Hawkes P.W., Spence J.C.H. (Eds.) Science of Microscopy. V.1 and 2
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Chapter 5 High-Speed Electron Microscopy 415
current of order 0.01–10 mA. Further, to obtain good results the illumi-
nation spot must be signifi cantly larger than the region of interest, and
some electrons will have to be thrown away at various apertures. So
even these enormously high currents may not be enough for some
applications. To squeeze 10 mA of current into a 1-µm radius spot with
a 1 milliradian convergence semiangle would require a normalized
brightness of order 4 % 10
9
A cm
−2
steradian
−1
(assuming a typical TEM
voltage of 200 keV). This is too high a brightness for a thermionic
cathode (pulsed or otherwise) and too much current for a typical fi eld
emission gun.
23
This brightness is just within reach for a pulsed pho-
tocathode-based system
24,25
and the required current can be attained
by using a somewhat larger emission area than is typical for TEM.
Beyond this point, signal levels must be traded off against spot size
and/or convergence angle—or invent a brighter electron source,
possibly using pulse compression (discussed in a later section).
Phenomena such as charge trapping
26
and space-charge repulsion
from previously emitted electrons (i.e., the effect that leads to Child’s
law in the steady-state case)
27
may also limit the number of electrons
in a pulse.
Bostanjoglo et al. report a maximum current of 20 mA in a 20-ns
pulse at 80 kV with a brightness of ∼2 % 10
6
A cm
−2
steradian
−1
, achieving
a factor of 50 enhancement in brightness by operating in a laser-driven
pulsed mode as compared to a DC thermionic mode.
27–29
The energy
spread was reported as 8.7 eV. Brightness is roughly proportional to
accelerating voltage in this regime; most of the reported increase of
brightness with voltage may be attributed to the conservation of nor-
malized brightness, which is ∼6 % 10
6
A cm
−2
steradian
−1
for this case.
Pulsed high-voltage RF photoelectron guns can signifi cantly exceed
this brightness and current,
24,25,30
but at the cost of very high energy
spreads ∼1% or more, which may limit their suitability for DTEM appli-
cations. An interesting near-future challenge would be to determine
whether and how to borrow design concepts from the RF guns to
improve the DTEM sources.
30
At any rate, in the DTEM every electron is precious. In static TEM it
is common practice to block most of the electrons at the condenser
apertures in order to attain good spatial co-herence and small lateral
emittance [emittance is, to within defi nition-dependent factors of order
unity,
31
the product of the spot size and convergence angle and is a
conserved purely geometric quantity in a linear electron-optical system
at constant voltage with no apertures (Reiser,
19
pp. 56ff)]. This allows
the production of very fi ne spots limited in size primarily by diffrac-
tion and spherical aberration, or of wider spots with very small inci-
dent angular spreads. This enables the familiar “nanoanalytical” and
“parallel-beam” modes of operation. In the DTEM one will typically
be electron-starved and will not have the luxury of using a small con-
denser aperture; the instrument performance will necessarily suffer.
A typical TEM that can attain atomic resolution lattice images and
subnanometer spot sizes in a static mode will simply not be able to
do these things in nanosecond pulses with any foreseeable electron
source.
416 W.E. King et al.
2.3 Materials for Photoelectron Guns
The ideal photo cathode should (1) be able to produce high current
densities of about 1 kA/cm
2
, (2) be resistant to residual gases in a high
vacuum (oxygen, water, hydrocarbons, carbon monoxide, and carbon
dioxide), (3) be tolerant to ion bombardment and thermal spikes due
to arcing. and (4) exhibit a low work function to relax requirements on
the driving photons.
32
To shift the negative space-charge-induced saturation of the electron
emission to higher current densities, the accelerating electric
fi eld should be as high as possible. This increases the chance for an
electric breakdown between the cathode and the accelerating anode.
