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 35

cartridge surrounded by the polepiece. The geometry of top entry

stages also precludes any EDX analysis and the provision of specimen

heating or cooling is extremely difﬁ cult (Watt, 1986). Despite these

limitations this design has inherent advantages in that the specimen

cartridge is mechanically and acoustically isolated from the external

environment and its cylindrical symmetry leads to high thermal

stability.

The more common side-entry design in which the specimen is

attached to a rod that is inserted into the goniometer allows far greater

ﬂ exibility but introduces potential mechanical and thermal instability.

However, this is outweighed for many applications by the ﬂ exibility of

this design (for example, in providing specimen heating and cooling)

and modern commercial implementations are sufﬁ ciently stable for

atomic resolution imaging. In the majority of cases commercial side-

entry goniometers provide high precision computer control of all ﬁ ve

axes of specimen movement with stepper motors or donator drives

(Emile et al., 1993). This design aspect has recently been further

improved to subnanometer precision in movement using additional

piezo controls for translation (Kondo et al., 1994a).

It is also worth noting that several designs for fully bakable UHV

HRTEM stages have been proposed in which a side-entry mechanism

is employed with a detachable tip that locates within the goniometer

allowing the transfer rod to be withdrawn (Kondo et al., 1994b).

3.4 Energy Filters

In many applications of HRTEM, energy ﬁ ltering is of great impor-

tance, particularly in improving quantiﬁ cation and interpretability of

images by removal of inelastically scattered electrons (Stobbs and

Saxton, 1987; Krivanek et al., 1990; Boothroyd, 1998, 2000; see Reimer,

1995, for reviews). At this point we note that the energy ﬁ lters outlined

here also ﬁ nd widespread application as spectrometers for electron

energy loss spectroscopy (EELS) and in energy ﬁ ltered imaging at

medium resolution (Reimer, 1995; Tsuno, 1999; Egerton, 1996). However,

this section is restricted to brieﬂ y reviewing their use for imaging.

Imaging ﬁ lters are best differentiated by the nature and geometry

of their constituent optical elements and can be classiﬁ ed as purely

electric, purely magnetic, and combined electric/magnetic (Uhlemann

and Rose, 1996). To separate electrons according to their energy loss

dipole ﬁ elds are required and hence all ﬁ lters have a curved optic axis

with the notable exception of the Wien ﬁ lter (Tsuno, 1993). For HRTEM

purely electric (see, for example, Henkelman and Ottensmeyer, 1979)

and combined electric/magnetic ﬁ lters have the disadvantage of being

limited to operation at relatively low accelerating voltages due to difﬁ -

culties associated with the large electric ﬁ eld strengths required at

higher voltage. However, as discussed later, these have found recent

alternative application as monochrometers in which the electrons have

an energy of only a few electronvolts (Rose, 1990).

Magnetic ﬁ lters have been constructed as both straight vision

(in-column) types (Rose and Plies, 1974; Tsuno et al., 1997) and in a

postcolumn geometry, which can be retroﬁ tted to conventional instru-

ments (Krivanek et al., 1992).

The former can be classiﬁ ed further as either Ω geometries in which

the deﬂ ection angles of the sector magnets cancels to zero and α ﬁ lters

in which the sum of the deﬂ ection angles equals 2π (Lanio, 1986). For

routine operation and ease of alignment of in-column ﬁ lters simple

ﬁ lter geometries are required while any residual aberrations should

not appreciably affect the quality of the ﬁ nal HRTEM image. This can

be achieved through careful choice of suitable symmetries of the mag-

netic deﬂ ection ﬁ eld and of the paraxial rays about the midplane of the

system by which second-order axial aberrations and distortions can be

canceled making these designs suitable for energy-ﬁ ltered HRTEM.

Based on this principle several commercial ﬁ lters have been success-

fully produced (Rose and Plies, 1974; Tsuno et al., 1997). A fully cor-

rected Ω geometry ﬁ lter has also been proposed and constructed,

initially using curved polepiece faces (Rose and Pejas, 1979) but subse-

quently revised to use straight polepiece faces and additional sextupole

elements (Lanio et al., 1986). Finally we note that a corrected in column

ﬁ lter, with an extremely high dispersion achieved through the use of

inhomogeneous magnetic ﬁ elds created by conical magnets has also

been described (Uhlemann and Rose, 1994).

