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Imaging a dense nanodot assembly by phase retrieval from TEM images
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Ultramicroscopy 100 (2004) 79–90
Imaging a dense nanodot assembly by phase retrieval from TEM images P. Donnadieua,*, M. Verdiera, G. Berthome! a, P. Murb a
LTPCM-INPG-CNRS-UJF, Domaine Universitaire BP75, 38402 Saint Martin d’H"eres, France b CEA-DRT - LETI/DTS - CEA/GRE, 17, rue des Martyrs, 38054 Grenoble cedex 9, France Received 7 May 2003; received in revised form 3 January 2004; accepted 26 January 2004
Abstract Phase retrieval is a classical inverse problem in many fields dealing with waves that is becoming of increasing interest in transmission electron microscopy (TEM). A non-interferometric approach is here applied to TEM images. Phase retrieval possibilities given by the transport intensity equation are compared to the ones deriving from the weak phase object approximation. In the limit of small angles, both methods lead to a similar equation between the phase and a set of defocus images. This equation can be solved by an image processing equivalent to using a specific filter in Fourier space. This processing leads to phase images with a spatial resolution here essentially limited by the defocus amount between images. A dense assembly of silicon nanodots is used as a model case to illustrate the interest of this approximate phase retrieval method which can be carried out on standard equipment. The dot heights estimated using the phase images are found to be in good agreement with ones measured by atomic force microscopy. Since image noise and large defocus values may strongly affect the solution given by the approximate method, an iterative phase retrieval method is also used as a test for working conditions. r 2004 Elsevier B.V. All rights reserved. PACS: 42.30; 61.48 Keywords: Phase retrieval; Defocus variation; Transport intensity equation; TEM; Nanodot
1. Introduction In the last decade, size reduction has become a key property in material research, the aim being either to push the limits of miniaturisation or to benefit from certain properties that appear below
*Corresponding author. Tel.: +00-33-4-7682-6686; fax: +00-33-4-7682-6644. E-mail address: [email protected] (P. Donnadieu).
some critical size. From fundamental physics to industrial applications, many fields are concerned with size reduction. Nevertheless, in this race to ultimate scale, microelectronics is the most emblematic one. Such an effort in size reduction requires, in parallel, to develop characterisation tools to both image and quantitatively study a microstructure at the nanometer scale. Regarding micro-nanoelectronics, there is a need for structural and chemical information as well as to be able to measure magnetic or electric fields in nanostructured materials of potential application.
0304-3991/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2004.01.007
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Due to its high resolution compared to other imaging methods transmission electron microscopy (TEM) remains, despite its age, a technique of choice for such a study. As attested by the development of electron holography, TEM offers various types of measurements: for instance, besides ultra-high-resolution imaging, holography allows the measurements of electric and magnetic fields. Such measurements, of course, are obtained owing to the holographic biprism inserted in the column. However, these measurements are made possible by the fact that the thin TEM sample acts as a phase object on the electron wave and that the phase shift is directly related to the sample inner potential. Indeed the key question is then to retrieve the phase which is a classical and high interest inverse problem in all the fields dealing with wave (optics, X-ray, neutrons, etc). Phase retrieval can be achieved by interferometric or non-interferometric methods. The former one requires interferometer as well as a high degree of coherency while the latter one needs more modest means to be carried out. In the limit of nanometer resolution, the non-interferometric approach applied to TEM imaging, especially with the transport intensity equation (TIE), seems powerful and of easy implementation. Recent papers have given very attractive examples of non-interferometric phase retrieval for Lorentz microscopy applied to magnetic samples [1–3] as well as TEM and optical studies of biological samples [4,5]. This paper aims at applying a non-interferometric phase retrieval method to experimental TEM images. Dense assemblies of nanodots elaborated in view of nanoelectronic application were used here as exemplar cases. These nanodots (o 10 nm) obtained by a low pressure chemical vapor deposition (LPCVD) process, are grown on an oxidised silicon substrate. In the present paper, the nanodot deposits being used as model cases, no specific information is given on elaboration and properties, for more details see Ref. [6]. Indeed, the nanodot assemblies very appropriately illustrate the interest for retrieving the phase since, though TEM can provide quite well-resolved plane views of the dots, the contrast interpretation remains limited as long as the
connection between contrast and dot height is not made. In the following, phase retrieval through the TIE solution will be considered with respect to the approach based on the weak phase object approximation (WPOA). Resolution limitation due to the method as well as the impact of noise will be also discussed.
2. Quantitative study of TEM images by phase retrieval In this section, we revise basic features of phase retrieval with particular emphasis on its application to TEM. Full details and deeper insights can be found in the references cited in this section. 2.1. About some non-interferometric phase retrieval methods Much work has been devoted in the last decades to the phase retrieval problem in optics, X-ray and electron microscopy. Non-interferometric methods are based on the acquisition of a set of images, further analysed using the propagation equation. Recently, the possibilities of the TIE early developed by Teague [7] have been considered as a solution in many field of applications. Besides theoretical aspects like unicity of solution, coherency requirements, robustness in presence of singularities, several successful applications of the TIE approach have been reported in X-ray projection microscopy, optical and transmission electron microscopy [4,5,8,9]. The transport intensity equation is a direct result from the description of diffracted waves under Fresnel diffraction conditions [7]. It comes out that the electron wave described by a timedependent plane wave function obeys the propagation equation: q r2 i þ þ k uz ðrÞ ¼ 0; ð1Þ qz 2k where z is the propagation direction, uz ðrÞ is the wave amplitude which depend only from rðx; yÞ and r the gradient operator in the ðx; yÞ plane, perpendicular to wave propagation. The
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measurable quantity is the image intensity Iz ðrÞ ¼ Juz ðrÞJ2 ; intensity which can be recorded at different z positions. Introducing the phase FðrÞ defined as uz ðrÞ ¼ ½Iz ðrÞ1=2 expðiFðrÞÞ allows Eq. (1) to be rewritten as a TIE having a convenient form with respect to the phase retrieval problem [7]: 2p qI ¼ r:ðIrFÞ: ð2Þ l qz In the limit of small intensity variation, the TIE simplifies as 2p DI ¼ Iðr; OÞr:rF l Dz with
in the following: in the small angle approximation, the WPOA leads to a relation between intensity and phase similar to the one resulting from the TIE approach (for detailed derivation on the WPOA, see Ref. [10]. The phase contrast object is accounted for by a transmission function in which, in the limit of small angles, the effect of defocusing is introduced in the expression of the transmission function through a Fresnel propagator term. This comes out for a weak phase object as a relation between the intensity at defocus Dz and the intensity at defocus zero. This relation more conveniently expressed in the reciprocal space writes as follows (the high order lense aberration term is here neglected):
DI ¼ Iðr; DzÞ Iðr; OÞ:
# DzÞ ¼ Iðq; # OÞð1 2Fðq; # DzÞsinðplDzq2 ÞÞ: Iðq;
ð3Þ
As demonstrated by Teague [7], the transport intensity equation is valid in the near-field approximation. The near-field condition which is basically a small angle condition writes in direct space as pr2 =lz51: Therefore the TIE provides an exact phase solution only when the working conditions meet the near-field requirement. In fact TEM rather relies on the Fraunhofer description since the plane of observation is not at finite distance. Under the Fraunhofer diffraction conditions, image formation is described by the Abbe imaging theory which relates by successive Fourier transforms object, diffraction and image. In most TEM cases, the thin sample can be considered as a phase object, most often appropriately described by the WPOA. In fact, as shown
ð4Þ
If we compare the WPOA equation to the simplified TIE rewritten for convenience in reciprocal space 2p # # DIðqÞ ¼ Iðq; 0Þ4pq2 FðqÞ lDz
ð5Þ
it comes out that Eqs. (4) and (5) are identical at very small defocus, i.e. under the near-field condition, since plDzq2 51; plDzq2 can be taken as an expansion of sinðplDzq2 Þ for all q: The TIE and WPOA approaches can be easily compared by plotting their respective transfer functions. Fig. 1a give the classical high-resolution transfer function, though calculated for defocus unusually large compared to the high-resolution imaging conditions. In high-resolution electron microscopy, the
Fig. 1. (a) Shows the classical TEM transfer function, at Dz=1000 nm, the simplified form TðqÞ=sin(plDzq2) is used, lens aberration and coherence damping function being neglected here as far only the small q range is concerned. (b) Report the TðqÞ transfer function together with the parabolic approximation T 0 ðqÞ ¼ plDzq2 which corresponds to the TIE approximation. The arrow indicates the intermediate field for a 1000 nm defocus ðplDzq2 ¼ 1Þ; in this range the transfer function TðqÞ only starts to depart from the simplified T 0 ðqÞ function.
