Accurate Structure Refinement and Measurementof Crystal Charge Distribution Using ConvergentBeam Electron DiffractionJ.M. ZUO*Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287

KEY WORDS energy filtering; imaging filter; 2-D detectors; structure factor

ABSTRACT The method for accurate structure refinement from energy-filtered convergent-beam electron diffraction (CBED) patterns is described with emphasis on recent progress in usingimaging filters and 2-D detectors. Details are given about the underlying theoretical model and thestatistical analysis of experimental data. The relationship between crystal potential and chargedensity is also derived for crystals at thermal equilibrium. The method is applied to the refinementof Si (111) and (222) structure factors using various goodness-of-fit (GOF) criteria. Results show thatthe refinement method is robust and highly accurate. The importance of the experimentallymeasured structure factors is illustrated through the study of the charge density in MgO. With themeasured structure factors, it is possible to obtain details about the charge redistribution due tocrystal bonding. Microsc. Res. Tech. 46:220–233, 1999. r 1999 Wiley-Liss, Inc.

INTRODUCTIONRefinement is an integral part of the structure deter-

mination process. Accurate information about atomicpositions, Debye-Waller (D-W) factors, and crystal po-tentials or charge distributions is obtained through therefinement process of comparing experimental andtheoretical intensities, after the initial estimation ofatomic positions using direct methods. While consider-able difficulties exist for using direct methods withelectron diffraction (ED) due to the multiple scattering,refinement of ED data is not only possible, but also canbe done with high accuracy. This is because the theoryfor dynamical ED is well developed, and, once thestructure model is known, theoretical intensities can becalculated relatively quickly with today’s fast comput-ers.

In addition to X-ray or neutron diffraction with theirestablished methods and sophisticated software pack-ages, the ED method is needed for the following rea-sons: (1) Electron beam can be focused down to a probeof sub-nanometer size. Such a small probe can be usedto characterize materials of similar dimensions incombination with other electron microscopic methods.(2) Electron beam also interacts with crystals differ-ently from X rays or neutrons; with its Coulomb interac-tion the electron probes both electrons and nuclei incrystals. (3) The interaction strength with matter is,104 times stronger for 100 kV electrons than for Xrays.

Recent developments show that it is possible toobtain accurate measurements of unit cell parameters(Tomokiyo et al., 1993; Zuo, 1992; Zuo et al., 1997a),structure factors (Holmestad et al., 1995; Saunders andBird 1995; Zuo and Spence 1991), atomic positions(Tsuda and Tanaka 1995), and D-W factors (Nuchter etal., 1995) using quantitative convergent beam electrondiffraction (CBED). This is due to the fact that mostapplications in materials science involve the character-

ization of structural modifications, for which refine-ment methods can be readily applied. It is also signifi-cant because there is a genuine need of methods foraccurate structural parameter measurement; e.g., todetermine the charge distribution due to crystal bond-ing. Accurate structure parameters, such as the distri-bution of charge, provide the indispensable test of thetheoretical approximations (Zuo et al., 1997b). Tradi-tional X-ray methods have been less successful inmeasuring charge distributions of crystals, especiallythe charge redistribution from bonding. A very smallfraction of charge redistributes when atoms form crys-tals; this can only be measured with highly accuratestructure factors. The accuracy of X-ray structure fac-tors obtained using the kinematical approximation islimited by extinction and absorption effects.

Although quantitative analysis of ED is a morerecent development, the history of electron crystallogra-phy dates to the early stages of the development ofelectron microscopy. The Bloch wave theory for electronmultiple scattering was originally formulated by H.Bethe (1928) in his thesis. The first CBED pattern wasobtained by Mollenstedt as early as 1939 (Mollenstedt,1939) from mica. MacGillavry (1940) first attemptedstructure-factor measurement from an experimentalCBED pattern by using the two-beam theory of Black-man (1939). During the 1970s, there were a number ofattempts to measure structure factors from CBEDpatterns using more sophisticated methods, most notice-ably by Cowley, Goodman, Moodie, Lemphful, andIchimiya (for a review, see Spence, 1993). Recently, thedramatic increase in computer speed and the develop-ment of new detectors have brought significant progress

Contract grant sponsor: NSF; Contract grant number DMR 9412146.*Correspondence to: J.M. Zuo, Dept. of Physics and Astronomy, Arizona State

University, Tempe, AZ 85287. E-mail address: zuo@asu.eduReceived 6 October 1997; accepted in revised form 5 April 1999

MICROSCOPY RESEARCH AND TECHNIQUE 46:220–233 (1999)

r 1999 WILEY-LISS, INC.

in this field. The fast computer makes it possible to do afull dynamical calculation of CBED intensities. Thecombination of the energy-filter (Krivanek et al., 1995;Lanio, 1986; Rose and Krahl, 1995; Tsuno, 1997) andtwo-dimensional (2-D) digital detectors—slow-scan CCDcameras (SSC) (Spence and Zuo, 1988) and imagingplates (IP) (Ogura et al., 1994)—provides accurateexperimental intensities with a speed that was notpossible before.

