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Electron resonance reflections from perfect crystal surfaces and surfaces with steps
Ultramicroscopy 27 (1989) 101-112 North-Holland, Amsterdam
101
ELECTRON RESONANCE REFLECTIONS FROM PERFECT CRYSTAL SURFACES AND SURFACES WITH STEPS Z.L. W A N G *, J. LIU, P I N G L U and J.M. C O W L E Y Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA Received 9 August 1988
Dynamical calculations are carried out to investigate the creation processes of reflection waves under the (660) specular reflection case of GaAs (110) in the geometry of reflection high energy electron diffraction (RHEED). It is shown that the resonance waves are localized at the top two to three atomic surface layers. The monolayer resonance characteristic happens only at some specified low angle incidence cases but not in general. Most of the reflection intensity in the R H E E D pattern is created while the electrons are propagating along the surface. The probability of immediate reflections of electrons from a surface is small. This is the reason that the resonance excitation of a surface can greatly enhance the total reflectance of a surface. The propagation of a electron beam at a crystal surface can be characterized by a mean travelling distance, which is 500 to 700 A for GaAs (110). It is pointed out that a column approximation may not be a good treatment for R H E E D calculations. Dynamical calculations for surfaces with steps show that the one-atom-high down-step can critically interrupt the resonance propagation of the resonance wave along the surface. The interrupted wave goes out of the surface and forms some "extra" spots in the RHEED pattern, which are observed in R H E E D experiments in a scanning transmission electron microscope. This confirms the existence of surface resonance waves and their oscillating propagations along crystal surfaces.
1. Introduction Investigations of dynamical electron scattering at a crystal surface under the geometry of reflection high energy electron diffraction ( R H E E D ) have been reported by many authors [1-7]. It has been predicted that there is a resonance wave propagating parallel or nearly parallel to the surface under some specified conditions. It has been found experimentally that the resonance propagation of this wave along the surface can greatly enhance the reflection intensity in the whole R H E E D pattern [8]. However, there is little theoretical calculations carried out to show how the resonance propagation of the electrons along a crystal surface can enhance the reflection intensity. Recently, Cowley has observed the surface resonance wave in the R H E E D pattern obtained
* Present address: Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK.
from an edge of a MgO single crystal [9], which shows the possibility of direct analysis of the surface resonance wave in R H E E D experiments. In this paper, we start from the full dynamical calculation to show how the resonance wave propagates along a crystal surface and enhances the reflection intensity. The predicted "extra" spots in a R H E E D pattern due to the disturbance of a down-step on the surface have been directly observed in R H E E D experiments and so confirm the existence of the surface resonance waves. All the studies in this paper are concentrated on the GaAs (110) surface.
2. Computation methods The multislice theory (elastic scattering theory) has been employed to carry out the calculation on G a A s (110) surfaces. A small beam with width d = 9.965 A is chosen to be incident at a glancing angle on the crystal surface of interest. In the calculations for flat surface cases, the GaAs (110)
0304-3991/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
surface is assumed to have a perfect structure, so that the atomic arrangement at the surface is the same as that in the bulk. No surface reconstruction or relaxation is considered. The multislice calculations are not limited to periodic arrangements of the surface layers and a surface step can be easily introduced at any desired slice position (see ref. [7] for details). The slice thickness is chosen as 2.827 A for the [001] azimuth, the electron propagation distance can be calculated by multiplying the slice number with 2.827 A. The size of the super cell is chosen to be large enough so that the electrons cannot hit the walls of the cell during the propagation. The incident glancing angle of the electrons is chosen as 23.6 mrad, which corresponds to the (660) specular reflection of the GaAs (110) surface after taking into account the refraction effect. The incident beam covers the surface area of 150 slices along the direction of the beam azimuth. The small beam mentioned in this paper means a small size in the direction perpendicular to the surface. This condition is selected for the convenience of the calculations. The finite dimension implies the incident beam is equivalent to a beam one-dimensionally convergent in the direction perpendicular to the surface. The calculations are performed for 100 kV electrons. An absorption factor has been added in the calculation according to the beam free path calculated for plasmon losses [11]. The intensity distributions of the electrons at the surfaces are output in the plane perpendicular to the [001] azimuth at particular slice positions, so that the build-up of the waves at a crystal surface can be seen through the comparison of the intensity distribution at different slices. The calculation results are output for only a few atomic cells near the surface to illustrate the basic scattering involved in the RHEED.
3. Calculation results
3.1. Resonance scattering of electrons from a perfect crystal surface and non-monolayer excitations Dynamical investigations of electron resonance propagations along a perfect crystal surface have
been reported by Wang et al. [6]. It has been shown that the propagation of electrons at a crystal surface is in a form of monolayer resonance between the first atomic layer and the surface potential barrier if the incident angle of the electrons is small, such as for the (440) specular reflection for GaAs (110) surface. In practical R H E E D experiments, however, with the increase of the incident angle the penetration depths of the electrons becomes greater, resulting in strong dynamical diffraction effects. Also the incident directions of the electrons are chosen to have a small azimuthal deviation parallel to the surface so that the (660) and (480) spots are almost equally excited in the (660) specular reflection case of GaAs (110). Due to the introduction of this deviation angle, the electrons are not incident along the atomic rows, and this will be seen to affect the resonance propagations of electrons at crystal surfaces. Fig. 1 shows the calculated electron intensity distributions at a GaAs (110) surface under the specular reflection of (660) with an azimuthal deviation angle ~ = 8.88 mrad. After cutting off the incident wave, the generated waves are divided into three streams. One stream is the transmission wave, which propagates into the crystal. The second stream is the surface resonance wave, which propagates along the surface. The last stream is the reflection wave, which can be seen in the vacuum and propagates away from the surface. It is noticed that the surface resonance wave is concentrated on the top two to three atomic layers instead of one monolayer. This calculation suggests that the monolayer resonance propagation of electrons happens only under some specific conditions. Generally speaking, the electron resonance is not concentrated on one monolayer if the incident angle is reasonably large. The transmission-reflection high energy electron diffraction ( T R H E E D ) pattern can be obtained by taking the Fourier transform of the wave function. The corresponding transmission R H E E D ( T R H E E D ) patterns for the waves displayed in fig. 1E are shown in fig. 1F. The intensity distribution in the spots along the line perpendicular to the surface is shown in fig. 2. Due to the convergent incidence of the small beam, the diffraction intensity in a spot does not
