I. INTRODUCTION
X-ray stress analysis is a popular and nondestructive tool in which residual and/or applied stresses are determined from measured residual strains using Hooke’s law (Noyan and Cohen, Reference Noyan and Cohen1987). In general, the sin2 ψ method has been commonly used for specimens satisfying the following conditions: the specimens consist of small crystallites randomly oriented and are in a uniaxial stress state within an area irradiated by X-rays. If ψ is taken as the angle between the normal of a lattice plane and that of the specimen surface, a linear relation exists between 2θ and sin2 ψ, and values of the stresses presented in the specimen can be determined from the tilt angle of such a linear relation. An advantage of the sin2 ψ method is that an accurate value of the unstressed unit-cell parameter for the thin-film material is not required.
For more than 10 years, there has been increasing interest in studying residual stresses in textured polycrystalline thin films of TiN, TiC, TiCN, Al2O3, etc. For a textured film, the well-known sin2 ψ method cannot be applied, because the diffraction intensities are not continuous with respect to the ψ values. Improvements of XRD stress analysis were shown by considering the elastic property of constituent crystallites of a specimen so as to be applicable to a textured polycrystalline material, in which Bragg reflections appear at discrete angles of ψ (Clemens and Bain, Reference Clemens and Bain1992; Tanaka and Akiniwa, Reference Tanaka and Akiniwa1998; Tanaka et al., Reference Tanaka, Akiniwa, Ito and Inoue1999; Hanabusa, Reference Hanabusa1999; Welzel et al., Reference Welzel, Ligot, Lamparter, Vermeulen and Mittemeijer2005). Besides the discrete ψ angles, a new continuous φ angle was introduced in expressing the measurement direction, φ being the angle around the normal of the specimen surface, so that an XRD measurement was extended to two-dimensional. As for the φ angles, however, they are specified to three angles of 0°, 45°, and 90°, because the analytical formulae to analyze the stress components are expressed in terms of the measurements along those three angles.
Methods for the determination of values of stress components and unit-cell parameter from observed X-ray diffraction angles using the least-squares refinement (LSR) method were reported in literatures (e.g., Welzel et al., Reference Welzel, Ligot, Lamparter, Vermeulen and Mittemeijer2005; Kumar et al., Reference Kumar, Welzel and Mittemeijer2006; Winholtz and Cohen, Reference Winholtz and Cohen1988; Noyan, Reference Noyan1985). However in the techniques reported by Welzel et al., Kumar et al., and Noyan, both the stress components and the unit-cell parameter are not directly refined from the observed shifts of Bragg angles, but a number of regression lines defined as slopes of the strains observed against sin2 ψ are refined to obtain the stress components. A direct refinement of stress components except the unit-cell parameter was reported by Winholtz and Cohen (Reference Winholtz and Cohen1988).
Recently, the XRD stress analysis of a textured polycrystalline specimen has been re-evaluated by Yokoyama and Harada (Reference Yokoyama and Harada2009) in a framework of the Reuss model (Reuss, Reference Reuss1929; Mura, Reference Mura1991) of elasticity. They suggested that, since the strain evaluated from any Bragg angle has a linear relationship with the stress components, a linear LSR method can be applied to analyze the stress components by fitting the strains evaluated with the calculations based on their formula. In order to confirm the LSR method proposed by Yokoyama and Harada (Reference Yokoyama and Harada2009), biaxial residual stress components presented in a cubic TiN film specimen with a 〈111〉 texture were determined by X-ray diffraction. TiN was chosen because it is cubic with the Laue class m 3m and the space group Fm 3m (Nye, Reference Nye1957; Burns and Glazer, Reference Burns and Glazer1978). In this paper, details of the linear LSR analysis and the results on the determination of three stress components and the unstressed unit-cell parameter of the 〈111〉 textured cubic TiN thin film are reported.
II. REPRESENTATION OF STRAIN FOR BIAXIAL STRESS
Let us consider an anisotropic residual stress σij presented in a polycrystalline specimen with a rectangular shape, in which thez-axis is perpendicular to the specimen surface in the x-y plane. The stress induces anisotropic strains εij in the specimen. Since the strain, 〈ε33L〉, measured by XRD is the average distortion of lattice spacing of constituent crystallites, and it is expressed as a function of β, φ, ψ and as 〈ε33L(β, φ, ψ)〉, where φ and ψ are angles representing the measurement direction in the polar coordinates with the z-axis normal, and the angle β is half of the angle between two lattice planes such as hkl and khl about the fibre axis, the [111] axis in the present case. Those two lattice planes are in the relation of mirror symmetry.