The principle of structure-factor measurement fromCBED patterns was outlined in the study of GaAs forstructure-factor amplitudes (Zuo et al., 1988) and CdSfor structure-factor phases (Zuo et al., 1989). Spence(1993) gave a detailed review and a survey of other EDtechniques. Since then, the method has been improvedsignificantly. Especially, by the combined use of animaging energy filter and 2-D detector, 2-D diffractionpatterns can be routinely recorded and used in fitting. A2-D diffraction pattern recorded in regular arrays givessignificantly more experimental information than a fewline scans. To take advantage of the 2-D informationavailable from the new detectors, we have developed anew algorithm (Zuo, 1997a). However, these new tech-nologies have also introduced a new set of problems.Especially, it is necessary to correct for the non-uniformity and the limited resolution in the filterand/or detector. It is also necessary to estimate theamount of noise introduced by the detector. We willaddress these problems in this paper.

This paper concentrates on the refinement methodand the measurement of individual structure factors. Itis also possible to refine structure parameters directlyfrom CBED patterns. Many works in this area havebeen published recently. Specifically, for the measure-ment of atomic positions see Tsuda and Tanaka (1995)and Tomokiyo et al. (1996); for the measurement of theD-W factor see Nuchter et al. (1995); and for themeasurement of lattice parameters using the programdescribed here see Zuo et al. (1997a). An attempt hasalso been made to measure atomic positions using themethod described here; for an initial report see Holm-estad et al. (1997). For the application of quantitativeCBED to other areas of crystallography and materialsproblems, see Holmestad et al. (1999). The methoddiscussed here is general and can be applied to bothsystematic and zone-axis cases. However, in a zone-axisorientation, additional benefits can be obtained usingsymmetry. For refinement using zone-axis patterns, seeSaunders and Bird (1995).

THEORYThe Bloch wave method is used to calculate theoreti-

cal intensities. For ED from perfect crystals withrelatively small unit cells, the Bloch wave method ispreferable because of its flexibility and accuracy. Themulti-slice method or other similar methods are best inthe case of diffraction from strained crystals or crystalscontaining defects. A more detailed description of theBloch wave method can be found in Spence and Zuo(1992). For high-energy electrons, the exchange andcorrelation between the beam and crystals electronscan be neglected (Rez, 1978a), and the problem of EDreduces to solving the Schrodinger equation for an

independent electron in a potential field:

[K 2 2 (k 1 g)2] Cg 1 oh

UghCh 5 0. (1)

Here we have

K 2 5 ko2 1 Uo, (2)

and

Ug 5 UgC 1 U 9g 1 iU8g, (3)

where

Ug 5 2m 0e 0Vg /h2 (4)

is the optical potential for beam electrons, which con-sists of the crystal potential V, absorption V8, and acorrection to the crystal potential due to virtual inelas-tic scattering V9. The theory of optical potentials for anelectron beam was developed by Yoshioka (1957). Themost important contribution to U8 and U9 for g Þ 0comes from inelastic phonon scattering. Both U’ and U9can be calculated, given the phonon distribution in thecrystal. Details on the evaluation of the absorptionpotential are given by Bird and King (1990), by Weick-enmeier and Kohl (1991) for atoms with isotropic D-Wfactors, and by Weickenmeier and Kohl (1998) and Peng(1997) for anisotropic D-W factors. The virtual scatter-ing term U9 has been estimated by Rez (1978b) forlimited cases, and more recently by Anstis (1996). TheU9 term is significantly smaller than U’ for siliconaccording to Rez. However, for heavy elements the U9may become important (Anstis, 1996).

For an electron beam with a sufficiently small diam-eter, the diffraction geometry can be approximated by acrystal slab with surface normal n. Letting

k 5 K 1 gn, (5)

where g is the dispersion of the wavevector inside thecrystal, and inserting equation 5 into 1 leads to K2 2(K 1 g)2 2 2Kng 2 g2 5 2KSg 2 2Kng 2 g2, inside thesquare bracket. Neglecting the backscattering term ofg2, we obtain from equation 1

2KSgCg 1 oh

UghCh 5 2Kn 11 1gn

Kn2 gCg. (6)

Here Kn 5 K · n and gn 5 g · n. Equation 6 reduces to aneigen equation by renormalizing the eigenvector(Metherell, 1975):

Bg 5 11 1gn

Kn21/2

Cg.