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Fig. 1. Calculated electron intensity distributions at a perfect GaAs (110) surface at different slices (as indicated at the corner of each output). The beam azimuth is [001], 0 = 23.5 mrad and ~ = 8.88 mrad. The projected atomic model is shown at the bottom of the output. a p p e a r to be of u n i f o r m d i s t r i b u t i o n . In the R H E E D case, one is p a r t i c u l a r l y interested in the reflected waves rather than the t r a n s m i s s i o n waves. To show the R H E E D pattern, an intensity filter is a d d e d to filter a w a y the waves p r o p a g a t i n g inside of the crystal. The r e m a i n i n g intensity is shown with a solid line in fig. 2. It is noticed that the oscillating tail of the (660) spot has been filtered away. R e s o n a n c e p r o p a g a t i o n s of electrons at a crystal surface can e n h a n c e the reflection intensities. Fig. 3A shows the c a l c u l a t e d total reflectance of the electrons from a G a A s (110) surface versus the p r o p a g a t i o n d i s t a n c e along b e a m direction. T h e i n c i d e n t b e a m is cut off at the 150th slice. A n i m p o r t a n t feature shown in fig. 3A is that the reflection intensity increases a l m o s t linearly after the b e a m is cut off. This indicates that the reson a n c e p r o p a g a t i o n of the electrons along the surface can generate strong reflection waves. The reflectance has b e e n increased b y a factor of two
103
at the 480th slice c o m p a r e d with that at 150th slice. T h e r e seems to exist a turning p o i n t at which the reflection rate (dR/dz) (see fig. 3B) decreases significantly. This p o i n t h a p p e n s at a b o u t the 300th slice. T h e m e a n travelling distance of the electrons along the surface can be e s t i m a t e d from fig. 3B as D = f z d R / f d R . It turns out to be about D=500 A to 700 A. This value is in r e a s o n a b l e a g r e e m e n t with the R E E L S measurem e n t s [10,11]. T h e r a p i d increase of dR/dz before c u t t i n g the i n c i d e n t b e a m (150th slice) is due to the c o n t r i b u t i o n s of the i m m e d i a t e reflection a n d the r e s o n a n c e reflection. A f t e r cutting off the incident b e a m , the o n l y c o n t r i b u t i o n to the reflectance is the r e s o n a n c e reflection. This is the reason that the dR/dz decreases after the 150th slice. The physical m e a n i n g of this D is the distance which most of the electrons travel before being scattered into vacuum. The i n t r o d u c t i o n of the c o n c e p t of the m e a n travelling d i s t a n c e is basically in a g r e e m e n t with the p r e d i c t e d results of K a m b e from the p o i n t view of optics [12]. A c t u a l l y , fig. 3A shows the v a l i d i t y of the G o o s e H~inchen effect in R H E E D . A n o t h e r effect shown in fig. 3 is that the contrib u t i o n of the " d i r e c t " reflections, i.e. the imm e d i a t e reflection of the electrons from the surface, is small. M o s t of the incident electrons travel a l o n g the surface for some distance before b e i n g scattered back into vacuum. This calculation result c o n f i r m s that the c o l u m n a p p r o x i m a t i o n is
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3.2. Electron resonance propagations at an imperfect crystal surface
Dynamical investigations of electron channeling at stepped crystal surfaces have been reported by W a n g [7]. The interruption of the surface resonance wave by a surface step has been proposed as the origin of double contrast contour observed in R E M images. A further study of the calculated R H E E D patterns involving the effect of disturbance by the surface steps will be described in this section. Fig. 4 shows a comparison of the calculated electron intensity distributions at a crystal surface without surface steps (no step, left column), with a step up (middle column) at the 200th slice, and with a step down (right column) at the 200th slice. The incident beam illuminates the first 150 slices. The step height is equal to one layer of atoms or 1.999 ,~. The electron intensities are output at the same thickness under the same incident conditions, so that the perturbation of the resonance wave by the surface steps can be clarified. C o m paring the intensity output in the left and the middle columns shows that a step up at 200th slice will not critically affect the propagation of the resonance wave. The resonance wave continues travelling along the " o l d " surface from which it was generated. The down step at the 200th slice
can effectively interrupt the propagation of the resonance wave along the surface (see figs. 4C and 4D, right column). The propagation tendency of the interrupted surface wave can be seen through the direction of shift of the intensity bands in the v a c u u m (right column fig. 4E). The propagation of the interrupted surface resonance waves in vacuum can be seen in the calculated T R H E E D pattern in fig. 4F, which introduces some "extra" spots (fig. 4F, right corner). This may be due to the fact that the propagation directions of the surface resonance waves are perturbed by the potential fields around the step edge. It is possible that the travelling direction of the interrupted wave is different from that of the Bragg scattering. The steps can significantly affect the reflectance of a surface. Fig. 5 shows a comparison of the line scan intensities across the (660) specular reflection spot for the three cases as illustrated in fig. 4. The surface with a down step seems to have largest reflectance, which is contributed from the interrupted surface resonance wave. The surface with an up step seems to be weak in reflectance, which is due to the blocking effect of the up step to the reflected wave; part of the blocked waves will be scattered back into the crystal by the step edge. Also it is indicated that the positions of the intensity peaks are slightly influenced by the surface steps. The differences in reflection intensities from a perfect crystal surface and a surface with a down step can be seen through a comparison of the
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Zig. 4. C o m p a r i s o n s of the calculated electron intensity d i s t r i b u t i o n s at different slice n u m b e r s at a G a A s (110) surface w i t h o u t steps (left column), w i t h a step up at 100th slice (middle column) a n d with a step d o w n (right column) at 200th slice. W i t h 0 = 23.5 m r a d and ,# = 8.88 m r a d a n d b e a m a z i m u t h [001]. The b e a m cut-off h a p p e n s at the 150th slice. (F) is the c a l c u l a t e d T R H E E D p a t t e r n s for the three cases.
