Graef M. Introduction to conventional transmission electron microscopy

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408 Systematic row and zone axis orientations

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-3 -2 -101

0.2

0.0

-0.2

-0.4

-0.6

-0.8

-3 -2 -10123

1.0

0.8

0.6

0.4

0.2

0.0

3.0

2.5

2.0

1.5

1.0

0.5

-3 -2 -10123

(nm

-1

)

- k

(nm

)

ⴛ10

(nm

)

/g k

(c)

(a)

(b)

(d)

(

Fig. 7.8. (a) and (b) eigenvalues γ

( j )

of the dynamical matrix for BaTiO

, 200 kV, nine-beam

101

systematic row simulation. (c) Bloch wave excitation coefﬁcients and (d) absorption

coefﬁcients (( j) = 1–6).

BaTiO

, the second Bloch wave excitation amplitude, labeled 2 in Fig. 7.8(c),

shows a small asymmetry near the center (arrows); this asymmetry is due to the

non-centrosymmetric nature of the structure. If we move the Ti atom to the center of

the cell, then the asymmetry disappears. Conversely, if we move the Ti atom further

away from the center, the asymmetry increases. The excitation amplitudes for the Cu

systematic row do not show any asymmetry since the structure is centrosymmetric.

The absorption components q

( j)

are shown in Figs 7.7(d) and 7.8(d); they are

again periodic along the systematic row since they are the imaginary parts of the

7.2 The systematic row case 409

-3 -2 -10123

0.0

0.1

0.2

0.1

0.2

0.1

0.2

0.1

0.2

000

202

404

606

202

404

606

Fig. 7.9. Systematic row bright ﬁeld and dark ﬁeld intensities for a 100 nm thick Cu foil,

oriented along the g

202

systematic row (see Fig. 7.7).

complex eigenvalues !

( j)

. The lowest index absorption coefﬁcients vary strongly

with beam orientation while the higher index coefﬁcients are nearly constant. The

absorption coefﬁcients contain both normal absorption and anomalous absorption

contributions.

Figures 7.9 and 7.10 show the bright ﬁeld and dark ﬁeld intensities for both

systematic rows for a 100 nm thick foil. The bright ﬁeld curve shows a lower in-

tensity between k

/g =−

and +

, and this would correspond to a dark bend

contour in a bright ﬁeld image. The +g and −g reﬂections give rise to symmet-

ric dark ﬁeld contours, and the width of the contour decreases rapidly with in-

creasing Miller indices. This can also be seen in the simulated bend contours of

Fig. 7.4. For BaTiO

, the 101 and

1 dark ﬁeld curves are asymmetric (arrows),

again a signature of the non-centrosymmetric nature of the crystal structure. A

careful comparison of experimental 101 and

1 bend contours may reveal this

asymmetry.

Using equation (5.43) on page 321, we can determine what each of the individual

Bloch waves looks like, similarly to the analysis that led to the type I and type II

Bloch waves for the two-beam case in Chapter 6. Figure 7.11(a) shows the modulus

squared of Bloch waves C

(1)

to C

(8)

for the g

202

systematic row in Cu, for identical

simulation parameters as in Figs 7.7 and 7.9. Two sets of curves are shown in relation

410 Systematic row and zone axis orientations

-3 -2 -10123

000

101

202

303

404

101

202

303

404

0.000

0.125

0.250

0.125

0.250

0.125

0.250

0.125

0.250

0.125

0.250

Fig. 7.10. Systematic row bright ﬁeld and dark ﬁeld intensities for a 100 nm thick BaTiO

foil, oriented along the g

101

systematic row (see Fig. 7.8).

to the lattice planes in the lower half of the ﬁgure:

†

the solid lines correspond to

the Bloch wave intensities for incident beam directions of the type k

/g = n, with

n = 0, ±1, ±2,..., i.e. for the Bragg orientation of the even multiples of g

202

. The

dotted curves show the same curves for the orientation k

/g = n/2, i.e. for the Bragg

orientation of the odd multiples of g

202

. These curves repeat themselves because of

the periodicity of the Bloch wave coefﬁcients C

( j)

and the eigenvalues !