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reciprocal space is explored up to the maximum distance allowed by the microscope information limit (i.e. typically qB5 nm1). Since the TIE is valid only for the small q values, the comparison of TIE and WPOA transfer functions shown in Fig. 1b is done only for qoB1 nm1. Hence, depending on the plDzq2 value, the phase can be retrieved more or less accurately using the TIE method instead of the WPOA. The approximation behind the transport intensity equation limits its application to the near-field conditions which means, in practice, either small defocus or large objects. On the other hand, the WPOA which is also valid, under small angles (because of the propagator expression), provides a convenient tool up to the intermediate field (plDzq2 E1). At small angles, rewriting of Eqs. (4) and (5) allows us to describe phase retrieval in terms of filtering in the reciprocal space. This filter description is convenient to discuss the resolution. For the TIE, the filter applied in reciprocal space is 2 1 plDzq for the WPOA the filter is while 1 sinðplDzq2 Þ : As illustrated by Fig. 1b the WPOA and TIE filters start to differ for plDzq2 E1: Though for plDzq2 ¼ 1; using the TIE equation, instead of the WPOA one, represent only an attenuation of the higher spatial frequency, i.e. a limited loss of resolution. In practice, plDzq2 ¼ 1 give a resolution limit. For distance below d ¼ ðplDzÞ1=2 ; the corresponding spatial frequencies are attenuated if the TIE filter is used instead of the WPOA one. In the following, the approximate method we use will be often referred to as a TIE type filtering. High resolution is not the objective of the present work which rather aims at a nanoscale characterisation. The main point is to improve the description of nanostructures (i.e lateral dimension as well as height for nanoobjects like nanodots). In that scope, phase retrieval achievable on standard microscope and based on the simplest approaches is of interest. It does not mean that high spatial resolution is not possible in phase retrieval, but more sophisticated equipment and treatments are required. Actually phase retrieval using an iterative method and the holographic one is the . basis behind the effort to reach sub angstrom resolution [11].
Beside resolution, noise is a serious drawback in the use of the TIE type filter. The development of the intensity, inferred by the Fresnel propagation approximation, implies that the first-order derivative of intensity is zero, which is usually not verified because of the noise. Through the successive Fourier transforms, this noise contribution is transferred back to the direct space on the phase image. The CCD camera being corrected from dark current and gain normalised, the main defect remains a signal noise characterised by high frequency 1in reciprocal space. The TIE type filter plDzq2 cuts the highest frequencies and therefore tends to produced smooth phase images free from rapid spatial fluctuation. Nevertheless the filter parabolic shape may have a too slow decay at high frequency, therefore there is a risk to introduce artefact details. Since the limit of resolution is known from the intermediate field conditions, any detail below this limit on the phase image is not relevant. Consequently these details are removed using appropriate masks in the image processing. The sensitivity to noise of the TIE method has been analysed in details by Allen and Oxley [12] who have discussed the possibilities and limit of several methods for getting the phase from focus series: *
* *
an approximate solution of the TIE by a Fourier method, an exact solution using multigrid methods, an iterative method assuming free space propagation between planes at different defocus.
They show that the Fourier method is efficient to rapidly obtain an approximate solution but can be safely used only in the vicinity of the in-focus plane in case of rapid phase variation. Phase discontinuities also prevent the application of the TIE method while the iterative method still works in presence of vortices [13]. Finally, the iterative method appears as the more robust one as well as being less sensitive to noise. Besides, according to Ref. [12], the amplitude of phase fluctuation is an important parameter concerning the choice of a method. In practice, using several methods is convenient to identify possible artefacts. In the present work, a TIE approach will be used on
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experimental images together with the iterative method provided by Allen and Oxley [12]. 2.2. Application TEM plane view images of silicon nanodots deposited on a substrate correspond to small thickness fluctuations which allows a phase object approximation. The nanodots sizes are expected to be in the 5–10 nm range, therefore a nanometric resolution method is of interest in the present case. According to the above analysis, two defocused images can be processed using the TIE type filter within some restriction on the spatial resolution. In practice, the expression sin plq2 approximated by plq2 leads typically to a spatial resolution limited to 1.7 nm (resp. 2.5 nm) for a defocus Dz=500 nm (resp. 1000 nm). It is worth noting that the value of the experimental defocus depends on the image contrast, hence the spatial resolution is somewhat dependent on the sample. The defocus amount is difficult to predict because it depends on the changes in image while defocusing. Besides, contrasted images and thus quite large defocus are required to test whether some drift has occurred. Finally, the defocus choice results from a compromise between the desired resolution and good working conditions in terms of contrast and noise. Being less sensitive than the TIE method, the iterative one will be used here to test whether the experimental conditions (defocus, noise level, amplitude of phase contrast) allow for phase retrieval with the TIE approach In the following, image processing using the TIE type filtering retrieves phase according to the equation: # # ðqÞ ¼ 1 1 DIðqÞ: F ð6Þ 2 Im 2plq Dz To avoid the divergence due to possible zero intensity points in I0 ; instead of setting a cut off intensity like [12], the intensity I0 is replaced by Im ; its transformation by a median filter of small radius which efficiently removes zero intensity points with small change in the whole image. Regarding the vicinity of q ¼ 0; a gaussian mask centred on q ¼ 0 is used, and its width must be
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chosen small enough to avoid dampening the thickness long range fluctuation. In the phase object approximation, the sample transmission is described by a phase shift which accounts for the interaction between electron and matter. In non-magnetic samples, the phase shift depends on composition and thickness and the specimen transfer function for a weak phase object writes as p f ðx; yÞ ¼ exp i Vt ðx; yÞ ; lE R where Vt ¼ V ðx; y; zÞ dz is the projected potential, l and E are the electron wavelength and the high voltage, respectively. In the case of interest here, the inner potential being constant, the phase shift measures thickness variation. Depending on the experimental conditions (i.e. whether the sample edge is in the recorded image field), absolute thickness measurement or only thickness fluctuations are derived. It is also worth noting that, the measured thickness being integrated values, the 3D phase representation shown further corresponds to actual topography only in case of simple topology and in absence of strain.
3. Phase images of nanodots 3.1. Experimental conditions Two types of nanodot deposits grown by LPCVD on Si substrate according to the method reported in Ref. [6] have been studied. Prior to the dot deposition, ultra thin oxides have been grown on the Si (1 0 0) surface by rapid thermal oxidation in slightly different conditions. According to ellipsometry measurements, the different oxidation conditions lead to oxide layer with different thickness (0.6 nm in sample 1, 1.2 nm in sample 2). As the oxidisation treatment affects only slightly the substrate surface, the dots elaborated in the same LPCVD conditions are expected to show only small differences. The dots deposited on Si oxidised wafers have been prepared for plane view TEM observations by argon ion beam bombardment on the substrate side.
