jmi 3353 lr - max planck society · ebsd: journalofmicroscopy,239,32–45

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 simulationfor electron backscatter diffraction

A . W I N K E L M A N NMax-Planck-Institut fur Mikrostrukturphysik, Halle (Saale), Germany

Key words. Electron backscatter diffraction, Kikuchi pattern, convergentbeam electron diffraction, dynamical electron diffraction

Summary

This paper presents a tutorial discussion of the principlesunderlying the depth-dependent Kikuchi pattern formation ofbackscattered electrons in the scanning electron microscope.To illustrate the connections between various electrondiffraction methods, the formation of Kikuchi bands in electronbackscatter diffraction in the scanning electron microscopeand in transmission electron microscopy are comparedwith the help of simulations employing the dynamicaltheory of electron diffraction. The close relationship betweenbackscattered electron diffraction and convergent beamelectron diffraction is illuminated by showing how both effectscan be calculated within the same theoretical framework.The influence of the depth-dependence of diffuse electronscattering on the formation of the experimentally observedelectron backscatter diffraction contrast and intensityis visualized by calculations of depth-resolved Kikuchipatterns. Comparison of an experimental electron backscatterdiffraction pattern with simulations assuming several differentdepth distributions shows that the depth-distribution ofbackscattered electrons needs to be taken into account inquantitative descriptions. This should make it possible toobtain more quantitative depth-dependent information fromexperimental electron backscatter diffraction patterns viacorrelation with dynamical diffraction simulations and MonteCarlo models of electron scattering.

Introduction

One of the most beautiful phenomena in electron diffractionis the appearance of Kikuchi patterns formed by electronsscattered by a crystalline sample (Kikuchi, 1928; Nishikawa& Kikuchi, 1928; Alam et al., 1954). These patterns exist asa network of lines and bands and can be thought of as being

Correspondence to: Aimo Winkelmann, Max-Planck-Institut fur

Mikrostrukturphysik Weinberg 2, D-06120 Halle (Saale), Germany. Tel: +49 345

5582 639; fax: +49 345 5511 223; e-mail: [email protected]

created by independent sources emitting divergent electronwaves from within the crystal (Cowley, 1995). Kikuchipatterns also appear in the scanning electron microscopewhen the angular distribution of backscattered electronsis imaged. Around this principle, the method of electronbackscatter diffraction (EBSD) has been developed (Schwarzer,1997; Wilkinson & Hirsch, 1997; Schwartz et al., 2000;Dingley, 2004; Randle, 2008). Because the Kikuchi patternsare tied to the local crystallographic structure in the probearea of the electron beam, EBSD can provide importantcrystallographic and phase information down to the nanoscalein materials science (Small & Michael, 2001; Small et al.,2002). The success of EBSD stems from the fact that themethod is conceptually simple: in principle only a phosphorscreen imaged by a sensitive CCD camera is needed. Also,the geometry of the Kikuchi line patterns can be explainedrelatively simply by tracing out the Bragg reflection conditionsfor a point source inside a crystal (Gajdardziska-Josifovska &Cowley, 1991). In principle, by such a procedure, a networkof interference cones perpendicular to reflecting lattice planesand with opening angles determined from the respectiveBragg angles can be projected onto the observation plane toanalyse the crystallographic orientation of a sample grain.However, this does not give direct information on the observedintensities, since a quantitative calculation of the backscattereddiffraction pattern needs to use the dynamical theory ofelectron diffraction that takes into account the localizationof the backscattering process of electrons in the crystal unitcell. The author has recently been able to show (Winkelmannet al., 2007; Winkelmann, 2008) that Kikuchi patterns inbackscattered electrons in the scanning electron microscopecan be successfully calculated using a Bloch-wave approachthat is usually applied for convergent beam electron diffraction(CBED) in the transmission electron microscope. Instead ofdivergent sources internal to the crystal, CBED patterns areformed by an external convergent probe sampling the sameBragg interference cones as the internal sources, and thusthe CBED patterns show line patterns of similar geometryto EBSD and other Kikuchi patterns. However, the intensity

C© 2009 The AuthorJournal compilation C© 2009 The Royal Microscopical Society

K I K U C H I P A T T E R N S I M U L A T I O N F O R E B S D 3 3

distributions in Kikuchi patterns and in CBED patterns arequalitatively different, because CBED patterns are ideallyformed by only those electrons which retain a fixed phase withrespect to the incident beam, whereas the Kikuchi patterns areformed by independent sources largely incoherent with respectto the primary beam.

The main purpose of this paper is to explain in detail howthe two types of problems are connected. Especially it willbe shown how the dynamical diffraction from completelyincoherent point sources (relevant to EBSD) can be treatedin exactly the same formalism as the dynamical diffraction inCBED. Close attention is paid to the rather different roles ofthe thickness parameter in coherent and localized incoherentscattering, because from many investigations in transmissionelectron microscopy it is known that the observed Kikuchipattern contrast is strongly depending on the sample thickness(Pfister, 1953; Reimer & Kohl, 2008). The previous theoreticalinvestigations of dynamical EBSD simulations (Winkelmannet al., 2007; Winkelmann, 2008) in a first approximation wereneglecting some specific details of the backscattered electrondepth distribution and assumed that the backscatteredelectrons were produced with equal intensity in a layer oflimited thickness near the surface, an approximation leadingto good agreement with a number of experimentally observedEBSD patterns. Based on observations of the width of measureddiffraction lines, the energy spread and correspondinglythe related depth sensitivity of electrons contributing to anEBSD pattern can be estimated. The depth sensitivity ofEBSD is generally assumed to be in the range between 10and 40 nm at 20 kV, with the lower values reached fordenser materials (Dingley, 2004). Experimental observationsof the disappearance of Kikuchi pattern diffraction contrastwhen depositing amorphous layers on crystalline samples areconsistent with this estimation (Yamamoto, 1977; Zaefferer,2007). It is clearly an important question how the depthdistribution of the backscattered electrons is quantitativelyinfluencing the EBSD patterns. The inclusion of the relevanteffects in dynamical simulations could possibly allow toextract additional information from experimental EBSDmeasurements. This is why we will analyse in detail how thedepth distribution of the backscattered and diffracted electronsis affecting the observed Kikuchi patterns in dynamical EBSDsimulations.

