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Connecting small-angle diffraction with real-space images by quantitative transmission electron microscopy of amorphous thin-films
Ultramicroscopy 74 (1998) 221—235
Connecting small-angle diffraction with real-space images by quantitative transmission electron microscopy of amorphous thin-films Peter D. Miller*,1, J. Murray Gibson Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801, USA Received 16 February 1998; received in revised form 26 May 1998
Abstract We have explored quantitative methods to study projected density fluctuations on the scale 0.5—2 nm in amorphous thin-films using weak phase object transmission electron microscopy. A useful method includes quantitative analysis of electron images and their Fourier amplitudes, essentially combining small-angle scattering analysis and high-resolution imaging from the same microscopic region. By comparing the Fourier spectra of differently prepared SiO specimens, we 2 show that clear differences between similar-looking samples can be detected quantitatively, and defect sizes and densities can be measured. Further development of this technique may hold significant value for researchers working to understand small-angle diffraction and for those investigating the performance of microelectronic thin oxides. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 81.05.Gc; 61.16.Bg; 61.14.Rq; 61.43.Dq Keywords: Transmission electron microscopy; Transmission electron microscopy examination of materials; Amorphous state; Semiconductor materials; Electron scattering
1. Introduction Characterizing the medium-range atomic structure of amorphous solids presents a significant challenge for experimentalists. Short-range properties are well characterized by a variety of techniques, including vibrational spectroscopies, * Corresponding author. 1Present address: Frederick Seitz Material Research Laboratory, University of Illinois at Urbana-Champaign, 104 S. Goodwin Avenue, Urbana, IL 61801, USA.
magnetic resonance, inelastic scattering, and diffraction. Diffraction, for example, allows measurement of nearest-neighbor bond lengths, bond angles, coordination, etc., but such detailed information is limited by averaging within the sample to correlations over a range of up to &0.5 nm [1]. Understanding medium-range structure is important for structural models of amorphous materials and for technological applications. This is the regime between which atomistic and continuum effects predominate, so a challenge arises in differentiating between natural randomness, which
0304-3991/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 9 8 ) 0 0 0 4 4 - 8
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occurs on length scales (&1—2 nm, and true structure arising from defects. This study deals with structure such as density fluctuations and voids, rather than medium-scale atomic correlations, which are usually referred to as medium range order. It is known that conventional diffraction and bright-field TEM imaging can be used to detect such medium-range structure, but that three- and four-body correlations which characterize medium range order [2] are invisible to these techniques; instead, such correlations must be probed using different techniques such as the novel variable coherence microscopy [3,4]. While detailed structural models of crystalline solids have existed for decades, amorphous solids have not lent themselves to such simple sums and ordering rules. Instead, modeling of amorphous solids relies on atomistic ball-and-stick arrangements and attempts to incorporate randomness in bond lengths and bond angles, the details of the models being determined by stoichiometric or energy constraints, or by “disorder parameters” which are determined by fitting the model to experimental data such as a radial distribution function. While such models are idealizations, understanding a disordered solid’s properties on different length scales is critical for differentiating between models and building a useful description of amorphous solids. Technologically, the defect structure of amorphous thin films such as SiO 2 layers used in metal-oxide-semiconductor devices plays an important role in current models for dielectric breakdown [5], yet few experiments which directly correlate structural properties with electronic properties have been performed. This is at least partially due to a lack of experimental probes. Several researchers have attempted to use the TEM for structural analyses of amorphous samples. Approaches ranging from straight-forward ultra-high resolution imaging [6] to clever qualitative approaches [7] have been tried, and contributed to our understanding of the limits and strengths of TEM in this area. Weak contrast and contrast reversals caused by the microscope’s contrast transfer function (CTF) make identification of medium-scale structure a difficult challenge, though not impossible. Indeed, objects as small as
single heavy atoms on amorphous substrates have been successfully observed, but identifying weakcontrast phenomena becomes a significant challenge on length scales of approximately 2 nm and below. A noteworthy success in this area was reported in a 1980 study in which Gibson and Dong pioneered a qualitative method to identify pores in ultrathin amorphous samples [8]. The present work may be regarded as a quantitative extension of the work of Gibson and Dong, taking advantage of recent improvements in computational power and imaging tools. The study described here includes a careful analysis of the image intensity in the weak-phase object approximation, wherein the image intensity is proportional to the specimen thickness and density. Provided that contrast reversals caused by the contrast transfer function can be adequately understood, mass thickness fluctuations such as voids will appear as higher-contrast image features which can be isolated by intensity thresholding. But determining whether features appearing in a threshold image are caused by natural fluctuations of a thermodynamically stable continuous random network [4] or by defects such as voids can be difficult. Since we do not know the “correct” structural model for amorphous solids, we cannot state how the image statistics of a “perfect” specimen compare with the statistics of a “defective” specimen. In this study, we use comparisons between a high-quality standard sample and samples with suspected defects to demonstrate that intensity thresholding and Fourier analysis of the images can reveal the defect structure of some amorphous specimens. In the weak-phase object approximation, a relatively good approximation to the image intensity at &1 nm resolution is given by [9]. I(r)+I M1!2pu(r)?FT[P(K)]sin[s(K)]]N, (1) 0 where I is the main beam intensity, p is an interac0 tion constant, u(r) is the phase shift caused by the specimen, ? denotes a convolution integral, and FT denotes a Fourier transform. The quantity being transformed is the contrast transfer function (CTF) and is described by a reciprocal space coordinate K, a damping envelope P(K) caused by microscope instabilities and limited coherence, and a phase shift function s(K). Eq. (1) is described in
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detail elsewhere [10], though some features should be emphasized. Information about the specimen potential is contained in the phase shift function via the relation
P
t(r) p» (r) dz, (2) *//%3 0 where » (r) is the local inner potential of the *//%3 specimen, t(r) is the local thickness of the specimen, and the beam is traveling in the #z direction. In a simple minded approximation it is assumed that for amorphous specimens, u(r) is isotropic and uncorrelated and that it obeys Gaussian or Poisson statistics due to a solid presumed to be assembled stochastially. Consequently, a row of pixels in an ideal image (an image unaffected by the microscope response function) will contain intensities which follow Gaussian or Poisson random statistics. The mean intensity will be related to the mean thickness of the specimen, and if the solid is defective, statistically significant departures from such an intensity distribution might be observed and isolated by intensity thresholding. The frequency spectrum of the phase shift can also be used to look for structure. If the image intensity can be represented as a series of random, uncorrelated pixel intensities its Fourier transform will be random and uncorrelated, with amplitudes distributed about a constant mean value. Structure due to defects in the specimen will cause the mean value to vary, producing, e.g., a peak in the Fourier spectrum whose width is related to the size of defects in the specimen and whose amplitude is related to the density of defects. This effect is clearly seen in simulations and is easily understood using the convolution theorem of Fourier transforms [11]. This is also the conceptual basis of small-angle kinematical scattering analyses which are widely used to study mediumrange structures such as nanometer-scale voids in a wide variety of materials [12]. Small-angle electron scattering has been available for mediumrange microscopic studies since 1960s [13,14], and it has been applied more recently to the study of craze formation [15,16]. In the craze studies, as in the early studies, high image contrast was available and the small-angle scattering offered a convenient means of determining average properties for a complicated image. In parallel with small-angle scatteru(r)"
223
ing, digital image analysis was performed to verify interpretation of the scattering data. In the current study, contrast by the features of interest is much weaker, and we are interested in higher-resolution analyses. Rather than detecting the small-angle scattering from, e.g., micrometer-sized features in a 50 lm region defined by a selected area aperture, we are interested in quantitative measurement of small-angle scattering from nanometer-scale features in a 50 nm region, and directly correlating the scattered intensity with image features. Unfortunately, the TEM does not provide simple, direct access to the phase shift function. Instead, the frequency spectrum of phase shifts is modulated by the CTF which reverses the contrast and changes the amplitudes of phase shift frequencies. The magnitude and sign of contrast transfer is determined by the microscope user via the defocus setting. For high-resolution work, the defocus is typically set for broad, non-reversed contrast transfer for as wide a range of frequency as possible which can be greater 5 nm~1 for the microscope used in this study. Such a setting is not ideal for the current study however, since the medium to low frequencies corresponding to medium-scale structure would be greatly attenuated. Instead, we choose to work with a moderately high defocus setting which centers a passband on the frequency range of interest. Reversed contrast at higher frequencies was eliminated by post-acquisition frequency filtering as described below. In the current experiment, we investigated the usefulness of the above-described framework for studying medium-range structure in amorphous specimens, paying careful attention to artifacts arising from imaging conditions. While we found that our simple-minded assumptions about the randomness of amorphous specimens do not coincide with observed intensity distributions, we nonetheless found that the technique can successfully be used to detect and quantitatively describe medium-range structure.
