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## Material Information- Title:
- The effect of crystal defects on microdiffraction patterns
- Creator:
- Warren, John Beverly, 1944-
Warren, John Beverly, 1944- - Language:
- English
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- vi, 175 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Crystals ( jstor )
Diameters ( jstor ) Diffraction patterns ( jstor ) Electron beams ( jstor ) Electrons ( jstor ) Geometric lines ( jstor ) Images ( jstor ) Silicon ( jstor ) Stacking faults ( jstor ) Wave diffraction ( jstor ) Dissertations, Academic -- Materials Science and Engineering -- UF Electrons -- Diffraction ( lcsh ) Materials Science and Engineering thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
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- Thesis--University of Florida.
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- Bibliography: leaves 173-174.
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- Vita.
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- by John Beverly Warren.
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THE EFFECT OF CRYSTAL DEFECTS ON MICRODIFFRACTION PATTERNS BY JOHN BEVERLY WARREN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978 i ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to his advisor, Dr. J. J. Hren, whose continuing encouragement during a decade spanning periods of graduate study, military service and employment in industry enabled the author to complete his degree. In addition, the advice of the supervisory committee, Dr. C. 5. Hartley, Dr. R. E. Reed-Hill, Dr. R. T. DeHoff and Dr. M. Eisenberg, was most useful in the final preparation of the manu- script. Support from the Department of Materials Science in the form of a research assistantship is also gratefully acknowledged. Finally, the author thanks his wife for her unending patience and encouragement. TABLE OF CONTENTS page ii v 1 6 6 10 18 22 29 47 47 61 65 77 81 88 ACKNOWLEDGMENTS .................. ................... ......... ABSTRACT. .................. ................... ............... CHAPTER I INTRODUCTION. ................... ................... 2 THE EFFECT OF A CONVERGENT ELECTRON BEAM ON ELECTRON SCATTERING ................... ............. 2.1 A Review of Electron Diffraction and the Reciprocal Lattice............................ 2.2 A Review of the Dynamical Theory of Electron Diffraction.... ...................... 2.3 The Effect of STEM on the Reciprocal Lattice Construction........... ............... 2.4 The Effect of STEM on the Dynamical Theory.... 2.5 A STEM Computer Simulation Program............ 3 MICRODIFFRACTION METHODS ................... ........ 3.1 A Review of the Effect of Inelastic Scattering on the Diffraction Pattern......... 3.2 The Use of Channeling Patterns to Characterize Polycrystalline Specimens........ 3.3 A Review of Ray Optics for Convergent Beam Diffraction ................... ........... 3.4 The Effect of Lens Aberrations ................ 3.5 A Review of the Rocking Beam Micro- diffraction Method ................... ......... 3.6 Transmission Channeling Patterns by the Rocking Beam Method ................... .... TABLE OF CONTENTS (continued) CHAPTER page 4 EXPERIMENTAL PROCEDURES AND MATERIALS............. 94 4.1 Ray Optics and Computations .................. 94 4.2 Sample Preparation............,....... ........ 105 4.3 The Character of the Dislocations Introduced................................. 111 4.4 Determination of Crystal Directions by Diffraction Patterns...................... 115 5 EXPERIMENTAL RESULTS ................... ........... 121 5.1 A Comparison of the Microdiffraction Me thods ................... ................... 121 5.2 Identification of Crystal Defects by Microdiffraction. ................... ......... 135 5.3 Computer Simulation of STEM Crystal Defect Images...................... ........., 143 5.4 Determination of Grain Orientation of a Fine-grained Superalloy Using Focused Condenser Aperture Micro- diffraction................................ 156 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH. ................... ................... ... 163 APPENDIX.......................................... 167 REFERENCES .......... ,...........,..................... 173 BIOGRAPHICAL SKETCH. ................... ................... .. 175 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE EFFECT OF CRYSTAL DEFECTS ON MICRODIFFRACTION PATTERNS by John Beverly Warren June, 1978 Chairman: Dr. J. J. Hren Major Department: Materials Science and Engineering The various modes of microdiffraction that can be performed with an electron microscope equipped with a scanning transmission attachment are examined from both an experimental and a theoretical standpoint. Particular attention is paid to the development of microdiffraction to analyze crystal defects such as dislocations and stacking faults. Th~e objective lens of the scanning transmission electron microscope forms a focused electron probe with a comparatively large convergence angle. Such a probe can produce a convergent beam diffraction pattern from an area on a crystalline specimen as small as 50 A in diameter. The scanning coils used to raster the probe over the specimen can also be used to rock a well- collimated electron beam over a region of the specimen in such a way that a microdiffraction pattern is formed on the cathode ray tube normally used to display the specimen image. By vary- ing either the convergence angle of the incident beam or the transmitted beam it is possible in both of these methods to produce microdiffraction patterns with high angular resolution or transmission channeling patterns with low angular resolution but excellent signal-to-noise ratios. In every case, the convergence angles of the incident and transmitted electron beams determine the appearance of the speci- men image and the associated diffraction pattern. The Howie-W~helan dynamical theory must be modified to correctly predict the intensi- ties of the scattered beans present in the diffraction pattern as well as the diffraction contrast image of crystal defects. Computer programs that simulate the electron beam-specimen interaction for the convergent beam case are developed and compared to experiment. A numerical method that utilizes transmission chan- neling patterns to precisely determine the orientation of a crys- talline specimen and an electron beam is also presented. CHAPTER 1 INTRODUCTION Since the advent of Scanning Transmission Electron Microscopy (STEM), several years ago, it was expected that STEM techniques would provide increased specimen penetration, electronically enhanced image quality, and improved capabilities to analyze areas a few nanometers in diameter by microdiffraction and energy dispersive X-ray micro- analysis. For crystalline materials where diffraction contrast imaging plays the dominant role, only the last of these, the analysis of a small region excited by a stationary electron probe, has enjoyed a measure of success. This thesis examines the ability of the STEM probe to form a microdiffraction pattern from a column of material roughly equal to its own diameter. A microdiffraction pattern formed by this method can be produced from an area more than two orders of magnitude smaller than the patterns from standard selected area diffraction. While this method is the most direct of several possible microdiffraction techniques, most work up to now has been limited by the poor angular resolution in the pattern resulting from the convergent TEM probe and by the high contamination rates associated with it. Another approach, rocking beam diffraction, circumvents the limitations of the stationary electron probe and is also studied in detail. Here, the deflection coils in the upper half of the microscope column, used to scan the probe across the specimen in the normal mode of operation, are adjusted to rock a well-collimated electron beam about a pivot-point level with the specimen, thus producing a time-dependent display of the variations of electron scattering as a function of the angle of rock. If an objective aperture of appro- priate size is placed below the specimen plane, the rocking motion of the incident beam results in the bright field and the associated dark field images being swept over the STEM detector. This motion forms a simulated diffraction pattern on the STEM cathode ray tube that corresponds precisely to the portion of the sample image "seen" by the detector. Since the incident beam is relatively well-collimated, contamination rates are much lower than for convergent beam STEM microdiffraction. Angular resolution is now controlled by the objec- tive aperture, and can be varied from values comparable to those found in standard selected area diffraction to values that result in low angular resolution but produce a scanning transmission channeling pattern with an excellent signal-to-noise ratio. For all methods of microdiffraction, the convergence angles of the incident and scattered beams are the crucial factors that deter- mine the manner in which the information in the microdiffraction pattern is presented. In convergent beam microdiffraction, gradually increasing cri, the semi-angle of incidence, changes the appearance of the pattern from the normal diffraction spot array to a transmission channeling pattern where discrete diffraction spots are no longer visible. Although the method of formation is quite different, the rocking beam method can produce a similar variety of effects if a , the exit semi-angle, is varied by changing the objective aperture size. Once the basic concepts of convergent beam and rocking beam nicrodiffraction can be quantitatively described, they can be used to solve problems of interest to the metallurgist. In this work, these techniques are used to examine dislocation strain fields, stack- ing fault images, and to provide very accurate information on the orientation of adjacent areas in fine-grain polycrystalline materials. The information provided by microdiffraction from small distorted regions of the specimen close to the defects is comparable in many respects to that provided by the computer simulation of defect dif- fraction contrast images. To review, the computer simulation method numerically integrates two or more simultaneous differential equations that describe the interaction of the transmitted and scattered beams as they pass through the crystal lattice. If the equations are solved for several thousand columns in the vicinity of a defect strain field, and the solutions (for example, the bright field intensity, IT 2, or the dark field intensity, 15 2) are plotted in a dot matrix to form a picture, a simulation of the experimental diffraction contrast image of the defect can be formed. The major limitation of the computer simulation technique is simply the time required to compute the interaction for each of several thousand columns before the simulation can be formed. Compu- tation time for n-beam situations, where there are several strongly excited beams, escalates accordingly. In practice, computer simula- tion is limited to the two-beam case, where information from only one diffraction plane is available in the image. Thus, several simulated images are needed to completely characterize a defect strain field. Microdiffraction can aid the computer simulation process by pro- viding the S 12 d12 value directly from the diffraction pattern re- sulting from a column passing through the dislocation strain field that is less than 100 A in diameter. By comparing the calculated ratios to the experimental ones for several colunns spaced along the dislocation, it should be possible to identify the dislocation with a much smaller expenditure of computation time. Such a procedure could be done with ordinary Transmission Electron Microscopy (TEM) techniques only by measuring the intensity in the bright field and dark field images at precisely the same point. This is not possible with ordinary TEM for two reasons. First, the process of obtaining a dark field image involves tilting the beam, and it is very difficult to get a dark field image with precisely the same Bragg deviation as the associated bright field image. Second, it is very difficult to locate exactly the same specimen point on different two-beam images due to the diffuse nature of the image itself. When the STEM probe is used to display the defect image on the Cathode Ray Tube (CRT) in the scanning mode, the probe convergence plays an important role in determining the appearance of the crystal defect image. The convergence effect must be carefully controlled if STE.M images are to be used for computer simulation, and this point is explored by examining the effect of beam convergence on computer simulated images of stacking faults in silicon. For crystal orientation problems, the electron-beam specimen orientation can be determined directly from the Kikuchi lines (or channeling lines for high convergence angles) present in the micro- diffraction pattern. The diffraction spots themselves persist over specimen tilts of several degrees, and cannot be used to determine the local electron beam direction. The location of both Kikuchi and channeling patterns is defined by the intersection of the Bragg diffracting cones of apex angle 1800-298 and the Ewald sphere. Each diffracting cone is normal to a particular diffracting plane and local lattice rotation caused by defect strain fields rotates each cone to a new position. Measure- ment of the relative shift of the lines in the microdiffraction pat- tern can be used to determine the precise crystal orientation of the crystal volume producing the pattern. While this approach can be accomplished with standard selected area diffraction, it has been limited to areas larger than 2 uim in diameter and to low atomic number materials where Kikuchi lines are more readily formed. Microdiffrac- tion, and in particular, transmission channeling formed by a highly convergent probe overcomes both of these limitations. In summary, this thesis uses several microdiffraction techniques to analyze individual crystal defects and also to solve orientation problems in polycrystalline materials. The advantages and disadvan- tages of each technique will be compared, but it will be shown that all of the phenomena in any type of microdiffraction pattern are critically dependent upon the convergence angle of the incident elec- tron beam. CHAPTER 2 THE EFFECT OF A CONVERGENT ELECTRON BEAM ON ELECTRON SCATTEING 2.1 A Review of Electron Diffraction and the Reciprocal Lattice Scanning Transmission Electron Microscopy (STEM) enables the production of microdiffraction patterns two orders of magnitude smaller than previous techniques (Geiss, 1975). However the short focal length objective lens used to produce a small, focused spot for STEM results in a far more convergent beam impinging on the spec- imen than standard TEM techniques. It is necessary to understand how a highly convergent electron beam effects diffraction contrast imaging if the STEM microdiffraction patterns are to be correctly interpreted. Thne difference between imaging with TEM and STEM can be readily understood with the aid of geometrical ray optics (Fig. 2.1). In TEM, the upper half of the objective lens in the Philips 301 electron microscope forms a collimated beam of radiation that impinges upon the specimen and is focused by the lower half of the objective lens field to a point in its back focal plane (Thompson, 1977). In STEM, the upper half of the objective lens produces a focused probe in the specimen plane. This probe diverges after passing through the speci- men but the lower objective lens field partially refocuses the probe and forms a stationary, convergent beam diffraction pattern in the sare objective focal plane as in TEM. (a) TEM (b) STEM C2 Len UpperC2 Aperture Upper - Specimen Lower i OBJ Diffraction i a Pattern 1 Figure 2.1 Ray diagrams for TEM and STEM conditions in the Philips 301. In STEM, the convergence of the beam is controlled by the second condenser aperture diameter, while the up- per half of the objective lens focuses the probe onto the specimen. In TEM, the second condenser lens offers an additional control of convergence. The effect of the large difference in beam convergence on elec- tron diffraction for STEM and TEM can be shown with the reciprocal lattice construction. Elastic scattering of electrons in the recip- rocal space is diagrammed in Fig. 2.2 where a vector Pt is drawn parallel to the transmitted electron beam and a number of vectors Ps are drawn parallel to the scattered beams. The magnitude of the transmitted vector is defined as --. Because scattering is assumed to be elastic, the wavelength of the electrons does not change upon diffraction and hT = 1 Inspection of Fig. 2.2a shows that the tips of all possible scattering vectors trace out a surface called the reflecting, or Ewald, sphere. The Laue conditions show that diffraction occurs only when the reflecting sphere intercepts a point in the reciprocal lattice that corresponds to a particular diffracting plane. This is written algebraically as (P -P ) t s (2.1) A =hkl where ghk1 is a vector drawn from the origin of the reciprocal lattice to any point hk1 that represents the set of hkL diffracting planes. For 100 KV, =27 o-1 units while I111 for Si = .32 4-1 Thus, for low index lattice planes, the radius of the reflecting sphere is 25 to 85 times greater than the spacing between the recip- rocal lattice. Even for these conditions, diffraction would rarely occur since the reciprocal lattice points for a perfect crystal of infinite size are considered to be dimensionless. However, if the Ewald Sphere j in the Reciprocal 1 1 1 1 1 1 Lattice (a) Foil Normal Figure 2.2. Reciprocal lattice diagram (a) showing that the reflect- ing sphere must intersect a rel-rod if the corresponding diffractive spot is to appear in the diffraction pattern. The Bragg deviation is defined in (b) as a vector drawn from the center of the rel-rod to its intersection with the reflecting sphere. total diffracted intensity is computed for only a finite number of unit cells, structure factor arguments (Edington, 1975) show the reciprocal lattice points can now be represented as a volume of di- mensions ,1 and where z is the specimen fail normal and N is the number of unit cells along a particular direction. For typical foil thicknesses, the dimension Nz is much less than Nx or Ny so the reciprocal lattice points are stretched into rods whose long axis is parallel to the foil normal of the specimen. As shown in Fig. 2.2a, the reflecting sphere can now intersect many of the elongated lattice points forming an electron diffraction pattern with numerous higher order diffraction spots. The distance between the diffraction vector and the reciprocal lattice point center is defined by the vector s, which shows the di- rection and magnitude of the deviation from the exact Bragg condition. In Fig. 2.2b, s is drawn parallel to the foil normal from the recip- rocal lattice center to the intersection point with the reflecting sphere. By convention, the positive direction of s is defined as anti-parallel to the electron beam direction. 2.2 A Rieview of the Dynamical Theory of Electron Diffraction STEM's effect on the formation of the diffraction contrast image can be understood if the Howie-Whelan dynamical theory is modified to account for convergent beam radiation (Whelan and Hirsch, 1957). To review, the dynamical theory states that the transmitted and scattered waves resulting from electron diffraction can no longer be treated independently. Now the scattered wave is considered to be continually rescattered back in the original direction of the trans- mitted beam as shown in Fig. 2.3. The transmitted and the scattered wave will interfere and the amplitudes, T and S, of the waves will oscillate with the penetration depth, z, into the crystal. This in- teraction is described by two, simultaneous linear differential equations: dT --= -n T + (i A)S dz dS dz-(i -A)T +(-n + 2is + 2nii)S Thle complete derivation of the equations, based on a quantum mechanical approach is discussed by Hirsch et al. (1965). In this chapter, it will be sufficient to explain the effects of the various tenms on the transmitted and scattered intensities IT 2 and S12~ since this is the primary type of information available in the STEM microdiffraction patterns. Ordinary absorption that controls the absolute intensities of the transmitted and scattered beams is defined as nl. This "normal" absorption simply lessens the intensity of both transmitted and scattered beams for crystalline sections of increasingly greater thickness. The absorption effect is thought to result from the in- elastic, or high-angle, scattering of electrons since actual absorp- tion of electrons does not occur (Hirsch, 1965). The anomalous Wasorption parameter, A, would actually be better described as "enhanced transmission" of the beam at certain positive deviations from the exact Bragg angle. A complete explana- tion of why the enhanced transmission phenomenon occurs can be made Figure 2.3 The multiple scattering assumption of the dynamic theory. Ray "a, elastically scattered in direction 5, can under- go additional elastic scattering to be rescattered in direction T, the original beam direction. only with extended reference to the derivation of the dynamical theory. In brief, one of the standing Bloch waves contributing to the solution distributes electronic charge in between the ion cores. Such a wave has lower average potential energy than a Bloch wave which distributes charge close to the ion cores and enhances the transmission of electrons through the lattice. The variable 5 or extinction distance, is the depth of the crystal necessary for either T or 5 to increase to their maximum amplitude and then decay to their minimum value. Thus, the wave- length of both T and S is 2( The extinction distance increases with increasing Bragg deviations, so the "effective" extinction dis- tance is usually written as W = 5_ s (Hirsch etal., 1965). The parameter describing the effect of a defect strain field, B, is written as B (g*R)S dz g (2.3) B describes the relative amount of lattice distortion produced by a - dR defect displacement field, R. The magnitude of strain, g dz, changes the effective value of s and thus plays a primary role in determining the diffraction contrast image resulting from the defect. The two-beam equations depend upon two important assumptions, both of which may have to be modified for the convergent beam STEM case. First, the two-beam assumption itself is strictly valid only for a completely collimated electron beam and crystals that are several extinction distances thick. For these conditions, a "thin- walled" Ewald sphere can be oriented such that is intercepts only one rel- rod that has a relatively short reciprocal length. For increas- ingly convergent beam radiation or for acceleration voltages much greater than 100 KV, additional scattered beams will tend to be ex- cited and each additional excited beam results in an additional dif- ferential equation. Second, the assumption that T and S vary significantly only in the z direction, such that 0, known as the column 3x ay ax ay approximation (Howie and Basinski, 1968), is less valid for STEM. Here, interactions between the transmitted and scattered electron waves are considered to be confined to a narrow column parallel to the incident electron beam. Interactions between electrons in adjacent columns are considered to be negligible. However, this is strictly true only if 6g is small and the beam is well collimated. While 6g does not change for STEM, the incident beam is now spread over a range of directions and it would seem that differentia- tion should take place along all possible electron beam directions contained in the convergent beam cone. These effects will be discussed further in the next section. At this point, it is helpful to solve the two-beam equation for the perfect crystal case and illustrate the effects of varying the anom- alous absorption and the Bragg deviation. Since B now equals zero, the equations become dT -- = -wT + w(i A)S dz (2.4) dS d-- = (i A)T + [-En + 2ni(w + Eg )s] Only for the perfect crystal can the equations be solved ana- lytically. After some manipulation the solution for the transmitted amplitude is (2.5) Y2 1 T = ---- exp (Y1Z) + --- exp (Y2Z) Y 2 1 2 where (2.6) Y1. = -un + in[w + vw + (i A) ] The amplitude of T thus oscillates not only with depth z but also the Bragg deviation. This is shown in Fig. 2.4 where T 12 is plotted versus s to form the rocking curve. The rocking curve has a maximum at s = 0 only when A is set equal to 0. Experimental evidence (Head et al., 1973) shows the proper value of A to vary between 0.07 and 0.13 according to the Bragg reflection used to form the scattered wave. The rocking curve closely resembles the contrast seen at a bend contour, where a section of a think fail bent at a constant radius results in a constant change in s. Since bend contours can be observed in both TEM and STEM, the experimental images offer a good test of numerical calculations. The variation of T 2 with z is shown in Fig. 2.5 where the period is 25g and the A/N ratio is clearly revealed as an attenuating effect to the wave. T and 5 are always 1800 out of phase, thus ex- plaining the contrast reversal seen in bright and dark-field images. This example also correctly predicts the presence of thickness fringes - I ii f ..* *' ,, ..* -. ..*** , ,*, **~, .-* ** . .. r **. . in ao Sa IIto O r F v Smm e = mlu loo -- m SO > 00 m~v md .....*** ~ **, **.. s .. ** * **, -***. *. ... o N O r QCl * m OO 00 mH 9 =0 0 c cc OE C O omma r( F: do 0 a *H < I' tJa~c- mtm 050r; cna0 0ta 0 -HO~ r0 ,C > E0 O ola */ O O II , I /: C * C commonly seen in a wedge-shaped fail. Convergent beam radiation will again affect the shape of these curves and will be discussed in the next section. 2.3 The Effect of STEM on the Reciprocal Lattice Construction The strongly convergent STEM beam must be represented as a volume in reciprocal space rather than a simple spherical surface. This volume is constructed geometrically from Fig. 2.6a, where the electron beam for STEM is represented by a cone-shaped bundle of rays. Each ray corresponds to a particular Pt direction in reciprocal space with its associated reflecting sphere. Addition of all of the possible reflecting spheres forms a "reflecting volume" (Fig. 2.6b) with a node at the origin of the reciprocal lattice. This volume could also be formed by rocking the reflecting sphere for the TEM case through all possible angles, o. are 5 x 104 ra nTM ad5x1 to 10~ rad in STEM. It is apparent from Fig. 2.6b that the STEM convergent beam alters the diffracting conditions in two ways. First, the Laue dif- fraction equation becomes increasingly relaxed for higher order spots as the rel-rods are more likely to be situated inside the reflecting volume at greater distances from the reciprocal lattice origin. Second, the deviation from the exact Bragg condition for a particular diffraction spot can no longer be assigned a specific value. It now corresponds to a range of values determined by the portion of the rel- rod's length that lies inside the reflecting volume. (a) INCIDENT ELECTRON BEAM WITH SEMI-ANGLE cr (b) CROSS SECTION OF RE- FLECTING VOLUME IN THE RECIPROCAL LATTICE Figure 2.6 Cross section of the reflecting volume for a convergent beam of semi-angle ai shown in (a) The reflecting volume gradually increases in thickness at greater dis- tances from the reciprocal lattice origin. Noe at 00) Analytical geometry can be used to describe the reflecting volume for the convergent beam case and provides quantitative esti- mates of both these effects. To begin, the central beam direction, B, is defined as parallel to the optic axis and is drawn from the origin of the reciprocal lattice anti-parallel to the electron beam. We then define the set of all vectors lying on the surface of the convergent beam cone at angle a.i away from B as B'. Since both B and B' are equal .to -, it follows that coscr. B' B (2.7) Any vector B' satisfying these conditions can act as radius for its associated reflecting sphere. The equation for each of the reflect- ing spheres is (X Bl)2 2 Y-B) + (2 B')2 (2.8 Once an initial B' is chosen, additional radii may be computed by specifying that each vector on the cone lies 4 degrees away from its neighbor. These additional vectors, B", are found by solving the simultaneous equations ...p cos4 X2 coscr B" B =---- (2.9) Each B", when substituted into Eq. 2.8, defines a segment of the outer surface of the reflecting volume for angle Q. The equation defining both the upper and lower surfaces of the reflecting volume is then 2 2 2 1_(2.10) (X t B")2 + (y + B")i + (z 1 8") Satisfaction of Bragg condition for any given reciprocal lattice point can now be checked by substituting its coordinates into x, y, and z of Eqn. 2.10 and solving for a given set of B". However, it has been shown that any reciprocal lattice point is stretched out to a rod parallel to the foil normal. For the con- vergent beam case, it is necessary to compute the ranges of s values for a particular operating reflection. If the rel-rod is arbitrarily divided into ten segments, the location of each segment in the recip- rocal lattice is given by (2.11) g' = g +s where s is a vector parallel to the foil normal (2.12) s = -- n (-5 < n< 5) Each rel-rod segment, defined by g', can now be checked with Eqn. 2.10 to tell whether it lies within the upper and lower surfaces of the reflecting volume. 