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Simulation and Interpretation of Images

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Scanning Transmission Electron Microscopy

Abstract

In this chapter we discuss the simulation and interpretation of both Z-contrast and electron energy-loss spectroscopy images in scanning transmission electron microscopy.

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Notes

  1. 1.

    The “reduced wave function” is obtained from the full wave function by factoring out a fast oscillating phase factor of \(\exp(2\pi {{\textrm i}} k_0 z)\). This is often referred to as the modulated plane wave ansatz (van Dyck 1985).

  2. 2.

    This expression is strictly only correct in the absence of absorption. In the presence of absorption \(C_{\bf g}^{i*}\) should be replaced by the appropriate element in the inverse matrix of eigenvalues (Allen and Rossouw 1989, Findlay 2005).

  3. 3.

    The phase-linked plane wave approach could equally well be applied in the multislice method, but, following Kirkland et al. (1987), all multislice treatments of STEM seem to deal with the entire wave function.

  4. 4.

    We must note that the model presented here for thermal absorption and HAADF imaging follows from the Hall and Hirsch description (1965), the derivation of which involves an analytic application of the frozen phonon concept, and in light of this the agreement between the models is not surprising. But as effectively the same potential has been derived by other means which do explicitly account for the inelastic transition (Weickenmeier and Kohl 1998), the assertion is not trivial.

  5. 5.

    Core loss on an atom-by-atom basis is justified because the initial bound state is meaningfully associated with a single atom, and so distinguishable from excitation of other atoms (Maslen 1987, Saldin and Rez 1987). Phonon excitation on an atom-by-atom basis is less obviously justified since it invokes an Einstein model and so neglects any correlated motion between the different atoms. For HAADF imaging, a detailed comparison was carried out by Muller et al. (2001) which showed that these two models give essentially the same predictions. We use that basis to justify the assumption.

  6. 6.

    Or, loosely speaking, cross-section expression since for high-energy electron scattering the two are related by constant factors.

  7. 7.

    To make connection with previous work by the authors and collaborators (Allen and Josefsson 1995, Allen et al. 2003, 2006, Findlay et al. 2005, Oxley et al. 2005, 2007, Rossouw et al. 2003), we note that Eq. (15) has generally been evaluated in reciprocal space. Defining inelastic scattering matrix elements \({\boldsymbol{\mu}}_{{\bf H},{\bf G}}\) via

    $$\frac{2\pi}{hv} W({\bf R},{\bf R}') = \frac{1}{A} \sum_{{\bf H},{\bf G}} {\boldsymbol{\mu}}_{{\bf H},{\bf G}} \exp\left(2\pi i{\bf H}\cdot{\bf R}\right) \exp\left(-2\pi i{\bf G}\cdot{\bf R}'\right)$$

    and substituting into Eq. (15), the nonlocal imaging expression may be rewritten as

    $$I({\bf R}_0) = \frac{1}{A} \sum_{{\bf H},{\bf G}} {\boldsymbol{\mu}}_{{\bf H},{\bf G}} \int^t_0 {\boldsymbol{\psi}}^*_0({\bf H},z,{\bf R}_0) {\boldsymbol{\psi}}_0({\bf G},z,{\bf R}_0) {\mathrm{d}} z \;.$$

    The area factor A is an artefact of the assumed normalization of the wave functions, which varies widely in the literature. Here we have assumed that the integral of the intensity of the wave function over a 2D plane is dimensionless, in both real and reciprocal space forms. This contrasts with the Bloch wave formulation, where the wave function itself is often taken to be dimensionless.

    The inelastic scattering matrix elements \({\boldsymbol{\mu}}_{{\bf H},{\bf G}}\) are closely related to the mixed dynamic form factor (Kohl and Rose 1985, Schattschneider et al. 2000), the difference being that the former further incorporates information about the detector geometry.

  8. 8.

    An extended version of this investigation may be found in Allen et al. (2008).

  9. 9.

    The 10 mrad case has the same shape as the test case of Oxley et al. (2005). However, since the approach based on Eq. (11) does not lend itself to integration over an energy window, a fixed energy loss of 10 eV above the threshold was chosen. This makes the units here slightly different to those in Oxley et al. (2005) since they are based on Eq. (15) where the integration over a 40 eV energy window was carried out over the effective scattering potential.

  10. 10.

    This is reminiscent of a very similar problem in atomic-resolution images in CTEM, referred to as the “Stobbs factor” problem (Howie 2004, Hÿtch 1994). If the cause were to lie in some insufficiently appreciated aspect of, say, the thermal scattering, the problems in CTEM and STEM could be connected. However, though the definitive evidence has not yet been published, it seems likely that the cause of the discrepancy in CTEM is due to the modulation transfer function of the detector (Thust 2008). If so, the problems are independent, the detector modulation transfer function having no direct analogue in the STEM set-up.

  11. 11.

    The outer detector semiangle for the calculations was ∼240 mrad, which is likely smaller than the experimental value. However, increasing the outer angle in the calculations to 400 mrad in the Bloch wave model (the upper experimental limit given by the detector dimensions) did not significantly affect the contrast. Sampling issues prohibit frozen phonon simulations being readily attempted with the larger detector range. Hence the use of the smaller outer angle, given the Bloch wave reassurance that the difference will be small.

  12. 12.

    It has recently been emphasized that the fractional occupancy method cannot fairly be applied in the frozen phonon model (Carlino and Grillo 2005). However, in the cross-section expression model it presents no serious inconsistencies, particularly when investigating only qualitative features rather than quantitative signals.

  13. 13.

    Calculations predict that the contribution of electrons that excite L 1 ionizations is over an order of magnitude smaller than those that excite L 2,3 ionizations, and so the former are neglected.

  14. 14.

    Confocal electron microscopy at lower resolutions having been previously established (Frigo et al. 2002).

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Acknowledgements

L. J. Allen acknowledges support by the Australian Research Council. S. D. Findlay is supported by the Japanese Society for the Promotion of Science (JSPS). M.P. Oxley was supported by the Office of Basic Energy Sciences, Materials Sciences and Engineering Division, U.S. Department of Energy. We would like to thank our following collaborators for their considerable inputs into various parts of the work summarized in this chapter: G. Behan, A. L. Bleloch, M. Bosman, E. C. Cosgriff, A. J. D’Alfonso, C. Dwyer, J. L. García-Muñoz, V. J. Keast, A. I. Kirkland, J. M. LeBeau, P. D. Nellist, S. Stemmer and P. Wang.

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Authors and Affiliations

  1. School of Physics, University of Melbourne, Melbourne, VIC, Australia

    Leslie J. Allen

  2. Institute of Engineering Innovation, The University of Tokyo, Tokyo, Japan

    Scott D. Findlay

  3. Department of Physics and Astronomy, Vanderbilt University, Nashville, TN, USA

    Mark P. Oxley

  4. Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN, USA

    Mark P. Oxley

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  1. Leslie J. Allen
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Correspondence to Leslie J. Allen .

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  1. Oak Ridge National Laboratory, 1 Bethel Valley Road, Oak Ridge, 37831-6056, Tennessee, USA

    Stephen J. Pennycook

  2. , Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, United Kingdom

    Peter D. Nellist

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Allen, L.J., Findlay, S.D., Oxley, M.P. (2011). Simulation and Interpretation of Images. In: Pennycook, S., Nellist, P. (eds) Scanning Transmission Electron Microscopy. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7200-2_6

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