High-Resolution Transmission Electron Microscopy on an Absolute Contrast Scale
A. Thust
Institute of Solid State Research, Research Centre Ju
¨
lich, D-52425 Ju
¨
lich, Germany
(Received 21 January 2009; published 4 June 2009)
A fully quantitative approach to high-resolution transmission electron microscopy requires a satisfac-
tory match between image simulations and experiments. While almost perfect agreement between
simulations and experiments is routinely achieved on a relative contrast level, a huge mutual discrepancy
in the absolute image contrast by a factor of 3 has been frequently reported. It is shown that a major reason
for this well-known contrast discrepancy, which is often called Stobbs-factor problem, lies in the neglect
of the detector modulation-transfer function in image simulations.
DOI: 10.1103/PhysRevLett.102.220801 PACS numbers: 07.78.+s, 42.30.Lr, 68.37.Og
The technique of high-resolution transmission electron
microscopy (HRTEM) offers nowadays the possibility to
study the atomic configuration of solid state objects with a
resolution of around 0.08 nm. Highly accurate information
about atomic positions and chemical occupancies at lattice
defects and interfaces can be obtained by the quantitative
use of this technique [1].
Because of the quantum-mechanical nature of the elec-
tron diffraction inside the object and due to the subsequent
electron-optical imaging process, any quantitative extrac-
tion of object information from HRTEM images requires a
justification by accompanying image simulations. Since
the introduction of digital image comparison to HRTEM
in the early 1990s, a satisfactory agreement between simu-
lation and experiment could be established only on a
relative basis by disregarding the absolute magnitude of
the image contrast [ 2]. The reason for this still common
practice was first published by Hy
¨
tch and Stobbs in 1994,
who found a remarkably strong discrepancy between the
magnitude of the simulated and the experimental image
contrast [3]. In their experiment, the dimensionless quan-
tity of the image contrast, which is defined as the standard
deviation of the image intensity distribution after normal-
ization of the image mean intensity to unity, was by a factor
of 3 too low when compared to image simulations. Since
the image motifs were nevertheless widely consistent
between simulation and experiment, the problem was re-
garded mainly as a scaling problem, and became promi-
nent as Stobbs-factor problem, factor of 3 problem, or
contrast-mismatch problem. A satisfactory explanation
for this long standing problem, which has been meanwhile
frequently reproduced and investigated, has not been found
so far [4–9].
The electron microscopic imaging process can be sub-
divided into three stages, of which each can be potentially
responsible for the contrast-mismatch problem: (i) the dif-
fraction of the incoming electron wave by the object,
(ii) the subsequent electron-optical transfer of the dif-
fracted wave by the electromagnetic lenses, and (iii) the
final image acquisition by a camera. Concerning the dif-
fraction part, plasmon and phonon scattering were explic-
itly ruled out as a reason for the problem [10–12], and
moreover, a remarkable discrepancy between measured
and simulated diffraction data was not found [6,8].
Concerning the optical part, the measurement of coherent
aberrations and of partially coherent contrast dampening
functions is possible with high accuracy and did not reveal
inconsistencies [13–15]. However, little attention has been
paid to the final image recording step, which involves the
frequency-transfer properties of the camera.
In the work of Hy
¨
tch and Stobbs the modulation-transfer
function (MTF) of the photographic film plates used at that
time was not considered [3]. It is also remarkable that the
most popular image simulation software packages
EMS and
MACTEMPAS do not allow us to incorporate actually mea-
sured MTFs [16,17]. In all, the actual MTFs of the mean-
while used CCD (charge coupled device) cameras were
often not considered in the literature related to the Stobbs-
factor problem [7,8]. Concerning the very rare implemen-
tation of actually measured MTFs into image simulations,
criticism is allowed: Besides the use of an MTF belonging
to a different camera than actually used [5], earlier MTF
measurements were often based on the noise method
[9,18]. However, the noise method is known to yield un-
realistic results compared to the knife-edge method, which
is the state-of-the-art technique for MTF measurement
[19–22].
