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STEM
Ultramicroscopy 36 (1991) 319-329 North-Holland
319
A study of small electron probe formation in a field emission gun TEM/STEM J.K. Weiss, R.W. Carpenter and A.A. Higgs Center for Solid State Science, Arizona State University, Tempe, A Z 85287-1704, USA Received 4 March 1991
The optics of the illuminating system of a TEM/STEM equipped with a high-brightness field emission source has been studied in order to determine the current density distributions of small electron probes used for microanalysis. The experimental distributions were measured from high-magnification TEM images of the small probes, and compared to theoretical distributions calculated using both coherent and incoherent optics. The microscope was shown to be capable of producing an electron probe with a FWHM diameter of 1.7 nm and a current of 1 hA. The size of the probe is limited by electrical and mechanical instabilities.
1. Introduction Analytical electron microscopes of the T E M / STEM or dedicated STEM type fitted with field emission electron sources have been successfully used to analyze phase transformation products of about 2 nm maximum dimension for materials science research [1] and are capable of producing analytical information from specimen areas with characteristic dimensions of 1 nm [2]. The lower limit for spatial resolution of spectroscopic methods in these microscopes is the incident probe size at the specimen plane, and the impressive spatial resolution of these microscopes results from the characteristic high brightness and small source size of field emission guns. The characteristic probe size which describes the probe diameter is variously defined as the full-width half-maximum (FWHM) of the current distribution [3], the diameter containing 70% of the total beam current [4], or the 20-80 width of the scattered intensity distribution from a probe scanned across an atomically abrupt interface [5]. These definitions were created to deal with the fact that direct measurements of the probe current density distributions are generally unavailable in
dedicated STEM's. In TEM/STEM microscopes it is easy to form real-space images of focussed probes at the specimen plane using the post-specimen lenses, and these images are very useful for aligning the probe-forming optics [6]. Measurement of these probe current distributions can be achieved by scanning a high-magnification realspace image of the focussed probe across the small entrance aperture of an electron detector [7] or they can be analyzed photographically [8]. The real-space probe image is subject to the same distortions from the imaging lenses (especially the post-specimen objective) as an ordinary specimen image, so the appropriate aberrations and defocus must be taken into account during the analysis. In this paper we report measured current density distributions for focussed probes formed under several different sets of electron-optical conditions in a Philips EM400ST-FEG TEM/STEM fitted with a low-spherical-aberration (Cs= 1.5 mm) supertwin objective lens and a field emission electron gun. The probe current distributions were measured by scanning high-magnification probe images across a small aperture at the entrance of a parallel-detecting EELS spectrometer using an external computer, which synchronized the posi-
0304-3991/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
320
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
tional scans with the detector readout. Theoretical calculations of probe current distributions using the electron-optical parameters of the microscope illuminating system were also carried out by methods of coherent and incoherent optics [9,10] in order to determine the best method for modeling and predicting probe current distributions over a wide range of operating conditions, but especially those best suited for nanospectroscopy.
2. Theoretical background The usual treatment of electron probe formation considers the electron probe at the specimen plane to be a demagnified image of the electron gun crossover as formed by the condenser lens(es) and the upper half of the symmetric objective twin lens [11]. If the gun crossover is assumed to have a Gaussian real-space current distribution [12], then the final image of the probe at the specimen plane will be a demagnified image of the Gaussian crossover convoluted with the point-spread functions of the probe-forming lenses [13]. A coherent optics approach considers the convolution of amplitudes and is therefore subject to coherent interference effects, whereas an incoherent optics approach uses the convolution of intensities and one does not need to consider interference effects [14].
2.1. Incoherent optics approach If the object for the image-forming lens is considered to be an extended self-luminous incoherent source, the formation of the image by an imperfect imaging system is described by convolving the magnified (perfect) image of the object with the point-spread function of the lens [15]. The point-spread function of the lens is a convolution of the point-spread functions of each of the significant lens aberrations, diffraction from the angle-limiting aperture, and defocus. Descriptions of the sources of these errors and the associated mathematical equations are given in several texts [10,16]. Only the effects of spherical aberration, diffraction, defocus and chromatic aberration will be considered here, since the effects of the other
aberrations can usually be minimized by proper microscope alignment. The intensity distribution due to diffraction from the circular probe-defining condenser aperture is described by the equation
[ Jl(2~rao/X) ]2 Ja(r) ec 21rroto/x
(1)
and has as its most prominent feature the so-called Airy disk of diameter 1.22X/a0, where a 0 is the convergence half-angle of the beam incident on the specimen plane and X is the electron wavelength. Although it is not possible to write an analytical expression for the intensity distribution due to spherical aberration and defocus, a simple expression for the current density distribution as a function of incidence angle a [appendix A] is given by
/0
1
J s ( a ) = ~ra°2 (C~a2_Az)(3C~a2_Az),
(2)
where the relationship between a and r is given by r(ct) = Csc~3 - Azct.
(3)
Cs is the sperical aberration coefficient of the probe-forming lens, Az is the underfocus from the Gaussian focal plane, and a 0 is the convergence half-angle. Using these expressions, it is straightforward to deduce some general properties of spherically aberrated probes. At the Gaussian focal plane (Az = 0), the current density distribution is given by Js(r)
10
31ra2C?/3
r-4/3.
(4)
For 0 < Az < 3Csa~/4 , there is a bright spot at the center surrounded by a bright ring of radius r = 2 [Aza/(3Cs)]I/2, superimposed on a dim disk of radius r = Csa 3 - Aza 0. The intensity distribution due to chromatic aberration is described [16] by a simple Gaussian
J~(r) = e -(~/~)~,
(5)
J.K. Weiss et al. / Small electron probe formation in T E M / S T E M
with
321
x 105
I
2.50
,
Af
I
I
rc = zCcao-f-L, 2.00 -
fr
1 + E/E o 1 + E/2E o '
where Cc is the chromatic aberration coefficient, A f / f is the focus spread and fr is a relativistic term. The convolution of the separate point-spread functions is carried out by the use of Fourier transforms and the convolution theorem [17]. The numerical evaluation of these transforms is easily carried out on a minicomputer using standard fast Fourier transform (FFT) algorithms coded in F O R T R A N [18]. Typical results of such calculations are shown in fig. l a for various values of a0, corresponding to several diffraction- and aberration-limited situations. The variation in the current density distribution with Az for an aberration-limited probe is shown in fig. lb; the characteristic features described above can be easily seen in these theoretical distributions.
I
a
:t
P'~",~~
'if
1.50 -
I q 1.00-
!~
........
2
.......
4
4,5 18
......
6
40.5
......... .....
10 15
112.5 253
- -
20
450
/" ! .-., i", !',
0•50 -
/ :..'I i
0.00
-5.00
. ",
i ~,
,: i"
I
I
-3.00
-1.00
r
I
1.00
3.00
5.00
Radius (rim) x 105 I
1.50
I
I
I
b 1.20 -
0.0
0.90 -
.....