22 2.4 The Effect of STEM on the Dynamical Theory The dynamic theory must be modified to take the convergent electron beam used in STEM into account. Section 2.3 has already shown that the reflecting sphere shape must be modified to account for convergent radiation. As was shown in Fig. 2.6b, the diffrac- tion excitation rules are relaxed for higher order reflections and thus the two-beam conditions should be more difficult to achieve for STEM. Of greater importance is the fact that even if two-beam con- ditions are achieved, the Bragg deviation is now spread over a range of values. Since the Howie-Whelan two-beam equations depend directly on s it can be seen immediately that both the rocking curve for IT 2 and the curve for T 12 versus z will be modified for STEM. For purposes of illustration, it is best to use a non-column- approximation set of two-beam equations, and see how beam convergence relates to the various terms. The work of Howie and Basinski shows that -+ y BT TT+i w exp [2ni (sz + g- ) 15 az z Bz B (2.13) 3T + Y T + Y T i T exp[-2xi(sz + g-R)] +t S Bzx x y ay 5 where (2.14) (Pt + 9), (Pt t-9)y X (p S y (lP + g + s)Z and the subscripts refer to the components of the vector sums for the x, y, and z directions. For the Bragg angles encountered in electron diffraction, CPt + or,, y 'C (t + g + S)z and the variation of s with respect to the x or y directions can be ignored. In STEM, 8B remains the same so this inequality should still hold. However, by Fig. 2.6b, Pt now has a range of directions determined by a i, the angle of in- cidence. Thus, for STEM, it is no longer a good approximation to differentiate in the z direction parallel to the optic axis of the microscope, and the two-beam equations must be solved repeatedly for a range of specific propagation directions determined by a.. For each propagation direction, however, the terms yx and yy are no larger than before and thus the form of the equations remains unchanged. For the beam convergences generally used in STEM crystal defect images, at is still only a fraction of the Bragg angle, and it is evident that varying s, the Bragg deviation, gives results equivalent to those obtained from differentiating over the range of propagation directions. For STEM conditions, the only necessary change in the two-beam equations is to replace T and 5 by Tn and S . dTn n=-WaT + (i A)S dz n n (2.15) dS x~(i -A)T, + [-rr + 2ni(w + S $)]5n where n indicates the equations must be solved repeatedly for n values of w + aw. The Bright-field intensity is then iT SEM n (2.16) In experimental practice, an additional control on the form of the STEM image is to, the angle subtended by the STEM detector. In the Philips 301 microscope, ao can be varied either with the final projector lens or by using the objective aperture to block part of the beam from striking the detector. In both cases reducing ao means that rays from only the central portion of the incident beam are used to form the image and the convergence can be equivalent to one where ai was reduced instead. Since ar offers the last opportunity in the column to control the convergence of the beam, Maher and Joy (1976) have defined the equation (2.17) gw as 5 where aw shows the extent of the variation from ideal TEM imaging with a perfectly collimated beam. As shown in Fig. 2.7, the range of Bragg deviations is proportional to ai and thus also to ao for the reasons outlined above. The equation for aw shows directly which electron optical con- ditions and which materials will show the greatest amount of image modification under STEM imaging conditions. As discussed by Booker et al. (1974), higher order reflections for STEM imaging produce situ- ations where the variation in the Bragg deviation range increases due 2 o(; >88 2 eg Figure 2.7 Beam convergence values for low cone angle STEM (a) and MBI (b) . to the greater thickness of the convergent beam reflecting volume. Low atomic number materials with relatively longer extinction dis- tances for a particular reflecting plane will also tend to show greater Aw values. For example, at 100 KV and a E 5 x 10 , nw = .97 for silicon but only .58 for Cu due to their different values for (11 and d11 The Aw relation can be used only in "low-cone angle" STEM where a. < 6 I~f this condition is not satisfied, the first order dif- fraction cones overlap the central beam as shown in Fig. 2.7b and it may be physically impossible to place the detector such that it is struck by only one beam. This case, known as Multi-Beam Imaging (faBI) was studied by Reimer (1976). As might be expected, in a case where both bright-field and dark-field images are used to simultan- cously form an image, crystal defect contrast is reduced but it also decreases rapidly with foil depth. Except for this feature, which allows easy determination of which end of a defect penetrates the top of the foil, MBI offers no advantage over standard TEM or low- cone angle STEM techniques. Reimer attempted to compare calculated profiles of stacking faults with experimental MBI images and obtained reasonable agreement by assuming a non-coherent source. If some degree of coherence of the incident electron beam is assumed, it is apparent from Fig. 2.7b that the primary transmitted and scattered waves could interact even if the multiple scattering phenomena used in the development of the dynamic theory are ignored. This does not occur with a standard hairpin filament because the tip of the filament from which electrons are emitted is approximately 15 pm in diameter and the 100 KV electron wavelength is only .037 A. For a much smaller source size, such as the field emission STEM, the dynamic theory may have to be modified to take increased coherence into account. If imaging is restricted to low-cone angle STEM, the aw equation provides a good indication of the extent of image modification. Joy and Maher (1975) examined bend contours in silicon for a range of ao -3 -7 -a o h 20 efeio B 02 r 7.5 x 10 rad to 3 x 10 rafo e<20relcin( =10 ad and found that alterations in fringe contrast that became negligible once to was less than 8 x 10- rad. Thus, if aw < 0.1, the STEM image should be virtually identical to conventional TEM. These conditions can be easily satisfied in the Philips 301 and Fig. 2.8 shows experi- Irental confirmation of Joy and Maher's results comparing a well colli- mated {ao = 5 x 10- rad) TEM image to that of STEM (to 1 x 10- rad) with g = <220> for an extrinsic stacking fault in silicon. Stacking faults and dislocation images are less susceptible to convergent beam imaging modification than bend contours or thickness fringes. Apparently, this occurs because defect displacement fields strongly affect transmitted and scattered beams only in a localized area, while a bent fail supplies a more gradual, but constant, modi- fication to the beam amplitude along the entire column. Since these features often obscure contrast from defects, STEM imaging with Aw = 1.0 may prove to be more advantageous than TEM in highly defrrmed spe camens. As explained in the last section, a bent fail of constant radius produces an intensity curve quite similar to the rocking curves in 28 3"-~p~~jl: 3 1, p~.e i ''r +~ ir 1 *i, a (a) TEM ~; T-i -9 Sf~L'cr r s .. r P 3~1 ~3 1 i .r ~'-~~qc : -.r r -- s. -6 ~a~ - -- a r~JI a.ri3tu;l.~---rn (b) STEM Figure 2.8 Extrinsic stacking faults in silic~a imaged for two-beam conditions in TEM with a, S x 10 rad (a) and in STEM with ct -3 1x10 (b). Fig. 2.4 that plot T 2 versus w. STEM conditions force the replace- ment of TEM rocking curve by a family of curves each slightly dis- placed along the w axis up to an amount -2. Figure 2.9 shows the results for both 4 and 6 curves that are added together and then normalized to produce the "STEM" rocking curve. As implied by Fig. 2.9, further summations beyond 4 curves improve the accuracy by only a small amount. The same procedure is used to produce the thickness fringe curve in Fig. 2.10, where curves are plotted for different values of s and then averaged to produce the STEM version of the IT 2 as function of thickness curve. In each case, the "square aperture approximation" developed by Fraser et al. (1976) is used for computation. Here, the convergent beam diffraction pattern is approximated by a square array (Fig. 2.11) and the range of dw is determined by the distance of the centroid of the square to the appropriate Kikuchi line. 2.5 A STEM Computer Simulation Program Since a crystal defect image is simply a greatly magnified image of the main beam or one of the diffraction spots, it follows that a microdiffraction pattern from the vicinity of the defect can contain no more information than the defect image itself. However, it will be shown in subsequent chapters that if a.i and the spot size are kept sufficiently small, it is possible to directly obtain in- tensity information such as lT 2 or lS 2 for a two-beam condition from an area as small as 50 A in diameter. *I .. V e ci m .. a r at eel O ~ O 40 *O . -m u~~~ l io CI CI O ~ I GC O q* I Em Y MI UA r(~~ 0 m -HE ,Q FHU cr s *c a~ - m id . ag -. v = 0 UNY *. *. 4 m -, *.c .*m . A ~~ I O~~l O OO 0 0a 09 O~r d d c 2 19c i1 S.. Orm II - *, 4 4 Ec ..-* g O >C cle *, me 01 W .*I a **...... p '~* -Ic idOm ..- r ar * m MMU Oa I I II 1 o COlO m 7 60 9 ~ c N r o a c' cO d d C Transmitted Beam Position First Order Diffraction Spot Position Kikuchi Line Pair Figure 2.11 The square aperture approximation (after Fraser) used to compute Fig. 2.9 and 2.10. The maximum and minimum w values are found by measuring the distance from the "diffraction square" center to the appropriate Kikuchi line. Any given w value does not vary in the x direc- tion. This would not be the case for a circular aper- ture that would produce a circular diffraction disc. The microdiffraction pattern provides, in one image, the direct T 2/ S 2 intensity radio (measured experimentally be a densitometer from the photographic plate) that could be obtained only by measuring intensities at exactly the same point in the bright-field and dark- field defect images. Since after tilting the beam to get a dark- field image it is impossible to keep the Bragg deviation precisely the same, obtaining IT 2 2s ratio is possible only with microdif- fraction. If the location of the incident beam can be precisely determined, experimental 1T 2 12 ratios can then be "simulated" by mathematically modeling the effect of the defect strain field, R, on the T-S interaction. The various experimental variables such as fail thickness or the Burgers vector can then be varied until the computed ratios are the same as the experimental model. This approach has been developed by Head et al. (1973) by matching defect images to a high degree. The microdiffraction ratio method potentially offers a method of characterizing the defect strain field with a much smaller expenditure of computer time than the image matching method, where several thousand columns must be computed to produce a simulated image. Ai program to simulate T 12/ S 2 ratios need not be as efficient as the one used by Head, since only a few columns should need to be computed before the defect is identified. While the approach used here was chosen for algebraic simplicity, it is sufficiently rapid that it can be used to produce defect image simulations if desired. Reference to Fig. 2.12 shows the two coordinate systems re- quired for the intensity calculations. The first, crystal coordinates, is used because necessary input data such as dislocation line direction, 34 BM BM x (u x BM) r vectors for one column U x BM | BM x (u x BM) U x BM Figure 2.12 Coordinate system in relation to disclocation and the image plane. fail normal and the g vector can be determined only by using the hkl coordinates found in the diffraction patterns. The second, image plane coordinates, are used to determine, R, the displacement field values necessary to solve the two-beam equations for a particular column . Once the necessary geometry transformations have been deter- mined, the intensity calculation is relatively simple. Each column, parallel to the electron beam direction, is divided into a number of segments, dz, and R, the defect displacement field, is computed for each segment. Then by using the approach of Th614~n (1970), each column segment is considered to be a slab of perfect crystal rotated a slight amount by the defect displacement field. Each segment can be considered to have a different Bragg deviation, and the analytic solution (Eqn. 2.5) is used to compute T or 5 for that segment. Th614n showed how to manipulate the analytic solution such that the amplitude T and S exiting from any random slab are related to the incident amplitude by a 2 x 2 matrix: T al al2(2.18) (out al 21 a22 in1 where (Y2 + an) (2.19) all (Yl- 2) 2-Yexp(Ylds) + ylexp(y2dz)] a(i -A) al2 = a21= (71- ~2 lexp(ylds) exp~y dz)] a2 (yl ) lexp(yldz) Yexp(y dz)] and .2.20) 712 inw I wi / +(i A)2 with this approach, the two-beam equations do not need to be numeri- cally integrated down a column of distorted crystal. Prediction of the amplitudes of T or 5 is now achieved by matrix multiplication, where each matrix corresponds to a slab of perfect crystal with thick- ness, dz, and a specific Bragg deviation. After the necessary trans- formations converting all crystal directions to image plane coordi- nates, the sequence for the numerical calculations of T 12 or S 2 proceeds in the following manner: First, the distance, r, between the dislocation line and the column segment is determined. The displacement field, R, is now a function of r, 6, and b, where b is the Burgers vector and 9 is the angle between r and the slip plane. With r known, compute R at the top and bottom of the segment. It is important to realize that R is a function of only two independent variables, r and 6, for any given Burgers vector. Large savings in computer time can result if R is pre-computed for a range of r and 6 values that correspond in size to an area somewhat larger in diameter than the specimen thickness. By storing g'R in matrix form, with subscripts corresponding to r and 6, repetitive calcula- tions of the same value of R can be greatly reduced. This idea, never used by Th614~n, is feasible because the dislocation displace- ment field is "two-dimensional," i.e*, an R value for a given r and 6, is the same along any point of the dislocation length, assuming the infinite solution. When g*R has been determined, the effect of the dislocation on each slab, dz, is (2.21) di (gR) =q(Rtop Rottom) . dz g ztop- bottom Now, the effective Bragg deviation w + B8( and a scattering matrix a.. of a specific value are known for the slab. Thijlin has shown that it is possible to pre-compute the scattering matrices for in- crements of w just as was done for R. B can then be used to select the appropriate pre-computed matrix needed for a particular slab. Once a scattering matrix has been selected for every slab in the column, the transmitted amplitude, T, at the top surface is set equal to I and S = 0. The amplitude of T and S at the bottom surface is then found by successively multiplying each scattering matrix times the exiting amplitude of T and S from the slab immediately above: (2.22) 5 =al a2 a3..a where an=1a2 out ina2 22 En practice, it is necessary to divide the foil thickness into 80 segments to compute IT 2 to a sufficient accuracy. Thus, 80 scattering matrices must be multiplied together to compute the T11 for a single column. The actual intensity T 12 of 5 12 can then be divided by the intensity found when r is set to a very large value, or to intensity computed for the perfect crystal where R = 0. The geometrical conversions can be much simplified if the ana- lytic geometry formulations for a line and a plane given rectalinear coordinates are used. Thus, for vector, A (2.23) C =Ci +Cj +C k the equation of a line parallel to the vector is (2.24) x -x y -y z - x y z and the equation of a plane normal to the vector is C (x- x + (y y + C(2 z )= C(2.25) x o y az o All important features in Fig. 2.12 such as the foil surfaces, or the dislocation line and column directions, can be represented by one of these equations. To begin the calculations, it is assumed that all necessary input vectors such as the beam direction, B, the fail normal, F, and diffracting vector, g, and the dislocation line direction, U, have been comp4uted from experimental hkl coordinates. The Bragg deviation and the analomous absorption are entered as dimensionless quantities while the foil thickness is entered in units of 5 the extinction distance. Units for b, the Burgers vector, and g are not important as all units cancel when g~b is calculated for a particular slab. An "image plane" coordinate system can now be defined where the vertical axis is parallel to B, and one horizontal axis is parallel to U x BM, and the final horizontal axis is parallel to BM x (u x BM) If the direction cosines between the hk1 coordinate system and the image coordinates are computed, a 3 x 3 matrix C.. is defined where (2.26) C., = x. U. C.. can be used immediately to transform all vectors to picture coordinates (2.27) U; C., U. B! = C.. B. b; = C.. b. Fi = C., F. g = Ci gi The dislocation is now situated such that its mid-point lies at the origin of the image coordinate axes. Taus the line equation for the dislocation is (2.28) To find equations for the upper and lower foil surfaces, x and y are set equal to 0 and TZ is set equal to TH/2, where TZ is the foil thickness in the beam direction. Equation 2.25 is then (2.29) TZ Once the constant, D, on the right hand side of the surface normal equation is known, Eqn. 2.25 can be re-arranged to solve for z, the fail surface height (2.30) Fx +F y TZ 1 2 2 Fj The starting point for any column can now be found by selecting any xoYo point in the image plane and solving Eqn. 2.30 for zo. Sub- sequent column segment coordinates are then x yo' and zo As where Az =t/80. Once the coordinates for any column segment have been deter- mined, the distance to the dislocation line must be calculated. To compute r, both a plane perpendicular to the dislocation line and passing through the origin and a line parallel to the dis- location must be determined. The vector r can then be drawn from the origin to the intersection of the plane and the line. From the de- finition of the coordinate system, the z component of the dislocation line is zero. Therefore, the equations of a line parallel to the dislocation and a plane perpendicular to the dislocation are (2.31) u y u z = C (line equation) z~ y (2.32) u y + u z = 0 (plane equation) Y z Equations 2.31 and 2.32 can be solved simultaneously to yield finally (2.33) z = -uC =-u (uzy u z) y = uZC = u (u y u z) The vector r is of the form: (2.34) r = (x xo~ ( o)j + (Z 2 )k but since it is drawn from the origin, the three components of r are (2.35) rx = x ry = uz(uzy u z) rz = -1(Uyugy z Now, the general solution for a dislocation of mixed Burgers vector is (Hirth and Loethe, 1968) (2.36) 1 *6 sin26 + xu1 -20 enlr+co2 2sr 4(1 u) 2(1 u) 4(1 -u) where be = the edge component of the Burgers vector normal to u 6 = an angle between r and the slip plane of the dislocation u = Poisson's ratio. As shown in Fig. 2.13,. a can be. found by taking the dot product between r and a vector lying at the intersection of the slip plane and the plane normal to the dislocation. The intersection vector is defined as (2.37) ux bx u Thus, (2.38) V-r cos6 and b, = (:b'V)V Slip Plane Figure 2.13 Radial coordinate system used for the calculation of R at point defined by r and 6 The angle 6 is always measured from the slip plane and varies 0 to 2rr. where V is a unit vector. Examination of Fig. 2.13, however, shows that 8, computed from Eqn. 2.37, could result from an r lying either above or below the slip plane. An additional test to determine the correct direction of I is performed by computing the unit vector (2.39) PP and then again computing the angle between P and r (2.40) cos6 = P r/ r By examining the two-dimensional coordinate system formed by P and V, it is seen that if the true position of r is in quadrant I, both cos6 and cos6 will be positive. If r lies in quadrant IV, cos6 will be positive, but 6 is now more than 900 and cos6 will be negative. If a similar test is applied for an r lying in each quadrant, it is found that Eqn. 2.37 gives the proper value for 6 only when cos6 is positive. The computer program calculates both 6 and 6 and changes 6 to 2n + 9, if cos6 is negative. Then the "sense" of 6 is always correct, regardless of the position of r. R can now be computed for every element in the column by the method discussed before. It is noteworthy that the linear distance between each precom- puted matrix element for R will vary slightly according to the dis- tance from the dislocation core as shown in Fig. 3.13. This is simply a consequence of defining R in terms of polar coordinates. Such a choice is convenient for the column calculation, because the r,6 matrix elements are packed most densely at the dislocation core, which is precisely where R varies most rapidly and the greatest ac- curacy is needed in the column calculation. With R chosen, the effective Bragg deviation for each segment, 8, is now calculated .. .. .(2.41) Sd (g.R) = g-(Rtop Rbottom) where Az dz az 80 As discussed by Th861n, since the a matrix depends only on w, it is far more efficient in terms of computer time to calculate B for incremental values of w and store the results. Since jw values higher than 3.5 imply such a large lattice distortion that the two- beam condition is no longer valid, the A matrix is calculated for a range of -3.5 < w < 3.5 in increments of .005 giving a total of 1401 2 x 2 matrices that must be stored and used as a "library" when any column is computed. Once the effective Bragg deviation is calculated, it can be converted to an integer by the formula (2.42) B = [--- x 700 .005 and the integer (from 1 to 80) is used to select the appropriate matrix for a particular slab. The sequence of the column calculation is then = 512...n 0(2.43) where 1, 2...n represents a sequence of 80 integers corresponding to each of the 80 column segments. Once computed and divided by the background intensity, jT 2 is compared to a series of 11 decimal values between 0 and 1 that comprise the grey scale used in printing the picture. The value closest to IT 2 is used by the program to select one of 11 symb~ols that repre- sent a particular shade of grey. This symbol is then printed by an IBM high speed printer to represent the intensity in the micrograph corresponding to a particular column in the specimen. If an image is not desired, ratios of 15 212 2 or any regular array of columns spaced along the dislocation can also be computed. In this work, experimental studies concentrated on extrinsic stacking faults in silicon. For this case, the displacement vector is confined to the plane of the stacking fault and does not vary with distance as for the dislocation. For this case, E = -<111> and is normal to the stacking fault which always lies on {lll}-type planes. It is shown by Whelan and Hirsch (1957) that the fault can be considered as a planar boundary separating two perfect crystals. The S wave must be adjusted by a phase factor exp(iar) as it passes through the fault while the T wave remains unchanged. The phase angle is computed by the equation: (2.44) ar = 2a g-R Therefore, a stacking fault is added to the program simply by locating its height in the specimen for a particular column and adjusting the amplitude of 5 at that particular point. Modification of either dislocation or stacking fault images for the effects of beam convergence in STEM is done in the program by adding a value nw (compeuted by nw = 1Sg U a) to the Bragg devia- tion, w, and then completing the column calculation as before. For 46 greatest accuracy, hdw must be divided into several segments and the column computation performed for a series of (w + q quantities. All of the lT 2 values for each nWsegment are then summed together to produce the total intensity for a particular column. CHAPTER 3 MICRODIFFRACTION METHODS 3.1 A Review of the Effect of Inelastic Scattering on the Diffraction Pattern In addition to the dynamic effects on diffraction spot inten- sities, convergent beam radiation also affects inelastic scattering which is the cause Kikuchi lines and transmission channeling patterns. Both of these phenomena can be understood by examining the surface formed by all vectors that are at an angle of 6g with the reflecting plane. This surface is a cone whose central axis is normal to the plane. For diffraction resulting from a perfectly collimated beam, only one vector lying on the surface of the cone is necessary to de- scribe the diffraction direction, and, although the actual diameter of the incident and exiting beams may be several microns in diameter, the beam can be focused to a point to forn the diffraction pattern in the back focal plane of the objective lens. If the beam is highly convergent, this is no longer the case. Now, only the part of the incident and reflected rays lay on the reflecting cone surface that defines the exact Bragg angle. The extent of the segment of the convergent beam that intersects the reflecting cone is shown in Fig. 3.1 to be (3.1) D = 2Lag 8z E o m00 w> ,CC Z a0 BO H LI~ P00 o501 C -O / 1i 0~a 'a / OO C] CO HO * nm~a c 0 0 = m a oac 'i O COO i C P) 8r where aj is the semi-angle of the incident beam and L is the effective camera length. The convergent beam diffraction pattern, as diagrammed in Fig. 3.2, now consists of an array of discs whose diameter, y, is propor- tional to 2a.. Only those electrons in the incident beam that strike the reflecting plane at exactly 6g form the part of the disc with zero Bragg deviation. It is obvious that if 2ai is increased suffi- ciently, this line of "zero Bragg deviation" can extend across several orders of reciprocal lattice spots. Calculation of ar from the convergent beam diffraction pattern is easily accomplished (Thompson, 1977). In Fig. 3.2, the convergent beam which subtends an angle of 2ac strikes the specimen and produces a main beam and several diffracted rays that are focused as discs in the back focal plane of the objective lens. Bragg's law shows that the distance between the main beam and any of the diffracted beams is proportional to 26 Also, it is evident that the width of the disc in the convergent beam diffraction pattern is proportional to the convergence angle of the incident beam. The law of similar tri- angles shows that the radius of one of the diffracted spots divided by the distance between the main beam and the diffracted beam is equal to convergence angle of the incident beam divided by the Bragg angle 28 x= B y i (3.2) y ag x OB Since x and y are easily measured on the diffraction pattern and 209 is known, the degree of collimation of the beam, a i, can now be computed from Eqn. 3.2. Note that as ai increases, the discs in the a c ct ~r ax a arv, cr~i oar rlu uo m rl aa cc mo u -rl +r rlm ru a i a arF car rl o ~u3 mel aar a arar u u cc arm tncl +1 .. O O 'TI E cc `C L a, E o c C OI (d0 o 1co u * * It I C~ diffraction pattern increase in size until they begin to overlap as shown in Fig. 3.3a. The camera length, which controls the size of the diffraction pattern, can be changed by adjusting the magnification lenses in the usual manner. The amount of spot overlap can be changed only by adjusting the size of the condenser aperture or the strength of the condenser lenses. Although their method of formation is different, both Kikuchi and transmission channeling lines that are observed in electron dif- fraction patterns obey the geometrical relations described above. Kikuchi lines, formed by elastically scattered electrons that pre- viously have undergone one or more inelastic collisions, are familiar in metallurgical electron microscopy and are often used as an aid to determine the exact orientation of the specimen with respect to the beam. Kikuchi lines can result even from a perfectly collimated beam; the inelastic scattering of the beam (once it enters the speci- men) causes it to diverge through an appreciable solid angle. The formation of these Kikuchi line patterns can be analyzed geometrically in Fig. 3.4. Figure 3.4a shows the scattering distri- bution for electrons that were inelastically scattered near the top of the specimen. Most of the electrons are deflected only slightly from the original path, and decreasingly fewer in number are scattered at greater angular deviations. Since the actual energy loss involved in a few 100 eV (Edington, 1975), some of these electrons will have been inelastically scattered in the proper direction for additional elastic Bragg scattering to occur. Such a scattering event for a v V H O *id -H 01 rO O rl rl ,Q (a) (b) Ray A~ Ray B (c) Figure 3.4 The formation mechanism and the geometrical construction for a Kikuchi line pattern. Fewer electrons are scattered along Ray B than Ray A. In three dimensions, Ray A forms the "excess" line (intensity higher than background) while Ray B forms the "deficit" line. particular set of planes is shown in Fig. 3.4b. By referring to the intensity scattering distribution in Fig. 3.4a, it is apparent that more electrons will be scattered along ray A than along ray B. In three dimensions, Fig, 3.4c shows that each reflected plane will produce two cones of scattered electrons. Due to the shape of the scattering distribution, one cone will have a higher intensity than the diffuse background radiation, while the other cone will have an intensity lower than background. These two cones intersect the Ewald sphere to produce hyperbolae. The radius of the Ewald sphere is large enough that they appear in the diffraction pattern plane as straight lines. TIhe importance of Kikuchi lines in determining speci- men orientation can be seen by examining the effect of the rotation of the specimen on both the diffracted spots and the Kukuchi lines. For example, in Fig. 2.2a, it can be seen that a small rotation of the sample may not change the spot pattern at all if the rel-rods are of a length normally produced by a crystal of average thickness. The reciprocal lattice is simply rotated about its origin in the same angular direction as the crystal. No new intersections between the Ewald sphere and rel-rods are formed after the tilt and the spot pattern remains changed. However, Fig. 3.4c shows that the same tilt would result in the pair of Kikuchi lines being shifted to the new position represented by the dotted rays. Such small shifts are easily detectable in experimental situations and the relation between the incident beam direction and the specimen can be calculated by methods described in Chapter IV. The form of transmission channeling patterns as shown in Fig. 3.5, appears geometrically quite similar to a Kikuchi pattern for the same beam direction. This similarity is no accident, as the angular relationship between the elastically scattered electrons and the reflecting planes must be exactly the same for both cases. For channeling, however, the angular range of the incident beam is pro- duced by the focusing action of the condenser lenses and no inelastic scattering need occur to produce the channel pattern. The rather simple kinematical models given here to describe the intensities and relative contrast present in these patterns are only an approximation and a precise description of the line intensi- ties must make use of the dynamic theory discussed in Chapter II. The need for the dynamic theory is obvious when it is realized that the scattered beans, regardless of their position on the reflecting cone, can undergo multiple Bragg scattering and interact with the portion of the incident beam parallel to the scattering direction, as shown previously in Fig. 2.4. A semi-quantitative theory developed by Thomas (1972) describes an n-beam solution for Kikuchi patterns simply by summing many-beam cocking curves for particular Bragg reflections. Although the effect of thickness on the rocking curve was ignored, the computer solution was able to correctly predict the relative intensities and widths of the excess-deficit Kikuchi line pairs. A more rigorous procedure (C. R. Hall, 1970) used the dynamical two-beam equations to show that contrast reversals could occur in a Kikuchi band as thicker areas of the crystal were examined. Fiur 3.5 Coprsno iuh trs n rnmsincanln patrsfo iio.Tedereo emcnegnei d Liffrn n(b n c Although image simulation techniques for channeling or Kikuchi patterns could be further developed, there seems no obvious advantage in using such a method for the examination of distorted crystals or defect strain fields. Any dynamic information in the channeling patterns resulting from defect-beam interactions should also be available in the convergent beam diffraction disc pattern in a more easily interpretable form. In addition, the use of diffraction in- formation from either convergent beam radiation or inelastically scattered electrons implies that a larger volume of crystal is irra- diated than would be the case for a STEM probe with a semi-angle less than 10 rad. As shown in the review paper by Goldstein and Williams (1977), inelastic scattering in thin fails results in x-ray production from a volume with an average diameter much larger than the original diameter of the probe. Thus, a Kikuchi pattern produced from a small STEM probe probably arises from a significantly greater volume than the region that gives rise to the dynamic bright-field intensity of the diffraction discs themselves. This follows from the fact that the bright-field intensity is restricted to elastic scat- tering and electrons suffering high angle collisions should not directly contribute the T 2 value of the diffraction disc, itself. For these reasons, only the geometrical information present in Kikuchi or channeling microdiffraction patterns can be used for examination of distorted foils. Even with this limitation, analysis of the line patterns can yield considerable information as long as g*R does not vary in the local volume illuminated by the electron beam. As a first example, an approach described by Cowley and Smith (1971) is used to determine the state of curvature of a small area of a foil. In Cowley and smith's paper, a convergent beam was used to produce a channeling pattern such as the ones shown in Fig. 3.5. Cowley and Smith discovered that by de-focusing the convergent beam such that the cross-over does not coincide with the specimen plane, each excess and deficit channel line pair can be split into two dis- tinct lines, where the width of the splitting is governed by the amount of defocus. Examples of the line splitting effect for a par- ticular channel line pair are marked by arrows in Fig. 3.5b and c. The explanation for the effect is shown in Figs. 3.6 and 3.7, which have been reproduced from Cowley and Smith's paper. Consider the incident and diffracted rays from opposite sides of the convergent cone for perfect crystal (Fig. 3.6). For this case, the incident rays that lie at the proper angle for Bragg dif- fraction are parallel to the scattered rays produced by the rays on the opposite of the cone. For the case of a bent crystal, shown in Fig. 3.7, this is no longer true. Now the transmitted rays deviate by angle O' is radius of specimen curvature. From the figure, it can be shown that x = 26 L (3.3) y2 = O'L where xL = the separation between a particular transmitted and scattered ray X2 = width of line split in the diffraction pattern Diffraction from Flat Foil Figure 3.6 Cowley diagram for diffraction from a flat crystal. Cowley and Smith (1971) Figure 3.7 Cowley diagram for diffraction from a bent crystal. Cowley and Smith (1971) L =camera length Manipulation of Eqns. 3.2 and 3.3 shows that the radius of curvature of the foil is (3.4) z(xl-x x2 where z is the amount of defocus, or the distance of the cross-over from the specimen. Since the effect is dependent upon a defocused probe, Cowley and Smith's method is not a true microdiffraction technique but it does provide an explanation for much of the fine detail seen in the channel- ing pattern. 