Fuzzy logic approaches to the analysis of HREM …Fuzzy logic approaches to the analysis of HREM...

Fuzzy logic approaches to the analysis of HREM images ofIII–V compounds

R. HILLEBRANDMax-Planck-Institut fur Mikrostrukturphysik, Weinberg 2, D-06120 Halle/Saale, Germany

Key words. Compositional mapping, edge detection, fuzzy logic, GaAs, high-resolution electron microscopy (HREM), III–V compounds, image processing.

Summary

It is known that high-resolution electron microscopy(HREM) can provide quantitative information on theproperties of crystalline materials. The HREM patterns oflayered structures of III–V semiconductors vary with thechemical composition of the latter within the sublattices,which is also influenced by interdiffusion. Local variationsof the crystal cell similarity are recorded for imageanalysis and compared with templates of known materialcomposition.

Of the advanced theories of data interpretation, the nowwell-established fuzzy logic is highly suited for correspond-ing image processing techniques. Combining neighbouringimage cell similarities, the underlying chemical compositionis evaluated by applying fuzzy logic criteria of inference tomasks of about 1 nm × 1 nm in size.

The new approach can be used to localize regions ofsignificant changes in composition, i.e. edge detection, andto determine the composition across the interface region.The methods introduced prove successfully applicable tosimulated as well as to experimental images of AlAs/AlxGa1¹xAs. Similarity/composition relations of nonlinearas well as nonmonotonic characteristics are studied toestablish an alternative fuzzy logic approach.

1. Introduction

Besides spectroscopic and analytical methods of micro-structure physics, electron microscopy (EM) can reflect thenature of semiconducting structures. Advanced techniquesof III–V compound crystal growth such as, e.g. molecularbeam epitaxy (MBE), allow one to produce III–V hetero-structures. The quality of the inherent interfaces has to becontrolled and optimized for subsequent opto-electronicapplications. As is well known, the conditions at theinterfaces are also influenced by interdiffusion processes inthe two sublattices. Heterostructures ranging from multi-layer systems to quantum dots are increasingly studied. For

physical reasons the samples are often preprocessed, e.g. bythermal treatment (induced diffusion). The GaAs-basedsemiconducting materials studied here are built up of cubiccrystal cells (ZnS-type sphalerite structure) with a latticespacing of about 0·56 nm. Depending on the orientation ofthe crystals prepared for electron microscopy (cf. (100) and(110) images below), different interference patterns of theelectrons scattered can be recorded.

High-resolution electron microscopy (HREM) is anappropriate technique to yield crystalline images whichcontain structural and chemical information with nearlyatomic resolution. The technique of chemical mappingdeveloped by Ourmazd et al. (1990) for the system(Al,Ga)As(100) is based on vector pattern recognitionmethods. Substantial contributions to the chemical map-ping of III–V compounds have also been made by De Jong &Van Dyck (1990), Thoma & Cerva (1991) and Walther et al.(1993). The more general QUANTITEM approach, devel-oped by Schwander et al. (1993), is not limited to specialcrystal orientations and chemical reflections; it enablespotential mapping to be done. Stenkamp & Strunk (1996)described a method of determining defocus, thickness andcomposition from high-resolution lattice images.

The present paper introduces very general fuzzy logic (FL)approaches, initiated by Tyan & Wang (1993), which alsogive near-atomic resolution. Based on simulated data aswell as experimental ones the principles of two FLapproaches are derived and described. In analysing andcombining the values of image cell similarities, the imageprocessing tools of edge detection and compositionalmapping will be applied to HREM patterns. It is our aimto quantitatively interpret HREM images of GaAs-basedmaterials by FL data analysis, especially with respect tocompositional barriers (interfaces) and to the chemicaldistributions of Ga,Al in the group III sublattice and As,P inthe group V sublattice. The FL schemes established forimage interpretation are of universal and flexible natureavoiding the need for analytical calculus. According to thephysical prerequisites, which will be discussed later, they

Journal of Microscopy, Vol. 190, Pts 1/2, April/May 1998, pp. 61–72.Received 10 March 1997; accepted 9 July 1997

61q 1998 The Royal Microscopical Society

can be adjusted to any known composition/similaritycharacteristics. These findings will be followed by a criticaldiscussion of experimental limitations and alternative FLapproaches for highly nonlinear relations between the III–Vchemistry and the HREM cell similarities.

Before the FL tools developed for analysing HREM imagesare introduced, some fundamentals of FL will briefly bepointed out. The ideas of FL have been proposed and mainlyworked out by Zadeh (1965). Fuzzy information that iscompared to prototype properties is described by linguisticvariables (solid, warm, fast, similar, . . .). The degree of truthcan be presented graphically by membership functions [0,1](triangles, Gaussian type functions, etc.). These membership

functions are combined by a set of fuzzy rules, e.g. of ‘if . . .,then’ type. The specific kind of membership functions aswell as the FL rule base, also called expert knowledge, haveto reflect the studied context. Set-theoretic operations andthe defuzzification (see below) allow one to draw quantita-tive conclusions. Fuzzy logic is typically applied to controltheory, approximative reasoning and FL decision making.

