research papers
Elastic constants of fibretextured thin films determined by Xray diffraction
^{a}Department of Materials Physics, University of Leoben and Erich Schmid Institute for Materials Science, Austrian Academy of Science, Jahnstrasse 12, 8700 Leoben, Austria, and ^{b}Department of Physical Metallurgy and Materials Testing and ChristianDoppler Laboratory for Advanced Hard Coatings, University of Leoben, Austria
^{*}Correspondence email: keckes@unileoben.ac.at
A new methodology is presented that allows the rapid determination of elastic constants of cubic fibretextured thin films by Xray diffraction. The theoretical concept is developed and tested on calculated examples of Cu and CrN films. The mechanical elastic constants are extrapolated from Xray elastic constants by taking into consideration crystal and macroscopic elastic anisotropy. The derived algorithm enables the determination of a reflection and the corresponding value of the Xray anisotropic factor Γ for which the Xray elastic constants are equal to their mechanical counterparts in the case of fibretextured cubic polycrystalline aggregates. The approach is independent of the crystal elastic anisotropy and depends on the fibretexture type, the texture sharpness, the number of randomly oriented crystallites and the supposed graininteraction model. In the experimental part, outofplane Young's moduli of 111 and 311 fibretextured Cu and CrN thin films deposited on monocrystalline Si(100) substrates are determined. The moduli are extrapolated from thinfilm experimental Xray elastic constants that are determined by a combination of Xray diffraction substrate curvature and sin^{2}ψ methods. For the calculation, the film macroscopic elastic anisotropy (texture) is considered. The advantage of the new technique lies in the fact that experimental moduli are determined nondestructively, using a static diffraction experiment, and represent volumeaveraged quantities.
Keywords: Xray diffraction; thin films; elastic constants; fibre texture.
1. Introduction
Xray elastic constants and diffraction stress factors are usually used to calculate residual stresses from experimental Xray elastic strains (Dölle, 1979; Noyan & Cohen, 1987; Welzel et al., 2005). In the case of a specimen with crystal elastic anisotropy, the Xray elastic constants differ for various hkl reflections and are, moreover, dependent on the texture, the graininteraction mechanism and the singlecrystal elastic constants (Barral et al., 1987; van Houtte & De Buyser, 1993; van Leeuwen et al., 1999; Leoni et al., 2001; Badawi et al., 2002; Welzel, 2002; Welzel et al., 2005).
Numerous experimental as well as computational methods have been developed in the past few decades to determine mechanical elastic moduli (or even singlecrystal elastic constants) from diffraction elastic constants (Hayakawa et al., 1985; Humbert & Diz, 1991; Wright, 1994; GnäupelHerold et al., 1998; Howard & Kisi, 1999; Badawi et al., 2002; Badawi & Villain, 2003). In the case of experimental characterization using Xrays or neutrons, polycrystalline samples (bulk or thin films, usually on a flexible substrate) are insitu strained, and diffraction elastic strains are recorded and correlated with the applied stress. To extract the mechanical elastic constants, Reuss, Voigt, Hill and Eshelby–Kröner graininteraction models are usually applied (Hayakawa et al., 1985; Wright, 1994; Howard & Kisi, 1999; Faurie et al., 2006). Although the graininteraction models represent only idealized theories, the elastic constants of quasiisotropic 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; GnäupelHerold et al., 1998; Howard & Kisi, 1999; Badawi et al., 2002; Badawi & Villain, 2003), however, the experimental elastic moduli or singlecrystal elastic constants are obtained from insitu 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}ψ and Xray diffraction substrate curvature techniques was proposed (Eiper et al., 2005, 2006; Keckes et al., 2007; Martinschitz et al., 2006). The new approach provides an opportunity to quantify experimental Xray 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; Segmüller et al., 1989). The experimental stress and strain can then be used to evaluate experimental Xray elastic constants and stress factors (Eiper et al., 2005, 2006; Martinschitz et al., 2006).
