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

Подождите немного. Документ загружается.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 25

is the symmetric part of the wave aberration function deﬁ ned earlier

with trial parameters C

, A

and ﬁ xed spherical aberration C

. This func-

tion measures the conjugate symmetry in the corrected image wave-

function and has a value of +1 when this is conjugate antisymmetric

and −1 when conjugate symmetric. Therefore under the weak phase

object approximation when the trial values of C

and A

are correct the

PCI

tends to 1 for all spatial frequencies, whereas for mismatched param-

eters, f

PCI

shows dark bands (Figure 1–11a–c). In practice, the correct

values of C

and A

are readily determined from a plot of the f

PCI

aver-

aged over k (Figure 1–11(d)).

This general method is best implemented experimentally using a

combined tilt-defocus dataset geometry as illustrated in Figure 1–12,

which comprises a set of short focal series recorded at a number of tilt

azimuths. This dataset is overredundant and can be used to calculate

values of C

and A

for each of the recorded tilt azimuths using the

PCF/PCI approach.

For all three measurement methods described, data recorded for a

suitable number of deﬁ ned illumination directions and magnitudes

provide the full set of aberration coefﬁ cients (Meyer, 2002) and these

Figure 1–11. Phase contrast index function f

PCI

. For mismatched values of C

(a) −50 nm, (b) −10 n m,

PCI

shows dark rings, whereas at the correct values of C

, f

PCI

is close to one (white) at all spatial fre-

quencies (c). (d) The f

PCI

averaged over k and plotted as a function of C

showing a sharp maximum

at the correct focus value of −128 nm. The values ∆C

given in (a) to (c) are relative to this focus.

can be conveniently and reliably determined in practice by least-squares

ﬁ tting of the parameters to be determined to the observations available.

Where the injected tilt is known accurately the dependence of the

observables on the aberration coefﬁ cients is linear yielding an easily

found, unique solution (Saxton, 1995; Meyer et al., 2004). Alternatively,

if the calibrations of the beam tilt coil magnitudes and orientations are

not known, these unknowns can be ﬁ tted in an additional nonlinear

iterative minimization of the residual misﬁ t from the linear ﬁ tting of

the aberration coefﬁ cients (Saxton, 1995).

2.4 Coherence

As already described, contrast in a high-resolution image formed in an

electron microscope arises from the interference of electron waves that

have been scattered by the specimen. The degree to which these waves

can interfere is often referred to as the degree of coherence (Born and

Wolf, 1999). Hence, the degree of coherence is vital in determining

high-resolution image contrast (for a review see Hawkes, 1978).

For all electron sources used in HRTEM imaging (Reimer, 1997), the

source is only partially coherent (Hawkes and Kasper, 1996; Hawkes,

1978). There are two aspects to this partial coherence: partial spatial

coherence, due to the ﬁ nite dimensions of the electron source (Born and

Wolf, 1999), and partial temporal coherence, which is associated with the

ﬁ nite energy distribution of the source and ﬂ uctuations in both the

accelerating voltage and the objective lens current (Hanßen and Trepte,

1971). The effect of this partial coherence of the source is to restrict the

amount of information that can be extracted from high-resolution

images (Frank, 1973; Wade and Frank, 1977; Hawkes, 1978). Thus, the

source and the associated condenser lens system represent the second

major optical component that directly affects HRTEM image contrast.

Frequently, the illumination system of the electron microscope is

treated as an incoherently ﬁ lled effective source (Hopkins, 1953, 1951)

with an intensity distribution S(q). Within this model the effects of

Figure 1–12. Schematic representation of the combined tilt/defocus dataset

used for both aberration determination using the PCF/PCI approach and tilt

azimuth reconstruction described in a subsequent section. A three member

focal series is recorded for each of six different tilt azimuth angles and also

for axial illumination. Image numbers refer to the experimental recording

sequence designed to minimize hysteresis in the objective lens and beam tilt

coils and to enable measurement of focus drift during acquisition.

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 27

partial spatial coherence on the HRTEM image intensity can be calcu-

lated by incoherent summation of intensities over all the incident

angles from the effective source.

For this source model the temporal partial coherence effect can also

be treated incoherently (Hanßen and Trepte, 1971) and it is generally

assumed that the spatial and temporal distributions of the source are

not correlated (for a detailed treatment of the case in which this is not

valid see Hawkes and Kasper, 1996). Under these conditions the image

intensity can be written as

(31)

in which the incoherent summations of the coherent image intensities

at incident angles q and focus spread levels, f are weighted by the

spatial and temporal intensity distributions S(q), F(f), respectively.

