Graef M. Introduction to conventional transmission electron microscopy

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368 Two-beam theory in defect-free crystals

(a)

(1)

(2)

(b)

Fig. 6.8. (a) Standard Ewald sphere construction and (b) an equivalent construction for the

empty crystal approximation.

then given by

(1)

= 0;

(2)

These eigenvalues are shown as thin straight lines in Fig. 6.6. The corresponding

wave vectors inside the crystal are:

(1)

= k

;

(2)

= k

n ≈ k

− s

The minus sign arises because the vector n points in a direction opposite to the

incident beam direction k

. In Bragg orientation, both vectors are equal since s

= 0.

The second equality is only valid when the incident beam is nearly parallel to the foil

normal. The vectors k

( j)

can be represented graphically as follows: on the Ewald

sphere construction of Fig. 6.8(a), superimpose two circles (spheres) with centers

at 0 and g, and radius equal to |k

|. The circles are labeled C

and C

, respectively

in Fig. 6.8(b). Then, draw a line through the center C of the Ewald sphere, parallel

to the foil normal n. This line intersects the two circles in the points a and b. The

point b coincides with the center of the Ewald sphere. Next, connect both points

6.4 The two-beam case: Bloch wave formalism 369

with the origin of reciprocal space; these are the vectors k

( j)

inside the “empty

crystal”. The scattered wave vectors k

( j)

can be drawn by connecting the points a

and b with the endpoint of k

+ g + s

Next, we can “turn on” the lattice potential and increase the Fourier coefﬁcient

from zero to its proper value. As illustrated in Fig. 6.6, the eigenvalues are no

longer linear, and, more importantly, they no longer intersect each other at Bragg

orientation. We can now transfer the information in this ﬁgure to a drawing similar

to that in Fig. 6.8(b). The two spheres around 0 and g must now be replaced by

two surfaces that do not intersect each other. The surfaces approach the spheres for

large excitation errors, and in Bragg orientation they are separated by a distance

1/ξ

. The resulting graphical construction is shown in Fig. 6.9(a) for exact Bragg

orientation and in Fig. 6.9(b) for a positive excitation error

. The wave vectors

( j)

are constructed in the same way as before, and now neither of them is equal to the

incident wave vector k

. The drawings in Fig. 6.9 are not to scale. It is straight-

forward, and left as an exercise for the interested reader, to draw the dispersion

surface construction for the case where the foil normal n is not perpendicular to the

reciprocal lattice vector g.

The ION-routine

tbbloch.pro on the website can be used to calculate the eigen-

values γ

( j)

for a given lattice potential Fourier coefﬁcient V

(in volts). The routine

produces two drawings, one using the same format as Fig. 6.6 and the second using

a different representation in terms of the z-component of the wave vectors. We

know that the tangential components of both wave vectors k

( j)

are equal to k

.Itis

(1)

(2)

(1)

(2)'

(1)

(2)

(1)'

(2)'

(2)

(1)

(a) (b)

Fig. 6.9. Dispersion surface construction for the two-beam case: (a) exact Bragg orientation

and (b) positive excitation error.

370 Two-beam theory in defect-free crystals

easy to show that the tangential component is also equal to

=−

− k

. (6.40)

Now we have two expressions for the wave vector k

( j)

= k

+ γ

( j)

= k

+ k

( j)

with k

( j)

normal to the tangential component. Equating the lengths of both vectors

we ﬁnd:

( j)

+ γ

( j)

(j)

(6.41)

with k

= k

− k

. It is then common to plot the value of k

( j)

− k

versus the

dimensionless orientation parameter −k

/g. In Bragg orientation we have k

−g/2; the minus sign stems from the fact that the wave vector points towards the

origin, not towards g. Example output of the

tbbloch ION routine is shown in

Fig. 6.10; the computation was performed for V

= 5 V, a mean inner potential of

-0.5 0.0 0.5 1.0 1.5

-k

0.10

0.05

0.00

-0.05

-0.10

(j)

(nm

)

-0.5 0.0 0.5 1.0 1.5

-k

0.05

0.00

-0.05

-0.10

-0.15

-0.20

(j)

-k

(nm

)

Fig. 6.10. Illustration of the output of the ION routine tbbloch.pro.

6.5 Numerical two-beam image simulations 371

25 V, and a microscope acceleration voltage of 200 kV. The dashed curve on the

right-hand side represents the sphere centered on the origin, with the vacuum wave

number as the radius (i.e. |K |). The thin curves represent the empty crystal results,

and the thicker curves form the dispersion surfaces.