An ion bombardment of the cathode results until the charge of the
capacitance between the cathode (usually at high negative potential)
and ground is exhausted. The ions are produced by ionization of evap-
orated anode material and desorbed/disintegrated layers (oxides,
hydrocarbons, water, air molecules) covering all surfaces in an ordi-
nary high vacuum system. If the capacitance between cathode and
ground is not small enough a high current arc can develop between
cathode and anode. The energy of the impinging ion pulse is large
enough to sputter and heat the cathode substantially. A weak but con-
tinuous ion bombardment derives from electron beam-induced ioniza-
tion of the residual gas. The sputtering and chemical reaction caused
by these ions are negligible, when the cathode is operated in ultrahigh
vacuum.
Materials suitable for photocathodes can be divided in semiconduct-
ing and metallic photoelectron emitters in view of their band structure
and associated current limitation.
2.3.1 Semiconducting Emitters
In semiconducting emitters valence electrons are photoexcited into the
conduction band. If the energy of the photons exceeds the sum of band
gap and electron affi nity, some excited electrons tunnel through the
surface potential barrier and escape to the vacuum. The rest are trapped
in surface states, increasing the surface barrier. Fortunately, they are
neutralized by thermionic and tunneling holes from the valence band,
which restore the equilibrium surface barrier. However, if the photo-
cathode is driven with nanosecond and shorter light pulses of high
intensity, the number of excited electrons that are trapped can become
too large to be compensated by the restoring hole currents. The surface
barrier is then increased, and the extractable charge saturates at a value
lower than the space charge limit. This surface charge limit effect is a
characteristic of nonmetallic emitters.
26
Cesium-based antimonides, such as Cs
2
Sb and K
2
CsSb, are among
the materials with low work function (about 2 eV).
33
But they are
extremely sensitive to contamination, thermal spikes, and ion bom-
bardment. For a higher work function, about 3.5 eV, there are materials
that are much more resistant to contamination by residual gases such
as Cs
2
Te and GaAs-based emitters.
33–36
Their thermal stability is poor,
however. Providing heavily p-doped GaAs or other III–V semiconduc-
tors with Cs-suboxide surface layers produces materials with a nega-
Chapter 5 High-Speed Electron Microscopy 417
tive electron affi nity.
34–37
These materials have large quantum effi ciencies
for spin-polarized electrons, 0.3% for wavelengths of 810–830 nm. The
heavy surface doping also enhances the restoring hole currents that
compensate trapped photoexcited electrons, avoiding the surface
charge limit phenomenon.
35–37
Unfortunately, the doped surface layers
make the negative electron affi nity cathodes again vulnerable to con-
tamination, ion bombardment, and thermal stressing.
In contrast to all the above semiconducting materials, p-doped
diamond and fullerene fi lms
38
are extremely wear and heat resistant.
The quantum effi ciencies, however, are distinctly lower (about 3 × 10
−4
for 217-nm photons).
2.3.2 Metallic Emitters
Metallic emitters can be subdivided into compounds with metallic
conductivity and metals.
Rare earth monosulfi des
39
and hexaborides,
40–42
borides and carbides
of refractory metals, are examples of compounds with a low work func-
tion and high conductivity.
40
The rare earth sulfi des, e.g., CeS, NdS, and
PrS, are distinguished by very low work functions of 1.1–1.4 eV, high
melting temperatures of 2400–2700 K, and high resistance to ion bom-
bardment and contamination. Thin fi lms of these materials are readily
produced by vacuum evaporation. However, their vapor pressure at
elevated temperatures is substantial, which prohibits thermionic
operation.
The rare earth hexaborides LaB
6
and CeB
6
are widely used in thermal
cathodes. They are thermally more stable than the monosulfi des, but
they are disintegrated by ion bombardment and have a higher work
function of about 2.6 eV.
Borides and carbides of refractory metals have even higher work
functions of 3.0–3.5 eV. But since their melting temperatures are
extremely high, e.g., 3800 K in the case of ZrC, these emitters readily
tolerate arcing. In addition, they are not sensitive to contamination and
can therefore be operated in ordinary high vacuum.
