Specimen thickness determination in transmission electron microscopy in the general case

Micron, 1973, 4:111-116

111

Specimen thickness determination in transmission electron microscopy in the general case M. von H E I M E N D A H L * Inorganic Materials Research Division, Lawrence Berkeley Laboratory and Department of Materials Science and Engineering, College of Engineering, University of California, Berkeley, California 94720, U.S.A. Manuscript received November 6, 1972

A technique is described to determine the foil thickness of transparent specimens as used in transmission electron microscopy. The foil thickness is determinedfrom the changes of the projected widths of pairs of latex balls on both surfaces of the specimens during tilting (large angles). The accuracy of the method is investigated critically. In routine work, a standard deviation of less than 4% is obtainable. Une mgthode pour la ddtermination de lYpaisseur des films minces utilisgs en microscopie dlectronique par transmission est d~crite. L'dpaisseur du film est ddtermin& gzpartir de variations des surfaces projetdes d' une paire de sph&es de latex enfonction d' angle de rotation. La prgcision de cette rrdthode est discutge de fo4on critique. En travail de routine un dcart-type inf~rieur 4% est gdrdralement obtenu. Es wird eine Methode beschrieben, welche die Ermittlung der Foliendicke transparenter Elektronenmikroskopproben erm6glicht. Dabei werden Veriinderungen in den Projektionsweiten von Latexkugelpaaren auf beiden Fliichen der Probe wiihrend einer Drehung durch grosse Winkel gemessen. Die Genauigkeit dieser Methode wird kritisch untersucht. Bei normaler Verwendung k6nnen Standardabweichungen yon weniger als 4% erzielt werden. INTRODUCTION The techniques to determine foil thicknesses as treated in some textbooks (Thomas, 1962; Hirsch, Howie, Nicholson, Pashley and Whelan, 1965; Reimer, 1967; Schimmel, 1969; yon Heimendahl, 1970) require special image features such as either traces (from dislocations or precipitates) or wedge fringes or bend contours. However, often there are none of these features in the particular area of interest. One method of thickness determination for this general case using extinction contours under 'divergent beam' conditions in the diffraction pattern was developed by Bell and Thomas (1969) (first introduced by Amelinckx, 1964). Sadhukhan (1970) suggested the use of latex balls to be applied on both sides of the foil. Its thickness t is derived from the changes of the projected ball distances after tilting the specimen through large angles. Additional cross shadowing is necessary to distinguish on which of the two surfaces the latex balls were located. Also Vingsbo (1970) used a high angle tilting technique together with inherent features extending from one foil surface to the other. The method described here can be used in the most general case of a transparent foil, i.e. no particular image features and no complicated shadow castings are necessary. This method may be considered as an independent alternative to the one of Bell and Thomas (1969). * Max-Kade fellow at the Department of Materials Science and Engineering, University of California. Present address: Institut ftir Werkstoffwissenschaften 1,852 Erlangen, Martensstr. 5, Germany.

1 12

yon HEIMENDAHL

A METHOD FOR DETERMINING SPECIMEN THICKNESS IN T H E G E N E R A L CASE

Experimental Latex balls as commercially available (diameter D .... 234 ± 2.6nm) are applied to the foil surfaces by simple dipping into a clean alcoholic suspension. The concentration of this latex suspension has to be tried out so that the number of spheres coming onto the surface is suitably low, ideally just one ball on each side in the field of view (one drop of the latex milk per 5cm 3 alcohol for first trial). Let the specimen dry by careful setting on filter paper in an oblique manner such that the suspension is allowed to flow down gently from either side. Text Figure 1 shows an example, in this case a disc specimen of an A1-4%Cu alloy obtained by twin-jet electropolishing in a commercial polisher (E. A. Fischione, Verona, Pennsylvania, U.S.A.). The sample was aged to produce large plate-like 0'-precipitate3 for an independent thickness determination by trace analysis. Text Figure 1 was taken at 100kV in a Philips EM 300 microscope fitted with goniometer stage which provides easy and accurate tilting in both directions up to ± 60 °. a =

a

24 °

=

0 ~

TILT AXIS

Text Figure 1. Electrolytically thinned specimen of an A1-4% Cu alloy with applied latex balls. Foil thickness t from equation (1) (see text), 324nm ~ 4O/o. ×25,600. Thickness determination In Text Figure 2 the three black dots should mark the centres of latex balls (which themselves are usually in the order of magnitude of the foil thickness t). The following treatment, therefore, yields t' ~ - t + D, the figure thus giving a cross section of a hypothetical foil, resulting from the real foil t plus twice the latex ball radiu~. At first