The second representation method (k

( j)

− k

versus −k

/g) has the advantage

that it can be extended readily to multi-beam representations, in particular to the

systematic row case which we will address in the next chapter. For the two-beam

case, the dispersion surfaces are surfaces of revolution around the reciprocal lattice

vector g;

†

because of the limited validity of the two-beam approximation, only the

region of the dispersion surface close to the Bragg orientation (i.e. close to the

Brillouin zone boundary at g/2) is of practical importance.

6.5 Numerical two-beam image simulations

In this section, we will discuss the basic principles of two-beam image simulations.

We start with an extensive discussion of the numerical computation of the various

parameters affecting the image contrast (excitation error, extinction distance, and

absorption length). Then we discuss the numerical implementation of the scattering

matrix formalism, and the two-beam Bloch wave formalism. We conclude this sec-

tion with example image simulations, for a variety of sample geometries, followed

by a brief discussion of a two-beam CBED method to determine the local sample

thickness.

6.5.1 Numerical computation of extinction distances and absorption lengths

The parameter q

is given by

+ i

iβ





−

sin β





+ i

cos β



, (6.42)

where we have written explicitly the real and imaginary parts. From the deﬁnition

of the extinction distance (equation 5.13) and using equation (2.38) on page 95 we

have

+ g|cos α

πλ

+ g|cos α

†

This is analogous to the conical surface on which the Bragg condition is satisﬁed, as illustrated in Fig. 2.2(b)

on page 96.

372 Two-beam theory in defect-free crystals

Since, to a very good approximation, we have |k

+ g|≈1/λ and cos α ≈ 1,

we ﬁnd

≈

|. (6.43)

The reader should be careful not to confuse the interaction constant σ in this equation

with the parameter σ introduced in equation (6.19).

Numerical computation of the extinction distance is now straightforward, since

we already have routines for the computation of both σ (CalcWaveLength on page 94)

and V

(CalcUcg, Section 2.6.7 on page 128).

The computation of the absorption length presents a whole new set of difﬁculties.

We have seen in Section 2.6.5 that the effects of absorption can be taken into account

in a phenomenological way, by adding an imaginary part to the electrostatic lattice

potential:

(r) = V (r) + iW (r). (6.44)

In the early days of TEM research, it was common practice to assume that the

two potential functions are in phase, with a constant ratio a = W/V . The resulting

complex potential can then be written as

(r) = (1 + ia)V (r).

It was found by comparison with experiment that reasonable values of a

are in the range 0.05–0.15 for monatomic solids (i.e. for the pure elements)

[HHN

77].

For more complex unit cells, in particular non-primitive cells, this simple ap-

proximation does not hold as well, and the absorption potential is no longer simply

proportional to the electrostatic potential. There is, in fact, no a priori reason why the

two potential functions should even be “in phase” for a non-centrosymmetric crys-

tal structure [GBA66]. We will follow Bird and King [BK90], and Weickenmeier

and Kohl [WK91], and assume that the absorption part of the potential is mostly

due to electron–phonon scattering, also known as thermal diffuse scattering

(TDS). This means that we ignore plasmon scattering and energy losses due to

core excitations. The derivation in this section serves as an example of how inelas-

tic scattering phenomena can be incorporated into the dynamical scattering theory.

For a more detailed discussion of inelastic scattering we refer to the book by

Z.L. Wang [Wan95].

We must further simplify the problem, since it is quite difﬁcult to compute the

electron–phonon interactions in general for an arbitrary crystal structure. The sim-

plest approximation to the phonon spectrum of a material is the Einstein dispersion

model, which assumes that the thermal vibrations of each atom are statistically

6.5 Numerical two-beam image simulations 373

independent. We have already made use of this approximation in the derivation of

the Debye–Waller factor (equation 2.101 in Chapter 2).

We follow Hall and Hirsch [HH65] and calculate the probability amplitude for a

Bloch wave in Bragg orientation to be scattered in the direction k by an assembly

of N identical atoms at positions r

. The electrostatic potential in the notation of

Chapter 2 is given by

V (r) = V

(r) ⊗



i=1

(

r − r

)

Using the two-beam approximation for the incident Bloch wave in Bragg orientation

(see page 365), we arrive at the following expressions for the two Bloch waves:

†



( j)

(r) =

√

2πik

·r

+ (−1)

j−1

2πi(k

+g)·r

The probability amplitude P

for scattering of this wave in the direction k is given

by the bra-ket:

2πik·r

V (r)



( j)

(r)

If we denote the scattering vector k − k

by S, then it is easy to show, using the

deﬁnition of the direct Fourier transform on page 104, that

√



[V (r)] + (−1)

j−1

S−g

[V (r)]



where the subscript on the Fourier transform operator indicates the reciprocal space

vector of the transform. Using the convolution theorem (2.54), we can rewrite the

Fourier transforms as

[

V (r)

]

= F

(r)



i=1

(

r − r

)

;

= f



i=1

2πiq.r

We have used the deﬁnition of the atomic scattering factor f (equation 2.70, drop-

ping the superscript e), and equation (2.50). The subscript to f indicates the scatter-

ing vector. In the kinematical approximation, the intensity scattered in the direction

†

The theory in this section is also valid for orientations away from the Bragg orientation, and the reader is

encouraged to work through the full derivation.

