Graef M. Introduction to conventional transmission electron microscopy

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428 Systematic row and zone axis orientations

(1)

(2)

(3)

[100]

(4)

Fig. 7.20. Three-dimensional representation of the 25 dispersion surfaces γ

( j )

of Cu for

the conditions of Fig. 7.16. Only one quadrant of the surfaces is shown.

vector range −1 ≤ k

200

≤+1 and −1 ≤ k

020

≤+1. Close to the zone axis

orientation, the ﬁrst and second Bloch waves are strongly excited. When the beam

tilt is increased, higher order Bloch waves begin to contribute, in a manner similar

to that shown in Figs 7.7(c) and 7.8(c). Figure 7.21(b) shows the corresponding

absorption coefﬁcients q

for the ﬁrst eight Bloch waves. The absorption coefﬁ-

cients vary smoothly for the ﬁrst Bloch wave, and show rapid variations for the

higher-order waves.

Finally, the position of each Bloch wave with respect to the atom positions can

also be computed, similar to the systematic row results in Fig. 7.11. Figure 7.22

shows the ﬁrst 14 Bloch waves for the [001] zone axis orientations of copper, along

with the projected atom positions (upper left). It is clear that the ﬁrst and second

Bloch waves are centered on the atom positions. Close to the zone axis orienta-

tion, these two waves have the strongest excitation coefﬁcients, as can be seen in

Fig. 7.21(a). Most of the beam electrons spend most of their time in the crystal

close to the projected atom positions (for orientations close to the zone axis orien-

tation) and one can draw an analogy with bound electron states in atoms [BLS78].

7.3 The zone axis case 429

(1) (2) (3) (4)

(5) (6) (7) (8)

(1) (2) (3) (4)

(5) (6) (7) (8)

= |g

200

(a)

(b)

Fig. 7.21. (a) The ﬁrst eight Bloch wave excitation coefﬁcients |α

( j )

; (b) the ﬁrst eight

Bloch wave absorption coefﬁcients q

( j )

for the conditions of Fig. 7.16.

This analogy lies at the basis of an alternative description of elastic electron scat-

tering in terms of the atomic column approximation introduced by Van Dyck and

Chen [VDC99]. Each projected atom position represents a column of atoms parallel

to the beam direction. Electrons travel through the foil and “channel” along these

columns. The probability of ﬁnding an electron inside the column oscillates with

depth in the crystal, and different Bloch waves have a different oscillation period

(i.e. a different extinction distance). We will return to this oscillation behavior in

Section 7.4.1 when we discuss the multi-slice method.

430 Systematic row and zone axis orientations

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Fig. 7.22. Bloch wave contour plots for the ﬁrst 14 Bloch waves for the [001] zone axis

orientation of copper. The computations used 25 Bloch waves and were carried out for

200 kV electrons.

7.3.2.3 CBED patterns

Zone axis convergent beam patterns can be computed using the same algorithms

discussed in the previous sections; the only two differences are the range of incident

beam directions, which is deﬁned according to equation (7.13), and the representa-

tion of the result as a set of (possibly overlapping) disks. The

MBCBED.f90 program

can be used to simulate zone axis CBED patterns based on the Bloch wave formal-

ism. The input parameters are the beam convergence angle θ

, the camera length L,

the zone axis direction [uvw], the location of the Laue center, and the microscope

acceleration voltage E. The output is a series of CBED patterns for increasing foil

thickness. The total number of incident beam directions to be considered depends

on the camera length and on the desired printing resolution. For a printing resolu-

tion of 300 dpi and a camera length of L = 1000 mm, a beam convergence angle of

= 8 mrad would correspond to a disk radius of 8 mm, which requires 189 pixels

or incident beam directions along a disk diameter. There is no point in computing

more beam directions than will be displayed in the ﬁnal pattern.

The results of two example calculations are shown in Figs 7.23 and 7.24.

Figure 7.23 shows two GaAs CBED patterns computed using the Bloch wave

approach of the

MBCBED.f90 program at 120 kV; 35 beams were used for foil

thicknesses of 50 and 120 nm. These images are to be compared with the experi-

mental images of Fig. 4.25 on page 290. The agreement between simulations and

7.3 The zone axis case 431

(a) (b)

Fig. 7.23. Simulated CBED patterns for the [110] zone axis orientation of GaAs. The

simulation parameters are 120 kV, θ

= 5.1 mrad, and 35 beams were used (Bloch wave

formalism). The foil thicknesses are 50 nm (left) and 120 nm (right). The patterns are to be

compared with the experimental patterns in Fig. 4.25 on page 290.

