Graef M. Introduction to conventional transmission electron microscopy

Подождите немного. Документ загружается.

458 Systematic row and zone axis orientations

multiply by the thickness. In addition, electrostatic contributions to the phase shift

will become important for the non-uniform thickness case. In the case of a foil with

uniform thickness the electrostatic phase shift is simply a constant. We will return

to the magnetic phase shift in Chapter 10 when we introduce image formation for

phase objects.

7.6 Recommended reading

There is a large amount of literature covering various aspects of multi-beam theory

and simulations. The slice methods are all based on the physical-optics approach

ﬁrst introduced by Cowley and Moodie in 1957 [CM57]. The ﬁrst use of the FFT

for multi-beam electron diffraction simulations was described by Ishizuka and

Uyeda in 1977 [IU77]. Various elastic scattering formalisms have been reviewed

by Goodman and Moodie [GM74]. Much of the initial literature (1970–1975) on

these simulation methods has appeared in journals such as Acta Crystallographica

A and Ultramicrocopy. Several simulation packages have then emerged, among

them SHRLI [OBI74], EMS [Sta87], NCEMS [OK89], MacTempas [Kil87], and

the real-space package [VDC84, CVD84a, CVD84b]. While these packages are

sufﬁcient for routine applications, the reader may ﬁnd it necessary to develop

new algorithms and code for more specialized situations. The interested reader

should also consult [KO89] for more information on simulation algorithms and

methods.

Source code for many applications can be found on the website of the Microscopy

Society of America at the URL

http://www.msa.microscopy.com. In many

cases one can make good use of routines and algorithms written by others (no

need to re-invent the wheel); a good starting point for general purpose numerical

procedures is the collection at

http://www.netlib.org, maintained by the Oak

Ridge National Laboratory. Combined with the Numerical Recipes book [PFTV89]

these resources should be sufﬁcient to tackle most computational problems.

Exercises

7.1 Consider a bright ﬁeld image containing a dark bend

contour of width

(this width

could be measured by locating two points on either side of the center of the contour

that have equal intensity). If the acceleration voltage of the microscope is increased

from V

to V

, what happens to the width of this contour (keeping all other parameters

constant)?

7.2 Show that π/λE

, with E

the total energy of the electron, is equal to the interaction

constant σ .

7.3 Use the

SRBW.f90 program as a starting point to write a new Bloch wave program

that will compute images similar to those shown in Fig. 7.6.

Exercises 459

7.4 The

SRCBED

.f90

program uses the scattering matrix approach for its dynamical in-

tensity computations. Rewrite this program using the Bloch wave approach. (You will

most likely be able to use signiﬁcant portions of the

SRBW.f90 program.)

7.5 Derive equations (7.9) and (7.10) for the two-beam case.

7.6 Implement equation (7.29) for the phase grating computation.

7.7 Show that for a wedge-shaped sample with constant slope the magnetic phase shift

depends on the square of the coordinate variable y (see equation 7.38).

Defects in crystals

8.1 Introduction

In this chapter, we will continue the discussion of the dynamical theory of electron

diffraction. We will cover several of the most important defects: dislocations, stack-

ing faults, and other planar boundaries, and we will discuss the diffraction contrast

caused by various types of interfaces, e.g. domain walls and twins. Finally, some

aspects of the weak beam technique will be covered. In all cases, we will describe

how image simulations can be carried out.

The survey of defects in this chapter is by necessity limited to only a few defects.

The reader will ﬁnd that the methods are sufﬁciently general, so that they may

be applied to other defects with only minimal modiﬁcations to the experimental

procedures or numerical algorithms.

8.2 Crystal defects and displacement ﬁelds

One of the most important applications of conventional TEM is the study of defects

in crystalline materials. As we have seen in Chapter 1, a crystal is characterized

by a periodic 3D lattice, decorated with a motif or asymmetric unit consisting of

one or more atoms. From space group theory, we know that the symmetry elements

of a crystal structure are either rotational,

†

translational, or a combination of both.

Since a space group is a group of inﬁnite order, and since real crystals are always

of ﬁnite size, we ﬁnd that a crystal cannot completely satisfy all elements of a

particular space group. In other words, space group symmetry must be broken at

one or more points in space. The most obvious case of broken symmetry is found at

the bounding surfaces of the crystal and we have already described the importance

of this type of symmetry breaking in Sections 2.6.3 (page 119) and 5.8 (page 328).

The surface truncates the inﬁnite crystal into a ﬁnite size and shape which can be

†

This includes both proper and improper rotations.

460

8.2 Crystal defects and displacement ﬁelds 461

represented by a shape function D(r), as described in Section 2.6.3. The existence

of relrods is a direct consequence of this type of symmetry breaking.