220
......... ......
450 66O
0.60 0.30 /
2.2. Coherent optics approach The use of coherent optics simplifies significantly the numerical calculations. The basis for this simplification is the existence of analytical expressions for the contribution of the objective lens imaging defect to the lens transfer function. This transfer function, which describes the selective transfer of various spatial frequencies to the final image [19], is then multiplied by the amplitude distribution in the back focal plane of the imaging lens. This distribution is given b y the Fourier transform of the object amplitude distribution, since it is equivalent to the Fraunhofer diffraction pattern formed by illuminating the object. Finally, this product gives the image amplitude distribution after inverse Fourier transformation. If it is assumed that the object intensity distribution is of Gaussian form with characteristic diameter 2rg, then the object amplitude distribu-
0.00 -7.00
.,.,
,, ....... i
I -4.20
..,. ..• ~ ...... -
I -1.40
I
I
1.40
4.20
7.00
Radius (nm) Fig. 1. Radial current density distributions calculated from incoherent optics• (a) Cs = 1.5 mm, rg = 0.5 nm, focussed at circle of least confusion• (b) Cs = 1.5 ram, rg = 0.3 nm, a 0 = 20 mrad.
tion will also be Gaussian, but with radius v~-rg. The Fourier transform will then be Gaussian, with radius qg=(V~-~rrs) -1. The transfer function is given by the expression T(o~) = e -ix(a) e -(Ira2rc/h)2 P(ot)
(6)
where X ( a ) is the wave aberration function given by =
q'/" ( C s o t
4 -
2Aza
2)
,
(7)
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
322
rc is related to the chromatic aberration coefficient by
x 105
I
7.50
I
I
I
a
1 r c = 4 ln~n-~ C~
........
f,
6.00 -
and P ( a ) is the pupil function which describes the effect of the objective aperture, and has a value of 1 f o r a < a 0 a n d 0 f o r a _ > a 0. A n analytical expression for the current density distribution can then be written in terms of the two-dimensional inverse Fourier transform of (6), J(r) =
Iffr(q)
~0 (mrnd) /~ (am)
4.50 -
3.00
4.5
4
18
6
40.5
10
112.5
15
253
20
450
i
1.50 --
t'i ;',,-,,:
i:"_".
0.00 -5.00
exp(2~riq- r ) d2ql 2,
2
I -3.00
I
I
I
1.00 Radius (nrn)
-1.00
3.00
5.00
which reduces to the one-dimensional integral x 106 J(r)=
+ r)~t4)]
Cfo a° exp[ ¢r2V(r2~2
I
I
I
b t~ (am)
×ex [-i I sO4-
1 . 2 0 --
0.0
0.90 --
2
x Jo (2 7r~r/X) ~ d ~
I
1.50
,
.....
220
.........
450
(8) 0.60 -
where the zero-order Bessel function is the result of the azimuthal integration [20]. The numerical evaluation of these expressions is again carried out o n a minicomputer using algorithms coded in F O R T R A N . The calculations for a coherent source which corresponds to the conditions used for figs. l a and l b are shown in figs. 2a and 2b.
2.3. Calculating the probe image T h e theoretical p r o b e calculations for incoherent probes shown in fig. l a allow one to distinguish clearly between diffraction-limited and aberration-limited probes. These curves are all calculated at the defocus corresponding to the circle of least confusion, Az = 0.75Csct 2. The curves for the smallest values of convergence used (2, 4 and 6 mrad) are diffraction-limited, and do not exhibit significant "tails" which are characteristic of the larger convergence aberration-limited probes. The curve corresponding to 10 m r a d convergence is near the o p t i m u m convergence angle at which the size of the Airy disk is equal to the size of the aberration disk. The curves calculated
0.30
-
0.00 -5.00
, -3.00
..... :i "/? .... -1.00
.... 1.00
3.00
5.00
Radius (nm) Fig. 2, Radial current density distributions calculated from coherent optics. (a) Cs = 1.5 ram, rg = 0.5 nm, focussed at circle of least confusion. (b) Cs = 1.5 mm, rg = 0.3 nm, a 0 = 20 mrad.
for a large convergence angle (15, 20 mrad) have a significant a m o u n t of current in their tails due to spherical aberration of the p r o b e - f o r m i n g lens. The current in the p r o b e increases as ct0 increases, thus large currents, corresponding to high spectral count rates, can easily be obtained by increasing % into the aberration-limited range, but with serious loss of spatial resolution. Calculated current distributions for an aberration-limited p r o b e are shown in fig. l b over a large range of defocus values. It is evident that significant tails remain a r o u n d the central maxim u m of the current distribution even at o p t i m u m
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
defocus (450 nm); at appreciably larger defocus even the central maximum disappears. A comparison of the coherent and incoherent calculations suggests that coherent destructive interference effects can destroy the bright ring which is expected to be present in the aberration-limited incoherent probes. Since this bright ring is always observed experimentally in the microscope for aberration-limited probes, it can be concluded that the electron-optical process of probe formation is not completely coherent, at least not for the aberration-limited probes. This is illustrated dearly in fig. 2b, where the calculated curves do not exhibit the strong tails which develop experimentally with the variation in defocus. For the non-aberrationlimited probes, the similarity between the incoherent and coherent calculations indicates that incoherent optics will adequately describe the probe-forming optics for all current microscope operating modes. The effects of aberrations of the other probeforming lenses (the condenser lenses) is included in the incoherent calculations by convoluting the point-spread functions for each of these lenses with the final image. An important consideration in this approach is that, due to the high angular magnification of the upper half of the objective lens (55 × ), the condenser lenses can be considered as operating entirely in the paraxial imaging mode such that, even though their aberration coefficients are an order of magnitude higher than those of the objective lens, the sizes of their point-spread functions are negligible compared to the probe size. This same assumption is used in high-resolution imaging, where the point-spread functions of the magnifying lenses are ignored in calculating the microscope resolution. The influence of the lower half of the objective lens on the image of the probe must be considered. Since the objective is considered a symmetric immersion lens, with the specimen at the center of the pole-piece gap, the point-spread function of the lower half can be considered identical to that of the upper half. In calculating probe images it is then necessary to convolute the current density distributions calculated at the specimen plane with the point-spread function of the lower half of the objective lens. This effect will be insignificant for
323
the non-aberration-limited cases, but for the aberration-limited probes, the total imaging error will be given by r t o t ( ~ ) = CsU~ 3 - AzUo~ q- Cslo¢3 - AZIo~,
(9)
where the superscripts u and 1 denote values for the upper and lower halves of the objective lens, respectively. Since the objective lens is symmetric, the aberration coefficients of the upper and lower halves are equal, and since the object plane for the lower objective moves the same distance and in the same relative direction as the image plane for the upper objective, the defocus values are also equal. Therefore, in the probe image, rtot (or) -- 2Csa3 - 2Azot,
(10)
which implies that the probe image should appear twice as large as the actual probe at the specimen plane for the aberration-limited probes.
