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TEM-simulation of amorphous carbon films: influence of supercell packaging
Ultramicroscopy 88 (2001) 111–125
TEM-simulation of amorphous carbon films: influence of supercell packaging Helga Schultricha,*, Bernd Schultrichb a
Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden, Mommsenstraße 13, D-01069 Dresden, Germany b Fraunhofer-Institut fu¨r Werkstoff- und Strahltechnik, Winterbergstraße 28, D-01277 Dresden, Germany Received 14 February 2000; received in revised form 14 November 2000
Abstract Recent developments in thin film technology allow to prepare deliberately amorphous carbon films with structures widely varying between graphite-like (sp2) and diamond-like (sp3) atomic bonds. This leads to amorphous structures with correspondingly varying densities. By periodically changing deposition conditions, nanometer multilayers may be prepared consisting of carbon layers of different density. Simulation of the electron microscopic imaging allows to differentiate between such real structural details (on the nanometer scale) and artefacts induced by the imaging procedure. But it must be assured that the modeled structure reflects the real one with sufficient accuracy. Thorough comparison of different simulation strategies shows that for the adequate simulation of TEM imaging of amorphous materials, the thickness of the layer with independently distributed atoms has to exceed a certain limit. Then, the statistical scattering of the randomly distributed atoms will be averaged. Otherwise, if the model of the transmission electron microscopy sample is constructed as iteration of thin identical supercells, the superposition of scattering waves with constant phase differences results in enhanced local fluctuations burying the multilayer structure. For thicker packages of supercells with independent random distributions, the effect of statistical atomic arrangements is more and more leveled off. Hence, nanometer structures based on regions with different density will be visible more distinctively in the random background. For carbon, this critical thickness amounts to about 4 nm. This is of special importance for the visualization of nanoscaled heterogeneities like multilayers or nanotube-like inclusions in amorphous matrices. # 2001 Elsevier Science B.V. All rights reserved. PACS: 61.16B; 61.43.D; 68.55; 68.56 Keywords: Simulation; Thin film; Carbon; Amorphous structure
1. Introduction
*Corresponding author. Tel.: +49-351-463-4881; fax: +49351-463-3199. E-mail address: [email protected] (H. Schultrich).
Amorphous carbon films are commonly used as carrier foils in transmission electron microscopy (TEM). Increasingly, the investigation of the properties of such films themselves has become a field of growing interest from the point of view of
0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 0 6 9 - 9
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materials science. Recent developments in thin film technology allow to prepare deliberately amorphous carbon films with widely varying structures and properties [1]. The nature of the atomic bonds may differ from dominating graphite-like (the socalled sp2) to dominating diamond-like (or sp3) bonding. Correspondingly, the atomic densities of amorphous carbon may vary between 100 nm3 (or mass density of 2.0 g/cm3) in graphite-like films up to 160 nm3 (or 3.2 g/cm3) in diamondlike bonded amorphous carbon films. Such differences can even be produced on a nanometer-scale in a highly controlled manner, e.g. to generate multilayer structures [2]. Related structures are represented by amorphous silicon– germanium multilayer [3]. TEM has approved as a powerful tool to reveal these structural inhomogeneities inside amorphous carbon films. However, specially for such highly amorphous structures it may be difficult to differentiate between real structural details and artefacts induced by the imaging procedure. Misinterpretation of the electron micrographs may be avoided by thorough simulation of the electron microscopic imaging. But it must be assured that the modeled structure reflects the real one with sufficient accuracy. The problems of structure modeling arise mainly from two sources: (1) insufficient knowledge of the underlying structure or (2) insufficient consideration of the structural details. Here, the different structural levels must be clearly differentiated: for the nanostructured carbon films, the relevant structures are represented by the inhomogeneities on the nanometer scale. The detailed atomic arrangement (with typical dimensions of a tenth nanometer) is only of interest as far as it influences the nanometer contrast. In amorphous films the missing longrange order leads to statistical speckle patterns according to the projection of uncorrelated and stochastic atom layers. Hence, only for very thin parts of the sample (as they may appear at the edges) the TEM images allow conclusions on the detailed atomic structures [4,5]. For usual thicknesses of some nanometers these high resolution features are overlaid by the random overlap [6]. For example, it was demonstrated that the
experimental images of amorphous silicon (which is structurally equivalent to tetrahedral-bonded amorphous carbon) can be simulated by a random phase distribution (white noise) [7]. Consequently, a random arrangement of carbon atoms provides a good enough approximation [8]. More promising than the evaluation of such atomic details is the extraction of nanoscaled structures from TEM images of amorphous materials. Apart from the identification of local regions of structural order in an amorphous matrix [9] or of amorphous regions in a crystalline matrix [10] the investigation of amorphous multilayers is of particular interest [3]. In the following, a basic problem of multilayer simulation shall be discussed in more detail: the relation between the deterministic multilayer structure (with given thickness and density of the layers) and the random arrangement inside the layer. The simulation has been carried out for the cross-section of a multilayered carbon film with amorphous layers of alternating density. The influence of the deterministic density difference between the layers and of the statistical
Fig. 1. Cross-section through a C/CNx multilayer of 44 periods prepared by pulsed laser deposition (C.-F. Meyer, Fraunhofer Institut fu¨r Werkstoff- und Strahltechnik, Dresden). Thickness of a double layer 8.5 nm (5.0 nm with density 2.6 g/cm3and 3.5 nm with density 3.1 g/cm3). Conventional TEM image taken at underfocus of D=1850 nm (K. Brand, Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden).