The individual coherent image intensities are thus given as the

modulus square of the convolution of the Fourier transform of speci-

men exit wavefunction ψ(k,q) at an incident angle q and the objective

lens aberration function W(k,C

+ f,q) with corresponding beam tilt and

defocus. General computation of the image intensity therefore involves

the full dynamical calculation of coherent image intensities over an

equally spaced mesh of beam tilts, q and focal spread values, f. This is

optimally tackled through the use of numerical Monte Carlo integra-

tion (Chang et al., 2005). A number of approximations may also be

made that make this calculation more tractable and that provide ana-

lytical solutions.

For small beam tilts of the incident wave and for modest specimen

thickness, the specimen exit wavefunction can be approximated as a

single exit wavefunction at the mean incident illumination angle.

A further approximation can also be made by expanding the aberra-

tion function to only ﬁ rst order which is valid if the beam tilt and focal

spread are small as typiﬁ ed by many modern instruments. Using

these approximations and assuming that the focal spread and beam

divergence distributions are Gaussian with e

−1

values of q

and

respectively the image intensity can be written as (Ishizuka, 1980)

(32)

with

(33)

For cases in which this basic assumption does not hold the mathematical

description of coherence is best treated in terms of the mutual dynamic object

spectrum and associated mutual coherence function. This treatment is outside

the scope of this section and an excellent overview can be found in Ernst and

Rühle (2003).

I(x) =

S(q)F (f )|F

−1

{ψ

(k; q) exp{−iW (k, C

+ f, q)}}|

I(k) = Σ

ψ(k + k

)ψ

∗

)T (k + k

, k

T (k

, k

) = exp{−i[W (k

; C

) − W (k

; C

)]} exp{−π

− W

]

}

× exp{−π

∆

[

∂W

∂C

k=k

−

∂W

∂C

k=k

]

Figure 1–13. Moduli of the transmission cross-coefﬁ cient (TCC) functions deﬁ ned in the text for

(b) for a strong object. In both case the functions are shown for defoci ranging from −1 to −9 Scherzer.

All calculations were carried out at 300 kV with C

= 0.6 nm and with e

−1

values of Gaussian beam

The Fourier transform of Eq. (33) is given by the integral over all

pairs of diffracted beams with T representing a transmission cross coef-

ﬁ cient (TCC) (Figure 1–13). Within this expression for the TCC the ﬁ rst

exponential term describes the spatial coherence for a Gaussian spatial

intensity distribution with an even aberration function. This term is a

A.I. Kirkland et al.

divergence and focal spread distributions of 0.28 mrad and 4.7 nm, respectively (Chang, 2004).

combined spatial and temporal coherence and calculated under (a) the weak object approximation and

Chapter 1 Atomic Resolution Transmission Electron Microscopy 29

function of the square difference of the gradient of the aberration func-

tion at two spatial frequencies and allows maximum transfer when the

slopes of the aberration function are identical at k

and k

. The second

exponential term describes the temporal coherence, and is a function

of the square difference of the derivative of the aberration function

with respect to the focus at two spatial frequencies. This function has

maximum transfer when |k

| = |k

only interference between the transmitted beam and diffracted beams

is considered, the TCC can be further simpliﬁ ed as (Wade and Frank,

1977)

In this limit both spatial and temporal coherence functions for weak

objects decrease with increasing spatial frequencies (Figure 1–13), and

can be recognized as the simple damping envelopes described in an

earlier section deﬁ ning the information limit.

3 Instrumentation

This section describes certain aspects of instrumental design that are

important for HRTEM in that they directly contribute to the resolution

attainable. Inevitably, due to limitations of space, not all aspects are

covered fully and the reader is referred to one of several dedicated texts

on HRTEM and electron optics for more detailed treatments of indi-

vidual areas (Spence, 2002; Hawkes, 1972, 1982; Reimer, 1984, 1995;

Hawkes and Kasper, 1989).

3.1 Electron Sources

For HRTEM electron sources have to fulﬁ ll a number of key require-

ments summarized as follows (Spence, 2002; Hawkes and Kasper,

1989):

1. High brightness and coherence.

2. High current efﬁ ciency.

3. Long life under available vacuum conditions.

4. Stable emission characteristics.

5. Low energy spread.

An ideal source would therefore provide independent control of the

illumination intensity and coherence for any given illuminated area, a

situation that is currently best approximated by ﬁ eld emission

sources.