3. Experimental details The values of the electron-optical parameters of the microscope were supplied by the manufacturer but were confirmed experimentally where feasible. The measured values were used whenever available in order to maintain experimental consistency. Table 1 lists values for the important parameters. The microscope was aligned by special methods to avoid off-axis aberrations, paying careful attention to the gun tilt and pre-specimen beam tilt alignments which, if misadjusted, lead to first-
Table 1 Electron-optical parameters of the pre-specimen objective lens in the Philips EM400ST Optical parameter
Measured
Cs (mm)
1.5 + 0.1
Manufacturer 1.5
c~ (mm)
1.1
Focal length (nun) Magnification (C O = 500): TEM mode 0.023 + 0.027 STEM mode 0.0084 ± 0.011 Defocus increment (nm): Step size 4 48 ± 7 Step size 5 220 ± 20
1.4
50 200
324
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
order axial astigmatism and large asymmetrical spherical aberration tails of the probe current distribution at the specimen plane. It is important to set the post-specimen objective stigmators correctly (for example, using the image of a thin amorphous specimen region) before the pre-specimen condenser stigmators are adjusted. Objective astigmatism can lead to a probe i m a g e which appears stigmatic while the actual probe at the specimen plane suffers from astigmatism. Such an effect can b e easily observed in the diffraction mode by observing the symmetry of the tangential circle of infinite magnification in, the convergentb e a m shadow image [21]. The probe current density distributions were measured by scanning high-magnification (1M X ) images of the focused probes across a 150 /~m aperture at the entrance of a G a t a n 666 parallel EELS (PEELS). The electron detector effective size was thus about 0.15 nm. The scans were produced by adding a small voltage from an analog-to-digital converter (ADC) to the pre-specimen beam-deflection coils. The A D C was driven by a microcomputer which synchronously collected an EELS spectrum (containing the zero-loss peak) at each scan position and, after subtracting the dark counts, summed the counts in the zeroloss peak of each spectrum [22]. The scan length was measured f r o m m i c r o g r a p h s taken at calibrated magnifications. The total probe current was measured by directing the probe image (at low magnification) into a Faraday cup mounted in the microscope viewing chamber and connected to a digital picoammeter.
4. Experimental results and discussion Current density distributions measured for each of the small-probe operating modes of the microscope are shown in fig. 3, and characteristic sizes calculated from each of these distributions are shown in table 2. High-magnification micrographs of each of these probes are shown in fig. 4. The agreement between experimental and theoretical probe current distributions is very good in all cases shown except fig. 3d, corresponding to a strongly aberration-limited STEM probe. An image of this probe is shown in fig. 4e. This discrepancy close to the probe center is most likely due to third-order astigmatism which is clearly observed in the high-magnification image as the three-fold symmetric figure at the probe center. The probes shown in fig. 4 can be roughly characterized as aberration-limited and non-aberration-limited. The aberration-limited probes (figs. 4a, 4d and 4e) are those formed with a convergence half-angle greater than 10 mrad. It is important to note that at the Gaussian image plane, the probe current density varies as r - 4 / 3 (eq. (4)) and is strongly peaked at the probe center. One can adjust the dark level of a STEM image so that the r a n d o m contributions to the signal from areas of the probe outside a radius of r0 do not contribute to the image. It is therefore possible to improve the spatial resolution of a STEM image to significantly lower than the actual probe diameter at the expense of losing image contrast. In that the observed spatial resolution is determined by the
Table 2 Summary of results of probe spatial current density distribution measurements at Gaussian focal plane (Az = 0) Mode TEM TEM TEM STEM STEM
Cond. Ap. dia. (/~m) 150 50 15 50 15
FWHM dia. (nm) 2.34+0.33 1.80 + 0.20 1.63 + 0.11 0.92 + 0.06 1.01 + 0.08
70% dia. (nm) 6.36+0.33 3.00 + 0.20 3.15 + 0.11 7.29 + 0.06 2.56 + 0.08
90% dia. (nm) 11.7 +0.33 5.40 + 0.20 5.53 + 0.11 8.05 + 0.06 4.73 + 0.08
20-80 wid. (nm) 2.68 + 0.33 1.60 + 0.20 1.52+0.11 2.50 + 0.06 1.24 + 0.08
326
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
Fig. 4. T E M i m a g e s of electron p r o b e s focussed at the circle of least c o n f u s i o n for a v a r i e t y of c o n v e r g e n c e half-angles. (a) a o = 15 m r a d , A z = 5 0 0 nm, T E M mode. (b) a o = 5 . 2 mrad, A z = 6 0 nm, T E M mode. (c) a 0 = 2 . 2 m r a d , A z ~ 1 0 rim, T E M mode. (d) a o = 74 mrad, Az = 12 /~m, S T E M mode. (e) a o = 23 m r a d , Az = 1.2 /~m, S T E M mode. (f) a o = 9.6 m r a d , Az = 200 nm, S T E M mode.
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
size and sharpness of the central current maximum, care must be taken in assuming that the total probe diameter determines the spatial resolution observed in a STEM image. The non-aberration-limited probes (figs. 4b, 4c and 4f) are subject to various limitations on probe size. The TEM probes suffer from a large Gaussian spot size, rg = 0.8 nm. The source of this problem appears to be instability in the reference voltages of the gun alignment coils. Monitoring of the alignment supply voltages showed a measureable ripple of magnitude less than 1 mV at 60 Hz frequency. Since a voltage of 100 mV was observed to produce a beam displacement of 400 nm, the positional instability of the probe due to the voltage ripple is estimated to be of the order of 1 nm. The smallest TEM probe (a 0 = 2 mrad) is also close to being diffraction-limited, although the Airy disk size is comparable to the demagnified Gaussian source size. The smallest STEM probe ( a 0 = 9.6 mrad) is not subject to the same limitation on probe size from the Gaussian spot size, due to the increased demagnification (by about a factor of 4) of the post-gun alignment coil lenses when operating in the STEM mode. The small STEM probe ( a 0 = 9.6 mrad) is the closest to having optimum convergence angle; it is almost equally aberration- and diffraction-limited. The available current and current density vary widely with the probe-defining parameters. The total beam current varies as the square of the condenser aperture size and as the inverse square of the demagnification of the illuminating system. The demagnification should be kept high in order to reduce the Gaussian spot size, and the convergence half-angle should be small in order to reduce the effects of spherical aberration. Both can only be done at the expense of beam current. The current density is also strongly dependent on convergence angle, defocus, and spherical aberration coefficient (see figs. 1 and 2). The most useful probe for microanalysis is the 5 mrad TEM probe. The current in this probe exceeds that of the small STEM probe by at least one order of magnitude, but the F W H M diameter is only about twice as large, and is limited by the instabilities mentioned above.