To simulate a CBED pattern, a coordinate system isneeded to specify each beam direction, K. We use thezone-axis coordinate system, in which the incidentbeam direction is specified by its tangential component(Kx, Ky) in the closest zone axis. For each point inside

221REFINEMENT USING CBED

the CBED disk of g, the intensity is given by:

Ig(Kx, Ky) 5 0fg(Kx, Ky) 02

5 0oi

ci(Kx, Ky) Cgi (Kx, Ky) exp [2pigi(Kx, Ky) t]0

2

,(7)

with eigenvalues gi and eigenvector Cgi from diagonaliz-

ing equation 6. Here ci is the first column of the inverseof the eigenvector matrix or the excitation coefficient asdetermined by the incident-beam boundary condition(Spence and Zuo, 1992). The calculations converge withincreasing number of beams included. Straightforwarduse of equation 6 with a large number of beams is oftenimpractical since the matrix diagonalization is verytime-consuming. The computer time needed to diagonal-ize an N 3 N matrix is proportional to N3. A solution tothis is to use the Bethe potential:

Ugeff 5 Ug 2 o

h

Ug2hUh

2KSh, (8)

and

2KSgeff 5 2KSg 2 o

h

Ug2hUh2g

2KSh(9)

for weak beams. Here the summation is overall weakbeams h. The Bethe potential accounts for the perturba-tion effects of weak beams. A weak beam is definedwhen

0KSg

Ug0 $ vmax. (10)

The vmax is selected for the best theoretical convergence(Birkeland et al., 1996; Zuo and Weickenmeier, 1995).Beams are selected subject to cutoff criteria of maxi-mum g length and their perturbation strength. Theseshould be tested to check their effects on the calculatedintensities.

For a 100-kV electron traveling at a speed 1⁄3 that oflight, the time it spends inside a TEM specimen isabout 10215 seconds. The typical phonon frequency isabout 1012 Hz. Thus, the fast electron sees individualfrozen configurations of a vibrating lattice. The experi-mentally observed diffraction pattern at a 1-secondexposure time is an average of individual diffractionpatterns from ,1012 instant configurations of the crys-tal. It can be shown that the average diffracted beam ata reciprocal lattice point is equal to that scattered by anaverage potential 7V(r)8 under both the kinematical anddynamical diffraction conditions. In general we have,

7V(r)8 50e 0

4peoe d3r8

7Z(r8) 2 r(r8)8

0r 2 r8 0, (11)

and

Vg 5 e d3r exp (2pig · r)7V(r)8

50e 0

4peoe d3 r8 exp (2pig · r8) 57Z(r8)8 2 7r(r8)86

· e d3rexp [2pig · (r 2 r8)]

0r 2 r8 0

(12)

50e 0

4p2eo g2(Zg 2 Fg). (13)

Here we have

Fq 51

N e d3r exp (2piq · r) 7r(r)8

51

N ol 5 1

N

exp (2piq · Rl) oi

fi(q) exp (2piq · ri)

3 7exp (2piq · ui)8

5 d(q 2 g) oi

fi(q) exp (2piq · ri)

3 7exp (2piq · ui)8,

(14)

and similarly,

Zq 5 d(q 2 g) oi

Zi exp (2piq · ri)

3 7exp (2piq ·ui)8.(15)

Here u is the displacement from the equilibrium atomicposition and g is a reciprocal lattice point. The quantity,7exp (2piq · ui)8, is the temperature factor. For har-monic vibrations, it can be shown that (Willis andPryor, 1975)

T(g) 5 7exp (2pig · u)8 5 exp (22p27(g · u)28). (16)

For isotropic vibrations, this reduces to the familiarD-W factor

T(g) 5 exp (22p2g27u28) 5 exp (2Bg2/4), (17)

with B 5 8p2 7u28. For the anisotropic case, see Willisand Pryor (1975). Equation 12 is the basic relationshipbetween potential and charge density as measured byelectrons or x-rays for the study of crystal bonding. It isclear from equation 13 that the thermal effects on bothpotential and charge density can be described by thesame temperature factors.

EXPERIMENTThe experimental setup using an in-column V energy-

filter is illustrated in Figure 1. A probe (P) is formed ontop of a thin crystal plate by de-magnifying the source

222 J.M. ZUO

using the condenser lens. A circular aperture is placedafter the condenser lens, which forms a cone of illumina-tion. A diffraction pattern consisting of disks is formedat the back focal plane of the objective lens. A point inthe CBED disk and all conjugate points in other disks(related by reciprocal lattice vectors) constitute a diffrac-tion pattern for one incident beam direction in theilluminating cone. In Figure 1, the filter accepts theimage of the probe formed by the objective lens. In theLEO 912 electron microscope, there is an additionalintermediate lens between the filter and objective lens.The filter forms a series of probe images at its exitplane, dispersed according to the beam energy. Anenergy-selection slit is placed around the zero-lossimage to obtain the energy-filtered CBED pattern,which is recorded with a 2-D detector. The condensoraperture determines the CBED disk size. The diskshould be comparable to the reciprocal-lattice spacingto avoid excessive overlap, which makes the interpreta-tion of experimental data difficult. For large unit-cellcrystals, the large angle CBED (LACBED) techniquemay be useful. LACBED selects a single diffractedbeam by using a small selected-area aperture. LACBED

also removes part of the large-angle inelastic scatteringdue to phonons, which is not possible with the currentenergy-filter (Morniroli et al., 1997).

