Graef M. Introduction to conventional transmission electron microscopy

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628 Phase contrast microscopy

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0.641.282.565.12

10 nm

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(a)

(b)

(c)

(d)

(e)

Fig. 10.23. Computed diffractograms for two different 200 kV microscopes: (a), (b) and

(e) conventional LaB

instrument, (c) and (d) FEG instrument. (a) and (c) represent the

ﬁrst four passband images for each machine (defocus indicated in upper right-hand corner).

The passbands are indicated by white arrows. (b) and (d) represent images with varying

amounts of astigmatism, indicated as a magnitude-angle pair in the upper right corner of

each diffractogram. The bottom row shows the effect of limiting the bandwidth of the image

with an aperture. The aperture diameter is indicated in nm

−1

10.2 The microscope as an information channel 629

10.2.5 Alignment and measurement of various imaging parameters

10.2.5.1 The importance of beam tilt

In the previous sections, we have implicitly assumed that the microscope column is

perfectly aligned, in other words, that the electron beam is incident along a direc-

tion for which aberrations such as coma are minimized or, ideally, absent. In reality,

this is not necessarily the case. When you begin observations on a sample, the mi-

croscope alignment will nearly always be less than perfect and it is important to

understand how an imperfect alignment affects the aberration function χ(q). Start-

ing from equation (10.33) we will ﬁrst rewrite the aberration function in Cartesian

coordinates, and then introduce a beam tilt. We will ignore four-fold astigmatism,

described by the term in H in equation (10.33), but include all other terms (defocus,

two-fold and three-fold astigmatism, coma, and spherical aberration).

The derivation below is based on [TOCL96] with a slightly different notation

and a different sign convention. Starting from equation (10.33), we introduce

Cartesian coordinates (u, v) ≡ q(cos φ,sin φ) and q

= u

+ v

. Making use

of standard trigonometric identities and deﬁning C

a,x

≡ C

cos 2φ

, C

a,y

≡

sin 2φ

, G

≡ G cos 3φ

, G

≡ G sin 3φ

, K

≡ K cos φ

, and K

≡

K sin φ

, it is easy to show that

χ(q)

2π

=−

λ(u

+ v

) f −

λ[C

a,x

− v

) + 2C

a,y

uv]

−

− 3uv

) + G

(3u

v − v

)] − λ

+ v

)[uK

+ vK

]

+ v

)

. (10.67)

Next, we introduce a beam tilt. Since a beam tilt in direct space is equivalent to

a translation in the diffraction pattern, a beam tilt is simply described by a trans-

lation vector t = (t

, t

); the aberration function for tilted illumination is derived

from equation (10.67) by the following substitution: u → u + t

; v → v + t

. The

mathematical operations are elementary but somewhat lengthy and we obtain the

following expression:

(q)

2π

=−

λ(u

+ v

) f −

a,x

− v

) + 2C

a,y

−

− 3uv

) + G

(3u

v − v

)

− λ

+ v

)

+ v K

+ v

)

+ uD

+ vD

, (10.68)

630 Phase contrast microscopy

with

 f ≡  f + 4λ(K

+ K

) − 2λ



+ t



;

a,x

≡ C

a,x

+ 2λ

(

+ K

)

+ t

− K

)

− λ



− t



;

a,y

≡ C

a,y

+ 2λ

+ K

) + t

(

− G

)

− 2λ

;

≡ K

− λC

;

≡ K

− λC

;

+ vD

≡−λ f [ut

+ vt

] − λ[C

a,x

(ut

− vt

) + C

a,y

(ut

+ vt

)]

− λ



− t



− 2vt



+ G



− t



+ 2ut

%

− λ



+ t



+ 2vt



+ K



+ t



+ 2ut

%

+ λ

(ut

+ vt

)



+ t



The ﬁnal two terms in equation (10.68) represent the image displacement that is

induced by the beam tilt. If the microscope is perfectly aligned (i.e. no coma,

no two-fold astigmatism, and corrected three-fold astigmatism) then the image

displacement depends only on the spherical aberration and the defocus, as we

have already seen in Section 4.4.2.4. The relation between image displacement,

defocus and C

can be used to determine the value of C

with very high precision,

as described in [KdJ91]; the method relies on the fact that the displacement is

described by a third-order polynomial in the beam tilt.

While the relations derived above are somewhat complicated, we can learn a lot

from them.

Coma can be removed by the appropriate beam tilt. If we tilt the beam so that t

= K

/λC

and t

= K

/λC

, then the effective coma coefﬁcients K

and K

vanish and the beam

is aligned along the so-called coma-free axis (see Section 10.2.5.2).