2. Experimental details Images of several different types of ultra-thin SiO samples were used for this study, though the 2
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results for only two specimens which adequately demonstrate the technique will be reported here. One set of specimens was taken from high-quality microelectronic oxides grown by Martin Greene at Bell Labs. Six nm thick oxides were grown on [0 0 1]-oriented Si in a vertical diffusion furnace at 850°C. The oxidizing ambient was N O which was 2 purified at the point of use to impurity levels of less than 1]10~9. TEM specimens were prepared by protecting the oxide layer with wax and thinning the sample from the back side both mechanically and chemically in a solution of 1 : 10 hydrofluoric acid (49%) : nitric acid. The specimens were cleaned by immersion in boiling trichloroethane, acetone, and methyl alcohol. Low-magnification images verified that the original oxide layer was successfully protected. A second set of native oxide specimens was prepared for comparison to the furnace-grown oxides. Native oxide specimens were prepared by thinning discs of Si [0 0 1] mechanically and chemically as described above, although the surface was not protected. After a hole was formed in the specimen, it was cleaned using the Ishizaka—Shiraki method [17], which was followed by dipping the specimen in a 1 : 10 solution of hydrofluoric acid : deionized water to strip away any remaining oxide. The resulting bare silicon surface was allowed to reoxidize in air for 11 days before being analyzed. The thickness of the oxide which we studied is estimated to be 40 A_ [18]. Experimental micrographs were obtained using the University of Illinois Frederic Seitz Materials Research Laboratory’s Hitachi H9000 transmission electron microscope operating at a voltage of 300 kV. The spherical aberration coefficient for this microscope is 1.1 mm. Various image sets were obtained, the vast majority at a defocus setting of approximately !200 nm which centers an imaging passband at &1 nm~1 to emphasize mediumrange length scales ranging from 0.8 to 2 nm. The objective aperture blade was retracted allowing the objective lens to define the aperture, and a 100 lm condenser aperture was used. To ensure that imaging conditions were sufficiently close for image comparisons among the different specimens, images were obtained at several different defocus and illumination angle (second condenser lens) settings. Images were acquired using a Gatan 679/5 slow-
scan CCD camera with a YAG scintillator. The magnification was set at 50 000] resulting in a length scaling of 1.8 A_ /pixel. Images, which typically contained approximately 700 counts/pixel, were modulation transfer function corrected using the noise method [19].
3. Spectrum analysis After acquisition, images were analyzed in both reciprocal space (the diffractogram) and in real space. Reciprocal space studies began with the image’s frequency spectrum, which was obtained by taking the Fourier transform of an image then averaging the amplitudes over annuli centered on the origin. The result is a one-dimensional array of amplitudes, proportional to the amplitude of the object’s Fourier spectrum. Fig. 1 shows the procedure for acquiring the Fourier spectrum which can be expressed mathematically as follows, with d(K) representing a dirac delta function: II (K)"I Md(K)#2pDuJ (K)D[P2(K)]sin2(s(K))]1@2N. 0 (3) Images were always normalized to the main beam intensity I (recorded with the specimen re0 moved) before extracting the Fourier spectrum. The Fourier spectrum contains essentially the same information as one retrieves from a diffraction experiment, i.e. the Fourier transform of the sample’s potential. By decreasing the magnification at which we acquire images, we restrict the images to lower frequencies or smaller scattering angles. Consequently, it is very straightforward using the TEM to access the “small angle” regime for both diffraction and image analyses. For comparison with SAXS measurements, we note that the SAXS intensity for a void-containing sample is approximated by Guinier [9] as exp(!4p2K2R2 ). I (K)" f 2(h)o2v2 D ! 1!35*#-% 4!
(4)
We recognize that this is the atomic scattering factor f (H) multiplied by the “shape function” 2 exp(!4p2K2R2 ), where R is the radius of gyraD D tion of a void. We can rewrite Eq. (4) in terms of the
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225
Fig. 1. Fourier spectrum. Each value in the Fourier spectrum is averaged over an annulus centered at the origin.
atomic potential’s fourier transform »I (K) as I (K)JD»I (K)D2]FT [void]. (5) 4! From this expression, it can be seen that I (K) 4! contains the same information that is in the image fourier transform (Eq. (3)), but the image transform also contains the microscope response function. In the image, the phase shift results from the atomic potential integrated over the thickness, so both the atomic potential »I (K) and the “shape function” exp(!4p2K2R2 ) are incorporated into uJ (K). In D
principle, the image Fourier transform could be used to extract data for a Guinier plot, but this has not been done because of the difficult challenge of deconvolving the microscope response function. We use the Fourier spectrum to help distinguish amorphous samples from one another, and as a first approach to doing this, we use a ratio technique which offers direct access to the phase shift without the complications of the CTF. With the ratio technique, the Fourier spectra are divided by one another. If the imaging conditions are
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sufficiently close, then in the weak-phase object regime the resulting quantity is simply the ratio of the two exit-face phase shift functions. Letting R represent this ratio for images 1 and 2, then d(K)#2pDuJ (K)D](P2(K)]sin2[s (K)])1@2 1 1 1 R" d(K)#2pDuJ (K)D](P2(K)]sin2[s (K)])1@2 2 2 2 DuJ (K)D " 1 . (6) DuJ (K)D 2 Zeroes in the transfer function are not problematic due to uniform background noise which prevents the transfer function from actually reaching zero. Noise is assumed to be the result of inelastic
scattering and possibly of residual dark current in the CCD. By acquiring sets of images over ranges of operating conditions, the contrast transfer functions could be divided away, yielding reliable values of this ratio, as depicted in Fig. 2. The Fourier ratio can be used to determine the sizes, and, in principle, the shapes of image features. Assuming that one specimen is non-defective and that the other contains defects with a characteristic size D, the Fourier ratio will display a peak (or a dip) whose width is proportional to 1/D. This analysis is based upon the convolution theorem of ourier transforms, and both simulation and experiment bear out our conclusion, as shown in Fig. 3.
Fig. 2. Fourier ratio preparation: (a) Fourier spectra for two different samples; and (b) the ratio of the spectra in (a).
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227
Fig. 3. Difference in spectral power. This comparison of a furnace-grown oxide with a native oxide shows a quantitative difference between similar-looking images. The peak in the ratio of Fourier amplitudes at 1 nm~1 does not appear when comparing, e.g., two furnace-grown oxide (see Fig. 2) Analyzing the images associated with these spectra should reveal the origin of the peak.
