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

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

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 sum of the incident beam wave vector K in the and the treatment of incoherent scattering in electron crystal and a surface normal component as k(j) = K + λ(j) n. diffraction techniques are discussed, including the role of the The eigenvalue λ(j) is complex in the general case. The reader thickness parameter. The unifying concepts are illustrated by can safely assume that a ‘black box’ Bloch-wave simulation dynamical model simulations, which are carried out with gives us the unknown parameters defining the wave function the same formalism and computer program simultaneously in Eq. (1) for a given incident beam direction K and electron for both coherent CBED patterns and incoherent Kikuchi acceleration voltage. patterns for molybdenum at 20 kV beam energy. Finally, The wave function (1) can be rearranged to show that an experimental EBSD pattern from a Mo single crystal is it can be seen as the sum of contributions of plane waves compared to full-scale dynamical simulations, where the exp[2πi(K + g) · r] moving into directions K + g and having

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

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EBSD: P only Kikuchi

"+" THEE KikuchiHEED 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 as it contains well-defined phase relationships (the surfaces coherent as indicated in Fig. 1(a). Instead, the electron of equal phase are spheres). This means that each of these waves can experience unknown, more or less random, independent waves will separately exhibit interference effects phase changes under scattering. Already in elastic on its way out of the crystal when it is elastically scattered backscattering, the electrons transfer recoil energy and by the surrounding atoms. In effect, the single incident plane momentum to the target atoms (Boersch et al., 1967; Went wave P from infinity (Fig. 1a) is replaced by a spherical wave & Vos, 2008), and if the corresponding atomic displacement from the inside of the crystal (Fig. 1b), which can be thought of is of the order of the incident electron wavelength this will asasuperpositionofaninfinitenumberplanewavesgoinginto lead to the effect that the phases of the backscattered waves all directions from S. If we know how to treat the diffraction of are not perfectly locked to each other. The discrete diffraction a single plane wave, we can in principle treat the diffraction of features disappear because the interference conditions are a combination of them. But it looks as though we have a much not spatially fixed anymore. In effect, each scattering atom more complicated problem to solve: a single initial plane wave scatters independently of the others and contributes a from P versus a huge number of initial plane waves starting continuous source intensity in all directions depending on from S that are diffracted. its differential scattering cross section. If this incoherent However, a major simplification arises if we take into scattering on average would take place homogeneously at account how we detect the diffraction pattern: the electron all places in the crystal, a continuous background results, intensity is detected basically at an infinite distance away from reflecting the atomic cross sections for different scattering the sample, where the electron wave travelling in a specific processes and the multiple inelastic and elastic scattering in all direction (corresponding to a point on our phosphor screen) directions. can be assumed to be a plane wave. For the problem of the By contrast, if the incoherent scattering remains point source S, we see that we actually do not need to know concentrated at specific sites in the unit cell, we are in a how much intensity is going into to all directions at the same situation, shown in Fig. 1(b), which does not look so different time, we only need to know how much is intensity is finally from Fig. 1(a): a single spherical wave is starting from a ending up in the plane wave component that is going towards point source S located at an atomic position. After the phase- our specified point on the screen. Of course, if we start our breaking incident, this spherical wave has lost memory of the waves from S, there is no way of knowing beforehand which phase of the incident parent plane wave P, so it cannot act part of these is ending up in the detection direction, because in concert with all the waves from the other atoms anymore. the outgoing waves are scattered elastically multiple times in But as an individual spherical wave, it is perfectly coherent all possible directions.