The effective defocus of the image is not only determined by the objective lens current.

A beam tilt changes the defocus by an amount proportional to the spherical aberration

constant (in the absence of coma). For instance, for a beam tilt θ

= 15 mrad and a spherical

aberration of C

= 0.65 mm, the induced defocus change is equal to −2C

=−292

nm. In a properly aligned, C

-corrected instrument, there would be no difference between

the effective defocus

 f and f , even for a large beam tilt

(assuming that higher-order

aberrations can be ignored).

It is always possible to use the objective stigmator coils to correct the effective two-

fold astigmatism, even in the presence of coma and beam tilt. Therefore, it is not sufﬁ-

cient to correct astigmatism of the image and then assume that the microscope is well

aligned.

The three-fold astigmatism coefﬁcients are not affected by beam tilt (assuming that higher-

order aberrations can be ignored). This means that it is possible to correct three-fold

astigmatism once and for all using special correction coils.

10.2 The microscope as an information channel 631

If coma and three-fold astigmatism are properly corrected, then the effective two-fold

astigmatism depends only on beam tilt and spherical aberration. If the astigmatism at

zero beam tilt is properly corrected, then the beam-tilt-induced astigmatism is given by

a,x

=−λ

cos 2φ

;

a,y

=−λ

sin 2φ

(10.69)

where the beam tilt is written as (t

, t

) = t(cos φ

, sin φ

). This means that one of the

major axes of the astigmatism diffractogram will always lie in the plane formed by the

tilted beam direction and the coma-free axis.

The aberrations in equation (10.68) fall into two categories: those that are even in the

coordinates u and v (defocus, two-fold astigmatism, spherical aberration), and those

that are odd (three-fold astigmatism and coma). The even aberrations can be visualized

using diffractograms from an amorphous region in the foil; odd aberrations do not affect

the diffractogram, and one must resort to the acquisition of a so-called Zemlin tableau,a

diffractogram series for a given beam tilt and varying beam azimuth (see Section 10.2.5.2).

It is obvious that a correct alignment of the microscope requires a detailed under-

standing of what it is we are trying to correct. The ideal aberration function χ(q)

contains only contributions from the defocus and spherical aberration terms. All

other terms are residual aberrations and their contributions must be kept as small

as possible through a proper alignment.

10.2.5.2 Coma-free alignment

When we introduced the Seidel aberrations in Chapter 2, we have seen that coma

is an aberration that “smears out” the image of a point object. This aberration can

be corrected by aligning the incident beam onto the so-called coma-free axis. Since

HREM images are often difﬁcult to interpret, even when we know what the crystal

structure looks like, it is nearly impossible to visually recognize the presence of

coma in an experimental image. A simple procedure for the correction of coma was

suggested by Zemlin and coworkers [ZWS

78] and involves a series of HREM

images of an amorphous object, recorded for a given beam tilt. The procedure for

determining whether or not coma is present works as follows.

Place an amorphous region in eucentric position, correct the astigmatism, and record an

image at an under-focus condition ( f ≈ 100 nm or so). The corresponding diffractogram

should show multiple concentric rings, as shown in the central diffractogram in Fig. 10.24.

The fact that the rings are concentric indicates that astigmatism was properly corrected

for this particular incident beam direction. It does not indicate that the microscope was

properly aligned.

Switch to dark ﬁeld mode and tilt the beam by an angle of 10–15 mrad. This requires

that the tilt controls have been calibrated. Record an HREM micrograph. Tilt the beam

by the same amount to the opposite direction and record another HREM micrograph.

632 Phase contrast microscopy

beam tilt: 15 mrad

120

150

180

210

240

270

300

330

Fig. 10.24. Zemlin tableau recorded using a 3 nm thick amorphous carbon ﬁlm on a Philips

CM30 UltraTwin ﬁeld emission microscope operated at 300 kV. The central diffractogram

shows concentric rings; the six tilt pairs were acquired in dark ﬁeld mode for a beam tilt

of 15 mrad. The asymmetry of the diagonal running from top left to bottom right indicates

that the central beam orientation does not coincide with the coma-free axis.

The diffractograms will appear to be astigmatic, but should have the same defocus

value. In other words, the pattern of distorted rings must be the same for opposite tilt

angles.

Repeat this procedure for several

additional pairs of beam tilts, with different azimuthal

angles for the tilt. If you record half a dozen or so different tilt directions, you can then

assemble all the diffractograms into a so-called Zemlin tableau, such as the one shown in

Fig. 10.24.