Fig. 7.24. Series of 15 [11.0] CBED patterns for Ti (200 kV, θ

= 2.6 mrad, 57 beam Bloch

wave simulation). The kinematically forbidden ±(00.1) reﬂections are arrowed in the ﬁrst

pattern. The thickness increases from 10 nm in steps of 10 nm from left to right and top to

bottom. (Intensities have been scaled exponentially to emphasize the weaker features.)

432 Systematic row and zone axis orientations

experimental observations is rather good. The absence of a vertical mirror plane

for the whole pattern symmetry is clear for the thicker foil and indicates that the

structure is non-centrosymmetric.

While it is easy to compare experimental and simulated CBED patterns visually,

it is possible to do a lot better than that. CBED patterns, in particular energy-

ﬁltered CBED patterns in which most of the inelastically scattered electrons have

been removed, can be used to determine with very high accuracy the amplitude

and phase of electron structure factors. For a detailed description of the procedures

involved we refer the interested reader to the book Electron Microdiffraction by

Spence and Zuo [SZ92]; a review of quantitative CBED can be found in [Spe93].

Figure 7.24 shows a series of 15 computed CBED patterns for the [11

20] zone

axis orientation of Ti. 57 beams were used at 200 kV, for a beam convergence angle

of 2.6 mrad. The thickness increases in steps of 10 nm from left to right and top

to bottom; the ﬁrst image on the top left has z

= 10 nm. A total of 6625 incident

beam directions were used for the central disk. The kinematically forbidden (00.l)

reﬂections (l = 2n + 1) are arrowed in the ﬁrst pattern. As the thickness increases

a faint intensity can be observed in the forbidden disks. The intensity distribution

shows a thin dark horizontal line for all thicknesses. Such a line in a kinematically

forbidden reﬂection is known as a Gj

onnes–Moodie line or G-M line [GM65]; the

origin of such lines will be discussed in Section 9.2.3. G-M lines can be used to

distinguish between different space groups, as illustrated in Section 9.5.

HOLZ reﬂections can also be taken into account in this type of CBED simula-

tions. This would turn the diffraction problem into a true three-dimensional prob-

lem, instead of simply using only the ZOLZ reﬂections. Computationally, HOLZ

reﬂections would increase signiﬁcantly the size of the dynamical matrix and hence

the computation time. If HOLZ reﬂections are to be included in the CBED com-

putation, the method of Bethe potentials, introduced in Section 7.3.3, should be

employed to reduce the size of the dynamical matrix.

7.3.2.4 Eades patterns

We have seen in Fig. 7.3, that a parallel beam incident on a bent foil is equivalent

to a convergent beam incident on a planar foil. If we had a method to acquire a

diffraction pattern for every incident beam orientation separately (somewhat like

serial acquisition of a convergent beam pattern), then we would be able to image

directly patterns similar to those shown in Fig. 7.16. In practice, this can be done by

using a narrow but parallel incident beam, similar to that used in microdiffraction

observations, and rocking the beam in two orthogonal directions. For each incident

orientation we measure the bright ﬁeld intensity and the dark ﬁeld intensity for

a particular Bragg angle (or range of angles). If your microscope has a scanning

attachment (STEM attachment) then you can set up the illumination conditions to

7.3 The zone axis case 433

obtain such double-rocking patterns or Eades patterns [Ead80]. If your microscope

has an annular dark ﬁeld detector, a ring-shaped detector that measures electron

current scattered into a given angular range, then you can determine for each incident

beam direction the total intensity scattered in between two conical surfaces. Plotting

the bright ﬁeld and dark ﬁeld signals as a function of beam position then results in

the Eades patterns.

Figure 7.25 shows a simulated Eades pattern for the [001] zone axis orientation of

Cu. The acceleration voltage is 200 kV, crystal thickness is 200 nm, and the annular

range includes the following families: {200}, {220}, {400}, and {420} (not shown as

Bright field Dark field sum

200 220 400

Fig. 7.25. Eades pattern simulation for the [001] zone axis orientation in Cu (200 kV,

200 nm thickness). The dark ﬁeld images of the 200, 220, and 400 reﬂections are copied

into the symmetrized patterns of the second row, which are then added together to form the

dark ﬁeld image.

