Elastic constants of fibre-textured thin films determined by X-ray diffraction

Supposing the Hill grain-interaction model, it is demonstrated that X-ray elastic constants can be used to determine mechanical elastic constants of cubic fibre-textured thin films. The new approach is demonstrated by the experimental characterization of out-of-plane moduli of fibre-textured Cu and CrN thin films.

Numerous experimental as well as computational methods have been developed in the past few decades to determine mechanical elastic moduli (or even single-crystal elastic constants) from diffraction elastic constants (Hayakawa et al., 1985;Humbert & Diz, 1991;Wright, 1994;Gnä upel-Herold et al., 1998;Howard & Kisi, 1999;Badawi et al., 2002;Badawi & Villain, 2003). In the case of experimental characterization using X-rays or neutrons, polycrystalline samples (bulk or thin films, usually on a flexible substrate) are in-situ strained, and diffraction elastic strains are recorded and correlated with the applied stress. To extract the mechanical elastic constants, Reuss, Voigt, Hill and Eshelby-Krö ner grain-interaction models are usually applied (Hayakawa et al., 1985;Wright, 1994;Howard & Kisi, 1999;Faurie et al., 2006). Although the grain-interaction models represent only idealized theories, the elastic constants of quasi-isotropic or textured samples, or samples with crystal elastic anisotropy, have been obtained with a relatively good accuracy (Howard & Kisi, 1999;Villain et al., 2002;Badawi & Villain, 2003;Goudeau et al., 2004). It was found that the anisotropic Hill model is a good approximation of the experimental data obtained from polycrystalline samples (usually within the precision of the techniques applied) and, moreover, the more sophisticated models were usually very close to the Hill approximation (Howard & Kisi, 1999, p. 632). In the majority of cases (Humbert & Diz, 1991;Wright, 1994;Gnä upel-Herold et al., 1998;Howard & Kisi, 1999;Badawi et al., 2002;Badawi & Villain, 2003), however, the experimental elastic moduli or single-crystal elastic constants are obtained from in-situ experiments coupling diffraction and sample loading, i.e. destructively, whereby it is necessary to use a tensile stage.
Recently, a new rapid experimental approach based on the simultaneous application of sin 2 and X-ray diffraction substrate curvature techniques was proposed (Eiper et al., 2005Keckes et al., 2007;Martinschitz et al., 2006). The new approach provides an opportunity to quantify experimental X-ray elastic strains and macroscopic stresses in thin films using a static diffraction experiment. The stresses applied on the film are determined from the geometrical changes of the elastically deformed substrate that is attached to the film (Stoney, 1909;Segmü ller et al., 1989). The experimental stress and strain can then be used to evaluate experimental X-ray elastic constants and stress factors (Eiper et al., 2005Martinschitz et al., 2006).
Mechanical elastic constants can be extrapolated from X-ray elastic constants considering crystal and macroscopic elastic anisotropy. In the case of cubic polycrystalline aggregates with macroscopic elastic isotropy (quasi-isotropic materials) that obey the Hill grain-interaction model, it was demonstrated that X-ray elastic constants correspond to their mechanical counterparts for À hkl = 0.2, where À hkl is the X-ray anisotropic factor according to the Reuss grain-interaction model, given by (hkl) are Miller indices of a crystallographic plane (Bollenrath et al., 1967). According to the Reuss (1929) model, X-ray elastic anisotropy is often expressed as a function of 3À hkl , and this formalism will be applied hereafter.
It is the aim of this paper to analyse under which conditions knowledge of X-ray elastic constants can be used to determine or estimate mechanical elastic constants of cubic fibretextured thin films that obey the Hill grain-interaction model. Firstly, the mechanical elastic constants of Cu and CrN will be calculated using a Hill model that represents a reasonable simplification of the problem (Hill, 1952;Bunge & Roberts, 1969;Bunge, 1982;Gnä upel-Herold et al., 1998;Howard & Kisi, 1999;Leoni et al., 2001;Welzel, 2002). As a next step, the mechanical values will be compared with calculated X-ray elastic constants. As a result, a 3À hkl -dependent selection rule will be derived, where the subscript hkl in 3À hkl denotes a reflection for which mechanical and X-ray elastic constants are equal. The approach will be demonstrated by experimental characterization of out-of-plane moduli of fibre-textured Cu and CrN thin films. The moduli are extracted from experimental X-ray elastic constants that are determined by a combination of X-ray diffraction substrate curvature and sin 2 methods in a static diffraction experiment. It should be noted that the methodology derived in this paper can be generally applied to any equibiaxially loaded or stressed polycrystalline aggregate with the fibre texture oriented perpendicular to the stress direction.
2. Mechanical elastic constants of thin films 2.1. Hill grain-interaction model Elastic behaviour of a thin film deposited on a solid substrate can be represented by Hooke's law: where h" S ij i is the mechanical elastic strain, hS S ijkl i expresses the mechanical elastic constants of the film and h S kl i represents the residual stress (Nye, 1976;Suresh & Freund, 2003). The brackets hi denote volume averages for all crystallites (i.e. mechanical averages; Welzel, 2002). The stress, strain and compliance tensors in equation (2) are expressed in the sample coordinate system (S) (Fig. 1).
In general, hS S ijkl i of a polycrystalline film depends on the texture, the single-crystal elastic constants and the graininteraction mechanism ( van Houtte & De Buyser, 1993). In practice, the Hill grain-interaction model can be used to evaluate hS S ijkl i of the film (Hill, 1952) using the arithmetic mean of the compliance tensors hS S ijkl i R and hS S ijkl i V obtained from the Reuss and Voigt grain-interaction models: Elastic constants according to the Reuss average hS S ijkl i R can be calculated as follows: In the case of the Voigt average, hS S ijkl i V can be determined according to where f(g) represents the orientation distribution function (ODF) of the crystallites in the film (Bunge, 1982;Huang & Weaver, 2005). S S ijkl and C S ijkl in equations (4) and (5) are singlecrystal elastic constants expressed in S, while f(g)dg indicates the volume fraction of the crystallites with the orientation g. The integration in equations (4) and (5) is carried out over the whole ODF space ( van Houtte & De Buyser, 1993).

