Graef M. Introduction to conventional transmission electron microscopy

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388 Two-beam theory in defect-free crystals

The thickness variations are generated through the superposition of 50 cosine waves

with random amplitude and phase. The microscope acceleration voltage is 200 kV,

002

= 48.4 nm, ξ



002

= 1489.2 nm, and ξ



= 576.7 nm. The foil orientation varies

from w =−1to+1 from left to right. This type of image contrast is sometimes

observed in thin foils, and is usually caused by specimen preparation artifacts.

The preceding examples, which took less than 0.3 s each to compute on a 512 ×

512 grid of pixels, show that it is quite straightforward to simulate two-beam images.

For a given reﬂection g, we only need to compute the relevant imaging parameters

(ξ

,ξ



,ξ



, and β

), provide an array of local thickness and excitation error values,

and feed this information to the

TBCalcInten routine. The reader is encouraged to

modify the sample geometries provided in the

BFDF.routines ﬁle, and/or to create

additional geometries.

6.5.5 Two-beam convergent beam electron diffraction

In Chapter 4 (Section 4.6.4), we have seen how to obtain a two-beam CBED pattern.

Such a pattern is essentially a parallel recording of many diffraction patterns for

a continuous range of incident beam directions. Consider the schematic drawing

in Fig. 6.15; the transmitted disk is centered on the optical axis, and each point in

the disk corresponds to a different tangential component k

of the incident wave

vector k

. We have also seen that a beam tilt is equivalent to a specimen tilt in

the opposite direction, so that a two-beam CBED pattern contains roughly the

same information as the bright ﬁeld image of a bend contour. Electrons which are

scattered into the diffracted beam g experience an elastic momentum transfer equal

|g|

transmitted

disk

diffracted

disk

-k

(b)

2θ

(a)

Fig. 6.15. (a) Schematic illustration of the geometry of a two-beam convergent beam elec-

tron diffraction pattern; (b) relation between equivalent locations in the bright ﬁeld and dark

ﬁeld diffraction disks.

6.5 Numerical two-beam image simulations 389

ω = π/2

(a)

π/2

K¢

ω = π/

2+α

(b)

Fig. 6.16. (a) Incident beam normal to g; (b) for the Bragg orientation a beam tilt (or sample

tilt) of α is needed to bring the reciprocal lattice point onto the Ewald sphere.

to hg, which means that the point in the diffracted disk equivalent to the point −k

in the transmitted disk is at the location g − k

The opening angle of the incident beam is the beam divergence angle θ

;itis

clear from the drawing in Fig. 6.15 that if θ

= θ

, with θ

the Bragg angle, then

the transmitted and diffracted disks will touch each other at the point g/2. From

the Bragg equation in reciprocal space, we also know that the Bragg condition is

satisﬁed when the center of the Ewald sphere lies on the perpendicular bisector

plane of g, which is equivalent to saying that the tangential component of the

incident wave vector must be equal to k

=−g/2. If the initial incident beam

direction k is perpendicular to g, as shown in Fig. 6.16(a), then g is not in Bragg

orientation. A beam tilt of α = θ

milliradians is needed to bring the reciprocal

lattice point g onto the Ewald sphere. The angles θ

and α are therefore independent

variables, both of which can be set by the microscope operator. Depending on the

microscope model, the angle θ

may have either a discrete range of settings or

a continuous range, whereas α can be continuously adjusted using the beam tilt

controls.

The intensity distribution inside the transmitted and diffracted disks can be un-

derstood easily from the theoretical derivations in this chapter. Since we are con-

sidering scattering from only one set of planes, described by a single g,wehave

rotational symmetry around g; in other words, the angle α is sufﬁcient to deﬁne

the orientation of the sample. The incident beam directions relative to the sample

therefore fall in the range [α − θ

,α+ θ

], which corresponds to the line segment

AB in Fig. 6.15(b). The distance between A and B is determined by θ

, and the mi-

croscope operator can change the orientation of the sample (i.e. α) independently.

It is an easy experiment to convince yourself that the intensity distribution inside

390 Two-beam theory in defect-free crystals

the disks can be changed by tilting the sample or the incident beam orientation. The

size and relative location of the disks do not change when α is changed.

In Exercise 2.9 on page 134 we have derived an expression for the excitation

error s

as a function of the angle ω between the transmitted beam direction and g.

Since ω = π/2 + α, we can use this expression to determine the angle α for which

= 0:

cos ω =

λ|g|

, (6.52)

from which follows

α = sin

−1



λ|g|



≈

λ|g|

= θ

. (6.53)

For this particular value of α, s

will be equal to zero at the center of the diffraction

disk. Moving away from the center towards A or B (Fig. 6.15b) the excitation error

becomes negative (towards B) or positive (towards A). The intensity proﬁle along

AB is therefore the dark ﬁeld rocking curve proﬁle derived earlier in this chapter.

