Computational comparisons between the conventional multislice method and the third-order multislice method for calculating high-energy electron diffraction and imaging

ult rmicroscopy ELSEVIER

Ultramicroscopy 69 (1997) 219-240

Computational comparisons between the conventional multislice method and the third-order multislice method for calculating high-energy electron diffraction and imaging J.H. Chen, D. Van Dyck*, M. Op de Beeckl, J. Van Landuyt EMAT, Unioersi~ of Antwerp (RUCA), Groenenborgerlaan

171, B-2020 Antwerp, Belgium

Received 21 March 1996; received in revised form 2 February 1997

Abstract

The third-order multislice method (TOMS) for the calculation of high-energy electron microscopic diffraction patterns and images, as proposed by Van Dyck, is tested by detailed computations. Results calculated by the TOMS and the conventional multislice method (CMS) with different slice thicknesses and dynamical apertures (g,,,., values) are compared with the accurate results. It is pointed out that for both the TOMS and the CMS there are basically two types of errors. One is the intrinsic error imposed by the order of the method in slice thickness, which appears as the pseudo HOLZ (high-order Laue zone) effect in the diffraction patterns, Another is the numerical error caused by the finite dynamical aperture, such as the aliasing error and the intensity-loss error. For zero-order Laue zone (ZOLZ) calculations, it is shown that the intrinsic errors are the dominant errors for both the TOMS and the CMS since the elimination of the intrinsic error leads to the disappearance of the numerical error, as long as the dynamical aperture is large enough to cover all the ZOLZ reflections. It is also shown that the TOMS has much smaller intrinsic error than the CMS for a large slice thickness and therefore is superior to the CMS with respect to accuracy vs. computational time for ZOLZ calculations. Nevertheless, the normalisation of the total intensity, as an error criterion, is more reliable for the TOMS than for the CMS. Hence, TOMS is a feasible and competitive procedure for dynamical calculations in high-energy electron diffraction (HEED). Possible error sources of practical multislice procedures are thoroughly discussed. PACS:

61.14.D~; 61.16.Bg

Keywords:

Multislice

methods;

Error sources; Normalisation

criterion

* Corresponding author. ’ Postdoc Researcher with the National Fund for Scientific Research (FWD), Belgium 0304-3991/97/$17.00 :Q 1997 Elsevier Science B.V. All rights reserved PI1 SO304-3991(97)00052-l

220

JH.

Chen et al. / lJ1tramicroscop.v 69 (1997) 219-240

1. Introduction

Since it was originally proposed by Cowley and Moodie from optical principles [l], the conventional multislice method (CMS) has extensively been used not only for HRTEM image simulations but also for microdiffraction pattern calculations [2,3]. In many of the areas of high-energy electron diffraction and imaging, it is currently the most efficient numerical procedure for dynamical calculations [4]. On the other hand, however, from the quantum mechanical point of view, the CMS formula is only a first-order method in the slice thickness if used for higher-order Laue zone (HOLZ) calculations and of second-order if only dealing with zero-order Laue zone (ZOLZ) reflections [S]. Hence it should be less effective than the higher-order multislice methods proposed by Van Dyck [6,7]. Of these methods, the second-order method has been shown to be more effective than the CMS procedure for calculating HOLZ effects [S, 91, but the third-order multislice method (TOMS) (for ZOLZ) has never been tested thus far for numerical calculations. In a sense, a test of this higher-order method is not only very useful for building up an efficient alternative for dynamical calculations but also helpful for understanding the sources of computational inaccuracy of the CMS which can be rather severe in some cases [S, 10, 111. Indeed, although the CMS has been used for decades and some successful algorithms, such as the FFTCMS (fast Fourier transformation CMS) [12], the RSCMS (real-space CMS) [6, 13, 141 and the anti-aliased CMS [15] have been developed to improve the computational speed and the accuracy of the CMS, the practical error sources of the CMS have not been well elucidated. This often causes some confusion in the use of the CMS: e.g., (i) the same CMS program will yield different results if different dynamical apertures are used for the calculation [ll], (ii) an intensity loss up to 10% is considered acceptable in the anti-aliased CMS [ 151 although this is obviously against the intrinsic property of the CMS, (iii) theoretically, the CMS converges to the quantum-mechanics solution only if the slice thickness approaches zero [16]. but in practice the slice thicknesses within l-2 A (or even

more) seem to be good enough [ 151 even in the case without including the Debye-Waller factors in the calculation, (iv) in which cases, the anti-aliased CMS and the non-anti-aliased CMS will converge to each other? In the present work, the validity of the TOMS as a practical procedure for calculating high-energy electron diffraction and imaging is checked by numerical calculations and the obtained results as well as the related errors are compared with those calculated by the CMS. It is shown that the TOMS is a feasible and competitive procedure since for the same computing time it gives more accurate results. Different error sources of practical multislice procedures are discussed in detail with respect to the accurate calculations.

2. Theory The multislice methods can be described by either the conventional physical optics approach in which the propagation of a wave is expressed as a propagator function convoluting the wavefunction [15], or the equivalent quantum mechanics expression which depicts the propagation by a propagator operator acting on the wavefunction [3]. For many cases, especially for understanding the validity of a multislice method, the latter proves more convenient [ 171. 2.1. Formulation

of CMS and TOMS

TOMS was originally developed by Van Dyck, based on the basic Schrodinger equation for highenergy electron diffraction (HEED) [7]. Since this procedure has never been programmed and used thus far for practical calculations, we repeat in more detail the formulations and functions which are necessary for this purpose. Nevertheless, for the sake of comparison, the CMS and the TOMS will be introduced in a systematic way, in which all the multislice methods are considered as numerical integration methods for solving the following modified-Schrodinger equation for HEED [ 171: F

= [d + V(r)]Y(r)

(1)

