Bloch wave treatment of electron channeling

NUCLEAR INSTRUMENTS AND METHODS

I32

(1976)

I4I-I48;

©

NORTH-HOLLAND

PUBLISHING

CO.

B L O C H WAVE T R E A T M E N T OF E L E C T R O N C H A N N E L I N G J. GJ~)NNES and J. T A F T O

Department of Physics, University of Oslo, Oslo, Norway Electron channeling patterns have been recorded at 100 kV and 1 MV from thick crystals of various substances using t h e electron microscope. The directions of high penetration near a zone axis appear as groups of diffuse spots which can be associated with simple antisymmetrical Bloch waves with very few plane wave components. Patterns around a [001] zone axis are considered in detail and compared with calculations based upon a simplified theory of multiple diffuse scattering, especially for patterns from silicon and uranium dioxide. Agreement between experimental and calculated patterns, using 54 beams, is good.

1. Introduction

Particle channeling and anomalous absorption of waves are closely related phenomena, as has been discussed by e.g. Humphreys and Lallyl), Ohtsuki2), Berry et al. 3) and Kambe et al.4). The latter effect which had been known for many years in X-rays as the Borrmann effect was demonstrated in the electron case by Honjo and MihamaS). The interpretation in terms of position dependent imaginary potentials, which lead to different absorption for different Bloch waves is well known, so are the many resulting effects in electron diffraction and microscopy, see Hirsch et al.6). Channeling of fast, charged particles from accelerators was discovered much later, first for protons and ions, later also for electrons [Uggerh6j and Andersen 7)]. This effect was studied under conditions when Bragg reflections are not resolved and was explained in terms of classical particle motion along strings of atoms, disregarding any interference between scattering at different strings [LindhardS)]. Although the validity and applicability of these two approaches has been discussed extensively, the wave description now appears to be recognized as the more complete. A unified derivation of both descriptions has been presented by Berry 9) for the electron case. But the wave treatment will often require a very large number of plane wave components in each Bloch wave and may hence seem impractical, especially for experiments in which individual Bragg reflections are not resolved. Furthermore, the wave treatment has often been restricted to the discrete Bragg beams, whereas channeling effects are most clearly seen under conditions of multiple diffuse scattering. This underlines, however, the need for a treatment of the transition region between particle and wave behaviour, which may be apparent in the case of moderate to high energy electrons. Here electron diffraction and electron beam channeling experiments are often performed in the same energy region. The trend towards the use

of higher voltages in electron microscopes on one hand and towards improvement of angular resolution in accelerator experiments on the other, may emphasize this need. From the channeling point of view more elaborate modes of particles motions, like the so called rosettemotion channeling a°'H) have been proposed; a quantized form has been presented by Tamura and Ohtsuki 12) and by Komaki and Fujimoto 1a). In such a treatment, in which different symmetries of electron orbits along atomic strings are introduced, the symmetry of the lattice and interplay of different strings is still neglected, however. The aim of the present study is therefore to present and apply a channeling description based upon a Bloch wave treatment which is simplified as much as possible but retains the effect of lattice symmetry. Since we are dealing with thick crystals, it is built upon a theory of multiple diffuse scattering ~4) of electrons. The calculations are compared with diffraction patterns from thick crystals at 100kV and 1 MV from substances of different atomic number, up to UO 2 . We have emphasized the development of simple concepts from which a qualitative understanding of the observed channeling effects can be obtained as well as of simple calculation procedures for sem!quantitative comparison with extensive and detailed patterns. It is found that Bloch wave symmetry plays an important role; the salient features of electron channeling near a zone axis can, in fact, be described by simple, antisymmetrical Bloch waves with very few plane wave components. The corresponding representation in direct space can be applied to the treatment of secondary effects, e.g. X-ray emission. 2. Experimental patterns

Several diffraction techniques in the electron microscope may be used to study channeling effects; for a review, see Fujimoto 15). In the present study we have IV. C H A N N E L I N G

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used diffraction patterns from quite thick crystals, with the incident beam along a zone axis. The thickness was chosen so large that the central spot and the diffraction spots had vanished. Due to symmetrical incidence and also to the quite flat diffuse background, the excess/ deficient Kikuchi lines also have disappeared into the background. The remaining contrast, as seen in figs. l a - h reveal directions of high penetration through the crystal.