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
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calculated R H E E D patterns, as shown in fig. 6. In the case of a perfect flat surface, the reflection intensities are mainly concentrated at the (660) and the (480) reflections. When a one-atom-high down step is introduced at 565.4 A away from the beam cut-off position, besides the regular (480)
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and (660) reflection spots an "extra" spot begins to appear at the arrowed position in fig. 6B. If the down step is set at 226.2 ,~ away from the beam cut-off, this extra spot becomes comparatively strong and splits into two. The splitting of this spot in the direction perpendicular to the surface shows that this wave comes from the small regions of the top few surface layers (fig. 6C). When the down step moves to 141.4 ,~ away from the beam cut-off, the "extra" spot becomes a single spot and is still very strong (fig. 6D). According to this calculation, the predicted "extra" spot should be seen in R H E E D patterns while scanning a narrow beam across a surface down step. The formation of this "extra" spot can be considered as the propagation of the resonance waves in vacuum after being interrupted by the step. The analysis of this wave should give the excitation information about the surface top few layers. Line scan intensity profiles across the calculated (660) spot for the several cases addressed in fig. 6 show that the position of the down step can significantly affect the intensity distribution in the (660) disc. The total intensity in the (660) disk can be considered to be composed of two parts. One part is the "direct" reflection, which should not be affected by the appearance of the step. The other is the resonance reflection, which is critically affected by the step position. This argument is in agreement with the early investigations [7,13]. The predicted "extra" spot can be split into two at some specified step positions. This splitting effect suggests that the surface resonance wave consists of two main beams, which could propagate in slightly different directions.
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Fig. 6. Calculated R H E E D patterns of GaAs (110) surfaces. (A) is from a perfect surface. (B), (C) and (D) are the calculations for GaAs (110) surfaces with a step down at 350th, 230th and 200th slices, respectively. The beam cut off happens at 150th slice.
3.3. Surface potential trapped resonance waves The formation of surface resonance waves at a crystal surface is critically affected by the incident conditions of an external beam as well as the surface structures. Fig. 7 shows a calculated electron intensity distribution for a 9.965 A beam with an incident angle of 23.5 mrad and q5 = 0 at GaAs (110). The beam cut-off is at the 150th slice. It is noticed that a strong resonance wave is propagating along the surface potential barrier between the G a and As atomic rows in the first layer. This
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wave propagates continuously along the potential barrier without any deflection. The decrease in intensity is due to the absorption effect. It is important to note that this wave is not channeling along the atomic rows but is on the vacuum side of the top layer of atoms. This phenomenon can be named as the surface potential trapped resonance propagation. The wave trapping propagation is very sensitive to the beam incident conditions. When the incident angle of the electrons is increased by 3 mrad (i.e. 0 = 0.26.5 mrad), this trapping wave disappears. When the incident beam size is increased to 20 A, this wave is not seen. It can be said that this trapping wave is a result of wave interference, which can be generated only under some specific conditions.
3.4. Steady state in R H E E D calculations
All the above investigations are carried out for a small incident beam, which introduces some beam size effects in the calculations. It is expected that, for infinite wide beam incidence, the distribution of the electrons at the crystal surface will not be affected by the beam size after the output position is beyond the mean travelling distance of the electrons at the surface. So the wave function should have a periodicity equal to the periodicity of the atomic unit cell (5.654 A) except for a constant phase difference. This state is named as the steady state. To show that this steady state can be simulated by using the multislice theory, a beam with size 80 in width perpendicular to the surface was cho-
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
108
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Fig. 8. Calculated steady state propagation of electrons at a GaAs (110) surface in the RHEED geometry. With 0 = 33.1 mrad and q~= 0. There is no beam cut off in the output regions.
at the 400th slice is identical to that at the 402nd slice but not to that at the 401st slice. The intensity distribution at the 401st slice is identical to that at the 403rd slice but not to that at the 402nd slice. Thus t he electron intensity distribution does have the periodicity of the atomic unit cell (2 slices = 5.654 A) in the direction of the beam. To confirm the calculation results of figs. 8A to 8D, the intensities at the 600th and 601th slices are shown in figs. 8E and 8F. Compared with the output at the 400th and the 401st slices, the identical intensity distributions have been repeated in 600th and 601st slices respectively. Therefore, these calculation results show that the steady state can be achieved in the calculation by using the multislice theory. Fig. 8 tells us that the calculated non-steady state results in the last few sections are due to small beam effects, and can conveniently be observed in a scanning microscope with an electron probe of small diameter. This will be illustrated in the next section. The appearance of the steady state happens at about the 300th slice. This is in reasonable agreement with the calculation in fig. 1, because the steady state can happen only after the detection position is at a distance from the first slice longer than the electron mean travelling distance, so that the perturbation effect of the sharp window edge of the incident wave becomes insignificant. For a large beam incident case, the total reflection intensity increases almost linearly after arriving at the steady state.
4. R H E E D studies of surface resonance in GaAs
sen to illuminate a surface area up to 2416.9 (855 slice for 0 = 33.1 mrad) along the beam direction, which is much larger than the mean travelling distance of the electron as predicted according to fig. 2. The electron intensity distributions are output at several successive slices to see the repeatability of the wave fields. Since the atomic unit cell of the GaAs crystal is cut into two slices, each of them having different atomic arrangement, it is expected that the intensity distribution of the electrons should repeat every other slice. Fig. 8 shows the calculated results. Detailed comparison shows that the intensity distribution
(110)
4.1. Experimental method Microdiffraction and scanning reflection electron microscopy (SREM) experiments were performed in a V G HB5 scanning transmission electron microscope (STEM), with a dedicated optical system which is capable of showing the microdiffraction patterns and the surface reflection images simultaneously [14]. The vacuum status at the specimen stage was about 10-8 Torr. The incident beam was chosen as 5 A in diameter in order to
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
Fig. 9. Microdiffraction R H E E D patterns with a 5 ~, beam scanning across a down step at GaAs (110) surface. (A) is obtained when the beam is far away from the step; (B) is obtained when the beam approaches the step; (C) is obtained when the beam is very close to the step and (D) is obtained just after the beam crosses the step.
study the small beam reflection effect from a surface. The diffraction patterns were recorded with a video system at TV rate while the beam was scanning across a step. The change of the R H E E D pattern with the change of the beam position with respect to the surface step can be observed directly. The scanning direction is perpendicular to the step direction. The smallest detector aperture was used to obtain good contrast in SREM images. The heights of the surface steps are predicted to be a few atomic sizes, because some of them are directly connected to the cores of screw dislocations [15]. The GaAs (110) surface was obtained by cleaving a bulk GaAs crystal in air and it was mounted on a double tilting cartridge. The microscope was operated at 100 kV with beam current 104 A / c m 2. The same type of experiment has been made for MgO cleavage faces by Cowley and Crozier [9].
