Graef M. Introduction to conventional transmission electron microscopy

Подождите немного. Документ загружается.

418 Systematic row and zone axis orientations

002

004

220

440

111

GaAs

V [Volt]V [Volt]

0.0

0.5 1.0

x [dimensionless]

111

004

440

111

002

220

(a)

(b)

Fig. 7.14. [1

10] projection of the structures of Si (a) and GaAs (b), along with the systematic

row potentials for three different rows.

intensities for As and Ga, i.e. (N/z

)

/(N/z

)

. The following observations

can be made:

for large beam tilts k

/g, the integrated intensity approaches a constant level for all

systematic row orientations;

since the (220) GaAs potential curve in Fig. 7.14(b) is symmetric, there is no difference

between the integrated intensities for the Ga and As sites; conversely, both (002) and

(111) GaAs potential curves show a difference between the Ga and As sites, and therefore

the integrated intensities for both sites are

also different;

for the (111) GaAs systematic row, there are virtually no beam tilts for which the ratio

equals unity; if the thickness integrated intensity varies strongly with incident beam

orientation, then the corresponding systematic row orientation can be used for ALCHEMI-

like observations.

We conclude that electron channeling near systematic row orientations may sig-

niﬁcantly alter the probability of electron-induced x-ray emission from particular

7.3 The zone axis case 419

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

0.0

0.2

0.4

0.6

0.8

1.0

-6 -4 -20246

1.0

0.0

2.0

1.0

0.0

1.0

0.0

Ga = As

1.0

0.0

1.0

0.0

-6 -4 -20 2 46

2.0

1.0

0.0

-6 -4 -20246

/g k

Silicon GaAs As/Ga

004 002

440 220

111 111

2.0

Fig. 7.15. Thickness integrated intensities for the systematic row orientations of Fig. 7.14,

for a 100 nm thick

foil, 200 kV, and

−6 < k

/g < 6 (29 beams). The last column shows

the ratio of the As curve to the Ga curve.

sites within the unit cell. For quantitative analytical observations, this orientation

dependence must be taken into account.

7.3 The zone axis case

In this section, we will review how the multi-beam dynamical diffraction problem

can be solved for the zone axis orientation. First, we will describe the geometry

of the problem, and to what extent it differs from the systematic row case. Then

we will introduce several simulation methods, equivalent to the

SR.f90 program

420 Systematic row and zone axis orientations

described for the systematic row case. We will describe how the lattice symmetry

can be used to reduce the number of multi-beam simulations needed for bend-center

and convergent beam computations. We will describe brieﬂy how channeling effects

determine the thickness-averaged electron-induced x-ray emission probabilities for

the zone axis case.

7.3.1 The geometry of the zone axis orientation

In Chapter 4, we have seen several examples of images or diffraction patterns ob-

tained in the zone axis orientation. Figure 4.18 shows a bend center in Ti; multiple

bend contours intersect at the location of the exact zone axis orientation. The result-

ing diffraction pattern is symmetric with respect to the central beam. Figure 4.25

shows a convergent beam electron diffraction pattern for GaAs, again for the exact

zone axis orientation.

Near a zone axis orientation, many reciprocal lattice points lie close to the Ewald

sphere. We can always select two short reciprocal lattice vectors g

and g

, and

express every other reciprocal lattice point in the same zone as a linear combination

of these two vectors:

g = mg

+ ng

where m and n are integers.

†

For computational purposes, we will use a single

index to label all the diffracted beams. We will either number the reciprocal lattice

points one row at a time, or number them by family {hkl}, with increasing numbers

indicating increasing |g

hkl

The basic multi-beam simulation proceeds as described on page 399, with es-

sentially no differences between the systematic row case and the zone axis case,

except for:

(i) the determination of the contributing beams (explained in the next paragraph), and

(ii) the graphical representation of the resulting intensities or Bloch wave coefﬁcients and

eigenvalues, which are now 2D functions of the incident beam orientation.

The reader will ﬁnd a zone axis multi-beam program MB.f90 on the website; this

program implements the zone axis case using the scattering matrix approach, the

differential equation approach, and the Bloch wave formalism.