Metals exhibit a wide range of work functions as well as thermal and
mechanical strengths. Candidates for high-performance photocath-
odes are reviewed by Travier.
43
Alkali and alkali earth metals are at
the low end of the work functions. They are chemically extremely
sensitive and mechanically and thermally weak. The mechanical
strength and tolerance to arcing of cathodes based on these high
quantum effi ciency metals can be substantially increased by implant-
ing their atoms into tungsten or other high melting metals.
44
Nonethe-
less, these cathodes remain sensitive to contamination and require
ultrahigh vacuum.
Rare earth metals have medium work functions of about 3.5 eV. They
are mechanically strong and high melting metals. But they vigorously
react with oxygen producing highly isolating oxide layers. Positive ions
attracted by the negative cathode settle on the oxide layer where they
produce a high electric fi eld, which in turn triggers a breakdown
(Malter effect). Accordingly, rare earth cathodes suffer from perpetual
electric breakdowns when they are operated in high vacuum.
418 W.E. King et al.
Most metals with a high thermal and elec trical stability and tolerat-
ing high vacuum have inconveniently large work functions. Zirconium
is a compromise. This metal combines a medium work function (3.8–
4.05 eV depending on impurities) with a high melting temperature
(2130 K), and it forms semiconducting suboxides in high vacuum,
which neither deteriorate the quantum effi ciency nor favor an electric
breakdown.
The infl uence of impurities on photoemission is very complicated.
Surface contamination by water, carbon monoxide, and carbon dioxide,
and hydrocarbons usually poisons photoemitters by producing hydrox-
ide or carbonaceous layers. Oxygen, for instance, can either increase or
reduce the work function, depending on the material, the emitting
crystal lattice plane, and the surface coverage. Oxidation of materials
with a low or medium work function usually decreases photoemission.
Zirconium is an exception, if oxidation is limited so that only a few
layers of the suboxide ZrO are formed.
45
Water, carbon monoxide,
carbon dioxide, and hydrocarbon molecules can poison photoemitters
by producing covering oxides or carbonaceous layers. Alkali and alkali
earth metals are chemically most reactive and therefore very sensitive
to the vacuum atmosphere. Cs
2
Tb is a favorable exception
33
as it is rela-
tively unaffected by carbon compounds abundant in conventional high
vacuum, such as CO, CO
2
, and CH
4
. If contamination is an issue, a
satisfactory lifetime (several hours) can be achieved only at pressures
below 10
−9
mbar.
The choice of a photocathode is a compromise between high quantum
effi ciency and long stable operation. Having selected the material and
the driving photon source, being almost exclusively a laser, the genera-
tion of electron pulses with high current densities can be improved in
several ways:
1. Using a photoelectron emitter with a rough surface enhances light
absorption, makes use of fi eld-assisted photoemission (Schottky effect),
and photo-assisted fi eld emission, increases the accelerating fi eld and
thus shifts the space charge-induced saturation to higher current den-
sities, and provides the correct orientation of many surface elements to
the electric fi eld vector of the light for a maximum electron yield.
Depending on the nature and degree of roughness the quantum effi -
ciencies can be increased by a factor of three
46
and more.
47
2. Producing a small plasma cloud with the driving laser pulse, the
saturation of the electron current due to negative space charge is avoided,
and the electron pulse charge is augmented by secondary electrons.
27
However, the pulse is stretched as trapped electrons are released when
the plasma decays.
3. Ultrashort driving laser pulses provide additional benefi ts. First,
if the emitted ultrashort electron bunch is shorter than the accelerating
gap, its space charge fi eld is approximately jτ/2ε
0
, with j the electron
current density, τ the pulse duration, and ε
0
the vacuum permittivity.
The current density saturates as this obstructing fi eld approaches the
accelerating fi eld, at values that may exceed the Child–Langmuir limit
of stationary space charge by orders of magnitude.
