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Pseudo-aberration free focus condition for atomic resolution electron microscope images
~
Pergamon PII: S0968-4328(98)00008-0
Micron Vol. 29, No. 2/3, pp. 113-121, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0968-4328/98 $19.00+0.00
Pseudo-aberration Free Focus Condition for Atomic Resolution Electron Microscope Images* H. HASHIMOTO,? H. ENDOH,5/M. HASHIMOTO,§ Z. P. LUO? and M. F. SONG?
?Okayama University of Science, Okayama, Japan ?iKyoto Institute of Technology, Sakyoku, Kyoto, Japan §Osaka Setsunan University, Neyagawa, Osaka, Japan (Received 10 October 1997;accepted9 January 1998)
Abstract--Spherical Aberration Free Focus (AFF)conditions for taking atomic resolution electron microscopeimages of single crystals and pseudo-AberrationFree Focus(~b-AFF)conditionsfor takingatomicimagesof morethan two adjoiningsinglecrystalsare discussedby using diagrams of Cs (spherical aberration coefficient)and Af(defocus). Calculated image contrast in those conditions is shownfor three individual crystalsof MgO,MgA1204,A1203and their adjoiningcrystalblocks. It is shownthat at least two kinds of Braggreflectedwaves can be in the AFFconditionin eachcase, and thusthe size of the aperturefor limitingthe wavesbecomesimportantin somecases. Cs can be changedby changingthe positionof the specimenalongthe objectivelens axis but highlycoherentilluminating wavesare needed for large Cs and Af © 1998Elsevier ScienceLtd. All rights reserved. Key words: AberrationFree Focus,pseudo-AberrationFree Focus,electronmicroscopeimagesof atoms, atomresolutionimagesof crystals.
value b = Rc~/ax 3/4, where R = 0.65~0.7, is generally used as the conventional resolution limit. With the improvement of the resolution of conventional It was shown (Hashimoto et al., 1977), however, that for a transmission electron microscopes (TEM) and the develop- thin crystalline specimen and an axial illuminating beam, ment of theories of image formation, many atomic structure the Bragg reflected waves sample the CTF in a discrete set of images of materials have been used for the better under- directions and, by a suitable choice of defocus and aberration, standing of the atomic arrangement in specimens. The the relative phase can be simply related to those for a zeroobserved electron microscope images with atomic resolu- aberration lens. The demonstration was carried out using Au tion, however, change with small changes in focus of the crystal in [100] orientation, in which dynamic scattering of objective lens. The change of image contrast by the defo- electrons in the crystal was taken into account. Since the cusing of the objective lens is well described by the phase phase shift of 7r/2 of diffracted waves (which is predominant contrast transfer function (CTF) (Scherzer, 1949). The for WPO) was taking place in the process of dynamic scatterphase of the waves leaving the specimen with some scatter- ing of electrons in reasonably thick crystals, the real part of ing angle is shifted by the spherical aberration of imaging CTF was used for the interpretation of image contrast. The lens but it can be compensated by underfocusing. Thus, the intensity distribution of electron waves at the bottom surface phase angle of the CTF is given by the difference of the was assumed to be projected to the image plane. terms of spherical aberration, 71-/2~k(Csot4),and defocus, 7r/ The method shown above is called the spherical Aberra2X(2hfc~2), where X, Cs, o~and h f are the wavelength, sphe- tion Free Focus (AFF) condition and it was shown (Hashirical aberration coefficient, electron scattering angle and moto et al., 1978-79) that this AFF condition is realized by defocusing. a defocusing of CsX2/2d 2 +- m(2d/X), where d is the spacing Scherzer (1949) proposed the optimum defocus value of of the appropriate lattice plane and 2dZ/X is equivalent to the h f = (CsX) 1/2 for weak phase objects (WPO) in which the period of Fourier images (Cowley and Moodie, 1957). For CTF can be approximated by its imaginary part and a wide simple crystal structure, such as [100], [110] and [111] proflat region appears close to the optical axis (small scattering jections of f.c.c, and b.c.c, structures and [001] projection of angle) in the oscillating CTF. The first zero of the oscillating h.c.p, lattice structure, it was shown (Hashimoto et al., CTF appears at the scattering angle of C~m= 1.41(Af/C~)112, 1978-79; Endoh, 1984) that there are many combinations Eliminating the oscillating region of the CTF (for scattered of Cs and Af for realizing the AFF-condition at 100 kV. It waves larger than am) by the objective aperture, the resolu- was thus shown that for Au in [110] orientation, when Cs tion limit becomes 6 = C~/4X3/4, which is now called con- and Af are 1.09mm and 134.8 nm, or 0.545 mm and ventionally the theoretical resolution limit. By adopting the 157.3 nm, respectively, 37 diffracted waves up to (440) allowance of the defocus which gives a nearly flat region in are in the same phase and sample the oscillating transfer the CTF, a higher resolution limit is obtainable and now the function as unity. Using the AFF condition, atomic structure images of Si *In honor of ProfessorM. J. Whelan. were successfully recorded (Izui et al., 1978-79; Izui et al., 113 INTRODUCTION
H. Hashimoto et al.
114
- 200
-1 O0
1oo
0
Af ('rim) 200
Fig. 1. C s-Af diagram giving AFF conditionfor {200} is given by the m l = 0 , ~ 1, --+2, "", line system and {220} by the m2 = 0, -+ 1, -+ 2, -" line system, for a MgO crystal in [001] projection. The crossing points of these two line systems (indicated by + ) are AFF conditions for both {200} and {220 }. 1987; Hutchison and Waddington, 1988). A similar approach, this time assuming kinematical behavior of thin crystals, has been used to characterize a vacancy-ordered superlattice structure in Ga2Te3 (Hutchison et al., 1994). Recently, many atomic resolution images of crystals on both sides of interfaces have been produced (Hutchison et al., 1986; Ichinose, 1987; Ko and Sinclair, 1994; Chen et al., 1994a, b. However, the image contrast of both crystals does not always coincide with the intensity distributions at the bottom surface of both respective crystals. Analyzing the Fourier image construction and considering the partial coherence of imaging electrons (Ishizuka, 1980), Hashimoto et al. (1996) proposed the best focus for obtaining the atomic resolution images by a 400-kV microscope with the lens of C~ = 0.1 nun, corresponding to the intensity distribution at the bottom surfaces of three crystals. They discussed this method by referring to the photograph taken by Hesse et al. (1994) of the crystal containing two interfaces and three crystal lattices of MgO/ MgA1204/A1203 and named this focus as pseudo-Aberration Free Focus (~b-AFF) condition. However, in this discussion, C~ and X were fixed and thus it was not exact but close to the ~b-AFF condition. In the present paper, the principle and the method to obtain the accurate AFF and ~b-AFF conditions by using a C ~ - A f diagram are treated for the case of the MgO/ MgA1204 / A1203 crystal system.
. . . .