J.H. Chen et al. / Ultramicroscopy 69 (1997) 219-240

with A = iA/47cv2 and V(r) = iaU(r), where 1 and 0 are, respectively, the electron wavelength and interaction constant. v2 is the Laplacian operator in the x-y plane and U(r) is the crystal potential. In the systematic approach to the multislice theory, a method is said to be of order n in the slice thickness E if the error is the order s”+ ‘. Solving Eq. (1) in the multislice scheme, in which the crystal is considered as a combination of a series of thin slices perpendicular (or nearly perpendicular) to the incident beam direction, the transmission of electron wave through one slice can then be described by the dynamical scattering operator (see, e.g., Ref. [S]) B,(R) = eEtd+“J,

(2)

where Eis the slice thickness and l/j = ioUj (R) with the average potential Uj(R) = l/s pG_ijE U(R, z) dz. The dynamical scattering through n slices can then be calculated by a repetitive action of operators of the type j for each slice: Y,(R,z=ns)=B,B,_,...B,‘Y,(R,z=O),

221

is relatively easy to obtain so that it is an ideal case for checking the validities of practical numerical procedures for dynamical calculations. Till now, many of the computational comparisons between different dynamical theories have been made in the pure ZOLZ case (see, e.g., Refs. [13, 15, 183). Although the operator of Eq. (2) can be calculated directly with the real-space algorithm [14,19,20] by expanding it to high-order terms (e.g.. up to the lOth-order term by Chen [21]), it is not suited for using the fast Fourier-transformation technique (FFT). In order to utilise the advantage of the FFT technique for fast calculation and at the same time preserve higher accuracy, Van Dyck [7] proposed to factorise the operator in terms of functions of A and Vj to higher orders in the slice thickness E, so that the CMS and the TOMS were introduced in a systematic way. However, it should be pointed out that the use of FFT to speed up the CMS calculation was not introduced in the original CMS paper [l] but later by Ishizuka and Uyeda [12].

(3)

where Yy,(R) is the incident wavefunction (or boundary condition). It is clear that Eq. (3) will yield the exact solution of Eq. (1) if the slice thickness approaches zero. For the exact solution, however, there are two cases which can be discussed separately: One is the general case, in which the HOLZ effects are included by using different slice potentials so as to describe potential variations of the crystal along the beam direction [S, 161. Another is the pure ZOLZ case in which all the slices are assumed to have the average

2.1.1. The second-order expansion - CMS The second-order expansion of the operator is e&(A + “J z e @/2)4e”“, eW2)d

(5)

By repeating the operator for each slice, we obtain the usual CMS formula except for a starting propagator and an ending propagator over half a slice thickness [5] yy,(R) = e’- W[e=’ e&vfi . . . e&Ae~“~]e(~/~)+/o(~)

potential o(R) = l/c f U(R, z) dz (c is the periodic0

ity along the z-axis) and therefore the exact solution is simplified as

The pure ZOLZ case is important and therefore often used in the HEED case for the following reasons: (i) in many cases of HEED, HOLZ effects are much weaker than ZOLZ effects so that the pure ZOLZ is a good approximation of the general case; (ii) in the pure ZOLZ case, the exact solution

= e&Ae&V8 . . . eEAe”“1 yo(~),

(6)

where eEAand eEA,respectively, are the well-known Fresnel propagator and the phase grating. So in this way, the CMS formula automatically becomes a second-order expansion of the operator, Eq. (2) by simply shifting the whole crystal by half a slice [7]. This does not influence the final results for the diffraction intensities and only causes a focal shift of half a slice for the high resolution images. For performing the calculation of Eq. (6) with the FFT, the following relation is used (see

222

J.H. Chen et al. / Ultramicroscopy

appendix):

69 (1997) 219-240

while the gratings (not phase gratings) in real space have the following forms:

F[e”“Y(R)] = e-irrh~K’@(K),

(7)

where F indicates the 2D Fourier transformation and Y(R) z Q(K).

(12)

Since Y(R), -eedy(R) = e(in”e)R”*

(8)

where the asterisk indicates a convolution, it is clear that the optical formulation of the CMS (using the convolution) is equivalent to the quantum mechanical description (using the operator). 2.1.2. The third-order expansion - TOMS The third-order expansion of operator (2) is [7]

a=:(3

*id),

b=l-a.

(9)

This expression has been established by identifying all the terms with the corresponding terms in the Taylor expansion up to third order. Only one solution in closed form meets these requirements. The TOMS procedure now consists of repeating this operator sequence for each slice: 'Y,(R)= ,-@a/W

fi eW2)4 e""YeW2M ebeY

[

j=l

1

In the CMS the phase-gratings as well as the propagators are unitary operators so that they keep the total intensity normalised (no matter how large the slice thickness is). But in the TOMS the normalisation is violated in the gratings and some of the propagators. Eqs. (lOH12) show that in the TOMS the gratings and some of the propagators will loose intensity while some of the others will gain. Theoretically, it can be expected that if the slice thickness is small enough, the intensity loss is compensated by the gain so that the normalisation of the wavefunction is maintained. In other words, in the TOMS normalisation naturally guarantees the correct slice thickness. 2.2. Errors

Assuming that the total crystal thickness is NC (c is the periodicity of the crystal in the z-direction), i.e. the crystal is N unit cell thick, and each of unit cells is divided into m slices, according to Ref. [S], the wavefunction at the exit face, Eq. (3) can be expressed as

x e(ba’2)d Y&t) = ew2)d

fi cad’,eW2MebeV,eWW e-‘““‘Wyo(~)~

j=l

1

(10)

In reciprocal space the propagators take the following forms F[e(E’2)d]= exp [ - (irchs/2)K2],

@/2)A]=exp[ FCe

-(nAc/4)(if$)lP],

(bfi2)d]=exp[ FCe

-(n&/4)(i&$)Kz],

(11)

_

[ec(d+v)

+

hnl

N‘y,(R)>

(13)

where v is the average potential of the crystal along the z-axis and ecfd+ ‘) represents the ZOLZ effects while &,,denotes the HOLZ effects in one unit cell. Ref. [S] shows that 6,,, is a complicated expression related to the potential variation along the z-axis and will reduce to zero if all the slices have the same potential (the pure ZOLZ case). Obviously, wavefunctions calculated by the CMS and the TOMS are not exactly the same as Eq. (13) because of the approximations, Eqs. (5) and (9). According to Van Dyck [7], the error

J.H. Chen et al. J Ultramicroscopy 69 (1997) 219-240

the TOMS can be viewed and compared clearly.

operator of slice j for the CMS is jp J

E3

= _-e*?” 24 J

+ . . . =&[A

GE4

4ms

mej

+

=$$A

“’

+ 21/.1’ [d + I/.1, [d ) V.]]] + J

...

