- Email: [email protected]
Annular dark-field imaging: Resolution and thickness effects
Ultramicroscopy 49 (1993) 14-25 North-Holland
!1~!'I¢!1~i~1
Annular dark-field imaging: resolution and thickness effects Sean Hillyard, Russell F. L o a n e a n d J o h n Silcox School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
Received 6 July 1992; at Editorial Office 11 August 1992
Simulations of annular dark-field zone-axis crystal images that explore the dependence of images with thickness are reported. Earlier work with digitally acquired experimental images and diffraction patterns provide excellent agreement with multi-slice-based, "frozen phonon" simulations and thus provide a foundation for this work. Additional experiments at 60 kV add support. Simulations of indium phosphide at 300 kV and 1.3 A resolution show evidence that the phosphorus image is weak. The results of the currently reported thickness simulations suggest that while the electron beam probe channels for long distances in low-Z-element materials such as silicon or phosphorus, the channeled probe travels only about 100 * in heavier elements such as germanium or indium. Such an effect has serious implications for quantitative analysis using annular dark-field imaging, electron energy loss spectroscopyand X-ray microanalysis.
I. Introduction To be of value, quantitative electron microscopy needs a well established theoretical base. Influential papers reaching towards this goal included critical work by Zeitler and Thompson [1] outlining the theory of bright-field phase-contrast images in scanning transmission electron microscopy (STEM) and the necessary estimates of elastic scattering cross-sections using the Moliere approximation by Zeitler and Olsen [2]. This work provided an excellent basis for discussion of the early single-atom images obtained by the Chicago group [3] as STEM was being developed and it provides the starting point for much later work. For annular dark-field (ADF) images the key idea is the incoherent imaging model [3,4]. Recently, high-resolution A D F images of zone-axisoriented crystals that demonstrate Z-contrast [5,6] and high resolution [7,8] have been reported. These are consistent with the earlier ideas and have attracted attention since they are simpler to interpret than bright-field phase-contrast images with no contrast reversals with focal changes. A picture of the process has been developed that suggests that the electron probe forms a channel-
ing peak on the atom columns and that the A D F image results from scattering by the atoms from this channeled peak [6,9,10]. Details of this picture are still sketchy in spite of a long series of papers on channeling [11-14]. In this paper we follow up earlier work exploring the validity of the incoherent model for zone-axis crystal imaging. STEM is intrinsically quantitative. The images are measured by linear detectors, are ideally suited for serial data recording due to the scanning mode and lose the quantitative element only at the display stage. If the intensities are recorded directly into a digital memory prior to display then they can be stored for extensive off-line analysis after the experimental session as well as being available for examination during operation. Image analysis can be pursued at a much more demanding quantitative level than with photographic recording. In previous work we have used a D E C VAX 3200 workstation interfaced by Kirkland [15] to a VG HB501 A STEM. This permits the acquisition of simultaneous dark-field and energy-filtered bright field images in perfect registration, and of energy-filtered diffraction patterns. In parallel
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
S. Hillyardet al. /Annular dark-field imaging: r6solutionand thicknesseffects with this, we have used multislice-based [16-18] simulations to provide model predictions of the intensity patterns [9,10], introduced a "frozen phonon" model to describe thermal diffuse scattering and demonstrated excellent agreement with experimental studies of silicon [19,20]. This was followed by a combined simulation and experimental study of imaging in a particular system, (100) indium phosphide [21]. In this case, variation with focus was under scrutiny and both the simulation and the experiment were found to behave in good agreement with the simple incoherent imaging model. In this paper, we extend this work in two areas. Questions concerning resolution are addressed by both simulation and experiment. Operation of our system at lower voltages offers the opportunity to vary electron-optical conditions permitting use of somewhat different angles and energies. In addition, the imminent availability of instruments at 300 kV and 1.3 ,~ resolution raised the question of whether such an instrument could image phosphorus in indium phosphide in distinction with the results reported at 100 kV [21]. The second area is the variation of STEM images with thickness. Simulations show interesting and very significant changes with thickness and atomic number so that, given the validity of the simulations as checked by the earlier experiments [19,20,21,22], it was considered advisable to include them here. Experimental verification of these conclusions is being sought. The paper begins by reviewing the background of the incoherent imaging model and the results established by the earlier work. This gives the foundation for the results reported here. Illustrations using experiments at 60 kV and simulations at 300 kV are presented. Finally, thickness effects in elemental silicon and germanium and in compound (100) indium phosphide are discussed with the aid of simulations. Both image intensities and the form of the channeling probe inside the model crystal are explored with a somewhat disturbing conclusion. The channeling peak inside the sample can be expected to vary significantly with depth and, for even moderate-atomic-number elements, to disappear at thicknesses as small as
100 h.
15
2. Background The incoherent imaging model [3,4,23] describes the image intensity field I(r) as the convolution of an instrumental probe function, h(r), with a specimen object function, s(r) where r is a two-dimensional positional vector perpendicular to the optic axis representing the probe position as follows: I(r) =h(r) ®s(r).
(1)
The probe function is determined by the coefficient of spherical aberration, Cs, and the lens defocus, Af. It is the square of the incident wavefunction ~O(r) which itself is the Fourier transform of the aperture function, A(k), multiplied by the exponential of the phase factor introduced by the lens (see ref. [21] for a fuller discussion). Thus gy(k) = A ( k ) exp[ix(k)], A(k)
=1 0
for k k o,
x ( k) = rrAk2(1/2CsA2k 2 - A f ) ,
(2) (3) (4)
where A is the electron wavelength and k is a two-dimensional vector perpendicular to the optic axis. This model provides a means of separating specimen characteristics from microscope-introduced artifacts. Earlier work [24-26] has used this model to identify optimum single focal settings for differing purposes. In this work [21] the thrust has been to exploit focal series, since rather more information can be extracted. Determination of C S and calibration of A f for conventional transmission electron microscopy has been possible for a long time [27] through the use of power spectra of bright-field phase-contrast images of thin carbon films. A parallel treatment for STEM has only recently been reported [28]. The critical step appears to have been digital image acquisition [15] so that the full intensity range was recorded with a sufficiently small collector aperture. Earlier work [29] exploited Ronchigrams recorded from large spacing samples to determine Cs and Af while recent obser-
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
16
2.!
B A
C
Fig. 1. Schematic of an indium phosphide (100) crystal. Columns of indium atoms are represented by gray circles, columns of phosphorus by white. Fringe maxima are represented by gray lines. After Loane et al. [16].
