Convergent beam electron microdiffraction from small crystals

Ultramicroscopy 6 (1981) 359-366 North-Holland Publishing Company

359

CONVERGENT BEAM ELECTRON MICRODIFFRACTION FROM SMALL CRYSTALS J.M. COWLEY and J.C.H. SPENCE Department of Physics, Arizona State University, Tempe, Arizona 85281, USA Received 17 March 1981

Diffraction spots in convergent beam electron diffraction patterns obtained with a coherent incident beam show a characteristic fine structure when the incident beam is close to a small crystal or the edge of a larger crystal. The fine structure may take the form of a splitting of the spot into two or more sharp spots or else a bright annulus with a dark spot in the middle. It is shown that this fine structure results from coherent interference effects for an asymmetrically placed crystal and reflects the sensitivity of the relative phases of the diffraction pattern amplitudes to the absolute position of the diffracting object.

1. Introduction The use of field emission guns in scanning transmission electron microscopes has recently made it feasible to obtain convergent beam diffraction patterns from very small regions of a thin specimen. If the incident beam is held stationary on the specimen the diffraction pattern comes from an area of diameter comparable with the STEM resolution limit which is currently of the order of 5 A for dedicated STEM instruments. By use of a two-dimensional detector system, such as the optical system described elsewhere [1 ] these convergent beam diffraction patterns may be seen clearly and recorded in a fraction of a second with reasonable resolution. In such patterns obtained from minimum specimen areas the diffraction spots are necessarily discs of large diameter. When such spots overlap, complicated interference effects take place [2] and the pattern is not easily interpreted. For many purposes it is convenient to insert a smaller objective aperture so that the diffraction spots are smaller and the diffraction pattern can be interpreted more directly by analogy with the familiar focussed parallel-beam spot patterns, even though the specimen region giving the pattern may then be as large as 10 A or more in diameter. It is common experience that for very thin crystals the diffraction spots in convergent beam diffraction (CBED) patterns are discs of uniform intensity. The 0304-3991/81/0000-0000/$02.50 © 1981 North-Holland

expectation is that for any thin, crystalline or noncrystalline specimen, the diffraction pattern intensity will be given by convoluting the parallel-beam diffraction pattern intensity with the incident beam angular intensity distribution. Such expectations are based on experience with instruments for which the incident convergent beam is almost ideally incoherent, i.e. for which electron waves incident at different angles on the specimen give intensity distributions which are added together to give the observed patterns. For a coherent incident convergent beam, such as is given by a field emission gun, it is the diffraction amplitudes due to waves incident at different angles which must be added together and a variety of interference effects may occur, changing the appearance of the diffraction pattern considerably. A number of these interference effects have been described elsewhere [3,4]. Here we concentrate on effects observed for relatively small, non-overlapping CBED spots obtained from small crystals or from crystal edges and not previously described. The basic observation is that, even for very thin, weakly scattering samples, the maxima of intensity in the diffraction patterns are often much smaller in their dimensions than the central beam spot size in the absence of a specimen, as determined by the objective aperture size, and so cannot be described by a convolution of any intensity distribution with the incident beam angular intensity distribution. This is

360

J.M. Cowley, J.CH. Spence / CBED from small crystals

°6C

Fig. 1. Microdiffraction patterns from small gold crystallites with the incident beam at the edge of the crystal. The beam convergence angle is about 3 × 10 -3 radian (beam diameter at specimen 15-20 A) for (a), (b) and (c) and 2 × 10 -2 radian for (d) and (e).

the case even for very thin fdms of nominally amorphous materials (silicon, germanium, carbon, alumina) for which microdiffraction patterns from small areas show irregular distributions of relatively small spots rather than continuous diffuse haloes [5]. The effect was noticed in more striking form in microdiffraction patterns from small gold particles, 2 0 - 1 0 0 A diameter, incorporated in thin films of amorphous polyester or alumina [6]. As shown in fig. I, the diffrac-

tion spots from the gold crystallites appeared to be subdivided into two or three relatively sharp spots, much smaller than the incident beam disc. Similar effects were later observed in patterns obtained from the edges of small crystals of MgO smoke (fig. 2) and from well-defined edges of other crystals. For thick crystals it is possible that fine structure within the diffraction spot discs may result from strong dynamical diffraction effects (see, e.g. ref. [2]) but for most of the cases under discussion the crystal thickness was much too small to allow such effects to be relevant. The explanation of our observation follows from a quite simple analysis for idealized nonrealistic cases. For situations representing practical experimental situations it is necessary to resort to computer calculations.