.

(15) So corresponding to Eq. (13), the wavefunction obtained by the CMS is

Y’$‘-(R) =

ec(d

+

P) + km

1 N

+ g

,i ,ip J-1

+ ...

ye(R)

(16)

1

(17)

and for the TOMS yim;““(R)=

3. Computations, programming

and that for the TOMS is J

more

+ 2T/i,[A,Vj]] + “‘3

(14)

jps =

223

eccd+ ‘) + &,

From these expressions as well as the expression of the HOLZ operator given in Ref. [S], it can be seen that when the slice thickness approaches zero, the errors reduce to zero whereas the calculated HOLZ effects converge to their real values. This is hy multislice procedures can be used to calculate not only ZOLZ but also HOLZ reflections. However, because the errors of the CMS and the TOMS due to the finite slice thickness will intuitively appear as pseudo HOLZ reflection rings (as it is shown later), which will always be located at the exact positions of real HOLZ reflection rings if, as people usually do, the slice thickness is taken as an integral fraction of the periodicity, most of the calculations in the present work will be made in the pure ZOLZ case where the projected potential of the crystal is used for all the slices and therefore no real HOLZ reflections are involved, i.e., the calculated wavefunctions will be the ZOLZ reflections plus errors. In this way the errors of the CMS and

numerical errors and techniques

The CMS and the TOMS are programmed using the FFT algorithm and calculations are carried out for YBazCu307 _-x [0 0 l] and Au[O 0 1) which is believed being a good example for testing dynamical theories [ 151. The lattice parameters and thermal Debye-Waller (DW) factors (at room temperature) for Au (FCC crystal), respectively, are 4.074 A and 0.56, and those for YBa#&O,_, crystal are the same as used,in Ref. [S]. In addition, the same accelerating voltage of 200 kV is used for all the calculations. Although it is strongly recommended that for practical calculations the DW factors for the atoms should always be included [S], many of the calculations in the present work are still made without including the DW factors in order not to hide the errors. This, however, means that for the simulation of a crystal at very low temperature the calculation errors can be more severe than at room temperature and therefore a thinner slice thickness will have to be used for multislice procedures. To compare the CMS with the TOMS we will mainly study the convergence and the increase in accuracy with decreasing slice thickness. However, for practical multislice procedures, there is another type of error which occurs when the dynamical aperture or gmaxused for the calculation is not large enough. In this work, we call intrinsic error, the error caused by the order of the method in the slice thickness, as described in Section 2.2, while we call numerical error, the error caused by a gmaxwhich is not large enough for the calculation. In most of the practical multislice calculations, these two types of errors are mixed up and have never been elucidated clearly. This causes confusion in choosing the proper gmaxvalue for the calculation and the proper technique for programming multislice procedures. The fact which causes such confusion is the following: in the pure ZOLZ case, for instance, the dynamical aperture needed to cover all the ZOLZ reflections is actually very limited. However, for

224

J.H. Chen et al. 1 Ultramicroscopy

4---

2 gmax+

(b) -22gmm---)

Fig. 1. Illustration of (a) the aliasing effects allowed in the TTCMS and (b) the cut-off effects allowed in the ATCMS.

practical multislice procedures, a much larger pseudo dynamical aperture will be imposed by the intrinsic error of the method since it appears as large-angle pseudo HOLZ reflections (see further calculated results). So it seems difficult to estimate the appropriate gmaxfor a multislice calculation. On the other hand, for programming a multislice procedure, there are two typical techniques for dealing with numerical errors: One is the traditional technique (TT), in which the dynamical aperture for the wavefunction (in reciprocal space) is the same as that for the phase-grating and therefore an aliasing error (or effect) can occur if the gmaxused for the calculation is not large enough to cover all

69 (1997) 219-240

the pseudo HOLZ effects (Fig. la) [12, 151. Another technique is the so-called anti-aliasing technique (AT) [15], in which the dynamical aperture for the wavefunction is only half (the one/one-half rule) or 4 (two-thirds/two-thirds rule) of that for the phase-grating array in reciprocal space but this is done during the calculation by repeatedly setting zeroes in the wavefunction array for the outer f or 3 reflections. In this way - by continuously cutting off the outer reflections (Fig. lb), an intensity-loss error will appear if the gmaxused for the calculation is not large enough to cover all the pseudo HOLZ effects. It is clear that the results obtained by a TTprogram will differ from those calculated by an AT-program if the gmaxused for the calculation is not large enough to cover all the pseudo HOLZ effects, but the TT-result and the AT-result will converge to each other when the gmaxbecomes large enough to cover all the pseudo HOLZ effects. So it seems again difficult to say which of the two techniques is better. However, for the purpose of comparing the intrinsic errors of the CMS and the TOMS, it is necessary to use the gmax as large as possible in order not to mix up the two types of errors. In this case, it is clear that using the TT for programming is better than using the AT since for the same result the AT requires a computer memory four times that for the TT and consequently much more computing time. In the present work, a gmaxas l:rge as 15.70~-‘forAu[OO1]andag,,,of16.48A-’for YBazCu307 --x [0 0 1) are used - corresponding to the sampling of 128 x 128 points in the x-y plane. But some results calculated for smaller gmaxvalues will also be presented in order to discuss the numerical errors for aliasing and intensity-loss (or cut-off). In most of the following calculations, the CMS and the TOMS are programmed with the TT. But we will compare the TT results with the AT results in Section 4.2.