48
Second, if the laser
Chapter 5 High-Speed Electron Microscopy 419
pulse duration is shorter than the electron–phonon relaxation time,
about 10 ps for metals, the electrons can be excited to a high tempera-
ture and be thermally emitted before losing substantial energy to the
lattice.
49
If photoemitters on substrates are to be subjected to elevated temper-
atures, e.g., for continuous thermionic emission to simplify alignment
and for cleaning and conditioning as in the case of dispenser cathodes,
50
the supporting material should neither disintegrate the photoemitter
nor produce a low melting eutectic. Substrate atoms rapidly penetrate
along grain boundaries and eutectic channels to the surface and deteri-
orate the quantum effi ciency if their work function is high.
51
In view of
this, for example, Zr on W or Re is a bad choice, as eutectics exist for
these systems. However, Zr on Ta works well since these metals have a
miscibility gap confi ning Ta that reaches the surface predominantly to
Zr grain boundaries (Figure 5–8). A Zr-covered Ta hairpin emitter can
be used as a conventional thermionic electron source and as a pulsed
high- current photocathode, both in a normal high vacuum (<10
−4
mbar).
Electron pulses with a current density of 700 A/cm
2
were produced by
a laser radiation with a wavelength of 266 nm.
51
In summary, the quantum effi ciencies (QE) of different less sensitive
metals are listed with references in Table 5–1 with the exciting wave-
length. Photocathodes used at TU Berlin had a brightness of ∼6 %
10
7
A/cm
2
sr at 100 kV, a quantum effi ciency of 1.2 % 10
−5
, and emitted
photoelectron pulses with current densities of up to 700 A/cm
2
from
an area of 20 µm diameter.
10
2.4 Lasers
Two lasers are required in the typical DTEM pump-probe experi-
ment—one each for the cathode and the sample. At TU Berlin, these
lasers must produce ∼6–10 ns pulses with spot diameters of 20–40 µm.
This is achieved with Q-switched Nd:YAG and Ti:Sapphire lasers. The
Figure 5–8. A Zr on Ta cathode after 6 h of operation as a thermionic cathode,
imaged by scanning electron microscopy with energy dispersive X-ray spec-
troscopy. The bright islands between the Zr grains (dark) are Ta, which has
penetrated the Zr layer. The quantum effi ciency (4% 10
−5
for light with a wave-
length of 266 nm) was not affected.
420 W.E. King et al.
Table 5 –1. Quantum effi ciencies of less sensitive metals.
l (nm)
193 211 213 248 263 266 308 355
QE (10
−5
)
Ag 28.4
a
2.6
a
2.0
b
Al 35.9
a
84
c
2.6
a
3.2
c
0.034
c
Au 16.2
a
40
c
1.1
a
1.3,
c
4.7
b
Cu 20
c
28.6
a
15
c
0.84
a
0.22,
c
14
a,b
0.016
c
Mg 62
b
Mo 14.4
a
0.49
a
Nb 45
d
0.3
d
Pd 1.2
b
Sm 6.8
a
0.47
a
72
b
0.016
c
Ta 12 .5
a
0.94
a
1.0
b
Tb 8.9
a
0.61
a
23
b
Y 13.9
a
1.4
a
0.27,
c
50
b
0.11
c
W(K+) 1.2
c
W(BaO) 35
e
14
e
2.3
e
W(Ba-Ca-Sr-O) 230
d
Zn 24.7
a
1.4
a
1.4
b
Zr 1.0
b
Li-Al 60
f
10
f
a
Anderson T., Tomov I.V., Rentzepis P.M. Laser-driven metal photocathodes for picosecond electron and X-ray
pulse generation. J Appl Phys 1992;71(10):5161–5167.
b
Srinivasan-Rao T., Fischer J., Tsang T. Photoemission studies on metals using picosecond ultraviolet-laser pulses.