The diffracted beams pass through an energy filter,which is placed after the final intermediate lens andbefore the projector lens of the microscope. Alterna-tively, a post-column energy filter can be placed belowthe viewing screen (Krivanek et al., 1995). With thein-column filter, it is possible to have a combination ofdifferent detectors, and it also takes the full advantageof the projector lens of the microscope. Uhlemann andRose (1994) made a theoretical comparison betweendifferent filters. Details about the V-filter can be foundin Lanio (1986), Rose and Krahl (1995), and Tsuno et al.(1997). The LEO 912 V filter uses the design of Lanio(1986). A description of the post-column Gatan ImagingFilter (GIF) is given by Krivanek et al. (1995). For the Vfilter, the filter takes an image (in diffraction mode) ofthe diffraction pattern at the entrance pupil plane andforms a diffraction pattern at the exit pupil plane andan image at the energy-slit aperture. Figure 2 shows azero-loss energy-filtered Si[111] CBED pattern and thecorresponding spectrum (see inset in Fig. 2) using the

Fig. 1. Schematic diagram of experimental setup using a in-column imaging energy filter.

223REFINEMENT USING CBED

LEO 912 electron microscope with an energy slit 16 eVwide. The envelope seen in Figure 2 is due to thenon-isochromaticity caused by the axial second-orderaberrations at the energy-selection plane (Rose andKrahl, 1995). The effects of this aberration make acircle in the object plane into an ellipse; the center ofwhich is displaced by a distance proportional to thesquare of the radius. This is illustrated in Figure 3. Adisk forms a flattened cone-shaped aberration figure.Figure 3c shows the experimentally observed aberra-tion figure for silicon. For a large object, the aberrationfigure overlaps on the energy-dispersion plane withthose for different energy losses. With a finite energyslit, the primary beam energy of the filtered patterndiffers at different radii from the optical axis (callednon-isochromaticity). By using a narrower energy slit,it is possible to reduce the contributions from electronsof different energies. However, this also results in asmaller field of view. For energy filtering with thezero-loss peak, the problem is most severe for metalswith large plasmon losses. In the CBED mode, theimage is made of diffracted beams. They are sufficientlyseparated in the Leo 912 V filter for the filtering of a fewreflections (see Fig. 2). We observed no significantdifference between the zero-loss disk intensities with 8-and 16-eV windows. The plasmon energy of silicon isabout 18 eV. Krivanek et al. (1995) reported a betterperformance for isochromatic imaging using the GIF.The field of view of the GIF is limited by the smallestcamera length (CL) available in the microscope and thesize of the CCD used for detection. The GIF has amagnification of 153, so to record a diffraction patternof 600 mm CL with the GIF, the microscope CL needs to

be 40 mm. At such a small CL, it is often difficult toadjust for lens distortions. Additionally, there are someresidual distortions in the GIF, which are difficult tocharacterize. The geometric distortion is almost absentin the V filter owing to its mid-plane symmetry (Roseand Krahl, 1995), in principle.

For quantitative analysis, the energy-filtered CBEDpattern is recorded digitally with an SSC or IP in anarray of pixels. Both detectors have excellent linearityand dynamic range in each pixel. For a comparisonbetween these two detectors, see Zuo (1996) and Zuo etal. (1996). Two important characteristics of a lineardetector for quantitative analysis are the point-spreadfunction (PSF) and detector quantum efficiency (DQE).The PSF determines the resolution of as-recorded im-ages, while DQE measures the amount of noise intro-duced in the detection process. Methods for measuringthese are given by Zuo (1996) and Weickenmeier et al.(1995). Figure 4 shows the measured modulation trans-fer function (MTF) and DQE for Gatan model 679 SSC.

In general, the recorded image can be expressed as

z 5 Hf 1 n. (18)

Here z and f are the recorded and original image,respectively, n is the noise, and H is the PSF operator.The effects of the PSF can be removed by deconvolution.However, the direct deconvolution of a recorded imageusing the inverse of the PSF leads to an excessiveamplification of noise. To overcome this, we use theconstrained image-restoration technique by employinga filter of type (Ren et al., 1997):

Q(u, v) 5H*(u, v)

H*(u, v) H(u, v) 1 gP(u, v)(19)

with P(u, v) as a smoothing function and g adjusted tosatisfy the constraint

00g 2 Hf 00 5 00Dn 00 . (20)

Here f is the deconvoluted image and Dn is the esti-mated additional noise introduced in the detectionprocess. There are a number of choices for P(u, v). Theoptimum in the least-square measure is the Wien filter,in which P(u, v) is the ratio of the power spectra ofsignal and noise. This requires a prior knowledge of thesignal, which is often unknown. Alternatively, we maytake P(u, v) 5 [24p(u2 1 v2)]2, which makes the re-stored image as smooth as possible. The value of g,which measures the degree of removal of the PSF, with0 as complete, depends on the amount of noise in therecorded image—the smaller the noise, the better thedeconvolution. Other procedures for deconvolution arealso available. The Richardson-Lucy method describedby Snyder et al. (1993) is specifically targeted forPoisson processes, which can be applied to CCD images.