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

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exit surface of the transmitted beam. No matter whether we a b 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 t0 less near to the sample surface for bulk samples. This depth range is determined in a complex way by the elastic scattering t1 t2 cross sections and the possible energy dissipation processes t3 (Reimer, 1998, Chapter 3). For samples with increasing t 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 Fig. 2. Comparison of the different roles of the thickness parameter t the entrance side, however, will simply stay nearly constant in the calculation of a coherent THEED diffraction pattern as compared beyond a certain thickness which can be considered as the to a Kikuchi pattern calculation: (a) in coherent THEED, the sample bulk limit. induces boundary conditions at depths t0 and t. (b) In Kikuchi patterns, contributions from sources in different depths have to be taken into At this stage, it might seem that the calculation for the account. The weight of each contribution (diameter of the dashed circles) diffuse pattern will become increasingly complicated because is given by the number of incoherently scattered electrons at that depth. we have to deal possibly with a huge number of scatterers at The purpose of the incident beam in (b) is in principle only to produce a different positions below the surface. However, the calculation depth distribution of incoherently scattered electrons, symbolized by the of our wave function (4) gives us the probability of going from incident beam in parentheses. 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 The role of the depth parameter depth. Instead, we simply weight each depth according to the The different role of the depth parameter in diffuse patterns relative number of electrons it scatters diffusely. This requires as compared to the coherent pattern is illustrated by Fig. 2. a simple depth integration, and if the depth distribution is fitted In part (a) of Fig. 2, it is shown that the coherent pattern is to a parameterized function, we need only the integral of that formed by the elastic scattering of all atoms from the entrance function to analytically incorporate all sources. plane at t0 up to the exit plane at t. The sample is transforming the incident beam into a set of Bloch waves, according to the Depth-resolved model calculations boundary conditions at t0, and from the Bloch waves, the diffracted beams are formed again at t. To illustrate the main effects of depth-dependent Kikuchi By contrast, the diffuse pattern from independent point band contrast, we will in the following apply a simple sources obviously depends on the depth of each individual model assuming that the observed backscattered electrons are source event beneath the exit surface, which is shown in created by single incoherent scattering events. After these part (b) of Fig. 2. In the experiment, we do not observe single incoherent events at localized point sources, the spherical electrons from a single event in a well-defined thickness but electron waves move through the crystal and are diffracted we collect all electrons from a range of depths. This is why we by the periodic part of the potential. The intensity variation in have to sum up all the diffuse patterns from all possible sources a Kikuchi band reflects the connection between the position at different depths according to the respective probability of of the localized scattering event within the crystal unit cell backscattering with a specified energy from a certain depth and the observed wave vector direction (i.e. a point on the below the surface. This depth-distribution of inelastic electrons phosphor screen). This can be visualized by explicit calculation cannot simply be inferred from the backscattered electron of the probability density distribution within a unit cell for spectrum in a direct way, but must be modelled analytically different positions along a Kikuchi band profile (Winkelmann, or obtained by Monte Carlo simulations (Werner, 2001). We 2009). The excited Bloch waves at observation angles smaller will assume here that we know how many electrons with than the Bragg angle for a relevant lattice plane sample the kinetic energy E are scattered from depth t below the surface. atomic planes, whereas the Bloch waves excited at angles This depth distribution can assume nontrivial shapes, that is, larger than the Bragg angle sample the space between it can have a maximum at a certain depth. This is shown these planes. Multiple inelastic scattering of the emitted in Fig. 2(b), where we draw the probability of backscattering backscattered waves will now tend to produce additional from different depths t as circles of different diameters around incoherent sources which are partly concentrated at the the independent scattering atoms. In Fig. 2(b), the inelastic atomic cores (e.g. phonon scattering), and partly distributed electron distribution has a maximum at the depth t2 from the over the whole unit cell (e.g. delocalized plasmon scattering).

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Eachoftheseadditionalsourcescanbethoughtofasproducing of the scattering processes. Instead, the overall removal of an individual Kikuchi diffraction pattern, just as the initial electrons from the diffracted channel with travelled distance spherical wave from the first incoherent backscattering event I diffraction(t) = I 0 exp(−t/lIMFP) is treated only in an average itself. The point sources between the atomic planes, however, way by a constant imaginary part of the crystal potential will produce inverted Kikuchi bands as compared to the point V 0i, corresponding to an inelastic mean free path (IMFP) 2 sources on the atomic planes, via the Bloch-wave unit-cell lIMFP = E/2me /eV0i (Spence & Zuo, 1992). Because sampling mechanism mentioned earlier. Thus, the intensity diffraction contrast can still be preserved after several inelastic variation over a Kikuchi band is partially cancelled due to the scatterings, and EBSD is averaging over a large range of redistribution of incoherent source positions within the unit energy losses, the IMFP is only a lower-limit estimation of cell by multiple inelastic scattering. This effectively results in the path length after which diffraction contrast disappears a smooth background from a part of the electrons that are in experiment. We calculate the diffraction patterns of a inelastically scattered on the way out of the crystal. Another transmitted elastic beam and the pattern of diffusely scattered part, which is effectively localized at the atomic positions, will electrons at the same time. By cutting out specific slices from contribute to the diffracted EBSD pattern as additional sources the crystal, we can then show how the diffraction pattern is from different depths as compared to the initial backscattered changing with thickness. We stress here that we can use one wave. As sources in different depths can also show inverted and the same calculation to get both types of patterns: after Kikuchi band contrast with respect to each other, this is solving the eigenvalue problem just as in a conventional CBED a further contribution to an effective decrease in intensity calculation (Spence & Zuo, 1992), we obtain the coherently variation of Kikuchi bands. transmitted intensity from Eq. (3) and the EBSD intensity from The multiple inelastic scattering and gradual loss of integrating Eq. (4) over the depth t (Rossouw et al., 1994). position specificness within the crystal unit cell is obviously The results of corresponding simulations for a molybdenum a complicated process which cannot be treated in our sample and a 20 keV electron energy are shown in Fig. 3. In simple approach without including the explicit properties 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.

