the original HRTEM image. In Fig. 5 the values are 0.75 nm as opposed to 0.6 nm; for Fig. 6, 0.8 and 0.65 nm. There are a number of reasons which can account for this, which will be discussed in turn. These are: grain boundary tilt, lattice fringe delocalisation, and diffuseness of the amorphous–crystalline interface. Firstly, tilt of the grain boundary away from the exactly edge-on will cause a difference in apparent width as already noted by Clarke [6]. Where lattice planes are imaged, they are likely to give stronger contrast than the amorphous material, and so, in an HRTEM image, any tilt will reduce the apparent thickness of the film (see, for example, Clarke [6]). Similarly, in the FF-HRTEM image, any tilt will result in an apparent broadening of the film. This is shown schematically in Fig. 7. The apparent widths in the HRTEM image and the FF-HRTEM image are w1 and w2 ; respectively, and may be shown to be w1 ¼
w w t tan y and w2 ¼ þ t tan y: cos y cos y
ð3Þ
ð4Þ
ð5Þ
where l is the microscope wavelength, Cs the spherical aberration and Df the defocus. The delocalisation is therefore reduced by increasing the accelerating voltage (and thus reducing l) or reducing Cs : It could only be totally eliminated in a spherical aberration corrected microscope [36]. In other microscopes it is then a strong function of both the spatial frequency and of the defocus, and becomes much worse for high spatial frequencies. The delocalisation can only be totally eliminated for one spatial frequency at any given focus. In Coene and Janssen [37] and Lichte [38] it was shown that the delocalisation is minimised for a range of spatial frequencies at a specific defocus Dfopt ¼ MCs l2 u2max
ð6Þ
where M is a factor between 0.75 and 1 and umax is the highest spatial frequency used in the image. In this case the delocalisation is minimised as DRmin ¼ Cs l3 u3max =4:
ð7Þ
This defocus is not simply related to the Scherzer defocus but is also a relatively low negative defocus. For instance, for the JEOL3010 used in this work with Cs ¼ 0:6 mm, and operated at a ( with alumihigh tension of 297 kV (l ¼ 0:0198 A) ( the na (0 0 0 6) planes with a spacing of 2.17 A, optimum defocus is Dfopt is 50 nm. For comparison, the Scherzer defocus is 41 nm. The
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Fig. 8. Plot of delocalisation versus defocus for the lattice fringes of Fig. 5(a) calculated individually for the main lattice fringes on either side of the boundary, and summed together to give the resulting total reduction in apparent film thickness at the boundary.
optimum defocus gives no delocalisation for the (0 0 0 6) planes and an average delocalisation for ( the lower spatial frequencies of DRmin ¼ 1:14 A. The minimum delocalisation possible for the two images shown in Fig. 5(a) and 6(a) was calculated using Eq. (5) for the fringes on both sides of the boundary, by adding the absolute values of delocalisation for each crystal corrected for the angle between the fringes and the boundary plane. The minimum possible sum of the out-of( for plane delocalisations was DR ¼ 0:95 A ( for Df ¼ 36:1 nm for Fig. 5(a) and DR ¼ 0:44 A Df ¼ 36:1 nm for Fig. 6(a). Fig. 6(a) shows a smaller effect since the fringes are at a larger angle to the boundary. Fig. 8 shows a graph of the perpendicular components of delocalisation plotted against defocus for both sets of lattice fringes in Fig. 1(a) and the sum of their absolute values. The absolute value of the defocus was not determined for either Fig. 5(a) or Fig. 6(a), although it was clearly small and underfocus in both cases. One can, however, conclude that image delocalisation had reduced the apparent width of ( and Fig. 6(a) by at least Fig. 5(a) by at least 0.95 A, ( It is highly likely that the focus was not 0.44 A. optimal and the effect could well have been larger. Thus fringe delocalisation could account for a
large part of the discrepancy between the thickness measurements from the HRTEM and the FF image. In order to minimise this effect, it is recommended to operate close to the Scherzer focus. Larger negative defoci (oDfScherzer ) are better for higher spatial frequencies and conversely smaller negative defoci (> DfScherzer ) are better for lower spatial frequencies. This should be calculated individually for each microscope to allow the optimum conditions to be determined. Operating at small negative defoci has the further advantage of minimising the appearance of Fresnel fringes, which may otherwise complicate the measurement of the film width on the images. Another possible source of discrepancies is atomic scale roughness, diffuseness or waviness of the crystal–glass interface. Such diffuseness has frequently been discussed in connection with the formation of Fresnel fringes [12–15], although strictly this is more to do with the form of the potential step and only indirectly the atomic structure. Such diffuseness would leave a region of uncertainty which would overlap crystalline and amorphous regions in the image, thus also resulting in a smaller apparent film thickness in the HRTEM image. In the FF-HRTEM image, if the film width is measured to the mid-point of the decay of the amorphous contrast at either edge of the film and not to the outer edges, it should give a more accurate value of the average film width, even if the width does display some local variations. Thus, a variety of effects contribute to the apparent thickness of the thin amorphous film in the HRTEM and FF-HRTEM images. These will, in general reduce the apparent thickness of the thin film in the HRTEM image. In the case of boundary tilt, this will also increase the apparent thickness in the FF-HRTEM image. Fringe delocalisation is a particularly serious problem in the measurement of film thicknesses, especially for the ultra-thin (o1 nm) films shown in this paper, unless the defocus is accurately known allowing the calculation of the delocalisation. Thus, measurements taken from the FF images are likely to be much more reliable than those taken from raw HRTEM images, especially if care is taken with exact edge-on alignment of the boundary.