F
L~
Gaussian image plane
PRINCIPLE The principle of AFF and pseudo-AFF conditions are described here systematically by citing the results published already (Hashimoto et al., 1978-79, 1996). The phase contrast transfer function, which is expressed as exp(i3`), where 3' = (Tr/2)(Cs)@d4-2Af ~ d 2 ) should satisfy exp(i3") = 1
(1)
for eliminating the effect of spherical aberration and defocus. This is expressed by assuming 3' = 2miTt, which produces (h3/4d~l)Cs - (~/2d2)A/= mt
(2)
for the lattice plane with spacing da, where ml = 0, - 1, ___2, • ... Eq. (2) is expressed as a set of lines in A f - C s diagrams,
Fig. 2. Ray diagram of the imaging condition, showingthe relationship between the AFF condition and Fourier images. The intensity distribution at the bottom surfaceof the specimencrystal, formedby the interference of the diffracted waves from the lattice plane with lattice spacing d j, is projected on a plane shifted from the Gausian image plane by CsXa/2d~ On both sides of this plane, Fourierimages with period of m(2d21/~) are produced as demonstratedat the bottom right by superposing three plane waves. Similar image shift and corresponding Fourier images for the smaller lattice planes d2, d3 (
Atomic Resolution Electron Microscope Images Table 2. A1203, dl = 0.3481 nm (11.2), d 2 = 0.2552 n m (11.4)
Table 1. MgO, dl = 0.2106 n m (200), d2 = 0.1480 n m (220) mi 2 1 0 --1 2 1 0 6
m2 5 3 1 --1 6 4 2 23
115
C s (mm)
A f (nm)
0.81116 0.83237 0.85359 0.87480 1.66475 1.68596 1.70718 9.26219
--83.212 --28.604 26.003 80.611 --57.209 --2.601 52.007 --41.609
ml
m2
Cs (mm)
15 8 1 1 -6 -13 16
28 15 2 3 -11 -24 30
0.75478 0.95304 1.15132 9.40880 1.34960 1.54788 1.90609
Af (nm) -2202.976 -1167.882 -134.588 -42.514 899.605 1933.799 -2337.565
Table 3. MgAI204, dl = 0.2857 nm (220), d2 = 0.2020 nm (400)
In this condition Af is given from Eq. (2) as A f = m I (2d~/X) +
(CsX2/2d2)
(3)
and shown schematically in Fig. 2. The diffracted waves from the lattice planes with the spacing d~ will not be focused on the Gaussian image plane but on a plane shifted by an amount Csh2/2d2. The intensity distribution at the bottom surface can be obtained by adjusting the focus to this plane, i.e. A f = c~xz/2d~. However, on both sides of this image plane, Fourier images with the period of m(2d~/X), where m 1 = 0, ----- 1, "+- 2, - . . , are also produced. Since the Fourier images have the same intensity distribution as the original image formed at (Cs~kZ/2d~), the defocus Af in Eq. (3) gives the focus which is equivalent to an exact focus for a zero-aberration lens. A similar relationship holds for another lattice plane, say (220) of MgO ( d 2 = 0.1480 nm), which is also shown in Fig. 1 as m2 = 0, -4- 1, + 2, "". The AFF conditions for two kinds of planes {200} and {220} are expressed as the crossing points of these two line systems. They are indicated with the combination of the numbers (mb m 2 ) and listed in Table 1. These AFF conditions correspond to the coincidence of two Fourier images of ml(2d~/X) and m2(2d22/X)formed on both sides of the image planes (CsX2/2d2) and (Cs~kZ/2d2), respectively, as shown in Fig. 2. For the MgO crystal in [001] orientation, four reflections from {200} and four from {220} are in the same phase in these conditions shown in Fig. 1 and Table 1, and produce the image which has the same intensity distribution of electrons at the bottom surface of the specimen crystal, if the objective aperture selects only these nine diffraction spots to contribute to the image contrast, as indicated in Fig. 3(a). Most of the conventional atomic-resolution electron
MgO
m2
C~ (mm)
z~f (nm)
0 1 1 2 2 1 0
1 3 4 6 7 5 2
3.00303 3.0002 6.0032 6.0003 9.0034 9.00625 6.00606
49.646 -49.837 -0.187 -99.668 -50.022 49.459 99.293
microscopes have resolution limits of around 6 = 0.140.25 nm, which correspond to apertures of 0 = 7 - 4 nm -l and thus, in the general case, it is possible to use the waves which are diffracted from two kinds of lattice planes with low index for producing the atomic structure images. For example, the atomic resolution images of MgAI204 and A1203 crystals can be produced with apertures of 4~3 = 5.0 nm -1 and 04 = 4.0 nm -1, respectively, as shown in Fig. 3(b) and (c). Table 2 shows the AFF condition for the image of an A1203 crystal projected onto the ac plane with an aperture of ~4 = 4.0 nm -1, whose Cs-Af diagram is shown in Fig. 4. Table 3 also shows the AFF condition for the image MgA1204 projected onto the ab plane with an aperture ~b3 = 5.0 nm -1 whose Cs-Afdiagram is shown in Fig. 5. Though there are many AFF and ~b-AFF conditions, not all of them are available if the partial coherence of electrons is taken into account. An example is shown in Fig. 6 for A1203 crystal. When the beam divergence is q = 0.1 mrd and chromatic defocus value A = 1 nm (AE/E = 10 6, Cc = 1 mm), the projected image has high contrast even though the values of Cs ( = 1.15 mm) and Aft = -- 134.6 nm) are rather large. However, for q = 1 mrd, and ~ = 10 nm, the image contrast becomes very low and can hardly be observed.
AI203
MgAI204
03
I
(a)
ml
(b)
I
0~
~
0
03
2
(c)
Fig. 3. Relationship between various sizes of apertures and electron diffraction patterns from (a) MgO, (b) M~A1204, (c) AI20 3 for ~bt = 7 n m -~ (d = 0.14 nm), ~2 6 n m 1 (d = 0.17 nm), 4~3 = 5 nm -I (d = 0.2 nm), and ~4 = 4 n m - (d = 0.25 nm). =
116
H. Hashimoto et al.
V/Z I/
/
U/
re
t ^dr
-200 -100 0 100 af(nrn) 200 Fig. 4. C s-Af diagram giving the AFF condition for {11.2}, m l = 0, _+ 1, _+2, "', and {11.4}, m2 0, + 1, + 2, "'" of an A1203crystal in [ 11.0] projection. =
j
ylo
// J
-200
-100
0
100
/ff rim) 200
Fig. 5. Cs-Afdiagram giving the AFF conditionfor {220},m l = 0, -+ 1, -+ 2, "", and {400},m2 = 0, -+ 1, -+ 2, "" of a MgAI204 crystal in [001] projection.
For applying the ~b-AFF condition to the system of MgO/ MgAleO4/A1203 (Hutchison and Waddington, 1988), the AFF diagrams made for MgO (Fig. 1), MgA1204 (Fig. 5) and AleO3 (Fig. 4) are superimposed as shown in Fig. 7. AFF conditions for each MgO, MgA1204 and A1203 crystals are expressed with the marks of + , ©, and D, respectively. The ~b-AFF conditions can be realized at the positions where these marks appear at a point or even very closely. They are found at the regions labelled A, B, C, D, ..., whose conditions are listed in Table 4. At the condition shown by A, AFF conditions of three crystals appear closely, which could be the best ff-AFF condition for taking images of the three crystals at the same time.
I M A G E C O N T R A S T C A L C U L A T E D A T ~b-AFF CONDITIONS Using multi-slice many-beam dynamical theory, the intensity distribution at the bottom surface of the MgO, MgA1204 and A1203 crystals of 50 nm thickness was calculated and compared with the images formed at the ~b-AFF condition. Though the partial coherence of the imaging electrons was taken into account in the present calculation, it will not be shown in the following because the effect can be known from Fig. 6 and changed by the system of illumination, for example, by using a field-emission gun. Fig. 8
shows the projection of the constituent atom position of the MgO, A1203 and MgAleO4 crystals in which some atoms are overlapped. Thin lines in each figure indicate the unit cell size. As can be seen in Fig. 3, the diffracted waves from the different crystals have different scattering angles. The optimum aperture radius for accepting {200} and {220} reflections from the MgO crystal is ~1 = 7 nm -1, which is too large to exclude the reflection {440} while accepting only {400} and {220} reflections from the MgA1204 crystal. However, as was shown already (Hashimoto et al., 197879) and will be discussed in the next chapter, the MgA1204 crystal belongs to the cubic system and higher-order reflections such as {440} can be in the AFF condition when {220} reflections are also in the AFF condition. Thus, when the reflections from {220} and {400} are in the AFF condition with the values (Cs = 9.003 mm, Af = 50.02 nm) for m1(220) = 2 and m2(400) = 7, as can be seen in Fig. 5 and Table 3, the reflections from {440} are within this AFF condition with the value m3(440) = 26. Though the AFF condition (Cs = 9.003 mm, Af = - 5 0 . 0 2 nm) for MgA1204 is slightly different from the AFF condition (Cs = 9.262 mm, Af = --41.61 nm) for MgO, the image contrast calculated by this condition, shown in Fig. 9, indicates that the image contrasts are almost exactly the same as the intensity distribution at the bottom surfaces, not only for the MgA1204 crystal but also for the MgO crystal. Mg, O and A1 atom positions appear as bright spots, as can