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TEM was carried out on a microscope Jeol 3010 (LaB6 filament) operating at 300 keV. Images are recorded by a Gatan Dual View camera (1030
1300 pixels2) fitted on the 35 mm port. In the TEM working conditions (magnification 300 000), the image sampling of images is 0.36 nm/pixel. The defocus value given by the constructor have been tested using HREM focal series. Images are formed by selection of the transmitted beam with a small objective aperture (diameter 1.2 nm1). Phase is retrieved on a personal computer either using the iterative method contained in the program PhaseRetrieval1.10 developed by Allen and Oxley [12] or by the TIE type filtering carried out with script written within the Digital Micrograph software [14]. 3.2. Results Classical plane view images are given in Figs. 2a and b for samples 1 and 2, respectively. Sample 1 is characterised by quite large dots (B10 nm) while sample 2 shows an homogeneous distribution of dots with sizes in the 3–8 nm range. In any case, sizes are difficult to ascertain since dots are not easy to identify. The mottled contrast of black and white dots is difficult to associate with bump or hole because the black and white contrast can be reversed by defocusing. The phase information is necessary to make the connection between the
image contrast and thickness fluctuation corresponding to dots. 3.2.1. Fourier and iterative methods Fig. 3a–c displays images of a same area, respectively, taken at+1000, 0 and 1000 nm on sample 1. Similar images have been also obtained at larger defocus (2000 nm). Note that, in the present case, images with good signal/noise ratio are obtained only for defocus larger than 1000 nm. A series of 5 images with defocus ranging from 2000 to 2000 nm has been analysed using the iterative method. Fig. 4a gives the resulting phase image which is characterised after several hundred of iterations by a 2 103 sum squared errors (SSE). According to the discussion in Ref. [12], in presence of a 5% noise, the SSE cannot be better than 103 while 105 SSE are obtained with noiseless simulated images. Therefore, a phase image with a 2 103 SSE can be here considered as a satisfactory solution for the iterative method. Fig. 4b give the phase image which has been derived from the 1000 nm defocused image using the TIE type filtering. For a 1000 nm defocus, the resolution being limited to 2.5 nm, all details below this limit have been removed by masks with gaussian edges during image processing. The phase image in Fig. 2b shows quite an agreement with the one obtained by the iterative method. This
Fig. 2. TEM plane view images taken at positive defocus on the nanodot samples. In sample 1, shown in (a), the dot size is not easy to determine: the largest features (10–15 nm) might be formed by the aggregation of smaller dots. In sample 2 (2b) the dots form an homogeneous assembly with a mean dot size in the 5 nm range.
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Fig. 3. Focus series of TEM plane view images of a nanodot sample. The series has been recorded on a CCD camera (1pixel=0.36 nm) at different focus: (a) Dz=1000 nm ; (b) Dz=0 ; (c) Dz=1000 nm.
are in agreement with details noticed on the plane view images indicating large dots with complex inside features (Fig. 5c). Note that the details shown in Fig. 5 are taken at the same scale but in different parts of sample 1.
Fig. 4. Comparison of the phase images retrieved from the series in Fig. 3 using two independent methods. On the phase images, the white contrast corresponds to the dots. The field of view imaged here is 90 nm, and with both methods the dot size is the 5 nm range: (a) Phase image obtained with the iterative method [12] on a series of 5 images (the 3 images in Fig. 1 plus images at Dz=2000 and 2000 nm). This phase image is obtained after several hundred iterations, stagnation occurs on this solution with a 2 103 sum squared errors (SSE). (b) Phase image obtained by the TIE type filtering. The resolution being limited by the method to 2.5 nm, details below this limit are removed by filtering.
agreement indicates that the noise level and the high working defocus do not prevent to use the most approximate approach. Fig. 5 illustrates the specific interest for nanodots to have thickness fluctuation mapping instead of the usual phase contrast plane view. Figs. 5a and b show a detail from the phase image and its 3D representation: the latter particularly points out that the dots were arranged in a 5-fold ring around central hole. This kind of dot aggregates
3.2.2. Quantitative measurements Figs. 6 and 7 display the phase images and their 3D representation obtained for samples 1 and 2. As mentioned above, since the sample edge is not in the recorded area, only phase fluctuations are measured here. These fluctuations have here two origins: the dots deposited on the substrate and the thickness fluctuations due to sample preparation. Measurements of phase can be made from Eq. (6) with only the knowledge of defocus and digitisation parameters. The scaling factor after the Fourier transformation processes is N 2 p2 =2plDz; where N is the image size in pixels, p the effective pixel size at the TEM operating magnification and Dz the defocus between images. This scaling factor is given here to point out how quantitative phase measurement relies on defocus and magnification calibrations. Inserts in Figs. 6 and 7 give typical phase profiles allowing for further thickness measurements. For silicon dots deposited on a substrate covered by a nanometer thick silica layer, the oxide layer being constant, the inner potential of interest is the silicon one. The thickness fluctuation t is then related to the phase fluctuation DF by: t ¼ lEDF=pV ; where V is the inner potential. The inner potential can be calculated knowing the
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Fig. 5. (a) Gives a high magnification detail of the phase image in Fig. 4b. The 3D representation in (b) Indicates that it can be interpreted as a ring of 5 dots around a central hole and (c) Gives high-magnification TEM plane view from sample 1 at the same scale as Fig. 5a. Such detail from untreated images is consistent with the existence of complex aggregate of small dots as revealed by the phase image.
Fig. 6. Phase image and its 3D representation obtained for the nanodot sample 1. The field of view is 180 nm.The dots which lateral size is in the 6 nm range form an irregular assembly. No large dots are observed but in frequent places rings of dots can be noticed. The phase (phase in radian as a function of distance in nanometer) profile given in the insert corresponds to dot height h1 E8 nm.The 3D representation shows long-range thickness fluctuation which are consistent with the ones expected from sample preparation by ion beam thinning under a 5 angle in the finishing step.
crystal structure and the atomic scattering factor given in classical books [15], for silicon V=17.6 V. According to the phase profiles, dot heights are h1 E8 nm and h2 E10 nm for samples 1 and 2, respectively. The dot lateral sizes estimated from the phase images are, in both samples, close to 6 nm.These values only aim at giving some size estimation for comparison with atomic force microscopy (AFM). According to the phase images, samples 1 and 2 shows similar features, essentially differing only by height and dot density. In sample 1 (resp. sample 2), the density estimated from the dot distance isE3 1011 dot/cm2 (resp.E1012 dot/cm2). These
values are actually approximately 2 times larger than the ones based on AFM or SEMFEG investigation. The next section reports the AFM study carried out on the same samples to test the reliability of measurements and examine the TEM phase image possibilities versus AFM.
4. Comparison with AFM imaging Atomic force microscopy were carried out on a Veeco D3100 AFM in tapping mode. Acquisition fields were saved only when drift effect was
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Fig. 7. Phase image and its 3D representation obtained for the nanodot sample 2. According to phase profiles as illustrated by the one in inset, the dot height (h2E10 nm) is in a similar range as in sample 1. Comparison of the 3D representation indicates that although the whole microstructure keeps same morphological features, the density is higher in sample 1 as well as the height/width ratio. The field imaged here is 180 nm wide and the dot lateral size is approximately 6 nm.According to the phase image, this dot assembly is characterised by a higher density than the sample 1 dot assembly.
