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Computer-simulated electron microdiffraction patterns from MgO crystal surfaces
Uitramicroscopy 26 (1988) 205-216 North-Ho|land, Amsterdam
205
COMPUTER-SIMULATED ELECTRON MICRODIFFI~oCTION PATI'ERNS FROM MgO CRYSTAL SURFACES M. PAN and J.M. COWLEY Department of Physics, Arizona State University. Tempe. Arizona 85287-1504, USA Received 18 March 1988; presented at Workshop January 1988
Dynamical diffraction calculations have been carried out for an incident 100 kV electron wavepacket (STEM case). 12/~ in diameter, running parallel to a flat (100) face of a MgO crystal. The microdiffraction patterns and the electron density distributions have been obtained. A surface potential field, representing the modified image force, was added to the crystal potential to study its effect on the small electron probe. The calculation results show that electron microdiffraction patterns are very sensitive to the surface potential for crystal thickness larger than 120 A,. The streakings of the central disk in the microdiffraction patterns can be accounted for as the contributions from surface channeling electrons and electrons being deflected in the surface potential field. Both contributions have their characteristic features in the streakings of the central spot. The forbidden spots, (100) and (100), which are experimentally observed, can be obtained only after the replacement of the vacuum potential by the modified surface image potential. With the incident beam outside the surface, the presence of the surface image potential field greatly enhances the electron surface monolayer channeling.
1. Introduction It has been reported that in scanning transmission electron microscopy (STEM) when a very small electron beam (10-15 A in diameter) is directed to run almost parallel to a flat crystal plane, the influence of surface potential on the microdiffraction patterns and the perturbation of electron energy loss spectroscopy (EELS) due to the surface excitations can be easily observed [1-3]. In the case of microdiffraction from flat single crystal surfaces of MgO and Au, the central spot was observed to be streaked in the direction which is perpendicular to the crystal surface, and diffrac,,v,. ovvta avv~a,c,~ at about .!. . . . . :.- . . . . r forbidden (100) and (100) reflections as the beam was moved from the vacuum towards the cr)stal surface. After the beam was completely inside the crystal these forbidden spots disappeared. Based on the calculation of electron paths in the surface potential field reasonable agreement, particularly in the streaking length, with the experimental observations was achieved for Au surfaces when it
was assumed that the surface potential field had the form [4] +(x)
=
x>0,
[-*0,
x<0,
where 40 is the inner potential of the crystal (30.2 V and 14.7 V for Au and MgO, respectively) and A, B and C are constants. This potential fidd represents the image force modified by a secondorder term. A theoretical analysis of the case of a fast electron moving parallel to the surface of a MgO crystal by Howie and Echenique [5,6] suggests that the image force potential should not have a form as given above and should be smaller by several orders of magnitude. However. the surface potential ,,,~,,'~;o'";r'"';"",~,,,,,,,,,may in practice be modified in various ways, such as by the adsorption of a monolayer of gas molecules. We have used the simple image-force model in order to evaluate the effects of an extended potential field. With the surface potential form given above, a simple ray-diagram was given by Cowley [1] to
0304-3991/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
206
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
explain the observed features of the microdiffraction patterns as due to the deflection of electrons outside the crystal surface a n d then channeling along the surface plane,_, giving rise to the appearance of (100) and (100) spots. This simple argument, however, could not a c c o u n t satisfactorily for some of the features, such as the intensities of (100) and (100) spots and diffraction spots with fractional indices along the line t h r o u g h the central spot p e r p e n d i c u l a r to the crystal surfaces. In S T E M microscopes the incident electron wavepacket can be a p p r o x i m a t e d as a complex Guassian with the width c o r r e s p o n d i n g to the diameter of the probe. In the case of an incident electron p r o b e running parallel to a flat crystal surface, w h e n there is a small o v e r l a p p i n g of the incident w a v e p a c k e t with the bulk crystal potential, there will be a small p o r t i o n of the incident electrons striking the crystal edge. If the Bragg angle for the a t o m i c planes parallel to the crystal surface is less t h a n the angle for total internal reflection, this p o r t i o n of incident electron b e a m can only leave the crystal at the exit face perpendicular to the incident beam. In other words, these electrons will be trapped inside the crystal and c h a n n e l e d along the surface. Therefore it is always expected that there will be surface channeling electrons in this situation. T h i s is the case for Au (111) planes a n d M g O (200) planes. In the simple geometric optics model this surface c h a n n e l i n g effect and the coherent interference effect have not been t a k e n into account. T o treat these effects more p r o p e r l y and accurately, a m a n y - b e a m dynamical diffraction calculation is required. The purpose of this paper is to report the use of dynamical diffraction theory to study these effects on the microdiffraction patterns.