434 Systematic row and zone axis orientations

separate images, but included in the dark ﬁeld sum). The dark ﬁeld image is the sum

of the symmetrized dark ﬁeld images of the individual reﬂections. For instance, after

application of the 4mm symmetry to the 200 dark ﬁeld image, the symmetrized

pattern on the second row is obtained. Summation of all such patterns for the

reﬂections inside the annular range (in this case from 10 to 32 mrad) then results in

the complete dark ﬁeld image on the bottom row. Simulations of Eades patterns are

identical to the simulations of CBED and zone axis bend center patterns. It is only the

ﬁnal representation for the dark ﬁeld image that is different. The dark ﬁeld images

for all reﬂections detected by the annular dark ﬁeld detector are added together in

the proper relative orientation and position. Other experimental techniques can be

used to obtain the same information; we refer the reader to the beautiful volumes on

convergent beam electron diffraction by Tanaka and coworkers [TT85], [TTK88],

and [TTT94]. The computation of Eades patterns has been implemented as part of

the

MBshow.f90 program.

7.3.2.5 Thickness-integrated intensities: zone axis case

The thickness-integrated intensity expression derived in Section 7.2.3 can also be

used for the zone axis case. The algorithm is identical, with the only difference

being that the resulting intensities are represented as 2D plots instead of 1D curves.

The computation has been implemented in the

MBTII.f90 program. An example of

the output of this program is shown in Fig. 7.26 for the [110] zone axis orientation

of GaAs. The acceleration voltage is 200 kV, foil thickness 50 nm, and 117 beams

(up to 18 nm

−1

) were taken into account for a total of 66 049 (= 257

) incident

beam directions (resulting in a computation time of slightly more than 5 days on a

666 MHz workstation). The maximum incident beam tilt is k

=−g

006

. The image

on the left shows the thickness-integrated intensity for the Ga positions; the intensity

varies between 0.223 and 2.296. The intensity proﬁle for As is very nearly the mirror

image (vertical mirror) of that for Ga, with an intensity range between 0.237 and

2.433. The third image shows the ratio of As to Ga; the intensity range is from

Ga As As/Ga ratio As+Ga

Fig. 7.26. Thickness-averaged integrated intensities for Ga and As in the [110] zone axis

orientation of GaAs. Simulation parameters are stated in the text.

7.3 The zone axis case 435

0.227 to 4.931. The ratio varies rapidly from one location to another, indicating

that channeling along the [110] zone axis orientation is strong and that this zone

axis should be avoided for quantitative EDS or EELS experiments. The sum of the

Ga and As integrated intensities is shown as the ﬁnal image and resembles the [110]

bright ﬁeld bend center image.

It should be clear by now that this type of multi-beam simulation can be rather

time-consuming, in particular when large numbers of beams are taken into ac-

count. In the following two sections, we will introduce brieﬂy two different ways

to increase the speed of the computations (other than running them on parallel

computers): Bethe potentials, which can reduce the size of the dynamical matrix

through the use of perturbation theory, and diffraction group symmetry, which can

be used to reduce the number of independent incident beam directions for which

the computations need to carried out.

7.3.3 Bethe potentials

The bend center computation in Fig. 7.19 shows that certain reﬂections only con-

tribute to a small region of the bright ﬁeld image (Bragg lines). Everywhere else

in the image their contribution is small. It is therefore not strictly necessary to take

exact account of the contributions of all these weak reﬂections to every pixel in

the image. We can use the following criterion to determine whether a reﬂection

is “strong” or “weak”: if the reciprocal lattice point is far enough from the Ewald

sphere, so that |s

| is signiﬁcantly larger than λ|U

|, then we can consider the con-

tribution of that reﬂection to be a perturbation to the overall scattered amplitude.