Fig. 8. AFM images obtained on sample 1 (a) and sample 2 (b). The scale ranges are reported on the z-axis for comparison with the phase image results and (c) gives an AFM plane view of sample 2 quite similar to the phase image in Fig. 7.
minimized between successive scan of the same area. Tips with a certified 10 nm radii were used. Major limitation in AFM is the lateral resolution generally estimated between 2 and 5 nm, the vertical resolution being in the subnanometer range. The lateral resolution falls in the range of the interdistance of the dense distribution of the plots studied here, therefore convolution effect with the tip are expected. For these samples, the exact shape of the nanodot can not be ascertain with this technique. It is however possible to determine the height of the nanoplots, and to demonstrate a variation in surface density of the plots between sample 1 and 2. Figs. 8a and b give 3D representation of the AFM images obtained on
samples 1 and 2, respectively. The AFM height range (h0 1=5 nm and h0 2=9.7 nm, are in the range of the results given by the phase method (h1E8 nm and h0 2E10 nm). The AFM plane view shown in Fig. 8c was taken in sample 2. It is rather similar to the phase image obtained for the same sample (Fig. 7), except for the dot density which seems smaller according to AFM. We attribute this difference to the limitation of lateral resolution due to the high dot density. Even with very small radius tip, such dot assemblies are difficult to image with AFM, for complete review of the AFM imaging possibilities in air with the tapping mode see Ref. [16].
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5. Discussion As illustrated by the nanodot samples, phase retrieval provides appropriate information to understand the TEM plane view images. Though it should be kept in mind that the phase is related to the integrated potential which corresponds to thickness only in a perfect homogeneous sample with simple topology and free of strain. Opposite to AFM, phase retrieval sees the sample ‘‘from the inside’’. Therefore the combination of AFM and phase retrieval from TEM plane view might be an interesting tool to tackle the question of strain in nanoobject. However such combination can be carried out only for systems in which AFM provides reliable shape information. In the present case, regarding the dot height, phase retrieval results are in fair agreement with the AFM ones. Still the comparison is limited since the system is typically difficult for AFM studies. In fact, the nanodot case was chosen because of this specific difficulty in order to point out the interest for direct method like TEM allowing to image a whole area at once. In the present work, the microscope resolution in the high-resolution mode is 0.25 nm.Because of the objective aperture we used, this resolution drops to 0.8 nm and finally the approximate phase retrieval method gives phase images with a 2.5 nm resolution. This illustrates a particular advantage on probe scan images: though the TEM phase retrieval resolution depends on the sample through the choice of working defocus, the resolution can be evaluated quantitatively for each working condition. The aim of the present paper was mainly to test the feasibility of thickness measurement on a dense nanoscale topography using standard equipment. Of course improvement in resolution can be achieved especially using a CCD camera dedicated to weak contrast imaging. Better contrast and noise reduction will allow a decrease in the defocus hence improving the spatial resolution. Regarding phase measurement, at the moment, the defocus has been recalibrated only on a limited range. A calibration for large defocus will be carried out using layers of known thickness. Better resolution can be obtained also by an improvement of the phase retrieval method, such as for instance
working with a series of defocused images. This method already applied to X-ray phase contrast [17] is worth trying. Though better image acquisition and processing should improve the resolution, the ultimate limit is determined by the source coherency which has not been mentioned up to now. In any case, the source related transfer function envelope will define the resolution limit reachable for a working defocus. The phase retrieval method used here allows the derivation of quantitative phase information at the nanometer scale. In addition the phase can be measured on large image field and unwrapped values are obtained. The TIE type solution to phase retrieval consisting in a Fourier image processing, the method is also advantageous in terms of rapidity and easy implementation. These features make this non-interferometric method an approach very complementary to the holographic one. Holography requires high degree of coherency and can image only a limited field of view but of course allows to get the atomic resolution information. Phase retrieval using the transport intensity equation on Lorentz microscopy images [1,2] has already shown the valuable complement it provides to holography as it leads to similar information but on a different scale. Phase retrieval already occupies an important place in TEM since it is the basis of a subangstrom resolution work project, through side band holography and focus series method [11]. However it could be an issue to introduce more systematically phase retrieval methods in more standard TEM not only as a complement to holography. As illustrated by recent works on Fresnel fringes [18,19], information on inner potential can be obtained in conventional microscopy conditions carried out on a standard microscope. In these interface studies, the inner potential was used to study roughness and chemical fluctuation. These studies point out how an access to the inner potential combined with appropriate post processing can address most relevant questions like separating chemical and geometrical effects. Among all the techniques which allow the measurement of the inner potential, phase retrieval through Fourier image processing is indeed one of the simplest to carry
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out on standard TEM equipment. As many CCD camera are now being installed, phase retrieval can be developed as a common on line analysis. Among other possibilities, phase retrieval in TEM may find interesting applications in 3D imaging because extracting the potential gives information on the thickness along the propagation direction. Similarly to phase contrast tomography which has been developed with hard X-ray at synchrotron [17], there could be a potential field in extending absorption electron tomography to phase contrast electron tomography. In that kind of application, getting the phase contrast with an efficient and rapid routine will be of high interest.
techniques in electron microscopy such as holography or focal series method which aim at the sub-angstrom resolution. In any case, at the nanometer scale, phase retrieval is achievable and can lead to information on the thickness or the chemical nature of the sample even on standard microscopes. The popularity of CCD cameras and the possibilities of remote control of the microscope functions provide a very appropriate context for the promotion of phase retrieval in TEM. The resolution can be easily improved with better acquisition conditions and image processing, though it will always remain limited by the degree of beam coherency.
6. Conclusion
Acknowledgements
Phase retrieval is an old question which has been given in the last years different solutions. In particular in TEM, phase can be retrieved using the WPOA approach which in the small angle approximation is equivalent to the solution given by the transport intensity equation. However both approaches are valid only in the limit of small angles which has a direct consequence on the resolution. Even though the atomic resolution cannot be reached by these approximate methods, they are worth using to extract information on inner potential at the nanometer resolution. We have illustrated the interest for approximate phase retrieval from defocused images on the specific example of nanodot assemblies. Here phase retrieval obviously provides the extra information necessary for interpreting the plane view TEM images. Comparison with AFM images on the same dot assembly points out than, even limited to a 2.5 nm resolution, the phase image gives information that are typically difficult to reach with probe scan methods. Even under the severe approximation we had made, the height measurements derived from the phase shows a good agreement with the AFM ones. The phase retrieval method used here is particular attractive in terms of speed and easy implementation. Of course, if the resolution is good in comparison with probe scan technique, it is modest in comparison with other phase retrieval
Samples were prepared in the frame of CEALETI/CPMA collaboration, with PLATO Organization teams and tools. The authors thank Thierry Baron for his contribution to the dot elaboration, Prof. L.J. Allen for kindly sending his program ‘‘Phase retrieval 1.10’ and Peter Cloetens for stimulating discussion.
References [1] M. De Graef, M.A. Willard, D.E. Laughlin, M.E. McHenry, IEEE Trans. Magn. 37 (4 Part 1) (2001) 2343. [2] M. De Graef, Y. Zhu, J. Appl. Phys. 89 (11, Part. 2) (2001) 7177. [3] W.O. Saxton, P.K. Somodi, R.E. Dunin-Borkowski, communication presented at the Joint Microscopy Meeting, Lille, France, 24–28 June 2002. [4] E.D. Barone-Nugent, A. Barty, K.A. Nugent, J. Micros. 206 (2002) 194. [5] P.J. McMahon, E.D. Barone-Nugent, B.E. Allman, K.A. Nugent, J. Micros. 206 (2002) 204. [6] T. Baron, F. Martin, P. Mur, C. Wyon, M. Dupuy, J. Cryst. Growth 209 (2000) 1004. [7] M.R. Teague, J. Opt. Soc. Am. 73 (11) (1983) 1434. [8] D. Paganin, K.A. Nugent, Phys. Rev. Lett. 80 (12) (1998) 2586. [9] S.C. Mayo, P.W. Miller, S.W. Wilkins, T.J. Davis, D. Gao, T.E. Gureyev, D. Paganin, D.J. Parry, A. Pogany, A.W. Stevenson, J. Micros. 207 (2002) 79. [10] J.M. Cowley, Diffraction Physics, 2nd Edition, NorthHolland, Amsterdam, ISBN 0 44486925 5. [11] D. Van Dyck, H. Lichte, K.D. Van der Mast, Ultramicroscopy 64 (1996) 1.