The methodology of fuzzy set theory and its applicationshave been studied and published by Zimmermann (1991).FL usage in control theory is discussed, e.g. by Pedrycz(1993); for a review see also Seraphin (1994). Questions of‘possibility’, ‘evidence’ and ‘probability’, which are of greatimportance to quantitative data analysis, are discussed, e.g.

Fig. 1. Simulated HREM contrast tableausof III–V crystals with varying sublatticecomposition; similarity profiles jnrm, relatedto the templates (right). (a) Alx-

Ga1¹xAs(100): t0 ¼ 17 nm, D ¼ 25 nm,vert. variation of Al/Ga, 30% noise; (b)GaAs1¹xPx (100): t0 ¼ 12 nm, D ¼ 30 nm,vert. variation of As/P, 40% noise. Acceler-ating voltage U ¼ 400 kV, aberrationCs ¼ 1 mm (insets: templates).

62 R. HILLEBRAND

q 1998 The Royal Microscopical Society, Journal of Microscopy, 190, 61–72

by Klir & Yuan (1995). In his book Bezdek (1981) discussesstatistical terms known for classical mathematics in view ofFL. Within the scope of the present paper, the reader isreferred to Bezdek (1981), where the terms ‘likelihood’,‘fuzzy-statistical criterion’, ‘Bayesian classifiers’ and ‘maxi-mum likelihood estimation’ are described. Here, the mainemphasis is on the practical points of deriving FL member-ship functions and fuzzy rules.

2. Method

For developing techniques of quantitative image analysis formaterial systems given, clear and comprehensive images aredesirable. Computer simulations of HREM contrast tableausare very helpful for systematically studying the contrastbehaviour of crystalline materials. The two physicalprocesses that have to be described numerically includethe dynamical electron diffraction within the crystal andthe nonlinear imaging process within the electron micro-scope. Nowadays, the following approximations are numeri-cally implemented (Stadelmann, 1987) and successfullyapplied:1 the multislice algorithm of dynamical electron diffraction,2 the contrast transfer formalism of partial coherence.

In Fig. 1 the stepwise chemical replacement of Ga by Al inthe group III sublattice as well as of As by P in the group Vsublattice are modelled and analysed. The statistical atomicexchange in the respective lattice positions can beapproximated by the so-called VCA interpolation, describedby Stenkamp & Jager (1993). Figure 1 shows HREMpatterns simulated from (100)-orientated AlxGa1¹xAs (a)and GaAs1¹xPx (b) crystals for an accelerating voltage of400 kV and for Cs ¼ 1 mm. These tableaus are made up ofcrystallographic cells 32 × 32 pixels in size with the crystalthickness linearly increasing from left to right (cf. lettering).To simulate selective imaging conditions, an underfocus ofabout 30 nm was predetermined. In the vertical direction,the Al/Ga and As/P ratios are stepwise varied by 10%. Forapproaching realistic experimental conditions, additiveamorphous noise is superimposed, which amounts to 30%(a) or 40% (b) of the total contrast.

There are clear differences between the GaAs cell imagesand the AlAs and GaP patterns. They are caused by thedifferent group III and group V components in the crystals.Contrary to the GaAs contrast features, for GaP the strong(200) reflections dominate the HREM basic patterns forextended parameter windows with bright dots at spacings of2·8 A. There are comparable results for the group IIIsublattice of (Al,Ga)As(100) in Fig. 1(a), where AlAs hasrather stronger polarity.

A systematic comparison of the resulting HREM patternsshows that their relative similarity correlates with thevariations of the chemical composition in the markedthickness ranges. For reviewing quantitative dissemblance

measures in digital pattern recognition the reader isreferred, e.g. to Diday & Simon (1976). Of the number ofsimilarity measures tested, including the normalized scalarproduct, I preferred the standard deviation (SD) of thedifference patterns of two compared images (Hillebrand &Hofmeister, 1995). The values obtained from a compo-sitional series proved to be closest to linearity. The measureof similarity S ¼ j(Id) of two images Iu and Iv of a cell size ofm × n pixels is calculated from the absolute differenceId

i,j ¼ |Iui,j - Iv

i,j| by

jðIdÞ ¼

��1

m·n

Xm

i¼1

Xn

j¼1

½Idi;j ¹ I0ÿ2

vuut ð1Þ

where j is standard deviation (SD), and I0 is the mean valueof Id.

Applying the SD measure with respect to the correspond-ing two internal templates of (Al,Ga)As and Ga(As,P) shownyields 2 × 2 series of similarity values along the markedcolumns. The corresponding profiles on the right of Fig. 1present the full-range normalized similarity/composition


Fig. 2. Simulated HREM image (20 × 11 cells) of AlxGa1¹xAs (100)with varying interface gradients of Al/Ga composition (cf. right)and 30% amorphous noise. Voltage U ¼ 400 kV, crystal thicknesst ¼ 17 nm, D ¼ 25 nm, Cs ¼ 1 mm.

FL A PPROACHES TO HREM OF III – V COMPOUNDS 63

characteristics jnrm ¼ j/jmax in the parameter windows fort ¼ t0 6 1·5 nm. The possibilities of studying diffusionphenomena in the group V sublattice by HREM imageanalysis are treated extensively elsewhere (Hillebrand et al.,1996b). As Fig. 1(b) indicates for a major experimentalwindow of t/D combinations, the compositional gradient ofGa(As,P) interfaces can be determined. The present paperwill focus on heterostructures of the III–V compoundsystem AlxGa1¹xAs.