2. Computing method D y n a m i c a l diffraction calculations have been carried out using the standard multislice program. In the calculation both the crystal surface and the small incident electron probe m u s t be treated as crystal " d e f e c t s " (non-periodic) [7]. Therefore it is necessary to use the periodic c o n t i n u a t i o n m e t h o d
to a p p r o x i m a t e diffraction from the crystal surface by such a small p r o b e of electrons. T h e size of the artificial unit cell has to be so large that the scattered waves f r o m each individual defect do n o t overlap. O n the other h a n d the c o m p u t i n g time is excessive if this super-unit cell is m a d e too large. In our calculation, with the e l e c t r o n p r o b e r u n n i n g along the [011] zone axis a n d parallel to (100) surface p l a n e of MgO, the u n i t cell was taken as a = 37.8 A, b = 35.6 ,~, c = 2.97 A and a - - f l = ) , = 90.0 ° with the c-axis e q u a l to the (01!) spacing. T h e (1130) surface p l a n e w a s formed by filling only h a l f of the unit cell with M g and O a t o m s at the p o s i t i o n s c o r r e s p o n d i n g to perfect crystals and leaving the other h a l f empty. The positions of the incident probe c a n be varied by changing its f r a c t i o n a l coordinates X a n d Y. To calculate the microdiffraction p a t t e r n s by the multislice p r o g r a m the first slice is replaced by the phase grating o f the incident p r o b e whose F o u r i e r coefficients of the; potential are given by g'°(u, v)=A(u, v)exp[ix(u, v)], where X(U, v) is the phase factor d e t e r m i n e d by defocus a n d spherical aberration c o n s t a n t of the objective lens a n d A(u, v) is the simple aperture function. If the incident probe is t r a n s l a t e d by XA a n d YB in b o t h x and y directions relative to the axis of the m i c r o s c o p e (as in the S T E M case) a n o t h e r phase factor will be a d d e d to ff,0(u, v) as
qP°(u, v ) = A ( u , v ) e x p [ i x ( u , v)] × e x p [ i 2 ~ r ( Xu + Yv)~ . T h e n the incident wavepacket at the specimen is obtained by F o u r i e r transform of g ' ° ( u , v), i.e.
t(x, y ) - F[ gt°(u, v ) ] , where F denotes the Fourier t r a n s f o r m . Here the incident electron p r o b e is treated as a coherent p r o b e at the specimen, and this has been shown to be a very good a p p r o x i m a t i o n for s o m e S T E M microscopes, such as the dedicated HB-5 S T E M microscope in o u r laboratory. In the simulation, the radius of the objective aperture, a, was chosen as 0.1 ~ - 1 c o r r e s p o n d i n g to the incident probe diameter of 12 A at the specimen. Since a is small compared with the { 111 }-type reflections of MgO
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
(g111 = 0.41 ,~-t) the phase factor X(u, v) can be ignored for the case of a well focussed beam. The crystal potential was synthesized by 9499 structure amplitudes and the dynamical diffraction calculation was made with 2000 beams. Slice thickness was chosen as 2.97 ,~ equal to the (011) lattice spacing. Then the computing time for each slice was about 5 min on a V A X l l / 7 5 0 computer.
O(x) eV 0.0
uco
crystal
[~.ool ¢
(100) vacuum
[ [.1003
Fig. 1. Notations used for the outward and inward surface normal directions.
[011]
M
l
/
-15.7
3. Results and discussion
In the simulation we have used two models of MgO crystal surfaces: (1) the crystal surface was formed by the simple termination of the bulk crystal structure; (2) the potential field outside the surface was replaced by the modified image potential field which varied as 1 / X approximately from the surface. In model (1) the surface potential range was very small, less than 1 ,~. In model (2), however, the surface potential had a long range over a distance of about 20 ,~. The results of the calculation show significant differences of the microdiffraction patterns for these two cases. For later reference, it may be noted that the outward and inward surface normal directions are referred to as the [100] and [100] directions, as shown in fig. 1. (1) Surface with very small potential range° The average potential along the surface normal direction is shown for this case in fig. 2 where the potential is seen to converge very rapidly to zero outside the surface. For the incident electron beam at a distance 3 A outside the surface a series of microdiffraction patterns corresponding to different thicknesses have been calculated as shown in fig. 3. For small thicknesses ( < 50 A) there seems no change in the central spot. Until about 120 ,~
207
I
0.0
1
1.78
nm
Fig. 2. Average potential along the surface normal direction for model ( I ).
the central spot appears to be streaked in the [100] direction. As thickness increases, up to 800 A, this streaking changes its length and another spot gradually appears at a position with fractional indices. Above 800 A the microdiffraction patterns are insensitive to the thickness. In no cases are spots seen to appear at the positions of forbidden (100) and (100) reflections. To study the electron scattering process in more detail the electron density distribution (modulus squared of the electron w=ve function) in real space is very helpful. Fig. 4 is an output of electron density distributions corresponding to the micrediffraction patterns in fig. 3. These distributions show that, as the streaking of the central spot appears, the electrons in real space are channeling along the atomic columns in the outermost atomic plane as shown in fig. 4b. More detailed analyses show that there are two excitation modes for the propagation of electrons in the surface region, i.e., elcctrons propagate either along the atomic columns or between the atomic plmnes which are parallel to the crystal surface plane. In the first mode electrons are trapped around the atomic columns by the atomic potentials and are seen to swing into and off of the atom stri.'-gs as shown in figs. 4f-4h. This oscillatory behavior,
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
208
--
-
~
-
"--
?
~00
000
200
t
~
:
(a)
1 2 . 8 nm
(b)
21.7nm
(c)
~.4
(d)
40.1 nm
(e)
5 3 . 8 nm
(f)
5 9 . 7 nm
(g)
6 7 . 4 nm
(h)
76.3
(i)
81.1 nm
nm
nm
Fig. 3. Simulated electron rater.diffraction patterns for various thicknesses. Elect,'on probe, 12 A, in diameter, was at a distance 3 ,~ outside the surface. Model (1).
however, h a p p e n s only in the surface normal direction, and the electron density peaks are on the atom strings in the direction parallel to the crystal surface. U n d e r certain circumstances the peak on a specific a t o m string, for instance the O - a t o m string in fig. 4e, can vanish. In the second m o d e the electron density distribution is one-dimensional in a single slice, parallel to the crystal surface. Electrons can be scattered from one m o d e to the other and also a m o n g the strings of atoms. The surface potential barrier (extremely narrow in
this case) prevents electrons from escaping from the crystal in the scattering process. Because the effective range o f the surface potential is small ( < I ,~,) there will be few electrons deflected by this surface potential field to enter the crystal through the surface which is parallel to the incident beam. Therefore the contribution of these deflected electrons to the microdiffraction pattern is negligible. In other words, the streaking of the central spct in the microdiffraction pattern is not related to the electron deflection in the surface
M. Pan, J.M. Cowley / Computer-simulated electron microdtffraction patterns from MgO
209
Fig. 4. Electron density distributions (modulus squared of the wavefunction) for various thicknesses. Electron probe of diameter of 12 was at a distance 3.4 outside the surface. Model (1). Long bar indicates the position of the outermost atomic plane.