Atomic Resolution Electron Microscope Images
117 Table 4.
(mL, m2)
(a)
(b)
A1203 MgO MgA1204
(1, 3) (6, 23) (2, 7)
9.409 9.262 9.003
-42.514 -41.609 -50.022
B
MgA1204 MgO
(2, 6) (5, 17)
6.0 5.9
-99.6 -91.0
C
MgA1204 MgO
(1, 3) (3, 10)
3.0 3.4
-49.8 -59.8
D
MgAI204 MgO
(0, 1) ( 1, 6)
3.0 3.4
49.6 49.4
E
A1203 MgO
(0, 1) (3, 16)
8.3 8.4
92.0 96.2
F
A1203 MgO
(1, 2) (3, 7)
1.1 0.8
-134.5 -137.8
(3, 8)
5.9 5.9
-199.1 -200.0
MgAI204 MgO
(c)
Cs (mm) 10
/ .... .- A~-7-4-~.-I -'/+ ~ . 4. 4-J ~."+/ ," +/ ~/ +"T
9 7 6
5
/--
2
-20{
, -/
I
/-/
/
./.+
/
+;. /)+-'
.:$ , 7 ~ /4". .'+"
4J
", A- "+"~ + "
+" c..~ v i +
l o
.+'"I
+/* +.-@E
f /
.+-"+.."',~"'+4 / /-r" +/' ,-4/ ;.+- ./~--'.~ .+.+~ 4: _94-, ~" +.." .,u . . . . .
7/.4" >G
-100
...-I-"/',,4-" .,
0
. +"
'-+'l
7.+_.{D i+" 4-" . +,/'/~ #.. ~ /s ..41l ",'/.+ ......-+".;".+ ....
,i> ..+ / /
...z+"
. +"/,|
100
..+J Af(nm)
200
Fig. 7. Superposition of Figs 1, 4 and 5. Relationships between the AFF conditions for MgO ( + ), MgA1204 (O) and A1203 (U]) are seen. The ~b-AFFcondition is realized at the positions A, B, C, D, E, F and G.
be seen by comparing Fig. 9 with Fig. 8. Thus, the condition C~ = 9.003 mm, A f = -- 50.02 nm, q~l = 7 nm -1 is a good ~k-AFF condition for MgO and MgAIzO4. For the images of A1203 crystal, the AFF condition is C~ = 9.409 m m and Af = - 42.514 nm for the planes m1{11.2}, m2{11.4} and available only by using the a p e r t u r e ~ 4 = 4 nm -1. In this condition, for the MgAI204 crystal only the diffracted waves from {220} contribute to the image contrast, while for MgO crystal no diffracted waves contribute to the image contrast, as can be seen in
Af (rim)
A
G
Fig. 6. Effect of partial coherence of the illuminating electrons on the image contrast of an A120 3 crystal formed at the AFF condition: C,~= 1.15 mm, Af = -- 134.59 nm and 4~2 = 6 nm-~. (a) Intensity at the bottom surface, (b) AFF for q = 0.1 mrad, A = 1 nm, (c) AFF for q = 1 mrad, A = 10nm.
C~ (mm)
(7, 21)
Fig. 3. The calculated image contrast using ~b4 = 4 nm -I for MgAI204 and A1203 crystals is shown in Fig. 10 together with the intensity distribution at the bottom surfaces of these crystals. Both image intensities agree well with the ones at the bottom surfaces and the intensity maxima appear at atomic positions for the A1203 crystal, even though the resolution is not very high. Thus, the condition Cs = 9.404 mm, Af = --42.514 nm, (~4 = 4 nm -l is a good ~AFF condition for MgA1204 and A1203. However, in MgA1204 crystal, the position of A1 and O atoms appear dark in Fig. 10(a) and (a'), though the positions of Mg atoms appear bright. This is seen to be due to the absence of {400} reflections from the imaging. For obtaining images from three crystals, MgO, MgA1204 and A1203 at the same time, the aperture ~b3 = 5 nm -1 and the condition, Cs = 9.0034 m m and ~ f = - 5 0 . 0 2 nm were applied to the calculations of image contrast and the intensity at the bottom surface, as shown in Fig. 11. The positions of the constituent atoms are well displayed for three crystals, though the fine structure of the image contrast and the intensity at the bottom for A1203 crystals are slightly different, as shown in (c) and (c'). Thus, the condition C~ = 9.0034 mm and Af = - 5 0 . 0 2 nm, ~ 3 • 5 nm -1 is a good ~-AFF condition for three crystals. Fig. 12 shows similar patterns to those of Fig. 11, but for the condition Cs = 9.409 mm, Af = - 4 2 . 5 1 4 nm (AFF condition for m l(11.2) -- 1, me(11.4) = 3) 00.6 spots are included using ~b3 = 5 nm -1. By comparing the image contrast (a), (b), (c) and bottom intensity (a'), (b'), (c'), some similarity is present but there are some differences. This is due to the large aperture ~b3 = 5 nm -1, which allowed the contribution of diffraction spots from {00-6}, and thus the AFF condition does not hold for ~b3 = 5 nm -1 but ~ 4 = 4 nm -1 for the A1203 crystal. For the images of the MgA1204 crystal, the AI and O atom positions appear at correct positions more clearly than in Fig. 10(a) due to the large aperture, but Mg atoms appear dark. Thus, this condition is not an AFF condition for the MgA1204 crystal. The above calculations suggest that the imaging condition of (C~ = 9.0034mm, Af = - - 5 0 . 0 2 n m and ~bs = 5 nm -l) is a ~b-AFF condition to take images of
118
H. Hashimotoet al.
(a) •
•
(b)
O Q I
I ~ t ~
Q O
O I
I Q •
e e ~Q
U Q
•
•
•
•
4)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
o
o
•
•
•
e e O Q
t Q
I i ~
g
Q
~
4
0
e
O
o
6
6
D
~
~
O
O
(a)
(a')
(b)
(b')
0
0
O O O J O O O e e
0 1
~ •
O ~
O I
e o
•
(b')
-: e Q
t
•
•
•
-.-
Q
•
4~ 0
-.-
•
e~
• •
•
g
e
~
t
B
o
e 0
~
O e O J o o J o j e e e o o o e
e
o e e w ~ o o e e o e e e o ~ o
4~
e e
t
o g e O e s o m e o o o m e
Fig. 9. (a) and (b) are the calculated image intensifiesof MgO and MgA!204 for C~= 9.003 mm, Af= - 50.22 nm and ~b] = 7 nm-1, which are exact AFF conditionsfor MgA1204. (a') and (b') are the intensitiesat the bottomsurfaces;400 kV, thickness t = 50 nm, A = q = 0. (AFF for MgO is Cs = 9.262 mm, Af= - 41.609 nm.)
4
(c)
(c')
Fig. 8. (a), (b) (b') and (c) (c') are the projectionsof the positionsof constituentatoms in MgO, MgA1204and A1203 crystals parallel to c, c and [110] directionsrespectively.