In addition to the free surfaces of a crystal, there are numerous other ways in

which space group symmetry may be broken. The broken symmetry may involve

only translational symmetry elements, as will be the case for the defects discussed in

this chapter, or it may involve rotational symmetry elements (a much rarer situation

in crystalline solids). Symmetry may be broken on a very local scale, such as a

missing atom (vacancy), or on a much more extended scale, as is the case for

planar defects.

When the translational symmetry of a crystal is broken, this generally means

that atoms are not found at the positions predicted by the Bravais lattice translation

vectors. The atoms are displaced from those positions by a vector R, which is in

general a function of position r. The vector ﬁeld R(r) is known as the displacement

ﬁeld. It relates the displaced position r



of the atom to the ideal (or reference)

position r by



= r + R(r). (8.1)

From a purely mathematical point of view, the vector ﬁeld R may be continuous

(and differentiable) or discontinuous. We will discuss examples of both in this

chapter.

Using the results of Section 2.5 (page 103) on Fourier transforms, it is easy to

see that a translation of a function in crystal space is equivalent to a phase shift in

reciprocal space. Mathematically, this is expressed as

f (r − a) = F

−1

−2πia·k

f (k)

In the case of a defect in a crystal, the translation vector a is replaced by the position-

dependent displacement ﬁeld R(r). The electrostatic lattice potential, V (r), takes

the place of the function f , and, hence, we expect to ﬁnd that the Fourier coefﬁcients

of the electrostatic lattice potential experience a phase shift near a translational

lattice defect. Let us cast this intuitive observation into more precise mathematical

terms.

Assume that the perfect crystal potential is given by V (r). A defect with displace-

ment ﬁeld R(r) deforms the crystal and the new potential at the deformed position,



(r + R(r)), is to a good approximation equivalent to the original potential at

the undistorted location. In other words, if displacements are small, then the local

environment of each atom remains approximately the same, or



(r + R(r)) = V (r). (8.2)

462 Defects in crystals

This is equivalent to the statement



(r) = V (r − R),

= V



2πig·(r−R)

= V



−2πig·R

2πig·r

. (8.3)

As anticipated, we ﬁnd that the defect modiﬁes the Fourier coefﬁcients of the

electrostatic lattice potential by a phase factor

→ V

−iα

(r)

with α

(r) ≡ 2πg · R(r). (8.4)

It is straightforward to incorporate this additional phase factor in the fundamental

equations of electron diffraction: it is left as an exercise for the reader to show that

the modiﬁed Darwin–Howie–Whelan equations (5.18) on page 310 are given by

dψ

− 2π is

= iπ





iθ

g−g



g−g



−iα

g−g



(r)



. (8.5)

If the displacement ﬁeld is continuous and differentiable, then it is convenient to

remove the r dependence from the phase factor on the right-hand side of equa-

tions (8.5). This can be achieved by the substitution:

= φ

−iα

(r)

This substitution does not change the intensity, since the modulus squared of a pure

phase factor equals 1. Using

g−g



(r) = α

(r) − α



(r),

this leads, after some simple mathematics, to the modiﬁed fundamental equations

of electron diffraction for a crystal containing a defect with a differentiable dis-

placement ﬁeld R(r):

dφ

− 2π i



+ g ·



= iπ





iθ

g−g



g−g



. (8.6)

Comparing this equation with that for a perfect crystal (5.18), we notice only

one difference: the excitation error s

has been replaced by an effective, position-

dependent excitation error:

eff

(r) = s

+ g ·

dR(r)

= s

d[g · R(r)]

, (8.7)

8.2 Crystal defects and displacement ﬁelds 463

where the derivative is taken parallel to the incident electron beam. Equation (8.7)

deserves some careful analysis.

If the displacement ﬁeld is constant in the z-direction, then the derivative vanishes and

the defect is invisible. In other words, what we can observe in the TEM is not the actual

displacement ﬁeld but the variation of the displacement ﬁeld with depth in the crystal.

If g · R(r) vanishes, then the effective excitation error s

eff

(r) is equal to the excitation

error for the perfect crystal, s

, and therefore the defect will be invisible, since the equa-

tions reduce to those for the perfect crystal. When g · R vanishes, this means that the

displacements are conﬁned to the planes represented by g (i.e. in-plane displacements).

The excitation error s

describes the orientation of the planes g with respect to the Ewald

sphere. A local bending of the lattice planes due to the defect is then described by the

second term in equation (8.7). Conversely, we can say that the presence of a defect changes

the shape of the reciprocal lattice point (or relrod).

In summary, a (translational) lattice defect described by a displacement ﬁeld R(r)

will only be visible for a particular two-beam condition if g · R varies along the

beam direction. This is known as the visibility criterion.

