Graef M. Introduction to conventional transmission electron microscopy

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618 Phase contrast microscopy

1.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

05101520

Spatial frequency q (nm

-1

)

(q)

d=0.05 nm

d=0.1 nm

u=0.05 nm

u=0.025 nm

Fig.

10.16. The drift envelope (thin line) for

d = 0.1

and 0

.05

nm and the vibration envelope

(thick line) for u = 0.05 and 0.025 nm.

that the corresponding damping envelope is given by

(q) = J

(2πq · u) ≈ e

−(πq·u)

, (10.61)

where J

(x) is a Bessel function of the ﬁrst kind. The vibration information limit can

be derived from the exponential approximation as

≡

πu

√

ln s

. (10.62)

The drift and vibration envelope functions have a similar functional shape, as shown

in Fig. 10.16 for d = 0.1 and 0.05 nm and u = 0.05 and 0.025 nm. The spatial frequen-

cies range from 0 to 20 nm

−1

, which corresponds to real space distances of 0.05 nm.

The meaning of these functions is obvious: if a high-resolution image is to contain

information at a length scale of 0.1 nm, then the experimental conditions must be such

that both the drift envelope E

(q) and the vibration envelope E

(q) are signiﬁcantly

larger than zero at q = 10 nm

−1

.Ifd = 0.1 nm, then the drift envelope reaches zero

at q = 10 nm

−1

, according to Fig. 10.16, and the highest spatial frequencies will be

absent from the image.

The drift and vibration

envelopes are partly determined by a proper design of the

microscope and the microscope room. The highest spatial frequencies (say around

q = 10 nm

−1

or higher) can only contribute to the image if they are not attenuated by

vibrations or drift. In practice, this means that the sample must be in thermal equilibrium

with its surroundings, which requires that the sample holder and the objective lens

cooling circuits are properly designed. The temperature in the room must also be

controlled to better than 1

◦

C, and there should be no air drafts caused by temperature

gradients and/or air conditioning systems. Vibrations can be kept to a minimum by a

proper design of the column suspension system (possibly with active vibration control)

10.2 The microscope as an information channel 619

and the ﬂoor on which the microscope is mounted. Often, the entire column sits on an

acoustically isolated concrete block. Finally, the microscope operator represents both

a heat source and a source of acoustic vibrations, so ideally the operator should control

the microscope from a remote location. While this is possible in the latest microscope

models, in most situations the operator must be in the same room as the instrument.

Recording images which contain high spatial frequency information therefore requires

some self-discipline from the operator: sounds must be minimized, and preferably the

operator should not move around at all since this will set up air currents and thermal

gradients. For more details on the environmental considerations of high-resolution

work we refer the reader to Chapters 9 and 10 in [Spe88]. A nice example of the

importance of drift and vibration considerations in the design of a microscopy facility

can be found in [Lic01].

(iii) Detector envelope function. The point spread function T (r) (PSF) of the microscope

is a sharply peaked (complex) function with oscillating tails which depend strongly

on the objective lens defocus value. An example PSF for a 400 kV microscope with

= 1 mm at Scherzer defocus is shown in Fig. 10.17(a). The real and imaginary

parts of both PSF and CTF are shown for a beam divergence angle of θ

= 0.8 mrad,

and a defocus spread  of 6 nm. Fig. 10.17(b) shows similar

curves for a 200 kV

instrument with identical spherical aberration, but with θ

= 0.4 mrad and  = 3 nm.

This latter machine has a more coherent electron source which is reﬂected in a wider

PSF. If the incoherent envelope functions extend to higher spatial frequencies,

then

the corresponding PSF will be wider, and this has consequences for digitally recorded

images.

Consider a CCD camera with M × M pixels. The camera captures an image for a

particular microscope setting, i.e. a particular PSF. The information originating from

a single point in the exit plane of the sample will be spread over an area determined

by the lateral extent of the PSF. The more coherent the source, the larger this area

becomes, i.e. the more delocalized the information becomes. This means that image

pixels which are close to the edge of the CCD camera will detect electrons which

originated from exit plane points that are not part of the acquired image. Figure 10.18

shows schematically that there is a central area on the CCD camera that detects no

electrons from outside the region of interest. The total width of the “incomplete” edge

region is N pixels. The ﬁnite size of the detector causes a decrease in the information in

the captured image, and hence the detector itself has an associated attenuation envelope.