(a)

(c)

Patterns were taken at different thicknesses and along several crystal axes. The discussion will be focussed on [001]-patterns from square projections, however. For light elements at 100 kV the patterns can be described as consisting of crossing deficient Kikuchi bands, see e.g. figs. la and lb from silicon and zinc blende; the central dark square is limited by the 220, 2]0 and 400, 040 bands respectively. At higher voltages

(b)

(d)

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BLOCH WAVE TREATMENT

(e)

(g)

(f)

(h)

Fig. 1. Channeling patterns from the [001]-projection of some cubic structures. Unless.otherwise stated, the accelerating voltage

is 100 kV. (a) Si, (b) ZnS, (c) VO0"s2, (d) Si at 1 MV, (e) Cu-13% Pt, (f) Au-50% Pd, (g) UO2 and (h) SrTiO3. or higher atomic numbers, these bands break up into groups of spots which are related to each other through Bragg reflections, as in fig. lc from VO0.s2 (where also some of the Bragg reflections are seen). The equivalence of increasing either the voltage (i.e. mass ratio m/mo) or the atomic number can be seen by comparing this pattern taken at 100 kV with the pattern of fig. ld from silicon at 1 MV. For thick crystals the patterns are relatively indepen-

dent of small angle tilt, although the intensity is increased when the direction of incidence coincides with a high penetration direction. There are also only slight variations in the contrast distribution as the thickness is varied. 3. Symmetry and kinetic energy of Bloch waves

It was found that many of the features in the patterns reproduced can be explained by very simple IV. C H A N N E L I N G

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arguments based upon properties of Bloch waves

bJ(r) = ~ CJh exp(ih, r) exp(ik j. r), h

where Chj are Bloch wave coefficients, h reciprocal lattice vectors and M eigenvalues. For a thin crystal, the appropriate solution of the electron scattering problem will be a superposition of such waves. At higher thickness, the electrons will have undergone diffuse scattering to the extent that the phase relationships between Bloch waves are lost; the propagation can then be described in terms of independent bJ's. F r o m the experimental patterns it is seen that the high penetration directions are found in regions around Brillouin zone corners16), let us therefore discuss briefly the Bloch wave properties at such special points. At such points in reciprocal space the Bloch waves can be classified according to their symmetry, as illustrated in fig. 5. The three upper rows show the symmetry and current density of the Bloch waves excited in a symmetrical four beam case [(020), (220) and (200) reflections at exact Bragg position]. It is seen that the first, symmetrical, wave has current density maxima at the atomic rows; whereas the third Bloch wave which is antisymmetrical with respect to the mirror planes (100) and (010) has maximum density in the channels between the atoms. The Bloch wave shown in the second row is degenerate, the fourth one is an antisymmetrical Bloch wave excited in an other direction, but belonging to the same point in the Brillouin zone. Symmetrical Bloch waves located near the atomic rows will be strongly absorbed, whereas the antisymmetrical waves will have zero density at atomic positions and be well transmitted. A simple estimate of the absorption coefficients /z~ of different Bloch waves can be obtained from their kinetic energies, as pointed out by Gj6nnes et al. 17) who proposed the expression /fi = ~ATJ + flUo,

(1)

where ~ and fl are constants and Uo the mean inner potential. A T j, the kinetic energies relative to the vacuum wave, are given by