109
existence of the "extra" spot as predicted in fig. 6. Only the (660) and (480) reflections are observed. When the b e a m is approaching the step, an "extra" spot appears near the edge of the surface, as arrowed in fig. 9B. While the beam moves very close to the step, this spot becomes very strong (fig. 9C). The location of this spot is close to the one predicted in fig. 6B. This spot disappears suddenly just after the beam crosses the step (fig. 9D). To confirm that the observed "extra" spot comes from the surface step, SREM images were taken by selecting the (660) and the "extra" spots, as shown in fig. 10. The step structures of the surface can be seen. Fairly good contrast can be observed at the steps but otherwise the surface seems to be atomically flat. In the scanning reflection image (SREM) taken from the "extra" spot, very strong bright lines are seen to be present, which indicates that the observed "extra" spot comes from the contribution of the surface steps. The position of the "extra" spot changes with the change of the incident beam position, as shown
4.2. Experimental results Fig. 9 shows a series of R H E E D patterns made while the b e a m is scanning across a surface down step. When the beam is far away from a down step (fig. 9A), there is almost no indication of the
Fig. 10. SREM images of the G a A s (110) surface examined in fig. 9. (A) and (B) are the surface images obtained from the (660) and the "extra" spot respectively.
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Fig. 11. Microdiffraction patterns show the variation of the positions of the "extra" spot while the beam position is changing, in fig. 11. The a m o u n t of shift of the "extra" spot is measured to be 0.18 ,~-a, which is about 6.4 mrad. As reported earlier [6], the propagation of the resonance waves can be considered simply as an oscillation between the atomic planes and the surface potential barrier, which means that the propagation direction of the resonance wave after being interrupted by a step depends on the position of the step with respect to the beam incidence. Therefore, the observed change of spot position in fig. 11 can be considered as due to the oscillatory propagation of the resonance wave at the surface. The "extra" spot can also be doubly split, as in fig. 12B. The pattern in fig. 12A is very close to the one calculated in fig. 6D. The splitting direction of the spot can be not only parallel to the surface but also perpendicular to the surface. This can be interpreted as the effect of beam convergence, which creates different surface resonance waves propagating in slightly different directions. Also it is shown that the resonance wave is not a single beam, but consists of two beams propagating in different directions. The calculated double split spots in fig. 6C are due to the onedimensional convergence of the incident b e a m postulated in the theoretical calculation. B
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Fig. 12. Microdiffraction patterns show the splitting effect of the "extra" spot at some definite incident positions with respect to the step.
The predicted " e x t r a " spot can also be observed for the (880) specular reflection case, as in fig. 13. W h e n the b e a m is far away from the step, no strong "extra" spot is observed near the crystal edge. While the b e a m moves close to the step, a strong spot is seen near the crystal edge (fig. 13A). Again, this spot disappears after the beam crosses the step. Similar S R E M images have been taken from the (880) and the " e x t r a " spots. The observation results are the same as in fig. 10. The R H E E D experiments show that an "extra" spot does appear near the edge of the crystal while the b e a m is scanning across a surface d o w n step. This observation agrees with the calculation results in fig. 6. This shows that the surface resonance wave does exist at the crystal surface top two to three atomic layers, as expected from the multislice calculations.
5. D i s c u s s i o n and conclusions
D y n a m i c a l diffraction calculations for electron propagation along a crystal surface have shown that the characteristic m o n o l a y e r resonance happens only at some specific conditions. Generally,
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Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
two to three atomic layers resonance excitations are expected. In R H E E D geometry, most of the reflection intensity is created from the resonance waves propagating along the surface. This is the reason why the intensity in a R H E E D pattern is enhanced greatly while the surface resonance conditions are satisfied. In surface non-resonance conditions, there is no resonance wave at the crystal surface and most of the electrons will penetrate into the crystal. Then the main process which contributes to the creation of reflection waves is the "direct" reflection of the electrons. The intensity created in this process is much weaker than that created from the resonance propagation. This is the reason why the reflection intensity is weak under the non-resonance conditions. The propagation of resonance electrons parallel to the surface can be characterized by a mean travelling distance, along which most of the electrons travel before being reflected back into vacuum. It is predicted through the calculation that this distance is about 500 to 750 ,~ for GaAs (660) reflection. This result shows that any calculation for R H E E D or R E M based on either column approximation or kinematic reflection might not reflect the real interaction of the incident electrons with surfaces. A full dynamical calculation is required to interpret the observed phenomena in R H E E D and REM. The possibility of achieving steady state propagation of electrons in R H E E D through the multislice calculation has been demonstrated. This establishes the basis for applying this method for R H E E D and R E M calculations. Any surface defect and imperfection can be simulated in this approach. It has been postulated recently by Wang that the inelastic scattering can be introduced in the multislice calculation by using the concept of an "effective" potential [16,17], which makes it possible to calculate the energy-filtered inelastic R H E E D patterns and R E M images [18]. The interruption of resonance waves by surface steps has been predicted and observed in scanning beam experiments. It is proved that the surface resonance wave does exist in a crystal surface under the R H E E D geometry. This wave is found to consist of two main beams propagating in
111
slightly different directions. The observed direction of this wave depends on the position of a down step with respect to the incident beam. Since this wave is localized in the top few surface layers, it is expected that the surface excitations can be detected sensitively from the analysis of this beam. The high sensitivity of the R E M technique to the atom-high surface steps in R E M could be attributed to the perturbation of the propagation of the resonance waves by the steps, resulting in a significant perturbation to the specular reflections. The surface down steps affect not only the phases but also the propagating directions to the resonance waves compared to the Bragg reflections.
Acknowledgements This work was supported by the office of Naval Research under grant N00014-86-k-0319 (MAPS program) and N S F G r a n t DMR-8514583 (P.L.) and made use of the resources of the ASU Facility for High Resolution Electron Microscopy, supported by N S F grant D M R 86-11609. The authors are grateful to Dr. D.J. Smith for his constant interest in this work.