Based upon this data we conclude that the native oxide sample described earlier contains defects approximately 0.7 nm in size. Describing defect shapes using the ratio could in principle be done by analyzing the shape of the peak. But, as the data in Fig. 3 show, noise in the spectrum due to a small sample thickness would make this procedure difficult. While one might argue that ratio techniques such as this are of limited usefulness since they only allow indirect conclusions about samples, the ratio technique has been valuable for our experiments. Taking away instrumental complications such as the contrast transfer function
and the detector’s modulation transfer function allows us to determine whether visually indistinguishable images contain “invisible” information about differences between the objects. It is also an unforgiving method of ensuring that experimental conditions are set identically. As we will see, small differences in illumination coherence or defocus setting show up clearly in the ratio. By taking ratios before proceeding with more detailed analyses, we know which images are best suited for further analysis. Also, this ratio method allows for easy simulation since we need only concern ourselves with the specimens, not with the microscope conditions.
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4. Identifying image artifacts Several artifacts which may be difficult to identify in an image or diffractogram show up clearly in the ratio of Fourier spectra. Identifying artifacts is important since the microscope settings strongly affect contrast transfer and can make differentiating between actual structures and meaningless artifacts quite difficult. A set of artifacts which can affect the appearance of small, weak-contrast features in images of disordered solids is presented here. Small differences in defocus are perhaps the easiest to visualize of these effects, since they cause the ratio to oscillate, as shown in Fig. 4. A small difference in defocus between two images may create the incorrect appearance that one image contains larger or smaller image features than the other. For this reason, many images taken at approximately equal defocus settings were acquired so that the images Fourier spectra could be compared and the best focus match be found. The best match was determined to be the pair of images for which the extrema of the two fourier spectra coincided most closely. Coherence artifacts also show up clearly. Limited beam coherence imposes a high-frequency damping function on the image, a function approximated by P(K)"exp[!p2D2j2K4/2 (7) !p2h2(C j2K3#KD)2/ln(2)], # 4 where D is the “defocus spread” due to a nonmonochromatic beam and electronic instabilities (measured [20] to be 10 nm), j is the wavelength, h is the beam convergence angle, and D is the # defocus setting. This expression assumes a gaussian-shaped source [21]. The strong dependence on wavevector imposes a severe damping envelope, and slight changes in operating conditions can produce significant changes in the contrast transfer. Fig. 5 shows the calculated ratio of the coherence functions when the illumination angle is changed by only 0.02 mrad1. This calculation very closely
1 For comparison, it may be noted that changing from the smallest (70 lm H) to the next-smallest (100 lm H) condenser aperture changes the value of h from 0.26 to 0.38 mrad. #
Fig. 4. Defocus effect on Fourier ratio. Slight differences in defocus cause the contrast transfer functions to fall out of phase, causing peaks and dips to appear in the ratio. Calculated radial Fourier scans for two images taken at defocus setting of !150 and !160 nm are shown in (a), with their ratio in (b). Experimental data for similar conditions is shown in (c).
matches experimental results acquired at similar conditions, also shown in Fig. 5. As a result of this sensitivity to beam convergence, the ratio technique allowed us to reproduce microscope illumination settings to within 0.005 mrad. Two other common image artifacts are easily identified visually in the Fourier transform of an image and in the Fourier ratio. First, astigmatism is a non-azimuthally-symmetric lens aberration which is described by adding an azimuth (/)
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Fig. 5. Coherence effects. By changing the angle of illumination upon the sample, the coherence of the electron beam is varied. The ratio of calculated CTFs at different beam convergences is shown in (a) and experimental data for images taken with focused and defocused illumination is shown in (a) and experimental data for images taken with focused and defocused illumination is shown in (b).
dependent term to the phase shift function, s(k):
A
B
K4 K2 s(K)"p K2Dj# C j3# C j sin(2/) . (8) 2 4 2 ! The anisotropy has the effect of averaging-out the higher frequency extrema in the radially averaged Fourier data, depressing the high-frequency radial Fourier spectrum, as shown in Fig. 6a. Specimen drift has an effect similar to astigmatism. Drift manifests itself as poor contrast transfer in the direction of drift, and it also shows up clearly in the high-frequency end of the Fourier spectrum. And, finally, uneven illumination across the specimen affects the low-frequency Fourier components. During an experiment, it is possible for the specimen to be non-uniformly illuminated as the user attempts to work with the highest possible current density by focusing the beam to a small spot. Sometimes, the beam is focused to a spot
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Fig. 6. (a) Astigmatism effect. A calculated Fourier ratio for astigmation : non-astigmatic transfer functions is shown. (b) The Fourier ratio for an image with a 10% illumunation bias and a uniformly illuminated specimen are shown. Illumination bias affects the very low-frequency Fourier components.
smaller than the CCD camera, resulting in uneven illumination. This error can be difficult to detect, since a bias as small as 10% across the image can dramatically affect Fourier amplitudes. Intuitively, we expect low-frequency consequences from such a mistake, and both simulation and experiment demonstrate its effect. Fig. 6b shows the effect of uneven illumination on the Fourier ratio.
5. Image analysis Our goal is to identify features in the real-space images of amorphous samples. Once we have Fourier spectra which tell us that the images contain structure information, we can invert the problem, returning to the images to directly visualize the structure. Parseval’s theorem describes the relationship between the spectral power (the Fourier transform amplitude) and the real-space power (the
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image intensity). If f(r) is a real-space function and g(k) is its Fourier transform, then
P
=
P
=
D f (r)D2 dr" Dg(k)D2 dk ~= ~=
(9)
If the Fourier transform of an image shows increased power over a spectral range, the frequencyfiltered image will also reflect that increase in power. For example, consulting Fig. 3, it can be seen that over the frequency range 0.2—1.2 nm~1, the native oxide specimen shows more power than the furnace oxide to which it was compared. Frequency filtering the images to include just those frequencies should highlight the differences.
6. Fourier filtering By thresholding the intensity of a Fourier filtered image, we can highlight features which contribute to the spectral power and limit the complications caused by an oscillating CTF. Images are Fourier filtered with a mask M(k) such that 1 M(k)" (1#eb(ki~k))(1#eb(k~k0))
(10)
which is essentially a radial step function (an annulus) with radii k ("0.4 nm~1) and k * 0 ("1.2 nm~1), and edges broadened by parameter b ("50 nm). This mask is multiplied by the image fourier transform. It should be noted that this mask nearly equals zero at k"0, so the mean of the filtered image will nearly equal zero as well. The resulting image, Fourier filtered and normalized to the main beam, is described by I (r )"2pu(r )?FT[CTF]M(k)]. & * 0
(11)
Since the phase shift depends on the mass thickness, local variations in density caused by defects will show up as variations in intensity. As discussed earlier, it is assumed that an ideal amorphous solid has a normally distributed mass thickness, with variations proportional to the square root of the number of atoms in a column through the sample. We analyze these Fourier filtered images using two separate techniques.
7. Feature counting In the first technique, we investigate the spatial distribution of intensities by applying an intensity threshold to the filtered images. Thresholding at intensity level I converts all intensities greater than 5 I to “1” and all those below I to “0”. If we set I to 5 5 5 a value in the high-end of the intensity distribution, the resulting image contains scattered features whose properties we can analyze further. We use a particle analysis program which indexes each feature, then determines each feature’s coordinates, area, ellipticity, average intensity (pre-threshold), and background intensity. To date, we have restricted our analysis to each feature’s area, constructing a histogram of feature sizes, though it should be noted that we could certainly expand the analysis to describe the shapes of individual defects. This would offer a useful complement to SAXS analyses of feature shapes which are sometimes limited by an excess of free parameters [22]. The choice of threshold is important for this analysis. Rather than setting our threshold levels to, e.g., 2, 3, and 4 times the standard deviation of each image, measures which would highlight anisotropy in the intensity distributions, we choose instead to threshold at fixed intensity values for comparison of two images. By Parseval’s theorem, these threshold values correspond to distinct levels of scattered intensity, and the histograms and filtered images show how the intensity is distributed. But we face the problem of knowing exactly where to set the threshold level. If the level is set too low, noise due to fluctuations in the sample (whether caused by defects or not) will dominate the threshold image; if the level is too high, only a few, also random, events will dominate the threshold image. Using a comparison of two images, the threshold can be varied and the absolute number of features counted. We observe that a maximum in the difference in the number of features appears at a certain threshold level (Fig. 7). Simulations show that 1 nm diameter voids (small, weak-contrast features) in 6 nm thick SiO films can be detected to an accu2 racy of approximately 80% using this method (Fig. 8). Consequently, we estimate a lower-limit defect density of 3$0.5]1013 cm~2 for the native oxide specimen described earlier (Fig. 9).