A T ~ = 2 K ? i - ~ ICgl 2 2KSh,

(2)

h

where 7J is the anpassun9, and Sh the exitation errors. By calculating the kinetic energy differences, A T j, at special points in the Brillouin zone one can establish that the characteristic groups of spots in the channeling

patterns can be associated with antisymmetrical Bloch waves as is illustrated in fig. 5, in which the alloy Au(Pd) and UO 2 are chosen as examples. From figs. I f and l g it is seen that the most striking difference between the two channeling patterns is the almost complete absence in the UO 2 pattern of the group of four strong spots inside the ring of eight spots which are about equally strong in both patterns. This is reflected in the calculated kinetic energies and is obviously related to the fact that the associated Bloch wave in the U O 2 case will be located at the oxygen atoms and hence is strongly absorbed, whereas Au(Pd) has no atom at the same position. As will be expected from this argument; the perovskite, SrTiO3 which has the same type of [t301] projection as to UO 2 produces a similar pattern (fig. lh).

4. A multiple scattering theory of channeling The qualitative arguments of the previous section are based upon the absorption properties of independent Bloch waves. A quantitative treatment must include the distribution of intensity between Bloch waves as it is brought about through multiple diffuse scattering in the crystal. Recently H6ier 14) has extended the previous theories of single diffuse scattering in the presence of Bragg reflections and used this extension to explain the variations of Kikuchi band contrast with scattering angle and thickness. The channeling case can be seen as a high thickness limit, in which the pattern appears stationary apart from an absorption factor which is constant over the whole pattern. Let us write the intensity distribution as a sum over independent Bloch waves, i.e.

I(s+h) = ~ [C~(s)[ 2 AJ(s, z),

(3)

J

where s now is a scattering vector within the first Brill ouin zone of the projection. The weight factors, AJ(s, z) are the result of scattering into and out of each Bloch wave:

dAJ(s,z) = _ pJ(s) A~(s,z) + dz + f~, X AJ'(s',z) PJ'J" (s,s') ds '

(4)

where pi, J' (s, s') are probabilities for scattering between the Bloch waves j', s' and j, s. The stationary state is obtained when dAJ/dz = O. Assuming the scattering into all states to be the same, that is the integral in eq. (4) to be a constant, we obtain

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BLOCH WAVE TREATMENT

AJ(s,z ) ~

cons_______t !(s+h) oc ~. ICg(s)12/#(s). /(s) (5)

A similar result is obtained from the multiple scattering expression given by H6ierl*), viz. l ( s + h ) = E iC~i2 {A~ ~ E F,(s,g,g') C~C~, + J

o,o"

+ ~ A~ ~ F.(s,g) n>O

IC~12),

(6)

scattering coefficients which include absorption. We note that this expression includes excess/deficient contrast through the variation of F,(s, g) with s+g. This contrast will decrease with increasing thickness, because the background becomes more flat, i.e. the variation of F,(s,g) with angle becomes slower. Approximating this with a constant and neglecting the first term, we obtain:

I(s+h) --, y y~ 2 IC~f IGI 2 A~ = y~ IGI 2 ~ A~.

g

J

where F~ are normalized scattering factors for nth order multiple diffuse scattering and A~ are multiple

j

n

From the expression given by H6ier for A~ it is indeed found that they approach lip j in the high thickness

(a)

(c)

6' n

(b)

(d)

Fig. 2. Si at 100 kV, [001J-projection. (a) Experimental channeling pattern, (b) calculated pattern, (c) and (d) calculated current density distribution in a repetition unit o f the projected structure for radius o f the aperture R = (h2+k2) ½= 2X/2 and R = 4V/2 respectively. The projected Si atoms lie at the corners of the unit. (The dark net comes from the oscilloscope screen.) IV. C H A N N E L I N G

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limit. This leads to the same intensity expression as above. This is certainly an approximation, but the resulting expression retains the most important mechanisms affecting the pattern, viz. the distribution o f intensity within each Bloch wave through C~ and their absorption properties through p J, and is also sufficiently simple to permit calculations to be c o m p a r e d with the extensive and detailed experimental patterns. The current distribution across the unit cell follows immediately from the same treatment:

I(r, z)

l(r,z) = f~ ~ AJ(s,z) lbJ(r,s)12ds,

(7)

where the integral over s again should be taken over one Brillouin zone. If the probability o f emission o f

(a)

B(r)

secondary radiation can be given as a function of coordinate within the unit cell, the intensity o f e.g. X-ray emission can also be calculated by performing the integral

f~,

B(r) l ( r , z ) d r d z .