Mechanical elastic constants can be extrapolated from Xray elastic constants considering crystal and macroscopic elastic anisotropy. In the case of cubic polycrystalline aggregates with macroscopic elastic isotropy (quasiisotropic materials) that obey the Hill graininteraction model, it was demonstrated that Xray elastic constants correspond to their mechanical counterparts for = 0.2, where is the Xray anisotropic factor according to the Reuss graininteraction model, given by
(hkl) are of a crystallographic plane (Bollenrath et al., 1967). According to the Reuss (1929) model, Xray elastic anisotropy is often expressed as a function of 3, and this formalism will be applied hereafter.
It is the aim of this paper to analyse under which conditions knowledge of Xray elastic constants can be used to determine or estimate mechanical elastic constants of cubic fibretextured thin films that obey the Hill graininteraction 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; GnäupelHerold et al., 1998; Howard & Kisi, 1999; Leoni et al., 2001; Welzel, 2002). As a next step, the mechanical values will be compared with calculated Xray elastic constants. As a result, a dependent selection rule will be derived, where the subscript hkl in denotes a reflection for which mechanical and Xray elastic constants are equal. The approach will be demonstrated by experimental characterization of outofplane moduli of fibretextured Cu and CrN thin films. The moduli are extracted from experimental Xray elastic constants that are determined by a combination of Xray diffraction substrate curvature and sin^{2}ψ 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 graininteraction model
Elastic behaviour of a thin film deposited on a solid substrate can be represented by Hooke's law:
where is the mechanical elastic strain, expresses the mechanical elastic constants of the film and represents the residual stress (Nye, 1976; Suresh & Freund, 2003). The brackets 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, of a polycrystalline film depends on the texture, the singlecrystal elastic constants and the graininteraction mechanism (van Houtte & De Buyser, 1993). In practice, the Hill graininteraction model can be used to evaluate of the film (Hill, 1952) using the arithmetic mean of the compliance tensors and obtained from the Reuss and Voigt graininteraction models:
Elastic constants according to the Reuss average can be calculated as follows:
In the case of the Voigt average, 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_{ijkl}^{S} and C_{ijkl}^{S} in equations (4) and (5) are singlecrystal elastic constants expressed in S, while f(g)dg indicates the 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).
The tensor [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 is usually used to express elastic behaviour of materials. The magnitude of in the direction can be obtained from the tensor :
The outofplane Young's modulus can be obtained from equation (7) using .
2.2. 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 singlecrystal 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 halfmaximum at the centre of the pole figure [which is usually measured experimentally using a ψ scan (Bunge, 1982)], hereafter denoted ψ_{FWHM}, was used (Fig. 2). Since the aim is to develop a simple laboratory method to determine elastic constants of thin films, ψ_{FWHM} was used as a measure of the texture sharpness [instead of using variables expressed in terms of the φ_{1}, Φ and φ_{2} angles (Fig. 2), which are usually needed to define ODF properties according the Bunge (1982) notation]. Numerous ODFs with ψ_{FWHM} in the range 0–180° with a step of 5° were generated in order to calculate [equation (2)] and subsequently the outofplane Young's modulus [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 outofplane Young's moduli of Cu and CrN thin films with 111 fibre texture are presented in Fig. 3. As parameters for the calculation, the texture sharpness ψ_{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 ψ_{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), with the Zener (1948) anisotropy ratio ZAR defined as
Since the direction in Cu is stiffer than all others, the films with less pronounced textures exhibit smaller moduli (Fig. 3). In CrN, the opposite situation has to be considered.
The results in Fig. 3 represent outofplane Young's moduli calculated from . In the case of fibretextured thin films, however, the elastic behaviour is inplane isotropic (i.e. independent of the angle φ) but dependent on the tilt angle ψ (Fig. 1). In order to demonstrate this situation, Young's moduli of CrN and Cu 111 fibretextured thin films (Fig. 2) were calculated as a function of angles φ and ψ 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 ψ = 0°, in contrast to the Cu dependence, which exhibits a maximum at the centre of the polar plot (Fig. 4).
