Elsevier

Ultramicroscopy

Volume 151, April 2015, Pages 11-22
Ultramicroscopy

Modelling the inelastic scattering of fast electrons

https://doi.org/10.1016/j.ultramic.2014.10.011Get rights and content

Highlights

  • Harald Rose was a pioneer of important work on atomic resolution imaging using inelastic scattering.

  • We review how the modelling of inelastic scattering has subsequently developed and been applied.

  • A software package μSTEM is introduced, capable of simulating various inelastic imaging modes.

Abstract

Imaging at atomic resolution based on the inelastic scattering of electrons has become firmly established in the last three decades. Harald Rose pioneered much of the early theoretical work on this topic, in particular emphasising the role of phase and the importance of a mixed dynamic form factor. In this paper we review how the modelling of inelastic scattering has subsequently developed and how numerical implementation has been achieved. A software package μSTEM is introduced, capable of simulating various imaging modes based on inelastic scattering in both scanning and conventional transmission electron microscopy.

Introduction

A landmark paper entitled “Theory of image formation by inelastically scattered electrons in the electron microscope” was published 30 years ago by Helmut Kohl and Harald Rose [1]. There they observed that “reviews on image formation treat the contribution of inelastically scattered electrons as a deleterious side effect” but suggested that “it seems worthwhile to examine image formation by inelastically scattered electrons in more detail”. This insight spurred the subsequent development of high resolution spectroscopic imaging modes [2], [3], [4], [5], but it would be fifteen years before the spectroscopic single atom imaging envisaged by Kohl and Rose was realised [6], and longer before atomic resolution spectroscopic imaging really came into its own [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].

In their seminal paper, Kohl and Rose [1] outline a quantum mechanical theory of imaging which considers both elastically and inelastically scattered electrons. Early on in their paper they address the importance of quantum mechanical phase in electron scattering and show how the concept of phase is related to a four-dimensional mixed dynamic form factor [19] encapsulating the essential physics of inelastic scattering. Kohl and Rose presented a clear conceptual picture to show that accounting for phase is essential, which we think worth repeating here. Consider the inelastic signal from an atom illuminated by an electron probe consisting of the coherent superposition of two plane waves with different incident angles, as shown in Fig. 1. A purely kinematic analysis might focus on the scattering angles between each of these plane waves and the detector. However, the relative phase of the two waves is critical in determining where the electron density in their interference pattern sits relative to the atom position, something physical intuition correctly identifies as being essential in determining the resultant signal. In essence, the mixed dynamic form factor describes the contribution to the signal due to the interference between pairs of plane waves (Fourier components) in the probe. Though less obvious, it turns out that knowing the contribution from each pair of plane wave components is sufficient to determine the total contribution from an arbitrarily shaped incident wave field. This is important in scanning transmission electron microscopy where the incident convergent probe contains a multitude of different plane wave components [20], [21].

Rose and co-workers generalised the work in Ref. [1] in terms of a mutual coherence function to also encapsulate incoherence in the probe electrons, in particular temporal incoherence [22]. They elucidated the basic governing equation for the mutual coherence function and discussed its solution via a generalised multislice formulation based on four dimensional propagators. Though the mutual coherence formalism has been used to gain conceptual insight into coherence in inelastic scattering (e.g. [23]), such an approach, involving four dimensional Fourier transforms, is demanding of computational resources – both in terms of memory and processing power. As we shall see in what follows, much of the intervening development has focused on ways of making calculations sufficiently tractable so that they may routinely be used to analyze experimental images.

Fundamental aspects of inelastic scattering in solids were addressed over a bit more than a decade, starting in the first half of the 1980s, by several authors [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. In particular, Dudarev et al. proceeded on the basis of a one particle density matrix, related to the mixed dynamic form factor, and its accompanying governing equation, the so-called kinetic equation [36]. Building on some of the earlier work, Allen and Josefsson presented a comprehensive theory of inelastic scattering, implicitly incorporating the mixed dynamic form factor, in a formulation based on Bloch waves [37]. The physical significance of the mixed dynamic form factor and its relation to the density matrix was comprehensively investigated by Schattschneider and co-workers [38], [39]. How the mixed dynamic form factor arises and its role in inelastic scattering will be discussed in Section 2.