MgO, MgA1204 and A1203 crystals at the same time. It can be emphasized that, for obtaining the ~b-AFF condition for more than two crystals, it is important to find appropriate values, not only A F F conditions for individual crystals but also the aperture size of selecting diffraction spots suitable to each crystal.
A F F I M A G E S F O R M E D BY M O R E T H A N T H R E E KINDS OF WAVES As shown in the introduction (Hashimoto et al., 197879), the A F F condition holds for many diffracted waves in a simple crystal structure. These results are checked by the C s - A f diagram used in the present paper. In Au crystal of 110 orientation, the A F F condition is given by Cs = 0.545 mm, Af = 157.3 nm. The A F F conditions for 111, 200, 220, 222, 311 are shown in Fig. 13. The AFF lines are crossing at one point (Cs = 0 . 5 4 5 8 m m , Af = 157.3 nm) for 100-kV electrons. In this condition, 17 diffracted waves are in the same phase and contribute to the correct imaging. The calculated image contrast at this condition is shown in Fig. 6 of Hashimoto et al. (1978-79).
(a)
(a')
(b)
(b')
Fig. 10. (a) and (b) are the calculatedimageintensifiesof MgA1204 and A1203for C~= 9.409 ram, Af= -42.514 nm and 4 4 = 4 nm-I, which are exact AFF conditions for A1203. (a') and (b') are the intensitiesat the bottom surfaces; 400 kV, t = 50 nm, A = q = 0.
CHANGING MECHANISM OF SPHERICAL ABERRATION COEFFICIENT The value of Cs can be changed by the change of the specimen height (Z-value) measured from the top surface of the bottom part of the pole-pieces of objective lens. The value of Cs, is in the order of m m and can be changed by a
Atomic Resolution Electron Microscope Images
119
(a)
(a')
(b)
(b')
(c)
(c')
Fig. 11. Calculated images of (a) MgO, (b) MgA1204, (c) A1203 for C, = 9.0034 mm, A f = -- 50.02 nm and ~b3 = 5 nm -1, which are exact AFF conditions for MgAI20~; 400 kV, t = 50 nm. (a'), (b') and (c') are the intensities at the bottom surfaces.
(a)
(a')
(b)
(b')
lob
R
-~i~ r 0
~aPPm
I m.
(c')
Fig. 12. Calculated images of (a) MgO, (b) MgA1204, (c) AlzO3 for C~ = 9.409 mm, At'= -- 42.514 nm and 4~3 = 5 nm i, which do not hold as AFF conditions for A1203 and MgA1204 due to the inclusion of 006 and 040 reflections in the aperture, respectively. (a'), (b') and (c') are the intensities at the bottom surfaces.
120
H. Hashimoto et al. Au 100kV spherical aberration Imml
/ /
+1.0 ~ 0.5458 mm
1
+0.5
1
j220 3
CONCLUSION
...... ,s
defocus
- 100
than 3.3 mm, the large values of Z and corresponding Cs can be obtained easily by changing the specimen position in the cartridge, for example using a simple attachment which can bring the specimen high into the cartridge to hold it at an optimum height to give AFF and ~ - A F F conditions.
/111 / 200
0
3
100
200
Into I
Fig. 13. c~-nf diagram for Au 110 orientationat 100kV, mill -3, m200= - 3, m220= 2, m222= 15, roll3 11, C~ = 0.545 mm, Af = 157.3 nm. =
=
4
.........~...........~..........~...........r..........r..........~...........r..........i..................
3.5
3
........ 4........... ',.......... i ........... i .......... i ........... i .......... i .......... i ........ i ......... ........
o 1.
~
0.5 ..............................................................................
0.6
0.9
1.2
1.5
1.8
2.1 2.4 Z (ram)
i---/
2.7
3
3.3
3.6
Fig. 14. The value of Cs (mm) against the specimen position Z (ram), measured from the top surface of the bottom part of the objective lens pole-pieces for a JEM-4000EX operated at 200, 300 and 400 kV.
Table 5. Z (mm) 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6
C~ (200 kV) (ram)
C~ (300 kV) (mm)
C~ (400 kV) (mm)
0.85 0.8 0.85 0.9 1.05 1.3 1.7 2.25 2.9 3.75
1.2 1 0.9 0.9 1.03 1.2 1.5 1.9 2.45 3.25
1.5 1.25 1.05 1 1.01 1.2 1.4 1.8 2.2 2.75
change of Z-value with the order of mm. Fig. 14 and Table 5 show the variation of C~ against the specimen position Z for a JEM-4000 EX. Z-value can be changed by +1.5 to - 0 . 5 m m relative to the position Z0 (1.8 mm) i.e. 3 . 3 1.3mm, which corresponds to Cs = 2 . 7 - 1 m m , for 400 kV and 3.7-0.8 m m for 200 kV. Since Cs increases almost linearly with increasing Z-value in the region larger
The present generation of conventional transmission electron microscopes have a resolution limit of around 0.14-0.25 nm, which enables us to record atomic structure images of many crystals. However, even by using these microscopes it is not always possible to take images which coincide to the intensity distribution of electron waves at the bottom faces of the crystals. Using Cs versus Af diagrams for single crystals of MgO, MgA1204, and A1203, it was shown that the correct atomic images can be formed by two kinds of Bragg reflected waves and there are many combinations of Cs and Af for obtaining the images without the effect of spherical aberration, i.e. without phase change of the electron waves. AFF conditions for simple crystals such as f.c.c., b.c.c, and h.c.p., which have already been discussed (Hashimoto et al., 1978-79), are shown in Fig. 13 as a C s - A f diagram. The simple method for obtaining the ff-AFF condition is as follows: (1) make AFF points in the diagrams of C s - A f for individual crystals, (2) superimpose them, (3) find coinciding AFF points, which are the ff-AFF conditions. The change in Cs of the microscope for the specimens concerned can be effected by changing the specimen position along the optical axis. Most conventional electron microscopes have such a capability. However, for large values of Cs and Af, highly coherent electron waves, for example field emission electrons, must be used for obtaining high-contrast images. Acknowledgements--It must be mentioned here that one of the present authors (H.H.) has been enormouslystimulatedand helped by many contacts with Prof. Mike Whelan since 1959. In particular, he should mention the fruitful collaborationwith Prof. Mike Whelan and Prof. Archie Howie in which they carried out the work on the anomalousabsorptionof electron waves appearingin the electronmicroscopeimages. This was organizedby the kind arrangementsof Prof. Sir James Menter, Prof. Sir Peter Hirsch and Prof. Dr Jack Nutting.It is a greathonourfor the presentauthorsto have this opportunityto contributethe present paper, coincidingwith the retirement of Prof. M. J. Whelan F.R.S. The authors are grateful to Dr Toshikazu Honda of JEOL for his helpful discussionsand the preparation of Fig. 14 and Table 5, and also to Prof. RobertFisherand Dr John Hutchisonfor their help with the English of this paper.
REFERENCES
Chen F. R., ChiouS. K., ChangL., Hong C. S., 1994. Ultramicroscopy, 54, 179-191. Chen L. J., Liang J. M., Liu C. S., Hsiech W. Y., Lee T. L., Wang M. H., Chen W. J., 1994. Ultramicroscopy, 54, 156-165. Cowley J. M., Moodie A. F., 1957. Proc. Phys. Soc., 70, 505. Endoh H., 1984 PhD thesis, Osaka University. Hutchison J. L., Chou C. T., Casanove M. J., Cherns D., Steeds J. W., Ashenford D. A., Lunn B., 1994. Ultramicroscopy, 53, 91-96. Hutchison J. L., WaddingtonG., 1988. Ultramicroscopy, 25, 93-95. HutchisonJ. L., WaddingtonG., Cullis A. G., Chew N. G., 1986../. Microscopy, 142, 153-162.