The alternative postcolumn ﬁ lter geometry based on a single 90°

sector magnet with curved polepiece faces has also been produced

commercially and has found found widespread application (Krivanek

et al., 1992). For energy-ﬁ ltered HRTEM this design has the disadvan-

tage of nonzero second-order distortions and chromatic aberration of

magniﬁ cation. These ﬁ lters therefore operate at a large intermediate

magniﬁ cation so that these do not appreciably affect the image quality.

However, due to the large magniﬁ cations required the third-order

distortions and higher order chromatic aberrations limit the number

of clearly resolved object elements. For these reasons a sequence of

quadrupole and hexapole elements is incorporated after the sector

magnet and alignment to third order is carried out by automated

computer control. A signiﬁ cant practical advantage, however, of this

geometry is that it can be retroﬁ tted to conventional microscopes.

Finally, we also note that investigations into the contribution of

phonon scattering to the Stobbs factor (Boothroyd, 2000) in off-axis

electron holograms provide compelling evidence for intrinsic elastic

energy ﬁ ltering (within experimental limitations) in this mode. This

issue has been rigorously treated (van Dyck et al., 2000) in a quantum

mechanical framework using a global Hamiltonian describing the elec-

tron, source and object to prove that although inelastic interference is

possible it is too small to be observed. Hence, electron holography

provides de facto perfect energy ﬁ ltering, which therefore suggests that

electron-optical ﬁ lters may be redundant in this geometry.

3.5 Detectors

Image recording should not impair the overall resolution of HRTEM

data. However, unless the detector used is carefully optimized with

respect to microscope operating conditions this component can have

a negative effect on overall performance. Historically, photographic

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 37

plates using one of several possible specialist emulsions were univer-

sally used for image recording (Kuo and Glaeser, 1975; Zeitler, 1992)

but these have now been largely superseded by digital detectors of

which slow-scan charge-coupled device (CCD) cameras are most com-

monly used (Spence and Zuo, 1988).

Particular beneﬁ ts associated with these detectors (Zuo, 2000; Fan

and Ellisman, 2000; de Ruijter, 1995; Spence and Zuo, 1988) are that the

images are instantly available, quantitatively in digital form, and the

camera response is practically linear over a large dynamic range. Their

sensitivity is also far higher than that of most photographic emulsions,

making single electron detection possible.

Though limited early experiments using CCD chips as direct TEM

electron detectors have been performed (Roberts et al., 1982), these

were not viable due to the sensitivity of the gate insulator to radiation

damage. For low energies (<10 keV), this damage can be avoided using

back-thinned CCDs (Ravel and Reinheimer, 1991), but higher energy

electrons penetrate through to the sensitive gate oxide on the front side.

Moreover, a primary electron of energy E generates E/3.64 eV electron-

hole pairs in silicon (Fiebiger and Müuller, 1972) leading to saturation

of the CCD well capacity after the detection of only a few electrons at

typical energies used in HRTEM.

For this reason indirect detection is employed in all cameras cur-

rently used for HRTEM by which the electrons impinge on a YAG

single crystal

or phosphor powder scintillator

and the generated

light is relayed to the CCD chip via a lens- or ﬁ ber optical coupling

(Figure 1–15). Within this complex coupling, scattering of both the

primary incident electrons and the emitted photons in the scintillator

material occurs. These processes blur the image, attenuating its high

spatial frequencies leading to relatively poor Modulation Transfer Func-

tions (MTFs) (Meyer and Kirkland, 1998, 2000; Meyer et al., 2000b).

Electron-sensitive imaging plates have also found application as an

alternative digital recording media (Zuo, 2000; Zuo et al., 1996). These

consist of a thin embedded layer (ca. 40 µm thick) of a photostimulable

phosphor. Luminescence is activated postexposure by a scanning laser

in a separate processing system that includes a photomultiplier to

convert the output light into a digitized electronic signal. The exposed

plate can be subsequently erased by exposure to a suitable light source.