4. Results and comparisons In this work, we present the calculated amplitudes of diffraction beams in two forms: (i) as amplitude-diffraction patterns at a certain thickness (Figs. 2 and 9) and (ii) as amplitudes plotted versus

J.H. Chen et al. J Ultramicroscopy 69 (1997) 219-240

Fig. 2. Amplitude-diffraction patterns (see the text for details), calculated by the CMS and the TOMS with different slice thickness rrolm Au[O 0 l] for a crystal thickness of 4Oc, (c = 4.074 A) and the gmvx value of 15.70A-‘; (b) from YBa2Cu,0,_, [00 (a) 1 a cr ‘yst al thickness of 4Oc (c = 11.6817 A) and the g,,, value of 16.48 A-‘.

225

c/n: for

226

J.H. Chen et al. / Ultramicroscopv

the crystal thickness. Since most of the pseudo HOLZ reflections (but not all of them) are very weak with respect to the main ZOLZ beams, and in order to intuitively show the pseudo HOLZ effects (or the intrinsic errors) in the form of diffraction patterns we choose the following amplitude range for displaying the amplitude-diffraction patterns: a zero amplitude value is set for all the reflections whose intensities are less than 9.0 x 10e6 and a fixed maximum amplitude value of 7.0 x lo-’ is set for those which have intensities larger than 4.9 x 10V3. For the plots, however, we present the amplitudes in absolute values. 4.1. Intrinsic errors and convergence of the CMS and the TOM? In order not to mix up intrinsic errors and numerical errors, we used very large gmax values to calculate ZOLZ reflections. Fig. 2a and Fig. 2b show that the intrinsic errors of the CMS and the TOMS appear as pseudo HOLZ reflections. In these patterns we can see the

69 (1997) 219-240

following. (i) upon decreasing the slice thickness the pseudo HOLZ effects can be driven out to largeangle positions and will then be damped down by the scattering factors of the atoms. So accurate numerical ZOLZ solution can be achieved by both the CMS and the TOMS if the slice thickness is small enough. (ii) For the same slice thickness the intrinsic errors of the TOMS are much less than those of the CMS. This can be seen more clearly in Fig. 3. The convergence of the ZOLZ reflections calculated by the CMS and the TOMS with different slice thicknesses are shown in Figs. 4 and 5. Fig. 4 shows that for YBazCu30,_, [0 0 l] the TOMS results with slice thicknesses c/8 and c/4 are much better than the CMS results with the same slice thicknesses and even better than those CMS results calculated with thinner slices. The faster convergence of the TOMS are more clearly shown in Fig. 5a and Fig. 5b, where we see that for Au[O 0 11 the TOMS results with slice thickness c/2 are almost exactly the same as the CMS results with slice thickness c/8.

YBCO[OOl]: gmax=16.48

’

0.015

,

’

-1

’

++p~;‘~

i

d d d

W

g

b/

pseudo HOLZ: 2000

0.010 -

cl

/ ;

$

P'

I 0

.I

I 10

, , , 20

.

I

I

30

I

40

50

THICKNESS(in c-unit) Fig. 3. Amplitudes of pseudo HOLZ reflection 20 0 0 from YBa2Cu30, CMS and the TOMS with the slice thickness c/4 (c = 11.6827 A).

_-I [0 0 l] plotted

against

crystal

thickness,

calculated

by the

69 (1997) 219-240

J.H. Chen et al. / Wramicroscopy

221

YBCO[COl]: gmax=16.48

t

0.8

1

-. 0.4 -

0.2 -

1

8 I

~

.,

I

0

,..,

I

I

,I..

30

20

10

1

..I.

..,.

1

iI 50

40

THICKNESS (in c-unit) YBCO[OOI]: gmax=16.48 x10

-4

II

’

”

6

”

I

”

-

”

TOMS: - + - TOMS: - -- - TOMS: -QCMS: --DCMS:

1.2 -

0

10

20

”

c/16 c/8 c/4 c/16 c/8

I

”

7 ’

I

”

I

t I! I!

,1

d.I

30

”

40

50

THICKNESS (in c-unit)

Fig. 4. Amplitudes of the ZOLZ reflections from YBa&u@_, the TOMS with different slice thicknesses c/n.

4.2. Convergence numerical errors

of the CA4S and the TOMS with

To see whether or not it is necessary to use very large pseudo dynamical apertures for the ZOLZ

[0 0 l] plotted against crystal thickness, calculated by the CMS and

calculation, we present the results calculated with gmax = 3.925 k1 for Au[O 0 l] (the sampling of 32 x 32 points), which is not large enough to cover all the pseudo HOLZ effects but quite sufficient to cover all the significant ZOLZ effects. In this

228

J.H. Chen et al. / Ultramicroscopy

69 (1997) 219-240

*“[MII.gmax=15 70

LO-

0

i

(a)

tm..,.i....i.,. ’ 10

Fig. 5. Amplitudes with different

20

8. 30

.,I

40

,..h

5”

THICKNESS ,I”c-u”,,) of the ZOLZ

slice thicknesses

reflections

’

”

(b) from Au[O 0 l] plotted

c/n: (a) reflection

0 0 0; (b) reflection

section, the TTCMS and the TTTOMS, respectively, refer to the CMS and the TOMS programmed with the TT, while the ATCMS refers the CMS programmed with the AT. The convergence of ATCMS and TTCMS with decreasing slice thickness is shown in Fig. 6aFig. 6c. From these we see the following: (i) when decreasing the slice thickness, the intensity-loss error approaches zero and the ATCMS results become convergent. (ii) the TTCMS results converge faster than the ATCMS results. However, Figs. 7 and 8 show that the TTTOMS converges better and again the TTTOMS results obtained with slice thickness c/2 are as accurate as the TTCMS results obtained with slice thickness c/8. It should be noticed that all the convergent results obtained by the ATCMS, the TTCMS and the TTOMS are not exactly the same as the most

against

crystal

thickness,

Xl 30 THKKNESS (1”E.U”II) calculated

40

5”

by the CMS and the TOMS

8 8 0.

