- Email: [email protected]
Residual stress determination in thin films by X-ray diffraction and the widespread analytical practice applying a biaxial stress model: An outdated oversimplification?
Applied Surface Science xxx (xxxx) xxx
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Residual stress determination in thin films by X-ray diffraction and the widespread analytical practice applying a biaxial stress model: An outdated oversimplification? ¨ck *, Harald Ko ¨stenbauer Peter Schoderbo Plansee SE, 6600 Reutte, Austria
A R T I C L E I N F O
A B S T R A C T
Keywords: Coatings Magnetron sputtering Method comparison Molybdenum Triaxial residual stress X-ray diffraction
Compressive stresses are commonly desired in thin films and the stress state in layers and coatings is in general evaluated assuming a biaxial nature of stress. But how thin is thick enough, or how many atom layers are necessary until this model assumption fails and a hydrostatic term joins in? In order to provide an answer molybdenum has been magnetron sputtered on display glass in layer thicknesses from 100 nm up to 2000 nm and afterwards investigated by X-ray diffraction to determine the residual stress state in the [1 1 0] fiber-textured sputtered films. It will be demonstrated that the use of a biaxial stress model might only be justified for thin layers if at all.
1. Introduction
2. Experimental details
Residual stresses are used to tailor the corresponding mechanical properties, and thereby the performance and durability of material components. In thin films they can improve the structural stiffness and strength. Furthermore, they can also damage the film, resulting in cracking in the presence of tensile stresses or in warping in the case of large compressive stresses. Many studies reported that the residual stresses in metallic films strongly depend on the deposition conditions [1–3] and the deposited film thickness [4]. Nevertheless, a general commonality of the applied determination methods is that in the ma jority of all cases, the analysis is performed simplifying the evaluation on a biaxial stress model with zero stress along the growth direction and no shear components [1,2,4,5]. But is this simplification really justified? Should the real state of stress not be treated as the sum of a hydrostatic component and of a biaxial stress [6]? Abadias and Tse [7] described a triaxial stress state in TiN hard coatings. Debelle et al. [8] also pointed to the fact that the hypothesis of a plain biaxial nature does not hold true. The present investigation, based on a systematic case study, has the same objective. The improvement of the resolution capabilities of modern laboratory diffractometers, enabling the detection of irregu larities and curvatures within the sin2Ψ plots, forces one to rethink and to adjust the model selection for evaluation.
2.1. Magnetron sputtering Molybdenum films were deposited onto display glass Corning Eagle XG substrates (quadratic discs, 5.1 × 5.1 cm2) applying such sputter conditions to achieve large film thicknesses up to 2000 nm without a failure of the coating due to the uncontrolled increase of the induced stresses (the different deposited coating thicknesses are listed later in 3. Results and Discussion). Unbalanced direct current (DC) magnetron sputtering from a Mo target (Ø100 × 8 mm, purity > 99.97%) was carried out in argon atmosphere (Ar pressure: 0.5 Pa) using a laboratory scale sputtering system. Prior to deposition the substrates have been etched using radio frequency (RF) Ar plasma for 60 s at 200 W. The distance between the target and the substrates was 59 mm. The sub strates were kept stationary during the deposition. The substrate was at room temperature and the power density at the target was 5 W/cm2. Using these conditions deposition rates of 1.39 nm/s have been ach ieved. All coating thicknesses exhibit the same typical columnar growth structure [1,4] as illustrated in Fig. 1. 2.2. X-ray diffraction X-ray diffraction has been carried out on a Bruker D8 Advance θ-θ
* Corresponding author. E-mail address: [email protected] (P. Schoderb¨ ock). https://doi.org/10.1016/j.apsusc.2020.148531 Received 16 July 2020; Received in revised form 21 September 2020; Accepted 17 November 2020 Available online 22 November 2020 0169-4332/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Peter Schoderböck, Harald Köstenbauer, Applied Surface Science, https://doi.org/10.1016/j.apsusc.2020.148531
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 1. Molybdenum deposited on display glass: SE image of the fracture cross section and the corresponding top view).
diffractometer (goniometer radius 280 mm), equipped with parallel ¨bel mirror, primary axial 2.5◦ soller, 1 mm anti-scatter beam optics (Go slit, secondary 0.2◦ collimator, 2.0 mm receiving slit) and Lynx Eye XET position sensitive detector (psd opening 2.94◦ ) using Cu-radiation (λKα = 0.15406 nm). The investigations with respect to residual stress have been realized using two different diffraction techniques. Measurements in ω mode [9] and in asymmetric diffraction [10] are feasible with the existing diffraction setup. Especially the defocusing effect correlated with the ω inclination can be circumvented by using a parallel beam geometry, which will also eliminate peak shifts by sample displacement. All experiments have been put into practice with a step size of 0.04◦ and 2 s counting time/step. In ω inclination, where ω represents the angular motion of the goniometer in the scattering plane, the detector has been used in 1D/high resolution mode. In this case the detector acts as a linear position-sensitive one with activated electronic background and CuKβ filtering. The signals from the single channels of the silicon strip are summed up when passing identic 2θ positions in a continuous scan and maximum intensity can be achieved. The measurements have been performed in 3Φ directions (0◦ ,45◦ ,90◦ ), at Ψ = 0 and with 9 positive ( +Ψ) and negative ( − Ψ) tilts respectively. The necessary negative tilts have been realized by an additional sample rotation in Φ of 180◦ . For the asymmetric scans at a fixed incidence angle α = 5◦ , the detector has been switched in 0D/high count rate mode and acts as point detector without electronic background suppression and CuKβ filtering. In general, the Xray penetration depth is calculated for depths at which the intensity of the incident radiation is attenuated to Gτ = 1–1/e ≅ 0.63. But depths which contribute about 95% or 99% to the intensities measured have already been brought up for discussion [11] (Gτ represents the fraction of the total intensity of the X-ray beam). Under the aspect of the almost complete attenuation of the beam, the X-ray penetration depth (τ) ex ceeds the dimensions (t) of the investigated coatings at all applied in clinations as well as over the whole scanned 2θ range. The sample caused attenuation, described by the exponential Lambert-Beer law I = I0 exp( − μl), is a function of the linear absorption coefficient (μMo = 1573 cm− 1) and the X-ray path length (l). The effective depth (τeff ) can be generally expressed by Eq. (1) [12] and in consequence for the applied diffraction geometries by Eqs. (2) and (3) [13,14]: ( ) ( ) ∫t − τz − τt dz ze te 0 ( ) τeff = = τ− ⎛ (1) ( )⎞ ∫ t − τz − τt ⎝ ⎠ dz e 1− e 0
− ln(1 − G )
τ τeff = ( 1 μ sin(θ+Ψ) + sin(θ−1
− ln(1 − G )
) for ω mode
limit
when
() t
τ
approaches zero. That means that especially for the ω