3.2 The Use of Channeling Patterns to Characterize Polycrystalline Specimenns A method of analysis of more general utility than Cowley and Smith's, using geometrical information from either Kikuchi patterns or channel patterns resulting from a focused probe, is developed below. Consider the case where a fine-grained polycrystalline foil is to be analyzed by microdiffraction. It is desired to characterize the crystalline structure by determining the difference in crystal orientation between grain 1 and 2, shown schematically in Fig. 3.8. If the microdiffraction pattern is recorded from each grain, the direction of the beam (Helfmeier and Feller-Kniepmeir, 1977) is --- --(3.5) 91 92 BM = L1 1 - L2f 2 f where Z =- gl x q2 = the approximate beam direction. Figure 3.8 If Kikuchi or channeling patterns are recorded for two adjacent grains, the orientation of the two grains can be determined from measuring the local beam direction for each grain for two different specimen tilts. L fl, L2, and f2 are measured from the diffraction pattern. A fuller discussion describing computation of the exact beam direction is discussed in Section 4.2. Let a superscript (*) designate the beam direction from grain 2 and a prime (') show the beam directions for both grains after the foil is tilted to a new position. For position 1, the relation between the two grains for any cubic crystalline structure is BM. = .. BM* (3.6) Similarly, for position 2 BM: = .. BM*' (3.7) where 1.. is the 3 x 3 direction cosine array describing the orienta- tion difference between the two grains. If each set of equations is expanded, the ones involving BMl are BM1 = 11 BM* + 11 BM* + 11 BM* (3.8) 1 11 1 B~' 12 B2 13 3M* BM' =11B*+12M*' +113B* and, it can be shown that 22 2 1 = 1 + 1 + 1 (3.9) 11 12 13 This is a set of three equations in three unknowns, 111, 112' and 113. Using the equations involving BM2, a similar set of equa- tions can be solved for 121, 122, and 123. The final three direction cosines are then determined with (3.10) 111 131 + 112 132 + 113 133 = 0 121 131 + 122 132 + 123 + 133 = 0 12 +12 + 12 1 31 32 33 where the underlined components are the remaining unknowns. This method can be used to describe the orientational relations between two adjacent areas of a distorted sample, low angle grain boundary, or two different grains of the same crystal phase. Simi- lar sets of equations can be used to solve the orientation between two phases of different crystal structure. It is important to note that this method can be used regardless of the orientation of the specimen with respect to the beam and is dependent only on an accu- rate computation of the beam direction using either Kikuchi or trans- mission channeling patterns. While this method is also applicable to standard selected area diffraction techniques, it can be used to the fullest extent only with microdiffraction techniques. Many structural features in metals are much less than 1 Fum in extent, and some, such as grain boundaries and stacking faults, are only a few atomic layers in thickness. For example, only a very narrow probe could be positioned close enough to an end-on grain boundary to pro- duce useful information about its local orientation and strain field. 3.3 A Review of Ray Optics for Convergent Beam Diffraction In the Philips 301, a strongly demagnifying upper objective lens field makes it possible to form a much smaller focused spot than was possible in earlier electron microscopes. However, the incident on the specimen is now much more convergent than before. For the TEM made in the Philips 301, the two condenser lenses and the upper half of the objective lens field are used to illuminate the specimen, while the lower half of the objective, diffraction, intermediate and projector lenses are used to form a magnified image on the phosphor screen. In Fig. 3.9, the filament image forced slightly below the electron gun assembly is damagnified by the first condenser lens (Cl (not shown)), and both the second condenser lens (C2) and the upper half of the objective lens are used to further damagnify the image on the specimen plane. Although the objective polepiece used for both TEM and STEM imaging strongly demagnifies the C2 cross-over, its excitation value is coupled to the lower objective polepiece and must be set to a certain value for a focused image. For the imagining mode, the C2 lens is adjusted so that its cross-over lies close to the upper focal plane of the upper objective polepiece (Fig. 3.9a). The cross-over then acts as a pseudo-point source which the upper objective lens field converts to a relatively well-collimated beam that illuminates the specimen. One advantage of the imaging mode in the TEM (mode) with the Philips 301 is that the C2 aperture can be focused on the sample in rl O m o"C~P, or ~Cn a la bS ~ OE rl (n(L r( L a O p, i: c, a, I I ;i to m uoc I a, aoia a ,, alL~ B o a a in E o- 1 u cn I\ S a, L~ rd Cg cQ id a, I r c I I rl LI o c, o b x w m c~n a ~ exactly the same way that the selected area diffraction aperture above the diffraction lens can be focused on the lower objective image plane. Consider Fig. 3.10, which shows the physical arrangement of the objective polepieces and the specimen as well as the ray diagram for a perfectly collimated beam. The specimen lies in the middle of the polepiece gap, near the center of the magnetic field formed by the polepieces. The actual magnetic field in the polepiece gap is great enough that it cannot be treated as a "thin" lens and the stan- dard ray tracing rules used in geometrical ray optics do not apply. The specimen, of course, is a real object, and only the portion of the lens field to the right of the specimen can be used to produce a magnified specimen image (Hall, 1966). Useful information from ray tracing can still be obtained if the objective lens field is consid- ered to be formed from two thin lenses, and this construction is used in the ray diagrams supplied by the manufacturer. Using this assumption, the objective lens field is treated as two thin lenses, and it can be seen from Fig. 3.11 that the upper objective lens forms a demagnified image of the C2 aperture at (a), while the lower objective lens forms a magnified image of the speci- men at (b). The method of selected area diffraction developed by Poole (1947), involves placing an aperture at (b), so that only the central part of the image is used to form the diffraction pattern at (c). By tracing the rays from (d) to (b), it can be seen that the selected area aperture allows only rays from the part of the specimen (e-e) to form the diffraction pattern that is recorded photographically. ~-Specimen Figure 3.10 Dimensions of the Objective Polepieces for the STEM Modification of the Philips 301. O r cl 0 ar , E 4 a , a On 0 0 did tO ar 0 0 CO r 0 ,5 a, CI O n O B\ S c a, E rl u a, a, rl CI arm [lln CL~~ 30 C 0 C However, if the specimen is raised to a non-eucentric position to coincide with the image of the condenser aperture, a "selected area" diffraction pattern will again be formed. In this case, the condenser aperture permits only a small area of the sample to be illuminated, and it is this area alone that forms the pattern. For the Philips 301 STEM polepiecer the condenser aperture is demagnified 38 times, while the selected area diffraction aperture is demagnified 14 times at the specimen plane. Because of spherical aberration, the utility of the focused condenser aperture method is even more favorable than suggested by these demagnification ratios. This point is explored further in the discussion on lens aberrations. To produce the convergent beam diffraction pattern, the strength of the C2 lens is reduced such that its cross-over would lie below the specimen if the objective lens were turned off (Fig. 3.9b). At the proper excitation strength the objective lens now acts to increase the convergence of the beam and focuses the rays to produce a cross- over at the specimen plane. The objective lens is always kept at the same excitation, so calculations involving spot size or conver- gence angle changes can be accomplished by considering the lens strength and the aperture sizes of the Cl and C2 condenser lenses alone (Fig. 3.12). Consider Fig. 3.13, where the distance from the gun cross-over to the Cl lens, 0, is fixed, but the lens to cross-over distance, I, depends on the strength of the Cl lens. Since the refractive index 0 *n ar o m 0 ,Q MG 0 oM z Z l C4V * *HQ m00 m3 c , e 00me 0000t -I , 0 * N C >a 9>#00 aQ OW OOr nO a, C100 U C1 LMmO rua 0 N -r0 0 L Od 90 SoND 0m 0 B3C O MM O m a -H .0 N m in d rl * 001 0 0 000 Ocr c9 0 z0J -HO UL u) 2' D 0b E-4 0 0 w H cO 000 4t ocm 0 a0 r *H &a OHdr 0 0 0000111 rdmm c40 *M 0nr .g00 90 o -HOA 0 N- r( mz CO MO N r N I 03 01 mtm me mm 3 C BsCO 3N OU 00 >cm 000 vlx iola OO 00a~ i -H 0 x0 D M 01 Iu ,C C c 0 >O HXO r r N m a, ..r a,00 a . I: HHna U vl Co ac0 O ,C m2-rc of electrons is the same on both sides of the Cl lens, fl = f2 = f and the thin lens equation is (3.11) -1 1 _1 As the Cl lens is increased in excitation the focal length must decrease and since 0 is fixed, I must decrease proportionally. The demagnification of the spot is given by the ratio (3.12) It can be seen that by increasing the Cl lens strength, the cross-over is brought closer to the right-hand side of the lens and the filament image is demagnified, thus forming a smaller spot. Once the spot size has been selected by a particular Cl lens excitation, only one possible setting for the C2 lens and the objective lens can be used if a focused spot is to be produced in the specimen plane. Thus, a specific Cl setting determines not only the spot size, but the convergence, ai, of the incident beam. The C2 lens can be used to alter a but at the expense of forming a defocused spot on the specimen plane. However, the convergence of the beam may still be increased by changing the size of the aperture used with the C2 lens. As was shown in Fig. 3.12, substituting a smaller aperture in C2 reduces ar while leaving the spot size essentially unchanged since all three lens excitations remain the same. It is important to see that angular resolution in the diffrac- tion pattern and the current density of the focused spot (and thus the intensity of the diffraction pattern) are inversely proportional. It has been shown (Broers, 1969), that the intensity of the beam at the specimen, ji, in amperes/cm2 is proportional to the intensity at the gun, j,, by (3.13) .2a eV sin I o a KT sin k But both ai and ak, and thus the intensity of the microdiffraction pattern, are controlled directly by the C2 aperture diameter. STEM microdiffraction methods differ from TEM convergent methods in that the specimen is still illuminated by a focused probe, but the image is formed sequentially as the probe is rastered over a rec- tangular area of the sample. To form a microdiffraction pattern from a specific region, all that is required is to stop the rastering motion produced by the deflection coils situated between the objective and C2 lenses, and manually adjust the deflection controls to place the probe over the desired area by watching the STEM image on the CRT. Regardless of the position of the probe, the microdiffraction patterns will be formed in the back focal plane of the lower half of the objective lens. Ray optics in STEM are described in Fig. 3.14. As in TEM, spot size can be varied by altering the C1 lens strength, but in STEM the C2 lens is turned completely off. The average convergence for a given probe size is now an order of magnitude greater than the TEM mode, but the spot sizes can be decreased proportionately. For example, the smallest spot size attainable with the standard hairpin filament is 400 A with the C2 lens on (TEM convergent beam mode), m O ,a o A a O am cme a .cl rl O rl ,Q OO 0 OrCI Caoa o -HO ~ma OIm LO OO 0 ,Q * E rla r *clM OmL >oa *n crv .0000 0 C -0 00 tur 000u 80 cmo~ * 4 0 * " ,Q 0500~ tola >0 OH 000~ 0 9 .0 E90~c ft) o m but 40 A with it off. As shown in the inset in Fig. 3.14, ct. is now controlled only by the C2 aperture. If the distances between the condenser lenses and the distance from the C2 aperture to the upper focal plane of the objective lens are known, a. can be easily calcu- lated. However, actual diameters of commercially available apertures may vary as much as 20% from the specification and it is more accurate to calculate convergence directly from the microdiffraction pattern as is done in Chapter IV. Once the Cl setting and C2 aperture size are chosen to select a particular spot size and current density, the STEM image is formed by using the wobbler coils to twice deflect the beam below the C2 aperture such that the beam pivots in the upper focal plane of the objective lens. As shown in Fig. 3.14, the upper objective field lens not only focuses the beam to a probe but converts the angular beam deflection to a deflection that remains parallel to the optic axis. Trimming potentiometers used to adjust the height of this pivot point must be carefully adjusted if the probe is to remain parallel to the optic axis over the entire raster scan. If this condition is not met, the Bragg deviation for any given hk1 plane will vary over different parts of the scanned area and the image contrast will not be comparable to that for a collimated beam used in TEM imaging. Once the probe passes through the specimen the diffraction pattern is formed in the back focal plane of the lower objective lens, just as it is in the standard TEM modes. The probe still re- mains parallel to the optic axis after passing through the specimen solid rays in 3.14) and it always converges to the same spot in the back focal plane after being focused by the lower objective lens re- gardless of its position in the scan. In the same fashion, dif- fracted rays (dotted rays) scattered through the same Bragg angle at different points of the scan are all focused to another point, forne ing a diffraction spot for a particular set of hkl planes. The remaining lenses below the objective lens, diffraction and projector lenses, only magnify the size of the microdiffraction pat- tern such that the diffraction discs subtend approximately the same angle as the solid state detector at the base of the column. The diffraction lens must be set at a fixed value so it can be focused on the back focal plane of the objective lens, so only the projector can be varied to adjust the camera length, or the size of the pattern appearing in the detector plane. By varying the projector lens strength, the size of the diffraction discs can be adjusted so that only the central portion of a disc strikes the detector, thus giving indirect control of a o, the exit angle of the electron beam leaving the specimen. Therefore the degree of coherence of the electron beam used to form the STEM image can be controlled by selecting only the central portion of the diffraction disc just as a selected area dif- fraction pattern is formed from only a portion of the bright- or dark-field image. 3.4 The Effect of Lens Aberrations The advantages of obtaining diffraction patterns from small areas are obvious. The region of interest, whether a small second phase particle or a localized strain field from a dislocation, is small in volume compared to the surrounding matrix. The smaller the total volume which is examined by diffraction, the greater the particle-matrix volume ratio and the better the signal to noise ratio in the diffraction pattern. What is not so obvious is why microdiffraction patterns cannot be obtained with standard selected area diffraction techniques where very small apertures are used to select a correspondingly small portion of the image. The answer lies in the nature of the aberration errors common to any electro-magnetic lens. The most important of these lens defects is spherical aberra- tion which results from electrons passing through the outer portion of a lens being focused more strongly than those electrons passing closer to the optic axis. The ray bundle cannot be brought to focus in a single focal plane, and an image of a point appears as a disc with a radius equal to (3.14) r = Cs3 where Cs, the spherical aberration constant equals 6 mm for the Philips 301 STEM polepiece and, 8, the lens aperture angle is control- led by the size of the objective aperture. Reducing the size of the objective aperture thus decreases 3 and the disc of confusion, r, but the process cannot be carried too far, or resolution again begins to decrease due to the diffraction effects caused by a very small aperture . For the TEM diffraction mode, the objective aperture must be removed. Now the diameter of the selected area diffraction aperture is the factor that limits the effectiveness of selected area diffrac- tion. Consider the formation of the selected area diffraction pat- tern shown in Fig. 3.15. The diffraction pattern from the entire area illuminated by the incident beam is formed in the back focal plane of the objective lens and the (000) beam is magnified to form the bright-field image. If the selected area aperture above the diffraction lens is placed such that it is in focus with the image, tracing the rays back to the specimen plane shows that the diffrac- tion pattern appearing on the phosphor screen results only from the portion of the image within the selected area diffraction aperture. This is true even though the diffraction pattern from the entire illuminated area is still present in the back focal plane of the objective lens. Diffracted beams, however, pass through the lower objective lens inclined at an angle 26B from the optic axis and are subject to spherical aberration. The diffracted beam that would be focused on a flat focal plane by a perfect lens is brought to focus on a spher- ical surface. By examining the image plane it can be seen that the iImge formed by the main beam and the diffracted beam do not coin- cide and this lack of coincidence increases as 26B increases. Thus, it is apparent that the diffraction spots in the selected area dif- fraction pattern are not formed from the same area of the specimen as the main beam. The extent of the error is described by the equation (3.15) x =Cs(26b 3 -0-., Curved -Perfect Lens Lens jj \ Field (a) (b) Figure 3.15 The spherical aberration of the objective lens forms a curved focal plane as shown in (a). For a perfect lens (b) the focal plane remains flat. where x is the distance on the image between the centers of the origin of the main beam and the diffracted beams. As the selected area at the specimen plane decreases in size below 0.5 Lla, the main and dif- fracted beam areas do not even overlap and the selected area techni- que is of little utility. With a STEM polepiece, the strongly de-magnifying upper objec- tive lens field allows the C2 aperture to be imaged on the specimen plane itself. For this method only a small portion of the sample is illuminated and only this part of the sample can produce the pattern. Thus the focused C2 aperture technique permits a quick and effective way to produce microdiffraction patterns from areas as small as 2000 A in diameter. The limitation in aperture diameter due to spherical aberration when the diffraction aperture is used no longer applies. Only standard TEM optics are required, and the angular re- solution in the diffraction pattern is equal to the older selected area method. The role of spherical aberration is confined to a slight distortion of the diffraction pattern, shown in exaggerated form in Fig. 3.16. According to Hirsch (1972), the distortion in- creases for higher order reflection (as these rays are deflected further from the optic axis) but is only about 1% for the 3rd order diffraction spots. 3.5 A Review of the Rocking Beam Microdiffraction Method A method that circumvents the contamination and angular reso- lution problems common to convergent beam techniques is the rocking beam microdiffraction method. As developed by Van Oostrum (1973) O O O O O O perfeat pattern jhift of spot Sdue to spherical aberration O Og O O a *0 O Figure 3.16 Barrel distortion in the diffraction pattern resulting from spherical aberration of the objective lens. This aberration is present for both the focused C2 aperture and the standard selected area diffraction techniques. and Geiss (1975), the rocking beam method makes use of the scanning coils to pivot the incident beam over the specimen surface in such a manner that the bright-field and dark-field images are swept across the detector at the base of the column in the Philips 301. The detector, of course, can sense nothing bout the character of the images that momentarily appear on the phosphor screen-it simply de- tects an increase in current density as each dark-field image comes in view. As the pivoting of the beam is repeated in a regular fash- ion, the increase in electrons striking the detector result in a momentary increase in the signal sent to the STEM unit electronics, forming a diffraction "spot" each time the incident beam is tilted at the appropriate angle. In spite of its unique method of formation, the rocking beam diffraction pattern closely resembles the patterns produced by selected area diffraction methods, and shows both dynamic and kinematic effects. The ray diagram for rocking beam diffraction is shown in Fig. 3.17. Consider the effect of the upper objective lens field on the imaging of the second condenser aperture. As discussed in the Section 3.3, the distance of the second condenser aperture is such that the upper objective lens forms a strongly demagnified image of the aperture in its back focal plane. Using the principles of ray tracing, it is apparent that any ray, regardless of its initial direction, that emanates from the aperture plane must still strike the same point in the plane to the right of the lens where the aper- ture image is formed. C2 Lens C2 Aperure mag Fiur .1 ayopis o rckn ba mcrdffatin.Ay a emnain fo te dg o heC2aprtr mstcoveg toteeqiaet poin ofteC2aet eiae A well-collimated beam used to form a normal TEM image can then be regularly deflected with the scanning coils such that the pivot point is co-planar with the aperture. As long as this condi- tion is satisfied, the aperture image will remain stationary in the image plane, regardless of the angular direction of the beam. When the specimen height is adjusted to coincide with the aperture image plane, the requirements for rocking beam diffraction are satisfied. Now, the same area of the sample remains illuminated during the beam rocking process. Although the specimen must be placed in a slightly non-eucentric position to coincide with the aperture image, the lens settings for the rocking beam method are the same as the focused condenser aper- ture method discussed in Section 3.3. The deflection of the beam produced by the scanning is no different from the deflection that occurs when a dark-field image is formed by manually tilting the incident beam to a specific position. The angular extent of the rocking beam pattern formed on the CRT is controlled by the angular deflection of the beam, and the de- flection itself is controlled by the magnification control on the STEM panel. Because the STEM imaging mode increases magnification by decreasing the scanned area, it follows that adjustment of the same control when in the rocking beam mode simply reduces the angle of rock. It is important to see that the rocking beam pattern is formed from only the portion of the image that strikes the detector. In standard selected area diffraction methods, the portion of the image that forms the diffraction pattern is chosen with the size of the diffraction aperture. In the rocking beam method, it is the diameter of the solid state detector that determines the portion of the image chosen. Thus, by increasing the magnification an increasingly smaller image segment is permitted to strike the detector and form the microdiffraction pattern. The detector diameter can also be re- duced by masking it with an aperture (Geiss, 1975) if even smaller portions of the image used to form the pattern are desired. For the Philips 301, the diameter of the detector is 4 mm and the maximum magnification at the detector height is 130,000X, so the minimum selected area is (3.16) D 4 mm A 308 A M 130,000 Geiss (1975) has reduced the diameter of the detector down to 1.0 mm and has obtained single crystal diffraction pattern from vapor depos- ited gold islands less than 100 ii in diameter. Signal strength de- creases as the active detector area is reduced or sample thickness is increased. A more practical limit for obtaining microdiffraction patterns with a reasonable signal to noise ratio from typical metal- lurgical thin fail specimens is 500 A. Angular resolution of the diffraction discs in the rocking beam pattern is controlled by the size of the objective aperture. A I-m objective aperture results in a pattern with resolution compa- rable to standard selected area diffraction, while removing the aperture entirely produces a rocking beam channeling pattern quite similar to the focused beam channeling patterns discussed previously. Patterns from a 2000 ii area in silicon for 4 different objective aperture sizes are shown in Fig. 3.18. F 0r a, ,C 00 m# c a0 4 * CO S.Q 00 *H DC co c rhese results are explained by Fig. 3.19, which shows that the diffraction pattern in the back focal plane of the objective lens must be swept across the objective aperture just as the image is swept over the detector. The objective aperture diameter d, limits the divergence of the rays in any diffracted beam. Thus, the maximum angular divergence is found by the ratio of the focal length of the objective lens and the objective aperture diameter: (3.17) P =d/f For the Philips 301, f = 3.7 mm, so an objective aperture of 5 Irm (the smallest practical size) gives a maximum angular resolu- tion of p = 1.35 x 103 rad. Thus, the angular resolution for rock- ing beam compares favorably with other microdiffraction techniques. 