vations [30] of amorphous carbon power spectra used too large a collector aperture. Because the large data set represented by an extensive focal series is recorded directly by the digital image acquisition system, analysis is straightforward and is not the formidable burden it is under the common photographic arrangement. For our first experiments [21], the (100) zone axis of indium phosphide was selected. This is a simple object in which the indium atoms line up in a square array with 2.9 .4, edges and the phosphorus atoms are at the square centers, i.e., 2.1 A from the indium atoms (see fig. 1). The sample therefore contains two atom species of substantially differing atomic number and two fundamental spatial frequencies both of which are within the resolution limits of our instrument. The results of these experiments demonstrate that although imaging in annular dark field is simpler than bright field in that no contrast reversals were evident, fringe contrast does vary with defocus, exhibiting oscillations in intensity with maxima at differing defoci for different spacings. The phosphorus atoms could not be resolved experimentally even though a smaller spacing had been resolved in other work [7,8]. This results from residual scattering of the heavy indium atom nearby which swamps the weak phosphorus signal. Thus, background scattering from other atoms in the sample can play a dominant role in Z-contrast imaging and must be considered, particularly near the resolution limits. These results are fully consistent with the incoherent imaging
model with the addition of a few extra experimental complications. The latter include a little astigmatism, specimen tilt and effects probably due to a finite source size. The same results emerge from an extensive simulation of the same images using computer codes developed for assessment of crystalline images [9,10]. The codes had earlier been tested for the ability to predict diffraction pattern intensities over a range of four orders of magnitude in intensity and with the addition of a "frozen phonon" Einstein model were able to match experimental intensities with great success [19,20]. In earlier work [22], correlation of the thicknesses of experimental specimens determined by matching diffraction patterns with model calculations and thicknesses determined by plasmon intensity measurements also give confidence that the simulations give good if not excellent predictions of the experimental observations. In the present case, the comparison was attempted not only with the experiment but also with the incoherent imaging mode. The result was that for this case at least, excellent agreement was found between the simulation and the model. This is not a trivial result. Electron scattering in specimens of the thickness considered is a very non-linear process and a clean proof of such a result seems highly unlikely. The results are also in good agreement with the experiments discussed above, but in this case since neither the simulation nor the model includes the experimental factors of astigmatism, tilt or finite source size, the same minor discrepancies with the data appear. As well as demonstrating that the simple model works reasonably well, this work implies that there are some complications arising from the choice of defocus since differing fringe spacings are at maximum contrast at different settings. Thus, it is by no means obvious that Scherzer settings are necessarily optimal as noted earlier [24]. A further conclusion is that care needs to be used to eliminate the differing weights placed on each spacing by the lens settings if a specimen object function is to be extracted from the image for quantitative work. In the following sections we discuss first the issue of resolution and expand upon these conclusions. Then we report some simulation re-
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
17
where T(q - k 1) and T(q - k 2) are the necessary complex scattering amplitudes and contain the requisite sample spatial information. The scattered intensity then becomes
Final Lens
14~(r, q) L2
Specimen
= IO(ka)lZlT(q-k~)[ 2 + IO(kz) l Z l T ( q - k 2 ) l ADF detector
+ 2 Re{0(k,)0*(k2) T(q-kl) × exp[i( k
Optic Axis
Fig. 2. Ray diagram showing incident beams of wave vector kl, k 2 scattering to wave vector q (eqs. (6) and (7)).
suits that carry the potential for substantial concern for simple interpretative schemes.
3. Spatial resolution A STEM A D F image provides intensity maps
2
1 -
T*(q-k2)
k2) ° r]},
(7)
where the notation * denotes complex conjugate. It is apparent that the first two terms contribute to the scattered intensity independent of the probe position and thus to a background intensity when integrated over the detector. It is the third term that depends on r and thus to variations in image intensity. Furthermore, at large scattering angles, so that the scattering is essentially kinematic, this term contributes to a Fourier component of the annular dark-field image with wave vector p = k 1 - k 2. In generalizing this discussion, we first consider the probe as a sum of incident waves so that the scattered intensity can be written
I(r) as a function of probe position r. The A D F detector integrates intensities ~b(r, q) scattered by the specimen onto a detector D(q) so that:
14~(r, q ) l 2
= (E~(kn)
I(r) = fD(q) 14~(r, q) L2 dq.
(5)
The dependence of 4~(r, q) on r is explicitly included here and fig. 2, which is a simple schematic of the scattering process, emphasizes that the position dependence of the scattered intensity arises from the coherent interference of the scattering of any two plane waves in the incident beam. Thus consider the two incident plane waves gt(kl) exp ik~'r and tb(kz) eXp ikz'r each of which is scattered by the specimen to the wave vector q. The resulting component of the scattered wave function th(r, q) is then ~(r,
q ) -----~O(kl) e ik,'r T ( q - k l ) -t- ~b(k2) e ik2"r T ( q - k 2 )
,
(6)
eik"r r ( q - k n ) )
n
= ~_. [ O ( k . ) 1 2 l T ( q - k . ) [ n=m + 2Re(.~mg,(kn)0*(k,.)
×T*(q-km)
2 T(q-k.)
exp[i(k n - k i n ) "r]},
(8)
where the wave vectors k n, k m are restricted to lie within the objective aperture (see eq. (3)). The interference term, i.e., the second sum, provides the position-dependent image intensity contributions. Wave vectors p = k n - k m up to the diameter of the objective aperture can be imaged. In
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
18
integral form, eqs. (5) and (8) can be put in the form
I(r) = f H( kl, p) S( k~, p) dk, exp(ip . r ) d p , (9) where
H(kl, p) = A ( k ) A*(k I - p ) e x p [ i ( x ( k )
-p))], s(k,, p) = fD(k) T(k-k,) T*(k-kl +v)dk, where H(k 1, p) is the contrast transfer function (CTF). In this formulation it appears as the weighting given by the instrument to Fourier components in the annular dark-field image within the integral in eq. (9). It is also the Fourier transform of the probe function h(r) of eq. (1) which is an approximate form of eq. (9) (for discussion see ref. [21]). Experimental evidence for elements of this discussion is provided by fig. 3. This provides two images of (100) indium phosphide recorded at 60 kV using the V G HBS01A STEM at Cornell under similar conditions as those used in our earlier work. In fig. 3a, the objective aperture was 40 izm corresponding approximately to Scherzer conditions for this voltage and lens (C s = 1.3 mm at 100 kV was estimated to correspond to C S = 1.0 mm at 60 kV). The power spectrum of this image is shown in fig. 3b and the peaks corresponding to the 2.9 ,~ spacing are clearly seen. The peaks corresponding to the 2.1 A spacing are absent and are not imaged with this aperture. This latter spacing is approximately equal to the diameter of the objective aperture as illustrated by the circles drawn around each spot in the power spectrum. The circles barely touch for the 2.1 A spots but significant overlap occurs for the 2.9 A spots. The image therefore contains just two superimposed sets of 2.9 A fringes. In contrast, fig. 3c shows an equivalent micrograph recorded with a 50 Izm objective aperture with the corresponding power spectrum in fig. 3d. The 2.1 A spacing appears in this image, consiso
tent with the fact that the 2.1 ,~ spatial frequency is now less than the objective aperture diameter as seen by the circles drawn around the spots. Significant overlap is observed for the 2.1 fringes unlike fig. 3b. In this case several overlaps occur and the image contains a small amount of three-beam interference. Thus, not surprisingly, a critical issue determining the resolution is the size of the objective aperture. The settings can be extended beyond the nominal limits somewhat by trading visibility against the resolution. Naturally, this cannot be pursued indefinitely and unless the lens is well characterized it may carry some interpretative hazards. The present experiments do give some indication of the elasticity in the limits and it may be worth noting that a number of the dark-field images in the literature are not recorded at Scherzer conditions. In fig. 4, we show contrast transfer functions (CTF) for annular dark-field relevant to the parameters of a modern STEM as discussed extensively in an earlier paper [21]. At Scherzer focus, the CTF is roughly triangular and weights each component differently. Unless some scheme is introduced to correct for these differences, then the intensities in the image will be distorted. Most high-resolution annular dark-field zone-axis images of crystalline materials so far published with STEM are simple images with only one main Fourier component in them. In this case the non-linearities introduced by the instrument are largely irrelevant and no serious harm has been done. Studies of defects or compounds (such as the indium phosphide) need to consider the problem. Ideally the operator will choose conditions that might be a " b e t t e r " i.e., more faithful, representation of the object. An ideal CTF would have a flat characteristic over much of the spatial frequency range and a somewhat more satisfactory CTF might be obtained, albeit at a cost in signal level. However, there are advantages in recording a focal series with a well characterized lens and appropriately combining the results corrected for lens distortion. The curve in fig. 4a was calculated with parameters estimated from 100 kV experience and is thought to represent the conditions (close to Scherzer) under which fig. 3a was recorded. An
S. Hillyard et aL / Annular dark-field imaging: resolution and thickness effects
estimate of the CTF corresponding to the micrograph of fig. 3c is also shown in fig. 4b. In this case, the contrast is lower but enough signal was apparently obtained to provide an image with more detail than the first by extending to higher
19
spatial frequencies. Nevertheless, the structure in this CTF indicates that in general this must only be carried out with well characterized lenses. The limiting factor (as always) will be the signal-tonoise ratio.