5. Comparison between experimental and calculated patterns Calculations o f intensity distribution for comparison with the observed patterns were performed mainly in the [001]-projection of UO2 and silicon using expression (5). The intensity was calculated at 66 points in the asymmetrical unit o f the Brillouin zone, 54 beams were used. The analytical expression for the scattering

(b)

(c) (d) Fig. 3. UO2 at 100 kV, [001]-projection. (a) Experimental channeling pattern, (b) calculated pattern, (c) and (d) calculated current density distribution in a repetition unit of the projected structure for radius of the aperture R = (h~+ k2)~ = 4 and R = 6 respectively. The oxygen atoms are at the center and the uranium atoms at the corners of the unit.

BLOCH

147

WAVE TREATMENT

amplitudes were taken from Doyle and Turner~8): the Debye-Waller factor was taken to be 0.5. The kinetic energy approximation (1) was used for the absorption parameters; the ratio ~/fl between the two parameters was fixed by trial. Calculations were performed on a CYBER 74, for the display routine a NORD-1 computer was used. The expression (5) does not include a fall-off of intensity with angle. Even for the thick crystals used here there is, of course, a radial decrease of the intensity

with angle. Therefore a small radial damping was introduced. Experimental and calculated patterns are compared in figs. 2, 3 and 4. For silicon at 100 kV (fig. 2) there is good agreement between calculation and experiment, apart from lack of fine details in the calculated patterns due to the limited number of points. For UO2, there is also broad agreement in the main features, but a distinct disagreement is seen in the central part of the pattern. The experimental pattern shows four faint diffuse spots at the coordinates (110) etc. whereas the calculation gives maxima at (100). A closer study of the Bloch waves involved at these points indicate that the absorption at the oxygen atoms may be overestimated in the calculations. Reduced absorption at the oxygen sites would lead to better penetration of the third Bloch wave shown in fig. 5 and hence to higher intensity at (110) etc. As a general rule the channeling patterns do not appear to be very sensitive to small changes in the a)

e)

b)

<

Eo1

AuPd

U0 2

0.326

0.206

(a) •

•

0

-0.096

- 0.0 8 6

-0.175

-0.034

-0.161

-0.118

[ 1 "

•

t

I

•

//

Ioo]

I

(b) Fig. 4. U 0 2 at ! MV, [O01]-projection. (a) Experimental channeling pattern, (b) calculated pattern.

Fig. 5. BIoch waves a r o u n d the fourfold [001]-axis. (a) Bloch wave s y m m e t r y , full a n d broken lines are s y m m e t r y a n d antis y m m e t r y lines respectively. T h e filled circles denote the reflections at the exact Bragg position. (b) Localization o f current density m a x i m a within a repetition unit o f the projected structure, open circles denote m a x i m a . T h e filled circles c o r r e s p o n d to metal a t o m s ; the hatched circles denote oxygen (in UO2). (c) T h e kinetic energy, ATJ, o f the Bloch waves at 100 kV for AuPd and UO2. IV. C H A N N E L I N G