For any source, the current, I, passing through an area, A, is propor-

tional to both A and to the solid angle subtended by the illumination

aperture at the source (Reimer, 1984). A beam brightness can be deﬁ ned

as the constant of proportionality, β, in the above relationship and as

the area and angle tend to zero this is given by

β = I/πAθ

(35)

2 2

1 3

3 3 2 2 2 2 2

T (k) = exp[−iW (k)] exp{−π q (C λk + C λ k ) } exp[−π ∆ (λk /2) ].

If a further assumption of a weak scattering object is made, such that

(34)

for an aperture semiangle θ.

The effect of an ideal lens (Hawkes, 1982; Reimer, 1984; Hawkes and

Kasper, 1989) below the source forming the illumination system is to

reduce the current density by M

but to increase the angular aperture

by 1/M

. Thus, in the absence of any lens aberrations, the brightness

remains constant at all conjugate planes in the microscope.

To compare different source types a theoretical brightness for a par-

ticular source can be expressed in terms of the emission current density

at the cathode (ﬁ lament), ρ, the cathode temperature, T, and the rela-

tivistic high voltage, V, as

= ρeV/πkT (36)

although this requires optimal operating conditions and in practice is

often reduced to lower values for thermionic sources of between 0.1

and 0.5 β

. Table 1–4 lists values of β

together with other physical and

operational characteristics for the four sources commonly used in

HRTEM instruments.

Tungsten hairpin and LaB

cathodes rely on thermionic emission of

electrons, where at suitably high temperatures electrons in the tail of

the Fermi distribution acquire sufﬁ cient kinetic energy to overcome the

workfunction of the cathode material. Tungsten hairpin sources are the

oldest electron sources in commercial use but are only marginally

useful for HRTEM and have largely been replaced by LaB

cathodes

(Ahmed and Broers, 1972) and more recently by ﬁ eld emission sources

(Honda et al., 1994; Otten and Coene, 1993). The choice of single crystal

LaB

cathodes as thermionic emitters for HRTEM is largely due to their

higher brightness while still being operable under relatively poor

vacuum conditions. They also have the advantage of operating at a

lower temperature than W ﬁ laments (due to a lower workfunction),

thus providing lower energy spreads. For these reasons until the rela-

tively recent availability of commercial ﬁ eld emission sources LaB

cathodes were most widely deployed for HRTEM.

Table 1– 4. Physical properties of modern electron sources important for HRTEM.

Measured

Virtual brightness Energy Melting Vacuum Emission

source at 100 kV FWHM point required current

Spource diameter (A cm

-2

-1

) (eV) (°C) (torr) (mA)

Heated ﬁ eld 100 nm 10

–10

0.8 3370 10

-8

50–100

emission ZrO/W 10

-9

Room temperature 2 nm 2 ¥ 10

0.3 3370 10

-10

ﬁ eld emission

Hair-pin ﬁ lament 30 mm 5 ¥ 10

0.8 3370 10

-5

100

LaB

5–10 mm 7 ¥ 10

1 2200 10

-6

(at 75 kV)

From Spence (2002).

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 31

The detailed physics of ﬁ eld emission and the design of ﬁ eld emis-

sion guns for HRTEM are beyond the scope of this section and the

article by Crewe (1971) and the chapter in Reimer (1997) provide

detailed reviews. These sources are now relatively common in HRTEM

instruments and consist of either a heated or, more rarely, an unheated

single crystal of tungsten in a 〈111〉 or 〈311〉 orientation. In operation

the tip of the source is held in a region of high electrostatic ﬁ eld

enabling electrons to tunnel through the lowered potential energy

barrier at the surface. In many HRTEM instruments the alternative

Schottky emitter comprising a ZrO-coated W tip (Tuggle and Swanson,

1985; Swanson and Schwind in Orloff, 1997), in which the workfunc-

tion is lowered to 2.8 eV (from 4.6 eV), is often used allowing electrons

to overcome the potential barrier at a lower temperature than for con-

ventional thermionic sources leading to a reduced energy spread. True

ﬁ eld emission sources operating at room temperature have been less

widely employed due to the need for stringent ultrahigh vacuum

(UHV) conditions at the tip region to avoid contamination (Von

Harrach, 1995). However, these sources provide the highest brightness

and lowest energy spread.

The above physical requirements of the source with respect to bright-

ness and energy spread affect HRTEM images via image recording

times and the resolution limit imposed by temporal coherence.

The detail in an HRTEM image is also strongly affected by the spatial

coherence of the electrons emitted from the source and certain physical

parameters contribute to this. A theoretical treatment of coherence and

its inﬂ uence on HRTEM images has been given in a previous section

and we therefore conﬁ ne discussion here to the relevant physical char-

acteristics of the source itself.