The criterion for a beam to be weak is derived from equation (5.51) in Chapter 5:

( j)



h=g

g−h

( j)

= 2k

( j)

If the prefactor of the ﬁrst term (k

|) is signiﬁcantly larger than the prefactor of

the second term (|U

g−h

|), then the coupling of g to h will be weak. Let us rewrite

this eigenvalue equation as follows:

( j)



h=g

g−h

( j)

= 0,

with

( j)

≡ 2

− k

( j)

We will denote weak beams by primed reciprocal lattice vectors, e.g. g



. We will

also assume that weak beams do not interact with each other, only with strong

436 Systematic row and zone axis orientations

beams. In that case, the eigenvalue equation for a weak beam becomes

( j)



( j)







−h

( j)

= 0,

where the summation runs only over the strong beams. This equation can now be

solved for the Bloch wave coefﬁcients of the weak beam:

( j)



=−

( j)







−h

( j)

. (7.14)

If the Bloch wave coefﬁcients of all strong beams are known, then this equation

can be used to determine the coefﬁcients of the weaker beams.

Next, we rewrite the eigenvalue equation for the strong beams by splitting the

summation over strong and weak beams:

( j)



h=g

g−h

( j)





g−h



( j)



= 0.

The Bloch wave coefﬁcients C

( j)



of the weak beams can be replaced by equa-

tion (7.14). This is left as an exercise for the reader. After rearranging the terms in

the resulting equation we arrive at

( j)

−





g−h



( j)



( j)



h=g

g−h

−





g−h



−h

( j)



( j)

= 0, (7.15)

where the factors η

( j)



in the denominators of the sums over the weak beams must

be approximated by 2k



. If we introduce the following notation:

¯η

( j)

≡ η

( j)

−





g−h



( j)



;

g−h

≡ U

g−h

−





g−h



−h

( j)



then the eigenvalue equation becomes

¯η

( j)



h=g

g−h

( j)

= 0, (7.16)

which has the same form as the original eigenvalue equation. We have replaced

the excitation errors of the strong beams by effective excitation errors; the Fourier

coefﬁcients of the electrostatic lattice potential have also been modiﬁed to take the

weaker beams into account. The coefﬁcients

g−h

are known as Bethe potentials

or dynamical potentials. This approximation was ﬁrst introduced by Bethe using

ﬁrst-order perturbation theory [Bet28].

7.3 The zone axis case 437

In all multi-beam computations described thus far, we have always explicitly

taken all beams into account. The advantage of doing this is that the off-diagonal

elements of the dynamical matrix need only be computed once. The disadvantage

is that the dynamical matrix has a large number of entries and perhaps only a small

number of beams with appreciable amplitude. We can reduce the size of the dynam-

ical matrix by incorporating all the weakest reﬂections by means of Bethe potential

coefﬁcients. The disadvantage of this approach would be that the off-diagonal ele-

ments of the dynamical matrix must be recomputed for every incident beam orien-

tation, but this is more than offset by the reduced size of the Bloch wave eigenvalue

problem. Bethe potentials have been implemented in the

CalcDynMat routine, which

is available from the

website. A detailed description of the implementation is also

available in PDF format as a web appendix (

appbethe.pdf) to this book.

The Bethe potential can be rewritten as follows:

= U

− λ





g−h



From this relation, we see that the Bethe potential can become equal to zero for a

particular wavelength λ, or, equivalently, for a particular acceleration voltage E.

The voltage for which a Bethe potential coefﬁcient vanishes is known as a critical

voltage E

, and the vanishing of

g−h

is known as the critical voltage effect.Ifthe

Bethe potential coefﬁcient for a particular reﬂection vanishes, then that reﬂection

will have zero intensity for all crystal thicknesses, even when the structure factor for

that reﬂection does not vanish. The voltage for which an allowed reﬂection vanishes

for all crystal thicknesses can be determined by varying the acceleration voltage over

a large range (e.g. from 100 to 1000 kV) and this, in turn, allows for the accurate

determination of the potential coefﬁcients U

g−h

. There is extensive literature on

the critical voltage effect and we refer the reader to [LHMF72] and [SZ92] for

more details.

7.3.4 Application of symmetry in multi-beam simulations

We have seen in Chapter 5 that the symmetry of the dynamical diffraction process

is described by the 31 diffraction groups. The diffraction groups relate the intensity

at a particular point of a diffraction disk to the intensity at one or more equivalent

points on other diffraction disks of the same family. It is possible to use this group

theoretical information in numerical computations to reduce the number of beam

directions for which the dynamical calculations must be performed. The imple-

mentation of the diffraction groups is rather technical and would interrupt the main

line of this text, so we refer the interested reader to a web appendix (

MBsym.pdf),

which deals with all of the details. For completeness we simply list in Table 7.2,