Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2

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Chapter 1 Atomic Resolution Transmission Electron Microscopy 15

Figure 1–6. Phase contrast transfer functions (PCTFs) plotted in one dimension (a and b) and two

dimensions (c and d) calculated for a modern 300-kV FEGTEM illustrating the interpretable resolution

) and information limits (d

). (a and c) calculated at the Scherzer defocus (−34.4 nm) (b and d) cal-

culated at the higher Lichte underfocus (−174.4 nm). In all cases CTFs are calculated for 300 kV, C

0.6 mm, ∆E = 0.8 eV, beam divergence = 0.1 mrad and the interpretable and information limits are

indicated.

As will be discussed in detail in a subsequent section the effects of

temporal and spatial coherence (see also Hawkes and Kasper, 1996) can

be treated through respective envelope functions of form

EqCC

) }

() exp{ ( )}kkk=− +πλλ

2 2 2 2

332

(20)

in which the expression for the spatial envelope includes only defocus

and spherical aberration, with q

, the standard deviation of a Gaussian

modeling the convergent cone of illumination at the specimen surface,

and where

∆= + +













σσσ

4() () ()

(21)

In Eq. (21) C

is the chromatic aberration, E

is the spread in electron

energies arising from the source, V

is the accelerating voltage, and I

is the objective lens current, which affects the objective lens magnetic

ﬁ eld (O’Keefe, 1992).

8,9

The form of these distributions is far less important than their width

and in general, as above, Gaussian distribution functions are used, as

this makes any further derivation analytically tractable.

10,11

The form of these two envelope functions leads to deﬁ nitions of

information limits in which the information transfer drops to a level

of exp(−2) or 13.5% given by (O’Keefe and Pitt, 1980).

dS S

=± +





















+−

πλ

πλπ

427

























C λ

(22)

However, it should be noted that the limit deﬁ ned by d

increases with

defocus and does not therefore deﬁ ne an absolute information limit

The last term is written here in terms of the objective lens current I

following convention. However, this is not strictly correct, particularly if the

lens is operated close to saturation where the magnetic ﬁ eld is not proportional

to the current.

It may seem surprising that the objective lens current inﬂ uences the

variation of focus only in the presence of chromatic aberration. However, this

is due to the general scaling rule that electron trajectories are identical when

energy and magnetic ﬁ eld are changed according to E′ = k

E and B′ = kB.

Hence, in a microscope corrected for chromatic aberration, the focus cannot

be changed by changing the currents in all lens elements by the same factor.

For the defocus spread, the terms due to voltage and lens current instabilities

are well described by Gaussian function, whereas the intrinsic source energy

spread would be more accurately described by a Maxwellian distribution for

thermionic emitters or the Fowler–Nordheim equation (Fowler and Nordheim,

1928) for ﬁ eld emitters.

The above deﬁ nition of δ assumes that the ﬂ uctuations are independent of

each other.

A.I. Kirkland et al.

k− π λ∆ (E (k) = exp 0.5{

Chapter 1 Atomic Resolution Transmission Electron Microscopy 17

per se. In particular, for the case of ﬁ eld emission sources where the

source size is small, the spatial coherence is not limiting and thus the

information limit due to temporal coherence determines the informa-

tion limit for all defoci.

2.3 The Wave Aberration Function

The key optical component affecting HRTEM image formation is the

objective lens and in this section we review its inﬂ uence in terms of

the wave aberration function (see also Hawkes and Kasper, 1996).

For an ideal lens, a point object at a position (x, y) in the object plane

leads to a spherical wavefront in the diffraction plane, contracting to a

conjugate point in the image plane.

However, all electromagnetic

lenses suffer from aberrations causing deviations from this ideal spher-

ical wavefront thus reducing the sharpness of an image point much

more severely than the diffraction limit.

For HRTEM, a wave aberration function W(u, v) is therefore deﬁ ned

that describes the distance between the ideal and actual wavefronts in

the diffraction plane as a function of the position of the point object in

the diffraction plane, (u, v) (Saxton, 1995) (Figure 1–7).

The wave aberration function, W(ω), written in terms of a complex

position variable ω = u + iv can be Taylor expanded to 3rd order in

terms of the axial aberrations as

Figure 1–7. Schematic diagram showing the origin of the wave aberration

function, W(u,v), which describes the complex deviation from an ideal spheri-

cally diffracted wave.

For a pure phase object this condition would, however, lead to lead to zero

contrast at the Gaussian focus.

More rigorously W is a function of both position of the point object in the

diffraction and image planes, i.e., W(x, y, u, v). However, HRTEM experiments

are carried out at high magnifcation and hence the feld of view is small

and the dependence of W on (x, y) can be neglected in the isoplanatic

approximation.