J Appl Phys 1991;69(5):3291–3296.
c
Chevallay E., Durand J., Hutchins S., Suberlucq G., Wurgel M. Photocathodes tested in the Dc gun of the Cern
Photoemission Laboratory. Nucl Instrum Meth A 1994;340(1):146–156.
d
Leblond B., Rajanoera G. Photoemission in the picosecond regime from a coated trioxide cathode. Nucl Instrum
Meth A 1994;340(1):195–198.
e
Leblond B. Short pulse photoemission from a dispenser cathode under the 2nd, 3rd and 4th harmonics of a
picosecond Nd-Yag laser. Nucl Instrum Meth A 1992;317(1–2):365–372.
f
Septier A., Sabary F., Dudek J.C., Bergeret H., Leblond B. A binary A1/Li alloy as a new material for the realization
of high-intensity pulsed photocathodes. Nucl Instrum Meth A 1991;304(1–3):392–395.
required pulse energy depends signifi cantly on the application and can
be as high as ∼200 µJ for driving a pulsed thermionic cathode.
29
Fre-
quency doubling and quadrupling is employed to more effectively
drive the required material transitions. This is especially important for
the photocathode guns, which require ultraviolet light and are driven
with a frequency-quadrupled Nd:YAG laser at 266 nm wavelength. The
laser spots have Gaussian profi les, which affects the spatial uniformity
of both the sample drive and the cathode emission.
The need for multiple-frame movies (with each frame illuminated
by its own cathode pulse) complicates the laser situation somewhat,
especially since different applications may require very different pulse
delays. The TU Berlin pulsed photocathode can produce single 7-ns
Chapter 5 High-Speed Electron Microscopy 421
electron pulses, but uniformity suffers as the number of pulses is
increased. This is because of the diffi culty of producing multiple closely
spaced identical pulses in a Q-switched nanosecond laser system. The
second and third pulses have durations of 10 and 11 ns, respectively,
with spacings ranging from 20 ns to 10 µs.
52
Producing more than three
nearly uniform pulses becomes progressively more diffi cult. The devel-
opment of many-frame movies will require that this challenge be fully
addressed.
3 Limitations
3.1 Time Resolution
The duration of the electron pulse at the sample determines the time
resolution of the DTEM in a normal imaging or diffraction mode. This
will be our primary concern in this section. Higher time resolution
has been demonstrated in a streak mode,
53,54
at the cost of having only
a one-dimensional image available at any given point in time. The
great majority of the electrons penetrating the sample are lost in these
cases, but the higher time resolution may be worthwhile for many
applications.
The electron pulse length is essentially equal to the cathode laser
pulse length in current instruments. The cathode laser pulses are
several nanoseconds long, whereas the mechanisms that would further
degrade the time resolution typically occur on a picosecond scale. The
time delay between a photon striking the cathode material and a pho-
toelectron exiting is much faster than a nanosecond (streak cameras
and ultrafast electron diffraction systems would not work otherwise),
provided the emission is purely via the photoelectric effect. Schafer and
Bostanjoglo
27
reported a pulse duration of 20 ns due to heating and
cooling of the fi lament and plasma relaxation processes, but this was
in a laser-driven thermionic source rather than a true photocathode; in
later work this limitation is bypassed.
52
Temporal expansion due to the
energy spread in the pulse in the initial acceleration region is of order
∆
∆
t
mE d
eV
=
(/)
/
2
12
(1)
with m and e the electron mass and charge, ∆E the energy spread, d
the cathode–anode distance, and V the accelerating voltage.
55
This
amounts to a few picoseconds or less for typical DTEM parameters.
After initial acceleration, the velocity spread is less than one part per
thousand and again is a negligible source of temporal spread. The
pulse of electrons is much longer than it is wide—a 1-ns, 200-keV pulse
will be ∼15 cm in length but a small fraction of a millimeter in diameter
(depending on lens settings and position within the column). In this
high-aspect-ratio limit, longitudinal space charge effects can be practi-
cally neglected except near the very leading and trailing edges of the
pulse. Image charges in the metal walls of the column may interact
with nonuniformities in the linear charge density to produce broaden-
ing and instabilities,
56–58
but it is fairly easy to estimate from the gov-
422 W.E. King et al.
erning equations that this phenomenon will have a negligible effect
over the few nanoseconds that the electrons are in the DTEM column.