SPECIMEN THICKNESS D E T E R M I N A T I O N

113

the primary b e a m P is considered to be perpendicular t o the foil in the untilted condition (deviations oJ from this condition are discussed later). First, one has to distinguish whether two balls needed for the analysis are, as required, on different sides. This m a y be easily performed as follows: two particles on the same side always reduce their projected widths d on the screen during tilting in both directions, whereas particles on different sides increase their d-value while tilting to one direction (a ~ 0 in Text Figure 2) and decrease d when tilting in the other direction

<0). Let us first suppose the foil is tilted in the positive direction as shown in Text Figure 2 from which follows cos /3 = t'/1, sin /3----dl/1, sin (a-+-/3) --d2/1. Eliminating 1 from the last two equations and applying the additive formula for sin (a ÷ / 3 ) yields dl (sin a cot/3 + cos a) -- d2. With cot/3 -- t'/dl the foil thickness t results as tollows: t =t'--D-~d2--dlc°sa--D ......... (1) sin a If the foil is tilted into the other direction the projected width becomes smaller and is denoted as d3 after tilt for better distinction. Taking the absolute value of a, an analysis analogous to Text Figure 2 yields in this case: t----t'--D

=dlc°sa--d3--D ......... (2) sin a The d's m a y be measured as centre-to-centre distances of the latex balls, measured in the direction perpendicular to the tilt axis, or more accurately as the corresponding edge distances as indicated in Text Figure 1. With d I z 3.7mm, d2 ---- 9.2mm, D = 6.0ram on the plate and a = 24 °, formula (1) yields t = 8.3ram or, divided by the

~

P

i_ 1-

_ d2

_1 -I

Text Figure 2. Cross section through a hypothetical foil of thickness t' ( z real foil thickness t -k diameter D of latex balls). Tilt axis perpendicular to drawing plane, a, tilt angle; 1, projection of ball distance upon drawing plane; dl and d2, projections of 1 on the plate before resp. after tilt; P, primary electron beam; co, see text.

114

yon H E I M E N D A H I ,

magnification 25,600 times, a foil thickness of 324nm. For comparison, thickness d e t e r m i n a t i o n by conventional trace analysis (see the textbooks mentioned above) using the 0'-plates in T e x t Figure 1 before and after tilt yields 347nm. ACCURACY OF THE METHOD Although the above m e t h o d is generally applicable, its accuracy is o f course limited as t h a t of a n y physical measurement. T h e following sources of error have been treated : (a) U n c e r t a i n t y of d l - m e a s u r e m e n t . Magnifying glasses or photometers are used; a l t h o u g h n o r m a l l y better, an error of 0 . 3 m m on the negative or original print is taken into a c c o u n t using a magnification o f 50,000 ×. (b) As above for d2-measurement. (c) T h e influence of an error of 1° in a is considered. I n crystalline samples ~ can be measured m u c h more accurately by use of Kikuchi line shifts (Thomas, 1970; von H e i m e n d a h l , 1971). H o w e v e r , the m e t h o d described here is also applicable to non-crystalline specimens like c a r b o n films etc. (d) Usually it is u n k n o w n whether P is exactly parallel to the foil n o r m a l in the start position (dl-measurement) as assumed in the above derivation. A slight deviation oa < 0 is considered in T e x t Figure 2 as follows: the foil is assumed to be fixed in the specimen holder forming co between P and the foil normal. T h e n the a p p e a r i n g (wrong) projected widths are seen not along P, but along the deviating direction (Text Figure 2) and can be w o r k e d out graphically or better numerically this w a y ; oa = -- 5 ° and -- 10 ° were taken as examples. TABLE

Relative error = (t* - - t ) / t of determined foil thickness in % as function of different error sources. Calculations performed for two thicknesses of t: 0.3 and l.O~m* B=0 Sources of errors

(a)

dl assumed to be 0.3mm too large measured on the plate ( X 50,000)

Foil thickness t 0.3 1.0

--

5.5

B = 40 °

/3 = 12 °

--

1.6

0.3

--

5.5

1.0

--

1.6

0.3

--

5.5

1.0ix

--

1.6

(b) d2 assumed to be 0.3mm too large measured on the plate ( × 50,000)

+5.9

+1.8

+5.9

+1.8

+5.9

+1.8

(c) a assumed to be 1° too large (21 ° instead of 20 °)

--8.1

--5.6

--7.5

--5.2

--5.6

--3.9

(d) co = -- 5 °

--0.7

--0.5

--4.0

--2.7

--

13.7

-- 9.5

(e) co = -- 10 °

--

--

--

--

-- 28.6

* T h e tilt a n g l e is a s s u m e d to be 20 °.