C 2009 The Author Journal compilation C 2009 The Royal Microscopical Society, Journal of Microscopy, 239, 32–45 KIKUCHI PATTERN SIMULATION FOR EBSD 39 patterns of Mo samples which are 1-, 5-, 10- and 50-nm thick. the electron waves are travelling on the atomic planes and This type of transmission pattern is usually called a Tanaka along the close-packed crystal directions, which is seen by the pattern or bright-field large-angle convergent-beam pattern maxima near zone axis directions in the EBSD pattern. By the (Eades, 1984; Morniroli, 2002). The patterns are shown in same extinction–distance mechanism, as for the transmission gnomonic projection, with the scattering angles reaching pattern, the EBSD pattern for the 1 nm sample becomes values from −45◦ to 45◦ in the maximum x-andy-directions, unsharp. Qualitatively, this means that a thin slice of crystal and the 111 directions are in the corners of the square areas cannot focus the electron waves sufficiently inside itself to shown. produce a large variation in diffraction probability from We begin by discussing the transmission simulation results different parts of the unit cell, which would be the basis of sharp in the upper row of Fig. 3. In agreement with expectations EBSD patterns. In this way, for EBSD, dynamical diffraction from dynamical theory, the transmission pattern for the theory implies a lower limit for the possible information thinnest (1 nm) sample is very unsharp. According to the two- depth on the order of a quarter of the extinction distance beam approximation, the diffracted beam acquires the same of the strongest reflections. Furthermore, we also see in the intensity as the direct beam after a quarter of the extinction middle row that the EBSD pattern intensity saturates after distance ξ which is determined by the Fourier amplitude of about 10 nm, shown by the number below each pattern. In the relevant reflection (Hirsch et al., 1965). In the simulation, our model simulation, 80% of the total diffracted intensity the extinction distances for the strongest reflections are 12 nm that is backscattered from the 50 nm sample is reached for the {110} and 16 nm for the {200} beams. Accordingly, already after 10 nm. This can be explained by the fact that it can be expected that a thickness of 3–4 nm is needed for in our simplified model, only electrons from depths not very the full development of dynamical effects in our considered much larger than the inelastic mean free path can contribute sample. Consistent with this expectation, it is seen that the to the Kikuchi diffraction pattern. As stated earlier, the transmission patterns increase in sharpness up to 10 nm, with inelastic mean free path is only a lower-limit estimation of contributions from weaker reflections. Beyond this thickness, the average distance after which electrons are removed absorption effects take over. The electron waves travelling from the diffracted channel. It remains to be explored alongtheatomicplanesaremoreeffectivelyabsorbed,whereas experimentally how many inelastic losses are necessary the waves between the atomic planes can still be transmitted. to remove all localized information from the initial The first type of wave is excited near the diagonals of backscattering process. Energy-filtered EBSD measurements the pattern, which appear completely dark for the 50 nm showed that significant contrast is still produced from sample. Symmetrically away from these diagonals, at angles electrons with energies down to about 80% of the primary which would correspond to angles slightly larger than the energy (Deal et al., 2008), and recently the influence of Bragg angle for the corresponding {110} reflections, we see plasmon losses on energy filtered backscattered Kikuchi band that high-intensity remains. These electrons can go farthest profiles was studied for Si(001), establishing that after several throughthecrystalbecausetheytravelbetweentheatomsand plasmon losses (4), significant Kikuchi band contrast is still thus are absorbed less. The total intensity in the whole pattern observed (Went et al., 2009). Assuming that electrons can is shown below each simulation in Fig. 3. A thickness of 0 nm still form Kikuchi patterns after a few plasmon losses, the would have intensity 1.0. As can be seen from Fig. 3, after relevant mean free path for the process of what might be 50 nm, intensity on the order of much less than a percent called absorption from the Kikuchi diffraction pattern to a smooth (0.002) remains. This is consistent with the imaginary part of background should be correspondingly larger than the inelastic the mean inner potential which corresponded to an inelastic mean free path. In this sense, the depth values in Fig. 3 are mean free path of lIMFP = 8 nm assumed in the calculation to be interpreted with caution, as they are related only to (Powell & Jablonski, 1999). Except for the unusually low the specific model we assumed. Experimental quantification of energy and very large angular extension of our simulated these effects should lead to useful models for the information bright-field large-angle convergent-beam patterns, these depth in EBSD patterns. simulations reproduce well known characteristic features of In the EBSD simulations of the middle row of Fig. 3, we such measurements. considered the integrated backscattered intensity from all Now we can directly compare what happens in the EBSD depths. Now we selectively pick out electrons from different pattern of the same samples. These are shown in the middle depths to separately analyse their contribution to the depth- panel (b), for exactly the same angular extension as the integrated pattern that is experimentally observed. This is transmission patterns above them. The EBSD patterns in shown in the lower row of Fig. 3 by EBSD patterns from 1-nm the middle row look clearly different from their transmission thick slices in increasing depth from the surface. The same counterparts. Looking at the 1-nm-thick sample, we see an number of diffusely scattered electrons is initially starting EBSDpatternthatlooksinvertedbycontrastwithrespecttothe from each slice, and the number below each picture shows transmission pattern. This can be understood from the decisive how much this slice contributes to the total intensity that role of absorption effects: backscattering is increased when is reaching the surface (consistent with the values in the