References [1] R. Colella, Act Cryst. A28 (1972) 11. [2] K. Britze and G. Meyer-Ehmsen, Surface Sci. 77 (1978) 131. [3] N. Masud and J.B. Pendry, J. Phys. C9 (1976) 1833. [4] H. Marten and G. Meyer-Ehmsen, Surface Sci. 151 (1985) 570, [5] L.M. Peng and J.M. Cowley, Acta Cryst. A42 (1986) 545. [6] Z.L. Wang, P. Lu and J.M. Cowley, Ultramicroscopy 23 (1987) 205. [7] Z.L. Wang, Ultramicroscopy 24 (1988) 371. [8] S. Miyake, K. Kohra and M. Takagi, Acta Cryst. 7 (1954) 393. [9] J.M. Cowley and P. Crozier, in: Proc. 46th Annual EMSA Meeting, Milwaukee, WI, 1988, Ed. G.W. Bailey (San Francisco Press, San Francisco, CA, 1988) p. 692. [10] Z.L. Wang and R.F. Egerton, Surface Sci. 205 (1988) 25. [11] Z.L. Wang, Ultramicroscopy 26 (1988) 321. [12] K. Kambe, Ultramicroscopy 25 (1988) 259.
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[13] G. Lehmpfuhl and Y. Uchida, in: Proc. 44th Annual EMSA Meeting, Albuquerque, NM, 1986, Ed. G.W. Bailey (San Francisco Press, San Francisco, CA, 1986) p. 376. [14] J.M. Cowley, J. Electron Microscopy Tech., in press. [15] R.H. Milne, A. Howie and M.G. Walls, submitted.
[16] Z.L. Wang and P. Lu, Ultramicroscopy 26 (1988) 217. [17] Z.L. Wang, in: Proc. 46th Annual EMSA Meeting, Milwaukee, WI, 1988, Ed. G.W. Bailey (San Francisco Press, San Francisco, CA, 1988) p. 818. [18] Z.L. Wang, Acta Cryst. A, in press.
101
ELECTRON RESONANCE REFLECTIONS FROM PERFECT CRYSTAL SURFACES AND SURFACES WITH STEPS Z.L. W A N G *, J. LIU, P I N G L U and J.M. C O W L E Y Department of Physics, Arizona State University, Tempe, Arizona 85287-1504, USA Received 9 August 1988
Dynamical calculations are carried out to investigate the creation processes of reflection waves under the (660) specular reflection case of GaAs (110) in the geometry of reflection high energy electron diffraction (RHEED). It is shown that the resonance waves are localized at the top two to three atomic surface layers. The monolayer resonance characteristic happens only at some specified low angle incidence cases but not in general. Most of the reflection intensity in the R H E E D pattern is created while the electrons are propagating along the surface. The probability of immediate reflections of electrons from a surface is small. This is the reason that the resonance excitation of a surface can greatly enhance the total reflectance of a surface. The propagation of a electron beam at a crystal surface can be characterized by a mean travelling distance, which is 500 to 700 A for GaAs (110). It is pointed out that a column approximation may not be a good treatment for R H E E D calculations. Dynamical calculations for surfaces with steps show that the one-atom-high down-step can critically interrupt the resonance propagation of the resonance wave along the surface. The interrupted wave goes out of the surface and forms some "extra" spots in the RHEED pattern, which are observed in R H E E D experiments in a scanning transmission electron microscope. This confirms the existence of surface resonance waves and their oscillating propagations along crystal surfaces.
1. Introduction Investigations of dynamical electron scattering at a crystal surface under the geometry of reflection high energy electron diffraction ( R H E E D ) have been reported by many authors [1-7]. It has been predicted that there is a resonance wave propagating parallel or nearly parallel to the surface under some specified conditions. It has been found experimentally that the resonance propagation of this wave along the surface can greatly enhance the reflection intensity in the whole R H E E D pattern [8]. However, there is little theoretical calculations carried out to show how the resonance propagation of the electrons along a crystal surface can enhance the reflection intensity. Recently, Cowley has observed the surface resonance wave in the R H E E D pattern obtained
* Present address: Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UK.
from an edge of a MgO single crystal [9], which shows the possibility of direct analysis of the surface resonance wave in R H E E D experiments. In this paper, we start from the full dynamical calculation to show how the resonance wave propagates along a crystal surface and enhances the reflection intensity. The predicted "extra" spots in a R H E E D pattern due to the disturbance of a down-step on the surface have been directly observed in R H E E D experiments and so confirm the existence of the surface resonance waves. All the studies in this paper are concentrated on the GaAs (110) surface.
2. Computation methods The multislice theory (elastic scattering theory) has been employed to carry out the calculation on G a A s (110) surfaces. A small beam with width d = 9.965 A is chosen to be incident at a glancing angle on the crystal surface of interest. In the calculations for flat surface cases, the GaAs (110)
0304-3991/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
surface is assumed to have a perfect structure, so that the atomic arrangement at the surface is the same as that in the bulk. No surface reconstruction or relaxation is considered. The multislice calculations are not limited to periodic arrangements of the surface layers and a surface step can be easily introduced at any desired slice position (see ref. [7] for details). The slice thickness is chosen as 2.827 A for the [001] azimuth, the electron propagation distance can be calculated by multiplying the slice number with 2.827 A. The size of the super cell is chosen to be large enough so that the electrons cannot hit the walls of the cell during the propagation. The incident glancing angle of the electrons is chosen as 23.6 mrad, which corresponds to the (660) specular reflection of the GaAs (110) surface after taking into account the refraction effect. The incident beam covers the surface area of 150 slices along the direction of the beam azimuth. The small beam mentioned in this paper means a small size in the direction perpendicular to the surface. This condition is selected for the convenience of the calculations. The finite dimension implies the incident beam is equivalent to a beam one-dimensionally convergent in the direction perpendicular to the surface. The calculations are performed for 100 kV electrons. An absorption factor has been added in the calculation according to the beam free path calculated for plasmon losses [11]. The intensity distributions of the electrons at the surfaces are output in the plane perpendicular to the [001] azimuth at particular slice positions, so that the build-up of the waves at a crystal surface can be seen through the comparison of the intensity distribution at different slices. The calculation results are output for only a few atomic cells near the surface to illustrate the basic scattering involved in the RHEED.