M. Pan. J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
210
q)(x)
uc, o J o l t ]
eV
0.0
same as those for model (1). T h e c e n t r a l spot in fig. 6a, for instance, appears to be streaking in the same direction with the same length as in the corresponding p a t t e r n , fig. 3a. This implies that the effect of the surface potential for crystal thickness less t h a n 120 A is negligible. So the streaking of the central spot in [100] direction is a characteristic feature of the surface c h a n n e l i n g electrons with little influence by the surface p o t e n t i a l field. This means t h a t the deflection of electrons in the surface p o t e n t i a l field is not a p p r e c i a b l e if the distance travelled by the electrons in the b e a m direction is less t h a n 120 ,~. As thickness increases, the further microdiffraction patterns a p p e a r to be d i f f e r e n t from the corresponding ones in fig. 3. T h e streaking of the central spot in the opposite direction, i.e., [100] direction, begins to show up (figs. 6b a n d 6c). This can be a t t r i b u t e d to the deflection of electrons in the long-range surface potential field as suggested by Cowley [1]. As electrons are b e i n g c o n t i n u o u s l y scattered a n d p r o p a g a t i n g up to 820 A, several new features g r a d u a l l y appear in the microdiffraction patterns. A t first a spot emerges at the position with fractional indices in the [100] direction. Later this spot g r a d u a l l y moves d o w n to the (100) position and c o r r e s p o n d i n g l y the streaking in the [100] direction shrinks in length a n d is concentrated a b o u t the (100) position. C o n s e q u e n t l y a pair of (100)-type spots are formed, which is in agreement with the experimental results. This leads to the conclusion t h a t the e x p e r i m e n t a l l y observed p h e n o m e n a are caused by the crystal surface potential. The electron density distributions are presented in fig. 7. The surface potential effects are obvious. First of all m o r e electrons outside the crystal are deflected in the long-range surface p o t e n t i a l field as seen in fig. 7j where a lmge intensity maxima exits outside the surface. This is w h y the streaking of the central spot in [100] direction a n d the pair of (100)-type spots appears at different thicknesses. Secondly, the surface m o n o l a y e r channeling effect is m u c h more e n h a n c e d by the presence of the surface image potential field. This conclusion easily derives from the c o m p a r i s o n with the density distributions calculated for m o d e l (1) (fig. 4). The density distributions in fig. 4 show that in o
t -15.7
rt 0.0
1.78
nm
Fig. 5. Average potential along the surface normal direction for model (2).
potential field in this case. So this streaking m u s t be a characteristic feature of the surface channeling electrons which come from the portion of the incident b e a m inside the crystal at the incident face. This conclusion will be verified in the following section where an image p o t e n t i a l field is associated with the crystal surface. (2) Surface with modified image potential field. The surface potential field in (1) was replaced by a slowly v a r y i n g potential, as discussed in the introduction section. In the c a l c u l a t i o n we chose B = 2.0 ,~,_, C = 0.2 tk, and A was determined by the condition t h a t the potential vary continuously from the value in v a c u u m to the value of inner potential inside the crystal across the snrface [8]. The inner potential of M g O calculated b y the multislice program was 15.7 V. Therefore A was fixed as 34.89 V A. Fig. 5 is a plot of the average potential along the surface n o r m a l direction. In order to investigate ,h,~ -~ •-,. o~n,°.., ,.,,,.,., ,..¢ - , ,th:,s 1,o,l~-ran~e surface potential field on microdiffraction p a t t e r n s and the electron surface c h a n n e l i n g behavior, calculation was d o n e under exactly the same c o n d i t i o n s as those used in model (1). Fig. 6 is a series of rnicrodiffraction p a t t e r n s c o r r e s p o n d i n g to various thicknesses. For thickness less t h a n 120 ~, the p a t t e r n s look almost the o
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns fram MgO
211
(c) 2 6 . 4 nm
(a) 1 2 . 8 nm
(b) 2 1 . 7 nm
(d) 40.1 nm
(e) 5 3 . 8 nm
(f) 59.7 nm
(g) 6 7 . 4 nm
(h) 7 6 . 3 nm
(i) 81.1 nm
Fig. 6. Simulated electron microdiffraction patterns for various thicknesses. Electron probe, 12 ,~, in diameter, was at a distance 3 ,~ outside the surface. Model (2).
the course of scattering some of the electrons are being gradually scattered into the bulk crystal, fig. 4j. The density distribution in fig. 7j, howe~er, indicates that most of the electrons are distributed in the outermost atomic plane and the region outside the surface, and few electrons are scattered into the buik crystal. This is the direct consequence of the slowly varying image potential outside th,_- surface. The channeling behavior is similar to that in model (1), i.e., electrons are trapped in the atomic columns by the atomic potential and in the course of propagation they oscillate about the centers of atomic columns in the surface normal directions.