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Fig. 7. Threshold variation. By varying the threshold and counting the remaining features, the number of voids can be estimated. The simulation plot (a) shows the difference for a sample which contained 100 1 nm voids compared with a sample containing no voids. Since the maximum difference in features counted is 80, this technique is a good estimator of the difference of sample features. The experimental plot (b) shows a similar result.
While the sensitivity of the technique as determined by image simulations allows us to measure a void density of approximately &5]1012 cm~2 for 1 nm voids in a 46]46 nm image, this is slightly outside the sensitivity needed to investigate highquality microelectronic oxides, where defect densities of &1]1012 cm~2 are of immediate interest. Improvement in this area may be possible by ana-
lyzing energy-filtered images or performing atomic force microscope experiments in parallel to correlate results. It is desirable to detect the shapes of features in the image, but one shortcoming of the feature analysis when used with Fourier filtering is that artificially small features must still be considered. Fourier filtering was intended in part to limit the image to
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Fig. 8. Feature counting simulation. I images of an amorphous specimen without (a) and with (b) voids are shown. After thresholding at the same level, image (a) displays fewer features than does (b) ((c) and (d) are the threshold images). By overlaying the positions where the voids were inserted, it can be seen that the extra above-threshold features in (d) correspond to the positions of the voids.
features with sizes ranging between 0.8 and 2.5 nm, or 4—13 pixels in the image. But as Fig. 8 shows, filtering and thresholding smooth edges and erode features such that after processing, features which
were originally more than 4 pixels wide are reduced to features as small as a single pixel in the threshold image. Such alterations do not affect our ability to count features, but they make describing the shapes
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Fig. 9. Feature counting experiment. Thresholding images of native (a) and furnace (b) oxides taken at identical microscope conditions shows a difference in the number of above-threshold features in (c) and (d), respectively. Subtraction of the number of features in the two threshold images provides an estimate of the number of voids in the native oxide specimen.
or sizes of features more difficult. This speaks to the difficulty of determining the detailed nature of density fluctuations in amorphous samples: The power of this combined technique is that it allows us to
measure the average feature size by Fourier analysis, and feature density by filtering and thresholding, but we would still have difficulty differentiating between, e.g., a larger, connected
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low-density region and isolated void-like features. A real-space filtering technique which coarsens the image to exclude high-frequency effects may be a desirable alternative. A pattern recognition algorithm, perhaps an automated one similar to that described by Fan and Cowley [23] could significantly strengthen the real-space analyses of defect shapes.
8. Histogram analysis As a second approach to understand the Fourier filtered images we can look directly at the histogram of intensities contained in an image and examine its properties. The width of the distribution describes the distribution of phase shifts and, hence, of projected potentials. Asymmetries in the histogram will highlight deviations from an ideal normal distribution of thicknesses, and could also help identify defective amorphous films. In our histogram analysis, we look for evidence that the distribution is normal by applying chi square and moments analyses. If the intensities are normally distributed, the probability distribution for an image’s intensities is described by
C
D
1 !(I!IM )2 P(I) dI" exp , 2p2 pJ2p
(12)
IM is the mean value of the intensity and p the standard deviation. Moments of this distribution can be calculated and compared with the moments of experimentally measured distributions as a figure of merit for the nature of the experimental intensity distribution. Moments are defined by the relation
P
m" n
=
(I!IM )nP(I) dI. ~=
(13)
Table 1 Moments of normal distribution, with moment measurements for two experimental images n
P
=
I@n
~= J2p
To standardize the comparison, the data are normalized such that the second moment (the standard deviation when IM "0), is equal to 1, by the following substitution: I!IM I@" . p
The odd moments are all equal to zero, and the even moments are summarized in Table 1. If the difference in contrast between two images is caused by voids and we are imaging with a negative defocus, the image intensity distribution should be asymmetric, with the void-containing sample’s distribution weighted toward the bright side. Such a distribution will have positive odd-numbered moments, and its fit to a normal distribution should not be good. This analysis proved ambiguous. The intensity distributions for a variety of amorphous samples show varying amounts of asymmetry, with no correlation between samples. For instance, some of the lower moments for similar furnace-grown specimens differ by an order of magnitude, while the moments of a furnace-grown oxide may differ from those of a native oxide by only 10%. Two possible explanations for this ambiguity are that thickness variations in amorphous specimen are not poisson distributed but are determined by non-random processes which shape the distribution function, or that the weak-phase object approximation is not strictly valid. While it has been shown that the weak-phase object approximation breaks down in some respects even under the most favorable conditions [9], we believe that the approximation is a good one for lower-resolution applications such as the study presented here. It is notable that the odd-numbered moments of all intensity distributions are positive, suggesting that defects in the sample are caused by density deficits, which is to say that they are void-like, rather than being composed of regions of high density such as
(14)
1 2 3 4 5 6 7
0 1 0 3 0 15 0
Native exp(!I@2/2)dI@ oxide 0.0 1.0 0.1 3.1 1.0 16.9 11.1
Furnace oxide 0.0 1.0 0.1 3.2 1.0 18.1 14.1
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atomic clusters. This agrees with our other results, with density measurements for amorphous solids, and with some model conclusions [24].
9. Conclusion Medium-range structural study of amorphous specimens by microscopy techniques has remained a challenging problem, with few techniques available to study the structure of bulk materials and virtually no techniques available to study the structure of thin films. The present study merges the advantages of a common medium-range probe, small-angle kinematical scattering, with the advantages of weak-phase object TEM imaging. We find that once the microscope’s response function is adequately treated, real-space transmission electron micrographs can yield detailed information about the amorphous specimens. This type of information may be particularly valuable for studies of microelectronic devices and for comparison with other new microscopy studies of medium-range order in amorphous thin films.
Acknowledgements The authors would like to thank Prof. J. Lee of the University of Texas, Austin and Dr. M. Greene at Bell Labs for providing furnace-grown oxide samples for this work. This material is based upon work supported by the Semiconductor Research Corporation under grant SRC96-BJ-3621, by the National Science Foundation under grant DMR89-20538, and by the Department of Energy under grant DEFG02-96ER45439.
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References [1] B.E. Warren, X-ray Diffraction, Dover, Toronto, 1990. [2] N.E. Cusack, The Physics of Structurally Disordered Matter, IOP, Bristol, 1987. [3] M.M.J. Treacy, J.M. Gibson, Acta Crystallogr. A 52 (1996) 212. [4] J.M. Gibson, M.M.J. Treacy, Phys. Rev. Lett. 78 (1997) 1074. [5] D.J. DiMaria, J.H. Stathis, Appl. Phys. Lett. 70 (1997) 2708. [6] D.J. Smith, W.M. Stobbs, W.O. Saxton, Philos Mag. B 43 (1981) 907. [7] O.L. Krivanek, P.H. Gaskell, A. Howie, Nature 262 (1976) 454. [8] J.M. Gibson, D.W. Dong, J. Electrochem. Soc. 127 (1980) 2722. [9] J.M. Gibson, Ultramicroscopy 56 (1994) 26. [10] J.H.C. Spence, Experimental High-Resolution Electron Microscopy, 2nd ed., Oxford University Press, Oxford, 1988. [11] L. Reimer, Transmission Electron Microscopy, 3rd ed., Springer, Berlin, 1993. [12] A. Guinier, X-ray Diffraction in Crystals, Imperfect Crystals, and Amorphous Bodies, Freeman, San Francisco, 1963. [13] H. Mahl, W. Weitsch, Z. Naturforsch. 15A (1960) 1051. [14] R.H. Wade, J. Silcox, Appl. Phys. Lett. 8 (1966) 7. [15] A.C.-M. Yang, E.J. Kramer, J. Mater. Sci. 21 (1986) 3601. [16] P. Miller, E.J. Kramer, J. Mater. Sci. 26 (1991) 4445. [17] A. Ishizaka, Y. Shiraki, J. Electrochem. Soc. 133 (1986) 666. [18] A.H. Cairm, R. Sinclair, J. Electrochem. Soc. 137 (1990) 279. [19] A.L. Weickenmeier, W. Nu¨chter, J. Mayer, Optik 4 (1995) 147. [20] T. Zheng, Ph.D. Thesis, University of Illinois at UrbanaChampaign, 1995. [21] J.H. Spence, Experimental High-Resolution Electron Microscopy, Oxford, New York, 1988. [22] M. Shibayama, S. Nomura, T. Hashimoto, E. Thomas, J. Appl. Phys. 66 (1989) 4188. [23] G.Y. Fan, J.M. Cowley, Ultramicroscopy 17 (1985) 345. [24] B. Tuttle, J.B. Adams, Phys. Rev. B 53 (1996) 16265.