The excitation error varies from s

(θ

− θ

)atA to s

(θ

+ θ

)atB. When θ

, and

hence the disk radius, is increased, a larger portion of the rocking curve proﬁle

becomes visible inside the diffraction disk.

Numerical simulation of a two-beam CBED pattern is now a straightforward

exercise. The ION routine

tbcbed.pro displays two-beam CBED patterns for six

different sample thicknesses (multiples of ξ

/2), for a given microscope acceleration

voltage V , beam divergence angle θ

, and beam tilt angle α. The user can also change

the length of the reciprocal lattice vector g. Example output of the

tbcbed routine

is shown in Fig. 6.17 for the following parameters: d = 0.2 nm, V = 120 kV,

= 100 nm, and the absorption ratio is equal to 15. Fig. 6.17 shows the output of

three separate runs of the

tbcbed routine, for the following pairs of (α, θ

): (5, 5),

(7, 6), and (11, 7) mrad. The Bragg angle for this conﬁguration is θ

= 8.373 mrad.

It is apparent from the examples in Fig. 6.17 that the details of the CBED pattern

are quite sensitive to the sample thickness. It is possible to use two-beam CBED to

estimate the local sample thickness z

and the extinction distance ξ

. This was ﬁrst

suggested by MacGillavry in 1940 [Mac40] and subsequently applied by, among

others, Kelly et al. [KJBN75], Allen [All81] and Allen and Hall [AH82].

From equation (6.22), we see that the scattered intensity vanishes (in the absence

of absorption) whenever

σ z



1 + w

= n, (6.54)

with n a positive integer. If there are N intensity minima on one side of the Bragg

orientation in the scattered disk, then there will also be N different values for n,

6.5 Numerical two-beam image simulations 391

Fig. 6.17. (a) Example output of the

tbcbed.pro ION-routine for the imaging parameters

stated in the text.

starting with n

= n, n

= n + 1, and so on. For each minimum, we also have a

corresponding excitation error s

. This results in

+ z

= (n + i)

. (6.55)

The parameters z

, z

, and n are unknown and must be determined. This can be

done in a number of different ways. Perhaps the most straightforward method is to

write down a χ

-type expression:

N −1



i=0

+ z

− (n + i)

= χ

, (6.56)

and minimize it with respect to the three parameters. In order for this method to work

there must be at least three intensity minima in the diffraction disk. Alternatively,

we could plot s

as a function of (n + i)

and through linear regression analysis

392 Two-beam theory in defect-free crystals

determine the parameters which provide the best ﬁt for the following relation:

(n + i)

−

. (6.57)

It is easy to show geometrically that the parameters s

are given by the expression

λ, (6.58)

where X is the distance between the centers of the two disks (conveniently measured

from the edge of one disk to the corresponding edge of the other disk), and X

is the distance between the ith minimum and the Bragg orientation. Since the

dark ﬁeld rocking curve is always symmetric, the Bragg orientation is readily

identiﬁable.

In practice, the following procedure may be used to estimate the local thickness

and the extinction distance. First, obtain a two-beam CBED pattern, as explained

in Section 4.6.4. Then, measure on the electron micrograph the distances X and

along the line connecting the disk centers, as shown in Fig. 6.18. Compute the

length of g from the crystallographic parameters. Measure the distance between the

center of the dark ﬁeld disk and the location of the Bragg orientation, and the radius

of the diffraction disks. Distances are negative on the outside of the dark ﬁeld disk,

and positive towards the central disk, as shown in Fig. 6.18. The reader will ﬁnd

the

tbthick.pro ION routine on the website. This routine takes all the experimental

...

positive

distances

negative

distances

Fig. 6.18. Illustration of the input parameters for the tbthick.pro ION routine.

6.5 Numerical two-beam image simulations 393

(a) (b)

(e)

close to [11.3]

00.0

11.2

01.1

10.1

40.4

00.0 11.2

(f)

Fig. 6.19. Experimental two-beam CBED patterns for the (11.

2) reﬂection of Ti (Philips

EM420 operated at 120 kV). The thickness increases from (a) to (d). The diffraction pattern

in (e) shows that in addition to the two main reﬂections, the (40.