For a given incident beam direction t

uvw

, it is rather straightforward to determine

two independent reciprocal lattice vectors which belong to the zone [uvw]. We can

†

The selection of the two shortest vectors must be made carefully. As we will see in Section 9.2.3 it is possible for

diffracted beams corresponding to “forbidden reﬂections” to have a non-zero intensity. This means that those

reciprocal lattice points must be taken into account, even when they correspond to a vanishing structure factor.

7.3 The zone axis case 421

simply select the following two vectors:

= va

∗

− ua

∗

;

= wa

∗

− ua

∗

It is easy to verify that both vectors satisfy the zone equation. If one or more indices

of the direction [uvw] are zero, then the corresponding reciprocal basis vector can

be selected as one of the two vectors. As an example, consider the direction [201]:

the second index is zero and therefore we select g

= a

∗

. The second vector can

then be obtained from the second relation above as g

= a

∗

− 2a

∗

. The algorithm

is straightforward to implement and the subroutine

ShortestG in the library ﬁle

symmetry.f90 implements the general and all special cases.

The vectors obtained from this procedure are not necessarily the shortest possible

reciprocal lattice vectors. If we denote the shortest vectors by g

and g

, then we

have

= n

+ m

;

= n

+ m

where n

, n

, m

, and m

are integers. It is then straightforward to determine these

integers such that the lengths of g

and g

are minimized. This is also implemented

in the subroutine

ShortestG. The vector g

is selected to fall on a symmetry element,

such as a mirror plane, if one is present. This is accomplished by computing the

families {g

} and {g

} and selecting as the ﬁnal g

the vector with the smallest

multiplicity.

If we do not use symmetry arguments to reduce the number of incident beam

directions for which the dynamical equations are to be solved, then we can simply

deﬁne a square grid of points, each point corresponding to an image pixel and also

to a particular value of k

(i, j) = i

+ j

⊥

, (7.11)

with  the grid spacing (in nm

−1

). The vector g

⊥

is normal to both g

and the

incident beam direction t, i.e.

⊥

= g

× t. (7.12)

Since directions are expressed with respect to the direct basis vectors, we must make

sure that the components of t are converted to reciprocal space before computation

of the cross product.

Once the tangential wave vector component is known, it is straightforward to

determine the complete wave vector, since we know that its length should be

422 Systematic row and zone axis orientations

equal to λ

−1

. For a convergent beam simulation, the range of the indices i and

j in equation (7.11) is determined by the illumination cone angle θ

and the grid

spacing :

+ j

≤



λ



. (7.13)

For a bend center computation, we can deﬁne the range of i and j by means of

the maximum length of k

(i, j) along the vector g

. If we represent the maximum

projection of k

onto g

by k

, and the number of grid points along g

by N

, then

the grid spacing (in nm

−1

)isgivenby

 =

The total number of beam orientations along the positive and negative g

vector is

then equal to N

= 2N

+ 1. The number of grid points N

along g

⊥

is taken to be

equal to N

, so as to obtain a square grid.

The routine

Calckvectors in the MB.f90 program implements the computation of

the incident wave vectors. The algorithm employs dynamical memory allocations

using pointers to store all incident beam directions in a so-called linked list. This

list is then passed on to the routine which computes the actual multi-beam solution

for each member of the list. While such a memory allocation scheme is not really

needed for this type of computation, it will become useful when we incorporate

symmetry arguments in the algorithm; the number of independent incident beam

directions is then not known a priori, and dynamical memory allocations are then

the only efﬁcient means to implement the multi-beam computation. Once the inci-

dent beam directions are known, it is also a trivial matter to compute the excitation

error of each scattered beam using the

Calcsg routine. The computation of all fac-

tors s

completes the diagonal of the crystal structure matrix A. The off-diagonal

elements can be precomputed before the iteration over all beam directions is

started.

7.3.2 Example simulations for the zone axis case

In this section, we will discuss several representative multi-beam computations for

the zone axis case. Once the exit plane wave function is known for all incident beam

directions, we can use several representation methods for the images: conventional

bright and dark ﬁeld images, convergent beam electron diffraction patterns, and

Eades patterns. Examples of each will be provided below. All multi-beam compu-

tations are carried out with the

MB.f90 program, and image output is generated with

MBshow.f90.