The drastic influence of the detector MTF on the ob-
tained image contrast is demonstrated here by a simple
experiment, where the same area of a SrTiO
3
crystal is
imaged with different magnification settings of a FEI Titan
80–300 microscope equipped with a 2k 2k Gatan
UltraScan 1000 CCD camera of 14 m pixel size. From
Table I, and from visual inspection of Fig. 1, the strong
dependence of the obtained image contrast on the detector
sampling rate is obvious. Although all images are recorded
with a sufficiently high sampling rate, a difference in the
obtained contrast by a factor of 2 arises between the lowest
PRL 102, 220801 (2009)
PHYSICAL REVIEW LETTERS
week ending
5 JUNE 2009
0031-9007=09=102(22)=220801(4) 220801-1 Ó 2009 The American Physical Society
and highest magnified images. Moreover, there is no in-
dication that this effect is already saturated at the highest
available magnification of 1 400 000.
A quantification of the described phenomenon requires a
measurement of the actual camera MTF, which is the
Fourier transform of the point-spread function PSFðx; yÞ.
The real-space convolution of the image intensity with the
point-spread function PSFðx; yÞ is described in Fourier
space as a product of the ideal intensity Iðu; vÞ and the
modulation-transfer function MTFðu; vÞ, yielding the de-
tected image intensity I
D
ðu; vÞ, with
I
D
ðu; vÞ¼Iðu; vÞMTFðu; vÞ: (1)
The modulation-transfer function MTFðu; vÞ itself can be
separated into a rotationally symmetric part MTF
S
ðu; vÞ
describing the image convolution by the scintillator and its
support, and a part describing the convolution over the
quadratic area of a single pixel, yielding
MTF ðu; vÞ¼MTF
S
ðu; vÞsincð0:5uÞsincð0:5vÞ; (2)
where the Fourier-space coordinates (u, v) are defined in
units of the Nyquist frequency.
The physical core principle of the MTF measurement
used in this Letter is identical to that of the knife-edge
method as presented in Refs. [19–21], but is extended from
the classical case of a one-dimensional step function to the
more general case of a sharply bounded two-dimensional
object. Instead of a knife edge positioned directly on the
detector, the beam stop located roughly 1 m above the
camera is used as a shadow-forming object. A calculation
of Fresnel diffraction for 300 kV electrons falling onto an
opaque edge shows that the resulting Fresnel fringing zone
at 1 m distance is much smaller than the pixel size of
14 m, thus justifying the assumption of a sharp shadow
on the detector. The assumption of full opacity is in turn
justified by Monte Carlo simulations, according to which
the fraction of registered electrons, which interacted with
the edge material, is negligibly small [20].
After correction of the experimental shadow image
I
E
ðx; yÞ for nonuniform illumination, a synthetic shadow
image I
S
ðx; yÞ is constructed in a three-step procedure:
First, the experimental image is n-fold up-sampled by
Fourier interpolation to obtain subpixel positional accu-
racy. Subsequently, this n-fold oversampled image is con-
verted into a sharp two-level image. Finally, this
intermediate image is n-fold down-binned to obtain a
synthetic shadow image I
S
ðx; yÞ of original size, which is
thereby already affected by the pixel-size sinc-convolution
of Eq. (2). The experimental and synthetic shadow images
differ thus only by the scintillator part MTF
S
of Eq. (2). By
changing to polar coordinates (k, ’) in Fourier-space one
can define a set of auxiliary functions, given by
f
E
ðk; ’Þ¼jI
E
ðk; ’Þj
2
N
2
ðkÞ; (3)
f
S
ðk; ’Þ¼jI
S
ðk; ’Þj
2
; (4)
f
ES
ðk; ’Þ¼RefI
E
ðk; ’Þ=I
S
ðk; ’Þg; (5)
where N
2
ðkÞ is an estimate for the noise power at Fourier-
space frequency k, and Re denotes the real part. The
estimate N
2
ðkÞ is obtained by evaluating the experimental
image I
E
ðk; ’Þ at positions (k, ’), where the synthetic
image I
S
ðk; ’Þ is known to have no signal contributions.
