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High-Resolution Transmission Electron Microscopy on an Absolute Contrast Scale

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High-Resolution Transmission Electron Microscopy on an Absolute Contrast Scale week ending PRL 102, 220801 (2009) PHYSICAL REVIEW LETTERS 5 JUNE 2009 High-Resolution Transmission Electron Microscopy on an Absolute Contrast Scale A. Thust Institute of Solid State Research, Research Centre Ju¨lich, D-52425 Ju¨lich, Germany (Received 21 January 2009; published 4 June 2009) A fully quantitative approach to high-resolution transmission electron microscopy requires a satisfac- tory match between image simulations and experiments. While almost perfect agreement between simulations and experiments is routinely achieved on a relative contrast level, a huge mutual discrepancy in the absolute image contrast by a factor of 3 has been frequently reported. It is shown that a major reason for this well-known contrast discrepancy, which is often called Stobbs-factor problem, lies in the neglect of the detector modulation-transfer function in image simulations. DOI: 10.1103/PhysRevLett.102.220801 PACS numbers: 07.78.+s, 42.30.Lr, 68.37.Og The technique of high-resolution transmission electron final image acquisition by a camera. Concerning the dif- microscopy (HRTEM) offers nowadays the possibility to fraction part, plasmon and phonon scattering were explic- study the atomic configuration of solid state objects with a itly ruled out as a reason for the problem [10–12], and resolution of around 0.08 nm. Highly accurate information moreover, a remarkable discrepancy between measured about atomic positions and chemical occupancies at lattice and simulated diffraction data was not found [6,8]. defects and interfaces can be obtained by the quantitative Concerning the optical part, the measurement of coherent use of this technique [1]. aberrations and of partially coherent contrast dampening Because of the quantum-mechanical nature of the elec- functions is possible with high accuracy and did not reveal tron diffraction inside the object and due to the subsequent inconsistencies [13–15]. However, little attention has been electron-optical imaging process, any quantitative extrac- paid to the final image recording step, which involves the tion of object information from HRTEM images requires a frequency-transfer properties of the camera. justification by accompanying image simulations. Since In the work of Hy¨tch and Stobbs the modulation-transfer the introduction of digital image comparison to HRTEM function (MTF) of the photographic film plates used at that in the early 1990s, a satisfactory agreement between simu- time was not considered [3]. It is also remarkable that the lation and experiment could be established only on a most popular image simulation software packages EMS and relative basis by disregarding the absolute magnitude of MACTEMPAS do not allow us to incorporate actually mea- the image contrast [2]. The reason for this still common sured MTFs [16,17]. In all, the actual MTFs of the mean- practice was first published by Hy¨tch and Stobbs in 1994, while used CCD (charge coupled device) cameras were who found a remarkably strong discrepancy between the often not considered in the literature related to the Stobbs- magnitude of the simulated and the experimental image factor problem [7,8]. Concerning the very rare implemen- contrast [3]. In their experiment, the dimensionless quan- tation of actually measured MTFs into image simulations, tity of the image contrast, which is defined as the standard criticism is allowed: Besides the use of an MTF belonging deviation of the image intensity distribution after normal- to a different camera than actually used [5], earlier MTF ization of the image mean intensity to unity, was by a factor measurements were often based on the noise method of 3 too low when compared to image simulations. Since [9,18]. However, the noise method is known to yield un- the image motifs were nevertheless widely consistent realistic results compared to the knife-edge method, which between simulation and experiment, the problem was re- is the state-of-the-art technique for MTF measurement garded mainly as a scaling problem, and became promi- [19–22]. nent as Stobbs-factor problem, factor of 3 problem, or The drastic influence of the detector MTF on the ob- contrast-mismatch problem. A satisfactory explanation tained image contrast is demonstrated here by a simple for this long standing problem, which has been meanwhile experiment, where the same area of a SrTiO3 crystal is frequently reproduced and investigated, has not been found imaged with different magnification settings of a FEI Titan so far [4–9]. 