accurate results obtained with a very large dynamical aperture and there are small differences between them. This is because there is still a very small amount of numerical error for this gmax value. 4.3. Results of the CA4S and the TOM!? upon including the D Wfactors Upon including the DW factors in the calculation, larger slice thicknesses can be used for both the CMS and the TOMS. This is shown in Figs. 9 and 10. However, with increasing slice thickness, the pseudo HOLZ rings move closer to the ZOLZ center. Since the pseudo HOLZ effects of the TOMS are still much weaker than those of the CMS (Figs. 9 and 1 l), a much larger slice thickness can be used for the TOMS to achieve an acceptable accuracy (Fig. 10): the accuracy of the TOMS with

J.H. Chen et al. J Ultramicroscopy

slice thickness c (11.6827 A for YBazCu@-, [0 0 11) is comparable with that of the CMS with slice thickness c/4 (2.9207 A). For calculating real HOLZ reflections with multislice methods, the potential variation along the beam direction has to be included by using very fine but different slices and the DW factors have to be included in the calculation [S]. Fig. 12 shows that the TOMS can calculate real HOLZ reflections similar to the CMS. 4.4. Normalisation

criterion for the CMS and the

TOMS

The normalisation of the calculated wavefunction for the TTCMS is always perfect and does not give any information about whether or not the slice thickness is taken small enough and the gmax is

69 (1997) 219-240

229

large enough. But the normalisation for the TTTOMS does give the information about the slice thickness. This can be seen in Table 1 and Figs. 13 and 14. Fig. 13 shows that a divergence of the TTTOMS (without including the DW factors) begins to occur approximately at the thickness of 30~ (350.48 A)0 when a large slice thickness of c/2 ( = 5.841 A) is used for the calculation. Table 1 lists the maximum deviation values of total intensity for the two procedures with different slice thicknesses, calculated for the range from 0 to 584.14 A (50~). It is shown that a perfect normalisation of the wavefunction can be achieved by the TTTOMS when a very fine slice thickness is used. It is also shown that the divergence of the TOMS with a large slice thickness can be overcome when DW factors are included. More clearly, Fig. 14 shows the relation between the accuracy of the TOMS

Fig. 6. Amplitudes of the ZOLZ reflections from Au[O 0 11 plotted against crystal thickness, calculated by the ATCMS and the TTCMS with the gmax value of 3.925 A-’ and different slice thicknesses c/n: (a) reflection 0 0 0; (b) reflection 2 2 0; (c) reflection 8 8 0. The intensity-loss errors for the ATCMS are indicated by the total intensity value ‘I = ’ at the crystal thickness 50~.

J.H. Chen et al. I Ultramicroscopy

-I

1

880

i 2

0

O

10

20

Id

m-s

33

40

50

cm c-unit1

Fig. 6. Continued.

results and the normalisation criterion. However, it is shown in Fig. 1.5 that the normalisation can still be achieved for the TTTOMS with a too small 9 maxwhich is not enough to cover the ZOLZ effects. In Fig. 6, however, we have seen that the normalisation for the ATCMS will also give informaslice thickness and a better tion about normalisation leads to a higher accuracy of the ATCMS results. But Fig. 15 shows that a perfect normalisation can also be achieved for the ATCMS with a too large slice thickness by using a very large dynamical aperture. So the normalisation criterion for the ATCMS seems to be very confusing.

5. Discussion In the pure ZOLZ along the z-direction

case, the potential is neglected and

variation the exact

69 (1997) 219-240

wavefunction at the exit surface (z = ns) can be expressed by Eq. (4) which yields only the ZOLZ effects. However, by using multislice approximations such as the CMS [Eq. (5)] and the TOMS [Eq. (9)] for the calculation of Eq. (4), an artificial periodicity E = c/m is created and this automatically results in pseudo HOLZ effects in the calculated diffraction patterns (Fig. 16). The pseudo HOLZ effects (or ‘false’ up-layer reflections [ 161 or pseudo up-layer lines [17]) are well known for the CMS as a kind of numerical phenomena which occurs when the phase of the Fresnel propagator in reciprocal space equals 2rtM (M = 1, 2, . ..) (e.g., Refs. [16,17,8]). However, for the TOMS the pseudo HOLZ effects appear when the phase of the Fresnel propagator in reciprocal space equals nM and they are much weaker than those in the CMS results. So the pseudo HOLZ effects of a multislice procedure should be understood as an intrinsic error of the method imposed by the order of the method in the slice thickness. Hence, for a certain slice thickness, using a higher-order method such as the available TOMS is the only way to achieve a higher accuracy in the calculation of ZOLZ reflections. On the other hand, for the same calculation accuracy, the TOMS allows to use a much larger slice thickness than the CMS does. This means that the TOMS is superior to the CMS with respect to accuracy vs. computational time. Since the TOMS uses two propagators and two gratings in a singlemultislice operation [see Eq. (lo)], for the computational time we have to compare this method with the CMS which uses half of the slice thickness. In the present work, most of the calculated results such as those given in Figs. $8 and 10 show that to achieve a reasonable accuracy the TOMS is twice as fast as the CMS. We have seen that for both the CMS and the TOMS the pseudo HOLZ rings can be eliminated from the central part of the diffraction patterns by using a smaller slice thickness and may finally be damped down by the atomic scattering factors for high-energy electron. When the slice thickness approaches zero the intrinsic errors of the CMS and the TOMS reduce to zero and the calculated results converge to the accurate solution. When the DW factors are included in the calcu-

J.H. Chen et al. / Ultramicroscope

231

69 (1997) 219-240

Au[OOl]: gmax=3.925

r

0.8 -

THICKNESS (in c-unit) Au[OOl]: gmax=3.925 x10

-4

3

t

“a z

2

$

1

0 0

10

20

30

40

50

THICKNESS (in c-unit) Fig. 7. Amplitudes of the ZOLZ reflections from Au[O 0 l] plotted gmar value of 3.925 A-’ and different slice thicknesses c/n.

