H. \!'AXG and I). h u : Foil Thickness Determination by CUED

phys. stat. sol. (a) '3'3, 47 (1987)

Subject classification: 61.14; $1.2; S4; S10.1

Bepartment of Physics, U'uhen ( i n i ~ e r s i t y ~ )

Foil Thickness Determination by CBED in the Kinematical Approximation7 BY K. WANO and I). LIU

A method of foil thickness determination is reported using the convergent beam electron diffraction (CBED) technique in the kinematical approximation. By selecting a.ppropriate 'ny systematic reflections so that the condition 1Egg2 2 4 holds and bringing t,he -3g disk to the Bragg position, t.he foil thickness t can be determined from the measured distance AE between positions of neigh. bouring minima in g-disk using the kinemat,ical approximation. The minimum measurable t.hick- ness is smaller than that measurable by the technique in the two-beam dynamicd condition.

Es wird eine Methode der Foliendickenbestimmung mgegeben, die die Konvergenzstrahl-Elek- tronenbeugung (CBED) in der kinematischen Naherung benutzt. Wahlt man geeignete systema- tische ng-Reflexe aus, so daB die Bedingung l&,g2 2 4 erfiillt wird, und bringt man die -39- Scheibe zur Bragg-Lage, so kann die Foliendicke t aus dem Abstand AZ zwischen benachbarter: Minima in der g-Scheibe mit der kinematischen Naherung berechnet werden. Nach dieser Met,hode ist die minimale meBbare Foliendicke geringer als die unter dynamisehen Zweistrahlbedingungen erhaltene.

1. Introduc.tio~i

Several methods of determining the crystal thickiiess by means of convergent heam electron diffraction (CBED) have been developed. The intensity distribution in a CBED p t t e r n of a specimen of thickness t corresponds directly to the rocking curve. By comparing the observed intensity distribution in CBEI) patterns of a MgO crystal n ith rocking curves calculated by n-beam dynamical theory for different thicknesses t , Goodman and Lehmpfuhl [l] obtained a n accuracy better than 1 yo in foil thickness determination. Based on the two-beam dynamical approximation MacGillavry [ Z ] derived the following relation between the deviation parameter s , of the i-th minimum, the extinction distance tff of reflection g and the integer TI,%:

( 1 1 A\ecoidiiig to ( I ) , llckerniatin 131 shoncd that the foi l thickness t cittl he obtained from t h e s l o p of a plot of sf V H I ~ S I I S t . Later, Kelly ( A t al. 141 pointed o i r t that (1 ) cnti 1" rearrnnged ns

(s? 4 1/53 t z = 1Lf .

( t 5 t / t / 7 ) 1 ( - - I / @ ( l / n , ) 2 , l / t A ( 2 )

arid the thickriess determined more accurately frorn the intercept of the &might litre of ( ~ , / n , ) ~ versus (l/n&)Z. Allen [ 5 ] derived an expression of the minimum measurable thickness with the technique of Kelly et al. For the lowcr-order reflection of A1 and Cu crystals, the ininimum thicknesses t,,,, are given in Table 1.

I ) Luo Jia Hill, Wuhan, People's Republic of China. 2) Projects supported by the Science Fund of the Chinese Academy of Sciences.

48 It. \V.\NC; irlld D. Lru

-2 I 111 57 200 47 220 24

20 15 8

C'U 111 60 200 34 340 19

16 12 6

[ii this payer, we shall ititr<duce a niethod of foil thickness deterinination from a Cl3EI) pattern using the kinematical approximation. Compared with the technique of Kelly et al. [4] this method needs neither simulation nor plotting, and reduces the loww. limit of thickncss which can be determined.

2. H i rw i r r r tied Approxi matioil

As we know, the intensity of the diffractioii beam in "o~ic-beani" kiuematical approxi- mation is giveii by

sin2 (nts,) I , = (iZt/&,)2 -. . (nts,)2

Here 8, represents the deviation paraineter of reflection g. Therefore, the period of oscillation in a y-disk is giveti by

Equation (3) holds only if I,, the inteiisity of the strongest diffracted beam, is much weaker than tha t of the incident beam. From (3) i t follows that

AS,, = l / t . (4)

1 I < " = (5)

Hence if (a&,( 2 p > 1, then (3) holds. ( .hers 161 chose arbitrarily the value of y = (%. Taft arid Metzger [7] suggested that for Bragg reflection with (sin @/A > > 10 mn-1, 100 keV electroiis can be treated quasi-kinematically for crystal thick- IICSSC'S less thaii 100 i in i . 111 order to explore conditions in which all minimum positions of II dyiiaiiiictil rocking ciirvc: coiticictc? with thc kiiicniut.icd positioiis, :I serics of thco- i*c:li cid rock i i ig cur w s wcro c i i h i lirtcd.