374 Two-beam theory in defect-free crystals

k is proportional to the modulus squared of P

, and we ﬁnd



i=1



j=1

2πiS·(r

−r

)

+ f

S−g



i=1



j=1

2πi(S−g)·(r

−r

)

+ 2(−1)

j−1

S−g



i=1



j=1

cos{2π[S · (r

− r

) + g · r

]}

Next, let us assume that the atoms vibrate independently of each other, and that the

instantaneous displacements are given by the vectors u

(t); we will not write the

time argument explicitly, as it is clear that only the u

depend on time. The ﬁrst

term on the right-hand side of the above equation then reads as



i=1



j=1

2πiS·(r

−r

)

2πiS·(u

−u

)

= N +



i, j



i=j

2πiS·(r

−r

)

2πiS·(u

−u

)

with similar expressions for the other terms. The time average can be computed in

the same way as the Debye–Waller factor in Chapter 2:

2πiS·(u

−u

)

= e

−2M

with M

= 2π

u

. Noting that

cos(2πg · u)=1 −

4π

u



+···=1 −

2π

u



+···=e

−M

and working out all time averages we ﬁnd for the intensity:

N + e

−2M



i, j



i=j

2πiS·(r

−r

)

+ f

S−g

N + e

−2M

S−g



i, j



i=j

2πi(S−g)·(r

−r

)

+2(−1)

j−1

S−g

N e

−M

+ e

−(M

S−g

)



i, j



i=j

cos{2π[S · (r

− r

) + g · r

]}

To determine the thermal diffuse scattering component of this expression, we must

subtract the Bragg scattering part, which can be obtained by repeating the com-

putation for a crystal with atomic scattering factors equal to f

−M

. The resulting

6.5 Numerical two-beam image simulations 375

expression for the thermal diffuse scattering intensity is

TDS

(k) =



1 − e

−2M



+ f

S−g



1 − e

−2M

S−g



+ 2(−1)

j−1

S−g



−M

+ e

−(M

S−g

)

%

. (6.45)

At this point, we should brieﬂy review what we have accomplished: our ﬁnal goal is

to come up with a method to compute the absorption length ξ



, or, equivalently, the

imaginary potential coefﬁcients V



. There are several contributions to the absorption

potential, and experience shows that thermal diffuse scattering is, for most materials,

the dominant contribution at room and elevated temperature. We have assumed a

crystal in the Bragg orientation, and computed the probability for the corresponding

Bloch waves to be scattered in an arbitrary direction k. After introducing atomic

vibrations and time-averaging the resulting expression, we ended up with the TDS

intensity in equation (6.45). Now we must connect this expression to the absorption

potential.

Electrons are absorbed when they no longer contribute to the Bragg-diffracted

beams, whatever the origin of the absorption process may be. If we integrate I

TDS

(k)

over all possible orientations of k, then we will have calculated the total number

of electrons that have undergone thermal diffuse scattering events. Those electrons

no longer contribute to the Bragg-diffracted intensity and therefore the integrated

TDS expression provides an estimate of the overall absorption. Consider the Bloch

wave absorption factor q

( j)

in exact Bragg orientation, deﬁned on page 366:

( j)



+ (−1)

j−1



If we compare this expression with equation (6.45), we ﬁnd that they have the same

structure: the ﬁrst two terms in (6.45) contribute to ξ



, and the last term in (6.45)

contributes to ξ



We can use an expression for the absorption Fourier coefﬁcient V



, which is

formally similar to that of V

(equation 2.102 on page 129):



0.047 878 01





j=1

−B



(s)



(D|t)

−2πig·(D|t)[r

]

. (6.46)

In this expression, f



(s) are the so-called absorptive form factors, which are atomic

quantities similar to the electron scattering factors f

(s). The expansion (2.76) of

(s) in terms of exponential functions is now advantageous because it turns out that

an analytical expression for the absorptive form factors f



(s) can be obtained; this

allows them to be calculated for every crystal structure, using an algorithm that is

not signiﬁcantly more complicated than that for the computation of f

(s) [WK91].