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[12] L.J. Allen, M.P. Oxley, Opt. Comm. 199 (2001) 65. [13] L.J. Allen, H.M.L. Faulkner, K.A. Nugent, M.P. Oxley, D. Paganin, Phys. Rev. E 63 (2001) 037602/1–037602/4. [14] Digital Micrograph (Gatan Inc), for script writing see the Gatan Reference scripting guide and http://www.gatan. com/Bsoftware/guide/. [15] P. Hirsh, A. Howie, R.B. Nicholson, T.W. Pashley, M.J. Whelan, Electron Microscopy of Thin Crystals, 2nd Edition, Butterworths, London, 1965.
[16] R. Garcia, R. Perez, Surf. Sci. Report 47 (2002) 197. [17] P. Cloetens, E. Boller, W. Ludwig, J. Baruchel, M. Schlenker, Europhys. News 32 (2001) 46. [18] R.E. Dunin-Borkowski, Ultramicroscopy 83 (2000) 193. [19] M.J. Hytch, M.G. Walls, J.-P. Chevalier, Ultramicroscopy 83 (2000) 217.
Ultramicroscopy 100 (2004) 79–90
Imaging a dense nanodot assembly by phase retrieval from TEM images P. Donnadieua,*, M. Verdiera, G. Berthome! a, P. Murb a
LTPCM-INPG-CNRS-UJF, Domaine Universitaire BP75, 38402 Saint Martin d’H"eres, France b CEA-DRT - LETI/DTS - CEA/GRE, 17, rue des Martyrs, 38054 Grenoble cedex 9, France Received 7 May 2003; received in revised form 3 January 2004; accepted 26 January 2004
Abstract Phase retrieval is a classical inverse problem in many fields dealing with waves that is becoming of increasing interest in transmission electron microscopy (TEM). A non-interferometric approach is here applied to TEM images. Phase retrieval possibilities given by the transport intensity equation are compared to the ones deriving from the weak phase object approximation. In the limit of small angles, both methods lead to a similar equation between the phase and a set of defocus images. This equation can be solved by an image processing equivalent to using a specific filter in Fourier space. This processing leads to phase images with a spatial resolution here essentially limited by the defocus amount between images. A dense assembly of silicon nanodots is used as a model case to illustrate the interest of this approximate phase retrieval method which can be carried out on standard equipment. The dot heights estimated using the phase images are found to be in good agreement with ones measured by atomic force microscopy. Since image noise and large defocus values may strongly affect the solution given by the approximate method, an iterative phase retrieval method is also used as a test for working conditions. r 2004 Elsevier B.V. All rights reserved. PACS: 42.30; 61.48 Keywords: Phase retrieval; Defocus variation; Transport intensity equation; TEM; Nanodot
1. Introduction In the last decade, size reduction has become a key property in material research, the aim being either to push the limits of miniaturisation or to benefit from certain properties that appear below
*Corresponding author. Tel.: +00-33-4-7682-6686; fax: +00-33-4-7682-6644. E-mail address: [email protected] (P. Donnadieu).
some critical size. From fundamental physics to industrial applications, many fields are concerned with size reduction. Nevertheless, in this race to ultimate scale, microelectronics is the most emblematic one. Such an effort in size reduction requires, in parallel, to develop characterisation tools to both image and quantitatively study a microstructure at the nanometer scale. Regarding micro-nanoelectronics, there is a need for structural and chemical information as well as to be able to measure magnetic or electric fields in nanostructured materials of potential application.
0304-3991/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2004.01.007
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Due to its high resolution compared to other imaging methods transmission electron microscopy (TEM) remains, despite its age, a technique of choice for such a study. As attested by the development of electron holography, TEM offers various types of measurements: for instance, besides ultra-high-resolution imaging, holography allows the measurements of electric and magnetic fields. Such measurements, of course, are obtained owing to the holographic biprism inserted in the column. However, these measurements are made possible by the fact that the thin TEM sample acts as a phase object on the electron wave and that the phase shift is directly related to the sample inner potential. Indeed the key question is then to retrieve the phase which is a classical and high interest inverse problem in all the fields dealing with wave (optics, X-ray, neutrons, etc). Phase retrieval can be achieved by interferometric or non-interferometric methods. The former one requires interferometer as well as a high degree of coherency while the latter one needs more modest means to be carried out. In the limit of nanometer resolution, the non-interferometric approach applied to TEM imaging, especially with the transport intensity equation (TIE), seems powerful and of easy implementation. Recent papers have given very attractive examples of non-interferometric phase retrieval for Lorentz microscopy applied to magnetic samples [1–3] as well as TEM and optical studies of biological samples [4,5]. This paper aims at applying a non-interferometric phase retrieval method to experimental TEM images. Dense assemblies of nanodots elaborated in view of nanoelectronic application were used here as exemplar cases. These nanodots (o 10 nm) obtained by a low pressure chemical vapor deposition (LPCVD) process, are grown on an oxidised silicon substrate. In the present paper, the nanodot deposits being used as model cases, no specific information is given on elaboration and properties, for more details see Ref. [6]. Indeed, the nanodot assemblies very appropriately illustrate the interest for retrieving the phase since, though TEM can provide quite well-resolved plane views of the dots, the contrast interpretation remains limited as long as the
connection between contrast and dot height is not made. In the following, phase retrieval through the TIE solution will be considered with respect to the approach based on the weak phase object approximation (WPOA). Resolution limitation due to the method as well as the impact of noise will be also discussed.