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fluctuations inside the amorphous layers has been compared. By systematic variation of the dimensions of the basic unit it is explored, which thickness of stacks of independent amorphous supercells should be necessary to assure the imaging of randomly arranged atoms and to avoid pseudo-crystalline effects due to iteration of thin identical supercells.
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2. Preparation and imaging of carbon multilayer structures As experimental reference structure, carbon multilayers of different densities were deposited on a silicon substrate by pulsed laser deposition (PLD) with an ArF-Excimer laser [11]. They consist of 44 double layers of amorphous carbon
Fig. 2. WTF (a) at Gabor focus D=30 nm, (b) at large underfocus D=2000 nm. The amplitude wave transfer function cos w (i.e. the real part of the complex WTF eiw ) starts with a value of 1.0 at 0 nm1 (corresponding to w ¼ 0), whereas the phase wave transfer function sin w (i.e. the imaginary part) vanishes at 0 nm1. At Gabor focus, the amplitude transfer function is close to 1 for the dominating spatial frequencies around 0.3 nm1 corresponding to optimum conditions for holographic imaging. At large underfocus the phase transfer function is close to 1 corresponding to optimum conditions for conventional electron microscopy.
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of alternating density: 3.0–3.2 g/cm2 (thickness 3.5 nm) and 2.6–2.7 g/cm3 (thickness 5.0 nm), respectively (Fig. 1). The deposition has been carried out in a high-vacuum chamber (basic pressure 105 Pa) under nitrogen (partial pressure 0.3 Pa). The density variation has been achieved by additional plasma activation. It leads to an incorporation of about 10% nitrogen in the growing film, favoring graphitic bonding with lower density. TEM preparation by cross-sectioning is performed by means of conventional thinning techniques. The imaging conditions correspond to the Philips CM200 FEG/ST electron microscope: acceleration voltage 200 kV, coefficient of spherical aberration Cs ¼ 1:3 mm, condensor aperture 0.1 mrad, focus spread 7 nm. The layer structure is of the dimension Dx 3 nm, corresponding to spatial frequencies in the range of about 0.3 nm1. In comparison to atomic structures this means large area contrast. The differing density of the layers is mainly reflected by a proportional phase shift, which may be directly revealed by electron holographic methods [12,13]. In this case the amplitude wave transfer function cos w (i.e. the real part of the complex wave transfer function (WTF)) should be close to 1. This case is already realized for smaller defocus values (Fig. 2a). For optimized holographic imaging the Gabor focus (D=39 nm in the considered case) is recommended [14]. The electron hologram of the carbon multilayer is represented in Fig. 3. In conventional electron microscopy, the phase modulations are transferred to amplitude modulations via the phase contrast transfer function sin w (i.e. the imaginary part of the WTF). To realize jsin wj 1 for the rather small spatial frequencies of special interest, a large defocus in the order of 2000 nm is preferable (Fig. 2b). The corresponding electron micrograph is given in Fig. 1.
3. Modeling of the multilayer structure and simulation of electron microscopic imaging Simulation of the electron microscopic imaging has been carried out with the program EMS of
Fig. 3. Electron hologram of cross-section through a C/CNx multilayer (see Fig. 1) at Gabor focus D=30 nm (a), reconstruction of phase (b) with linescan (c) (H. Banzhof, Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden).
Pierre Stadelmann, Lausanne based on the multislice method [15]. Here, supercells are periodically continued infinitely in the lateral directions. The given sample thickness is realized by a corresponding number of iterations in normal direction. In
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the used program every such supercell iteration is treated as a single slice. Hence, for reliable calculations of the electron imaging the supercell thickness should not exceed some tenth nanometers. The cross-section of the layered structure is modeled by rectangular supercells with regions of
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different atomic density. Apart from this nanoscaled superstructure a disordered atomic arrangement has to be setup. The minimum lateral dimensions are determined by the layer structure with a characteristic length much larger than the atomic distances. Compared to this large-scaled lateral periodicity, the periodic continuation of the
Fig. 4. (a) Scheme of modeling a thick sample by iteration of supercell packages of different thicknesses with the limiting cases of completely independent atomic arrangements by using different supercells (right) and of pseudo-periodicity by using the identical supercell (left). (b) Scheme of modeling samples with increasing thickness by using different supercells leading to completely independent atomic arrangements (bottom) and by using the identical supercell leading to pseudo-periodicity (top).
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Fig. 5. Comparison of phase images of the C/C multilayer cross-section for different kinds of modeling (Gabor focus D=30 nm, radius of objective aperture 8 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample thickness 11.8 nm realized by 30 iterations of the same thin supercell. (c) Sample thickness 11.8 nm realized by a single thick supercell package consisting of a series of 30 supercells with different random distributions. In (c) the boundary between layers of different density clearly shows up. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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supercell in irradiation direction is much more critical. Due to the slice conditions thin supercells must be used with a correspondingly large number of iterations. For the case of completely identical supercells, this may lead to a (possibly smallscaled) artificial periodicity in irradiation direction, where the periodicity length could be comparable even to the atomic distances. Its influence on the resulting image is the main topic of this investigation. The investigation aims at the nanometer-scaled layered structure. Correspondingly, the electron microscopic imaging is well above the atomic resolution, and the exact consideration of the atomic arrangement is of no importance at this stage. In a first step, the two layers of different atomic density (150 and 130 nm3, respectively) were modeled by ordered diamond structures with correspondingly modified lattice parameters a. The in-plane dimensions of such a supercell amounts to Lx =8.436 nm (=5.125 nm (low density)+3.311 nm (high density)) and Ly =5.519 nm in direction perpendicular and parallel to the layer boundary, respectively and Lz =0.3942 nm in normal direction. By dividing the supercell in several subslices, it was proved that the supercell thickness of 0.3942 nm was small enough for a reliable application of the multislice algorithm. Such a supercell contains 2536 atoms. In a second step, all these ordered atomic positions are displaced randomly in all three directions between Lz /4 and +Lz /4, where atoms displaced beyond the boundaries of the supercell are positioned correspondingly on the opposite side. These supercells with a thickness of 0.39 nm represent the building stones for the constructions of thicker samples by combining them up to the desired thickness. Then the following question arises: Is it possible to iterate simply the same supercell or is it necessary to use different supercells with independent random arrangements? In the following, the effect of the supercell choice shall be considered along two lines: (1) a constant sample thickness of 11.8 nm (corresponding to about 76 000 atoms) realized by an iteration of supercell packages of increasing thickness, where each such package consists of a series of supercells
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with different random distributions (Fig. 4a), (2) a series of samples with increasing thickness up to 11.8 nm, realized by iteration of identical thin supercells (Fig. 4b, top) and by a series of different thin supercells (Fig. 4b, bottom) up to the corresponding thickness.