Since the width of the PSF is determined by the spatial incoherent envelope function

(q), it should come as no surprise that the information limit imposed by the detector

depends on the same factors as the spatial incoherence information limit. For a detailed

derivation we refer the interested reader to [dJVD93]. The detector envelope function is

given by

(q) = exp

−2π





, (10.63)

620 Phase contrast microscopy

Re(CTF)

Im(CTF)

Re(PSF)

Im(PSF)

-1

0.4

0.2

0.0

-0.4

-0.2

0.4

0.2

0.0

-0.4

-0.2

0246810-4 -2024

Re(CTF)

Im(CTF)

Re(PSF)

Im(PSF)

-1

0.4

0.2

0.0

-0.4

-0.2

0.4

0.2

0.0

-0.4

-0.2

0246810

-4 -2024

q (nm

-1

) x (nm)

Fig. 10.17. Real and imaginary parts of the CTF and PSF functions for (a) a 400 kV

instrument with C

= 1 mm, θ

= 0.8 mrad, and  = 6 nm, and (b) a 200 kV instrument

with C

= 1 mm, θ

= 0.4 mrad and  = 3 nm.

where N is the number of detector pixels in the unusable border region. The information

limit corresponding to the detector envelope is then given by

≡



√

2πa

√

ln s



1/4

. (10.64)

Typical values of N are in the range of 200–400 pixels.

10.2 The microscope as an information channel 621

Fig. 10.18. Illustration of the effect of delocalization on the reliable area of an image

acquired by a CCD camera. The circles represent the extent of the PSF; only for points

inside the central gray square is the delocalized information completely captured in the

image.

10.2.3 Plug in the numbers

Most of the derivations in the preceding sections were expressed in terms of di-

mensionless quantities, which makes the equations essentially independent of any

particular microscope model. Now it is time to actually plug some realistic numbers

into the equations. We will consider three 200 kV instruments: I

is equipped with

a LaB

source, I

with a ﬁeld emission source, and I

has a ﬁeld emission source

and a spherical aberration corrector.

†

The electron optical parameters of all three

machines along with several derived and dimensionless quantities are shown in

Table 10.1.

Figures 10.19 and 10.20 show the function E

(q)E

(q) sin χ (q) for the frequency

range q = [0, 12] nm

−1

and the defocus range f = [−100, 400] nm for all three

microscopes I

, I

, and I

. The horizontal white lines indicate the location of the

Scherzer frequency q

and the chromatic information limit ρ

. The vertical lines

indicate the Scherzer defocus  f

and the optimum defocus  f

opt

(for q

max

12 nm

−1

). The dash-dotted line corresponds to the Gaussian defocus  f = 0.

The following observations can be made regarding the transfer functions in

Figs. 10.19 and 10.20.

The higher coherence of a ﬁeld emission source increases the frequency range trans-

ferred by the microscope (compare I

with I

). The defocus spread, determined by the

chromatic aberration of the objective lens and the energy spread of the source according

to equation (3.61), sets the chromatic aberration limit, while the beam divergence angle

determines the spatial incoherence envelope. The source energy spread and hence the

†

Commercial spherical aberration correctors have only recently become available, and it is anticipated that many

instruments will be equipped with a C

-corrector in coming years.

622 Phase contrast microscopy

Table 10.1. Electron optical and operational parameters for

three hypothetical E = 200 kV microscopes.

Parameter I

[mm] 1.00.5 Variable

[mm] 1.61.01.0

σ (E ) [eV] 1.00.40.4

σ (V )/V 10

−6

σ (I )/I 10

−6

[mrad] 0.60.20.2

ln s 2.03.53.5

Derived quantities

σ (E )/E 5 × 10

−6

2 × 10

−6

2 × 10

−6

 [nm] 8.83.03.0

[nm

−1

]0.239 0.079 0.079

[nm] 0.227 0.191 Variable

 f

[nm] 61.33 43.37 Variable

[nm] 0.186 0.094 0.094

 f

opt

[nm] 136.36 266.95 266.95

Dimensionless

quantities

 0.175 0.085 0.085

0.085 0.023 0.023

0-100 200 400 0 200 400

∆f

opt

∆f

opt

q (nm

-1

)

∆f (nm) ∆f (nm)

Fig. 10.19. Attenuated phase contrast transfer function E

(q)E

(q) sin χ (q) for the micro-

scopes I

and I

from Table 10.1. See the text for more details.

10.2 The microscope as an information channel 623

0-100 200 400

∆f

opt

q (nm

-1

)

∆f

(

)

Fig. 10.20. Same as Fig. 10.19 for microscope I

with C

= 50 µm.

defocus spread can be reduced by means of

a post-source

monochromator. There are

several candidate monochromator designs available [Ros90, Tie99,MK00] and only time

will tell which monochromator(s) will become commercially available. The main effect

of a monochromator will be an improved chromatic information

limit, in addition to a

better energy resolution for electron energy loss observations. The reader is referred to

the literature cited for more details.

A decrease of the spherical aberration improves the point resolution ρ

but the improve-

ment is not as dramatic as that of the information limit when the beam divergence angle is

changed.