2. Observations Fig. 2. Microdiffraction patterns from the edge of a small MgO smoke crystal set so that the incident beam was parallel to a (100) face. The crystal thickness was small (-100 A) for (a) and larger for (b).

Microdiffraction patterns were obtained by use of an HB-5 STEM instrument fitted with a two-dimensional detector system consisting o f a fluorescent screen, an image intensifier, an optical analyzer sys-

J.M. Cowley, J. CH. Spence / CBEDfrom small crystals tern and a low light level TV camera [1 ]. The patterns were recorded either by photographing directly the TV display screen with exposure times of 1/8 to 1/2 s, or else by recording the TV output on videotape and then playing it back and storing one frame at a time on an image converter (Vistascan) so that it could be photographed using the photographic cathode ray screen on the HB-5 console. The specimens containing small gold crystallites were prepared by co-sputtering gold with polyester or alumina [6]. Many of the gold particles, including those 20 ,~ or less in diameter, showed evidence of multiple twinning which complicated the diffraction patterns with doubling of the spot.arrays by reflection across the twin planes and also a streaking of the spots. These effects are discussed elsewhere [7]. For the most part they could be clearly distinguished from the subdivisions of the diffraction spots which we treat here. Patterns such as those in figs. l a - l c were obtained with an objective aperture size of 10/am and a focal length of 3 mm (beam convergence angle 3 × 10 -~ radian;beam size on the specimen 12-15 A). Frequently, as the incident beam was moved across a small gold crystallite, the diffraction spots first appeared weakly, split into two or three relatively sharp peaks. Then they became stronger and unsplit and finally weaker and again subdivided as the beam left the crystal (see fig. 3 of ref. [6]). As is evident in figs. l a - l c , all spots from one crystal are equally split. The patterns of figs. 1d and 1 e were obtained from similar gold crystaUites with a larger, 60/am diameter, objective aperture, and illustrate the fact that the subdivision of the diffraction spots is evident for all aperture sizes and frequently involves the formation of an almost complete annulus. In the case of an objective aperture as large as 60/am, spherical aberration effects are present to such an extent that the spot shapes may be regarded as being produced by strongly distorted imaging of the crystal by the marginal rays within the aperture (see, for example, ref. [3]). However similar annular spots appear in many cases even for small objective aperture sizes for which the spherical aberration effects are negligible. For example, in fig. 2 diffraction spots obtained with a 10/am objective aperture when the incident beam is close to the edge of a small MgO smoke crys-

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tal may be split into two sharp components or else may form almost complete rings. In this case the incident beam was parallel to a (100) face of the crystal. Fig. 2a was obtained near a corner, where the crystal was very thin in the beam direction. Fig. 2b was obtained where the thickness was of the order of 500 A.

3. Theoretical description

3.1. General The wave incident on the specimen in a scanning transmission electron microscope with a stationary beam is given by the Fourier transform of the objective lens transfer function. For a thin specimen the modification of this wave by the object may be described by multiplication by a transmission function q ( x , y ) . Thus the wave leaving the specimen is

tPo(x,y) = q ( x , y ) t ( x , y ) ,

(1)

where

t ( x , y ) = ~[A(u, v) exp~ix(u, v))] . Here A(u, v) is the aperture function and X(u, v) is the phase change due to defocus A, and lens aberrations, usually limited to the spherical aberration with coefficient Cs, so that

×(u, v) --- rrAX(u 2 + 02) + ½nCsXa(u 2 + v2): ,

(2)

where u, o are the reciprocal space coordinates of magnitude (sin ¢)/X and ¢ is the scattering angle.. The diffraction pattern intensity seen on the detector plane and recorded by means of the optical system is given by Fourier transform o f ( l ) as

l ( x y ) = IQ(u, v) * [A(u, v) exp(ix(u, v)}] 12 ,

(3)

where Q(u, v) is the Fourier transform of q (xy). For the more usual case of an incoherent addition of intensities for incident beams in different directions the intensity distribution would be

li(xy ) = IQ(u,v)[ 2 * A 2 ( u , v ) ;