327
5. Conclusions The measured current density distributions of small focussed probes agreed well with the distributions calculated with standard incoherent optics, and for small convergence angles, coherent optics. It is therefore possible to characterize the probeforming optics of the microscope in order to fully describe the electron probe at the specimen plane, and to describe the probe current density distributions with a spatial resolution of better than 0.2 am. Optimum operation of the microscope for microanalysis depends on both the spatial resolution and the sensitivity desired. Increased current density necessary for high sensitivity can be obtained by decreasing the demagnification of the illuminating system which increases the collection solid angle at the gun. The current density for the small T E M probes ( a 0 < 10 mrad) depends inversely on the illuminating system magnification, while the probe size depends linearly on the magnification. If the goal of a study is to detect trace amounts of an element in a matrix, the current density can be increased at the expense of degrading the spatial resolution, and even then, one must consider the effects of intense electron beam irradiation on the sample [23,24]. In all cases, accurate knowledge of the actual probe current density distribution at the specimen plane is necessary to evaluate the interpretable resolution available to high-spatial-resolution scanning probe techniques [25].
Acknowledgements The authors wish to acknowledge P. Rez for assistance in programming the on-line acquisition computer and P. Perkes for assistance in off-line processing and programming. This research was supported by a grant from the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences grant # D E - F G 0 2 - 8 7 E R 45305; the experimental work was carried out in the National Facility for High Resolution Electron Microscopy within the Center for Solid State Science at Arizona State University.
J.K. Weiss et aL / Small electron probe formation in TEM / STEM
328
Appendix A. Derivation of the spherical aberration current density distribution W e start with a p o i n t source in the o b j e c t p l a n e of a G a u s s i a n lens (see fig. 5) a n d a s s u m e t h a t a c o n s t a n t a n g u l a r c u r r e n t d e n s i t y Ja = d I / d I 2 passes t h r o u g h t h a t point. If an a p e r t u r e limits the m a x i m u m t r a n s m i t t e d angle to a 0 on the i m a g i n g side o f the lens a n d a total c u r r e n t I 0 passes t h r o u g h t h a t aperture, the a n g u l a r c u r r e n t d e n s i t y will be given b y Ja = Io/~2, where the t o t a l solid angle s u b t e n d i n g the a p e r t u r e is, using the smallangle a p p r o x i m a t i o n , I2 = rra 2. A small solid angle d~2 = sin a d a dq~ = a d a d ~ c o n t a i n s a c u r r e n t dI I0 d I = ~ - ~ d I 2 = - - - 2 a d a dq~, ~ra 0
Co--Len
Condenser Aperture
~
I
) fl
!
~
d A = r d r ddp
= ( Csot3 - Aza)(3Csot2 - A z ) dot ddp,
(A.2)
w h e r e d A has b e e n w r i t t e n in (A.2) as a f u n c t i o n of the c o n v e r g e n c e angle a. T h e c u r r e n t d e n s i t y J = dI/dA can now be written Js(a) =
I0 rta~
1 (Csa=-Az)(3C~a2-Az)
'
(A.3)
w h i c h c a n n o t b e w r i t t e n explicitly in terms o f the r a d i u s r.
(A.1)
w h i c h will b e s p r e a d i n t o a finite a r e a d A in the i m a g e p l a n e of the lens d u e to spherical a b e r r a t i o n a n d defocus. T h e p o s i t i o n a l d e v i a t i o n f r o m the o p t i c axis d u e to the i m a g e defect is given b y eq. (3), where the values of Cs a n d Az are given for the i m a g i n g side of the lens ( a n d are r e l a t e d to the
" %, £
values o n the o b j e c t side of the lens b y the lens m a g n i f i c a t i o n M ) . T h e a r e a d A c a n t h e n b e written as
Objective
Lens
Gaussian Focal Plane --Cst~ 3- Aztxo
Csct o Fig. 5. Ray diagram showing image formation of a point source by a lens with spherical aberration.
References [1] R.W, Carpenter, Y.L. Chen, M.J. Kim and J.C. Barry, in: Proc. Microscopy of Semiconducting Materials Conf., Oxford, UK, 1989, Inst. Phys. Conf. Ser. 100, Section 7, p. 543. [2] C. CoUiex, J.L. Maurice and D. Ugarte, Ultramicroscopy 29 (1989) 88. [3] J.R. Michael and D.B. Williams, J. Microscopy 147 (1987) 289. [4] C. Mory, C. Colliex and J.M. Cowley, Ultramicroscopy 21 (1987) 171. [5] J.A. Venables and A.P. Janssen, Ultramicroscopy 5 (1980) 297. [6] R.W. Carpenter and J.C.H. Spence, J. Microscopy 136 (1984) 165. [7] C. Colliex and C. Mory, Optik 59 (1981) 311. [8] G. Cliff and P.B. Kenway, in: Microbeam Analysis (San Francisco Press, San Francisco, 1982) p. 107. [9] C. Mory, M. Tence and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 381. [10] O.C. Wells, in: Scanning Electron Microscopy (McGrawHill, New York, 1974) p. 69. [11] A.V. Crewe, Ultramicroscopy 23 (1987) 159. [12] J.C. Wiesner and T.E. Everhart, J. Appl. Phys. 44 (1973) 2140. [13] A.V. Crewe and D.B. Salzman, Ultramicroscopy 9 (1982) 373. [14] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, UK, 1980). [15] C.S. Williams and O.A. Becklund, Introduction to the Optical Transform Function (Wiley, New York, 1989). [16] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [17] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1984) p. 34.
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
[18] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge University Press, New York, 1986. [19] P.M. Duffieux, The Fourier Transform and its Application to Optics (Wiley, New York, 1983). [20] C.R. Wylie and L.C. Barrett, Advanced Engineering Mathematics (McGraw-Hill, New York, 1982) p. 589. [21] R.W. Carpenter, I.Y.T. Chan and J.M. Cowley, in: Proc. 40th Annu. EMSA Meeting (Claitor's, New Orleans, 1982) p. 696.
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[22] J.K. Weiss, P. Rez and A.A. Higgs, to be published. [23] M.R. McCartney, P.A. Crozier, J.K. Weiss and D.J. Smith, Vacuum 42 (1990) 301. [24] K. Das Chowdhury, R.W. Carpenter, J.K. Weiss and A.A. Higgs, in: Proc. 47th Annu. EMSA Meeting (San Francisco Press, San Francisco, 1989) p. 428. [25] J.K. Weiss and R.W. Carpenter, to be published.