4. Results of imaging simulation The simulated TEM images of a thin supercell (thickness 0.39 nm) are presented in Figs. 5a, 6a, and Figs. 8a, 9a, for phase (at Gabor focus) and intensity (at a large defocus of 2000 nm), respectively, at different objective apertures. For an aperture radius of 8 nm1 the higher spatial frequencies are cut off by the WTF envelope function, whereas for a radius of 1 nm1 it is determined by the aperture. In the highly resolved phase images (large aperture), the boundary between the two carbon layers with differing density is nearly undetectable (Fig. 5a). With smaller apertures, the speckle from the randomly distributed positions of the atoms, which is here of no importance, is suppressed and the boundary becomes clearly visible (Fig. 6a). Under large area contrast conditions, i.e. at large defocus, the interface contrast is only slightly improved (Figs. 8a, 9a). In the first simulation series (corresponding to Fig. 4a), a realistic TEM sample with thickness 11.8 nm was built up by iteration of supercell packages of different thickness. In such a supercell package the atoms are independently distributed. To achieve the given thickness, this random arrangement is repeated several times up to the sample thickness: 30 times the 0.39 nm package, 15 times a 2 0.39 nm package, and so on. For brevity, in Figs. 5, 6, 8, 9 only the limiting cases are given. In general, any contrast is enhanced if the imaging electron beam passing through the sample, repeatedly feels an identical atomic distribution. This amplification is clearly visible from the intensity profiles. Repeating the same thin supercell again and again, the features of both the deterministic layer structure and the random atomic distribution are enhanced. Consequently, the visibility of the boundary is even worsened by
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Fig. 6. Comparison of phase images of the C/C multilayer cross section for different kinds of modeling (Gabor focus D=30 nm, radius of objective aperture 1 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same thin supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. Under these conditions, the multilayer structure is reflected very clearly. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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the prevailing statistical pattern in comparison to the thin slice (compare (a) and (b) in Figs. 5, 6, 8, 9). If, however, one uses only one thick supercell package with independently distributed atoms along the whole sample thickness, the effect of the random atomic distributions essentially averages out and the layer structure is more pronounced (compare (b) and (c) in Figs. 5, 6, 8, 9). This holds especially for small apertures, which additionally average the small-scale variations. The results of the simulation series are summarized in Figs. 7 and 10. The diagrams present the dependence of the relative contrast (image contrast/image mean) on the thickness of the supercell package at constant sample thickness (11.8 nm). The contrast is determined by the fluctuations inside the layer and by the density differences between the layers. For comparison, the contrast curves for a homogeneous amorphous material with the mean atomic density of 134 nm3 are included. In the second simulation series (corresponding to Fig. 4b) the sample thickness was varied between 0.39 nm and 11.8 nm. The sample was built up by corresponding iteration of the same thin supercell (0.39 nm thickness) and by a series of different supercells. Now, with increasing thickness, the statistical contrast is enhanced for the
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case of supercell iteration. In the opposite case of independent supercells, the random fluctuations are more and more blurred, and the deterministic layer structure appears more distinctively (compare (a) and (c) in Figs. 5, 6, 8, 9). The results are comprehended in Figs. 11 and 12 presenting the dependence of the relative contrast on the sample thickness for these two complementary cases of modeling.
5. Discussion The comparison of the TEM images}and in a more quantitative way, the comparison of the respective line profiles}clearly reveals the large influence of the kind of modeling. If thicker amorphous films are constructed in the simplest way by iterating the same thin supercell, i.e. by repeating the same statistical layer, then the image shows large unwanted fluctuations. Such a construction corresponds to a random atomic arrangement in the plane perpendicular to the incident electron beam, but in the beam direction the atomic arrangement shows a periodical order. Hence, the scattering waves are accumulated with well defined phase differences. Consequently, fluctuations are enhanced with increasing number
Fig. 7. Dependence of the relative contrast (image contrast / image mean) in the phase image (Gabor focus D=30 nm) of the C/C multilayer cross-section (sample thickness 11.8 nm) on the thickness of the supercell package (normalized to the supercell thickness of 0.39 nm). Radius of objective aperture 8 and 1 nm1. For comparison, the relative contrast for a homogeneous carbon layer with a mean density of 134 nm3 is included.