Let k, hc the coittpcment of the iucic1t:irt \vuv(: vector prtrallel to t h c t diffrautioti vcctor g, and k,, the valiie of k whcn the Uragg reflectioii coiiditioii for y is exactly xatisfitd (Itzf, = -0.Bg). Yrom the reflection sphere cotixtriictioti in Nig. 1, it follows that

wherc: i, is the wa\&iigth of tlic clectroiis. Tlie value of y iiiay be expressed as

where ( 1 ) is the miiiimum value of I (k, - klur)/gl in the case of coincideiice between the dytiatriical a d kinematical (cf. (3)) miaimum positions.

p = a t 8 2 (7 )

Equation (3) suggests that the variatioti of l , / ( ~ . l / ~ , ) ~ as a function of (k, - k,)/g only depends ~ I I the parameter tIg2, but not 011 the individual values of g, I, or t. Itockirig curves for oiie row of systematic refleotioiis were calculated for tAg2 = 1, 10/9, 5/4, 10/7, 5/3, 2, 5/2, 10/3, and 5 using seven-beam dynamical theory altering, with chatige of orieotatioti, the beams involved [S]. The rocking curves were calculated for the (I 1 l), (200), and (220) systematic rows of Al, Cu, and MgO crystals at accelerat- ing voltages of 80, 100, and 120 kV. Fig. 2 and 3 show parts of calculation results with tIg2 = 1 and 2, respectively. Table 2 shows the results of three different systematic rows of aluminium at 120 kV. From one computed curve, we can only get a range of the value of m. For instance, in the case of (111) systematic reflections of an A1 crystal a t 120 kV (see full line in Fig. 2), the (k, - k,)/g value of the first minimum is not that of the kinematical approximation ((k, - k,)/g = 1.0) while the second is. It follows that w must lie between 1 and 2. Each actual value of w in column 7 of Table 2 is taken as the common value of several w ranges for different thicknesses t but the same value of Itpg2, owing to (7).

Summarizing the calculated results, we conclude that the minima of the dynamical curves coincide with kinematically calculated positions if the following condition is fulfilled :

(8) &$2 yc;& 1 =; Is,€vl 2 6 - This corresponds to I, 0.028 ( p = 6).

3. Thickness Determination in the Hinerncrtical Approximation Because of (S), if a systematic row is selected so that the condition A[,g2 2 4 holds, the positions of the mitiinla in y diffracted disk are the same as those in the kinematical approximatioil wheii

If the central part of the -39 disk of a systematic row is at its Bragg position, then 1 < k,/g < 2 and hence condition (9) holds since k,, = -0.5g. Using these conditions from the measured distalice A1 on the diffractioii pattern between positions of neigh- houritig minima in the g-disk, the foil thickness t can be calculated according to the relation

(10) where d (= l/g) is the spacing of the reflection planes and ZB the distance between successive orders of Bragg reflection.

t = ( @ / I ) ls/Al ,

4 physic& (a) 99/1

Pig. 2 . Rocking curves calculated using dynaniical theory for WyL - 1 (iiitciisity in < ~ [ b . ~i l i l s ) . a) Ai. ( 1 1 1 ) ; b) 8 1 , 120 kV; c) ( l l l ) , 120 kV, d) (200), 120 kV

Ibil 'l'hi(4kness Uctermination by C U I W in the Kineniatical Appproxiination 51

0.70 I I I C

0.50 1 ' 1 ' ' d

kx/9 - B'ig. 3. Rocking curves calaculated using dynamical theory for t l g 2 = 2 (intensity in arb. units). a)Al,{111);b)Al,120kV;c)(111),120kV;d)(200),120kV 4*

I’ ti kl y f g ( m i ) t i @ t (nm) 1 1 )

KLrlgc vitlur

1 16.3 lop 18.1 4 4 “0.4

1 1 1 4.277 59.4 513 27.2

6/2 40.8 10/3 54.4

10/7 23.3

2 32.7

5 81.ti

1 . 0 ~ 2.f) 0.9 I .8 0.8 1.15 0.7 - I .4 1.2 -1.8 1 .s 4.7 1 . 0 1.5 1.2 1 . 6 1.2- 1.5 1.2 - 1.4

1 12.2 10/9 13.6 514 15.3

10/7 17.5 200 4.938 71.4 5/3 20.4

2 24.5 6/2 30.6 10/3 40.8 5 61.2

1.0 -2.0 0.9 - 1.8 0.8- 1.6 0.7 ~ 1.4 1.2 -1.8 1 . 1 t1.4 0.5-- 1 .o l.2-l.ti 0.9-1.2 1.0-1.2

1 6.1 1 O/Y 6.8 314 7.7

10/7 8.5

512 16.3

-- ““(J 6.984 113.,5 5/3 10.2 2 12.2

10/3 20.4 5 30.6

< 1.0 (0.9 (0.8 <0.7 <0.6 <O.% x3.7 (0.5 (0.4 (0.3 <0.2

[Jsiiig this kirieiiiatical approxintation, the miiiimrini measurable thickness ti,,,,, O C ~ ~ I ~ S mhen the g-disk inchidcs two niiriiiria of the rocking ciirve (II: 2 1.3 AZ), i.c.

t;,,,, 1 1.3d”A ( 1 1) Iiich is about om-tliird of the \ aliie of t , , , , , for tlic t\\o-l)ealn ~iiet~hocl, cf. ’l’i~ble 1.