As already described an effective source can be deﬁ ned as an imagi-

nary electron emitter ﬁ lling the illuminating aperture (Hopkins, 1951,

1953). In this case each point within the aperture represents a point

source of electrons giving an emergent spherical wave, becoming

approximately plane at the specimen. Thus, at the specimen plane each

electron can be speciﬁ ed by the direction of an incident plane wave.

Hence increasing the size of the illuminating aperture increases the

size of the central diffraction spot and decreases the spatial coherence.

This can also be described as a coherence width, which deﬁ nes the

transverse distance at the object plane over which the illuminating

radiation may be treated coherently. Thus waves scattered from atoms

separated by less than this distance will interfere coherently and their

complex amplitudes must be added, whereas atoms separated by more

than this distance scatter incoherently and the intensities of the scat-

tered radiation are added. For sources to fulﬁ ll the commonly held

assumption that the illuminating source can be replaced by an incoher-

ently ﬁ lled aperture (see earlier) the coherence width in the plane of

this aperture must therefore be small compared to its size, thus making

the degree of coherence dependent only on the size of the illuminating

aperture. The effect of source size on coherence hence becomes

important only for sources smaller than ca. 1 µm, e.g., the smallest LaB

or pointed tungsten sources available. Field emission sources do not

conform to this model and for FEG HRTEM instruments the degree of

coherence is critically dependent on both the source size and the excita-

tion of the condenser and gun lens system.

3.2 Optics

The optical column of a modern HRTEM instrument typically consists

of two or three condenser lenses, an objective lens, and up to six imaging

lenses below the specimen. The science of magnetic lens design dates

back to the late 1920s (Busch, 1926) when it was realized that rotation-

ally symmetric magnetic ﬁ elds could be used to focus electrons and

that to produce a lens of high refractive power the magnetic ﬁ eld along

the axis of rotational symmetry needed to be conﬁ ned to a small region

in which the ﬁ eld strength was sufﬁ ciently high. This and other elegant

early work using algebraic expressions for the lens ﬁ eld has been exten-

sively reviewed elsewhere (Hall, 1966; Hawkes, 1972, 1982; Hawkes and

Kasper, 1989; Orloff, 1997) and due to limitations of space is not consid-

ered further. Modern lens design is guided by computed solutions of

the Laplace equation (see, for example, Septier, 1967, for a review) and

subsequent numerical solution of the ray equation (Mulvey and Wall-

ington, 1973). The practical design of electron optics has also been

extensively reviewed elsewhere (Hawkes, 1982; Grivet, 1965; Tsuno,

1999; Hawkes and Kasper, 1989) (see also the chapter by Tsuno in Orloff,

1997) including not only the design of the lens polepiece itself but also

that of the overall magnetic circuit. For these reasons we provide here

only a brief non-mathematically rigorous summary of selected aspects

of electron optics relevant to HRTEM and further restrict our discussion

to the properties of the objective lens only.

For all lenses the aberrations important for HRTEM increase with

angle and it is largely the optical properties of the objective lens that

dominate the ﬁ nal quality of the image formed. These can be treated

analytically through either the Eikonal (Sturrock, 1955) or the Trajec-

tory methods (for collected references to these see Hawkes, 1982,

Chapter 1). Alternatively, a numerical approach has been described by

(Munro, 1973; Orloff, 1997) based on ﬁ nite element calculation (FEM)

of the magnetic ﬂ ux density distribution for a trial model lens geome-

try (see also Lencova in Orloff, 1997, for details). Subsequently this can

be used to calculate the axial magnetic ﬁ eld distributions and the par-

axial rays, which can be ﬁ nally combined to calculate the aberration

integrals (for further details see Tsuno, 1999, and Orloff, 1997).

The most important lens aberrations for HRTEM are spherical and

chromatic aberration and the effects of these are illustrated in Figure

1–14.

Lenses which use the image formed by a preceding lens as an object are

classed as projector lenses and include “intermediate” lenses in modern

instruments. These are thus distinguished from the objective lens which uses

the physical specimen as its object.

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 33

Spherical aberration occurs when rays leaving an object at large

angle, θ

, are refracted too strongly in the outer regions of the lens and

are brought to a focus before the Gaussian image plane. If all rays from

such an object are considered then the radius of the circle of least con-

fusion due to the spherical aberration is r = MC

in the Gaussian

image plane, with M the magniﬁ cation. The value of C

depends on the

lens geometry and excitation and on the object position within the lens

ﬁ eld and typically takes values between between 0.5 and 2 mm for

modern HRTEM instruments and sets the limit to the interpretable

resolution (see earlier) in the absence of corrector elements.