(4)

i.e. the parallel-beam intensity distribution would be smeared out by the aperture function. Because we use a field emission electron source, there is a coherent addition of amplitudes on the detector plane and the

ZM. Cowley, J.CH. Spence / CBED from small crystals

362

intensity (3) is sensitive to both the phase variations in the parallel-beam diffraction amplitude, Q(u, o), and the phase variations in the transfer function due to defocus and aberrations. If the specimen includes a small object, such as a small crystal at a distance b from the origin (the axis of the microscope), the transmission function may be written qo(x - b,y), so that

0.6

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Fig. 3. Intensity distribution across a diffraction spot for an idealized one-dimensional m o d e l

a(uo) = Qo(uo) exp{2rribu} . Then the diffraction pattern intensity will be sensitive to the position of the object relative to the center of the incident beam. If a small objective aperture is used, as in the cases of figs. 1 a - l c and fig. 2, the transfer function is insensitive to Cs and the intensity will vary only slowly with defocus. Hence for a well-focussed beam, it is possible to use the simplification that X(uo) = 0. Even for this case it is possible to evaluate (3) without extensive calculations only for a few ideal cases. Here we present the analytical analysis for only one non-realistic, one-dimensional case in order to illustrate the relevant factors and then present the results of calculations for more relevant cases.

This function is plotted in fig. 3 for several values of the displacement, b, of the object from the beam axis. There will be two distinct maxima, separated by a distanceA + a - b -1 i f b > ( 2 a ) -~. F o r b = a -1 the intensity between the maxima will be zero. For b > 3 (2a) -x each of the maxima will be subdivided. The factor b -2 in (6), however, ensures that the intensity in this case will be small. If the object is a small crystal having a shape function

q(x) = Orx)-' sin(~ax), the transmission function will be multiplied by

3. 2. Simple example

F h exp{2~'ihx/c} , h

We consider an object of transmission function

q(x) = (rrx) - l sin(rrax) for whi'i:h the amplitude distribution is Q(u) = 1 if l ul ~ a[2. The aperture function is assumed to be A (u) = 1 if lul <~A/2 and 0 if lul > A[2. The diffraction pattern intensity given when the object is displaced by a distance b, is

l(u) = Ira(u) exp(2rribU) A(u - U) dUI 2 .

(5)

where c is the periodicity and c < < a -1 . Then the intensity distribution (6) will be placed, with relative intensity IF h 12, at each of the diffraction spot positions, u = h/e. Thus each diffraction spot will appear to be split into two components if the crystal is sufficiently far from the beam axis, and the separation of the two components will be slightly greater than the spot size due to the incident beam convergence (since we have assumed A > a). This conclusion is consistent with the observations.

Assuming A > a, integration gives

3.3. Calculations

I(u) = (rib2) -1 sin ~ {rib [u + (,4 + a)/2]} if ( - A - a) < 2u < (-,4 + a) = (,rb~) -~ sin s Orab)

if -(.4 - a) < 2u < (A - a) = Orb=) - ' sin 2 {,rb [u - (A + a)12]) if (,4 - a) < 2u < (A + a ) .

(6)

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odic function is represented by a finite (but large) number of sampling points in reciprocal space ("beams"). For the one-dimensional calculations leading to the results of fig. 4, it was assumed that a small gold crystal of thickness 40 A and width 16 or 18 A was illuminated by a convergent incident beam of width (between the first zeroes of the wave function) of 6 A. The diffraction pattern intensities were calculated for various positions of the beam relative to the crystal and for various amounts of defocus and spherical aberration. The diffraction in the crystal was calculated using the multi-slice method with 81 beams. In each part of fig. 4, the configuration assumed is sketched in an insert. The convergent beam diffraction "discs" which would be given by an infinite thin crystal are indicated by the hatched areas at the bottom. In fig. 4a, with the beam at the center of the

crystal, the expected discs of intensity are found, but with some fine structure due to the finite crystal dimensions. In figs. 4b and 4c, calculated for the cases in which the incident beam is centered at the edge, or 3 A outside, the crystal, the diffraction patterns show two sharp peaks at the edges of the diffraction discs and relatively small intensities between them. This effect is clearly not a consequence of spherical aberration, since for fig. 4b, Cs = 2 mm and for fig. 4c,

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gold bar, 30 A thick and 20 A. wide as shown in the figure. The calculations were made with a total of 4000 Fourier coefficients. In the calculated diffraction pattern the darker parts are of higher intensity. The annulus of high intensity around each diffraction spot is clearly visible.