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A study of small electron probe formation in a field emission gun TEM/STEM J.K. Weiss, R.W. Carpenter and A.A. Higgs Center for Solid State Science, Arizona State University, Tempe, A Z 85287-1704, USA Received 4 March 1991
The optics of the illuminating system of a TEM/STEM equipped with a high-brightness field emission source has been studied in order to determine the current density distributions of small electron probes used for microanalysis. The experimental distributions were measured from high-magnification TEM images of the small probes, and compared to theoretical distributions calculated using both coherent and incoherent optics. The microscope was shown to be capable of producing an electron probe with a FWHM diameter of 1.7 nm and a current of 1 hA. The size of the probe is limited by electrical and mechanical instabilities.
1. Introduction Analytical electron microscopes of the T E M / STEM or dedicated STEM type fitted with field emission electron sources have been successfully used to analyze phase transformation products of about 2 nm maximum dimension for materials science research [1] and are capable of producing analytical information from specimen areas with characteristic dimensions of 1 nm [2]. The lower limit for spatial resolution of spectroscopic methods in these microscopes is the incident probe size at the specimen plane, and the impressive spatial resolution of these microscopes results from the characteristic high brightness and small source size of field emission guns. The characteristic probe size which describes the probe diameter is variously defined as the full-width half-maximum (FWHM) of the current distribution [3], the diameter containing 70% of the total beam current [4], or the 20-80 width of the scattered intensity distribution from a probe scanned across an atomically abrupt interface [5]. These definitions were created to deal with the fact that direct measurements of the probe current density distributions are generally unavailable in
dedicated STEM's. In TEM/STEM microscopes it is easy to form real-space images of focussed probes at the specimen plane using the post-specimen lenses, and these images are very useful for aligning the probe-forming optics [6]. Measurement of these probe current distributions can be achieved by scanning a high-magnification realspace image of the focussed probe across the small entrance aperture of an electron detector [7] or they can be analyzed photographically [8]. The real-space probe image is subject to the same distortions from the imaging lenses (especially the post-specimen objective) as an ordinary specimen image, so the appropriate aberrations and defocus must be taken into account during the analysis. In this paper we report measured current density distributions for focussed probes formed under several different sets of electron-optical conditions in a Philips EM400ST-FEG TEM/STEM fitted with a low-spherical-aberration (Cs= 1.5 mm) supertwin objective lens and a field emission electron gun. The probe current distributions were measured by scanning high-magnification probe images across a small aperture at the entrance of a parallel-detecting EELS spectrometer using an external computer, which synchronized the posi-
0304-3991/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
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J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
tional scans with the detector readout. Theoretical calculations of probe current distributions using the electron-optical parameters of the microscope illuminating system were also carried out by methods of coherent and incoherent optics [9,10] in order to determine the best method for modeling and predicting probe current distributions over a wide range of operating conditions, but especially those best suited for nanospectroscopy.
2. Theoretical background The usual treatment of electron probe formation considers the electron probe at the specimen plane to be a demagnified image of the electron gun crossover as formed by the condenser lens(es) and the upper half of the symmetric objective twin lens [11]. If the gun crossover is assumed to have a Gaussian real-space current distribution [12], then the final image of the probe at the specimen plane will be a demagnified image of the Gaussian crossover convoluted with the point-spread functions of the probe-forming lenses [13]. A coherent optics approach considers the convolution of amplitudes and is therefore subject to coherent interference effects, whereas an incoherent optics approach uses the convolution of intensities and one does not need to consider interference effects [14].
2.1. Incoherent optics approach If the object for the image-forming lens is considered to be an extended self-luminous incoherent source, the formation of the image by an imperfect imaging system is described by convolving the magnified (perfect) image of the object with the point-spread function of the lens [15]. The point-spread function of the lens is a convolution of the point-spread functions of each of the significant lens aberrations, diffraction from the angle-limiting aperture, and defocus. Descriptions of the sources of these errors and the associated mathematical equations are given in several texts [10,16]. Only the effects of spherical aberration, diffraction, defocus and chromatic aberration will be considered here, since the effects of the other
aberrations can usually be minimized by proper microscope alignment. The intensity distribution due to diffraction from the circular probe-defining condenser aperture is described by the equation
[ Jl(2~rao/X) ]2 Ja(r) ec 21rroto/x
(1)
and has as its most prominent feature the so-called Airy disk of diameter 1.22X/a0, where a 0 is the convergence half-angle of the beam incident on the specimen plane and X is the electron wavelength. Although it is not possible to write an analytical expression for the intensity distribution due to spherical aberration and defocus, a simple expression for the current density distribution as a function of incidence angle a [appendix A] is given by
/0
1
J s ( a ) = ~ra°2 (C~a2_Az)(3C~a2_Az),
(2)
where the relationship between a and r is given by r(ct) = Csc~3 - Azct.
(3)
Cs is the sperical aberration coefficient of the probe-forming lens, Az is the underfocus from the Gaussian focal plane, and a 0 is the convergence half-angle. Using these expressions, it is straightforward to deduce some general properties of spherically aberrated probes. At the Gaussian focal plane (Az = 0), the current density distribution is given by Js(r)
10
31ra2C?/3
r-4/3.
(4)
For 0 < Az < 3Csa~/4 , there is a bright spot at the center surrounded by a bright ring of radius r = 2 [Aza/(3Cs)]I/2, superimposed on a dim disk of radius r = Csa 3 - Aza 0. The intensity distribution due to chromatic aberration is described [16] by a simple Gaussian
J~(r) = e -(~/~)~,
(5)
J.K. Weiss et al. / Small electron probe formation in T E M / S T E M
with
321
x 105
I
2.50
,
Af
I
I
rc = zCcao-f-L, 2.00 -
fr
1 + E/E o 1 + E/2E o '
where Cc is the chromatic aberration coefficient, A f / f is the focus spread and fr is a relativistic term. The convolution of the separate point-spread functions is carried out by the use of Fourier transforms and the convolution theorem [17]. The numerical evaluation of these transforms is easily carried out on a minicomputer using standard fast Fourier transform (FFT) algorithms coded in F O R T R A N [18]. Typical results of such calculations are shown in fig. l a for various values of a0, corresponding to several diffraction- and aberration-limited situations. The variation in the current density distribution with Az for an aberration-limited probe is shown in fig. lb; the characteristic features described above can be easily seen in these theoretical distributions.
I
a
:t
P'~",~~
'if
1.50 -
I q 1.00-
!~
........
2
.......
4
4,5 18
......
6
40.5
......... .....
10 15
112.5 253
- -
20
450
/" ! .-., i", !',
0•50 -
/ :..'I i
0.00
-5.00
. ",
i ~,
,: i"
I
I
-3.00
-1.00
r
I
1.00
3.00
5.00
Radius (rim) x 105 I
1.50
I
I
I
b 1.20 -
0.0
0.90 -
.....