3. Calculation results
3.1. Resonance scattering of electrons from a perfect crystal surface and non-monolayer excitations Dynamical investigations of electron resonance propagations along a perfect crystal surface have
been reported by Wang et al. [6]. It has been shown that the propagation of electrons at a crystal surface is in a form of monolayer resonance between the first atomic layer and the surface potential barrier if the incident angle of the electrons is small, such as for the (440) specular reflection for GaAs (110) surface. In practical R H E E D experiments, however, with the increase of the incident angle the penetration depths of the electrons becomes greater, resulting in strong dynamical diffraction effects. Also the incident directions of the electrons are chosen to have a small azimuthal deviation parallel to the surface so that the (660) and (480) spots are almost equally excited in the (660) specular reflection case of GaAs (110). Due to the introduction of this deviation angle, the electrons are not incident along the atomic rows, and this will be seen to affect the resonance propagations of electrons at crystal surfaces. Fig. 1 shows the calculated electron intensity distributions at a GaAs (110) surface under the specular reflection of (660) with an azimuthal deviation angle ~ = 8.88 mrad. After cutting off the incident wave, the generated waves are divided into three streams. One stream is the transmission wave, which propagates into the crystal. The second stream is the surface resonance wave, which propagates along the surface. The last stream is the reflection wave, which can be seen in the vacuum and propagates away from the surface. It is noticed that the surface resonance wave is concentrated on the top two to three atomic layers instead of one monolayer. This calculation suggests that the monolayer resonance propagation of electrons happens only under some specific conditions. Generally speaking, the electron resonance is not concentrated on one monolayer if the incident angle is reasonably large. The transmission-reflection high energy electron diffraction ( T R H E E D ) pattern can be obtained by taking the Fourier transform of the wave function. The corresponding transmission R H E E D ( T R H E E D ) patterns for the waves displayed in fig. 1E are shown in fig. 1F. The intensity distribution in the spots along the line perpendicular to the surface is shown in fig. 2. Due to the convergent incidence of the small beam, the diffraction intensity in a spot does not
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103
at the 480th slice c o m p a r e d with that at 150th slice. T h e r e seems to exist a turning p o i n t at which the reflection rate (dR/dz) (see fig. 3B) decreases significantly. This p o i n t h a p p e n s at a b o u t the 300th slice. T h e m e a n travelling distance of the electrons along the surface can be e s t i m a t e d from fig. 3B as D = f z d R / f d R . It turns out to be about D=500 A to 700 A. This value is in r e a s o n a b l e a g r e e m e n t with the R E E L S measurem e n t s [10,11]. T h e r a p i d increase of dR/dz before c u t t i n g the i n c i d e n t b e a m (150th slice) is due to the c o n t r i b u t i o n s of the i m m e d i a t e reflection a n d the r e s o n a n c e reflection. A f t e r cutting off the incident b e a m , the o n l y c o n t r i b u t i o n to the reflectance is the r e s o n a n c e reflection. This is the reason that the dR/dz decreases after the 150th slice. The physical m e a n i n g of this D is the distance which most of the electrons travel before being scattered into vacuum. The i n t r o d u c t i o n of the c o n c e p t of the m e a n travelling d i s t a n c e is basically in a g r e e m e n t with the p r e d i c t e d results of K a m b e from the p o i n t view of optics [12]. A c t u a l l y , fig. 3A shows the v a l i d i t y of the G o o s e H~inchen effect in R H E E D . A n o t h e r effect shown in fig. 3 is that the contrib u t i o n of the " d i r e c t " reflections, i.e. the imm e d i a t e reflection of the electrons from the surface, is small. M o s t of the incident electrons travel a l o n g the surface for some distance before b e i n g scattered back into vacuum. This calculation result c o n f i r m s that the c o l u m n a p p r o x i m a t i o n is
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3.2. Electron resonance propagations at an imperfect crystal surface
Dynamical investigations of electron channeling at stepped crystal surfaces have been reported by W a n g [7]. The interruption of the surface resonance wave by a surface step has been proposed as the origin of double contrast contour observed in R E M images. A further study of the calculated R H E E D patterns involving the effect of disturbance by the surface steps will be described in this section. Fig. 4 shows a comparison of the calculated electron intensity distributions at a crystal surface without surface steps (no step, left column), with a step up (middle column) at the 200th slice, and with a step down (right column) at the 200th slice. The incident beam illuminates the first 150 slices. The step height is equal to one layer of atoms or 1.999 ,~. The electron intensities are output at the same thickness under the same incident conditions, so that the perturbation of the resonance wave by the surface steps can be clarified. C o m paring the intensity output in the left and the middle columns shows that a step up at 200th slice will not critically affect the propagation of the resonance wave. The resonance wave continues travelling along the " o l d " surface from which it was generated. The down step at the 200th slice
can effectively interrupt the propagation of the resonance wave along the surface (see figs. 4C and 4D, right column). The propagation tendency of the interrupted surface wave can be seen through the direction of shift of the intensity bands in the v a c u u m (right column fig. 4E). The propagation of the interrupted surface resonance waves in vacuum can be seen in the calculated T R H E E D pattern in fig. 4F, which introduces some "extra" spots (fig. 4F, right corner). This may be due to the fact that the propagation directions of the surface resonance waves are perturbed by the potential fields around the step edge. It is possible that the travelling direction of the interrupted wave is different from that of the Bragg scattering. The steps can significantly affect the reflectance of a surface. Fig. 5 shows a comparison of the line scan intensities across the (660) specular reflection spot for the three cases as illustrated in fig. 4. The surface with a down step seems to have largest reflectance, which is contributed from the interrupted surface resonance wave. The surface with an up step seems to be weak in reflectance, which is due to the blocking effect of the up step to the reflected wave; part of the blocked waves will be scattered back into the crystal by the step edge. Also it is indicated that the positions of the intensity peaks are slightly influenced by the surface steps. The differences in reflection intensities from a perfect crystal surface and a surface with a down step can be seen through a comparison of the
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calculated R H E E D patterns, as shown in fig. 6. In the case of a perfect flat surface, the reflection intensities are mainly concentrated at the (660) and the (480) reflections. When a one-atom-high down step is introduced at 565.4 A away from the beam cut-off position, besides the regular (480)
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and (660) reflection spots an "extra" spot begins to appear at the arrowed position in fig. 6B. If the down step is set at 226.2 ,~ away from the beam cut-off, this extra spot becomes comparatively strong and splits into two. The splitting of this spot in the direction perpendicular to the surface shows that this wave comes from the small regions of the top few surface layers (fig. 6C). When the down step moves to 141.4 ,~ away from the beam cut-off, the "extra" spot becomes a single spot and is still very strong (fig. 6D). According to this calculation, the predicted "extra" spot should be seen in R H E E D patterns while scanning a narrow beam across a surface down step. The formation of this "extra" spot can be considered as the propagation of the resonance waves in vacuum after being interrupted by the step. The analysis of this wave should give the excitation information about the surface top few layers. Line scan intensity profiles across the calculated (660) spot for the several cases addressed in fig. 6 show that the position of the down step can significantly affect the intensity distribution in the (660) disc. The total intensity in the (660) disk can be considered to be composed of two parts. One part is the "direct" reflection, which should not be affected by the appearance of the step. The other is the resonance reflection, which is critically affected by the step position. This argument is in agreement with the early investigations [7,13]. The predicted "extra" spot can be split into two at some specified step positions. This splitting effect suggests that the surface resonance wave consists of two main beams, which could propagate in slightly different directions.
