Principles of depth-resolved Kikuchi pattern simulation for electron backscatter diffraction

Transcription

1 Journal of Microscopy, Vol. 239, Pt , pp Received 1 April 2009; accepted 29 October 2009 doi: /j x Principles of depth-resolved Kikuchi pattern simulation for electron backscatter diffraction A. WINKELMANN Max-Planck-Institut für Mikrostrukturphysik, Halle (Saale), Germany Key words. Electron backscatter diffraction, Kikuchi pattern, convergent beam electron diffraction, dynamical electron diffraction Summary This paper presents a tutorial discussion of the principles underlying the depth-dependent Kikuchi pattern formation of backscattered electrons in the scanning electron microscope. To illustrate the connections between various electron diffraction methods, the formation of Kikuchi bands in electron backscatter diffraction in the scanning electron microscope and in transmission electron microscopy are compared with the help of simulations employing the dynamical theory of electron diffraction. The close relationship between backscattered electron diffraction and convergent beam electron diffraction is illuminated by showing how both effects can be calculated within the same theoretical framework. The influence of the depth-dependence of diffuse electron scattering on the formation of the experimentally observed electron backscatter diffraction contrast and intensity is visualized by calculations of depth-resolved Kikuchi patterns. Comparison of an experimental electron backscatter diffraction pattern with simulations assuming several different depth distributions shows that the depth-distribution of backscattered electrons needs to be taken into account in quantitative descriptions. This should make it possible to obtain more quantitative depth-dependent information from experimental electron backscatter diffraction patterns via correlation with dynamical diffraction simulations and Monte Carlo models of electron scattering. Introduction One of the most beautiful phenomena in electron diffraction is the appearance of Kikuchi patterns formed by electrons scattered by a crystalline sample (Kikuchi, 1928; Nishikawa & Kikuchi, 1928; Alam et al., 1954). These patterns exist as a network of lines and bands and can be thought of as being Correspondence to: Aimo Winkelmann, Max-Planck-Institut für Mikrostrukturphysik Weinberg 2, D Halle (Saale), Germany. Tel: ; fax: ; winkelm@mpi-halle.mpg.de created by independent sources emitting divergent electron waves from within the crystal (Cowley, 1995). Kikuchi patterns also appear in the scanning electron microscope when the angular distribution of backscattered electrons is imaged. Around this principle, the method of electron backscatter diffraction (EBSD) has been developed (Schwarzer, 1997; Wilkinson & Hirsch, 1997; Schwartz et al., 2000; Dingley, 2004; Randle, 2008). Because the Kikuchi patterns are tied to the local crystallographic structure in the probe area of the electron beam, EBSD can provide important crystallographic and phase information down to the nanoscale in materials science (Small & Michael, 2001; Small et al., 2002). The success of EBSD stems from the fact that the method is conceptually simple: in principle only a phosphor screen imaged by a sensitive CCD camera is needed. Also, the geometry of the Kikuchi line patterns can be explained relatively simply by tracing out the Bragg reflection conditions for a point source inside a crystal (Gajdardziska-Josifovska & Cowley, 1991). In principle, by such a procedure, a network of interference cones perpendicular to reflecting lattice planes and with opening angles determined from the respective Bragg angles can be projected onto the observation plane to analyse the crystallographic orientation of a sample grain. However, this does not give direct information on the observed intensities, since a quantitative calculation of the backscattered diffraction pattern needs to use the dynamical theory of 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 et al., 2007; Winkelmann, 2008) that Kikuchi patterns in backscattered electrons in the scanning electron microscope can be successfully calculated using a Bloch-wave approach that is usually applied for convergent beam electron diffraction (CBED) in the transmission electron microscope. Instead of divergent sources internal to the crystal, CBED patterns are formed by an external convergent probe sampling the same Bragg interference cones as the internal sources, and thus the CBED patterns show line patterns of similar geometry to EBSD and other Kikuchi patterns. However, the intensity Journal compilation C 2009 The Royal Microscopical Society

2 KIKUCHI PATTERN SIMULATION FOR EBSD 33 distributions in Kikuchi patterns and in CBED patterns are qualitatively different, because CBED patterns are ideally formed by only those electrons which retain a fixed phase with respect to the incident beam, whereas the Kikuchi patterns are formed by independent sources largely incoherent with respect to the primary beam. The main purpose of this paper is to explain in detail how the two types of problems are connected. Especially it will be shown how the dynamical diffraction from completely incoherent point sources (relevant to EBSD) can be treated in exactly the same formalism as the dynamical diffraction in CBED. Close attention is paid to the rather different roles of the thickness parameter in coherent and localized incoherent scattering, because from many investigations in transmission electron microscopy it is known that the observed Kikuchi pattern contrast is strongly depending on the sample thickness (Pfister, 1953; Reimer & Kohl, 2008). The previous theoretical investigations of dynamical EBSD simulations (Winkelmann et al., 2007; Winkelmann, 2008) in a first approximation were neglecting some specific details of the backscattered electron depth distribution and assumed that the backscattered electrons were produced with equal intensity in a layer of limited thickness near the surface, an approximation leading to good agreement with a number of experimentally observed EBSD patterns. Based on observations of the width of measured diffraction lines, the energy spread and correspondingly the related depth sensitivity of electrons contributing to an EBSD pattern can be estimated. The depth sensitivity of EBSD is generally assumed to be in the range between 10 and 40 nm at 20 kv, with the lower values reached for denser materials (Dingley, 2004). Experimental observations of the disappearance of Kikuchi pattern diffraction contrast when depositing amorphous layers on crystalline samples are consistent with this estimation (Yamamoto, 1977; Zaefferer, 2007). It is clearly an important question how the depth distribution of the backscattered electrons is quantitatively influencing the EBSD patterns. The inclusion of the relevant effects in dynamical simulations could possibly allow to extract additional information from experimental EBSD measurements. This is why we will analyse in detail how the depth distribution of the backscattered and diffracted electrons is affecting the observed Kikuchi patterns in dynamical EBSD simulations. The paper is structured as follows. First, the theoretical framework is summarized, then the implications of coherence and the treatment of incoherent scattering in electron diffraction techniques are discussed, including the role of the thickness parameter. The unifying concepts are illustrated by dynamical model simulations, which are carried out with the same formalism and computer program simultaneously for both coherent CBED patterns and incoherent Kikuchi patterns for molybdenum at 20 kv beam energy. Finally, an experimental EBSD pattern from a Mo single crystal is compared to full-scale dynamical simulations, where the characteristic influence of the assumed depth distribution of the diffracted backscattered electrons on the dynamical EBSD patterns can be clearly sensed. Theoretical background The fundamental building block of our understanding