mode the information of thin layers is limited to a small region in the middle of the film. Already Genzel has discussed this “paradox” [12]. The observed stress value σobs (Eq. (4) [15]) becomes a weighed sum of true stresses over the penetration depth τ and the corresponding depth z. The deeper the level the less the contribution. ( ) ∫t σ(z)exp − τz dz 0 ( ) (4) σ obs = ∫ t exp − τz dz 0 The condition σ33 = 0, must be fulfilled at the free surface. This circumstance implies that a stress gradient along the growth direction is present [16]. Malhotra et al. [17,18] have confirmed this for poly crystalline Mo thin films by the application of high-resolution X-ray diffraction. Within the presented study the classical sin2Ψ method and the grazing incidence based multiple hkl-method [19] have been utilized to investigate the tendency of residual stress evolution as function of the coating dimension. The analysis of the diffraction data derived in ω mode has been realized with the software Diffracplus LEPTOS S V7.8 (Bruker AXS, Karlsruhe, Germany) using the implemented routines (peak smoothing, background correction, CuKα2 stripping, absorption and polarization correction). The peak positions have been calculated applying the sliding gravity method. The evaluation of the stress states within the classical technique is based on the solution of the residual stress Eq. (5) [20] applying least square minimization according to Winholtz and Cohen [21]. 1 1 2 2 + σ 23 sinΦsin(2Ψ) + σ33 cos2 Ψ] + S1 (σ11 + σ22 + σ 33 )}
2 2 2 εhkl ΦΨ = { S2 [σ 11 cos Φ + σ 12 sin(2Φ) + σ 22 sin Φ]sin Ψ + S2 [σ 13 cosΦsin(2Ψ)
(5) S1 and 12S2 are the diffraction elastic constants (MPa− 1), σ ii and σ ij are normal and shear stresses (MPa), Φ is the azimuthal angle (angle be tween the projected diffracting planes normal to the sample’s surface with respect to the x-axis of the sample) (◦ ) and Ψ is the tilt angle be tween the normal of the diffracting plane and the normal of the sample’s surface (◦ ). The residual stress magnitudes from the datasets measured in asymmetric diffraction geometry have been extracted by “combined analysis” exploiting the capabilities of the profile and structure analysis software TOPAS 5.0 [22] (Bruker AXS, Karlsruhe, Germany). Details about this procedure, the basic requirements and the underlying X-ray diffraction pattern decomposition according to the multiple hkl-method, its implementation with respect to the calculation of triaxial stress states considering the elastic crystal-anisotropy and the texture of the spec imen are explained in detail in preceding articles [23–25] and the cited literature therein. Mo has a bcc structure, further the elastic anisotropy is small (Zener’s ratio AZ = (C112C− 44C12 ) = 0.91). On the basis of the Voigt
(2)
Ψ)
τ ) for asymmetric diffraction τeff = ( 1 μ sin1 α + sin(2θ− α)
t 2
(3)
and Reuss schemes to average the single-crystal elastic constants for the polycrystalline behaviour, the degree of elastic anisotropy can also be ]/[ ] [ 3(AZ − 1)2 +25AZ [26]. A* rated by A* (in%) = 100 3(AZ − 1)2
For the case (τ < t) the penetration depth τ corresponds approxi mately τeff . Additionally, it can be demonstrated that τeff reaches the
equals 0 for elastic isotropic crystals and is 1 for an anisotropic crystal; 2
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 2. Comparison of the {3 2 1} reflection: (a) recorded for all investigated samples by the classical sin2Ψ technique in Φ = 0◦ and at Ψ = 0◦ , the intensity of the powder has been downscaled by factor 4 to facilitate comparison; illustration of the peak shifts at Φ = 0◦ in dependence of the tilt angle Ψ: the 200 nm Mo film (b) and the 2000 nm coating (c).
A*Mo = 0.12. The lattice parameter of the unstressed state has been determined on the initial molybdenum powder (a0 = 0.31471 nm) by a Bragg-Brentano diffraction experiment and evaluated by pattern decomposition in line with the full parameter approach [27]. Due to the specific stress nature in magnetron sputtered coatings, the general valid evaluation procedure by “combined analysis” [23,24] can be significantly simplified. In light of past experience shear stress components do not develop during the deposition process [17,18]. This is proven by the the classical sin2 Ψ method (see the corresponding results in 3. Results and Discussion). Therefore, the residual stress Eq. (5) can be reduced to a triaxial prin cipal stress situation. Further the stress state is characterized by a rotationally symmetric behavior with respect to the in-plane compo nents (σ11 = σ 22 ). With these boundary conditions it is possible to simplify Eq. (5) to Eq. (6), considering for asymmetric diffraction by Eq. (7). 1 1 Δ2θ = − 2tanθ{ S2 (σ 11 − σ 33 )sin2 Ψ + S1 (2σ 11 + σ 33 ) + S2 (σ33 )} 2 2
(6)
Ψ = (θ − α)
(7)
Δ2θ defines the residual stress induced reflection shifts to be compen sated in the refinement. The corresponding X-ray elastic constants have been calculated by the Neerfeld-Hill approach [28,29] starting from the single-crystal elastic coefficients of molybdenum (c11 = 441.6 GPa, c12 = 172.7 GPa, c44 = 121.9 GPa) as reported by Behnken and Hauk [30]. Details about and a collection of the basic equations for cubic, tetragonal, trigonal, and hexagonal Laue classes are summarized in an introductory article [31]. Polycrystalline thin films usually exhibit a macroscopic anisotropic elastic behavior. But the aforementioned tiny elastic anisotropy of molybdenum is an indicator, that despite the [1 1 0] fiber-texture in the sputtered coatings, the specimens can be treated applying macroscopic elastic quasi-isotropy [32]. The neglection of
Fig. 3. In ω tilt on the {3 2 1} reflection measured sin2ψ plots: (a) the1000 nm coating in Φ = 0◦ ( ), Φ = 45◦ ( ) and Φ = 90◦ ( ) (+ψ□), (− ψ◇); (b) sin2ψ plots inΦ = 0◦ direction for a selection of coating thicknesses: 100 nm ( ), 600 nm( ), 1500 nm ( ), 2000 nm ( ) and the initial molybdenum powder (■),) (+ψ open markers; − ψ filled markers); the corresponding linear regressions (dashed lines), all with a confidence of ≥95%.
3
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 4. The asymmetric diffraction data (α = 5◦ ): (a) for the different coating thicknesses derived patterns ( ) in comparison to the initial Mo pow der; enlargements of selected reflections in the subgraphs; offset between the single data ranges. 200/ 50 counts respectively. The corresponding decom position ( ) for the 200 nm film (b) and 2000 nm coating (c) according to “combined analysis”; dif ference curve (─, scaling factor ×2); The back ground from the glass substrate in (b) has been fitted by a split pseudo-Voigt function, goodness of fit [39] GOF: 1.3, 1.48 respectively; CuKβ marked by arrow.
direction-dependent grain interactions is justified by the resulting linear sin2Ψ functions (missing presence of elastic sample anisotropy induced wavy functions [33]). The reasonable treatment of [1 1 0] fiber-textured molybdenum as an isotropic material has been confirmed by Zaouali, et al [34]. Tanaka et al. [35] also pointed to the fact that the sin2Ψ func tions remain linear for the cases of [1 0 0], [1 1 0], and [1 1 1] fiber tex tures if the elastic anisotropy is low. The Reuss and Voigt model provide
nearly identical results. For in-plane rotationally symmetric stress states without shear components, it is also possible to reduce the experimental effort of data acquisition [23,24] for the application of the evaluation procedure “combined analysis”. Due to the fact that Eq. (6) is independent from the Φ angle, it is sufficient to measure the sample, including sample spin ning, with the side benefit of achieving an excellent grain statistic. The 4
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 5. In the unit cell embedded spherical harmonic solutions: the [1 1 0] fiber-texture in the 200 nm (a) and 2000 nm (b) sputtered coatings in comparison to the initial molybdenum powder (c) extracted from the corresponding coupled 2θ/θ scans to obtain a representative averaged overall view.
