High precision x-ray metrology

High precision x-ray metrology P. Seyfried, P. Becker and D. Windisch* Highly pure silicon crystals with almost perfect lattice structure constitute a powerful metrological tooL Lattice spacings of the order of O. 1 nm are reproduced by nature to better than 1 part in 107. These have been applied as length standards in the sub-nanometre region using the scanning x-ray interferometer. Crystallographic angles between different sets of lattice planes within the same crystal differ only very little ( 1 x 10-9 tad) from values calculated for the perfect crystal. Double-crystal x-ray diffraction has been used to provide a single-crystal polygon.

Keywords: x-ray interferometry, metrology, silicon crystals

X-ray i n t e r f e r o m e t r y The scanning x-ray interferometer according to Bonse and Hart 1 provides the practical means to compare lattice spacings in single crystals with the unit of length. In the past, in t w o experiments of this kind 2'3, the d220 spacing in very pure and almost perfect single crystals of silicon has been measured, achieving standard deviations of 1 × 10 7 and 5.8 x 10 -8, respectively. The experimental set-up of the PTB (see Fig 1 ) consists of a triple Laue-case x-ray interferometer cut into two pieces, one comprising the beam splitter S and the mirror M lamellae, the other the analyser lamella A. Both crystal parts have steel balls attached to their bases, and the silicon surface is polished to form optical mirrors which are parallel, within 10 arcsec, to the lattice planes used for the displacement measurements. The analyser is moved by a double parallel spring translation stage made from a single steel plate. Translation is achieved by a piezoelectric element acting upon the translation stage via a lever, generating displacements up to 200 #m with guiding errors of less than 1 nrad. The displacement is measured with a polarization interferometer as described by Curtis et al 4. When the analyser crystal is moved, the displacement is measured with two different scales 5. At the output of the optical interferometer a modulation is observed with a period of 2 / 2 , while the x-ray interferometer signal exhibits a modulation with the period d of the lattice planes (see Fig 2). Thus a displacement of 2 / 2 corresponds to a displacement of nd, where n is the number of x-ray fringes. In the general case, n is not an integer; the fractional part is obtained from phase measurements between the optical and x-ray signals with the highest accuracy. The optical signal is therefore differentiated and amplified, giving a bipolar, rectangular waveform with steep zero crossings (see upper diagram in * Physikalisch-TechnischeBundesanstalt,D 3300 Braunschweig, FederalRepublic of Germany

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Fig 2). The fractional part of the x-ray period at the optical zero point is taken at the beginning (see lower diagram in Fig 2) and at the end of the measured interval, and both parts are added to the integer of n. In this way n = 1648.281 626 has been determined with 2 = (632 991 41 5 _+ 5) fm, resulting in d220 = ( 192 015.560 _+0.012) fm (in vacuum at 22.5°C) for the PTB standard crystal. This crystal can be removed from the translation stage in order to be used in lattice spacing comparison experiments as a transfer standard, a unique feature of the PTB arrangement. Before an x-ray interferometer can be put into operation, the three lamellae must be aligned extremely accurately. Distances between M and S, and between M and A, must be equal within 1 #m; the lattice planes must be parallel within a few nanorad and below 1 #rad with respect to the pand 8-axes (see Fig 1 ); these conditions must be maintained during movement. Relative vibration amplitudes between the movable lamella and the others are not permitted to exceed more than a quarter of a lattice plane spacing. This condition sets very strict constructional requirements for appropriate translation stages (eg stiffness). Most of the problems are avoided if x-ray interferometer and translation stage are manufactured as an integral combination from one single crystal as shown in Fig 3. When properly machined and etched to about 50 #m to remove the surface damage, no additional alignment is necessary. Vibrations due to pickup from residual noise on commonly used supporting tables ( A s t ~ 1 #m) must be kept to a minimum, cot and Ost being the respective resonance frequencies of the table and the translation stage, the vibration amplitude of the stage Asst is given by Asst = (~t/~)st)2Ast Assuming (ot = 4 Hz, COst= 1000 Hz and Ast = 1 #m, ASst = 0.016 nm is obtained; this value corresponds to d / 1 2 . Integrated interferometers similar to that in Fig 3 have been successfully used for the calibration of

0141-6359/88/010035-08/S03.00 © 1988 Butterworth & Co (Publishers) Ltd

35

Seyfried et al--high precision x-ray metrology

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36

translation stage displacement sensors 6 and roughness measurements 7. A linear and sensitive interpolation of fringes in optical interferometers is another interesting application for precision length measurement devices.