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3.3. Surface potential trapped resonance waves The formation of surface resonance waves at a crystal surface is critically affected by the incident conditions of an external beam as well as the surface structures. Fig. 7 shows a calculated electron intensity distribution for a 9.965 A beam with an incident angle of 23.5 mrad and q5 = 0 at GaAs (110). The beam cut-off is at the 150th slice. It is noticed that a strong resonance wave is propagating along the surface potential barrier between the G a and As atomic rows in the first layer. This
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wave propagates continuously along the potential barrier without any deflection. The decrease in intensity is due to the absorption effect. It is important to note that this wave is not channeling along the atomic rows but is on the vacuum side of the top layer of atoms. This phenomenon can be named as the surface potential trapped resonance propagation. The wave trapping propagation is very sensitive to the beam incident conditions. When the incident angle of the electrons is increased by 3 mrad (i.e. 0 = 0.26.5 mrad), this trapping wave disappears. When the incident beam size is increased to 20 A, this wave is not seen. It can be said that this trapping wave is a result of wave interference, which can be generated only under some specific conditions.
3.4. Steady state in R H E E D calculations
All the above investigations are carried out for a small incident beam, which introduces some beam size effects in the calculations. It is expected that, for infinite wide beam incidence, the distribution of the electrons at the crystal surface will not be affected by the beam size after the output position is beyond the mean travelling distance of the electrons at the surface. So the wave function should have a periodicity equal to the periodicity of the atomic unit cell (5.654 A) except for a constant phase difference. This state is named as the steady state. To show that this steady state can be simulated by using the multislice theory, a beam with size 80 in width perpendicular to the surface was cho-
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
108
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Fig. 8. Calculated steady state propagation of electrons at a GaAs (110) surface in the RHEED geometry. With 0 = 33.1 mrad and q~= 0. There is no beam cut off in the output regions.
at the 400th slice is identical to that at the 402nd slice but not to that at the 401st slice. The intensity distribution at the 401st slice is identical to that at the 403rd slice but not to that at the 402nd slice. Thus t he electron intensity distribution does have the periodicity of the atomic unit cell (2 slices = 5.654 A) in the direction of the beam. To confirm the calculation results of figs. 8A to 8D, the intensities at the 600th and 601th slices are shown in figs. 8E and 8F. Compared with the output at the 400th and the 401st slices, the identical intensity distributions have been repeated in 600th and 601st slices respectively. Therefore, these calculation results show that the steady state can be achieved in the calculation by using the multislice theory. Fig. 8 tells us that the calculated non-steady state results in the last few sections are due to small beam effects, and can conveniently be observed in a scanning microscope with an electron probe of small diameter. This will be illustrated in the next section. The appearance of the steady state happens at about the 300th slice. This is in reasonable agreement with the calculation in fig. 1, because the steady state can happen only after the detection position is at a distance from the first slice longer than the electron mean travelling distance, so that the perturbation effect of the sharp window edge of the incident wave becomes insignificant. For a large beam incident case, the total reflection intensity increases almost linearly after arriving at the steady state.
4. R H E E D studies of surface resonance in GaAs
sen to illuminate a surface area up to 2416.9 (855 slice for 0 = 33.1 mrad) along the beam direction, which is much larger than the mean travelling distance of the electron as predicted according to fig. 2. The electron intensity distributions are output at several successive slices to see the repeatability of the wave fields. Since the atomic unit cell of the GaAs crystal is cut into two slices, each of them having different atomic arrangement, it is expected that the intensity distribution of the electrons should repeat every other slice. Fig. 8 shows the calculated results. Detailed comparison shows that the intensity distribution
(110)
4.1. Experimental method Microdiffraction and scanning reflection electron microscopy (SREM) experiments were performed in a V G HB5 scanning transmission electron microscope (STEM), with a dedicated optical system which is capable of showing the microdiffraction patterns and the surface reflection images simultaneously [14]. The vacuum status at the specimen stage was about 10-8 Torr. The incident beam was chosen as 5 A in diameter in order to
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
Fig. 9. Microdiffraction R H E E D patterns with a 5 ~, beam scanning across a down step at GaAs (110) surface. (A) is obtained when the beam is far away from the step; (B) is obtained when the beam approaches the step; (C) is obtained when the beam is very close to the step and (D) is obtained just after the beam crosses the step.
study the small beam reflection effect from a surface. The diffraction patterns were recorded with a video system at TV rate while the beam was scanning across a step. The change of the R H E E D pattern with the change of the beam position with respect to the surface step can be observed directly. The scanning direction is perpendicular to the step direction. The smallest detector aperture was used to obtain good contrast in SREM images. The heights of the surface steps are predicted to be a few atomic sizes, because some of them are directly connected to the cores of screw dislocations [15]. The GaAs (110) surface was obtained by cleaving a bulk GaAs crystal in air and it was mounted on a double tilting cartridge. The microscope was operated at 100 kV with beam current 104 A / c m 2. The same type of experiment has been made for MgO cleavage faces by Cowley and Crozier [9].