The interface model of Fig. 2 has been established toderive and demonstrate the FL schemes of quantitativeimage interpretation. For comparison with recorded experi-mental images, for the (Al,Ga)As specimen model a stepwisechemical conversion from Al0.4 Ga0.6As to AlAs has beenassumed. The simulated image of Fig. 2 is calculated fora<100> crystal orientation and appropriate imagingconditions (crystal thickness t ¼ 17 nm, defocusD ¼ 25 nm). Five interface structures with amorphous noiseadded having different barrier gradients of chemical

composition are simulated (see right-hand side of Fig. 2),ranging from a smooth linear profile (bottom) to an abruptAl/Ga step (top). For image analysis, all cells of the HREMimage simulated are compared with the respective cells oftwo internal reference templates, namely Al0.4 Ga0.6As (left)and pure AlAs (right), yielding two arrays of similarity.

Figure 3(a) displays the similarity distribution of thesimulated HREM images around the vertical interface. Thedark areas present a strong dissimilarity, i.e. white cellsmark the identity with the corresponding template. Fromleft to right at the bottom of Fig. 3(a) there is the expectedsmooth transition of grey values. The effect of the 30% ofamorphous noise on such similarity matrices is distinct,because the size of the template is chosen to be 1/2 × 1/2crystal cell (asymmetric unit). Figure 3(b) displays twohistograms of the normalized similarity values related to thegiven templates. They show two pronounced maxima,which refer to the image areas far from the interface. Thesimilarity values between are related to the interface

Fig. 3. (a). Similarities (j/cell) in Fig. 2,related to templates. Note the differences.(b) Histograms of similarities of the differ-ence patterns (40 × 22 cells of 16 × 16 pix-els) with examples of triangular FLmembership functions s_A and s_B.

64 R. HILLEBRAND


positions. The separation of the peaks and their half-widthsare determined mostly by the size of the model and theamorphous noise. In addition, the height and the shape of thepeaks vary with the share of the corresponding image areas.

Fuzzy logic membership functions are shown schemati-cally in the histograms of Fig. 3(b), and will be described inmore detail below. For edge detection, the two fuzzy sets ofsimilarity s_A and s_B have to classify the histogram values ofsimilarity in the closed intervals S [ [0,1]. In the region ofintersection of both triangular membership functions thecorresponding image cells are members of A and Bsimultaneously. The two fuzzy sets are symmetric, with theirdegree of intersection varying. (The reader is referred to Fig. 7for the experimental data sets of cell similarities as well as theFL membership functions used for compositional mapping.)

The idea of the FL approach to compositional edgedetection consists of the evaluation of masks of neighbour-ing image cells. Their degree of similarity with respect to thetemplates is transformed into membership values, which areinterpreted by FL rules to infer the properties of the crystalcell (i,j). The size of masks necessary to identify distinctimage changes was studied, e.g. by Wang (1985) fordifferent kinds of objects and recording techniques. Figure4(a) shows schematically the search for similarity edges inHREM images. In the image analysis both fuzzy sets s_A ands_B are applied to each of the six cells inside the 2 × 3 mask,yielding the fuzzy variables

Si;j¹1;Si;j ;Si;jþ1; and Siþ1;j¹1;Siþ1;jSiþ1;jþ1:

The unusual size of 2 × 3 masks was chosen (Hillebrand etal., 1996a) in order to maintain as much lateral imageinformation as possible of the cell-type micrographs.According to the fuzzy rules, graphically represented foredges parallel to the image borders (1) to (4), and for edgesin diagonal direction (5) to (8), fuzzy sets of composition areobtained.


Fig. 4. Scheme of 2 × 3 matrices of neighbouring image cells, repre-senting the FL approach to ‘edge detection’ (a). Schemes of 2 × 1matrices of neigbouring image cells (related to A and B) represent-ing the FL approach to ‘compositional mapping’ (b).

ð1Þ if Si;j¹1 is A and Si;j is A and Siþ1;j¹1 is B and Siþ1;j is B; then Cði; jÞ is a

ð2Þ if Si;j¹1 is A and Si;j is B and Siþ1;j¹1 is A and Siþ1;j is B; then Cði; jÞ is a

ð3Þ if Si;j¹1 is B and Si;j is A and Siþ1;j¹1 is B and Siþ1;j is A; then Cði; jÞ is a

ð4Þ if Si;j¹1 is B and Si;j is B and Siþ1;j¹1 is A and Siþ1;j is A; then Cði; jÞ is a

ð5Þ if Si;j¹1 is A and Si;j is B and Siþ1;j is A and Siþ1;jþ1 is B; then Cði; jÞ is a

ð6Þ if Si;j¹1 is B and Si;j is A and Siþ1;j is B and Siþ1;jþ1 is A then Cði; jÞ is a

ð7Þ if Si;j is A and Si;jþ1 is B and Siþ1;j¹1 is A and Siþ1;j is B; then Cði; jÞ is a

ð8Þ if Si;j is B and Si;jþ1 is A and Siþ1;j¹1 is B and Siþ1;j is A; then Cði; jÞ is a:


The fuzzy variable of composition C(i,j) represents the resultof the inference. For variable C the two fuzzy sets a(ltered)and u(nchanged) are defined, where set a identifies thepositions of edge-type similarity and composition, respec-tively. With the degree of intersection (cf. Fig. 3a, bottom)even widely varying, only one crystal cell in a monotonicgradient of composition is identified as an edge. Theformulated fuzzy rules exclude each other with respect tothe conclusions. For this reason, defuzzification may becarried out by the fast ‘maximum method’ (Kahlert &Frank, 1993), i.e. the highest degree of membershipdetermines the result.