of Kikuchi pattern formation will be the prototypical example of transmission electron diffraction: the dynamical diffraction of an incident plane wave beam by a thin crystal sample, which leads to the formation of a transmitted discrete spot diffraction pattern. For perfect crystals, the Bloch-wave approach is a method often used to describe this process. For the purposes of this paper, we actually do not need to understand the mathematical details of this method. We will simply assume that we have a working method at hand to calculate from a given crystal structure and from the incident beam direction and energy the electron wave field inside the sample and the transmitted diffraction pattern. The Bloch-wave approach has been shown to lead to very convincing agreement between calculated and measured electron backscatter diffraction patterns (Winkelmann et al., 2007; Day, 2008; Maurice & Fortunier, 2008; Winkelmann, 2008; Villert et al., 2009). The same approach is used for quantitative convergent beam electron diffraction (Spence & Zuo, 1992) and thus we have a consistent framework to describe Kikuchi pattern formation in relation to the coherent elastic diffraction. The main idea behind the Bloch-wave approach can be summarized in a very compact way by noting that it seeks the wave function in a specific form. This form is known from Bloch s theorem for a translationally invariant scattering potential (Humphreys, 1979): (r) = j c j exp[2πik ( j) r] g C ( j) g exp[2πig r] (1) The Bloch-wave calculation then finds the coefficients c j,c (j) g, and the vectors k (j) by solving a matrix eigenvalue problem derived from the Schrödinger equation by limiting the wave-function expansion to a number of Fourier coefficients labelled by the respective reciprocal lattice vectors g, eachof which couples the incident beam to a diffracted beam. The eigenvalues λ (j) appear when the Bloch-wave vector k (j) is written as the sum of the incident beam wave vector K in the crystal and a surface normal component as k (j) = K + λ (j) n. The eigenvalue λ (j) is complex in the general case. The reader can safely assume that a black box Bloch-wave simulation gives us the unknown parameters defining the wave function in Eq. (1) for a given incident beam direction K and electron acceleration voltage. The wave function (1) can be rearranged to show that it can be seen as the sum of contributions of plane waves exp[2πi(k + g) r] moving into directions K + g and having

3 34 A. WINKELMANN a depth dependent amplitude φ g (t): (r) = φ g (t) exp[2πi(k + g) r] g = { } C g ( j) c j exp[2πiλ ( j) n r] exp[2πi(k + g) r] g j (2) Here,wenotethatn r = t is the surface normal component (depth) of the point r,witht = 0 at the entrance surface. If all the wave-function parameters are known after solving the eigenvalue problem for the complex λ (j), we can in principle calculate the intensity that is moving in the plane-wave beams diffracted into the directions K + g after transmission through a crystal of thickness t: I g (K, t) = j,l c j cl C g ( j) C g (l) exp [ 2πi(λ ( j) λ (l) )t ] (3) and the wave function also gives the probability density P(r) = (r) (r) at every point inside the crystal by straightforward application of Eq. (2): P(r) = g,h j,l c j cl C g ( j) C (l) h exp [ 2πi(λ ( j) λ (l) )t ] exp[2πi(g h) r] (4) Please note that Eq. (2) contains plane waves in directions K + g only. These correspond to the diffracted beams that form a spot diffraction pattern. By itself, Eq. (2) does not provide an explicit mechanism by which inelastically scattered waves can appear in a general direction K + k. It is important to realize that the conventional procedure of introducing an imaginary potential to account for electron absorption handles only the reduction of the beam intensities in the limited set of directions K + g (the diffraction spots), although it does not describe the details of the redistribution of this intensity into all the other directions K + k (the initially black space between the diffraction spots). This redistribution, however, is fundamental in the formation of the diffuse scattering patterns, as the scattered electrons can reappear with a changed energy in a different direction. While the coherent elastic scattering from a periodic crystal allows a change from the primary beam wave vector K only by discrete reciprocal lattice vectors g, inelastic or diffuse incoherent scattering, symbolized by an operator Ŝ k g, allows in addition to the limited discrete set of waves with wave vectors K + g scattered waves K + k moving into arbitrary directions with in principle any k, thus producing a continuous background in addition to discrete diffraction spots. The exact details of the various possible processes which here have been only very schematically symbolized by Ŝ k g are treated by explicit dynamical theories of Kikuchi band formation (Kainuma, 1955; Chukhovskii et al., 1973; Rez et al., 1977; Dudarev et al., 1995), and include, for example, the description of phonon, plasmon and core-electron excitations. An explicit recent treatment of Kikuchi pattern formation, including a discussion of various approximations and simulations for high-energy transmission electron microscopy can be found in (Omoto et al., 2002). The exact modelling of the inelastic scattering has important implications for quantitative structure analysis based on experimental diffraction patterns, since the problem of coherent and incoherent scattering must be treated on the same level (Wang, 1995; Peng et al., 2004). In the discussion later, we will analyse the implications of the extreme cases of complete coherence or incoherence with respect to the incident beam. For the simulation of EBSD patterns in this paper, we will assume that practically all intensity is incoherently scattered from all the crystal atoms and isotropically emitted into all directions. What results is a collection of independent point sources in crystalline order, and the diffraction of the spherical waves emanating into all directions from these sources produces the Kikuchi patterns. This model will allow us to analyse the fundamental dynamical diffraction physics behind thicknessdependent Kikuchi pattern formation, although neglecting the exact details of inelastic scattering (most importantly, the incoherently and inelastically scattered electrons are not distributed isotropically in reality but are dominantly scattered in the forward direction). It will be shown in this paper how in principle any method that describes the scattering of a single plane wave by a crystal into a set of diffracted beams can be turned into a method for the calculation of Kikuchi patterns under the earlier assumptions. Principle of calculation for incoherent point sources We will now discuss the close connection between the spot diffraction pattern of a transmitted beam and the Kikuchi pattern from incoherent point sources. In Fig. 1(a), we show symbolically how the spot diffraction pattern (e.g. in microdiffraction in the transmission electron microscope) of transmitted beams is formed. The incident beam enters from the top side of the sample which is assumed to be a perfect crystal of constant thickness. A part of the incident electron plane wave is then scattered coherently by all atoms in the interaction volume, meaning that the waves emanating from the scattering centres all have a perfectly known fixed phase relationship with respect to the incident beam and thus with respect to each other. Wave theory tells us then that there will be well-defined constructive and destructive interference between the waves coming from different scatterers. It turns out that only in directions corresponding to wave vector changes equal to reciprocal lattice vectors will we have resulting intensity in the form of a discrete spot pattern, whereas destructive interference prohibits electrons from going into all the other scattering directions. The presence of the discrete spot diffraction pattern is shown in the lower part of Fig. 1(a).