Figure 5 shows the relative importance of energy-filtering and image deconvolution. The example used isthe TiA1 (200) systematics. The top is the unfilteredCBED pattern, which shows a large amount of back-ground in the line profile. The filtering removes most ofthe background. The remainder is mostly due to the

Fig. 2. Energy-filtered Si [111] zone axis pattern using the V-filterof Leo 912 electron microscope at 120 kV with 15 eV energy window.The corresponding spectrum is shown at lower right corner. The whitelines indicate the position of the energy window. Arrows indicate theenergy-filtering envelope. (The size of the envelope can be enlargedwith free lens control).

224 J.M. ZUO

limited resolution of the detector, which is removed bydeconvolution. The deconvolution also makes edgessharper. For comparison, we show the fitted theoreticalpattern at the bottom. The effects of filtering anddeconvolution are large compared to the amount ofnoise expected from the detector.

The quality of a CBED pattern depends on thesample and the microscope. Critical alignment of themicroscope is required. Special attention is needed forthe optimum objective lens setting, specimen z-posi-tion, correction for projector lens distortion, and thechoice of probe size. For the use of an SSC, a high-quality image is needed to correct for the gain varia-tions. For a relatively flat sample with no defects, such

as silicon, acquiring a good-quality CBED pattern in anon-zone-axis orientation for quantitative analysis isfairly easy. Acquiring a good zone-axis pattern is consid-erably more difficult; the strong dynamical channelingcondition is very sensitive to defects and surface imper-fections or contamination. Distortions of CBED pat-terns due to crystal faults such as stacking faults anddislocations are easy to spot. The deficiency line tendsto split owing to the strain field associated with thesefaults. Thickness variation under the probe also dis-torts the CBED pattern; this can be reduced with asmaller probe. The ultimate check of the quality ofCBED patterns is the refinement itself, which mea-sures the degree of agreement between theory and

Fig. 3. The aperture aberration in uncorrected V energy filter. a: Aberration figure in energy selectionplane. b: Corresponding circle in the object plane for ellipse in a. c: Experimentally observed spectrum forSilicon in image mode.

225REFINEMENT USING CBED

experiment. Additional assessment of experimentaldiffraction pattern quality can be made in a zone-axisorientation, where the symmetry-related points can beused to detect distortions.

REFINEMENTAutomated refinement from CBED intensities is

achieved by finding the best fit between experiment andtheory. The best fit is defined by using a goodness-of-fit(GOF) parameter:

GOF 5 e oi

h(I iexp, si, I i

t). (21)

Here the function h measures the agreement betweenthe experimental intensity, Iexp, with estimated uncer-tainty s and the theoretical intensity, It. The theoreticalintensity is calculated with parameters, p. e is thenormalization coefficient. In the maximum-likelihoodapproach, h is the logarithm of the probability distribu-

tion of (I t 2 I exp)/s. Box and Tiao (1992) defined a classof probability distributions

r(I exp 0p, s) ~

exp 125G[3(1 1 b)/2]

G[(1 1 b)/2] 61/(11b)

0Iexp 2 I t( p)

s 02/(11b)

2 ,(22)

with 21 , b # 1. For b 5 0, equation 22 reduces to theGaussian distribution. On the basis of equation 22, wecan define a series of GOF’s using different values of b:

GOF 5 e oi0I i

exp 2 I it( p)

si02/(11b)

. (23)

For constant standard error, s, the lowest GOF givesthe maximum likelihood of the probability distributionof equation 22. The definition of equation 23 encom-passes the two most commonly used measures in

Fig. 4. Experimentally measured (a) MTF and (b) DQE of Gatan model 679 SSC. The dashed lines inb are projected DQE with zero noise in gain image.

226 J.M. ZUO

crystallography, the x2 (b 5 0) and normalized R-factor(b 5 1). For large b, the probability distribution falls offslower than the Gaussian, which gives a more robustmeasure against systematic errors in theory and experi-ment due to its long-tailed distribution. For caseswhere differences between theory and experiment aremostly normally distributed, except for a few outlierpoints, the x2 refinement can be distorted owing to thepresence of these outlier points (for discussion on this,see Huber, 1981). Huber proposed an alternative mea-sure considering a small percentage of outlier points:

M 5 e oi

hi 1 0I iexp 2 I i

t( p)

si0 2 , (24)

with

h(t) 5 5 t2 t , a

2at 2 a2 t $ a(25)

This definition adopts the normal distribution for thecenter and exponential distribution for the tail. Theparameter a is selected so that the majority of points

(for example 95%) is covered by the normal distribu-tion. In next section we will show how the choice of GOFaffects the refinement results using a test case ofsilicon.