Atomic Resolution Electron Microscope Images Hashimoto H., Endoh H., Tanji T., Ono A., Watanabe E., 1977. J. Phys. Soc. Japan, 42, 1073-1074. Hashimoto, H., Endoh, H., Takai, Y., Tomita, H. and Yokota, Y., 1978-79. Chemica Scripta 14, 23-31. Hashimoto H., Makita Y., Endoh H., 1996. Materials Chemistry and Physics, 46, 7-14. Hesse H., Senz S., Scholz R., Wemer P., Heydenreich J., 1994. Interface Science, 2, 221-273.
121
Ichinose H., 1987. J. Electron Microscopy, 36, 82. Ishizuka K., 1980. Ultramicroscopy, 5, 55. Izui K., Furuno S., 1987. Ultramicroscopy, 21, 399-402. Izui, K., Furuno, S., Nishida, T. and Otsu, H., 1978-79. Chem. Scripta 14, 99-108. Ko D. H., Sinclair R., 1994. Ultramicroscopy, 54, 166-178. Scherzer O., 1949.3. AppL Phys., 20, 20.
Pergamon PII: S0968-4328(98)00008-0
Micron Vol. 29, No. 2/3, pp. 113-121, 1998 © 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0968-4328/98 $19.00+0.00
Pseudo-aberration Free Focus Condition for Atomic Resolution Electron Microscope Images* H. HASHIMOTO,? H. ENDOH,5/M. HASHIMOTO,§ Z. P. LUO? and M. F. SONG?
?Okayama University of Science, Okayama, Japan ?iKyoto Institute of Technology, Sakyoku, Kyoto, Japan §Osaka Setsunan University, Neyagawa, Osaka, Japan (Received 10 October 1997;accepted9 January 1998)
Abstract--Spherical Aberration Free Focus (AFF)conditions for taking atomic resolution electron microscopeimages of single crystals and pseudo-AberrationFree Focus(~b-AFF)conditionsfor takingatomicimagesof morethan two adjoiningsinglecrystalsare discussedby using diagrams of Cs (spherical aberration coefficient)and Af(defocus). Calculated image contrast in those conditions is shownfor three individual crystalsof MgO,MgA1204,A1203and their adjoiningcrystalblocks. It is shownthat at least two kinds of Braggreflectedwaves can be in the AFFconditionin eachcase, and thusthe size of the aperturefor limitingthe wavesbecomesimportantin somecases. Cs can be changedby changingthe positionof the specimenalongthe objectivelens axis but highlycoherentilluminating wavesare needed for large Cs and Af © 1998Elsevier ScienceLtd. All rights reserved. Key words: AberrationFree Focus,pseudo-AberrationFree Focus,electronmicroscopeimagesof atoms, atomresolutionimagesof crystals.
value b = Rc~/ax 3/4, where R = 0.65~0.7, is generally used as the conventional resolution limit. With the improvement of the resolution of conventional It was shown (Hashimoto et al., 1977), however, that for a transmission electron microscopes (TEM) and the develop- thin crystalline specimen and an axial illuminating beam, ment of theories of image formation, many atomic structure the Bragg reflected waves sample the CTF in a discrete set of images of materials have been used for the better under- directions and, by a suitable choice of defocus and aberration, standing of the atomic arrangement in specimens. The the relative phase can be simply related to those for a zeroobserved electron microscope images with atomic resolu- aberration lens. The demonstration was carried out using Au tion, however, change with small changes in focus of the crystal in [100] orientation, in which dynamic scattering of objective lens. The change of image contrast by the defo- electrons in the crystal was taken into account. Since the cusing of the objective lens is well described by the phase phase shift of 7r/2 of diffracted waves (which is predominant contrast transfer function (CTF) (Scherzer, 1949). The for WPO) was taking place in the process of dynamic scatterphase of the waves leaving the specimen with some scatter- ing of electrons in reasonably thick crystals, the real part of ing angle is shifted by the spherical aberration of imaging CTF was used for the interpretation of image contrast. The lens but it can be compensated by underfocusing. Thus, the intensity distribution of electron waves at the bottom surface phase angle of the CTF is given by the difference of the was assumed to be projected to the image plane. terms of spherical aberration, 71-/2~k(Csot4),and defocus, 7r/ The method shown above is called the spherical Aberra2X(2hfc~2), where X, Cs, o~and h f are the wavelength, sphe- tion Free Focus (AFF) condition and it was shown (Hashirical aberration coefficient, electron scattering angle and moto et al., 1978-79) that this AFF condition is realized by defocusing. a defocusing of CsX2/2d 2 +- m(2d/X), where d is the spacing Scherzer (1949) proposed the optimum defocus value of of the appropriate lattice plane and 2dZ/X is equivalent to the h f = (CsX) 1/2 for weak phase objects (WPO) in which the period of Fourier images (Cowley and Moodie, 1957). For CTF can be approximated by its imaginary part and a wide simple crystal structure, such as [100], [110] and [111] proflat region appears close to the optical axis (small scattering jections of f.c.c, and b.c.c, structures and [001] projection of angle) in the oscillating CTF. The first zero of the oscillating h.c.p, lattice structure, it was shown (Hashimoto et al., CTF appears at the scattering angle of C~m= 1.41(Af/C~)112, 1978-79; Endoh, 1984) that there are many combinations Eliminating the oscillating region of the CTF (for scattered of Cs and Af for realizing the AFF-condition at 100 kV. It waves larger than am) by the objective aperture, the resolu- was thus shown that for Au in [110] orientation, when Cs tion limit becomes 6 = C~/4X3/4, which is now called con- and Af are 1.09mm and 134.8 nm, or 0.545 mm and ventionally the theoretical resolution limit. By adopting the 157.3 nm, respectively, 37 diffracted waves up to (440) allowance of the defocus which gives a nearly flat region in are in the same phase and sample the oscillating transfer the CTF, a higher resolution limit is obtainable and now the function as unity. Using the AFF condition, atomic structure images of Si *In honor of ProfessorM. J. Whelan. were successfully recorded (Izui et al., 1978-79; Izui et al., 113 INTRODUCTION
H. Hashimoto et al.
114
- 200
-1 O0
1oo
0
Af ('rim) 200
Fig. 1. C s-Af diagram giving AFF conditionfor {200} is given by the m l = 0 , ~ 1, --+2, "", line system and {220} by the m2 = 0, -+ 1, -+ 2, -" line system, for a MgO crystal in [001] projection. The crossing points of these two line systems (indicated by + ) are AFF conditions for both {200} and {220 }. 1987; Hutchison and Waddington, 1988). A similar approach, this time assuming kinematical behavior of thin crystals, has been used to characterize a vacancy-ordered superlattice structure in Ga2Te3 (Hutchison et al., 1994). Recently, many atomic resolution images of crystals on both sides of interfaces have been produced (Hutchison et al., 1986; Ichinose, 1987; Ko and Sinclair, 1994; Chen et al., 1994a, b. However, the image contrast of both crystals does not always coincide with the intensity distributions at the bottom surface of both respective crystals. Analyzing the Fourier image construction and considering the partial coherence of imaging electrons (Ishizuka, 1980), Hashimoto et al. (1996) proposed the best focus for obtaining the atomic resolution images by a 400-kV microscope with the lens of C~ = 0.1 nun, corresponding to the intensity distribution at the bottom surfaces of three crystals. They discussed this method by referring to the photograph taken by Hesse et al. (1994) of the crystal containing two interfaces and three crystal lattices of MgO/ MgA1204/A1203 and named this focus as pseudo-Aberration Free Focus (~b-AFF) condition. However, in this discussion, C~ and X were fixed and thus it was not exact but close to the ~b-AFF condition. In the present paper, the principle and the method to obtain the accurate AFF and ~b-AFF conditions by using a C ~ - A f diagram are treated for the case of the MgO/ MgA1204 / A1203 crystal system.