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Fig. 8. Comparison of intensity images of the C/C multilayer cross-section for different kinds of modeling (large underfocus D=2000 nm, radius of objective aperture 8 nm1 ). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. The boundary is rather blurred due to the large defocus. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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Fig. 9. Comparison of intensity images of the C/C multilayer cross-section for different kinds of modeling (large underfocus D=2000 nm, radius of objective aperture 1 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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Fig. 10. Dependence of the relative contrast (image contrast/image mean) in the intensity image (underfocus D=2000 nm) of the C/C multilayer cross section (sample thickness 11.8 nm) on the thickness of the supercell package (normalized to the supercell thickness of 0.39 nm). Radius of objective aperture 8 nm1 and 1 nm1. For comparison, the relative contrast for a homogeneous carbon layer is included. For modeling by many iterations of thin identical supercells the contrast is mainly determined by the artificially enhanced statistical fluctuations. For thicker supercell packages above 2 nm (corresponding to a normalized thickness above 5), the contrast reflects in the main the density step.
of iterations, i.e. for higher film thickness. On the contrary, if the sample is modeled by a truly random arrangement, then the phases of the scattered waves are not correlated and, as desirable and correct, their effect is more and more leveled off with increasing film thickness. The same trend is observed for given foil thickness, if the thickness of the supercell packages is varied: Already a doubling of the random layer leads to a strong reduction of the artificially induced fluctuations. Above 2 nm there is only a moderate further decrease. The critical thickness necessary to avoid modeling artefacts depends on the special imaging conditions: In the case of carbon foils for amplitude imaging and/or for imaging with small apertures, a thickness of 4 nm of the supercell package with independently distributed atoms is sufficient to level off the statistical fluctuations. The comparison with a homogeneous material without a density jump across the sample shows that the remaining contrast is mainly determined by the deterministic difference between the high density and the lower density
region of the TEM sample. On the contrary, for the phase imaging at large aperture the contrast is dominated by the statistics even at 11 nm thickness.
6. Conclusions For the adequate simulation of TEM imaging of amorphous materials, the thickness of the supercell packages with independent random distributions has to exceed a certain limit. Then the statistical scattering from the disordered atoms will be averaged. Otherwise, if the model of the TEM sample is constructed as a stack of very thin identical layers, the superposition of scattering waves with constant phase differences results in enhanced local fluctuations burying the nanometer scaled structure to be investigated. For thicker films, the effect of statistical atomic arrangements is more and more leveled off. Hence, deterministic nanometer-structures due to differing densities will be visible more distinctively in the random background. For carbon, this critical thickness
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Fig. 11. Dependence of the relative contrast (image contrast/image mean) in the phase image (Gabor focus D=30 nm) of the C/C multilayer cross-section on the sample thickness (normalized to the supercell thickness of 0.39 nm) realized by repeated iteration of the same thin supercell (supercell thickness 0.39 nm) and by a series of independent supercells. (a) Radius of aperture objective 8 nm1. (b) Radius of aperture objective 1 nm1. For thicker samples modeled by different supercells, the contrast decreases towards the limit given by the density step. When the sample is built up by repetition of identical supercells, the contrast increases by superposition of the effects of the random atomic arrangement, same for all layers.
amounts to about 4 nm (at a mean density of about 2.7 g/cm3). The guidelines of the right choice of supercell stack dimensions discussed above will be applied in a forthcoming paper on a more complex structure of high interest: the investigation of thin carbon films with nanotube-like inclusions embedded in an amorphous matrix [16].
Acknowledgements Dr. C.-F. Meyer is greatly acknowledged for the preparation of the layered carbon samples, Dr. H. Banzhof and Dr. K. Brand for the TEM investigations. The authors thank Prof. H. Lichte for stimulating discussions. The work was partially supported by Deutsche Forschungsgemeinschaft, Grant no. Li 346/13-1.
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Fig. 12. Dependence of the relative contrast (image contrast/image mean) in the intensity image (underfocus D=2000 nm) of the C/C multilayer cross section on the sample thickness (normalized to the supercell thickness of 0.39 nm) realized by repeated iteration of the same thin supercell (supercell thickness 0.39 nm) and by a series of independent supercells. (a) Radius of objective aperture 8 nm1. (b) Radius of objective aperture 1 nm1. For thicker samples modeled by different supercells, the contrast increases slowly proportional to the thickness. When the sample is built up by repetition of identical supercells, the contrast increases steeply by superposition of the effects of the same random atomic arrangements.
References [1] J.J. Pouch, S.A. Alterovitz (Eds.), Properties and characterization of amorphous carbon films, Materials Science Forum 52&53, Trans Tech Publications, Aedermannsdorf/Switzerland, 1990. [2] H. Banzhof, K. Brand, H. Lichte, A. Luft, C.-F. Meyer, H.-J. Scheibe, B. Schultrich, H. Ziegele, Mater. Sci. Forum 287–288 (1998) 243. [3] P. Henning, L.R. Wallenberg, K. Ja¨rrendahl, L. Hultman, L.K.L. Falk, J.-E. Sundgren, Ultramicroscopy 66 (1996) 221. [4] A. Howie, J. Non-Cryst. Solids 31 (1978) 41.
[5] V. Schmidt, R. Hillebrand, R. Albrecht, W. Neumann, B. Mu¨ller, Ultramicroscopy 17 (1985) 357. [6] K. Scheerschmidt, R. Hillebrand, J. Heydenreich, Optik 77 (1987) 115. [7] G.Y. Fan, J.M. Cowley, Ultramicroscopy 21 (1987) 125. [8] C.B. Boothroyd, Ultramicroscopy 83 (2000) 159. [9] R.S. Timsit, W.G. Waddington, C. Gallerneault, Ultramicroscopy 45 (1992) 65. [10] M.L. Miller, R.C. Ewing, Ultramicroscopy 48 (1993) 203. [11] C.F. Meyer, Pra¨paration von CNx-Schichten mittels PLD Final Report BMBF-Project 03N 5002 E5 ‘‘Herstellung, Struktur und Eigenschaften von CNx-Schichten’’, April 1998.