The experimentally recorded intensity in each detec-tor pixel (i, j) is the sum of all intensities within thepixel. Intensity mixing between different beam direc-tions is also expected from the finite spatial resolutionof the detector and energy window of the filter; inparticular, both small- and large-angle thermal diffusescattering are not removed by the energy filter. Ageneral expression for the theoretical intensity consid-ering all these factors is:

I i, jt 5 ee dx8dy8t(x8, y8) C(xi 2 x8, yj 2 y8)

1 B(xi, yj).(26)

Here, the diffracted intensity, t, convoluted with theresponse function, C, and the background intensity, B,are integrated over the pixel. Simple models of C arethe delta and Gaussian functions, which can be imple-mented easily. The background function is often as-

Fig. 5. A comparison between the effects of energy-filtering and deconvolution. For reference, thebottom shows the fitted theoretical pattern.

227REFINEMENT USING CBED

sumed constant within the CBED disk (Saunders andBird, 1995), which seems sufficient for the refinementprocess.

Depending on the experimental intensities collected,the parameters can be structure-factor amplitudesand/or phases, absorption coefficients, lattice con-stants, atomic position parameters, and D-W factors.Other parameters include crystal orientation, thick-ness, electron accelerating voltage, detector function,and residual diffuse background intensities. For struc-ture-factor measurements, we can choose between sys-tematic, three-beam, or zone-axis diffraction condi-tions. We prefer the systematic condition for itssimplicity, short computational time, and good accu-racy.

Estimation of errors in refined parameters is quiteinvolved in the case that the distribution is not normal.For discussions on this, see Box and Tiao (1992). In thenormal case, the precision of the measured parametersis given by

sak

2 5 x2Dkk21, (27)

and

Dkl 5 oi

1

si2 1­I i

t

­pk2 1­I i

t

­pl2 . (28)

The partial derivative of the intensity with respect to aparameter can be calculated numerically or by using

perturbation theory. General formulas using the first-order perturbation method for these derivatives aregiven by Zuo (1997b).

There are a number of refinement algorithms (Zuoand Spence, 1991; Nuchter et al., 1995; Saunders andBird, 1995; Birkeland 1997). The REFINE program byZuo and Spence (1991) uses line scans of intensities.Modifications of this program have also been made toinclude non-parallel line scans (Nuchter et al., 1995;Birkeland 1997). The algorithm of Saunders and Bird(1995) uses a Fourier-transform method for the evalua-tion of derivatives and has been mostly applied tozone-axis patterns.

We have developed an new algorithm EXTAL (Elec-tron crys(X)TALlography) for quantitative analysis of2-D CBED patterns (Zuo, 1997a; Zuo, 1998b). Thesimplified flowchart of the program is shown in Figure6. This program is capable of dealing with multipleCBED patterns by separating the structural param-eters of structure factors, atomic positions, and D-Wfactors from the geometrical parameters of crystalorientation, image position, camera length, and samplethickness. Since ED patterns are generally two-dimensional owing to the short wavelength of the

Fig. 7. a: Experimentally recorded energy-filtered Si (1 1 1) sys-tematic CBED pattern. b: Intensities along lines indicated in a and thebest theoretical fit. c: Difference between theory and experimentplotted in b normalized by the estimated error in experimentalintensities.

Fig. 6. A simplified flow chart of EXTAL program. Two optimization cycles are employed for structuraland geometric parameters respectively. The optimization for geometric parameters is repeated for eachindividual diffraction pattern.

228 J.M. ZUO

a

b

Fig. 7.

incident electrons, using multiple diffraction patternsin different projections enables the measurement ofthree-dimensional parameters such as atomic positionsand the parameters of the three-dimensional unit cell.The separation between structural and geometricalparameters is achieved using two optimization cycles.For each trial in the structural parameter optimization,an independent optimization is carried out for thegeometrical parameters. This is repeated for each dif-fraction pattern included. Considerable saving of com-puter time is achieved by fixing the beam direction inthe theoretical calculation and locating the correspond-ing directions in the 2-D pattern as suggested byGribelyuk and Ruhle (1993). When additional imageprocessing is applied to both experimental and theoreti-cal CBED patterns, the program has shown to be veryeffective in measuring lattice parameters in the pres-ence of dynamical effects. The advantage of this pro-gram, compared with the REFINE program, is thatmultiple line or area scans can be taken, which allowsthe user flexibility in designating the experiment forthe maximum sensitivity to the measured structuralparameters. An example of using this program forstructure-factor measurement is shown next. For anexample of measuring lattice parameters, see Zuo et al.(1997a).

REFINEMENT OF SI (111) AND (222)STRUCTURE FACTORS

As an example, we describe the structure factormeasurements of Si (111) and (222) using the methodsdescribed here. Silicon is a good test case because ofexisting X-ray pendellosung measurements. The experi-mental CBED pattern of silicon (1 11) systematics wasrecorded by tilting about 12° off the [110] zone at roomtemperature and with a beam energy of 196.7 kV. Thisis shown in Figure 7a. The pattern was taken on aPhilips CM200 with a field-emission gun and equippedwith a GIF (BEML, China). The characteristics of theCCD were measured following the procedures describedby Zuo (1996). Figure 7a was deconvoluted using themethod described in section 3 (Ren et al., 1997). Theorientation was estimated by indexing the diffraction pat-tern using the IdealMicroscope program (Zuo et al., 1993).