Connecting small-angle diffraction with real-space images by quantitative transmission electron microscopy of amorphous thin-films Peter D. Miller*,1, J. Murray Gibson Department of Physics, University of Illinois at Urbana-Champaign, 1110 W. Green St., Urbana, IL 61801, USA Received 16 February 1998; received in revised form 26 May 1998
Abstract We have explored quantitative methods to study projected density fluctuations on the scale 0.5—2 nm in amorphous thin-films using weak phase object transmission electron microscopy. A useful method includes quantitative analysis of electron images and their Fourier amplitudes, essentially combining small-angle scattering analysis and high-resolution imaging from the same microscopic region. By comparing the Fourier spectra of differently prepared SiO specimens, we 2 show that clear differences between similar-looking samples can be detected quantitatively, and defect sizes and densities can be measured. Further development of this technique may hold significant value for researchers working to understand small-angle diffraction and for those investigating the performance of microelectronic thin oxides. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 81.05.Gc; 61.16.Bg; 61.14.Rq; 61.43.Dq Keywords: Transmission electron microscopy; Transmission electron microscopy examination of materials; Amorphous state; Semiconductor materials; Electron scattering
1. Introduction Characterizing the medium-range atomic structure of amorphous solids presents a significant challenge for experimentalists. Short-range properties are well characterized by a variety of techniques, including vibrational spectroscopies, * Corresponding author. 1Present address: Frederick Seitz Material Research Laboratory, University of Illinois at Urbana-Champaign, 104 S. Goodwin Avenue, Urbana, IL 61801, USA.
magnetic resonance, inelastic scattering, and diffraction. Diffraction, for example, allows measurement of nearest-neighbor bond lengths, bond angles, coordination, etc., but such detailed information is limited by averaging within the sample to correlations over a range of up to &0.5 nm [1]. Understanding medium-range structure is important for structural models of amorphous materials and for technological applications. This is the regime between which atomistic and continuum effects predominate, so a challenge arises in differentiating between natural randomness, which
0304-3991/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 9 8 ) 0 0 0 4 4 - 8
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occurs on length scales (&1—2 nm, and true structure arising from defects. This study deals with structure such as density fluctuations and voids, rather than medium-scale atomic correlations, which are usually referred to as medium range order. It is known that conventional diffraction and bright-field TEM imaging can be used to detect such medium-range structure, but that three- and four-body correlations which characterize medium range order [2] are invisible to these techniques; instead, such correlations must be probed using different techniques such as the novel variable coherence microscopy [3,4]. While detailed structural models of crystalline solids have existed for decades, amorphous solids have not lent themselves to such simple sums and ordering rules. Instead, modeling of amorphous solids relies on atomistic ball-and-stick arrangements and attempts to incorporate randomness in bond lengths and bond angles, the details of the models being determined by stoichiometric or energy constraints, or by “disorder parameters” which are determined by fitting the model to experimental data such as a radial distribution function. While such models are idealizations, understanding a disordered solid’s properties on different length scales is critical for differentiating between models and building a useful description of amorphous solids. Technologically, the defect structure of amorphous thin films such as SiO 2 layers used in metal-oxide-semiconductor devices plays an important role in current models for dielectric breakdown [5], yet few experiments which directly correlate structural properties with electronic properties have been performed. This is at least partially due to a lack of experimental probes. Several researchers have attempted to use the TEM for structural analyses of amorphous samples. Approaches ranging from straight-forward ultra-high resolution imaging [6] to clever qualitative approaches [7] have been tried, and contributed to our understanding of the limits and strengths of TEM in this area. Weak contrast and contrast reversals caused by the microscope’s contrast transfer function (CTF) make identification of medium-scale structure a difficult challenge, though not impossible. Indeed, objects as small as
single heavy atoms on amorphous substrates have been successfully observed, but identifying weakcontrast phenomena becomes a significant challenge on length scales of approximately 2 nm and below. A noteworthy success in this area was reported in a 1980 study in which Gibson and Dong pioneered a qualitative method to identify pores in ultrathin amorphous samples [8]. The present work may be regarded as a quantitative extension of the work of Gibson and Dong, taking advantage of recent improvements in computational power and imaging tools. The study described here includes a careful analysis of the image intensity in the weak-phase object approximation, wherein the image intensity is proportional to the specimen thickness and density. Provided that contrast reversals caused by the contrast transfer function can be adequately understood, mass thickness fluctuations such as voids will appear as higher-contrast image features which can be isolated by intensity thresholding. But determining whether features appearing in a threshold image are caused by natural fluctuations of a thermodynamically stable continuous random network [4] or by defects such as voids can be difficult. Since we do not know the “correct” structural model for amorphous solids, we cannot state how the image statistics of a “perfect” specimen compare with the statistics of a “defective” specimen. In this study, we use comparisons between a high-quality standard sample and samples with suspected defects to demonstrate that intensity thresholding and Fourier analysis of the images can reveal the defect structure of some amorphous specimens. In the weak-phase object approximation, a relatively good approximation to the image intensity at &1 nm resolution is given by [9]. I(r)+I M1!2pu(r)?FT[P(K)]sin[s(K)]]N, (1) 0 where I is the main beam intensity, p is an interac0 tion constant, u(r) is the phase shift caused by the specimen, ? denotes a convolution integral, and FT denotes a Fourier transform. The quantity being transformed is the contrast transfer function (CTF) and is described by a reciprocal space coordinate K, a damping envelope P(K) caused by microscope instabilities and limited coherence, and a phase shift function s(K). Eq. (1) is described in
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detail elsewhere [10], though some features should be emphasized. Information about the specimen potential is contained in the phase shift function via the relation
P
t(r) p» (r) dz, (2) *//%3 0 where » (r) is the local inner potential of the *//%3 specimen, t(r) is the local thickness of the specimen, and the beam is traveling in the #z direction. In a simple minded approximation it is assumed that for amorphous specimens, u(r) is isotropic and uncorrelated and that it obeys Gaussian or Poisson statistics due to a solid presumed to be assembled stochastially. Consequently, a row of pixels in an ideal image (an image unaffected by the microscope response function) will contain intensities which follow Gaussian or Poisson random statistics. The mean intensity will be related to the mean thickness of the specimen, and if the solid is defective, statistically significant departures from such an intensity distribution might be observed and isolated by intensity thresholding. The frequency spectrum of the phase shift can also be used to look for structure. If the image intensity can be represented as a series of random, uncorrelated pixel intensities its Fourier transform will be random and uncorrelated, with amplitudes distributed about a constant mean value. Structure due to defects in the specimen will cause the mean value to vary, producing, e.g., a peak in the Fourier spectrum whose width is related to the size of defects in the specimen and whose amplitude is related to the density of defects. This effect is clearly seen in simulations and is easily understood using the convolution theorem of Fourier transforms [11]. This is also the conceptual basis of small-angle kinematical scattering analyses which are widely used to study mediumrange structures such as nanometer-scale voids in a wide variety of materials [12]. Small-angle electron scattering has been available for mediumrange microscopic studies since 1960s [13,14], and it has been applied more recently to the study of craze formation [15,16]. In the craze studies, as in the early studies, high image contrast was available and the small-angle scattering offered a convenient means of determining average properties for a complicated image. In parallel with small-angle scatteru(r)"
223
ing, digital image analysis was performed to verify interpretation of the scattering data. In the current study, contrast by the features of interest is much weaker, and we are interested in higher-resolution analyses. Rather than detecting the small-angle scattering from, e.g., micrometer-sized features in a 50 lm region defined by a selected area aperture, we are interested in quantitative measurement of small-angle scattering from nanometer-scale features in a 50 nm region, and directly correlating the scattered intensity with image features. Unfortunately, the TEM does not provide simple, direct access to the phase shift function. Instead, the frequency spectrum of phase shifts is modulated by the CTF which reverses the contrast and changes the amplitudes of phase shift frequencies. The magnitude and sign of contrast transfer is determined by the microscope user via the defocus setting. For high-resolution work, the defocus is typically set for broad, non-reversed contrast transfer for as wide a range of frequency as possible which can be greater 5 nm~1 for the microscope used in this study. Such a setting is not ideal for the current study however, since the medium to low frequencies corresponding to medium-scale structure would be greatly attenuated. Instead, we choose to work with a moderately high defocus setting which centers a passband on the frequency range of interest. Reversed contrast at higher frequencies was eliminated by post-acquisition frequency filtering as described below. In the current experiment, we investigated the usefulness of the above-described framework for studying medium-range structure in amorphous specimens, paying careful attention to artifacts arising from imaging conditions. While we found that our simple-minded assumptions about the randomness of amorphous specimens do not coincide with observed intensity distributions, we nonetheless found that the technique can successfully be used to detect and quantitatively describe medium-range structure.