The comparison of the calculation results shows that (1) the streaking of the central spot in [100] direction is associated with the surface channeling electrons and not affected by the surface potential if the crystal thickness is less than 120 A, and (2) the presence of the long-range surface image potential causes the streaking of the central spot in [100] and the pair of the (100)-type reflectio~ls. To see this more clearly another set of calculations was carried out with the incident beam being 6 A, outside the crystal surface. The results are shown in fig. 8 and fig. 9 foi nficrodiffraction patterns and electron density distributions respectively. When the incident beam. 12 A in diameter, was
212
M. Pan, J.M. Cowley / Computer-simulated electron microd~ffraction patterns from MgO
Fig. 7. Electron density distributions for various thicknesses. E l e c t r o n probe of 12 ,~ in d i a m e t e r was at a d i s t a n c e 3 ,~ outside the surface. M o d e l (2). Long bar indicates the position of the o u t e r m o s t a t o m i c plane.
m o v e d to 6 ,~ outside the s u r f a c e there w a s n o o v e r l a p p i n g of the electron p r o b e w i t h the c r y s t a l p o t e n t i a l as s h o w n in fig. 9a. In this case the
m i c r o d i f f r a c t i o n p a t t e r n s in fig. 8 s h o w n o streaking of the c e n t r a l spot in [100] d i r e c t i o n . As the thickness i n c r e a s e s f u r t h e r (to - 950 ~,) the p a i r
M. Pan, J.M. Cowley / Computer-simulated electron mlcrodiffraction patterns from MgO
213
r-
- -
200
@ ooo --
O
m
I
200
(a) 12.8 nm
(b) 53.8 nm
(c) 6 2 . 7 nm
(d) 7 6 . 3 nm
(e) 81.1 nm
(f)
94.1 nm
Fig. 8. Simulated electron microdiffraction patterns for different thicknesses, Electron probe o r diameter of 12 ,~ was at a distance 6 A outside the surface. Model (2).
of (100)-type reflections appear (fig. 8f). The electron density distribution in fig. 9f shows great similarity to fig. 7j, indicating again that the surface monolayer channeiing is enhanced by the surface image potential field. The two excitation modes for the propagation of electrons inside the crystals are consistent with the descriptions of bound Bloch waves by Kambe et al. [9]. Their results of many-beam calculations, for electrons transmitted along the [110] axis of MgO crystals, showed that at 50 keV two Bloch waves were strongly excited and bound to the rows of atoms; the tbjrd Bloch wave was not bound strongly and had intensity maxima between the atomic planes. The most common type of electron channeling was represented by the strongly bound Bloch waves. In the present calculation it is shown that in the course of scattering electrons can be scattered among these Bloch wave states. The oscillatory nature of the strongly bound Bloch states about the centers of the atomic columns may be caused by the presence of the crystal
surface. Because electrons are mainly channeling along the atomic columns it is more likely for these electrons to be scattered inelastically. This inelastic scattering effect has not been incorporated into the multislice program successfully. Therefore the effect of the inelastic scattering on electron microdiffraction patterns cannot be calculated. Due to the fact that electron density distributions are oscillating at the positions of atomic columns in the surface normal direction the probability of total excitatien for any atomic column, which is important for microanalysis, must be found by the integration over all slices. More detailed discussion was given by Wang et a!. [!0].
4. Conclusion
Electron microdiffraction patterns, produced by the coherent incident electron probe in some STEM microscopes (e.g. HB-5 STEM), can be calculated by use of the multislice program. The
214
M. Pan, J. 21/1.Cowley / Computo'-s~mulated electron microdiffraction patterns from MgO
Fig. 9. Electron density distributions for different thicknesses. Electron probe of diameter of 12 ~, was at a distance 6 ,~, outside the surlace. Model (2). Long bar indicates the position of the outermost atomic plane.
observed streaking of the central spot, from a flat MgO crystal surface, can be interpreted as contributions from surface channeling electrons and electrons being deflected by the surface potential field. The streaking direct:on is opposite for each of these contributions. For crystal thickness less than 120 2~ the effect of surface potential is negligible and the streaking is associated with the surface channeling process and appears in the surface outward normal direction. For crystal thickness more than 120 A the streaking appearing in the surface inward normal direction is caused
by the surface potential field. With larger crystal thickness ( < 800 A) the pair of (100)-type forbidden reflections also results from the surface potential field and surface channeling effect. The calculation results show that microdiffraction patterns from crystal surface are very sensitive to the surface potential for crystal thickness larger than 120 A. An adsorbed surface monolayer may affect the surface potential and the future experiments should be carried out in the U HV conditions. The presence of the surface potential field greatly enhances the surface monolayer channelO
M. Pan. J.M. Cowley / Computer-simulated electron ,ucrodiffraction patterns/rom MgO
ing phenomenon. The inelastic scattering effect was not taken into account in the calculation.
Acknowledgements We are grateful to Dr. J.C.H. Spence for his advice and encouragement in the computer simulation. This work was supported by DOE grant DE-FG-02-86-ER45228 and made use of ASU HREM Facility, supported by NSF grant D M R 8611609.
References [1] J.M. Cowley, Ultramicroscopy 7 (1981) 181. [2] C.S. Tan and J.M. Cowley, Ultramicroscopy 12 (1984) 333.
[3] [4] [5] [6]
215
J.M. Cowley, Ultramicroscopy 9 (1982) 231. P,H. Cuher and J.C. Davis, Surface Sci. 1 11964) 104, A. Howie, Ultramicroscopy 11 (1983) 141. P,M. Echenique and A. Howle, Uhramicroscop), 16 (19S5) 269. [7] J.C.H. Spence, Acta Cryst. (1978) A34, 112. [8] J.M. Cowley and Z.L. Wang, Ultramicroscopy 19 (1986) 217. [9] K. Kambe, G. Leh:'apfuhl and F. Fuiimoto, Z. Naturforsch. 29a (19"/4) 1034. [10] Z.L. Wang, P. Lu and J.M. Cowley, UItramicroscopy 23 (1987) 205.