H. Schultrich, B. Schultrich / Ultramicroscopy 88 (2001) 111–125 [12] H. Lichte, Ultramicroscopy 47 (1992) 223. [13] H. Lichte, E. Vo¨lkl, K. Scheerschmidt, Ultramicroscopy 47 (1992) 231. [14] H. Lichte, Ultramicroscopy 38 (1991) 13. [15] P.A. Stadelmann, Ultramicroscopy 21 (1987) 131.
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[16] B. Schultrich, H.-J. Scheibe, H. Mai, Herstellung und Charakterisierung nanodisperser Kohlenstoffschichten, in: K. Schulte, K.U. Kainer (Eds.), Verbundwerkstoffe und Werkstoffverbunde, Wiley VCH Verlag GmbH, Weinheim, 1999, pp. 671–676.
TEM-simulation of amorphous carbon films: influence of supercell packaging Helga Schultricha,*, Bernd Schultrichb a
Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden, Mommsenstraße 13, D-01069 Dresden, Germany b Fraunhofer-Institut fu¨r Werkstoff- und Strahltechnik, Winterbergstraße 28, D-01277 Dresden, Germany Received 14 February 2000; received in revised form 14 November 2000
Abstract Recent developments in thin film technology allow to prepare deliberately amorphous carbon films with structures widely varying between graphite-like (sp2) and diamond-like (sp3) atomic bonds. This leads to amorphous structures with correspondingly varying densities. By periodically changing deposition conditions, nanometer multilayers may be prepared consisting of carbon layers of different density. Simulation of the electron microscopic imaging allows to differentiate between such real structural details (on the nanometer scale) and artefacts induced by the imaging procedure. But it must be assured that the modeled structure reflects the real one with sufficient accuracy. Thorough comparison of different simulation strategies shows that for the adequate simulation of TEM imaging of amorphous materials, the thickness of the layer with independently distributed atoms has to exceed a certain limit. Then, the statistical scattering of the randomly distributed atoms will be averaged. Otherwise, if the model of the transmission electron microscopy sample is constructed as iteration of thin identical supercells, the superposition of scattering waves with constant phase differences results in enhanced local fluctuations burying the multilayer structure. For thicker packages of supercells with independent random distributions, the effect of statistical atomic arrangements is more and more leveled off. Hence, nanometer structures based on regions with different density will be visible more distinctively in the random background. For carbon, this critical thickness amounts to about 4 nm. This is of special importance for the visualization of nanoscaled heterogeneities like multilayers or nanotube-like inclusions in amorphous matrices. # 2001 Elsevier Science B.V. All rights reserved. PACS: 61.16B; 61.43.D; 68.55; 68.56 Keywords: Simulation; Thin film; Carbon; Amorphous structure
1. Introduction
*Corresponding author. Tel.: +49-351-463-4881; fax: +49351-463-3199. E-mail address: [email protected] (H. Schultrich).
Amorphous carbon films are commonly used as carrier foils in transmission electron microscopy (TEM). Increasingly, the investigation of the properties of such films themselves has become a field of growing interest from the point of view of
0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 1 ) 0 0 0 6 9 - 9
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materials science. Recent developments in thin film technology allow to prepare deliberately amorphous carbon films with widely varying structures and properties [1]. The nature of the atomic bonds may differ from dominating graphite-like (the socalled sp2) to dominating diamond-like (or sp3) bonding. Correspondingly, the atomic densities of amorphous carbon may vary between 100 nm3 (or mass density of 2.0 g/cm3) in graphite-like films up to 160 nm3 (or 3.2 g/cm3) in diamondlike bonded amorphous carbon films. Such differences can even be produced on a nanometer-scale in a highly controlled manner, e.g. to generate multilayer structures [2]. Related structures are represented by amorphous silicon– germanium multilayer [3]. TEM has approved as a powerful tool to reveal these structural inhomogeneities inside amorphous carbon films. However, specially for such highly amorphous structures it may be difficult to differentiate between real structural details and artefacts induced by the imaging procedure. Misinterpretation of the electron micrographs may be avoided by thorough simulation of the electron microscopic imaging. But it must be assured that the modeled structure reflects the real one with sufficient accuracy. The problems of structure modeling arise mainly from two sources: (1) insufficient knowledge of the underlying structure or (2) insufficient consideration of the structural details. Here, the different structural levels must be clearly differentiated: for the nanostructured carbon films, the relevant structures are represented by the inhomogeneities on the nanometer scale. The detailed atomic arrangement (with typical dimensions of a tenth nanometer) is only of interest as far as it influences the nanometer contrast. In amorphous films the missing longrange order leads to statistical speckle patterns according to the projection of uncorrelated and stochastic atom layers. Hence, only for very thin parts of the sample (as they may appear at the edges) the TEM images allow conclusions on the detailed atomic structures [4,5]. For usual thicknesses of some nanometers these high resolution features are overlaid by the random overlap [6]. For example, it was demonstrated that the
experimental images of amorphous silicon (which is structurally equivalent to tetrahedral-bonded amorphous carbon) can be simulated by a random phase distribution (white noise) [7]. Consequently, a random arrangement of carbon atoms provides a good enough approximation [8]. More promising than the evaluation of such atomic details is the extraction of nanoscaled structures from TEM images of amorphous materials. Apart from the identification of local regions of structural order in an amorphous matrix [9] or of amorphous regions in a crystalline matrix [10] the investigation of amorphous multilayers is of particular interest [3]. In the following, a basic problem of multilayer simulation shall be discussed in more detail: the relation between the deterministic multilayer structure (with given thickness and density of the layers) and the random arrangement inside the layer. The simulation has been carried out for the cross-section of a multilayered carbon film with amorphous layers of alternating density. The influence of the deterministic density difference between the layers and of the statistical
Fig. 1. Cross-section through a C/CNx multilayer of 44 periods prepared by pulsed laser deposition (C.-F. Meyer, Fraunhofer Institut fu¨r Werkstoff- und Strahltechnik, Dresden). Thickness of a double layer 8.5 nm (5.0 nm with density 2.6 g/cm3and 3.5 nm with density 3.1 g/cm3). Conventional TEM image taken at underfocus of D=1850 nm (K. Brand, Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden).
