*Corresponding author. Tel.: #81-52-789-4457; fax: #81-52-789-3724.

E-mail address: a41263a@nucc.cc.nagoya-u.ac.jp (N. Tanaka)

Ultramicroscopy 80 (1999) 1}5

Ultramicroscopy Letter

A study of the validity of the image deconvolution methodon the basis of channelling theory for thicker crystals

J.J. Hu, N. Tanaka*

Department of Applied Physics, Graduate School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan

Received 24 June 1998; received in revised form 14 April 1999

Abstract

The image deconvolution method in high-resolution electron microscopy is theoretically studied for the applications inmaterials science. The weak-phase object approximation is replaced by a new formulation of di!raction wave functionbased on the channelling theory [D. Van Dyck, M. Op de Beeck, Ultramicroscopy 64 (1996) 99], which is valid for ratherthicker crystals. It is shown that in some special cases, the phases of two-dimensional structure factors can be reliablydetermined by the compensation for contrast transfer function on the images of thicker crystals. Furthermore, theprojected atomic columns are always displayed at the correct positions in the deconvoluted images. In contrast tothe weak phase object approximation, however, the projected column image intensities are non-linearly related to theprojected atomic potential. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Image deconvolution method; Channeling theory; HREM; Thicker crystals

High-resolution electron microscope (HREM)images are in#uenced by the non-optimal para-meters of electron optics such as defocus, sphericalaberration, etc., which can be faithfully describedby a contrast transfer function (CTF). There arevarious kinds of image processing methods on theelectron micrographs, concerning the correction ofCTF [1}5]. The phases of two-dimensional crystalstructure factors F(h, k) thus obtained from HREMimages can be used as initial values for the iterativecalculations such as phase extension and re"ne-ment. The "nal image resolution can achieve thedi!raction limit which is higher than the resolution

determined by lenses [6}8]. These programs havebeen paid attention for the crystal structure deter-minations of nano-phase materials where onlymicro-crystalline specimens are available for thedata collection. The compensation for CTF by im-age deconvolutions in HREM is ordinarily basedon the weak-phase object approximation (WPOA),which does not hold when the experimental imagesare taken from thicker specimens. However, theimage deconvolution method has been shown to berather e!ective in the structure determination ofinorganic compounds recently [9}11].

The channelling theory is recommended as a use-ful tool to intuitively interpret the images for largercrystal thickness. In that case, the `locala dynam-ical di!raction phenomenon comes from the chan-nelling of electrons along the atomic columnsparallel to the beam direction. A simple analytical

0304-3991/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 3 9 9 1 ( 9 9 ) 0 0 0 8 0 - 7

expression for the exit wave function had beenproposed for the electron channelling in isolatedcolumns [12]. That theory is related to the experi-mental reconstruction of exit waves in real space.Our purpose here is to theoretically reconsider theimage deconvolution method by employinga Fourier expression of the above channelled wavefunction below realistically thicker crystals.

According to the channelling theory of electronstrapped in the electrostatic crystal potential ofatomic columns, the dynamical di!raction wavefunction in the back focal plane can be written as[12]

W(u, z)"d(u)#+jCexp G!ip

Ej

E0

kzH!1D]f

j(u) expM!i2pu ) R

jN, (1)

where u is the scattering vector, z, the crystal thick-ness, d(u), the Dirac-delta function, k, the wavenumber, and E

0, the incident electron energy. At

the jth atomic column, Ej

is the eigen-energy re-lated to the `weighta of the atoms in the column,fj(u) the Fourier transform (FT) of the eigenfunction

Uj(R), and R

j, the two-dimensional column position

vector in the projection plane. The presumptionson the expression (1) include only bound stateslocalized near the column cores and one radialsymmetric state excited for one type of atomiccolumns, like the 1S state of an atom, and hencefj(u)"f

j(!u). The validity of the above treatment

can be proven reliably by the periodical variationof amplitudes and phases of W(u, z) with increasingz [12], that is a familiar fact with mono-atomiccrystals. In comparison with the simple kinematicalexpression under WPOA, we can de"ne a dynam-ical structure factor of the columns as

F(u, z)"!iE

0pkz

+jCexp G!ip

Ej

E0

kzH!1D]f

j(u) expM!i2pu ) R

jN. (2)

Furthermore, combining the terms of identicaltype columns, it becomes

F(u, z)"!i2E

0pkz

+n

sinGpE

n2E

0

kzH]exp G!i C

p

2#

pEn

2E0

kzDH fn(u), (3)

where n denotes the nth type of atomic columns,and f

n(u) is the sum of f

j(u)expM!i2pu ) R

jN for all

the nth type columns and must be a real factor forcentro-symmetric projected structures. Then ex-pression (1) can be rewritten as

W(u, z)"d(u)#ipkz

E0

F(u, z)

"d(u)#2+n

sinGpE

n2E

0

kzH]exp G!i C

p2#

pEn

2E0

kzDH fn(u). (4)

The above formulae will be quite useful for manypurposes in the "eld of dynamical electron di!rac-tion. Besides the inverse solution of the electrondi!raction problem, they can be used to correct thecrystal thickness e!ect of measured electron di!rac-tion intensities for the direct phase calculations inthe electron crystallography. Expression (4) canconvincingly replace the formal wave function de-rived by employing the Sayre's relation and phaseobject approximation, on which we had givena multi-beam description to the `ellipsea concept inHREM previously [13].