2. Quantitative study of TEM images by phase retrieval In this section, we revise basic features of phase retrieval with particular emphasis on its application to TEM. Full details and deeper insights can be found in the references cited in this section. 2.1. About some non-interferometric phase retrieval methods Much work has been devoted in the last decades to the phase retrieval problem in optics, X-ray and electron microscopy. Non-interferometric methods are based on the acquisition of a set of images, further analysed using the propagation equation. Recently, the possibilities of the TIE early developed by Teague [7] have been considered as a solution in many field of applications. Besides theoretical aspects like unicity of solution, coherency requirements, robustness in presence of singularities, several successful applications of the TIE approach have been reported in X-ray projection microscopy, optical and transmission electron microscopy [4,5,8,9]. The transport intensity equation is a direct result from the description of diffracted waves under Fresnel diffraction conditions [7]. It comes out that the electron wave described by a timedependent plane wave function obeys the propagation equation: q r2 i þ þ k uz ðrÞ ¼ 0; ð1Þ qz 2k where z is the propagation direction, uz ðrÞ is the wave amplitude which depend only from rðx; yÞ and r the gradient operator in the ðx; yÞ plane, perpendicular to wave propagation. The
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measurable quantity is the image intensity Iz ðrÞ ¼ Juz ðrÞJ2 ; intensity which can be recorded at different z positions. Introducing the phase FðrÞ defined as uz ðrÞ ¼ ½Iz ðrÞ1=2 expðiFðrÞÞ allows Eq. (1) to be rewritten as a TIE having a convenient form with respect to the phase retrieval problem [7]: 2p qI ¼ r:ðIrFÞ: ð2Þ l qz In the limit of small intensity variation, the TIE simplifies as 2p DI ¼ Iðr; OÞr:rF l Dz with
in the following: in the small angle approximation, the WPOA leads to a relation between intensity and phase similar to the one resulting from the TIE approach (for detailed derivation on the WPOA, see Ref. [10]. The phase contrast object is accounted for by a transmission function in which, in the limit of small angles, the effect of defocusing is introduced in the expression of the transmission function through a Fresnel propagator term. This comes out for a weak phase object as a relation between the intensity at defocus Dz and the intensity at defocus zero. This relation more conveniently expressed in the reciprocal space writes as follows (the high order lense aberration term is here neglected):
DI ¼ Iðr; DzÞ Iðr; OÞ:
# DzÞ ¼ Iðq; # OÞð1 2Fðq; # DzÞsinðplDzq2 ÞÞ: Iðq;
ð3Þ
As demonstrated by Teague [7], the transport intensity equation is valid in the near-field approximation. The near-field condition which is basically a small angle condition writes in direct space as pr2 =lz51: Therefore the TIE provides an exact phase solution only when the working conditions meet the near-field requirement. In fact TEM rather relies on the Fraunhofer description since the plane of observation is not at finite distance. Under the Fraunhofer diffraction conditions, image formation is described by the Abbe imaging theory which relates by successive Fourier transforms object, diffraction and image. In most TEM cases, the thin sample can be considered as a phase object, most often appropriately described by the WPOA. In fact, as shown
ð4Þ
If we compare the WPOA equation to the simplified TIE rewritten for convenience in reciprocal space 2p # # DIðqÞ ¼ Iðq; 0Þ4pq2 FðqÞ lDz
ð5Þ
it comes out that Eqs. (4) and (5) are identical at very small defocus, i.e. under the near-field condition, since plDzq2 51; plDzq2 can be taken as an expansion of sinðplDzq2 Þ for all q: The TIE and WPOA approaches can be easily compared by plotting their respective transfer functions. Fig. 1a give the classical high-resolution transfer function, though calculated for defocus unusually large compared to the high-resolution imaging conditions. In high-resolution electron microscopy, the
Fig. 1. (a) Shows the classical TEM transfer function, at Dz=1000 nm, the simplified form TðqÞ=sin(plDzq2) is used, lens aberration and coherence damping function being neglected here as far only the small q range is concerned. (b) Report the TðqÞ transfer function together with the parabolic approximation T 0 ðqÞ ¼ plDzq2 which corresponds to the TIE approximation. The arrow indicates the intermediate field for a 1000 nm defocus ðplDzq2 ¼ 1Þ; in this range the transfer function TðqÞ only starts to depart from the simplified T 0 ðqÞ function.
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reciprocal space is explored up to the maximum distance allowed by the microscope information limit (i.e. typically qB5 nm1). Since the TIE is valid only for the small q values, the comparison of TIE and WPOA transfer functions shown in Fig. 1b is done only for qoB1 nm1. Hence, depending on the plDzq2 value, the phase can be retrieved more or less accurately using the TIE method instead of the WPOA. The approximation behind the transport intensity equation limits its application to the near-field conditions which means, in practice, either small defocus or large objects. On the other hand, the WPOA which is also valid, under small angles (because of the propagator expression), provides a convenient tool up to the intermediate field (plDzq2 E1). At small angles, rewriting of Eqs. (4) and (5) allows us to describe phase retrieval in terms of filtering in the reciprocal space. This filter description is convenient to discuss the resolution. For the TIE, the filter applied in reciprocal space is 2 1 plDzq for the WPOA the filter is while 1 sinðplDzq2 Þ : As illustrated by Fig. 1b the WPOA and TIE filters start to differ for plDzq2 E1: Though for plDzq2 ¼ 1; using the TIE equation, instead of the WPOA one, represent only an attenuation of the higher spatial frequency, i.e. a limited loss of resolution. In practice, plDzq2 ¼ 1 give a resolution limit. For distance below d ¼ ðplDzÞ1=2 ; the corresponding spatial frequencies are attenuated if the TIE filter is used instead of the WPOA one. In the following, the approximate method we use will be often referred to as a TIE type filtering. High resolution is not the objective of the present work which rather aims at a nanoscale characterisation. The main point is to improve the description of nanostructures (i.e lateral dimension as well as height for nanoobjects like nanodots). In that scope, phase retrieval achievable on standard microscope and based on the simplest approaches is of interest. It does not mean that high spatial resolution is not possible in phase retrieval, but more sophisticated equipment and treatments are required. Actually phase retrieval using an iterative method and the holographic one is the . basis behind the effort to reach sub angstrom resolution [11].
Beside resolution, noise is a serious drawback in the use of the TIE type filter. The development of the intensity, inferred by the Fresnel propagation approximation, implies that the first-order derivative of intensity is zero, which is usually not verified because of the noise. Through the successive Fourier transforms, this noise contribution is transferred back to the direct space on the phase image. The CCD camera being corrected from dark current and gain normalised, the main defect remains a signal noise characterised by high frequency 1in reciprocal space. The TIE type filter plDzq2 cuts the highest frequencies and therefore tends to produced smooth phase images free from rapid spatial fluctuation. Nevertheless the filter parabolic shape may have a too slow decay at high frequency, therefore there is a risk to introduce artefact details. Since the limit of resolution is known from the intermediate field conditions, any detail below this limit on the phase image is not relevant. Consequently these details are removed using appropriate masks in the image processing. The sensitivity to noise of the TIE method has been analysed in details by Allen and Oxley [12] who have discussed the possibilities and limit of several methods for getting the phase from focus series: *
* *
an approximate solution of the TIE by a Fourier method, an exact solution using multigrid methods, an iterative method assuming free space propagation between planes at different defocus.
They show that the Fourier method is efficient to rapidly obtain an approximate solution but can be safely used only in the vicinity of the in-focus plane in case of rapid phase variation. Phase discontinuities also prevent the application of the TIE method while the iterative method still works in presence of vortices [13]. Finally, the iterative method appears as the more robust one as well as being less sensitive to noise. Besides, according to Ref. [12], the amplitude of phase fluctuation is an important parameter concerning the choice of a method. In practice, using several methods is convenient to identify possible artefacts. In the present work, a TIE approach will be used on
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experimental images together with the iterative method provided by Allen and Oxley [12]. 2.2. Application TEM plane view images of silicon nanodots deposited on a substrate correspond to small thickness fluctuations which allows a phase object approximation. The nanodots sizes are expected to be in the 5–10 nm range, therefore a nanometric resolution method is of interest in the present case. According to the above analysis, two defocused images can be processed using the TIE type filter within some restriction on the spatial resolution. In practice, the expression sin plq2 approximated by plq2 leads typically to a spatial resolution limited to 1.7 nm (resp. 2.5 nm) for a defocus Dz=500 nm (resp. 1000 nm). It is worth noting that the value of the experimental defocus depends on the image contrast, hence the spatial resolution is somewhat dependent on the sample. The defocus amount is difficult to predict because it depends on the changes in image while defocusing. Besides, contrasted images and thus quite large defocus are required to test whether some drift has occurred. Finally, the defocus choice results from a compromise between the desired resolution and good working conditions in terms of contrast and noise. Being less sensitive than the TIE method, the iterative one will be used here to test whether the experimental conditions (defocus, noise level, amplitude of phase contrast) allow for phase retrieval with the TIE approach In the following, image processing using the TIE type filtering retrieves phase according to the equation: # # ðqÞ ¼ 1 1 DIðqÞ: F ð6Þ 2 Im 2plq Dz To avoid the divergence due to possible zero intensity points in I0 ; instead of setting a cut off intensity like [12], the intensity I0 is replaced by Im ; its transformation by a median filter of small radius which efficiently removes zero intensity points with small change in the whole image. Regarding the vicinity of q ¼ 0; a gaussian mask centred on q ¼ 0 is used, and its width must be
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chosen small enough to avoid dampening the thickness long range fluctuation. In the phase object approximation, the sample transmission is described by a phase shift which accounts for the interaction between electron and matter. In non-magnetic samples, the phase shift depends on composition and thickness and the specimen transfer function for a weak phase object writes as p f ðx; yÞ ¼ exp i Vt ðx; yÞ ; lE R where Vt ¼ V ðx; y; zÞ dz is the projected potential, l and E are the electron wavelength and the high voltage, respectively. In the case of interest here, the inner potential being constant, the phase shift measures thickness variation. Depending on the experimental conditions (i.e. whether the sample edge is in the recorded image field), absolute thickness measurement or only thickness fluctuations are derived. It is also worth noting that, the measured thickness being integrated values, the 3D phase representation shown further corresponds to actual topography only in case of simple topology and in absence of strain.