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fluctuations inside the amorphous layers has been compared. By systematic variation of the dimensions of the basic unit it is explored, which thickness of stacks of independent amorphous supercells should be necessary to assure the imaging of randomly arranged atoms and to avoid pseudo-crystalline effects due to iteration of thin identical supercells.
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2. Preparation and imaging of carbon multilayer structures As experimental reference structure, carbon multilayers of different densities were deposited on a silicon substrate by pulsed laser deposition (PLD) with an ArF-Excimer laser [11]. They consist of 44 double layers of amorphous carbon
Fig. 2. WTF (a) at Gabor focus D=30 nm, (b) at large underfocus D=2000 nm. The amplitude wave transfer function cos w (i.e. the real part of the complex WTF eiw ) starts with a value of 1.0 at 0 nm1 (corresponding to w ¼ 0), whereas the phase wave transfer function sin w (i.e. the imaginary part) vanishes at 0 nm1. At Gabor focus, the amplitude transfer function is close to 1 for the dominating spatial frequencies around 0.3 nm1 corresponding to optimum conditions for holographic imaging. At large underfocus the phase transfer function is close to 1 corresponding to optimum conditions for conventional electron microscopy.
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of alternating density: 3.0–3.2 g/cm2 (thickness 3.5 nm) and 2.6–2.7 g/cm3 (thickness 5.0 nm), respectively (Fig. 1). The deposition has been carried out in a high-vacuum chamber (basic pressure 105 Pa) under nitrogen (partial pressure 0.3 Pa). The density variation has been achieved by additional plasma activation. It leads to an incorporation of about 10% nitrogen in the growing film, favoring graphitic bonding with lower density. TEM preparation by cross-sectioning is performed by means of conventional thinning techniques. The imaging conditions correspond to the Philips CM200 FEG/ST electron microscope: acceleration voltage 200 kV, coefficient of spherical aberration Cs ¼ 1:3 mm, condensor aperture 0.1 mrad, focus spread 7 nm. The layer structure is of the dimension Dx 3 nm, corresponding to spatial frequencies in the range of about 0.3 nm1. In comparison to atomic structures this means large area contrast. The differing density of the layers is mainly reflected by a proportional phase shift, which may be directly revealed by electron holographic methods [12,13]. In this case the amplitude wave transfer function cos w (i.e. the real part of the complex wave transfer function (WTF)) should be close to 1. This case is already realized for smaller defocus values (Fig. 2a). For optimized holographic imaging the Gabor focus (D=39 nm in the considered case) is recommended [14]. The electron hologram of the carbon multilayer is represented in Fig. 3. In conventional electron microscopy, the phase modulations are transferred to amplitude modulations via the phase contrast transfer function sin w (i.e. the imaginary part of the WTF). To realize jsin wj 1 for the rather small spatial frequencies of special interest, a large defocus in the order of 2000 nm is preferable (Fig. 2b). The corresponding electron micrograph is given in Fig. 1.
3. Modeling of the multilayer structure and simulation of electron microscopic imaging Simulation of the electron microscopic imaging has been carried out with the program EMS of
Fig. 3. Electron hologram of cross-section through a C/CNx multilayer (see Fig. 1) at Gabor focus D=30 nm (a), reconstruction of phase (b) with linescan (c) (H. Banzhof, Institut fu¨r Angewandte Physik, Technische Universita¨t Dresden).
Pierre Stadelmann, Lausanne based on the multislice method [15]. Here, supercells are periodically continued infinitely in the lateral directions. The given sample thickness is realized by a corresponding number of iterations in normal direction. In
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the used program every such supercell iteration is treated as a single slice. Hence, for reliable calculations of the electron imaging the supercell thickness should not exceed some tenth nanometers. The cross-section of the layered structure is modeled by rectangular supercells with regions of
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different atomic density. Apart from this nanoscaled superstructure a disordered atomic arrangement has to be setup. The minimum lateral dimensions are determined by the layer structure with a characteristic length much larger than the atomic distances. Compared to this large-scaled lateral periodicity, the periodic continuation of the
Fig. 4. (a) Scheme of modeling a thick sample by iteration of supercell packages of different thicknesses with the limiting cases of completely independent atomic arrangements by using different supercells (right) and of pseudo-periodicity by using the identical supercell (left). (b) Scheme of modeling samples with increasing thickness by using different supercells leading to completely independent atomic arrangements (bottom) and by using the identical supercell leading to pseudo-periodicity (top).