220
......... ......
450 66O
0.60 0.30 /
2.2. Coherent optics approach The use of coherent optics simplifies significantly the numerical calculations. The basis for this simplification is the existence of analytical expressions for the contribution of the objective lens imaging defect to the lens transfer function. This transfer function, which describes the selective transfer of various spatial frequencies to the final image [19], is then multiplied by the amplitude distribution in the back focal plane of the imaging lens. This distribution is given b y the Fourier transform of the object amplitude distribution, since it is equivalent to the Fraunhofer diffraction pattern formed by illuminating the object. Finally, this product gives the image amplitude distribution after inverse Fourier transformation. If it is assumed that the object intensity distribution is of Gaussian form with characteristic diameter 2rg, then the object amplitude distribu-
0.00 -7.00
.,.,
,, ....... i
I -4.20
..,. ..• ~ ...... -
I -1.40
I
I
1.40
4.20
7.00
Radius (nm) Fig. 1. Radial current density distributions calculated from incoherent optics• (a) Cs = 1.5 mm, rg = 0.5 nm, focussed at circle of least confusion• (b) Cs = 1.5 ram, rg = 0.3 nm, a 0 = 20 mrad.
tion will also be Gaussian, but with radius v~-rg. The Fourier transform will then be Gaussian, with radius qg=(V~-~rrs) -1. The transfer function is given by the expression T(o~) = e -ix(a) e -(Ira2rc/h)2 P(ot)
(6)
where X ( a ) is the wave aberration function given by =
q'/" ( C s o t
4 -
2Aza
2)
,
(7)
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
322
rc is related to the chromatic aberration coefficient by
x 105
I
7.50
I
I
I
a
1 r c = 4 ln~n-~ C~
........
f,
6.00 -
and P ( a ) is the pupil function which describes the effect of the objective aperture, and has a value of 1 f o r a < a 0 a n d 0 f o r a _ > a 0. A n analytical expression for the current density distribution can then be written in terms of the two-dimensional inverse Fourier transform of (6), J(r) =
Iffr(q)
~0 (mrnd) /~ (am)
4.50 -
3.00
4.5
4
18
6
40.5
10
112.5
15
253
20
450
i
1.50 --
t'i ;',,-,,:
i:"_".
0.00 -5.00
exp(2~riq- r ) d2ql 2,
2
I -3.00
I
I
I
1.00 Radius (nrn)
-1.00
3.00
5.00
which reduces to the one-dimensional integral x 106 J(r)=
+ r)~t4)]
Cfo a° exp[ ¢r2V(r2~2
I
I
I
b t~ (am)
×ex [-i I sO4-
1 . 2 0 --
0.0
0.90 --
2
x Jo (2 7r~r/X) ~ d ~
I
1.50
,
.....
220
.........
450
(8) 0.60 -
where the zero-order Bessel function is the result of the azimuthal integration [20]. The numerical evaluation of these expressions is again carried out o n a minicomputer using algorithms coded in F O R T R A N . The calculations for a coherent source which corresponds to the conditions used for figs. l a and l b are shown in figs. 2a and 2b.
2.3. Calculating the probe image T h e theoretical p r o b e calculations for incoherent probes shown in fig. l a allow one to distinguish clearly between diffraction-limited and aberration-limited probes. These curves are all calculated at the defocus corresponding to the circle of least confusion, Az = 0.75Csct 2. The curves for the smallest values of convergence used (2, 4 and 6 mrad) are diffraction-limited, and do not exhibit significant "tails" which are characteristic of the larger convergence aberration-limited probes. The curve corresponding to 10 m r a d convergence is near the o p t i m u m convergence angle at which the size of the Airy disk is equal to the size of the aberration disk. The curves calculated
0.30
-
0.00 -5.00
, -3.00
..... :i "/? .... -1.00
.... 1.00
3.00
5.00
Radius (nm) Fig. 2, Radial current density distributions calculated from coherent optics. (a) Cs = 1.5 ram, rg = 0.5 nm, focussed at circle of least confusion. (b) Cs = 1.5 mm, rg = 0.3 nm, a 0 = 20 mrad.
for a large convergence angle (15, 20 mrad) have a significant a m o u n t of current in their tails due to spherical aberration of the p r o b e - f o r m i n g lens. The current in the p r o b e increases as ct0 increases, thus large currents, corresponding to high spectral count rates, can easily be obtained by increasing % into the aberration-limited range, but with serious loss of spatial resolution. Calculated current distributions for an aberration-limited p r o b e are shown in fig. l b over a large range of defocus values. It is evident that significant tails remain a r o u n d the central maxim u m of the current distribution even at o p t i m u m
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
defocus (450 nm); at appreciably larger defocus even the central maximum disappears. A comparison of the coherent and incoherent calculations suggests that coherent destructive interference effects can destroy the bright ring which is expected to be present in the aberration-limited incoherent probes. Since this bright ring is always observed experimentally in the microscope for aberration-limited probes, it can be concluded that the electron-optical process of probe formation is not completely coherent, at least not for the aberration-limited probes. This is illustrated dearly in fig. 2b, where the calculated curves do not exhibit the strong tails which develop experimentally with the variation in defocus. For the non-aberrationlimited probes, the similarity between the incoherent and coherent calculations indicates that incoherent optics will adequately describe the probe-forming optics for all current microscope operating modes. The effects of aberrations of the other probeforming lenses (the condenser lenses) is included in the incoherent calculations by convoluting the point-spread functions for each of these lenses with the final image. An important consideration in this approach is that, due to the high angular magnification of the upper half of the objective lens (55 × ), the condenser lenses can be considered as operating entirely in the paraxial imaging mode such that, even though their aberration coefficients are an order of magnitude higher than those of the objective lens, the sizes of their point-spread functions are negligible compared to the probe size. This same assumption is used in high-resolution imaging, where the point-spread functions of the magnifying lenses are ignored in calculating the microscope resolution. The influence of the lower half of the objective lens on the image of the probe must be considered. Since the objective is considered a symmetric immersion lens, with the specimen at the center of the pole-piece gap, the point-spread function of the lower half can be considered identical to that of the upper half. In calculating probe images it is then necessary to convolute the current density distributions calculated at the specimen plane with the point-spread function of the lower half of the objective lens. This effect will be insignificant for
323
the non-aberration-limited cases, but for the aberration-limited probes, the total imaging error will be given by r t o t ( ~ ) = CsU~ 3 - AzUo~ q- Cslo¢3 - AZIo~,
(9)
where the superscripts u and 1 denote values for the upper and lower halves of the objective lens, respectively. Since the objective lens is symmetric, the aberration coefficients of the upper and lower halves are equal, and since the object plane for the lower objective moves the same distance and in the same relative direction as the image plane for the upper objective, the defocus values are also equal. Therefore, in the probe image, rtot (or) -- 2Csa3 - 2Azot,
(10)
which implies that the probe image should appear twice as large as the actual probe at the specimen plane for the aberration-limited probes.
