*Corresponding author. Tel.: #81 52 789 4457; fax: #81 52789 3724; e-mail: a41263a@nucc.cc.nagoya-u.ac.jp.

Ultramicroscopy 74 (1998) 105—111

An approximate multi-beam form of the “ellipse”in high-resolution transmission electron microscopy

J.J. Hu, N. Tanaka*Department of Applied Physics, Graduate School of Engineering, Nagoya University, Chikusa-ku, Nagoya 464-8603, Japan

Received 9 October 1997; received in revised form 17 March 1998

Abstract

The concept of “ellipse”, appearing in recent studies of high-resolution transmission electron microscopy, is newlydefined as the curves in the Fourier transform patterns of the images caused by many beams by plotting the intensities ofdiffracted beams vs. the transmitted beam. The present approximate ellipsoidal path can also be derived by using thephase-grating approximation and Sayre’s relation, but without the two-Bloch-waves assumption. Its validity is examinedfrom the data obtained by Fourier transforming the multi-slice simulated images of silicon S1 1 0T, and gives a valuableissue for quantitative image analysis in diffraction space. ( 1998 Elsevier Science B.V. All rights reserved.

PACS: 41.80.!y

Keywords: Multi-beam images; ‘‘ellipse’’ form; HREM; Phase-grating approximation

In the vector pattern for quantitative recognitionof lattice images of semiconductor super-latticessuch as Al

xGa

1~xAs [1], the image of each unit cell

is divided into an n]n pixel array and is represent-ed by an n2-dimensional vector, whose componentsare the intensity values measured at each of thepixels. Then the angular position of an image unitcell vector (see Fig. 4 of Ref. [1]) may be somewhatrelated to the composition of the materials. Thepath described by the image vectors of wedge-shaped samples is expressed in the form of ellipseson the two-dimensional plane defined by templatevectors [2]. For the quantitative high-resolution

transmission electron microscopy (HRTEM), themathematical relationship between the ellipse andimaging parameters, such as the amount of defocusand sample thickness, is established under theimaging theory of three-beam [3] and five-beam[4] cases, respectively. The HRTEM image inten-sities are assumed to consist of the background,single-periodic part with a period of 1/g anddouble-periodic part with a period of 1/2g in realspace [3], or of the linear and nonlinear compo-nents in reciprocal space (noted J

1and J

2, respec-

tively, in Ref. [4]). Then an ellipse is obtained byplotting the path of the components by increasingsample thickness in the formula. Two recent worksfurther show the feasibility of ellipse description inthe digital analysis of the images [5,6], though the

0304-3991/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 3 9 9 1 ( 9 8 ) 0 0 0 3 2 - 1

experimental images are hardly taken under thetwo-Bloch-waves condition. Since the multiplescatterings between reflections in samples are usu-ally inevitable and exclude the use of the two-Bloch-waves dynamical theory [7], it is necessaryto construct the ellipse for the cases of more thantwo Bloch waves.

In this note, a new form of the ellipse is directlydefined using the intensities of diffracted beams inthe Fourier transform patterns (FTPs), i.e. the op-tical diffraction patterns, of the images. It is pointedout here that the ellipsoidal path is, in fact, anapproximate concept for multi-beam imaging. Theconcept is interpreted also under the phase-gratingapproximation (PGA), Sayre’s relation and the lin-ear imaging theory. The multi-slice simulated im-ages of silicon S1 1 0T are employed in order toexamine the present definition, and furthermore, tocheck the effects of defocus, misalignment, chemicalcomposition and structural type to the ellipse.

The PGA has been a simplest dynamical formu-lation available for application to electron diffrac-tions [7]. First we apply the PGA to dynamicalscattering in crystals, in order to theoretically ex-plain the present new definition of the ellipse. Thedynamical exit-wave function from a sample ofthickness t is approximately given by assumingPGA as

º(k)"FK Mexp[ip/(r)t]N

"FK M1#ip/(r)t!12¦

p2/2(r)t2! *3¦

p3/3(r)t3

#14¦

p4/4(r)t4#2N , (1)

where FK denotes an operation of two-dimensionalFourier transform (FT), p"p/j», » is the acceler-ating voltage, j is the wavelength and /(r) is theprojected potential per unit thickness. The k andr are two-dimensional vectors in reciprocal and realspaces, respectively.

Based on the Sayre’s equation of F(k)"h(k)+

k{F(k@#k)F(k@) [8], in which h(k) is a known

function of scattering angle, and the two-dimen-sional structure factor F(k)"FK M/(r)N, the FT of/n`1(r) is approximately presented by a function ofF(k)/hn(k) due to the theorem of FT of convolution.The physical meaning is the similarity between thefunction /(r) and its nth power, if there is an enoughseparation between the projected atomic peaks.

The power peaks become sharper and sharperwhile n increases. The actual reliability of such anapproach is backward demonstrated by the follow-ing simulation results on silicon S1 1 0T crystals.Then the exit-wave function can be given by ex-panding the exponential function and FT into re-ciprocal space as

º(k)"d(k)#ih(k)F(k)Cpt

h(k)!