Figure 1
The definition of the two coordinate systems used for the characterization of in-plane elastic strains using the sin 2 method: sample system S and laboratory system L (Noyan & Cohen, 1987). The X-ray elastic strain along the direction L 3 (which is parallel to the diffraction vector Q hkl ) is characterized by measuring the reflection hkl. The orientation of the vector Q hkl with respect to S i is defined by the angles ' and . The direction cosines a ij in equation (6) represent a transformation from S to L coordinate systems.
The tensor hS S ijkl i [equation (2)] represents the elastic behaviour of the material in the sample coordinate system S ( Fig. 1) (Nye, 1976) and can be expressed in the L system using where a ij represent the direction cosines between the L and S systems ( Fig. 1; Noyan & Cohen, 1987). In practice, Young's modulus hEi is usually used to express elastic behaviour of materials. The magnitude of hEi in the direction L 3 can be obtained from the tensor hS L ijkl i '; : The out-of-plane Young's modulus hEi '; ¼0 can be obtained from equation (7) using hS L 3333 i '; ¼0 .

Calculation of mechanical elastic constants
Using the procedure from the previous section, Young's moduli of Cu and CrN thin films with various fibre textures were calculated numerically, applying single-crystal elastic constants from Table 1 and various types of ODFs.
In Fig. 2, an example of a 111 pole figure, the distribution of the intensity across the pole figure and the corresponding ODF demonstrate a strong 111 fibre texture with a 10% fraction of randomly oriented crystallites in a cubic thin film.
As a parameter for the ODF calculation, the full width at half-maximum at the centre of the pole figure [which is usually measured experimentally using a scan (Bunge, 1982)], hereafter denoted FWHM , was used ( Fig. 2). Since the aim is to develop a simple laboratory method to determine elastic constants of thin films, FWHM was used as a measure of the texture sharpness [instead of using variables expressed in terms of the ' 1 , È and ' 2 angles (Fig. 2), which are usually needed to define ODF properties according the Bunge (1982) notation]. Numerous ODFs with FWHM in the range 0-180 with a step of 5 were generated in order to calculate hS S ijkl i [equation (2)] and subsequently the out-of-plane Young's modulus hEi '; ¼0 [equation (7)]. This calculation was performed for various uvw fibre textures, where the subscript uvw represents the indices of the (uvw) crystallographic planes oriented preferably parallel to the sample surface. Additionally, it was supposed that the films also contain crystallites with a random orientation (hereafter denoted as ISO) in the range 0-100%.
As an example of the procedure, calculated out-of-plane Young's moduli hE 111 i '; ¼0 of Cu and CrN thin films with 111 fibre texture are presented in Fig. 3. As parameters for the calculation, the texture sharpness FWHM and the number of randomly oriented crystallites ISO were applied. The threedimensional plots document that the moduli of the film exhibit relatively strong maxima or minima for small FWHM and ISO but converge to the moduli of isotropic Cu and CrN when one of the parameters increases.
Cu and CrN possess different types of crystal elastic anisotropy (Table 1) Table 1 Single-crystal elastic constants (in 10 À3 GPa À1 ) of Cu and CrN at room temperature and the Zener (1948) anisotropy ratio (ZAR) defined by equation (8) (Suresh & Freund, 2003;Birkholz, 2006   Since the hhhhi direction in Cu is stiffer than all others, the films with less pronounced hhhhi textures exhibit smaller moduli (Fig. 3). In CrN, the opposite situation has to be considered.
The results in Fig. 3 represent out-of-plane Young's moduli calculated from hS S ijkl i. In the case of fibre-textured thin films, however, the elastic behaviour is in-plane isotropic (i.e. independent of the angle ') but dependent on the tilt angle ( Fig. 1). In order to demonstrate this situation, Young's moduli of CrN and Cu 111 fibre-textured thin films (Fig. 2) were calculated as a function of angles ' and using equations (3)-(7) and are presented in polar coordinates in Fig. 4. The difference in crystal elastic anisotropy causes the CrN modulus to possess a minimum at = 0 , in contrast to the Cu dependence, which exhibits a maximum at the centre of the polar plot (Fig. 4).