4 KIKUCHI PATTERN SIMULATION FOR EBSD 35 P EBSD: only Kikuchi S P* THEE Kikuchi "+" HEED Kikuchi Fig. 1. (a) Coherent scattering of an incident beam (plane wave P) in transmission high energy electron diffraction (THEED). This can lead, for example, to a microdiffraction pattern or to convergent beam electron diffraction (CBED) disks if the incident beam is convergent. (b) Incoherent emission from point sources S. Only a single point source is shown, but emission proceeds independently from various possible sites. Continuous Kikuchi pattern intensities are observed. (c) Time reversed process of (b) showing that the Kikuchi pattern can be in principle simulated with any method that is able to handle the problem (a), with the modification that we have to calculate the intensity at the point S inside the crystal and not the transmitted diffraction pattern which is put in parentheses. (d) Combination of the effects (a) and (b) in a real experiment. The transmitted pattern now shows contributions of both the discrete diffraction of the incident beam, as well as continuous intensity from incoherent sources. To describe this combined pattern, a general treatment of the coupling of (a) to (b) by scattering processes is necessary. In the case of EBSD (upper part), a much simpler situation is present if all electrons have lost coherence with the incident beam, making possible a separate treatment of (b) only. In reality, not all the scattering from the atoms will be coherent as indicated in Fig. 1(a). Instead, the electron waves can experience unknown, more or less random, phase changes under scattering. Already in elastic backscattering, the electrons transfer recoil energy and momentum to the target atoms (Boersch et al., 1967; Went & Vos, 2008), and if the corresponding atomic displacement is of the order of the incident electron wavelength this will lead to the effect that the phases of the backscattered waves are not perfectly locked to each other. The discrete diffraction features disappear because the interference conditions are not spatially fixed anymore. In effect, each scattering atom scatters independently of the others and contributes a continuous source intensity in all directions depending on its differential scattering cross section. If this incoherent scattering on average would take place homogeneously at all places in the crystal, a continuous background results, reflecting the atomic cross sections for different scattering processes and the multiple inelastic and elastic scattering in all directions. By contrast, if the incoherent scattering remains concentrated at specific sites in the unit cell, we are in a situation, shown in Fig. 1(b), which does not look so different from Fig. 1(a): a single spherical wave is starting from a point source S located at an atomic position. After the phasebreaking incident, this spherical wave has lost memory of the phase of the incident parent plane wave P, so it cannot act in concert with all the waves from the other atoms anymore. But as an individual spherical wave, it is perfectly coherent as it contains well-defined phase relationships (the surfaces of equal phase are spheres). This means that each of these independent waves will separately exhibit interference effects on its way out of the crystal when it is elastically scattered by the surrounding atoms. In effect, the single incident plane wave P from infinity (Fig. 1a) is replaced by a spherical wave from the inside of the crystal (Fig. 1b), which can be thought of as a superposition of an infinite number plane waves going into all directions from S. If we know how to treat the diffraction of a single plane wave, we can in principle treat the diffraction of a combination of them. But it looks as though we have a much more complicated problem to solve: a single initial plane wave from P versus a huge number of initial plane waves starting from S that are diffracted. However, a major simplification arises if we take into account how we detect the diffraction pattern: the electron intensity is detected basically at an infinite distance away from the sample, where the electron wave travelling in a specific direction (corresponding to a point on our phosphor screen) can be assumed to be a plane wave. For the problem of the point source S, we see that we actually do not need to know how much intensity is going into to all directions at the same time, we only need to know how much is intensity is finally ending up in the plane wave component that is going towards our specified point on the screen. Of course, if we start our waves from S, there is no way of knowing beforehand which part of these is ending up in the detection direction, because the outgoing waves are scattered elastically multiple times in all possible directions.