whole determination procedure becomes very timesaving. The resolu tion and separation of the in-plane contribution σ11 = σ22 and of the zcomponent σ 33 can easily be achieved by the applied X-ray diffraction patten decomposition [24,25]. Additionally, a refraction correction [13,14] (Eqs. (8a) and (8b)) must be applied, despite the quite large incidence angle: Δ2θ = δ[cotα + cot(2θ − α) + 2tanθ]
(8a)
cosαc = 1 − δ
(8b)
more affected is the error in calculation. The X-ray penetration depth and the tilt angle are directly correlated. Further, alterations in ω vary the irradiated volume segment. The cur vature is an indicator for the presence of a σ33 stress field and also visible in the measurements of Malhotra et al. [18]. The impact of σ 33 contri ¨lle butions on the shape of sin2Ψ plots has already been introduced by Do [33]. The effect of gradients in the other stress components and the associated curvatures in the sin2Ψ distributions are also illustrated there. Genzel [38] has discussed in detail the interrelation of σ 33 and the depthdependent alterations of the interplanar spacing d. The detected cur vature becomes more and more distinct with increasing coating thick ness; the thicker the coating the more pronounced becomes the stress gradient. Notice for comparison also the plot of the powder sample, which remains linear over the whole range. The trend is similar in the case of the 100 nm coating. The numeric residual stress results from the regression analysis are summarized in Table 1. The in the last decades significantly improved resolution capabilities of modern laboratory diffractometers (goniometers with large radii and high angular accu racy, detector technology etc.) enable, in combination with the basic requirements for residual stress determination (e.g. diffractometer alignment; sample preparation; choice of counting time, step size and number of tilt angles) [9] the exact determination even of tiny alter ations of the interplanar spacing. All patterns derived in asymmetric diffraction geometry are collected in Fig. 4a. The {1 1 0}, {2 1 1} and {3 2 1} reflections are enlarged in separated sub-graphs to illustrate the reflection shifts in dependence of the coating dimensions and the associated stress state. The analysis of those in asymmetric diffraction geometry derived diffraction patterns on the basis of “combined analysis” is illustrated at the example of the 200 nm and 2000 nm coating in Fig. 4b+c. The X-ray diffraction pattern decomposition procedure considers directly for texture by the applica ¨bel mirror, the tion of spherical harmonics [24]. Despite the use of a Go appearance of the CuKβ emission line (Fig. 4c) has to be considered within the implemented CuKα5 Berger emission profile [27] when the detector is operated in the high count-rate mode. Regarding X-ray diffraction pattern decomposition, it is necessary to recall the fact that the X-ray beam penetrates the complete coating thickness within the whole scanned 2θ range. As a consequence of the recording of the integral residual stress information σ obs (Eq. (4)), those by Genzel [12] described residual stress gradient induced reflection asymmetries cannot be completely compensated at the large layer di mensions. The implementation of an anisotropic peak broadening model [24] is satisfying, but insufficient to achieve a perfect profile match
αc represents the critical angle of the total external reflection (αMo = c 0.34◦ [17]) and δ introduces the decrement of the refractive index representing the dispersion for X-rays. This approach provides a suffi cient description as long as an incidence angle far away from αc has been chosen [36]. The influence of the refraction correction on the resulting stress magnitudes as a function of the incidence angle has been demonstrated by Genzel [12] and in a preceding article of one of the authors [37]. All measurements executed in grazing incidence geometry have been corrected for absorption to compensate asymmetric reflection broadening and the associated peak shifts [31]. 3. Results and discussion The results for both applied diffraction techniques are summarized in the following. A collection of the {3 2 1} reflection measured in ω tilt at Ψ = 0◦ /Φ=0◦ for the coatings investigated and the initial molybdenum √̅̅ powder is given in Fig. 2a. The figure has been created as I-plot to make the large differences in measured intensities (I) better comparable. The reflection shift for all listed coatings towards lower 2θ angles, larger d spacing is obvious even at Ψ = 0◦ . By altering the tilt angle, the residual stress induced shift of the {3 2 1} reflection can easily be uncovered and subsequently be converted into a d versus sin2Ψ plot. The manifestation of a negative slope in the sin2Ψ plots is already reflected in the general peak shift tendency (Fig. 2b+c). Due to the in-plane rotationally symmetric stress state the appearances of the sin2Ψ functions for the different measured Φ di rections are identical (see Fig. 3a). Further, it can be seen that indeed no shear-stress components are involved (missing Ψ-splitting). The sin2Ψ plots for a selected number of coating thicknesses and the initial mo lybdenum powder are collected in Fig. 3b. For the linear regressions only the sin2Ψ ranges >0.1 have been used and the regions of curvature have been excluded. But it has to be mentioned that a regression over the whole range does not change the residual stress results significantly, 5
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
[23,25]. The differences in the calculated stress magnitudes might result from the unequal resolution capabilities of both applied methods and from the parameter correlation of the principal stress components within the least square minimization procedure [23,24]. The differing integral weighting of the depth-information, averaging over the stress gradient, and the aforementioned insufficiently described reflection asymmetries also contribute to this circumstance. But both techniques show the residual stress evolution as a function of the coating di mensions in the same way. The fact that the in-plane stresses for sputtered molybdenum films exhibit in comparison to the z-component opposite sign has already been described [8] and explained by the continuous bombardment of the growing film surface with highly energetic particles. The XRD software LEPTOS provides an illustrative feature [42] to display a graphical view of the spatial residual stress distribution, the so called “lame figure”. The development from an in-plane stress state into a spherical stress distri bution can be clearly depicted (Fig. 