376 Two-beam theory in defect-free crystals

The explicit expression is too long to include here and we refer to the appendix of

Weickenmeier and Kohl’s paper for the details. The procedure involves integration

of the TDS intensity over all scattering vectors S:



(g) =

2π

S−g



−M

+ e

−(M

S−g

)



dS. (6.47)

The numerical computation of the absorptive form factor has been implemented

in the Fortran routine

CalcUcg. The absorptive form factors for individual atoms

can be plotted using the ION routine

scatfac.pro, available from the website, and

discussed in more detail in Chapter 2.

The reader should realize that many assumptions are involved in the computation

of both the extinction distances and the absorption lengths. The Fourier coefﬁcients

of the electrostatic lattice potential are calculated on the basis of an isolated spher-

ical atom approximation for the electron scattering factors; since there are quite

a few parameterizations available for the scattering factors, the precise value of a

Fourier coefﬁcient V

will depend on the particular model used. In addition, the

parameterizations ignore bonding contributions.

The Debye–Waller factors are unknown for most crystal structures. They can be

calculated if a model for the phonon dispersion relation is available. In most cases

this is not feasible, and the parameters will need to be estimated. This results in

some uncertainty in the values of the Fourier coefﬁcients of the electrostatic lattice

potential. Moreover, it is probably not a good idea to use an isotropic Debye–Waller

factor in all situations.

If experimental values for the Debye–Waller factors are unavailable, then one

could use the elemental values as rough estimates. Peng and coworkers [PRDW96]

list the Debye–Waller factors as a function of temperature for 42 different elemental

crystals. From their Table 1 it can be seen that, in general, the Debye–Waller

factor is larger for elements in the leftmost columns of the periodic table. At room

temperature, values around 0.1nm

would be quite reasonable for ﬁrst column

elements (Li, Na, K, Rb, Cs), whereas for second column elements (Be, Mg, Ca, Sr,

Ba) values around 0.01–0.03 nm

are reasonable. For most other elements, values in

the range 0.003–0.007 are acceptable. The Debye–Waller factors for the elemental

solids decrease down a column of the periodic table. If experimental values for

B(T ) are available, then those should be used instead of the estimated values. At

liquid nitrogen temperature, the Debye–Waller factors are typically about one-third

of their room-temperature values.

Because of these uncertainties, one should treat the calculated values for V



(using CalcUcg or any other similar routine) as ﬁrst-order estimates. Reﬁnements

based on experimental observations should then lead to more accurate values; this

is currently an active area of research and we refer the reader to [SZ92] for a

6.5 Numerical two-beam image simulations 377

more detailed description. In recent years, there has been a dramatic increase of

interest in quantitative TEM observations, and the measurement of V

and related

parameters certainly falls under this umbrella. Since this is not the topic of this

book, we will assume hence forth that the extinction distance and absorption length

can be calculated in one way or the other.

The Fortran program

qg.f90 can be used to compute the real and imaginary parts

of 1/q

for an arbitrary crystal structure. The program calls the

CalcUcg routine

to compute both the real and imaginary parts of the potential Fourier coefﬁcients

for a given plane (hkl). As an example, Table 6.2 shows the results for three face-

centered cubic materials (Al, Cu, and Au), along with the non-centrosymmetric

study material GaAs.

Table 6.3 shows computed parameters for the non-centrosymmetric tetragonal

form of BaTiO

, and the hexagonal materials Ti, ZnS, and BeO, along with exper-

imental (reﬁned) values for the (002) reﬂection of BeO [SZ92]. The information in

this table can be generated for individual planes by means of the

qg.f90 program,

or for all independent families of planes using the

xtalinfo.f90 program introduced

in Chapter 4.

6.5.2 The two-beam scattering matrix

Matrix methods were introduced in 1961 by Howie and Whelan [HW61] to facilitate

contrast computations, and they have since been used by many researchers, e.g.

[GVLA65,Man62,Th¨o70]. We will closely follow [Th¨o70], but adapt the notation

to that used throughout this chapter. The theory described below will prove to be

very useful for the description of defect image contrast.

In obtaining the solutions (6.20) and (6.21) to the two-beam equations we have

made use of the following initial conditions:

(

s, z = 0

)

= 1;

(

s, z = 0

)

= 0.

A second special solution can be obtained by using the following independent initial

conditions:



(

s, z = 0

)

= 0;



(

s, z = 0

)

= 1;

in other words, all the incident amplitude is in the direction of the scattered beam

rather than the transmitted beam. Proceeding in the same way as in Section 6.3.3.2