4) reﬂection is also strongly

excited; this gives rise to the diagonal dark lines running through both bright ﬁeld and

dark ﬁeld disks in (a)–(d). The pattern in (f) is the output of the tbthick.pro routine on the

website for the parameters of (d).

parameters and minimizes the χ

expression to obtain ξ

, z

, and n. The routine also

displays the computed two-beam CBED pattern for the ﬁtted parameters, so that a

comparison with the experimental image may be carried out. The source-code for

the algorithm may be found on the

website.

As an example of a thickness determination, consider the two-beam CBED pat-

terns shown in Figs 6.19(a)–(d); the patterns were obtained in a Philips EM420,

operated at 120 kV, using a Ti foil oriented close to the [11.3] zone axis.

Figure 6.19(e) shows the corresponding diffraction pattern. The thickness of the foil

increases from (a) to (d). The parameters deﬁned in Fig. 6.18 were measured on an

enlarged print of Fig. 6.19(d), and are given by (all in mm): X = 68.5, T = 16.5,

Y = 0.0, X

= 3.0, X

= 6.0, X

= 8.8, and X

= 11.2. The d-spacing of the

(11.

2) planes is 0.1248 nm. The ﬁtting algorithm of the

tbthick.pro routine then

returns the following parameters: z

= 134.9 nm, ξ

= 87.2 nm, and n = 2. This

compares reasonably well with the theoretical extinction distance for the {2

1.2}

394 Two-beam theory in defect-free crystals

family of ξ

= 86.47 nm (at 120 kV; computed using the xtalinfo.f90 program). The

simulated CBED pattern is shown in Fig. 6.19(f).

The two-beam CBED method provides a simple way to estimate the local thick-

ness of the thin foil and the extinction distance. Comparison of the measured ξ

with a calculated value may reveal signiﬁcant differences between the two val-

ues. The measured values are only reliable to the same extent that the two-beam

approximation is reliable. In other words, if a crystal structure has a large lattice

parameter along the direction g, then the two-beam approximation will fail since

the intensity in other reﬂections of the type ng may not be negligible. The accuracy

of the thickness and extinction distance determination improves as the length of g

increases, or, equivalently, when higher-order planes are selected for the acquisi-

tion of the CBED pattern. The accuracy also improves with decreasing microscope

acceleration voltage, since the increased curvature of the Ewald sphere improves

the validity of the two-beam approximation. Two-beam CBED thickness determi-

nations should therefore always be regarded as rough estimates. A more detailed

and accurate thickness determination is only possible if the complete intensity pro-

ﬁle and possibly the effects of other diffracted beams are taken into account. For

a detailed comparison of various two-beam methods to determine the local foil

thickness and the extinction error we refer to the paper by Delille et al. [DPVC01].

We will discuss the contributions of multi-beam scattering in the next chapter.

Exercises

6.1 Repeat the dispersion surface construction of Fig. 6.9 for the case where the foil normal

n is not perpendicular to the reciprocal lattice vector g.

6.2 The

TBBFDF.f90 program is provided on the website along with a list of possible

sample geometries (

BFDF.routines). Create a new sample geometry for a sample that

has facets on its top and bottom surfaces. The facets correspond to surface steps of a

few nanometers, and can be triangular, rectangular, or lozenge-shaped. Compute bright

ﬁeld and dark ﬁeld images, and determine for which step height

the steps still give rise

to detectable image contrast.

6.3 Derive the explicit equations (6.24) and (6.25) from the Bloch wave formalism.

6.4 Show geometrically that the excitation error corresponding to a minimum intensity in

the dark ﬁeld CBED disk is given by equation (6.58).

Systematic row and zone axis orientations

7.1 Introduction

In the previous chapter, we analyzed in great detail how the bright ﬁeld and dark

ﬁeld image contrast for the two-beam case depend on the sample thickness and

the crystal orientation. We deﬁned two dimensionless parameters (w and z

)in

terms of which the intensities can be expressed in the absence of absorption. Two

additional parameters appear in the presence of absorption: the normal absorption

length ξ



and the anomalous absorption length ξ



. The former causes an exponential

damping with increasing sample thickness, common to both bright ﬁeld and dark

ﬁeld intensities; the latter produces an asymmetry between the bright ﬁeld intensities

for positive and negative excitation errors. We used the Bloch wave approach to

explain the origin of this anomalous absorption, and we introduced thermal-diffuse

scattering as one of the sources of the absorption process.