5 36 A. WINKELMANN However, instead of going from the source to the detector, we can apply the powerful reciprocity theorem (Pogany & Turner, 1968) and go backwards in time from our detector to the source: turn around our final plane wave, let it propagate from the detection point towards the crystal and look how this singleplanewaveisdiffractedbythecrystalandhowmuchofit finally ends up at the source. Instead of keeping track of all the waves in all possible directions from S, we are dealing only with the waves that are important for our detected direction, and we can do this with exactly the same theory that we use for a single plane wave hitting a crystal from infinity. This is shown in Fig. 1(c), where an arrow represents the time-reversed plane wave P travelling backwards from a specified direction on the screen. The time-reversed wave is in the same situation as the forward travelling wave P in Fig. 1(a), the only difference being that in Fig. 1(c) we are interested in the diffracted wave functionatpointsinthecrystalandwearenotinterestedinthe transmitted diffraction pattern that is described by exactly the same formalism. Both types of information are simultaneously contained in the wave function (2), resulting in Eq. (3) that describes the diffraction pattern, whereas Eq. (4) describes the diffracted electron wave field inside the crystal, resulting from the dynamical interaction of a plane wave incident from infinity. We stress that any energy change of the wave originating from S which might happen due to inelastic scattering is not a necessary difference between the situations of Fig. 1(a) and (b): the loss of a fixed phase with respect to the incident beam is the defining characteristic. In reality, of course, inelastic scattering does generally change the energy of the waves emitted from all the possible placess in the crystal, and thus the corresponding change in wavelength will need to be taken into account. If the inelastic sources are completely independent, this can be achieved by carrying out the procedure of Fig. 1(c) for the whole spectrum of electron kinetic energies that are picked up by the detector. Finally, we can summarize this section by pointing to the general situation depicted in Fig. 1(d). In the real experiment of a beam transmitted through a thin enough sample, we will see a discrete diffraction pattern formed by the coherent scattering of the crystal, combined with a background due to elastic scattering starting from incoherent point sources and additionally a background from non-localized inelastic scattering effects. The exact treatment of coherent and incoherent, elastic and inelastic scattering is the most general and most difficult problem in electron diffraction (Peng et al., 2004), and its solution certainly is not attempted here. However, by help of Fig. 1(d), we can argue that the degree of difficulty of treating EBSD dynamically is considerably reduced compared to the most general situation of the combined diffraction problem of waves that are in varying degrees coherent to the incident beam. In EBSD, the number of electrons scattered coherently with respect to the incident beam is usually negligible, and the corresponding spot patterns are not observable. This is why in EBSD, it turns out to be a good approximation to treat the scattering from incoherent point sources only and neglect the coherent diffraction from the incident beam. In Fig. 1(d), this is symbolized on the top side of the sample, corresponding to a backscattering geometry. Summarizing this section, we have seen how one of the standard problems of transmission electron diffraction, the diffraction of a plane wave incident on a crystal, can be viewed by reciprocity as providing also the intensity from incoherent point sources localized inside the crystal. In the next section, we will demonstrate by explicit dynamical simulations how the depth distribution of inelastic scattering is manifesting itself in the intensity distribution and contrast of Kikuchi bands. Diffusely scattered electrons and their depth distribution The contrast in Kikuchi patterns which are observed in transmission electron microscope investigations depends on the thickness of the sample (Uyeda & Nonoyama, 1967, 1968; Uyeda, 1968; Reimer & Kohl, 2008 Fig. 7.26, p. 324). For a thin sample, the transmitted Kikuchi bands are high in intensity for angles smaller than the Bragg angle away from the relevant lattice plane (the middle of the Kikuchi band). With increasing thickness of the sample, the bands become dark in the middle. Already at this point, we note that the contrast for thin samples in transmission is the same as is usually observed under standard EBSD conditions: increased intensity in the middle of the band. This type of contrast can be explained by the fact that backscattering takes places near the atomic positions. For angles smaller than the Bragg angle, those Bloch waves dominate which are located at the atomic positions (type I waves), thus providing an efficient transfer channel for the backscattered electrons. At angles larger than the Bragg angle, the Bloch waves are located between the atomic planes (type II waves), and the backscattered electrons cannot couple efficiently to the outgoing plane wave. This effect can be visualized in real space (Winkelmann, 2009), and is at work in several diffraction techniques based on localized emitters (Winkelmann et al., 2008). With increasing depth, the effect of anomalous absorption takes over, because electrons that move along the atomic planes in type I Bloch waves are also inelastically scattered more often and thus absorbed more efficiently. The maximum intensity in the middle of a band turns into a minimum for thicker samples in transmission. The electrons moving between the atomic planes are the only ones that survive beyond a certain thickness and these electrons are found in the type II Bloch wave which is excited at the outer edges of a Kikuchi band. In EBSD, this contrast reversal process is usually not observable, because the backscattered electrons from small depths dominate. In a simulation, however, the depth effects can be analysed by assuming artificial depth distributions of backscattered electrons as is shown later.

6 KIKUCHI PATTERN SIMULATION FOR EBSD 37 t a t 0 t 1 Fig. 2. Comparison of the different roles of the thickness parameter t in the calculation of a coherent THEED diffraction pattern as compared to a Kikuchi pattern calculation: (a) in coherent THEED, the sample induces boundary conditions at depths t 0 and t. (b) In Kikuchi patterns, contributions from sources in different depths have to be taken into account. The weight of each contribution (diameter of the dashed circles) is given by the number of incoherently scattered electrons at that depth. The purpose of the incident beam in (b) is in principle only to produce a depth distribution of incoherently scattered electrons, symbolized by the incident beam in parentheses. The role of the depth parameter The different role of the depth parameter in diffuse patterns as compared to the coherent pattern is illustrated by Fig. 2. In part (a) of Fig. 2, it is shown that the coherent pattern is formed by the elastic scattering of all atoms from the entrance plane at t 0 up to the exit plane at t. The sample is transforming the incident beam into a set of Bloch waves, according to the boundary conditions at t 0, and from the Bloch waves, the diffracted beams are formed again at t. By contrast, the diffuse pattern from independent point sources obviously depends on the