The paper is structured as follows. First, the theoreticalframework is summarized, then the implications of coherenceand the treatment of incoherent scattering in electrondiffraction techniques are discussed, including the role of thethickness parameter. The unifying concepts are illustrated bydynamical model simulations, which are carried out withthe same formalism and computer program simultaneouslyfor both coherent CBED patterns and incoherent Kikuchipatterns for molybdenum at 20 kV beam energy. Finally,an experimental EBSD pattern from a Mo single crystal iscompared to full-scale dynamical simulations, where the

characteristic influence of the assumed depth distribution ofthe diffracted backscattered electrons on the dynamical EBSDpatterns can be clearly sensed.

Theoretical background

The fundamental building block of our understanding ofKikuchi pattern formation will be the prototypical exampleof transmission electron diffraction: the dynamical diffractionof an incident plane wave beam by a thin crystal sample, whichleads to the formation of a transmitted discrete spot diffractionpattern. For perfect crystals, the Bloch-wave approach is amethod often used to describe this process. For the purposesof this paper, we actually do not need to understand themathematical details of this method. We will simply assumethat we have a working method at hand to calculate from agiven crystal structure and from the incident beam directionand energy the electron wave field inside the sample and thetransmitted diffraction pattern. The Bloch-wave approach hasbeen shown to lead to very convincing agreement betweencalculated and measured electron backscatter diffractionpatterns (Winkelmann et al., 2007; Day, 2008; Maurice &Fortunier, 2008; Winkelmann, 2008; Villert et al., 2009).The same approach is used for quantitative convergent beamelectron diffraction (Spence & Zuo, 1992) and thus we havea consistent framework to describe Kikuchi pattern formationin relation to the coherent elastic diffraction.

The main idea behind the Bloch-wave approach can besummarized in a very compact way by noting that it seeksthe wave function in a specific form. This form is knownfrom Bloch’s theorem for a translationally invariant scatteringpotential (Humphreys, 1979):

�(r) =∑

j

c j exp[2π ik( j ) · r]∑

g

C ( j )g exp[2π ig · r] (1)

The Bloch-wave calculation then finds the coefficientscj,C

(j)g , and the vectors k(j) by solving a matrix eigenvalue

problem derived from the Schrodinger equation by limiting thewave-function expansion to a number of Fourier coefficientslabelled by the respective reciprocal lattice vectors g, each ofwhich couples the incident beam to a diffracted beam. Theeigenvalues λ(j) appear when the Bloch-wave vector k(j) iswritten as the sum of the incident beam wave vector K in thecrystal and a surface normal component as k(j) = K + λ(j) n.The eigenvalue λ(j) is complex in the general case. The readercan safely assume that a ‘black box’ Bloch-wave simulationgives us the unknown parameters defining the wave functionin Eq. (1) for a given incident beam direction K and electronacceleration voltage.

The wave function (1) can be rearranged to show thatit can be seen as the sum of contributions of plane wavesexp[2π i(K + g) · r] moving into directions K + g and having

C© 2009 The AuthorJournal compilation C© 2009 The Royal Microscopical Society, Journal of Microscopy, 239, 32–45

3 4 A . W I N K E L M A N N

a depth dependent amplitude φg(t):

�(r) =∑

g

φg(t) exp[2π i(K + g) · r]

=∑

g

{ ∑j

C ( j )g c j exp[2π iλ( j )n · r]

}exp[2π i(K + g) · r]

(2)

Here, we note that n · r = t is the surface normal component(depth) of the point r, with t = 0 at the entrance surface.

If all the wave-function parameters are known after solvingthe eigenvalue problem for the complexλ(j), we can in principlecalculate the intensity that is moving in the plane-wave beamsdiffracted into the directions K + g after transmission througha crystal of thickness t:

Ig(K, t) =∑

j ,l

c j c∗l C ( j )

g C (l)∗g exp

[2π i(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 straightforwardapplication of Eq. (2):

P (r) =∑g,h

∑j ,l

c j c∗l C ( j )

g C (l)∗h

× exp[2π i(λ( j ) − λ(l)∗)t

]exp[2π i(g − h) · r] (4)

Please note that Eq. (2) contains plane waves in directionsK + g only. These correspond to the diffracted beams that forma spot diffraction pattern. By itself, Eq. (2) does not provide anexplicit mechanism by which inelastically scattered waves canappear in a general direction K + k′. It is important to realizethat the conventional procedure of introducing an imaginarypotential to account for ‘electron absorption’ handles only thereduction of the beam intensities in the limited set of directionsK + g (the diffraction spots), although it does not describethe details of the redistribution of this intensity into all theother directions K + k′ (the initially black space between thediffraction spots). This redistribution, however, is fundamentalin the formation of the diffuse scattering patterns, as thescattered electrons can reappear with a changed energy in adifferent direction. While the coherent elastic scattering froma periodic crystal allows a change from the primary beamwave vector K only by discrete reciprocal lattice vectors g,inelastic or diffuse incoherent scattering, symbolized by anoperator Sk′g, allows in addition to the limited discrete setof waves with wave vectors K + g scattered waves K + k′

moving into arbitrary directions with in principle any k′, thusproducing a continuous background in addition to discretediffraction spots. The exact details of the various possibleprocesses which here have been only very schematicallysymbolized by Sk′g are treated by explicit dynamical theoriesof Kikuchi band formation (Kainuma, 1955; Chukhovskiiet al., 1973; Rez et al., 1977; Dudarev et al., 1995), andinclude, for example, the description of phonon, plasmonand core-electron excitations. An explicit recent treatment of

Kikuchi pattern formation, including a discussion of variousapproximations and simulations for high-energy transmissionelectron microscopy can be found in (Omoto et al., 2002).