3. X-ray elastic constants of thin films 3.1. X-ray elastic moduli In X-ray diffraction, Hooke's law relates X-ray elastic strain f" L 33 g hkl '; measured in the direction L 3 by scanning the reflection hkl, X-ray elastic compliances fS L 33ij g hkl '; and the macroscopic stress h L ij i expressed in the L coordinate system ( Fig. 1) as follows: where fS L 33ij g hkl '; depends generally on the texture, the graininteraction mechanism, the reflection hkl, the single-crystal elastic constants, and the angles ' and (Dö lle, 1979; van Houtte & De Buyser, 1993). The brackets fg denote volumeweighted averages for all diffracting crystallites (i.e. diffraction averages; Welzel, 2002). For simplicity, fS L 33ij g hkl '; can be calculated using the Hill grain-interaction model as follows (Serruys, 1988;van Houtte & De Buyser, 1993):    The X-ray elastic compliances fS L 33ij g V '; represent an elastic behaviour of the film according the Voigt grain-interaction model (V) and can be calculated as where the hS S klmn i V tensor was obtained using equation (5) ( van Houtte & De Buyser, 1993).
X-ray elastic compliances according the Reuss grain-interaction model (R) can be obtained by integration over the crystal orientations g for which the diffraction vector Q hkl is parallel to the direction L 3 ( van Houtte & De Buyser, 1993): Considering fibre-textured cubic thin films with the fibre axis oriented perpendicular to the sample surface, it will be supposed that (i) The mechanical state of the films is biaxial and in-plane Moreover, shear stresses h S 12 i and h S 21 i, shear strains h" S 12 i, h" S 13 i and h" S 23 i, and out-of-plane stresses h S i3 i can be neglected on the macroscopic scale.
(ii) The thin films are in-plane elastic isotropic, and not only the distribution of crystallites but also the grain-interaction mechanism possess a rotational symmetry. The elastic properties of the films are therefore not dependent on the azimuth angle ', with hS L ijkl i '; = hS L ijkl i . The above implies that equation (9) can be expressed as follows (Stickforth, 1966): In the case of the experimental dependence of f" L 33 g hkl on sin 2 , the term fS L 3311 g hkl þ fS L 3322 g hkl corresponds to the intercept on the f" L 33 g hkl axis and the term ðfS L 3333 g hkl À fS L 3311 g hkl Þ sin 2 þ fS L 3313 g hkl sin 2 is responsible for the curvature in the sin 2 plots. The term fS L 3313 g hkl sin 2 vanishes, however, under certain conditions (cf. Stickforth, 1966;van Houtte & De Buyser, 1993;Welzel, 2002).
Since the tensor components fS L 33ij g hkl in equation (13) change as a function of the orientation of the diffraction vector Q hkl , they can be used to determine diffraction elastic constants as a function of (hkl) and . For example, the diffraction modulus fEg hkl along the direction L 3 reads On condition that the fS L 3333 g hkl components are independent of the angles and ' and the material is quasi-isotropic, equation (13) can be written as in which the symbols fs 1 g hkl and fs 2 =2g hkl represent isotropic X-ray elastic constants (Dö lle, 1979;Noyan & Cohen, 1987). The constants are sometimes substituted as (Noyan & Cohen, 1987) fs 1 g hkl ¼ À fg hkl fEg hkl ; The symbol fg hkl represents the diffraction Poisson number determined by the measurement of reflection hkl. In the case of macroscopic elastic isotropic aggregates, fEg hkl and fg hkl can be calculated using equations (15) and (16) provided f" L 33 g hkl and h S i are known.

Calculation of diffraction elastic moduli
The X-ray elastic compliances fS L 33ij g hkl express the elastic behaviour of the aggregate along the diffraction vector Q hkl . In Fig. 5, Young's moduli of 111 fibre-textured Cu and CrN thin films are presented as a function of the tilt angle .
The data in Fig. 5 document that the mechanical moduli hE 111 i of the 111 fibre-textured films lie always between diffraction moduli fE 111 g 111 and fE 111 g 100 . The diffraction moduli fE 111 g hkl represent the elastic response of diffracting crystallites in the direction of diffraction vector Q hkl (Fig. 2). The mechanical moduli hE 111 i represent the elastic response of all crystallites in the direction of diffraction vector Q hkl . For = 0 the out-of-plane mechanical modulus hE 111 i ¼0 approaches the diffraction modulus fE 111 g 111 ¼0 (which can be obtained by the characterization of the 111 reflection and f" L 33 g 111 ¼0 ) because of the specific texture type (Fig. 5).  The moduli are expressed as a function of the tilt angle , which defines also the orientation of the diffraction vector Q hkl (Fig. 1).

A comparison of mechanical and X-ray elastic constants 4.1. General considerations
The results in Fig. 5 demonstrate that the mechanical elastic constants hS L 33ij i are constrained by the X-ray elastic constants fS L 33ij g hkl . It is therefore obvious that, by considering a specific ODF, tilt angle and single-crystal elastic constants, it is always possible to determine a reflection hkl and a corresponding X-ray anisotropy factor 3À Ã hkl for which the X-ray elastic constants are equal to their mechanical counterparts (Fig. 5). 3À Ã hkl will therefore be used to denote conditions in accordance with the Hill model, under which