An estimate for MTF
S
ðkÞ can then be obtained in two
alternative ways, namely, by
MTF
S
ðkÞðhf
E
i=hf
S
iÞ
1=2
; (6)
MTF
S
ðkÞhf
ES
i; (7)
where hfðk; ’Þi denotes a weighted average over the azi-
muthal coordinate ’. Within the averaging process, a
weighting factor wðk; ’Þ is applied in order to (i) select
Fourier-space regions of strong signal known from the
synthetic image, (ii) to deselect regions affected by aliasing
artifacts following the principle of Ref. [23], and (iii) to
deselect regions affected by the periodic continuation arti-
facts inherent to the Fourier transform. The estimates for
FIG. 1. Edge of a [110]-oriented SrTiO
3
crystal recorded with
a FEI Titan 80–300 microscope, which is equipped with a Gatan
UltraScan 1000 CCD camera. The underlying image was re-
corded at a nominal electron-optical magnification of 670 000,
the inset image at a magnification of 1 400 000. For comparison,
the images were resampled to identical magnification and are
overlaid within the same absolute intensity range between 0.5
(black) and 1.9 (white).
TABLE I. Contrast obtained with a FEI Titan 80–300 micro-
scope equipped with a Gatan UltraScan 1000 camera from an
identical area of SrTiO
3
using four different microscope mag-
nifications. Nominal magnification, sampling rate, inverse sam-
pling rate, and contrast are listed.
Nominal
magnification
Sampling
rate
[1=
A]
I. Sampling
rate
[nm] Contrast
670 000 6.709 0.014 91 0.082
850 000 8.532 0.011 72 0.111
1 100 000 11.033 0.009 06 0.136
1 400 000 14.243 0.007 02 0.165
PRL 102, 220801 (2009)
PHYSICAL REVIEW LETTERS
week ending
5 JUNE 2009
220801-2
MTF
S
in Eqs. (6) and (7) differ in the sequence of averag-
ing and division, and in the way noise is treated: The
estimate of Eq. (6) uses an explicit noise subtraction via
Eq. (3), whereas the estimate of Eq. (7) profits from a noise
reduction via the real-part projection of Eq. (5). A notice-
able difference between the two evaluation modes was not
found. A correct noise treatment is crucial, since otherwise
noise would be falsely interpreted as signal, yielding a too
optimistic MTF.
The result for MTF
S
, as averaged from ten single mea-
surements, is shown in Fig. 2(c). The most prominent
feature of MTF
S
is its strong falloff already at low frequen-
cies, which is reflected in the very low central value of the
related point-spread function PSF
S
shown in Fig. 2(d). The
function PSF
S
exhibits a wide-ranging tail, which exceeds
the area shown in Fig. 2(d), and which is hardly visible
there due to its very low level. The ratio s between the
central values of the measured PSF and an ideal PSF, which
corresponds to the well-known Strehl ratio in light optics
[24], adopts a remarkably low value of s ¼ 0:11. The rapid
falloff of the MTF, and the resulting broad PSF, are known
to be caused by electrons being backscattered from the
scintillator support [19,25].
A focal series of [110]-oriented SrTiO
3
was acquired
with a spherical-aberration corrected FEI Titan 80–300
microscope operated at 300 kV and is compared with
image simulations including the measured MTF. This
comparison is intentionally held compact in a sense that
a well-known variety of effects, which are known to reduce
the image contrast by less than roughly 10% [4,11,12], like
absorption, amorphous cover layers, or phonon scattering,
are neglected in the simulations. It is essential to state that a
neglect of such effects in simulations has always a ten-
dency to further promote a possibly existing contrast-
mismatch problem. The series of 30 images was acquired
with a spherical aberration of C
S
¼25 m at an exces-
sively high sampling rate of 9:71 pixels=
A. Image simula-
tions were carried out with the
EMS package [16], using an
average Debye-Waller factor of 0:006 nm
2
for all atoms
within the SrTiO
3
unit cell. A beam semiconvergence of
0.25 mrad and a defocus spread of 3.3 nm were input to the
simulations [15]. Special software was devised to inject
residual lens aberrations to the
EMS wave function [13,14].