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It should therefore be stressed that most film thickness measurements from HRTEM images will tend to slightly underestimate the true value. This may be one reason why a small discrepancy between the results gained from HRTEM measurements and those from Fresnel fringe measurements has sometimes been observed [6,7]. 3.4. Why use HRTEM with lattice fringes whatsoever? It is clear then, that the image processing of a HRTEM image of a grain boundary to remove the fringes from the two crystals is highly advantageous for the accurate measurement of the film thickness. One could therefore conclude that it would perhaps be simpler to orient the grain boundary so that neither crystal is oriented for diffraction with the grain boundary edge-on. This should, in principle give a similar image to those generated in this paper by FF. In practice, however, this is problematic. It is fairly straightforward to orient a grain boundary of a ceramic with a relatively large unit cell so that just one grain is diffracting, but to orient it so that both grains are non-diffracting with the grain boundary still edge-on is challenging. In alumina or similar materials having a relatively large unit cell, there is almost always some weak diffraction regardless of orientation. Secondly, orienting the grain boundary without the help of the crystal fringes is also somewhat challenging. In this case, there is little contrast to help the user whilst tilting. Nevertheless, Fresnel fringes would be of assistance here since they only appear symmetrically about the boundary when it is truly edge-on. It is clearly an alternative possibility to the method suggested here of conventional HRTEM imaging with lattice fringes and then Fourier filtering, although its application may be practically difficult. Up to now, the author has been unable to match the quality of the FF images shown above using this method. 3.5. Other applications of subtractive Fourier filtering In addition to the imaging of thin intergranular films, there are other possible applications of this
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Fourier filtering method. In particular, it may be noted in Fig. 1, that surface relief on the sample is more clearly revealed in the FF image. Some such applications could therefore be the imaging of surface features such as steps, texturing, damage or even thin amorphous surface films after the removal of the fringes from the crystal. In the case of features which may follow specific crystalline directions, in particular steps, it may well be possible to image such features using FF-HRTEM and by comparison to the original HRTEM image, be able to determine the crystallographic orientation of the features. More generally, such a subtractive approach could possibly be used to image things other than just amorphous or nonperiodic features. For example, by subtraction of the strong crystalline reflections other more subtle features could perhaps be more clearly revealed, for example: surface reconstructions, incommensurate periodicities, or nanoscale ordered domains in a disordered matrix.
4. Conclusions It is shown that for the alumina specimens of this study, Fresnel fringe analysis gives confusing results, showing fringes both for boundaries having amorphous films, and for film-free boundaries. Those at film-free boundaries probably arose as a result of a Y-doping induced expansion of the boundary resulting in a reduced density and reduced inner potential at the boundary. It should therefore be stressed that the film thickness determined by the indirect method of Fresnel fringe analysis should always be backed up with direct observations of these films using HRTEM. This is particularly important in materials where film-free boundaries occur and segregation of dopants or impurities to the boundaries is strong (e.g. Al2O3, NiO). It is shown that the visibility of non-periodic information in HRTEM micrographs can be greatly enhanced by the use of subtractive Fourier filtering method to remove the crystalline fringes. This is very advantageous for the imaging of extremely thin films (o0:8 nm thick) which otherwise show rather low contrast. Consideration of
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the effects of boundary tilt, fringe delocalisation and diffuseness/waviness of the amorphous/crystalline interfaces shows that the film thickness measured directly from a HRTEM image will tend to slightly underestimate the thickness, often by in ( In many cases, the worst effect will excess of 1 A. be due to image delocalisation although with the advent of spherical aberration corrected microscopes, this may well be less of a problem in the future. In contrast to this, if care is taken to minimise the boundary tilt and to measure from a midpoint on the two amorphous–crystalline interfaces, then the average boundary thickness is very well represented in the FF-HRTEM image. Similar results could in principle be achieved by orienting a grain boundary edge-on with neither grain in a diffracting orientation, although this may be challenging to achieve in practice. It was suggested that such subtractive Fourier filtering could be fruitfully applied in the imaging of a number of other non-periodic or incommensurate features in HRTEM images. These include imaging of surface relief and reconstructions, as well as incommensurate periodicities, and small ordered regions in a disordered matrix.
Acknowledgements Helpful discussions with Profs. M. Ruhle, . R.M. Cannon and H. Fuess, and Drs. M.A. Gulg . un . and G. Miehe are gratefully acknowledged. Of particular help in the preparation of this manuscript were the detailed and constructive criticisms of one of the reviewers, to whom I am greatly indebted. The author is very thankful to Dr. R. Voytovych for the preparation of the alumina ceramic used in this work and to Mrs. M. Sycha for preparing TEM specimens from the ceramic. The continuing support of the Fonds der Chemischen Industrie is gratefully acknowledged.
References [1] D.R. Clarke, G. Thomas, J. Am. Ceram. Soc. 61 (1977) 491.