depth of each individual source event beneath the exit surface, which is shown in part (b) of Fig. 2. In the experiment, we do not observe single electrons from a single event in a well-defined thickness but we collect all electrons from a range of depths. This is why we have to sum up all the diffuse patterns from all possible sources at different depths according to the respective probability of backscattering with a specified energy from a certain depth below the surface. This depth-distribution of inelastic electrons cannot simply be inferred from the backscattered electron spectrum in a direct way, but must be modelled analytically or obtained by Monte Carlo simulations (Werner, 2001). We will assume here that we know how many electrons with kinetic energy E are scattered from depth t below the surface. This depth distribution can assume nontrivial shapes, that is, it can have a maximum at a certain depth. This is shown in Fig. 2(b), where we draw the probability of backscattering from different depths t as circles of different diameters around the independent scattering atoms. In Fig. 2(b), the inelastic electron distribution has a maximum at the depth t 2 from the b t 3 t 2 exit surface of the transmitted beam. No matter whether we observe in a transmission or in a backscattering geometry, the multiple elastic and inelastic scattering of the incident beam leads to a limited interaction region which is located more or less near to the sample surface for bulk samples. This depth range is determined in a complex way by the elastic scattering cross sections and the possible energy dissipation processes (Reimer, 1998, Chapter 3). For samples with increasing thickness, one can thus see very pronounced changes in the Kikuchi patterns on the transmitted side of the sample because the scattering region will effectively recede deeper and deeper into the crystal and asymptotically disappear when no electrons are transmitted anymore. The situation on the entrance side, however, will simply stay nearly constant beyond a certain thickness which can be considered as the bulk limit. At this stage, it might seem that the calculation for the diffuse pattern will become increasingly complicated because we have to deal possibly with a huge number of scatterers at different positions below the surface. However, the calculation of our wave function (4) gives us the probability of going from point r in the crystal to the point on the observation screen, for all possible points r in the crystal in a single run. We thus do not have to do a separate dynamical calculation for each depth. Instead, we simply weight each depth according to the relative number of electrons it scatters diffusely. This requires a simple depth integration, and if the depth distribution is fitted to a parameterized function, we need only the integral of that function to analytically incorporate all sources. Depth-resolved model calculations To illustrate the main effects of depth-dependent Kikuchi band contrast, we will in the following apply a simple model assuming that the observed backscattered electrons are created by single incoherent scattering events. After these incoherent events at localized point sources, the spherical electron waves move through the crystal and are diffracted by the periodic part of the potential. The intensity variation in a Kikuchi band reflects the connection between the position of the localized scattering event within the crystal unit cell and the observed wave vector direction (i.e. a point on the phosphor screen). This can be visualized by explicit calculation of the probability density distribution within a unit cell for different positions along a Kikuchi band profile (Winkelmann, 2009). The excited Bloch waves at observation angles smaller than the Bragg angle for a relevant lattice plane sample the atomic planes, whereas the Bloch waves excited at angles larger than the Bragg angle sample the space between these planes. Multiple inelastic scattering of the emitted backscattered waves will now tend to produce additional incoherent sources which are partly concentrated at the atomic cores (e.g. phonon scattering), and partly distributed over the whole unit cell (e.g. delocalized plasmon scattering).

7 38 A. WINKELMANN Each of these additional sources can be thought of as producing an individual Kikuchi diffraction pattern, just as the initial spherical wave from the first incoherent backscattering event itself. The point sources between the atomic planes, however, will produce inverted Kikuchi bands as compared to the point sources on the atomic planes, via the Bloch-wave unit-cell sampling mechanism mentioned earlier. Thus, the intensity variation over a Kikuchi band is partially cancelled due to the redistribution of incoherent source positions within the unit cell by multiple inelastic scattering. This effectively results in a smooth background from a part of the electrons that are inelastically scattered on the way out of the crystal. Another part, which is effectively localized at the atomic positions, will contribute to the diffracted EBSD pattern as additional sources from different depths as compared to the initial backscattered wave. As sources in different depths can also show inverted Kikuchi band contrast with respect to each other, this is a further contribution to an effective decrease in intensity variation of Kikuchi bands. The multiple inelastic scattering and gradual loss of position specificness within the crystal unit cell is obviously a complicated process which cannot be treated in our simple approach without including the explicit properties of the scattering processes. Instead, the overall removal of electrons from the diffracted channel with travelled distance I diffraction (t) = I 0 exp( t/l IMFP ) is treated only in an average way by a constant imaginary part of the crystal potential V 0i, corresponding to an inelastic mean free path (IMFP) l IMFP = 2 E/2m e /ev 0i (Spence & Zuo, 1992). Because diffraction contrast can still be preserved after several inelastic scatterings, and EBSD is averaging over a large range of energy losses, the IMFP is only a lower-limit estimation of the path length after which diffraction contrast disappears in experiment. We calculate the diffraction patterns of a transmitted elastic beam and the pattern of diffusely scattered electrons at the same time. By cutting out specific slices from the crystal, we can then show how the diffraction pattern is changing with thickness. We stress here that we can use one and the same calculation to get both types of patterns: after solving the eigenvalue problem just as in a conventional CBED calculation (Spence & Zuo, 1992), we obtain the coherently transmitted intensity from Eq. (3) and the EBSD intensity from integrating Eq. (4) over the depth t (Rossouw et al., 1994). The results of corresponding simulations for a molybdenum sample and a 20 kev electron energy are shown in Fig. 3. In the top panel (a) of Fig. 3, we see the hypothetical transmission Fig. 3. Comparison of simulated, thickness-dependent transmission and EBSD patterns at 20 kev beam energy. The values at the lower right of each picture indicate the relative intensity for each pattern and (in arbitrary units, consistent separately within CBED and EBSD, see text). (Upper row) Simulated [001] bright-field large-angle CBED pattern from Mo samples of the indicated thickness. (Middle row) Backscattered EBSD patterns for the same samples as in the upper row. (Lower row) Depth-resolved EBSD patterns produced by slices of 1 nm thickness.