3.6 Transmission Channeling Patterns by the Rocking Beam Method In Fig. 3.18d, it is seen that completely removing the objec- tive aperture produces a transmission channeling pattern similar to the channeling patterns described in Section 3.1. These patterns, however, have been formed with a well collimated beam. The expla- nation is shown by Fig. 3.19, where a completely collimated beam is rocked through the necessary solid angles such that the Bragg angle for a particular diffracting plane is satisfied at opposite sides of the optic axis. Since the rocking occurs in three dimensions the cone that is formed is similar in angular extent to the incident beam cone shown in Fig. 3.1. Upper OBJ Specimen Plane Lower OBJ OBJ Focal Plane OBJ Aperture ImaBge Plane Figure 3.19 Rocking beam ray diagram showing how the objective aper- ture controls angular resolution. Both cases produce exactly the same channeling line pair in the back focal plane of the objective lens. For the convergent beam case, the pattern is formed all at one time since different segments of the beam form different angles with the diffracting planes. For a rocking beam, the pattern is formed sequentially in time, and a well-collimated beam is now positioned by the scanning coils at pre- cisely the right angle to form one segment of the channel line seen in the diffraction pattern. Channeling patterns formed from solid samples with the scanning electron microscope have been studied for several years (Coates, 1970) and can be formed either by backscattered electrons or speci- ment current. These SEM channeling patterns are limited to solid samples. In addition, the longer focal lengths of the final probe forming lenses typical of commerical SEMs results in much greater spherical aberration than the corresponding STEM lens, and the mini- mum specimen area that can be studied is only about 2 um in diameter (Van Essen and Schulson). Transmission channeling patterns were first produced by Fujimoto et al. (1972) using JEOL 100B STEM but no experimental images were shown and the minimum area that could be used to form a pattern was not described. Aside from the paper by Fujimoto, transmission channeling patterns produced by rocking beam methods have not been discussed in the scientific literature. The spherical aberration of both the upper and lower objective lens fields determinesthe minimum area that can be used to form the rocking beam microdiffraction pattern. For the rocking beam method, spherical aberration effects must be considered any time the beam has been rocked to a position non-parallel to the optic axis (dotted lines in Fig. 3.19). For the upper objective lens, Geiss has shown that spherical aberration causes the image of the C2 aperture on the specimen plane to shift a linear distance: (3.18) x =M Cs a where M is the demagnification of the C2 aperture, Cs is the coeffi- cient of spherical aberration for the objective lens and 6 is the angle of rock. M is comnputed in Chapter 5 to be 1/38X, Cs is 6.3 mm and the maximum angle of rock is approximately 10. At this angle, the shift of the C2 aperture image would be approximately 90 A which causes no problem during the rocking beam process as the illuminated area defined by the C2 aperture is several microns in diameter. However, the same equation can be used to determine the shift of the specimen image itself. This shift is much more crucial since it causes the specimen image to sweep across the detector, thus limiting the minimum area from which the pattern can be obtained. While the upper objective lens field reduces the shift of the illu- ninated area by 38X, the lower objective lens field increases the shift of the specimen image by 14X. For a la rocking angle: (3.19) x = 14 (6.7 mm) (.017 rad) x = 5000 A Thus, the minimum usable area needed for a transmission rocking beam channeling pattern is approximately 1 ipm. However, note that the shift is much reduced at smaller rocking angles. If the maximum angle of rock is limited to 0.50, x is reduced to 620 A. Thus, chan- neling patterns out to about the second order diffraction spots can be produced with a very high level of accuracy. An example of the shift of the image resulting from spherical aberration is shown in Fig. 3.20. It is important to realize that since transmission channeling patterns are found with the objective aperture removed, the current density striking the detector is much higher. Thus, the signal to noise ratio is much higher than any other type of microdiffraction pattern. If it were not for the effect of spherical aberration, transmission channeling would be a very powerful tool for extracting diffraction information from the specimen . Ti~-. r i Figure 3.20 Image shift for rocking beam optics. At 40,000, sample image shifted approximately 3500 A, for a 10 angle of rock. I -; ~J~Yi`t.. *c i CHAPTER 4 EXPERIMENTAL PROCEDURES AND MATERIALS 4.1 Ray Optics and Computations Since electron beam spot size and beam convergence determine the volume of the crystal producing the microdiffraction pattern, an experimental means of spot size measurement is of importance. Such a measurement may be performed directly in the Philips 301 by using the following procedure. An easily observed sample is imaged at high magnification in the standard TEM mode and objects of appropriate size, such as the small gold particles in the interior of the shadowed region in Fig. 4.1a, are recorded photographically. Then, with the sample in place, the microscope is switched to standard STEM conditions. With the stationary STEM diffraction pattern in view, the diffraction lens strength is now changed such that a raster pattern formed by the flying spot is imaged on the TEM phosphor screen as shown in Fig. 4.1b. By adjusting the specimen height, both sample image and raster pattern can be brought into sharp focus and observed simultaneously. The spot size can now be measured by stopping the scan, removing the specimen, and comparing the spot image diameter to gold particle size. When the intermediate lens is set at zero strength for the STEM mode, the magnification reading on the microscope panel is |

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PAGE 1 THE EFFECT OF CRYSTAL DEFECTS ON MICRODIFFRACTION PATTERNS BY JOHN BEVERLY WARREN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1978 PAGE 2 ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to his advisor, Dr. J. J. Hren, whose continuing encouragement during a decade spanning periods of graduate study, military service and employment in industry enabled the author to complete his degree. In addition, the advice of the supervisory committee, Dr. C. S. Hartley, Dr. R. E. Reed-Hill, Dr. R. T. DeHoff and Dr. M. Eisenberg, was most useful in the final preparation of the manuscript. Support from the Department of Materials Science in the form of a research assistantship is also gratefully acknowledged. Finally, the author thanks his wife for her unending patience and encouragement. PAGE 3 TABLE OF CONTENTS page ACKNOWLEDGMENTS ii ABSTRACT v CHAPTER 1 INTRODUCTION 1 2 THE EFFECT OF A CONVERGENT ELECTRON BEAM ON ELECTRON SCATTERING 6 2.1 A Review of Electron Diffraction and the Reciprocal Lattice 6 2.2 A Review of the Dynamical Theory of Electron Diffraction 10 2.3 The Effect of STEM on the Reciprocal Lattice Construction 18 2.4 The Effect of STEM on the Dynamical Theory . 22 2.5 A STEM Computer Simulation Program 29 3 MICRODIFFRACTION METHODS 47 3.1 A Review of the Effect of Inelastic Scattering on the Diffraction Pattern 47 3.2 The Use of Channeling Patterns to Characterize Polycrystalline Specimens 61 3.3 A Review of Ray Optics for Convergent Beam Diffraction 65 3 . 4 The Effect of Lens Aberrations 77 3.5 A Review of the Rocking Beam Microdiffraction Method 81 3.6 Transmission Channeling Patterns by the Rocking Beam Method 88 PAGE 4 TABLE OF CONTENTS (continued) CHAPTER page 4 EXPERIMENTAL PROCEDURES AND MATERIALS 94 4 . 1 Ray Optics and Computations 94 4 . 2 Sample Preparation 105 4.3 The Character of the Dislocations Introduced Ill 4.4 Determination of Crystal Directions by Diffraction Patterns 115 5 EXPERIMENTAL RESULTS 121 5.1 A Comparison of the Microdif fraction Methods 121 5.2 Identification of Crystal Defects by Microdiffraction 135 5.3 Computer Simulation of STEM Crystal Defect Images 143 5.4 Determination of Grain Orientation of a Fine-grained Superalloy Using Focused Condenser Aperture Microdiffraction 156 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 163 APPENDIX 167 REFERENCES 173 BIOGRAPHICAL SKETCH 175 PAGE 5 Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE EFFECT OF CRYSTAL DEFECTS ON MICRODIFFRACTION PATTERNS by John Beverly Warren June, 1978 Chairman: Dr. J. J. Hren Major Department: Materials Science and Engineering The various modes of microdif fraction that can be performed with an electron microscope equipped with a scanning transmission attachment are examined from both an experimental and a theoretical standpoint. Particular attention is paid to the development of microdiffraction to analyze crystal defects such as dislocations and stacking faults. The objective lens of the scanning transmission electron microscope forms a focused electron probe with a comparatively large convergence angle. Such a probe can produce a convergent beam diffraction pattern from an area on a crystalline specimen o as small as 50 A in diameter. The scanning coils used to raster the probe over the specimen can also be used to rock a wellcollimated electron beam over a region of the specimen in such a way that a microdiffraction pattern is formed on the cathode ray tube normally used to display the specimen image. By varying either the convergence angle of the incident beam or the transmitted beam it is possible in both of these methods to PAGE 6 produce microdiffraction patterns with high angular resolution or transmission channeling patterns with low angular resolution but excellent signal-to-noise ratios. In every case, the convergence angles of the incident and transmitted electron beams determine the appearance of the specimen image and the associated diffraction pattern. The Howie-Whelan dynamical theory must be modified to correctly predict the intensities of the scattered beams present in the diffraction pattern as well as the diffraction contrast image of crystal defects. Computer programs that simulate the electron beam-specimen interaction for the convergent beam case are developed and compared to experiment. A numerical method that utilizes transmission channeling patterns to precisely determine the orientation of a crystalline specimen and an electron beam is also presented. PAGE 7 CHAPTER 1 INTRODUCTION Since the advent of Scanning Transmission Electron Microscopy (STEM), several years ago, it was expected that STEM techniques would provide increased specimen penetration, electronically enhanced image quality, and improved capabilities to analyze areas a few nanometers in diamater by microdiffraction and energy dispersive X-ray microanalysis. For crystalline materials where diffraction contrast imaging plays the dominant role, only the last of these, the analysis of a small region excited by a stationary electron probe, has enjoyed a measure of success. This thesis examines the ability of the STEM probe to form a microdiffraction pattern from a column of material roughly equal to its own diameter. A microdiffraction pattern formed by this method can be produced from an area more than two orders of magnitude smaller than the patterns from standard selected area diffraction. While this method is the most direct of several possible microdiffraction techniques, most work up to now has been limited by the poor angular resolution in the pattern resulting from the convergent TEM probe and by the high contamination rates associated with it. Another approach, rocking beam diffraction, circumvents the limitations of the stationary electron probe and is also studied in detail. Here, the deflection coils in the upper half of the microscope PAGE 8 column, used to scan the probe across the specimen in the normal mode of operation, are adjusted to rock a well-collimated electron beam about a pivot-point level with the specimen, thus producing a time-dependent display of the variations of electron scattering as a function of the angle of rock. If an objective aperture of appropriate size is placed below the specimen plane, the rocking motion of the incident beam results in the bright field and the associated dark field images being swept over the STEM detector. This motion forms a simulated diffraction pattern on the STEM cathode ray tube that corresponds precisely to the portion of the sample image "seen" by the detector. Since the incident beam is relatively well-collimated, contamination rates are much lower than for convergent beam STEM microdif fraction. Angular resolution is now controlled by the objective aperture, and can be varied from values comparable to those found in standard selected area diffraction to values that result in low angular resolution but produce a scanning transmission channeling pattern with an excellent signal-to-noise ratio. For all methods of microdif fraction, the convergence angles of the incident and scattered beams are the crucial factors that determine the manner in which the information in the microdif fraction pattern is presented. In convergent beam microdif fraction, gradually increasing a^, the semi-angle of incidence, changes the appearance of the pattern from the normal diffraction spot array to a transmission channeling pattern where discrete diffraction spots are no longer visible. Although the method of formation is quite different, the rocking beam method can produce a similar variety of effects if a , PAGE 9 the exit semi-angle, is varied by changing the objective aperture size. Once the basic concepts of convergent beam and rocking beam microdif fraction can be quantitatively described, they can be used to solve problems of interest to the metallurgist. In this work, these techniques are used to examine dislocation strain fields, stacking fault images , and to provide very accurate information on the orientation of adjacent areas in fine-grain polycrystalline materials. The information provided by microdiffraction from small distorted regions of the specimen close to the defects is comparable in many respects to that provided by the computer simulation of defect diffraction contrast images. To review, the computer simulation method numerically integrates two or more simultaneous differential equations that describe the interaction of the transmitted and scattered beams as they pass through the crystal lattice. If the equations are solved for several thousand columns in the vicinity of a defect strain field, i |2 and the solutions (for example, the bright field intensity, |T| , or i 1 2 the dark field intensity, |S| ) are plotted in a dot matrix to form a picture, a simulation of the experimental diffraction contrast image of the defect can be formed. The major limitation of the computer simulation technique is simply the time required to compute the interaction for each of several thousand columns before the simulation can be formed. Computation time for n-beam situations, where there are several strongly excited beams, escalates accordingly. In practice, computer simulation is limited to the two-beam case, where information from only one PAGE 10 diffraction plane is available in the image. Thus, several simulated images are needed to completely characterize a defect strain field. Microdiffraction can aid the computer simulation process by providing the |s| /|t| value directly from the diffraction pattern resulting from a column passing through the dislocation strain field o that is less than 100 A in diameter. By comparing the calculated ratios to the experimental ones for several columns spaced along the dislocation, it should be possible to identify the dislocation with a much smaller expenditure of computation time. Such a procedure could be done with ordinary Transmission Electron Microscopy (TEM) techniques only by measuring the intensity in the bright field and dark field images at precisely the same point. This is not possible with ordinary TEM for two reasons. First, the process of obtaining a dark field image involves tilting the beam, and it is very difficult to get a dark field image with precisely the same Bragg deviation as the associated bright field image. Second, it is very difficult to locate exactly the same specimen point on different two-beam images due to the diffuse nature of the image itself. When the STEM probe is used to display the defect image on the Cathode Ray Tube (CRT) in the scanning mode, the probe convergence plays an important role in determining the appearance of the crystal defect image. The convergence effect must be carefully controlled if STEM images are to be used for computer simulation, and this point is explored by examining the effect of beam convergence on computer simulated images of stacking faults in silicon. PAGE 11 5 For crystal orientation problems, the electron-beam specimen orientation can be determined directly from the Kikuchi lines (or channeling lines for high convergence angles) present in the microdiffraction pattern. The diffraction spots themselves persist over specimen tilts of several degrees , and cannot be used to determine the local electron beam direction. The location of both Kikuchi and channeling patterns is defined by the intersection of the Bragg diffracting cones of apex angle 180Â°-29 and the Ewald sphere. Each diffracting cone is normal to a a particular diffracting plane and local lattice rotation caused by defect strain fields rotates each cone to a new position. Measurement of the relative shift of the lines in the microdiffraction pattern can be used to determine the precise crystal orientation of the crystal volume producing the pattern. While this approach can be accomplished with standard selected area diffraction, it has been limited to areas larger than 2 ym in diameter and to low atomic number materials where Kikuchi lines are more readily formed. Microdiffraction, and in particular, transmission channeling formed by a highly convergent probe overcomes both of these limitations. In summary, this thesis uses several microdiffraction techniques to analyze individual crystal defects and also to solve orientation problems in polycrystalline materials. The advantages and disadvantages of each technique will be compared, but it will be shown that all of the phenomena in any type of microdiffraction pattern are critically dependent upon the convergence angle of the incident electron beam. PAGE 12 CHAPTER 2 THE EFFECT OF A CONVERGENT ELECTRON BEAM ON ELECTRON SCATTERING 2 .1 A Review of Electron Diffraction and the Reciprocal Lattice Scanning Transmission Electron Microscopy (STEM) enables the production of mi crodif fraction patterns two orders of magnitude smaller than previous techniques (Geiss, 1975). However the short focal length objective lens used to produce a small, focused spot for STEM results in a far more convergent beam impinging on the specimen than standard TEM techniques. It is necessary to understand how a highly convergent electron beam effects diffraction contrast imaging if the STEM mi crodif fraction patterns are to be correctly interpreted. The difference between imaging with TEM and STEM can be readily understood with the aid of geometrical ray optics (Fig. 2.1). In TEM, the upper half of the objective lens in the Philips 301 electron microscope forms a collimated beam of radiation that impinges upon the specimen and is focused by the lower half of the objective lens field to a point in its back focal plane (Thompson, 1977) . In STEM, the upper half of the objective lens produces a focused probe in the specimen plane. This probe diverges after passing through the specimen but the lower objective lens field partially refocuses the probe and forms a stationary, convergent beam diffraction pattern in the same objective focal plane as in TEM. PAGE 13 (a) TEM (b) STEM C2 Lens Upper OBJ Specimen Lower OBJ Diffraction Pattern C2 Aperture i i f Figure 2.1 Ray diagrams for TEM and STEM conditions in the Philips 301. In STEM, the convergence of the beam is controlled by the second condenser aperture diameter, while the upper half of the objective lens focuses the probe onto the specimen. In TEM, the second condenser lens offers an additional control of convergence. PAGE 14 The effect of the large difference in beam convergence on electron diffraction for STEM and TEM can be shown with the reciprocal lattice construction. Elastic scattering of electrons in the reciprocal space is diagrammed in Fig. 2.2 where a vector P is drawn parallel to the transmitted electron beam and a number of vectors P are drawn parallel to the scattered beams. The magnitude of the transmitted vector is defined as Â— . Because scattering is assumed to be elastic, the wavelength of the electrons does not change upon diffraction and A = A . Inspection of Fig. 2.2a shows that the tips of all possible scattering vectors trace out a surface called the reflecting, or Ewald, sphere. The Laue conditions show that diffraction occurs only when the reflecting sphere intercepts a point in the reciprocal lattice that corresponds to a particular diffracting plane. This is written algebraically as ( VV (2.1, A g hkl where g is a vector drawn from the origin of the reciprocal lattice to any point hkl that represents the set of hkl diffracting planes. For 100 KV, = 27 A -1 units while Ig..., I for Si = .32 I" 1 . A 111 Thus, for low index lattice planes, the radius of the reflecting sphere is 25 to 85 times greater than the spacing between the reciprocal lattice. Even for these conditions, diffraction would rarely occur since the reciprocal lattice points for a perfect crystal of infinite size are considered to be dimensionless. However, if the PAGE 15 Ewald Sphere I Rel-Rods in the Reciprocal | | | | Lattice (a) Foil Normal (b) Figure 2.2. Reciprocal lattice diagram (a) showing that the reflecting sphere must intersect a rel-rod if the corresponding diffractive spot is to appear in the diffraction pattern. The Bragg deviation is defined in (b) as a vector drawn from the center of the rel-rod to its intersection with the reflecting sphere. PAGE 16 10 total diffracted intensity is computed for only a finite number of unit cells, structure factor arguments (Edington, 1975) show the reciprocal lattice points can now be represented as a volume of di2 2 2 mensions Â— , Â— and ~where z is the specimen foil normal and N is x y z the number of unit cells along a particular direction. For typical foil thicknesses, the dimension NÂ„ is much less than N or N so the z x y reciprocal lattice points are stretched into rods whose long axis is parallel to the foil normal of the specimen. As shown in Fig. 2.2a, the reflecting sphere can now intersect many of the elongated lattice points forming an electron diffraction pattern with numerous higher order diffraction spots. The distance between the diffraction vector and the reciprocal lattice point center is defined by the vector s", which shows the direction and magnitude of the deviation from the exact Bragg condition. In Fig. 2.2b, s" is drawn parallel to the foil normal from the reciprocal lattice center to the intersection point with the reflecting sphere. By convention, the positive direction of s is defined as anti-parallel to the electron beam direction. 2.2 A Review of the Dynamical Theory of Electron Diffraction STEM's effect on the formation of the diffraction contrast image can be understood if the Howie-Whelan dynamical theory is modified to account for convergent beam radiation (Whelan and Hirsch, 1957) . To review, the dynamical theory states that the transmitted and scattered waves resulting from electron diffraction can no longer be treated independently. Now the scattered wave is considered to be PAGE 17 11 continually rescattered back in the original direction of the transmitted beam as shown in Fig. 2.3. The transmitted and the scattered wave will interfere and the amplitudes, T and S, of the waves will oscillate with the penetration depth, z, into the crystal. This interaction is described by two, simultaneous linear differential equations : dT Â— = -n T + (i A)S Â— = (i A)T + (-n + 2isÂ£ + 2iTiB)S dz g The complete derivation of the equations, based on a quantum mechanical approach is discussed by Hirsch et_ al . (1965) . In this chapter, it will be sufficient to explain the effects of the various i i2 i i2 terms on the transmitted and scattered intensities |T| and |S| , since this is the primary type of information available in the STEM mi crodif fraction patterns. Ordinary absorption that controls the absolute intensities of the transmitted and scattered beams is defined as n. This "normal" absorption simply lessens the intensity of both transmitted and scattered beams for crystalline sections of increasingly greater thickness. The absorption effect is thought to result from the inelastic, or high-angle, scattering of electrons since actual absorption of electrons does not occur (Hirsch, 1965) . The anomalous absorption parameter, A, would actually be better described as "enhanced transmission" of the beam at certain positive deviations from the exact Bragg angle. A complete explanation of why the enhanced transmission phenomenon occurs can be made PAGE 18 12 Figure 2.3 The multiple scattering assumption of the dynamic theory. Ray "a," elastically scattered in direction S, can undergo additional elastic scattering to be rescattered in direction T, the original beam direction. PAGE 19 13 only with extended reference to the derivation of the dynamical theory. In brief, one of the standing Bloch waves contributing to the solution distributes electronic charge in between the ion cores. Such a wave has lower average potential energy than a Bloch wave which distributes charge close to the ion cores and enhances the transmission of electrons through the lattice. The variable 5 , or extinction distance, is the depth of the g crystal necessary for either T or S to increase to their maximum amplitude and then decay to their minimum value. Thus, the wavelength of both T and S is 2E, . The extinction distance increases g with increasing Bragg deviations, so the "effective" extinction distance is usually written as W = E, s (Hirsch et al., 1965). The parameter describing the effect of a defect strain field, 3, is written as 6 =Â£''Â•Â« Â«9 (2.3, 3 describes the relative amount of lattice distortion produced by a defect displacement field, R. The magnitude of strain, g Â• Â—, changes the effective value of s and thus plays a primary role in determining the diffraction contrast image resulting from the defect. The two-beam equations depend upon two important assumptions, both of which may have to be modified for the convergent beam STEM case. First, the two-beam assumption itself is strictly valid only for a completely collimated electron beam and crystals that are several extinction distances thick. For these conditons, a "thinwalled" Ewald sphere can be oriented such that is intercepts only one PAGE 20 14 relrod that has a relatively short reciprocal length. For increasingly convergent beam radiation or for acceleration voltages much greater than 100 KV, additional scattered beams will tend to be excited and each additional excited beam results in an additional differential equation. Second, the assumption that T and S vary significantly only in 3t St 3 s 3 s the z direction, such that rÂ— = tÂ— =:t-=tÂ— = 0, known as the column 9x 3y 3x 3y approximation (Howie and Basinski, 1968), is less valid for STEM. Here, interactions between the transmitted and scattered electron waves are considered to be confined to a narrow column parallel to the incident electron beam. Interactions between electrons in adjacent columns are considered to be negligible. However, this is strictly true only if 8 is small and the beam is well collimated. While 8 does not change for STEM, the incident beam is now B spread over a range of directions and it would seem that differentiation should take place along all possible electron beam directions contained in the convergent beam cone. These effects will be discussed further in the next section. At this point, it is helpful to solve the two-beam equation for the perfect crystal case and illustrate the effects of varying the anomalous absorption and the Bragg deviation. Since 6 now equals zero, the equations become dT -Â— = -tttiT + IT (l A)S dz (2.4) -Â— = Tf(i a)t + [-Tin + 2Tfi (w + 5 )s] dz g PAGE 21 15 Only for the perfect crystal can the equations be solved analytically. After some manipulation the solution for the transmitted amplitude is Y + irn y, + irn T = exp (y.Z) + exp (y_Z) Y-, Y-, L Y-, Y-, ^ (2.5) where (2.6) / 2 2 Y, = Â— ttti + iir[w Â± /w + (l A) ] The amplitude of T thus oscillates not only with depth z but also n the Bragg deviation. This is shown in Fig. 2.4 where |t| is plotted versus s to form the rocking curve. The rocking curve has a maximum at s = only when A is set equal to 0. Experimental evidence (Head et al. , 197 3) shows the proper value of A to vary between 0.07 and 0.13 according to the Bragg reflection used to form the scattered wave. The rocking curve closely resembles the contrast seen at a bend contour, where a section of a think foil bent at a constant radius results in a constant change in s. Since bend contours can be observed in both TEM and STEM, the experimental images offer a good test of numerical calculations. The variation of |t| with z is shown in Fig. 2.5 where the period is 2? and the A/N ratio is clearly revealed as an attenuating effect to the wave. T and S are always 180Â° out of phase, thus explaining the contrast reversal seen in bright and dark-field images. This example also correctly predicts the presence of thickness fringes PAGE 22 16 Â• Â•Â• ' +J PAGE 23 17 .