1

3!

p3t3

h3(k)#2D

#h(k)F(k)CA1!1

2!

p2t2

h2(k)#2B!1D , (2)

where d(k) denotes the Dirac-delta function. Re-placing the power series of pt/h(k) by its sine andcosine functions, respectively, the exit-wave func-tion of Eq. (1) is finally rewritten as

º(k)"d(k)#ih(k)F(k) sin Apt

h(k)B#h(k)F(k) Ccos A

pt

h(k)B!1D"d(k)

#2h(k)F(k) sin Apt

2h(k)B expp

2i A1#

pt

ph(k)B .

(3)

The exit-wave amplitudes for varying samplethickness are

Dº(0)D"Jº(0) )º*(0)

"C1#4h(0)F(0) sin Apt

2h(0)B cos Ap

2#

pt

2h(0)B#4h2(0)F2(0) sin2A

pt

2h(0)BD1@2

"Ccos2Apt

2h(0)B#[1!2h(0)F(0)]2sin2A

pt

2h(0)BD1@2

,

Dº(k)D"2h(k) KF(k) sin Apt

2h(k)BK . (4)

Though F(k) is the kinematical structure factor ofa projected crystallographic unit cell, no term isneglected during the above derivations. The present

106 J.J. Hu, N. Tanaka / Ultramicroscopy 74 (1998) 105—111

formulation is in the same order as phase-gratingapproximation, and its validity can be checked bycomparing with the “pendulum” variations of dif-fraction intensities vs. sample thickness given bymulti-slice dynamical simulations. The value of h(k)would be directly determined by experimentallymeasuring the extinction distance in samples, orfrom the “pendulum” variations calculated by thedynamical simulations.

In the linear imaging theory in the case of cen-trosymmetric electron sources and projected crys-tal structures, the Fourier spectra of imageintensities are approximately given, neglecting dif-fuse scattering, by

I(k"0)"FK MI(r)NDk/0"+

k

Dº(k)D2

"Dº(0)D2# +kE0

Dº(k)D2

"1!4h(0)F(0) sin2Apt

2h(0)B#4h2(0)F2(0) sin2A

pt

2h(0)B#4 +

kE0

h2(k)KF(k) sin Apt

2h(k)BK2

"1!4h(0)F(0) sin2Apt

2h(0)B#4+

k

h2(k)KF(k) sin Apt

2h(k)BK2,

I(kO0)"4 KC(k)º(0)h(k)F(k) sin Apt

2h(k)BK]sin M*g(k, t)#s(k)N#(cross terms),

(5)

where +kDº(k)D2 corresponds to a backgroundintensity in the image, *g(k, t) is the relative phaseshift between the zero-beam and k-beam, C(k) ands(k) are the amplitude and phase of the contrasttransfer function (CTF).

In an ideal case, the values of h(k) could bealmost a constant for zero-beam and k-beam (e.g.,see Fig. 7 of Ref. [9]). Based on the formulism ofEq. (5), the ellipse for imaging due to many beams

would be obtained by plotting the intensities of anyk-beam vs. zero beam in the FTPs of the images.The angular frequency of ellipsoidal traversing isdetermined by the “Lissajous” of diffraction inten-sities vs. thickness, if the cross terms (interferencebetween diffracted beams on the imaging plane) arenegligible, i.e. the weak scattering approximation.However, for the reasons as electron optical aberra-tions and tilts, the paths of I(k) vs. thickness havehardly the same periodicity, as shown in the twographs at the top of Fig. 1. Since the non-synchron-ism of the paths of I(k) vs. thickness between thezero-beam and k-beam would make the “Lis-sajous” traversing away from an ellipsoidal pathwith increasing thickness, it would be more exact toconsider the concept of the ellipse as an approxi-mated quantitative description of the images.

It would be rather difficult to give analyticalpredictions for ellipsoidal paths without actual cal-culation, even though the previous theories of twoBloch waves and the present linear expression ofEq. (5) were confided. It is noted again that thoseapproximate theories are just employed to theoret-ically introduce the ellipse concept in HRTEM.The exact shape of ellipsoidal paths for the digitalanalysis of experimental images should be deter-mined by a series of dynamical simulations.

In order to actually check the present ellipseconcept by using the diffracted beams vs. the trans-mitted beam in the FTPs of the images, the multi-slice method is employed to simulate the imageintensities. Fig. 1 shows the curves by plotting dif-fracted beam 1 1 1, 0 0 2 and 2 2 0 intensities vs.sample thickness (first row) and vs. the intensity oftransmitted beam (second to fourth rows) in theFTPs of the multi-slice simulated images of a sili-con crystal along the S1 1 0T zone-axis. The calcu-lation conditions are the accelerating voltage,»"200 kV, the spherical aberration coefficient,C

4"0.5 mm, the divergence semi-angle of incident

beam, b"1.0 mrad, and the standard deviation ofthe Gaussian distribution for the spread in focus,D"8.0 nm. The defocus values of left and rightcolumns are !50 and !100 nm (negative forunderfocus), respectively. The sample thicknessranges from 1.0 to 51.0 nm with a step of 2.0 nm, asshown by the markers on the curves. The firstmarker on the left corresponds to a thickness of