8 KIKUCHI PATTERN SIMULATION FOR EBSD 39 patterns of Mo samples which are 1-, 5-, 10- and 50-nm thick. This type of transmission pattern is usually called a Tanaka pattern or bright-field large-angle convergent-beam pattern (Eades, 1984; Morniroli, 2002). The patterns are shown in gnomonic projection, with the scattering angles reaching values from 45 to 45 in the maximum x-andy-directions, and the 111 directions are in the corners of the square areas shown. We begin by discussing the transmission simulation results in the upper row of Fig. 3. In agreement with expectations from dynamical theory, the transmission pattern for the thinnest (1 nm) sample is very unsharp. According to the twobeam approximation, the diffracted beam acquires the same intensity as the direct beam after a quarter of the extinction distance ξ which is determined by the Fourier amplitude of the relevant reflection (Hirsch et al., 1965). In the simulation, the extinction distances for the strongest reflections are 12 nm for the {110} and 16 nm for the {200} beams. Accordingly, it can be expected that a thickness of 3 4 nm is needed for the full development of dynamical effects in our considered sample. Consistent with this expectation, it is seen that the transmission patterns increase in sharpness up to 10 nm, with contributions from weaker reflections. Beyond this thickness, absorption effects take over. The electron waves travelling along the atomic planes are more effectively absorbed, whereas the waves between the atomic planes can still be transmitted. The first type of wave is excited near the diagonals of the pattern, which appear completely dark for the 50 nm sample. Symmetrically away from these diagonals, at angles which would correspond to angles slightly larger than the Bragg angle for the corresponding {110} reflections, we see that high-intensity remains. These electrons can go farthest through the crystal because they travel between the atoms and thus are absorbed less. The total intensity in the whole pattern is shown below each simulation in Fig. 3. A thickness of 0 nm would have intensity 1.0. As can be seen from Fig. 3, after 50 nm, intensity on the order of much less than a percent (0.002) remains. This is consistent with the imaginary part of the mean inner potential which corresponded to an inelastic mean free path of l IMFP = 8 nm assumed in the calculation (Powell & Jablonski, 1999). Except for the unusually low energy and very large angular extension of our simulated bright-field large-angle convergent-beam patterns, these simulations reproduce well known characteristic features of such measurements. Now we can directly compare what happens in the EBSD pattern of the same samples. These are shown in the middle panel (b), for exactly the same angular extension as the transmission patterns above them. The EBSD patterns in the middle row look clearly different from their transmission counterparts. Looking at the 1-nm-thick sample, we see an EBSD pattern that looks inverted by contrast with respect to the transmission pattern. This can be understood from the decisive role of absorption effects: backscattering is increased when the electron waves are travelling on the atomic planes and along the close-packed crystal directions, which is seen by the maxima near zone axis directions in the EBSD pattern. By the same extinction distance mechanism, as for the transmission pattern, the EBSD pattern for the 1 nm sample becomes unsharp. Qualitatively, this means that a thin slice of crystal cannot focus the electron waves sufficiently inside itself to produce a large variation in diffraction probability from different parts of the unit cell, which would be the basis of sharp EBSD patterns. In this way, for EBSD, dynamical diffraction theory implies a lower limit for the possible information depth on the order of a quarter of the extinction distance of the strongest reflections. Furthermore, we also see in the middle row that the EBSD pattern intensity saturates after about 10 nm, shown by the number below each pattern. In our model simulation, 80% of the total diffracted intensity that is backscattered from the 50 nm sample is reached already after 10 nm. This can be explained by the fact that in our simplified model, only electrons from depths not very much larger than the inelastic mean free path can contribute to the Kikuchi diffraction pattern. As stated earlier, the inelastic mean free path is only a lower-limit estimation of the average distance after which electrons are removed from the diffracted channel. It remains to be explored experimentally how many inelastic losses are necessary to remove all localized information from the initial backscattering process. Energy-filtered EBSD measurements showed that significant contrast is still produced from electrons with energies down to about 80% of the primary energy (Deal et al., 2008), and recently the influence of plasmon losses on energy filtered backscattered Kikuchi band profiles was studied for Si(001), establishing that after several plasmon losses ( 4), significant Kikuchi band contrast is still observed (Went et al., 2009). Assuming that electrons can still form Kikuchi patterns after a few plasmon losses, the relevant mean free path for the process of what might be called absorption from the Kikuchi diffraction pattern to a smooth background should be correspondingly larger than the inelastic mean free path. In this sense, the depth values in Fig. 3 are to be interpreted with caution, as they are related only to the specific model we assumed. Experimental quantification of these effects should lead to useful models for the information depth in EBSD patterns. In the EBSD simulations of the middle row of Fig. 3, we considered the integrated backscattered intensity from all depths. Now we selectively pick out electrons from different depths to separately analyse their contribution to the depthintegrated pattern that is experimentally observed. This is shown in the lower row of Fig. 3 by EBSD patterns from 1-nm thick slices in increasing depth from the surface. The same number of diffusely scattered electrons is initially starting from each slice, and the number below each picture shows how much this slice contributes to the total intensity that is reaching the surface (consistent with the values in the

9 40 A. WINKELMANN middle row, that is, adding up all 1 nm slices from 0 to 50 nm would give 0.19; please note that the total intensities of the transmission and the EBSD patterns cannot be quantitatively compared with each other since we do not know the absolute efficiency of diffuse vs. coherent scattering in our model). We see clear differences in the contrast of the EBSD patterns with increasing depth of the slice: compared to the 4- to 5-nm slice, the slices at larger depths begin to change contrast, and for the slice extending from 49 to 50 nm, we see that it contributes with a contrast that is inverted with respect to the slices nearer to the surface. However, the absolute contribution of electrons from these depths to the final pattern is negligible (0.014 contributed from 4 to 5 nm vs from 49 to 50 nm), if electrons start with equal probability from each depth. The contrast reversal of Kikuchi bands with thickness is theoretically well understood in the transmission case (Høier, 1973, and references therein). Experimentally, we can increase the contribution of the slices at larger depths by changing the incidence conditions. Choosing an incidence angle nearer to the surface normal direction results in a deeper penetration of primary electrons into the sample. Now, if we observe those electrons that are backscattered at shallow angles with respect