Â» "*Â° *4. Â„** a O PAGE 25 19 (a) INCIDENT ELECTRON BEAM WITH SEMI-ANGLE a(b) CROSS SECTION OF REFLECTING VOLUME IN THE RECIPROCAL LATTICE Node at (000) Â» I I I I I I Figure 2.6 Cross section of the reflecting volume for a convergent beam of semi-angle a^ shown in (a) . The reflecting volume gradually increases in thickness at greater distances from the reciprocal lattice origin. PAGE 26 20 Analytical geometry can be used to describe the reflecting volume for the convergent beam case and provides quantitative estimates of both these effects. To begin, the central beam direction, B, is defined as parallel to the optic axis and is drawn from the origin of the reciprocal lattice anti-parallel to the electron beam. We then define the set of all vectors lying on the surface of the convergent beam cone at angle a. away from B as B'. Since both B and B' are equal to Â— , it follows that X cosa . Â— Â— 1 B' Â• B = Â— (2.7) |b' I = |i| Any vector B' satisfying these conditions can act as radius for its associated reflecting sphere. The equation for each of the reflecting spheres is 2 2 1 (2 ' 8) (x B^P + (y B^) + (z B^r = -+2 Once an initial B' is chosen, additional radii may be computed by specifying that each vector on the cone lies $ degrees away from its neighbor. These additional vectors, B" , are found by solving the simultaneous equations Â— ,, Â— . cos PAGE 27 21 Each B", when substituted into Eq. 2.8, defines a segment of the outer surface of the reflecting volume for angle PAGE 28 22 2.4 The Effect of STEM on the Dynamical Theory The dynamic theory must be modified to take the convergent electron beam used in STEM into account. Section 2.3 has already shown that the reflecting sphere shape must be modified to account for convergent radiation. As was shown in Fig. 2.6b, the diffraction excitation rules are relaxed for higher order reflections and thus the two-beam conditions should be more difficult to achieve for STEM. Of greater importance is the fact that even if two-beam conditions are achieved, the Bragg deviation is now spread over a range of values . Since the Howie-Whelan two-beam equations depend directly on s it can be seen immediately that both the rocking curve for | T | and the curve for |t| versus z will be modified for STEM. For purposes of illustration, it is best to use a non-columnapproximation set of two-beam equations , and see how beam convergence relates to the various terms. The work of Howie and Basinski shows that 3T 3T 3T Tri iri ,Â„ . , 4 , 3z" +Y z3z" +Y y37 = ^ T + ^ exp[2ll(sz + g R)]S (2.13) 3T 3T 3T n r Â„ . , iri Â„ tÂ— + y T~ + Y T~ = T~ T exp[-2in sz + g-R) ] + Â— S 3z 'x 3x 'y 3y Â£_. ^ l v Â£_ where (2.14) (P t + g) x (P t + g). (P. + g + s) Y (P. + g + s) t z t PAGE 29 23 and the subscripts refer to the components of the vector sums for the x, y, and z directions. For the Bragg angles encountered in electron diffraction, (P + g) + << (P + g + s) and the variation of s with respect to the x or y directions can be ignored. In STEM, 8 remains the same so this inequality should still hold. However, by Fig. 2.6b, P now has a range of directions determined by a. , the angle of incidence. Thus, for STEM, it is no longer a good approximation to differentiate in the z direction parallel to the optic axis of the microscope, and the two-beam equations must be solved repeatedly for a range of specific propagation directions determined by a . . For i each propagation direction, however, the terms y and y are no x y larger than before and thus the form of the equations remains unchanged. For the beam convergences generally used in STEM crystal defect images, a. is still only a fraction of the Bragg angle, and it is evident that varying s, the Bragg deviation, gives results equivalent to those obtained from differentiating over the range of propagation directions. For STEM conditions, the only necessary change in the two-beam equations is to replace T and S by T and S . dT r. = -ttdT + (i A)S dz dS Â— Â— = tt (i A)T + [-im + 27ri(w + Z 8)]S dz n n g r (2.15) PAGE 30 24 where n indicates the equations must be solved repeatedly for n values of w + Aw. The Bright-field intensity is then I 1 1 STEM n ' n ' In experimental practice, an additional control on the form of the STEM image is a , the angle subtended by the STEM detector. In the Philips 301 microscope, a can be varied either with the final projector lens or by using the objective aperture to block part of the beam from striking the detector. In both cases reducing a means that rays from only the central portion of the incident beam are used to form the image and the convergence can be equivalent to one where a. was reduced instead. Since a Q offers the last opportunity in the column to control the convergence of the beam, Maher and Joy (1976) have defined the equation (2.17) Aw = g \ E a go where Aw shows the extent of the variation from ideal TEM imaging with a perfectly collimated beam. As shown in Fig. 2.7, the range of Bragg deviations is proportional to aand thus also to a for the reasons outlined above. The equation for Aw shows directly which electron optical conditions and which materials will show the greatest amount of image modification under STEM imaging conditions. As discussed by Booker et al . (1974) , higher order reflections for STEM imaging produce situations where the variation in the Bragg deviation range increases due PAGE 31 25 Figure 2.7 Beam convergence values for low cone angle STEM (a) and MB I (b) . PAGE 32 26 to the greater thickness of the convergent beam reflecting volume. Low atomic number materials with relatively longer extinction distances for a particular reflecting plane will also tend to show % -3 greater Aw values. For example, at 100 KV and a. ^ 5 x 10 , Aw = .97 for silicon but only .58 for Cu due to their different values for Â£iii and di,i. The Aw relation can be used only in "low-cone angle" STEM where a. < 8 . If this condition is not satisfied, the first order difi B fraction cones overlap the central beam as shown in Fig. 2.7b and it may be physically impossible to place the detector such that it is struck by only one beam. This case, known as Multi-Beam Imaging (MBI) was studied by Reimer (1976). As might be expected, in a case where both bright-field and dark-field images are used to simultaneously form an image, crystal defect contrast is reduced but it also decreases rapidly with foil depth. Except for this feature, which allows easy determination of which end of a defect penetrates the top of the foil, MBI offers no advantage over standard TEM or lowcone angle STEM techniques. Reimer attempted to compare calculated profiles of stacking faults with experimental MBI images and obtained reasonable agreement by assuming a non-coherent source. If some degree of coherence of the incident electron beam is assumed, it is apparent from Fig. 2.7b that the primary transmitted and scattered waves could interact even if the multiple scattering phenomena used in the development of the dynamic theory are ignored. This does not occur with a standard hairpin filament because the tip of the filament from which electrons PAGE 33 27 are emitted is approximately 15 ym in diameter and the 100 KV electron o wavelength is only .037 A. For a much smaller source size, such as the field emission STEM, the dynamic theory may have to be modified to take increased coherence into account. If imaging is restricted to low-cone angle STEM, the Aw equation provides a good indication of the extent of image modification. Joy and Maher (1975) examined bend contours in silicon for a range of a = o -3 -7 -2 7.5 x 10 rad to 3 x 10 rad for the <220> reflection (9 =10 rad) B and found that alterations in fringe contrast that became negligible -4 once a was less than 8 x 10 rad. Thus, if Aw < 0.1, the STEM image should be virtually identical to conventional TEM. These conditions can be easily satisfied in the Philips 301 and Fig. 2.8 shows experimental confirmation of Joy and Maher 's results comparing a well colli-4 _3 mated (a = 5 x 10 rad) TEM image to that of STEM (a = 1 x 10 rad) o o with g = <220> for an extrinsic stacking fault in silicon. Stacking faults and dislocation images are less susceptible to convergent beam imaging modification than bend contours or thickness fringes. Apparently, this occurs because defect displacement fields strongly affect transmitted and scattered beams only in a localized area, while a bent foil supplies a more gradual, but constant, modification to the beam amplitude along the entire column. Since these features often obscure contrast from defects, STEM imaging with Aw = 1.0 may prove to be more advantageous than TEM in highly deformed specimens. As explained in the last section, a bent foil of constant radius produces an intensity curve quite similar to the rocking curves in PAGE 34 28 (a) TEM (b) STEM Figure 2.8 Extrinsic stacking faults in silicon imaged for two-beam conditions in TEM with a Q = 5 x 10 " rad (a) and in STEM with a = 1 x 10" (b). PAGE 35 29 2 Fig. 2.4 that plot T versus w. STEM conditions force the replacement of TEM rocking curve by a family of curves each slightly disAw placed along the w axis up to an amount -r-. Figure 2.9 shows the results for both 4 and 6 curves that are added together and then normalized to produce the "STEM" rocking curve. As implied by Fig. 2.9, further summations beyond 4 curves improve the accuracy by only a small amount. The same procedure is used to produce the thickness fringe curve in Fig. 2.10, where curves are plotted for different values of s and then averaged to produce the STEM version of the i |2 |T| as a function of thickness curve. In each case, the "square aperture approximation" developed by Fraser _et _al. (1976) is used for computation. Here, the convergent beam diffraction pattern is approximated by a square array (Fig. 2.11) and the range of Aw is determined by the distance of the centroid of the square to the appropriate Kikuchi line. 2. 5 A STEM Computer Simulation Program Since a crystal defect image is simply a greatly magnified image of the main beam or one of the diffraction spots, it follows that a microdiffraction pattern from the vicinity of the defect can contain no more information than the defect image itself. However, it will be shown in subsequent chapters that if a. and the spot size are kept sufficiently small, it is possible to directly obtain in2 tensity information such as |t| or |s| for a two-beam condition o from an area as small as 50 A in diameter. PAGE 36 30 PAGE 37 31 H 0) PAGE 38 32 Transmitted Beam Position First Order Diffraction Spot Position Kikuchi Line Pair Figure 2.11 The square aperture approximation (after Fraser) used to compute Fig. 2.9 and 2.10. The maximum and minimum w values are found by measuring the distance from the "diffraction square" center to the appropriate Kikuchi line. Any given w value does not vary in the x direction. This would not be the case for a circular aperture that would produce a circular diffraction disc. PAGE 39 33 The microdif fraction pattern provides, in one image, the direct i 1 2 i 1 2 |T| /|S| intensity radio (measured experimentally be a densitometer from the photographic plate) that could be obtained only by measuring intensities at exactly the same point in the bright-field and darkfield defect images. Since after tilting the beam to get a darkfield image it is impossible to keep the Bragg deviation precisely the same, obtaining |t| /|s| ratio is possible only with microdiffraction. If the location of the incident beam can be precisely determined, experimental |T| /|s| ratios can then be "simulated" by mathematically modeling the effect of the defect strain field, R, on the T-S interaction. The various experimental variables such as foil thickness or the Burgers vector can then be varied until the computed ratios are the same as the experimental model. This approach has been developed by Head et al. (1973) by matching defect images to a high degree. The microdif fraction ratio method potentially offers a method of characterizing the defect strain field with a much smaller expenditure of computer time than the image matching method, where several thousand columns must be computed to produce a simulated image . i 1 2 i 1 2 A program to simulate |T| /|s| ratios need not be as efficient as the one used by Head, since only a few columns should need to be computed before the defect is identified. While the approach used here was chosen for algebraic simplicity, it is sufficiently rapid that it can be used to produce defect image simulations if desired. Reference to Fig. 2.12 shows the two coordinate systems required for the intensity calculations. The first, crystal coordinates, is used because necessary input data such as dislocation line direction, PAGE 40 34 BMx (u x BM) r vectors for one column U x BM BM x (u x BM) U x BM Figure 2.12 Coordinate system in relation to disclocation and the image plane. PAGE 41 35 foil normal and the g vector can be determined only by using the hkl coordinates found in the diffraction patterns. The second, image plane coordinates, are used to determine, R, the displacement field values necessary to solve the two-beam equations for a particular column. Once the necessary geometry transformations have been determined, the intensity calculation is relatively simple. Each column, parallel to the electron beam direction, is divided into a number of segments, dz, and R, the defect displacement field, is computed for each segment. Then by using the approach of Tholen (1970) , each column segment is considered to be a slab of perfect crystal rotated a slight amount by the defect displacement field. Each segment can be considered to have a different Bragg deviation, and the analytic solution (Eqn. 2.5) is used to compute T or S for that segment. Tholen showed how to manipulate the analytic solution such that the amplitude T and S exiting from any random slab are related to the incident amplitude by a 2 x 2 matrix: out ra il a i2 21 22 (2.18) Â© where (Y 2 + i"l) (2.19) l ll = JZ Â— v ) t-Y 2 exp(Y 1 dz) + Y 1 exp(Y 2 dz)] a 12 = a 21 = ]y[ Y 2 ) [ex P PAGE 42 36 and (2.20) Y, 2 iff(w Â± /w + (i A) ) With this approach, the two-beam equations do not need to be numerically integrated down a column of distorted crystal. Prediction of the amplitudes of T or S is now achieved by matrix multiplication, where each matrix corresponds to a slab of perfect crystal with thickness, dz, and a specific Bragg deviation. After the necessary transformations converting all crystal directions to image plane coordinates , the sequence for the numerical calculations of | T | or | S | proceeds in the following manner : First, the distance, r, between the dislocation line and the column segment is determined. The displacement field, R, is now a function of r, 8, and b, where b is the Burgers vector and 9 is the angle between r and the slip plane. With r known, compute R at the top and bottom of the segment. It is important to realize that R is a function of only two independent variables, r and 0, for any given Burgers vector. Large savings in computer time can result if R is pre-computed for a range of r and 9 values that correspond in size to an area somewhat larger in diameter than the specimen thickness. By storing g'R in matrix form, with subscripts corresponding to r and 9, repetitive calculations of the same value of R can be greatly reduced. This idea, never used by Tholen, is feasible because the dislocation displacement field is " two-dimensional," i.e., an r value for a given and 9, is the same along any point of dislocation length, assuming infinite solution. PAGE 43 37 When gÂ»R has been determined, the effect of the dislocation on each slab, dz, is (2.21) 3 -Â£(^1)5 top bottom dz g z z, g top bottom Now, the effective Bragg deviation w + 35 and a scattering matrix g a. . of a specific value are known for the slab. Tholen has shown that it is possible to pre-compute the scattering matrices for increments of w just as was done for R. 3 can then be used to select the appropriate pre-computed matrix needed for a particular slab. Once a scattering matrix has been selected for every slab in the column, the transmitted amplitude, T, at the top surface is set equal to 1 and S = 0. The amplitude of T and S at the bottom surface is then found by successively multiplying each scattering matrix times the exiting amplitude of T and S from the slab immediately above: (2.22) (J = a, a a ...a [ where a = V ^ 12 3 nV. n out in a a 11 12 21 22 In practice, it is necessary to divide the foil thickness into 80 segments to compute |t| to a sufficient accuracy. Thus, 80 i |2 scattering matrices must be multiplied together to compute the |T| ,,2 i i2 for a single column. The actual intensity |T| of |S| can then be divided by the intensity found when r is set to a very large value, or to intensity computed for the perfect crystal where R = 0. PAGE 44 38 The geometrical conversions can be much simplified if the analytic geometry formulations for a line and a plane given rectalinear coordinates are used. Thus, for vector, A C=Ci+Cj+Ck x y z the equation of a line parallel to the vector is x x y y o o (2.23) (2.24) C Y and the equation of a plane normal to the vector is C(x-x) +C(y-y) + C (z z ) = C x o y o z o (2.25) All important features in Fig. 2.12 such as the foil surfaces, or the dislocation line and column directions, can be represented by one of these equations. To begin the calculations, it is assumed that all necessary input vectors such as the beam direction, B, the foil normal, F, and diffracting vector, g, and the dislocation line direction, U, have been computed from experimental hkl coordinates. The Bragg deviation and the analomous absorption are entered as dimensionless quantities while the foil thickness is entered in units of Â£ , the extinction g distance. Units for b, the Burgers vector, and g are not important as all units cancel when gÂ«b is calculated for a particular slab. An "image plane" coordinate system can now be defined where the vertical axis is parallel to B, and one horizontal axis is parallel PAGE 45 39 to U x BM, and the final horizontal axis is parallel to BM x (u x BM) If the direction cosines between the hkl coordinate system and the image coordinates are computed, a 3 x 3 matrix C. . is defined where (2.26) C. . = x. Â• U. 13 i 3 C. . can be used immediately to transform all vectors to picture coordinates (2.27) u: = c . u. i 13 3 B! = C. . B. i 13 3 b! = C. . b. 1 13 i F! = C. . F. 1 13 3 g! = C. . g. The dislocation is now situated such that its mid-point lies at the origin of the image coordinate axes. Thus the line equation for the dislocation is (2.28) x _ _y_ _ z u ~ u ~ u x y z To find equations for the upper and lower foil surfaces, x and y are set equal to and TZ is set equal to TH/2 , where TZ is the foil thickness in the beam direction. Equation 2.25 is then (2.29) V? Â» Â» Once the constant, D, on the right hand side of the surface normal PAGE 47 41 The vector r is of the form: r = (x x )i + (y y )j + (z z Q )k but since it is drawn from the origin, the three components of r are (2.35) r = x x r y = V U z y " U y z) r Z = -Uy(" Z y " U y Z) Now, the general solution for a dislocation of mixed Burgers vector is (Hirth and Loethe, 1968) (2.36) tT 1 ' ;vTq j_ u sin26 , rÂ— r 1 2u nil. cos28 -i -, R = -Â— U30 + b p + b x ulÂ— Â— Â£n r + Â— Â— -I } 2it e 4(1 u) ^2(1 u) ' ' 4(1 u) J where b e = the edge component of the Burgers vector normal to u 8 = an angle between r and the slip plane of the dislocation u = Poisson's ratio. As shown in Fig. 2.23.',. can be found by taking the dot product between r and a vector lying at the intersection of the slip plane and the plane normal to the dislocation. The intersection vector is defined as (2.37) Â— _ u x b x u I V I Thus, (2.38) V-r Â— cos6 = Â— Â— and bÂ„ = (b V)V PAGE 48 42 * Slip Plane Figure 2.13 Radial coordinate system used for the calculation of R at point defined by r and . The angle 9 is always measured from the slip plane and varies to 2ir. PAGE 49 43 where V is a unit vector. Examination of Fig. 2.13, however, shows that 8, computed from Eqn. 2.37, could result from an r lying either above or below the slip plane. An additional test to determine the correct direction of r is performed by computing the unit vector (2.39) p = v * u |P| and then again computing the angle between P and r (2.40) cos 6 = P Â• r/ r By examining the two-dimensional coordinate system formed by P and V, it is seen that if the true position of r is in quadrant I, both cos6 and cosS will be positive. If r lies in quadrant IV, cos8 will be positive, but 5 is now more than 90Â° and cos6 will be negative. If a similar test is applied for an r lying in each quadrant, it is found that Eqn. 2.37 gives the proper value for 9 only when co'so is positive. The computer program calculates both 9 and 6 and changes 8 to 2tt + 9, if cos6 is negative. Then the "sense" of 9 is always correct, regardless of the position of r. R can now be computed for every element in the column by the method discussed before. It is noteworthy that the linear distance between each precomputed matrix element for R will vary slightly according to the distance from the dislocation core as shown in Fig.. 3.13. This is simply a consequence of defining R in terms of polar coordinates. Such a choice is convenient for the column calculation, because the r,9 matrix elements are packed most densely at the dislocation core, PAGE 50 44 which is precisely where R varies most rapidly and the greatest accuracy is needed in the column calculation. With R chosen, the effective Bragg deviation for each segment, 3, is now calculated _ _ _ (2.41) g(R R, ) _ a .Â— Â— . top bottom , . TZ 3 = Â— g-R) = c Â— ; where Az = Â— dz Az 80 As discussed by Tholen, since the a matrix depends only on w, it is far more efficient in terms of computer time to calculate a for incremental values of w and store the results. Since |w| values higher than 3.5 imply such a large lattice distortion that the twobeam condition is no longer valid, the A matrix is calculated for a range of -3.5 < w < 3.5 in increments of .005 giving a total of 1401 2x2 matrices that must be stored and used as a "library" when any column is computed. Once the effective Bragg deviation is calculated, it can be converted to an integer by the formula t fi . (2 ' 42) B = -2 Â— J x 700 \005 J and the integer (from 1 to 80) is used to select the appropriate matrix for a particular slab. The sequence of the column calculation is then (2.43) = h 5 2--a nÂ® where 1, 2...n represents a sequence of 80 integers corresponding to each of the 80 column segments. i 1 2 Once computed and divided by the background intensity, |T| is compared to a series of 11 decimal values between and 1 that comprise PAGE 51 45 the grey scale used in printing the picture. The value closest to l |2 |T| is used by the program to select one of 11 symbols that represent a particular shade of grey. This symbol is then printed by an IBM high speed printer to represent the intensity in the micrograph corresponding to a particular column in the specimen. If an image i 1 2 i 1 2 is not desired, ratios of |S| /|T| for any regular array of columns spaced along the dislocation can also be computed. In this work, experimental studies concentrated on extrinsic stacking faults in silicon. For this case, the displacement vector is confined to the plane of the stacking fault and does not vary with distance as for the dislocation. For this case, R = Â— <111> and is normal to the stacking fault which always lies on {ill}type planes. It is shown by Whelan and Hirsch (1957) that the fault can be considered as "a planar boundary separating two perfect crystals. The S wave must be adjusted by a phase factor exp(ia) as it passes through the fault while the T wave remains unchanged. The phase angle is computed by the equation: (2.44) a = 2ir gÂ«R Therefore, a stacking fault is added to the program simply by locating its height in the specimen for a particular column and adjusting the amplitude of S at that particular point. Modification of either dislocation or stacking fault images for the effects of beam convergence in STEM is done in the program by adding a value Aw (computed by Aw = |g|C a.) to the Bragg deviag 1 tion, w, and then completing the column calculation as before. For PAGE 52 46 greatest accuracy, Aw must be divided into several segments and the column computation performed for a series of (w + Â— ) quantities. All of the |t| values for each Â— segment are then summed together to produce the total intensity for a particular column. PAGE 53 CHAPTER 3 MICRODIFFRACTION METHODS 3.1 A Review of the Effect of Inelastic Scattering on the Diffraction Pattern In addition to the dynamic effects on diffraction spot intensities, convergent beam radiation also affects inelastic scattering which is the cause Kikuchi lines and transmission channeling patterns. Both of these phenomena can be understood by examining the surface formed by all vectors that are at an angle of 9 with the reflecting B plane. This surface is a cone whose central axis is normal to the plane. For diffraction resulting from a perfectly collimated beam, only one vector lying on the surface of the cone is necessary to describe the diffraction direction, and, although the actual diameter of the incident and exiting beams may be several microns in diameter, the beam can be focused to a point to form the diffraction pattern in the back focal plane of the objective lens. If the beam is highly convergent, this is no longer the case. Now, only the part of the incident and reflected rays lay on the reflecting cone surface that defines the exact Bragg angle. The extent of the segment of the convergent beam that intersects the reflecting cone is shown in Fig. 3.1 to be (3.1) D = 2La.j 47 PAGE 54 48 E-i W U Z 23 PAGE 55 49 where a. is the semi-angle of the incident beam and L is the effective camera length . The convergent beam diffraction pattern, as diagrammed in Fig. 3.2, now consists of an array of discs whose diameter, y, is proportional to 2a.. Only those electrons in the incident beam that strike the reflecting plane at exactly 6 form the part of the disc with B zero Bragg deviation. It is obvious that if 2a. is increased sufficiently, this line of "zero Bragg deviation" can extend across several orders of reciprocal lattice spots. Calculation of a^ from the convergent beam diffraction pattern is easily accomplished (Thompson, 1977). In Fig. 3.2, the convergent beam which subtends an angle of 2astrikes the specimen and produces a main beam and several diffracted rays that are focused as discs in the back focal plane of the objective lens. Bragg 's law shows that the distance between the main beam and any of the diffracted beams is proportional to 26 . Also, it is evident that the width of the B disc in the convergent beam diffraction pattern is proportional to the convergence angle of the incident beam. The law of similar triangles shows that the radius of one of the diffracted spots divided by the distance between the main beam and the diffracted beam is equal to convergence angle of the incident beam divided by the Bragg angle 26 a. x = B % _ "1 (3.2) y a i x 9 B Since x and y are easily measured on the diffraction pattern and 28 B is known, the degree of collimation of the beam, a. , can now be computed from Eqn. 3.2. Note that as a, increases, the discs in the PAGE 56 50 tt CD OD Â•b. PAGE 57 51 diffraction pattern increase in size until they begin to overlap as shown in Fig. 3.3a. The camera length, which controls the size of the diffraction pattern, can be changed by adjusting the magnification lenses in the usual manner. The amount of spot overlap can be changed only by adjusting the size of the condenser aperture or the strength of the condenser lenses. Although their method of formation is different, both Kikuchi and transmission channeling lines that are observed in electron diffraction patterns obey the geometrical relations described above. Kikuchi lines, formed by elastically scattered electrons that previously have undergone one or more inelastic collisions , are familiar in metallurgical electron microscopy and are often used as an aid to determine the exact orientation of the specimen with respect to the beam. Kikuchi lines can result even from a perfectly collimated beam; the inelastic scattering of the beam (once it enters the specimen) causes it to diverge through an appreciable solid angle. The formation of these Kikuchi line patterns can be analyzed geometrically in Fig. 3.4. Figure 3.4a shows the scattering distribution for electrons that were inelastically scattered near the top of the specimen. Most of the electrons are deflected only slightly from the original path, and decreasingly fewer in number are scattered at greater angular deviations. Since the actual energy loss involved in a few 100 eV (Edington, 1975) , some of these electrons will have been inelastically scattered in the proper direction for additional elastic Bragg scattering to occur. Such a scattering event for a PAGE 58 52 CD CD V V CD CD 22 U l O ^ H O C rH 8* CD CD Al Â•H B tn C -^ Â•H U in c nm 0) I P o -P rH a X 3 a? nÂ» a M in Â» H ,Q Q Â«- PAGE 59 53 Ray A (a) (b) (c) Figure 3.4 The formation mechanism and the geometrical construction for a Kikuchi line pattern. Fewer electrons are scattered along Ray B than Ray A. In three dimensions, Ray A forms the "excess" line (intensity higher than background) while Ray B forms the "deficit" line. PAGE 60 54 particular set of planes is shown in Fig. 3.4b. By referring to the intensity scattering distribution in Fig. 3.4a, it is apparent that more electrons will be scattered along ray A than along ray B. In three dimensions, Fig. 3.4c shows that each reflected plane will produce two cones of scattered electrons. Due to the shape of the scattering distribution, one cone will have a higher intensity than the diffuse background radiation, while the other cone will have an intensity lower than background. These two cones intersect the Ewald sphere to produce hyperbolae. The radius of the Ewald sphere is large enough that they appear in the diffraction pattern plane as straight lines. The importance of Kikuchi lines in determining specimen orientation can be seen by examining the effect of the rotation of the specimen on both the diffracted spots and the Kukuchi lines. For example, in Fig. 2.2a, it can be seen that a small rotation of the sample may not change the spot pattern at all if the rel-rods are of a length normally produced by a crystal of average thickness. The reciprocal lattice is simply rotated about its origin in the same angular direction as the crystal. No new intersections between the Ewald sphere and rel-rods are formed after the tilt and the spot pattern remains unchanged. However, Fig. 3.4c shows that the same tilt would result in the pair of Kikuchi lines being shifted to the new position represented by the dotted rays. Such small shifts are easily detectable in experimental situations and the relation between the incident beam direction and the specimen can be calculated by methods described in Chapter IV. PAGE 61 55 The form of transmission channeling patterns as shown in Fig. 3.5, appears geometrically quite similar to a Kikuchi pattern for the same beam direction. This similarity is no accident, as the angular relationship between the elastically scattered electrons and the reflecting planes must be exactly the same for both cases. For channeling, however, the angular range of the incident beam is produced by the focusing action of the condenser lenses and no inelastic scattering need occur to produce the channel pattern. The rather simple kinematical models given here to describe the intensities and relative contrast present in these patterns are only an approximation and a precise description of the line intensities must make use of the dynamic theory discussed in Chapter II. The need for the dynamic theory is obvious when it is realized that the scattered beams, regardless of their position on the reflecting cone, can undergo multiple Bragg scattering and interact with the portion of the incident beam parallel to the scattering direction, as shown previously in Fig. 2.4. A semi-quantitative theory developed by Thomas (1972) describes an n-beam solution for Kikuchi patterns simply by summing many-beam rocking curves for particular Bragg reflections. Although the effect of thickness on the rocking curve was ignored, the computer solution was able to correctly predict the relative intensities and widths of the excess-deficit Kikuchi line pairs. A more rigorous procedure (C. R. Hall, 1970) used the dynamical two-beam equations to show that contrast reversals could occur in a Kikuchi band as thicker areas of the crystal were examined. PAGE 62 56 Figure 3.5 Comparison of Kikuchi patterns and transmission channeling patterns from silicon. The degree of beam convergence is different in (b) and (c) . PAGE 63 57 Although image simulation techniques for channeling or Kikuchi patterns could be further developed, there seems no obvious advantage in using such a method for the examination of distorted crystals or defect strain fields. Any dynamic information in the channeling patterns resulting from defect-beam interactions should also be available in the convergent beam diffraction disc pattern in a more easily interpretable form. In addition, the use of diffraction information from either convergent beam radiation or inelastically scattered electrons implies that a larger volume of crystal is irradiated than would be the case for a STEM probe with a semi-angle less than 10 rad. As shown in the review paper by Goldstein and Williams (1977) , inelastic scattering in thin foils results in x-ray production from a volume with an average diameter much larger than the original diameter of the probe. Thus, a Kikuchi pattern produced from a small STEM probe probably arises from a significantly greater volume than the region that gives rise to the dynamic bright-field intensity of the diffraction discs themselves. This follows from the fact that the brightfield intensity is restricted to elastic scattering and electrons suffering high angle collisions should not i 1 2 directly contribute the |T| value of the diffraction disc, itself. For these reasons, only the geometrical information present in Kikuchi or channeling mi crodif fraction patterns can be used for examination of distorted foils. Even with this limitation, analysis of the line patterns can yield considerable information as long as g*R does not vary in the local volume illuminated by the electron beam. PAGE 64 58 As a first example, an approach described by Cowley and Smith (1971) is used to determine the state of curvature of a small area of a foil. In Cowley and Smith's paper, a convergent beam was used to produce a channeling pattern such as the ones shown in Fig. 3.5. Cowley and Smith discovered that by defocusing the convergent beam such that the cross-over does not coincide with the specimen plane, each excess and deficit channel line pair can be split into two distinct lines, where the width of the splitting is governed by the amount of defocus. Examples of the line splitting effect for a particular channel line pair are marked by arrows in Fig. 3.5b and c. The explanation for the effect is shown in Figs. 3.6 and 3.7, which have been reproduced from Cowley and Smith's paper. Consider the incident and diffracted rays from opposite sides of the convergent cone for perfect crystal (Fig. 3.6). For this case, the incident rays that lie at the proper angle for Bragg diffraction are parallel to the scattered rays produced by the rays on the opposite of the cone. For the case of a bent crystal, shown in Fig. 3.7, this is no longer true. Now the transmitted rays deviate by angle 0' is radius of specimen curvature. From the figure, it can be shown that x = 29 L 1 B (3.3) y 2 e-L where x = the separation between a particular transmitted and scattered ray x = width of line split in the diffraction pattern PAGE 65 59 ion from Figure 3.6 Cowley diagram for diffraction from a flat crystal. Cowley and Smith (1971) PAGE 66 60 LD IS N^RD R L 7777777777 7 ^^///^^///^/^/y/^^yy////////y//// Figure 3.7 Cowley diagram for diffraction from a bent crystal. Cowley and Smith (1971) PAGE 67 61 L = camera length Manipulation of Eqns. 3.2 and 3.3 shows that the radius of curvature of the foil is (3.4) z(x X ) r = where z is the amount of defocus, or the distance of the cross-over from the specimen. Since the effect is dependent upon a defocused probe, Cowley and Smith's method is not a true microdif fraction technique but it does provide an explanation for much of the fine detail seen in the channeling pattern. 