205
COMPUTER-SIMULATED ELECTRON MICRODIFFI~oCTION PATI'ERNS FROM MgO CRYSTAL SURFACES M. PAN and J.M. COWLEY Department of Physics, Arizona State University. Tempe. Arizona 85287-1504, USA Received 18 March 1988; presented at Workshop January 1988
Dynamical diffraction calculations have been carried out for an incident 100 kV electron wavepacket (STEM case). 12/~ in diameter, running parallel to a flat (100) face of a MgO crystal. The microdiffraction patterns and the electron density distributions have been obtained. A surface potential field, representing the modified image force, was added to the crystal potential to study its effect on the small electron probe. The calculation results show that electron microdiffraction patterns are very sensitive to the surface potential for crystal thickness larger than 120 A,. The streakings of the central disk in the microdiffraction patterns can be accounted for as the contributions from surface channeling electrons and electrons being deflected in the surface potential field. Both contributions have their characteristic features in the streakings of the central spot. The forbidden spots, (100) and (100), which are experimentally observed, can be obtained only after the replacement of the vacuum potential by the modified surface image potential. With the incident beam outside the surface, the presence of the surface image potential field greatly enhances the electron surface monolayer channeling.
1. Introduction It has been reported that in scanning transmission electron microscopy (STEM) when a very small electron beam (10-15 A in diameter) is directed to run almost parallel to a flat crystal plane, the influence of surface potential on the microdiffraction patterns and the perturbation of electron energy loss spectroscopy (EELS) due to the surface excitations can be easily observed [1-3]. In the case of microdiffraction from flat single crystal surfaces of MgO and Au, the central spot was observed to be streaked in the direction which is perpendicular to the crystal surface, and diffrac,,v,. ovvta avv~a,c,~ at about .!. . . . . :.- . . . . r forbidden (100) and (100) reflections as the beam was moved from the vacuum towards the cr)stal surface. After the beam was completely inside the crystal these forbidden spots disappeared. Based on the calculation of electron paths in the surface potential field reasonable agreement, particularly in the streaking length, with the experimental observations was achieved for Au surfaces when it
was assumed that the surface potential field had the form [4] +(x)
=
x>0,
[-*0,
x<0,
where 40 is the inner potential of the crystal (30.2 V and 14.7 V for Au and MgO, respectively) and A, B and C are constants. This potential fidd represents the image force modified by a secondorder term. A theoretical analysis of the case of a fast electron moving parallel to the surface of a MgO crystal by Howie and Echenique [5,6] suggests that the image force potential should not have a form as given above and should be smaller by several orders of magnitude. However. the surface potential ,,,~,,'~;o'";r'"';"",~,,,,,,,,,may in practice be modified in various ways, such as by the adsorption of a monolayer of gas molecules. We have used the simple image-force model in order to evaluate the effects of an extended potential field. With the surface potential form given above, a simple ray-diagram was given by Cowley [1] to
0304-3991/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
206
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
explain the observed features of the microdiffraction patterns as due to the deflection of electrons outside the crystal surface a n d then channeling along the surface plane,_, giving rise to the appearance of (100) and (100) spots. This simple argument, however, could not a c c o u n t satisfactorily for some of the features, such as the intensities of (100) and (100) spots and diffraction spots with fractional indices along the line t h r o u g h the central spot p e r p e n d i c u l a r to the crystal surfaces. In S T E M microscopes the incident electron wavepacket can be a p p r o x i m a t e d as a complex Guassian with the width c o r r e s p o n d i n g to the diameter of the probe. In the case of an incident electron p r o b e running parallel to a flat crystal surface, w h e n there is a small o v e r l a p p i n g of the incident w a v e p a c k e t with the bulk crystal potential, there will be a small p o r t i o n of the incident electrons striking the crystal edge. If the Bragg angle for the a t o m i c planes parallel to the crystal surface is less t h a n the angle for total internal reflection, this p o r t i o n of incident electron b e a m can only leave the crystal at the exit face perpendicular to the incident beam. In other words, these electrons will be trapped inside the crystal and c h a n n e l e d along the surface. Therefore it is always expected that there will be surface channeling electrons in this situation. T h i s is the case for Au (111) planes a n d M g O (200) planes. In the simple geometric optics model this surface c h a n n e l i n g effect and the coherent interference effect have not been t a k e n into account. T o treat these effects more p r o p e r l y and accurately, a m a n y - b e a m dynamical diffraction calculation is required. The purpose of this paper is to report the use of dynamical diffraction theory to study these effects on the microdiffraction patterns.
2. Computing method D y n a m i c a l diffraction calculations have been carried out using the standard multislice program. In the calculation both the crystal surface and the small incident electron probe m u s t be treated as crystal " d e f e c t s " (non-periodic) [7]. Therefore it is necessary to use the periodic c o n t i n u a t i o n m e t h o d
to a p p r o x i m a t e diffraction from the crystal surface by such a small p r o b e of electrons. T h e size of the artificial unit cell has to be so large that the scattered waves f r o m each individual defect do n o t overlap. O n the other h a n d the c o m p u t i n g time is excessive if this super-unit cell is m a d e too large. In our calculation, with the e l e c t r o n p r o b e r u n n i n g along the [011] zone axis a n d parallel to (100) surface p l a n e of MgO, the u n i t cell was taken as a = 37.8 A, b = 35.6 ,~, c = 2.97 A and a - - f l = ) , = 90.0 ° with the c-axis e q u a l to the (01!) spacing. T h e (1130) surface p l a n e w a s formed by filling only h a l f of the unit cell with M g and O a t o m s at the p o s i t i o n s c o r r e s p o n d i n g to perfect crystals and leaving the other h a l f empty. The positions of the incident probe c a n be varied by changing its f r a c t i o n a l coordinates X a n d Y. To calculate the microdiffraction p a t t e r n s by the multislice p r o g r a m the first slice is replaced by the phase grating o f the incident p r o b e whose F o u r i e r coefficients of the; potential are given by g'°(u, v)=A(u, v)exp[ix(u, v)], where X(U, v) is the phase factor d e t e r m i n e d by defocus a n d spherical aberration c o n s t a n t of the objective lens a n d A(u, v) is the simple aperture function. If the incident probe is t r a n s l a t e d by XA a n d YB in b o t h x and y directions relative to the axis of the m i c r o s c o p e (as in the S T E M case) a n o t h e r phase factor will be a d d e d to ff,0(u, v) as
qP°(u, v ) = A ( u , v ) e x p [ i x ( u , v)] × e x p [ i 2 ~ r ( Xu + Yv)~ . T h e n the incident wavepacket at the specimen is obtained by F o u r i e r transform of g ' ° ( u , v), i.e.