6b). The segments on the ellipsoid lame show the principal stress values in the coordinate system of the sample. Please consider that the absolute sizes of the figures are not directly comparable, only the proportions within, reflected by the shape of the spatial functions. All samples exhibit in-plane compressive stresses. Only for the 100 nm coating the assumption of the absence of a σ33 component is justified. In this case, within both investigation methods, the error in calculation is dispro portionately large in comparison to the extracted magnitude of the zcomponent (see Table 1). Please recall also the information presented in Fig. 2b. For the thinnest investigated coating the sin2Ψ plot tend, in contrast to the others, to remain linear at small tilt angles. This fact additionally implies the absence of resolvable σ 33 contributions. In correlation with the increasing layer thickness, the compressive in-plane residual stress contributions are successively decreasing [4], while the magnitude of the tensile z-component becomes larger and finally tend to remain constant. Considering the interplay of all principal stress com ponents, the general trend of the overall residual stress state is very much towards tensile stress as the total film thickness increases. Similar results, but with much greater effort (synchrotron radiation), have been found by Malhotra et al. [17]. The mechanism of stress generation
Table 1 Summary of the residual stress results (in MPa) from the sin2Ψ method and those by pattern decomposition extracted ones including the associated uncertainties from the least-squares minimization (Levenberg–Marquardt algorithm). Mo-coating thickness
sin2Ψ σ11 =σ22
100 nm 200 nm 400 nm 600 nm 800 nm 1000 nm 1250 nm 1500 nm 1750 nm 2000 nm
− − − − − − − − − −
897 ± 60 947 ± 47 723 ± 53 675 ± 36 514 ± 36 457 ± 34 412 ± 31 350 ± 35 381 ± 31 200 ± 25
σ33
“combined analysis” σ11 =σ22
35 ± 28 105 ± 22 151 ± 27 102 ± 17 153 ± 17 140 ± 16 227 ± 14 209 ± 16 251 ± 14 202 ± 11
− − − − − − − − − −
871 ± 64 1015 ± 36 789 ± 27 690 ± 21 569 ± 29 485 ± 24 389 ± 22 361 ± 21 414 ± 20 191 ± 15
σ33
23 ± 24 117 ± 14 178 ± 11 113 ± 9 139 ± 12 174 ± 10 277 ± 10 311 ± 10 287 ± 9 288 ± 8
(notice the small corrugation in the difference plot of the 2000 nm coating in the high 2θ range, particularly pronounced on the {3 1 0}, {2 2 2} and {3 2 1} reflection (Fig. 4c)). The qualitative information of the involved texture is illustrated in Fig. 5 in terms of spherical harmonics and reflects the pronounced outof-plane [1 1 0] fiber texture within the sputtered Mo layers [4,40,41]. The resultant texture is similar for all investigated samples. This fact is also reflected in the relative intensity proportions within the diffraction patterns (Fig. 4a). A generally beneficial effect is that the more distinct the texture, the more accurate becomes the determination of the stresses [34]. The resulting residual stress magnitudes are listed in Table 1 and are additionally compared with those of the sin2Ψ method in Fig. 6. In accordance with theory, when comparing the measured peak positions with the corresponding peaks of the initial molybdenum powder, the tensile σ 33 component intensifies the peak shift of the {1 1 0} reflection towards larger d spacing (see Fig. 4a). For details concerning the impact of the stress state on the individual reflection shifts within the multiple hkl-method, it is referred to cited articles
Fig. 6. The residual stress magnitudes: (a) comparison of the results derived by the sin2Ψ method ( ) and those by “combined analysis” ( ) evaluated ones as function of the coating dimensions. (b) Illustration of the spatial organization of the stress components by the ellipsoid lame feature for a selection of coating dimensions. The directions of the acting stresses are denoted by the arrows: compressive stress (red), tensile stress (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
6
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
considering the interrelation of substrate and deposited coating, the role of entrapped Ar and the balancing of the principal stress components during the growth of thin polycrystalline Mo films have been briefly addressed by Debelle et al. [8]. Some other mechanistic aspects are discussed by Magnf¨ alt et al. [43].
[7] G. Abadias, Y.Y. Tse, Diffraction stress analysis in fiber-textured TiN thin films grown by ionbeam sputtering: Application to (001) and mixed (001)+(111) texture, J. Appl. Phys. 95 (2004) 2414–2428, https://doi.org/10.1063/1.1646444. [8] A. Debelle, G. Abadias, A. Michel, C. Jaouen, Stress field in sputtered thin films: Ion irradiation as a tool to induce relaxation and investigate the origin of growth stress, Appl. Phys. Lett. 84 (2004) 5034–5035, https://doi.org/10.1063/1.1763637. [9] M.E. Fitzpatrick, A.T. Fry, P. Holdway, F.A. Kandil, J. Shackleton, L. Suominen, Measurement Good practice guide No. 52: determination of residual stresses by Xray diffraction - Issue 2. National Physical Laboratory, Materials Science Centre, Manchester, 2005. [10] A. Baczmanski, C. Braham, W. Seiler, N. Shiraki, Multireflection method and grazing incident geometry used for stress measurement by Xray diffraction, Surf. Coat. Technol. 182 (2004) 43–54, https://doi.org/10.1016/j. surfcoat.2003.07.005. [11] V.M. Hauk, E. Macherauch, A useful guide for X-ray stress evaluation (XSE), Adv. X-ray Anal. 27 (1983) 81–99, https://doi.org/10.1154/S0376030800016992. [12] Ch. Genzel, X-ray residual stress analysis in thin films under grazing incidence – basic aspects and applications, Mat. Sci. Tech. 21 (1) (2005) 10–18, https://doi. org/10.1179/174328405X14100. [13] S. Wronski, K. Wierzbanowski, A. Baczma´ nski, A. Lodini, C. Braham, W. Seiler, Xray grazing incidence technique-corrections in residual stress measurement - a review, Powder Diffr. 