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Fig. 5. Comparison of phase images of the C/C multilayer cross-section for different kinds of modeling (Gabor focus D=30 nm, radius of objective aperture 8 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample thickness 11.8 nm realized by 30 iterations of the same thin supercell. (c) Sample thickness 11.8 nm realized by a single thick supercell package consisting of a series of 30 supercells with different random distributions. In (c) the boundary between layers of different density clearly shows up. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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supercell in irradiation direction is much more critical. Due to the slice conditions thin supercells must be used with a correspondingly large number of iterations. For the case of completely identical supercells, this may lead to a (possibly smallscaled) artificial periodicity in irradiation direction, where the periodicity length could be comparable even to the atomic distances. Its influence on the resulting image is the main topic of this investigation. The investigation aims at the nanometer-scaled layered structure. Correspondingly, the electron microscopic imaging is well above the atomic resolution, and the exact consideration of the atomic arrangement is of no importance at this stage. In a first step, the two layers of different atomic density (150 and 130 nm3, respectively) were modeled by ordered diamond structures with correspondingly modified lattice parameters a. The in-plane dimensions of such a supercell amounts to Lx =8.436 nm (=5.125 nm (low density)+3.311 nm (high density)) and Ly =5.519 nm in direction perpendicular and parallel to the layer boundary, respectively and Lz =0.3942 nm in normal direction. By dividing the supercell in several subslices, it was proved that the supercell thickness of 0.3942 nm was small enough for a reliable application of the multislice algorithm. Such a supercell contains 2536 atoms. In a second step, all these ordered atomic positions are displaced randomly in all three directions between Lz /4 and +Lz /4, where atoms displaced beyond the boundaries of the supercell are positioned correspondingly on the opposite side. These supercells with a thickness of 0.39 nm represent the building stones for the constructions of thicker samples by combining them up to the desired thickness. Then the following question arises: Is it possible to iterate simply the same supercell or is it necessary to use different supercells with independent random arrangements? In the following, the effect of the supercell choice shall be considered along two lines: (1) a constant sample thickness of 11.8 nm (corresponding to about 76 000 atoms) realized by an iteration of supercell packages of increasing thickness, where each such package consists of a series of supercells
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with different random distributions (Fig. 4a), (2) a series of samples with increasing thickness up to 11.8 nm, realized by iteration of identical thin supercells (Fig. 4b, top) and by a series of different thin supercells (Fig. 4b, bottom) up to the corresponding thickness.
4. Results of imaging simulation The simulated TEM images of a thin supercell (thickness 0.39 nm) are presented in Figs. 5a, 6a, and Figs. 8a, 9a, for phase (at Gabor focus) and intensity (at a large defocus of 2000 nm), respectively, at different objective apertures. For an aperture radius of 8 nm1 the higher spatial frequencies are cut off by the WTF envelope function, whereas for a radius of 1 nm1 it is determined by the aperture. In the highly resolved phase images (large aperture), the boundary between the two carbon layers with differing density is nearly undetectable (Fig. 5a). With smaller apertures, the speckle from the randomly distributed positions of the atoms, which is here of no importance, is suppressed and the boundary becomes clearly visible (Fig. 6a). Under large area contrast conditions, i.e. at large defocus, the interface contrast is only slightly improved (Figs. 8a, 9a). In the first simulation series (corresponding to Fig. 4a), a realistic TEM sample with thickness 11.8 nm was built up by iteration of supercell packages of different thickness. In such a supercell package the atoms are independently distributed. To achieve the given thickness, this random arrangement is repeated several times up to the sample thickness: 30 times the 0.39 nm package, 15 times a 2 0.39 nm package, and so on. For brevity, in Figs. 5, 6, 8, 9 only the limiting cases are given. In general, any contrast is enhanced if the imaging electron beam passing through the sample, repeatedly feels an identical atomic distribution. This amplification is clearly visible from the intensity profiles. Repeating the same thin supercell again and again, the features of both the deterministic layer structure and the random atomic distribution are enhanced. Consequently, the visibility of the boundary is even worsened by
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Fig. 6. Comparison of phase images of the C/C multilayer cross section for different kinds of modeling (Gabor focus D=30 nm, radius of objective aperture 1 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same thin supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. Under these conditions, the multilayer structure is reflected very clearly. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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the prevailing statistical pattern in comparison to the thin slice (compare (a) and (b) in Figs. 5, 6, 8, 9). If, however, one uses only one thick supercell package with independently distributed atoms along the whole sample thickness, the effect of the random atomic distributions essentially averages out and the layer structure is more pronounced (compare (b) and (c) in Figs. 5, 6, 8, 9). This holds especially for small apertures, which additionally average the small-scale variations. The results of the simulation series are summarized in Figs. 7 and 10. The diagrams present the dependence of the relative contrast (image contrast/image mean) on the thickness of the supercell package at constant sample thickness (11.8 nm). The contrast is determined by the fluctuations inside the layer and by the density differences between the layers. For comparison, the contrast curves for a homogeneous amorphous material with the mean atomic density of 134 nm3 are included. In the second simulation series (corresponding to Fig. 4b) the sample thickness was varied between 0.39 nm and 11.8 nm. The sample was built up by corresponding iteration of the same thin supercell (0.39 nm thickness) and by a series of different supercells. Now, with increasing thickness, the statistical contrast is enhanced for the
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case of supercell iteration. In the opposite case of independent supercells, the random fluctuations are more and more blurred, and the deterministic layer structure appears more distinctively (compare (a) and (c) in Figs. 5, 6, 8, 9). The results are comprehended in Figs. 11 and 12 presenting the dependence of the relative contrast on the sample thickness for these two complementary cases of modeling.