This is due to the fact that the point resolution depends only on

the

-power of

. Alternatively, the point resolution could be improved by increasing the accelerating

voltage which in turn decreases the electron wavelength. This is the main reason for the

existence of intermediate-voltage and high-voltage transmission electron microscopes.

The improved resolution of such machines may come at the expense of higher irradiation

damage to the sample.

A reduced C

also “straightens out” the curved contours of constant sin χ , as can be seen by

comparison of Fig. 10.19 (I

) with Fig. 10.20 for which C

= 50 µm. This corresponds to

the fact that the transfer function oscillations are pushed to higher frequencies and would

disappear altogether for C

= 0 and D = 0. Spherical aberration correctors can be used

to select a given C

value (either positive or negative) so that the point resolution can be

made equal to the information limit. We refer the interested reader to the following papers

for more details on C

correctors: [HU97, KDL99].

The moduli of the point spread functions corresponding to the microscopes de-

scribed in Table 10.1 are shown in Fig. 10.21. The delocalization effect associated

with the ﬁeld emission source (I

) is clearly visible. C

-correction removes part of

the delocalization and sharpens the PSF. The two curves on the right are for the

third set of microscope parameters, with C

= 50 µm and C

= 5 µm, respectively.

It is clear that the PSF becomes more sharply peaked and approaches the ideal

δ-function shape.

624 Phase contrast microscopy

= 50 µm)

= 5 µm)

-4 -20 2 4-4 -2024

|PSF|

x (nm) x (nm)

Fig. 10.21. Modulus of the point spread functions for the microscopes described in

Table 10.1 (arbitrary vertical scale, Scherzer defocus); the microscope I

PSF is shown

for two different values of the spherical aberration constant.

Example 10.5 Consider microscope I

of Table 10.1. What is the maximum ac-

ceptable vibration amplitude u for this instrument to be used out to its chromatic

information limit?

Answer: The chromatic information limit equals ρ

= 0.094 nm. If sample vibration

is to be small enough so that the vibration information limit ρ

is smaller than ρ

then we have

u <

√

ln s

≈ 0.6ρ

= 0.056 nm.

The vibration amplitude must therefore be kept below about 0.5

A to keep the

vibration information limit smaller than ρ

The contrast transfer function can be displayed using the

ctf1.pro routine, avail-

able on the

website. The routine displays the real and imaginary parts of the atten-

uated CTF, along with a 2D representation similar to Fig. 10.12, and the modulus

of the point spread function. A second routine,

ctf2.pro, can be used to display

grayscale images of E(q) sin χ (q), where E(q) is the complete damping envelope.

An example of the output of

ctf2.pro is shown in Fig. 10.22: the effect of two-fold,

three-fold, and four-fold astigmatism and axial coma is illustrated for a 200 kV

instrument operated at Scherzer defocus (C

= 1 mm, C

= 1 mm, E = 1eV,

10.2 The microscope as an information channel 625

Table 10.2. Total number of lattice planes with d-spacings larger

than a given point resolution ρ

for a few technologically important

cubic materials.

Material a [nm] S.G. 0.20 nm 0.15 nm 0.10 nm

Fe 0.2866 Im

3m 1 1 4

GaAs 0.5653 F

43m 2 5 10

Al 0.3572 Pm

3m 3 5 12

CuZn 0.2959 Pm

3m 2 3 7

BaTiO

0.4012 Pm

3m 4 6 15

2-fold astigmatism

3-fold astigmatism

4-fold astigmatism

axial coma

0 µm

= 0.1 µm

G = 10 µm

H = 1000 µm

K = 10 µm

Fig. 10.22. Illustration of the effect of two-fold, three-fold, four-fold astigmatism, and axial

coma on the imaginary part of the attenuated transfer function for a 200 kV LaB

instrument

operated at Scherzer defocus (other parameters in text).

= 0.5 mrad). The maximum values of the aberrations are C

= 0.1 µm, G =

µm, H = 1 mm, and K = 10 µm.

The reader may ask the question: why do we need a microscope with an infor-

mation limit of better than 0.1 nm? The answer to this question is rather simple:

consider the technologically important cubic materials listed in Table 10.2. The table

lists for each material how many families of lattice planes have a d-spacing larger

than a given point resolution. For pure Fe, a rather important material, only one

single d-spacing can be resolved on a microscope with ρ

= 0.2nm(d

110

= 0.203

nm and d

200

= 0.143 nm). This means that the only zone axis orientation for which

a high-resolution image can be obtained on a machine with an information limit of

= 0.15 nm is the [111] zone axis, which contains three independent reﬂections

of the {110} family. Similarly, for GaAs there are only three families of planes

for which the d-spacing is larger than 0.2 nm: {11

1}, {111}, and {200}, which

corresponds to only two different d-spacings. Microscopes with a point resolution

626 Phase contrast microscopy

close to or better than 0.1 nm allow us to obtain high-resolution images of many

different zone axis orientations, which is useful for the study of the structure of

defects and interfaces. One must always bear in mind that the maximum num-

ber of degrees of freedom that can be transferred by the microscope is of the

order of three per area ρ

; this means that it may not be possible to determine

the atomic structure of a defect if the density of projected atom positions is too

high.