Double-crystal arrangements Lattice spacings of crystals w h i c h differ by no more

J A N U A R Y 1988 V O L 10 NO 1

Seyfried et al--high precision x-ray metrology Table 1

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than 10 -5 d are compared with the two-crystal arrangements in Fig 4 using two x-ray tubes. If the Bragg angles of sample and reference crystal are equal (88 --- 8~) and their lattice planes are perfectly parallel to each other, detector 1 and detector 2 will record maximum x-ray intensities for the same angle ¢. If the Bragg angles of sample and reference crystal are different, the t w o maxima of the rocking curves R 1 (cz) and R2(0~) of the two x-ray beams occur at different angles =. The offset A~ is twice the difference of the Bragg angles:/~¢ = 2 ( e B - e~). From Bragg's equation, with A d / d = ( ds - dR) / dR, we derive:

comparison experiments, for which there are two main objectives. The first objective is to obtain information on the perfection of the lattice within one single crystal and to quantitatively derive the real shape and the volume of the unit cell. The complete set of 220 lattice planes was therefore measured, and the six distortion parameters (three lateral distortions: ~a, %, Sc; three angular: ~=, ~p, ~ ) of the slightly deformed unit cell (see Fig 5) were calculated. Results are c3jven in Table 2. From the mean value d220, the volume Vo of the unit cell was calculated: Vo = ( 0 . 1 6 0 1 9 3 259 + 0.000 000 044) nm 3. The second objective is to study the influence of impurities (eg C, 0) on the lattice parameter in order to be able to specify the lattice parameter of the pure and perfect silicon crystal. From latticespacing comparisons of samples with different, known impurity concentrations Nx (a measurement for carbon is shown as an example in Fig 6), influence coefficients/~ and atomic radii rex were derived which agree well with theoretical predictions 9 rth. These results are compiled in Table 3. The homogeneity of lattice spacings over larger regions in crystals is investigated with the twocrystal arrangement in Fig 7. The x-ray beam of the source is reflected at the lattice planes of the

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1 A d / d = - Z~,~¢ cot 8B To achieve an uncertainty 6 ( A d / d ) = ___1 x 10 -8, an uncertainty 6 (z~=) = ___2x 10 -8 rad (4 marcsec) for 8 B = 45 ° must be reached. The method was used to establish secondary standards of lattice spacings. These are compiled in Table 1. One of them is the reference crystal which will be used in further

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Fig 5 Scheme for a distorted unit cell for deriving the distortion parameters of lengths Ea, Sb, ~c and of angles ~, ~, ~ (from Ref 8)

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Seyfried et al--high precision x-ray metrology Table 2 Unit cell distortions of a highly pure silicion crystal: ( a ) lateral and ( b ) angular aberrations from t h e cubic shape a

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reference crystal CR at the Bragg angle O R, The surface of this crystal is cut to form an angle ([DR with the lattice planes. This asymmetric reflection transforms the incident narrow X-ray beam into a wide beam illuminating a larger area of the sample crystal Cs; the reflected beam is recorded by a photographic plate. The sample crystal is adjusted around the s-axis to one flank of the rocking curve; in this way small local differences A d / d in the sample give rise to local intensity variations in the optical density of the plate. A topographical record of a silicon crystal with a carbon concentration of Nc = 1.3 x 1016 cm -3 is shown in the lower section in Fig 7. Periodic variations A d / d are observed. From records of crystals with differing carbon contents a strong correlation between the amplitude of the periodic A d / d variations and the carbon concentration was found. The variations of

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Two examples of rocking curves are given in Fig 8 to demonstrate the high angle resolution which can be reached using x-ray diffraction. Very narrow curves are obtained with a high diffraction order at the expense of low intensity. The peak part of the Ag K ~ ( 1 2 0 0 ) reflection curve (upper diagram in Fig 8) has a half width of not much more than one hundredth of an arcsec; the peak can therefore easily be reproduced to 1 marcsec. The more intense Mo Ka(220) reflection shown in Fig 8 was used to measure the guidance errors of a translation stage for an x-ray interferometer. For this purpose one crystal of the usual two-crystal arrangement (see Fig 4) was mounted on the stage. A slight misalignment of the crystal defines the operating point of the flank of the very narrow peak as indicated, so that small angle variations during displacement cause high intensity changes. With the slope of the curve m = (3( N / N o ) / ~ 0 the guidance errors A0 are obtained from:

