Modelling high-resolution electron microscopy based on core-loss spectroscopy
Introduction
Structure determination based on inelastic thermal scattering has been successfully implemented in both conventional transmission electron microscopy (CTEM), via Rutherford back-scattering [1], and scanning transmission electron microscopy (STEM), via high-angle annular dark field imaging [2], [3], [4]. For thermal scattering, a local approximation for the effective inelastic interaction proves adequate [5]. Structure determination can also be based upon inelastic core-loss signals since these are suitably “localized” [6], providing chemical information in addition to spatial information. For core-loss spectroscopy the effective inelastic interaction is better described by a full nonlocal model [7], [8], [9], [10]. We illustrate this unambiguously using high angular resolution electron channelling electron spectroscopy (HARECES) on graphite [11], [12].
The veracity of the full nonlocal model thus demonstrated, we turn our attention to core-loss imaging in STEM where we predict that nonlocal effects in imaging may become more pronounced with decreasing probe size. We discuss and elaborate on the prediction that nonlocality in STEM can lead to imaging at a distance, whereby signal can be obtained from a part of the specimen which is not directly illuminated by the electron probe [13]. This is not the well-known cross-talk phenomenon where the signal results from the probe spreading to adjacent columns [14], [15], [16], [17], [18]. To illustrate the effects consider Fig. 1(a), showing a simulated carbon K-shell electron energy-loss spectroscopy (EELS) image of a thick SiC crystal, viewed along the [0 1 1] zone axis using 200 keV electrons (we assume the spread in energy is negligible throughout this paper). The probe is aberration-free (all the probe forming lens aberrations are zero) with a 50 mrad semi-angle, an aperture size anticipated in the next generation of microscopes. The detector integrates over a 40 eV energy window above the ionization threshold and has a 10 mrad semi-angle. The brightest features in the carbon K-shell image do not occur upon the carbon columns, but rather upon the silicon columns. When the probe is on the silicon column there is virtually no flux on the carbon column [13].
Not only can finer probes increase the importance of nonlocal effects, they also increase the degree of thermal scattering occurring when a probe is situated upon a heavy column, compared to when it is situated between columns. This can lead to complications in interpreting images in terms of the crystal structure. A simulation of this possibility is shown in Fig. 1 (b) for a thick Ag crystal, imaged along the [0 0 1] zone axis using 100 keV electrons. The probe is aberration balanced to offset an assumed fifth order spherical aberration , as is done experimentally when we are able to correct spherical aberration to third order. The probe, with semi-angle 20 mrad, has and . The image represents the silver M-shell signal integrated over a 40 eV window above ionization threshold using a detector with 20 mrad semi-angle. As can be seen the signal is strongest between the columns of atoms.
These two cases will be examined in greater detail, with a view to understanding how they arise. In the case arising from nonlocality, we will explore the experimental conditions needed to optimize the visual correspondence between images and structure.
Section snippets
Theory of inelastic electron imaging
Making the common assumption that only excitations from the ground state contribute significantly to the scattering, the Schrödinger equation can be written in the form [9], [13]where is the elastic wave function in the sample, is the electron wave number, denotes elastic scattering, and the complex nonlocal kernel representing inelastic scattering of the fast electron is given by
Experimental evidence for nonlocal effects in CTEM
The awareness that nonlocality can affect electron microscopy images dates back over two decades [7], [9], [10], [18], [28], [34]. And yet there has been no experimental result which can be explained in a fully nonlocal model but where the local approximation is clearly inadequate. In this section we will describe a CTEM experiment which makes this distinction unambiguously.
Fig. 2(a) shows a portion of a HARECES data set from graphite, as a function of energy loss and incident orientation,
Nonlocality in STEM imaging
Using a 200 keV probe with semi-angle 50 mrad, a detector with a 10 mrad semi-angle, and integrating over a 40 eV energy window above the carbon K-shell edge, Oxley et al. [13] presented simulations showing that the signal from SiC [0 1 1] would display a larger signal on the silicon column than it would on the carbon column. Though arising specifically as a result of the very fine probe used, they demonstrated that this is not a cross-talk effect in the sense that it did not result from spreading of
Conditions for direct image interpretation
The results of the dependence on detector size study in the previous section show the most promise for providing a way to obtain images which may be directly interpreted. Those results suggest that for detectors with a suitably large collection angle the local approximation becomes valid, and within that approximation we are able to use the simpler forms of Eqs. (10) and (11) to assay the localization of the interaction and the fidelity of the images in indicating the sample structure. Rossouw
The role of thermal scattering
Single atom imaging simulations provide an estimate for the resolution which may be expected when a particular probe is used in conjunction with a particular EELS detector arrangement [27]. If the width of a single atom STEM image was greater than the typical inter-column separation in the related specimen then column-by-column resolution would be difficult. However, this estimate alone does not give complete predicting power because it does not account for the evolution of the probe in the
Conclusion
We have presented clear evidence for the nonlocal character of the effective ionization interaction in core-loss EELS in the context of HARECES data on graphite. Having established the importance and validity of a model describing these nonlocal effects, we discussed the prediction that, with the realization of fine probes in STEM, the nexus between structure and image may not be straightforward. Imaging at a distance was demonstrated and we explored the experimental conditions under which a
Acknowledgements
L.J. Allen acknowledges support by the Australian Research Council. This research was sponsored in part by the Laboratory Directed Research and Development Program of ORNL, managed by UT-Battelle, LLC, for the US Department of Energy under Contract No. DEAC05-00OR22725 and by appointment to the ORNL Postdoctoral Research Program administered jointly by ORNL and ORISE. N.J. Zaluzec was supported by the US DoE BES under contract MS W-31-109-Eng-38 at ANL.
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