. . . .
F
L~
Gaussian image plane
PRINCIPLE The principle of AFF and pseudo-AFF conditions are described here systematically by citing the results published already (Hashimoto et al., 1978-79, 1996). The phase contrast transfer function, which is expressed as exp(i3`), where 3' = (Tr/2)(Cs)@d4-2Af ~ d 2 ) should satisfy exp(i3") = 1
(1)
for eliminating the effect of spherical aberration and defocus. This is expressed by assuming 3' = 2miTt, which produces (h3/4d~l)Cs - (~/2d2)A/= mt
(2)
for the lattice plane with spacing da, where ml = 0, - 1, ___2, • ... Eq. (2) is expressed as a set of lines in A f - C s diagrams,
Fig. 2. Ray diagram of the imaging condition, showingthe relationship between the AFF condition and Fourier images. The intensity distribution at the bottom surfaceof the specimencrystal, formedby the interference of the diffracted waves from the lattice plane with lattice spacing d j, is projected on a plane shifted from the Gausian image plane by CsXa/2d~ On both sides of this plane, Fourierimages with period of m(2d21/~) are produced as demonstratedat the bottom right by superposing three plane waves. Similar image shift and corresponding Fourier images for the smaller lattice planes d2, d3 (
Atomic Resolution Electron Microscope Images Table 2. A1203, dl = 0.3481 nm (11.2), d 2 = 0.2552 n m (11.4)
Table 1. MgO, dl = 0.2106 n m (200), d2 = 0.1480 n m (220) mi 2 1 0 --1 2 1 0 6
m2 5 3 1 --1 6 4 2 23
115
C s (mm)
A f (nm)
0.81116 0.83237 0.85359 0.87480 1.66475 1.68596 1.70718 9.26219
--83.212 --28.604 26.003 80.611 --57.209 --2.601 52.007 --41.609
ml
m2
Cs (mm)
15 8 1 1 -6 -13 16
28 15 2 3 -11 -24 30
0.75478 0.95304 1.15132 9.40880 1.34960 1.54788 1.90609
Af (nm) -2202.976 -1167.882 -134.588 -42.514 899.605 1933.799 -2337.565
Table 3. MgAI204, dl = 0.2857 nm (220), d2 = 0.2020 nm (400)
In this condition Af is given from Eq. (2) as A f = m I (2d~/X) +
(CsX2/2d2)
(3)
and shown schematically in Fig. 2. The diffracted waves from the lattice planes with the spacing d~ will not be focused on the Gaussian image plane but on a plane shifted by an amount Csh2/2d2. The intensity distribution at the bottom surface can be obtained by adjusting the focus to this plane, i.e. A f = c~xz/2d~. However, on both sides of this image plane, Fourier images with the period of m(2d~/X), where m 1 = 0, ----- 1, "+- 2, - . . , are also produced. Since the Fourier images have the same intensity distribution as the original image formed at (Cs~kZ/2d~), the defocus Af in Eq. (3) gives the focus which is equivalent to an exact focus for a zero-aberration lens. A similar relationship holds for another lattice plane, say (220) of MgO ( d 2 = 0.1480 nm), which is also shown in Fig. 1 as m2 = 0, -4- 1, + 2, "". The AFF conditions for two kinds of planes {200} and {220} are expressed as the crossing points of these two line systems. They are indicated with the combination of the numbers (mb m 2 ) and listed in Table 1. These AFF conditions correspond to the coincidence of two Fourier images of ml(2d~/X) and m2(2d22/X)formed on both sides of the image planes (CsX2/2d2) and (Cs~kZ/2d2), respectively, as shown in Fig. 2. For the MgO crystal in [001] orientation, four reflections from {200} and four from {220} are in the same phase in these conditions shown in Fig. 1 and Table 1, and produce the image which has the same intensity distribution of electrons at the bottom surface of the specimen crystal, if the objective aperture selects only these nine diffraction spots to contribute to the image contrast, as indicated in Fig. 3(a). Most of the conventional atomic-resolution electron
MgO
m2
C~ (mm)
z~f (nm)
0 1 1 2 2 1 0
1 3 4 6 7 5 2
3.00303 3.0002 6.0032 6.0003 9.0034 9.00625 6.00606
49.646 -49.837 -0.187 -99.668 -50.022 49.459 99.293
microscopes have resolution limits of around 6 = 0.140.25 nm, which correspond to apertures of 0 = 7 - 4 nm -l and thus, in the general case, it is possible to use the waves which are diffracted from two kinds of lattice planes with low index for producing the atomic structure images. For example, the atomic resolution images of MgAI204 and A1203 crystals can be produced with apertures of 4~3 = 5.0 nm -1 and 04 = 4.0 nm -1, respectively, as shown in Fig. 3(b) and (c). Table 2 shows the AFF condition for the image of an A1203 crystal projected onto the ac plane with an aperture of ~4 = 4.0 nm -1, whose Cs-Af diagram is shown in Fig. 4. Table 3 also shows the AFF condition for the image MgA1204 projected onto the ab plane with an aperture ~b3 = 5.0 nm -1 whose Cs-Afdiagram is shown in Fig. 5. Though there are many AFF and ~b-AFF conditions, not all of them are available if the partial coherence of electrons is taken into account. An example is shown in Fig. 6 for A1203 crystal. When the beam divergence is q = 0.1 mrd and chromatic defocus value A = 1 nm (AE/E = 10 6, Cc = 1 mm), the projected image has high contrast even though the values of Cs ( = 1.15 mm) and Aft = -- 134.6 nm) are rather large. However, for q = 1 mrd, and ~ = 10 nm, the image contrast becomes very low and can hardly be observed.
AI203
MgAI204
03
I
(a)
ml
(b)
I
0~
~
0
03
2
(c)
Fig. 3. Relationship between various sizes of apertures and electron diffraction patterns from (a) MgO, (b) M~A1204, (c) AI20 3 for ~bt = 7 n m -~ (d = 0.14 nm), ~2 6 n m 1 (d = 0.17 nm), 4~3 = 5 nm -I (d = 0.2 nm), and ~4 = 4 n m - (d = 0.25 nm). =
116
H. Hashimoto et al.
V/Z I/
/
U/
re
t ^dr
-200 -100 0 100 af(nrn) 200 Fig. 4. C s-Af diagram giving the AFF condition for {11.2}, m l = 0, _+ 1, _+2, "', and {11.4}, m2 0, + 1, + 2, "'" of an A1203crystal in [ 11.0] projection. =
j
ylo
// J
-200
-100
0
100
/ff rim) 200
Fig. 5. Cs-Afdiagram giving the AFF conditionfor {220},m l = 0, -+ 1, -+ 2, "", and {400},m2 = 0, -+ 1, -+ 2, "" of a MgAI204 crystal in [001] projection.
For applying the ~b-AFF condition to the system of MgO/ MgAleO4/A1203 (Hutchison and Waddington, 1988), the AFF diagrams made for MgO (Fig. 1), MgA1204 (Fig. 5) and AleO3 (Fig. 4) are superimposed as shown in Fig. 7. AFF conditions for each MgO, MgA1204 and A1203 crystals are expressed with the marks of + , ©, and D, respectively. The ~b-AFF conditions can be realized at the positions where these marks appear at a point or even very closely. They are found at the regions labelled A, B, C, D, ..., whose conditions are listed in Table 4. At the condition shown by A, AFF conditions of three crystals appear closely, which could be the best ff-AFF condition for taking images of the three crystals at the same time.