For compositional mapping (cf. Fig. 4b) three fuzzy sets ofsimilarity, similar, medium and different are defined,abbreviated to s, m, d. They are to classify the values ofsimilarity gained by comparing the image cells to templatesA and B, respectively, as applied to the experimentalexample in Fig. 7. The masks of neighbouring cells (i,j)and (i þ 1,j) are pre-orientated perpendicular to the inter-face. The image analysis is based on a tailored formalismcombining the fuzzy sets of one A-related and one B-relatedmask. The following fuzzy variables are defined:

Si;j A Siþ1;j B A ¹ rel : AlAs

Siþ1;j A Siþ1;j B B ¹ rel : Al0:4Ga0:6As:

The fuzzy variable composition C(i,j) represents theresult of the inference. For variable C the fuzzy sets: [c_A,c_Am, c_m, c_mB, c_B] are defined. Figure 4(b) showssymbolically all useful combinations of fuzzy sets ofsimilarity and the fuzzy rules that allow one to concludethe chemical composition. The fuzzy rules for composi-tional mapping that have been numerically implementedare:

If there are combinations of cell similarity values, which donot fulfil any of the FL rules (1)–(13), no composition canbe concluded. Further fuzzy rules may be added to rendersome kind of drop-out correction by allowing one of the fourFL conditions per rule to be violated.

For practical applications of this FL rule base, the shape,the base size and the intersections of the actual membershipfunctions have to reproduce the similarity/compositioncharacteristics of the III–V system under investigation.The quasilinear case of (100)-orientated GaAs1¹xPx hasbeen analysed using triangular fuzzy sets of similarity(Hillebrand et al., 1996b) for varying chemical gradients inthe group V sublattice. In this paper, both quasilinear andhighly nonlinear similarity/composition characteristics ofAlxGa1¹xAs are studied in detail using trapezoidal andtriangular membership functions of similarity (for experi-mental details see Fig. 7).

The two FL approaches introduced are briefly demon-strated by means of the AlxGa1¹xAs interface model of Fig.2. In Fig. 5(a–c), the edge detection technique is applied tothe similarity distribution, which in Fig. 3 is shown for the(Al,Ga)As interface. The blurred contours are most dis-tinctly there. In Fig. 5(a), all eight FL rules of the algorithmestablished (cf. Fig. 4a) are applied to analyse the data ofsimilarity. The interface course is clearly visible with littledropouts, where the different model types join each other.They can be explained by the intentionally very small size ofthe 2 × 3 mask. The plots of Fig. 5(b,c) show the excerptsfrom both the parallel and the diagonal selective rule base. Itis obvious that they both contribute to the detection ofpronounced composition changes.

In Fig. 5(d,e) the FL interpretation scheme of compositionmapping is applied to the matrices of cell similarity shown

ð1Þ if Si;j A is s and Siþ1;j A is s and Si;j B is d and Siþ1;j B is d; then Cði; jÞ is c A

ð2Þ if Si;j A is s and Siþ1;j A is m and Si;j B is d and Siþ1;j B is m; then Cði; jÞ is c Am

ð3Þ if Si;j A is m and Siþ1;j A is s and Si;j B is m and Siþ1;j B is d; then Cði; jÞ is c Am

ð4Þ if Si;j A is s and Siþ1;j A is m and Si;j B is m and Siþ1;j B is d; then Cði; jÞ is c Am

ð5Þ if Si;j A is m and Siþ1;j A is s and Si;j B is d and Siþ1;j B is m; then Cði; jÞ is c Am

ð6Þ if Si;j A is m and Siþ1;j A is m and Si;j B is m and Siþ1;j B is m; then Cði; jÞ is c m

ð7Þ if Si;j A is s and Siþ1;j A is d and Si;j B is d and Siþ1;j B is s; then Cði; jÞ is c m

ð8Þ if Si;j A is d and Siþ1;j A is s and Si;j B is s and Siþ1;j B is d; then Cði; jÞ is c m

ð9Þ if Si;j A is m and Siþ1;j A is d and Si;j B is m and Siþ1;j B is s; then Cði; jÞ is c mB

ð10Þ if Si;j A is d and Siþ1;j A is m and Si;j B is s and Siþ1;j B is m; then Cði; jÞ is c mB

ð11Þ if Si;j A is m and Siþ1;j A is d and Si;j B is s and Siþ1;j B is m; then Cði; jÞ is c mB

ð12Þ if Si;j A is d and Siþ1;j A is m and Si;j B is m and Siþ1;j B is s; then Cði; jÞ is c mB