The amount of experimental data in figure 7a farexceeds the number of parameters to be determined, sowe select only some of the pixels in the experimentalpattern for fitting. The typical number of data points isabout a few thousand. We chose these data points byline scans. These lines are drawn with considerationgiven to their sensitivity to certain parameters. Forexample, intensities along a line across a diffracted diskare sensitive to structure factors and specimen thick-ness, intensities along a line across the zero disk aresensitive to absorption parameters, and intensitiesalong a line across ZOLZ and HOLZ lines are sensitiveto orientation and magnification. The line scans used

for silicon (1 11) systematics are shown in Figure 7a.The refinement program EXTAL was used with differ-ent GOF criteria. Theoretical intensity was calculatedwith 279 beams, selected using the beam selectioncriteria described in Zuo and Weickenmeier (1995). Therefinement program was run repeatedly with differentstarting points to ensure that the real minimum wasfound.

The refinement depends on the GOF criterion se-lected. To check the effects of this and to see whichcriterion may suit our purpose best, we refined Figure7a with various GOF defined in Refinement. Table 1shows the refinement results with different b for theGOF of equation 23. The estimated error is quoted inparentheses for the normal distribution (b 5 0). Therefinement results depends on the selected GOF or bvalue. The column ‘‘Mean’’ shows the average of fivedifferent refinements and its standard deviation. Thedifference between the (222) structure factors deter-mined using the x2 and R-factor is somewhat largecompared with the estimated error.

Table 2 compares the results using the M estimator ofHuber with different cutoff values of a. As a decreases,the results approach the ones obtained with the R-factor. Compared with the X-ray value, the mean fromTable 1 gives the best agreement. Of the individualGOFs, the R-factor seems the best. The results from thestandard x2 are acceptable within the estimated error;although the (222) is on the low side compared with theX-ray value.

To check the success of the selected GOF, we look atthe distribution of the residual difference betweenexperiment and theory. Figure 8 shows the histogram ofthe normalized difference obtained with x2, the Huberestimator of a 5 2.0, and the normalized R-factor. Forcomparison, we also fitted the histogram with Gaussian(Fig. 8a and b) and exponential (Fig. 8c) distributions.The R-factor resulted in the largest number of pointshaving deviations within the estimated error. In obtain-ing the mean from different distributions in Table 1, weassumed an equal probability for the distribution. Amore rigorous averaging using Bayesian statistics wouldrequire the evaluation of the probability for the distribu-tion with the given data. This would require integrationin the parameter space, which is difficult to do in thiscase. For a full discussion on the robust statistics inBayesian approach, see Box and Tiao (1992).

TABLE 2. Dependence of Si (1 1 1) and (2 2 2) Refinement Resultson the M-Estimator of Equation 24 and Comparison

With the Experimental X-Ray Results*

Å2 A 5 3.0 A 5 2.0 A 5 1.5 X ray

U(1 1 1) 1021 0.4726 0.4731 0.4738 0.4736(4)U8(1 1 1) 1023 0.822 0.823 0.825U(2 2 2) 1023 0.867 0.914 0.978 0.943(5)

*For a list of X-ray structure factors of Si, see Zuo et al. (1997b).

TABLE 1. Dependence of Si (1 1 1) and (2 2 2) Refinement Results on the Probability Distributions of Equation 22

Å2 b 5 0 b 5 0.25 b 5 0.5 b 5 0.75 b 5 1.0 Mean

U(1 1 1) 1021 0.4732(5) 0.4737 0.4746 0.4727 0.4738 0.4736(7)U8(1 1 1) 1023 0.832 0.872 0.807 0.829 0.825 0.833(23)U(2 2 2) 1023 0.893(54) 0.912 0.862 0.990 0.949 0.921(50)

230 J.M. ZUO

The results here show that the refinement method ishighly accurate for the structure-factor measurementof both strong and weak reflections. The accuracy forthe Si (111) reflection is comparable to the X-raypendellosung method. Unlike the X-ray method, the EDmethod discussed here is general. With the small probeof CBED, it can be applied to most crystals with smallunit cells. There is no fit-for-all magic solution inchoosing the best GOF criteria. Results here show thatthe refinement does depend on the GOF within theestimated error bar. The R-factor gives better resultsthan x2 in this case.