4. ISxpcrimentnl llcsults

4.1 Conyurisoti with the teclitcique o f Kelly et ul. 141 Big. 4 shows CBED patterns of ail aluniiniu~u speciiriier~ ill an oricntatiori \I hero tlic: (200) systematic row was strongly excited. 111 Fig. 4a the (200) reflection is at the exact Bragg position, arid a thickness of 72.5 nm was obtained by using the method of Kelly et al. (see Fig. 5). Fig. 4 b is the pattern when the Bragg condition is satisfied at the ceritre of the (800) reflection. From the spacing of the fringes in the (200) disk, we can get t = 72.7 nm by using (10). While the two values of thickness agree within the cstiriiated errors the latter method is much simpler.

Foil Thickness Determination by (!BED in the Kinrmntirnl Approximation

a

53

b

Fig. 4. CBED patterns. Al, (200) systematic reflections. :I) (200) a t Rmgg position. h ) (BOO) a t Rrngp position

-1.2 ;l.feaszri*eiizent of a thicliitess thinnet- thatt tnlin

Fig. 6 shows two CBED patterns when the A1 specimen is in a (111) systematic reflec- tion orientation. The thickness cannot be determined by the technique of Kelly et al. when the centre of the (111) reflection is a t the exact Bragg position because only one value of st in (2) can he determined in this case (see Fig. 6a). However, the thickness t is determined as 53.0 nm using the kinematical approximation method when (335) is a t its Bragg position (see Fig. 6 h).

4.3 C'ompavisoti of different methods for thirkness detwmination

Fig. 7 shows CBET) patterns obtained from three regions with different thicknesses near a stacking fault of an austenitie stainless steel specimen. They are all cases of ( 1 1 1 ) systematic row reflections and the ccntral parts of the (3%) disks arp a t their Bragg conditions (1 < k,/g < 2 ) . Columns 2 , 3, and 4 i n Table 3 arc respectively t h c foil thicknesses obtained by using trace analysis of the contrast image of the stacking fault, by fitting the rocking curws c.alciilatled iising dpnamical theory with the CBEI)

54 R. WANG and D. LIU

a

1) Fig. 6. CBED patterns. Al, (111) systematic reflections. a ) (111) a t Rragg position), b) (333) at l3r;tgg position

a

b

c Fig. 7. CBED patterns. Austenitic stainless steel, (111) systematic reflections. (333) a t Brngg positions. a) t - - 47, b) 37, and c) 31 nm

patterns, and by using the kinematical approximation. The results are ill good ngree- merit. Errors of the third method might originate from the difficulty of locating the positions of the minima.

T a b l e 3 Measured foil thicknesses (in nm) by three methods

region method

trace anal. dynam. calc. lrinern. npprox. fitting

1 47.6 47.0 47.7 2 37.3 37.0 36.6 3 31.0 31.0 31 .0

Foil Thickness Determination by CBED in the Kinematiral Approximation 55

5. Conclusion

I r i tlic present study we have shown that convergent beam electron diffraction pat- terns from systematic rows of reflection can be used to determine specimen thicknesses using the kinematical approximation. The technique has two main advantages: (1) tjhe procedure of analysis is very simple and (2) the minimum measurable thickness is ahoiit 1/3 that of the technique developed by Kelly et al. The disadvantages of thc kineinatical method are that i t cannot be used to determine the extinction distance tCr and that the minimum moasiirahle thickness tinin is larger than that obtained hy large-angle CBED techniqiic in [9].

4 r1~)ioic~ledg~in~mit.u

We should like to thank Prof. Dr. J . W. Steeds for correcting the maniiscript and Dr. H. %ou and 1)r. (2. T,u for helpful suggestions.

Referoneos

[L] P. GOODMAN and G. LEHMPBUHL, Acts cryst. 22, 14 (1967). [2] C. H. M A C G I L L ~ ~ V R Y , Physica 7, 429 (1940). [4] I. ACKERMANN, Ann. Phys. 6, 19 (1948). [4] P. h2. KELLY, A. JOSTSONS, R. G. BLAKE, and J. G . NAPIER, phys. stat. sol. (a) 31, 771 (1975). [51 S. 11. ALLEN, Phil. Mag. A43,325 (1981). r6] R. GEVERS, Kinematical Theory of Electron Diffraction, in: Diffraction and Imaging Tech-

niques in Material Science, Ed. S. AMELINCKX e t al., North-Holland Publ. Co., Amsterdam 1978 (p. 9).

171 J. TAFTO and T. H. METZGER, J. appl. Cryst. 18, 110 (1985). 181 D. LIU and R. Wawc, Chinese Phys. Letters 2, 381 (1985). [S] D. LIU, R. \VAUG, and Y. HUANG. phys. stat. sol. (a) !)!), No. 2 (1987).

(Rewired August 2.5, I H R f i i