In the present chapter, we will consider the complete N-beam dynamical diffrac-

tion equations in several of the formalisms described in Chapter 5, still under the

assumption of a perfect crystal. We will discuss a number of numerical techniques

for solving the dynamical equations, and provide illustrations of various contrast

features which are commonly observed in experiments but cannot be explained

by the two-beam theory. We will discuss matrix methods, the differential equation

approach, the Bloch wave approach, the multi-slice method, and the real-space

method. For each of those methods we will introduce source code in the form

of pseudo code descriptions. While many of the multi-beam simulations are too

computationally demanding to perform in real-time, the

website has several ION

routines which illustrate various aspects of the multi-beam simulations. We will,

along the way, introduce a number of techniques to speed up the computations,

mostly through the use of symmetry arguments. For instance, for a beam direction

slightly inclined to a zone axis there are usually additional beam directions, related

to the ﬁrst one by the symmetry operations of the crystal structure, which will give

395

396 Systematic row and zone axis orientations

rise to an identical intensity distribution at the exit plane of the crystal (identical

apart from a symmetry operation). It is then computationally more efﬁcient to

ﬁrst identify equivalent beam orientations, and then carry out the calculation for

only one of the equivalent orientations. A detailed description of the application of

diffraction group symmetry to multi-beam dynamical calculations is available as a

web-based appendix to this book.

7.2 The systematic row case

7.2.1 The geometry of a bend contour

The solutions to the two-beam dynamical diffraction equations can be expressed in

terms of the two dimensionless parameters w and z

, as illustrated in Section 6.3.3.2.

This effectively means that two-beam image contrast is independent of the partic-

ular material being studied. A visual comparison of the bright ﬁeld images for the

four study materials shown in Chapter 4 immediately shows that there are signif-

icant differences in the details of bend contour and bend center contrast, and that

therefore different materials do give rise to different image contrast. The two-beam

theory represents the idealized case, and, with very few exceptions, it is also the

only case that can be solved analytically.

†

One of the most striking differences between, for instance,

the bend contour in

Fig. 4.19 (page 281) and the bright ﬁeld image for the standard parameter space

shown in Fig. 6.3 (page 355) is the fact that in the experimental contour there are two

regions near the foil edge where the bright ﬁeld contrast shows intensity oscillations

(arrows in Fig. 4.19). In addition, the darker image contrast is conﬁned to a narrow

band (the actual bend contour), whereas in the standard parameter space image all

orientations with negative w display reduced contrast due to anomalous absorption.

The two-beam theory, therefore, cannot explain all of the details of bend contour

contrast.

It is, however, not very difﬁcult to understand the image contrast associated with

a bend contour. Consider the schematic in Fig. 7.1. A bent foil of constant thickness

is oriented such that on the left-hand side of the diagram the planes with plane

normal −g are close to the Bragg orientation; on the right-hand side diffraction

occurs from the opposite side of the planes −g, or, equivalently, from the planes g.

Consider the excitation error for the planes −g: towards the left of the drawing, there

is a location in the foil for which s

−g

= 0 (location 2). At this point in the foil, g lies

rather far outside of the Ewald sphere (s

< 0), so it is a good approximation to use

the two-beam theory for the image contrast. The two-beam bright ﬁeld and dark

†

There are certain multi-beam cases that also have analytical solutions (e.g. [Fuk66]), but the general multi-beam

dynamical diffraction equations have no known analytical solutions.

7.2 The systematic row case 397

-g

> 0

-g

= 0

> 0

= 0

-gg

< 0

= s

-g

Fig. 7.1. Illustration of the geometry giving rise to a bend contour. The orientation of the

lattice planes −g (1) varies such that the reﬂection −g approaches the Ewald sphere (2),

falls outside

of the Ewald sphere (3), and then the situation repeats for the lattice planes

(4) and (5).

ﬁeld contrast around the location s

−g

= 0 is shown on the lower left-hand corner

of the ﬁgure, for an arbitrary reduced thickness z

= 4. When s

−g

< 0, anomalous

absorption occurs and the bright ﬁeld image will have a rather low intensity in this

region.

The same situation occurs on the other side of the foil: the planes g are in exact

Bragg orientation at location 4. Since s

−g

is large and negative, we can again use

the two-beam theory to describe the image contrast, as shown on the lower right-

hand side. Note that this time the bright ﬁeld curve is reversed, and the anomalous

absorption occurs on the side for which s

< 0. In the central region both s

−g

and

are negative, which means that anomalous absorption will occur for scattering

from both sides of the planes. The situation depicted at location 3 will thus give

rise to a low transmitted intensity and this is what makes a bend contour look dark

in bright ﬁeld images.

At location 3, both −g and g have the same excitation error, and the two-beam

approximation must break down. A computation of the image contrast at this loca-

tion will therefore involve at least three beams. We say “at least” because there may

be more than three beams involved; consider location 5 in Fig. 7.1. The reciprocal

lattice point g is well inside the Ewald sphere, but the point 2g is close to the Ewald