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absorption parameters, however. Calculations for silicon, using Radis 19) values for the imaginary part of the scattering amplitudes revealed only slight changes from those based upon the kinetic energy approximation. For UO2 at 1 MV, which corresponds to primary energies in the 3-5 MV range for lighter elements, the agreement between the calculated and the experimental pattern is distinctly poorer. This may partly be due to an insufficient number of beams in the calculation. However, several of the most important features of the experimental patterns are reproduced by the calculations, in particular the group of eight spots which denote the first Bloch wave with antisymmetry about (100), (010), (110) and (110), i.e. the fourth row in fig. 5. This agrees with the conclusion of a previous note 16) in which it was shown that this particular Bloch wave is expected to retain its simple form up to quite high energies. The corresponding density distributions across the projected unit cell are shown in c and d of figs. 2 and 3. The calculations, according to eq. (7) were performed with two different limiting apertures. The effect of aperture size is seen to be small, indicating that the distribution is governed mainly by the groups, or rings, of beams around the centre of the pattern. As expected, there is little evidence for bound states around the atomic strings. 6. Conclusions

The present study shows that the channeling patterns recorded around a zone axis in the electron microscope can be explained to considerable detail by a distribution of independent Bloch waves in the energy range 100 kV to 1 MV. The distribution has been calculated using a simplified theory of multiple diffuse scattering in thick crystals and the kinetic energy approximation for the Bloch wave absorption. The agreement between experimental and calculated pattern is considered to be good, especially in view of the relatively few beams included. This is evidently connected with the fact that the main channeling features are associated with very

simple Bloch waves with few plane wave components. Some disagreements between calculations and experiments were noted for heavier elements and high voltage, however. This may indicate need for including more beams as the interaction is raised and, possibly for a more detailed model for the absorption. The Bloch wave distribution can also be used for calculation of the distribution of electron current density within the projected unit cell - and hence to calculate emission probabilities for secondary radiation from thick crystals. The authors are indebted to Dr Y. Uchida of Japan Electron Optics Laboratory for his kind assistance in taking the 1 MV channeling patterns and to Mr E. Sbrbrbden and Mr R. P. S6vik for their help in developing the program for the display routine. References 1) C.J. Humphreys and J. S. Lally, J. Appl. Phys. 41 (1970) 232. 2~ y . H. Ohtsuki, Phys. Stat. Sol. (b) 59 (1973) 303. 3) M.V. Berry, B. F. Buxton and A. M. Ozorio de Almeida, Rad. Effects 20 (1973) 1. 4) K. Kambe, G. Lehmpfuhl and F. Fujimoto, Z. Naturforsch. 29a (1974) 1034. 5) G. Honjo and K. Mihama, J. Phys. Soc. Japan 9 (1954) 184. 6) p. B. Hirsch, A. Howie, R. B. Nicholson, D. W. Pashley and M. J. Whelan, Electron microscopy of thin crystals (Butterworths, London, 1965). 7) E. Uggerh6j and J. U. Andersen, Can. J. Phys. 46 (1968) 543. 8) j. Lindhard, Kgl. Danske Vidensk. Selskab. Mat. Fys. Medd. 34 (1965) no. 14. 9) M. V. Berry, J. Phys. C 4 (1971) 697. 10) H. J. Kreiner, F. Bell, R. Sizmann, D. Harder and W. Htittl, Phys. Lett. 33A (1970) 135. 11) H. Kumm, F. Bell, R. Sizmann, H . J . Kreiner and D. Harder, Rad. Effects 12 (1972) 53. 1~) A. Tamura and Y. H. Ohtsuki, Phys. Stat. Sol. (b) 62 (1974) 477. 13) K. Komaki and F. Fujimoto, Phys. Lett. 49A (1974) 445. 14) R_ H6ier, Acta Cryst. A29 (1973) 663. 15) F. Fujimoto, S. Takagi, K. Komaki, H. Koike and Y. Uchida, Rad. Effects 12 (1972) 153. 16) j. Gj6nnes and J. Taft,5, Phys. Lett. 54A (1975) 55. 17) j. Gj6nnes, O. A. Hafnor and R. H6ier, Jernkont. Ann. 155 (1971) 471. 18) p. A. Doyle and P. S. Turner, Acta Cryst. A24 (1968) 390. 19) G. Radi, Acta Cryst. A26 (1970) 41.