The exact modelling of the inelastic scattering has importantimplications for quantitative structure analysis based onexperimental diffraction patterns, since the problem ofcoherent and incoherent scattering must be treated onthe same level (Wang, 1995; Peng et al., 2004). In thediscussion later, we will analyse the implications of theextreme cases of complete coherence or incoherence withrespect to the incident beam. For the simulation of EBSDpatterns in this paper, we will assume that practicallyall intensity is incoherently scattered from all the crystalatoms and isotropically emitted into all directions. Whatresults is a collection of independent point sources incrystalline order, and the diffraction of the spherical wavesemanating into all directions from these sources producesthe Kikuchi patterns. This model will allow us to analyse thefundamental dynamical diffraction physics behind thickness-dependent Kikuchi pattern formation, although neglectingthe exact details of inelastic scattering (most importantly,the incoherently and inelastically scattered electrons are notdistributed isotropically in reality but are dominantly scatteredin the forward direction). It will be shown in this paper how inprinciple any method that describes the scattering of a singleplane wave by a crystal into a set of diffracted beams can beturned into a method for the calculation of Kikuchi patternsunder the earlier assumptions.

Principle of calculation for incoherent point sources

We will now discuss the close connection between the spotdiffraction pattern of a transmitted beam and the Kikuchipattern from incoherent point sources. In Fig. 1(a), weshow symbolically how the spot diffraction pattern (e.g. inmicrodiffraction in the transmission electron microscope) oftransmitted beams is formed. The incident beam enters fromthe top side of the sample which is assumed to be a perfectcrystal of constant thickness. A part of the incident electronplane wave is then scattered coherently by all atoms in theinteraction volume, meaning that the waves emanating fromthe scattering centres all have a perfectly known fixed phaserelationship with respect to the incident beam and thus withrespect to each other. Wave theory tells us then that therewill be well-defined constructive and destructive interferencebetween the waves coming from different scatterers. It turnsout that only in directions corresponding to wave vectorchanges equal to reciprocal lattice vectors will we haveresulting intensity in the form of a discrete spot pattern,whereas destructive interference prohibits electrons fromgoing into all the other scattering directions. The presenceof the discrete spot diffraction pattern is shown in the lowerpart of Fig. 1(a).



THEE HEED

"+"

Kikuchi Kikuchi

S

P

P*

EBSD:only 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, toa microdiffraction pattern or to convergent beam electron diffraction (CBED) disks if the incident beam is convergent. (b) Incoherent emission from pointsources S. Only a single point source is shown, but emission proceeds independently from various possible sites. Continuous Kikuchi pattern intensitiesare 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 theproblem (a), with the modification that we have to calculate the intensity at the point S inside the crystal and not the transmitted diffraction patternwhich is put in parentheses. (d) Combination of the effects (a) and (b) in a real experiment. The transmitted pattern now shows contributions of both thediscrete diffraction of the incident beam, as well as continuous intensity from incoherent sources. To describe this combined pattern, a general treatmentof 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 havelost coherence with the incident beam, making possible a separate treatment of (b) only.

In reality, not all the scattering from the atoms will becoherent as indicated in Fig. 1(a). Instead, the electronwaves can experience unknown, more or less random,phase changes under scattering. Already in elasticbackscattering, the electrons transfer recoil energy andmomentum to the target atoms (Boersch et al., 1967; Went& Vos, 2008), and if the corresponding atomic displacementis of the order of the incident electron wavelength this willlead to the effect that the phases of the backscattered wavesare not perfectly locked to each other. The discrete diffractionfeatures disappear because the interference conditions arenot spatially fixed anymore. In effect, each scattering atomscatters independently of the others and contributes acontinuous source intensity in all directions depending onits differential scattering cross section. If this incoherentscattering on average would take place homogeneously atall places in the crystal, a continuous background results,reflecting the atomic cross sections for different scatteringprocesses and the multiple inelastic and elastic scattering in alldirections.

By contrast, if the incoherent scattering remainsconcentrated at specific sites in the unit cell, we are in asituation, shown in Fig. 1(b), which does not look so differentfrom Fig. 1(a): a single spherical wave is starting from apoint source S located at an atomic position. After the phase-breaking incident, this spherical wave has lost memory of thephase of the incident parent plane wave P, so it cannot actin 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 surfacesof equal phase are spheres). This means that each of theseindependent waves will separately exhibit interference effectson its way out of the crystal when it is elastically scatteredby the surrounding atoms. In effect, the single incident planewave P from infinity (Fig. 1a) is replaced by a spherical wavefrom the inside of the crystal (Fig. 1b), which can be thought ofas a superposition of an infinite number plane waves going intoall directions from S. If we know how to treat the diffraction ofa single plane wave, we can in principle treat the diffraction ofa combination of them. But it looks as though we have a muchmore complicated problem to solve: a single initial plane wavefrom P versus a huge number of initial plane waves startingfrom S that are diffracted.

However, a major simplification arises if we take intoaccount how we detect the diffraction pattern: the electronintensity is detected basically at an infinite distance away fromthe sample, where the electron wave travelling in a specificdirection (corresponding to a point on our phosphor screen)can be assumed to be a plane wave. For the problem of thepoint source S, we see that we actually do not need to knowhow much intensity is going into to all directions at the sametime, we only need to know how much is intensity is finallyending up in the plane wave component that is going towardsour specified point on the screen. Of course, if we start ourwaves from S, there is no way of knowing beforehand whichpart of these is ending up in the detection direction, becausethe outgoing waves are scattered elastically multiple times inall possible directions.