(23)

Converting to polar notation, with ω = ke

iφ

, A

= |A

iα

and B

iβ

, Eq. (23) can be more conveniently rewritten as

(24)

which makes the azimuthal and radial dependence of the various aber-

ration terms more apparent.

Table 1–2 lists the aberration coefﬁ cients important in HRTEM ignor-

ing those with a chromatic dependence.

All of the aberration coefﬁ cients listed in Table 1–2 except C

and C

are due to lens imperfections (either mechanical or electrical) and

would not appear in a perfect round lens (Scherzer, 1949). However,

they can be corrected using combinations of multipole ﬁ elds of appro-

priate symmetry and in recent years this approach has enabled the

Table 1–2. Isoplanatic aberration coefﬁ cients important for

HRTEM.

Aberration Order in w Azimuthal symmetry Name and description

1 1 Image shift

2 2 Two-fold astigmatism

3 3 Three-fold astigmatism

4 4 Four-fold astigmatism

3 1 Axial coma

4 2 Axial star

2 • Defocus

4 • Spherical aberration

The seemingly counterintuitive notation (e.g., C

in Table 1–2 for a second-

order term in the wave aberration fucntion stems from the ray-optical theory

of Seidel aberrations (Hawkes and Kasper, 1989), which are described in

terms of displacements of ray-path intersections with the image plane as a

function of (u,v). These displacements are proportional to the gradient of the

wave aberration function and hence an nth-order Seidel aberration corresponds

to a term of order n + 1 in the wave aberration function.

A.I. Kirkland et al.

W (ω) = <{

2π

λω

∗

∗2

∗

∗3

∗2

∗4

∗3

ω +

∗2

+ . . .}}

W (k, φ) =

2π

{|A

|λk cos(φ − α

)

|λ

cos 2(φ − α

) +

|λ

cos 3(φ − α

) +

|λ

cos(φ − β

)

| λ

cos 4(φ − α

) +

|λ

cos 2(φ − σ

)

+ . . .}

Chapter 1 Atomic Resolution Transmission Electron Microscopy 19

Table 1–3. Accuracy of aberration coefﬁ cients.

Resolution Accuracy

max

(/nm

-1

) d

min

(nm) A

, C

(nm) A

, B

(nm) A

, S

, C

(mm) Tilt (mrad)

11 0.09 0.5 35 2.1 21

10 0.1 0.6 47 3.1 27

9 0.11 0.8 64 4.8 38

8 0.125 1.0 92 7.6 54

7 0.14 1.3 137 13 80

Accuracy to which the aberration coefﬁ cients need to be determined such that each of them causes a maximum

RMS error in the wave aberration function of less that λ/16, i.e., a phase change of less than π/8 for a given target

resolution. The values are calculated for an accelerating voltage of 300 kV (λ ≈ 2 pm). The necessary accuracy for

the beam tilt τ is calculated from that for B

using ∆τ = −∆B

/(3C

) with C

= 0.57 mm.

correction of C

using long hexapole ﬁ elds (see later) (Rose, 1981, 1990;

Haider et al., 1998c; Urban et al., 1999).

For HRTEM imaging the effects of the coefﬁ cients of the wave aber-

ration function are to introduce phase shifts in the image given by

multiplying each term in W by 2 π/λ. These phase shifts can also be

used to estimate the maximum tolerable value in any coefﬁ cient for a

particular target resolution (Table 1–3).

Few of the aberration coefﬁ cients deﬁ ned above are directly observ-

able under axial illumination and their determination therefore relies

on measurements taken as a function of known injected beam tilts.

If the illumination is tilted by an angle τ the new aberration coefﬁ -

cients up to C

(marked with a prime) are given by

′

=+ + + + + +

′

AAA A C B B C

AA A

001 2

11 2

ττ τ τ ττ ττ

ττ

* +++

′

=+ +

′

11 2 3

22 3

CC B C

BB C

ττ

τττ

R(*) *

(25)

From Eq. (25) it is immediately apparent that the shift A′

− A

between two images taken at beam tilts differing by τ depends on all

the other aberration coefﬁ cients. Hence, measuring the image shifts

induced by a suitable set of beam tilts provides a method for measuring

all coefﬁ cients.

Alternatively, the tilt-induced changes, C′

− C

and A′

− A

in defocus

or two-fold astigmatism measured from diffractograms (Figure 1–8),

depend on A

, C

, A

, B

, and C

, thus providing an alternative measure

of these coefﬁ cients. If the spherical aberration is independently deter-

mined then measurement of the orientations only of diffractograms also

provides another simple measure of A

, A

, and B

(Saxton, 2000).