Isotropic case
In the case of polycrystalline materials with crystal elastic isotropy or with negligible macroscopic elastic anisotropy, fEg hkl and fg hkl as well as fs 1 g hkl and fs 2 =2g hkl are independent of the angle , and equation (15) supposes a linear dependence of f" L 33 g hkl on sin 2 (Stickforth, 1966;Noyan & Cohen, 1987;van Houtte & De Buyser, 1993). Provided that the elastic strain f" L 33 g hkl and the macroscopic stress h S i can be determined by experiment, the isotropic X-ray elastic constants fs 1 g hkl and fs 2 =2g hkl , and subsequently also fEg hkl and fg hkl , can be obtained by solving a system of linear equations of the same type as equation (15) when f" L 33 g hkl is known for different (Ortner, 1986a,b).
An example of this procedure is presented in Fig. 6. Considering the single-crystal elastic constants from Table 1 and an in-plane isotropic stress h S i = 100 MPa, calculated diffraction strains f" L 33 g hkl for a quasi-isotropic Cu thin film are presented in Fig. 6(a).
According to equation (15), the slopes in Fig. 6(a) correspond to fs 2 =2g hkl and the intercepts on the f" L 33 g hkl axis can be correlated with the magnitude of fs 1 g hkl . In practice, the X-ray elastic constants are obtained by fitting the experimental data from Fig. 6(a) using equation (15). The reciprocal diffraction elastic moduli 1/fEg hkl in Fig. 6(c) can then be determined from fs 1 g hkl and fs 2 =2g hkl (Fig. 6b) as follows: The reciprocal mechanical modulus 1=fEg M = 0.81 Â 10 À11 Pa À1 was extrapolated from the reciprocal diffraction moduli 1=fEg hkl supposing 1=hEi = 1=fEg hkl for 3À Ã hkl = 0.6, as predicted by the Hill grain-interaction model for quasiisotropic materials (Bollenrath et al., 1967). The mechanical modulus hEi is therefore 123.45 GPa. This procedure is, however, valid only in the case of elastic isotropic aggregates.

Fibre-textured thin films
The procedure described in Fig. 6 is an often used simplification. Polycrystalline thin films are, however, usually macroscopic elastic anisotropic, and therefore the extrapolation of the mechanical modulus from X-ray elastic constants for 3À Ã hkl = 0.6 would provide incorrect results.
In the majority of cases, polycrystalline thin films possess a certain uvw fibre texture with the fibre axis oriented perpendicular to the substrate surface. In that case, the mechanical and X-ray elastic compliances are dependent on the angle  (a) Calculated X-ray elastic strains in a quasi-isotropic Cu thin film with equibiaxial stress of 100 MPa. (b) X-ray elastic constants fs 1 g hkl and fs 2 =2g hkl refined from (a), plotted as a function of 3À hkl . (c) Reciprocal diffraction Young's moduli 1/fEg hkl obtained from (b). The mechanical modulus hEi can be extrapolated for 3À Ã hkl = 0.6, resulting in a value of 123.45 GPa. (Figs. 4 and 5). In order to determine the experimental hS L 33ij i from fS L 33ij g hkl it is necessary to know the exact value of 3À Ã hkl , which is also dependent on , as demonstrated in Fig. 5. We discuss below the possibilities for determining hS L 33ij i and hEi from the experimental fS L 33ij g hkl by applying Hooke's law [equation (13)].
(i) In the case of in-plane elastic isotropic films fS L 3311 g hkl ¼0 is equal to fS L 3322 g hkl ¼0 for = 0 and equation (13) reduces to f" L 33 g hkl ¼0 = 2fS L 3311 g hkl ¼0 . fS L 3311 g hkl ¼0 can be determined experimentally by evaluating the intercept of the sin 2 dependence on the f" L 33 g hkl axis when h S i is known. The dependence of fS L 3311 g hkl ¼0 on 3À hkl could then be used to determine the thinfilm mechanical compliance hS S 3311 i. (ii) By comparing the intercepts 2fS L 3311 g hkl ¼0 and the slopes fS L 3333 g hkl À fS L 3311 g hkl of the sin 2 curves for ! 0 [and by simultaneously neglecting the term fS L 3313 g hkl sin 2 since fS S 3313 g hkl ¼0 ¼ fS L 3313 g hkl ¼0 ¼ 0 for hexagonal macroscopic symmetry of the sample (Martinschitz, 2008)], equation (13) can be used to extract fS L 3333 g hkl and the diffraction out-ofplane modulus E f g hkl ¼0 , as in x4.2. By considering the macroscopic elastic anisotropy, the knowledge of E f g hkl ¼0 can be used to determine the mechanical Young's modulus hEi ¼0 or the term hS S 3333 i. (iii) By evaluating the intercepts on the f" L 33 g hkl axis for ¼ 90 , equation (13) can be used to determine the term fS L 3333 g hkl ¼90 þ fS L 3322 g hkl ¼90 , which, in this special case, can be used to quantify the in-plane biaxial modulus of the thin film hS S 1111 i þ hS S 1122 i. In order to quantify the parameters hS S 3311 i, hS S 3333 i and hS S 1111 i þ hS S 1122 i, the macroscopic elastic anisotropy of the film must be considered. Furthermore, the determination of outof-plane moduli hEi ¼0 = 1/hS S 3333 i from the X-ray elastic constants fS L 3333 g hkl ¼0 will be discussed.