ð13Þ if Si;j A is d and Siþ1;j A is d and Si;j B is s and Siþ1;j B is s; then Cði; jÞ is c B:

66 R. HILLEBRAND


in Fig. 3. The corresponding membership functions arediscussed in connection with the more relevant experi-mental applications in Fig. 7. The Al/Ga composition reliefsdetermined clearly reflect the tendency of the chemicalgradients in the model. For the composition graph 5d, theinterface model of Fig. 2 was analysed using a cell size ofexactly one crystallographic unit cell. During the HREMimage simulations 30% of amorphous noise was super-imposed, limiting the resolution attainable. It is obvious thatfor a cell size of ½ × ½ unit cell, assumed in Fig. 5(e) for the3D composition graph, the noise becomes critical. Thelinear composition gradient in front of the model is nolonger unambiguously detectable.

3. Applications

In interpreting simulated HREM patterns of interfaces oneshould consider that the crystal cell sizes and the cellpositions in the image field are exactly known. The effect oflateral cell/template misalignments is discussed elsewhere(Hillebrand et al., 1995a, 1996b). Practical aspects of

QUANTITEM, also with respect to the crystal cell detectionin experimental images, are discussed by Kisielowski et al.(1995). Contrary to III–V compound crystals of (100)orientation, (Al,Ga)As(110) HREM images are known tohave only small experimental t/D windows, with the cellsimilarity varying almost proportionally to the group IIIcomposition.

Figure 6 shows part of an experimental HREM image ofan AlAs/Al0.4Ga0.6As interface in cross-section. The imageof the MBE-grown material was taken in a JEOL microscopeat an accelerating voltage of 400 kV. Studying the quality ofthe interface as well as extracting the Al/Ga profiles requirereference templates to be established. Parts of them areshown on the left and on the right of Fig. 6. HREM images ofthe system (Al,Ga)As are assumed to be nearly free of latticemismatch and strain. For crystallographic reasons, the twointernal templates have to be obtained by Fourier filteringand/or accumulation techniques, applied to image areas farfrom the interface. Their chemical composition is gained bygrowth and physical reference methods (e.g. SIMS). Thevalues of similarity are calculated according to Eq. (1), with


Fig. 5. Results of the FL approaches, appliedto the five-gradient interface model. (a–c)Edge detection: (a) all eight fuzzy rules; (b)parallel rules; (c) diagonal rules. Note thatdark cells mark distinct changes in compo-sition. (d,e) Compositional mapping: (d) cellsize 32 × 32; (e) cell size 16 × 16.


the complete HREM image being subtracted from themultiplied full-range template images. The matrices ofsimilarity analysed in the next figures are calculated cell bycell (16 × 16 pixels); the error bar is 7% from statistical noise.

The histograms of cell similarity for both difference mapsare represented in Fig. 7(a). The graphs clearly reveal twowell-distinguishable peaks with a range of similarity transi-tions (interface) between. Unlike the Al0.4 Ga0.6As template(B), the AlAs template (A) provides a good symmetry withrespect to both parts of the HREM image. In detail, Fig. 7(b)illustrates the fuzzy sets of similarity (s, m, d) defined forneighbouring cells and related to A (left) and B (right),respectively. The trapezoidal membership functions referringto the fuzzy sets ‘similar’ and ‘different’ ensure that extendedimage areas of nearly template type and micrographs of littlenoise (narrow peaks) are properly evaluated. In Fig. 7(c) thefive triangular fuzzy sets of composition are defined.

The four values of similarity measured in comparison toboth templates provide degrees of membership (y-axis) forthe corresponding functions (cf. arrows). The fuzzy rules(1–13) control the decisions on membership in theresulting sets of composition (bottom). The conjunction ofthe four logical ands in the fuzzy rules is provided by thecorresponding minimum of memberships. The defuzzifica-tion of the resulting composition scheme, indicated by anarrow in Fig. 7(c), is carried out by the ‘centre of gravitymethod’ (Zimmermann, 1991), which has a weightedinterpolating effect. Physical prerequisites such as the con-servation of matter in the sublattices and the monotonicity of

chemical gradients are considered in this approach to HREMimage analysis.

The results of the FL image analysis of the Al/GaAsmicrograph are summarized in Fig. 8. Part (a) shows theoutput of the FL edge detection procedure. In practice, thefuzzy rules (1–8) can provide a closed trace of edge-typesimilarity/composition variation as shown in the imageplane. It is possible to derive clearly parameters of interestfor HREM investigations of III–V materials such as theposition and course of interfaces in micrographs. Theinterface analysed here is relatively smooth. Nevertheless,the FL methodology allows one to distinguish two parts ofthe interface that have different features.

In the 3D compositional map of Fig. 8(b), the two matricesof similarity values are utilized and evaluated by FL inference(see Fig. 7). As can be expected from the edge detection, thechemical gradient in the group III sublattice is not constantover the whole interface region studied. The data analysed inthis graph provide interpretable and valuable profiles of thechemical diffusion gradient as shown in Fig. 8(c). Forcomputing the profile in the two interface segments, thetwo axes of the plot have to be calibrated:1 the magnification, the scale or the number of cells in theimage (x-axis),2 the chemical composition of the two templates (plateausin the y-axis).