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 tothe source: turn around our final plane wave, let it propagatefrom the detection point towards the crystal and look how thissingle plane wave is diffracted by the crystal and how much of itfinally ends up at the source. Instead of keeping track of all thewaves in all possible directions from S, we are dealing only withthe waves that are important for our detected direction, andwe can do this with exactly the same theory that we use for asingle plane wave hitting a crystal from infinity. This is shownin Fig. 1(c), where an arrow represents the time-reversed planewave P ∗ travelling backwards from a specified direction onthe screen. The time-reversed wave is in the same situation asthe forward travelling wave P in Fig. 1(a), the only differencebeing that in Fig. 1(c) we are interested in the diffracted wavefunction at point S in the crystal and we are not interested in thetransmitted diffraction pattern that is described by exactly thesame formalism. Both types of information are simultaneouslycontained in the wave function (2), resulting in Eq. (3) thatdescribes the diffraction pattern, whereas Eq. (4) describesthe diffracted electron wave field inside the crystal, resultingfrom the dynamical interaction of a plane wave incident frominfinity.

We stress that any energy change of the wave originatingfrom S which might happen due to inelastic scattering is nota necessary difference between the situations of Fig. 1(a) and(b): the loss of a fixed phase with respect to the incident beamis the defining characteristic. In reality, of course, inelasticscattering does generally change the energy of the wavesemitted from all the possible places S in the crystal, and thus thecorresponding change in wavelength will need to be taken intoaccount. 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 arepicked up by the detector.

Finally, we can summarize this section by pointing to thegeneral situation depicted in Fig. 1(d). In the real experimentof a beam transmitted through a thin enough sample, wewill see a discrete diffraction pattern formed by the coherentscattering of the crystal, combined with a background dueto elastic scattering starting from incoherent point sourcesand additionally a background from non-localized inelasticscattering effects. The exact treatment of coherent andincoherent, elastic and inelastic scattering is the most generaland 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 ofdifficulty of treating EBSD dynamically is considerably reducedcompared to the most general situation of the combineddiffraction problem of waves that are in varying degreescoherent to the incident beam. In EBSD, the number ofelectrons scattered coherently with respect to the incident beamis usually negligible, and the corresponding spot patterns are

not observable. This is why in EBSD, it turns out to be a goodapproximation to treat the scattering from incoherent pointsources only and neglect the coherent diffraction from theincident beam. In Fig. 1(d), this is symbolized on the top sideof the sample, corresponding to a backscattering geometry.

Summarizing this section, we have seen how one of thestandard problems of transmission electron diffraction, thediffraction of a plane wave incident on a crystal, can be viewedby reciprocity as providing also the intensity from incoherentpoint sources localized inside the crystal. In the next section,we will demonstrate by explicit dynamical simulations how thedepth distribution of inelastic scattering is manifesting itself inthe intensity distribution and contrast of Kikuchi bands.

Diffusely scattered electrons and their depth distribution

The contrast in Kikuchi patterns which are observed intransmission electron microscope investigations depends onthe thickness of the sample (Uyeda & Nonoyama, 1967, 1968;Uyeda, 1968; Reimer & Kohl, 2008 Fig. 7.26, p. 324). Fora thin sample, the transmitted Kikuchi bands are high inintensity for angles smaller than the Bragg angle away fromthe relevant lattice plane (the middle of the Kikuchi band).With increasing thickness of the sample, the bands becomedark in the middle. Already at this point, we note that thecontrast for thin samples in transmission is the same as isusually observed under standard EBSD conditions: increasedintensity in the middle of the band. This type of contrast canbe explained by the fact that backscattering takes places nearthe atomic positions. For angles smaller than the Bragg angle,those Bloch waves dominate which are located at the atomicpositions (type I waves), thus providing an efficient transferchannel for the backscattered electrons. At angles larger thanthe Bragg angle, the Bloch waves are located between theatomic planes (type II waves), and the backscattered electronscannot couple efficiently to the outgoing plane wave. Thiseffect can be visualized in real space (Winkelmann, 2009),and is at work in several diffraction techniques based onlocalized emitters (Winkelmann et al., 2008). With increasingdepth, the effect of anomalous absorption takes over, becauseelectrons that move along the atomic planes in type I Blochwaves are also inelastically scattered more often and thusabsorbed more efficiently. The maximum intensity in themiddle of a band turns into a minimum for thicker samplesin transmission. The electrons moving between the atomicplanes are the only ones that survive beyond a certainthickness and these electrons are found in the type II Blochwave which is excited at the outer edges of a Kikuchiband. In EBSD, this contrast reversal process is usually notobservable, because the backscattered electrons from smalldepths dominate. In a simulation, however, the depth effectscan be analysed by assuming artificial depth distributions ofbackscattered electrons as is shown later.



a

tt3

t0

t1t2

b

Fig. 2. Comparison of the different roles of the thickness parameter tin the calculation of a coherent THEED diffraction pattern as comparedto a Kikuchi pattern calculation: (a) in coherent THEED, the sampleinduces boundary conditions at depths t0 and t. (b) In Kikuchi patterns,contributions from sources in different depths have to be taken intoaccount. 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 adepth distribution of incoherently scattered electrons, symbolized by theincident beam in parentheses.

The role of the depth parameter

The different role of the depth parameter in diffuse patternsas compared to the coherent pattern is illustrated by Fig. 2.In part (a) of Fig. 2, it is shown that the coherent pattern isformed by the elastic scattering of all atoms from the entranceplane at t0 up to the exit plane at t. The sample is transformingthe incident beam into a set of Bloch waves, according to theboundary conditions at t0, and from the Bloch waves, thediffracted beams are formed again at t.

