- Email: [email protected]
Detector geometry, thermal diffuse scattering and strain effects in ADF STEM imaging
ultramicroscopy ELSEVIER
Ultramicroscopy58 (1995) 6-17
Detector geometry, thermal diffuse scattering and strain effects in ADF STEM imaging Scan Hillyard, John Silcox School of Applied and Engineering Physics, Cornell University, Ithaca, N Y 14850, USA
Received 17 April 1994; at the Editorial Office 20 September 1994
Abstract
Intensities of atomic-resolution Annular Dark Field Scanning Transmission Electron Microscopy (ADF STEM) images of zone-axis-oriented specimens change with defocus at rates that depend on lattice spacing. Thickness and strain effects on the intensities have been demonstrated. In this paper, image simulations (with some experimental basis) are presented that consider the dimensions of the ADF detector. Changing the inner radius of the detector seems to have relatively small effect on the image except to lower the detected intensity. Probe size was explored and a case identified where multiple scattering was important in the image. Thermal diffuse scattering (TDS) is important in high-angle scattering at room temperatures but it does not seem to alter the image appearance markedly. Finally, the image arising from the strain field around a single boron atom has been simulated and the results suggest increased scattering in agreement with observations. This mechanism may be adequate for single impurity atom detection at low temperatures and with special detector angles.
I. Introduction
Annular dark field imaging [1] is an important tool in electron microscopy both as a high-resolution imaging technique [2-4] and as a way to locate a specimen probe on the sample while simultaneously recording analytical data such as electron energy loss spectra [5-7]. It is intrinsically quantitative since image data can be recorded directly from linear detectors into digital memory [8], then analyzed and displayed subsequent to the recording process. However, while extensive modeling of the image formation process has been carried out [9-13], relatively little experimental testing of the models to check assertions concerning the importance of factors such as the detector inner angle and thermal diffuse
scattering has been published. In this paper, we use a particular simulation approach [14,15] based on multislice theory [16,17] and tested earlier by experiment [18-23] to explore issues raised by the modeling studies. In earlier papers, a "frozen phonon" approximation based on the Einstein approximation [19,20] introduced into the multislice simulations provided a good fit with diffraction experiments including the thermal diffuse background. This has been used to simulate images recorded experimentally as a function of defocus. The good fit observed [21] between the experiment and the calculation provides additional confidence in the simulations. The incoherent imaging model gives a good description of the focal variation of zoneaxis atomic-resolution experimental and simu-
0304-3991/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-3991(94)00173-1
S. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6 - 1 7
lated images. In subsequent work [22], this approach was refined to include incoherent factors in simulating the probe shape giving again excellent agreement. The virtual gun source size is the most significant of these in limiting high-spatialfrequency components in the image. Considerable effects are predicted in the depth dependence of the images due to the growth and subsequent decay of channeling peaks along the zone axes. Experimental observations [22,23] confirm these effects and provide further evidence that "frozen phonon", multislice-based simulations provide an excellent basis for predicting and analyzing the images and diffraction patterns found in STEM. In this paper, we use simulation to explore issues that have been widely discussed. Diffraction-contrast effects in ADF STEM images were troublesome in early studies of Pt catalysis particles on alumina substrates prompting Howie to propose [24] use of a detector with an inner angle > 40 mrads. In later work [25] the value suggested for this angle was extended to larger values, e.g., 80 to 100 mrad to ensure that all the scattering was incoherent, i.e., thermal diffuse scattering. It was noted [26], however, that HOLZ line scattering was kinematic and would also give rise to an incoherent image. Nevertheless, arguments for the use of high detector inner angles in atomic-resolution ADF STEM appear [12] and are widely accepted. Using simulation, we look at this question in atomic-resolution imaging together with the importance of thermal diffuse scattering. Other factors explored include the probe size and the effects of strain which have recently been unambiguously identified as a source of contrast in ADF STEM [27].
2. Background The image-formation process can be divided into four stages. These are: formation of the incident probe by the microscope, propagation of the probe through the sample to form a wave function at the exit surface, evolution of that wave function to the detector plane (forming a diffraction pattern in the process) and integration
7
of the diffraction pattern over the detector plane to give the intensity at one pixel in the image. The simulation procedure [14,15] follows those stages. The incident probe is h ( r ) = I qZ(r) l 2, where the electron wave function, ~(r), is the Fourier transform of qt(k), k is the two-dimensional Fourier conjugate to r, the two-dimensional position vector, and ~ ( k ) is given by: qt(k) = A ( k ) e x p ( i x ( k ) ) ,
(1)
A(k)=A o fork
for k > k o,
x ( k ) = ~Ak2(~Csa2k 2 - AT), where k o is the objective aperture, C s is the spherical aberration of the objective lens and A f is the lens defocus. The quantities Cs and A f can be determined by Krivanek analysis [28] of STEM bright field phase contrast images of carbon films [29]. This determines the probe width, a, which depends on defocus, objective aperture and the spherical aberration. As pointed out elsewhere [22,30], the maximum spatial frequency that can be formed in the image is given by 2k o, the diameter of the objective aperture. In real space, this condition translates to a < d, where d is the smallest observable spacing. Propagation of the probe through the specimen is modeled using the multislice theory of Cowley and Moodie [16]. This first entails calculation of the phase shift of the electron beam as it passes through the atom and is determined from the Mott formula using modified [31] X-ray scattering factors [32]. The second step is Fresnel propagation of the phase-shifted electron probe to the next layer of atoms where the procedure is repeated until the probe reaches the exit surface. Atomic positions are determined from the lattice positions with slight random displacements added in accordance with the "frozen phonon" approximation [19]. The Fourier transform of the exit waveform provides the diffraction pattern at the detector plane and this is integrated over the detector dimensions. The calculation has to be repeated for each pixel in the image. Within the precision of the calculation (i.e., depending on the computer resources available)
8
S. Hillyard, J. Silcox/Ultramicroscopy 58 (1995) 6-17
this calculation is valid for relatively thick samples and includes dynamical effects, channeling, scattering to fractional Bragg angles, HOLZ scattering and arbitrary specimen structure. It includes multiple elastic and thermal diffuse scattering to all orders but does not include inelastic scattering due to electronic excitation. Since the instrument has an electron spectrometer, the experiments can be set up to match these conditions. Excellent matches between observed data and the simulations are found. The phonon scattering is based on random atomic displacements which are given by a Gaussian probability distribution. The root-mean-square (RMS) amplitude has been treated as a free parameter with good agreement [20] with X-ray determinations [33]. For many purposes it is useful to consider ADF STEM images as arising from an incoherent imaging mechanism. This assumes that the image is given by an equation of the form:
I(r) = h ( r ) ® s ( r ) ,
(2)
where I(r) is the image intensity and is formed by the convolution, ®, of h(r), the intensity distribution within the scanned probe, with s(r), a specimen function that describes the details of the interaction of the electron beam with the sample. The image-forming mechanism does not necessarily rely on incoherent electron scattering although this is often assumed. Provided that the scattering process taking the electron to the detector actually removes the electron from the probe, i.e., it no longer participates in the probeformation process, then the scattering can be either coherent or incoherent. In testing the model accuracy, the Fourier transform of Eq. (2) has been found to be very useful, i.e.,
I(k) =H(k)S(k),
(3)
where I(k), H(k) and S(k) are the Fourier transforms respectively of I(r), h(r) and s(r). For a crystalline specimen, the image is periodic and intensity arises only at reciprocal lattice points, which are few in number since the largest sample spacings are close to the instrumental resolution limits. For both simulated and experimental images the intensities in the peaks of the power
spectrum provide a simple means of image evaluation. Using this approach, the incoherent image model was found to describe the observed focal variation of zone-axis images relatively well even though the calculation as carried out calculates the amplitude of scattering before determining the intensity incident on the detector [21]. It should be noted that there is now evidence [22,23] using this approach that the form of the probe changes as a result of the channeling peaks formed down a zone axis and that at each layer the shape of the probe is different. The ADF signal is the integral of the scattering from each layer in the sample. Thus, although Eq. (2) describes the dependence of the image on defocus well, it should not be assumed that the actual probe inside the sample is given by Eq. (1) as a simple interpretation of Eq. (2) might suppose. In particular, interpretation of s(r) as a set of 3functions at the atomic positions weighted by an atomic scattering strength (proportional, for example, with our annular detector to Z 17) has the consequence that the atomic arrangement may generate systematic absences in S(k). Hence careful analysis of the intensities that shows the appearance of forbidden peaks in the power spectrum of an image is prima facie evidence of the breakdown of the simple model.
3. Simulation details
Typical calculations followed patterns reported in earlier calculations [19,21]. For convenience, we review these here. For indium phosphide [100] samples (i.e., with the electron beam incident along [100]) the slice dimensions are 512 x 512 pixels, 6 x 6 unit cells and 35.2 x 35.2 A dimensions. These choices yield a maximum included scattering angle of 179 mrad after bandwidth limiting. All detector inner angles at a spacing of the angular resolution (1.05 mrad) were used to produce a range of ADF STEM signals for each thickness. Examples are shown at selected inner radii of 30, 50, 90 and 120 mrad in later sections of the paper. In the line scans, simulated image points are typically 0.15 A apart. All calculations for InP [100] were performed
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
with four 1.47 ,~ (one plane of atoms) thick slices. For frozen phonon calculations, several iterations, corresponding to different atomic displacements were averaged. The atoms were displaced from the lattice positions by a random amount with a RMS amplitude equal to that of the phonon RMS amplitude. The different configurations within an iteration were obtained by randomly shifting slices horizontally by a lattice vector. At least 4 different iterations were used to adequately represent the ensemble and usually 10 were needed to obtain a 1% RMS relative error as determined by sampling tests. The appropriate RMS vibration amplitudes for indium phosphide are unknown. An estimate was made by adjusting the known silicon vibration amplitude at room temperature of 0.076 .~ [33] for a higher mean atomic mass and lower vibration frequency [21]. If the vibration amplitudes are inversely proportional to the square root of the atomic mass, an amplitude of 0.05 ,~ results for the indium atoms and 0.096 ,~ for the phosphorus atoms. For silicon [100] samples, the comparable dimensions were 32.6 X 32.6 A and a 1.13 mrad angular resolution. Four atomic slices 1.36 ,~ apart and then repeated comprised the sample. For silicon [110] samples, the comparable dimensions were 32.6 x 30.7 A with 1.20 mrad angular resolution. Two atomic slices 1.92 A apart and then repeated formed the specimen. Three electron-optical conditions have been used in this work to simulate three different probe sizes at 100 keV. A relatively standard
o.2 5 :
~
.