Elastic modulus of 111 fibre-textured Cu thin film
In Fig. 7, calculated sin 2 dependencies for a Cu thin film with a strong 111 fibre texture (Fig. 2) are presented. The plots were calculated supposing an in-plane isotropic stress of h S i = 100 MPa and using the single-crystal elastic constants from Table 1.
The data in Fig. 7(a) were evaluated according the procedure described in x4.2 point (ii), and fS L 3333 g hkl ¼0 values were determined (Fig. 7b). Using the ODF from Fig. 2, the out-ofplane mechanical compliance hS L 3333 i ¼0 was also calculated [equations (3)-(7)] with hE 111 i ¼0 = 174 GPa. Comparison of the out-of-plane X-ray and mechanical compliances showed that hS S 3333 i ¼ fS L 3333 g hkl ¼0 for 3À Ã hkl = 0.937. This result demonstrates that, in order to determine hS L 3333 i ¼0 from fS L 3333 g hkl

¼0
[i.e. to apply an opposite algorithm flow to that in Fig. 7(b)], it is necessary to know the value of 3À Ã hkl , which is strongly texture dependent.

3C * hkl -3C uvw plot
In the case of cubic uvw fibre-textured films with the fibre axis oriented perpendicularly to the substrate surface, the texture type will be further described using the parameters À uvw defined as (Bollenrath et al., 1967;Huang & Weaver, 2005) Supposing various uvw fibre textures (i) with a texture sharpness FWHM in the range 0-60 ( Fig. 2), (ii) with 3À uvw in the range 0-1 and (iii) with ISO in the range 0-100%, numerous ODFs were generated. Following the algorithm from x4.3 point (ii), fS L 3333 g hkl ¼0 and hS L 3333 i ¼0 values were calculated numerically for materials with Zener's anisotropy ratio in the range 0.36-9.95 (corresponding to KCl and Na). Then the mechanical and X-ray elastic constants were compared, with the aim of finding out for which 3À Ã hkl value hS S 3333 i ¼ fS L 3333 g hkl ¼0 . As a result 3À Ã hkl -3À uvw plots were constructed, indicating how 3À Ã hkl depends on the uvw fibretexture type, on FWHM (Fig. 8b) and on ISO (Fig. 8a).   refined from (a), plotted as a function of 3À hkl . Since, for this special type of texture, the mechanical compliance hS L 3333 i ¼0 = 0.575 Â 10 À11 Pa À1 , 3À Ã hkl = 0.937 was extrapolated from the fS L 3333 g hkl ¼0 dependence on 3À hkl .
The 3À Ã hkl -3À uvw plots in Fig. 8 do not depend on the crystal elastic anisotropy of the thin-film material and represent therefore a certain type of universal plot valid for all materials. In the case of isotropic materials (like tungsten) where ZAR ffi 1, the choice of À Ã hkl is arbitrary. In Fig. 8(a), one can recognize that, for very strong uvw fibre textures with FWHM < 10 and a small or no fraction of randomly oriented crystallites, the X-ray elastic constants correspond approximately to the mechanical constants for 3À Ã hkl = 3À uvw . In other words, in order to determine the out-ofplane modulus of a thin film with a very strong uvw texture one has to characterize the X-ray elastic constants of the uvw reflections. For not very pronounced fibre textures, the 3À Ã hkl value must be selected from the intervals h3À uvw ; 0:6i or h0:6; 3À uvw i for thin films with À uvw smaller or larger than 0.6, respectively. When the fraction of randomly oriented crystallites ISO increases, however, X-ray elastic constants of the hkl reflections for which 3À Ã hkl ! 0:6 should be quantified (x4.2). Similarly, in Fig. 8(b), the decrease of the texture sharpness results in behaviour that is typical for elastic isotropic materials and 3À Ã hkl ! 0:6. In the case of sharp uuu or u00 fibre textures the search for an exact 3À Ã hkl value is extremely important, because the application of the procedure from x4.2 (valid for elastic isotropic materials) could result in large errors when determining the out-of-plane moduli. For films with uvw fibre textures with 3À uvw ffi 0:6, the procedure from x4.2 can still provide relevant results.
The results in Figs. 8(a) and 8(b) represent an example of the À Ã hkl -À uvw selection rule. In order to express the dependence of 3À Ã hkl on 3À uvw , on FWHM and on ISO generally and in a 'user-friendly' way, the following empirical equation was derived: where A = ( FWHM Â 8.8 + ISO Â 5.8 À FWHM Â ISO Â 0.083)/1000. Equation (19) provides an easy way to determine 3À Ã hkl values considering fibre-texture parameters. The parameters ISO and FWHM in equation (19) can be obtained from pole figure data (Fig. 2), or they can be extracted from an ODF analysis of experimental pole figures. The ODF analysis is recommended especially in the case of strong mixed textures or texture gradients. It is important to note that in the quantification of the 3À Ã hkl value using equation (19) the crystal elastic anisotropy does not play a role.
It is obvious that the considerations of xx4.1-4.5 can be extended to determine other mechanical elastic constants of thin films (e.g. in-plane biaxial moduli). Therefore, there is a need for a general approach when comparing hS L 33ij i and fS L 33ij g hkl for various fibre-texture types and angles. The derived 3À Ã hkl dependence on the texture parameters [equation (19)] based on the comparison of hS L 33ij i and fS L 33ij g hkl depends obviously on the supposed grain-interaction model. In the present case, the Hill grain-interaction model was used, and therefore the approach should be applied only to fibre-textured films that are assumed to obey that model. In the next section, the approach is demonstrated on the experimental characterization of fibre-textured Cu and CrN thin films.