The effect is far more important for large-scale particle accelerators.
Space charge thus might spread out the tails of the temporal response
curve but will have very little effect on its FWHM; the pulse is for the
most part steady state. Timing jitter in laser pump-probe experiments
is similarly on the picosecond scale or faster. Thus until DTEM pulses
get into the picosecond range (which is likely to be very diffi cult for
the above reasons), the time resolution will be controlled entirely by
the photocathode pulse length.
Picosecond DTEM will also be extremely challenging purely on the
basis of signal levels. As we have seen, the DTEM is already electron
starved in the nanosecond regime, so that the optimal time resolution
is dictated by (1) the required fl uence at the sample, (2) the maximum
tolerable convergence angle, and (3) the brightness of the electron
source. Pushing the time resolution to the picosecond range will only
be possible either with an extremely bright electron source or with very
unchallenging imaging requirements. It will likely be some time before
this is accomplished.
In summary, the time resolution for a single shot is dictated by the
electron fl uence requirements and directly controlled by the photocath-
ode laser pulse duration. What then determines the interframe time
resolution for a multiple-shot experiment? Single-electron-sensitive
high-resolution high-dynamic-range cameras with nanosecond readout
times are somewhat beyond current technology. This means multiple
shots must be fi t into a single CCD image, which is read out once after
all exposures are complete. This is achieved with the previously
described “movie mode,” implemented with a stepped electrostatic
defl ector system. The minimum interframe time is ∼20 ns, dictated by
the ring-down time of the high-voltage switching circuitry.
With the principle proven, the problem of producing better movies
(with more frames and/or reduced interframe times) is a nontrivial
matter of high-voltage/high-speed electrical engineering. The driving
system must produce a series of preprogrammed voltage plateaus at
very precise times with as little ring-down as possible. It needs to be
fl exible, precise, and as free of parasitic inductance and capacitance as
possible. Some advantage may be gained by using a more recent TEM
column design with more than one intermediate lens. Such columns
offer more degrees of freedom to optimize the “leverage” (displace-
ment at the camera plane per unit defl ection voltage). It may also be
possible to increase the frame count by using a two-dimensional defl ec-
tor, although this greatly complicates the electrical design problem.
Ongoing developments in multiframe camera technology
59–61
may
improve the frame counts, particularly if combined with an advanced
electrostatic frame shifter. Each buffered CCD frame could thereby
contain multiple images, and the CCD’s interframe time could be much
longer than the DTEM’s interframe time. This would unfortunately
force the high-voltage switching system to repeat its performance mul-
tiple times with very little delay, not to mention the challenges in pro-
ducing the required cathode laser pulses. If the engineering problems
Chapter 5 High-Speed Electron Microscopy 423
turn out to be surmountable, we may (although certainly not very
soon) see DTEM movies with hundreds of frames and interframe times
perhaps as short as 1 ns.
It is instructive to compare the time resolution in DTEM with that
of conventional in situ TEM. Conventional systems are typically limited
to video rate imaging, ∼30 ms per frame. DTEM thus opens up a regime
of more than six orders of magnitude in time resolution. This advan-
tage is currently offset by the stringent limitations on multiframe
DTEM movies. DTEM is therefore complementary to conventional in
situ techniques. Video-rate TEM is too slow to capture many nanoscale
solid state processes—often a signifi cant part of the dynamics occurs
between two frames of video.
62
DTEM offers an opportunity to help
fi ll in these gaps of time, provided issues of timing and frame count
limitations can be addressed.
3.2 Image Resolution
Our present understanding of the spatial resolution limits in DTEM is
limited, due to a lack of systematic studies on both empirical and theo-
retical fronts. This section must therefore be considered as a sort of
snapshot of our current state of understanding in a very young fi eld.
Still, we can make some statements on general principles and identify
future areas of research.