[2] D.R. Clarke, J. Appl. Phys. 49 (1978) 2407. [3] M. Ruhle, . E. Bischoff, O. David, Ultramicroscopy 14 (1984) 37. [4] D.W. Susnitzky, C.B. Carter, J. Am. Ceram. Soc. 73 (1990) 2485. [5] O.L. Krivanek, T.M. Shaw, G. Thomas, J. Appl. Phys. 50 (1979) 4223. [6] D.R. Clarke, Ultramicroscopy 4 (1979) 33. [7] M.K. Cinibulk, H.-J. Kleebe, M. Ruhle, . J. Am. Ceram. Soc. 76 (1991) 426. [8] Q. Jin, D.S. Wilkinson, G.C. Weatherly, J. Eur. Ceram. Soc. 18 (1998) 2281. [9] Q. Jin, D.S. Wilkinson, G.C. Weatherly, J. Eur. Ceram. Soc. 18 (1998) 2281. [10] L. Reimer, Transmission Electron Microscopy, Springer, Berlin, Germany, 1989, pp. 63–69. [11] L.K.L. Falk, J. Eur. Ceram. Soc. 18 (1998) 2263. [12] J.N. Ness, W.M. Stobbs, T.F. Page, Philos. Mag. A 54 (1986) 679. [13] D.R. Rasmussen, C.B. Carter, Ultramicroscopy 32 (1990) 337. [14] F.M. Ross, W.M. Stobbs, Philos. Mag. A 63 (1991) 1. [15] F. Kara, R.E. Dunin-Borkowski, C.B. Boothroyd, J.A. Little, W.M. Stobbs, Ultramicroscopy 66 (1996) 59. [16] R.E. Dunin-Borkowski, Ultramicroscopy 83 (2000) 193. [17] K.L. Merkle, J.F. Reddy, C.L. Wiley, Ultramicroscopy 18 (1985) 281. [18] Y.K. Simpson, C.B. Carter, K.J. Morrissey, P. Angelini, J. Bentley, J. Mater. Sci. 21 (1986) 2689. [19] W. Coene, A.F. de Jong, D. van Dyck, G. van Tendeloo, J. van Landuyt, Phys. Stat. Sol. (A) 107 (1988) 521. [20] A.F. de Jong, W. Coene, D. van Dyck, Ultramicroscopy 27 (1989) 53. [21] P. Faust, M. Brandst.attner, A. Ding, Z. Phys. D 21 (1991) 285. [22] G. Nihoul, Microsc. Microanal. Microstruct. 2 (1991) 637. [23] M.J. H.ytch, M. Gandais, Philos. Mag. A 72 (1995) 619. [24] M.J. H.ytch, Microsc. Microanal. Microstruct. 8 (1997) 41. [25] E. Snoeck, B. Warot, H. Ardhuin, A. Rocher, M.J. Casanove, R. Kilaas, M.J. H.ytch, Thin Solid Films 319 (1998) 157. [26] M.J. H.ytch, E. Snoeck, R. Kilaas, Ultramicroscopy 74 (1998) 131. [27] R. Hillebrand, P. Werner, H. Sieber, D. Hesse, Microsc. Microanal. Microstruct. 8 (1997) 89. [28] M.J. H.ytch, L. Potez, Philos. Mag. A 76 (1997) 1119. [29] O.Yu. Gorbenko, I.E. Graboy, A.R. Kaul, H.W. Zandbergen, J. Magn. Magn. Mater. 211 (2000) 97. . en, Mater. Charact. 44 (2000) 441. [30] Y. Yao, A. Thol! [31] M.A. Gulg . un, . R. Voytovych, I. MacLaren, M. Ruhle, . R.M. Cannon, Interface Sci. 10 (2002) 99. [32] R. Voytovych, I. MacLaren, M.A. Gulg . un, . R.M. Cannon, M. Ruhle, . Acta Mater. 50 (2002) 3453. [33] I. MacLaren, R.M. Cannon, R. Voytovych, M.A. Gulg . un, . N. Popescu-Pogrion, C. Scheu, U. T.affner, M. Ruhle, . J. Am. Ceram. Soc. 86 (2002) 650.
ARTICLE IN PRESS I. MacLaren / Ultramicroscopy 99 (2004) 103–113 [34] S. Fabris, C. Els.asser, Acta Materialia 51 (2003) 71. [35] D.B. Williams, C.B. Carter, Transmission Electron Microscopy, Plenum Press, New York, USA, 1996, pp. 473–475. [36] M. Haider, H. Rose, S. Uhlemann, E. Schwan, B. Kabius, K. Urban, Ultramicroscopy 75 (1998) 53.
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[37] W. Coene, A. Janssen, in: P.W. Hawkes (Ed.), Signal and Image Processing in Microscopy and Microanalysis, Scanning Microscopy Supplement, Vol. 6, SEM Inc., AMF, O’Hare, IL, USA, 1992, p. 379. [38] H. Lichte, Ultramicroscopy 38 (1991) 13.