( j)

The Bloch waves for BaTiO

are shown in Fig. 7.11(b). The asymmetry caused

by the displacement of the Ti atom from the center of the unit cell is clearly visible

for the Bloch waves (2), (3), and (4). It is no longer straightforward to deﬁne a type I

and type II wave, but it is clear from the simulations in Fig. 7.11 that several waves

have their maxima on the projected atom positions, whereas others have maxima

in between the atoms.

†

Since the systematic row is a 1D section of the reciprocal lattice, the structures in Figs 7.11(a) and (b) must be

projected onto the [101] direction.

7.2 The systematic row case 411

[101]

(202) systematic row (101) systematic row

Cu BaTiO

(a) (b)

Fig. 7.11. Bloch wave intensities of the ﬁrst eight Bloch waves for (a) the g

202

systematic

row in Cu and (b) the g

101

systematic row in BaTiO

, both at 200 kV, nine-beam simulation.

The waves are drawn in the correct position relative to the structure drawings in the lower

half.

7.2.2.7 Systematic row convergent beam electron diffraction

Figure 7.2 shows a typical systematic row convergent beam electron diffraction

(SRCBED) pattern. Several diffraction disks have signiﬁcant intensity in them,

mostly in the form of parallel fringes, similar to those discussed for the two-beam

case in Section 6.5.5. The simulation of a systematic row CBED pattern proceeds

along similar lines as for the two-beam CBED pattern: for a given incident beam

direction (represented by k

), beam divergence angle θ

, and g-vector, the N -beam

rocking curve is computed and displayed across diffraction disks located at the

proper relative positions. Pseudo code for the algorithm is shown in

PC-17 .

The algorithm is implemented in the

SRCBED.f90 program; example output is

shown in Fig. 7.12 for the (200) systematic row in copper, 200 kV, nine beams.

The ﬁgure shows the intensity distributions for the reﬂections ±(400), ±(200) and

(000), for ﬁve different incident beam directions. The beam direction changes from

200

= 1.0, for which

400 is in Bragg orientation, to k

200

=−1.0, corre-

sponding to s

400

= 0. The SRCBED pattern contains the same information as

412 Systematic row and zone axis orientations

Pseudo Code PC-17 Systematic row CBED simulation.

Input: PC-2

Output: systematic row CBED pattern

ask user for number of beams

repeat

ask for k

, α, g and θ

compute geometrical parameters

set up crystal structure matrix A

use Taylor expansion to compute S

compute scattered amplitudes

create individual diffraction disks

check for disk overlap

if coherent illumination and disk overlap then

add amplitudes to row image

else

add intensities to row image

end if

dump image to PostScript ﬁle

until no further patterns requested

Fig. 7.9, but represented in a different form. Note that each diffraction disk acts

as a “window” through which a portion of the corresponding bright ﬁeld or dark

ﬁeld rocking curve can be observed. Increasing the diameter of the condensor aper-

ture

†

results in a larger window; moving the aperture reveals different portions of

the rocking curves. Moving the aperture is equivalent to tilting the conical incident

beam, so we can also use the beam tilt to translate the disks in the plane of the pattern.

Any of the systematic row algorithms discussed in the previous sections may

be used for this computation; the

SRCBED.f90 implementation available from the

ebsite

uses the scattering matrix (Taylor expansion) approach. It is left as an

exercise for the reader to implement the SRCBED algorithm using the Bloch wave

approach.

7.2.3 Thickness integrated intensities

To conclude this section on systematic row multi-beam dynamical scattering

we will apply the Bloch wave formalism to the computation of the orientation

†

Some microscopes, like the JEOL 3000 series, have an optional α-selector, which permits a ﬁner control of the

beam convergence angle, so that the discrete condensor aperture diameters no longer determine the only disk

diameters that can be obtained.

7.2 The systematic row case 413

000 202 404202404

202

1.0

0.5

0.0

-0.5

-1.0

5 nm

-1

Fig. 7.12. Simulated systematic row CBED patterns for the (202) row in copper; 200 kV,

foil thickness 100 nm, nine-beam simulation (these conditions are identical to those of

Fig. 7.9). The radius of the diffraction disks corresponds to a convergence angle of

9.0 mrad.

dependence of electron-induced x-ray emission from selected sites in the unit cell.

Although this book does not deal with the multitude of analytical methods based on

inelastic scattering processes, it is important for the microscope user who wishes to

employ these analytical techniques to understand how dynamical elastic scattering

processes can affect the results of analytical observations.