Fig. 3. Experimental A D F STEM images of indium phosphide (100) taken with the high-resolution pole piece at 60 kV (C s = 1.0 mm), and corresponding power spectra. (a) A 12 mrad objective aperture and an estimated defocus of 800 A. (c) A 15 mrad objective aperture and a defocus of 1600 A. (b, d) Corresponding power spectra. The circles in the power spectra are equal to the size of the objective apertures used.
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects
20
The significance of the objective lens diameter in these images arises from the way in which the scattered electrons carry the spatial information. As in eq. (8) the intensity is a sum of terms arising from one plane wave only (and carry no spatial information) and those arising from the product of one plane wave component interfering with another that do carry the spatial information. This mechanism has been discussed by Spence and Cowley [31] for Bragg reflection in particular but it applies to any scattering process. The maximum spatial frequency that can be carried by these waves is given by the diameter of the objective aperture. It appears in the CTF as the point at which the CTF finally goes to zero. A consequence of this unambiguous point is that the probe in real space never can go exactly to zero although it is equally surely not a Gaussian. Given the imminent availability of 300 kV STEM instruments and the success the simulations have enjoyed in modeling the diffraction patterns and images so far, two simulations have been carried out to see if the phosphorus atoms would be visible with these machines unlike the 100 kV case. One simulation models the 1 mm C s objective lens instrument currently available while the other anticipates one of 0.5 mm C S. In this case, six Fourier components will appear so that six different object points are necessary in the simulation [21]. Two images simulated at Scherzer conditions are shown in fig. 5. Imaging phosphorus is still not that easy. The signal is weak relative to the indium atom and if it was not already known to be there, it could easily be missed. This large difference demonstrates why the low-Z element can be swamped by neighboring high-Z elements. The invention of methods to explore effectively for such elements may be needed and the simultaneous use of bright-field phase contrast to look for light elements is likely to be necessary.
4.
T h i c k n e s s
effects
Following the procedures outlined by Loane et al. [21], the scattering intensity as a function of thickness at different points in the image can be
recovered. Given the size of the objective aperture the image contains only three independent Fourier components. Simulations were carried out at five points in the unit cell. The results confirm that only three Fourier components are necessary and provide these as a function of thickness. It remains to relate these values to the scattered intensities at each of the atomic columns and the background. As given in eq. (1), s(r) reflects the propagation of the incident electron wave packet h(r) through the sample. The packet changes shape and magnitude as it passes through the sample depending on the initial position [10]. However, the sample is periodic in directions perpendicular
1
, l l l l r , , l l r ~ l
0.8 ~ 0.6
%p=12mrad Af=800A ~ E=60kV
0.4
2.9 A ~
0.2
,
2. IA
IILIIIL~!III
0
0.10.20.30.40.50.60.7 Spatial F r e q u e n c y ( A ) ' t
(a)
1
°
Cs=1.0ram
i
,
,
,
,
,
,
i
,
,
,
0.8
aap=l5mrad M=I600A
0.6
E=60kV
,
Cs=l,0mm
0.4
0.2 0
(b)
I
0
i
I
I
1
I
~
I
I
I~
0.1 0.2 0.3 0.4 0.5 0.6 Spatial F r e q u e n c y ( A ) "l
I
0.7
Fig. 4. STEM A D F contrast transfer functions for the high resolution pole piece at 60 kV (C s = 1 mm). (a) For an objective aperture of 12 mrad and a defocus of 800 A, Close to the Scherzer conditions (O~ap= 12 mrad and A f = 700). (b) For an aperture of 15 mrad and a defocus of 1600 A. The 2.9 ,~ and 2.1 ,~ spacings of InP (100) are marked. The annular
aperture size was 60 to 370 mrads.
S. Hillyard et aL /Annular dark-field imaging: resolution and thickness effects
21
to the zone axis so that s(r) is periodic. Accordingly,
tions situated at the columns and multiplied by column partial cross sections, as in eq. (12):
8(r) = ~ S ( g ) e ig'r,
S(r)
(10)
g
°'° +
= A7
~(r-r,n
°tin rln
) +crpy'~(r-rp). rp
(12) where g is a 2D reciprocal lattice vector and, from eq. (1), I(g) =H(g)
S(g),
(11)
where l(g) is the intensity in the image corresponding to Fourier component g and H(g) is the weighting factor of the CTF for that component. Given both I(g) and H(g), then S(g) can be determined. At each thickness a focal series (about thirty images) simulation permits the retrieval of the two fringe amplitudes (i.e., $2.1 and $2.9) and the background S O using a least-squares routine with eq. (11). To proceed further, we assume that the specimen object function consists of delta func-
Fig. 5. Simulated A D F STEM objective aperture = 11 mrad, linescan passing through both logarithm of the intensity.
Here, A c is the area of the primitive unit cell, ~r0, ~ln, ~rp are partial cross sections for the background, indium column and phosphorus columns respectively, and r~,, r e are the lattice positions of the indium and phosphorus atoms. The Fourier transform of this provides the connection with the fringe amplitudes recovered from the focal series calculations, i.e., O'ln = A c ( S 2 . 9 + S 2 . 1 ) / 2 , ~rp = A t ( S 2 . 1
- $2.~)/2 ,
<, = A c ( S 0 - $2.,),
(13)
where So, $2.1 and S2. 9 a r e the respective Fourier components.
images of InP (100) at 300 kV. Both were calculated at the Scherzer conditions: (a) C s = 0.5 mm, A f = 320 A. and (b) C~ = 1 mm, objective aperture = 9.5 mrad, A f = 450. The inset is a diagonal the easily seen indium columns and the barely discernible phosphorus columns that displays the Shot noise was added to the image simulations corresponding to a dose of 7 × 104 electrons [10].
22
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects ADF
STEM
Signal
c r e a s e s no m o r e after such a short distance. Since the signal is the result of the i n t e g r a t e d s c a t t e r i n g from all the layers t h r o u g h which the p r o b e has
35 SA
-
30
25 20
15 10 !
o o
(a)
100
200 300 Thickness
Column
Partial
I
~
0.8
ffln
i
400 in A
0.4~ ' .
.
600
Cross-sections L
I
I
o,
_
, , "~" " " "
0.6
.
.
.
t~
"~-
.
0.2
~p /
0
./
/-"( 0
(b)
500
4 100
I
k
I
200 300 Thickness
400 in A
500
600
Fig. 6. Simulated ADF STEM signal and column partial cross-sections as function of specimen thickness. Incident probe modeled the Cornell STEM (E = 100 kV) with the high resolution pole piece (Cs = 1.3 mm) near Scherzer focus (a~p = 10.5 mrad and Af = 800 ~,).
In fig. 6 we show t h e results of this calculation. Fig. 6a shows t h e s i m u l a t e d a n n u l a r d a r k field signal at A, on the i n d i u m column, at B, on the p h o s p h o r u s atom, a n d at C, b e t w e e n the two i n d i u m a t o m s as a function o f thickness. In fig. 6b, t h e effects o f the i n s t r u m e n t a l w e i g h t i n g factors have b e e n r e m o v e d a n d t h e c o n t r i b u t i o n s o f the b a c k g r o u n d a n d c o l u m n s s e p a r a t e d . It is clearly n e c e s s a r y to include t h e b a c k g r o u n d t e r m which p r o b a b l y arises from the c h a n n e l i n g effect. Even at very small thicknesses, e l e c t r o n s a r e att r a c t e d to t h e n u c l e u s a n d s c a t t e r e d to large angles even for t h e p r o b e l o c a t e d b e t w e e n the atoms. This will c o n t r i b u t e to t h e b a c k g r o u n d . It is n o t e w o r t h y t h a t t h e scattering, f r o m the i n d i u m c o l u m n s a t u r a t e s after roughly 100 ~, a n d in-
~4
_, XgnSXL~o~L -
~- •
lc
Fig. 7. Simulated channeled intensity thickness series of indium phosphides (100) for two probe positions. Probe positioned on indium (left) and phosphorus (right). Thicknesses
are (a,b) 35 ,~, (c,d) 70 ~,, (e,f) 106 A,, and (g,h) 211 ~,. Incident probe parameters are the same as in fig. 6.