~
.
~
~
~
~
3o
A.
4. Results
The best documented examples of zone-axis A D F STEM image simulations using the multislice approach are for silicon [14] and InP [15]. Using parameters and simulations that have successfully modelled the observed intensity distributions in CBED patterns, Fig. 1 shows simulated intensity line scans along a [011] direction for an electron beam incident along the [100] direction in silicon for a specimen of 192 ,~ thickness and a 2.2 ,~ probe. The basic intensity patterns are shown in Fig. l a while corresponding scaled intensities are seen in Fig. lb. No change in the overall visibility of the features are seen as a result of the changes in the dark field detector inner angle but a noticeable decrease in the overall intensity is noted as the inner angle is increased from 30 to 120 mrad. The images include two basic Fourier compo-
°3 i
0.25
0.2 ~ ;~ •~ 0 . 1 5 ~
:50
.
=
probe of 2.2 ,~ (given by 6 = 0.43A3/4C 1/4) is simulated by an objective-lens spherical aberration C s of 1.3 mm, an objective aperture of 10.5 mrad, i.e., 2k o = A/0.021 = 1.76 A. A higher resolution of 6 = o1.8 A and a smallest detectable spacing of 1.39 A is obtained with a C s of 0.5 mm, an objective aperture of 13.3 mrad and a defocus of 430 A. Finally an optimistic 1 ,~ probe is simulated by a C s of 0.05 mm, an objective aperture of 23 mrad and a defocus of 136 A. The corresponding smallest detectable spacing is 0.8
E
F
""
0.1
~0.15
= _
o.05 0
. . . . . . . . L i,
0 (a)
,I,
1
,i
2
~
,
3
I,,,,I,,,
4
. . . . I
5
Angstroms
6
,,,i
,
7
X90
_~ o.1
d120
0.05
3 8
0
(b)
I
~.,
0
,~, ,i ,i,,,
1
2
,1~,
,~,,,
,~,,,
3 4 5 Angstroms
,],,,
,I,,,
6
7
Fig. 1. Simulated A D F intensity line scans along the [110] direction for an electron probe of 2.2 ,~ size (for details see text) incident down a 200 ,~ thick [100] silicon sample. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
10
0 . 5 ~ ,
.......
, ....
,,,
'~o.o2~ ~
....
I ..........
'~ 0 . 4 ~
~o.~ ~ !
~
so
.......
0.015
0,2
0.01
....
90
"~ 0
100 200 300 400 500 Thickness in Angstroms
(a)
~
~ o.005
,.
_
~,~.-....
_
~
0 0
600
100 200 300 400 500 T h i c k n e s s in A n g s t r o m s
(b)
0.04
O0
!
-~ 0 . 0 2
L ~,
0
100 200 300 400 500 T h i c k n e s s in A n g s t r o m s
(c)
600
Fig. 2. Background (a) and fringe strength (b) for the four apertures shown in Fig. 1 plotted as a function of thickness. T h e ratio (c) shows the little difference within the noise levels (higher at small thicknesses) in the relative fringe strengths for the four detector geometries.
nents, a background (with k = 0) and a fringe of spacing 1.9 A. In Fig. 2 the intensity of the background and the 1.9 ,~ fringe strength are displayed as a function of thickness. The background fringe grows monotonically (but not lin-
0.6
....
,
-
0.5
!
so
;~ 0 . 4
=
0.6
so /~
~
early) for all inner detector angles. The 1.9 fringe shows oscillations with thickness (presumably due to dynamical diffraction) for the detector with an inner radius of 30 mrad but no such effects at higher inner detector angles. Since the
0.5 ,~ 0 . 4
0.3
'3
0.2
~ o.2
0.1
0.1
0
(a)
,
0
,
~ ;
5
,
i
, ~ ; ,
10 Angstroms
, ,
,
15
,
,
O. 3
0
,J
2
(b)
I
0
5
10 Angstroms
15
20
Fig. 3. Simulated A D F intensity line scans along the [110] direction for an electron probe of 1 ,~ size (for details see text) incident on a 205 A [100] InP sample. Four detector inner angles of 30, 50, 90 and 120 mrad are seen in (a). T h e same data scaled to superimpose on the indium atoms are seen in (b).
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
Si
(II0)
Projected Lattice ®
©
©
®
©
@
2.72A
@
@
1.3aA~ --@
--® 14 O
3.84A,
,k O
©
®
@
©
Reciprocal Lattice
O
o
Zero
Forbidden
Fig. 4. Projected lattice of Si [110] showing the dumbbells and the associated reciprocal lattice. Note that the 2.72 A (200) spacing (open circles) along the dumbbells is a forbidden spatial frequency.
30 mrad inner detector angle borders on the zero-layer reflections this seems reasonable. Within the noise level, the ratio of these two signals (which is what is plotted in later figures) is
~'.
J-
F
. . . .
i
i;
. . . .
I
~
r
~
'
I
'
the same for all inner angles but is not constant with thickness. Similar data to Fig. 1 for a 205 ,~ thick [100] InP sample is seen in Fig. 3 for images simulated with a 1 ,~ probe. In this case, since there are two elements present, Z-contrast ( Z is the atomic number) is seen and changes with the inner angle of the dark field detectoL Fig. 3a shows intensity scans along a [110] direction as the beam passes successively over indium and then phosphorus atom columns. The same data scaled to superimpose the intensities of the indium atoms is then seen in Fig. 3b. The relative visibility of the phosphorus atom shrinks as the detector inner angle grows with the phosphorus becoming almost undetectable at 90 to 120 mrad. This effect is due to the smaller angular scattering of the phosphorus atom relative to the indium atom. For example, if we take the Wentzel model of the atom [34], then a screening angle 6)0 can be defined as 6)0 = A/2~rR0 where a is the electron wavelength and R o is aH/g 1/3 where a H is the Bohr radius. The screening angles are 41 mrad for the indium and 27.5 mrad for the phosphorus. Over the range of angles appropriate for the detector inner angles considered here, a significant difference occurs. Note, however, that if the goal is to image the phosphorus atoms, then this result suggests that a smaller, rather than a larger, inner detector angle is recommended. To observe effects due to probe size, i.e., limiting interference on the lateral scale, or lateral incoherence, it is convenient to :tudy the scatter-
0.18
'
3o
i1
1
.
.
.
.
.
~ T
0.16
0.15
0.14
•~ 0.12
0.1
o
I:I
0.I
-=0.08
0.05
0.06 0
0.04 0
(a)
5
10 15 20 Angstroms
25
30 (b)
~' 0
~ 5
-
.......... 10 15 20 Angstroms
' 25
30
Fig. 5. Simulated ADF intensity line scans along the [100] direction for an electron probe of 2.2 A size (for details see text) incident down a 192 A thick [110] silicon sample. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
S. Hillyard, J. Silcox / UItramicroscopy 58 (1995) 6-17
12 0.25 ~
,
30
T
t
0.2
0.25
i
. . . .
1 ' ' ¸ 7
. . . .
i
,
,
0.2
~0.15
~0.15
0.1
~ 0.1
0.05
0.05
0
(a)
0
5
10 15 20 Angstroms
25
30
0 (bl
5
10 15 20 Angstroms
25
30
Fig. 6. Simulated A D F intensity line scans along the [100] direction for an electron probe of 1 ,~ size (for details see text) incident down a 192 ,~ thick [110] silicon sample for comparison with Fig. 5. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
ing from a probe incident along the [011] direction. This is also a common system for experimental studies. In this direction, the atoms form arrays that are staggered in depth and that project into an array of two columns 1.36 ,~ apart (see Fig. 4). The column arrays are 5.4 ,~ apart. Thus for a 2.2 ,~ probe, both columns fit well within the probe size whereas for a 1 A probe the two columns are separated. In the first case interference between the two scattered beams is expected but not in the second. This also holds true if the smallest spacings that can be recorded by these probes are considered, viz., 1.76 A for the larger probe and 0.8 ,~ for the smaller probe. Fig. 5 shows both the unscaled (a) and scaled (b) intensity line scans for the larger probe. This situation is similar to the earlier figures, i.e., the primary effect of the change in the detector inner
angle is to lower the detected intensity but it does not affect the overall atomic structure image. Thus we conclude that in this case the two columns are sufficiently close laterally that no significant effects due to interference occur. The thickness dependence of the fringe strengths has been extensively explored earlier [22,23]. In Fig. 6, intensity scans for the same situation with the 1 ,~ probe show that the atomic column spacing of 1.36 ,~ is now well resolved. The effect of increasing the inner radius of the detector aperture is again to reduce the intensity as is demonstrated in Fig. 6b. The scaled intensities superimpose except for the smallest aperture (30 mrad) where there is some slight deviation from the scaled curve. As noted earlier, this aperture borders on the zero-layer reflections, so this could be expected. The results reported so far suggest
0.09
0.1 L i
. . . .
i
. . . .