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Imaging and thickness measurement of amorphous intergranular films using TEM I. MacLaren* Institut fur . Materialwissenschaft, Technische Universitat . Darmstadt, Petersenstr. 23, D-64287 Darmstadt, Germany Received 26 February 2003; received in revised form 16 September 2003; accepted 17 October 2003
Abstract Fresnel fringe analysis is shown to be unreliable for grain boundaries in yttrium-doped alumina: the determined thicknesses do not agree well with those measured from high resolution transmission electron microscopy (HRTEM), the asymmetry between under- and overfocus is very large, and Fresnel fringes are sometimes shown at boundaries which contain no amorphous film. An alternative approach to the analysis of HRTEM images of grain boundary films is demonstrated: Fourier filtering is used to remove the lattice fringes from the image thereby significantly enhancing the visibility of the intergranular films. The apparent film thickness shows a discrepancy between measurements from the original HRTEM image and the filtered image. It was shown that fringe delocalisation and diffuseness of the amorphous/crystalline interfaces will lead to a significant underestimate of the thickness in unprocessed HRTEM images. In contrast to this, the average thickness can be much more accurately measured from the Fourier-filtered image, provided the boundary is oriented accurately edge-on. r 2003 Elsevier B.V. All rights reserved. PACS: 61.16.B; 61.72.M; 07.05.P Keywords: High resolution transmission electron microscopy (HRTEM); Image processing; Fourier filtering; Fresnel fringe analysis; Amorphous films
1. Introduction 1.1. Imaging and thickness measurement of grain boundary films The presence of thin amorphous films at grain boundaries in ceramics was first demonstrated in Si3N4 by Clarke and Thomas [1] and these have since been found in many different materials, *Corresponding author. E-mail address: [email protected] (I. MacLaren).
especially ceramics such as Si3N4-derivatives, ZnO varistors [2], ZrO2 [3] and Al2O3[3,4]. Three main methods have been developed for the imaging of such films in transmission electron microscopy (TEM). Firstly, a dark field image may be created using the objective aperture to select part of the first diffuse diffraction ring for the amorphous material, the so-called diffuse dark field (DDF) method. If care is taken to ensure that no crystalline reflections come within the aperture then the only regions to show bright contrast are those which are amorphous [5]. Secondly, the
0304-3991/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2003.10.002
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difference in atomic density or composition at the grain boundary leads to a change in inner potential. When imaged in the TEM this leads to the appearance of Fresnel fringes at the boundary in over- or underfocus conditions. The fringe spacing can be measured for various defocus values, plotted on a graph and extrapolated to Gaussian focus, giving a value for the film thickness [6]. Finally, films can be imaged using high resolution TEM (HRTEM) and the thickness directly measured from the micrograph [1,5,6]. In the latter two cases, the boundary must be oriented edge-on to the electron beam. Even for the dark field imaging method, one finds better contrast when the boundary is approximately edge-on, and this condition must be fulfilled if measurements of film thickness are to be made. In the case of HRTEM there is the additional and more exacting requirement that the specimen must be so oriented that lattice planes in both crystals are also edge on, so as to give a clear high resolution lattice image of both grains. These three methods have been compared by Cinibulk et al. [7]. It was shown that the HRTEM method gave the smallest thickness, that the Fresnel fringe method gave a thickness about 20% larger, and that the diffuse dark field method gave a thickness 50–100% larger than the HRTEM method. It was concluded that the DDF method is good for revealing amorphous films, but unsuited for thickness measurements, the Fresnel method is widely applicable but may overestimate the thickness slightly (although see Ref. [8]), and the HRTEM method is the most accurate for thickness measurements, but is also very exacting and therefore time consuming. Jin et al. [9] showed widespread applicability of the Fresnel fringe method for grain boundaries and noted less of a discrepancy with HRTEM measurements. Nevertheless, the Fresnel fringe spacing does not always obey the proposed functional dependence on defocus W ¼ W0 þ cDf 1=2 ;
ð1Þ
where c ¼ ð3lÞ1=2 :
ð2Þ
W is the observed fringe spacing, W0 is the film thickness, Df is the defocus, and l the electron wavelength [10]. Jin et al. [8] found that the constant of proportionality, c; was often smaller than expected; however, no explanation was given for this discrepancy. Moreover, an anisotropy is typically observed between the under- and overfocus plots [7,8,11,12]. More detailed studies of Fresnel fringe formation at interfaces including detailed mathematical and computational calculations have been performed by other workers [12–16]. It is clear from these studies that the amorphous-crystalline interface and thus the change in internal potential is rarely atomically sharp, and that grain boundary grooving as a result of preferential etching during specimen preparation is common; both of these effects affect the form of the fringes. Moreover, space charge has also been shown to affect the form of the fringes [15,16]. It is therefore no great surprise if the functional form of the dependence of the fringe spacing on defocus is not always as simple as first derived by Clarke on the assumption of sharp interfaces [6]. What is perhaps more worrying, however, is that the boundary region can have a reduced atomic density even when fully crystalline, resulting in the appearance of Fresnel fringes even in the absence of any amorphous film [3,17,18]. 1.2. Fourier filtering of HRTEM micrographs Fourier filtering has been used for the processing of HRTEM images almost ever since computers became powerful enough to perform fast Fourier transforms in a reasonable time period. For example, the group of van Dyck at the University of Antwerp were among the first to demonstrate the wide variety of possibilities of such an approach in image processing. This included noise removal from HRTEM images, observation of small deviations from periodicity at stacking faults, epitaxial interfaces, and antiphase boundaries, and the imaging of short range order in Au4Cr [19,20]. Since then it has been used for a wide variety of applications. For example, improvement of the visibility of small crystalline clusters or small particles [21–23],
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imaging displacement and strain in HRTEM images [24–26], generation of thickness maps in certain special cases [27], characterisation of antiphase domains and domain boundaries [28] and the localisation of dislocations in HRTEM images [25,29,30]. Whilst the periodic information or small deviations from periodicity at defects, steps etc. have been widely used in Fourier filtering, the nonperiodic information has generally been thrown away. In the present work, use is made of a somewhat neglected suggestion of Coene et al. [19] where Fourier filtering is used to do the opposite and remove the periodic information from HRTEM micrographs in order to allow the better observation of non-periodic information. In that case, it was used for the imaging of Pt particles on a crystalline support. In the current work, a similar procedure is used for the imaging of thin amorphous films at grain boundaries. The results of this approach are compared with those of standard TEM techniques for analysis of such grain boundary films.