Residual aberrations, which can occur unintentionally due
to a limited lifetime of the perfectly corrected optical state
even with a spherical-aberration corrected microscope,
were measured directly after recording the series. The
contrast dampening effects of the MTF, and of object drift
during exposure time, were applied in a final step to the
EMS images. Since the effect of object drift is marginal
compared to that of the MTF, it will not be separately
discussed in the following. The main simulation steps are
visualized by four exemplary images in Fig. 3, where an
almost perfect match between simulation and experiment
is obtained on an absolute contrast scale for an object
thickness of 2.8 nm. A quantitative contrast comparison
for the complete series is shown in Fig. 4. It can be seen
there that the residual aberrations, which change the image
motifs in a drastic way, have a comparatively small impact
on the image contrast. The major contrast reducing effect is
clearly due to the MTF.
FIG. 2. (a) Shadow image of beam stop acquired with 300 kV
electrons by a Gatan UltraScan 1000 camera. (b) Line scan
through the vertical section labeled by ‘‘S’’ in the shadow image.
Abscissa in pixel units. (c) 1-dim section of modulation-transfer
function MTF
S
derived from shadow image. Abscissa in units of
the Nyquist frequency. (d) Central 9 9 ‘‘pixels’’ of the related
point-spread function PSF
S
. The bar at the edge shows the height
of an ideal -like PSF.
FIG. 3. Image patches comprising 3 3 SrTiO
3
unit cells
projected along the [110] zone axis. The patches 1, 5, 12, and
30, belong to different focal values and are shown exemplarily
for a series of 30 images. All patches are displayed on the same
gray scale extending between intensity values 0.35 (black) and
2.2 (white). (a) Simulation for an object thickness of 2.8 nm,
(b) simulation including additionally residual lens aberrations,
(c) simulation including residual aberrations and additionally the
measured MTF, (d) experimental images.
PRL 102, 220801 (2009)
PHYSICAL REVIEW LETTERS
week ending
5 JUNE 2009
220801-3
It is important to confirm that the object thickness of
2.8 nm, which was used for the simulations, is indeed
unique. Otherwise one could argue that simulated high-
contrast images could exist, which result from larger object
thickness values, and which could again cause a Stobbs-
factor problem. Fortunately, the image motifs resulting
from higher simulated object thickness values do not re-
produce the experimentally recorded motifs. Since the
thickness change of the image motifs cannot be shown
here due to space restrictions, a more compact way is
chosen to confirm the correct thickness value. Curves
displaying the simulated image contrast in dependence of
defocus have been calculated for different object thick-
nesses, including the effect of residual aberrations, and of
the MTF. Such exemplary contrast-defocus curves
(CD curves) are displayed in Fig. 5(a). These CD curves
change their shape in a very characteristic way with object
thickness and serve thus as a unique fingerprint of the
respective thickness. The shape change of the CD curves
is caused by the different scattering behavior of the in-
volved Sr-O, Ti, and O atom columns with object thick-
ness. A comparison between the experimentally recorded
CD curve and the simulated CD curves based on the mutual
cross covariance, which neglects explicitly the absolute
contrast scale, and which is purely focused on the curve
shape, reveals a maximum curve resemblance at an object
thickness of 2.8 nm [Fig. 5(b)]. Absolute and relative
contrast measures are thus fully consistent.
In summary, the existence of a general contrast-
mismatch problem of remarkable size in HRTEM cannot
be confirmed. On the contrary, an almost perfect consis-
tency between simulation and experiment is found when
including the correct detector MTF in image simulations.
The author is grateful to Dr. J. Barthel for leaving him
the focal series and for recording the shadow images.
[1] K. W. Urban, Science 321, 506 (2008).
[2] A. Thust and K. Urban, Ultramicroscopy 45, 23 (1992).
[3] M. J. Hy
¨
tch and W. M. Stobbs, Ultramicroscopy 53, 191
(1994).
[4] C. B. Boothroyd, J. Microsc. 190, 99 (1998).
[5] C. B. Boothroyd, Ultramicroscopy 83, 159 (2000).
[6] R. Rosenfeld, Ph.D. thesis, Berichte des
Forschungszentrums Ju
¨
lich, 2001.