I
'¸
0.08
30
0.07
~o.o6 0.05
~ 0.04
~ 0.04
0.03 0.02
0 (a)
5
10 15 20 Angstroms
25
30
0 ' 0 1 0 . . . . . .5. . . . 1 0 . . . .1.5. . . . . .2. .0. . . 2 5 (b) Angstroms
Fig. 7. Simulated A D F intensity line scans along the [100] direction for an electron probe of 1.8 A size (for details see text) incident down a 192 ,~ thick [110] silicon sample for comparison with Figs. 5 and 6. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b),
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
that the primary effect of increasing the inner angle of the detector is to lower the detected signal (presumably at the cost of the signal-tonoise ratio) while altering the details in the image by very little. In an interesting intermediate situation, resolution of the 1.36 .A column spacing in (110) silicon at a 10% signal drop between the two columns has been demonstrated by Liu and Cowley [35] with a probe size of 1.8 A. This situation is a case in which interference or diffraction effects may become visible since the lateral spacing between the atoms is less than but approaching the probe size. The 1.36 A spacing is just outside the smallest possible spacing that can be imaged for this situation (estimated above to be 1.39 ,~) and given the likelihood of incoherent source size effects [21-23] direct imaging should be considered unlikely. In the simulated line scan along the [100] direction seen in Fig. 7a, the two atoms are seen resolved at the 10% level and the visibility again shows little dependence on the aperture inner diameter as shown by the superimposed data in Fig. To. The thickness dependence of the fringes making up the image show interesting effects as seen in Fig. 8a. Specifically, a (200) intensity fringe with a spacing of 2.7 A emerges in the imagoe after the probe has traveled approximately 60 A
into the sample. This is also seen in the power spectrum of the Liu and Cowley [35] experimental image. If the projected lattice undergoes a Fourier transformation, this fringe is forbidden as seen in Fig. 4. Thus the observation is an example of the breakdown of the simple t3-function model for the specimen function, s(r). The 2.7 .A fringe is seen in the simulations at relatively small excitation levels for both the 2.2 ,~ probe and in Fig. 8b for the 1 A probe. In the latter case, the (400) fringe giving the 1.36 A spacing directly is seen. Returning to consideration of the 1.8 A probe simulation, it is the combination of the forbidden (200) fringe with the (100)-type fringe that gives rise to the observed 1.36 ,~ splitting. The 5.4 A (100) fringe in this direction arises from a superposition of (lll)-type fringes in the two-dimensional image. It is important to appreciate that the (200) fringe does not arise immediately on entrance to the sample where a simple convolution of h(r) with a 6-function model would hold but it develops after the beam has passed about 50 A into the specimen. This suggests that it is a multiple scattering effect. In Fig. 9, the shape of the probe inside the sample is illustrated showing two well resolved channeling peaks. However, as the probe size, a, gets bigger than the inter column spacing, d, then interference becomes possible between electrons
-a 0.6
,~ 0.5
13
I
I
I
=
,~ 0.4
P 0.5
5.4A
.o 0.4
.o
o
o
I
0.3
0.2
~ 0.3 8
t 2.7A
e!
1
o.1 o
=
0
(a)
-1.8A I
50 100 150 Depth in Angstroms
1
-~.36A
20O
g o.2 e~
1 1.8A 1
~ 0.1
~.TsA
o
1 2.7A
0
(b)
0
50 100 150 Depth in Angstroms
O0
Fig. 8. (a) Thickness dependence of the lineoscans of Fig. 7 as resolved into four spatial frequencies: (1/5.4) ,~-1, (1/2.7) ,~-1, (1/1.8) A - 1 and (1/1.36) A-1. The (1/5.4) A-1 intensity is seen to start out and remain high, whereas the (1/2.7) ,~-1 intensity begins in the noise but then increases at approximately 60 ,~ into the sample. The (1/1.8) A -1 and (1/1.36) ,~-1 remain small and are outside the resolution of the probe. This effect is thought to be responsible for the dip associated with the resolution of the 1.36 ,~ spacing seen in Fig. 7. The corresponding thickness analysis for the data of Fig. 6 is seen in (b). The scale bar represents approximate noise levels.
S. Hillyard, Z Silcox / UItramicroscopy 58 (1995) 6-17
14
Fig. 9. Shape of the electron probe of initial size 1.8 ,~ at 100 A into the [110] silicon sample. Notice the double peak formed on each atom of the dumbbell.
scattered by the two atomic columns (of 0.7 ,~ channeling diameter) within the 2.2 A diameter probe. Given a silicon screening angle of approxi-
mately 27 mrad, the distance at which the scattering from one atom in the dumbbell can affect the second would be 1.36/0.027 ,~ or 50 ,~, i.e., the depth within the sample at which the forbidden (200) reflection appears. Channel-to-channel interference thus may be the underlying cause of the appearance of this forbidden spatial frequency and its thickness dependence. At larger probe sizes, the scattering contains yet more columns and more diffraction and the incoherent nature of the image disappears. Thermal diffuse scattering has been modeled successfully with the frozen phonon approximation and has been included where appropriate in the simulations. Comparisons of the zone-axis images with various levels of phonon disorder can be readily made. The conclusion is that although thermal diffuse scattering strongly modifies the high-angle scattering at room temperature for most materials, the primary effect is to alter the
0.2 Room
Temperature
0.15
0.15
/~
"3 0.1 0.05
" 0.05
h°
o k
0
5
(a)
0.07
~
-,
0.06 ~
10 15 20 Angstroms
25
30
'No Phonor~s~ ~'
'
i
..........
i
(b)
.......
0
0.07
5
'
,i ¸¸
,. . . .
~. . . . . . .
10 15 20 Angstroms
, ~
i ....
i ....
~ - ..... 2 25
i
30
,,,
0.06
0.05
•~ 0.04
: ~
'r~ 0.04
o03
o.03
0.02
- 0.02
0.01 0
/~ I /
\~
"
,Tempe 10 15 20 Angstroms
,~i ,
R
0.01 0
10 15 20 25 30 Angstroms Fig. 10. Comparison of the result of incorporating thermal diffuse scattering into an image simulation. (a) Unscaled linescan of Si(ll0) with a 2.2 ]k and a 30 mrad inner detector radius. The solid line includes phonons while the dashed line does not. In (b) the two curves have been scaled and very little difference is seen. (c, d) Unscaled and scaled linescans, respectively, for a 1 A probe and a 120 mrad inner detector radius. Note that in (a, b) the effect of TDS is to increase the intensity while in (c, d) the TDS reduces the intensity. 0
(c)
5
25
0
30
(d)
5
S. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6-17
intensity with which an image is scaled. The detailed structure of the image is not changed. For example, in Fig. 10 very little difference is seen between a scaled simulated intensity line scan with and without the thermal scattering component. At lower temperatures, less of the intensity is in the thermal diffuse scattering component and more is in the higher-order Laue zone rings (see Fig. 4 of Ref. [19]). This scattering is essentially kinematic [26] so that the contribution to the image does not change. Simulated image intensity levels at the lower temperatures show a marked drop as the inner detector radius moves out and passes across a Laue ring. More intensity is now embodied in these features and less is in the relatively continuous thermal diffuse scattering. Nevertheless, there is very little difference between the low- and high-temperature simulated images except for the intensity level. Most experimental observations are carried out at room temperature where the differences between the Einstein approximation in the simulation and the real displacements are not expected to be large. Provided that the appropriate rootmean-square displacement is used, only slight differences occur between the observed and simulated patterns in the diffraction pattern comparisons (see Refs. [19,20]). The intensities have to be integrated over the detector dimensions before becoming image points and the slight differ-
15
ences then tend to average out. For almost all current observations, the influence of TDS on the detailed structure of the scaled image is minor. The effects of strain fields due to defects have been clearly identified in scattering from point defects (boron and arsenic) in silicon [27]. Strain scattering from a single boron atom can be readily simulated with the insertion of an elastic displacement field given by: (4)
u = c r / r 3,
where u is the vector elastic displacement of the atomic positions and c is a constant found to be 1.4 ~3 for boron in silicon [36]. The scattering from this strain field occurs in an angular range from 40 to 60 mrad and, for one atom, is overshadowed at room temperature by the thermal diffuse scattering. Several atoms superimposed in the specimen depth will add up to significant scattering at room temperature. These points are illustrated in Fig. 11, where the angular dependence of the scattering relative to the thermal diffuse scattering in undoped silicon at low temperature (a RMS phonon amplitude of 0.031 roughly equal to the zero point motion was used) is shown in Fig. l l a . The magnitude of the maximum scattering difference (which occurs with approximately a 40 mrad inner detector radius) with temperature is shown in Fig. l l b . These latter simulations suggest that a single boron atom may
10 10
I
i
I
8 U
6 - ;-~ ~' 4
................... ,: ~..................
"d
...... [
i
?.
2 '
0 -2 {fl) 0
_:'
.............
"'~
2
-
0 20
40 60 80 Detector Inner Angle
100
(b) o
50
:Minimum
",
:--
De~eetlble
",~ Signal i . . . . ~i
100 lS0 200 T e m p e r a t u r e (K)
250
300
Fig. 11. Strain field scattering from a single b o r o n atom impurity situated in the middle of a 200 A thick [100] silicon sample c o m p a r e d with u n d o p e d sample at low t e m p e r a t u r e is shown in (a) as a function of detector inner angle. In (b) the signal is shown as a function of t e m p e r a t u r e for a 40 m r a d inner detector angle. Probe p a r a m e t e r s were C s = 3.3 mm, an objective a p e r t u r e of 8.2 mrad, and A f = 1100 ,&, creating a p r o b e of 2.8 ,& in size.