to the surface plane, we can expect that these have to traverse the largest amount of material, and thus they experience a large effective thickness for dynamical interactions. This is why a contrast reversal is observed first for the electrons with the largest angles with respect to the surface normal, because effectively they come from larger depths, as viewed along their path. This explains the observation, already in the early experiments, of Kikuchi band contrast reversal in EBSD patterns when going to steeper incidence angles (Alam et al., 1954). Combination of coherent and incoherent diffraction The combined treatment of coherent elastic scattering and incoherent scattering in transmission electron diffraction experiments in general is a complicated problem. Energy filtering is one way to remove the inelastically scattered electrons from the observed pattern. However, even with modern high-resolution energy filters (Brink et al., 2003; van Aken et al., 2007), the thermal diffuse scattering cannot be removed, and so it is important to include this effect in quantitative simulations. A successful way to include the diffuse scattering of electrons by thermal vibrations is the frozen phonon approach in multislice calculations (Loane et al., 1991; Muller et al., 2001), where the nonperiodicity in the crystal induced by thermal vibrations is explicitly included via correspondingly displaced atomic coordinates. The calculation is then carried out for a number of different atomic arrangements. This approach has been shown to give very good agreement with experimental measurements (Van Dyck, 2009). At first sight, a similar procedure seems to be necessary for EBSD simulation since the incident beam can in principle scatter coherently as well as incoherently to produce the backscattered diffraction pattern. As we have seen, a major simplification of the problem is possible by noticing that usually one does not observe significant signs of scattering that is coherent with the incident beam, which would be indicated by the appearance of diffraction spots. These spots are observed only under rather special circumstances in the standard EBSD setup in the SEM: at grazing incidence and exit angles, one can realize a reflection high energy diffraction type of experiment and spot patterns are observed(baba-kishi, 1990). In this case, a unified treatment of the coherent high energy diffraction spot pattern and the Kikuchi pattern in the background would be necessary to achieve a quantitative description of the relative intensities of coherently and incoherently scattered intensity by dynamical high energy diffraction theory (Korte & Meyer- Ehmsen, 1993; Korte, 1997). However, this would be much more information than we actually need for the simulation of an EBSD pattern. In a standard EBSD setup with large scattering angles, the coherent part is practically absent, and we are left with finding the relative angular intensity variations within the incoherent part itself (containing the Kikuchi pattern). As we have shown earlier, this is possible by a standard CBED-type calculation via application of the reciprocity principle, leading to good agreement with experimental EBSD patterns (Winkelmann et al., 2007; Winkelmann, 2008). We actually do not need to know exactly how large is the coherent part relative to the incoherent part, if experimentally the coherent part is negligible. This explains the surprising success of dynamical EBSD simulations assuming effectively a complete incoherence between the incident beam and the backscattered electron waves. If it is possible to approximate experimentally observed transmission patterns as a weighted sum of coherent and incoherent contributions, this provides a way to approximately include thermal diffuse scattering, for example in dynamical CBED Bloch-wave calculations (Omoto et al., 2002). Both coherent and incoherent contributions can be derived from the solution of exactly the same eigenvalue problem in the Bloch-wave calculation. The close relationship of coherent dynamical scattering of a convergent incident beam and the thermal diffuse scattering from internal divergent incoherent sources is shown in model calculations of a thought-experiment in Fig. 4. We calculated simultaneously a bright-field [001] CBED disk for Molybdenum at 20 kv in Fig. 4(a), and the scattering from incoherent point sources in Fig. 4(b). This is a thought-experiment in the sense that CBED measurements are conventionally not carried out at 20 kv, but usually at energies of 100 kv or more. By comparison with EBSD measurements, we know that the incoherent diffuse scattering is correctly reproduced, so that we can assume that also the CBED pattern is a plausible representation. Fig. 4(c) shows qualitatively how diffuse scattering is influencing the coherent CBED disk. It is arbitrarily assumed that the

10 KIKUCHI PATTERN SIMULATION FOR EBSD 41 Fig. 4. Simulation of an idealized experiment illustrating the influence of coherent and incoherent scattering in the formation of a general diffraction pattern. The simulation is for a 10-nm-thick Mo (001) single crystal sample, with the [001] surface normal pointing out of in the paper plane, for 20 kev electrons, gnomonic projection out to the 111 directions. (a) A calculated large-angle bright-field Mo [001] convergent beam electron diffraction (CBED) disk. (b) The diffraction pattern of electrons spherically emitted with equal intensity from the Mo atomic positions in all depths (0 10 nm), all sources emitting incoherently. (c) Arbitrarily weighted sum of (a) and (b) showing that intensity appears in the dark parts of the coherent disk pattern from (a) and that this incoherent intensity adds to the to the coherent intensity in the CBED disk as a complex structured background. Higher order CBED disks are not shown. coherent scattering is dominant, and thus only a relatively low incoherent intensity is added. Qualitatively, we see that intensity appears in the formerly dark region of the coherent pattern, and that the measured intensity in the disk itself is also influenced by a structured background(see also Fig. 7). Because of this possible structured background, it is necessary to have a correct model of the incoherent scattering in quantitative analyses of CBED measurements for structural investigations (Saunders, 2003). Comparison to experiment The general results of the previous sections are now applied to a comparison of experimental EBSD pattern with dynamical simulations assuming different semi-realistic depth profiles for the diffracted backscattered electrons. To obtain model depth distributions, the Monte Carlo program CASINO was used (Drouin et al., 2007). In Fig. 5, we show simulated average depth distributions of backscattered electrons from a 20 kv primary beam incident at 70 onto a Mo sample. Since the dominating diffraction contrast in the EBSD pattern is produced by electrons which have lost energy of up to about a few hundred ev (Deal et al., 2008; Winkelmann, 2009), two cases were considered: first, only backscattered electrons that have lost not more than 500 ev ( kev) and, second, electrons having lost not more than 1500 ev ( kev). The latter group can be expected to originate on average from larger depths than the former, which is reproduced by the simulation. It is stressed here that the simulated depth distributions serve simply as theoretical model assumptions in order to analyse their influence on the dynamical calculation whereas they cannot be expected to quantitatively reproduce details of the real depth distribution. The theoretical model used in the CASINO code is the continuous slowing down approximation, which for example, does not take into account the quantized loss of energy and thus cannot be expected to Fig. 5. Simulated depth-profiles of backscattered electrons from Mo using the CASINO Monte Carlo code (Drouin et al., 2007) and fitted to analytical models with parameters t m (see text). accurately reproduce the backscattered energy spectrum and depth distribution in the quasi-elastic regime that is relevant for the diffraction contrast in EBSD. Keeping these serious limitations in mind, the Monte Carlo simulated depth profiles were fitted to two analytical models: a Poisson distribution I P x exp( x/t m ), and an exponential decay I E exp( x/t m ). The Poisson distribution reflects the statistics of a single large-angle backscattering event after a mean path t m and is seen to agree well with the Monte Carlo simulations for the 500 ev loss group, whereas the agreement for the larger energy losses is quantitatively less good, but qualitatively still consistent. The exponential decay model is included for comparison to analyse the implications of the neglect of the finite penetration depth before backscattering and the