109
existence of the "extra" spot as predicted in fig. 6. Only the (660) and (480) reflections are observed. When the b e a m is approaching the step, an "extra" spot appears near the edge of the surface, as arrowed in fig. 9B. While the beam moves very close to the step, this spot becomes very strong (fig. 9C). The location of this spot is close to the one predicted in fig. 6B. This spot disappears suddenly just after the beam crosses the step (fig. 9D). To confirm that the observed "extra" spot comes from the surface step, SREM images were taken by selecting the (660) and the "extra" spots, as shown in fig. 10. The step structures of the surface can be seen. Fairly good contrast can be observed at the steps but otherwise the surface seems to be atomically flat. In the scanning reflection image (SREM) taken from the "extra" spot, very strong bright lines are seen to be present, which indicates that the observed "extra" spot comes from the contribution of the surface steps. The position of the "extra" spot changes with the change of the incident beam position, as shown
4.2. Experimental results Fig. 9 shows a series of R H E E D patterns made while the b e a m is scanning across a surface down step. When the beam is far away from a down step (fig. 9A), there is almost no indication of the
Fig. 10. SREM images of the G a A s (110) surface examined in fig. 9. (A) and (B) are the surface images obtained from the (660) and the "extra" spot respectively.
110
Z.L. Wang et al. / Electron resonance reflections frorn crystal surfaces
[3
B
660 ~:@,
Fig. 11. Microdiffraction patterns show the variation of the positions of the "extra" spot while the beam position is changing, in fig. 11. The a m o u n t of shift of the "extra" spot is measured to be 0.18 ,~-a, which is about 6.4 mrad. As reported earlier [6], the propagation of the resonance waves can be considered simply as an oscillation between the atomic planes and the surface potential barrier, which means that the propagation direction of the resonance wave after being interrupted by a step depends on the position of the step with respect to the beam incidence. Therefore, the observed change of spot position in fig. 11 can be considered as due to the oscillatory propagation of the resonance wave at the surface. The "extra" spot can also be doubly split, as in fig. 12B. The pattern in fig. 12A is very close to the one calculated in fig. 6D. The splitting direction of the spot can be not only parallel to the surface but also perpendicular to the surface. This can be interpreted as the effect of beam convergence, which creates different surface resonance waves propagating in slightly different directions. Also it is shown that the resonance wave is not a single beam, but consists of two beams propagating in different directions. The calculated double split spots in fig. 6C are due to the onedimensional convergence of the incident b e a m postulated in the theoretical calculation. B
A
Fig. 12. Microdiffraction patterns show the splitting effect of the "extra" spot at some definite incident positions with respect to the step.
The predicted " e x t r a " spot can also be observed for the (880) specular reflection case, as in fig. 13. W h e n the b e a m is far away from the step, no strong "extra" spot is observed near the crystal edge. While the b e a m moves close to the step, a strong spot is seen near the crystal edge (fig. 13A). Again, this spot disappears after the beam crosses the step. Similar S R E M images have been taken from the (880) and the " e x t r a " spots. The observation results are the same as in fig. 10. The R H E E D experiments show that an "extra" spot does appear near the edge of the crystal while the b e a m is scanning across a surface d o w n step. This observation agrees with the calculation results in fig. 6. This shows that the surface resonance wave does exist at the crystal surface top two to three atomic layers, as expected from the multislice calculations.
5. D i s c u s s i o n and conclusions
D y n a m i c a l diffraction calculations for electron propagation along a crystal surface have shown that the characteristic m o n o l a y e r resonance happens only at some specific conditions. Generally,
C
880
Fig. 13. Observations of the "extra" spot while the beam is crossing a surface down step at the specular reflection case of (880).
Z.L. Wang et al. / Electron resonance reflections from crystal surfaces
two to three atomic layers resonance excitations are expected. In R H E E D geometry, most of the reflection intensity is created from the resonance waves propagating along the surface. This is the reason why the intensity in a R H E E D pattern is enhanced greatly while the surface resonance conditions are satisfied. In surface non-resonance conditions, there is no resonance wave at the crystal surface and most of the electrons will penetrate into the crystal. Then the main process which contributes to the creation of reflection waves is the "direct" reflection of the electrons. The intensity created in this process is much weaker than that created from the resonance propagation. This is the reason why the reflection intensity is weak under the non-resonance conditions. The propagation of resonance electrons parallel to the surface can be characterized by a mean travelling distance, along which most of the electrons travel before being reflected back into vacuum. It is predicted through the calculation that this distance is about 500 to 750 ,~ for GaAs (660) reflection. This result shows that any calculation for R H E E D or R E M based on either column approximation or kinematic reflection might not reflect the real interaction of the incident electrons with surfaces. A full dynamical calculation is required to interpret the observed phenomena in R H E E D and REM. The possibility of achieving steady state propagation of electrons in R H E E D through the multislice calculation has been demonstrated. This establishes the basis for applying this method for R H E E D and R E M calculations. Any surface defect and imperfection can be simulated in this approach. It has been postulated recently by Wang that the inelastic scattering can be introduced in the multislice calculation by using the concept of an "effective" potential [16,17], which makes it possible to calculate the energy-filtered inelastic R H E E D patterns and R E M images [18]. The interruption of resonance waves by surface steps has been predicted and observed in scanning beam experiments. It is proved that the surface resonance wave does exist in a crystal surface under the R H E E D geometry. This wave is found to consist of two main beams propagating in
111
slightly different directions. The observed direction of this wave depends on the position of a down step with respect to the incident beam. Since this wave is localized in the top few surface layers, it is expected that the surface excitations can be detected sensitively from the analysis of this beam. The high sensitivity of the R E M technique to the atom-high surface steps in R E M could be attributed to the perturbation of the propagation of the resonance waves by the steps, resulting in a significant perturbation to the specular reflections. The surface down steps affect not only the phases but also the propagating directions to the resonance waves compared to the Bragg reflections.
Acknowledgements This work was supported by the office of Naval Research under grant N00014-86-k-0319 (MAPS program) and N S F G r a n t DMR-8514583 (P.L.) and made use of the resources of the ASU Facility for High Resolution Electron Microscopy, supported by N S F grant D M R 86-11609. The authors are grateful to Dr. D.J. Smith for his constant interest in this work.
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[13] G. Lehmpfuhl and Y. Uchida, in: Proc. 44th Annual EMSA Meeting, Albuquerque, NM, 1986, Ed. G.W. Bailey (San Francisco Press, San Francisco, CA, 1986) p. 376. [14] J.M. Cowley, J. Electron Microscopy Tech., in press. [15] R.H. Milne, A. Howie and M.G. Walls, submitted.
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