7.2.3.1 Theoretical derivation

Consider a cubic crystal structure with two atom types, say A and B. For simplicity

we take the structure to be the CsCl structure, i.e. atom A is located in the origin

and atom B in the cell center. We know from structure factor considerations that for

the disordered structure all reﬂections for which h + k + l = 2n + 1 are forbidden.

For the ordered structure, these reﬂections have |F

hkl

|=|f

− f

|. We can now

ask the question: what is the probability that the beam electrons will induce x-ray

emission by A (or B) atoms, and how does this probability depend on the sample

orientation?

The answer to this question is not straightforward. We need to compute the

probability that an incident electron will transfer energy to an atom at a certain site

414 Systematic row and zone axis orientations

in the unit cell, for a given orientation of the sample with respect to the incident

beam. We do not concern ourselves with the computation of the inelastic scattering

probability itself; that would require detailed considerations of various inelastic

scattering mechanisms and is beyond the scope of this book. We are, however,

in a position to compute a second factor of importance to the inelastic scattering

probability: the probability that a beam electron will actually travel near a given

atom type. The total inelastic scattering probability from an atom A is then given by

the product of the inelastic scattering cross-section σ

, which is a quantity derived

from time-dependent quantum mechanics, and the probability that a beam electron

is actually in a position to undergo this inelastic scattering event. It is this second

factor that can be computed directly as a by-product of a multi-beam Bloch wave

simulation.

The probability that a beam electron will be found at a certain site r

in the

unit cell is given by the modulus squared |(r

of the electron wave function

(r) evaluated at that site. Since this probability depends upon the thickness of

the sample, we will compute the average probability over the sample thickness.

Mathematically, this is done as follows [Kri88, Wan95]: if S denotes the set of

equivalent points in the unit cell for which we wish to evaluate the probability

of inelastic scattering, then the number of electrons per unit thickness which may

undergo inelastic scattering events by atoms in the set S is given by



i∈S





∗

)(r

)dz



, (7.4)

where z

is the foil thickness. Using the Bloch wave formalism, the electron wave

function can be expressed as

(r) =



( j)



( j)

2πi[k

( j )

+g]·r

;



( j)



( j)

2πi[k

+γ

( j )

n+iq

( j )

n+g]·r

If we only consider normal incidence, i.e. n · r = z, then the wave function can be

rewritten as

(r) =



( j)

−2πq

( j )



( j)

2πiγ

( j )

2πi(k

+g)·r

Substitution in equation (7.4) (exercise) results in



i∈S

2πi(h−g)·r



(i)∗

(k)

( j)∗

(k)

7.2 The systematic row case 415

where

−2π(α

+iβ

dz,

and

= q

( j)

+ q

(k)

;

= γ

( j)

− γ

(k)

The integration is elementary and results in

1 − e

−4πq

( j )

4πq

( j)

; (7.5)

1 − e

−2π(α

+iβ

2π(α

+ iβ

( j = k). (7.6)

The thickness-averaged intensity can then be expressed as the product of two factors,

one determined by the atom positions and the active reﬂections, and one by the

crystal orientation through the Bloch wave coefﬁcients:







, (7.7)

with

≡



i∈S

2πi(h−g)·r

;

≡



( j)∗

(k)

The matrix elements S

can be precomputed once for a given subset S of atom

positions in the unit cell, whereas the matrices L

must be calculated for each

crystal orientation, or, equivalently, for each incident beam direction k

It is interesting to note that the off-diagonal terms of the matrix I (equation 7.6)

have an oscillatory character; with increasing crystal thickness, the off-diagonal el-

ements will become less important, and only the diagonal elements I

will remain.