16
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
be detected at temperatures of 77 K or below provided the detector is carefully chosen. Note that in this case of distinguishing static from dynamic diffuse scattering, the choice of inner detector angle is important.
5. Discussion and conclusions
These simulations suggest that the emphasis on using a large inner detector aperture to minimize coherent scattering may be misplaced. Use of a very high inner angle detector may make imaging unnecessarily difficult by reducing the visibility of lower atomic number elements in the presence of heavy ones and by discarding valuable intensity that could be used to improve the signal-to-noise ratio of the images. The original suggestion (Howie [24]) that a detector of about 40 mrad inner angle would remove dynamical diffraction effects seems a better choice for atomic structure imaging than the later modification of that proposal to use a very high inner angle [25]. Within the original context this proposal was made, i.e., the study of small particles on a small grained substrate use of the larger inner detector angle may still be appropriate. The work reported here argues the need for a flexible annular detector that is adaptable to a range of different samples (e.g., Ref. [37]). Thermal diffuse scattering, while clearly the dominant part of the angular ADF signal in the detector region does not seem to alter the details of the image markedly, i.e., calculations that do not include thermal diffuse scattering (TDS) are relatively close to those calculations that do include TDS. The primary error in image structure arising from neglect of thermal diffuse scattering is a change in the apparent thickness (at the 10% level). Unless the thickness is independently known and an accurate comparison of experiment and simulation made, it would be impossible to distinguish the images. Some of these points have been made earlier [19,20,38] but are re-emphasized here. Localized static strain does affect the image considerably, as observed, and gives rise to annular dark field effects that can counter those of Z-contrast [26]. The simulations are consistent
with this and suggest that the relative reduction of phonon amplitude versus static strain may allow imaging of the latter for a single impurity atom at low temperatures. The essential idea suggested by these results (and indeed by much of our earlier work, e.g. Refs. [19,21,22,38]) is that once the probe size, a, is significantly less than the intercolumn distance, d, of the atomic columns in the specimen, then scattering at any instant will be dominated by a single column, with little interference from the small scattered beams from neighboring atom strings. Thermal vibrations will blur the atomic positions by about 0.1 ,~, the RMS phonon amplitude at room temperature. The resultant thermal diffuse scattering may change the intensity level of the ADF signal, depending on the choice of detector. At room temperature where the thermal diffuse scattering is relatively strong, changing the detector inner radius causes the signal to drop smoothly as the radius is increased. At low temperatures, where the high-angle scattering is more strongly located in HOLZ rings, discontinuous jumps in intensity will occur as the detector radius is increased beyond them. However, as long as channeling preserves the localness of the probe, then the effects on the image of thermal displacements and the aperture are going to be minor as suggested by the simulations. This scheme shows up in the simulations of the Si (110) images. With a probe size small enough to illuminate the columns in the 1.36 A spacing individually, the allowed (400) spatial frequency in the image yields resolved 1.36 .~ detail. However, in a probe that is clearly too large, no such detail appears. In an intermediate sized probe the very interesting forbidden (200) reflection (observed experimentally [35]) develops only after propagating through the sample to a 50 A depth thus suggesting the importance of multiple scattering. Such observations suggest the need for reservations over the interpretation of digitally enhanced images until the image mechanisms are thoroughly established. The inner detector radius should be that angle at which the detected electrons no longer interfere with the original probe. If Bragg reflections are excited, then the extinction distance for that
s. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6-17
reflection should be m u c h g r e a t e r t h a n the specim e n thickness to e n s u r e k i n e m a t i c scattering. Inc o h e r e n t contrast is observed for H O L Z rings [26] for which this c o n d i t i o n holds. S i m u l a t i o n s of [100] silicon C B E D p a t t e r n s [19] with probes larger t h a n the i n t e r a t o m i c spacing suggests that relatively small i n n e r detector radii w o u l d suffice, e.g., 40 m r a d . W h i l e i n d e e d it is fair to claim that a n n u l a r dark field images are simpler to i n t e r p r e t t h a n phase c o n t r a s t bright field images a n d are p o t e n tially of higher resolution, the simulations rep o r t e d here a n d in o u r earlier work a n d of others [21,22,26] suggest that there are c o n s i d e r a b l e complications. T h e thickness results [22] a n d the simulations r e p o r t e d here relating to detector i n n e r angles, t h e r m a l diffuse scattering a n d multiple scattering effects with probes close to the r e s o l u t i o n limit all suggest that care is still necessary. It is n o t yet a r o u t i n e approach.
Acknowledgments T h e a u t h o r s acknowledge v a l u a b l e conversations with E.J. Kirkland, T.-C. Lee, P. Crozier, D.A. M u l l e r a n d J.C.H. Spence. This research was s u p p o r t e d by the award of a g r a d u a t e fellowship to S.H. by the D e p a r t m e n t of E d u c a t i o n a n d by the D e p a r t m e n t of E n e r g y ( D E F G 0 2 8 7 E R 45322). C a l c u l a t i o n s were p e r f o r m e d at the Cornell Materials Science C e n t e r c o m p u t e r facility s u p p o r t e d by the N S F (DMR-9121654).
References [1] A.V. Crewe, J.P. Langmore and M.S. Isaacson, in: Physical Aspects of Electron Microscopy, Eds. B.M. Siegel and D.K. Beaman (Wiley, New York, 1975) p. 47. [2] S.J. Pennycook and L.A. Boatner, Nature 336 (1988) 565. [3] S.J. Pennycook, Ultramicroscopy 30 (1989) 58. [4] P. Xu, E.J. Kirk/and, J. Silcox and R. Keyse, Ultramicroscopy 32 (1990) 93. [5] N.D. Browning, M.F. Chisholm and S.J. Pennycook, Nature 366 (1993) 143. [6] P.E. Batson, Nature 366 (1993) 727. [7] D.A. Muller, Y. Tzou, R. Raj and J. Silcox, Nature 366 (1993) 725. [8] E.J. Kirk/and, Ultramicroscopy 32 (1990) 349.
17
[91 S.J. Pennycook and D.E. Jesson, Ultramicroscopy 37 (1991) 14. Z.L. Wang and J.M. Cowley, Ultramicroscopy 31 (1989) [10] 437. [11] Z.L. Wang and J.M. Cowley, Ultramicroscopy 32 (1990) 275. [12] D.E. Jesson and S.J. Pennycook, Proc. Roy. Soc. (London) A 441 (1993) 261. [13] M.M.J. Treacy and J.M. Gibson, Ultramicroscopy 52 (1993) 31. [141 E.J. Kirkland, R.F. Loane and J. Silcox, Ultramicroscopy 23 (1987) 77. [15] R.F. Loane, E.J. Kirk/and and J. Silcox, Acta Cryst. A 44 (1988) 912. J.M. Cowley and A.F. Moodie, Acta. Cryst. 10 (1957) [16] 609. [171 K. Ishizuka and N. Uyeda, Acta Cryst. A 33 (1977) 740. [18] E.J. Kirk/and, R.F. Loane, P. Xu and J. Silcox, in: Computer Simulation of Electron Microscope Diffraction and Images, Eds. W. Krakow and M. O'Keefe (Minerals, Metals and Materials Society, New York, 1989) p. 1315. [19] R.F. Loane, P. Xu and J. Silcox, Acta Cryst. A 47 (1991) 267. [20] P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 38 (1991) 127. [21] R.F. Loane, P. Xu and J. Silcox, Ultramicroscopy 40 (1992) 121. [22] S. Hillyard, R.F. Loane and J. Silcox, Ultramicroscopy 49 (1993) 14. [23] S. Hillyard and J. Silcox, Ultramicroscopy 52 (1993) 325. [24] A. Howie, J. Microscopy 117 (1979) 11. [25] M.M.J. Treacy, A. Howie and S.J. Pennycook, Inst. Phys. Conf. Ser. 52 (1980) 261. [26] J.C.H. Spence, J.M. Zuo and J. Lynch, Ultramicroscopy 31 (1989) 233. [27] D.D. Perovic, C.J. Rossouw and A. Howie, Ultramicroscopy 52 (1993) 353. [28] O.L. Krivanek, Optik 45 (1976) 97. [29] K. Wong, E.J. Kirk/and, P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 40 (1992) 139. [30] J.C.H. Spence and J.M. Cowley, Optik 50 (1978) 129. [31] L.M. Peng and J.M. Cowley, Acta Cryst. A 44 (1988) 1. [32] International Tables for X-ray Crystallography, Vol. IV (revised and supplementary tables) (International Union of Crystallographers and Kynoch, Birmingham, 1974) ch. 2.2. [33] S.J.E. Alldred and M. Hart, Proc. Roy. Soc. (London) A 332 (1973) 239. [34] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1989) p. 196. [35] J. Liu and J.M. Cowley, Ultramicroscopy 52 (1993) 335. [36] C.R. Hall, P.B. Hirsch and G.R. Booker, Phil. Mag. 14 (1966) 979. [37] M. Mankos, S.Y. Wang, J.K. Weiss and J.M. Cowley, in: Proc. 50th Annual EMSA Meeting (San Francisco Press, San Francisco, 1992) p. 102. [38] J. Silcox, P. Xu and R.F. Loane, Ultramicroscopy 47 (1993) 173.