For a known lens geometry with a deﬁ ned magnetic ﬁ eld distribu-

tion values for the spherical aberration can be obtained from either

computed solutions of the full (nonparaxial) ray equation (Liebmann,

1949) or through the use of the expression below due to Glaser

(1956):

dB z

Bz Bz

zz3

{}

+−

′

{}









()

() ()

()





∫

hzdz

()

(37)

where h(z) is the paraxial trajectory of a ray that leaves an axial object

with unit slope and B

(z) is the magnetic ﬁ eld.

The presence of deriva-

tives in Eq. (37) explains why the value of C

is sensitive to the shape

of the ﬁ eld. For this reason the detailed geometry of the polepiece

(Munro, 1973) and the location of the specimen within the polepiece

gap are key design parameters of the objective lens (Tsuno, 1999).

Chromatic aberration arises from variations in the lens focal length

with wavelength and hence electron energy. As all electron sources are

polychromatic to a greater or lesser extent (see earlier) a series of in-

focus images are formed on a set of planes normal to the optic axis,

one for each wavelength present in the illumination. A geometrical

optics treatment, as a ﬁ rst approximation, can be used to determine the

effects of this focal length variation, which leads to an extended disk

Figure 1–14. Illustration of certain lens aberrations. (a) A perfect lens focuses

a point source to a single image point. (b) Spherical aberration causes rays at

higher angles to be overfocused. (c) Chromatic aberration causes rays at dif-

ferent energies (indicated by color) to be focused differently. (See color plate.)

The spherical aberration can be measured using either diffractograms of

image shifts as detailed earlier.

in the Gaussian image plane (of a point object) with radius, r =

Mθ

[(∆V

) − (2∆I/I)]. Typical values for C

in modern instruments

are similar to those for C

and contribute to setting the absolute infor-

mation limit for HRTEM (see earlier).

In a similar fashion to the spherical aberration the value of the chro-

matic aberration can also be evaluated if the ﬁ eld distribution is known

(Glaser, 1952) as

Bzhzdz

∫

()()

(38)

For applications to HRTEM both of these aberrations can be reduced

by introducing the specimen into the ﬁ eld of an immersion objective

lens for which the magnetic ﬁ eld on the illumination side of the elec-

tron optical column (the preﬁ eld) does not directly affect image forma-

tion. This symmetrical (condenser objective) geometry also has the

signiﬁ cant practical advantage of enabling switching from HRTEM

using a relatively broad parallel beam to probe-forming modes for

analysis or convergent beam electron diffraction from the same area of

the sample and has largely replaced earlier asymmetric “top entry”

lens designs.

The spherical and chromatic aberration are sensitive to the overall

geometry of the polepiece, which is often described in terms of the key

dimensions deﬁ ning the upper and lower polepiece faces and bores

and the gap between the two polepieces. However, the behavior of both

and C

(and of focal length) can be readily calculated for a given

geometry as a function of the lens excitation (Spence, 2002; Tsuno, 1999;

Hawkes, 1982) and specimen position enabling these coefﬁ cients to be

optimized, subject to constraints imposed by machining tolerances

and requirements for specimen movement and tilt.

3.3 Specimen Stages

The most important purely mechanical components of modern HRTEM

instruments are the specimen holder and the goniometer into which it

is ﬁ tted. A number of complex designs for these, retaining maximum

ﬂ exibility in specimen movement and access to peripheral devices

under constraints of extremely high mechanical and thermal stability,

have been proposed and are reviewed elsewhere (Valdrè, 1979; Valdrè

and Goringe, 1971; Watt, 1986; Turner et al., 2005). Most recently facili-

ties for nanoscale specimen manipulation and electrical measurement

have also been included (Wang, 2003).

The original designs for instruments used for high resolution were

based on “top entry” goniometers into which a conical cartridge was

inserted ﬁ tting into the upper tapered bore of the objective lens pole-

piece with the specimen supported in a cup at the base of the cartridge.

However, this arrangement leads to restrictions in objective lens design

as the polepiece bore must be grossly asymmetric with an upper bore

of sufﬁ cient diameter to allow the specimen to pass (Tsuno, 1999).

Translate and tilt movements also require complex, high precision

mechanical mechanisms as the specimen is located at the base of the

A.I. Kirkland et al.