7.3 The zone axis case 423

220 220

220

020

200

2mm

4mm

2mm

Fig. 7.16. [001] zone axis bright ﬁeld and dark ﬁeld images for a spherically bent copper

foil at 200 kV; the foil is 100 nm thick, and the Bloch wave computation was performed for

49 beams.

7.3.2.1 Bend center simulations

Figure 7.16 shows a bright ﬁeld (center) and eight dark ﬁeld images for a spherically

bent copper foil oriented along the [001] zone axis. The computation was performed

using the

MB.f90 program for 49 beams, 512 × 512 pixels, a crystal thickness of

100 nm, and a microscope acceleration voltage of 200 kV. The Laue center position

ranges from −3to+3 along both g

200

and g

020

systematic rows. The bright ﬁeld

image shows a rich variety of fringes and contrast variations. Note that the dark

ﬁeld images for opposite reﬂections, e.g. 200 and

200, are displaced with respect

to the center of the corresponding bend contour. This corresponds to the scenario

described in Fig. 7.1, where the Bragg orientation is satisﬁed for each individual

reﬂection at locations with opposite excitation errors. The symmetry of the patterns

is also indicated in Fig. 7.16: the bright ﬁeld image has an obvious 4mm symmetry,

424 Systematic row and zone axis orientations

(a) (b)

(c)

(d)

(g)

(h)(e)

(f)

(i)

(j) (k) (l)

Fig. 7.17. Bright ﬁeld zone axis patterns corresponding to Fig.7.16, for various locations

of the selected area aperture,

and the resulting diffraction patterns.

whereas both {200} and {220} patterns have 2mm symmetry, in agreement with the

entries for the 4mm1

diffraction group in Table 5.1 and Fig. 5.11.

†

One might wonder about the usefulness of this type of image simulation. After

all, bend contours are not usually considered to be of great interest in conventional

microscopy observations. There is, however, a distinct advantage in understanding

the nature and origin of bend contours, and how they relate to local diffraction

conditions. Suppose that you have a foil of copper which is bent and oriented in

such a way that you observe an intersection of bend contours, similar to the one

shown in Fig. 7.17(a). If you were to insert a selected area aperture which would

enclose the entire region, the corresponding diffraction pattern would be that shown

in Fig. 7.17(b), where the intensity in each individual beam is an average over the

entire area included in the SA aperture. There is not much difference between

the intensities of different beams, other than perhaps a general decrease for higher-

order reﬂections. For a smaller aperture centered on the zone axis, (c), the diffraction

pattern, (d), shows an average over a smaller area, and intensity differences between

†

Cu has point group m

3m, which corresponds to the diffraction group 4mm1

according to Fig. 5.11. For

this diffraction group the bright ﬁeld image has symmetry 4mm, whereas the special G reﬂections have 2mm

symmetry. Both {200} and {220} are special reﬂections because they lie on mirror planes.

7.3 The zone axis case 425

diffracted beams are more pronounced. If we move the aperture along one of the

individual bend contours, towards the upper right-hand corner, (e) and (g), then

the corresponding diffraction patterns, (f) and (h), show an off-axis pattern, since the

Laue center of the selected area no longer coincides with the optical axis. Note that

when the aperture is centered on an individual bend contour, the diffraction pattern

will reveal symmetric intensities for g and −g reﬂections, as is apparent from the

systematic row patterns in (f), (h), and (j). If the aperture is moved away from

the center of the bend contour, as in (k), then one reciprocal lattice point will

move closer to the Ewald sphere, and the diffraction pattern approaches that of

a true two-beam situation; the arrowed reﬂection in (l) is in nearly exact Bragg

orientation.

The bend center image depends strongly on the foil thickness, as shown in

Fig. 7.18 for a 49-beam Bloch wave computation for pure copper (200 kV, [001]).

The thickness increases in steps of 25 nm. For small foil thicknesses, the contrast

features are a bit diffuse, and intensity oscillations are broad. For larger thicknesses,

the number of intensity oscillations increases, similar to the increase of the number

of thickness fringes in the two-beam case (see Fig. 6.4). Experimental observations

of bend centers are often not as detailed as the simulated images. There are several

possible reasons for this difference: a high defect density may blur the smallest

features of the image, and the foil is not necessarily symmetrically bent, giving

rise to various distortions of the bend center image. In addition, inelastic scattering

often causes a signiﬁcant amount of blurring of the ﬁnest details. Zero-loss energy

ﬁltering may remove most of the inelastic scattering, and it may be necessary to

25 nm

50 nm 75 nm

100 nm

200 nm

175 nm150 nm

125 nm

Fig. 7.18. Bright ﬁeld zone axis patterns for increasing foil thickness. Computations were

carried out for the same conditions as Fig. 7.16.