Ultramicroscopy58 (1995) 6-17
Detector geometry, thermal diffuse scattering and strain effects in ADF STEM imaging Scan Hillyard, John Silcox School of Applied and Engineering Physics, Cornell University, Ithaca, N Y 14850, USA
Received 17 April 1994; at the Editorial Office 20 September 1994
Abstract
Intensities of atomic-resolution Annular Dark Field Scanning Transmission Electron Microscopy (ADF STEM) images of zone-axis-oriented specimens change with defocus at rates that depend on lattice spacing. Thickness and strain effects on the intensities have been demonstrated. In this paper, image simulations (with some experimental basis) are presented that consider the dimensions of the ADF detector. Changing the inner radius of the detector seems to have relatively small effect on the image except to lower the detected intensity. Probe size was explored and a case identified where multiple scattering was important in the image. Thermal diffuse scattering (TDS) is important in high-angle scattering at room temperatures but it does not seem to alter the image appearance markedly. Finally, the image arising from the strain field around a single boron atom has been simulated and the results suggest increased scattering in agreement with observations. This mechanism may be adequate for single impurity atom detection at low temperatures and with special detector angles.
I. Introduction
Annular dark field imaging [1] is an important tool in electron microscopy both as a high-resolution imaging technique [2-4] and as a way to locate a specimen probe on the sample while simultaneously recording analytical data such as electron energy loss spectra [5-7]. It is intrinsically quantitative since image data can be recorded directly from linear detectors into digital memory [8], then analyzed and displayed subsequent to the recording process. However, while extensive modeling of the image formation process has been carried out [9-13], relatively little experimental testing of the models to check assertions concerning the importance of factors such as the detector inner angle and thermal diffuse
scattering has been published. In this paper, we use a particular simulation approach [14,15] based on multislice theory [16,17] and tested earlier by experiment [18-23] to explore issues raised by the modeling studies. In earlier papers, a "frozen phonon" approximation based on the Einstein approximation [19,20] introduced into the multislice simulations provided a good fit with diffraction experiments including the thermal diffuse background. This has been used to simulate images recorded experimentally as a function of defocus. The good fit observed [21] between the experiment and the calculation provides additional confidence in the simulations. The incoherent imaging model gives a good description of the focal variation of zoneaxis atomic-resolution experimental and simu-
0304-3991/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-3991(94)00173-1
S. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6 - 1 7
lated images. In subsequent work [22], this approach was refined to include incoherent factors in simulating the probe shape giving again excellent agreement. The virtual gun source size is the most significant of these in limiting high-spatialfrequency components in the image. Considerable effects are predicted in the depth dependence of the images due to the growth and subsequent decay of channeling peaks along the zone axes. Experimental observations [22,23] confirm these effects and provide further evidence that "frozen phonon", multislice-based simulations provide an excellent basis for predicting and analyzing the images and diffraction patterns found in STEM. In this paper, we use simulation to explore issues that have been widely discussed. Diffraction-contrast effects in ADF STEM images were troublesome in early studies of Pt catalysis particles on alumina substrates prompting Howie to propose [24] use of a detector with an inner angle > 40 mrads. In later work [25] the value suggested for this angle was extended to larger values, e.g., 80 to 100 mrad to ensure that all the scattering was incoherent, i.e., thermal diffuse scattering. It was noted [26], however, that HOLZ line scattering was kinematic and would also give rise to an incoherent image. Nevertheless, arguments for the use of high detector inner angles in atomic-resolution ADF STEM appear [12] and are widely accepted. Using simulation, we look at this question in atomic-resolution imaging together with the importance of thermal diffuse scattering. Other factors explored include the probe size and the effects of strain which have recently been unambiguously identified as a source of contrast in ADF STEM [27].
2. Background The image-formation process can be divided into four stages. These are: formation of the incident probe by the microscope, propagation of the probe through the sample to form a wave function at the exit surface, evolution of that wave function to the detector plane (forming a diffraction pattern in the process) and integration
7
of the diffraction pattern over the detector plane to give the intensity at one pixel in the image. The simulation procedure [14,15] follows those stages. The incident probe is h ( r ) = I qZ(r) l 2, where the electron wave function, ~(r), is the Fourier transform of qt(k), k is the two-dimensional Fourier conjugate to r, the two-dimensional position vector, and ~ ( k ) is given by: qt(k) = A ( k ) e x p ( i x ( k ) ) ,
(1)
A(k)=A o fork
for k > k o,
x ( k ) = ~Ak2(~Csa2k 2 - AT), where k o is the objective aperture, C s is the spherical aberration of the objective lens and A f is the lens defocus. The quantities Cs and A f can be determined by Krivanek analysis [28] of STEM bright field phase contrast images of carbon films [29]. This determines the probe width, a, which depends on defocus, objective aperture and the spherical aberration. As pointed out elsewhere [22,30], the maximum spatial frequency that can be formed in the image is given by 2k o, the diameter of the objective aperture. In real space, this condition translates to a < d, where d is the smallest observable spacing. Propagation of the probe through the specimen is modeled using the multislice theory of Cowley and Moodie [16]. This first entails calculation of the phase shift of the electron beam as it passes through the atom and is determined from the Mott formula using modified [31] X-ray scattering factors [32]. The second step is Fresnel propagation of the phase-shifted electron probe to the next layer of atoms where the procedure is repeated until the probe reaches the exit surface. Atomic positions are determined from the lattice positions with slight random displacements added in accordance with the "frozen phonon" approximation [19]. The Fourier transform of the exit waveform provides the diffraction pattern at the detector plane and this is integrated over the detector dimensions. The calculation has to be repeated for each pixel in the image. Within the precision of the calculation (i.e., depending on the computer resources available)
8
S. Hillyard, J. Silcox/Ultramicroscopy 58 (1995) 6-17
this calculation is valid for relatively thick samples and includes dynamical effects, channeling, scattering to fractional Bragg angles, HOLZ scattering and arbitrary specimen structure. It includes multiple elastic and thermal diffuse scattering to all orders but does not include inelastic scattering due to electronic excitation. Since the instrument has an electron spectrometer, the experiments can be set up to match these conditions. Excellent matches between observed data and the simulations are found. The phonon scattering is based on random atomic displacements which are given by a Gaussian probability distribution. The root-mean-square (RMS) amplitude has been treated as a free parameter with good agreement [20] with X-ray determinations [33]. For many purposes it is useful to consider ADF STEM images as arising from an incoherent imaging mechanism. This assumes that the image is given by an equation of the form:
I(r) = h ( r ) ® s ( r ) ,
(2)
where I(r) is the image intensity and is formed by the convolution, ®, of h(r), the intensity distribution within the scanned probe, with s(r), a specimen function that describes the details of the interaction of the electron beam with the sample. The image-forming mechanism does not necessarily rely on incoherent electron scattering although this is often assumed. Provided that the scattering process taking the electron to the detector actually removes the electron from the probe, i.e., it no longer participates in the probeformation process, then the scattering can be either coherent or incoherent. In testing the model accuracy, the Fourier transform of Eq. (2) has been found to be very useful, i.e.,
I(k) =H(k)S(k),
(3)
where I(k), H(k) and S(k) are the Fourier transforms respectively of I(r), h(r) and s(r). For a crystalline specimen, the image is periodic and intensity arises only at reciprocal lattice points, which are few in number since the largest sample spacings are close to the instrumental resolution limits. For both simulated and experimental images the intensities in the peaks of the power
spectrum provide a simple means of image evaluation. Using this approach, the incoherent image model was found to describe the observed focal variation of zone-axis images relatively well even though the calculation as carried out calculates the amplitude of scattering before determining the intensity incident on the detector [21]. It should be noted that there is now evidence [22,23] using this approach that the form of the probe changes as a result of the channeling peaks formed down a zone axis and that at each layer the shape of the probe is different. The ADF signal is the integral of the scattering from each layer in the sample. Thus, although Eq. (2) describes the dependence of the image on defocus well, it should not be assumed that the actual probe inside the sample is given by Eq. (1) as a simple interpretation of Eq. (2) might suppose. In particular, interpretation of s(r) as a set of 3functions at the atomic positions weighted by an atomic scattering strength (proportional, for example, with our annular detector to Z 17) has the consequence that the atomic arrangement may generate systematic absences in S(k). Hence careful analysis of the intensities that shows the appearance of forbidden peaks in the power spectrum of an image is prima facie evidence of the breakdown of the simple model.
3. Simulation details
Typical calculations followed patterns reported in earlier calculations [19,21]. For convenience, we review these here. For indium phosphide [100] samples (i.e., with the electron beam incident along [100]) the slice dimensions are 512 x 512 pixels, 6 x 6 unit cells and 35.2 x 35.2 A dimensions. These choices yield a maximum included scattering angle of 179 mrad after bandwidth limiting. All detector inner angles at a spacing of the angular resolution (1.05 mrad) were used to produce a range of ADF STEM signals for each thickness. Examples are shown at selected inner radii of 30, 50, 90 and 120 mrad in later sections of the paper. In the line scans, simulated image points are typically 0.15 A apart. All calculations for InP [100] were performed
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
with four 1.47 ,~ (one plane of atoms) thick slices. For frozen phonon calculations, several iterations, corresponding to different atomic displacements were averaged. The atoms were displaced from the lattice positions by a random amount with a RMS amplitude equal to that of the phonon RMS amplitude. The different configurations within an iteration were obtained by randomly shifting slices horizontally by a lattice vector. At least 4 different iterations were used to adequately represent the ensemble and usually 10 were needed to obtain a 1% RMS relative error as determined by sampling tests. The appropriate RMS vibration amplitudes for indium phosphide are unknown. An estimate was made by adjusting the known silicon vibration amplitude at room temperature of 0.076 .~ [33] for a higher mean atomic mass and lower vibration frequency [21]. If the vibration amplitudes are inversely proportional to the square root of the atomic mass, an amplitude of 0.05 ,~ results for the indium atoms and 0.096 ,~ for the phosphorus atoms. For silicon [100] samples, the comparable dimensions were 32.6 X 32.6 A and a 1.13 mrad angular resolution. Four atomic slices 1.36 ,~ apart and then repeated comprised the sample. For silicon [110] samples, the comparable dimensions were 32.6 x 30.7 A with 1.20 mrad angular resolution. Two atomic slices 1.92 A apart and then repeated formed the specimen. Three electron-optical conditions have been used in this work to simulate three different probe sizes at 100 keV. A relatively standard
o.2 5 :
~
.