Measurements of diffractograms (Krivanek, 1976; Krivanek and

Leber, 1994; Zemlin et al., 1978; Zemlin, 1979; Typke and Dierksen, 1995;

Pan, 1998; Saxton, 1995) or image shifts (Typke and Dierksen, 1995;

Figure 1–8. Individual axial diffractograms calculated from HRTEM images of a thin amorphous

germanium foil showing the effects odefocus (C

) and two-fold astigmatism (A

). (a) Axial image

recorded at large underfocus with the two-fold astimatism corrected. (b) Axial image in the presence

of a small amount of two-fold astigmatism giving rise to an elliptical diffractogram. (c) As for (b) but

with a larger two-fold astigmatism giving rise to a “Maltese Cross”-shaped diffractogram showing

overfocus and underfocus along the axes indicated. (d) Representative tableau of diffractograms

recorded for a set of differently titled illumination conditions illustrating the variation in A

and C

with illumination tilt. Each experimental diffractogram is shown merged with a simulated diffracto-

gram calculated from the ﬁ tted values of A

and C

. The illumination tilt axes are indicated.

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 21

Koster et al., 1987, 1989; Koster, 1989; Koster and de Ruijter, 1992; Saxton,

1995), however, have their own particular experimental advantages

and disadvantages.

A practical difﬁ culty with the application of tilt-induced displace-

ments arises from the measurement of the image shifts using the peak

position in the cross-correlation function deﬁ ned as

XCF =

−

FT c c

[

]

(26)

The tilt-induced change in A

introduces a linear phase variation in

the cross spectrum c*

, which leads to a displacement of the XCF peak

to a position given by the shift vector between the images. As already

shown, the other imaging parameters also change as the beam is tilted

and this causes the phase variation to become nonlinear at higher

spatial frequencies and leads to distorted cross-correlation peaks.

However, when the imaging conditions in both images are approxi-

mately known, these nonlinear phase shifts can be compensated and

a sharp XCF peak can be recovered (Saxton, 1994; Kirkland et al., 1995).

Shift measurements also fail for clean perfectly periodic specimens in

which image positions differing by any integer multiple of a lattice

vector cannot be distinguished. A more severe problem associated

with the tilt-induced shift method is that any displacement due to

specimen drift is indistinguishable from the tilt-induced displacement

required and hence this approach is most frequently used at low reso-

lution or for initial coarse alignment at high resolution.

Conversely, diffractogram measurements require the presence of an

area of thin disordered material and are thus less generally applicable.

However, they are insensitive to specimen drift and their measurement

is relatively straightforward even under tilted illumination conditions.

Hence defocus and astigmatism measurements are best suited to ﬁ ne

adjustment at high resolution (Pan, 1998).

Historically the use of diffractograms for the determination of aber-

rations was ﬁ rst suggested by Thon (1966) for the measurement of

defocus from ring positions in an optically generated diffractogram.

This method was later extended through the use of diffractogram tab-

leaus acquired with different beam tilt azimuths (Figure 1–8) to the

measurement of the spherical aberration C

, axial coma B

, and three-

fold astigmatism A

by Krivanek (1976) and Zemlin et al. (1978).

However, the diffractogram tableau method was computationally too

demanding for routine use at this time and was used only to demonstrate

that the alignment achieved from current reversal or voltage centering

was inadequate as an alignment for HRTEM (Saxton et al., 1983).

The current reversal center alignment involves reversing the current of the

objective lens and is no longer practical with the strong lenses used in modern

instruments. It should not be confused with the current center alignment,

where the objective lens current is oscillated by a small amount. Similarly, in

the voltage center alignment, the high tension is oscillated. Generally, the axes

found by the three methods are distinct and the voltage center provides a

workable approximation to the coma-free axis, but for HRTEM coma-free

alignment is preferred.

Fitting C

and A

to match simulated diffractogram patterns with

experimental ones can be done manually to relatively high accuracy

(Figure 1–8d; Chand, 1997). This, however, is a lengthy and tedious

process and automation of this task is highly desirable. This problem

has been addressed by Baba et al. (1987) and Fan and Krivanek (1990)

in an algorithm where the diffractogram is divided into 32 sectors and

the defocus along the directions in each sector is determined by cross-

correlating the rotational sector average with an array of theoretical

diffractograms. The sinusoidal focus variation as a function of the

azimuth angle obtained is subsequently ﬁ tted to the measured focus

values to determine C

and A

and automatic alignment can be achieved

using a tableau with as few as four tilt azimuths.