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3. Results and analysis 3.1. Fresnel fringe analyses of grain boundaries in Y-doped alumina A grain boundary was tilted to an essentially non-diffracting orientation and a defocal series recorded to produce Fresnel fringe images. Three images from this series are shown in Fig. 1:
2. Experimental procedure An alumina ceramic doped with 500 wt ppm Y2O3 was prepared as described previously [31–33] and was sintered at 1450 C for 96 h and then annealed at 1650 C for 12 h. As a result, abnormal growth of some grains was observed as has been reported in detail elsewhere [33]. TEM specimens were prepared using a standard procedure of slicing, disc cutting, mechanical polishing, dimpling and ion milling. A final treatment of 30 min low energy ion-beam polishing using Ar+ ions accelerated by just 500 V was used to remove surface damage from ion-beam thinning and resulted in a sample having very little amorphous material at the edges of the hole in the specimen. The sample was examined using a JEOL3010 TEM equipped with a high resolution objective lens polepiece (Cs ¼ 0:6 mm) and operated at 297 kV; images were recorded using the CCD camera of the attached Gatan image filter. Processing of these images was performed using the Digital Micrograph software (Gatan Inc., Pleasanton, CA, USA).
Fig. 1. Fresnel images of a grain boundary (an HRTEM image of the same boundary is shown in (Fig. 5): (a) Df E 480 nm; (b) Df E0; (c) Df E þ 480 nm; (d) Intensity profiles across the boundary for each image, averaged 500 pixels along the boundary length.
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480 nm, Gaussian focus, and +480 nm, together with an intensity profile across the boundary for each image (averaged in each case from 500 pixels along the length of the boundary). As is normal, a bright central fringe surrounded by two dark fringes is seen in underfocus, and a dark central fringe surrounded by two bright fringes is seen in overfocus. At Gaussian focus a dark line was also seen at the boundary. This may be attributed to increased scattering due to the segregation of relatively heavy Y ions at the boundary, as is normally observed in these materials [31,33]. Fringe spacings were measured from the intensity profile for each image and plotted against defocus resulting in Fig. 2. This graph shows the commonly observed asymmetry between under- and over-focus [7,8,11,12]. The overfocus- and underfocus curves can both be fitted using a line of the form given in Eq. (1). In the case of the underfocus series, W0 ¼ 0:35 nm and c ¼ 0:053 nm0.5. For the overfocus series, W0 ¼ 0:5 nm and c ¼ 0:071 nm0.5. For comparison according to Eq. (2), c should equal 0.78 nm0.5. Taking the two series together, however, one would expect a film thickness under 0.5 nm from these results. The same film was examined by HRTEM and Fourier filtering as . . . . . . .
Fig. 2. Plot of Fresnel fringe spacings against defocus for the boundary shown in Figs. 1 and 5. Curves which fit the fringe spacings have been drawn on, the parameters for which are in the text. In dashed lines are curves drawn according to Eq. (1) using c as defined in Eq. (2) and W0 ¼ 0:7 nm.
Fig. 3. High resolution images of a grain boundary showing Fresnel fringes in under- and overfocus conditions, but containing no detectable amorphous film: (a) Df EF48 nm; (b) Approximately Gaussian focus; (c) Df E þ 64 nm.