Sample preparation
Cu and CrN thin films were deposited on Si(100) using the Balzers RCS coating system. In order to induce a measurable substrate curvature and to avoid a substrate plastic deformation, monocrystalline Si(100) wafers with thicknesses of 140 and 400 mm and lateral dimensions of 30 Â 8 mm were chosen for the deposition of Cu and CrN films, respectively. The substrates were ultrasonically cleaned in acetone and alcohol, and Ar etched prior to the deposition. The Cu was deposited in an argon atmosphere at room temperature and then annealed at 673 K for 10 min in order to increase the residual stress (and substrate curvature) magnitude. The 3 mm-thick CrN thin film was deposited at a temperature of 623 K. The thicknesses of the Cu and CrN thin films (0.6 and 3 mm, respectively) were determined from the film cross sections using a scanning electron microscope. The thickness of the substrate was measured mechanically using a micrometre gauge with a precision of better than 1 mm.

Diffraction setup
The substrate curvature, elastic strain and texture of Cu and CrN on Si(100) were characterized in laboratory conditions using a Seifert 3000 PTS four-circle diffractometer. The setup comprised Cu K radiation, polycapillary optics on the primary side, vertical Soller slits, a graphite monochromator and a scintillation detector on the secondary side. For the elastic strain and curvature characterization, beam sizes of 3.0 and 0.5 mm in diameter were chosen. The relatively large beam in the case of strain measurements enabled the assessment of volume-averaged properties. The elastic strains were determined with precision better than AE10%. The limited pole figure characterization was performed using the Schultz reflectivity technique with a beam of 2 mm in diameter, with the range set to 0-80 . The rectangular samples were glued with just one of their narrower sides onto sample holders, in order to allow for free bending when the strain and the curvature were characterized in the diffractometer. For comparison, the texture of the films was also characterized using a Bruker GADDS system equipped with a two-dimensional detector, and the pole figures were identical to those obtained using the Seifert system.

Thin-film texture
The texture in Cu and CrN thin films was characterized using pole figure measurements (Figs. 9 and 10). The orientation distribution function was then calculated from the experimental data in order to assess the proportion of randomly oriented crystallites ISO. The ODF analysis was performed using the commercial software LaboTex applying the ADC (arbitrarily defined cells) method (LaboSoft, 2006;Pawlik, 1986). In the case of Cu, one can easily identify a sharp 111 fibre texture (Fig. 9) with a width at half-maximum FWHM at the centre of the 111 pole figure of 14 and an ISO of 10%. For CrN, a 311 texture is visible in Fig. 10, with FWHM = 12 and an ISO of 13%.
The experimental parameters FWHM and ISO were used to determine 3À Ã hkl using equation (19). For the Cu and CrN thin films from Figs. 9 and 10, it was found that 3À Ã hkl is equal to 0.89 and 0.51, respectively.

Macroscopic stress characterized by the X-ray diffraction substrate curvature technique
The pole figure measurements confirmed that the thin films are in-plane elastic isotropic. Since the films were unpassivated, the residual stress h S i in the plane of the films was considered as equibiaxial and the out-of-plane components h S i3 i were neglected. The volume-averaged macroscopic stresses in Cu and CrN polycrystalline thin films were determined using the X-ray diffraction substrate curvature method (Stoney, 1909;Segmü ller et al., 1989;Zhao et al., 2002;Keckes et al., 2007). The quantification of the curvature was performed by the measurement of rocking curves of Si 400 reflections at different sample positions x i as described in our previous work . In Fig. 11, the relative positions of the rocking curves (expressed through angle !) on Áx are presented for the Cu/Si(100) and CrN/Si(100) samples. The plots in Fig. 11 indicate a homogeneous curvature and residual stress across the sample. In practice, provided the sample homogeneity is not questionable, it would be enough to quantify the curvature from just a few measurement points.
The data in Fig. 11 were used to calculate the radius of curvature R according to where @Á!/@Áx represents the slope of the linear dependencies . Applying R, it was possible to determine the macroscopic in-plane isotropic residual stress h S i in the films using the Stoney (1909) formula where h s and h f denote the substrate and film thicknesses, respectively, and the term E/(1 À ) = 181 GPa is the biaxial modulus of the silicon substrate (Suresh & Freund, 2003). The macroscopic stress h S i in the Cu and CrN films (Table 2) was determined with a precision of about AE5%.