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects
traveled this result implies that no more scattering arises after 100 .~. To verify this conclusion the shape of the electron probe inside the crystal was provided by the simulation (see also [10]). Fig. 7 shows the result. When the probe is located over the phosphorus atom column it forms a channeling peak of 0.7 ,~ diameter at half maximum intensity and stays prominent through over 200 ,~ of the material. On the other hand when the probe is located over the indium column it rapidly grows to a channeling peak of 0.4 ,~ half width and then very quickly disappears. If the intensity in the channeling peak is integrated out to a radius of 0.5; ,~ for each peak for each thickness, the behavior with thickness closely matches the intensity behavior seen in fig. 5b. This is shown in fig. 8. We conclude that the channeling peak on the indium atom is completely scattered away by the point it has reached 100 ,~ into the crystal, an effect presumably reflecting anomalous absorption. No information can be obtained from regions deeper in the specimen using this channel. We see no evidence of the probe for a depth of several 100 ,~ but have not explored beyond that point. In fig. 9 we show the results of charmeling calculations for silicon and germanium. As in our earlier work [10] the probe in silicon forms a peak of 0.7 A half width and channels for a lon~g distance. In germanium the peak is now 0.4 A Integrated |
Channeled
"l
a
~°
g 2o
b
N ~o
jQ ~o
g
; y£oS . .,tx . o
"
%
~oO
Intensity i
v
O'ln
0.8
~o
23
0.6
g N
0.4 0.2 0 0
100
200 300 Thickness
400
500
600
in
Fig. 8. Match between simulated column partial cross-sections from fig. 6 and the integral through the specimen thickness of the intensity under the channeling peaks located on the indium atoms (lln) and the phosphorus atoms (Iv). Incident probe parameters are the same as fig. 6.
Fig. 9. Simulated channeled intensity thickness series of Ge(100) (left) and Si(100) (right) for probe centered on atomic column. Thicknesses are (a, b) 33 and 34 ,~., (c, d) 65 and 68 ,~, (e, f) 152 and 158 ,~, and (g, h) 239 and 249 ,~. Incident probe parameters are the same as in fig. 6.
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
24 250
i
200
,
5. Conclusions l
s
~
150
50
100 150 200 250 300 Thickness in A
50
Fig. 10. Integrated channeling peak intensities for Ge(100) and Si(100) as a function of thickness. Incident probe parameters are the same as in fig. 6.
diameter and again disappears after a short distance. In fig. 10 we show the calculations of the intensities on the silicon and germanium atoms as a function of thickness analogous to those shown in fig. 6 for indium and phosphorus. This result is consistent with the Bloch wave calculations of Pennycook and Jesson [32] in which the silicon signal steadily increased to large depth whereas the germanium signal saturated after a short distance as in the calculations reported here. The simulations suggest that the saturation is associated with disappearance of the probe after penetrating only a short distance into the sample. The phenomenon appears to be associated with anomalous absorption and should be widespread among higher-atomic-number elements. The effect identified here is very likely that pointed out by Cherns, Howie and Jacobs [33] in studies of X-ray emission of electron beams oriented at crystal zone axes. If allowance is made for different thickness normalizations in figs. 6 and 10, these results are consistent with the X-ray results suggesting that the latter also reflect disappearance of a channeling probe after only a short propagation distance. The Bloch wave picture thus needs to include the appropriate anomalous absorption [32,33] to describe the channeling peak appropriately. A background is also necessary for quantitative evaluation. Finally, we note that electron energy loss studies also need to take these effects into account.
Continued exploration of the conditions under which atomic resolution images are obtained in A D F STEM confirms the earlier conclusion [21] that the incoherent imaging model is a good description of the A D F images of zone-axis crystalline film specimens. In particular, this makes possible the correction of images for the different weights that the instrument applies to Fourier components of differing spatial frequencies. It is then possible to obtain the intensities associated with various atom columns in the sample provided a simple model can be assumed for the specimen scattering function. In the present instance, for example, it has provided assessments of the scattering of columns of atoms as a function of the column thickness as well as a background intensity. The result of the simulation study of column scattering is that while the electron beam propagates relatively large distances through lowatomic-number elements such as silicon or phosphorus this does not appear to be the case for high-atomic-number elements such as indium or germanium. In the latter elements, the channeled electron probe disappears after as little distance as 100 A. This effect probably arises from anomalous absorption. Thus annular dark-field images of such a column would appear to come only from the first 100 A of the sample. This effect clearly is very important for quantitative analysis of the images and is likely to be true also for electron energy loss studies and X-ray microanalysis recorded under these conditions. Efforts arc under way to seek experimental confirmation of these simulations. Simulations of images at 300 kV at 1.3 resolution suggest that images of low-Z elements will be weak. In specimens with nearby heavy elements, the low-Z elements can be identified only with care. Simultaneous bright-field phase contrast might be of great value.
Acknowledgements Special thanks to Dr. E.J. Kirkland for his implementation of the STEM data acquisition
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
system and to M. Thomas for upkeep and careful maintenance of the STEM. This research was supported by the Department of Energy (DEFG02-87ER45322). Calculations were carried out on the Cornell Materials Science Center computer facility. Funding for the purchase (DMR8314265) and operation of the STEM through the Materials Science Center (DMR-8818558) was provided by the National Science Foundation.
References [1] E. Zeitler and M.G.R. Thompson, Optik 31 (1970) 258. [2] E. Zeitler and H. Olsen, Phys. Rev. 162 (1967) 1439. [3] 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. [4] G. Black and E.H. Linfoot, Proc. Roy. Soc. (London) A 239 (1957) 522. [5] S.J. Pennycook and L.A. Boatner, Nature 336 (1988) 565. [6] S.J. Pennycook, Ultramicroscopy 30 (1989) 58. [7] D.H. Shin, E.J. Kirkland and J. Silcox, Appl. Phys. Len. 55 (1989) 2456. [8] P. Xu, E.J. Kirkland, J. Silcox and R. Keyse, Ultramicroscopy 32 (1990) 93. [9] E.J. Kirkland, R.F. Loane and J. Silcox, Ultramicroscopy 23 (1987) 77. [10] R.F. Loane, E.J. Kirkland and J. Silcox, Acta. Cryst. A 44 (1988) 912. [11] A. Howie, Phil. Mag. 14 (1966) 223. [12] K. Kambe, G. Lempfuhl and F. Fujimoto, Z. Naturforsch. A 29 (1974) 1034.
25
[13] B.F. Buxton, J.E. Loveluck and J.W. Steeds, Phil. Mag. A 38 (1978) 259. [14] J. Fertig and H. Rose, Optik 59 (1981) 407. [15] E.J. Kirkland, Ultramicroscopy 32 (1990) 349. [16] J.M. Cowley and A.F. Moodie, Acta. Cryst. 10 (1957) 609. [17] P. Goodman and A.F. Moodie, Acta. Cryst. A 3 (1974) 280. [18] K. Ishizuka and N. Uyeda, Acta. Cryst. A 33 (1977) 740. [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] E.J. Kirkland, 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. 13. [23] D.A. Kopf, Optik 59 (1981) 89. [24] C. Mory, M. Tence and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 381. [25] C. Mory and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 389. [26] C. Mory, C. Colliex and J.M. Cowley, Ultramicroscopy 21 (1987) 171. [27] O.L. Krivanek, Optik 45 (1976) 97. [28] K. Wong, E.J. Kirkland, P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 40 (1992) 139. [29] J.A. Lin and J.M. Cowley, Ultramicroscopy 19 (1986) 31. [30] M. Hammel, C. Colliex, C. Mory, H. Kohl and M. Rose, Ultramicroscopy 34 (1990) 257. [31] J.C.H. Spence and J.M. Cowley, Optik 50 (1978) 129. [32] S.J. Pennycook and D.E. Jesson, Ultramicroscopy 37 (1991) 14. [33] D. Cherns, A. Howie and M.H. Jacobs, Z. Naturforsch. 28a (1973) 565.