described later and a film thickness of E0.75 nm was determined. It may be noted that the FWHM of the dark line at the boundary at zero defocus (Figs. 1(b) and (d)) was measured as 0.71 nm, in much better agreement with the HRTEM and Fourier filtering measurement. Fig. 3 shows HREM images of a grain boundary recorded at defocus values of (a) 48 nm, (b) E0 nm, and (c) +64 nm. None of these images shows any sign of an amorphous film at the grain boundary. Nevertheless, Fresnel fringes are apparent in Figs. 3(a) and (c). In Fig. 3a, a bright fringe is found at the centre surrounded by two dark fringes. In Fig. 3(c), a dark fringe is at the boundary centre, with bright fringes on either side. Both appear just as would be expected in the presence of an amorphous film. The bright central fringe in underfocus and dark central fringe in overfocus would seem rather to indicate a reduced GB inner potential [16]. This is perhaps surprising in view of the fact that all grain boundaries in such an yttrium-doped alumina have yttrium segregated to the boundary at levels roughly 3–9 cat nm2 (E0.25–0.75 monolayer) [31,33]. This not-insignificant concentration of high-Z yttrium segregated at the boundary could perhaps be expected to increase the inner potential of the grain boundary region. A recently published
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theoretical study investigated the effect of doping on grain boundary structure in alumina using a variety of different dopant ions including Sc, Y and La [34]. Both in special (prismatic S3) and more general (S13) boundaries it was shown that the dopants led to a significant GB expansion. In the case of Y at similar doping levels to those seen here, this was found to result in an expansion perpendicular to the boundary of about 6.5%. This increase in GB volume and consequent reduction in density probably more than offsets the increased potential of the large-Z yttrium atoms. Thus, grain boundary segregation induces an increase in grain boundary volume and therefore a reduction in density and thus inner potential as previously concluded by Ruhle . et al. [3]. As a further example, a plot of Fresnel Fringe spacings is shown in Fig. 4 for another boundary between small grains where no amorphous film would be expected [33] (although in this case this could not be confirmed by HRTEM as the material was too thick in this place). In this case there is a huge anisotropy between the under- and overfocus results. For the underfocus series, W0 ¼ 0:1 nm whereas for the overfocus series, W0 ¼ 0:75 nm. Meanwhile, the FWHM of the dark line at Gaussian focus was 0.72 nm. It is difficult in such a case with such huge underfocus–
. . . . . .
Fig. 4. Plot of Fresnel fringe spacings against defocus for another boundary not expected to have any amorphous film, showing significant anisotropy between under- and overfocus.
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overfocus anisotropy, to reach a definite conclusion about whether an amorphous film is present, and if so, how thick it is. Nevertheless, this illustrates very well the difficulties in performing reliable Fresnel fringe analysis on this material. Thus, it can be concluded that Fresnel fringe analysis of amorphous film thickness at grain boundaries in yttria-doped alumina is rather unreliable. Widths of grain boundary films measured from Fresnel fringes are significantly less than those measured by HRTEM. This could be a consequence of diffuse crystalline-amorphous interfaces or of compositional gradients across the film itself. In particular, Fresnel fringes are found at segregated grain boundaries which do not possess disordered cores, probably due to grain boundary expansion as a result of the Y-doping. Thus, the use of Fresnel fringe measurements cannot be considered a trustworthy technique for either the detection of amorphous films or the measurement of their thickness in alumina, in agreement with earlier studies [18]. 3.2. Fourier filtering of HRTEM images to reveal amorphous material The principle of the method here is very simple. Instead of using Fourier filtering to enhance periodic information, it is used to remove it [19]. This significantly enhances the visibility of all nonperiodic information, in particular that from amorphous phases, although surface relief is also more clearly revealed. Fig. 5(a) shows an HRTEM image of a grain boundary at which a thin disordered film is apparent. A fast Fourier transform (FFT) of this image is shown in Fig. 5(b). The spots belonging to crystalline reflections were then blanked out using circular masks, resulting in Fig. 5(c). This was then inverse Fourier transformed to produce the image shown in Fig. 5(d). This last image shows no lattice fringes, except as image artefacts at the edges, and shows clearly the amorphous core of the grain boundary. A certain amount of nonperiodic information in the form of surface relief over the whole picture area is also visible. In such a case where the amorphous film is very thin
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Fig. 5. Fourier filtering of an HRTEM image of a grain boundary with a thin amorphous film: (a) original HRTEM image; (b) Fourier transform; (c) Fourier transform with crystalline reflections masked out; (d) reverse filtered image, now showing the amorphous film at the boundary as well as other non-periodic information.
Fig. 6. HRTEM images of a second grain boundary: (a) original HRTEM image; (b) Fourier filtered version showing only the grain boundary film and other non-periodic features.
(o1 nm), the Fourier filtered image shows the film much more clearly than the original HRTEM image. Fig. 6(a) shows a HRTEM image of another grain boundary, this time cut so that only the
boundary is shown (a square image is preferable for the FFT). Fig. 6(b) shows the Fourier-filtered (FF) image, also cut to just show the boundary region. This shows clearly the improvement in visibility that can be achieved using this method.