In view of the interference of di!racted waves inthe image plane, when the waves are rather weak incomparison with the transmission wave, only the"rst-order terms of the wave function W(u, z) maymake greater contributions to the intensity of lat-tice and structure images. For the zone-axis inci-dent illumination and centro-symmetric projectedstructures, the image intensity can be approxim-ately expressed as [14,15]

I(R, z)"constant#4+u

+n

sinGpE

n2E

0

kzH]f

n(u)DCTF(g)DsinMs(g)#g(E

n, z)N

]exp(2piu ) R), (5)

where g"DuD, g(En, z) is the relative deviation of the

kinematical phase of exp M!i(pEn/2E

0)kzN, and

DW(0, z)D is normalized as `1a. The wave aberrationof CTF is given by

s(g)"!pjg2(*f!0.5C4j2g2), (6)

where j is the electron wave length, *f the defocusvalue (positive for underfocus), and C

4the spherical

2 J.J. Hu, N. Tanaka / Ultramicroscopy 80 (1999) 1}5

aberration coe$cient of the objective lens. The FTspectrum of the image intensity can be written as

¹(u, z)"d(u)#4+n

sinGpE

n2E

0

kzH]f

n(u)DCTF(g)DsinMs(g)#g(E

n, z)N. (7)

The above weak scattering beam approximation, ingeneral, is not the same as WPOA, though it is alsothe "rst-order approximation. It is a weaker restric-tion than the WPOA to the specimen thickness inexperiments.

In case of mono-atomic crystals or crystals con-sisting of the same `weighta columns, all E

nare

identical. Expression (7) can be simply reduced as

¹(u, z)"d(u)#4

En

sinGpE

n2E

0

kzH]DCTF(g)DsinMv(g)#g(E

n, z)N+

n

Enfn(u),

(8)

where +nM!E

nfn(u)N is the FT of the projected

crystal potential [12], i.e., a projected kinematicalstructure factor that can be de"ned as F(u, 0)"lim

z?0F(u, z). For gO0 and sinMs(g)#

g(En, z)NO0, we can "nally get an equation as

2

En

sin GpE

n2E

0

kzH F(u, 0)

"

¹(u, z)

!2DCTF(g)DsinMs(g)#g(En, z)N

. (9)

The right-hand side of Eq. (9) is the same as theformula of the image deconvolution method de-rived by assuming WPO [3]. The e!ect of g(E

n, z)

results in a deviation of defocus that can be for-mally expressed as g(E

n, z)/pjg2 as we had pre-

viously pointed out [15]. The two-dimensionalstructure factor obtained from the images, (2/E

n)

sinM(pEn/2E

0)kzN F(u, 0) on the left-hand side of Eq.

(9), is equal to the modulated structure factor. It isconcluded from those dynamical e!ects that theimage deconvolution result varies with E

nand

thickness z of the crystal.However, (2/E

n) sinM(pE

n/2E

0) kzN is a positive

number when (pDEnD/2E

0) kz(p, i.e., the crystal is

thinner than a maximum thickness with a value of

z.!9

((2E0/DE

nDk). Hence, it could not in#uence the

sign of the phases of the structure factors obtainedfrom the image deconvolution. The value of z

.!9in-

creases with the accelerating voltage approximately

as E0k~1JJE

0, but decreases with the energy

Enrelated to the `weighta of atoms in the column.

According to Eq. (3), the WPOA is valid only when(pDE

nD/2E

0) kz;1, i.e., z

810;(2E

0/pDE

nDk). There-

fore, the value of z.!9

is at least three times largerthan z

810. Consequently, the present expression of

the image deconvolution method can be used be-yond the limitation of WPOA.

Next let us consider crystals consisting of atomiccolumns with, respectively, di!erent `weightsa. ForgO0, we can only get an equation for processingby expression (7) as

2+n

sinGpE

n2E

0

kzH fn(u)sinMs(g)#g(E

n, z)N

"

¹(u, z)

2DCTF(g)D, (10)

where the values of sinM(pEn/2E

0)kzN vary from one

to another columns. From the mathematical view-point, we may "nd a mean value of g6 (z), whichsatis"es

sinMs(g)#g6 (z)N+n

sinGpE

n2E

0

kzH fn(u)

"+n

sinGpE

n2E

0

kzH fn(u)sinMs(g)#g(E

n, z)N. (11)

When sinMs(g)#g6 (z)NO0, we can "nally get anequation as

+n

2

En

sin GpE

n2E

0

kzH M!Enfn(u)N

"