24 (S1) (2009) 11–15, https://doi.org/10.1154/1.3139054. [14] S.J. Skrzypek, A. Baczma´ nski, Progress in X-ray diffraction of residual macro-stress determination related to surface layer gradients and anisotropy, Adv. X-Ray Anal. 44 (2001) 134–145. http://www.icdd.com/assets/resources/axasearch/VOL44 /v44_022.pdf. [15] L. Suominen, D. Carr, Selected methods of evaluating residual stress gradients measured by X-ray diffraction, traditional, full tensor, wavelet, Adv. X-ray Anal. 43 (2000) 21–30. http://www.icdd.com/resources/axa/vol43/v43_003.pdf. [16] I.C. Noyan, J.B. Cohen, Residual stress: measurement by diffraction and interpretation, in: B. Ilschner, N.J. Grant (Eds.), Springer Series on Materials Research and Engineering, Springer-Verlag, New York-Berlin-Heidelberg-LondonParis-Tokyo 1987, pp. 140–144. [17] S.G. Malhotra, Z.U. Rek, S.M. Yalisove, J.C. Bilello, Depth dependence of residual strains in polycrystalline Mo thin films using high-resolution X-ray diffraction, J. Appl. Phys. 79 (1996) 6872–6879, https://doi.org/10.1063/1.361509. [18] S.G. Malhotra, Z.U. Rek, S.M. Yalisove, J.C. Bilello, Analysis of thin film stress measurement techniques, Thin Solid Films 301 (1997) 45–54, https://doi.org/ 10.1016/S0040-6090(96)09569-7. [19] M. Marciszko, A. Baczma´ nski, K. Wierzbanowski, M. Wr´ obel, C. Braham, J. P. Chopart, A. Lodini, J. Bonarski, L. Tarkowski, N. Zazi, Application of multireflection grazing incidence method for stress measurements in polished AlMg alloy and CrN coating, Appl. Surf. Sci. 266 (2012) 256–267, https://doi.org/ 10.1016/j.apsusc.2012.12.005. [20] H. D¨ olle, V. Hauk, R¨ ontgenographische Spannungsermittlung für Eigenspannungssysteme allgemeiner Orientierung, H¨ art. Tech. Mitt. 31 (1976) 165–168. https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail &idt=PASCAL7780124177. [21] R.A. Winholtz, J.B. Cohen, Generalised least-squares determination of triaxial stress states by X-ray diffraction and the associated errors, Aust. J. Phys. 41 (1988) 189–199, https://doi.org/10.1071/PH880189. [22] A.A. Coelho, TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++, J. Appl. Cryst. 51 (2018) 210–218, https://doi.org/10.1107/S1600576718000183. [23] P. Schoderb¨ ock, Investigation of complex residual stress states in the near-surface region: Evaluation of the complete stress tensor by X-ray diffraction pattern decomposition, Appl. Surf. Sci. 466 (2019) 151–164, https://doi.org/10.1016/j. apsusc.2018.10.010. [24] P. Schoderb¨ ock, X-ray diffraction assessment of complex sample states on the basis of “combined analysis”, Appl. Surf. Sci. 509 (144870) (2020) 1–10, https://doi. org/10.1016/j.apsusc.2019.144870. [25] P. Schoderb¨ ock, Powder metallurgy and process induced residual stress states: a case distinction by X-ray diffraction, Materialia 12C (2020) 100714, https://doi. org/10.1016/j.mtla.2020.100714. [26] D.H. Chung, W.R. Buessem, The elastic anisotropy of crystals, J. Appl. Phys. 38 (1967) 2010–2012, https://doi.org/10.1063/1.1709819. [27] R.W. Cheary, A.A. Coelho, J.P. Cline, Fundamental parameters line profile fitting in laboratory diffractometers, J. Res. Natl. Inst. Stand. Technol. 109 (2004) 1–25. http s://www.ncbi.nlm.nih.gov/pmc/articles/PMC4849620/. [28] H. Neerfeld, Zur Spannungsberechnung aus r¨ ontgenographischen Dehnungsmessungen, Mitt. K.-Wilh.-Inst. Eisenforschg. 24 (1942) 61–70. https ://www.worldcat.org/title/zur-spannungsberechnung-aus-rontgenographische n-dehnungsmessungen/oclc/73010304&referer=brief_results. [29] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. London 65 (1952) 349–354, https://doi.org/10.1088/0370-1298/65/5/307. [30] H. Behnken, V. Hauk, Berechnung der r¨ ontgenographischen Elastizit¨ atskonstanten (REK) des Vielkristalls aus den Einkristalldaten für beliebige Kristallsymmetrie, Z. Metallkd. 77 (1986) 620–626. http://pascal-francis.inist.fr/vibad/index.php? action=getRecordDetail&idt=7756106. [31] P. Schoderb¨ ock, P. Leibenguth, M. Tkadletz, Pattern decomposition for residual stress analysis: a generalization taking into consideration elastic anisotropy and extension to higher-symmetry Laue classes, J. Appl. Cryst. 50 (2017) 1011–1020, https://doi.org/10.1107/S1600576717006616. [32] U. Welzel, J. Ligot, P. Lamparter, A.C. Vermeulen, E.J. Mittemeijer, Stress analysis of polycrystalline thin films and surface regions by X-ray diffraction, J. Appl. Cryst. 38 (2005) 1–29, https://doi.org/10.1107/S0021889804029516.
4. Conclusions For the sake of simplicity, the z-component σ33 is often neglected in the residual stress analysis of layers and coatings. But within this sys tematic study, the usual assumption of a biaxial stress nature turns out to be only applicable in individual cases and is justified only for thin layer thicknesses. Nevertheless, even in such cases, the question can be raised if a hydrostatic term is indeed not present? More sophisticated diffrac tion techniques show opposite results [17,18]. Does the detection of the σ 33 component in thin layers exceed the resolution capabilities of con ventional diffraction methods and the associated evaluation proced ures? Do the corresponding contributions get lost in the noise? However, within the investigation presented it was possible to demonstrate that the overall stress state in magnetron sputtered Mo films exceeding a certain thickness unequivocally consists of rotationally symmetric biaxial compressive in-plane stresses and a tensile hydrostatic compo nent, resulting in an overall triaxial stress situation. Author contributions P.S. carried out all X-ray diffraction experiments and the data anal ysis. H.K. was responsible for magnetron sputtering. Both authors summarized and discussed the results and wrote the manuscript. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Data availability statement The raw/processed data required to reproduce these findings cannot be shared at this time due to legal reasons (company policy). Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] S.-F. Chen, S.-J. Wang, W.-D. Lee, M.-H. Chen, C.-N. Wei, H.-Y.Y. Bor, Preparation and characterization of molybdenum thin films by direct-current magnetron sputtering, Atlas J. Mat. Sci. 2 (1) (2015) 54–59, https://doi.org/10.5147/ajms. v2i1.123. [2] H. Cao, C. Zhang, J. Chu, The effect of working gas pressure and deposition power on the properties of molybdenum films deposited by DC magnetron sputtering, Sci. China Technol. Sci. 57 (2014) 947–952, https://doi.org/10.1007/s11431-0145537-x. [3] C. Paturaud, G. Farges, M.C. Sainte Catherine, J. Machet, The influence of particle energies on the properties of magnetron sputtered tungsten films, Surf. Coat. Technol. 98 (1998) 1257–1261, https://doi.org/10.1016/S0257-8972(97) 00404-0. [4] A.M. Hofer, J. Schlacher, J. Keckes, J. Winkler, C. Mitterer, Sputtered molybdenum films: Structure and property evolution with film thickness, Vacuum 99 (2014) 149–152, https://doi.org/10.1016/j.vacuum.2013.05.018. [5] G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G.B. Thompson, E. Barthel, G. L. Doll, C.E. Murray, C.H. Stoessel, L. Martinu, Stress in thin films and coatings: Current status, challenges, and prospects, J. Vac. Sci. Technol. A 36 (2) (2018) 020801, https://doi.org/10.1116/1.5011790, 1-48. [6] J.-D. Kamminga, Th.H. de Keijser, R. Delhez, E.J. Mittemeijer, On the origin of stress in magnetron sputtered TiN layers, J. Appl. Phys. 88 (2000) 6332, https:// doi.org/10.1063/1.1319973.