2. Experimental details Images of several different types of ultra-thin SiO samples were used for this study, though the 2
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results for only two specimens which adequately demonstrate the technique will be reported here. One set of specimens was taken from high-quality microelectronic oxides grown by Martin Greene at Bell Labs. Six nm thick oxides were grown on [0 0 1]-oriented Si in a vertical diffusion furnace at 850°C. The oxidizing ambient was N O which was 2 purified at the point of use to impurity levels of less than 1]10~9. TEM specimens were prepared by protecting the oxide layer with wax and thinning the sample from the back side both mechanically and chemically in a solution of 1 : 10 hydrofluoric acid (49%) : nitric acid. The specimens were cleaned by immersion in boiling trichloroethane, acetone, and methyl alcohol. Low-magnification images verified that the original oxide layer was successfully protected. A second set of native oxide specimens was prepared for comparison to the furnace-grown oxides. Native oxide specimens were prepared by thinning discs of Si [0 0 1] mechanically and chemically as described above, although the surface was not protected. After a hole was formed in the specimen, it was cleaned using the Ishizaka—Shiraki method [17], which was followed by dipping the specimen in a 1 : 10 solution of hydrofluoric acid : deionized water to strip away any remaining oxide. The resulting bare silicon surface was allowed to reoxidize in air for 11 days before being analyzed. The thickness of the oxide which we studied is estimated to be 40 A_ [18]. Experimental micrographs were obtained using the University of Illinois Frederic Seitz Materials Research Laboratory’s Hitachi H9000 transmission electron microscope operating at a voltage of 300 kV. The spherical aberration coefficient for this microscope is 1.1 mm. Various image sets were obtained, the vast majority at a defocus setting of approximately !200 nm which centers an imaging passband at &1 nm~1 to emphasize mediumrange length scales ranging from 0.8 to 2 nm. The objective aperture blade was retracted allowing the objective lens to define the aperture, and a 100 lm condenser aperture was used. To ensure that imaging conditions were sufficiently close for image comparisons among the different specimens, images were obtained at several different defocus and illumination angle (second condenser lens) settings. Images were acquired using a Gatan 679/5 slow-
scan CCD camera with a YAG scintillator. The magnification was set at 50 000] resulting in a length scaling of 1.8 A_ /pixel. Images, which typically contained approximately 700 counts/pixel, were modulation transfer function corrected using the noise method [19].
3. Spectrum analysis After acquisition, images were analyzed in both reciprocal space (the diffractogram) and in real space. Reciprocal space studies began with the image’s frequency spectrum, which was obtained by taking the Fourier transform of an image then averaging the amplitudes over annuli centered on the origin. The result is a one-dimensional array of amplitudes, proportional to the amplitude of the object’s Fourier spectrum. Fig. 1 shows the procedure for acquiring the Fourier spectrum which can be expressed mathematically as follows, with d(K) representing a dirac delta function: II (K)"I Md(K)#2pDuJ (K)D[P2(K)]sin2(s(K))]1@2N. 0 (3) Images were always normalized to the main beam intensity I (recorded with the specimen re0 moved) before extracting the Fourier spectrum. The Fourier spectrum contains essentially the same information as one retrieves from a diffraction experiment, i.e. the Fourier transform of the sample’s potential. By decreasing the magnification at which we acquire images, we restrict the images to lower frequencies or smaller scattering angles. Consequently, it is very straightforward using the TEM to access the “small angle” regime for both diffraction and image analyses. For comparison with SAXS measurements, we note that the SAXS intensity for a void-containing sample is approximated by Guinier [9] as exp(!4p2K2R2 ). I (K)" f 2(h)o2v2 D ! 1!35*#-% 4!
(4)
We recognize that this is the atomic scattering factor f (H) multiplied by the “shape function” 2 exp(!4p2K2R2 ), where R is the radius of gyraD D tion of a void. We can rewrite Eq. (4) in terms of the
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225
Fig. 1. Fourier spectrum. Each value in the Fourier spectrum is averaged over an annulus centered at the origin.
atomic potential’s fourier transform »I (K) as I (K)JD»I (K)D2]FT [void]. (5) 4! From this expression, it can be seen that I (K) 4! contains the same information that is in the image fourier transform (Eq. (3)), but the image transform also contains the microscope response function. In the image, the phase shift results from the atomic potential integrated over the thickness, so both the atomic potential »I (K) and the “shape function” exp(!4p2K2R2 ) are incorporated into uJ (K). In D
principle, the image Fourier transform could be used to extract data for a Guinier plot, but this has not been done because of the difficult challenge of deconvolving the microscope response function. We use the Fourier spectrum to help distinguish amorphous samples from one another, and as a first approach to doing this, we use a ratio technique which offers direct access to the phase shift without the complications of the CTF. With the ratio technique, the Fourier spectra are divided by one another. If the imaging conditions are
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sufficiently close, then in the weak-phase object regime the resulting quantity is simply the ratio of the two exit-face phase shift functions. Letting R represent this ratio for images 1 and 2, then d(K)#2pDuJ (K)D](P2(K)]sin2[s (K)])1@2 1 1 1 R" d(K)#2pDuJ (K)D](P2(K)]sin2[s (K)])1@2 2 2 2 DuJ (K)D " 1 . (6) DuJ (K)D 2 Zeroes in the transfer function are not problematic due to uniform background noise which prevents the transfer function from actually reaching zero. Noise is assumed to be the result of inelastic
scattering and possibly of residual dark current in the CCD. By acquiring sets of images over ranges of operating conditions, the contrast transfer functions could be divided away, yielding reliable values of this ratio, as depicted in Fig. 2. The Fourier ratio can be used to determine the sizes, and, in principle, the shapes of image features. Assuming that one specimen is non-defective and that the other contains defects with a characteristic size D, the Fourier ratio will display a peak (or a dip) whose width is proportional to 1/D. This analysis is based upon the convolution theorem of ourier transforms, and both simulation and experiment bear out our conclusion, as shown in Fig. 3.