3. Xray elastic constants of thin films
3.1. Xray elastic moduli
In Xray diffraction, Hooke's law relates Xray elastic strain measured in the direction by scanning the reflection hkl, Xray elastic compliances and the macroscopic stress expressed in the L coordinate system (Fig. 1) as follows:
where depends generally on the texture, the graininteraction mechanism, the reflection hkl, the singlecrystal elastic constants, and the angles φ and ψ (Dölle, 1979; van Houtte & De Buyser, 1993). The brackets { } denote volumeweighted averages for all diffracting crystallites (i.e. diffraction averages; Welzel, 2002). For simplicity, can be calculated using the Hill graininteraction model as follows (Serruys, 1988; van Houtte & De Buyser, 1993):
The Xray elastic compliances represent an elastic behaviour of the film according the Voigt graininteraction model (V) and can be calculated as
where the tensor was obtained using equation (5) (van Houtte & De Buyser, 1993).
Xray elastic compliances according the Reuss graininteraction model (R) can be obtained by integration over the crystal orientations g for which the diffraction vector is parallel to the direction (van Houtte & De Buyser, 1993):
Considering fibretextured 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 inplane isotropic with and . Moreover, shear stresses and , shear strains , and , and outofplane stresses can be neglected on the macroscopic scale.
(ii) The thin films are inplane elastic isotropic, and not only the distribution of crystallites but also the graininteraction mechanism possess a rotational symmetry. The elastic properties of the films are therefore not dependent on the azimuth angle φ, with = .
The above implies that equation (9) can be expressed as follows (Stickforth, 1966):
In the case of the experimental dependence of on , the term corresponds to the intercept on the axis and the term is responsible for the curvature in the plots. The term vanishes, however, under certain conditions (cf. Stickforth, 1966; van Houtte & De Buyser, 1993; Welzel, 2002).
Since the tensor components in equation (13) change as a function of the orientation of the diffraction vector , they can be used to determine diffraction elastic constants as a function of (hkl) and ψ. For example, the diffraction modulus along the direction reads
On condition that the components are independent of the angles ψ and φ and the material is quasiisotropic, equation (13) can be written as
in which the symbols { s_{1} } ^{hkl} and { s_{2}/2 } ^{hkl} represent isotropic Xray elastic constants (Dölle, 1979; Noyan & Cohen, 1987). The constants are sometimes substituted as (Noyan & Cohen, 1987)
The symbol represents the diffraction Poisson number determined by the measurement of reflection hkl. In the case of macroscopic elastic isotropic aggregates, { E} ^{hkl} and can be calculated using equations (15) and (16) provided and are known.
3.2. Calculation of diffraction elastic moduli
The Xray elastic compliances express the elastic behaviour of the aggregate along the diffraction vector . In Fig. 5, Young's moduli of 111 fibretextured Cu and CrN thin films are presented as a function of the tilt angle ψ.
The data in Fig. 5 document that the mechanical moduli of the 111 fibretextured films lie always between diffraction moduli and . The diffraction moduli represent the elastic response of diffracting crystallites in the direction of diffraction vector (Fig. 2). The mechanical moduli represent the elastic response of all crystallites in the direction of diffraction vector . For ψ = 0 the outofplane mechanical modulus approaches the diffraction modulus (which can be obtained by the characterization of the 111 reflection and ) because of the specific texture type (Fig. 5).
4. A comparison of mechanical and Xray elastic constants
4.1. General considerations
The results in Fig. 5 demonstrate that the mechanical elastic constants are constrained by the Xray elastic constants . It is therefore obvious that, by considering a specific ODF, tilt angle ψ and singlecrystal elastic constants, it is always possible to determine a reflection hkl and a corresponding Xray anisotropy factor 3 for which the Xray elastic constants are equal to their mechanical counterparts (Fig. 5). 3 will therefore be used to denote conditions in accordance with the Hill model, under which = .
4.2. Isotropic case
In the case of polycrystalline materials with crystal elastic isotropy or with negligible macroscopic elastic anisotropy, { E} ^{hkl} and as well as { s_{1} } ^{hkl} and { s_{2}/2 } ^{hkl} are independent of the angle ψ, and equation (15) supposes a linear dependence of on sin^{2}ψ (Stickforth, 1966; Noyan & Cohen, 1987; van Houtte & De Buyser, 1993). Provided that the elastic strain and the macroscopic stress can be determined by experiment, the isotropic Xray elastic constants { s_{1} } ^{hkl} and { s_{2} /2} ^{hkl}, and subsequently also { E} ^{hkl} and , can be obtained by solving a system of linear equations of the same type as equation (15) when is known for different ψ (Ortner, 1986a,b).