~
.
~
~
~
~
3o
A.
4. Results
The best documented examples of zone-axis A D F STEM image simulations using the multislice approach are for silicon [14] and InP [15]. Using parameters and simulations that have successfully modelled the observed intensity distributions in CBED patterns, Fig. 1 shows simulated intensity line scans along a [011] direction for an electron beam incident along the [100] direction in silicon for a specimen of 192 ,~ thickness and a 2.2 ,~ probe. The basic intensity patterns are shown in Fig. l a while corresponding scaled intensities are seen in Fig. lb. No change in the overall visibility of the features are seen as a result of the changes in the dark field detector inner angle but a noticeable decrease in the overall intensity is noted as the inner angle is increased from 30 to 120 mrad. The images include two basic Fourier compo-
°3 i
0.25
0.2 ~ ;~ •~ 0 . 1 5 ~
:50
.
=
probe of 2.2 ,~ (given by 6 = 0.43A3/4C 1/4) is simulated by an objective-lens spherical aberration C s of 1.3 mm, an objective aperture of 10.5 mrad, i.e., 2k o = A/0.021 = 1.76 A. A higher resolution of 6 = o1.8 A and a smallest detectable spacing of 1.39 A is obtained with a C s of 0.5 mm, an objective aperture of 13.3 mrad and a defocus of 430 A. Finally an optimistic 1 ,~ probe is simulated by a C s of 0.05 mm, an objective aperture of 23 mrad and a defocus of 136 A. The corresponding smallest detectable spacing is 0.8
E
F
""
0.1
~0.15
= _
o.05 0
. . . . . . . . L i,
0 (a)
,I,
1
,i
2
~
,
3
I,,,,I,,,
4
. . . . I
5
Angstroms
6
,,,i
,
7
X90
_~ o.1
d120
0.05
3 8
0
(b)
I
~.,
0
,~, ,i ,i,,,
1
2
,1~,
,~,,,
,~,,,
3 4 5 Angstroms
,],,,
,I,,,
6
7
Fig. 1. Simulated A D F intensity line scans along the [110] direction for an electron probe of 2.2 ,~ size (for details see text) incident down a 200 ,~ thick [100] silicon sample. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
10
0 . 5 ~ ,
.......
, ....
,,,
'~o.o2~ ~
....
I ..........
'~ 0 . 4 ~
~o.~ ~ !
~
so
.......
0.015
0,2
0.01
....
90
"~ 0
100 200 300 400 500 Thickness in Angstroms
(a)
~
~ o.005
,.
_
~,~.-....
_
~
0 0
600
100 200 300 400 500 T h i c k n e s s in A n g s t r o m s
(b)
0.04
O0
!
-~ 0 . 0 2
L ~,
0
100 200 300 400 500 T h i c k n e s s in A n g s t r o m s
(c)
600
Fig. 2. Background (a) and fringe strength (b) for the four apertures shown in Fig. 1 plotted as a function of thickness. T h e ratio (c) shows the little difference within the noise levels (higher at small thicknesses) in the relative fringe strengths for the four detector geometries.
nents, a background (with k = 0) and a fringe of spacing 1.9 A. In Fig. 2 the intensity of the background and the 1.9 ,~ fringe strength are displayed as a function of thickness. The background fringe grows monotonically (but not lin-
0.6
....
,
-
0.5
!
so
;~ 0 . 4
=
0.6
so /~
~
early) for all inner detector angles. The 1.9 fringe shows oscillations with thickness (presumably due to dynamical diffraction) for the detector with an inner radius of 30 mrad but no such effects at higher inner detector angles. Since the
0.5 ,~ 0 . 4
0.3
'3
0.2
~ o.2
0.1
0.1
0
(a)
,
0
,
~ ;
5
,
i
, ~ ; ,
10 Angstroms
, ,
,
15
,
,
O. 3
0
,J
2
(b)
I
0
5
10 Angstroms
15
20
Fig. 3. Simulated A D F intensity line scans along the [110] direction for an electron probe of 1 ,~ size (for details see text) incident on a 205 A [100] InP sample. Four detector inner angles of 30, 50, 90 and 120 mrad are seen in (a). T h e same data scaled to superimpose on the indium atoms are seen in (b).
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
Si
(II0)
Projected Lattice ®
©
©
®
©
@
2.72A
@
@
1.3aA~ --@
--® 14 O
3.84A,
,k O
©
®
@
©
Reciprocal Lattice
O
o
Zero
Forbidden
Fig. 4. Projected lattice of Si [110] showing the dumbbells and the associated reciprocal lattice. Note that the 2.72 A (200) spacing (open circles) along the dumbbells is a forbidden spatial frequency.
30 mrad inner detector angle borders on the zero-layer reflections this seems reasonable. Within the noise level, the ratio of these two signals (which is what is plotted in later figures) is
~'.
J-
F
. . . .
i
i;
. . . .
I
~
r
~
'
I
'
the same for all inner angles but is not constant with thickness. Similar data to Fig. 1 for a 205 ,~ thick [100] InP sample is seen in Fig. 3 for images simulated with a 1 ,~ probe. In this case, since there are two elements present, Z-contrast ( Z is the atomic number) is seen and changes with the inner angle of the dark field detectoL Fig. 3a shows intensity scans along a [110] direction as the beam passes successively over indium and then phosphorus atom columns. The same data scaled to superimpose the intensities of the indium atoms is then seen in Fig. 3b. The relative visibility of the phosphorus atom shrinks as the detector inner angle grows with the phosphorus becoming almost undetectable at 90 to 120 mrad. This effect is due to the smaller angular scattering of the phosphorus atom relative to the indium atom. For example, if we take the Wentzel model of the atom [34], then a screening angle 6)0 can be defined as 6)0 = A/2~rR0 where a is the electron wavelength and R o is aH/g 1/3 where a H is the Bohr radius. The screening angles are 41 mrad for the indium and 27.5 mrad for the phosphorus. Over the range of angles appropriate for the detector inner angles considered here, a significant difference occurs. Note, however, that if the goal is to image the phosphorus atoms, then this result suggests that a smaller, rather than a larger, inner detector angle is recommended. To observe effects due to probe size, i.e., limiting interference on the lateral scale, or lateral incoherence, it is convenient to :tudy the scatter-
0.18
'
3o
i1
1
.
.
.
.
.
~ T
0.16
0.15
0.14
•~ 0.12
0.1
o
I:I
0.I
-=0.08
0.05
0.06 0
0.04 0
(a)
5
10 15 20 Angstroms
25
30 (b)
~' 0
~ 5
-
.......... 10 15 20 Angstroms
' 25
30
Fig. 5. Simulated ADF intensity line scans along the [100] direction for an electron probe of 2.2 A size (for details see text) incident down a 192 A thick [110] silicon sample. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
S. Hillyard, J. Silcox / UItramicroscopy 58 (1995) 6-17
12 0.25 ~
,
30
T
t
0.2
0.25
i
. . . .
1 ' ' ¸ 7
. . . .
i
,
,
0.2
~0.15
~0.15
0.1
~ 0.1
0.05
0.05
0
(a)
0
5
10 15 20 Angstroms
25
30
0 (bl
5
10 15 20 Angstroms
25
30
Fig. 6. Simulated A D F intensity line scans along the [100] direction for an electron probe of 1 ,~ size (for details see text) incident down a 192 ,~ thick [110] silicon sample for comparison with Fig. 5. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b).
ing from a probe incident along the [011] direction. This is also a common system for experimental studies. In this direction, the atoms form arrays that are staggered in depth and that project into an array of two columns 1.36 ,~ apart (see Fig. 4). The column arrays are 5.4 ,~ apart. Thus for a 2.2 ,~ probe, both columns fit well within the probe size whereas for a 1 A probe the two columns are separated. In the first case interference between the two scattered beams is expected but not in the second. This also holds true if the smallest spacings that can be recorded by these probes are considered, viz., 1.76 A for the larger probe and 0.8 ,~ for the smaller probe. Fig. 5 shows both the unscaled (a) and scaled (b) intensity line scans for the larger probe. This situation is similar to the earlier figures, i.e., the primary effect of the change in the detector inner
angle is to lower the detected intensity but it does not affect the overall atomic structure image. Thus we conclude that in this case the two columns are sufficiently close laterally that no significant effects due to interference occur. The thickness dependence of the fringe strengths has been extensively explored earlier [22,23]. In Fig. 6, intensity scans for the same situation with the 1 ,~ probe show that the atomic column spacing of 1.36 ,~ is now well resolved. The effect of increasing the inner radius of the detector aperture is again to reduce the intensity as is demonstrated in Fig. 6b. The scaled intensities superimpose except for the smallest aperture (30 mrad) where there is some slight deviation from the scaled curve. As noted earlier, this aperture borders on the zero-layer reflections, so this could be expected. The results reported so far suggest
0.09
0.1 L i
. . . .
i
. . . .