4. Conclusions The detailed calculations confirm the interpretation of the split spots as being a result of interference effects involving the phase differences associated with

the relative positions of the crystal and the incident beam. It is to be expected that diffraction spots from a crystalline region limited by any discontinuity will be split whenever an incident coherent convergent beam is close to the discontinuity. The splitting may take the form of two sharp spots separated by approximately the incident beam spot diameter, of more than two spots or o f • complete annulus, depending on the precise geometry of the crystal discontinuities. The artificial example used above in section 3.2, chosen for the convenience of an easy analytical solution, serves the useful purpose of allowing ready eval.uat•on of the effects of other variables on the details of

366

J.M. Cowley, J.C.H. Spence / CBED from small crystals

the diffraction patterns. In particular an estimate can be made of the effect of partial coherence of the incident beam on the spot splitting. If, instead of a point source, an incoherent source of finite dimensions is used, the diffraction pattern will be given by adding the intensities of the patterns produced by all points of the source separately. In the one-dimensional example, each source point will correspond to a different value of b. From eq. (6) and fig. 3 it is seen that distinct maxima will appear, in approximately the same positions, for a range ofb from a -1 to about 3a -1 . Hence split spots will be visible in this case for an incoherent broadening of the beam by an amount approximately equal to the crystal dimensions or (since A > a) by an amount approaching that of the width of the incident beam. Experimental observations verify this conclusion: The width of the incident beam was broadened incoherently by applying a fast two-dimensional raster scan, such as used to obtain STEM images, to the incident beam while the diffraction pattern was being recorded. For a coherent incident beam diameter of about 15 A,, the spot splitting was observed with reduced contrast when a scan of amplitude 8 - 1 0 A (with the "reduced area" scan at the 25 X 106 magnification setting) was applied. With a scan of amplitude 2 5 - 3 0 A., the spot splitting was of such low contrast that it could be distinguished only with difficulty. The spot splitting described here is observed frequently in microdiffraction patterns obtained with the field emission gun of the HB5 microscope. Because the coherence requirements are not exacting, it should be observable with any instrument capable of producing an incident beam diameter of 20 A or less. It is possible that this effect has not been recognized

in the past because if the diffraction conditions are not well specified, it may readily be confused with fine structure due to dynamical diffraction effects from crystalline regions of sufficiently large thickness (100 A or more) to twinning effects or else to the formation of shadow images of the specimen in the diffraction spots when the diffraction patterns are grossly de focussed.

Acknowledgements

The authors are grateful to Dr. A.J. Craven for discussion on the means to vary incident beam coherence. This work was supported by NSF Grants DMR07926460 and DMR8002108 and made use of the resources of the Facility for High Resolution Electron Microscopy supported by the NSF Regional Instrumentation Facilities Program, Grant No. CHE7916098.

References

[1] J.M. Cowleyand J.C.H. Spence, Ultramicroscopy 3 (1979) 433. [2] J.C.H. Spence and J.M. Cowley, Optik 50 (1978) 129. [3] J.M. Cowley, Ultramicroscopy4 (1979) 413. [4] J.M. Cowley, Ultramicroscopy4 (1979) 435. [5] J.M. Cowley,in: Diffraction Studies of Non-Crystalline Substances, Eds. I. Hargittai and W.J. Orville Thomas, in press. [6] R.A. Roy, R. Messier and J.M. Cowley, Thin Solid Films, in press. [7] J.M. Cowley and R.A. Roy in Scanning Electron Microscopy, 1981, Ed. O. Johari, in press. [8] J.C.H. Spenee, Aeta Cryst. A34 (1978) 112.