According to the method presented by Yokoyama and Harada (Reference Yokoyama and Harada2009), strain 〈ε33L(β, φ, ψ)〉 is a linear function of applied stress components σij because of Hooke’s law. Their coefficients are given in terms of the elastic compliance constants sij, the measurement directions of β, φ, and ψ. The angles β and ψ, however, can be determined, once the unit-cell parametera 0 and the hkl indices are selected. In biaxial stress (σ11, σ22, and σ12≠0) such as the present case, the strain equation can be reduced to
where θ0 is the Bragg angle for the unstressed state, s 0 is the anisotropy index defined by s 0=s 11−s 12−s 44/2, the upper and lower suffixes, and L and 33 are the notations indicating that the observed direction corresponds to the z-axis in laboratory coordinates. For simplification, 〈ε33L(β, φ, ψ)〉 is expressed by 〈εX(β, φ, ψ)〉, where X denotes the strain determined by XRD.
III. CRYSTALLOGRAPHIC CONSIDERATION OF MEASUREMENT POINTS
Figure I shows the reciprocal space presentation of four Bragg reflections, 111, 222, 331, and 420, observable for the 〈111〉 textured TiN specimen using Cu Kα radiation. As shown in Figure 1, the (111) and (222) Bragg reflections appear as spots, while the (331) and (420) reflections are represented by continuous rings around the [111] axis in the reciprocal space for the 〈111〉 textured cubic TiN specimen. This is the case for an ideal stress-free specimen.
When the 〈111〉 textured specimen is placed in a stress field, however, the rings will be deformed due to strains induced into the crystallites in the specimen. The degree of deformation of a ring depends on the stress components σij. An interesting point to be noted from Eq. is that the deformation 〈εX(β, φ, ψ)〉 due to biaxial stress has a twofold symmetry around the [111] axis, while the crystallites are in the crystallographic threefold symmetry around the [111] axis.
IV. EXPERIMENTAL
A. Specimen preparation
In this study, a TiN thin film with a thickness of about 1 μm, which was obtained by sputtering TiN on a polyimide film of a 6 mm×40 mm surface with thickness of 125 μm, was used. This is the same specimen used in the previous XRD study (Yokoyama et al., Reference Yokoyama, Harada and Akiniwa2009). The TiN film specimen was previously confirmed to be [111] preferred oriented along the direction normal to the film surface.
The film specimen is, however, bent with a radius of curvature R≃100 mm because of the residual stress induced by the difference between the thermal expansion coefficients
Figure 1. Reciprocal space representation of a TiN film with 〈111〉 fibre texture. The 111 and 222 points are located on the [111] axis. The 331 and 420 reflections consist of continuous rings distributed around the [111] fibre axis. The two reciprocal lattice points of 111 and 222 and also the three reciprocal lattice points with φ=0°, 45°, and 90° are illustrated for each of the 331 and 420 rings. K331 is a scattering vector of the 331 reflection.
for TiN (αTiN=8.9×10−6 K−1) (Aigner et al., Reference Aigner, Lengauer, Rafaja and Ettmayer1994) and polyimide (αpolyimide=27×10−6 K−1) (Du Pont-TORAY Kapton500H).
B. XRD measurements and data reduction
The diffractometer used in this study is the same four-circle diffractometer, a Rigaku ATX-E, used in the previous study by Yokoyama et al. (Reference Yokoyama, Harada and Akiniwa2009), except with a modification of its incident beam optics. In this study a parabolic multilayer mirror, taking away the original Ge (220) channel-cut monochromator, was used to collimate the divergent X-rays from the Cu target into a quasi-parallel incident beam. This setup gives high intensities and also has high signal to noise ratios. Since the incident X-ray beam has both Cu Kα1 and Kα2 components, the observed profile of each reflection has two peaks. The instrumental resolution of this configuration was estimated to be 0.094 (0.010)° obtained from the full width at half maximum (FWHM) of a LaB6 reference standard (SRM 660a). Although the profile of each peak was rather broad, it was symmetric because of the quasi-parallel beam.
Line broadenings were detected in all observed diffraction peaks because of small particle sizes presented in the TiN film. The values of FWHM were more than two times wider than the value 0.094° for the (222) reflection and about four times larger along the direction of the (420) reflection (Yokoyama et al., Reference Yokoyama, Harada and Akiniwa2009). All the profiles observed were symmetrical so that it was relatively straightforward to strip off the Cu Kα2 peaks. The reproducibility of the 2θ angles for the observed Cu Kα1 peaks was found to be within ±0.01°.
The (111), (222), (331), and (420) reflections were selected because of their favourable geometrical conditions determined by the incident Cu Kα X-rays and by the brightness of the diffracted X-ray beams. The [111] fibre axis of the specimen was set along the z-axis of the quarter-circle goniometer so that those reflections could be observed by setting the three angles of the four-circle diffractometer to 2Θ=2θ, Ω=θ, and χ=ψ. Measurement directions can be changed by simply rotating the Θ-circle (e.g., φ=0°, 45°, and 90°).