!1~!'I¢!1~i~1
Annular dark-field imaging: resolution and thickness effects Sean Hillyard, Russell F. L o a n e a n d J o h n Silcox School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA
Received 6 July 1992; at Editorial Office 11 August 1992
Simulations of annular dark-field zone-axis crystal images that explore the dependence of images with thickness are reported. Earlier work with digitally acquired experimental images and diffraction patterns provide excellent agreement with multi-slice-based, "frozen phonon" simulations and thus provide a foundation for this work. Additional experiments at 60 kV add support. Simulations of indium phosphide at 300 kV and 1.3 A resolution show evidence that the phosphorus image is weak. The results of the currently reported thickness simulations suggest that while the electron beam probe channels for long distances in low-Z-element materials such as silicon or phosphorus, the channeled probe travels only about 100 * in heavier elements such as germanium or indium. Such an effect has serious implications for quantitative analysis using annular dark-field imaging, electron energy loss spectroscopyand X-ray microanalysis.
I. Introduction To be of value, quantitative electron microscopy needs a well established theoretical base. Influential papers reaching towards this goal included critical work by Zeitler and Thompson [1] outlining the theory of bright-field phase-contrast images in scanning transmission electron microscopy (STEM) and the necessary estimates of elastic scattering cross-sections using the Moliere approximation by Zeitler and Olsen [2]. This work provided an excellent basis for discussion of the early single-atom images obtained by the Chicago group [3] as STEM was being developed and it provides the starting point for much later work. For annular dark-field (ADF) images the key idea is the incoherent imaging model [3,4]. Recently, high-resolution A D F images of zone-axisoriented crystals that demonstrate Z-contrast [5,6] and high resolution [7,8] have been reported. These are consistent with the earlier ideas and have attracted attention since they are simpler to interpret than bright-field phase-contrast images with no contrast reversals with focal changes. A picture of the process has been developed that suggests that the electron probe forms a channel-
ing peak on the atom columns and that the A D F image results from scattering by the atoms from this channeled peak [6,9,10]. Details of this picture are still sketchy in spite of a long series of papers on channeling [11-14]. In this paper we follow up earlier work exploring the validity of the incoherent model for zone-axis crystal imaging. STEM is intrinsically quantitative. The images are measured by linear detectors, are ideally suited for serial data recording due to the scanning mode and lose the quantitative element only at the display stage. If the intensities are recorded directly into a digital memory prior to display then they can be stored for extensive off-line analysis after the experimental session as well as being available for examination during operation. Image analysis can be pursued at a much more demanding quantitative level than with photographic recording. In previous work we have used a D E C VAX 3200 workstation interfaced by Kirkland [15] to a VG HB501 A STEM. This permits the acquisition of simultaneous dark-field and energy-filtered bright field images in perfect registration, and of energy-filtered diffraction patterns. In parallel
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
S. Hillyardet al. /Annular dark-field imaging: r6solutionand thicknesseffects with this, we have used multislice-based [16-18] simulations to provide model predictions of the intensity patterns [9,10], introduced a "frozen phonon" model to describe thermal diffuse scattering and demonstrated excellent agreement with experimental studies of silicon [19,20]. This was followed by a combined simulation and experimental study of imaging in a particular system, (100) indium phosphide [21]. In this case, variation with focus was under scrutiny and both the simulation and the experiment were found to behave in good agreement with the simple incoherent imaging model. In this paper, we extend this work in two areas. Questions concerning resolution are addressed by both simulation and experiment. Operation of our system at lower voltages offers the opportunity to vary electron-optical conditions permitting use of somewhat different angles and energies. In addition, the imminent availability of instruments at 300 kV and 1.3 ,~ resolution raised the question of whether such an instrument could image phosphorus in indium phosphide in distinction with the results reported at 100 kV [21]. The second area is the variation of STEM images with thickness. Simulations show interesting and very significant changes with thickness and atomic number so that, given the validity of the simulations as checked by the earlier experiments [19,20,21,22], it was considered advisable to include them here. Experimental verification of these conclusions is being sought. The paper begins by reviewing the background of the incoherent imaging model and the results established by the earlier work. This gives the foundation for the results reported here. Illustrations using experiments at 60 kV and simulations at 300 kV are presented. Finally, thickness effects in elemental silicon and germanium and in compound (100) indium phosphide are discussed with the aid of simulations. Both image intensities and the form of the channeling probe inside the model crystal are explored with a somewhat disturbing conclusion. The channeling peak inside the sample can be expected to vary significantly with depth and, for even moderate-atomic-number elements, to disappear at thicknesses as small as
100 h.
15
2. Background The incoherent imaging model [3,4,23] describes the image intensity field I(r) as the convolution of an instrumental probe function, h(r), with a specimen object function, s(r) where r is a two-dimensional positional vector perpendicular to the optic axis representing the probe position as follows: I(r) =h(r) ®s(r).
(1)
The probe function is determined by the coefficient of spherical aberration, Cs, and the lens defocus, Af. It is the square of the incident wavefunction ~O(r) which itself is the Fourier transform of the aperture function, A(k), multiplied by the exponential of the phase factor introduced by the lens (see ref. [21] for a fuller discussion). Thus gy(k) = A ( k ) exp[ix(k)], A(k)
=1 0
for k
x ( k) = rrAk2(1/2CsA2k 2 - A f ) ,
(2) (3) (4)
where A is the electron wavelength and k is a two-dimensional vector perpendicular to the optic axis. This model provides a means of separating specimen characteristics from microscope-introduced artifacts. Earlier work [24-26] has used this model to identify optimum single focal settings for differing purposes. In this work [21] the thrust has been to exploit focal series, since rather more information can be extracted. Determination of C S and calibration of A f for conventional transmission electron microscopy has been possible for a long time [27] through the use of power spectra of bright-field phase-contrast images of thin carbon films. A parallel treatment for STEM has only recently been reported [28]. The critical step appears to have been digital image acquisition [15] so that the full intensity range was recorded with a sufficiently small collector aperture. Earlier work [29] exploited Ronchigrams recorded from large spacing samples to determine Cs and Af while recent obser-
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
16
2.!
B A
C
Fig. 1. Schematic of an indium phosphide (100) crystal. Columns of indium atoms are represented by gray circles, columns of phosphorus by white. Fringe maxima are represented by gray lines. After Loane et al. [16].
vations [30] of amorphous carbon power spectra used too large a collector aperture. Because the large data set represented by an extensive focal series is recorded directly by the digital image acquisition system, analysis is straightforward and is not the formidable burden it is under the common photographic arrangement. For our first experiments [21], the (100) zone axis of indium phosphide was selected. This is a simple object in which the indium atoms line up in a square array with 2.9 .4, edges and the phosphorus atoms are at the square centers, i.e., 2.1 A from the indium atoms (see fig. 1). The sample therefore contains two atom species of substantially differing atomic number and two fundamental spatial frequencies both of which are within the resolution limits of our instrument. The results of these experiments demonstrate that although imaging in annular dark field is simpler than bright field in that no contrast reversals were evident, fringe contrast does vary with defocus, exhibiting oscillations in intensity with maxima at differing defoci for different spacings. The phosphorus atoms could not be resolved experimentally even though a smaller spacing had been resolved in other work [7,8]. This results from residual scattering of the heavy indium atom nearby which swamps the weak phosphorus signal. Thus, background scattering from other atoms in the sample can play a dominant role in Z-contrast imaging and must be considered, particularly near the resolution limits. These results are fully consistent with the incoherent imaging
model with the addition of a few extra experimental complications. The latter include a little astigmatism, specimen tilt and effects probably due to a finite source size. The same results emerge from an extensive simulation of the same images using computer codes developed for assessment of crystalline images [9,10]. The codes had earlier been tested for the ability to predict diffraction pattern intensities over a range of four orders of magnitude in intensity and with the addition of a "frozen phonon" Einstein model were able to match experimental intensities with great success [19,20]. In earlier work [22], correlation of the thicknesses of experimental specimens determined by matching diffraction patterns with model calculations and thicknesses determined by plasmon intensity measurements also give confidence that the simulations give good if not excellent predictions of the experimental observations. In the present case, the comparison was attempted not only with the experiment but also with the incoherent imaging mode. The result was that for this case at least, excellent agreement was found between the simulation and the model. This is not a trivial result. Electron scattering in specimens of the thickness considered is a very non-linear process and a clean proof of such a result seems highly unlikely. The results are also in good agreement with the experiments discussed above, but in this case since neither the simulation nor the model includes the experimental factors of astigmatism, tilt or finite source size, the same minor discrepancies with the data appear. As well as demonstrating that the simple model works reasonably well, this work implies that there are some complications arising from the choice of defocus since differing fringe spacings are at maximum contrast at different settings. Thus, it is by no means obvious that Scherzer settings are necessarily optimal as noted earlier [24]. A further conclusion is that care needs to be used to eliminate the differing weights placed on each spacing by the lens settings if a specimen object function is to be extracted from the image for quantitative work. In the following sections we discuss first the issue of resolution and expand upon these conclusions. Then we report some simulation re-
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
17
where T(q - k 1) and T(q - k 2) are the necessary complex scattering amplitudes and contain the requisite sample spatial information. The scattered intensity then becomes
Final Lens
14~(r, q) L2
Specimen
= IO(ka)lZlT(q-k~)[ 2 + IO(kz) l Z l T ( q - k 2 ) l ADF detector
+ 2 Re{0(k,)0*(k2) T(q-kl) × exp[i( k
Optic Axis
Fig. 2. Ray diagram showing incident beams of wave vector kl, k 2 scattering to wave vector q (eqs. (6) and (7)).