The visibility criterion lies at the basis of defect identiﬁcation and suggests a

simple experimental procedure: look for two-beam conditions for which the defect

contrast vanishes. For these g vectors, the dot product g · R vanishes or is constant,

which provides a number of linear equations from which the displacement vector

can be determined. This is especially useful when the displacement ﬁeld is only

weakly dependent upon position, or when the ﬁeld can be deﬁned in terms of a

single vector, e.g. the Burgers vector of a dislocation, or the displacement vector of

a planar fault.

We have seen in Chapter 6 that the strongest bright ﬁeld image contrast occurs for

excitation errors close to zero. This remains true for the effective excitation error,

eff

; the strongest image contrast for defects will also occur in those regions of the

foil where the effective excitation error is close to zero. If we orient a foil such that a

bend contour is visible in the bright ﬁeld image, then we know from Chapters 6 and

7 that the edges of the bend contour, where the contrast varies rapidly, correspond to

±g

≈ 0 (Fig. 8.1a). If we move away from the bend contour to a region with positive

, and a defect is present in this region, then the local, effective excitation error will

become small wherever the second term in equation (8.7) becomes comparable in

magnitude to s

, but opposite in sign, as illustrated in Fig. 8.1(a). An experimental

dark ﬁeld image for the

20.1 bend contour in Ti is shown in Figs 8.1(b) and (c). This

foil has a rather high defect density, and many dislocations can be observed near the

bend contour. The dislocations close to the bend contour have broad contrast (black

arrows), whereas dislocations slightly removed from the contour (white arrows) are

visible as much narrower white lines.

464 Defects in crystals

Bend Contour

-g

= 0

> 0

-g

> 0

d(g

< 0

defect

(a)

500 nm

(b)

(c)

Fig. 8.1. (a) Schematic illustration of a defect near a bend contour; the defect displacement

ﬁeld may locally contribute a negative d(g ·R)/dz to the positive s

, resulting in a near-zero

effective excitation error and hence strong defect image contrast. The titanium (

21.0) dark

ﬁeld image, obtained on a JEOL 2000EX microscope operated at 200 kV, clearly shows

dislocation contrast well outside of the bend contour. Part (c) is a magniﬁed view of the

region outlined in (b).

This variation of the apparent width of the dislocation contrast can be understood

as follows: as we move away from the bend contour, the excitation error s

increases

to larger positive values. Only those regions near the defect where the second term

of equation (8.7) is equally large but negative will give rise to strong image contrast.

Large derivatives of g ·R only occur near the core of the defect, where the atom

displacements are largest, and this core region is rather narrow.

†

The resulting

†

We will see in Section 8.4.2 that the displacement ﬁeld of an edge dislocation in an elastically isotropic medium

depends on ln r , with r the distance to the dislocation line. The derivative of the logarithmic function is largest

for small r -values.

8.3 Numerical simulation of defect contrast images 465

image (bright ﬁeld or dark ﬁeld) will show a narrow contrast feature located close

to the actual defect core. You can easily verify this behavior yourself, by looking at

the image contrast of a defect while tilting the sample such that the bend contour

moves away from the defect; the region of strong image contrast will become

narrower.

In Fig. 4.13(b) we have seen that, if we bring the reﬂection g to the optical axis

to obtain a dark ﬁeld image, then the reﬂection 3g will be in Bragg orientation

(assuming that g is in Bragg orientation before we tilt the beam). If s

≈ 0, then

must be large and positive, which is exactly the diffraction condition necessary

to obtain a narrow dislocation image. The g–3g diffraction condition, also known

as the weak beam imaging condition, can therefore be used to obtain defect images

which reveal the location(s) around the defect where the derivative of g · R is large;

i.e. the core of the defect. Continuing this intuitive argument along the same lines,

we also ﬁnd that a gradual increase of s

(i.e. by tilting the crystal such that s

becomes positive) will lead to contrast in an increasingly narrower region around

the dislocation core. However, simultaneously the overall intensity of the image

will decrease so that a longer exposure time is needed.

8.3 Numerical simulation of defect contrast images

In this section, we will discuss general principles of image contrast simulations for

a crystal containing a defect with displacement ﬁeld R(r). It is assumed throughout

that the displacement ﬁeld is available in analytical or numerical form; in other

words, that its components can be computed at an arbitrary point in the crystal.

Nearly all defect contrast simulations make use of the column approximation,

introduced in Chapter 6. While explicit formal expressions are available for the two-

beam transmitted and scattered amplitudes in a perfect crystal, analytical solutions

are no longer possible in the general case of a crystal with an arbitrary defect.

Equations (8.5) must be integrated numerically along each column, taking into

account the variation of the displacement ﬁeld along the center line of the column.

Each column again corresponds to one pixel in the computed bright or dark ﬁeld

image.