lation, the intrinsic errors larger slice thicknesses. understand why, for the slice thicknesses within

can So, use l-2

be damped down for from these, we can of the CMS, finite A (or even larger)

against

the crystal

thickness,

calculated

by the TTTOMS

with the

may be good enough for most of the practical cases of high-resolution transmission electron microscopy (HRTEM) image simulation [15], even without including the DW factors: this is

J.H. Chew el al. I Ultranzicroscop~~ 69 (I 9~7) 2 I Y 230

232

Au[OOl]: gmax=3.925

I

I

”

I,

”

I

“I,

-Q

I

‘,

I,

I.,

i

- TTCMS: c/8

- -D TTCMS: cl4 - --p - TTCMS: c/2 - -- - TTTOMS: c/2 I

1.1

1 1.0 g

2 3

0.9 -

8 0.8 -

0.7 -

0

IO

30

20

40

50

40

50

THICKNESS (in c-unit) Au[OGl]: gmax=3.925 x10

-4

-+I- TTCMS:c/4 --v- - TTCMS: c/2 - -r - T’ITOMS: c/2 3

0

10

20

30

THICKNESS (in c-unit) Fig. 8. Amplitudes of the ZOLZ reflections from Au[OO l] plotted against crystal TTTOMS with the qmax value of 3.925 A-’ and different slice thicknesses c/n.

because the intrinsic error of the CMS appears as large-angle pseudo HOLZ effects, which have weak intensities and do not directly contribute to the simulated images, in the same way as real large-

thickness.

calculated

by the TTCMS

and the

angle resonant HOLZ effects are not important for the HRTEM image simulation [lo, 51. However, for calculating real HOLZ effects with the CMS (and the TOMS as well). much thinner slice

J.H. Chen et al. J Ultramicroscopy

69 (1997) 219-240

233

Fig. 9. Amplitude-diffraction patterns (see the text for details) from YBa&u,O,_, [0 0 l] for a crystal thickness of 4Oc (c = 11.6827 A) and the gmar value of4.12 A-‘, calculated by the CMS and the TOMS with different slice thicknesses c/n including the DW factors in the calculation.

thickness as well as the DW factors have to be used [S]. We have seen that for accurate ZOLZ results, the dynamical aperture (or gmax value) is very limited. But using the CMS and the TOMS with finite slice thicknesses for the calculation of ZOLZ reflections, a very large pseudo dynamical aperture will automatically be imposed by the intrinsic errors (pseudo HOLZ reflections) of the calculation methods. This fact causes the following confusion in the use of the CMS: (i) For a certain slice thickness the same CMS program will yield results which change upon changing the gmax, even when all the gmax values are large enough to cover the ZOLZ area. (ii) for the same slice thickness and gmax(large enough to cover the ZOLZ area), different CMS programs such as the TTCMS and the ATCMS programs will give different results. The reason for case (i) is that

different gmax values lead to different amounts of numerical errors, while the reason for case (ii) is that different programs allow different numerical errors, e.g., the TTCMS causes aliasing errors but the ATCMS leads to intensity-loss error (or artificial cut-off error). However, all those errors arise from the intrinsic error of the CMS as long as the gmax used is large enough to cover the ZOLZ area: decreasing the intrinsic error, e.g., by decreasing the slice thickness, the amount of numerical error will be reduced and when the intrinsic error (or pseudo HOLZ effects) reduces to zero, there will be nothing to be aliased or cut-off and all those calculations should yield the same results. So for accurate ZOLZ calculation with the CMS or the TOMS it is not necessary to use the very large pseudo dynamical range and the AT for programming which requires a computer memory of four times that for

234

J.H. Chen et al. 1 Ultramicroscopy

69 (1997) 219-240

YEKO[OOl]: gmax=4.12; with DW factors

t

“.

’

“,

.

”

”

”

”

”

”

‘i

I

TOMS: c/4 - -- - TOMS: c/l -+CMS: c/g -QCMS: cl4 -13CMS: cl2 --wCMS: c/l

i i I

=i 8

0.02

0.01

0.00 0

10

20

30

40

50

THICKNESS (in c-unit)

YBCO[OOl]: gmax=4.12; with DW factors x10

1

-:-

0

10

20

30

40

50

THICKNESS (in c-unit) Fig. 10. The same as for Fig. 4 but with the gmax value of 4.12 k

the TT (for the one/one-half rule) and consequently much more computing time, as long as the gmax used is large enough to cover the ZOLZ area and the slice thickness is small enough to remove

’ and including

the DW factors.

significant pseudo HOLZ effects. It should be noticed that for the same g,,, the TOMS will have less numerical errors than the CMS since it has smaller intrinsic error than the CMS.

235

J.H. Chen et al. / Ultramicroscopy 69 (1997) 219-240

YBCO[OOl]:gmax=4.12; with DW factors -3

x10

I

”

I

’

1

“,

‘,

I

1

1

’

1

x

,-I

!

I

‘9 - -- - TOMS: c/l CMS: cl2 CMS: c/l

:

--+--v--

8

0.8 -

$

0.6 -

3

:

1’

?

;

\

\ :

‘\

I

Y

\

1

1’

:

I

I I’ ;

pseudo HOLZ: 14 0 0

0.4 r-T\ 0.2

-

30

20

10

0

50

40

THICKNESS (in c-unit)

Fig. 11. Amplitudes of pseudo HOLZ reflection 14 0 0 from YBa,Cu,O,_, [O 0 l] plotted against crystal thickness, calculated by the CMS and the TOMS with different slice thicknesses c/n (c = 11.6827 A) and including the DW factors in the calculation.

YBCO[tXll]: gmax=8.24; with DW factors x10-3 11

”

--+8 _ -

0

1

”

”

”

”

I””

”

“‘1

?