I M A G E C O N T R A S T C A L C U L A T E D A T ~b-AFF CONDITIONS Using multi-slice many-beam dynamical theory, the intensity distribution at the bottom surface of the MgO, MgA1204 and A1203 crystals of 50 nm thickness was calculated and compared with the images formed at the ~b-AFF condition. Though the partial coherence of the imaging electrons was taken into account in the present calculation, it will not be shown in the following because the effect can be known from Fig. 6 and changed by the system of illumination, for example, by using a field-emission gun. Fig. 8
shows the projection of the constituent atom position of the MgO, A1203 and MgAleO4 crystals in which some atoms are overlapped. Thin lines in each figure indicate the unit cell size. As can be seen in Fig. 3, the diffracted waves from the different crystals have different scattering angles. The optimum aperture radius for accepting {200} and {220} reflections from the MgO crystal is ~1 = 7 nm -1, which is too large to exclude the reflection {440} while accepting only {400} and {220} reflections from the MgA1204 crystal. However, as was shown already (Hashimoto et al., 197879) and will be discussed in the next chapter, the MgA1204 crystal belongs to the cubic system and higher-order reflections such as {440} can be in the AFF condition when {220} reflections are also in the AFF condition. Thus, when the reflections from {220} and {400} are in the AFF condition with the values (Cs = 9.003 mm, Af = 50.02 nm) for m1(220) = 2 and m2(400) = 7, as can be seen in Fig. 5 and Table 3, the reflections from {440} are within this AFF condition with the value m3(440) = 26. Though the AFF condition (Cs = 9.003 mm, Af = - 5 0 . 0 2 nm) for MgA1204 is slightly different from the AFF condition (Cs = 9.262 mm, Af = --41.61 nm) for MgO, the image contrast calculated by this condition, shown in Fig. 9, indicates that the image contrasts are almost exactly the same as the intensity distribution at the bottom surfaces, not only for the MgA1204 crystal but also for the MgO crystal. Mg, O and A1 atom positions appear as bright spots, as can
Atomic Resolution Electron Microscope Images
117 Table 4.
(mL, m2)
(a)
(b)
A1203 MgO MgA1204
(1, 3) (6, 23) (2, 7)
9.409 9.262 9.003
-42.514 -41.609 -50.022
B
MgA1204 MgO
(2, 6) (5, 17)
6.0 5.9
-99.6 -91.0
C
MgA1204 MgO
(1, 3) (3, 10)
3.0 3.4
-49.8 -59.8
D
MgAI204 MgO
(0, 1) ( 1, 6)
3.0 3.4
49.6 49.4
E
A1203 MgO
(0, 1) (3, 16)
8.3 8.4
92.0 96.2
F
A1203 MgO
(1, 2) (3, 7)
1.1 0.8
-134.5 -137.8
(3, 8)
5.9 5.9
-199.1 -200.0
MgAI204 MgO
(c)
Cs (mm) 10
/ .... .- A~-7-4-~.-I -'/+ ~ . 4. 4-J ~."+/ ," +/ ~/ +"T
9 7 6
5
/--
2
-20{
, -/
I
/-/
/
./.+
/
+;. /)+-'
.:$ , 7 ~ /4". .'+"
4J
", A- "+"~ + "
+" c..~ v i +
l o
.+'"I
+/* +.-@E
f /
.+-"+.."',~"'+4 / /-r" +/' ,-4/ ;.+- ./~--'.~ .+.+~ 4: _94-, ~" +.." .,u . . . . .
7/.4" >G
-100
...-I-"/',,4-" .,
0
. +"
'-+'l
7.+_.{D i+" 4-" . +,/'/~ #.. ~ /s ..41l ",'/.+ ......-+".;".+ ....
,i> ..+ / /
...z+"
. +"/,|
100
..+J Af(nm)
200
Fig. 7. Superposition of Figs 1, 4 and 5. Relationships between the AFF conditions for MgO ( + ), MgA1204 (O) and A1203 (U]) are seen. The ~b-AFFcondition is realized at the positions A, B, C, D, E, F and G.
be seen by comparing Fig. 9 with Fig. 8. Thus, the condition C~ = 9.003 mm, A f = -- 50.02 nm, q~l = 7 nm -1 is a good ~k-AFF condition for MgO and MgAIzO4. For the images of A1203 crystal, the AFF condition is C~ = 9.409 m m and Af = - 42.514 nm for the planes m1{11.2}, m2{11.4} and available only by using the a p e r t u r e ~ 4 = 4 nm -1. In this condition, for the MgAI204 crystal only the diffracted waves from {220} contribute to the image contrast, while for MgO crystal no diffracted waves contribute to the image contrast, as can be seen in
Af (rim)
A
G
Fig. 6. Effect of partial coherence of the illuminating electrons on the image contrast of an A120 3 crystal formed at the AFF condition: C,~= 1.15 mm, Af = -- 134.59 nm and 4~2 = 6 nm-~. (a) Intensity at the bottom surface, (b) AFF for q = 0.1 mrad, A = 1 nm, (c) AFF for q = 1 mrad, A = 10nm.
C~ (mm)
(7, 21)
Fig. 3. The calculated image contrast using ~b4 = 4 nm -I for MgAI204 and A1203 crystals is shown in Fig. 10 together with the intensity distribution at the bottom surfaces of these crystals. Both image intensities agree well with the ones at the bottom surfaces and the intensity maxima appear at atomic positions for the A1203 crystal, even though the resolution is not very high. Thus, the condition Cs = 9.404 mm, Af = --42.514 nm, (~4 = 4 nm -l is a good ~AFF condition for MgA1204 and A1203. However, in MgA1204 crystal, the position of A1 and O atoms appear dark in Fig. 10(a) and (a'), though the positions of Mg atoms appear bright. This is seen to be due to the absence of {400} reflections from the imaging. For obtaining images from three crystals, MgO, MgA1204 and A1203 at the same time, the aperture ~b3 = 5 nm -1 and the condition, Cs = 9.0034 m m and ~ f = - 5 0 . 0 2 nm were applied to the calculations of image contrast and the intensity at the bottom surface, as shown in Fig. 11. The positions of the constituent atoms are well displayed for three crystals, though the fine structure of the image contrast and the intensity at the bottom for A1203 crystals are slightly different, as shown in (c) and (c'). Thus, the condition C~ = 9.0034 mm and Af = - 5 0 . 0 2 nm, ~ 3 • 5 nm -1 is a good ~-AFF condition for three crystals. Fig. 12 shows similar patterns to those of Fig. 11, but for the condition Cs = 9.409 mm, Af = - 4 2 . 5 1 4 nm (AFF condition for m l(11.2) -- 1, me(11.4) = 3) 00.6 spots are included using ~b3 = 5 nm -1. By comparing the image contrast (a), (b), (c) and bottom intensity (a'), (b'), (c'), some similarity is present but there are some differences. This is due to the large aperture ~b3 = 5 nm -1, which allowed the contribution of diffraction spots from {00-6}, and thus the AFF condition does not hold for ~b3 = 5 nm -1 but ~ 4 = 4 nm -1 for the A1203 crystal. For the images of the MgA1204 crystal, the AI and O atom positions appear at correct positions more clearly than in Fig. 10(a) due to the large aperture, but Mg atoms appear dark. Thus, this condition is not an AFF condition for the MgA1204 crystal. The above calculations suggest that the imaging condition of (C~ = 9.0034mm, Af = - - 5 0 . 0 2 n m and ~bs = 5 nm -l) is a ~b-AFF condition to take images of
118
H. Hashimotoet al.
(a) •
•
(b)
O Q I
I ~ t ~
Q O
O I
I Q •
e e ~Q
U Q
•
•
•
•
4)
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
o
o
•
•
•
e e O Q
t Q
I i ~
g
Q
~
4
0
e
O
o
6
6
D
~
~
O
O
(a)
(a')
(b)
(b')
0
0
O O O J O O O e e
0 1
~ •
O ~
O I
e o
•
(b')
-: e Q
t
•
•
•
-.-
Q
•
4~ 0
-.-
•
e~
• •
•
g
e
~
t
B
o
e 0
~
O e O J o o J o j e e e o o o e
e
o e e w ~ o o e e o e e e o ~ o
4~
e e
t
o g e O e s o m e o o o m e
Fig. 9. (a) and (b) are the calculated image intensifiesof MgO and MgA!204 for C~= 9.003 mm, Af= - 50.22 nm and ~b] = 7 nm-1, which are exact AFF conditionsfor MgA1204. (a') and (b') are the intensitiesat the bottomsurfaces;400 kV, thickness t = 50 nm, A = q = 0. (AFF for MgO is Cs = 9.262 mm, Af= - 41.609 nm.)