By contrast, the diffuse pattern from independent pointsources obviously depends on the depth of each individualsource event beneath the exit surface, which is shown inpart (b) of Fig. 2. In the experiment, we do not observe singleelectrons from a single event in a well-defined thickness butwe collect all electrons from a range of depths. This is why wehave to sum up all the diffuse patterns from all possible sourcesat different depths according to the respective probability ofbackscattering with a specified energy from a certain depthbelow the surface. This depth-distribution of inelastic electronscannot simply be inferred from the backscattered electronspectrum in a direct way, but must be modelled analyticallyor obtained by Monte Carlo simulations (Werner, 2001). Wewill assume here that we know how many electrons withkinetic 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 shownin Fig. 2(b), where we draw the probability of backscatteringfrom different depths t as circles of different diameters aroundthe independent scattering atoms. In Fig. 2(b), the inelasticelectron distribution has a maximum at the depth t2 from the

exit surface of the transmitted beam. No matter whether weobserve in a transmission or in a backscattering geometry, themultiple elastic and inelastic scattering of the incident beamleads to a limited interaction region which is located more orless near to the sample surface for bulk samples. This depthrange is determined in a complex way by the elastic scatteringcross sections and the possible energy dissipation processes(Reimer, 1998, Chapter 3). For samples with increasingthickness, one can thus see very pronounced changes inthe Kikuchi patterns on the transmitted side of the samplebecause the scattering region will effectively recede deeperand deeper into the crystal and asymptotically disappearwhen no electrons are transmitted anymore. The situation onthe entrance side, however, will simply stay nearly constantbeyond a certain thickness which can be considered as thebulk limit.

At this stage, it might seem that the calculation for thediffuse pattern will become increasingly complicated becausewe have to deal possibly with a huge number of scatterers atdifferent positions below the surface. However, the calculationof our wave function (4) gives us the probability of going frompoint r in the crystal to the point on the observation screen,for all possible points r in the crystal in a single run. We thusdo not have to do a separate dynamical calculation for eachdepth. Instead, we simply weight each depth according to therelative number of electrons it scatters diffusely. This requiresa simple depth integration, and if the depth distribution is fittedto a parameterized function, we need only the integral of thatfunction to analytically incorporate all sources.

Depth-resolved model calculations

To illustrate the main effects of depth-dependent Kikuchiband contrast, we will in the following apply a simplemodel assuming that the observed backscattered electrons arecreated by single incoherent scattering events. After theseincoherent events at localized point sources, the sphericalelectron waves move through the crystal and are diffractedby the periodic part of the potential. The intensity variation ina Kikuchi band reflects the connection between the positionof the localized scattering event within the crystal unit celland the observed wave vector direction (i.e. a point on thephosphor screen). This can be visualized by explicit calculationof the probability density distribution within a unit cell fordifferent positions along a Kikuchi band profile (Winkelmann,2009). The excited Bloch waves at observation angles smallerthan the Bragg angle for a relevant lattice plane sample theatomic planes, whereas the Bloch waves excited at angleslarger than the Bragg angle sample the space betweenthese planes. Multiple inelastic scattering of the emittedbackscattered waves will now tend to produce additionalincoherent sources which are partly concentrated at theatomic cores (e.g. phonon scattering), and partly distributedover the whole unit cell (e.g. delocalized plasmon scattering).



Each of these additional sources can be thought of as producingan individual Kikuchi diffraction pattern, just as the initialspherical wave from the first incoherent backscattering eventitself. The point sources between the atomic planes, however,will produce inverted Kikuchi bands as compared to the pointsources on the atomic planes, via the Bloch-wave unit-cellsampling mechanism mentioned earlier. Thus, the intensityvariation over a Kikuchi band is partially cancelled due to theredistribution of incoherent source positions within the unitcell by multiple inelastic scattering. This effectively results ina smooth background from a part of the electrons that areinelastically scattered on the way out of the crystal. Anotherpart, which is effectively localized at the atomic positions, willcontribute to the diffracted EBSD pattern as additional sourcesfrom different depths as compared to the initial backscatteredwave. As sources in different depths can also show invertedKikuchi band contrast with respect to each other, this isa further contribution to an effective decrease in intensityvariation of Kikuchi bands.

The multiple inelastic scattering and gradual loss ofposition specificness within the crystal unit cell is obviouslya complicated process which cannot be treated in oursimple approach without including the explicit properties

of the scattering processes. Instead, the overall removal ofelectrons from the diffracted channel with travelled distanceI diffraction(t) = I 0 exp(−t/l IMFP) is treated only in an averageway by a constant imaginary part of the crystal potentialV 0i, corresponding to an inelastic mean free path (IMFP)lIMFP =

√�2 E/2me/e V0i (Spence & Zuo, 1992). Because

diffraction contrast can still be preserved after several inelasticscatterings, and EBSD is averaging over a large range ofenergy losses, the IMFP is only a lower-limit estimation ofthe path length after which diffraction contrast disappearsin experiment. We calculate the diffraction patterns of atransmitted elastic beam and the pattern of diffusely scatteredelectrons at the same time. By cutting out specific slices fromthe crystal, we can then show how the diffraction pattern ischanging with thickness. We stress here that we can use oneand the same calculation to get both types of patterns: aftersolving the eigenvalue problem just as in a conventional CBEDcalculation (Spence & Zuo, 1992), we obtain the coherentlytransmitted intensity from Eq. (3) and the EBSD intensity fromintegrating Eq. (4) over the depth t (Rossouw et al., 1994).The results of corresponding simulations for a molybdenumsample and a 20 keV electron energy are shown in Fig. 3. Inthe 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 eachpicture 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 samesamples as in the upper row. (Lower row) Depth-resolved EBSD patterns produced by slices of 1 nm thickness.



patterns of Mo samples which are 1-, 5-, 10- and 50-nm thick.This type of transmission pattern is usually called a Tanakapattern or bright-field large-angle convergent-beam pattern(Eades, 1984; Morniroli, 2002). The patterns are shown ingnomonic projection, with the scattering angles reachingvalues from −45◦ to 45◦ in the maximum x- and y-directions,and the 〈111〉 directions are in the corners of the square areasshown.