80–300 microscope equipped with a 2k  2k Gatan The electron microscopic imaging process can be sub- UltraScan 1000 CCD camera of 14 m pixel size. From divided into three stages, of which each can be potentially Table I, and from visual inspection of Fig. 1, the strong responsible for the contrast-mismatch problem: (i) the dif- dependence of the obtained image contrast on the detector fraction of the incoming electron wave by the object, sampling rate is obvious. Although all images are recorded (ii) the subsequent electron-optical transfer of the dif- with a sufficiently high sampling rate, a difference in the fracted wave by the electromagnetic lenses, and (iii) the obtained contrast by a factor of 2 arises between the lowest 0031-9007=09=102(22)=220801(4) 220801-1 Ó 2009 The American Physical Society week ending PRL 102, 220801 (2009) PHYSICAL REVIEW LETTERS 5 JUNE 2009 TABLE I. Contrast obtained with a FEI Titan 80–300 micro- The physical core principle of the MTF measurement scope equipped with a Gatan UltraScan 1000 camera from an used in this Letter is identical to that of the knife-edge SrTiO identical area of 3 using four different microscope mag- method as presented in Refs. [19–21], but is extended from nifications. Nominal magnification, sampling rate, inverse sam- the classical case of a one-dimensional step function to the pling rate, and contrast are listed. more general case of a sharply bounded two-dimensional Sampling I. Sampling object. Instead of a knife edge positioned directly on the Nominal rate rate detector, the beam stop located roughly 1 m above the magnification [1=A ] [nm] Contrast camera is used as a shadow-forming object. A calculation 670 000 6.709 0.014 91 0.082 of Fresnel diffraction for 300 kV electrons falling onto an 850 000 8.532 0.011 72 0.111 opaque edge shows that the resulting Fresnel fringing zone 1 100 000 11.033 0.009 06 0.136 at 1 m distance is much smaller than the pixel size of 1 400 000 14.243 0.007 02 0.165 14 m, thus justifying the assumption of a sharp shadow on the detector. The assumption of full opacity is in turn justified by Monte Carlo simulations, according to which and highest magnified images. Moreover, there is no in- the fraction of registered electrons, which interacted with dication that this effect is already saturated at the highest the edge material, is negligibly small [20]. available magnification of 1 400 000. After correction of the experimental shadow image I x; y A quantification of the described phenomenon requires a Eð Þ for nonuniform illumination, a synthetic shadow I x; y measurement of the actual camera MTF, which is the image Sð Þ is constructed in a three-step procedure: n Fourier transform of the point-spread function PSFðx; yÞ. First, the experimental image is -fold up-sampled by The real-space convolution of the image intensity with the Fourier interpolation to obtain subpixel positional accu- n point-spread function PSFðx; yÞ is described in Fourier racy. Subsequently, this -fold oversampled image is con- space as a product of the ideal intensity Iðu; vÞ and the verted into a sharp two-level image. Finally, this n modulation-transfer function MTFðu; vÞ, yielding the de- intermediate image is -fold down-binned to obtain a synthetic shadow image ISðx; yÞ of original size, which is tected image intensity IDðu; vÞ, with thereby already affected by the pixel-size sinc-convolution IDðu; vÞ¼Iðu; vÞMTFðu; vÞ: (1) of Eq. (2). The experimental and synthetic shadow images differ thus only by the scintillator part MTFS of Eq. (2). By k ’ The modulation-transfer function MTFðu; vÞ itself can be changing to polar coordinates ( , ) in Fourier-space one separated into a rotationally symmetric part MTFSðu; vÞ can define a set of auxiliary functions, given by 2 2 describing the image convolution by the scintillator and its fEðk; ’Þ¼jIEðk; ’Þj À N ðkÞ; (3) support, and a part describing the convolution over the quadratic area of a single pixel, yielding 2 fSðk; ’Þ¼jISðk; ’Þj ; (4) MTF ðu; vÞ¼MTFSðu; vÞsincð0:5uÞsincð0:5vÞ; (2) fESðk; ’Þ¼RefIEðk; ’Þ=ISðk; ’Þg; (5) where the Fourier-space coordinates (u, v) are defined in where N2ðkÞ is an estimate for the noise power at Fourier- units of the Nyquist frequency. space frequency k, and Re denotes the real part. The estimate N2ðkÞ is obtained by evaluating the experimental image IEðk; ’Þ at positions (k, ’), where the synthetic image ISðk; ’Þ is known to have no signal contributions. An estimate for MTFSðkÞ can then be obtained in two alternative ways, namely, by 1=2 MTF SðkÞðhfEi=hfSiÞ ; (6) MTF SðkÞhfESi; (7) hfðk; ’Þi SrTiO where denotes a weighted average over the azi- FIG. 1. Edge of a [110]-oriented 3 crystal recorded with muthal coordinate ’. Within the averaging process, a a FEI Titan 80–300 microscope, which is equipped with a Gatan weighting factor wðk; ’Þ is applied in order to (i) select UltraScan 1000 CCD camera. The underlying image was re- corded at a nominal electron-optical magnification of 670 000, Fourier-space regions of strong signal known from the the inset image at a magnification of 1 400 000. For comparison, synthetic image, (ii) to deselect regions affected by aliasing the images were resampled to identical magnification and are artifacts following the principle of Ref.
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