suits that carry the potential for substantial concern for simple interpretative schemes.
3. Spatial resolution A STEM A D F image provides intensity maps
2
1 -
T*(q-k2)
k2) ° r]},
(7)
where the notation * denotes complex conjugate. It is apparent that the first two terms contribute to the scattered intensity independent of the probe position and thus to a background intensity when integrated over the detector. It is the third term that depends on r and thus to variations in image intensity. Furthermore, at large scattering angles, so that the scattering is essentially kinematic, this term contributes to a Fourier component of the annular dark-field image with wave vector p = k 1 - k 2. In generalizing this discussion, we first consider the probe as a sum of incident waves so that the scattered intensity can be written
I(r) as a function of probe position r. The A D F detector integrates intensities ~b(r, q) scattered by the specimen onto a detector D(q) so that:
14~(r, q ) l 2
= (E~(kn)
I(r) = fD(q) 14~(r, q) L2 dq.
(5)
The dependence of 4~(r, q) on r is explicitly included here and fig. 2, which is a simple schematic of the scattering process, emphasizes that the position dependence of the scattered intensity arises from the coherent interference of the scattering of any two plane waves in the incident beam. Thus consider the two incident plane waves gt(kl) exp ik~'r and tb(kz) eXp ikz'r each of which is scattered by the specimen to the wave vector q. The resulting component of the scattered wave function th(r, q) is then ~(r,
q ) -----~O(kl) e ik,'r T ( q - k l ) -t- ~b(k2) e ik2"r T ( q - k 2 )
,
(6)
eik"r r ( q - k n ) )
n
= ~_. [ O ( k . ) 1 2 l T ( q - k . ) [ n=m + 2Re(.~mg,(kn)0*(k,.)
×T*(q-km)
2 T(q-k.)
exp[i(k n - k i n ) "r]},
(8)
where the wave vectors k n, k m are restricted to lie within the objective aperture (see eq. (3)). The interference term, i.e., the second sum, provides the position-dependent image intensity contributions. Wave vectors p = k n - k m up to the diameter of the objective aperture can be imaged. In
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
18
integral form, eqs. (5) and (8) can be put in the form
I(r) = f H( kl, p) S( k~, p) dk, exp(ip . r ) d p , (9) where
H(kl, p) = A ( k ) A*(k I - p ) e x p [ i ( x ( k )
-p))], s(k,, p) = fD(k) T(k-k,) T*(k-kl +v)dk, where H(k 1, p) is the contrast transfer function (CTF). In this formulation it appears as the weighting given by the instrument to Fourier components in the annular dark-field image within the integral in eq. (9). It is also the Fourier transform of the probe function h(r) of eq. (1) which is an approximate form of eq. (9) (for discussion see ref. [21]). Experimental evidence for elements of this discussion is provided by fig. 3. This provides two images of (100) indium phosphide recorded at 60 kV using the V G HBS01A STEM at Cornell under similar conditions as those used in our earlier work. In fig. 3a, the objective aperture was 40 izm corresponding approximately to Scherzer conditions for this voltage and lens (C s = 1.3 mm at 100 kV was estimated to correspond to C S = 1.0 mm at 60 kV). The power spectrum of this image is shown in fig. 3b and the peaks corresponding to the 2.9 ,~ spacing are clearly seen. The peaks corresponding to the 2.1 A spacing are absent and are not imaged with this aperture. This latter spacing is approximately equal to the diameter of the objective aperture as illustrated by the circles drawn around each spot in the power spectrum. The circles barely touch for the 2.1 A spots but significant overlap occurs for the 2.9 A spots. The image therefore contains just two superimposed sets of 2.9 A fringes. In contrast, fig. 3c shows an equivalent micrograph recorded with a 50 Izm objective aperture with the corresponding power spectrum in fig. 3d. The 2.1 A spacing appears in this image, consiso
tent with the fact that the 2.1 ,~ spatial frequency is now less than the objective aperture diameter as seen by the circles drawn around the spots. Significant overlap is observed for the 2.1 fringes unlike fig. 3b. In this case several overlaps occur and the image contains a small amount of three-beam interference. Thus, not surprisingly, a critical issue determining the resolution is the size of the objective aperture. The settings can be extended beyond the nominal limits somewhat by trading visibility against the resolution. Naturally, this cannot be pursued indefinitely and unless the lens is well characterized it may carry some interpretative hazards. The present experiments do give some indication of the elasticity in the limits and it may be worth noting that a number of the dark-field images in the literature are not recorded at Scherzer conditions. In fig. 4, we show contrast transfer functions (CTF) for annular dark-field relevant to the parameters of a modern STEM as discussed extensively in an earlier paper [21]. At Scherzer focus, the CTF is roughly triangular and weights each component differently. Unless some scheme is introduced to correct for these differences, then the intensities in the image will be distorted. Most high-resolution annular dark-field zone-axis images of crystalline materials so far published with STEM are simple images with only one main Fourier component in them. In this case the non-linearities introduced by the instrument are largely irrelevant and no serious harm has been done. Studies of defects or compounds (such as the indium phosphide) need to consider the problem. Ideally the operator will choose conditions that might be a " b e t t e r " i.e., more faithful, representation of the object. An ideal CTF would have a flat characteristic over much of the spatial frequency range and a somewhat more satisfactory CTF might be obtained, albeit at a cost in signal level. However, there are advantages in recording a focal series with a well characterized lens and appropriately combining the results corrected for lens distortion. The curve in fig. 4a was calculated with parameters estimated from 100 kV experience and is thought to represent the conditions (close to Scherzer) under which fig. 3a was recorded. An
S. Hillyard et aL / Annular dark-field imaging: resolution and thickness effects
estimate of the CTF corresponding to the micrograph of fig. 3c is also shown in fig. 4b. In this case, the contrast is lower but enough signal was apparently obtained to provide an image with more detail than the first by extending to higher
19
spatial frequencies. Nevertheless, the structure in this CTF indicates that in general this must only be carried out with well characterized lenses. The limiting factor (as always) will be the signal-tonoise ratio.
Fig. 3. Experimental A D F STEM images of indium phosphide (100) taken with the high-resolution pole piece at 60 kV (C s = 1.0 mm), and corresponding power spectra. (a) A 12 mrad objective aperture and an estimated defocus of 800 A. (c) A 15 mrad objective aperture and a defocus of 1600 A. (b, d) Corresponding power spectra. The circles in the power spectra are equal to the size of the objective apertures used.