C 2009 The Author Journal compilation C 2009 The Royal Microscopical Society, Journal of Microscopy, 239, 32–45 40 A. WINKELMANN middle row, that is, adding up all 1 nm slices from 0 to 50 nm scatter coherently as well as incoherently to produce the would give 0.19; please note that the total intensities of the backscattered diffraction pattern. As we have seen, a major transmission and the EBSD patterns cannot be quantitatively simplification of the problem is possible by noticing that compared with each other since we do not know the absolute usually one does not observe significant signs of scattering that efficiency of diffuse vs. coherent scattering in our model). We is coherent with the incident beam, which would be indicated see clear differences in the contrast of the EBSD patterns with by the appearance of diffraction spots. These spots are observed increasing depth of the slice: compared to the 4- to 5-nm slice, only under rather special circumstances in the standard EBSD the slices at larger depths begin to change contrast, and for the setup in the SEM: at grazing incidence and exit angles, one can slice extending from 49 to 50 nm, we see that it contributes realize a reflection high energy diffraction type of experiment with a contrast that is inverted with respect to the slices nearer andspotpatternsareobserved(Baba-Kishi,1990).Inthiscase, to the surface. However, the absolute contribution of electrons a unified treatment of the coherent high energy diffraction spot from these depths to the final pattern is negligible (0.014 pattern and the Kikuchi pattern in the background would be contributed from 4 to 5 nm vs. 0.00004 from 49 to 50 nm), if necessary to achieve a quantitative description of the relative electrons start with equal probability from each depth. intensities of coherently and incoherently scattered intensity The contrast reversal of Kikuchi bands with thickness by dynamical high energy diffraction theory (Korte & Meyer- is theoretically well understood in the transmission case Ehmsen, 1993; Korte, 1997). (Høier, 1973, and references therein). Experimentally, we However, this would be much more information than we can increase the contribution of the slices at larger depths actually need for the simulation of an EBSD pattern. In a by changing the incidence conditions. Choosing an incidence standardEBSDsetupwithlargescatteringangles,thecoherent angle nearer to the surface normal direction results in a deeper part is practically absent, and we are left with finding the penetration of primary electrons into the sample. Now, if relative angular intensity variations within the incoherent we observe those electrons that are backscattered at shallow part itself (containing the Kikuchi pattern). As we have shown angles with respect to the surface plane, we can expect that earlier, this is possible by a standard CBED-type calculation these have to traverse the largest amount of material, and via application of the reciprocity principle, leading to good thus they experience a large effective thickness for dynamical agreement with experimental EBSD patterns (Winkelmann interactions. This is why a contrast reversal is observed first for et al., 2007; Winkelmann, 2008). We actually do not need the electrons with the largest angles with respect to the surface to know exactly how large is the coherent part relative to normal, because effectively they come from larger depths, as the incoherent part, if experimentally the coherent part is viewedalongtheirpath.Thisexplainstheobservation,already negligible. This explains the surprising success of dynamical in the early experiments, of Kikuchi band contrast reversal in EBSDsimulationsassumingeffectivelyacompleteincoherence EBSD patterns when going to steeper incidence angles (Alam between the incident beam and the backscattered electron et al., 1954). waves. If it is possible to approximate experimentally observed Combination of coherent and incoherent diffraction transmission patterns as a weighted sum of coherent The combined treatment of coherent elastic scattering and and incoherent contributions, this provides a way to incoherent scattering in transmission electron diffraction approximately include thermal diffuse scattering, for example experiments in general is a complicated problem. Energy in dynamical CBED Bloch-wave calculations (Omoto et al., filtering is one way to remove the inelastically scattered 2002). Both coherent and incoherent contributions can be electrons from the observed pattern. However, even with derived from the solution of exactly the same eigenvalue modern high-resolution energy filters (Brink et al., 2003; problem in the Bloch-wave calculation. The close relationship van Aken et al., 2007), the thermal diffuse scattering cannot of coherent dynamical scattering of a convergent incident be removed, and so it is important to include this effect in beam and the thermal diffuse scattering from internal quantitative simulations. A successful way to include the divergent incoherent sources is shown in model calculations of diffuse scattering of electrons by thermal vibrations is the a thought-experiment in Fig. 4. We calculated simultaneously “frozen phonon” approach in multislice calculations (Loane a bright-field [001] CBED disk for Molybdenum at 20 kV in et al., 1991; Muller et al., 2001), where the nonperiodicity Fig. 4(a), and the scattering from incoherent point sources in in the crystal induced by thermal vibrations is explicitly Fig. 4(b). This is a thought-experiment in the sense that CBED included via correspondingly displaced atomic coordinates. measurementsareconventionallynotcarriedoutat20kV,but The calculation is then carried out for a number of different usually at energies of 100 kV or more. By comparison with atomic arrangements. This approach has been shown to give EBSD measurements, we know that the incoherent diffuse very good agreement with experimental measurements (Van scattering is correctly reproduced, so that we can assume that Dyck, 2009). also the CBED pattern is a plausible representation. Fig. 4(c) At first sight, a similar procedure seems to be necessary shows qualitatively how diffuse scattering is influencing for EBSD simulation since the incident beam can in principle the coherent CBED disk. It is arbitrarily assumed that the