‘IEMS: c/32 C&Is: c/64

10

30 20 THICKNESS (in c-unit)

40

50

Fig. 12. Amplitudes of real HOLZ reflections 5 9 1, 14 0 2 and 17 0 3 from YBaaCu,O,_, [O 0 l] pJotted against crystal thickness, calculated by the CMS and the TOMS, respectively, with the slice thickness c/64 and c/32 (c = 11.6827 A). The DW factors for atoms are included and the qmaxvalue of 8.24 A-’ is used in the calculations.

236

J.H. Chen et al. 1 Ultramicroscopy

69 (1997) 219-240

Table 1 Cases of the normalization of wavefunctions with different slice thicknesses

from YBa&u,O,_,

[0 0 l] diffraction,

)I - I(,,,

Slicing

cl2

cl4

Without DW With DW Without DW With DW

Divergency 7.0x 1o-3 < lo-‘2 < lo-r2

3.1 3.3 < <

TOMS

procedure method

CMS method

calculated

by the TOMS

cl8 x 1o-2 x lo-” lo-‘2 lo-r2

4.3 2.0 < i

x 10-Z x 10-h lo-l2 lo-”

and CMS procedures

c/16

c/32

9.2 6.0 < <

1.0 x 1o-6

x 10-j x lo-’ lo-” lo-‘Z

< lo-‘* < lo-‘2

YBCO[OOl]: gmax=8.24 1”‘.

:’

I

““I’,

,I, . I :

rxd

2.’ .

I-

rC~-TrC.~--r*.---rrC~--rr,t---_C_*~--rr I 0

Zr

I

I

I

,

10

20

30

40

THICKNESS

I 50

( in c-unit)

Fig. 13. The total intensity of the TOMS with a large slice thickness c/2 (c = 11.6827 8) plotted against crystal thickness, calculated for YBa,CusO, --x [0 0 l] and without including the DW factors in the calculation. showing the computational divergence of the TOMS for a too large slice thickness.

However, if the gmax values used are not sufficient to cover the ZOLZ area, the numerical error can never be removed even when the intrinsic error has been reduced to zero, since in this case the numerical error arises not only from the pseudo HOLZ but the ZOLZ itself. In practice, it is hard to determine the size of ZOLZ area: the ZOLZ area for a thin crystal seems to be larger than that for the thick crystal. But from the calculated results we see that a gmax as large as 4 A-’ will be appropriate even for atoms as heavy as Au.

For choosing the slice thickness for the ZOLZ calculation, we have seen that a slice thickness of 2 A may be appropriate for the TOMS to obtain a reasonable accuracy, even for heavy-atom materials such as Au[O 0 11. Upon including the DW factors at room temperature, the slice thickness can be two or three times of the value. For the CMS to achieve a similar accuracy, the slice thickness should be one-fourth the slice thickness of the TOMS. Indeed, theoretically, the normalisation of the calculated wavefuction is the intrinsic property of

J.H. Chen et al. / Ultramicroscopy

237

69 (1997) 219-240

YBCO[OOl]:gmax=4.12; with DW factors x104

I

”

”

I

’

“,

I,

‘,

1,.

.

I

I

“,

TOM% c/4,1max=1.00004 - Q - TOM% c/2, Imax=1.0093 --r - TOMS: c/l; Imax=1.042

4600 5 h d 2-

10

0

20 30 THICKNESS (in c-unit)

40

Fig. 14. Amplitudes of reflection 6 0 0 from YBa2Cu307_x [0 0 l] plotted against crystal thickness, calculated by the TOMS with the gmsr value of 4.12 A-’ and different slice thicknesses c/n (c = 11.6827 A), showing the relation between the normalisation criterion and the accuracy of the calculated results. YBCO[OOl]:with the DW factors

I

“.

I

”

,

’

“,

I

”

*

I

”

”

I

1.0

.z? 33 0.9 a z 0.8

0

10

20 30 THICKNESS (in c-unit)

40

50

Fig. 15. Total intensities plotted against the crystal thickness, calculated by the ATCMS with a large slice thickness c (c = 11.6827 A) and different gm.. values and the TTTOMS with a small slice thickness and a small grnaxvalue, for YBa#taO, _-x{O0 l] and including the DW factors in the calculation, showing the normalisation of the ATCMS for a too large slice thickness and that of theTTTOMS for a too small gm.. value.

238

J.H. Chen et al. / Ultramicroscopy

69 (1997) 219-240

NC

Fig. 16. Illustration of the error source of the multislice procedure: an artificial periodicity which equals the slice thickness multislicing and this results in pseudo HOLZ effects in the calculated diffraction patterns.

the CMS because both the phase grating and the Fresnel propagator are unitary operators and there is no mechanism which allows the CMS to lose intensity, how ever large the slice thickness is and therefore how ever serious the intrinsic error will be. But, for the TOMS, the normalisation can be achieved accurately only if the slice thickness used is small enough and the intrinsic error approaches zero. However, by artificially cutting off the intrinsic error outside a certain dynamical aperture, the ATCMS relates its normalisation to the slice thickness since the amount of intrinsic error which will be cut off changes upon using different slice thicknesses (Fig. 6). When the slice thickness approaches zero and therefore the intrinsic error (pseudo HOLZ) reduces to zero, there will be nothing to be cut off and then the normalisation can be achieved

is created

by

as long as the gmax is large enough to cover the ZOLZ area. But the difficulty with the normalisation of the ATCMS is that it can also be achieved by increasing the gmax to cover all the pseudo HOLZ effects in spite of a too large slice thickness (Fig. 15). A useful advantage of the normalisation for the ATCMS is the determination of the minimum dynamical aperture needed for the calculation: As we have pointed out before, if the gmaxused is smaller than the ZOLZ area, not only the intrinsic error but also part of ZOLZ reflections will be cut off. So if we first remove the intrinsic error by decreasing the slice thickness to a small enough value, the normalisation will be related directly to the gmaxwhich is just enough to cover the ZOLZ area. So we can see that for the ATCMS the normalisation can be achieved in two ways: (i) increasing the gmaxalone, and (ii) first decreasing

J.H. Chen et al. / Ultramicroscopy 69 (1997) 219-240

the slice thickness and then (if it is still not achieved) increasing the gmax.But only the second normalisation leads to the correct results. For the TTTOMS, the normalisation automatically guarantees an appropriate slice thickness, but it cannot choose the proper gmax, e.g., the normalisation can still be achieved by decreasing the slice thickness even when the gmaxis smaller than the ZOLZ area since the part of ZOLZ reflections outside the dynamical aperture will be aliased (see the discussion of Ref. [ 1.51)(Fig. 15). However, from the advantage of the AT in the determination of the minimum dynamical aperture needed for the calculation, as we have just discussed for the ATCMS, we can expect that the normalisation for the ATTOMS (anti-alising-technique TOMS) will always lead to the proper slice thickness and the proper dynamical range as well.