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects
20
The significance of the objective lens diameter in these images arises from the way in which the scattered electrons carry the spatial information. As in eq. (8) the intensity is a sum of terms arising from one plane wave only (and carry no spatial information) and those arising from the product of one plane wave component interfering with another that do carry the spatial information. This mechanism has been discussed by Spence and Cowley [31] for Bragg reflection in particular but it applies to any scattering process. The maximum spatial frequency that can be carried by these waves is given by the diameter of the objective aperture. It appears in the CTF as the point at which the CTF finally goes to zero. A consequence of this unambiguous point is that the probe in real space never can go exactly to zero although it is equally surely not a Gaussian. Given the imminent availability of 300 kV STEM instruments and the success the simulations have enjoyed in modeling the diffraction patterns and images so far, two simulations have been carried out to see if the phosphorus atoms would be visible with these machines unlike the 100 kV case. One simulation models the 1 mm C s objective lens instrument currently available while the other anticipates one of 0.5 mm C S. In this case, six Fourier components will appear so that six different object points are necessary in the simulation [21]. Two images simulated at Scherzer conditions are shown in fig. 5. Imaging phosphorus is still not that easy. The signal is weak relative to the indium atom and if it was not already known to be there, it could easily be missed. This large difference demonstrates why the low-Z element can be swamped by neighboring high-Z elements. The invention of methods to explore effectively for such elements may be needed and the simultaneous use of bright-field phase contrast to look for light elements is likely to be necessary.
4.
T h i c k n e s s
effects
Following the procedures outlined by Loane et al. [21], the scattering intensity as a function of thickness at different points in the image can be
recovered. Given the size of the objective aperture the image contains only three independent Fourier components. Simulations were carried out at five points in the unit cell. The results confirm that only three Fourier components are necessary and provide these as a function of thickness. It remains to relate these values to the scattered intensities at each of the atomic columns and the background. As given in eq. (1), s(r) reflects the propagation of the incident electron wave packet h(r) through the sample. The packet changes shape and magnitude as it passes through the sample depending on the initial position [10]. However, the sample is periodic in directions perpendicular
1
, l l l l r , , l l r ~ l
0.8 ~ 0.6
%p=12mrad Af=800A ~ E=60kV
0.4
2.9 A ~
0.2
,
2. IA
IILIIIL~!III
0
0.10.20.30.40.50.60.7 Spatial F r e q u e n c y ( A ) ' t
(a)
1
°
Cs=1.0ram
i
,
,
,
,
,
,
i
,
,
,
0.8
aap=l5mrad M=I600A
0.6
E=60kV
,
Cs=l,0mm
0.4
0.2 0
(b)
I
0
i
I
I
1
I
~
I
I
I~
0.1 0.2 0.3 0.4 0.5 0.6 Spatial F r e q u e n c y ( A ) "l
I
0.7
Fig. 4. STEM A D F contrast transfer functions for the high resolution pole piece at 60 kV (C s = 1 mm). (a) For an objective aperture of 12 mrad and a defocus of 800 A, Close to the Scherzer conditions (O~ap= 12 mrad and A f = 700). (b) For an aperture of 15 mrad and a defocus of 1600 A. The 2.9 ,~ and 2.1 ,~ spacings of InP (100) are marked. The annular
aperture size was 60 to 370 mrads.
S. Hillyard et aL /Annular dark-field imaging: resolution and thickness effects
21
to the zone axis so that s(r) is periodic. Accordingly,
tions situated at the columns and multiplied by column partial cross sections, as in eq. (12):
8(r) = ~ S ( g ) e ig'r,
S(r)
(10)
g
°'° +
= A7
~(r-r,n
°tin rln
) +crpy'~(r-rp). rp
(12) where g is a 2D reciprocal lattice vector and, from eq. (1), I(g) =H(g)
S(g),
(11)
where l(g) is the intensity in the image corresponding to Fourier component g and H(g) is the weighting factor of the CTF for that component. Given both I(g) and H(g), then S(g) can be determined. At each thickness a focal series (about thirty images) simulation permits the retrieval of the two fringe amplitudes (i.e., $2.1 and $2.9) and the background S O using a least-squares routine with eq. (11). To proceed further, we assume that the specimen object function consists of delta func-
Fig. 5. Simulated A D F STEM objective aperture = 11 mrad, linescan passing through both logarithm of the intensity.
Here, A c is the area of the primitive unit cell, ~r0, ~ln, ~rp are partial cross sections for the background, indium column and phosphorus columns respectively, and r~,, r e are the lattice positions of the indium and phosphorus atoms. The Fourier transform of this provides the connection with the fringe amplitudes recovered from the focal series calculations, i.e., O'ln = A c ( S 2 . 9 + S 2 . 1 ) / 2 , ~rp = A t ( S 2 . 1
- $2.~)/2 ,
<, = A c ( S 0 - $2.,),
(13)
where So, $2.1 and S2. 9 a r e the respective Fourier components.
images of InP (100) at 300 kV. Both were calculated at the Scherzer conditions: (a) C s = 0.5 mm, A f = 320 A. and (b) C~ = 1 mm, objective aperture = 9.5 mrad, A f = 450. The inset is a diagonal the easily seen indium columns and the barely discernible phosphorus columns that displays the Shot noise was added to the image simulations corresponding to a dose of 7 × 104 electrons [10].
22
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects ADF
STEM
Signal
c r e a s e s no m o r e after such a short distance. Since the signal is the result of the i n t e g r a t e d s c a t t e r i n g from all the layers t h r o u g h which the p r o b e has
35 SA
-
30
25 20
15 10 !
o o
(a)
100
200 300 Thickness
Column
Partial
I
~
0.8
ffln
i
400 in A
0.4~ ' .
.
600
Cross-sections L
I
I
o,
_
, , "~" " " "
0.6
.
.
.
t~
"~-
.
0.2
~p /
0
./
/-"( 0
(b)
500
4 100
I
k
I
200 300 Thickness
400 in A
500
600
Fig. 6. Simulated ADF STEM signal and column partial cross-sections as function of specimen thickness. Incident probe modeled the Cornell STEM (E = 100 kV) with the high resolution pole piece (Cs = 1.3 mm) near Scherzer focus (a~p = 10.5 mrad and Af = 800 ~,).
In fig. 6 we show t h e results of this calculation. Fig. 6a shows t h e s i m u l a t e d a n n u l a r d a r k field signal at A, on the i n d i u m column, at B, on the p h o s p h o r u s atom, a n d at C, b e t w e e n the two i n d i u m a t o m s as a function o f thickness. In fig. 6b, t h e effects o f the i n s t r u m e n t a l w e i g h t i n g factors have b e e n r e m o v e d a n d t h e c o n t r i b u t i o n s o f the b a c k g r o u n d a n d c o l u m n s s e p a r a t e d . It is clearly n e c e s s a r y to include t h e b a c k g r o u n d t e r m which p r o b a b l y arises from the c h a n n e l i n g effect. Even at very small thicknesses, e l e c t r o n s a r e att r a c t e d to t h e n u c l e u s a n d s c a t t e r e d to large angles even for t h e p r o b e l o c a t e d b e t w e e n the atoms. This will c o n t r i b u t e to t h e b a c k g r o u n d . It is n o t e w o r t h y t h a t t h e scattering, f r o m the i n d i u m c o l u m n s a t u r a t e s after roughly 100 ~, a n d in-
~4
_, XgnSXL~o~L -
~- •
lc
Fig. 7. Simulated channeled intensity thickness series of indium phosphides (100) for two probe positions. Probe positioned on indium (left) and phosphorus (right). Thicknesses
are (a,b) 35 ,~, (c,d) 70 ~,, (e,f) 106 A,, and (g,h) 211 ~,. Incident probe parameters are the same as in fig. 6.