7
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
[33] H. D¨ olle, The influence of multiaxial stress states, stress gradients and elastic anisotropy on the evaluation of (residual) stresses by X-rays, J. Appl. Cryst. 12 (1979) 489–501, https://doi.org/10.1107/S0021889879013169. [34] M. Zaouali, J.L. Lebrun, P. Gergaud, X-ray diffraction determination of texture and internal stresses in magnetron PVD molybdenum thin films, Surf. Coat. Technol. 50 (1) (1991) 5–10, https://doi.org/10.1016/0257-8972(91)90185-Y. [35] K. Tanaka, Y. Akiniwa, T. Ito, K. Inoue, Elastic constants and X-ray stress measurement of cubic thin films with fiber texture, Jpn. Soc. Mech. Eng. Int. J. Ser. A 42 (2) (1999) 224–234, https://doi.org/10.1299/jsmea.42.224. [36] A. Benediktovich, I. Feranchuk, A. Ulyanenkov, Theoretical concepts of X-Ray nanoscale analysis, Springer Series in Materials Science vol. 183 (2014). [37] P. Schoderb¨ ock, J. Brechbühl, Application of the whole powder pattern decomposition procedure in the residual stress analysis of layers and coatings, Thin Solid Films 31 (2015) 419–426, https://doi.org/10.1016/j.tsf.2015.05.071. [38] Ch. Genzel, A self-consistent method for X-ray diffraction analysis of multiaxial residual-stress fields in the near-surface region of polycrystalline materials,
[39] [40] [41] [42] [43]
8
I. Theoretical concept, J. Appl. Cryst. 32 (1999) 770–778, https://doi.org/ 10.1107/S0021889899005506. B.H. Toby, R Factors in Rietveld Analysis: How good is good enough, Powder Diffr. 21 (2006) 67–70, https://doi.org/10.1154/1.2179804. O.P. Karpenko, J.C. Bilello, S.M. Yalisove, Combined transmission electron microscopy and X-ray study of the microstructure and texture in sputtered Mo films, J. Appl. Phys. 76 (1994) 4610–4617, https://doi.org/10.1063/1.357295. L. Chen, P. Shimpi, T.-M. Lu, G.-C. Wang, Fiber texture of sputter deposited molybdenum films and structural zone model, Mater. Chem. Phys. 145 (2014) 288–296, https://doi.org/10.1016/j.matchemphys.2014.02.010. A. Ulyanenkov, LEPTOS: a universal software for X-ray reflectivity and diffraction, in: Proc. SPIE 5536, Advances in Computational Methods for X-Ray and Neutron Optics, (Denver, Colorado, USA, 21 October 2004), https://doi.org/10.1117/12.563302. D. Magnf¨ alt, G. Abadias, K. Sarakinos, Atom insertion into grain boundaries and stress generation in physically vapor deposited films, Appl. Phys. Lett. 103 (5) (2013) 051910, https://doi.org/10.1063/1.4817669.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Full Length Article
Residual stress determination in thin films by X-ray diffraction and the widespread analytical practice applying a biaxial stress model: An outdated oversimplification? ¨ck *, Harald Ko ¨stenbauer Peter Schoderbo Plansee SE, 6600 Reutte, Austria
A R T I C L E I N F O
A B S T R A C T
Keywords: Coatings Magnetron sputtering Method comparison Molybdenum Triaxial residual stress X-ray diffraction
Compressive stresses are commonly desired in thin films and the stress state in layers and coatings is in general evaluated assuming a biaxial nature of stress. But how thin is thick enough, or how many atom layers are necessary until this model assumption fails and a hydrostatic term joins in? In order to provide an answer molybdenum has been magnetron sputtered on display glass in layer thicknesses from 100 nm up to 2000 nm and afterwards investigated by X-ray diffraction to determine the residual stress state in the [1 1 0] fiber-textured sputtered films. It will be demonstrated that the use of a biaxial stress model might only be justified for thin layers if at all.
1. Introduction
2. Experimental details
Residual stresses are used to tailor the corresponding mechanical properties, and thereby the performance and durability of material components. In thin films they can improve the structural stiffness and strength. Furthermore, they can also damage the film, resulting in cracking in the presence of tensile stresses or in warping in the case of large compressive stresses. Many studies reported that the residual stresses in metallic films strongly depend on the deposition conditions [1–3] and the deposited film thickness [4]. Nevertheless, a general commonality of the applied determination methods is that in the ma jority of all cases, the analysis is performed simplifying the evaluation on a biaxial stress model with zero stress along the growth direction and no shear components [1,2,4,5]. But is this simplification really justified? Should the real state of stress not be treated as the sum of a hydrostatic component and of a biaxial stress [6]? Abadias and Tse [7] described a triaxial stress state in TiN hard coatings. Debelle et al. [8] also pointed to the fact that the hypothesis of a plain biaxial nature does not hold true. The present investigation, based on a systematic case study, has the same objective. The improvement of the resolution capabilities of modern laboratory diffractometers, enabling the detection of irregu larities and curvatures within the sin2Ψ plots, forces one to rethink and to adjust the model selection for evaluation.
2.1. Magnetron sputtering Molybdenum films were deposited onto display glass Corning Eagle XG substrates (quadratic discs, 5.1 × 5.1 cm2) applying such sputter conditions to achieve large film thicknesses up to 2000 nm without a failure of the coating due to the uncontrolled increase of the induced stresses (the different deposited coating thicknesses are listed later in 3. Results and Discussion). Unbalanced direct current (DC) magnetron sputtering from a Mo target (Ø100 × 8 mm, purity > 99.97%) was carried out in argon atmosphere (Ar pressure: 0.5 Pa) using a laboratory scale sputtering system. Prior to deposition the substrates have been etched using radio frequency (RF) Ar plasma for 60 s at 200 W. The distance between the target and the substrates was 59 mm. The sub strates were kept stationary during the deposition. The substrate was at room temperature and the power density at the target was 5 W/cm2. Using these conditions deposition rates of 1.39 nm/s have been ach ieved. All coating thicknesses exhibit the same typical columnar growth structure [1,4] as illustrated in Fig. 1. 2.2. X-ray diffraction X-ray diffraction has been carried out on a Bruker D8 Advance θ-θ
* Corresponding author. E-mail address: [email protected] (P. Schoderb¨ ock). https://doi.org/10.1016/j.apsusc.2020.148531 Received 16 July 2020; Received in revised form 21 September 2020; Accepted 17 November 2020 Available online 22 November 2020 0169-4332/© 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Peter Schoderböck, Harald Köstenbauer, Applied Surface Science, https://doi.org/10.1016/j.apsusc.2020.148531
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 1. Molybdenum deposited on display glass: SE image of the fracture cross section and the corresponding top view).