t(x, y ) - F[ gt°(u, v ) ] , where F denotes the Fourier t r a n s f o r m . Here the incident electron p r o b e is treated as a coherent p r o b e at the specimen, and this has been shown to be a very good a p p r o x i m a t i o n for s o m e S T E M microscopes, such as the dedicated HB-5 S T E M microscope in o u r laboratory. In the simulation, the radius of the objective aperture, a, was chosen as 0.1 ~ - 1 c o r r e s p o n d i n g to the incident probe diameter of 12 A at the specimen. Since a is small compared with the { 111 }-type reflections of MgO
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
(g111 = 0.41 ,~-t) the phase factor X(u, v) can be ignored for the case of a well focussed beam. The crystal potential was synthesized by 9499 structure amplitudes and the dynamical diffraction calculation was made with 2000 beams. Slice thickness was chosen as 2.97 ,~ equal to the (011) lattice spacing. Then the computing time for each slice was about 5 min on a V A X l l / 7 5 0 computer.
O(x) eV 0.0
uco
crystal
[~.ool ¢
(100) vacuum
[ [.1003
Fig. 1. Notations used for the outward and inward surface normal directions.
[011]
M
l
/
-15.7
3. Results and discussion
In the simulation we have used two models of MgO crystal surfaces: (1) the crystal surface was formed by the simple termination of the bulk crystal structure; (2) the potential field outside the surface was replaced by the modified image potential field which varied as 1 / X approximately from the surface. In model (1) the surface potential range was very small, less than 1 ,~. In model (2), however, the surface potential had a long range over a distance of about 20 ,~. The results of the calculation show significant differences of the microdiffraction patterns for these two cases. For later reference, it may be noted that the outward and inward surface normal directions are referred to as the [100] and [100] directions, as shown in fig. 1. (1) Surface with very small potential range° The average potential along the surface normal direction is shown for this case in fig. 2 where the potential is seen to converge very rapidly to zero outside the surface. For the incident electron beam at a distance 3 A outside the surface a series of microdiffraction patterns corresponding to different thicknesses have been calculated as shown in fig. 3. For small thicknesses ( < 50 A) there seems no change in the central spot. Until about 120 ,~
207
I
0.0
1
1.78
nm
Fig. 2. Average potential along the surface normal direction for model ( I ).
the central spot appears to be streaked in the [100] direction. As thickness increases, up to 800 A, this streaking changes its length and another spot gradually appears at a position with fractional indices. Above 800 A the microdiffraction patterns are insensitive to the thickness. In no cases are spots seen to appear at the positions of forbidden (100) and (100) reflections. To study the electron scattering process in more detail the electron density distribution (modulus squared of the electron w=ve function) in real space is very helpful. Fig. 4 is an output of electron density distributions corresponding to the micrediffraction patterns in fig. 3. These distributions show that, as the streaking of the central spot appears, the electrons in real space are channeling along the atomic columns in the outermost atomic plane as shown in fig. 4b. More detailed analyses show that there are two excitation modes for the propagation of electrons in the surface region, i.e., elcctrons propagate either along the atomic columns or between the atomic plmnes which are parallel to the crystal surface plane. In the first mode electrons are trapped around the atomic columns by the atomic potentials and are seen to swing into and off of the atom stri.'-gs as shown in figs. 4f-4h. This oscillatory behavior,
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
208
--
-
~
-
"--
?
~00
000
200
t
~
:
(a)
1 2 . 8 nm
(b)
21.7nm
(c)
~.4
(d)
40.1 nm
(e)
5 3 . 8 nm
(f)
5 9 . 7 nm
(g)
6 7 . 4 nm
(h)
76.3
(i)
81.1 nm
nm
nm
Fig. 3. Simulated electron rater.diffraction patterns for various thicknesses. Elect,'on probe, 12 A, in diameter, was at a distance 3 ,~ outside the surface. Model (1).
however, h a p p e n s only in the surface normal direction, and the electron density peaks are on the atom strings in the direction parallel to the crystal surface. U n d e r certain circumstances the peak on a specific a t o m string, for instance the O - a t o m string in fig. 4e, can vanish. In the second m o d e the electron density distribution is one-dimensional in a single slice, parallel to the crystal surface. Electrons can be scattered from one m o d e to the other and also a m o n g the strings of atoms. The surface potential barrier (extremely narrow in
this case) prevents electrons from escaping from the crystal in the scattering process. Because the effective range o f the surface potential is small ( < I ,~,) there will be few electrons deflected by this surface potential field to enter the crystal through the surface which is parallel to the incident beam. Therefore the contribution of these deflected electrons to the microdiffraction pattern is negligible. In other words, the streaking of the central spct in the microdiffraction pattern is not related to the electron deflection in the surface
M. Pan, J.M. Cowley / Computer-simulated electron microdtffraction patterns from MgO
209
Fig. 4. Electron density distributions (modulus squared of the wavefunction) for various thicknesses. Electron probe of diameter of 12 was at a distance 3.4 outside the surface. Model (1). Long bar indicates the position of the outermost atomic plane.