The stated resolution in the TU Berlin instruments in pulsed mode
is ∼200 nm
52
(in some of the images features as small as ∼100 nm seem
to be coming through
63
). This resolution limit is attributed to a combi-
nation of the point spread function of the camera and the SNR limiting
the pixel size for a given fl uence
52
(recall Figure 5–4). Both limits could
in principle be improved by increasing magnifi cation and reducing the
illuminated area, but these will incur various tradeoffs. For example,
time-dependent radiation sensitivity of the sample may limit the
current density in specifi c cases, so that the practical resolution limit
may be very strongly dependent on the robustness of the sample under
the electron beam. Also, the uniformity of the illumination area may
suffer due to the large beam emittance coupled with condenser lens
aberrations and nonuniform emission at the cathode.
In any case, if we assume that the sample can survive the required
fl uence, then the fundamental resolution limit of the machine must
lie in the electron optics. The ability of static TEM to attain Angstrom-
scale resolution is dependent on several factors including brightness,
energy spread, objective lens aberrations, the quality of the sample, and
the faithful magnifi cation of the image by the intermediate/projector
lens system. All of these may have to be compromised in the DTEM.
Further, electron–electron interactions including space charge, Boersch
effects, and trajectory displacement effects
64–66
will be much more
important in the DTEM, with its extremely high currents compared to
static TEM. The DTEM has at least one advantage, in that low-frequency
vibrations and high-voltage drift that would harm the resolution in
the conventional TEM will have no effect on a nanosecond-exposure
image.
424 W.E. King et al.
Let us consider the electron-optical factors in some detail. First,
consider the need for a bright electron source. As we have discussed,
high brightness is needed to attain high fl uence with nearly parallel
illumination. Any angular spread in the condensed beam will tend to
wash out the high-spatial-frequency information.
67
The necessarily
large emittance of the DTEM electron source will tend to compromise
this performance. Also, the energy spread ∆E of the electrons will be
quite large in the DTEM, due most likely to an enhanced Boersch effect
at high currents.
52
∆E is about 8 eV for the TU Berlin instruments.
28
The
energy spread will interact with the objective lens chromatic aberration
C
C
to limit resolution to
dC
E
E
CC
=θ
∆
(2)
where θ is the angular deviation of the electron path from the optic
axis.
67
If we make the simplifying assumption of normal incidence,
then θ is related to the feature spacing d via the usual formula
d = λ/θ (3)
(assuming coherent single scattering and small angles), and this implies
a chromatic resolution cutoff of order
dC
E
E
CC
=λ
∆
(4)
In addition, spherical aberration limits resolution to a value of order
67
d
S
= 0.66(C
S
λ
3
)
1/4
(5)
This rounds out the list of electron-optical effects that usually limit
resolution in a conventional TEM.
These effects all interact with one another, and it is best to use TEM
simulation software (such as Java EMS,
68
used here) to understand
their combined effects. The square root dependence in Eq. (4), for
example, is very much complicated by beam convergence effects. Con-
sider Figure 5–9, which presents amplitude and phase contrast transfer
functions for a set of imaging parameters that are similar to those that
occur in a DTEM. Parameters comparable to the ones used in this cal-
culation were measured
28
near, at, or below the sample position and
thus include all effects that take place in the gun, accelerator, and con-
denser systems. It would appear that atomic resolution (via phase
contrast, at spatial frequencies ∼3 cycles/nm or higher) will be extremely
diffi cult, but that it may well be possible to achieve usable contrast up
to a resolution limit of ∼1–2 cycles/nm, particularly through amplitude
contrast mechanisms. The rough square root dependence on energy
spread helps to minimize the impact of chromatic aberration; Eq. (4)
predicts a cutoff of 1.4 cycles/nm. Spherical aberration [Eq. (5)] predicts
a roughly comparable 2.1 cycles/nm. The lack of spatial coherence
(represented in the simulation as an incident angular spread) interacts
with the other parameters in a complicated way, broadening the CTF
pass band but signifi cantly reducing its amplitude. Thus it may be