3. Experimental details The values of the electron-optical parameters of the microscope were supplied by the manufacturer but were confirmed experimentally where feasible. The measured values were used whenever available in order to maintain experimental consistency. Table 1 lists values for the important parameters. The microscope was aligned by special methods to avoid off-axis aberrations, paying careful attention to the gun tilt and pre-specimen beam tilt alignments which, if misadjusted, lead to first-
Table 1 Electron-optical parameters of the pre-specimen objective lens in the Philips EM400ST Optical parameter
Measured
Cs (mm)
1.5 + 0.1
Manufacturer 1.5
c~ (mm)
1.1
Focal length (nun) Magnification (C O = 500): TEM mode 0.023 + 0.027 STEM mode 0.0084 ± 0.011 Defocus increment (nm): Step size 4 48 ± 7 Step size 5 220 ± 20
1.4
50 200
324
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
order axial astigmatism and large asymmetrical spherical aberration tails of the probe current distribution at the specimen plane. It is important to set the post-specimen objective stigmators correctly (for example, using the image of a thin amorphous specimen region) before the pre-specimen condenser stigmators are adjusted. Objective astigmatism can lead to a probe i m a g e which appears stigmatic while the actual probe at the specimen plane suffers from astigmatism. Such an effect can b e easily observed in the diffraction mode by observing the symmetry of the tangential circle of infinite magnification in, the convergentb e a m shadow image [21]. The probe current density distributions were measured by scanning high-magnification (1M X ) images of the focused probes across a 150 /~m aperture at the entrance of a G a t a n 666 parallel EELS (PEELS). The electron detector effective size was thus about 0.15 nm. The scans were produced by adding a small voltage from an analog-to-digital converter (ADC) to the pre-specimen beam-deflection coils. The A D C was driven by a microcomputer which synchronously collected an EELS spectrum (containing the zero-loss peak) at each scan position and, after subtracting the dark counts, summed the counts in the zeroloss peak of each spectrum [22]. The scan length was measured f r o m m i c r o g r a p h s taken at calibrated magnifications. The total probe current was measured by directing the probe image (at low magnification) into a Faraday cup mounted in the microscope viewing chamber and connected to a digital picoammeter.
4. Experimental results and discussion Current density distributions measured for each of the small-probe operating modes of the microscope are shown in fig. 3, and characteristic sizes calculated from each of these distributions are shown in table 2. High-magnification micrographs of each of these probes are shown in fig. 4. The agreement between experimental and theoretical probe current distributions is very good in all cases shown except fig. 3d, corresponding to a strongly aberration-limited STEM probe. An image of this probe is shown in fig. 4e. This discrepancy close to the probe center is most likely due to third-order astigmatism which is clearly observed in the high-magnification image as the three-fold symmetric figure at the probe center. The probes shown in fig. 4 can be roughly characterized as aberration-limited and non-aberration-limited. The aberration-limited probes (figs. 4a, 4d and 4e) are those formed with a convergence half-angle greater than 10 mrad. It is important to note that at the Gaussian image plane, the probe current density varies as r - 4 / 3 (eq. (4)) and is strongly peaked at the probe center. One can adjust the dark level of a STEM image so that the r a n d o m contributions to the signal from areas of the probe outside a radius of r0 do not contribute to the image. It is therefore possible to improve the spatial resolution of a STEM image to significantly lower than the actual probe diameter at the expense of losing image contrast. In that the observed spatial resolution is determined by the
Table 2 Summary of results of probe spatial current density distribution measurements at Gaussian focal plane (Az = 0) Mode TEM TEM TEM STEM STEM
Cond. Ap. dia. (/~m) 150 50 15 50 15
FWHM dia. (nm) 2.34+0.33 1.80 + 0.20 1.63 + 0.11 0.92 + 0.06 1.01 + 0.08
70% dia. (nm) 6.36+0.33 3.00 + 0.20 3.15 + 0.11 7.29 + 0.06 2.56 + 0.08
90% dia. (nm) 11.7 +0.33 5.40 + 0.20 5.53 + 0.11 8.05 + 0.06 4.73 + 0.08
20-80 wid. (nm) 2.68 + 0.33 1.60 + 0.20 1.52+0.11 2.50 + 0.06 1.24 + 0.08
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J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
Fig. 4. T E M i m a g e s of electron p r o b e s focussed at the circle of least c o n f u s i o n for a v a r i e t y of c o n v e r g e n c e half-angles. (a) a o = 15 m r a d , A z = 5 0 0 nm, T E M mode. (b) a o = 5 . 2 mrad, A z = 6 0 nm, T E M mode. (c) a 0 = 2 . 2 m r a d , A z ~ 1 0 rim, T E M mode. (d) a o = 74 mrad, Az = 12 /~m, S T E M mode. (e) a o = 23 m r a d , Az = 1.2 /~m, S T E M mode. (f) a o = 9.6 m r a d , Az = 200 nm, S T E M mode.
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
size and sharpness of the central current maximum, care must be taken in assuming that the total probe diameter determines the spatial resolution observed in a STEM image. The non-aberration-limited probes (figs. 4b, 4c and 4f) are subject to various limitations on probe size. The TEM probes suffer from a large Gaussian spot size, rg = 0.8 nm. The source of this problem appears to be instability in the reference voltages of the gun alignment coils. Monitoring of the alignment supply voltages showed a measureable ripple of magnitude less than 1 mV at 60 Hz frequency. Since a voltage of 100 mV was observed to produce a beam displacement of 400 nm, the positional instability of the probe due to the voltage ripple is estimated to be of the order of 1 nm. The smallest TEM probe (a 0 = 2 mrad) is also close to being diffraction-limited, although the Airy disk size is comparable to the demagnified Gaussian source size. The smallest STEM probe ( a 0 = 9.6 mrad) is not subject to the same limitation on probe size from the Gaussian spot size, due to the increased demagnification (by about a factor of 4) of the post-gun alignment coil lenses when operating in the STEM mode. The small STEM probe ( a 0 = 9.6 mrad) is the closest to having optimum convergence angle; it is almost equally aberration- and diffraction-limited. The available current and current density vary widely with the probe-defining parameters. The total beam current varies as the square of the condenser aperture size and as the inverse square of the demagnification of the illuminating system. The demagnification should be kept high in order to reduce the Gaussian spot size, and the convergence half-angle should be small in order to reduce the effects of spherical aberration. Both can only be done at the expense of beam current. The current density is also strongly dependent on convergence angle, defocus, and spherical aberration coefficient (see figs. 1 and 2). The most useful probe for microanalysis is the 5 mrad TEM probe. The current in this probe exceeds that of the small STEM probe by at least one order of magnitude, but the F W H M diameter is only about twice as large, and is limited by the instabilities mentioned above.