C. XRD data analysis
The parameters determined by the linear LSR method were the three stress components (i.e., σ11, σ22, and σ12) of the biaxial stress as well as the unstressed unit-cell parameter, α0, of the TiN film. Consequently eight 2θ angles of the
Figure 2. A profile of the 420 reflection observed at φ=0° with the α1α2 doublet structure overlapped with two single profiles. The profile was separated with the ratio of intensity of Cu Kα2 line to that of Cu Kα2 line to be 2 with the Rachinger method.
(111), (222), (331), and (420) reflections were measured and used for the refinement of those four parameters. The (111) and (222) reflections were only observed at φ=0° because these lattice planes were parallel to the film surface. The lattice planes of (331) and (420) reflections were observed in the three different directions of φ=0°, 45°, and 90° at ψ=22°and 39°, respectively.
Since the incident X-ray beam had both Cu Kα1 and Kα2 components, the two observed diffraction peaks in each reflection were separated on the basis of the Rachinger correction (Rachinger, Reference Rachinger1994) by using a profile fitting program with a Lorentzian function. An example showing the separation of Cu Kα1 and Kα2 diffraction peaks of the (420) reflection observed at φ=0° is given in Figure 1. The diffraction profile is seen to be well modelled by two symmetric Lorentzian functions. The fit provided two values for the lattice spacings with d 420(Kα1)=0.946 47±0.000 10 Å for the observed 2θ=108.95° and d 420(Kα2)=0.946 47±0.000 10 Å for the observed 2θ=109.35°, where the X-ray wavelengths of 1.540 59 and 1.544 43 Å; were used for Cu Kα1 and Cu Kα2 radiations, respectively (International Tables for X-ray Crystallography, Reference Prince2004). The excellent agreement between the two d 420 spacings indicates the high quality of the data. The reflection angles, 2θ (obs.), obtained from the Cu Kα1 peaks were used as observed data and tabulated in Table I for all eight reflections.
TABLE I. Residual strain results for the cubic TiN thin film with 〈111〉 texture.
a 2θ0 (calc.), 〈εobsX〉, and 〈εX〉 are estimated by using the optimized unit-cell parameter of a 0=4.2332 Å. ΔεX=〈εobsX〉−〈εX〉.
TABLE II. Residual stress components refined by LSR.
V. ANALYSIS OF THE RESIDUAL STRESS COMPONENTS IN TiN
Equation shows that the strain, 〈εχ(β, φ, ψ)〉, is a linear function of three stress components(σ11, σ22, and σ12) and linear LSR can be used to obtain those stress components from 〈εX(β, φ, ψ)〉. However, values of 〈εX(β, φ, ψ)〉 need to be determined first from the observed Bragg angles, 2θ (obs.). An initial unit-cell parameter a 0 for the unstressed TiN film was needed to start the linear LSR analysis and the value of a 0=2433 Å from Yen et al. (Reference Yen, Toth and Shy1967) was used. The values of the strains determined from the 2θ (obs.)s were then used to obtain the stress components, σ11, σ22, and σ12, from Eq. by linear LSR. The reliability of the LSR analysis is evaluated by the reliability factor defined as
In the LSR analysis, the elastic compliance constants s 11=2.17, s 12=−0.38, and s 44=5.95/Tpa (Perry, Reference Perry1989) were used and assumed to be the same for all values of the unit-cell parameter a 0 used for TiN. This assumption is valid because the errors in the elastic compliance constants should have relatively smaller effects on the LSR results than that by the error in the value of the unit-cell parameter.
Values of the three stress components, σ11, σ22, and σ12, and the R-factor obtained by LSR using the initial unit-cell parameter of a 0=4.2433 Åare listed in the left-hand side of Table II. It should be noted that very large stress components and R-factor of 0.521 were obtained suggesting that the unit-cell parameter a 0 was incorrect. Thus, the LSR analysis was
Figure 3. R-factor plotted against the unit-cell parameter a 0. The stress components σ11, σ22, and σ12 were optimized for every unit-cell parameter plotted. The R-factor with 0.521 at the initial unit-cell parameter of a 0=4.2433 Å was much larger than that of the optimized unit-cell parameter of a 0=4.2332 Å at which the R-factor shows a minimum with 0.255.
repeated by changing the value of the unit-cell parameter, a 0, within a range of values for TiN. The LSR values of the three stress components, σ11, σ22, and σ12, as well as their R-factors determined from three other unit-cell parameters of a 0=4.2326, 4.2332, and 4.2338 Å are also listed in Table II.
The R-factors obtained from various values of a 0 are plotted in Figure 3 showing that the R-factor reaches a minimum of 0.255 at a 0=4.2332 Å. Values of 2θ0 calculated from the optimized unit-cell parameter of a 0=4.2332 Å, the observed strains 〈εobsX〉 calculated from the 2θ0 (calc.)s are also listed in Table I.