An example of this procedure is presented in Fig. 6. Considering the singlecrystal elastic constants from Table 1 and an inplane isotropic stress = 100 MPa, calculated diffraction strains for a quasiisotropic Cu thin film are presented in Fig. 6(a).
According to equation (15), the slopes in Fig. 6(a) correspond to { s_{2}/2 } ^{hkl} and the intercepts on the axis can be correlated with the magnitude of { s_{1} }^{hkl}. In practice, the Xray elastic constants are obtained by fitting the experimental data from Fig. 6(a) using equation (15). The reciprocal diffraction elastic moduli 1/{ E} ^{hkl} in Fig. 6(c) can then be determined from { s_{1} }^{hkl} and { s_{2}/2 } ^{hkl} (Fig. 6b) as follows:
The reciprocal mechanical modulus = 0.81 × 10^{−11} Pa^{−1} was extrapolated from the reciprocal diffraction moduli supposing = 1/{ E } ^{hkl} for 3 = 0.6, as predicted by the Hill graininteraction model for quasiisotropic materials (Bollenrath et al., 1967). The mechanical modulus is therefore 123.45 GPa. This procedure is, however, valid only in the case of elastic isotropic aggregates.
4.3. Fibretextured 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 Xray elastic constants for 3 = 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 Xray elastic compliances are dependent on the angle ψ (Figs. 4 and 5). In order to determine the experimental from it is necessary to know the exact value of 3, which is also dependent on ψ, as demonstrated in Fig. 5. We discuss below the possibilities for determining and from the experimental by applying Hooke's law [equation (13)].
(i) In the case of inplane elastic isotropic films is equal to for ψ = 0 and equation (13) reduces to = . can be determined experimentally by evaluating the intercept of the sin^{2}ψ dependence on the axis when is known. The dependence of on 3Γ_{hkl} could then be used to determine the thinfilm mechanical compliance .
(ii) By comparing the intercepts and the slopes of the sin^{2}ψ curves for [and by simultaneously neglecting the term since for hexagonal macroscopic symmetry of the sample (Martinschitz, 2008)], equation (13) can be used to extract and the diffraction outofplane modulus , as in §4.2. By considering the macroscopic elastic anisotropy, the knowledge of can be used to determine the mechanical Young's modulus or the term .
(iii) By evaluating the intercepts on the axis for , equation (13) can be used to determine the term , which, in this special case, can be used to quantify the inplane biaxial modulus of the thin film .
In order to quantify the parameters , and , the macroscopic elastic anisotropy of the film must be considered. Furthermore, the determination of outofplane moduli = 1/ from the Xray elastic constants will be discussed.
4.4. Elastic modulus of 111 fibretextured Cu thin film
In Fig. 7, calculated sin^{2}ψ dependencies for a Cu thin film with a strong 111 fibre texture (Fig. 2) are presented. The plots were calculated supposing an inplane isotropic stress of = 100 MPa and using the singlecrystal elastic constants from Table 1.
The data in Fig. 7(a) were evaluated according the procedure described in §4.2 point (ii), and values were determined (Fig. 7b). Using the ODF from Fig. 2, the outofplane mechanical compliance was also calculated [equations (3)–(7)] with = 174 GPa. Comparison of the outofplane Xray and mechanical compliances showed that for 3 = 0.937. This result demonstrates that, in order to determine from [i.e. to apply an opposite algorithm flow to that in Fig. 7(b)], it is necessary to know the value of 3, which is strongly texture dependent.