Fig. 2. Fourier ratio preparation: (a) Fourier spectra for two different samples; and (b) the ratio of the spectra in (a).
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Fig. 3. Difference in spectral power. This comparison of a furnace-grown oxide with a native oxide shows a quantitative difference between similar-looking images. The peak in the ratio of Fourier amplitudes at 1 nm~1 does not appear when comparing, e.g., two furnace-grown oxide (see Fig. 2) Analyzing the images associated with these spectra should reveal the origin of the peak.
Based upon this data we conclude that the native oxide sample described earlier contains defects approximately 0.7 nm in size. Describing defect shapes using the ratio could in principle be done by analyzing the shape of the peak. But, as the data in Fig. 3 show, noise in the spectrum due to a small sample thickness would make this procedure difficult. While one might argue that ratio techniques such as this are of limited usefulness since they only allow indirect conclusions about samples, the ratio technique has been valuable for our experiments. Taking away instrumental complications such as the contrast transfer function
and the detector’s modulation transfer function allows us to determine whether visually indistinguishable images contain “invisible” information about differences between the objects. It is also an unforgiving method of ensuring that experimental conditions are set identically. As we will see, small differences in illumination coherence or defocus setting show up clearly in the ratio. By taking ratios before proceeding with more detailed analyses, we know which images are best suited for further analysis. Also, this ratio method allows for easy simulation since we need only concern ourselves with the specimens, not with the microscope conditions.
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4. Identifying image artifacts Several artifacts which may be difficult to identify in an image or diffractogram show up clearly in the ratio of Fourier spectra. Identifying artifacts is important since the microscope settings strongly affect contrast transfer and can make differentiating between actual structures and meaningless artifacts quite difficult. A set of artifacts which can affect the appearance of small, weak-contrast features in images of disordered solids is presented here. Small differences in defocus are perhaps the easiest to visualize of these effects, since they cause the ratio to oscillate, as shown in Fig. 4. A small difference in defocus between two images may create the incorrect appearance that one image contains larger or smaller image features than the other. For this reason, many images taken at approximately equal defocus settings were acquired so that the images Fourier spectra could be compared and the best focus match be found. The best match was determined to be the pair of images for which the extrema of the two fourier spectra coincided most closely. Coherence artifacts also show up clearly. Limited beam coherence imposes a high-frequency damping function on the image, a function approximated by P(K)"exp[!p2D2j2K4/2 (7) !p2h2(C j2K3#KD)2/ln(2)], # 4 where D is the “defocus spread” due to a nonmonochromatic beam and electronic instabilities (measured [20] to be 10 nm), j is the wavelength, h is the beam convergence angle, and D is the # defocus setting. This expression assumes a gaussian-shaped source [21]. The strong dependence on wavevector imposes a severe damping envelope, and slight changes in operating conditions can produce significant changes in the contrast transfer. Fig. 5 shows the calculated ratio of the coherence functions when the illumination angle is changed by only 0.02 mrad1. This calculation very closely
1 For comparison, it may be noted that changing from the smallest (70 lm H) to the next-smallest (100 lm H) condenser aperture changes the value of h from 0.26 to 0.38 mrad. #
Fig. 4. Defocus effect on Fourier ratio. Slight differences in defocus cause the contrast transfer functions to fall out of phase, causing peaks and dips to appear in the ratio. Calculated radial Fourier scans for two images taken at defocus setting of !150 and !160 nm are shown in (a), with their ratio in (b). Experimental data for similar conditions is shown in (c).
matches experimental results acquired at similar conditions, also shown in Fig. 5. As a result of this sensitivity to beam convergence, the ratio technique allowed us to reproduce microscope illumination settings to within 0.005 mrad. Two other common image artifacts are easily identified visually in the Fourier transform of an image and in the Fourier ratio. First, astigmatism is a non-azimuthally-symmetric lens aberration which is described by adding an azimuth (/)
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Fig. 5. Coherence effects. By changing the angle of illumination upon the sample, the coherence of the electron beam is varied. The ratio of calculated CTFs at different beam convergences is shown in (a) and experimental data for images taken with focused and defocused illumination is shown in (a) and experimental data for images taken with focused and defocused illumination is shown in (b).
dependent term to the phase shift function, s(k):
A
B
K4 K2 s(K)"p K2Dj# C j3# C j sin(2/) . (8) 2 4 2 ! The anisotropy has the effect of averaging-out the higher frequency extrema in the radially averaged Fourier data, depressing the high-frequency radial Fourier spectrum, as shown in Fig. 6a. Specimen drift has an effect similar to astigmatism. Drift manifests itself as poor contrast transfer in the direction of drift, and it also shows up clearly in the high-frequency end of the Fourier spectrum. And, finally, uneven illumination across the specimen affects the low-frequency Fourier components. During an experiment, it is possible for the specimen to be non-uniformly illuminated as the user attempts to work with the highest possible current density by focusing the beam to a small spot. Sometimes, the beam is focused to a spot
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Fig. 6. (a) Astigmatism effect. A calculated Fourier ratio for astigmation : non-astigmatic transfer functions is shown. (b) The Fourier ratio for an image with a 10% illumunation bias and a uniformly illuminated specimen are shown. Illumination bias affects the very low-frequency Fourier components.
smaller than the CCD camera, resulting in uneven illumination. This error can be difficult to detect, since a bias as small as 10% across the image can dramatically affect Fourier amplitudes. Intuitively, we expect low-frequency consequences from such a mistake, and both simulation and experiment demonstrate its effect. Fig. 6b shows the effect of uneven illumination on the Fourier ratio.
5. Image analysis Our goal is to identify features in the real-space images of amorphous samples. Once we have Fourier spectra which tell us that the images contain structure information, we can invert the problem, returning to the images to directly visualize the structure. Parseval’s theorem describes the relationship between the spectral power (the Fourier transform amplitude) and the real-space power (the
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image intensity). If f(r) is a real-space function and g(k) is its Fourier transform, then
P
=
P
=
D f (r)D2 dr" Dg(k)D2 dk ~= ~=
(9)
If the Fourier transform of an image shows increased power over a spectral range, the frequencyfiltered image will also reflect that increase in power. For example, consulting Fig. 3, it can be seen that over the frequency range 0.2—1.2 nm~1, the native oxide specimen shows more power than the furnace oxide to which it was compared. Frequency filtering the images to include just those frequencies should highlight the differences.
6. Fourier filtering By thresholding the intensity of a Fourier filtered image, we can highlight features which contribute to the spectral power and limit the complications caused by an oscillating CTF. Images are Fourier filtered with a mask M(k) such that 1 M(k)" (1#eb(ki~k))(1#eb(k~k0))
(10)
which is essentially a radial step function (an annulus) with radii k ("0.4 nm~1) and k * 0 ("1.2 nm~1), and edges broadened by parameter b ("50 nm). This mask is multiplied by the image fourier transform. It should be noted that this mask nearly equals zero at k"0, so the mean of the filtered image will nearly equal zero as well. The resulting image, Fourier filtered and normalized to the main beam, is described by I (r )"2pu(r )?FT[CTF]M(k)]. & * 0
(11)
Since the phase shift depends on the mass thickness, local variations in density caused by defects will show up as variations in intensity. As discussed earlier, it is assumed that an ideal amorphous solid has a normally distributed mass thickness, with variations proportional to the square root of the number of atoms in a column through the sample. We analyze these Fourier filtered images using two separate techniques.