Section snippets

The mixed dynamic form factor

In this section we will show how the mixed dynamic form factor arises. Let us take as our starting point the following equation describing an inelastic scattering event that occurs at a specific depth zi into the specimen [40], [41], measured from the entrance surface of the specimen and along the optical axis:ψn(P,r,zi)=iσnHn0(r,zi)ψ0(P,r,zi).The probe wave function ψ0 at the depth zi depends on the co-ordinate r in a plane perpendicular to the optical axis. The functional dependence denoted

Thermal diffuse scattering

A beam of fast electrons incident on a crystal produces a diffraction pattern which exhibits several well-known features including Bragg peaks, a diffuse background, higher-order-Laue-zone rings and Kikuchi bands [54]. Phonon excitation (thermal scattering) makes an important contribution to many of these features, in particular the diffuse background and Kickuchi lines [55]. Thermal scattering also makes the essential contribution to high-angle annular dark field measurements in scanning

Inner-shell ionisation

Two important modes of imaging are based upon the inelastic scattering associated with ionisation within the specimen. Elemental mapping in two dimensions at atomic resolution using electron energy-loss spectroscopy (EELS) based on inner-shell ionisation has evolved since it was demonstrated in 2007 [9], [10], [11], [12], [74] and is now at the point where it can be used to solve problems of technological interest [14]. However, unless the detector collection angle is very large, EELS is a

Software to model imaging based on inelastic scattering

Software has been developed to simulate inelastic scattering using some of the ideas outlined in this paper and applications made by several authors. The book by Kirkland [64] can be consulted for details of simulations using the frozen phonon model and a small selection of other relevant papers are Refs. [41], [79], [80], [20], [81].

As already pointed out, propagators in 4D are numerically intensive, both in terms of memory requirements and in terms of processing power. Propagating the

Summary and conclusions

Starting with the early pioneering work of Harald Rose, we have reviewed progress in the imaging at atomic resolution based on the inelastic scattering of electrons over the last three decades, in particular the theoretical aspects. Applications to high-angle annular dark-field imaging and elemental mapping using electron energy loss spectroscopy or energy dispersive x-ray analysis have been discussed in some depth. The software package μSTEM 2.0, capable of simulating various imaging modes

Acknowledgements

This research was supported under the Australian Research Councils Discovery Projects funding scheme (Projects DP110101570 and DP110102228) and its DECRA funding scheme (Project DE130100739). The authors acknowledge the important contribution of our collaborators and colleagues in the electron microscopy community to this work. We would like to acknowledge the crucial contributions of Mark Oxley and Chris Rossouw to the theoretical and numerical aspects of the work discussed here. Eireann

References (87)

  • N.R. Lugg et al.

    Energy-filtered transmission electron microscopy based on inner-shell ionization

    Ultramicroscopy

    (2010)
  • C. Dwyer et al.

    Scattering of Å-scale electron probes in silicon

    Ultramicroscopy

    (2003)
  • S.J. Pennycook et al.

    High-resolution Z-contrast imaging of crystals

    Ultramicroscopy

    (1991)
  • C.B. Boothroyd et al.

    Contribution of phonon scattering to high-resolution images measured by off-axis holography

    Ultramicroscopy

    (2004)
  • R.F. Loane et al.

    Incoherent imaging of zone axis crystals with ADF STEM

    Ultramicroscopy

    (1992)
  • S.D. Findlay et al.

    Modelling imaging based on core-loss spectroscopy in scanning transmission electron microscopy

    Ultramicroscopy

    (2005)
  • H.L. Xin et al.