We begin by discussing the transmission simulation resultsin the upper row of Fig. 3. In agreement with expectationsfrom dynamical theory, the transmission pattern for thethinnest (1 nm) sample is very unsharp. According to the two-beam approximation, the diffracted beam acquires the sameintensity as the direct beam after a quarter of the extinctiondistance ξ which is determined by the Fourier amplitude ofthe relevant reflection (Hirsch et al., 1965). In the simulation,the extinction distances for the strongest reflections are 12 nmfor the {110} and 16 nm for the {200} beams. Accordingly,it can be expected that a thickness of 3–4 nm is needed forthe full development of dynamical effects in our consideredsample. Consistent with this expectation, it is seen that thetransmission patterns increase in sharpness up to 10 nm, withcontributions from weaker reflections. Beyond this thickness,absorption effects take over. The electron waves travellingalong the atomic planes are more effectively absorbed, whereasthe waves between the atomic planes can still be transmitted.The first type of wave is excited near the diagonals ofthe pattern, which appear completely dark for the 50 nmsample. Symmetrically away from these diagonals, at angleswhich would correspond to angles slightly larger than theBragg angle for the corresponding {110} reflections, we seethat high-intensity remains. These electrons can go farthestthrough the crystal because they travel between the atoms andthus are absorbed less. The total intensity in the whole patternis shown below each simulation in Fig. 3. A thickness of 0 nmwould have intensity 1.0. As can be seen from Fig. 3, after50 nm, intensity on the order of much less than a percent(0.002) remains. This is consistent with the imaginary part ofthe mean inner potential which corresponded to an inelasticmean free path of l IMFP = 8 nm assumed in the calculation(Powell & Jablonski, 1999). Except for the unusually lowenergy and very large angular extension of our simulatedbright-field large-angle convergent-beam patterns, thesesimulations reproduce well known characteristic features ofsuch measurements.

Now we can directly compare what happens in the EBSDpattern of the same samples. These are shown in the middlepanel (b), for exactly the same angular extension as thetransmission patterns above them. The EBSD patterns inthe middle row look clearly different from their transmissioncounterparts. Looking at the 1-nm-thick sample, we see anEBSD pattern that looks inverted by contrast with respect to thetransmission pattern. This can be understood from the decisiverole of absorption effects: backscattering is increased when

the electron waves are travelling on the atomic planes andalong the close-packed crystal directions, which is seen by themaxima near zone axis directions in the EBSD pattern. By thesame extinction–distance mechanism, as for the transmissionpattern, the EBSD pattern for the 1 nm sample becomesunsharp. Qualitatively, this means that a thin slice of crystalcannot focus the electron waves sufficiently inside itself toproduce a large variation in diffraction probability fromdifferent parts of the unit cell, which would be the basis of sharpEBSD patterns. In this way, for EBSD, dynamical diffractiontheory implies a lower limit for the possible informationdepth on the order of a quarter of the extinction distanceof the strongest reflections. Furthermore, we also see in themiddle row that the EBSD pattern intensity saturates afterabout 10 nm, shown by the number below each pattern. Inour model simulation, 80% of the total diffracted intensitythat is backscattered from the 50 nm sample is reachedalready after 10 nm. This can be explained by the fact thatin our simplified model, only electrons from depths not verymuch larger than the inelastic mean free path can contributeto the Kikuchi diffraction pattern. As stated earlier, theinelastic mean free path is only a lower-limit estimation ofthe average distance after which electrons are removedfrom the diffracted channel. It remains to be exploredexperimentally how many inelastic losses are necessaryto remove all localized information from the initialbackscattering process. Energy-filtered EBSD measurementsshowed that significant contrast is still produced fromelectrons with energies down to about 80% of the primaryenergy (Deal et al., 2008), and recently the influence ofplasmon losses on energy filtered backscattered Kikuchi bandprofiles was studied for Si(001), establishing that after severalplasmon losses (�4), significant Kikuchi band contrast is stillobserved (Went et al., 2009). Assuming that electrons canstill form Kikuchi patterns after a few plasmon losses, therelevant mean free path for the process of what might becalled absorption from the Kikuchi diffraction pattern to a smoothbackground should be correspondingly larger than the inelasticmean free path. In this sense, the depth values in Fig. 3 areto be interpreted with caution, as they are related only tothe specific model we assumed. Experimental quantification ofthese effects should lead to useful models for the informationdepth in EBSD patterns.

In the EBSD simulations of the middle row of Fig. 3, weconsidered the integrated backscattered intensity from alldepths. Now we selectively pick out electrons from differentdepths to separately analyse their contribution to the depth-integrated pattern that is experimentally observed. This isshown in the lower row of Fig. 3 by EBSD patterns from 1-nmthick slices in increasing depth from the surface. The samenumber of diffusely scattered electrons is initially startingfrom each slice, and the number below each picture showshow much this slice contributes to the total intensity thatis reaching the surface (consistent with the values in the



middle row, that is, adding up all 1 nm slices from 0 to 50 nmwould give 0.19; please note that the total intensities of thetransmission and the EBSD patterns cannot be quantitativelycompared with each other since we do not know the absoluteefficiency of diffuse vs. coherent scattering in our model). Wesee clear differences in the contrast of the EBSD patterns withincreasing depth of the slice: compared to the 4- to 5-nm slice,the slices at larger depths begin to change contrast, and for theslice extending from 49 to 50 nm, we see that it contributeswith a contrast that is inverted with respect to the slices nearerto the surface. However, the absolute contribution of electronsfrom these depths to the final pattern is negligible (0.014contributed from 4 to 5 nm vs. 0.00004 from 49 to 50 nm), ifelectrons start with equal probability from each depth.