6. Conclusions Based on the systematic quantum-mechanical approach to the multislice theory for calculating HEED, the CMS and the TOMS, their errors, their calculation results as well as the normalisation criterion can consistently be compared and therefore be well discussed. In summary, concerning the use of the CMS and the TOMS for calculating HEED and imaging, the following can be concluded: (1) For both the TOMS and the CMS there are basically two types of errors: One is the intrinsic error imposed by the order of the method in the slice thickness, which appears as the pseudo HOLZ effects in the diffraction patterns; Another is the numerical error caused by the finite dynamical aperture, such as the aliasing error and the intensityloss error. For ZOLZ calculations, it is shown that the intrinsic errors are the dominant errors for both the TOMS and the CMS since the elimination of the intrinsic error leads to the disappearance of the numerical error, as long as the dynamical aperture is large enough to cover all the ZOLZ reflections. (2) For ZOLZ calculations, the TOMS has a much smaller intrinsic error than the CMS for a large slice thickness and therefore is superior to the CMS with respect to accuracy vs. computational time. For the TOMS to achieve a good

239

accuracy in ZOLZ calculations, a slice thickness of 2 A will be small enough for most of the materials whose average potentials along the beam direction are not larger than that of Au[OO 11. But if the thermal DW factors are included in the calculation, thicker slices can be used. For the CMS to obtain a similar accuracy, 4 of the slice thickness for the TOMS should be used. However, for the calculation of HOLZ reflections, the TOMS and the CMS are in the same order and for both much thinner and different slices have to be taken so as to describe the potential variation along the z-axis. (3) For accurate dynamical calculations, once the gmaxis taken large enough to cover all the real (but not pseudo) reflections, it is not necessary to use the AT programs since there will be nothing to be aliased as the slice thickness is taken small enough. However, the AT can be useful through the normalisation criterion to automatically choose the proper gmaxneeded to cover all the real reflections. (4) The normalisation criterion is reliable for the ATTOMS, not always reliable for the TTTOMS and the ATCMS and useless for the TTCMS.

Acknowledgements

This paper presents research results partly sponsored by the Belgian programme on InterUniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office of Science Policy Programming. The scientific responsibility is assumed by the authors. One of the authors (JHC) is grateful to Prof. G. Van Tendeloo for his enduring support.

Appendix A.

Eq. (5) can be proved as follows. Since d =.$+$), @(K)e2”iK.R dK

AY = A

s =

(- irciKZ)@(K)e2”‘K’RdK. s

240

J.H. Chen et al. / Ultramicroscop~v 69 (I 997) 219-240

ZrriK’

F

=

S[

z

(-

iniK’)“]B(K)e’“‘”

‘R dK

.

de _

R~K

-nilcK”qK)e2aiK.R

dK.

We therefore have F[e”“Y] = e-ltinaX’@(K).

References

Ul

J.M. Cowley. A.F. Moodie, Acta Crystallogr. 10 (1957) 609. PI J.C.H. Spence, J.M. Zuo. Electron Microdiffraction, Plenum Press, New York, 1992, p. 309. c31 J.H. Chen, D. Van Dyck, M. Op De Beeck, J. Broeckx, J. Van Landuyt, Phys. Stat. Sol. (a) 150 (1995) 13. M Z.L. Wang, Elastic and Inelastic Scattering in Electron Diffraction and Imaging, Plenum Press, New York, 1995 (Char. 3 and 10).

[S] J.H. Chen, M. Op De Beeck, D. Van Dyck, Microsc. Microanal. Microstruct. 7 (1996) 27. [6] D. Van Dyck, J. Microsc. 119 (1980) 141. [7] D. Van Dyck, Phys. Stat. Sol. (a) 72 (1979) 283. [S] R. Kilaas. M.A. O’Keefe, K.M. Krishnan, Ultramicroscopy 21 (1987) 47. [9] J.H. Chen, Ph.D. Thesis, University of Antwerp, 1997. [lo] L.C. Qin, K. Urban, Ultramicroscopy 33 (1990) 159. [l l] D. Van Dyck. M. Op De Beeck. Ultramicroscopy 55 (1994) 435. [I23 K. Ishizuka, N. Uyeda, Acta Crystallogr. A 33 (1977) 740. [13] W. Coene. D. Van Dyck. Ultramicroscopy 15 (1984) 41. [14] Y.M. Wang, J.H. Chen, Phil. Mag. A 58 (1988) 818. [15] P.G. Self, M.A. O’Keefe, P.R. Buseck. A.E.C. Spargo. Ultramicroscopy 11 (1983) 35. [ 161 P. Goodman, A.F. Moodie. Acta Crystallogr. A 30 (1974) 280. [ 171 D. Van Dyck, Adv. Elec. Elec. Phys. (1985) p. 295. [18] Y. Ma, L.D. Marks, Acta Crystallogr. A 46 (1990) 11. [I91 S.X. Zhu. Y.M. Wang. S.Q. Wang, Phil. Mag. Lett. 67 (1993) 293. [20] J.H. Chen, Y.M. Wang, X.J. Luo, L.Y. Ding, X.L. Cheng, Phil. Mag. Lett. 71 (1995) 33. [21] J.H. Chen, unpublished work, 1993.