The parameters of physical interest for the III–V diffusionstudies (see also Tan et al., 1991) such as the slope can clearlybe derived for the two identified segments of the interface.

Fig. 6. Experimental HREM image of an AlAs/Al0·4 Ga0·6As interface. U ¼ 400 kV, Cs ¼ 1 mm, crystal orientation <110> (512 × 512 pixels, 8bits); left and right: templates.

68 R. HILLEBRAND


Practical problems of template fit, caused by distortedHREM images of strained crystals, are discussed elsewhere(Hillebrand et al., 1996b). It proved to be complicated toderive templates representative of the two materials in thecorresponding images. While the edge detection algorithmclearly could identify an interface although only onetemplate was of sufficient quality, the FL chemical mappingtechnique required both matrices of cell similarity to beappropriate. This fact does not restrict the FL technique ofimage analysis; it is a problem of data acquisition inpreprocessing. Improving the sensitivity of the approach toexperimental applications requires the individual cell by cellfit of the templates to compensate geometrical imperfectionsof the crystalline images, as also reported by Ourmazd(1993).

In section 4 analysis of simulated HREM images of(Al,Ga)As(110) interfaces for highly nonlinear and non-monotonic similarity/composition relations are described. Itis the aim to generalize this FL approach of compositionalmapping for these cases and to discuss the necessaryphysical prerequisites.

4. Discussion of alternative FL approaches

The FL interpretation scheme introduced proved applicableto the HREM images of chemically modified III–V materialseven if remarkable deviations from linear composition/similarity characteristics arise. Then the base size (support)of the corresponding FL membership functions has to followthe similarity/composition characteristics of the considered


Fig. 7. (a) Histograms of template similarities at the Al,GaAs inter-face. (b) Similarity values Si,j_A, Siþ1,j_A and Si,j_B, Siþ1,j_B ofneighbouring cells marked by ↑ in the membership functions s, mand d. The FL rules fire to the five fuzzy sets of composition (c)for defuzzification.

Fig. 8. Results of the FL approaches, applied to the (Al,Ga)As micro-graph. (a) Edge detection: the nearly smooth interface is identified,marked by ridge. (b) The composition relief shows the two regionsS1, S2 of different slope. (c) Two mean concentration profiles S1and S2 of Al/Ga are found.


III–V system. However, if one of the two profiles of similarityis no longer monotonic, revealing an extremum somewherebetween the two template compositions, the rule base and/or the membership functions of similarity introduced areviolated. This is studied for III–V compounds of <110>orientation under practically relevant imaging conditions,followed by a discussion of possible solutions.

Before simulating the HREM images of the next interfacemodel, t/D tableaus of AlAs and Al0.4Ga0.6As for the polar(110) crystal orientation were computed by the EMS package(Stadelmann, 1987). The simulated contrast tableaus of(Al,Ga)As(110) allow one to identify experimental thickness/defocus combinations close to t ¼ 28 nm and D¼ 60 nm,which provide optimum conditions to distinguish the HREMpatterns of AlAs and Al0.4Ga0.6As.

Figure 9 shows HREM simulations of (Al,Ga)As(110)interface models, varying the Al/Ga slope from an abrupt

step to a smooth linear mode (middle), as the model profileson the right show. Each of the III–V crystal cells is representedby an array of 22 × 32 pixels. For the imaging parametersspecified in the figure captions, variations in composition bysteps of 10% are used to establish the HREM barrier image.The matrices of cell similarities, obtained by the introducedtemplate comparisons, are presented as half-tone graphs inFig. 9(b). The graph related to AlAs (right) clearly shows thatthe gradation in the central linear part of the model is nolonger monotonous (cf. darkest area). The transitions betweenthe different classes of similarity result from the 20% ofamorphous noise assumed and from the 3D plot routine used.

Inspecting the corresponding similarity profiles inFig. 10(a) (right) reveals that the AlAs-related curve has aprominent maximum, whereas the slope of the other profileis very low for medium compositions. Physically, this parti-cular behaviour is due to phase- and reference origin-relatedproblems of nonsymmetrical N-beam cases. Using theestablished fuzzy rules and the standard FL membershipfunctions here implies contradictions. The FL rules do not fitall those pairs of similarity that violate the criterion ofmonotonocity. If none of the FL rules is fulfilled, it isimpossible to conclude any composition (Fig. 10a, left). Inthe profile gained for linear composition changes the tendencyof Al/Ga variations is not reproduced at all. Basically, in the FLfield two objects can be adapted to the problem:1 the set of FL rules,2 the membership functions.

Before doing that, a preprocessing technique of thesimilarity data is physically introduced. It can be assumedfor one interface that the composition gradients in both ofthe III–V sublattices are always monotonous. In otherwords, diffusion does not occur uphill (without additionaldriving forces). For this reason it can be concluded that allchanges of composition and consequently also those of‘rectified similarities’ have to proceed in one sense. For theFL analysis of this <110> system of (Al,Ga)As, the AlAs-related similarity profile is to be rectified.