diffractometer (goniometer radius 280 mm), equipped with parallel ¨bel mirror, primary axial 2.5◦ soller, 1 mm anti-scatter beam optics (Go slit, secondary 0.2◦ collimator, 2.0 mm receiving slit) and Lynx Eye XET position sensitive detector (psd opening 2.94◦ ) using Cu-radiation (λKα = 0.15406 nm). The investigations with respect to residual stress have been realized using two different diffraction techniques. Measurements in ω mode [9] and in asymmetric diffraction [10] are feasible with the existing diffraction setup. Especially the defocusing effect correlated with the ω inclination can be circumvented by using a parallel beam geometry, which will also eliminate peak shifts by sample displacement. All experiments have been put into practice with a step size of 0.04◦ and 2 s counting time/step. In ω inclination, where ω represents the angular motion of the goniometer in the scattering plane, the detector has been used in 1D/high resolution mode. In this case the detector acts as a linear position-sensitive one with activated electronic background and CuKβ filtering. The signals from the single channels of the silicon strip are summed up when passing identic 2θ positions in a continuous scan and maximum intensity can be achieved. The measurements have been performed in 3Φ directions (0◦ ,45◦ ,90◦ ), at Ψ = 0 and with 9 positive ( +Ψ) and negative ( − Ψ) tilts respectively. The necessary negative tilts have been realized by an additional sample rotation in Φ of 180◦ . For the asymmetric scans at a fixed incidence angle α = 5◦ , the detector has been switched in 0D/high count rate mode and acts as point detector without electronic background suppression and CuKβ filtering. In general, the Xray penetration depth is calculated for depths at which the intensity of the incident radiation is attenuated to Gτ = 1–1/e ≅ 0.63. But depths which contribute about 95% or 99% to the intensities measured have already been brought up for discussion [11] (Gτ represents the fraction of the total intensity of the X-ray beam). Under the aspect of the almost complete attenuation of the beam, the X-ray penetration depth (τ) ex ceeds the dimensions (t) of the investigated coatings at all applied in clinations as well as over the whole scanned 2θ range. The sample caused attenuation, described by the exponential Lambert-Beer law I = I0 exp( − μl), is a function of the linear absorption coefficient (μMo = 1573 cm− 1) and the X-ray path length (l). The effective depth (τeff ) can be generally expressed by Eq. (1) [12] and in consequence for the applied diffraction geometries by Eqs. (2) and (3) [13,14]: ( ) ( ) ∫t − τz − τt dz ze te 0 ( ) τeff = = τ− ⎛ (1) ( )⎞ ∫ t − τz − τt ⎝ ⎠ dz e 1− e 0
− ln(1 − G )
τ τeff = ( 1 μ sin(θ+Ψ) + sin(θ−1
− ln(1 − G )
) for ω mode
limit
when
() t
τ
approaches zero. That means that especially for the ω
mode the information of thin layers is limited to a small region in the middle of the film. Already Genzel has discussed this “paradox” [12]. The observed stress value σobs (Eq. (4) [15]) becomes a weighed sum of true stresses over the penetration depth τ and the corresponding depth z. The deeper the level the less the contribution. ( ) ∫t σ(z)exp − τz dz 0 ( ) (4) σ obs = ∫ t exp − τz dz 0 The condition σ33 = 0, must be fulfilled at the free surface. This circumstance implies that a stress gradient along the growth direction is present [16]. Malhotra et al. [17,18] have confirmed this for poly crystalline Mo thin films by the application of high-resolution X-ray diffraction. Within the presented study the classical sin2Ψ method and the grazing incidence based multiple hkl-method [19] have been utilized to investigate the tendency of residual stress evolution as function of the coating dimension. The analysis of the diffraction data derived in ω mode has been realized with the software Diffracplus LEPTOS S V7.8 (Bruker AXS, Karlsruhe, Germany) using the implemented routines (peak smoothing, background correction, CuKα2 stripping, absorption and polarization correction). The peak positions have been calculated applying the sliding gravity method. The evaluation of the stress states within the classical technique is based on the solution of the residual stress Eq. (5) [20] applying least square minimization according to Winholtz and Cohen [21]. 1 1 2 2 + σ 23 sinΦsin(2Ψ) + σ33 cos2 Ψ] + S1 (σ11 + σ22 + σ 33 )}
2 2 2 εhkl ΦΨ = { S2 [σ 11 cos Φ + σ 12 sin(2Φ) + σ 22 sin Φ]sin Ψ + S2 [σ 13 cosΦsin(2Ψ)
(5) S1 and 12S2 are the diffraction elastic constants (MPa− 1), σ ii and σ ij are normal and shear stresses (MPa), Φ is the azimuthal angle (angle be tween the projected diffracting planes normal to the sample’s surface with respect to the x-axis of the sample) (◦ ) and Ψ is the tilt angle be tween the normal of the diffracting plane and the normal of the sample’s surface (◦ ). The residual stress magnitudes from the datasets measured in asymmetric diffraction geometry have been extracted by “combined analysis” exploiting the capabilities of the profile and structure analysis software TOPAS 5.0 [22] (Bruker AXS, Karlsruhe, Germany). Details about this procedure, the basic requirements and the underlying X-ray diffraction pattern decomposition according to the multiple hkl-method, its implementation with respect to the calculation of triaxial stress states considering the elastic crystal-anisotropy and the texture of the spec imen are explained in detail in preceding articles [23–25] and the cited literature therein. Mo has a bcc structure, further the elastic anisotropy is small (Zener’s ratio AZ = (C112C− 44C12 ) = 0.91). On the basis of the Voigt
(2)
Ψ)
τ ) for asymmetric diffraction τeff = ( 1 μ sin1 α + sin(2θ− α)
t 2
(3)
and Reuss schemes to average the single-crystal elastic constants for the polycrystalline behaviour, the degree of elastic anisotropy can also be ]/[ ] [ 3(AZ − 1)2 +25AZ [26]. A* rated by A* (in%) = 100 3(AZ − 1)2
For the case (τ < t) the penetration depth τ corresponds approxi mately τeff . Additionally, it can be demonstrated that τeff reaches the
equals 0 for elastic isotropic crystals and is 1 for an anisotropic crystal; 2
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 2. Comparison of the {3 2 1} reflection: (a) recorded for all investigated samples by the classical sin2Ψ technique in Φ = 0◦ and at Ψ = 0◦ , the intensity of the powder has been downscaled by factor 4 to facilitate comparison; illustration of the peak shifts at Φ = 0◦ in dependence of the tilt angle Ψ: the 200 nm Mo film (b) and the 2000 nm coating (c).
A*Mo = 0.12. The lattice parameter of the unstressed state has been determined on the initial molybdenum powder (a0 = 0.31471 nm) by a Bragg-Brentano diffraction experiment and evaluated by pattern decomposition in line with the full parameter approach [27]. Due to the specific stress nature in magnetron sputtered coatings, the general valid evaluation procedure by “combined analysis” [23,24] can be significantly simplified. In light of past experience shear stress components do not develop during the deposition process [17,18]. This is proven by the the classical sin2 Ψ method (see the corresponding results in 3. Results and Discussion). Therefore, the residual stress Eq. (5) can be reduced to a triaxial prin cipal stress situation. Further the stress state is characterized by a rotationally symmetric behavior with respect to the in-plane compo nents (σ11 = σ 22 ). With these boundary conditions it is possible to simplify Eq. (5) to Eq. (6), considering for asymmetric diffraction by Eq. (7). 1 1 Δ2θ = − 2tanθ{ S2 (σ 11 − σ 33 )sin2 Ψ + S1 (2σ 11 + σ 33 ) + S2 (σ33 )} 2 2
(6)
Ψ = (θ − α)
(7)
Δ2θ defines the residual stress induced reflection shifts to be compen sated in the refinement. The corresponding X-ray elastic constants have been calculated by the Neerfeld-Hill approach [28,29] starting from the single-crystal elastic coefficients of molybdenum (c11 = 441.6 GPa, c12 = 172.7 GPa, c44 = 121.9 GPa) as reported by Behnken and Hauk [30]. Details about and a collection of the basic equations for cubic, tetragonal, trigonal, and hexagonal Laue classes are summarized in an introductory article [31]. Polycrystalline thin films usually exhibit a macroscopic anisotropic elastic behavior. But the aforementioned tiny elastic anisotropy of molybdenum is an indicator, that despite the [1 1 0] fiber-texture in the sputtered coatings, the specimens can be treated applying macroscopic elastic quasi-isotropy [32]. The neglection of
Fig. 3. In ω tilt on the {3 2 1} reflection measured sin2ψ plots: (a) the1000 nm coating in Φ = 0◦ ( ), Φ = 45◦ ( ) and Φ = 90◦ ( ) (+ψ□), (− ψ◇); (b) sin2ψ plots inΦ = 0◦ direction for a selection of coating thicknesses: 100 nm ( ), 600 nm( ), 1500 nm ( ), 2000 nm ( ) and the initial molybdenum powder (■),) (+ψ open markers; − ψ filled markers); the corresponding linear regressions (dashed lines), all with a confidence of ≥95%.
