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Principles of Depth-Resolved Kikuchi Pattern Simulation for Electron Backscatter Diffraction

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Principles of Depth-Resolved Kikuchi Pattern Simulation for Electron Backscatter Diffraction Journal of Microscopy, Vol. 239, Pt 1 2010, pp. 32–45 doi: 10.1111/j.1365-2818.2009.03353.x Received 1 April 2009; accepted 29 October 2009 Principles of depth-resolved Kikuchi pattern simulation for electron backscatter diffraction A. WINKELMANN Max-Planck-Institut fur¨ Mikrostrukturphysik, Halle (Saale), Germany Key words. Electron backscatter diffraction, Kikuchi pattern, convergent beam electron diffraction, dynamical electron diffraction Summary created by independent sources emitting divergent electron waves from within the crystal (Cowley, 1995). Kikuchi This paper presents a tutorial discussion of the principles patterns also appear in the scanning electron microscope underlying the depth-dependent Kikuchi pattern formation of when the angular distribution of backscattered electrons backscattered electrons in the scanning electron microscope. is imaged. Around this principle, the method of electron To illustrate the connections between various electron backscatter diffraction (EBSD) has been developed (Schwarzer, diffractionmethods,theformationofKikuchibandsinelectron 1997; Wilkinson & Hirsch, 1997; Schwartz et al., 2000; backscatter diffraction in the scanning electron microscope Dingley, 2004; Randle, 2008). Because the Kikuchi patterns and in transmission electron microscopy are compared are tied to the local crystallographic structure in the probe with the help of simulations employing the dynamical area of the electron beam, EBSD can provide important theory of electron diffraction. The close relationship between crystallographicandphaseinformationdowntothenanoscale backscattered electron diffraction and convergent beam in materials science (Small & Michael, 2001; Small et al., electron diffraction is illuminated by showing how both effects 2002). The success of EBSD stems from the fact that the can be calculated within the same theoretical framework. method is conceptually simple: in principle only a phosphor The influence of the depth-dependence of diffuse electron screen imaged by a sensitive CCD camera is needed. Also, scattering on the formation of the experimentally observed the geometry of the Kikuchi line patterns can be explained electron backscatter diffraction contrast and intensity relatively simply by tracing out the Bragg reflection conditions is visualized by calculations of depth-resolved Kikuchi for a point source inside a crystal (Gajdardziska-Josifovska & patterns. Comparison of an experimental electron backscatter Cowley, 1991). In principle, by such a procedure, a network diffraction pattern with simulations assuming several different of interference cones perpendicular to reflecting lattice planes depth distributions shows that the depth-distribution of and with opening angles determined from the respective backscattered electrons needs to be taken into account in Bragg angles can be projected onto the observation plane to quantitative descriptions. This should make it possible to analyse the crystallographic orientation of a sample grain. obtain more quantitative depth-dependent information from However, this does not give direct information on the observed experimental electron backscatter diffraction patterns via intensities,sinceaquantitativecalculationofthebackscattered correlation with dynamical diffraction simulations and Monte diffraction pattern needs to use the dynamical theory of Carlo models of electron scattering. electron diffraction that takes into account the localization of the backscattering process of electrons in the crystal unit cell. The author has recently been able to show (Winkelmann Introduction et al., 2007; Winkelmann, 2008) that Kikuchi patterns in backscattered electrons in the scanning electron microscope One of the most beautiful phenomena in electron diffraction can be successfully calculated using a Bloch-wave approach is the appearance of Kikuchi patterns formed by electrons that is usually applied for convergent beam electron diffraction scattered by a crystalline sample (Kikuchi, 1928; Nishikawa (CBED) in the transmission electron microscope. Instead of & Kikuchi, 1928; Alam et al., 1954). These patterns exist as divergent sources internal to the crystal, CBED patterns are a network of lines and bands and can be thought of as being formed by an external convergent probe sampling the same Correspondence to: Aimo Winkelmann, Max-Planck-Institut fur¨ Bragg interference cones as the internal sources, and thus Mikrostrukturphysik Weinberg 2, D-06120 Halle (Saale), Germany. Tel: +49 345 the CBED patterns show line patterns of similar geometry 5582 639; fax: +49 345 5511 223; e-mail: [email protected] to EBSD and other Kikuchi