3. Phase images of nanodots 3.1. Experimental conditions Two types of nanodot deposits grown by LPCVD on Si substrate according to the method reported in Ref. [6] have been studied. Prior to the dot deposition, ultra thin oxides have been grown on the Si (1 0 0) surface by rapid thermal oxidation in slightly different conditions. According to ellipsometry measurements, the different oxidation conditions lead to oxide layer with different thickness (0.6 nm in sample 1, 1.2 nm in sample 2). As the oxidisation treatment affects only slightly the substrate surface, the dots elaborated in the same LPCVD conditions are expected to show only small differences. The dots deposited on Si oxidised wafers have been prepared for plane view TEM observations by argon ion beam bombardment on the substrate side.
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TEM was carried out on a microscope Jeol 3010 (LaB6 filament) operating at 300 keV. Images are recorded by a Gatan Dual View camera (1030
1300 pixels2) fitted on the 35 mm port. In the TEM working conditions (magnification 300 000), the image sampling of images is 0.36 nm/pixel. The defocus value given by the constructor have been tested using HREM focal series. Images are formed by selection of the transmitted beam with a small objective aperture (diameter 1.2 nm1). Phase is retrieved on a personal computer either using the iterative method contained in the program PhaseRetrieval1.10 developed by Allen and Oxley [12] or by the TIE type filtering carried out with script written within the Digital Micrograph software [14]. 3.2. Results Classical plane view images are given in Figs. 2a and b for samples 1 and 2, respectively. Sample 1 is characterised by quite large dots (B10 nm) while sample 2 shows an homogeneous distribution of dots with sizes in the 3–8 nm range. In any case, sizes are difficult to ascertain since dots are not easy to identify. The mottled contrast of black and white dots is difficult to associate with bump or hole because the black and white contrast can be reversed by defocusing. The phase information is necessary to make the connection between the
image contrast and thickness fluctuation corresponding to dots. 3.2.1. Fourier and iterative methods Fig. 3a–c displays images of a same area, respectively, taken at+1000, 0 and 1000 nm on sample 1. Similar images have been also obtained at larger defocus (2000 nm). Note that, in the present case, images with good signal/noise ratio are obtained only for defocus larger than 1000 nm. A series of 5 images with defocus ranging from 2000 to 2000 nm has been analysed using the iterative method. Fig. 4a gives the resulting phase image which is characterised after several hundred of iterations by a 2 103 sum squared errors (SSE). According to the discussion in Ref. [12], in presence of a 5% noise, the SSE cannot be better than 103 while 105 SSE are obtained with noiseless simulated images. Therefore, a phase image with a 2 103 SSE can be here considered as a satisfactory solution for the iterative method. Fig. 4b give the phase image which has been derived from the 1000 nm defocused image using the TIE type filtering. For a 1000 nm defocus, the resolution being limited to 2.5 nm, all details below this limit have been removed by masks with gaussian edges during image processing. The phase image in Fig. 2b shows quite an agreement with the one obtained by the iterative method. This
Fig. 2. TEM plane view images taken at positive defocus on the nanodot samples. In sample 1, shown in (a), the dot size is not easy to determine: the largest features (10–15 nm) might be formed by the aggregation of smaller dots. In sample 2 (2b) the dots form an homogeneous assembly with a mean dot size in the 5 nm range.
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Fig. 3. Focus series of TEM plane view images of a nanodot sample. The series has been recorded on a CCD camera (1pixel=0.36 nm) at different focus: (a) Dz=1000 nm ; (b) Dz=0 ; (c) Dz=1000 nm.
are in agreement with details noticed on the plane view images indicating large dots with complex inside features (Fig. 5c). Note that the details shown in Fig. 5 are taken at the same scale but in different parts of sample 1.
Fig. 4. Comparison of the phase images retrieved from the series in Fig. 3 using two independent methods. On the phase images, the white contrast corresponds to the dots. The field of view imaged here is 90 nm, and with both methods the dot size is the 5 nm range: (a) Phase image obtained with the iterative method [12] on a series of 5 images (the 3 images in Fig. 1 plus images at Dz=2000 and 2000 nm). This phase image is obtained after several hundred iterations, stagnation occurs on this solution with a 2 103 sum squared errors (SSE). (b) Phase image obtained by the TIE type filtering. The resolution being limited by the method to 2.5 nm, details below this limit are removed by filtering.
agreement indicates that the noise level and the high working defocus do not prevent to use the most approximate approach. Fig. 5 illustrates the specific interest for nanodots to have thickness fluctuation mapping instead of the usual phase contrast plane view. Figs. 5a and b show a detail from the phase image and its 3D representation: the latter particularly points out that the dots were arranged in a 5-fold ring around central hole. This kind of dot aggregates
3.2.2. Quantitative measurements Figs. 6 and 7 display the phase images and their 3D representation obtained for samples 1 and 2. As mentioned above, since the sample edge is not in the recorded area, only phase fluctuations are measured here. These fluctuations have here two origins: the dots deposited on the substrate and the thickness fluctuations due to sample preparation. Measurements of phase can be made from Eq. (6) with only the knowledge of defocus and digitisation parameters. The scaling factor after the Fourier transformation processes is N 2 p2 =2plDz; where N is the image size in pixels, p the effective pixel size at the TEM operating magnification and Dz the defocus between images. This scaling factor is given here to point out how quantitative phase measurement relies on defocus and magnification calibrations. Inserts in Figs. 6 and 7 give typical phase profiles allowing for further thickness measurements. For silicon dots deposited on a substrate covered by a nanometer thick silica layer, the oxide layer being constant, the inner potential of interest is the silicon one. The thickness fluctuation t is then related to the phase fluctuation DF by: t ¼ lEDF=pV ; where V is the inner potential. The inner potential can be calculated knowing the
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Fig. 5. (a) Gives a high magnification detail of the phase image in Fig. 4b. The 3D representation in (b) Indicates that it can be interpreted as a ring of 5 dots around a central hole and (c) Gives high-magnification TEM plane view from sample 1 at the same scale as Fig. 5a. Such detail from untreated images is consistent with the existence of complex aggregate of small dots as revealed by the phase image.
Fig. 6. Phase image and its 3D representation obtained for the nanodot sample 1. The field of view is 180 nm.The dots which lateral size is in the 6 nm range form an irregular assembly. No large dots are observed but in frequent places rings of dots can be noticed. The phase (phase in radian as a function of distance in nanometer) profile given in the insert corresponds to dot height h1 E8 nm.The 3D representation shows long-range thickness fluctuation which are consistent with the ones expected from sample preparation by ion beam thinning under a 5 angle in the finishing step.
crystal structure and the atomic scattering factor given in classical books [15], for silicon V=17.6 V. According to the phase profiles, dot heights are h1 E8 nm and h2 E10 nm for samples 1 and 2, respectively. The dot lateral sizes estimated from the phase images are, in both samples, close to 6 nm.These values only aim at giving some size estimation for comparison with atomic force microscopy (AFM). According to the phase images, samples 1 and 2 shows similar features, essentially differing only by height and dot density. In sample 1 (resp. sample 2), the density estimated from the dot distance isE3 1011 dot/cm2 (resp.E1012 dot/cm2). These
values are actually approximately 2 times larger than the ones based on AFM or SEMFEG investigation. The next section reports the AFM study carried out on the same samples to test the reliability of measurements and examine the TEM phase image possibilities versus AFM.