4.5. 3Γ^{*}_{hkl}–3Γ_{uvw} plot
In the case of cubic uvw fibretextured films with the fibre axis oriented perpendicularly to the substrate surface, the texture type will be further described using the parameters Γ_{uvw} defined as (Bollenrath et al., 1967; Huang & Weaver, 2005)
Supposing various uvw fibre textures (i) with a texture sharpness ψ_{FWHM} in the range 0–60° (Fig. 2), (ii) with in the range 0–1 and (iii) with ISO in the range 0–100%, numerous ODFs were generated. Following the algorithm from §4.3 point (ii), and 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 Xray elastic constants were compared, with the aim of finding out for which 3 value . As a result 3– plots were constructed, indicating how 3 depends on the uvw fibretexture type, on ψ_{FWHM} (Fig. 8b) and on ISO (Fig. 8a).
The – plots in Fig. 8 do not depend on the crystal elastic anisotropy of the thinfilm material and represent therefore a certain type of universal plot valid for all materials. In the case of isotropic materials (like tungsten) where ZAR 1, the choice of is arbitrary.
In Fig. 8(a), one can recognize that, for very strong uvw fibre textures with ψ _{FWHM} < 10° and a small or no fraction of randomly oriented crystallites, the Xray elastic constants correspond approximately to the mechanical constants for 3 = 3Γ_{uvw}. In other words, in order to determine the outofplane modulus of a thin film with a very strong uvw texture one has to characterize the Xray elastic constants of the uvw reflections. For not very pronounced fibre textures, the 3 value must be selected from the intervals or for thin films with Γ_{uvw} smaller or larger than 0.6, respectively. When the fraction of randomly oriented crystallites ISO increases, however, Xray elastic constants of the hkl reflections for which 3 should be quantified (§4.2). Similarly, in Fig. 8(b), the decrease of the texture sharpness results in behaviour that is typical for elastic isotropic materials and 3.
In the case of sharp uuu or u00 fibre textures the search for an exact 3 value is extremely important, because the application of the procedure from §4.2 (valid for elastic isotropic materials) could result in large errors when determining the outofplane moduli. For films with uvw fibre textures with 3, the procedure from §4.2 can still provide relevant results.
The results in Figs. 8(a) and 8(b) represent an example of the – selection rule. In order to express the dependence of 3 on 3, on ψ_{FWHM} and on ISO generally and in a `userfriendly' way, the following empirical equation was derived:
where A = (ψ_{FWHM} × 8.8 + ISO × 5.8 − ψ_{FWHM} × ISO × 0.083)/1000.
Equation (19) provides an easy way to determine 3 values considering fibretexture parameters. The parameters ISO and ψ_{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 3 value using equation (19) the crystal elastic anisotropy does not play a role.
It is obvious that the considerations of §§4.1–4.5 can be extended to determine other mechanical elastic constants of thin films (e.g. inplane biaxial moduli). Therefore, there is a need for a general approach when comparing and for various fibretexture types and ψ angles.
The derived 3 dependence on the texture parameters [equation (19)] based on the comparison of and depends obviously on the supposed graininteraction model. In the present case, the Hill graininteraction model was used, and therefore the approach should be applied only to fibretextured films that are assumed to obey that model. In the next section, the approach is demonstrated on the experimental characterization of fibretextured Cu and CrN thin films.
5. Experimental procedure
5.1. 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 µm and lateral dimensions of 30 × 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 µmthick CrN thin film was deposited at a temperature of 623 K. The thicknesses of the Cu and CrN thin films (0.6 and 3 µm, 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 µm.
5.2. 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 fourcircle diffractometer. The setup comprised Cu Kα radiation, polycapillary optics on the primary side, vertical Soller slits, a graphite monochromator and a 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 volumeaveraged properties. The elastic strains were determined with precision better than ±10%. The limited pole figure characterization was performed using the Schultz reflectivity technique with a beam of 2 mm in diameter, with the ψ 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 twodimensional detector, and the pole figures were identical to those obtained using the Seifert system.
5.3. Thinfilm 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 halfmaximum ψ_{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 ψ_{FWHM} = 12° and an ISO of 13%.
The experimental parameters ψ_{FWHM} and ISO were used to determine 3 using equation (19). For the Cu and CrN thin films from Figs. 9 and 10, it was found that 3 is equal to 0.89 and 0.51, respectively.