3.2 The Use of Channeling Patterns to Characterize Polycrystalline Specimens A method of analysis of more general utility than Cowley and Smith's, using geometrical information from either Kikuchi patterns or channel patterns resulting from a focused probe, is developed below. Consider the case where a fine-grained polycrystalline foil is to be analyzed by microdif fraction. It is desired to characterize the crystalline structure by determining the difference in crystal orientation between grain 1 and 2, shown schematically in Fig. 3.8. If the microdif fraction pattern is recorded from each grain, the direction of the beam (Helfmeier and FellerKniepmeir, 1977) is s ifr Vi TFT " Va T7T where Z = g x g = the approximate beam direction. PAGE 68 62 t Â• Â• Figure 3.8 If Kikuchi or channeling patterns are recorded for two adjacent grains, the orientation of the two grains can be determined from measuring the local beam direction for each grain for two different specimen tilts . PAGE 69 63 L , f , L , and f are measured from the diffraction pattern. A fuller discussion describing computation of the exact beam direction is discussed in Section 4.2. Let a superscript (*) designate the beam direction from grain 2 and a prime ( ' ) show the beam directions for both grains after the foil is tilted to a new position. For position 1, the relation between the two grains for any cubic crystalline structure is BM. = 1. . BM* (3.6) i ID J Similarly, for position 2 BMI = 1. . BM*' (3.7) r i: ] where 1. . is the 3x3 direction cosine array describing the orientation difference between the two grains. If each set of equations is expanded, the ones involving BM are BM 1 = L ll BM 1 + 1 12 BM 2 + 1 13 BM 3 (3 ' 8) BM' = 1 BM*' +1 BM*' + 1 1 11 1 12 2 13 BM*' and, it can be shown that 1 = L L + X 12 + L 13 (3 ' 9) This is a set of three equations in three unknowns, 1 , ^--i?' and 1Â«-. Using the equations involving BM , a similar set of equations can be solved for 1 , 1 and 1Â„_The final three direction PAGE 70 64 cosines are then determined with (3.10) Hi X 31 + X 12 X 32 + X 13 X 33 = Â° 1 21 1 31 + 1 22 L 32 + 1 23 + 1 33 ~ Â° 2 2 2 1 + 1 +1 =1 31 32 33_ where the underlined components are the remaining unknowns. This method can be used to describe the orientational relations between two adjacent areas of a distorted sample, low angle grain boundary, or two different grains of the same crystal phase. Similar sets of equations can be used to solve the orientation between two phases of different crystal structure. It is important to note that this method can be used regardless of the orientation of the specimen with respect to the beam and is dependent only on an accurate computation of the beam direction using either Kikuchi or transmission channeling patterns. While this method is also applicable to standard selected area diffraction techniques, it can be used to the fullest extent only with microdif fraction techniques. Many structural features in metals are much less than 1 urn in extent, and some, such as grain boundaries and stacking faults, are only a few atomic layers in thickness. For example, only a very narrow probe could be positioned close enough to an end-on grain boundary to produce useful information about its local orientation and strain field. PAGE 71 65 3.3 A Review of Ray Optics for Convergent Beam Diffraction In the Philips 301, a strongly demagnifying upper objective lens field makes it possible to form a much smaller focused spot than was possible in earlier electron microscopes. However, the incident on the specimen is now much more convergent than before. For the TEM mode in the Philips 301, the two condenser lenses and the upper half of the objective lens field are used to illuminate the specimen, while the lower half of the objective, diffraction, intermediate and projector lenses are used to form a magnified image on the phosphor screen. In Fig. 3.9, the filament image formed slightly below the electron gun assembly is demagnified by the first condenser lens (CI(not shown)), and both the second condenser lens (C2) and the upper half of the objective lens are used to further demagnify the image on the specimen plane. Although the objective polepiece used for both TEM and STEM imaging strongly demagnifies the C2 cross-over, its excitation value is coupled to the lower objective polepiece and must be set to a certain value for a focused image. For the imagining mode, the C2 lens is adjusted so that its cross-over lies close to the upper focal plane of the upper objective polepiece (Fig. 3.9a). The cross-over then acts as a pseudo-point source which the upper objective lens field converts to a relatively well-collimated beam that illuminates the specimen. One advantage of the imaging mode in the TEM (mode) with the Philips 301 is that the C2 aperture can be focused on the sample in PAGE 72 66 I u Â•H id n 2 H E* PAGE 73 67 exactly the same way that the selected area diffraction aperture above the diffraction lens can be focused on the lower objective image plane. Consider Fig. 3.10, which shows the physical arrangement of the objective polepieces and the specimen as well as the ray diagram for a perfectly collimated beam. The specimen lies in the middle of the polepiece gap, near the center of the magnetic field formed by the polepieces. The actual magnetic field in the polepiece gap is great enough that it cannot be treated as a "thin" lens and the standard ray tracing rules used in geometrical ray optics do not apply. The specimen, of course, is a real object, and only the portion of the lens field to the right of the specimen can be used to produce a magnified specimen image (Hall, 1966) . Useful information from ray tracing can still be obtained if the objective lens field is considered to be formed from two thin lenses, and this construction is used in the ray diagrams supplied by the manufacturer. Using this assumption, the objective lens field is treated as two thin lenses, and it can be seen from Fig. 3.11 that the upper objective lens forms a demagnif ied image of the C2 aperture at (a) , while the lower objective lens forms a magnified image of the specimen at (b) . The method of selected area diffraction developed by Poole (1947) , involves placing an aperture at (b) , so that only the central part of the image is used to form the diffraction pattern at (c) . By tracing the rays from (d) to (b) , it can be seen that the selected area aperture allows only rays from the part of the specimen (e-e) to form the diffraction pattern that is recorded photographically . PAGE 74 68 Upper Objective Polepiece Lower Ob j ective Polepiece Specimen Objective Aperture Figure 3.10 Dimensions of the Objective Polepieces for the STEM Modification of the Philips 301. PAGE 75 69 T! PAGE 76 70 However, if the specimen is raised to a non-eucentric position to coincide with the image of the condenser aperture, a "selected area" diffraction pattern will again be formed. In this case, the condenser aperture permits only a small area of the sample to be illuminated, and it is this area alone that forms the pattern. For the Philips 301 STEM polepiece, the condenser aperture is demagnified 38 times, while the selected area diffraction aperture is demagnified 14 times at the specimen plane. Because of spherical aberration, the utility of the focused condenser aperture method is even more favorable than suggested by these demagnification ratios. This point is explored further in the discussion on lens aberrations. To produce the convergent beam diffraction pattern, the strength of the C2 lens is reduced such that its cross-over would lie below the specimen if the objective lens were turned off (Fig. 3.9b) . At the proper excitation strength the objective lens now acts to increase the convergence of the beam and focuses the rays to produce a crossover at the specimen plane. The objective lens is always kept at the same excitation, so calculations involving spot size or convergence angle changes can be accomplished by considering the lens strength and the aperture sizes of the Cl and C2 condenser lenses alone (Fig. 3.12) . Consider Fig. 3.13, where the distance from the gun cross-over to the Cl lens, 0, is fixed, but the lens to cross-over distance, I, depends on the strength of the Cl lens. Since the refractive index PAGE 77 71 PAGE 78 72 Â•H S PAGE 79 73 of electrons is the same on both sides of the CI lens, f = f = f 1 2 and the thin lens equation is (3.11) I f As the CI lens is increased in excitation the focal length must decrease and since is fixed, I must decrease proportionally. The demagnification of the spot is given by the ratio (3.12) It can be seen that by increasing the Cl lens strength, the cross-over is brought closer to the right-hand side of the lens and the filament image is demagnified, thus forming a smaller spot. Once the spot size has been selected by a particular Cl lens excitation, only one possible setting for the C2 lens and the objective lens can be used if a focused spot is to be produced in the specimen plane. Thus, a specific Cl setting determines not only the spot size, but the convergence, otj, of the incident beam. The C2 lens can be used to alter a., but at the expense of forming a defocused spot on the specimen plane. However, the convergence of the beam may still be increased by changing the size of the aperture used with the C2 lens. As was shown in Fig. 3.12, substituting a smaller aperture in C2 reduces a. while leaving the spot size essentially unchanged since all three lens excitations remain the same. It is important to see that angular resolution in the diffraction pattern and the current density of the focused spot (and thus the intensity of the diffraction pattern) are inversely proportional. PAGE 80 74 It has been shown (Broers, 19 69), that the intensity of the beam at 2 the specimen, j^, in amperes/cm is proportional to the intensity at the gun, j Q , by . 2a (3 ' 13 > eV sin Â• JÂ± ~ 3 Â° .2a KT sin k But both a. and a, , and thus the intensity of the microdif fraction pattern, are controlled directly by the C2 aperture diameter. STEM microdiffraction methods differ from TEM convergent methods in that the specimen is still illuminated by a focused probe , but the image is formed sequentially as the probe is rastered over a rectangular area of the sample. To form a microdiffraction pattern from a specific region, all that is required is to stop the rastering motion produced by the deflection coils situated between the objective and C2 lenses, and manually adjust the deflection controls to place the probe over the desired area by watching the STEM image on the CRT. Regardless of the position of the probe, the microdiffraction patterns will be formed in the back focal plane of the lower half of the objective lens. Ray optics in STEM are described in Fig. 3.14. As in TEM, spot size can be varied by altering the Cl lens strength, but in STEM the C2 lens is turned completely off. The average convergence for a given probe size is now an order of magnitude greater than the TEM mode, but the spot sizes can be decreased proportionately. For example, the smallest spot size attainable with the standard hairpin o filament is 400 A with the C2 lens on (TEM convergent beam mode) , PAGE 81 75 PAGE 82 76 but 40 A with it off. As shown in the inset in Fig. 3.14, a. is now l controlled only by the C2 aperture. If the distances between the condenser lenses and the distance from the C2 aperture to the upper focal plane of the objective lens are known, a. can be easily calculated. However, actual diameters of commercially available apertures may vary as much as 20% from the specification and it is more accurate to calculate convergence directly from the microdif fraction pattern as is done in Chapter IV. Once the CI setting and C2 aperture size are chosen to select a particular spot size and current density, the STEM image is formed by using the wobbler coils to twice deflect the beam below the C2 aperture such that the beam pivots in the upper focal plane of the objective lens. As shown in Fig. 3.14, the upper objective field lens not only focuses the beam to a probe but converts the angular beam deflection to a deflection that remains parallel to the optic axis. Trimming potentiometers used to adjust the height of this pivot point must be carefully adjusted if the probe is to remain parallel to the optic axis over the entire raster scan. If this condition is not met, the Bragg deviation for any given hkl plane will vary over different parts of the scanned area and the image contrast will not be comparable to that for a collimated beam used in TEM imaging. Once the probe passes through the specimen the diffraction pattern is formed in the back focal plane of the lower objective lens, just as it is in the standard TEM modes. The probe still remains parallel to the optic axis after passing through the specimen PAGE 83 77 solid rays in 3.14) and it always converges to the same spot in the back focal plane after being focused by the lower objective lens regardless of its position in the scan. In the same fashion, diffracted rays (dotted rays) scattered through the same Bragg angle at different points of the scan are all focused to another point, forming a diffraction spot for a particular set of hkl planes. The remaining lenses below the objective lens, diffraction and projector lenses, only magnify the size of the microdif fraction pattern such that the diffraction discs subtend approximately the same angle as the solid state detector at the base of the column. The diffraction lens must be set at a fixed value so it can be focused on the back focal plane of the objective lens, so only the projector can be varied to adjust the camera length, or the size of the pattern appearing in the detector plane. By varying the projector lens strength, the size of the diffraction discs can be adjusted so that only the central portion of a disc strikes the detector, thus giving indirect control of a Q , the exit angle of the electron beam leaving the specimen. Therefore the degree of coherence of the electron beam used to form the STEM image can be controlled by selecting only the central portion of the diffraction disc just as a selected area diffraction pattern is formed from only a portion of the brightor dark-field image. 3.4 The Effect of Lens Aberrations The advantages of obtaining diffraction patterns from small areas are obvious. The region of interest, whether a small second PAGE 84 78 phase particle or a localized strain field from a dislocation, is small in volume compared to the surrounding matrix. The smaller the total volume which is examined by diffraction, the greater the particle-matrix volume ratio and the better the signal to noise ratio in the diffraction pattern. What is not so obvious is why microdiffraction patterns cannot be obtained with standard selected area diffraction techniques where very small apertures are used to select a correspondingly small portion of the image. The answer lies in the nature of the aberration errors common to any electro-magnetic lens. The most important of these lens defects is spherical aberration which results from electrons passing through the outer portion of a lens being focused more strongly than those electrons passing closer to the optic axis. The ray bundle cannot be brought to focus in a single focal plane, and an image of a point appears as a disc with a radius equal to (3.14) r = c s 3 3 where C s , the spherical aberration constant equals 6 mm for the Philips 301 STEM polepiece and, 3, the lens aperture angle is controlled by the size of the objective aperture. Reducing the size of the objective aperture thus decreases $ and the disc of confusion, r, but the process cannot be carried too far, or resolution again begins to decrease due to the diffraction effects caused by a very small aperture. PAGE 85 79 For the TEM diffraction mode, the objective aperture must be removed. Now the diameter of the selected area diffraction aperture is the factor that limits the effectiveness of selected area diffraction. Consider the formation of the selected area diffraction pattern shown in Fig. 3.15. The diffraction pattern from the entire area illuminated by the incident beam is formed in the back focal plane of the objective lens and the (000) beam is magnified to form the bright-field image. If the selected area aperture above the diffraction lens is placed such that it is in focus with the image, tracing the rays back to the specimen plane shows that the diffraction pattern appearing on the phosphor screen results only from the portion of the image within the selected area diffraction aperture. This is true even though the diffraction pattern from the entire illuminated area is still present in the back focal plane of the objective lens. Diffracted beams, however, pass through the lower objective lens inclined at an angle 29from the optic axis and are subject to spherical aberration. The diffracted beam that would be focused on a flat focal plane by a perfect lens is brought to focus on a spherical surface. By examining the image plane it can be seen that the image formed by the main beam and the diffracted beam do not coincide and this lack of coincidence increases as 29g increases. Thus, it is apparent that the diffraction spots in the selected area diffraction pattern are not formed from the same area of the specimen as the main beam. The extent of the error is described by the equation x = C s (26 b ) (3-15) PAGE 86 80 f J Â— Perfect Lens (a) (b) Figure 3.15 The spherical aberration of the objective lens forms a curved focal plane as shown in (a) . For a perfect lens (b) the focal plane remains flat. PAGE 87 81 where x is the distance on the image between the centers of the origin of the main beam and the diffracted beams. As the selected area at the specimen plane decreases in size below 0.5 ym, the main and diffracted beam areas do not even overlap and the selected area technique is of little utility. With a STEM polepiece, the strongly de-magnifying upper objective lens field allows the C2 aperture to be imaged on the specimen plane itself. For this method only a small portion of the sample is illuminated and only this part of the sample can produce the pattern. Thus the focused C2 aperture technique permits a quick and effective way to produce microdif fraction patterns from areas as small as o 2000 A in diameter. The limitation in aperture diameter due to spherical aberration when the diffraction aperture is used no longer applies. Only standard TEM optics are required, and the angular resolution in the diffraction pattern is equal to the older selected area method. The role of spherical aberration is confined to a slight distortion of the diffraction pattern, shown in exaggerated form in Fig. 3.16. According to Hirsch (1972) , the distortion increases for higher order reflection (as these rays are deflected further from the optic axis) but is only about 1% for the 3rd order diffraction spots. 3.5 A Review of the Rocking Beam Microdiffraction Method A method that circumvents the contamination and angular resolution problems common to convergent beam techniques is the rocking beam microdiffraction method. As developed by Van Oostrum (1973) PAGE 88 82 O O O O O m O m g O #C > perfect ^pattern shift of spot /due to spherical O^ QP Q *T) ms~\ aberration o # o # 8 Â•< o Â° o a *o o Figure 3.16 Barrel distortion in the diffraction pattern resulting from spherical aberration of the objective lens. This aberration is present for both the focused C2 aperture and the standard selected area diffraction techniques . PAGE 89 33 and Geiss (1975) , the rocking beam method makes use of the scanning coils to pivot the incident beam over the specimen surface in such a manner that the bright-field and dark-field images are swept across the detector at the base of the column in the Philips 301. The detector, of course, can sense nothing about the character of the images that momentarily appear on the phosphor screen-it simply detects an increase in current density as each dark-field image comes in view. As the pivoting of the beam is repeated in a regular fashion, the increase in electrons striking the detector result in a momentary increase in the signal sent to the STEM unit electronics, forming a diffraction "spot" each time the incident beam is tilted at the appropriate angle. In spite of its unique method of formation, the rocking beam diffraction pattern closely resembles the patterns produced by selected area diffraction methods, and shows both dynamic and kinematic effects. The ray diagram for rocking beam diffraction is shown in Fig. 3.17. Consider the effect of the upper objective lens field on the imaging of the second condenser aperture. As discussed in the Section 3.3, the distance of the second condenser aperture is such that the upper objective lens forms a strongly demagnified image of the aperture in its back focal plane. Using the principles of ray tracing, it is apparent that any ray, regardless of its initial direction, that emanates from the aperture plane must still strike the same point in the plane to the right of the lens where the aperture image is formed. PAGE 90 84 C2 Lens C2 Aperture C2 Aperture Image Figure 3.17 Ray optics for rocking beam microdif fraction. Any ray emanating from the edge of the C2 aperture must converge to the equivalent point of the C2 aperture image. PAGE 91 85 A well-collimated beam used to form a normal TEM image can then be regularly deflected with the scanning coils such that the pivot point is co-planar with the aperture. As long as this condition is satisfied, the aperture image will remain stationary in the image plane, regardless of the angular direction of the beam. When the specimen height is adjusted to coincide with the aperture image plane, the requirements for rocking beam diffraction are satisfied. Now, the same area of the sample remains illuminated during the beam rocking process. Although the specimen must be placed in a slightly non-eucentric position to coincide with the aperture image, the lens settings for the rocking beam method are the same as the focused condenser aperture method discussed in Section 3.3. The deflection of the beam produced by the scanning is no different from the deflection that occurs when a darkfield image is formed by manually tilting the incident beam to a specific position. The angular extent of the rocking beam pattern formed on the CRT is controlled by the angular deflection of the beam, and the deflection itself is controlled by the magnification control on the STEM panel. Because the STEM imaging mode increases magnification by decreasing the scanned area, it follows that adjustment of the same control when in the rocking beam mode simply reduces the angle of rock. It is important to see that the rocking beam pattern is formed from only the portion of the image that strikes the detector. In standard selected area diffraction methods, the portion of the image that forms the diffraction pattern is chosen with the size of the PAGE 92 86 diffraction aperture. In the rocking beam method, it is the diameter of the solid state detector that determines the portion of the image chosen. Thus, by increasing the magnification an increasingly smaller image segment is permitted to strike the detector and form the microdif fraction pattern. The detector diameter can also be reduced by masking it with an aperture (Geiss, 1975) if even smaller portions of the image used to form the pattern are desired. For the Philips 301, the diameter of the detector is 4 mm and the maximum magnification at the detector height is 130,000X, so the minimum selected area is (3.16) A = Â° 4 "" = 308 A M 130,000 Geiss (1975) has reduced the diameter of the detector down to 1.0 mm and has obtained single crystal diffraction pattern from vapor deposited gold islands less than 100 A in diameter. Signal strength decreases as the active detector area is reduced or sample thickness is increased. A more practical limit for obtaining microdif fraction patterns with a reasonable signal to noise ratio from typical metallurgical thin foil specimens is 500 A. Angular resolution of the diffraction discs in the rocking beam pattern is controlled by the size of the objective aperture. A um objective aperture results in a pattern with resolution comparable to standard selected area diffraction, while removing the aperture entirely produces a rocking beam channeling pattern quite similar to the focused beam channeling patterns discussed previously. Patterns from a 2000 A area in silicon for 4 different objective aperture sizes are shown in Fig. 3.18. PAGE 93 87 fcs PAGE 94 These results are explained by Fig. 3.19, which shows that the diffraction pattern in the back focal plane of the objective lens must be swept across the objective aperture just as the image is swept over the detector. The objective aperture diameter d, limits the divergence of the rays in any diffracted beam. Thus , the maximum angular divergence is found by the ratio of the focal length of the objective lens and the objective aperture diameter: (3.17) P d/f For the Philips 301, f = 3.7 mm, so an objective aperture of 5 pm (the smallest practical size) gives a maximum angular resolution of p = 1.35 x 10 rad. Thus, the angular resolution for rocking beam compares favorably with other microdif fraction techniques. 