Elastic strain in thin films
In Figs. 12(a) and 12(b) the X-ray elastic strains f" L 33 g hkl in the Cu and CrN films for different hkl reflections are presented as a function of the sample tilt angle . The different crystal elastic anisotropy has the result that @f" L 33 g 200 ¼0 =@ sin 2 > @f" L 33 g hkl ¼0 =@ sin 2 for Cu and @f" L 33 g 111 ¼0 = @ sin 2 > @f" L 33 g hkl ¼0 =@ sin 2 for CrN in Fig. 12. In the case of Cu, the dependencies are nearly linear, whereas for CrN films one can observe nonlinearities which can be attributed to the experimental errors and to gradients of strain, texture or unstressed lattice parameters. Especially in the case of hard coatings like CrN, nonlinearities (Fig. 12b) are usual (cf. Donohue et al., 1999;Gö bel et al., 2001). Although it has often been observed that polycrystalline samples do not exhibit perfectly linear experimental sin 2 dependencies for h00 and hhh reflections, the application of the anisotropic Hill graininteraction model to assess the mechanical behaviour of such samples has generally provided satisfactory results (van Houtte & De Buyser, 1993;Gnä upel-Herold et al., 1998;Howard & Kisi, 1999). It is also known that results obtained using other more sophisticated models such as those of Krö ner or Vook-Witt (Krö ner, 1958;Leoni et al., 2001;Welzel, 2002;Welzel et al., 2005) are very close to those of Hill (van Houtte & De Buyser, 1993;Gnä upel-Herold et al., 1998;Howard & Kisi, 1999, Welzel, 2002. Moreover, since the grain-interaction models represent idealized theories, it is very difficult to distinguish which model is applicable in the case of experimental data obtained from real materials. The precision of research papers Figure 11 Plots of the Á! dependence on Áx for the Cu/Si(100) and CrN/Si(100) samples. The results indicate different radii of curvature R of 2.193 and 3.572 m for Cu and CrN. The convex and the concave bending correspond to tensile and compressive stresses of 275.9 and À1415.9 MPa in Cu and CrN, respectively . Table 2 An experimental algorithm to determine out-of-plane mechanical moduli of fibre-textured thin films is presented.
The macroscopic stress h S i was determined using the curvature measurement (x5.4 and Fig. 11). The elastic strain f" L 33 g hkl dependencies on sin 2 (x5.5 and Fig. 12) were analysed in order to evaluate the intercepts fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 and slopes fS L 3333 g hkl !0 À fS L 3311 g hkl !0 for ! 0. The factor 3À Ã hkl indicates for which value of the X-ray anisotropic factor the X-ray and mechanical elastic constants are equal (x5.3). The compliances and the moduli were then determined using equation (22). diffraction techniques is, moreover, often insufficient to distinguish between different models (Howard & Kisi, 1999). This is the case here also, since the precision of the X-ray elastic strain characterization was not better than 10%. Since the films were polycrystalline, the methodology based on the anisotropic Hill grain-interaction model was used to extract mechanical elastic constants applying the formalism from xx2-4.
The plots in Fig. 12 illustrate that it was possible to perform lattice spacing measurements and to determine X-ray elastic strains at every sample tilt angle , even for the Cu film with the strong 111 fibre texture. This fact indicates that there was a nonzero fraction of randomly oriented crystallites in the films. The lattice spacing measurements at arbitrary angle were possible, however, only after the polycapillary optics and vertical Soller slits were installed and used (Welzel & Leoni, 2002). Therefore, the use of parallel X-ray optics seems to be an important prerequisite to apply successfully the method described in this work.
Another important prerequisite for the use of the new method is the fact that the strains should be analysed using a relatively large beam (3 mm in diameter in the present case). Only then can representative information on the average X-ray elastic strain be obtained.

Experimental Young's moduli
The X-ray elastic constants fS L 33ij g hkl [equation (13) and x4.3] can be obtained by a numerical fitting of the experimental X-ray elastic strains f" L 33 g hkl from Fig. 12, applying the macroscopic stress values h S i from Table 2. This type of analysis was performed in order to evaluate (i) fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 from the intercepts on the f" L 33 g hkl axis and (ii) fS L 3333 g hkl !0 À fS L 3311 g hkl !0 from the slopes in Fig. 12 X-ray elastic constants fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 and fS L 3333 g hkl !0 À fS L 3311 g hkl !0 obtained by fitting equation (13) (x3.1) to the data from Fig. 12 and by evaluating the intercepts (a) and the slopes (b) under the consideration of the macroscopic stress h S i (x5.4).

Figure 12
Measured X-ray elastic strains f" L 33 g hkl in Cu (a) and CrN (b) thin films as a function of the sample tilt angle . Positive (a) and negative (b) slopes indicate tensile and compressive stresses in Cu and CrN, respectively. The strains were determined with a precision better than AE10%.
Figs. 13(a) and 13(b), the fitted parameters fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 and fS L 3333 g hkl !0 À fS L 3311 g hkl !0 from Fig. 12 are presented as a function of 3À hkl for the Cu and CrN thin films. These parameters differ for various hkl reflections, which is the consequence of the crystal elastic anisotropy.
The fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 and fS L 3333 g hkl !0 À fS L 3311 g hkl !0 dependencies on 3À hkl from Fig. 13 were approximated by linear dependencies and the results are presented in Table 2. By easy calculus it was possible to derive also the dependence of fS L 3333 g hkl !0 on 3À hkl (Table 2). Considering the macroscopic elastic anisotropy and by applying the 3À Ã hkl values from x5.3 one can determine an inverse out-of-plane X-ray elastic modulus fS L 3333 g 3À Ã ¼0 which is equal to the mechanical compliance hS L 3333 i ¼0 . The out-of plane Young's modulus can then be easily determined as follows: The experimental out-of-plane Young's moduli of Cu and CrN thin films were found to be 169.40 and 244.87 GPa. The results are comparable to the experimental data obtained using other techniques (Hong et al., 2005;Sue et al., 1994;Lee et al., 2008).