5.4. Macroscopic stress characterized by the Xray diffraction substrate curvature technique
The pole figure measurements confirmed that the thin films are inplane elastic isotropic. Since the films were unpassivated, the residual stress in the plane of the films was considered as equibiaxial and the outofplane components were neglected. The volumeaveraged macroscopic stresses in Cu and CrN polycrystalline thin films were determined using the Xray diffraction substrate curvature method (Stoney, 1909; Segmüller 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 (Martinschitz et al., 2006). In Fig. 11, the relative positions of the rocking curves (expressed through angle ω) on Δx 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 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 / represents the slope of the linear dependencies (Martinschitz et al., 2006). Applying R, it was possible to determine the macroscopic inplane isotropic residual stress 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 − ν) = 181 GPa is the biaxial modulus of the silicon substrate (Suresh & Freund, 2003). The macroscopic stress in the Cu and CrN films (Table 2) was determined with a precision of about ±5%.
5.5. Elastic strain in thin films
In Figs. 12(a) and 12(b) the Xray elastic strains in the Cu and CrN films for different hkl reflections are presented as a function of the sample tilt angle ψ. The different crystal elastic anisotropy has the result that > for Cu and > 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 parameters. Especially in the case of hard coatings like CrN, nonlinearities (Fig. 12b) are usual (cf. Donohue et al., 1999; Göbel et al., 2001). Although it has often been observed that polycrystalline samples do not exhibit perfectly linear experimental sin^{2}ψ 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; GnäupelHerold et al., 1998; Howard & Kisi, 1999). It is also known that results obtained using other more sophisticated models such as those of Kröner or Vook–Witt (Kröner, 1958; Leoni et al., 2001; Welzel, 2002; Welzel et al., 2005) are very close to those of Hill (van Houtte & De Buyser, 1993; GnäupelHerold et al., 1998; Howard & Kisi, 1999, Welzel, 2002). Moreover, since the graininteraction 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 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 Xray elastic strain characterization was not better than 10%.
Since the films were polycrystalline, the methodology based on the anisotropic Hill graininteraction model was used to extract mechanical elastic constants applying the formalism from §§2–4.
The plots in Fig. 12 illustrate that it was possible to perform spacing measurements and to determine Xray elastic strains at every sample tilt angle ψ, 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 spacing measurements at arbitrary ψ 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 Xray 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 Xray elastic strain be obtained.
5.6. Experimental Young's moduli
The Xray elastic constants [equation (13) and §4.3] can be obtained by a numerical fitting of the experimental Xray elastic strains from Fig. 12, applying the macroscopic stress values from Table 2. This type of analysis was performed in order to evaluate (i) from the intercepts on the axis and (ii) from the slopes in Fig. 12. In Figs. 13(a) and 13(b), the fitted parameters and from Fig. 12 are presented as a function of 3Γ_{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 and dependencies on 3 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 on 3 (Table 2). Considering the macroscopic elastic anisotropy and by applying the 3 values from §5.3 one can determine an inverse outofplane Xray elastic modulus which is equal to the mechanical compliance . The outof plane Young's modulus can then be easily determined as follows:
The experimental outofplane 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).
5.7. Error discussion
The accuracy with which the outofplane 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}ψ and Xray 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}ψ 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 ±15%. 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 ±10%
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 ±5%. 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 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 and on 3 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}ψ and Xray 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 Δx range (Martinschitz et al., 2006), 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 3. 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 3 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 and .
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 graininteraction 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 –sin^{2}ψ dependencies are often an important indication that a polycrystalline film does not obey the Hill graininteraction model. As already mentioned, nonlinearities can be attributed also to gradients of strain, texture or unstressed 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 –sin^{2}ψ 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).
6. Conclusions
A new method to determine elastic moduli of thin films in a contactless manner using Xray light was proposed. Supposing the Hill graininteraction model, it was demonstrated that Xray elastic constants can be used to determine mechanical elastic constants of cubic thin films with strong fibre textures. For this purpose, numerically calculated Xray 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 fibretexture type, the texture sharpness, the number of randomly oriented crystallites in the polycrystalline aggregate and the assumed graininteraction model. For this purpose, a universal plot (and equation) was derived. The method was used to quantify outofplane Young's moduli of Cu and CrN fibretextured thin films with satisfactory results.
Acknowledgements
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'.
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