3.6 Transmission Channeling Patterns by the Rocking Beam Method In Fig. 3.18d, it is seen that completely removing the objective aperture produces a transmission channeling pattern similar to the channeling patterns described in Section 3.1. These patterns, however, have been formed with a well collimated beam. The explanation is shown by Fig. 3.19, where a completely collimated beam is rocked through the necessary solid angles such that the Bragg angle for a particular diffracting plane is satisfied at opposite sides of the optic axis. Since the rocking occurs in three dimensions the cone that is formed is similar in angular extent to the incident beam cone shown in Fig. 3.1. PAGE 95 89 \ \ \ \ PAGE 96 90 Both cases produce exactly the same channeling line pair in the back focal plane of the objective lens. For the convergent beam case, the pattern is formed all at one time since different segments of the beam form different angles with the diffracting planes. For a rocking beam, the pattern is formed sequentially in time, and a well-collimated beam is now positioned by the scanning coils at precisely the right angle to form one segment of the channel line seen in the diffraction pattern. Channeling patterns formed from solid samples with the scanning electron microscope have been studied for several years (Coates , 1970) and can be formed either by backscattered electrons or speciment current. These SEM channeling patterns are limited to solid samples. In addition, the longer focal lengths of the final probe forming lenses typical of commerical SEMs results in much greater spherical aberration than the corresponding STEM lens, and the minimum specimen area that can be studied is only about 2 ym in diameter (Van Essen and Schulson) . Transmission channeling patterns were first produced by Fujimoto et_ al. (1972) using JEOL 100B STEM but no experimental images were shown and the minimum area that could be used to form a pattern was not described. Aside from the paper by Fujimoto, transmission channeling patterns produced by rocking beam methods have not been discussed in the scientific literature. The spherical aberration of both the upper and lower objective lens fields determines the minimum area that can be used to form the rocking beam microdif fraction pattern. For the rocking beam method, spherical aberration effects must be considered any time the beam has PAGE 97 91 been rocked to a position non-parallel to the optic axis (dotted lines in Fig. 3.19). For the upper objective lens, Geiss has shown that spherical aberration causes the image of the C2 aperture on the specimen plane to shift a linear distance: (3.18) x = M C s 6 where M is the demagnification of the C2 aperture, C is the coefficient of spherical aberration for the objective lens and 8 is the angle of rock. M is computed in Chapter 5 to be 1/38X, C is 6.3 mm s and the maximum angle of rock is approximately 1Â°. At this angle, o the shift of the C2 aperture image would be approximately 90 A which causes no problem during the rocking beam process as the illuminated area defined by the C2 aperture is several microns in diameter. However, the same equation can be used to determine the shift of the specimen image itself. This shift is much more crucial since it causes the specimen image to sweep across the detector, thus limiting the minimum area from which the pattern can be obtained. While the upper objective lens field reduces the shift of the illuminated area by 38X, the lower objective lens field increases the shift of the specimen image by 14X. For a 1Â° rocking angle: (3.19) x = 14 (6.7 mm) (.017 rad) x = 5000 A Thus, the minimum usable area needed for a transmission rocking beam PAGE 98 92 channeling pattern is approximately 1 ym. However, note that the shift is much reduced at smaller rocking angles. If the maximum o angle of rock is limited to 0.5Â°, x is reduced to 620 A. Thus, channeling patterns out to about the second order diffraction spots can be produced with a very high level of accuracy. An example of the shift of the image resulting from spherical aberration is shown in Fig. 3.20. It is important to realize that since transmission channeling patterns are found with the objective aperture removed, the current density striking the detector is much higher. Thus, the signal to noise ratio is much higher than any other type of microdif fraction pattern. If it were not for the efffect of spherical aberration, transmission channeling would be a very powerful tool for extracting diffraction information from the specimen. PAGE 99 93 (a) (b) Figure 3.20 Image shift for rocking beam optics. At 40,000, sample image shifted approximately 3500 A, for a 1Â° angle of rock. PAGE 100 CHAPTER 4 EXPERIMENTAL PROCEDURES AND MATERIALS 4.1 Ray Optics and Computations Since electron beam spot size and beam convergence determine the volume of the crystal producing the microdif fraction pattern, an experimental means of spot size measurement is of importance. Such a measurement may be performed directly in the Philips 301 by using the following procedure. An easily observed sample is imaged at high magnification in the standard TEM mode and objects of appropriate size, such as the small gold particles in the interior of the shadowed region in Fig. 4.1a, are recorded photographically. Then, with the sample in place, the microscope is switched to standard STEM conditions. With the stationary STEM diffraction pattern in view, the diffraction lens strength is now changed such that a raster pattern formed by the flying spot is imaged on the TEM phosphor screen as shown in Fig. 4.1b. By adjusting the specimen height, both sample image and raster pattern can be brought into sharp focus and observed simultaneously. The spot size can now be measured by stopping the scan, removing the specimen, and comparing the spot image diameter to gold particle size. When the intermediate lens is set at zero strength for the STEM mode, the magnification reading on the microscope panel is 94 PAGE 101 95 (a) (b) Figure 4.1 Images taken with Philips 301 plate camera of 0.25 ym latex balls shadowed with gold (a) and the STEM raster pattern (b) . The gold islands in the interior of the shadow range from 20 to 100 A in diameter. PAGE 102 96 incorrect, and the spot size can only be determined by comparison to the sample image. It is thus quite important that the beam crossover and specimen are at the same height in the column, or errors in measurement will result. Once this comparison is made, spot sizes, controlled by the first condenser (CI) lens, can be recorded for several CI settings as shown in Fig. 4.2. Measured spot sizes are plotted versus the CI lens current in Fig. 4.3 and the results compare closely to data published in the Philips STEM manual. It is interesting to show how spot size measurements can be used to determine the characteristics of various lenses in the microscope column. Because spot size is controlled by the CI lens alone in the STEM mode, increasing the lens current to reduce spot size must also reduce the CI lens focal length. Since the object distance, (in this case, the distance from the CI lens to the gun cross-over) , is fixed, only the image distance, I, changes with the CI lens strength. The formula for magnification for a simple, "thin" lens (4.1) Â»-* Now, the thin lens formula relating focal length, f, to the object and image distance, and I, is I f By rearranging terms and substituting M can be defined in terms of the object distance and the focal length PAGE 103 97 Figure 4.2 Plate camera images of the stationary STEM probe for four different settings of CI, the first condenser lens. Magnification is 1,000,000X. PAGE 104 98 TO in csi o CN m m CD PAGE 105 99 (4.3) f-0 In the STEM mode with the C2 lens turned off, the spot size (ignoring aberrations) at the specimen plane is found by multiplying the demagnification factors of the CI and objective (OBJ) lens times the original spot size, d Q , of the gun cross-over: (4.4) d = M x M OBJ x d o Except for minor focusing corrections, M remains fixed for any spot size. Also, the CI lens, being of relatively long focal length, can be considered thin and the focal length of Cl can be related to the lens current (Lifshin, 1969) : (4.5) f -* V (NI) 2 where K is a constant, V the accelerating voltage, and NI the number of ampere turns. By substituting Eqns. 4.5 and 4.3 into 4.4, an equation relating the spot size to Cl lens current can be obtained: (4.6) d [1 0(NI) ]M d KV OBJ Since M is known (see Chapter 3 ) , the spot size can be computed OBJ and compared to the experimental values in Fig. 4.3. As discussed in Chapter 3 , the beam convergence angle , or the incident semi-angle, a^, directly controls the angular resolution of the microdif fraction pattern. For both TEM and STEM convergent beam diffraction modes, a^ is determined by the ratio of the diffraction PAGE 106 100 disc diameter to the Bragg angle as derived in Chapter 3 : 6 B y (4.7) 1 x For STEM, a^ is determined solely by the second condenser (C2) aperture size and can be computed by measuring x and y from diffraction patterns such as those shown in Fig. 4.4. The convergence angle is linearly proportional to the C2 aperture diameter so a plot of a^ versus aperture size should be linear as shown in Fig. 4.5. Errors can result both from measurement of diffraction disc diameters as well as from actual aperture diameters that differ by as much as 20% from the stated size. When high accuracy is needed, the best method is to measure the aperture in a scanning electron microscope. For the range of aperture sizes shown, a 10 ym diameter was found to present the best compromise between a highly collimated probe and enough current density to portray a diffraction contrast image on the STEM CRT with an acceptable signal-to-noise ratio. The 10 pm aperture, with a^ = 10 rad, easily satisfies Maher and Joy's criterion for Aw, and thus produces a STEM image very similar to a TEM image formed with a well-collimated beam. For the TEM convergent beam mode, where the C2 lens remains on, calculation of a^ remains exactly the same as for STEM, but control of convergence is accomplished by changing either the C2 aperture size or the C2 lens strength. For this case, convergence is controlled by using the C2 lens to place the beam cross-over above or below the upper focal plane of the upper objective lens. When the C2 crossover is set exactly at the upper focal plane of the objective lens, PAGE 107 101 Figure 4.4 Convergent beam diffraction patterns (STEM mode) for the <111> direction in silicon. As the C2 (second condenser) aperture size decreases in diameter, the angular resolution of the diffraction pattern improves in linear proportion. PAGE 108 102 a. en 03 b 03 CM q Q. < CM o Jo o o rd PAGE 109 103 a large area of the specimen is illuminated with a well-collimated beam. However, if C2 is now defocused sufficiently that the crossover would lie before the plane of the upper objective lens, a raypath such as that shown in Fig. 3.9b occurs, producing a focused spot at the specimen plane. Thus, once C2 has been used to focus the spot for microdif fraction, a^ is fixed and can be adjusted onlychanging the C2 aperture size as is done in the STEM mode. These principles are illustrated in Fig. 4.6. Twelve microdiffraction patterns were taken with the same series of C2 lens settings for three different aperture sizes. Then, the calculated convergence was plotted versus C2 lens current. Although the C2 lens control can be used to vary a, only one setting corresponds to the focused spot. If the angular resolution in the diffraction pattern is insufficient for a given C2 lens strength, the only remaining alternative is to use a smaller C2 aperture. As discussed in the last chapter, resolution for the rocking beam microdif fraction pattern is controlled by the diameter of the objective aperture. Using the formula of Geiss, the angular resolution is (4.8) P = d/f 0BJ For the STEM polepiece, f = 3.7 mm and the objective aperOBJ ture size, d, can range from 5 to 50 urn. Fig. 3.18 shows rocking beam patterns for objective sizes of 5, 20, 50, and no aperture, respectively. The angular resolution of the pattern is directly proportional to the size of the diffraction disc formed on the CRT and PAGE 110 104 Â«4 Q Â• O -p PAGE 111 105 Eqn. 4.7 can be used to find p. Plotted values of p versus objective aperture diameter are shown in Fig. 4.7, and the diffraction disc diameter can also be plotted versus diameter on the same graph. If the objective aperture diameters are precise, the slope of the curves should be the same. 4.2 Sample Preparation The equipment needed to produce stacking faults in silicon consists of a standard metallurgical tube furnace and the accompanying apparatus necessary to force steam over the samples at a controlled rate. As shown in Fig. 4.8, a "steam bubbler" was constructed by connecting a 500 ml Erlenmeyer flask to a Vycor tube lying in the furnace cavity. Distilled water in the flask was then maintained at 99Â°C and heating tape was used to prevent any condensation in the coupling between the flask and the furnace tube. A Teflon boiling chip placed in the bottom of the flask aided bubble formation and encouraged the steam to flow through the hot zone at an essentially constant rate. For a typical temperature of 1100 Â°C, the steam bubbler consumed approximately 100 ml of water per hour. Since the growth rate of the standing faults is quite temperature sensitive, samples were placed on a Vycor spatula that was carefully set in exactly the same position in the furnace tube to insure reproducibility from one run to another. Both temperature and time could be checked by noting the color of the oxide film found on the silicon sample at the completion of the run. It has been shown (Runyan, 1965) that the oxide color is the result of PAGE 112 106 M o PAGE 113 107 optical interference, and that the color is an accurate indication of the oxide thickness. Runyan's results show that a bright green film at 1100Â°C should be equivalent to a 5000 A layer. Forty-five minute annealing times were found to consistently produce a green oxide layer, and, if the sample was first abraded with diamond paste, generous numbers of stacking faults. Starting material for the oxidation experiments consisted of silicon wafers that were mechanically polished on one side to a mirror finish. Undoped, or intrinsic, material was supplied by the Ventron Corporation, while lft-cm n-type material was obtained from the Wacker Chemical Corporation. All wafers were scored with a diamond cutter and then broken into 3 mm squares, a size that fits conveniently in the Philips 301 doubletilt holder. To insure that all contaminants were removed before heat treatment, the individual squares were washed in sulfuric and nitric acids, methanol, and finally rinsed in deionized water. The polishing procedure used for electron microscopy must produce a thin region less than 0.5 pm below this surface. This is accomplished with a chemical polishing procedure that removes material from the back side of the sample while the polished side is protected with a wax mask. The appearance of the sample cross-section is diagrammed in Fig. 4.9. First, the oxide layer is removed by placing the sample in hydrofluoric acid for five minutes and the sample is rinsed in deionized water. The sample is then placed polished side down on a 5 mm square glass cover slide, and both glass and specimen are PAGE 114 108 Figure 4.8 Apparatus used to grow extrinsic stacking faults i silicon. in PAGE 115 109 completely coated with liquid beeswax and allowed to cool until the wax has hardened. The wax is then scraped away from the center of the nonpolished specimen side such that about one-third of the area is exposed. This assembly is then placed in a plastic beaker containing 10 ml of solution of 3 HF:5 HNO3 : 3 HAC. Only the HN0 3 actually attacks the silicon surface, but it must be used in combination with HF, which attacks the Si02 layer that would otherwise reform and halt the polishing process. HAC, or acetic acid, plays no active role in polishing but acts as a buffering agent to control the pH of the solution (Booker and Stickler, 1962). After approximately five minutes of chemical etching a polished depression that extends approximately one-third of the way into the sample appears. Although the step could be allowed to proceed to completion, it was found that this formed a hole with sides so steep that the thin area varied rapidly in thickness and was too small to be of use for electron imaging. If, however, additional wax was scraped away such that two-thirds of the sample surface is exposed after the first depression was made, and the sample was reimmersed in the polishing solution, the depression had much shallower sides. The electron-transparent region in the center was often 0.5 mm across and was of nearly constant thickness. The exact point at which to stop the process is of crucial importance. If carried on too long, the thinnest area is consumed before the solution can be rinsed away. If stopped too soon, the process cannot be restarted without clouding the mirror surface of PAGE 116 110 u PAGE 117 Ill the polished region, as it takes several seconds for equilibrium polishing conditions to be re-established. However, as noted by Booker and Stickler (1962), silicon sections thinner than 0.5 urn are optically transparent and appear pale yellow in transmitted light. Such a thickness is easily penetrated by 100 KV electrons. Thus, the final stage of polishing was completed by placing the beaker containing the waxed sample over a piece of frosted glass and observing it through a microscope with transmitted light. As the sample is thinned, it becomes transparent in the center, changing in hue from deep red to pale yellow. At the appropriate instant, the polishing process was stopped by quenching the entire contents of the beaker into 1000 ml of distilled water. The wax was then dissolved in warm trichlorethylene and the sample cleaned in several rinses of absolute methanol and deionized water. Finally, the sample was glued in the Philips double-tilt holder with a colloidal graphite solution. 4.3 The Character of the Dislocations Introduced Crystal defects introduced into silicon by the methods of the preceding section have been widely studied, and a consensus has begun to emerge on how nucleation, growth, and annihilation of these defects occur. The primary kinds of defects, extrinsic stacking faults surrounded by partial dislocations of Burger's vector Â— <111>, are shown in Fig. 4.10. The stacking faults apparently form at stress concentrations near scratches in the original abraded specimen surface and grow in front of the SiO -Si interface that gradually advances into the silicon matrix during oxidation. PAGE 118 112 Figure 4.10 Extrinsic stacking faults in silicon grown from a polished surface abraded with diamond paste. The specimen is approximately 0.7 ym thick. The curved partial dislocation surrounding the fault is pure edge in character and does not intersect the bottom surface of the foil. PAGE 119 113 The actual mucleation method of the partial is in dispute. One possible mechanism is the dissociation of a perfect dislocation into a Shockley and a Frank partial (Booker and Tunstall, 1966): 1 1-1 (4 ' 9) -[110] -> Â£[112] + Â£[111] 2 6 3 where the Frank partial would be free to undergo climb and create the extrinsic stacking fault. However, the reaction is not energetically favorable and it is doubtful that it actually occurs. A more probable nucleation mechanism (Mahajan, 1976) involves the diffusion of oxygen atoms supplied by the SiO oxide layer to form Si-0 complexes on {ill} planes which compress the adjoining lattice along the <111> directions. The silicon atoms displaced by the oxygen can now "plate out" to form an additional (111) plane that forms an extrinsic stacking fault bounded by a Â—[111] dislocation. As this process continues, additional interstitials diffuse to the core of the bounding partial dislocation, and the stacking fault increases in area by climb of the bounding partial. This method explains how a perfect, single atomic plane of SiO could be created in the diamond cubic silicon lattice. Further evidence in support of this model is supplied by Ravi (1974) , who showed that stacking faults in wafers with surfaces normal to <112> preferentially nucleated on the (111) and (111) plane. Both of these planes lie at an angle of 62Â° to the surface, and have the greatest resolved shear stress of the {lll}-type planes. On the other hand, when the wafer surface is not abraded, faults preferentially grow on (111) planes that are at the shallowest angle to the PAGE 120 114 surface since it is the closest to the oxide-matrix interface, the source of excess interstitials necessary for growth. In the specimens used in this study, with a [111] surface normal, the (111) -(111)(111) planes all make equal angles with the surface, and the density of faults is the same. The growth rates of individual faults was found to obey an Arrhenius relationship (Murarka, 1976) : (4.10) L = At n exp(-Q/KT) where n = .85 eV and Q = 2.55 eV. This equation was found to hold in both n-type and p-type materials. However, the density of faults was on an order of magnitude higher in n-type material. Once the faults reach a length of several microns, additional annealing in vacuum or extended oxidation treatments (Ravi, 1974) result in the annihilation of the stacking faults and the formation of perfect Â— <110> dislocations. In this mechanism, the faults shrink by an exact reversal of the process discussed for fault growth. Here, oxygen could diffuse away from the fault plane, leaving vacancies with the silicon lattice to interact with the partial dislocation core and shrink the extra plane by positive climb. Faults can also be destroyed by dislocation reactions, such as a process involving the nucleation and passage of two Shockley partials over the extra fault to remove the faulted layer. A possible reaction 1 1 1 1Â—1 (4 U) j[lll] + | [112] + Â£[114] + Â£[211] -> jtlOl] PAGE 121 115 This reaction leaves the remains of a perfect dislocation. Only this unfaulting reaction leads to the formation of perfect 1 r<110> dislocations. The strong tendency of the perfect dislocations to lie along <110> directions is caused by the large Pieirls force in diamond cubic materials (Hirth and Loethe , 1968). The Pieirls force is the tendency of dislocations to minimize their line energy by lying along certain preferred directions in the crystal lattice. In summary, defects produced by the annealing treatments discussed above result in extrinsic stacking faults bounded by partial edge dislocations with a Burgers vector b = Â— <111>. These directions are sessile and move only by climb. Upon much longer anneal times, an unfaulting reaction results in a reduction in fault density and the formation of numerous perfect dislocations where the Burgers vector is either parallel to, or lies at a 60Â° angle from, the dislocation line. 4. 4 Determination of Crystal Directions by Diffraction Patterns Identification of crystal defects can proceed only when the orientation between defect, diffracting plane, and electron beam has been precisely determined. In particular, the dislocation line direction, specimen foil normal, and electron beam direction must be repeatedly calculated for every two-beam condition. To proceed, a series of diffraction experiments is performed as in Fig. 4.11, where a two-beam image of a defect is shown with its associated selected area diffraction pattern. The exact beam direction, expressed in hkl coordinates, can be found by measuring the relative positions PAGE 122 116 Figure 4.11 Extrinsic stacking fault in silicon and selected area diffraction pattern from surrounding area. After correction for rotation of the image due to the lenses , the g vector (g" = <111> for the <112> pole) can be drawn on the image as shown. The angle between the partial dislocation direction, u, and g can then be measured directly on the photograph. PAGE 123 117 of Kikuchi line pairs on the diffraction pattern. Once the exact beam direction for a particular two-beam condition is found, the projection of the defect line direction in the image plane (i.e., the actual dislocation image, seen in Fig. 4.11a is used in conjunction with the beam direction and the g vector to compute the actual defect line direction and the specimen foil normal. The beam direction will be needed in subsequent calculations and must be computed first. Consider the orientation shown in Fig. 4.12 between g, the two-beam diffraction vector, B, the true beam direction defined as anti-parallel to the electron beam, and A, the low-index pole direction that is determined by inspection from the diffraction pattern. It can be seen that B must be perpendicular to g, and that A and B must be separated by <5, the angle of tilt necessary to produce the two-beam condition. Since the center of the Kikuchi pattern can be thought of as rigidly attached to the crystal, 6 is proportional to the linear distance from the main beam position to the Kikuchi pattern center. Since the distance between each Kikuchi pair is proportional to 29, 6 is proportional to X/29 where B " B X is the distance drawn from the main beam to the Kikuchi pattern center. The vectors A, g, and g x A form a right-handed coordinate system. Since B is perpendicular to g, it must lie in the plane formed by A and g x A at 5 degrees away from A. If A and (g x A) are assigned unit lengths, we can thus write the equation for the true beam direction as B = |cos a| A + |cos(90Â° a) | g x A (4.12) PAGE 124 118 Kikuchi Line Pair Figure 4.12 Angular relation between the two-beam direction, BM (determined from measurement of Kikuchi line pairs) , and A, the low index pole direction. Both BM and A are assumed perpendicular to the g~ vector. PAGE 125 119 With B computed for two different two-beam images, we then use Fig. 4.13 to compute the defect line direction, u. For both of the two-beam images, the projected directions, u and p, are co-planar with the g vector used in each case. If the acute angle between p and g is measured and a coordinate system of B, g, and g x B is formed, then p is _ _ < 4 -!3) p = | cos 3 1 g + | cos (90Â° g) | (g x B) As shown in Fig. 4.13, each p and its associated beam direction, B, forms a plane. The normals to each of the planes are (4.14) n = p x B n 1 = p 1 x B' Now, the interactions of the planes must be the dislocation line directions, so u is the cross product of the two normals. (4.15) u= (pxB) x (p 1 xB') The foil normal computation is an extension of the computation for the line direction. To proceed, the directions of two lines drawn from mid-point to mid-point of three conveniently spaced dislocations are computed by the preceding methods. This forms a triangle whose surface is perpendicular to the foil normal. If two edges of the triangle are labeled C and D, then the foil normal is again found by computing the cross product: (4.16) F = C x D PAGE 126 120 Figure 4.13 The dislocation image, as recorded by the plate camera on the STEM CRT, is a projection along the beam direction to a plane containing the g vector. The projected directions p and p ' form planes with there associated beam direction that intersect at u, the actual dislocation line direction. PAGE 127 CHAPTER 5 EXPERIMENTAL RESULTS 5.1 A Comparison of the Microdiffraction Methods It should be apparent that there is considerable overlap among the various microdiffraction techniques and there are often several possible choices to extract information from a crystalline specimen. The methods themselves can be roughly categorized according to ray optics. In the table below, the four basic methods are compared to standard selected area diffraction techniques. Table 5.1 Microdiffraction Methods Method Ray Optics Minimum Area from Which Diffraction Angular Resolution Pattern is Formed Selected PAGE 128 122 It would seem all of the methods are capable of producing patterns from very small areas with a reasonable degree of angular resolution. In addition, the rocking beam and all focused spot techniques are capable of producing high quality transmission o channeling patterns from an area approximately 2000 A in diameter. Nevertheless, the quality of the results obtained by the different methods can vary widely, particularly with regard to the signal-to-noise ratio in the pattern, sample damage, and time spent in obtaining acceptable results. The following examples in this section are used to demonstrate the advantages and disadvantages of the methods listed in the table. Certainly the easiest method of obtaining microdiffraction patterns is the focused C2 aperture technique. Illumination of the sample is the same as the imaging mode and the condenser lenses do not have to be adjusted to obtain the pattern. Although the minimum area is rather larger than the other methods, angular resolution is just as good as selected area diffraction and the contamination, problems of the focused spot techniques are avoided. One drawback of the focused C2 aperture approach is that the sample must be raised for both the image and the aperture edge to be sharply in focus. The specimen position is then non-eucentric, that is, it cannot be tilted with the goniometer stage without the image sweeping rapidly out of the field of view. For the STEM polepiece with the Philips 301, however, the focusing points for specimen and aperture nearly coincide and diffraction patterns can be obtained PAGE 129 123 from the eucentric position if one is willing to allow a slight fuzziness in the shadow image of the C2 aperture. Examples of microdif fraction patterns produced by this technique are shown in Figs . 5.1 to 5.3 which compare patterns taken with Selected Area Diffraction (SAD) techniques to the focused C2 aperture method. In Fig. 5.1, a polycrystalline gold foil with an average grain size of 0.5 \im was imaged (Fig. 5.1a) and the pattern recorded for a 20 \im SAD aperture. Diffraction patterns using an aperture less than 20 ym in diameter would be useless due to spherical aberration, so standard techniques were not sufficient to produce single crystal diffraction patterns (Fig. 5.1b) . The size of the 20 urn in the SAD position was recorded with the plate camera (Fig. 5.2). The same aperture was then transferred to the C2 aperture position and the image again recorded. By comparing the diameters of the aperture in the two positions with the plate magnification it was found that the objective lens prefield de-magnified the aperture by a factor of 38, while the lower half of the objective lens magnified by a factor of 14. The illuminated o specimen area was thus 5200 A for the 20 ym C2 aperture and single crystal diffraction patterns were easily recorded. Although the pattern is crisp with good angular resolution (Fig. 5.3a), note that Kikuchi lines are very difficult to distinguish. This effect, characteristic of high atomic materials, means that the crystal structure can be identified but the exact electron beam specimen orientation cannot be determined. This problem can be overcome only by using one of the focused spot microdiffraction techniques. PAGE 130 124 (a) (b) Figure 5.1 Image of polycrystalline gold foil and a selected area diffraction pattern obtained from a 20 um aperture in the SAD position. PAGE 131 125 Figure 5.2 Size of the focused images of the C2 aperture and the diffraction aperture at the specimen plane. Aperture diameter is 20 urn for each case. PAGE 132 126 (a) (b) Figure 5.3 Microdif fraction patterns from polycrystalline gold foil with a focused 20 urn C2 aperture (a) and the focused TEM spot convergent beam (b) where the probe diameter was 200 PAGE 133 127 Figure 5 . 