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For small values of y these can be approximated to w1 ¼ w ty and w2 ¼ w þ ty:
Fig. 7. Schematic diagram showing the effect of tilt of the boundary away from the edge-on orientation on the apparent width in the HRTEM image, w1 ; and the apparent width after Fourier filtering, w2 :
Thus, for y ¼ 0:5 ; t ¼ 20 nm, ty =0.17 nm. For w ¼ 1 nm, this would give w1 ¼ 0:83 nm and w2 ¼ 1:17 nm. It is therefore clear that even very small tilts can give rise to significant discrepancies between the thicknesses estimated from each image. In this case, the true film thickness would then be the mean of the two measurements. A further complication can arise because of delocalisation of the lattice fringe images [5,7,35] causing the lattice fringes to extend over the intergranular film reducing its apparent width. The delocalisation, DR; of a set of lattice fringes with the spatial frequency, u, is given by the equation: DR ¼ Cs l3 u3 þ Df lu;
3.3. Thickness measurement of amorphous grain boundary films It may be noted, however, that the apparent thickness of the amorphous film is consistently larger in the FF-HRTEM image than in the original HRTEM image. In Fig. 5 the values are 0.75 nm as opposed to 0.6 nm; for Fig. 6, 0.8 and 0.65 nm. There are a number of reasons which can account for this, which will be discussed in turn. These are: grain boundary tilt, lattice fringe delocalisation, and diffuseness of the amorphous–crystalline interface. Firstly, tilt of the grain boundary away from the exactly edge-on will cause a difference in apparent width as already noted by Clarke [6]. Where lattice planes are imaged, they are likely to give stronger contrast than the amorphous material, and so, in an HRTEM image, any tilt will reduce the apparent thickness of the film (see, for example, Clarke [6]). Similarly, in the FF-HRTEM image, any tilt will result in an apparent broadening of the film. This is shown schematically in Fig. 7. The apparent widths in the HRTEM image and the FF-HRTEM image are w1 and w2 ; respectively, and may be shown to be w1 ¼
w w t tan y and w2 ¼ þ t tan y: cos y cos y
ð3Þ
ð4Þ
ð5Þ
where l is the microscope wavelength, Cs the spherical aberration and Df the defocus. The delocalisation is therefore reduced by increasing the accelerating voltage (and thus reducing l) or reducing Cs : It could only be totally eliminated in a spherical aberration corrected microscope [36]. In other microscopes it is then a strong function of both the spatial frequency and of the defocus, and becomes much worse for high spatial frequencies. The delocalisation can only be totally eliminated for one spatial frequency at any given focus. In Coene and Janssen [37] and Lichte [38] it was shown that the delocalisation is minimised for a range of spatial frequencies at a specific defocus Dfopt ¼ MCs l2 u2max
ð6Þ
where M is a factor between 0.75 and 1 and umax is the highest spatial frequency used in the image. In this case the delocalisation is minimised as DRmin ¼ Cs l3 u3max =4:
ð7Þ
This defocus is not simply related to the Scherzer defocus but is also a relatively low negative defocus. For instance, for the JEOL3010 used in this work with Cs ¼ 0:6 mm, and operated at a ( with alumihigh tension of 297 kV (l ¼ 0:0198 A) ( the na (0 0 0 6) planes with a spacing of 2.17 A, optimum defocus is Dfopt is 50 nm. For comparison, the Scherzer defocus is 41 nm. The
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Fig. 8. Plot of delocalisation versus defocus for the lattice fringes of Fig. 5(a) calculated individually for the main lattice fringes on either side of the boundary, and summed together to give the resulting total reduction in apparent film thickness at the boundary.
optimum defocus gives no delocalisation for the (0 0 0 6) planes and an average delocalisation for ( the lower spatial frequencies of DRmin ¼ 1:14 A. The minimum delocalisation possible for the two images shown in Fig. 5(a) and 6(a) was calculated using Eq. (5) for the fringes on both sides of the boundary, by adding the absolute values of delocalisation for each crystal corrected for the angle between the fringes and the boundary plane. The minimum possible sum of the out-of( for plane delocalisations was DR ¼ 0:95 A ( for Df ¼ 36:1 nm for Fig. 5(a) and DR ¼ 0:44 A Df ¼ 36:1 nm for Fig. 6(a). Fig. 6(a) shows a smaller effect since the fringes are at a larger angle to the boundary. Fig. 8 shows a graph of the perpendicular components of delocalisation plotted against defocus for both sets of lattice fringes in Fig. 1(a) and the sum of their absolute values. The absolute value of the defocus was not determined for either Fig. 5(a) or Fig. 6(a), although it was clearly small and underfocus in both cases. One can, however, conclude that image delocalisation had reduced the apparent width of ( and Fig. 6(a) by at least Fig. 5(a) by at least 0.95 A, ( It is highly likely that the focus was not 0.44 A. optimal and the effect could well have been larger. Thus fringe delocalisation could account for a
large part of the discrepancy between the thickness measurements from the HRTEM and the FF image. In order to minimise this effect, it is recommended to operate close to the Scherzer focus. Larger negative defoci (oDfScherzer ) are better for higher spatial frequencies and conversely smaller negative defoci (> DfScherzer ) are better for lower spatial frequencies. This should be calculated individually for each microscope to allow the optimum conditions to be determined. Operating at small negative defoci has the further advantage of minimising the appearance of Fresnel fringes, which may otherwise complicate the measurement of the film width on the images. Another possible source of discrepancies is atomic scale roughness, diffuseness or waviness of the crystal–glass interface. Such diffuseness has frequently been discussed in connection with the formation of Fresnel fringes [12–15], although strictly this is more to do with the form of the potential step and only indirectly the atomic structure. Such diffuseness would leave a region of uncertainty which would overlap crystalline and amorphous regions in the image, thus also resulting in a smaller apparent film thickness in the HRTEM image. In the FF-HRTEM image, if the film width is measured to the mid-point of the decay of the amorphous contrast at either edge of the film and not to the outer edges, it should give a more accurate value of the average film width, even if the width does display some local variations. Thus, a variety of effects contribute to the apparent thickness of the thin amorphous film in the HRTEM and FF-HRTEM images. These will, in general reduce the apparent thickness of the thin film in the HRTEM image. In the case of boundary tilt, this will also increase the apparent thickness in the FF-HRTEM image. Fringe delocalisation is a particularly serious problem in the measurement of film thicknesses, especially for the ultra-thin (o1 nm) films shown in this paper, unless the defocus is accurately known allowing the calculation of the delocalisation. Thus, measurements taken from the FF images are likely to be much more reliable than those taken from raw HRTEM images, especially if care is taken with exact edge-on alignment of the boundary.