A coma-free alignment essentially requires that the pairs of diffractograms across

the Zemlin tableau be identical with respect to the defocus. If they are not identical,

10.2 The microscope as an information channel 633

as is the case for the tableau shown in Fig. 10.24, then the central beam orien-

tation with respect to which the tableau was recorded does not coincide with the

coma-free axis. The coma-free alignment then involves changing the beam tilt of

the central image while maintaining zero astigmatism until the entire tableau is

symmetric with respect to the defocus. Many modern microscopes have special

alignment controls for this type of alignment, and we refer to the standard operat-

ing procedures for more details. The resulting Zemlin tableau should look similar

to that shown in Fig. 10.25: the ring pattern appears to be astigmatic, and the main

axis of the elliptical distortion points towards the central diffractogram. If three-

fold astigmatism is present then the elliptical patterns will all look alike, but will

be oriented at different angles with respect to the central diffractogram. For the

particular microscope used to obtain Figs 10.24 and 10.25, three-fold astigmatism

beam tilt: 15 mrad

120

150

180

210

240

270

300

330

Fig. 10.25. Same as Fig. 10.24, but with the correct coma-free orientation of the incident

beam.

634 Phase contrast microscopy

was corrected by the insertion of a properly excited correction coil, and all ellipses

point towards the center diffractogram. The apparent astigmatism present in the

tilted diffractograms is due to spherical aberration; in a C

-corrected microscope

the Zemlin tableau would show circular patterns for all tilt angles.

The manual acquisition of a Zemlin tableau is somewhat tedious. If the micro-

scope can be controlled by a computer, then a script can be used to automatically

acquire the diffractograms, determine the location of the coma-free axis, correct the

beam tilt and acquire a ﬁnal tableau. Additional procedures for various alignments

can be found in [SSE82] and [Spe88].

10.2.5.3 Measurement of electron optical parameters

When the residual aberrations have been corrected, the aberration function χ(Q)

only contains the defocus ( f ) and spherical aberration (C

) contributions. Assum-

ing that drift and vibration can be kept reasonably small, the only other parameters

needed for a full characterization of the image formation process are the beam

convergence angle θ

and the defocus spread . The spherical aberration and de-

focus can be determined using ring diffractograms from an amorphous area. If the

magniﬁcations and camera lengths of the microscope have been accurately cali-

brated, then the location of the zeros of the ring pattern can be measured. The zeros

occur whenever the total phase shift of equation (10.9)

is equal to a multiple of

π,or

n =−λ fq

; (10.70)

if we introduce the new variable p ≡ λq

, then we have the following relation:

=− f +

λp. (10.71)

The left-hand side is represented graphically by a hyperbola, the right-hand

side is a straight line. Figure 10.26 shows the hyperbolic curves for values

of n =−1,... ,−20. Superimposed are nine straight lines for defocus values

 f

= 100 × i and C

= 1 mm. The electron wavelength is λ = 2.508 pm (i.e.

200 kV). The straight lines have slope

λ and intercepts − f . At each intersec-

tion of a straight line with a hyperbola the corresponding value of p indicates the

location q =

√

p/λ of the zero of the diffractogram.

Krivanek [Kri76] proposed in 1976 to use this relation in a least-squares ﬁt

to determine C

and  f , given a series of zeros for a single or multiple defo-

cus values. Coene and Denteneer [CD91] improved the method and used several

diffractograms and a modiﬁed least-squares procedure to obtain a better ﬁt for the

10.2 The microscope as an information channel 635

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0.01 0.02 0.03 0.04 0.05

n/p

slope = C

λ/2

intercept

-∆f

Fig. 10.26. Graphical representation of Krivanek’s method to determine the spherical aber-

ration coefﬁcient and defocus from a series of ring diffractograms.

spherical aberration coefﬁcient. The improved method uses the parameter r = q

and the relation

n =−λ fr +

(10.72)

to determine the intersections of a series of straight lines (left-hand side) and a

parabola (right-hand side). The explicit least-squares relations can be found in

[CD91]. The method can result in a standard deviation of less than 1% for the

spherical aberration coefﬁcient.

To improve the accuracy of these ﬁtting methods the diffractograms must be

obtained for the coma-free incident beam orientation. The accuracy of the zero

locations is improved if the diffractograms are rotationally averaged to increase

the signal-to-noise ratio. It is not difﬁcult to write a program to carry out the least-

squares ﬁt; we refer the reader to the “Numerical Recipes” book for details on

least-squares algorithms [PFTV89].