426 Systematic row and zone axis orientations

cool the sample down to a low temperature to enable observation of the smallest

details in Fig. 7.18.

It is relatively easy to incorporate more realistic foil shapes into the computations.

The ﬁgure on the cover of this book was computed for the same foil shape as

Fig. 7.16, and in addition the foil thickness increases linearly from 0 to 300 nm

(bottom to top). The computation used a 77-beam Bloch wave simulation (|g|≤

28.5nm

−1

), 200 kV, [001] zone axis orientation, 2100 × 3000 image pixels. A

second example of a bend center simulation is shown in Fig. 7.19: an experimental

500 nm

(a)

(b)

Experiment

Simulation

Fig. 7.19. Experimental (a) and simulated (b) bend center for [211] copper at 200 kV. The

foil thickness increases from 40 (left) to 110 nm (right). The incident beam tangential com-

ponent varies from −7 ≤ k

≤+7 horizontally and vertically. A 47-beam Bloch wave

simulation was carried out for 750 × 525 image pixels. The normal absorption parameter



was reduced by a factor of 10 to enhance the contrast in the thicker region of the foil and

a γ brightness correction of 1.5 was applied. (Experimental image courtesy of S. Tandon.)

7.3 The zone axis case 427

[211] bend center in a thin copper foil (a) can be compared with a simulated image

(b). The simulation uses 47 beams (|g|≤25.0nm

−1

), an acceleration voltage of

200 kV, and the thickness increases from 40 nm on the left to 110 nm on the right.

The tangential component of the wave vector ranges from −7|g

| to +7|g

along both vertical and horizontal directions. The horizontal bend contour is the

1) contour, the vertical contour corresponds to (0

22). The agreement with the

experimental image is remarkably good, down to the smallest details, even allowing

for the distortions due to the local foil shape. Along the horizontal bend contour,

the experimental image is somewhat blurred due to the presence of dislocations

which destroy the local lattice symmetry.

The reader should be warned that this type of image simulation can be very time

and memory consuming; the cover page illustration took about 76 hours of CPU time

on a 660 MHz Compaq TRU64 UNIX workstation, using the

cover.f90 program,

available from the

website. This type of computation lends itself to implementation

on parallel computers, or on distributed workstation clusters, since each image pixel

is independent of all the others. In addition, the use of symmetry arguments can

signiﬁcantly speed up the computation. In the case of the cover ﬁgure, we used

the vertical mirror plane to reduce the number of computations by a factor of 2.

Furthermore, points at equal distance from the horizontal center line have Bloch

wave solutions related to each other by symmetry; only the foil thickness is different

for these pairs of points. This provided a further reduction by a factor of 2.

7.3.2.2 Bloch wave simulations

The results of a Bloch wave simulation can also be displayed as 2D images, 3D

surfaces or contour plots. Figure 7.20 shows a 3D rendering of the dispersion

surface for the Cu simulation of Fig. 7.16. The tangential component of the wave

vector varies from 0 ≤ k

200

≤

along both 200 and 020 systematic rows. The

vertical bar on the left indicates the position of the bend center (i.e. the incident

beam is parallel to the zone axis). In the absence of the electrostatic lattice potential,

the empty crystal approximation, the dispersion surface would consist of a set of

intersecting spheres, centered on each of the contributing reciprocal lattice points.

In the presence of a lattice potential the various branches of the dispersion surface

are asymptotic to these spheres. While the spheres for the systematic row are all

aligned along the row and give rise to a relatively simple dispersion surface drawing

(e.g. Fig. 7.7b), for the zone axis case the geometry of the dispersion surface is

signiﬁcantly more complicated. The dispersion surface construction of Fig. 6.9 on

page 369 remains valid for the zone axis case.

The Bloch wave excitation amplitudes α

( j)

can also be represented as intensity

distributions. The ﬁrst eight coefﬁcients are shown in Fig. 7.21(a) for the wave