I
'¸
0.08
30
0.07
~o.o6 0.05
~ 0.04
~ 0.04
0.03 0.02
0 (a)
5
10 15 20 Angstroms
25
30
0 ' 0 1 0 . . . . . .5. . . . 1 0 . . . .1.5. . . . . .2. .0. . . 2 5 (b) Angstroms
Fig. 7. Simulated A D F intensity line scans along the [100] direction for an electron probe of 1.8 A size (for details see text) incident down a 192 ,~ thick [110] silicon sample for comparison with Figs. 5 and 6. Four detector inner angles of 30, 50, 90 and 120 mrad are shown in (a). The same data scaled to superimpose are shown in (b),
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
that the primary effect of increasing the inner angle of the detector is to lower the detected signal (presumably at the cost of the signal-tonoise ratio) while altering the details in the image by very little. In an interesting intermediate situation, resolution of the 1.36 .A column spacing in (110) silicon at a 10% signal drop between the two columns has been demonstrated by Liu and Cowley [35] with a probe size of 1.8 A. This situation is a case in which interference or diffraction effects may become visible since the lateral spacing between the atoms is less than but approaching the probe size. The 1.36 A spacing is just outside the smallest possible spacing that can be imaged for this situation (estimated above to be 1.39 ,~) and given the likelihood of incoherent source size effects [21-23] direct imaging should be considered unlikely. In the simulated line scan along the [100] direction seen in Fig. 7a, the two atoms are seen resolved at the 10% level and the visibility again shows little dependence on the aperture inner diameter as shown by the superimposed data in Fig. To. The thickness dependence of the fringes making up the image show interesting effects as seen in Fig. 8a. Specifically, a (200) intensity fringe with a spacing of 2.7 A emerges in the imagoe after the probe has traveled approximately 60 A
into the sample. This is also seen in the power spectrum of the Liu and Cowley [35] experimental image. If the projected lattice undergoes a Fourier transformation, this fringe is forbidden as seen in Fig. 4. Thus the observation is an example of the breakdown of the simple t3-function model for the specimen function, s(r). The 2.7 .A fringe is seen in the simulations at relatively small excitation levels for both the 2.2 ,~ probe and in Fig. 8b for the 1 A probe. In the latter case, the (400) fringe giving the 1.36 A spacing directly is seen. Returning to consideration of the 1.8 A probe simulation, it is the combination of the forbidden (200) fringe with the (100)-type fringe that gives rise to the observed 1.36 ,~ splitting. The 5.4 A (100) fringe in this direction arises from a superposition of (lll)-type fringes in the two-dimensional image. It is important to appreciate that the (200) fringe does not arise immediately on entrance to the sample where a simple convolution of h(r) with a 6-function model would hold but it develops after the beam has passed about 50 A into the specimen. This suggests that it is a multiple scattering effect. In Fig. 9, the shape of the probe inside the sample is illustrated showing two well resolved channeling peaks. However, as the probe size, a, gets bigger than the inter column spacing, d, then interference becomes possible between electrons
-a 0.6
,~ 0.5
13
I
I
I
=
,~ 0.4
P 0.5
5.4A
.o 0.4
.o
o
o
I
0.3
0.2
~ 0.3 8
t 2.7A
e!
1
o.1 o
=
0
(a)
-1.8A I
50 100 150 Depth in Angstroms
1
-~.36A
20O
g o.2 e~
1 1.8A 1
~ 0.1
~.TsA
o
1 2.7A
0
(b)
0
50 100 150 Depth in Angstroms
O0
Fig. 8. (a) Thickness dependence of the lineoscans of Fig. 7 as resolved into four spatial frequencies: (1/5.4) ,~-1, (1/2.7) ,~-1, (1/1.8) A - 1 and (1/1.36) A-1. The (1/5.4) A-1 intensity is seen to start out and remain high, whereas the (1/2.7) ,~-1 intensity begins in the noise but then increases at approximately 60 ,~ into the sample. The (1/1.8) A -1 and (1/1.36) ,~-1 remain small and are outside the resolution of the probe. This effect is thought to be responsible for the dip associated with the resolution of the 1.36 ,~ spacing seen in Fig. 7. The corresponding thickness analysis for the data of Fig. 6 is seen in (b). The scale bar represents approximate noise levels.
S. Hillyard, Z Silcox / UItramicroscopy 58 (1995) 6-17
14
Fig. 9. Shape of the electron probe of initial size 1.8 ,~ at 100 A into the [110] silicon sample. Notice the double peak formed on each atom of the dumbbell.
scattered by the two atomic columns (of 0.7 ,~ channeling diameter) within the 2.2 A diameter probe. Given a silicon screening angle of approxi-
mately 27 mrad, the distance at which the scattering from one atom in the dumbbell can affect the second would be 1.36/0.027 ,~ or 50 ,~, i.e., the depth within the sample at which the forbidden (200) reflection appears. Channel-to-channel interference thus may be the underlying cause of the appearance of this forbidden spatial frequency and its thickness dependence. At larger probe sizes, the scattering contains yet more columns and more diffraction and the incoherent nature of the image disappears. Thermal diffuse scattering has been modeled successfully with the frozen phonon approximation and has been included where appropriate in the simulations. Comparisons of the zone-axis images with various levels of phonon disorder can be readily made. The conclusion is that although thermal diffuse scattering strongly modifies the high-angle scattering at room temperature for most materials, the primary effect is to alter the
0.2 Room
Temperature
0.15
0.15
/~
"3 0.1 0.05
" 0.05
h°
o k
0
5
(a)
0.07
~
-,
0.06 ~
10 15 20 Angstroms
25
30
'No Phonor~s~ ~'
'
i
..........
i
(b)
.......
0
0.07
5
'
,i ¸¸
,. . . .
~. . . . . . .
10 15 20 Angstroms
, ~
i ....
i ....
~ - ..... 2 25
i
30
,,,
0.06
0.05
•~ 0.04
: ~
'r~ 0.04
o03
o.03
0.02
- 0.02
0.01 0
/~ I /
\~
"
,Tempe 10 15 20 Angstroms
,~i ,
R
0.01 0
10 15 20 25 30 Angstroms Fig. 10. Comparison of the result of incorporating thermal diffuse scattering into an image simulation. (a) Unscaled linescan of Si(ll0) with a 2.2 ]k and a 30 mrad inner detector radius. The solid line includes phonons while the dashed line does not. In (b) the two curves have been scaled and very little difference is seen. (c, d) Unscaled and scaled linescans, respectively, for a 1 A probe and a 120 mrad inner detector radius. Note that in (a, b) the effect of TDS is to increase the intensity while in (c, d) the TDS reduces the intensity. 0
(c)
5
25
0
30
(d)
5
S. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6-17
intensity with which an image is scaled. The detailed structure of the image is not changed. For example, in Fig. 10 very little difference is seen between a scaled simulated intensity line scan with and without the thermal scattering component. At lower temperatures, less of the intensity is in the thermal diffuse scattering component and more is in the higher-order Laue zone rings (see Fig. 4 of Ref. [19]). This scattering is essentially kinematic [26] so that the contribution to the image does not change. Simulated image intensity levels at the lower temperatures show a marked drop as the inner detector radius moves out and passes across a Laue ring. More intensity is now embodied in these features and less is in the relatively continuous thermal diffuse scattering. Nevertheless, there is very little difference between the low- and high-temperature simulated images except for the intensity level. Most experimental observations are carried out at room temperature where the differences between the Einstein approximation in the simulation and the real displacements are not expected to be large. Provided that the appropriate rootmean-square displacement is used, only slight differences occur between the observed and simulated patterns in the diffraction pattern comparisons (see Refs. [19,20]). The intensities have to be integrated over the detector dimensions before becoming image points and the slight differ-
15
ences then tend to average out. For almost all current observations, the influence of TDS on the detailed structure of the scaled image is minor. The effects of strain fields due to defects have been clearly identified in scattering from point defects (boron and arsenic) in silicon [27]. Strain scattering from a single boron atom can be readily simulated with the insertion of an elastic displacement field given by: (4)
u = c r / r 3,
where u is the vector elastic displacement of the atomic positions and c is a constant found to be 1.4 ~3 for boron in silicon [36]. The scattering from this strain field occurs in an angular range from 40 to 60 mrad and, for one atom, is overshadowed at room temperature by the thermal diffuse scattering. Several atoms superimposed in the specimen depth will add up to significant scattering at room temperature. These points are illustrated in Fig. 11, where the angular dependence of the scattering relative to the thermal diffuse scattering in undoped silicon at low temperature (a RMS phonon amplitude of 0.031 roughly equal to the zero point motion was used) is shown in Fig. l l a . The magnitude of the maximum scattering difference (which occurs with approximately a 40 mrad inner detector radius) with temperature is shown in Fig. l l b . These latter simulations suggest that a single boron atom may
10 10
I
i
I
8 U
6 - ;-~ ~' 4
................... ,: ~..................
"d
...... [
i
?.
2 '
0 -2 {fl) 0
_:'
.............
"'~
2
-
0 20
40 60 80 Detector Inner Angle
100
(b) o
50
:Minimum
",
:--
De~eetlble
",~ Signal i . . . . ~i
100 lS0 200 T e m p e r a t u r e (K)
250
300
Fig. 11. Strain field scattering from a single b o r o n atom impurity situated in the middle of a 200 A thick [100] silicon sample c o m p a r e d with u n d o p e d sample at low t e m p e r a t u r e is shown in (a) as a function of detector inner angle. In (b) the signal is shown as a function of t e m p e r a t u r e for a 40 m r a d inner detector angle. Probe p a r a m e t e r s were C s = 3.3 mm, an objective a p e r t u r e of 8.2 mrad, and A f = 1100 ,&, creating a p r o b e of 2.8 ,& in size.
