Modelling the inelastic scattering of fast electrons
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: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
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