The contrast reversal of Kikuchi bands with thicknessis theoretically well understood in the transmission case(Høier, 1973, and references therein). Experimentally, wecan increase the contribution of the slices at larger depthsby changing the incidence conditions. Choosing an incidenceangle nearer to the surface normal direction results in a deeperpenetration of primary electrons into the sample. Now, ifwe observe those electrons that are backscattered at shallowangles with respect to the surface plane, we can expect thatthese have to traverse the largest amount of material, andthus they experience a large effective thickness for dynamicalinteractions. This is why a contrast reversal is observed first forthe electrons with the largest angles with respect to the surfacenormal, because effectively they come from larger depths, asviewed along their path. This explains the observation, alreadyin the early experiments, of Kikuchi band contrast reversal inEBSD patterns when going to steeper incidence angles (Alamet al., 1954).

Combination of coherent and incoherent diffraction

The combined treatment of coherent elastic scattering andincoherent scattering in transmission electron diffractionexperiments in general is a complicated problem. Energyfiltering is one way to remove the inelastically scatteredelectrons from the observed pattern. However, even withmodern high-resolution energy filters (Brink et al., 2003;van Aken et al., 2007), the thermal diffuse scattering cannotbe removed, and so it is important to include this effect inquantitative simulations. A successful way to include thediffuse scattering of electrons by thermal vibrations is the“frozen phonon” approach in multislice calculations (Loaneet al., 1991; Muller et al., 2001), where the nonperiodicityin the crystal induced by thermal vibrations is explicitlyincluded via correspondingly displaced atomic coordinates.The calculation is then carried out for a number of differentatomic arrangements. This approach has been shown to givevery good agreement with experimental measurements (VanDyck, 2009).

At first sight, a similar procedure seems to be necessaryfor EBSD simulation since the incident beam can in principle

scatter coherently as well as incoherently to produce thebackscattered diffraction pattern. As we have seen, a majorsimplification of the problem is possible by noticing thatusually one does not observe significant signs of scattering thatis coherent with the incident beam, which would be indicatedby the appearance of diffraction spots. These spots are observedonly under rather special circumstances in the standard EBSDsetup in the SEM: at grazing incidence and exit angles, one canrealize a reflection high energy diffraction type of experimentand spot patterns are observed (Baba-Kishi, 1990). In this case,a unified treatment of the coherent high energy diffraction spotpattern and the Kikuchi pattern in the background would benecessary to achieve a quantitative description of the relativeintensities of coherently and incoherently scattered intensityby dynamical high energy diffraction theory (Korte & Meyer-Ehmsen, 1993; Korte, 1997).

However, this would be much more information than weactually need for the simulation of an EBSD pattern. In astandard EBSD setup with large scattering angles, the coherentpart is practically absent, and we are left with finding therelative angular intensity variations within the incoherentpart itself (containing the Kikuchi pattern). As we have shownearlier, this is possible by a standard CBED-type calculationvia application of the reciprocity principle, leading to goodagreement with experimental EBSD patterns (Winkelmannet al., 2007; Winkelmann, 2008). We actually do not needto know exactly how large is the coherent part relative tothe incoherent part, if experimentally the coherent part isnegligible. This explains the surprising success of dynamicalEBSD simulations assuming effectively a complete incoherencebetween the incident beam and the backscattered electronwaves.

If it is possible to approximate experimentally observedtransmission patterns as a weighted sum of coherentand incoherent contributions, this provides a way toapproximately include thermal diffuse scattering, for examplein dynamical CBED Bloch-wave calculations (Omoto et al.,2002). Both coherent and incoherent contributions can bederived from the solution of exactly the same eigenvalueproblem in the Bloch-wave calculation. The close relationshipof coherent dynamical scattering of a convergent incidentbeam and the thermal diffuse scattering from internaldivergent incoherent sources is shown in model calculations ofa thought-experiment in Fig. 4. We calculated simultaneouslya bright-field [001] CBED disk for Molybdenum at 20 kV inFig. 4(a), and the scattering from incoherent point sources inFig. 4(b). This is a thought-experiment in the sense that CBEDmeasurements are conventionally not carried out at 20 kV, butusually at energies of 100 kV or more. By comparison withEBSD measurements, we know that the incoherent diffusescattering is correctly reproduced, so that we can assume thatalso the CBED pattern is a plausible representation. Fig. 4(c)shows qualitatively how diffuse scattering is influencingthe coherent CBED disk. It is arbitrarily assumed that the



Fig. 4. Simulation of an idealized experiment illustrating the influence of coherent and incoherent scattering in the formation of a general diffractionpattern. 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 keVelectrons, 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 sourcesemitting 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 disksare not shown.

coherent scattering is dominant, and thus only a relativelylow incoherent intensity is added. Qualitatively, we see thatintensity appears in the formerly dark region of the coherentpattern, and that the measured intensity in the disk itself is alsoinfluenced by a structured background (see also Fig. 7). Becauseof this possible structured background, it is necessary to havea correct model of the incoherent scattering in quantitativeanalyses of CBED measurements for structural investigations(Saunders, 2003).

Comparison to experiment

The general results of the previous sections are now applied toa comparison of experimental EBSD pattern with dynamicalsimulations assuming different semi-realistic depth profilesfor the diffracted backscattered electrons. To obtain modeldepth distributions, the Monte Carlo program CASINO wasused (Drouin et al., 2007). In Fig. 5, we show simulatedaverage depth distributions of backscattered electrons froma 20 kV primary beam incident at 70◦ onto a Mo sample.Since the dominating diffraction contrast in the EBSD patternis produced by electrons which have lost energy of up to abouta few hundred eV (Deal et al., 2008; Winkelmann, 2009), twocases were considered: first, only backscattered electrons thathave lost not more than 500 eV (20–19.5 keV) and, second,electrons having lost not more than 1500 eV (20–18.5 keV).The latter group can be expected to originate on averagefrom larger depths than the former, which is reproduced bythe simulation. It is stressed here that the simulated depthdistributions serve simply as theoretical model assumptions inorder to analyse their influence on the dynamical calculationwhereas they cannot be expected to quantitatively reproducedetails of the real depth distribution. The theoretical modelused in the CASINO code is the continuous slowing downapproximation, which for example, does not take into accountthe quantized loss of energy and thus cannot be expected to