    Is there a Stobbs factor in atomic-resolution STEM-EELS mapping?

    Ultramicroscopy

    (2014)
  • D. Van Dyck

    Is the frozen phonon model adequate to describe inelastic phonon scattering?

    Ultramicroscopy

    (2009)
  • C. Dwyer

    Atomic-resolution core-level spectroscopy in the scanning transmission electron microscope

    Adv. Imaging Electron Phys.

    (2013)
  • K.J. Dudeck et al.

    Quantitative statistical analysis, optimization and noise reduction of atomic resolved electron energy loss spectrum images

    Micron

    (2012)
  • C. Dwyer

    Simulation of scanning transmission electron microscope images on desktop computers

    Ultramicroscopy

    (2010)
  • S. Löffler et al.

    A pure state decomposition of the mixed dynamic form factor for mapping atomic orbitals

    Ultramicroscopy

    (2013)
  • J. Verbeeck et al.

    Image simulation of high resolution energy filtered TEM images

    Ultramicroscopy

    (2009)
  • A.J. D׳Alfonso et al.

    Volcano structure in atomic resolution core-loss images

    Ultramicroscopy

    (2008)
  • N.D. Browning et al.

    Atomic-resolution chemical analysis using a scanning transmission electron microscope

    Nature

    (1993)
  • N.D. Browning et al.

    Corrigendum

    Nature

    (2006)
  • D.A. Muller et al.

    Mapping sp2 and sp3 states of carbon at sub-nanometre spatial resolution

    Nature

    (1993)
  • P.E. Batson

    Simultaneous STEM imaging and electron energy-loss spectroscopy with atomic-column sensitivity

    Nature

    (1993)
  • K. Suenaga et al.

    Element-selective single atom imaging

    Science

    (2000)
  • L.J. Allen et al.

    Atomic-resolution electron energy loss spectroscopy imaging in aberration corrected scanning transmission electron microscopy

    Phys. Rev. Lett.

    (2003)
  • M. Varela et al.

    Spectroscopic imaging of single atoms within a bulk solid

    Phys. Rev. Lett.

    (2004)
  • M. Bosman et al.

    Two-dimensional mapping of chemical information at atomic resolution

    Phys. Rev. Lett.

    (2007)
  • K. Kimoto et al.

    Element-selective imaging of atomic columns in a crystal using STEM and EELS

    Nature

    (2007)
  • D.A. Muller et al.

    Atomic-scale chemical imaging of composition and bonding by aberration-corrected microscopy

    Science

    (2008)
  • L.J. Allen

    New directions for chemical maps

    Nat. Nanotechnol.

    (2008)
  • O.L. Krivanek, N. Dellby, M.F. Murfitt, M.F. Chisholm, T.J. Pennycook, S. Kazutomo, N.V., Gentle STEM: ADF imaging and...
  • S. Trasobares et al.

    Chemical imaging at atomic resolution as a technique to refine the local structure of nanocrystals

    Angew. Chem. Int. Edition

    (2011)
  • A.J. D׳Alfonso et al.

    Atomic resolution chemical mapping using energy dispersive x-ray spectroscopy

    Phys. Rev. B

    (2010)
  • M.W. Chu et al.

    Emergent chemical mapping at atomic-column resolution by energy-dispersive x-ray spectroscopy in an aberration-corrected electron microscope

    Phys. Rev. Lett.

    (2010)
  • L.J. Allen et al.

    Chemical mapping at atomic resolution using energy-dispersive x-ray spectroscopy

    MRS Bull.

    (2012)
  • G. Kothleitner et al.

    Quantitative elemental mapping at atomic resolution using x-ray spectroscopy

    Phys. Rev. Lett.

    (2014)
  • H. Rose

    Image formation by inelastically scattered electrons in electron microscopy

    Optik

    (1976)
  • H. Müller et al.

    A coherence function approach to image simulation

    J. Microsc.

    (1998)
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