4. Comparison with AFM imaging Atomic force microscopy were carried out on a Veeco D3100 AFM in tapping mode. Acquisition fields were saved only when drift effect was
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Fig. 7. Phase image and its 3D representation obtained for the nanodot sample 2. According to phase profiles as illustrated by the one in inset, the dot height (h2E10 nm) is in a similar range as in sample 1. Comparison of the 3D representation indicates that although the whole microstructure keeps same morphological features, the density is higher in sample 1 as well as the height/width ratio. The field imaged here is 180 nm wide and the dot lateral size is approximately 6 nm.According to the phase image, this dot assembly is characterised by a higher density than the sample 1 dot assembly.
Fig. 8. AFM images obtained on sample 1 (a) and sample 2 (b). The scale ranges are reported on the z-axis for comparison with the phase image results and (c) gives an AFM plane view of sample 2 quite similar to the phase image in Fig. 7.
minimized between successive scan of the same area. Tips with a certified 10 nm radii were used. Major limitation in AFM is the lateral resolution generally estimated between 2 and 5 nm, the vertical resolution being in the subnanometer range. The lateral resolution falls in the range of the interdistance of the dense distribution of the plots studied here, therefore convolution effect with the tip are expected. For these samples, the exact shape of the nanodot can not be ascertain with this technique. It is however possible to determine the height of the nanoplots, and to demonstrate a variation in surface density of the plots between sample 1 and 2. Figs. 8a and b give 3D representation of the AFM images obtained on
samples 1 and 2, respectively. The AFM height range (h0 1=5 nm and h0 2=9.7 nm, are in the range of the results given by the phase method (h1E8 nm and h0 2E10 nm). The AFM plane view shown in Fig. 8c was taken in sample 2. It is rather similar to the phase image obtained for the same sample (Fig. 7), except for the dot density which seems smaller according to AFM. We attribute this difference to the limitation of lateral resolution due to the high dot density. Even with very small radius tip, such dot assemblies are difficult to image with AFM, for complete review of the AFM imaging possibilities in air with the tapping mode see Ref. [16].
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5. Discussion As illustrated by the nanodot samples, phase retrieval provides appropriate information to understand the TEM plane view images. Though it should be kept in mind that the phase is related to the integrated potential which corresponds to thickness only in a perfect homogeneous sample with simple topology and free of strain. Opposite to AFM, phase retrieval sees the sample ‘‘from the inside’’. Therefore the combination of AFM and phase retrieval from TEM plane view might be an interesting tool to tackle the question of strain in nanoobject. However such combination can be carried out only for systems in which AFM provides reliable shape information. In the present case, regarding the dot height, phase retrieval results are in fair agreement with the AFM ones. Still the comparison is limited since the system is typically difficult for AFM studies. In fact, the nanodot case was chosen because of this specific difficulty in order to point out the interest for direct method like TEM allowing to image a whole area at once. In the present work, the microscope resolution in the high-resolution mode is 0.25 nm.Because of the objective aperture we used, this resolution drops to 0.8 nm and finally the approximate phase retrieval method gives phase images with a 2.5 nm resolution. This illustrates a particular advantage on probe scan images: though the TEM phase retrieval resolution depends on the sample through the choice of working defocus, the resolution can be evaluated quantitatively for each working condition. The aim of the present paper was mainly to test the feasibility of thickness measurement on a dense nanoscale topography using standard equipment. Of course improvement in resolution can be achieved especially using a CCD camera dedicated to weak contrast imaging. Better contrast and noise reduction will allow a decrease in the defocus hence improving the spatial resolution. Regarding phase measurement, at the moment, the defocus has been recalibrated only on a limited range. A calibration for large defocus will be carried out using layers of known thickness. Better resolution can be obtained also by an improvement of the phase retrieval method, such as for instance
working with a series of defocused images. This method already applied to X-ray phase contrast [17] is worth trying. Though better image acquisition and processing should improve the resolution, the ultimate limit is determined by the source coherency which has not been mentioned up to now. In any case, the source related transfer function envelope will define the resolution limit reachable for a working defocus. The phase retrieval method used here allows the derivation of quantitative phase information at the nanometer scale. In addition the phase can be measured on large image field and unwrapped values are obtained. The TIE type solution to phase retrieval consisting in a Fourier image processing, the method is also advantageous in terms of rapidity and easy implementation. These features make this non-interferometric method an approach very complementary to the holographic one. Holography requires high degree of coherency and can image only a limited field of view but of course allows to get the atomic resolution information. Phase retrieval using the transport intensity equation on Lorentz microscopy images [1,2] has already shown the valuable complement it provides to holography as it leads to similar information but on a different scale. Phase retrieval already occupies an important place in TEM since it is the basis of a subangstrom resolution work project, through side band holography and focus series method [11]. However it could be an issue to introduce more systematically phase retrieval methods in more standard TEM not only as a complement to holography. As illustrated by recent works on Fresnel fringes [18,19], information on inner potential can be obtained in conventional microscopy conditions carried out on a standard microscope. In these interface studies, the inner potential was used to study roughness and chemical fluctuation. These studies point out how an access to the inner potential combined with appropriate post processing can address most relevant questions like separating chemical and geometrical effects. Among all the techniques which allow the measurement of the inner potential, phase retrieval through Fourier image processing is indeed one of the simplest to carry
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out on standard TEM equipment. As many CCD camera are now being installed, phase retrieval can be developed as a common on line analysis. Among other possibilities, phase retrieval in TEM may find interesting applications in 3D imaging because extracting the potential gives information on the thickness along the propagation direction. Similarly to phase contrast tomography which has been developed with hard X-ray at synchrotron [17], there could be a potential field in extending absorption electron tomography to phase contrast electron tomography. In that kind of application, getting the phase contrast with an efficient and rapid routine will be of high interest.
techniques in electron microscopy such as holography or focal series method which aim at the sub-angstrom resolution. In any case, at the nanometer scale, phase retrieval is achievable and can lead to information on the thickness or the chemical nature of the sample even on standard microscopes. The popularity of CCD cameras and the possibilities of remote control of the microscope functions provide a very appropriate context for the promotion of phase retrieval in TEM. The resolution can be easily improved with better acquisition conditions and image processing, though it will always remain limited by the degree of beam coherency.
6. Conclusion
Acknowledgements
Phase retrieval is an old question which has been given in the last years different solutions. In particular in TEM, phase can be retrieved using the WPOA approach which in the small angle approximation is equivalent to the solution given by the transport intensity equation. However both approaches are valid only in the limit of small angles which has a direct consequence on the resolution. Even though the atomic resolution cannot be reached by these approximate methods, they are worth using to extract information on inner potential at the nanometer resolution. We have illustrated the interest for approximate phase retrieval from defocused images on the specific example of nanodot assemblies. Here phase retrieval obviously provides the extra information necessary for interpreting the plane view TEM images. Comparison with AFM images on the same dot assembly points out than, even limited to a 2.5 nm resolution, the phase image gives information that are typically difficult to reach with probe scan methods. Even under the severe approximation we had made, the height measurements derived from the phase shows a good agreement with the AFM ones. The phase retrieval method used here is particular attractive in terms of speed and easy implementation. Of course, if the resolution is good in comparison with probe scan technique, it is modest in comparison with other phase retrieval
Samples were prepared in the frame of CEALETI/CPMA collaboration, with PLATO Organization teams and tools. The authors thank Thierry Baron for his contribution to the dot elaboration, Prof. L.J. Allen for kindly sending his program ‘‘Phase retrieval 1.10’ and Peter Cloetens for stimulating discussion.
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