Figure 10(b) illustrates the realization of this kind ofsimilarity preprocessing perpendicular to the interface, markedon the right by arrows. On the left, Fig. 10(b) shows the FLcomposition profile obtained from similar values of the linearcomposition gradient. For ranges of medium composition thegradient of the composition profile is too weak, resulting fromthe only slight slope in the similarity characteristics.

If all the composition/similarity information requiredis available to characterize one III–V interface, themembership functions can be adapted to optimize the FLinference scheme. The bottom part of Fig. 10 shows how themembership functions of similarity are fitted to the centre ofthe similarity distribution (S ¼ 0·65) by a linear offset. Theform and the intersection of the membership functions aretaken over. The two different types of symbols in thebottom scheme represent both similarity profiles for linear

Fig. 9. (a) Simulated HREM image (9 × 10 cells) of AlxGa1¹xAs(110) with varying interface gradients of Al/Ga composition (cf.right) and 20% noise. Voltage U ¼ 400 kV, crystal thicknesst ¼ 28 nm, D ¼ 63 nm, Cs ¼ 1 mm. (b) Half-tone plots of templatesimilarities. Notice the dark central area in the right-hand plot.

70 R. HILLEBRAND


composition changes. The set of fuzzy rules and the fuzzysets of composition also remain unchanged. As the finalresult (Fig. 10c, left), the FL output approaches the linearityof the model, shown in the central part of Fig. 9. An FLscheme adapted to specific III–V data is suitable to analyseany other interfaces of the material system considered.

For further improvement of FL approaches, the method-ology of neuro-fuzzy can be proposed to optimize the size(geometrical form, support, intersection) of the membershipfunctions for a given rule base. In principle, the similarityvalues and the target function of composition behind theprofiles of similarity are mapped into a neural network withthe aim of generating perfect membership functions. In thisway it is possible to compensate final nonlinearities in an FLinterpretation scheme of III–V HREM images. In apreliminary study (Hillebrand et al., 1995b) the support ofeach fuzzy set of similarity has been defined in directproportion to the SD step found in the calibration curve oflinear composition changes. Experimental noise andmeasuring errors can be expressed by a correspondingdegree of intersection of the single fuzzy sets (ambiguity) ofsimilarity. The resulting fuzzy sets of composition are definedby a support proportional to the ‘mean absolute value’ of thetwo corresponding SD steps, i.e. the rectification is implicitlydone via the membership functions of composition. In thisway, the principle of chemical monotonicity is implementedand strong changes in the similarities provide the reactionrequired in the inference of composition. In the correspondingrelief graph of chemical composition, the defuzzification

provides solutions for each of the direct combinations of theAlAs and Al0.4 Ga0.6As data of similarity.

After introducing and applying the FL approach ofquantitative image analysis to simulated as well asexperimental examples of III–V materials, possibilitieshave been discussed of adapting the FL inference schemeto more general similarity/composition characteristics.These alternative approaches will be verified experimentallyby making use of the monotonicity criterion of diffusion.

5. Conclusions

HREM images of GaAs-related heterosystems are interpretedquantitatively with respect to their III–V sublattice compo-sition. The introduced image processing tools of edgedetection and compositional mapping are based on theanalysis of the similarity of image cells by comparing themwith templates of known composition. Traditional techni-ques of image preprocessing and data acquisition providematrices of similarity values. To profit from the potentialityof FL (see also Hillebrand et al., 1997), the HREM imagesof III–V semiconductor interfaces are analysed by FLinterpretation schemes.

The FL rules of edge detection derived for neighbouringimage cells (2 × 3 masks) combine two triangular member-ship functions of similarity to identify image positions ofdistinctive changes. The FL rule base, established forcompositional mapping, allows one to analyse the similarityof neighbouring image cells (2 × 1 masks) relative to two


Fig. 10. FL compositional analysis of simulated (Al,Ga)As(110) patterns. (a) Original fuzzy rules and fuzzy sets, right: nonmonotonic simi-larity profile for a linear composition variation. (b) Similarity profiles rectified for physical reasons. (c) Centred membership functions of simi-larity. (Compare the different results of FL composition determination.)


templates of known composition. The values of similarityare transformed into trapezoidal and triangular member-ship functions to draw inference on the local composition ofone of the sublattices. The resolution attainable is in theorder of one crystallographic lattice parameter. Theapproaches have been applied successfully to interpretsimulated contrast tableaus of interfaces as well asexperimental micrographs of MBE-grown layered structuresof (Al,Ga)As (Al/Ga variation).

If the similarity/composition characteristics are known tobe highly nonlinear, the base size of the fuzzy sets can beadapted. Physical prerequisites to FL schemes of nonmono-tonic similarity/composition ratios are discussed and thecorresponding attempt is applied to simulated Al,Ga(110)barrier models.

It can be concluded that the FL methodology is aneffective and useful tool for solving problems of quantitativeHREM image analysis. Tailored FL rules and adequatemembership functions basically allow one to analyse evenother image features of HREM micrographs.

Acknowledgments

I am grateful to P. P. Wang (Durham, NC, U.S.A.) forsuggestions as to theoretical image processing and FL. Iwould like to thank R. Hey, Paul Drude Institute Berlin, forthe MBE-grown samples of AlGaAs heterostructures, andP. Werner for the excellent experimental work in the field ofHREM. Many thanks are due to D. Hesse for criticallyreading the manuscript.