[7] A. Howie, Ultramicroscopy 98, 73 (2004).
[8] W. Qu, C. Boothroyd, and A. Huan, Appl. Surf. Sci. 252,
3984 (2006).
[9] K. Du, K. v. Hochmeister, and F. Phillipp, Ultra-
microscopy 107, 281 (2007).
[10] C. B. Boothroyd, Scanning Microsc. 11, 31 (1997).
[11] C. B. Boothroyd and M. Yeadon, Ultramicroscopy 96, 361
(2003).
[12] C. B. Boothroyd and R. E. Dunin-Borkowski, Ultra-
microscopy 98, 115 (2004).
[13] A. Thust et al., Ultramicroscopy 64, 249 (1996).
[14] S. Uhlemann and M. Haider, Ultramicroscopy 72, 109
(1998).
[15] J. Barthel and A. Thust, Phys. Rev. Lett. 101, 200801
(2008).
[16] P. A. Stadelmann, Ultramicroscopy 21, 131 (1987).
[17] M. A. O’Keefe and R. Kilaas, Scanning Microsc. Suppl. 2,
225 (1988).
[18] J. M. Zuo, Microsc. Res. Tech. 49, 245 (2000).
[19] R. R. Meyer and A. Kirkland, Ultramicroscopy 75,23
(1998).
[20] R. R. Meyer and A. I. Kirkland, Microsc. Res. Tech. 49,
269 (2000).
[21] R. R. Meyer et al., Ultramicroscopy 85, 9 (2000).
[22] R. Henderson et al.,inProceedings of the 14th European
Microscopy Congress, Aachen, Germany, 2008 (Springer-
Verlag, Berlin, Heidelberg, 2008), Vol. 1, p. 71.
[23] W. J. de Ruijter, Micron 26, 247 (1995).
[24] K. Strehl, Z. Instrumentenkd. 22, 213 (1902).
[25] G. McMullan et al.,inProceedings of the 14th European
Microscopy Congress, Aachen, Germany, 2008 (Springer-
Verlag, Berlin, Heidelberg, 2008), Vol. 1, p. 73.
FIG. 5. (a) Simulated image contrast of SrTiO
3
including
residual aberrations and MTF versus defocus (CD curves). The
curves belong to object thickness values of 0.6, 1.7, 3.3, and
6.1 nm (bottom to top). (b) Cross covariance between experi-
mental CD curve (squares in Fig. 4) and CD curves simulated for
the object thickness values indicated on the abscissa.
FIG. 4. Solid line: Image contrast simulated for an object
thickness of 2.8 nm and for the focal values of the experimental
SrTiO
3
series. Dotted line: Simulated contrast including addi-
tionally residual lens aberrations. Dashed line: Simulated con-
trast including residual aberrations and additionally the MTF.
Squares: Experimental image contrast. Image numbers within
focal series are indicated.
PRL 102, 220801 (2009)
PHYSICAL REVIEW LETTERS
week ending
5 JUNE 2009
220801-4
![FIG. 1. Edge of a [110]-oriented SrTiO3 crystal recorded with a FEI Titan 80–300 microscope, which is equipped with a Gatan UltraScan 1000 CCD camera. The underlying image was recorded at a nominal electron-optical magnification of 670 000, the inset image at a magnification of 1 400 000. For comparison, the images were resampled to identical magnification and are overlaid within the same absolute intensity range between 0.5 (black) and 1.9 (white).](/figures/fig-1-edge-of-a-110-oriented-srtio3-crystal-recorded-with-a-12x5q8cu.webp)
![FIG. 3. Image patches comprising 3 3 SrTiO3 unit cells projected along the [110] zone axis. The patches 1, 5, 12, and 30, belong to different focal values and are shown exemplarily for a series of 30 images. All patches are displayed on the same gray scale extending between intensity values 0.35 (black) and 2.2 (white). (a) Simulation for an object thickness of 2.8 nm, (b) simulation including additionally residual lens aberrations, (c) simulation including residual aberrations and additionally the measured MTF, (d) experimental images.](/figures/fig-3-image-patches-comprising-3-3-srtio3-unit-cells-2hap6n45.webp)