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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 influencedbyastructured background(seealsoFig.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 Fig. 5. Simulated depth-profiles of backscattered electrons from Mo using average depth distributions of backscattered electrons from the CASINO Monte Carlo code (Drouin et al., 2007) and fitted to analytical ◦ a 20 kV primary beam incident at 70 onto a Mo sample. models with parameters tm (see text). 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 accurately reproduce the backscattered energy spectrum and cases were considered: first, only backscattered electrons that depth distribution in the quasi-elastic regime that is relevant have lost not more than 500 eV (20–19.5 keV) and, second, for the diffraction contrast in EBSD. Keeping these serious electrons having lost not more than 1500 eV (20–18.5 keV). limitations in mind, the Monte Carlo simulated depth profiles The latter group can be expected to originate on average were fitted to two analytical models: a Poisson distribution from larger depths than the former, which is reproduced by I P ∼ x · exp(−x/tm), and an exponential decay I E ∼ the simulation. It is stressed here that the simulated depth exp(−x/tm). The Poisson distribution reflects the statistics of distributions serve simply as theoretical model assumptions in a single large-angle backscattering event after a mean path tm order to analyse their influence on the dynamical calculation and is seen to agree well with the Monte Carlo simulations for whereas they cannot be expected to quantitatively reproduce the 500 eV loss group, whereas the agreement for the larger details of the real depth distribution. The theoretical model energy losses is quantitatively less good, but qualitatively used in the CASINO code is the continuous slowing down still consistent. The exponential decay model is included for approximation, which for example, does not take into account comparison to analyse the implications of the neglect of the quantized loss of energy and thus cannot be expected to the finite penetration depth before backscattering and the