11 42 A. WINKELMANN corresponding local maximum in the depth distribution. As is seen in Fig. 5, this is the main qualitative difference between the two analytical models. Dynamical EBSD simulations were then carried out for bcc Mo (a = Å) at 20 kv, assuming the analytical depth distributions with fit parameters t m that are listed in Fig. 5 (t m are the mean depths of backscattering in both models). In the dynamical simulation, 925 reflections with minimum lattice spacing d hkl = 0.35 Å were included, the Debye Waller factor was taken as B = 0.25 Å 2 (Peng et al., 1996). In Fig. 6, the results of the dynamical simulations are compared to an experimental EBSD pattern from a Mo single crystal measured at 20 kv (Langer & Däbritz, 2007). The intensity in the dynamical simulations is scaled from the minimum to the maximum calculated value in each simulation separately, neglecting a possible background that is present in experimental patterns due to delocalized and inelastically scattered electrons. In connection with the fact that the dynamical simulations are restricted to a single energy, this leads to a generally higher contrast and sharpness of the simulated patterns compared to the experimental EBSD patterns. Apart from this limitation, we can see an overall convincing agreement of all the dynamical simulations with the experimental pattern, including the pattern fine structure and the presence of higher-order Laue zone (HOLZ) rings (Michael & Eades, 2000). Fig. 6. Experimental EBSD pattern from Mo at 20 kv (Langer & Däbritz, 2007) and dynamical simulations corresponding to the analytical depth-profiles shown in Fig. 5. The numbers correspond to the depth profile parameter t m in the Poisson model and in the exponential decay model (see text).

12 KIKUCHI PATTERN SIMULATION FOR EBSD 43 Comparing the dynamical simulations in detail, we see a noticeable influence of the different assumed depth distributions on the simulated patterns. First, we compare the patterns for lower average depth of backscattering (a,b) with the ones for the larger mean depth (c,d). Patterns (c) and (d) appear with slightly lower contrast than patterns (a) and (b), which can be explained with the larger range of thickness that contributes to (c) and (d): as we have seen in Fig. 3(c), the contrast of Kikuchi bands tends to reduce and invert for layers in larger depths. Looking more closely at the differences between the Poisson model and the exponential decay model, we notice that the exponential model produces a locally higher intensity in the major zone axes, which is most clearly visible for the 4-fold [001] zone axis. Again, this can be nicely reconciled with our depth-resolved simulations in Fig. 3(c): since the very low depths dominate in the exponential decay model much more than in the Poisson model, the corresponding patterns in the exponential model contain more contributions from patterns qualitatively like the unsharp 1 nm pattern in Fig. 3(c). These contributions are characterized mainly by high intensity intensity in the zone axes, without other fine structure due to the extinction distance effects discussed earlier. Accordingly, the simulated Mo patterns (b) and (d) locally show higher intensity in the [001] direction than their Poisson model counterparts (a) and (c). If we tentatively conclude that pattern (c) shows the worst agreement with experiment (mainly based on the intensity in the [001] zone axis), the simulations would lead to the interpretation that the electrons from depths up to about 10 nm characteristic for models (a), (b) and (d) dominate in the experimental pattern. These low depths will be further emphasized by the additional inelastic and incoherent scattering in the outgoing path for electrons from larger depths. We see that the dynamical simulations exhibit a detectable influence of the assumed depth profile of backscattered electrons on EBSD patterns, enabling us to draw conclusions about the probable depth distribution of backscattered electrons which are consistent with previous estimations (Dingley, 2004). Compared to the simplified Monte Carlo model considered here mainly for illustration, the use of more realistic models of the energy- and angle-dependent backscattering process (Werner, 2001) should lead to a more accurate description of the backscattering depth profile and could provide additional information concerning the depthsensitivity of EBSD measurements in variable geometries. Conclusions In this paper, we illustrated the interconnections between the Bloch-wave simulation frameworks for convergent beam electron diffraction in the transmission electron microscope and electron backscatter diffraction in the scanning electron microscope. The dynamical simulation of EBSD patterns is Fig. 7. Visualization of the connection between coherent and incoherent scattering as discussed in this paper. The patterns of electrons which are scattered by a crystalline sample into all possible directions can be mapped on spheres. Imagine a small crystal sample in the center of the spheres, from where electrons are emitted and made visible when they hit the surrounding spheres from the inside. Left Ball: simulated electron backscatter diffraction (EBSD) pattern from Molybdenum at 20 kev electron energy, Right Ball: simulated full solid angle Convergent Beam Electron Diffraction (CBED) bright field pattern of a coherently transmitted beam. Middle Ball: A combination of coherent scattering (light circle, contributed by the right ball) and incoherent scattering (dark background, contributed by the left ball) is observed in real transmission experiments. greatly simplified if coherent scattering with respect to the incident beam can be neglected. By explicit simultaneous simulation of CBED and EBSD patterns with the same computer program, it was shown how the calculation of the EBSD pattern from incoherent point sources is related to a brightfield transmission calculation for a probe of convergence angle corresponding to the field of view of the EBSD phosphor screen. Depth-resolved dynamical EBSD simulations exhibit a detectable influence of the assumed backscattering depth profile on the intensity distribution in EBSD patterns, which provides an additional information channel in studies applying comparisons of experimental and simulated EBSD patterns. Acknowledgements The author thanks E. Langer (Technical University Dresden, Germany) for supplying the experimental EBSD pattern. References Alam, M.N., Blackman, M. & Pashley, D.W. (1954) High-angle Kikuchi patterns.proc. R. Soc. Lond. A 221,