5. Discussion The comparison of the TEM images}and in a more quantitative way, the comparison of the respective line profiles}clearly reveals the large influence of the kind of modeling. If thicker amorphous films are constructed in the simplest way by iterating the same thin supercell, i.e. by repeating the same statistical layer, then the image shows large unwanted fluctuations. Such a construction corresponds to a random atomic arrangement in the plane perpendicular to the incident electron beam, but in the beam direction the atomic arrangement shows a periodical order. Hence, the scattering waves are accumulated with well defined phase differences. Consequently, fluctuations are enhanced with increasing number
Fig. 7. Dependence of the relative contrast (image contrast / image mean) in the phase image (Gabor focus D=30 nm) of the C/C multilayer cross-section (sample thickness 11.8 nm) on the thickness of the supercell package (normalized to the supercell thickness of 0.39 nm). Radius of objective aperture 8 and 1 nm1. For comparison, the relative contrast for a homogeneous carbon layer with a mean density of 134 nm3 is included.
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Fig. 8. Comparison of intensity images of the C/C multilayer cross-section for different kinds of modeling (large underfocus D=2000 nm, radius of objective aperture 8 nm1 ). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. The boundary is rather blurred due to the large defocus. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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Fig. 9. Comparison of intensity images of the C/C multilayer cross-section for different kinds of modeling (large underfocus D=2000 nm, radius of objective aperture 1 nm1). (a) Sample thickness corresponding to one thin supercell with thickness of 0.39 nm. (b) Sample with thickness 11.8 nm realized by 30 iterations of the same supercell. (c) Sample with thickness 11.8 nm realized by one thick supercell package consisting of a series of 30 supercells with different random distributions. The intensity profiles give the absolute values in contrary to the images where the contrast range is automatically adapted to the optimum range.
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Fig. 10. Dependence of the relative contrast (image contrast/image mean) in the intensity image (underfocus D=2000 nm) of the C/C multilayer cross section (sample thickness 11.8 nm) on the thickness of the supercell package (normalized to the supercell thickness of 0.39 nm). Radius of objective aperture 8 nm1 and 1 nm1. For comparison, the relative contrast for a homogeneous carbon layer is included. For modeling by many iterations of thin identical supercells the contrast is mainly determined by the artificially enhanced statistical fluctuations. For thicker supercell packages above 2 nm (corresponding to a normalized thickness above 5), the contrast reflects in the main the density step.
of iterations, i.e. for higher film thickness. On the contrary, if the sample is modeled by a truly random arrangement, then the phases of the scattered waves are not correlated and, as desirable and correct, their effect is more and more leveled off with increasing film thickness. The same trend is observed for given foil thickness, if the thickness of the supercell packages is varied: Already a doubling of the random layer leads to a strong reduction of the artificially induced fluctuations. Above 2 nm there is only a moderate further decrease. The critical thickness necessary to avoid modeling artefacts depends on the special imaging conditions: In the case of carbon foils for amplitude imaging and/or for imaging with small apertures, a thickness of 4 nm of the supercell package with independently distributed atoms is sufficient to level off the statistical fluctuations. The comparison with a homogeneous material without a density jump across the sample shows that the remaining contrast is mainly determined by the deterministic difference between the high density and the lower density
region of the TEM sample. On the contrary, for the phase imaging at large aperture the contrast is dominated by the statistics even at 11 nm thickness.
6. Conclusions For the adequate simulation of TEM imaging of amorphous materials, the thickness of the supercell packages with independent random distributions has to exceed a certain limit. Then the statistical scattering from the disordered atoms will be averaged. Otherwise, if the model of the TEM sample is constructed as a stack of very thin identical layers, the superposition of scattering waves with constant phase differences results in enhanced local fluctuations burying the nanometer scaled structure to be investigated. For thicker films, the effect of statistical atomic arrangements is more and more leveled off. Hence, deterministic nanometer-structures due to differing densities will be visible more distinctively in the random background. For carbon, this critical thickness
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Fig. 11. Dependence of the relative contrast (image contrast/image mean) in the phase image (Gabor focus D=30 nm) of the C/C multilayer cross-section on the sample thickness (normalized to the supercell thickness of 0.39 nm) realized by repeated iteration of the same thin supercell (supercell thickness 0.39 nm) and by a series of independent supercells. (a) Radius of aperture objective 8 nm1. (b) Radius of aperture objective 1 nm1. For thicker samples modeled by different supercells, the contrast decreases towards the limit given by the density step. When the sample is built up by repetition of identical supercells, the contrast increases by superposition of the effects of the random atomic arrangement, same for all layers.
amounts to about 4 nm (at a mean density of about 2.7 g/cm3). The guidelines of the right choice of supercell stack dimensions discussed above will be applied in a forthcoming paper on a more complex structure of high interest: the investigation of thin carbon films with nanotube-like inclusions embedded in an amorphous matrix [16].
Acknowledgements Dr. C.-F. Meyer is greatly acknowledged for the preparation of the layered carbon samples, Dr. H. Banzhof and Dr. K. Brand for the TEM investigations. The authors thank Prof. H. Lichte for stimulating discussions. The work was partially supported by Deutsche Forschungsgemeinschaft, Grant no. Li 346/13-1.
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Fig. 12. Dependence of the relative contrast (image contrast/image mean) in the intensity image (underfocus D=2000 nm) of the C/C multilayer cross section on the sample thickness (normalized to the supercell thickness of 0.39 nm) realized by repeated iteration of the same thin supercell (supercell thickness 0.39 nm) and by a series of independent supercells. (a) Radius of objective aperture 8 nm1. (b) Radius of objective aperture 1 nm1. For thicker samples modeled by different supercells, the contrast increases slowly proportional to the thickness. When the sample is built up by repetition of identical supercells, the contrast increases steeply by superposition of the effects of the same random atomic arrangements.
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