More generally, the major challenges for implementing a robust auto-

mated diffractogram ﬁ tting procedure are as follows:

1. The most abundant amorphous material in the microscope, carbon,

is a weak scatterer. Hence, particularly at high spatial frequencies,

the signal is weak compared to the noise background affecting

overall diffractogram quality.

2. The strength of the observed signal depends on the object as well

as on the phase contrast transfer function.

3. Diffractograms taken close to Scherzer or Gaussian defocus show

few rings and are difﬁ cult to ﬁ t.

4. In the presence of large two-fold astigmatism and at close to Gauss-

ian defocus the diffractograms are cross- rather than ring-shaped,

which leads to difﬁ culties for algorithms based on the evaluation of

rotationally averaged sectors of the diffractogram.

5. It is difﬁ cult to distinguish between overfocus and underfocus.

6. Most automated algorithms fail when a signiﬁ cant amount of crys-

talline material is present, leading to strong reﬂ ections at positions

unrelated to the ring pattern.

More recently the advent of C

-corrected microscopes (see later) has

made automated aberration measurement more important because the

nonround lens elements introduce a multitude of high-order aberra-

tions (up to six-fold astigmatism) that require correction in an elaborate

alignment procedure. Uhlemann and Haider (1998) have implemented

an algorithm that can evaluate the apparent defocus and astigmatism

from a diffractogram in less than 400 ms based on a comparison of the

experimental diffractogram with an extensive library of precalculated

diffractograms and is robust to disturbances from the presence of

crystalline material.

However, given the limitations of both of these traditional methods

an alternative method has also been proposed, which is both applicable

to a wide range of specimens and sufﬁ ciently accurate for use in atomic

resolution imaging (Meyer, 2002; Meyer et al., 2002, 2004).

In this approach a phase correlation function (PCF) (Figure 1–9) (Kuglin

and Hines, 1975) is initially calculated (deﬁ ned as the conventional XCF

with the modulus set to unity):

PCF FT F

() ()

()* ()













−1

(27)

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 23

where c

(k) are the image Fourier transforms and F(k) is a rotationally

symmetric weighting factor used to suppress high-frequency noise.

The modulus normalization is essential as it suppresses the crystal

reﬂ ections that make the conventional XCF periodic, thus allowing

aberration measurement in the absence of amorphous material.

The PCF calculated between two images recorded at different defocus

levels consists of a centrosymmetric ring pattern, the exact form of

which depends on the relative defocus between them (assuming that

the other aberration coefﬁ cients remain constant between the two

images) (Figure 1–9). It is possible to compensate for the phase shifts

giving rise to this ring pattern by the application of a phase factor

dependent on the defocus difference, thus deﬁ ning a phase compen-

sated PCF

PCF FT F

Wcc

() ()

cos[ ( )]

()()

|cos[ ( )]

()(

kkk

−1

))|+





















(28)

Figure 1–9. Comparison

of a (a) cross- and (b) phase

correlation functions (XFC

and PCF) between two

images of a crystalline

material with a defocus

difference of 69 nm. Due to

the periodicity of the crys-

talline specimen, the XCF

peak repeats periodically

but the PCF dose not show

this repetitive pattern and

consists of a single peak

broadened into a concen-

tric ring pattern due to the

defocus difference.

in which the very small positive number, h, prevents a zero denomina-

tor and where W

(k) = π∆C

λ|k|

describes the propagation from the

ﬁ rst to the second image in the presence of only a defocus difference

∆C

. When the value of ∆C

matches the actual focus difference the

phase compensated PCF collapses to a sharp localized correlation

peak. Hence the relative defocus difference between two images can

be determined by simply maximizing peak height in the phase com-

pensated PCF as a function of the compensated defocus difference

(Figure 1–10), which with subsequent reﬁ nement can yield relative

defoci to an accuracy better than 1 nm (Meyer et al., 2002).

In the second step an initial restored image wavefunction (see later

for details of the restoration process), ψ

(k), in the plane of a reference

image is restored and the absolute values of A

and C

are determined

using a phase contrast index function (PCI) f

PCI

given by

fCA

WCA

PCI si si

()cos[arg(())] arg[ ( )]

()

kkk

=− + −

ψψ

(29)

where

(30)

Figure 1–10. Peak height of the PCF (solid line) and XCF (dashed line) (shown

against the left hand side and right hand side scales) between two images as

a function of compensated focus difference showing a sharp maximum at the

correct compensating focus difference in the PCF.

A.I. Kirkland et al.

W (k, C

, A

) =

|λk

cos 2(φ − α

) +

λk