2.7

1.9

9.3

6.4

-- 19.8

SPECIMEN THICKNESS DETERMINATION

115

In all cases equation (1) was applied and the resulting thicknesses infected with the errors are called t*. It is of particular interest to know these errors as a function of fl since in practice one has often a choice between particle pairs more or less close together. Table I, therefore, lists the results of (a)-(d) for fi = 0 °, 12 ° and 40 °. It can be seen from Table I that the errors (a) to (c) are small to medium, do not depend strongly on the choice of fl and partially cancel out each other. Larger errors may appear only from oJ # 0 /f fl is large. From this point of view, it is therefore recommended to work with small ft. Unfortunately, an oJ # 0 cannot be identified by using the two different tilt directions : for a given w > 0 in both cases, using a > 0 and < 0, t* comes out too large, and for ~o < 0 in both cases too small. Also a oJ # 0 condition cannot be checked by using two particles on one side of the foil: although basically these should have a maximum distance while tilting at oJ -~ 0, this effect is much too small for practical use due to the extremely slow variation of the cos-function near zero. It is one more advantage of disc specimens as obtained from electrolytic jet stream polishing that they usually have only a negligibly small deviation parameter a,. The dependence on the assumed foil thickness in Table I is twofold and easily understood: first, with larger thickness, the projected widths become also larger and therefore the relative dl and d2-accuracy becomes better, keeping the absolute width accuracy constant (0.3mm on the plate). This is in agreement with similar observation ofVingsbo (1970) ; second, a larger thickness t means generally a better relative accuracy since in (1) and (2) the constant diameter D has to be subtracted (which is in the order of t). Therefore, the relative errors from a- and co- deviations in Table I also become smaller the thicker the foil. Four or five measurements with different tilt angles a are usually possible under good contrast conditions. In practice, by averaging these independent measurements it was always possible to determine t with a standard deviation of less than 4%. Largest deviation of a single measurement was 8%. This is about in agreement with the calculations of Table I. Strongly wedge-shaped specimens need a special geometric treatment which is not intended to be dealt with in this paper. The derivation of equation (1) requires a plain-parallel specimen which is usually provided within the small area of view also in electropolished thin metal foils. The comparison with trace analysis value for t taken from the 0'-platelets as calculated above for the example of Text Figure 1 yielded agreement within 7 %. However, trace analysis results in the current investigation were less accurate due to irregular plate edges giving poor resolution of the intersection between plate and surface. For reliable and accurate results it is therefore preferable to use the method described in this paper. ACKNOWLEDGEMENTS The author would like to thank Dr. K. Schneider and Mr. H.-J. Hausselt, Dipl. Phys., University of Erlangen, for fruitful discussions and Professor G. Thomas, University of California, Berkeley, for critically reading this manuscript. The author would also like to acknowledge gratefully the award of a grant from the Max-Kade Foundation in partial support of this investigation. The work was carried out under the auspices of the U.S. Atomic Energy Commission while the author was on leave from the University of Erlangen-Niirnberg, Germany.

116

yon HEI MENI)AHI. REFERENCES

AMELmCKX, S., 1964. Direct Observation of Dislocations. Academic Press, New York, U.S.A. BELL, W. L. and THOMAS,G., 1969. Foil thickness determination from single crystal diffraction patterns. In: Proc. 28th Annual Meeting EMSA, Houston, Arceneaux, C. J. (ed.), Claitors, Baton Rouge, Louisiana, U.S.A., 158-159. HEIMENDAHL, M. yoN, 1970. Einffihrung in die Elektronenmikroskopie. Macherauch, E. and Gerold, V. (eds.), Vieweg Verlag, Braunschweig, Vol. 1. HEIMENDAHL, M. yoN, 1971. Precise orientation determination by electron diffraction single-pole Kikuchi patterns. Phys. Stat. Sol. (a), 5:137-146. HIRSCH, P. B., HOWIE, A., NICHOLSON,R. B., PASHLEY,D. W. and WHELAN,M. J., 1965. Electron Microscopy of Thin Crystals. Butterworths, London. REIMER, L., 1967. Elektronenmikroskopische Untersuchungs-und Prilparationsmethoden. Springer Verlag, Berlin, 2nd Ed. SADHUKHAN,P., 1970. Thickness measurement of supporting films. Philips Bulletin, Eindhoven, Netherlands, No. EM54. SCHIMMEL,G., 1969. Elektronenmikroskopische Methodik. Springer Verlag, Berlin, Heidelberg and New York. THOMAS, G., 1962. Transmission Electron Microscopy of A4etals. Wiley, New York and London. THOMAS, G., 1970. Kikuchi electron diffraction and applications. In: Modern Diffraction and Imaging Techniques in Material Science, Amelinckx, S., Gevers, R., Remant, G. and Van Landuyt, J. (eds.), North-Holland, Amsterdam and London, 159-185. VINGSBO, O., 1970. A stereo method for specimen thickness determination in transmission electron microscopy. In: Microscopie I~lectronique, Proc. 7th International Conference on Electron Microscopy, Grenoble, France, Favard, P. (ed.), Soci~td Fran~aise de Microscopic Electronique, Paris, 1: 325-326,