These systems provide excellent recording linearity over a wide dynamic

range with a sensitivity about three times that of photographic emul-

sions, but which is voltage dependent. However, their performance also

depends strongly on electron dose and drops rapidly at low dose levels.

Hence, the most promising application of this technology is in record-

ing quantitative diffraction information (Zuo, 2000).

Very recent experiments by several groups (Faruqi et al., 1994,

2005a,b; Faruqi and Cattermole, 2005; Fan et al., 1998; Milazzo et al.,

YAG: yttrium-aluminum-garnet Y

, a transparent crystal that is made

scintillating by doping with impurity atoms, usually europium (YAG:Eu) or

cerium (YAG:Ce).

A wide range of materials is available for powder scintillators. Popular

choices in CCD cameras include P22 (Y

S:Eu) and P43 (Gd

S:Tb).

2005) suggest that detectors for HRTEM are on the verge of a revolution

in which direct injection of beam electrons into solid-state devices may

be achieved leading to substantial increases in both resolution and

sensitivity. At the time of writing a number of pixellated CMOS-based

devices have been fabricated (albeit with initially limited array sizes)

including active pixel sensors (Milazzo et al., 2005) in which the readout

electronics and ampliﬁ cation are integrated at each pixel. These have

demonstrated direct electron imaging with performance characteris-

tics exceeding indirect CCD devices in cryoelectron microscopy and

diffraction from radiation-sensitive biological material (Faruqi et al.,

1994, 2005a,b; Faruqi and Cattermole, 2005).

3.5.1 Detector Characterization

All digital detectors can be characterized in terms of three key param-

eters that, in combination, describe their overall performance. These

involve measurement of the gain, resolution, and detector quantum

efﬁ ciency (DQE). The linearity and uniformity of the response may

also be important, but in currently available CCD devices are generally

extremely well controlled.

Detector resolution is expressed through the MTF, which describes the

spatial-frequency-dependent signal transfer, which is affected by both

electron and photon scattering within the detection chain. This parame-

ter can be evaluated by integration (over the scintillator area) of the ratio

of the output to the input signal in reciprocal space (u, v), normalized to

unity at zero spatial frequency (Meyer and Kirkland, 1998).

More formally

MTF u v G g u v d() (),,=

∫

µ (39)

where the Fourier transform of the light intensity (number of photons

per unit area) collected at a particular position on the detector in its

Figure 1–15. Schematic diagram of a CCD detector used for HRTEM. The incoming high-energy

electrons are scattered and along their trajectory within the scintillator where they give rise to photon

emission. Electrons that penetrate through the scintillator may be back-scattered into the scintillator

from the ﬁ ber plate or a mechanical support layer. A lens- or ﬁ ber-optical coupling system conveys

the generated light to the CCD chip where the photons generate electron-hole pairs. The electrons of

these pairs are collected in pixelated potential wells. (Taken from Meyer et al., 2002.)

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 39

focal plane is given by gˆ

(u, v), dµ deﬁ nes the probability of a particular

electron hitting the scintillator at this point, and G is the total gain,

i.e., the average number of detectable CCD well electrons per primary

electron given by

Gg d=

(

)

∫

µ00, (40)

The inverse Fourier transform of this MTF is the point spread func-

tion (PSF)

PSF , ,xy G g xyd

(

)

(

)

∫

µ (41)

The DQE measures the statistical performance of radiation detectors

(Hermann and Krahl, 1982) generally deﬁ ned as the quotient of the

squared signal-to-noise ratio (SNR) at the output and input of the detec-

tor [in general as a function of spatial frequency (de Ruijter, 1995)].

This can also be expressed in terms of experimentally accessible

quantities as

DQE u v

pG uv

Vuv

(

)

(

)

(

)

[]

(

)

MTF

(42)

where pˆ(0, 0) is the total number of electrons recorded in each expo-

sure, V (u, v) is the variance, (in units of “number of CCD well electrons

squared”) and G is the gain deﬁ ned previously. It should be noted that

unless the dose is very low, in which case dose-independent noise

sources such as readout noise become important, the observed vari-

ance V (u, v) is proportional to the electron dose and therefore the DQE

is dose independent.