Error discussion
The accuracy with which the out-of-plane Young's moduli were determined using the new algorithm is influenced by numerous factors. The approach is based on the combined application of well established techniques, sin 2 and X-ray diffraction substrate curvature, the experimental accuracies of which have been discussed in numerous papers (e.g. Noyan & Cohen, 1987;Winholz & Cohen, 1988;Zhao et al., 2002). The combination of the two techniques can in the worst case result in the accumulation of experimental errors.
The accuracy of the sin 2 technique was assessed by Noyan & Cohen (1987) and by Winholz & Cohen (1988). Depending on numerous parameters, such as sample quality, diffraction system, number of measured reflections, scattering intensity and measurement time (Noyan & Cohen, 1987;Winholz & Cohen, 1988), the precision is usually below AE15%. Moreover, the exactness of the elastic strain characterization can be improved by increasing the number of measured reflections (Fig. 13). In the present case, since the measurements were performed using a commercial diffractometer, the precision was about AE10% The exactness of the macroscopic stress characterization is extremely important since the stress is used to divide the experimental strain values. For this reason, not only the substrate curvature but also the film and the substrate thickness must be determined with a high precision. In the present case, the macroscopic stresses were determined with a precision of AE5%. Experimental elastic strain and macroscopic stress data are combined in Fig. 13, whereby the linear dependencies provided coefficients of determination R 2 larger than 0.9.
The data in Fig. 13 were used to extract hS L 3333 i ¼0 parameters and finally also elastic moduli. It can be therefore supposed that the moduli in Table 2 were determined in the present case with a precision better than 15%. Fig. 13 implies good control over the reliability of the technique proposed here. If the dependence of fS L 3311 g hkl ¼0 þ fS L 3322 g hkl ¼0 and fS L 3333 g hkl !0 À fS L 3311 g hkl !0 on 3À hkl is not linear and the R 2 factors are smaller than 0.8, the technique will not provide reliable data.
Moreover, the simultaneous application of sin 2 and X-ray diffraction curvature techniques should be performed on a representative sample region without strong gradients in microstructure and in stresses. By extending the curvature characterization to a large Áx range , it is possible to analyse if the curvature and the stress are homogeneous. The strain measurements should be performed in the region for which the curvature as well as the film and the substrate thickness are known. When performing the strain characterization, however, a large beam ensuring good statistics is required.
Another source of error could reside in the parameter 3À Ã hkl . The parameter can be quantified exactly using a numerical ODF analysis of the texture data or estimated from the pole figure plots. The higher the crystal elastic anisotropy of the materials, the more significantly the 3À Ã hkl inaccuracy will contribute to the errors when determining the moduli.
Similarly, the present approach supposes that thin films possess a certain type of fibre texture, which occurs, for instance, in annealed metallic films (where the film thickness is comparable to the crystallite thickness; Eiper et al., 2007). Since some thin films possess very complex fibre textures (with strong gradients), a careful ODF analysis must be performed before comparing hS L 33ij i and fS L 33ij g hkl . Incorrect values of moduli will be obtained when the monocrystalline substrate under the film is plastically deformed. In that case the Stoney formula does not hold. For this reason, it is important to pay significant attention to the sample preparation.
An important assumption made in this work is that the mechanical behaviour of the polycrystalline films can be described using the Hill grain-interaction model. This is generally not the case (for instance, in epitaxial thin films where behaviour according the Voigt model can be expected). Therefore, the proposed method cannot be applied automatically. It is always important to analyse the microstructure of the film. In the case of film materials with unknown graininteraction mechanism, it is recommended to perform a comparative characterization using other techniques, such as nanoindentation. Nonlinearities in the f" L 33 g hkl -sin 2 dependencies are often an important indication that a polycrystalline film does not obey the Hill grain-interaction model. As already mentioned, nonlinearities can be attributed also to gradients of strain, texture or unstressed lattice parameters, or to plasticity in the film. In the case of strong nonlinearities, the method should not be applied. Although in the present case (Fig. 12b) the f" L 33 g hkl -sin 2 dependencies are not perfectly linear, the results obtained from the CrN film are comparable to results obtained using other techniques (Sue et al., 1994;Lee et al., 2008).

Conclusions
A new method to determine elastic moduli of thin films in a contactless manner using X-ray light was proposed. Supposing the Hill grain-interaction model, it was demonstrated that X-ray elastic constants can be used to determine mechanical elastic constants of cubic thin films with strong fibre textures. For this purpose, numerically calculated X-ray elastic constants of polycrystalline films were compared with their mechanical counterparts. The results demonstrate that the algorithm to determine the mechanical elastic constants strongly depends on the fibre-texture type, the texture sharpness, the number of randomly oriented crystallites in the polycrystalline aggregate and the assumed grain-interaction model. For this purpose, a universal plot (and equation) was derived. The method was used to quantify out-of-plane Young's moduli of Cu and CrN fibre-textured thin films with satisfactory results. This work was supported by the Austrian NANO Initiative via a grant from the Austrian Science Fund FWF within the project 'StressDesign -Development of Fundamentals for Residual Stress Design in Coated Surfaces'.