327
5. Conclusions The measured current density distributions of small focussed probes agreed well with the distributions calculated with standard incoherent optics, and for small convergence angles, coherent optics. It is therefore possible to characterize the probeforming optics of the microscope in order to fully describe the electron probe at the specimen plane, and to describe the probe current density distributions with a spatial resolution of better than 0.2 am. Optimum operation of the microscope for microanalysis depends on both the spatial resolution and the sensitivity desired. Increased current density necessary for high sensitivity can be obtained by decreasing the demagnification of the illuminating system which increases the collection solid angle at the gun. The current density for the small T E M probes ( a 0 < 10 mrad) depends inversely on the illuminating system magnification, while the probe size depends linearly on the magnification. If the goal of a study is to detect trace amounts of an element in a matrix, the current density can be increased at the expense of degrading the spatial resolution, and even then, one must consider the effects of intense electron beam irradiation on the sample [23,24]. In all cases, accurate knowledge of the actual probe current density distribution at the specimen plane is necessary to evaluate the interpretable resolution available to high-spatial-resolution scanning probe techniques [25].
Acknowledgements The authors wish to acknowledge P. Rez for assistance in programming the on-line acquisition computer and P. Perkes for assistance in off-line processing and programming. This research was supported by a grant from the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences grant # D E - F G 0 2 - 8 7 E R 45305; the experimental work was carried out in the National Facility for High Resolution Electron Microscopy within the Center for Solid State Science at Arizona State University.
J.K. Weiss et aL / Small electron probe formation in TEM / STEM
328
Appendix A. Derivation of the spherical aberration current density distribution W e start with a p o i n t source in the o b j e c t p l a n e of a G a u s s i a n lens (see fig. 5) a n d a s s u m e t h a t a c o n s t a n t a n g u l a r c u r r e n t d e n s i t y Ja = d I / d I 2 passes t h r o u g h t h a t point. If an a p e r t u r e limits the m a x i m u m t r a n s m i t t e d angle to a 0 on the i m a g i n g side o f the lens a n d a total c u r r e n t I 0 passes t h r o u g h t h a t aperture, the a n g u l a r c u r r e n t d e n s i t y will be given b y Ja = Io/~2, where the t o t a l solid angle s u b t e n d i n g the a p e r t u r e is, using the smallangle a p p r o x i m a t i o n , I2 = rra 2. A small solid angle d~2 = sin a d a dq~ = a d a d ~ c o n t a i n s a c u r r e n t dI I0 d I = ~ - ~ d I 2 = - - - 2 a d a dq~, ~ra 0
Co--Len
Condenser Aperture
~
I
) fl
!
~
d A = r d r ddp
= ( Csot3 - Aza)(3Csot2 - A z ) dot ddp,
(A.2)
w h e r e d A has b e e n w r i t t e n in (A.2) as a f u n c t i o n of the c o n v e r g e n c e angle a. T h e c u r r e n t d e n s i t y J = dI/dA can now be written Js(a) =
I0 rta~
1 (Csa=-Az)(3C~a2-Az)
'
(A.3)
w h i c h c a n n o t b e w r i t t e n explicitly in terms o f the r a d i u s r.
(A.1)
w h i c h will b e s p r e a d i n t o a finite a r e a d A in the i m a g e p l a n e of the lens d u e to spherical a b e r r a t i o n a n d defocus. T h e p o s i t i o n a l d e v i a t i o n f r o m the o p t i c axis d u e to the i m a g e defect is given b y eq. (3), where the values of Cs a n d Az are given for the i m a g i n g side of the lens ( a n d are r e l a t e d to the
" %, £
values o n the o b j e c t side of the lens b y the lens m a g n i f i c a t i o n M ) . T h e a r e a d A c a n t h e n b e written as
Objective
Lens
Gaussian Focal Plane --Cst~ 3- Aztxo
Csct o Fig. 5. Ray diagram showing image formation of a point source by a lens with spherical aberration.
References [1] R.W, Carpenter, Y.L. Chen, M.J. Kim and J.C. Barry, in: Proc. Microscopy of Semiconducting Materials Conf., Oxford, UK, 1989, Inst. Phys. Conf. Ser. 100, Section 7, p. 543. [2] C. CoUiex, J.L. Maurice and D. Ugarte, Ultramicroscopy 29 (1989) 88. [3] J.R. Michael and D.B. Williams, J. Microscopy 147 (1987) 289. [4] C. Mory, C. Colliex and J.M. Cowley, Ultramicroscopy 21 (1987) 171. [5] J.A. Venables and A.P. Janssen, Ultramicroscopy 5 (1980) 297. [6] R.W. Carpenter and J.C.H. Spence, J. Microscopy 136 (1984) 165. [7] C. Colliex and C. Mory, Optik 59 (1981) 311. [8] G. Cliff and P.B. Kenway, in: Microbeam Analysis (San Francisco Press, San Francisco, 1982) p. 107. [9] C. Mory, M. Tence and C. Colliex, J. Microsc. Spectrosc. Electron. 10 (1985) 381. [10] O.C. Wells, in: Scanning Electron Microscopy (McGrawHill, New York, 1974) p. 69. [11] A.V. Crewe, Ultramicroscopy 23 (1987) 159. [12] J.C. Wiesner and T.E. Everhart, J. Appl. Phys. 44 (1973) 2140. [13] A.V. Crewe and D.B. Salzman, Ultramicroscopy 9 (1982) 373. [14] M. Born and E. Wolf, Principles of Optics (Pergamon Press, Oxford, UK, 1980). [15] C.S. Williams and O.A. Becklund, Introduction to the Optical Transform Function (Wiley, New York, 1989). [16] L. Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [17] J.M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1984) p. 34.
J.K. Weiss et al. / Small electron probe formation in TEM / S T E M
[18] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge University Press, New York, 1986. [19] P.M. Duffieux, The Fourier Transform and its Application to Optics (Wiley, New York, 1983). [20] C.R. Wylie and L.C. Barrett, Advanced Engineering Mathematics (McGraw-Hill, New York, 1982) p. 589. [21] R.W. Carpenter, I.Y.T. Chan and J.M. Cowley, in: Proc. 40th Annu. EMSA Meeting (Claitor's, New Orleans, 1982) p. 696.
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[22] J.K. Weiss, P. Rez and A.A. Higgs, to be published. [23] M.R. McCartney, P.A. Crozier, J.K. Weiss and D.J. Smith, Vacuum 42 (1990) 301. [24] K. Das Chowdhury, R.W. Carpenter, J.K. Weiss and A.A. Higgs, in: Proc. 47th Annu. EMSA Meeting (San Francisco Press, San Francisco, 1989) p. 428. [25] J.K. Weiss and R.W. Carpenter, to be published.