16
S. Hillyard, J. Silcox / Ultramicroscopy 58 (1995) 6-17
be detected at temperatures of 77 K or below provided the detector is carefully chosen. Note that in this case of distinguishing static from dynamic diffuse scattering, the choice of inner detector angle is important.
5. Discussion and conclusions
These simulations suggest that the emphasis on using a large inner detector aperture to minimize coherent scattering may be misplaced. Use of a very high inner angle detector may make imaging unnecessarily difficult by reducing the visibility of lower atomic number elements in the presence of heavy ones and by discarding valuable intensity that could be used to improve the signal-to-noise ratio of the images. The original suggestion (Howie [24]) that a detector of about 40 mrad inner angle would remove dynamical diffraction effects seems a better choice for atomic structure imaging than the later modification of that proposal to use a very high inner angle [25]. Within the original context this proposal was made, i.e., the study of small particles on a small grained substrate use of the larger inner detector angle may still be appropriate. The work reported here argues the need for a flexible annular detector that is adaptable to a range of different samples (e.g., Ref. [37]). Thermal diffuse scattering, while clearly the dominant part of the angular ADF signal in the detector region does not seem to alter the details of the image markedly, i.e., calculations that do not include thermal diffuse scattering (TDS) are relatively close to those calculations that do include TDS. The primary error in image structure arising from neglect of thermal diffuse scattering is a change in the apparent thickness (at the 10% level). Unless the thickness is independently known and an accurate comparison of experiment and simulation made, it would be impossible to distinguish the images. Some of these points have been made earlier [19,20,38] but are re-emphasized here. Localized static strain does affect the image considerably, as observed, and gives rise to annular dark field effects that can counter those of Z-contrast [26]. The simulations are consistent
with this and suggest that the relative reduction of phonon amplitude versus static strain may allow imaging of the latter for a single impurity atom at low temperatures. The essential idea suggested by these results (and indeed by much of our earlier work, e.g. Refs. [19,21,22,38]) is that once the probe size, a, is significantly less than the intercolumn distance, d, of the atomic columns in the specimen, then scattering at any instant will be dominated by a single column, with little interference from the small scattered beams from neighboring atom strings. Thermal vibrations will blur the atomic positions by about 0.1 ,~, the RMS phonon amplitude at room temperature. The resultant thermal diffuse scattering may change the intensity level of the ADF signal, depending on the choice of detector. At room temperature where the thermal diffuse scattering is relatively strong, changing the detector inner radius causes the signal to drop smoothly as the radius is increased. At low temperatures, where the high-angle scattering is more strongly located in HOLZ rings, discontinuous jumps in intensity will occur as the detector radius is increased beyond them. However, as long as channeling preserves the localness of the probe, then the effects on the image of thermal displacements and the aperture are going to be minor as suggested by the simulations. This scheme shows up in the simulations of the Si (110) images. With a probe size small enough to illuminate the columns in the 1.36 A spacing individually, the allowed (400) spatial frequency in the image yields resolved 1.36 .~ detail. However, in a probe that is clearly too large, no such detail appears. In an intermediate sized probe the very interesting forbidden (200) reflection (observed experimentally [35]) develops only after propagating through the sample to a 50 A depth thus suggesting the importance of multiple scattering. Such observations suggest the need for reservations over the interpretation of digitally enhanced images until the image mechanisms are thoroughly established. The inner detector radius should be that angle at which the detected electrons no longer interfere with the original probe. If Bragg reflections are excited, then the extinction distance for that
s. Hillyard, J. Silcox /Ultramicroscopy 58 (1995) 6-17
reflection should be m u c h g r e a t e r t h a n the specim e n thickness to e n s u r e k i n e m a t i c scattering. Inc o h e r e n t contrast is observed for H O L Z rings [26] for which this c o n d i t i o n holds. S i m u l a t i o n s of [100] silicon C B E D p a t t e r n s [19] with probes larger t h a n the i n t e r a t o m i c spacing suggests that relatively small i n n e r detector radii w o u l d suffice, e.g., 40 m r a d . W h i l e i n d e e d it is fair to claim that a n n u l a r dark field images are simpler to i n t e r p r e t t h a n phase c o n t r a s t bright field images a n d are p o t e n tially of higher resolution, the simulations rep o r t e d here a n d in o u r earlier work a n d of others [21,22,26] suggest that there are c o n s i d e r a b l e complications. T h e thickness results [22] a n d the simulations r e p o r t e d here relating to detector i n n e r angles, t h e r m a l diffuse scattering a n d multiple scattering effects with probes close to the r e s o l u t i o n limit all suggest that care is still necessary. It is n o t yet a r o u t i n e approach.
Acknowledgments T h e a u t h o r s acknowledge v a l u a b l e conversations with E.J. Kirkland, T.-C. Lee, P. Crozier, D.A. M u l l e r a n d J.C.H. Spence. This research was s u p p o r t e d by the award of a g r a d u a t e fellowship to S.H. by the D e p a r t m e n t of E d u c a t i o n a n d by the D e p a r t m e n t of E n e r g y ( D E F G 0 2 8 7 E R 45322). C a l c u l a t i o n s were p e r f o r m e d at the Cornell Materials Science C e n t e r c o m p u t e r facility s u p p o r t e d by the N S F (DMR-9121654).
References [1] A.V. Crewe, J.P. Langmore and M.S. Isaacson, in: Physical Aspects of Electron Microscopy, Eds. B.M. Siegel and D.K. Beaman (Wiley, New York, 1975) p. 47. [2] S.J. Pennycook and L.A. Boatner, Nature 336 (1988) 565. [3] S.J. Pennycook, Ultramicroscopy 30 (1989) 58. [4] P. Xu, E.J. Kirk/and, J. Silcox and R. Keyse, Ultramicroscopy 32 (1990) 93. [5] N.D. Browning, M.F. Chisholm and S.J. Pennycook, Nature 366 (1993) 143. [6] P.E. Batson, Nature 366 (1993) 727. [7] D.A. Muller, Y. Tzou, R. Raj and J. Silcox, Nature 366 (1993) 725. [8] E.J. Kirk/and, Ultramicroscopy 32 (1990) 349.
17
[91 S.J. Pennycook and D.E. Jesson, Ultramicroscopy 37 (1991) 14. Z.L. Wang and J.M. Cowley, Ultramicroscopy 31 (1989) [10] 437. [11] Z.L. Wang and J.M. Cowley, Ultramicroscopy 32 (1990) 275. [12] D.E. Jesson and S.J. Pennycook, Proc. Roy. Soc. (London) A 441 (1993) 261. [13] M.M.J. Treacy and J.M. Gibson, Ultramicroscopy 52 (1993) 31. [141 E.J. Kirkland, R.F. Loane and J. Silcox, Ultramicroscopy 23 (1987) 77. [15] R.F. Loane, E.J. Kirk/and and J. Silcox, Acta Cryst. A 44 (1988) 912. J.M. Cowley and A.F. Moodie, Acta. Cryst. 10 (1957) [16] 609. [171 K. Ishizuka and N. Uyeda, Acta Cryst. A 33 (1977) 740. [18] E.J. Kirk/and, R.F. Loane, P. Xu and J. Silcox, in: Computer Simulation of Electron Microscope Diffraction and Images, Eds. W. Krakow and M. O'Keefe (Minerals, Metals and Materials Society, New York, 1989) p. 1315. [19] R.F. Loane, P. Xu and J. Silcox, Acta Cryst. A 47 (1991) 267. [20] P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 38 (1991) 127. [21] R.F. Loane, P. Xu and J. Silcox, Ultramicroscopy 40 (1992) 121. [22] S. Hillyard, R.F. Loane and J. Silcox, Ultramicroscopy 49 (1993) 14. [23] S. Hillyard and J. Silcox, Ultramicroscopy 52 (1993) 325. [24] A. Howie, J. Microscopy 117 (1979) 11. [25] M.M.J. Treacy, A. Howie and S.J. Pennycook, Inst. Phys. Conf. Ser. 52 (1980) 261. [26] J.C.H. Spence, J.M. Zuo and J. Lynch, Ultramicroscopy 31 (1989) 233. [27] D.D. Perovic, C.J. Rossouw and A. Howie, Ultramicroscopy 52 (1993) 353. [28] O.L. Krivanek, Optik 45 (1976) 97. [29] K. Wong, E.J. Kirk/and, P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 40 (1992) 139. [30] J.C.H. Spence and J.M. Cowley, Optik 50 (1978) 129. [31] L.M. Peng and J.M. Cowley, Acta Cryst. A 44 (1988) 1. [32] International Tables for X-ray Crystallography, Vol. IV (revised and supplementary tables) (International Union of Crystallographers and Kynoch, Birmingham, 1974) ch. 2.2. [33] S.J.E. Alldred and M. Hart, Proc. Roy. Soc. (London) A 332 (1973) 239. [34] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1989) p. 196. [35] J. Liu and J.M. Cowley, Ultramicroscopy 52 (1993) 335. [36] C.R. Hall, P.B. Hirsch and G.R. Booker, Phil. Mag. 14 (1966) 979. [37] M. Mankos, S.Y. Wang, J.K. Weiss and J.M. Cowley, in: Proc. 50th Annual EMSA Meeting (San Francisco Press, San Francisco, 1992) p. 102. [38] J. Silcox, P. Xu and R.F. Loane, Ultramicroscopy 47 (1993) 173.