¹(u, z)

!2DCTF(g)DsinMs(g)#g6 (z)N. (12)

Although it is still in a similar form as Eq. (9), thereis no evidence that the deconvoluted structure fac-tors as a sum of atomic columns on the left-handside of Eq. (12) will ever keep the same sign as thekinematical F(u, 0) as shown in Eq. (9). But itdoes not exclude that in some cases the correctphases can be given by image deconvolutions. An

J.J. Hu, N. Tanaka / Ultramicroscopy 80 (1999) 1}5 3

easily applicable situation is that the coe$cientM!E

nfn(u)N of di!erent column types has an identi-

cal sign at a certain scattering vector of u, and allthe values of (2/E

n) sinM(pE

n/2E

0)kzN are positive.

Then the phase of deconvoluted structure factor ofu must be the same as F(u, 0). Such an assumptionof sign-identical M!E

nfn(u)N depends only on the

equivalent sites of projected atomic columns ofdi!erent types. Those sites may be rightly deter-mined in the deconvoluted images as shown by thefollowing expression (13). Eq. (9) based on mono-atomic crystals is the simplest example of Eq. (12).For all the terms satisfying (pDE

nD/2E

0) kz(p, the

maximum thickness z.!9

must be less than(2E

0/DE

n, .!9Dk), where E

n, .!9is the energy of the

`heaviesta column.For example, the energies of Mn and Au columns

in a Au4Mn alloy are 80 and 250 eV, which corres-

pond to the periodic values of DsinM(pEn/2E

0) kzND in

depth z of 13 and 4 nm [12], respectively. Thespecimen thickness of Au

4Mn must be much less

than 1nm for the WPOA, but the valid thickness ofEq. (12) can approach up to 4 nm.

The above-discussed considerations may be ex-tended to the materials where only one or a fewtypes of projected atomic columns with some sign-identical coe$cients of M!E

nfn(u)N dominantly

contribute to the amplitudes and phases of u dif-fraction beams. In that case a relatively practicalprocessing for crystal structures should be that Eq.(12) is approximately applied only for those domi-nant atomic columns. The phases of the u di!rac-tion waves could be correctly obtained after thecompensation for CTF.

Finally, we would like to point out another im-portant result obtained from the above theory. Nomatter what the phase is, the deconvoluted imageI$%#

(R, z) always shows the correct column positionRj, according to the FT of the left-hand side of Eq.

(12) as

I$%#

(R, z)"F~1G+n

2

En

sinGpE

n2E

0

kzH M!Enfn(u)NH

"+j

!2 sinGpE

j2E

0

kzHUj(R!R

j), (13)

derived from expressions (1)}(3), where F~1

denotes the inverse FT. The spatial distance be-

tween projected columns must be indeed largerthan the best resolution of the microscope. Sincethe intensity of the jth column is proportional tosinM(pE

j/2E

0) kzNU

j(R!R

j), the dynamical e!ect

results in that the image intensities of the projectedatomic columns may be changed to some extent.The absolute values of sinM(pE

j/2E

0) kzN of heavy

atomic columns will go down with the depthof z within (p/2)((pDE

jD/2E

0) kz(p, while those

values of light atomic columns go up because ofa small value of DE

jD. Therefore, the relative inten-

sities of light atomic columns will be enhanced incomparison with heavy atomic ones within a cer-tain range of crystal depth. The relative `weightavariation between Ge and Si atomic columns ob-served in the images of Ge/(0 0 1) Si interfaces canbe interpreted successfully by using the presentformula [16].

In conclusion, the validity of the image decon-volution method is theoretically studied byemploying the channelling theory of dynamicalelectron di!raction. On the premise of identicalsigns of coe$cient M!E

nfn(u)N of di!erent atomic

column types, or existing such dominant columnsin crystals, the phases of two-dimensional structurefactors of u can be reliably determined using theimage deconvolution method for thicker crystals.In this note, the e!ects of crystal thickness andatomic columns on the deconvolution results arediscussed in detail. Particularly, within the imageresolution, the projected atomic column positionsare always correctly reconstructed by the imagedeconvolution processing. The present conclusionis helpful to dispel the `WPOA doubta to the struc-ture determinations through the images of inor-ganic specimens, and proves reliable for somecrystal structures solved by employing the imagesas reported. It is clari"ed that the WPOA is nota sole theoretical basis of the image deconvolutionmethod in HREM.

Acknowledgements

One of the present authors (J.J.H.) acknowledgesthe support from a Japanese Science PromotionSociety (JSPS) Research Fellowship (P96232). Thiswork was partly supported by the `Research for the

4 J.J. Hu, N. Tanaka / Ultramicroscopy 80 (1999) 1}5

Futurea program of JSPS (dJSPS-RFTF96R13101). We also wish to thank Professor DirkVan Dyck of the University of Antwerp for hiscareful reading of our paper and his helpful refereecomments.

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