J.J. Hu, N. Tanaka / Ultramicroscopy 74 (1998) 105—111 107

108 J.J. Hu, N. Tanaka / Ultramicroscopy 74 (1998) 105—111

b

Fig. 1. Curves of diffracted beam (1 1 1), (0 0 2) and (2 2 0) intensities vs. sample thickness (first row) and vs. the intensity of thetransmitted beam (second to fourth rows), given by the FT of multi-slice simulated images of silicon crystals along S1 1 0T zone-axis,under the conditions of »"200 kV, C

4"0.5 mm, b"1.0 mrad and D"8.0 nm. The thickness is ranging from 1.0 to 51.0 nm with

a step of 2.0 nm. The defocus values of the left and right columns are !50 and !100 nm (underfocus), respectively. The parts below theindicated thickness are the approximate ellipsoidal paths on the curves, and the arrows would indicate the direction of the ellipsoidalpaths if they still kept on traversing with the thickness increasing.

Fig. 2. The effect of misalignment on the intensity curves of the (1 1 1) beam vs. the transmitted beam under the same simulationconditions as Fig. 1. The solid lines correspond to a lens tilt of 2.0 mrad and the dashed lines correspond to the 0.0 mrad tilt case.

1.0 nm, and from left to right, each marker repres-ents a thickness increasing of 2.0 nm. The thicknessvalues indicated on the curves show limiting pointsof the above approximate ellipse description. Thesignificant departures beyond the thickness implylimitation of the approximate ellipse description.The arrows in Fig. 1 would indicate the direction ofthe ellipsoidal paths if they still kept on traversingwith the crystal thickness increasing.

An obvious fact is that the phase change in themost graphs in Refs. [2,3] is smaller than 4.0 rad.For a closed ellipse, the phase change should be 2p(6.283) rad. Though one period of the path wasproposed in terms of the extinction distance in theabove papers, the errors between the actual phasesand ideal lines would evidently tend upwards inthinner or thicker sample areas (see e.g., theGa

0.6Al

0.4As(1 1 0) and AlAs(1 1 0) plots in Fig.

7 of Ref. [3]).Since the intensity of zero beam given in Eq. (5) is

constant at a certain thickness, the value of I(k) in

relationship with F(k) and CTF will change theellipsoidal paths of different diffracted beams, asshown in Fig. 1, because they are the functions of k.The effect of crystal structures to the ellipsoidalpaths is in proportion to the value of F(k), and theeffect of CTF is represented by its imaginary part,sin(*g#s). The value of CTF is determined withthe defocus, spherical aberration, astigmatism, mis-alignment and so on [10]. The effect of defocus tothe curves is shown in the left and right columns ofFig. 1. The effect of misalignment is shown inFig. 2, in which the solid and dash curves corres-pond to a lens tilt of 2.0 and 0.0 mrad tilt, respec-tively. An important phenomenon is the dramaticsplit in tilt cases beyond the limited thickness. Thesimulation results for germanium S1 1 0T are givenin Fig. 3 by solid lines to show the chemical com-position effect on the curves. It should be noticedthat the actual complexity due to mixed effectsmakes it difficult to directly determine the samplethickness and electron optical parameters from the

J.J. Hu, N. Tanaka / Ultramicroscopy 74 (1998) 105—111 109

Fig. 3. The effect of composition on the intensity curves of the (1 1 1) beam vs. the transmitted beam under the same simulationconditions as Fig. 1, by changing Si atoms to Ge atoms with the same crystallographic unit cell. The solid lines correspond togermanium element and the dashed lines correspond to silicon element.

Fig. 4. The effect of crystallographic structure type on the intensity curves of the (1 1 1) beam vs. the transmitted beam under the samesimulation conditions as Fig. 1. The solid lines correspond to a hypothetical f.c.c.-structured silicon and the dashed lines correspond tothe diamond-structured silicon.

ellipsoidal paths as ideally suggested in Refs. [3,4].Hence the “trial and error” fitting techniques maybe necessary to make quantitative conclusions.Moreover, a possible effect due to the changes ofcrystallographic structure type is examined by a hy-pothetical f.c.c.-structured model of silicon, asshown in Fig. 4.

Concluding, we note that the curves by plottingthe intensities of diffracted beams vs. transmitted

beam in the FTPs of the images obtained are provedto approximately create a practical form of the el-lipse. The approximate ellipse definition that wehave proposed here can be applied to analyze theimages of many beams, and is examined by theFourier calculations of the multi-slice simulated im-ages of silicon S1 1 0T, while the effects of defocus,misalignment, composition and crystal structure areclarified. A remarkable advantage of the present

110 J.J. Hu, N. Tanaka / Ultramicroscopy 74 (1998) 105—111

form of the ellipse should be its practicability forquantitative characterization of the actual images.

One of the authors (J.J.H.) is supported by theJapan Society for the Promotion of Science (JSPS)for working in Japan (Pg6232). The study is partlysupported by a Grand-In-Aid from the Ministry ofEducation, Science, Sports and Culture(d09236212, d10136216), and by a Special Projectof JSPS (dJSPS-RFTF96-R13101).

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