10.2.4 Image formation for an amorphous thin foil

To illustrate the effect of the transfer function on the electron wave function exiting

a given thin foil, it is useful to start from a wave function which has all spatial

frequencies present with about the same weight. Such a wave function would result

from scattering from a random 3D arrangement of identical atoms; in a 2D pro-

jection (any projection) all possible projected interatomic spacings will be present,

and therefore the frequency spectrum of such a projected structure will be rather

ﬂat. To get as many electrons as possible to scatter at large angles, we must make an

amorphous foil with the heaviest possible atom, ideally U. In practice, it is easier

to make amorphous foils out of Ge, or, occasionally, W. Amorphous carbon is very

easy to obtain by evaporation methods, but it has a low atomic number and does not

scatter strongly enough at the spatial frequencies corresponding to the information

limit of modern microscopes.

Every microscopy laboratory will probably have a few amorphous calibration

foils available for measurement of various microscope parameters. The basic nu-

merical tool used for the analysis of images of amorphous objects is the fast Fourier

transform. The FFT of a high-resolution image of an amorphous thin foil will al-

low us to determine many microscope parameters, as described in more detail in

Section 10.2.5. The modulus squared of the FFT of a high-resolution image is known

as a diffractogram. In the early days of high-resolution observations, diffractograms

were created by optical means, using a laser and an optical bench [GMWW86].

Nowadays, digital image recording makes it possible to observe diffractograms in

real time.

To create a physically realistic structure model for the amorphous ﬁlm, we would

need to use, in principle, a real radial distribution function. Fan and Cowley [FC87]

have shown that an amorphous object function can be constructed using a simple

uniform random number generator: for each image pixel assign a unit amplitude

and a random phase in the interval [−π, +π], i.e. create a weak phase object with

randomly varying phase,

ψ(r) = e

iφ(r)

10.2 The microscope as an information channel 627

The Fourier transform of this weak phase object is then multiplied by the microscope

transfer function, and after an inverse Fourier transform the image intensity is

given by

I (r) =

−1

F[e

iφ(r)

]E(q)e

−iχ(q)

, (10.65)

where E(q) represents the complete damping envelope. The modulus-squared

of the Fourier transform of this image intensity is then known as the diffracto-

gram

†

(q) ≡|F[I (r)]|

. (10.66)

To obtain a more realistic version of the diffractogram, the Fourier transform F[ψ]

could be multiplied by the electron scattering factor f

(q) for the particular element

that makes up the amorphous ﬁlm.

Figure 10.23 shows several computed diffractograms for an amorphous thin foil,

created using the random phase algorithm. Images for two different microscopes are

represented: the ﬁrst microscope is equivalent to I

of Table 10.1 (with θ

= 1 mrad),

and the second is identical to I

. Row (a) of Fig. 10.23 shows four diffractograms

and corresponding images for the ﬁrst four passbands  f

for machine I

. The

passband is visible for the ﬁrst two defocus values but rapidly disappears below

the noise level for higher-order passbands. This attenuation is mostly due to the

large beam divergence angle θ

. For machine I

(row c), the higher-order passbands

are still clearly visible in the diffractogram. The half or quarter circles indicate 6,

8 and 10 nm

−1

, respectively.

Rows (b) and (d) in Fig. 10.23 reveal the effect of two-fold astigmatism on

the diffractogram. The astigmatism values are indicated as magnitude–angle pairs

(magnitude in nm) in the upper right-hand corner of each diffractogram. The astig-

matism clearly affects the shape of the diffractogram rings, whereas it is not always

clear in the corresponding images. For the images with higher beam coherence

(row d), astigmatism becomes harder to recognize visually.

The bottom row of Fig. 10.23 shows the effect of limiting the bandwidth of

the transmitted signal. The diffractograms are computed for machine I

with de-

creasing aperture radii indicated in nm

−1

in the upper right-hand corner of each

diffractogram. The effect of frequency truncation on the image contrast and detail

is dramatic.

Experimental diffractograms of amorphous thin foils can be used to align the

microscope and to determine various electron optical parameters. In the next section,

we will discuss in detail how microscope alignment can be performed and also how

electron optical parameters can be measured.

†

A diffractogram is also known as a Thon diagram, after F. Thon who ﬁrst employed them in the area of electron

optics [Tho66].