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where [V/N o is the normalized counting rate. Curves 1 to 4 in Fig 9 represent guidance errors for different states of adjustment, resulting in AO/As = + 3 , + 2 , + 0 . 2 and - 3 marcsec/40/~m, respectively. Even smaller guidance errors ( + 0 . 1 6 nrad) have been observed for a second translation stage during a displacement of 160 #m. In pure high quality single crystals the angles between lattice planes are very close to those of a perfect crystal; this is shown in Table 2. So crystals can be used as angle standards. In Fig 10 the standard stereographic projection for the ( 1 0 0 ) and (111 ) orientation of the diamond-type structure of silicon is shown. Angles of 90 ° and 60 ° are distinguished for lattice planes with h 2 4- k 2 4 - / 2 = 1 and 2, respectively. Quite a number of other angles and combinations of angles can be obtained. In the lower diagram in Fig 10 a practical form of a silicon polygon used for the calibration of an optical polygon is shown. Six thin crystal wafers

JANUARY 1988 VOL 10 NO 1

Seyfried et al--high precision x-ray metrology \

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have been cut perpendicular to the lattice planes (eg (220)). One of the wafers and the reference crystal form a two-crystal diffractometer as illustrated in Fig 4. From the maximum of the rocking curve observed for one wafer to that of the next one, the table has been turned by an angle equal to the crystallographic angle between the two sets of lattice planes (eg 60°).

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which is given by A0, since A d / d = 0 when using the same crystal. The same wavelength (A/l/~ = 0) can be used to calibrate angles. The three-crystal arrangement in Fig 11 is a practical means of measuring the large differences in lattice plane spacings usually observed between crystals of different species, eg when comparing

JANUARY 1988 VOL 10 NO 1

Seyfried et a l - - h i g h precision x-ray metrology Diamond (111) 60 °

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a0(Si ) with a0(Ge ). The crystals C1 and C2 define the wavelength and direction of the radiation incident on crystal C 3. When this is turned about the 0-axis as indicated, Bragg reflection is observed for two positions symmetric to the beam direction. The angle between both positions, which is equal to 2OB (as can easily be derived from the geometry), is measured. For the comparison of lattice spacings, the u n k n o w n and reference crystals are used in the position of C 3. From the two Bragg equations n~. = 2dxsin Ox and n~ = 2dRSin O R w e obtain: sin OR dx = sin O~-~dR The uncertainty is determined by the uncertainty of the angle measurement; the wavelength need not be known, but must be the same during both measurements. This condition is easily fulfilled when, instead of two crystals, t w o different sets of

41

lattice planes in the same crystal are used for the Bragg reflection. X-ray t e c h n i q u e s make length and angle measurements w i t h extremely high resolution possible. The examples given demonstrate o n l y a f e w of the possible a p p l i c a t i o n s and were developed in the course of a n e w determination of A v o g a d r o ' s constant at the PTB. The t e c h n i q u e s are still very delicate and much w o r k remains to be done before x-ray interferometry becomes a w i d e l y applicable tool.

4 Curtis I., Morgan I., Hart M. and Milne A. D. A new determination of Avogadro's number. In: Precision Measurement and Fundamental Constants, edited by D. N. Langenberg and B. N. Taylor, National Bureau of Standards Spec. PubL No. 343, 1971, (US GPO, Washington DC), 285-289 5 HanKen K.-J., Ade G., Lucas W., Siegert H. and Backer P. Berichte ~ber Arbeiten an R~)ntgenverschiebeinterferometer, Tail II1: Erste Bestimmung des Netzebenenabstandes (220) yon Silicon, PTB-Report APh- 14, 1981, Physikalisch- Technische Bundesanstalt Braunschweig 6 C h e t w y n d D. G., Siddons, D. P. and Bowen D. K. X-ray

interferometer calibration of microdisplacement transducers. J. Phys. E, 1983, 16, 871-874

References 1 Bonse U. and Hart M. Principle and design of Laue-case Xray interferometers. Z. Phys., 1965, 188, 154-164 2 Deslattes R. D. and Henins A. X-ray to visible wavelength ratios. Phys. Rev. Lett., 1973, 31,972 975 3 Backer P., D o r e n w e n d t K., Ebeling G., Lauer R., Lucas W., Probst R., Rademacher H.-J., Reim G.,

Seyfried P. and Siegert H. Absolute measurement of the (220) lattice plane spacing in a silicon crystal. Phys. Rev. Lett., 1981, 46, 1540-1543

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7 Franks A. Pr~zisionsmechanikim r6ntgenographischen Ger~itebau. Feinger~tetechnik, 1986, 35, 531-535 8 Siegert H., Backer, P. and Sayfried P. Determination of silicion unit cell parameters by precision measurements of Bragg plane spacings. Z. Phys., 1984, B 56, 273-278 9 Pauling L. The Nature of the Chemical Bond. Ithaca, New York, 1960

10 Windisch D. and Backer P. Lattice distortions induced by carbon in silicon, subm. to Phil. Mag. B, 1987 11 Becket P., Seyfried P. and Siegert H. Translation stage for a scanning X-ray optical interferometer. Rev. Sci. Instrum., 1987, 58, 207 211

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