Buddinger and Glaeser [BG76] proposed a method based on image displacements

between bright ﬁeld and dark ﬁeld images to measure C

; an accuracy of 3% was

reported. The image displacement method was subsequently improved by Koster

and de Jong [KdJ91] to a precision of better than 1%.

636 Phase contrast microscopy

(a) (b) (c)

(d)

4 nm

0.235 nm

0.144 nm

111

220

111

220

Fig. 10.27. (a) High-resolution micrograph of Au islands on an amorphous Ge ﬁlm (Philips

CM30 Ultra-Twin, operated at 300 kV); (b) diffractogram of (a); (c) rotationally averaged

diffractogram and (d) diffractogram proﬁle, with Au 111 and 220 spacings indicated.

As an example, we determined the spherical aberration constant of a Philips

CM30 FEG microscope with an Ultra-Twin objective lens, operated at 300 kV. The

manufacturer-provided value for C

is 0.65 mm for the eucentric sample position.

We used an amorphous Ge sample with Au islands to record a through-focus series

of 50 images with a defocus step size of around 4 nm. A single image of the series

is shown in Fig. 10.27(a). Several Au particles are visible, and two lattice spacings

are indicated. The diffractogram is shown in Fig. 10.27(b). The rotational average

is shown in both (c) and (d); from measurements of the 111 and 220 Au peaks

on several diffractograms, we can deduce that the diffractogram has a scale of

0.0491 ± 0.0007 nm

−1

per pixel.

Using the weighted least-squares method proposed by Coene and Denteneer

[CD91], we used 189 zero crossings in 50 diffractograms to reﬁne the initial esti-

mates of C

= 0.65 mm, the ﬁrst defocus value of −10 nm, and the defocus step

size of 4 nm. The resulting values are: C

= 0.663 ± 0.038 mm, initial defocus

−5.86 nm, and defocus step size 3.47 nm. The error bars on C

are entirely due

to the standard deviation on the diffractogram scale. A better magniﬁcation cal-

ibration would result in a narrower range for C

. The ﬁtted zero-crossing curves

are shown in Fig. 10.28, along with the experimental data points for the zeros of

the diffractograms. The inset in the upper left-hand corner is a combination of all

50 diffractogram proﬁles. The Scherzer defocus (44.2 nm), and Scherzer spatial

frequency (q

= 5.86 nm

−1

) are also indicated.

Measurement of the defocus spread and the beam divergence angle are a bit more

complicated. The chromatic aberration coefﬁcient C

for the CM30 microscope is

1.14 mm. For an energy spread of 0.9 eV, and voltage and current stabilities of 10

−6

10.2 The microscope as an information channel 637

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0246810

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100

150

200

Defocus ∆f [nm]

|q| [nm

-1

]

∆f

Fig. 10.28. Weighted least-squares ﬁt of the zero-crossings of 50 experimental diffrac-

tograms (Philips CM30, 300 kV). The diamonds indicate the experimental measurements,

the solid lines are the results of the ﬁtting procedure.

the defocus spread (equation 3.61) amounts to 4.26 nm. The chromatic information

limit ρ

(equation 10.56) is then approximately equal to 0.1 nm (at 300 kV). A direct

measurement of ρ

is possible in principle, if we use an amorphous ﬁlm which has

a sufﬁciently large electron scattering factor out to q = 10 nm

−1

. An amorphous

germanium ﬁlm may do the trick, but it is better to use a heavier element, such as

tungsten.

The information limit may be estimated using the Young’s fringes technique,

ﬁrst introduced by J. Frank and coworkers in 1970 [FBLH70, Fra75, Fra76]. The

method uses two electron micrographs of the same region of the sample, with one

translated intentionally by a small amount with respect to the ﬁrst one. The Fourier

transform of the sum of the two images then contains a sinusoidal modulation,

which can be used to estimate which part of the diffractogram contains meaningful

information. An example is shown in Fig. 10.29 for a Ge amorphous ﬁlm with Au

particles. The images were obtained at 300 kV in a Philips CM30 microscope. The

region of the diffractogram where the fringes are clearly visible is indicated by

an ellipse in Figs 10.29(b) and (c). In the absence of sample drift and vibrations,

the ellipse should become a circle, so that the information limit is isotropic. The

information limits corresponding to the major and minor axes of the ellipse are

1/8.4 = 0.119 nm and 1/6.4 = 0.156 nm, respectively; both limits are larger than

the chromatic limit derived above.

The beam divergence angle can in principle be determined by analyzing the de-

focus dependence of the extent of the Young’s fringes. This analysis is based on