M. Pan. J.M. Cowley / Computer-simulated electron microdiffraction patterns from MgO
210
q)(x)
uc, o J o l t ]
eV
0.0
same as those for model (1). T h e c e n t r a l spot in fig. 6a, for instance, appears to be streaking in the same direction with the same length as in the corresponding p a t t e r n , fig. 3a. This implies that the effect of the surface potential for crystal thickness less t h a n 120 A is negligible. So the streaking of the central spot in [100] direction is a characteristic feature of the surface c h a n n e l i n g electrons with little influence by the surface p o t e n t i a l field. This means t h a t the deflection of electrons in the surface p o t e n t i a l field is not a p p r e c i a b l e if the distance travelled by the electrons in the b e a m direction is less t h a n 120 ,~. As thickness increases, the further microdiffraction patterns a p p e a r to be d i f f e r e n t from the corresponding ones in fig. 3. T h e streaking of the central spot in the opposite direction, i.e., [100] direction, begins to show up (figs. 6b a n d 6c). This can be a t t r i b u t e d to the deflection of electrons in the long-range surface potential field as suggested by Cowley [1]. As electrons are b e i n g c o n t i n u o u s l y scattered a n d p r o p a g a t i n g up to 820 A, several new features g r a d u a l l y appear in the microdiffraction patterns. A t first a spot emerges at the position with fractional indices in the [100] direction. Later this spot g r a d u a l l y moves d o w n to the (100) position and c o r r e s p o n d i n g l y the streaking in the [100] direction shrinks in length a n d is concentrated a b o u t the (100) position. C o n s e q u e n t l y a pair of (100)-type spots are formed, which is in agreement with the experimental results. This leads to the conclusion t h a t the e x p e r i m e n t a l l y observed p h e n o m e n a are caused by the crystal surface potential. The electron density distributions are presented in fig. 7. The surface potential effects are obvious. First of all m o r e electrons outside the crystal are deflected in the long-range surface p o t e n t i a l field as seen in fig. 7j where a lmge intensity maxima exits outside the surface. This is w h y the streaking of the central spot in [100] direction a n d the pair of (100)-type spots appears at different thicknesses. Secondly, the surface m o n o l a y e r channeling effect is m u c h more e n h a n c e d by the presence of the surface image potential field. This conclusion easily derives from the c o m p a r i s o n with the density distributions calculated for m o d e l (1) (fig. 4). The density distributions in fig. 4 show that in o
t -15.7
rt 0.0
1.78
nm
Fig. 5. Average potential along the surface normal direction for model (2).
potential field in this case. So this streaking m u s t be a characteristic feature of the surface channeling electrons which come from the portion of the incident b e a m inside the crystal at the incident face. This conclusion will be verified in the following section where an image p o t e n t i a l field is associated with the crystal surface. (2) Surface with modified image potential field. The surface potential field in (1) was replaced by a slowly v a r y i n g potential, as discussed in the introduction section. In the c a l c u l a t i o n we chose B = 2.0 ,~,_, C = 0.2 tk, and A was determined by the condition t h a t the potential vary continuously from the value in v a c u u m to the value of inner potential inside the crystal across the snrface [8]. The inner potential of M g O calculated b y the multislice program was 15.7 V. Therefore A was fixed as 34.89 V A. Fig. 5 is a plot of the average potential along the surface n o r m a l direction. In order to investigate ,h,~ -~ •-,. o~n,°.., ,.,,,.,., ,..¢ - , ,th:,s 1,o,l~-ran~e surface potential field on microdiffraction p a t t e r n s and the electron surface c h a n n e l i n g behavior, calculation was d o n e under exactly the same c o n d i t i o n s as those used in model (1). Fig. 6 is a series of rnicrodiffraction p a t t e r n s c o r r e s p o n d i n g to various thicknesses. For thickness less t h a n 120 ~, the p a t t e r n s look almost the o
M. Pan, J.M. Cowley / Computer-simulated electron microdiffraction patterns fram MgO
211
(c) 2 6 . 4 nm
(a) 1 2 . 8 nm
(b) 2 1 . 7 nm
(d) 40.1 nm
(e) 5 3 . 8 nm
(f) 59.7 nm
(g) 6 7 . 4 nm
(h) 7 6 . 3 nm
(i) 81.1 nm
Fig. 6. Simulated electron microdiffraction patterns for various thicknesses. Electron probe, 12 ,~, in diameter, was at a distance 3 ,~ outside the surface. Model (2).
the course of scattering some of the electrons are being gradually scattered into the bulk crystal, fig. 4j. The density distribution in fig. 7j, howe~er, indicates that most of the electrons are distributed in the outermost atomic plane and the region outside the surface, and few electrons are scattered into the buik crystal. This is the direct consequence of the slowly varying image potential outside th,_- surface. The channeling behavior is similar to that in model (1), i.e., electrons are trapped in the atomic columns by the atomic potential and in the course of propagation they oscillate about the centers of atomic columns in the surface normal directions.