Fig. 5. Simulated depth-profiles of backscattered electrons from Mo usingthe CASINO Monte Carlo code (Drouin et al., 2007) and fitted to analyticalmodels with parameters tm (see text).

accurately reproduce the backscattered energy spectrum anddepth distribution in the quasi-elastic regime that is relevantfor the diffraction contrast in EBSD. Keeping these seriouslimitations in mind, the Monte Carlo simulated depth profileswere fitted to two analytical models: a Poisson distributionI P ∼ x · exp(−x/tm), and an exponential decay I E ∼exp(−x/tm). The Poisson distribution reflects the statistics ofa single large-angle backscattering event after a mean path tm

and is seen to agree well with the Monte Carlo simulations forthe 500 eV loss group, whereas the agreement for the largerenergy losses is quantitatively less good, but qualitativelystill consistent. The exponential decay model is included forcomparison to analyse the implications of the neglect ofthe finite penetration depth before backscattering and the



corresponding local maximum in the depth distribution. Asis seen in Fig. 5, this is the main qualitative difference betweenthe two analytical models.

Dynamical EBSD simulations were then carried out forbcc Mo (a = 3.147 Å) at 20 kV, assuming the analyticaldepth distributions with fit parameters tm that are listed inFig. 5 (tm are the mean depths of backscattering in bothmodels). In the dynamical simulation, 925 reflections withminimum lattice spacing dhkl = 0.35 Å were included, theDebye–Waller factor was taken as B = 0.25 Å2 (Peng et al.,1996). In Fig. 6, the results of the dynamical simulationsare compared to an experimental EBSD pattern from a Mosingle crystal measured at 20 kV (Langer & Dabritz, 2007).

The intensity in the dynamical simulations is scaled fromthe minimum to the maximum calculated value in eachsimulation separately, neglecting a possible background thatis present in experimental patterns due to delocalized andinelastically scattered electrons. In connection with the factthat the dynamical simulations are restricted to a singleenergy, this leads to a generally higher contrast and sharpnessof the simulated patterns compared to the experimental EBSDpatterns. Apart from this limitation, we can see an overallconvincing agreement of all the dynamical simulations withthe experimental pattern, including the pattern fine structureand the presence of higher-order Laue zone (HOLZ) rings(Michael & Eades, 2000).

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



Comparing the dynamical simulations in detail, we seea noticeable influence of the different assumed depthdistributions on the simulated patterns. First, we comparethe 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 rangeof thickness that contributes to (c) and (d): as we haveseen in Fig. 3(c), the contrast of Kikuchi bands tends toreduce and invert for layers in larger depths. Looking moreclosely at the differences between the Poisson model and theexponential decay model, we notice that the exponential modelproduces 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-resolvedsimulations in Fig. 3(c): since the very low depths dominate inthe exponential decay model much more than in the Poissonmodel, the corresponding patterns in the exponential modelcontain more contributions from patterns qualitatively likethe unsharp 1 nm pattern in Fig. 3(c). These contributionsare characterized mainly by high intensity intensity in thezone axes, without other fine structure due to the extinctiondistance effects discussed earlier. Accordingly, the simulatedMo 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) showsthe worst agreement with experiment (mainly based on theintensity in the [001] zone axis), the simulations wouldlead to the interpretation that the electrons from depths upto about 10 nm characteristic for models (a), (b) and (d)dominate in the experimental pattern. These low depths will befurther emphasized by the additional inelastic and incoherentscattering in the outgoing path for electrons from largerdepths.

We see that the dynamical simulations exhibit a detectableinfluence of the assumed depth profile of backscatteredelectrons on EBSD patterns, enabling us to draw conclusionsabout the probable depth distribution of backscatteredelectrons which are consistent with previous estimations(Dingley, 2004). Compared to the simplified Monte Carlomodel considered here mainly for illustration, the use ofmore realistic models of the energy- and angle-dependentbackscattering process (Werner, 2001) should lead to a moreaccurate description of the backscattering depth profile andcould provide additional information concerning the depth-sensitivity of EBSD measurements in variable geometries.

Conclusions

In this paper, we illustrated the interconnections betweenthe Bloch-wave simulation frameworks for convergent beamelectron diffraction in the transmission electron microscopeand electron backscatter diffraction in the scanning electronmicroscope. The dynamical simulation of EBSD patterns is

Fig. 7. Visualization of the connection between coherent and incoherentscattering as discussed in this paper. The patterns of electrons whichare scattered by a crystalline sample into all possible directions can bemapped on spheres. Imagine a small crystal sample in the center of thespheres, from where electrons are emitted and made visible when they hitthe surrounding spheres from the inside. Left Ball: simulated electronbackscatter diffraction (EBSD) pattern from Molybdenum at 20 keVelectron energy, Right Ball: simulated “full solid angle” Convergent BeamElectron Diffraction (CBED) bright field pattern of a coherently transmittedbeam. 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 theincident beam can be neglected. By explicit simultaneoussimulation of CBED and EBSD patterns with the same computerprogram, it was shown how the calculation of the EBSDpattern from incoherent point sources is related to a bright-field transmission calculation for a probe of convergence anglecorresponding to the field of view of the EBSD phosphorscreen. Depth-resolved dynamical EBSD simulations exhibita detectable influence of the assumed backscattering depthprofile on the intensity distribution in EBSD patterns, whichprovides an additional information channel in studies applyingcomparisons of experimental and simulated EBSD patterns.

Acknowledgements

The author thanks E. Langer (Technical University Dresden,Germany) for supplying the experimental EBSD pattern.

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