It is then easy to show that

≈



|α

( j)

( j)∗

( j)

1 − e

−4πq

( j )

4πq

( j)

(for large z

). (7.8)

We can apply the above equation to the two-beam situation for our model

A–B crystal structure. Using the two-beam Bloch wave results from the previous

416 Systematic row and zone axis orientations

2.0

1.5

1.0

0.5

0.0

-3

-2 -1012 3

N/t

-3 -2 -1012 3

(N/t)

/ (N/t)

(b)(a)

Fig. 7.13. (a) Thickness-integrated intensity (two-beam case) for the A and B sites of the

CsCl structure; (b) ratio of the two curves in (a).

chapter we ﬁnd for the thickness averaged expression above (in the thick crystal

approximation without absorption):





= 1 −

sin(2β) = 1 −

1 + w

; (7.9)





= 1 −

sin(2β) cos π (h + k +l) = 1 −

w cos π(h + k + l)

1 + w

. (7.10)

The resulting curves are shown in Fig. 7.13; we ﬁnd that, for negative s

100

, the

induced x-ray photons are more likely to come from the A-sites than from the

B-sites. The opposite is true for a positive excitation error. Figure 7.13 also shows

the ratio of the A-curve to the B-curve. Note that this ratio is only dependent on the

excitation error for the superlattice reﬂections, for which h + k + l = 2n + 1; for

fundamental reﬂections the thickness integrated scattering probability for A and B

sites is identical for all excitation errors (since cos(2πn) =+1), and the ratio is

always equal to 1. At w =−1, the probability of electron-induced x-ray emission

from the A-sites is three times as large as for the B-sites, and a simple change of

the orientation to w =+1 changes the ratio to

The importance of this orientation dependence of the electron-induced x-ray

emission probability can not be overstated. The geometrical component of the

inelastic scattering probability for AB compounds with the CsCl structure de-

pends in a sensitive way on the foil orientation when superlattice reﬂections are

close to (two-beam) Bragg orientation. Nearly all analytical observation meth-

ods (energy dispersive x-ray spectroscopy (EDAX or EDS), electron energy

7.2 The systematic row case 417

loss spectroscopy (EELS), etc.) performed on crystalline samples aim to deter-

mine the local chemistry and electronic structure. It is crucial that the effect

of sample orientation on analytical TEM observations be taken into account

properly, otherwise the analytical results may be incorrectly interpreted. In

fact, one can make good use of the channeling phenomena that give rise to

this orientation dependence; in 1982, Taft¨o and Liliental [TL82] used the ori-

entation dependence of the electron-induced x-ray emission to determine the

cation distribution in ZnCr

2−x

spinels. This technique was then nicknamed

ALCHEMI (Atom Location by CHanneling Enhanced MIcro analysis) by Spence

and Taft¨o [ST83]. The ALCHEMI method is now used by many researchers

to determine site occupations and substitutional atom types in a large variety

of materials. The basic computation underlying the ALCHEMI technique uses

equation (7.7).

For more details on the ALCHEMI method and related theory we refer the inter-

ested reader to the following references: [MR84, RM84, Kri88, Pen88, RTW88,

AR93, JHJF99]. An early discussion of the channeling effect can be found in

[KLF74]. Since the two-beam approach of Fig. 7.13 is only a crude approximation

to the complete multi-beam scattering process, we will next perform more realistic

multi-beam computations for the systematic row orientation.

7.2.3.2 Thickness integrated intensities: systematic row case

As an example of the thickness integrated intensity for the systematic row case, let us

consider three g vectors belonging to the [1

10] zone of Si and GaAs. Figure 7.14(a)

shows the Si structure projected along [1

10]. The (111), (004), and (440) planes are

indicated by solid lines. Using the

SRpot.f90 program, we can compute the potential

of the systematic row for each of these planes; the resulting potential curves are

shown in Fig. 7.14(a). The curves were computed for a 29-beam case, and are

displayed for one repeat period (i.e. the distance x is drawn in units of the interplanar

spacing d

hkl

). The potentials were computed using the Fourier coefﬁcients V

the systematic row. Figure 7.14(b) shows the same curves for the [1

10] projection

of GaAs. In this case, the structure is non-centrosymmetric, and the (220) and (002)

reﬂections are allowed. The potential curves clearly show the difference between

the Ga and As positions.

Next, we apply equations (7.5)–(7.7) to all six systematic rows of Fig. 7.14.

The equations are implemented in the

SRTII.f90 program, along with the sys-

tematic row Bloch wave code of the

SR.f90 program. The resulting thickness

integrated intensities for a 100 nm thick foil, 200 kV acceleration voltage, and

−6 < k

/g < 6 are shown in Fig. 7.15. The ﬁrst column shows the results for

Si, the center column for GaAs, and the last column shows the ratio of the