The comparison of the calculation results shows that (1) the streaking of the central spot in [100] direction is associated with the surface channeling electrons and not affected by the surface potential if the crystal thickness is less than 120 A, and (2) the presence of the long-range surface image potential causes the streaking of the central spot in [100] and the pair of the (100)-type reflectio~ls. To see this more clearly another set of calculations was carried out with the incident beam being 6 A, outside the crystal surface. The results are shown in fig. 8 and fig. 9 foi nficrodiffraction patterns and electron density distributions respectively. When the incident beam. 12 A in diameter, was
212
M. Pan, J.M. Cowley / Computer-simulated electron microd~ffraction patterns from MgO
Fig. 7. Electron density distributions for various thicknesses. E l e c t r o n probe of 12 ,~ in d i a m e t e r was at a d i s t a n c e 3 ,~ outside the surface. M o d e l (2). Long bar indicates the position of the o u t e r m o s t a t o m i c plane.
m o v e d to 6 ,~ outside the s u r f a c e there w a s n o o v e r l a p p i n g of the electron p r o b e w i t h the c r y s t a l p o t e n t i a l as s h o w n in fig. 9a. In this case the
m i c r o d i f f r a c t i o n p a t t e r n s in fig. 8 s h o w n o streaking of the c e n t r a l spot in [100] d i r e c t i o n . As the thickness i n c r e a s e s f u r t h e r (to - 950 ~,) the p a i r
M. Pan, J.M. Cowley / Computer-simulated electron mlcrodiffraction patterns from MgO
213
r-
- -
200
@ ooo --
O
m
I
200
(a) 12.8 nm
(b) 53.8 nm
(c) 6 2 . 7 nm
(d) 7 6 . 3 nm
(e) 81.1 nm
(f)
94.1 nm
Fig. 8. Simulated electron microdiffraction patterns for different thicknesses, Electron probe o r diameter of 12 ,~ was at a distance 6 A outside the surface. Model (2).
of (100)-type reflections appear (fig. 8f). The electron density distribution in fig. 9f shows great similarity to fig. 7j, indicating again that the surface monolayer channeiing is enhanced by the surface image potential field. The two excitation modes for the propagation of electrons inside the crystals are consistent with the descriptions of bound Bloch waves by Kambe et al. [9]. Their results of many-beam calculations, for electrons transmitted along the [110] axis of MgO crystals, showed that at 50 keV two Bloch waves were strongly excited and bound to the rows of atoms; the tbjrd Bloch wave was not bound strongly and had intensity maxima between the atomic planes. The most common type of electron channeling was represented by the strongly bound Bloch waves. In the present calculation it is shown that in the course of scattering electrons can be scattered among these Bloch wave states. The oscillatory nature of the strongly bound Bloch states about the centers of the atomic columns may be caused by the presence of the crystal
surface. Because electrons are mainly channeling along the atomic columns it is more likely for these electrons to be scattered inelastically. This inelastic scattering effect has not been incorporated into the multislice program successfully. Therefore the effect of the inelastic scattering on electron microdiffraction patterns cannot be calculated. Due to the fact that electron density distributions are oscillating at the positions of atomic columns in the surface normal direction the probability of total excitatien for any atomic column, which is important for microanalysis, must be found by the integration over all slices. More detailed discussion was given by Wang et a!. [!0].
4. Conclusion
Electron microdiffraction patterns, produced by the coherent incident electron probe in some STEM microscopes (e.g. HB-5 STEM), can be calculated by use of the multislice program. The
214
M. Pan, J. 21/1.Cowley / Computo'-s~mulated electron microdiffraction patterns from MgO
Fig. 9. Electron density distributions for different thicknesses. Electron probe of diameter of 12 ~, was at a distance 6 ,~, outside the surlace. Model (2). Long bar indicates the position of the outermost atomic plane.
observed streaking of the central spot, from a flat MgO crystal surface, can be interpreted as contributions from surface channeling electrons and electrons being deflected by the surface potential field. The streaking direct:on is opposite for each of these contributions. For crystal thickness less than 120 2~ the effect of surface potential is negligible and the streaking is associated with the surface channeling process and appears in the surface outward normal direction. For crystal thickness more than 120 A the streaking appearing in the surface inward normal direction is caused
by the surface potential field. With larger crystal thickness ( < 800 A) the pair of (100)-type forbidden reflections also results from the surface potential field and surface channeling effect. The calculation results show that microdiffraction patterns from crystal surface are very sensitive to the surface potential for crystal thickness larger than 120 A. An adsorbed surface monolayer may affect the surface potential and the future experiments should be carried out in the U HV conditions. The presence of the surface potential field greatly enhances the surface monolayer channelO
M. Pan. J.M. Cowley / Computer-simulated electron ,ucrodiffraction patterns/rom MgO
ing phenomenon. The inelastic scattering effect was not taken into account in the calculation.
Acknowledgements We are grateful to Dr. J.C.H. Spence for his advice and encouragement in the computer simulation. This work was supported by DOE grant DE-FG-02-86-ER45228 and made use of ASU HREM Facility, supported by NSF grant D M R 8611609.
References [1] J.M. Cowley, Ultramicroscopy 7 (1981) 181. [2] C.S. Tan and J.M. Cowley, Ultramicroscopy 12 (1984) 333.
[3] [4] [5] [6]
215
J.M. Cowley, Ultramicroscopy 9 (1982) 231. P,H. Cuher and J.C. Davis, Surface Sci. 1 11964) 104, A. Howie, Ultramicroscopy 11 (1983) 141. P,M. Echenique and A. Howle, Uhramicroscop), 16 (19S5) 269. [7] J.C.H. Spence, Acta Cryst. (1978) A34, 112. [8] J.M. Cowley and Z.L. Wang, Ultramicroscopy 19 (1986) 217. [9] K. Kambe, G. Leh:'apfuhl and F. Fuiimoto, Z. Naturforsch. 29a (19"/4) 1034. [10] Z.L. Wang, P. Lu and J.M. Cowley, UItramicroscopy 23 (1987) 205.