patterns. However, the intensity C 2009 The Author Journal compilation C 2009 The Royal Microscopical Society KIKUCHI PATTERN SIMULATION FOR EBSD 33 distributions in Kikuchi patterns and in CBED patterns are characteristic influence of the assumed depth distribution of qualitatively different, because CBED patterns are ideally the diffracted backscattered electrons on the dynamical EBSD formed by only those electrons which retain a fixed phase with patterns can be clearly sensed. respect to the incident beam, whereas the Kikuchi patterns are formedbyindependentsourceslargelyincoherentwithrespect to the primary beam. Theoretical background The main purpose of this paper is to explain in detail how the two types of problems are connected. Especially it will The fundamental building block of our understanding of be shown how the dynamical diffraction from completely Kikuchi pattern formation will be the prototypical example incoherent point sources (relevant to EBSD) can be treated of transmission electron diffraction: the dynamical diffraction in exactly the same formalism as the dynamical diffraction in ofanincidentplanewavebeambyathincrystalsample,which CBED. Close attention is paid to the rather different roles of leads to the formation of a transmitted discrete spot diffraction the thickness parameter in coherent and localized incoherent pattern. For perfect crystals, the Bloch-wave approach is a scattering, because from many investigations in transmission method often used to describe this process. For the purposes electron microscopy it is known that the observed Kikuchi of this paper, we actually do not need to understand the pattern contrast is strongly depending on the sample thickness mathematical details of this method. We will simply assume (Pfister, 1953; Reimer & Kohl, 2008). The previous theoretical that we have a working method at hand to calculate from a investigations of dynamical EBSD simulations (Winkelmann given crystal structure and from the incident beam direction et al., 2007; Winkelmann, 2008) in a first approximation were and energy the electron wave field inside the sample and the neglecting some specific details of the backscattered electron transmitted diffraction pattern. The Bloch-wave approach has depth distribution and assumed that the backscattered been shown to lead to very convincing agreement between electrons were produced with equal intensity in a layer of calculated and measured electron backscatter diffraction limited thickness near the surface, an approximation leading patterns (Winkelmann et al., 2007; Day, 2008; Maurice & to good agreement with a number of experimentally observed Fortunier, 2008; Winkelmann, 2008; Villert et al., 2009). EBSD patterns. Based on observations of the width of measured The same approach is used for quantitative convergent beam diffraction lines, the energy spread and correspondingly electron diffraction (Spence & Zuo, 1992) and thus we have the related depth sensitivity of electrons contributing to an a consistent framework to describe Kikuchi pattern formation EBSD pattern can be estimated. The depth sensitivity of in relation to the coherent elastic diffraction. EBSD is generally assumed to be in the range between 10 The main idea behind the Bloch-wave approach can be and 40 nm at 20 kV, with the lower values reached for summarized in a very compact way by noting that it seeks denser materials (Dingley, 2004). Experimental observations the wave function in a specific form. This form is known of the disappearance of Kikuchi pattern diffraction contrast from Bloch’s theorem for a translationally invariant scattering when depositing amorphous layers on crystalline samples are potential (Humphreys, 1979): consistent with this estimation (Yamamoto, 1977; Zaefferer, 2007). It is clearly an important question how the depth = π ( j) · ( j) π · (r) c j exp[2 ik r] C g exp[2 ig r](1) distribution of the backscattered electrons is quantitatively j g influencing the EBSD patterns. The inclusion of the relevant effects in dynamical simulations could possibly allow to The Bloch-wave calculation then finds the coefficients (j) (j) extract additional information from experimental EBSD cj,C g , and the vectors k by solving a matrix eigenvalue measurements. This is why we will analyse in detail how the problem derived from the Schrodinger¨ equation by limiting the depth distribution of the backscattered and diffracted electrons wave-function expansion to a number of Fourier coefficients is affecting the observed Kikuchi patterns in dynamical EBSD labelled by the respective reciprocal lattice vectors g,eachof simulations. which couples the incident beam to a diffracted beam. The The paper is structured as follows. First, the theoretical eigenvalues λ(j) appear when the Bloch-wave vector k(j) is framework is summarized, then the implications of coherence written as the
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