4 Transmission channeling pictures from polycrystalline gold foil. Spot size was 400 A and the C2 aperture diameter was 300 >jm. PAGE 134 128 4 ^KgixOi ay (a) (b) (c) Figure 5.5 Polycrystalline molybdenum is imaged in (a). As is characteristic of high atomic number materials, the SAD patterns show few Kikuchi lines . PAGE 135 129 Figure 5.6 Double exposure of moybdenum transmission channeling patterns and selected area diffraction patterns shown on previous page. PAGE 136 130 (a) (b) Figure 5.7 Channeling pattern produced by rocking beam compared with a rocking beam microdiffraction pattern from the same area. No aperture was used for (a) while a 20 um aperture was used for (b) . PAGE 137 131 In Fig. 5.3b, for example, Cl is adjusted to reduce the spot size to approximately 1000 A and channeling lines begin to appear. The low index pole position can now be computed (using the methods of Chapter IV) with precision. If ctj_ is increased with a larger C2 aperture the complete transmission channeling pattern appears as shown in Fig. 5.4. Since a^ is now much greater, the channeling pictures are produced at the expense of much greater spherical aberration which distorts the edges of the image. Nevertheless, spherical aberration is still sufficiently low that channeling patterns from single grains less than 0.5 ym areas are achievable. One interesting modification of focused spot TEM microdiffraction is to record a diffraction spot pattern using either focused spot or SAD methods, and then directly compare it with channeling information by making a double exposure from two separate negatives. This method was used for a molybdenum foil, where it is again difficult to produce Kikuchi lines (Fig. 5.5). The exact beam direction for diffraction patterns with as little as 3 or 4 excited reflections can now easily be determined with the aid of the channeling lines (Fig. 5.6) . The results for molybdenum can be duplicated with the rocking beam procedure for approximately the same areas needed to produce a pattern. In Fig. 5.7, two rocking beam patterns were taken with a 10 ym objective aperture (Fig. 5.7b) and the objective aperture removed (Fig. 5.7a). The images are noisier than those obtained from PAGE 138 132 Figure 5.8 Transmission channeling patterns from (111) direction in silicon produced by rocking beam technique with objective aperture removed. PAGE 139 133 <013> <112> <011> Figure 5.9 STEM microdif fraction patterns from gold islands ^ 200 A in diameter. PAGE 140 134 the focused spot procedure but contamination was minimal since the beam remains defocused for rocking beam. The "noisiness" of the image formed on the CRT is a constant problem with this method, but high quality results (Fig. 5.8) can still be obtained for thin sections or low atomic materials such as silicon that are more transparent to the electron beam. o If diffraction patterns from areas less than 400 A must be obtained, STEM methods that form the image on the CRT and record the microdif fraction pattern with the plate camera is the only possibility. With the STEM polepiece, spot sizes as small as 30 A can be used and, since beam spreading should involve only inelastically scattered electrons, diffraction patterns from a column of crystal of approximately the same diameter can be recorded. Although angular resolution can be quite high if a C2 aperture of 10 urn diameter is used, the intensity of the pattern may be quite low. High beam current densities, fast emulsions, and long exposure times may help to overcome these limitations but contamination and drift of the probe due to electronic instabilities are continuing problems. Probably the best aid to good results is using a very short camera length so that the entire diffraction pattern is concentrated in the center of the negative. With this procedure, exposure times o for a 50 A STEM probe can be less than 1 second, much reducing probe drift and specimen contamination. Three excellent examples are shown in Fig. 5.9, where easily recognizable <011>, <112>, and <103> patterns were produced from gold islands formed by vapor deposition on a formvar film. The quality of these images compare favorably to rocking beam patterns done by Geiss (1976) using a similar sample. PAGE 141 135 A second example (Fig. 5.10) shows that STEM microdif fraction methods can often reduce the complexity of solving complicated orientation problems with its ability to spatially resolve areas only a few hundred Angstroms apart. Here, two 100 A microdif fraction patterns from adjacent twins in an ordered Ti-Al alloy can be analyzed much more easily than the typical SAD pattern taken from a much larger area. Several pairs of such patterns should allow the major twinning directions to be easily determined. 5. 2 Identification of Crystal Defects by Microdif fraction Complete characterization of dislocations or stacking faults by using microdif fraction patterns alone was not achieved. Nevertheless, it can be shown very clearly that it is possible in principle to carry out such an experiment. The primary limitations with the equipment, probe instabilities and high contamination rates, may be possibly overcome by new generation electron microscopes in a short time. For example, the successor to the Philips EM 301, the EM 400, has a vacuum of 10 torr as opposed to 10" 5 torr for the 301 and a o guaranteed maximum drift rate of less than 10 A/5 minutes. Even with the 301, it is relatively easy to demonstrate the sort of microdiffraction conditions required for patterns that could be adapted to the computer simulation approach. Ideally, a defect identification procedure using microdiffraction should contain the following steps: 1) In the STEM mode, image a defect such as the stacking fault image seen in Fig. 5.11, preferably under n-beam conditions. PAGE 142 136 Q PAGE 143 137 Figure 5.11 Silicon stacking fault imaged for N-beam conditions and its associated SAD diffraction pattern from a 20 urn aperture. PAGE 144 138 2) Stop the scanning motion, and position the probe along several positions close to the dislocation core. Record the microdif fraction pattern with the plate camera and the position of each probe position with respect to the ends of the dislocation. Isl 2 3) Compute the J Â— ' Â— value using the methods discussed in kl 2 Chapter II and compare them to the experimental values obtained by a densitometer directly from the microdiffraction patterns . As the core of the dislocation may be displaced a significant amount from the position implied by the center of the diffraction contrast image, it is probable that the computer simulated column would have to be "moved" several times to obtain a good fit with the experimental data. However, only one column need be computed for each test and, even with the increase in computation time required for n-beam conditions, it would seem that many possible Burgers vectors could be eliminated without resorting to computing the entire computed simulated defect image. It should be emphasized that microdif fraction could probably never completely supplant the image simulation approachthe information it provides is the same but the area of sample from which it comes must be determined with an extremely high degree of accuracy . STEM microdiffraction patterns that could be used for defect identification are shown for a stacking fault in silicon with n-beam conditions . Once the fault image was displayed on the CRT the raster was stopped and an 80 A diameter probe was slowly moved across the PAGE 145 139 Â•Â•Â•Â• Figure 5.12 STEM microdif fraction patterns from areas spaced roughly 100 A apart over the stacking fault fringes shown in the previous figure. The relative intensities of the spots are diagrammed next to each pattern. PAGE 146 140 stacking fault fringes in the direction shown in Fig. 5.11. Each time an intensity change was visually detected the pattern was recorded with the plate camera. The results shown in Fig. 5.12 show clearly that several intensity changes can be seen even with a relatively "simple" contrast image such as fringes produced by a stacking fault. It is also important to note that there is more information in an n-beam microdiffraction pattern than the equivalent point on the bright-field image. This is because the bright-field intensity, i 1 2 |T| , is |t| 2 =i| Sl | 2 |s 2 | 2 . . . |s n | 2 The intensities for the scattered beams could be obtained from their dark-field images only one at a time but they are all available directly in a single microdiffraction pattern. An estimate of the number of microdiffraction experiments necessary to identify a defect is shown in Fig. 5.13. Here, computer simulations of six dislocations with different Burgers vectors are imaged for exactly the diffraction conditions. If microdiffraction patterns were taken from the positions marked by the four circles o (roughly 70 A in diameter) on each of the six dislocations, the dark i 1 2 areas of the image would indicate a high | S | value and the lighter i |2 areas a nigh |T| value. For a rough comparison, then, let a . ,2,1 .2 |S| /|T| ratio be symbolized by a "+" for a dark area and a "0" for a lighter one. For each of the 4 circled areas, the six dislocations have the values: PAGE 147 141 [on; PAGE 148 142 [110] Figure 5.13 Computer simulated two-beam images of dislocations in (cont.) silicon. The beam and foil normal are (112), g = (111), and s = 0.0. Images are shown for six different Burgers vectors. PAGE 149 143 Table 5.2 Dislocation Identification b 12 3 A PAGE 150 144 results (Fig. 5.14 to 5.17), show contrast variations similar to the thickness fringes seen experimentally in a wedge-shaped foil. As discussed in Chapter II, the STEM modification is of the form: (5.2) WÂ„ = W Â± AW STEM where AW = Igl Â£ a. '*' ^g l The correction factor represents a range of values between and AW where each value is weighted appropriately. Thus, a STEM calculation for a column consists of several matrix multiplication sequences, each with a different value of AW. The results for each set of multiplications is then added together to get the final result: l T ltotal = l T l! 2+ l T 2| 2 + Â• ' " Kl 2 Where the initial amplitude of T n is: T total ^^ If W = 0.5, for example, and the AW range was divided into 3 equal segments, then 4 curves would be plotted with AW values equal i 1 2 to -0.5, -0.516, +0.5/6 and +0.5. Intensity curves (|T| ) versus thickness curves were plotted for as many as 6 AW values (Fig. 5.14). In general, modifications to the curve were quite minor if more than 4 AW values were used and, for reasons of economy, all STEM image simulations were plotted with a 4 AW value average. The effect of the STEM modification is shown in Figures 5.15, 5.16 and 5.17, where a value of a= 5 x 10 rad, is added to |t| versus thickness curves for W = -0.5, 0.0 and +0.5. In each case, PAGE 151 145 PAGE 152 146 PAGE 153 147 \ :1 \ * \ Â» I.' f / \ a / M Â• / * * ( 0i ia u m o o u I PAGE 154 148 t ; / Â• i Â• i Â• t Â• W \. \. iÂ» a r w . m '* A a /. 1 /Â•Â• t Â• V '. \ . \ . \Â» Â• Â• _.-* PAGE 155 149 the STEM modification damps-out the oscillations in |t| and lowers the average height of the curve. This implies that an image of thickness fringes in a wedge-shaped foil would exhibit reduced contrast, as well as a lower absolute intensity than would be the case for TEM conditions. This result must be interpreted with care. At certain thicknesses and Bragg deviations the |t| value for STEM may be increased over the TEM value. This is true in Fig. 5.17 for t = 3/2 E, where g i , 2 ,,2 T c-tom = Â°3 and T mM = Â°15 In Fi 9 s 5 15 and 5.16, where STEM ' ' TEM W > 0.0, this occurs less frequently. It is clear that the STEM correction changes the wavelength of the |t| curve, particularly when W < 0. This implies that the width of extinction fringes in a wedgeshaped foil will vary with the magnitude of a . . The extent of the variation should be greatest for negative Bragg deviations. In Figs. 5.18, 5.19 and 5.20, extrinsic stacking faults silicon, where R = 1/3 <111>, are imaged for 3 different g vectors. Here the decrease in intensity from STEM is of minor importance, since the gain control can be used to increase the strength of the overall signal on the CRT. Contrast does decrease as before, but it is noteworthy that modification of the image is most severe in the central regions of the stacking fault. As predicted by Joy and Maher (1975) , defect images with the shorter extinction distances such as the g = <111> image (Fig. 5.18) show the least amount of modification from the STEM correction, while those with higher g values such as g = [311] show strong changes even for low a. values. For example, in image (b) PAGE 156 150 (a) a. = (b) a. = 2 x 10 rad Figure 5.18 TEM and STEM simulations of stacking fault in silicon. BM = (112), FN (112), g = (111), zero Bragg deviation. PAGE 157 151 (a) a = (b) -3 rr^==: a. = 2 x 10 rad si (c) a. = 8 x 10 rad Figure 5.19 TEM and STEM simulations of fault shown in Fig. 5.18. g vector changed to (311) . PAGE 158 152 Figure 5.20 TEM and STEM simulations of fault shown in Fig. 5.18. g vector changed to (220) and BM and FN changed to (111). PAGE 159 153 of Figs. 5.18, 5.19 and 5.20, (a. = 2 x 10~ rad) only the g = <311> image undergoes significant modification. The STEM simulations are compared to experimental TEM and STEM Â— -4 images in Fig. 5.21. Here, g = <311> and a= 5 X 10 rad for TEM (a), and 8 X 10 rad for STEM (b) . In the STEM image, the central fault fringes exhibit reduced contrast as was predicted by the computer simulation in Fig. 5.20. A final point concerning the effect of convergence or diffraction contrast imagining is shown in Fig. 5.22. Here, the TEM plate camera was used to record two-beam fault images for two widely different values of a^. Although a. is changed by a factor of 50 the images appear identical. The reason is that the fault itself for the highly convergent beam does not lie in the specimen plane. The convergence angle of any portion of the beam that forms a particular point on the image is much smaller that the convergence angle of the entire beam. This is not the case for STEM imaging, where the probe cross-over lies at the specimen plane. Thus, the actual distance of the focal point of the electron beam from the specimen as well as the convergence angle must be considered to correctly determine the extent of image modification by convergent beam radiation. PAGE 160 154 WHgBWBWjiMpgpni Â» TEM i-.^\..*>^ii'^,^ STEM Figure 5.21 Comparison of STEM and TEM two-beam fault images for g = (311) PAGE 161 155 (a) (b) -2 Figure 5.22 Two-beam image of fault with g = (220) and a^ = 10 (a) and a. = 5 x 10" (b) . The images are nearly identical. PAGE 162 156 5.4 Determination of Grain Orientation of a Fine-Grained Superalloy Using Focused Condenser Aperture Microdif fraction A good example of the use of focused C2 aperture microdif fraction is the analysis of intergranular orientation of the nickel superalloy shown in Fig. 5.23. The image shows many grains less than lpm in diameter, less than the minimum size that allows normal selected area diffraction methods to produce single crystal diffraction patterns. However, focusing the demagnified image of a 10ym C2 aperture at the specimen plane allows microdiffraction patterns o to be formed from areas only 2600 A in diameter. Now, single crystal patterns can be obtained as shown in Fig. 5.24 and 5.25. These patterns can be used to determine the orientation of adjacent grains using the method described in Section 3.2. To proceed, microdiffraction patterns from each of the two grains in the center of the image were recorded for two different tilts of the goniometer stage. For this alloy, the Kikuchi lines were sufficiently sharp to allow the electron beam direction, with respect to each grain, to be determined to an accuracy of about 1/2Â°. If this had not been the case, channeling lines, produced either by the focused spot or the rocking beam methods , could have been used for the same purpose. In Fig. 5.24 and 5.25, two prominent pairs of Kikuchi lines are diagrammed along with the first-order spots for each of the four diffraction patterns. The linear distance between the Kikuchi pattern center and the zero-order beam can now be used to compute a, the angle through which the major pole direction must be tilted to form PAGE 163 157 Figure 5.23 TEM image of Ni Al(Mo) alloy showing fine grain structure and intergranular precipitates. Many of the grains are less than lum in diameter. The two grains used for the analysis are marked A and B. PAGE 164 158 Figure 5.24 Focused C2 aperture microdif fraction patterns for grain "A." Kikuchi lines pairs used to determine exact beam direction are diagrammed beside the diffraction patterns. Tilt angle between patterns is 26 c PAGE 165 159 Figure 5.25 Microdif fraction patterns for grain "B" taken at same goniometer tilts as Figure 5.24. PAGE 166 160 the diffraction pattern. Using Eqn. 4.12, the exact beam direction is calculated for each microdif fraction pattern and listed below. PAGE 167 161 ax + by + cz = d ex + fy + gz = h 2 x 2 J. 2 x + y + z =1 With a solution of the form x = C3 C4*z y = CI C2*z (5.6) (5.7) Z = BB + BB 4*AA*CC 2*AA where CI PAGE 168 162 To check for accuracy, 1. . can now be used to transform any beam direction in one grain for a given tilt into the beam direction of the adjacent grain. Thus, from Table 5.3 / .Â« PAGE 169 CHAPTER 6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH Results of this study show that the interpretation of any microdif fraction method depends upon the use of a convergent, rather than a parallel, or well-collimated, electron beam. The convergence angle, as defined by a . , directly affects: a) signal to noise ratio in the diffraction pattern (larger a. values increase signal strength), b) contamination (larger a. values increase contamination rates) , c) form of information in the diffraction pattern (larger a. values form channel-type patterns while smaller ones produce discrete diffraction spots composed of small diameter discs) . Convergence angles can be calculated directly from microdif fraction patterns as described in Section 4.1, so exact quantitative comparisons of the effect of a . on the parameters listed above can be found and optimum a. values can be repeated in subsequent experiments. It is suggested that convergence angles be listed in microdif fraction studies just as "two-beam conditions" are always recorded in selected area diffraction experiments. The most efficient way to categorize microdiffraction methods is according to the electron-optic lens configuration used to form 163 PAGE 170 164 the diffraction pattern. The methods described in this work can be summarized as: a) Focused Spot Â— The image can be viewed directly or displayed on a CRT in STEM, but the microdif fraction patterns are always recorded with the plate camera at the base of the microscope column. b) Focused Condenser Aperture Â— Image and diffraction pattern formed in a conventional manner, but a small (<10 ym) condenser is focused on the sample and illuminates only a small area in the diffraction mode. c) Rocking Beam Â— Image formed in a conventional manner but the beam is rocked over the sample, the detector size selects a portion of the image to form diffraction pattern on CRT. In addition, both the focused spot and rocking beam techniques can form transmission channeling patterns for large convergence angles. All of the three methods have advantages and disadvantages. o Focused spot methods can examine areas as small as 50 A in diameter but have high contamination rates and cannot be used on sensitive materials . Focused condenser aperture methods have excellent angular resolution in the diffraction pattern and low contamination o rates, but the minimum area that can be examined is 500 A (for a 2 \aa C2 aperture) . The rocking beam technique offers complete control of angular resolution and contamination no higher than conventional TEM, but its signal-to-noise ratio is the poorest of the three methods . PAGE 171 165 Results from the computer simulation studies in Chapter 5 show that STEM crystal defect images can be simulated in the same manner as those formed by conventional TEM as long as convergence effects are taken into account. The stacking fault simulations verify that the extent that a STEM image will differ from its TEM counterpart is predicted by the equation AW = a , E, | g | . Thus , STEM images undergo the greatest modification for highly convergent probes and higher order reflections. The primary effect of STEM appears to be a loss of contrast, particularly in regions where the defect passes close to the central and lower parts of the specimen. Since the STEM simulations (just as TEM) consists of thousands of column calculations, there appears no immediate advantage to studying diffraction contrast images with a STEM-equipped electron microscope. However, this work does show in a semi-quantitative fashion that STEM differs from TEM only by the convergence angle effect. As a result, the same mathematical formalism developed Howie and Whelan can now be used with confidence to simulate the intensity of n-beam microdif fraction patterns. The same basic type of information is contained in microdiffraction patterns and selected area diffraction patterns. Microdiffraction has only two advantages over selected area techniques. First, it improves the signal-to-noise ratio of the diffraction pattern from small precipitates. In other words, the precipitate/ matrix volume ratio increases because a smaller portion of the sample is used to form the diffraction pattern. Second, the reduction in effective excited specimen volume enables the analysis of PAGE 172 166 structures that vary periodically over distances of a few hundred Angstroms . In this work, most of the emphasis was placed on the second class of problem. More data, consisting of at least several microdiffraction patterns, are required to solve a problem, and an efficient means of mathematical analysis has been shown to be essential. It has been shown that slight modifications to the dynamical theory to account for convergence and a few simple relations from analytical geometry can be used to solve a wide variety of problems in metallurgical microscopy. Probably the single most promising avenue for future research is the solution of the n-beam problems to simulate microdif fraction patterns that have been modified by three-dimensional (non-symmetric) strain fields. Computer simulation of the images of these defects is still far too wasteful of computer time. The possibility of characterizing a defect strain field with only a few microdif fraction patterns, each corresponding to one column of distorted crystal, should be of interest to many investigators . PAGE 173 APPENDIX The program listed below can be used to compute | T | or | S | values for rocking curves, intensity versus thickness curves, and dislocation and stacking fault images. The Tholen method of matrix multiplication described in Chapter 2 is used in each case. 167 PAGE 174 168 DIMENSION BM(3) ,F(3) ,U(3) ,G(3) ,BR(3) ,BXU(3) ,V(3) ,P(3) ,BRE(3) DIMENSION DUM(3) ,GTHETA(5000) ,GDOTR(5000) ,X(3) ,Y(3) ,Z(3) DIMENSION A(3,3),IIROW(121),IIROWl(121),ROW(121) DIMENSION ISYM(11),JSYM(11),SYM(11) DIMENSION Fl(3) ,G1(3) ,BMl(3) ,Ul(3) ,BRl(3) DATA PI, POIS,TWOPl/3. 14159,0.33,6.28318/ DATA ISYM/' *, , , , , : , , , l , , , = , J "","", , H , , ,,,, , , H , , , B7 DATA JSYM/3*' ',',','','.','',"/*, 'U' , 'Z ', '7o7 COMPLEX All(1401) ,A12(1401) ,A22(1401) ,SQ,C1,C2,II,T,S COMPLEX TEM,C3,C4,DENOM DATA SYM/1. 15,. 93,. 78,. 69,. 63,. 55 ,.48,. 42,. 36,. 31,. 05/ T=(l. 0,0.0) S=(0. 0,0.0) READ(5,1) BM(1) ,BM(2) ,BM(3) ,F(1) ,F(2) ,F(3) READ(5,1) U(1),U(2),U(3),G(1),G(2),G(3) READ(5,1) BR(1) ,BR(2) ,BR(3) ,DUM(1) ,DUM(2) ,DUM(3) 1 FORMAT(6(F4.1)) READ(5,2) ANO,WF,TH 2 FORMAT(3(F4.2)) READ ( 5 , 3 ) NUM , GEXT , ALI 3 FORMAT(12,F4.1,F6.4) T=(1.0,0.0)/NUM S=(0. 0,0.0) VAL=0 DELWF=GEXT*ALI INSURE U & BM ARE ACUTE CALL NORM(U,BM,DUM) CUB=U(1)*BM(1)+U(2)*BM(2)+U(3)*BM(3) IF(CUB.GE.0.0)GO TO 8 U(l)=-U(l) U(2)=-U(2) U(3)=-U(3) 8 CONTINUE COMPUTE PICTURE COOD. & NORMALIZE DO 10 1=1,3 10 Z(I)=BM(I) CALL XPROD(X,U,BM) CALL XPROD(Y,Z,X) CALL NORM(X,Y,Z) DEFINE TRANSFORMATION MATRIX DO 20 1=1,3 A(1,I)=X(I) A(2,I)=Y(I) 20 A(3,I)=Z(I) PAGE 175 169 TRANSFORM INPUT VECTORS TO TO PICTURE COOD. DO 22 1=1,3 BMl(I)=BM(I) Fl(I)-F(I) Ul(I)=U(I) G1(I)-G(1) 22 BR1(I)=BR(I) DO 25 1=1,3 BM(I)=A(I,1)*BM1(1)+A(I,2)*BM1(2)+A(I,3)*BM1(3) F(I)=A(I,l)*Fl(l)+A(I,2)*Fl(2)+A(I,3)*Fl(3) U(I)=A(I,1)*U1(1)+A(I,2)*U1(2)+A(I,3)*U1(3) 25 G(I)=A(I,l)*Gl(l)+A(I,2)*Gl(2)+A(I,3)*Gl(3) C NORMALIZE, BM,F, & U ONLY CALL NORM(BM,F,U) DOT=BM(l)*F(l)+BM(2)*F(2)+BM(3)*F(3) DIG=ABS(DOT-l.) PRINT, U(1),U(2),U(3) C C COMPUTE VECTORS PREPENDICULAR TO DISL. & NORMALIZE CALL XPROD(BXU,BR,U) CALL XPROD(V,U,BXU) CALL XPROD(P,V,U) CALL NORM(P,V,BXU) C C COMPUTE BURGER'S EDGE COMPONENT DO 30 1=1,3 30 BRE(I)=(BR(1)*V(1)+BR(2)*V(2)+BR(3)*V(3))*V(I) C C COMPUTE FOIL THICKNESS SEEN BY BEAM,DZ, & YO ANORM=ANO/2. TZ=TH/ABS(BM(1)*F(1)+BM(2)*F(2)+BM(3)*F(3)) DZ=TZ/40 YO=(U(2)*F(3)*TZ)/(2*(U(2)*F(2)+U(3)*F(3))) YO=ABS(YO) CR= 5000/(10. 0*TZ) IB=700 IQQ=100 C C COMPUTE CONSTANTS FOR COLUMN CALCULATION UA=U(2)**2 UB=U(2)*U(3) UC=U(3)**2 C C COMPUTE CONSTANTS FOR DISPLACEMENT C AL'N CA=(G(1)*BR(1)+G(2)*BR(2)+G(3)*BR(3))/(2*PI) CB=(G(l)*BRE(l)-KK2)*BRE(2)+G(3)*BRE(3))/(8*PI*(l-POIS) DUMMY=G(1)*BXU(1)+G(2)*BXU(2)+G(3)*BXU(3) CC=DUMMY*((1-2*P0IS)/(4*PI*(1-P0IS))) CD=DUMMY / ( 8*PI* ( 1 -POI S ) PAGE 176 170 C C COMPUTE & STORE THE DISPLACEMENT MATRIX DO 35 1=1,5000 GAM=(2*PI/5000)*I R=(10.0*TZ/5000)*I GTHE TA ( I ) =CA*GAM+CB*S IN ( 2*GAM) +CD*C0S ( 2*GAM) GDOTR(I)=CC*(ALOG(R)) 35 CONTINUE C COMPUTE & STORE THE 1400 SCATTERING MATRICES II=(0. 0,1.0) DO 5 1=1,1401 IJ=I-1 W=-.3.5+.005*LJ SQ=CSQRT( 1 .+W**2-AN0**2+2 . *II*AN0) DENOM= (TWOPI*II*SQ) C1=PI*II*(W+SQ) C2=PI*II*(W-SQ) C3=PI*(II*W-ANORM)+PI*II*SQ C4=PI*(II*W-AN0RM) -PI*II*SQ A11(I)=(-C2*CEXP(C3*DZ)+C1*CEXP(C4*DZ))/DEN0M A12(I)=(CEXP(C3*DZ)-CEXP(C4*DZ))/(DENOM/(PI*(II-ANO))) A22(I)=(C1*CEXP(C3*DZ)-C2*CEXP(C4*DZ))/DEN0M TEST=W-WF IF(TEST.GE..001)GO TO 5 TINTB=(CABS((-C2*CEXP(C3*TZ)+Cl*CEXP(C4*TZ))/DENOM))**2 5 CONTINUE C WRITE(6,63) 63 FORMAT (1H1) C BEGIN PICTURE CAL'N & COMPUTE CONSTANTS FOR INNER LOOP DO 40 1=1,60 QM=I-0.5 XX=.618*YO*QM/30-.618*YO DO 45 J=l, 121,2 QJ=J-0.5 YY=(YO*QJ)/60-YO C C COMPUTE POSITION OF COLUMN TOP IF(DIG.LT. .02)Z0=TZ/2 Z0=TZ/2-(F(l)*XX+F(2)*YY)/F(3) C C START COLUMN CAL'N DO 51 N=1,NUM WF=WF+DELWF* (N/NUM-O . 5) DO 50 K=1,41 KK=K-1 ZZ=Z0-DZ*KK PAGE 177 171 C C FIND DISTANCE BETWEEN DISLOCTION & DZ R2=UC*YY-UB*ZZ R3=UA*ZZ-UB*YY R=SQRT(XX**2+R2**2+R3**2) THETA=ARCOS ( (XX*V( 1)+R2*V(2)+R3*V(3) ) ) /R) COSAP=(XX*P(l)+R2*P(2)+R3*P(3) ) /R IF (COSAP . LT. . 0)THETA=TWOPI-THETA C C FIND DISP. MATRIX CORRESPONDING TO R & THETA IR=CR*R IF(IR.EQ.0)IR=1 IT=(5000*THETA) /(TWOPI) IF(IT.EQ.0)IT=5000 IF(K.EQ.l)GO TO 49 C COMPUTE BETA 6c CONVERT TO SBSCRIPT FORM BETA=(GDOTR(IR)+GTHETA(IT)-HOLD)/DZ + WF IF(BETA.GT.3.5)BETA=3.5 IF(BETA.LT.-3.5)BETA=-3.494 IB=(BETA+3.5)/.005 IF(K.EQ.2)ITEST=IB I TES T= LAB S ( IB IHOLD ) IF (ITEST.GE. IQQ) IB=IHOLD C C PAGE 178 172 PRINT ONE ROW OF THE PICTURE DO 54 J=l,121 DO 55 M=l,ll IF(ROW(J)-SYM(M))55,60,60 55 CONTINUE M=ll 60 IIROW(J)=ISYM(M) 54 IIR0W1(J)=JSYM(M) WRITE(6,65)IIROW WRITE(6,66)IIROWl 65 FORMAT (IX, 12 1A1) 66 FORMAT (1H+, 12 1A1) 40 CONTINUE STOP END SUBROUTINE XPROD(A,B,C) DIMENSION A(3),B(3),C(3) A(1)=B(2)*C(3)-B(3)*C(2) A(2)=B(3)*C(1)-B(1)*C(3) A(3)=B(1)*C(2)-B(2)*C(1) RETURN END SUBROUTINE NORM(A,B,C) DIMENSION A(3),B(3),C(3) SQA=SQRT(A(1)**2+A(2)**2+A(3)**2) SQB=SQRT(B(1)**2+B(2)**2+B(3)**2) SQC=SQRT(C(1)**2+C(2)**2+C(3)**2) DO 7 1=1,3 A(I)=A(I)/SQA B(I)=B(I)/SQB 7 C(I)=C(I)/SQC RETURN END PAGE 179 REFERENCES Booker, G. R. , D. C. Joy, J. P. Spencer and H. Graf von Harrach (1974); IITRI, Chicago, 1974, 225. Booker, G. R. and R. Stickler (1962); Brit. J. Appl. Phys. 13 , 446. Booker, G. R. and W. J. Tunstall (1966) ; Phil. Mag. 55_, 71. Broers, A. N. (1969); J. Sci. Instr. 2_, 273. Cowley, J. M. and D. J. Smith (1973); J. Appl. Cryst. 4_, 482. Edington, J. W. (1975); Electron Diffraction in the Electron Microscope (Macmillan, New York) p. 9. Fraser, H. L. , M. H. Loretto, and I. P. Jones (1976); IITRI, Chicago, 1976, 329. Fujimoto, F. K. Komaki and S. Takagi (1972); Z. NaturForsch. 27 , 441. Geiss, R. H. (1975); Appl. Phys. L. 27_, 174. Geiss, R. H. (1976); IITRI, Chicago, 1976, 337. Goldstein, J. I. and D. B. Williams (1977); IITRI, Chicago, 1977, 172. Hall, C. E. (1966); Introduction to Electron Microscopy (McGrawHill, New York) p. 52. Hall, C. R. (1970); Phil. Mag. 60, 63. Head, A. K. , and P. Humble, L. M. Clarbrough, A. J. Morton, C. T. Forwood (1973); Computed Electron Micrographs and Defect Indentification (NorthHolland, New York) . Helfmeier, H. and M. Feller-Kniepmeier (1977); Appl. Phys. L. 48 , 3997. Hirsch, P. B. (1972); Phil. Mag. 26, 193. 173 PAGE 180 174 Hirsch, P. B. and A. Howie, R. B. Nicholson, D. W. Pashly, M. J. Whelan (1965); Electron Microscopy of Thin Cyrstals (Butlerworths , Washington) . Hirth, J. P. and J. Loethe (1968); Theory of Dislocations (McGrawHill, New York), p. 202. Howie, A. and Z. S. Basinski (1968); Phil. Mag. 17_, 1039. Joshi, M. L. (1966); Acta Met. 14_, 1157. Joy, D. C. and D. M. Maher (1975); 33rd EMSA Proc. (G. W. Bailey, Las Vegas) , p. 204. Lifshin, E. , W. G. Morris, and R. B. Bolon (1969); Jour, of Met. 44, 227. Mahajan, S., G. A. Rozgongi, and D. Brasen (1977); Appl. Phys. L. 30, 73. Maher, D. M. and D. C. Joy (1976); Mic. Soc. of Canada 3_, 18. Muraka, S. P. (1977); J. Vac. Sci. Tech. 14, 47. Poole, J. B. (1947); Philips Techn. Rev. 9_, 33. Ravi, K. V. (1974); Phil. Mag. 30_, 1081. Ravi, K. V. (1974); Phil. Mag. 31_, 405. Runyan, B. J. (1965) ; The Metallurgy of Silicon (Texas Instruments Technical Report, Dallas, Texas) . Reimer, L. and P. Aagemann (1976); IITRI, Chicago, 1976, 321. Tholen, A. R. (1970); Phil. Mag. 22, 175. Thomas, L. E. (1972); Phil. Mag. 24, 1447. Thompson, M. N. (1977) ; A Review of TEM Microdif fraction Techniques (Philips Technical Report, Eindhoven, Netherlands). Van Ooostrum, K. J., A. Leenhouts, and A. Jore (1973); Appl. Phys. L. 23_, 283. Whelan, M. J. and P. B. Hirsch (1957); Phil. Mag. 2, 1303. PAGE 181 BIOGRAPHICAL SKETCH John Beverly Warren was born October 29, 1944, in Omaha, Nebraska. He attended Cornell University in Ithaca, New York, and graduated in 1967, with the Bachelor of Science degree in Materials Science. After receiving the Master of Engineering from the Department of Materials Science and Engineering at the University of Florida in 1968, he entered the United States Army and served in Boston and Korea. At the conclusion of military service , the author re-entered the Department of Materials Science and completed his graduate studies in 1976. John Beverly Warren in married to the former Barbara-Jean Panessa of New York City. He is now employed at the Brookhaven National Laboratory. 175 PAGE 182 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. L JxShn J. Hren, Chairman 'rbfessor of Materials Science ind Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Robert E. Reed -Hill Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / Craig S. Hartley Professor of Materials Science and Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. PAGE 183 I certify that I have read this study and that in ray opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Â•5? Y. Martin Eisenberg '' Professor of Engineering Sciences This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June, 1978 lUjoJk Dean, College of Engineering Dean, Graduate School PAGE 184 UNIVERSITY OF FLORIDA 3 1262 08553 2850 |