References

Bezdek, J.C. (1981) Pattern Recognition with Fuzzy Objective FunctionAlgorithms (ed. by M. Nadler), pp. 60, 216. Plenum Press, NewYork.

De Jong, A.F. & Van Dyck, D. (1990) Composition determinationfrom HREM images of substitutional alloys. Ultramicroscopy, 33,269–279.

Diday, E. & Simon, J.C. (1976) Digital Pattern Recognition (ed. by K.S. Fu), p. 51. Springer-Verlag, Berlin.

Hillebrand, R. & Hofmeister, H. (1995) Quantitative analysis ofHREM images: measures of similarity. Phys. Stat. Solidi (a), 150,65–76.

Hillebrand, R., Hofmeister, H., Werner, P. & Gosele, U. (1995a)Quantitative high resolution electron microscopy of III-V com-pounds: a fuzzy logic approach. Appl. Phys. Lett. 67, 1763–1765.

Hillebrand, R., Wang, P.P. & Goesele, U. (1997) Fuzzy logic imageprocessing applied to electron micrographs of semiconductors.Proceedings of the Third Joint Conference on Information Sciences (ed.by P. P. Wang), Vol. I, pp. 55–57. Duke University, Durham.

Hillebrand, R., Wang, P.P. & Gosele, U. (1996a) A fuzzy logicapproach to edge detection in HREM images of III-V crystals.Information Sci. 93, 321–338.

Hillebrand, R., Werner, P., Hofmeister, H. & Gosele, U. (1995b)A fuzzy logic approach to quantitative HREM. Microscopy ofSemiconducting Materials. Inst. Phys. Conf. Series, 146, 57–60.

Hillebrand, R., Werner, P., Hofmeister, H. & Gosele, U. (1996b) Afuzzy logic approach to image analysis of HREM micrographs ofIII-V compounds. Ultramicroscopy, 66, 73–88.

Kahlert, J. & Frank, H. (1993) Fuzzy-Logik und Fuzzy-Control, pp.89–110. Vieweg, Braunschweig/Wiesbaden.

Kiesielowski, C., Schwander, P., Baumann, F.H., Seibt, M., Kim, Y. &Ourmazd, A. (1995) An approach to quantitative high-resolu-tion transmission electron microscopy of crystalline materials.Ultramicroscopy, 58, 131–155.

Klir, G.J. & Yuan, B. (1995) Fuzzy Sets and Fuzzy Logic: Theory andApplications. Prentice Hall, Englewood Cliffs, NJ.

Ourmazd, A. (1993) Chemical mapping and its application tointerfaces, point defects and materials processing. Mater. Sci. Rep.9, 201–250.

Ourmazd, A., Baumann, F.H., Bode, M. & Kim, Y. (1990)Quantitative chemical lattice imaging: theory and practice.Ultramicroscopy, 34, 237–255.

Pedrycz, W. (1993) Fuzzy Control and Fuzzy Systems, pp. 1, 14. J.Wiley, New York.

Schwander, P., Kisielowski, C., Seibt, M., Baumann, F.H., Kim, Y. &Ourmazd, A. (1993) Mapping projected potential, interfacialroughness, and composition in general crystalline solids byquantitative transmission electron microscopy. Phys. Rev. Lett.71, 4150–4153.

Seraphin, M. (1994) Neuronale Netze und Fuzzy-Logik, pp. 115,134. Franzis’, Munchen.

Stadelmann, P. (1987) EMS – A software package for electrondiffraction analysis and HREM simulation in materials science.Ultramicroscopy, 2, 131–146.

Stenkamp, D. & Jager, W. (1993) Compositional and structuralcharacterization of SixGe1-x alloys and heterostructures by highresolution electron microscopy. Ultramicroscopy, 50, 321–354.

Stenkamp, D. & Strunk, H.P. (1996) Quantitative determination ofdefocus, thickness and composition from high-resolution trans-mission electron microscopy lattice images. Appl. Phys. A, 62,369–372.

Tan, T.Y., Gosele, U. & Yu, S. (1991) Crit. Rev. Solid State Mater. Sci.17, 47–106.

Thoma, S. & Cerva, H. (1991) The influence of non-linearinterference processes on the HREM contrast of AlGaAs in<100> projection. Ultramicroscopy, 35, 77–97.

Tyan, C.-Y. & Wang, P.P. (1993) Image processing – enhancement,filtering and edge detection using the fuzzy logic approach. Proc.2nd IEEE Conf. on Fuzzy Systems (San Francisco), pp. 660–665.

Walther, T., Gerthsen, D., Carius, R., Forster, A. & Urban, K. (1993)Correlation between the structural and optical properties ofAlAs/GaAs quantum well structures. Microscopy of Semiconduct-ing Materials. Inst. Phys. Conf. Series, 134, 449–454.

Wang, C.Y. (1985) Edge detection using template matching. PhDdissertation, Duke University Durham, NC.

Zadeh, L.A. (1965) Fuzzy sets. Information and Control, 8, 338–353.Zimmermann, H.-J. (1991) Fuzzy Set Theory – and its Applications.

Kluwer, Boston.

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