In the late 1950s and throughout the 1960s, when the ﬁrst defect image simula-

tions were carried out, computers were primitive and slow compared with contem-

porary machines. Random access memory (RAM) was expensive and limited, to the

extent that the memory of most computers in those days was measured in kilobytes

rather than megabytes. This meant that image simulations (or any other type of

numerical simulation) had to be optimized to run in the limited available memory.

Furthermore, there were no graphics terminals and laser printers, so that the repro-

duction quality of computed images was rather poor: grayscale images were often

466 Defects in crystals

printed on line-printers, where the number and type of characters printed on one

single location would determine the gray level of that location.

Nowadays, RAM is no longer expensive and even laptop computers can be

equipped with hundreds of megabytes of RAM. This means that image simulations

can now be carried out easily on a laptop. It is interesting to note, however, that

the “older” programs, those written when the available memory was rather small,

still work ﬁne on modern machines, and that they tend to run much faster than

programs written more recently. Perhaps memory restrictions forced programmers

to optimize their code to run in the available space, whereas nowadays we rely more

on the optimization capabilities of compilers and we need not worry too much about

how much memory is allocated for a particular simulation.

For the numerical simulation of defect contrast, in both two-beam and multi-

beam cases, we must perform two basic operations: (1) set up the geometry of the

problem, and (2) solve the dynamical equations. We will discuss both topics in the

following subsections.

8.3.1 Geometry of a thin foil containing a defect

We will assume that the image simulation is carried out for an array of pixels,

each pixel corresponding to a single “column” through the thin foil, parallel to the

beam direction (see Fig. 6.1). The number of columns is then not important for our

discussion, and is usually limited by the available computing power. All columns

have the same width and a square cross-section normal to the electron beam. The

position of a column in the image is determined by the integer pair (i, j). The width

of a column is described by L (expressed in nanometers).

With every column (i, j ) we can associate a number of parameters which fully

describe the crystal geometry and properties in the absence of a defect.

(i) Foil thickness. The local sample thickness can change from one column to the next,

and is described by an array of values (or a mathematical function, if available) z

(i, j).

The local foil thickness is not necessarily equal to the length of the column, since the

entire foil may be inclined with respect to the beam direction. We will refer to the

crystal thickness along

the beam direction by the symbol

(i, j).

(ii) Diffraction condition. The local diffraction condition may also vary from one col-

umn to the next. In the two-beam case, this would simply require an array of values

for the excitation error s

(i, j). In the multi-beam case, we must compute the ex-

citation error of each diffracted beam based on the actual orientation of the crystal

with respect to the Ewald sphere (i.e. the tangential component of the incident wave

vector).

(iii) Lattice potential. The local electrostatic lattice potential can also differ from one col-

umn to the neighboring one. This could be due to the presence of an interface between

8.3 Numerical simulation of defect contrast images 467

the two columns, an interface separating two regions with the same crystal structure

but different orientation. Alternatively, the local chemistry might change gradually (i.e.

a composition gradient), which would also give rise to a position-dependent potential

and therefore position-dependent parameters q

(i, j).

It is clear that the description of these parameters requires a number of independent

reference frames. We have introduced the crystal reference frames in Chapter 1:

the direct and reciprocal reference frames (or Bravais reference frames, described

by the direct and reciprocal metric tensors g

and g

∗

), and the corresponding

Cartesian reference frame (using the direct and reciprocal structure matrices a

and b

). To distinguish between the various reference frames introduced in the

following paragraphs, we will identify the reference frame by a single lowercase

superscript. The Cartesian reference frame deﬁned in Chapter 1 is then labeled by

the superscript c. We will continue to use an asterisk ∗ to indicate quantities in the

reciprocal reference frame, and quantities in the direct Bravais reference frame will

have no superscript.

Next we have the foil reference frame, which is ﬁxed with respect to the thin

foil. For convenience we select the foil normal (F with respect to the direct Bravais

reference frame) to lie along the z-direction of the foil reference frame. The x-

and y-directions are both normal to F and to each other. This means that they

must belong to the zone F. We will assume that the foil normal for the untilted

foil is parallel to the incident beam direction k

, but points towards the electron

gun, i.e.

F =−

where the ˆ hat indicates a unit vector. Then we select for the x-direction a reciprocal

space vector q that lies in the foil plane, such that F · q = 0, and which points

towards the goniometer airlock along the primary tilt axis.

The vector q has non-integer components in the general case, and can be deter-

mined based on the calibration procedure described in Section 4.5.2. Since the foil

normal may vary from one point of the foil to another, we will take the foil normal

F to be for the central pixel of the image simulation. The Cartesian foil reference

frame, labeled by the superscript f , is then constructed as follows:

|q|

;

|F|

;

= e

× e











(8.8)