The above expression can be conveniently rewritten through a simple

rescaling of units into the digital numbers (DN) that are read out

directly from the camera to give the expression

DQE u v

IG uv

Vuv

MTF ,

DN DN

(

)

(

)

[]

(

)

(43)

where G

is the gain in digital numbers per incident electron, I

pˆ(0, 0)G

is the total intensity in digital numbers, and V

(u, v) is the

variance in (DN)

Experimentally these performance characteristics can be readily

measured from a series of controlled exposures of a deterministic input

signal (such as a knife edge) (Meyer et al., 2000b) for a range of experi-

mental conditions (Figure 1–16).

3.6 Aberration Correctors

One of the long standing goals in electron optics has been the correc-

tion of the positive spherical aberration that is present in all round

lenses (Scherzer, 1949). The essential optical elements used in all cor-

rectors are nonround (multipole) lenses as originally proposed by

Scherzer (1947).

Subsequently designs for various correctors have

For a more detailed treatment of spherical aberration correctors see Chapter

10 by Hawkes in this volume.

been proposed (for reviews see Rose, 1990; Hawkes and Kasper, 1989).

However, it is only relatively recently that these components have been

developed to a level at which improvements to the highest performance

uncorrected instruments can be made. For the most part this has been

due to the extreme mechanical precision and electrical stability required

in the corrector elements and the need for sophisticated computer

control of the corrector alignment. However, in the 1990s signiﬁ cant

improvements in resolution for both HRTEM (Haider et al., 1998a,b)

and Scanning Transmission Electron Microscopy (STEM) (Krivanek et

al., 1999) were demonstrated, although for the purpose of this section

we will not consider the latter further. We also note that very recently

proposals for correction of both spherical and chromatic aberration

have also been detailed (Rose, 2005) based on a combination of elec-

tromagnetic and electrostatic multipole elements.

Figure 1–16. (a) MTFs measured for 1024 × 1024 and 2048 × 2048 CCD cameras. Solid line, 1024 × 1024;

short dashed line, 2048 × 2048; long dashed line, 2048 × 2048 with 2 × 2 pixel binning. (b) DQEs corre-

sponding to (a). In all cases plots extend from 0 spatial frequency to the spatial frequency at the corner

of Fourier space. (Adapted from Meyer et al., 2002.)

A.I. Kirkland et al.

Chapter 1 Atomic Resolution Transmission Electron Microscopy 41

Correctors for HRTEM are based on a pair of strong electromagnetic

hexapole elements together with four additional lenses (Haider et al.,

1995). Correction is achieved due to the fact that the primary, non-

rotationally symmetric second-order aberrations of the ﬁ rst hexapole (a

strong three-fold astigmatism) are exactly compensated by the second

hexapole element. Because of their nonlinear diffraction power, the two

hexapoles additionally induce a residual secondary, third-order spherical

aberration that is rotationally symmetric (Beck, 1979) and proportional to

the square of the hexapole strength. This aberration has a negative sign,

thus canceling the positive spherical aberration of the objective lens. For

HRTEM applications it is essential that the corrector is aplanatic to provide

a sufﬁ ciently large ﬁ eld of view, which is achieved by matching the coma-

free plane of the objective lens to that of the corrector using a round

transfer lens doublet. To reduce the azimuthal (anisotropic) component

of the off-axial coma the current direction of the ﬁ rst transfer lens doublet

is opposite to that in the objective lens. In addition to these primary

optical elements commercial correctors (Haider et al., 1998a; Hutchison

et al., 2005) contain a number of additional multipole elements for align-

ment and correction of any residual parasitic aberrations.

The main difﬁ culties associated with the practical operation of these

complex electron optical components are the requirements for sophis-

ticated computer control of the various elements and a systematic

alignment procedure providing rapid, accurate measurement of the

coefﬁ cients of the wave aberration function. In practice, this is achieved

using measurements of the two-fold astigmatism, A

, and defocus, C

from tableaus of diffractograms recorded at known beam tilts (see

earlier). These measured values are subsequently used to calculate the

aberrations present in the corrector and the appropriate currents to the

optical elements are then applied under direct computer control.