C 2009 The Author Journal compilation C 2009 The Royal Microscopical Society, Journal of Microscopy, 239, 32–45 42 A. WINKELMANN corresponding local maximum in the depth distribution. As The intensity in the dynamical simulations is scaled from is seen in Fig. 5, this is the main qualitative difference between the minimum to the maximum calculated value in each the two analytical models. simulation separately, neglecting a possible background that Dynamical EBSD simulations were then carried out for is present in experimental patterns due to delocalized and bcc Mo (a = 3.147 Å) at 20 kV, assuming the analytical inelastically scattered electrons. In connection with the fact depth distributions with fit parameters tm that are listed in that the dynamical simulations are restricted to a single Fig. 5 (tm are the mean depths of backscattering in both energy, this leads to a generally higher contrast and sharpness models). In the dynamical simulation, 925 reflections with of the simulated patterns compared to the experimental EBSD minimum lattice spacing dhkl = 0.35 Å were included, the patterns. Apart from this limitation, we can see an overall Debye–Waller factor was taken as B = 0.25 Å2 (Peng et al., convincing agreement of all the dynamical simulations with 1996). In Fig. 6, the results of the dynamical simulations the experimental pattern, including the pattern fine structure are compared to an experimental EBSD pattern from a Mo and the presence of higher-order Laue zone (HOLZ) rings single crystal measured at 20 kV (Langer & Dabritz,¨ 2007). (Michael & Eades, 2000).

Fig. 6. Experimental EBSD pattern from Mo at 20 kV (Langer & Dabritz,¨ 2007) and dynamical simulations corresponding to the analytical depth-profiles shown in Fig. 5. The numbers correspond to the depth profile parameter tm in the Poisson model and in the exponential decay model (see text).

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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 exponentialdecaymodel,wenoticethattheexponentialmodel 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 Fig. 7. Visualization of the connection between coherent and incoherent distance effects discussed earlier. Accordingly, the simulated scattering as discussed in this paper. The patterns of electrons which Mo patterns (b) and (d) locally show higher intensity in the are scattered by a crystalline sample into all possible directions can be [001] direction than their Poisson model counterparts (a) mapped on spheres. Imagine a small crystal sample in the center of the and (c). If we tentatively conclude that pattern (c) shows spheres, from where electrons are emitted and made visible when they hit the surrounding spheres from the inside. Left Ball: simulated electron the worst agreement with experiment (mainly based on the backscatter diffraction (EBSD) pattern from Molybdenum at 20 keV intensity in the [001] zone axis), the simulations would electron energy, Right Ball: simulated “full solid angle” Convergent Beam lead to the interpretation that the electrons from depths up Electron Diffraction (CBED) bright field pattern of a coherently transmitted to about 10 nm characteristic for models (a), (b) and (d) beam. Middle Ball: A combination of coherent scattering (light circle, dominate in the experimental pattern. These low depths will be contributedbytherightball)andincoherentscattering(darkbackground, further emphasized by the additional inelastic and incoherent contributed by the left ball) is observed in real transmission experiments. scattering in the outgoing path for electrons from larger depths. greatly simplified if coherent scattering with respect to the We see that the dynamical simulations exhibit a detectable incident beam can be neglected. By explicit simultaneous influence of the assumed depth profile of backscattered simulationofCBEDandEBSDpatternswiththesamecomputer electrons on EBSD patterns, enabling us to draw conclusions program, it was shown how the calculation of the EBSD about the probable depth distribution of backscattered pattern from incoherent point sources is related to a bright- electrons which are consistent with previous estimations field transmission calculation for a probe of convergence angle (Dingley, 2004). Compared to the simplified Monte Carlo corresponding to the field of view of the EBSD phosphor model considered here mainly for illustration, the use of screen. Depth-resolved dynamical EBSD simulations exhibit more realistic models of the energy- and angle-dependent a detectable influence of the assumed backscattering depth backscattering process (Werner, 2001) should lead to a more profile on the intensity distribution in EBSD patterns, which accurate description of the backscattering depth profile and providesanadditionalinformationchannelinstudiesapplying could provide additional information concerning the depth- comparisons of experimental and simulated EBSD patterns. sensitivity of EBSD measurements in variable geometries. Acknowledgements Conclusions The author thanks E. Langer (Technical University Dresden, Germany) for supplying the experimental EBSD pattern. In this paper, we illustrated the interconnections between the Bloch-wave simulation frameworks for convergent beam electron diffraction in the transmission electron microscope References and electron backscatter diffraction in the scanning electron Alam, M.N., Blackman, M. & Pashley, D.W. (1954) High-angle Kikuchi microscope. The dynamical simulation of EBSD patterns is patterns. Proc. R. Soc. Lond. A 221, 224–242.

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