The values of observed strains, 〈εobsX〉, and least-squares refined strains, 〈εX〉, obtained with a 0=4.2433 and 4.2332 Å are plotted in Figures 4(a) and 4(b), respectively. Figure 4(a) shows large differences between the observed and the calculated strains for four out of the eight reflections. On the other hand, Figure Figure 4(b) shows very good agreements between the
Figure 4. Comparisons with the observation and calculation of strains: (a) by using the initial unit-cell parameter of a 0=4.2433 Å; (b) by using the optimized of a 0=4.2332 Å.
observed and calculated strains for all eight reflections. Consequently, the linear LSR results on the values of the three stress components obtained with a 0=4.2332 Å are more reliable than those obtained with a 0=4.2433 Å.
VI. DISCUSSION
A. Determination of stress by LSR
Values of the three residual stress components of σ11, σ22, and σ12 determined by the linear LSR method are listed in Table II. It shows that the values of the residual stress obtained with the initial unit-cell parameter of a 0=4.2433 Å are very large: σ11=2454, σ22=3285, and σ12=−2304 MPa, with STD of about 1000 MPa, and R-factor=0.521. On the other hand, the stress components obtained with the optimized unit-cell parameter, a 0=4.2332 Å, are much more reasonable with σ11=397±88, σ22=401±88, and σ12=−108±103 MPa, and a significantly lower R-factor of 0.255. A closer examination of the results listed in Table II shows that the two principal stress components σ11 and σ22 are almost the same within their STDs, and the value of the shearing stress σ12 after taking into account of its STD is relatively small.
In the previous Sec. IV B, it was stated that the reproducibility of the 2θ peak positions was within the precision of ±0.01° suggesting that the precision in which the lattice parameter can be determined within an error of Δa 0=±0.0006 Å. This means that even if lattice parameter a 0 was determined to be 4.2332 Å by the present linear LSR analysis, it is better to report the value of a 0 to be 4.2332±0.0006 Å. We also tested how the stress components differ by changing the values of the lattice parameter. For comparison, the results for two extreme cases of a 0=4.2326 (=4.2332+0.0006) Åand 4.2338 (=4.2332−0.0006) Åare shown in Table II. It is seen that the stress components refined by using those two unit-cell parameters are essentially within the STDs obtained by using a 0=4.2332 Å. This suggests that the use of the linear LSR method to obtain the optimized value of the unit-cell parameter and the stress components in the biaxial stress state worked well for the 〈111〉 textured cubic TiN film.
The minimum value of R-factor=0.255 obtained by the linear LSR analysis is significantly higher than a typical R-factor value of less than 0.10 for nonlinear LSR in X-ray crystal-structure analysis. One of the reasons could be that the number of reflections utilized in this study was only eight while there were four refined parameters. Another possible reason is the limited measurement precision of the Bragg angles of ±0.01°, and this was limited by line broadening due to small particle sizes presented in the TiN thin film.
In the XRD analysis of residual stress components shown in Sec. V, linear LSR was repeated by changing the unit-cell parameter one step at a time. This process should give essentially the same results obtained by nonlinear LSR, which simultaneously refines all three stress components together with the unit-cell parameter of the TiN film.
B. Unit-cell parameter of titanium nitride
The value of the unit-cell parameter a 0 of titanium nitride is known to be affected by lattice defects and nonstoichiometry due to diffusion of Ti and/or N atoms into or out of the titanium nitride lattice. Therefore, there is no surprise that the optimized unit-cell parameter, a 0=4.2332 Å, for the TiN film used in this study is different from that of a 0=4.2433 Å reported by Yen et al. (Reference Yen, Toth and Shy1967).
VII. CONCLUDING REMARKS
It was found from the present LSR method based on the formula given by Yokoyama and Harada (Reference Yokoyama and Harada2009) that the optimum values of the stress components and the unit-cell parameter in a biaxial stress state can be obtained for a 〈111〉 textured cubic polycrystalline material. Thus, a long standing problem in stress analysis, in which the stress-free unit-cell parameter a 0 of a compound material is always considerably difficult to determine, has been solved in the biaxial stress state within the framework of the Reuss model.
The present LSR technique can also be applicable to other noncubic specimens such as tetragonal, trigonal, or hexagonal, and with or without preferred orientation, if stress and strain formulae are provided for those systems. Although the diffraction angles observed at φ=0°, 45°, and 90° have been used in this study, there is no restriction on which set of angles to be used in a linear LSR analysis.
It is worth to note that the development of two-dimensional detectors for measurements of X-ray diffraction has made significant advances in recent years. The use of a two-dimensional detector can drastically reduce the experimental time for simultaneously recording all diffraction peaks and therefore significantly improve the speed of the residual stress analysis of a specimen (Taguchi, Reference Taguchi2008).