4
(c)
(c')
Fig. 8. (a), (b) (b') and (c) (c') are the projectionsof the positionsof constituentatoms in MgO, MgA1204and A1203 crystals parallel to c, c and [110] directionsrespectively.
MgO, MgA1204 and A1203 crystals at the same time. It can be emphasized that, for obtaining the ~b-AFF condition for more than two crystals, it is important to find appropriate values, not only A F F conditions for individual crystals but also the aperture size of selecting diffraction spots suitable to each crystal.
A F F I M A G E S F O R M E D BY M O R E T H A N T H R E E KINDS OF WAVES As shown in the introduction (Hashimoto et al., 197879), the A F F condition holds for many diffracted waves in a simple crystal structure. These results are checked by the C s - A f diagram used in the present paper. In Au crystal of 110 orientation, the A F F condition is given by Cs = 0.545 mm, Af = 157.3 nm. The A F F conditions for 111, 200, 220, 222, 311 are shown in Fig. 13. The AFF lines are crossing at one point (Cs = 0 . 5 4 5 8 m m , Af = 157.3 nm) for 100-kV electrons. In this condition, 17 diffracted waves are in the same phase and contribute to the correct imaging. The calculated image contrast at this condition is shown in Fig. 6 of Hashimoto et al. (1978-79).
(a)
(a')
(b)
(b')
Fig. 10. (a) and (b) are the calculatedimageintensifiesof MgA1204 and A1203for C~= 9.409 ram, Af= -42.514 nm and 4 4 = 4 nm-I, which are exact AFF conditions for A1203. (a') and (b') are the intensitiesat the bottom surfaces; 400 kV, t = 50 nm, A = q = 0.
CHANGING MECHANISM OF SPHERICAL ABERRATION COEFFICIENT The value of Cs can be changed by the change of the specimen height (Z-value) measured from the top surface of the bottom part of the pole-pieces of objective lens. The value of Cs, is in the order of m m and can be changed by a
Atomic Resolution Electron Microscope Images
119
(a)
(a')
(b)
(b')
(c)
(c')
Fig. 11. Calculated images of (a) MgO, (b) MgA1204, (c) A1203 for C, = 9.0034 mm, A f = -- 50.02 nm and ~b3 = 5 nm -1, which are exact AFF conditions for MgAI20~; 400 kV, t = 50 nm. (a'), (b') and (c') are the intensities at the bottom surfaces.
(a)
(a')
(b)
(b')
lob
R
-~i~ r 0
~aPPm
I m.
(c')
Fig. 12. Calculated images of (a) MgO, (b) MgA1204, (c) AlzO3 for C~ = 9.409 mm, At'= -- 42.514 nm and 4~3 = 5 nm i, which do not hold as AFF conditions for A1203 and MgA1204 due to the inclusion of 006 and 040 reflections in the aperture, respectively. (a'), (b') and (c') are the intensities at the bottom surfaces.
120
H. Hashimoto et al. Au
/ /
+1.0 ~ 0.5458 mm
1
+0.5
1
j220 3
CONCLUSION
...... ,s
defocus
- 100
than 3.3 mm, the large values of Z and corresponding Cs can be obtained easily by changing the specimen position in the cartridge, for example using a simple attachment which can bring the specimen high into the cartridge to hold it at an optimum height to give AFF and ~ - A F F conditions.
/111 / 200
0
3
100
200
Into I
Fig. 13. c~-nf diagram for Au 110 orientationat 100kV, mill -3, m200= - 3, m220= 2, m222= 15, roll3 11, C~ = 0.545 mm, Af = 157.3 nm. =
=
4
.........~...........~..........~...........r..........r..........~...........r..........i..................
3.5
3
........ 4........... ',.......... i ........... i .......... i ........... i .......... i .......... i ........ i ......... ........
o 1.
~
0.5 ..............................................................................
0.6
0.9
1.2
1.5
1.8
2.1 2.4 Z (ram)
i---/
2.7
3
3.3
3.6
Fig. 14. The value of Cs (mm) against the specimen position Z (ram), measured from the top surface of the bottom part of the objective lens pole-pieces for a JEM-4000EX operated at 200, 300 and 400 kV.
Table 5. Z (mm) 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 3.3 3.6
C~ (200 kV) (ram)
C~ (300 kV) (mm)
C~ (400 kV) (mm)
0.85 0.8 0.85 0.9 1.05 1.3 1.7 2.25 2.9 3.75
1.2 1 0.9 0.9 1.03 1.2 1.5 1.9 2.45 3.25
1.5 1.25 1.05 1 1.01 1.2 1.4 1.8 2.2 2.75
change of Z-value with the order of mm. Fig. 14 and Table 5 show the variation of C~ against the specimen position Z for a JEM-4000 EX. Z-value can be changed by +1.5 to - 0 . 5 m m relative to the position Z0 (1.8 mm) i.e. 3 . 3 1.3mm, which corresponds to Cs = 2 . 7 - 1 m m , for 400 kV and 3.7-0.8 m m for 200 kV. Since Cs increases almost linearly with increasing Z-value in the region larger
The present generation of conventional transmission electron microscopes have a resolution limit of around 0.14-0.25 nm, which enables us to record atomic structure images of many crystals. However, even by using these microscopes it is not always possible to take images which coincide to the intensity distribution of electron waves at the bottom faces of the crystals. Using Cs versus Af diagrams for single crystals of MgO, MgA1204, and A1203, it was shown that the correct atomic images can be formed by two kinds of Bragg reflected waves and there are many combinations of Cs and Af for obtaining the images without the effect of spherical aberration, i.e. without phase change of the electron waves. AFF conditions for simple crystals such as f.c.c., b.c.c, and h.c.p., which have already been discussed (Hashimoto et al., 1978-79), are shown in Fig. 13 as a C s - A f diagram. The simple method for obtaining the ff-AFF condition is as follows: (1) make AFF points in the diagrams of C s - A f for individual crystals, (2) superimpose them, (3) find coinciding AFF points, which are the ff-AFF conditions. The change in Cs of the microscope for the specimens concerned can be effected by changing the specimen position along the optical axis. Most conventional electron microscopes have such a capability. However, for large values of Cs and Af, highly coherent electron waves, for example field emission electrons, must be used for obtaining high-contrast images. Acknowledgements--It must be mentioned here that one of the present authors (H.H.) has been enormouslystimulatedand helped by many contacts with Prof. Mike Whelan since 1959. In particular, he should mention the fruitful collaborationwith Prof. Mike Whelan and Prof. Archie Howie in which they carried out the work on the anomalousabsorptionof electron waves appearingin the electronmicroscopeimages. This was organizedby the kind arrangementsof Prof. Sir James Menter, Prof. Sir Peter Hirsch and Prof. Dr Jack Nutting.It is a greathonourfor the presentauthorsto have this opportunityto contributethe present paper, coincidingwith the retirement of Prof. M. J. Whelan F.R.S. The authors are grateful to Dr Toshikazu Honda of JEOL for his helpful discussionsand the preparation of Fig. 14 and Table 5, and also to Prof. RobertFisherand Dr John Hutchisonfor their help with the English of this paper.
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