3.6.1 Aberration-Corrected Imaging Conditions for HRTEM

The optimization of imaging conditions for corrected instruments and

the interpretation of the images obtained have also required renewed

and the defocus is also set to zero, then the phase contrast transfer

set by partial coherence, while the amplitude contrast transfer function

equals 1.

Under these conditions HRTEM can be carried out under

pure amplitude contrast conditions in a mode that is not available in a

standard uncorrected TEM (Lentzen, 2004).

The above treatment ignores the effects of higher order aberrations such as

, which also contribute to high resolution phase contrast.

analysis. It is possible to identify three conditions for HRTEM imaging

Alternatively, phase contrast HRTEM images can be obtained using

when spherical aberration is a variable parameter (Lentzen, 2004).

several conditions in which the spherical aberration and defocus

If the spherical aberration of the objective lens is exactly corrected

are balanced against either C or the chromatic aberration coefficient,

function equals 0 for all spatial frequencies up to the information limit

C (Chang et al., 2003; Chang et al., 2006).

For the ﬁ rst of these conditions, in the presence of C

, restricting the

phase shifts due to the wave aberration function to lie between 0 and

π/2 gives optimum values for C

and C

= 1.56(C

)

1/3

= −2.88(C

λ)

1/3

(44)

with a corresponding point resolution, limited by C

, given by

d = (C

)

1/6

/1.47 (45)

Under circumstances in which C

= 0 the choice of C

reduces to the

conventional Scherzer defocus (C

= −(C

λ)

1/2

) (for ﬁ nite positive C

)

with a corresponding point resolution (as deﬁ ned previously) of

()

Matching the ﬁ rst zero of the phase contrast transfer function to the

information limit of the instrument, determined by C

, and again con-

straining the phase shifts due to the aberration function to lie between

0 and π/2, yields alternative optimum values of

=−

()

∆

(46)

where ∆ is the focal spread deﬁ ned in a previous section.

By analogy with the C

limited condition when C

is zero the above

=−

∆

∆()

(47)

For either of these conditions when C

is zero or has a negligible

effect at the experimentally available resolution, the signs of both C

and C

can be inverted, and this has been shown to differentially affect

the transfer of the linear and nonlinear components of the image

intensity in images of complex oxides, providing higher contrast at the

weakly scattering anion sites (Lentzen et al., 2002).

3.7 Monochromators

As already discussed, one of the limiting factors in determining the

information limit for HRTEM is the energy spread of the source.

One possible route to reducing the energy spread is in the use of

high brightness sources with intrinsically low energy spreads such as

nanometer-sized cold ﬁ eld emitters (Morin and Fink, 1994; Purcell

et al., 1995) or more ambitiously, ballistic emission sources (Borgonjen

et al., 1997), photocathodes (Baum et al., 1995a,b), and metal ion sources

(Purcell and Bihn, 2001). However, none of these has currently been

In addition to the beneﬁ ts of a reduced energy spread for HRTEM there are

also substantial beneﬁ ts in EELS and related techniques where a lower energy

spread improves the spectral energy resolution.

A.I. Kirkland et al.

reduce to (O’Keefe, 2000; Chang et al., 2006)

Chapter 1 Atomic Resolution Transmission Electron Microscopy 43

manufactured with sufﬁ cient brightness and stability for routine use

in HRTEM.

For this reason monochromation of the electron beam produced

from conventional Schottky or cold ﬁ eld emission sources using dedi-

cated electron optical components provides an attractive alternative

that has now been realized experimentally using several different

designs operating at 120 kV and 200 kV (Mook and Kruit, 1999; Rose,

1999; Batson, 1986).