CHARGE DENSITY IN MgOOne of the important applications of quantitative

CBED is the study of crystal bonding by accuratemeasurement of low-order structure factors. An ex-

ample of this is the study of the charge density in MgO(Zuo et al., 1997c; Zuo 1997b). For applications ofcharge-density measurements to important materialsproblems, see Holmestad et al. (1999). Magnesiumoxide is a prototype of ionic bonding with its NaClstructure. There is a general lack of experimentalknowledge on the charge distribution in ionic crystals.In studying MgO, we want to see whether the chargedistribution of MgO can be described in the traditionalpicture of ionic bonding with a charge transfer of twoelectrons from Mg to O. We also want to see the chargedistribution of oxygen. With oxides playing an increas-ingly important role in materials with new properties,such as high-Tc superconductivity and colossal magneto-resistence, understanding oxygen states and their de-pendence on crystal structure can provide an importantclue to the mechanism for these exotic phenomena.

Fig. 8. Histogram of residual difference, (I t 2 I exp)/s, as obtained from (a) chi-square fit (b) M fit witha 5 2.0, and (c) R-factor fit. For details, see text.

231REFINEMENT USING CBED

The MgO specimen was prepared by burning magne-sium ribbons in air and collecting MgO smoke on acopper grid covered with a thin carbon film. A platelet ofmicrometer size and about 80 nm thick was selectedfrom different shapes of MgO particles. The experimentwas performed on a Philips 400T electron microscope.The intensities were measured by deflecting the CBEDpattern over the entrance aperture of a Gatan 607serial energy-loss spectrometer. This was done beforethe energy filters and parallel detectors were available.This setup still offers unsurpassed resolution and signal-to-noise ratio when conditions in the microscope arestable during the scan. Details about the measurementhave been published (Zuo, 1997b). Table 3 lists allexperimentally measured structure factors togetherwith theoretical models for comparison.

The theoretical structure factors are calculated usingdifferent models. The Dirac-Fock (DF) spherical ionsare calculated using the multi-configurational DF pro-gram of Rez et al., 1994 and a Watson 12 well foroxygen, which is an atomic calculation. In this model,we found that the radius of 1.2 Å gives the best fit toexperimental structure factors. Theoretical structure

factors are calculated using the full-potential linearizedaugmented plane wave (LAPW) method as imple-mented in the WIEN95 package (Blaha et al., 1990) forcrystals. The calculation uses approximated function-als for the exchange and correlation energies. Twoapproximations are used. One is the local densityapproximation (LDA) and the other is the generalizedgradient approximation (GGA). The LDAgives a slightlybetter fit than the GGA, but the difference is small.

The deformation map, formed by the difference be-tween the crystal charge density and the charge densityof superimposed atoms, is used to show the bondingcharge distributions. Figure 9 shows the difference incharge density between the experimental data andsuperimposed neutral atoms. The continuous linesshow excess electrons and the broken lines show thedepletion of electrons. Figure 9 clearly shows thecharge transfer from Mg to O. The charge distributionaround the oxygen is slightly non-spherical. A multipleanalysis of the MgO charge density shows a smallhexadecapole component. It was found that the chargedistribution in MgO can be in the full ionic form.However, the distribution of transferred electrons issignificantly broader than predicted by theory. Thissuggests some effects due to the polarization of chargein the ionic lattice due to thermal vibrations. Fordetails about this, see Zuo et al. (1997c).

CONCLUSIONSRecent developments in quantitative ED have shown

that it is possible to extract quantitative structureinformation from CBED patterns. The success relies onquantitative analysis of experimental and theoreticalintensities. The experimental setup for the measure-ment of intensities was found sufficient. The theoreticalintensity is calculated taking the full account of dynami-cal effects. Experimentally measured structure factorsprovide an accurate description of charge distributions

Fig. 9. MgO charge density difference map between crystal and superimposed neutral atoms in a-bplane for (a) experiment and (b) theory using LAPW and LDA.

TABLE 3. Listing of Present Measured Low-Order Structure Factorsof MgO and Comparison With Theory*

hkl Present DF-N1 DF-ION2 LDA3 GGA4

111 11.142(20) 12.389 11.090 11.175 11.082200 52.89(3) 52.030 53.040 52.765 52.918220 40.68(8) 41.073 41.062 40.953 41.072311 12.41(12) 12.309 12.633 12.356 12.401222 33.75(12) 34.005 33.800 33.777 33.865400 29.01(8) 28.993 28.790 28.949 29.012331 10.06(10) 9.6401 9.7352 9.6025 9.6533420 25.1(2) 25.256 25.065 25.205 25.256422 22.7(3) 22.371 22.222 22.348 22.390R-factor .011 .0072 .0063 .0067

*Superimposed spherical 1atoms and 2ions calculated using Dirac-Fock method;Crystal structure factor calculated using LAPW and the 3LDA or the 4GGA.

232 J.M. ZUO

in real crystals. For MgO, the experimental chargedensity can be approximated by a superposition ofspherical Mg21 and O22 ions. The additional differencebetween theory and experiment is due to the details ofthe charge distribution around oxygen.

ACKNOWLEDGMENTSMany thanks to Dr. R. Holmestad for providing

Silicon diffraction patterns and Prof. John Spence formany discussions.

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233REFINEMENT USING CBED