7. Feature counting In the first technique, we investigate the spatial distribution of intensities by applying an intensity threshold to the filtered images. Thresholding at intensity level I converts all intensities greater than 5 I to “1” and all those below I to “0”. If we set I to 5 5 5 a value in the high-end of the intensity distribution, the resulting image contains scattered features whose properties we can analyze further. We use a particle analysis program which indexes each feature, then determines each feature’s coordinates, area, ellipticity, average intensity (pre-threshold), and background intensity. To date, we have restricted our analysis to each feature’s area, constructing a histogram of feature sizes, though it should be noted that we could certainly expand the analysis to describe the shapes of individual defects. This would offer a useful complement to SAXS analyses of feature shapes which are sometimes limited by an excess of free parameters [22]. The choice of threshold is important for this analysis. Rather than setting our threshold levels to, e.g., 2, 3, and 4 times the standard deviation of each image, measures which would highlight anisotropy in the intensity distributions, we choose instead to threshold at fixed intensity values for comparison of two images. By Parseval’s theorem, these threshold values correspond to distinct levels of scattered intensity, and the histograms and filtered images show how the intensity is distributed. But we face the problem of knowing exactly where to set the threshold level. If the level is set too low, noise due to fluctuations in the sample (whether caused by defects or not) will dominate the threshold image; if the level is too high, only a few, also random, events will dominate the threshold image. Using a comparison of two images, the threshold can be varied and the absolute number of features counted. We observe that a maximum in the difference in the number of features appears at a certain threshold level (Fig. 7). Simulations show that 1 nm diameter voids (small, weak-contrast features) in 6 nm thick SiO films can be detected to an accu2 racy of approximately 80% using this method (Fig. 8). Consequently, we estimate a lower-limit defect density of 3$0.5]1013 cm~2 for the native oxide specimen described earlier (Fig. 9).
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Fig. 7. Threshold variation. By varying the threshold and counting the remaining features, the number of voids can be estimated. The simulation plot (a) shows the difference for a sample which contained 100 1 nm voids compared with a sample containing no voids. Since the maximum difference in features counted is 80, this technique is a good estimator of the difference of sample features. The experimental plot (b) shows a similar result.
While the sensitivity of the technique as determined by image simulations allows us to measure a void density of approximately &5]1012 cm~2 for 1 nm voids in a 46]46 nm image, this is slightly outside the sensitivity needed to investigate highquality microelectronic oxides, where defect densities of &1]1012 cm~2 are of immediate interest. Improvement in this area may be possible by ana-
lyzing energy-filtered images or performing atomic force microscope experiments in parallel to correlate results. It is desirable to detect the shapes of features in the image, but one shortcoming of the feature analysis when used with Fourier filtering is that artificially small features must still be considered. Fourier filtering was intended in part to limit the image to
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Fig. 8. Feature counting simulation. I images of an amorphous specimen without (a) and with (b) voids are shown. After thresholding at the same level, image (a) displays fewer features than does (b) ((c) and (d) are the threshold images). By overlaying the positions where the voids were inserted, it can be seen that the extra above-threshold features in (d) correspond to the positions of the voids.
features with sizes ranging between 0.8 and 2.5 nm, or 4—13 pixels in the image. But as Fig. 8 shows, filtering and thresholding smooth edges and erode features such that after processing, features which
were originally more than 4 pixels wide are reduced to features as small as a single pixel in the threshold image. Such alterations do not affect our ability to count features, but they make describing the shapes
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Fig. 9. Feature counting experiment. Thresholding images of native (a) and furnace (b) oxides taken at identical microscope conditions shows a difference in the number of above-threshold features in (c) and (d), respectively. Subtraction of the number of features in the two threshold images provides an estimate of the number of voids in the native oxide specimen.
or sizes of features more difficult. This speaks to the difficulty of determining the detailed nature of density fluctuations in amorphous samples: The power of this combined technique is that it allows us to
measure the average feature size by Fourier analysis, and feature density by filtering and thresholding, but we would still have difficulty differentiating between, e.g., a larger, connected
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low-density region and isolated void-like features. A real-space filtering technique which coarsens the image to exclude high-frequency effects may be a desirable alternative. A pattern recognition algorithm, perhaps an automated one similar to that described by Fan and Cowley [23] could significantly strengthen the real-space analyses of defect shapes.
8. Histogram analysis As a second approach to understand the Fourier filtered images we can look directly at the histogram of intensities contained in an image and examine its properties. The width of the distribution describes the distribution of phase shifts and, hence, of projected potentials. Asymmetries in the histogram will highlight deviations from an ideal normal distribution of thicknesses, and could also help identify defective amorphous films. In our histogram analysis, we look for evidence that the distribution is normal by applying chi square and moments analyses. If the intensities are normally distributed, the probability distribution for an image’s intensities is described by
C
D
1 !(I!IM )2 P(I) dI" exp , 2p2 pJ2p
(12)
IM is the mean value of the intensity and p the standard deviation. Moments of this distribution can be calculated and compared with the moments of experimentally measured distributions as a figure of merit for the nature of the experimental intensity distribution. Moments are defined by the relation
P
m" n
=
(I!IM )nP(I) dI. ~=
(13)
Table 1 Moments of normal distribution, with moment measurements for two experimental images n
P
=
I@n
~= J2p
To standardize the comparison, the data are normalized such that the second moment (the standard deviation when IM "0), is equal to 1, by the following substitution: I!IM I@" . p
The odd moments are all equal to zero, and the even moments are summarized in Table 1. If the difference in contrast between two images is caused by voids and we are imaging with a negative defocus, the image intensity distribution should be asymmetric, with the void-containing sample’s distribution weighted toward the bright side. Such a distribution will have positive odd-numbered moments, and its fit to a normal distribution should not be good. This analysis proved ambiguous. The intensity distributions for a variety of amorphous samples show varying amounts of asymmetry, with no correlation between samples. For instance, some of the lower moments for similar furnace-grown specimens differ by an order of magnitude, while the moments of a furnace-grown oxide may differ from those of a native oxide by only 10%. Two possible explanations for this ambiguity are that thickness variations in amorphous specimen are not poisson distributed but are determined by non-random processes which shape the distribution function, or that the weak-phase object approximation is not strictly valid. While it has been shown that the weak-phase object approximation breaks down in some respects even under the most favorable conditions [9], we believe that the approximation is a good one for lower-resolution applications such as the study presented here. It is notable that the odd-numbered moments of all intensity distributions are positive, suggesting that defects in the sample are caused by density deficits, which is to say that they are void-like, rather than being composed of regions of high density such as
(14)
1 2 3 4 5 6 7
0 1 0 3 0 15 0
Native exp(!I@2/2)dI@ oxide 0.0 1.0 0.1 3.1 1.0 16.9 11.1
Furnace oxide 0.0 1.0 0.1 3.2 1.0 18.1 14.1
P.D. Miller, J.M. Gibson / Ultramicroscopy 74 (1998) 221—235
atomic clusters. This agrees with our other results, with density measurements for amorphous solids, and with some model conclusions [24].
9. Conclusion Medium-range structural study of amorphous specimens by microscopy techniques has remained a challenging problem, with few techniques available to study the structure of bulk materials and virtually no techniques available to study the structure of thin films. The present study merges the advantages of a common medium-range probe, small-angle kinematical scattering, with the advantages of weak-phase object TEM imaging. We find that once the microscope’s response function is adequately treated, real-space transmission electron micrographs can yield detailed information about the amorphous specimens. This type of information may be particularly valuable for studies of microelectronic devices and for comparison with other new microscopy studies of medium-range order in amorphous thin films.
Acknowledgements The authors would like to thank Prof. J. Lee of the University of Texas, Austin and Dr. M. Greene at Bell Labs for providing furnace-grown oxide samples for this work. This material is based upon work supported by the Semiconductor Research Corporation under grant SRC96-BJ-3621, by the National Science Foundation under grant DMR89-20538, and by the Department of Energy under grant DEFG02-96ER45439.
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