The number of possible monochromator geometries that have been

proposed and tested can broadly be divided into those with straight

optical axes (typically Wien ﬁ lters) (Mook and Kruit, 1999; Rose, 1987;

Tsuno, 1999) and those with curved axes (typiﬁ ed by Ω geometries)

(Rose, 1999; Kahl and Rose, 1998). The former require both electromag-

netic and electrostatic elements to satisf y the Wien condition and to

achieve the required energy dispersion, whereas the latter can be con-

structed from purely electromagnetic or electrostatic elements. Further

classiﬁ cation of monochromators can be made by considering whether

the monochromator and/or the energy selecting slit are operated at

high potential or at ground and whether the monochromator is a single

dispersion or double dispersion design. Finally, monochromators can

be operated in either short ﬁ eld (deﬂ ector) mode, in which the mono-

chromator ﬁ eld length is short compared to the distance to the beam

focus, or in long ﬁ eld mode, in which the monochromator also acts as

a focusing element. Within these various classes there are advantages

and disadvantages associated with each conﬁ guration that have been

extensively reviewed by Mook and Kruit (1999).

For all of the above design possibilities the monochromator must

provide sufﬁ cient current density for HRTEM at a given energy resolu-

tion determined by the energy selecting slit width. Monochromation

inevitably reduces brightness as it ﬁ lters electrons with unwanted ener-

gies and thereby reduces the current. However, the optical design of

the monochromator itself also affects the brightness through the aber-

rations introduced by the ﬁ lter and effects due to Coulomb interactions

within the monochromator (Rose, 1999; Tsuno, 1999; Mook and Kruit,

1999).

4 Exit-Wave Reconstruction

Although HRTEM now routinely achieves atomic resolution, the rela-

tionship of the image contrast to the underlying object structure for a

single image is complex due to the nature of the scattering and imaging

processes, as already outlined. A number of alternative approaches to

inversion of the imaging process to recover the specimen exit-plane

wavefunction, including holography, ﬁ rst proposed by Gabor (1948)

and more recently realized at atomic resolution (Lichte, 1991; Orchowski

et al., 1995; Lehmann et al., 1999; Tonamura, 1987, 1999). The particular

approach described in this section combines the complementary infor-

mation available from images taken under different conditions in

indirect exit-wave reconstruction, which is directly related to the more

general problem of coherent detection (Spence, 2002).

As already discussed, the wave aberration function of the objective

lens deﬁ nes a passband of well transmitted spatial frequencies up to

the instrument point resolution (at the Scherzer defocus) and beyond

this limit the transfer function oscillates. At higher values of defocus

this passband extends to higher frequencies, which suggests, in prin-

ciple, that the addition of several images recorded at different defoci

should enable the deconvolution of the effects of the objective lens

transfer function and provide a wider passband extending to the axial

information limit. This forms the basis of focal series restoration (recon-

struction). A simpler alternative using only a single image has also

been proposed (O’Keefe et al., 2001) but suffers from difﬁ culties associ-

ated with the positions of the zero crossings in the transfer function

that prevent deconvolution and from ampliﬁ cation of noise.

4.1 Theory

The recorded image plane intensity can be written as

I(x) = |ψ

(x)|

= 1 + ψ

(x) + ψ*

(x) + |ψ

(x)|

(48)

For conditions in which the quadratic term is small then the Fourier

transform of the image contrast (the fractional intensity deviation) is

given as

c(k) = ψ

(k) + ψ*

(−k) (49)

the objective lens) and image waves are related by the previously

described phase shifting wave aberration function and thus

(k) = ψ

(k)w(k) (50)

where

w(k) = exp {−iW(k)} (51)

Hence in terms of the object wave (in the back focal plane of the

rewritten as

d d

image.

The essence of all reconstructions is now to ﬁ nd an estimate ψ′

(k)

of ψ

(k) given a set of observed image contrast Fourier transforms c(k)

and measurements of the individual w(k).

Given data from several differently aberrated images (obtainable

from focal or tilt azimuth dataset geometries as described later), an

Throughout this section we use k and x as Fourier space and real-space

variables rather than (u, v) and (x, y) as in Section 2.

A.I. Kirkland et al.

c(k) = ψ (k)w(k) + ψ*(−k)w(−k) + n(k) (52)

The Fourier transforms of the object wave (in the back focal plane of

objective lens) the Fourier transform of the image contrast can be

in which the term n(k) represents the observational noise in the