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It should therefore be stressed that most film thickness measurements from HRTEM images will tend to slightly underestimate the true value. This may be one reason why a small discrepancy between the results gained from HRTEM measurements and those from Fresnel fringe measurements has sometimes been observed [6,7]. 3.4. Why use HRTEM with lattice fringes whatsoever? It is clear then, that the image processing of a HRTEM image of a grain boundary to remove the fringes from the two crystals is highly advantageous for the accurate measurement of the film thickness. One could therefore conclude that it would perhaps be simpler to orient the grain boundary so that neither crystal is oriented for diffraction with the grain boundary edge-on. This should, in principle give a similar image to those generated in this paper by FF. In practice, however, this is problematic. It is fairly straightforward to orient a grain boundary of a ceramic with a relatively large unit cell so that just one grain is diffracting, but to orient it so that both grains are non-diffracting with the grain boundary still edge-on is challenging. In alumina or similar materials having a relatively large unit cell, there is almost always some weak diffraction regardless of orientation. Secondly, orienting the grain boundary without the help of the crystal fringes is also somewhat challenging. In this case, there is little contrast to help the user whilst tilting. Nevertheless, Fresnel fringes would be of assistance here since they only appear symmetrically about the boundary when it is truly edge-on. It is clearly an alternative possibility to the method suggested here of conventional HRTEM imaging with lattice fringes and then Fourier filtering, although its application may be practically difficult. Up to now, the author has been unable to match the quality of the FF images shown above using this method. 3.5. Other applications of subtractive Fourier filtering In addition to the imaging of thin intergranular films, there are other possible applications of this
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Fourier filtering method. In particular, it may be noted in Fig. 1, that surface relief on the sample is more clearly revealed in the FF image. Some such applications could therefore be the imaging of surface features such as steps, texturing, damage or even thin amorphous surface films after the removal of the fringes from the crystal. In the case of features which may follow specific crystalline directions, in particular steps, it may well be possible to image such features using FF-HRTEM and by comparison to the original HRTEM image, be able to determine the crystallographic orientation of the features. More generally, such a subtractive approach could possibly be used to image things other than just amorphous or nonperiodic features. For example, by subtraction of the strong crystalline reflections other more subtle features could perhaps be more clearly revealed, for example: surface reconstructions, incommensurate periodicities, or nanoscale ordered domains in a disordered matrix.
4. Conclusions It is shown that for the alumina specimens of this study, Fresnel fringe analysis gives confusing results, showing fringes both for boundaries having amorphous films, and for film-free boundaries. Those at film-free boundaries probably arose as a result of a Y-doping induced expansion of the boundary resulting in a reduced density and reduced inner potential at the boundary. It should therefore be stressed that the film thickness determined by the indirect method of Fresnel fringe analysis should always be backed up with direct observations of these films using HRTEM. This is particularly important in materials where film-free boundaries occur and segregation of dopants or impurities to the boundaries is strong (e.g. Al2O3, NiO). It is shown that the visibility of non-periodic information in HRTEM micrographs can be greatly enhanced by the use of subtractive Fourier filtering method to remove the crystalline fringes. This is very advantageous for the imaging of extremely thin films (o0:8 nm thick) which otherwise show rather low contrast. Consideration of
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the effects of boundary tilt, fringe delocalisation and diffuseness/waviness of the amorphous/crystalline interfaces shows that the film thickness measured directly from a HRTEM image will tend to slightly underestimate the thickness, often by in ( In many cases, the worst effect will excess of 1 A. be due to image delocalisation although with the advent of spherical aberration corrected microscopes, this may well be less of a problem in the future. In contrast to this, if care is taken to minimise the boundary tilt and to measure from a midpoint on the two amorphous–crystalline interfaces, then the average boundary thickness is very well represented in the FF-HRTEM image. Similar results could in principle be achieved by orienting a grain boundary edge-on with neither grain in a diffracting orientation, although this may be challenging to achieve in practice. It was suggested that such subtractive Fourier filtering could be fruitfully applied in the imaging of a number of other non-periodic or incommensurate features in HRTEM images. These include imaging of surface relief and reconstructions, as well as incommensurate periodicities, and small ordered regions in a disordered matrix.
Acknowledgements Helpful discussions with Profs. M. Ruhle, . R.M. Cannon and H. Fuess, and Drs. M.A. Gulg . un . and G. Miehe are gratefully acknowledged. Of particular help in the preparation of this manuscript were the detailed and constructive criticisms of one of the reviewers, to whom I am greatly indebted. The author is very thankful to Dr. R. Voytovych for the preparation of the alumina ceramic used in this work and to Mrs. M. Sycha for preparing TEM specimens from the ceramic. The continuing support of the Fonds der Chemischen Industrie is gratefully acknowledged.
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