S. Hillyard et al. / Annular dark-field imaging." resolution and thickness effects
traveled this result implies that no more scattering arises after 100 .~. To verify this conclusion the shape of the electron probe inside the crystal was provided by the simulation (see also [10]). Fig. 7 shows the result. When the probe is located over the phosphorus atom column it forms a channeling peak of 0.7 ,~ diameter at half maximum intensity and stays prominent through over 200 ,~ of the material. On the other hand when the probe is located over the indium column it rapidly grows to a channeling peak of 0.4 ,~ half width and then very quickly disappears. If the intensity in the channeling peak is integrated out to a radius of 0.5; ,~ for each peak for each thickness, the behavior with thickness closely matches the intensity behavior seen in fig. 5b. This is shown in fig. 8. We conclude that the channeling peak on the indium atom is completely scattered away by the point it has reached 100 ,~ into the crystal, an effect presumably reflecting anomalous absorption. No information can be obtained from regions deeper in the specimen using this channel. We see no evidence of the probe for a depth of several 100 ,~ but have not explored beyond that point. In fig. 9 we show the results of charmeling calculations for silicon and germanium. As in our earlier work [10] the probe in silicon forms a peak of 0.7 A half width and channels for a lon~g distance. In germanium the peak is now 0.4 A Integrated |
Channeled
"l
a
~°
g 2o
b
N ~o
jQ ~o
g
; y£oS . .,tx . o
"
%
~oO
Intensity i
v
O'ln
0.8
~o
23
0.6
g N
0.4 0.2 0 0
100
200 300 Thickness
400
500
600
in
Fig. 8. Match between simulated column partial cross-sections from fig. 6 and the integral through the specimen thickness of the intensity under the channeling peaks located on the indium atoms (lln) and the phosphorus atoms (Iv). Incident probe parameters are the same as fig. 6.
Fig. 9. Simulated channeled intensity thickness series of Ge(100) (left) and Si(100) (right) for probe centered on atomic column. Thicknesses are (a, b) 33 and 34 ,~., (c, d) 65 and 68 ,~, (e, f) 152 and 158 ,~, and (g, h) 239 and 249 ,~. Incident probe parameters are the same as in fig. 6.
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
24 250
i
200
,
5. Conclusions l
s
~
150
50
100 150 200 250 300 Thickness in A
50
Fig. 10. Integrated channeling peak intensities for Ge(100) and Si(100) as a function of thickness. Incident probe parameters are the same as in fig. 6.
diameter and again disappears after a short distance. In fig. 10 we show the calculations of the intensities on the silicon and germanium atoms as a function of thickness analogous to those shown in fig. 6 for indium and phosphorus. This result is consistent with the Bloch wave calculations of Pennycook and Jesson [32] in which the silicon signal steadily increased to large depth whereas the germanium signal saturated after a short distance as in the calculations reported here. The simulations suggest that the saturation is associated with disappearance of the probe after penetrating only a short distance into the sample. The phenomenon appears to be associated with anomalous absorption and should be widespread among higher-atomic-number elements. The effect identified here is very likely that pointed out by Cherns, Howie and Jacobs [33] in studies of X-ray emission of electron beams oriented at crystal zone axes. If allowance is made for different thickness normalizations in figs. 6 and 10, these results are consistent with the X-ray results suggesting that the latter also reflect disappearance of a channeling probe after only a short propagation distance. The Bloch wave picture thus needs to include the appropriate anomalous absorption [32,33] to describe the channeling peak appropriately. A background is also necessary for quantitative evaluation. Finally, we note that electron energy loss studies also need to take these effects into account.
Continued exploration of the conditions under which atomic resolution images are obtained in A D F STEM confirms the earlier conclusion [21] that the incoherent imaging model is a good description of the A D F images of zone-axis crystalline film specimens. In particular, this makes possible the correction of images for the different weights that the instrument applies to Fourier components of differing spatial frequencies. It is then possible to obtain the intensities associated with various atom columns in the sample provided a simple model can be assumed for the specimen scattering function. In the present instance, for example, it has provided assessments of the scattering of columns of atoms as a function of the column thickness as well as a background intensity. The result of the simulation study of column scattering is that while the electron beam propagates relatively large distances through lowatomic-number elements such as silicon or phosphorus this does not appear to be the case for high-atomic-number elements such as indium or germanium. In the latter elements, the channeled electron probe disappears after as little distance as 100 A. This effect probably arises from anomalous absorption. Thus annular dark-field images of such a column would appear to come only from the first 100 A of the sample. This effect clearly is very important for quantitative analysis of the images and is likely to be true also for electron energy loss studies and X-ray microanalysis recorded under these conditions. Efforts arc under way to seek experimental confirmation of these simulations. Simulations of images at 300 kV at 1.3 resolution suggest that images of low-Z elements will be weak. In specimens with nearby heavy elements, the low-Z elements can be identified only with care. Simultaneous bright-field phase contrast might be of great value.
Acknowledgements Special thanks to Dr. E.J. Kirkland for his implementation of the STEM data acquisition
S. Hillyard et al. / Annular dark-field imaging: resolution and thickness effects
system and to M. Thomas for upkeep and careful maintenance of the STEM. This research was supported by the Department of Energy (DEFG02-87ER45322). Calculations were carried out on the Cornell Materials Science Center computer facility. Funding for the purchase (DMR8314265) and operation of the STEM through the Materials Science Center (DMR-8818558) was provided by the National Science Foundation.
References [1] E. Zeitler and M.G.R. Thompson, Optik 31 (1970) 258. [2] E. Zeitler and H. Olsen, Phys. Rev. 162 (1967) 1439. [3] 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. [4] G. Black and E.H. Linfoot, Proc. Roy. Soc. (London) A 239 (1957) 522. [5] S.J. Pennycook and L.A. Boatner, Nature 336 (1988) 565. [6] S.J. Pennycook, Ultramicroscopy 30 (1989) 58. [7] D.H. Shin, E.J. Kirkland and J. Silcox, Appl. Phys. Len. 55 (1989) 2456. [8] P. Xu, E.J. Kirkland, J. Silcox and R. Keyse, Ultramicroscopy 32 (1990) 93. [9] E.J. Kirkland, R.F. Loane and J. Silcox, Ultramicroscopy 23 (1987) 77. [10] R.F. Loane, E.J. Kirkland and J. Silcox, Acta. Cryst. A 44 (1988) 912. [11] A. Howie, Phil. Mag. 14 (1966) 223. [12] K. Kambe, G. Lempfuhl and F. Fujimoto, Z. Naturforsch. A 29 (1974) 1034.
25
[13] B.F. Buxton, J.E. Loveluck and J.W. Steeds, Phil. Mag. A 38 (1978) 259. [14] J. Fertig and H. Rose, Optik 59 (1981) 407. [15] E.J. Kirkland, Ultramicroscopy 32 (1990) 349. [16] J.M. Cowley and A.F. Moodie, Acta. Cryst. 10 (1957) 609. [17] P. Goodman and A.F. Moodie, Acta. Cryst. A 3 (1974) 280. [18] K. Ishizuka and N. Uyeda, Acta. Cryst. A 33 (1977) 740. [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] E.J. Kirkland, 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. 13. [23] D.A. Kopf, Optik 59 (1981) 89. [24] C. Mory, M. Tence and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 381. [25] C. Mory and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 389. [26] C. Mory, C. Colliex and J.M. Cowley, Ultramicroscopy 21 (1987) 171. [27] O.L. Krivanek, Optik 45 (1976) 97. [28] K. Wong, E.J. Kirkland, P. Xu, R.F. Loane and J. Silcox, Ultramicroscopy 40 (1992) 139. [29] J.A. Lin and J.M. Cowley, Ultramicroscopy 19 (1986) 31. [30] M. Hammel, C. Colliex, C. Mory, H. Kohl and M. Rose, Ultramicroscopy 34 (1990) 257. [31] J.C.H. Spence and J.M. Cowley, Optik 50 (1978) 129. [32] S.J. Pennycook and D.E. Jesson, Ultramicroscopy 37 (1991) 14. [33] D. Cherns, A. Howie and M.H. Jacobs, Z. Naturforsch. 28a (1973) 565.