3
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 4. The asymmetric diffraction data (α = 5◦ ): (a) for the different coating thicknesses derived patterns ( ) in comparison to the initial Mo pow der; enlargements of selected reflections in the subgraphs; offset between the single data ranges. 200/ 50 counts respectively. The corresponding decom position ( ) for the 200 nm film (b) and 2000 nm coating (c) according to “combined analysis”; dif ference curve (─, scaling factor ×2); The back ground from the glass substrate in (b) has been fitted by a split pseudo-Voigt function, goodness of fit [39] GOF: 1.3, 1.48 respectively; CuKβ marked by arrow.
direction-dependent grain interactions is justified by the resulting linear sin2Ψ functions (missing presence of elastic sample anisotropy induced wavy functions [33]). The reasonable treatment of [1 1 0] fiber-textured molybdenum as an isotropic material has been confirmed by Zaouali, et al [34]. Tanaka et al. [35] also pointed to the fact that the sin2Ψ func tions remain linear for the cases of [1 0 0], [1 1 0], and [1 1 1] fiber tex tures if the elastic anisotropy is low. The Reuss and Voigt model provide
nearly identical results. For in-plane rotationally symmetric stress states without shear components, it is also possible to reduce the experimental effort of data acquisition [23,24] for the application of the evaluation procedure “combined analysis”. Due to the fact that Eq. (6) is independent from the Φ angle, it is sufficient to measure the sample, including sample spin ning, with the side benefit of achieving an excellent grain statistic. The 4
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
Fig. 5. In the unit cell embedded spherical harmonic solutions: the [1 1 0] fiber-texture in the 200 nm (a) and 2000 nm (b) sputtered coatings in comparison to the initial molybdenum powder (c) extracted from the corresponding coupled 2θ/θ scans to obtain a representative averaged overall view.
whole determination procedure becomes very timesaving. The resolu tion and separation of the in-plane contribution σ11 = σ22 and of the zcomponent σ 33 can easily be achieved by the applied X-ray diffraction patten decomposition [24,25]. Additionally, a refraction correction [13,14] (Eqs. (8a) and (8b)) must be applied, despite the quite large incidence angle: Δ2θ = δ[cotα + cot(2θ − α) + 2tanθ]
(8a)
cosαc = 1 − δ
(8b)
more affected is the error in calculation. The X-ray penetration depth and the tilt angle are directly correlated. Further, alterations in ω vary the irradiated volume segment. The cur vature is an indicator for the presence of a σ33 stress field and also visible in the measurements of Malhotra et al. [18]. The impact of σ 33 contri ¨lle butions on the shape of sin2Ψ plots has already been introduced by Do [33]. The effect of gradients in the other stress components and the associated curvatures in the sin2Ψ distributions are also illustrated there. Genzel [38] has discussed in detail the interrelation of σ 33 and the depthdependent alterations of the interplanar spacing d. The detected cur vature becomes more and more distinct with increasing coating thick ness; the thicker the coating the more pronounced becomes the stress gradient. Notice for comparison also the plot of the powder sample, which remains linear over the whole range. The trend is similar in the case of the 100 nm coating. The numeric residual stress results from the regression analysis are summarized in Table 1. The in the last decades significantly improved resolution capabilities of modern laboratory diffractometers (goniometers with large radii and high angular accu racy, detector technology etc.) enable, in combination with the basic requirements for residual stress determination (e.g. diffractometer alignment; sample preparation; choice of counting time, step size and number of tilt angles) [9] the exact determination even of tiny alter ations of the interplanar spacing. All patterns derived in asymmetric diffraction geometry are collected in Fig. 4a. The {1 1 0}, {2 1 1} and {3 2 1} reflections are enlarged in separated sub-graphs to illustrate the reflection shifts in dependence of the coating dimensions and the associated stress state. The analysis of those in asymmetric diffraction geometry derived diffraction patterns on the basis of “combined analysis” is illustrated at the example of the 200 nm and 2000 nm coating in Fig. 4b+c. The X-ray diffraction pattern decomposition procedure considers directly for texture by the applica ¨bel mirror, the tion of spherical harmonics [24]. Despite the use of a Go appearance of the CuKβ emission line (Fig. 4c) has to be considered within the implemented CuKα5 Berger emission profile [27] when the detector is operated in the high count-rate mode. Regarding X-ray diffraction pattern decomposition, it is necessary to recall the fact that the X-ray beam penetrates the complete coating thickness within the whole scanned 2θ range. As a consequence of the recording of the integral residual stress information σ obs (Eq. (4)), those by Genzel [12] described residual stress gradient induced reflection asymmetries cannot be completely compensated at the large layer di mensions. The implementation of an anisotropic peak broadening model [24] is satisfying, but insufficient to achieve a perfect profile match
αc represents the critical angle of the total external reflection (αMo = c 0.34◦ [17]) and δ introduces the decrement of the refractive index representing the dispersion for X-rays. This approach provides a suffi cient description as long as an incidence angle far away from αc has been chosen [36]. The influence of the refraction correction on the resulting stress magnitudes as a function of the incidence angle has been demonstrated by Genzel [12] and in a preceding article of one of the authors [37]. All measurements executed in grazing incidence geometry have been corrected for absorption to compensate asymmetric reflection broadening and the associated peak shifts [31]. 3. Results and discussion The results for both applied diffraction techniques are summarized in the following. A collection of the {3 2 1} reflection measured in ω tilt at Ψ = 0◦ /Φ=0◦ for the coatings investigated and the initial molybdenum √̅̅ powder is given in Fig. 2a. The figure has been created as I-plot to make the large differences in measured intensities (I) better comparable. The reflection shift for all listed coatings towards lower 2θ angles, larger d spacing is obvious even at Ψ = 0◦ . By altering the tilt angle, the residual stress induced shift of the {3 2 1} reflection can easily be uncovered and subsequently be converted into a d versus sin2Ψ plot. The manifestation of a negative slope in the sin2Ψ plots is already reflected in the general peak shift tendency (Fig. 2b+c). Due to the in-plane rotationally symmetric stress state the appearances of the sin2Ψ functions for the different measured Φ di rections are identical (see Fig. 3a). Further, it can be seen that indeed no shear-stress components are involved (missing Ψ-splitting). The sin2Ψ plots for a selected number of coating thicknesses and the initial mo lybdenum powder are collected in Fig. 3b. For the linear regressions only the sin2Ψ ranges >0.1 have been used and the regions of curvature have been excluded. But it has to be mentioned that a regression over the whole range does not change the residual stress results significantly, 5
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
[23,25]. The differences in the calculated stress magnitudes might result from the unequal resolution capabilities of both applied methods and from the parameter correlation of the principal stress components within the least square minimization procedure [23,24]. The differing integral weighting of the depth-information, averaging over the stress gradient, and the aforementioned insufficiently described reflection asymmetries also contribute to this circumstance. But both techniques show the residual stress evolution as a function of the coating di mensions in the same way. The fact that the in-plane stresses for sputtered molybdenum films exhibit in comparison to the z-component opposite sign has already been described [8] and explained by the continuous bombardment of the growing film surface with highly energetic particles. The XRD software LEPTOS provides an illustrative feature [42] to display a graphical view of the spatial residual stress distribution, the so called “lame figure”. The development from an in-plane stress state into a spherical stress distri bution can be clearly depicted (Fig. 6b). The segments on the ellipsoid lame show the principal stress values in the coordinate system of the sample. Please consider that the absolute sizes of the figures are not directly comparable, only the proportions within, reflected by the shape of the spatial functions. All samples exhibit in-plane compressive stresses. Only for the 100 nm coating the assumption of the absence of a σ33 component is justified. In this case, within both investigation methods, the error in calculation is dispro portionately large in comparison to the extracted magnitude of the zcomponent (see Table 1). Please recall also the information presented in Fig. 2b. For the thinnest investigated coating the sin2Ψ plot tend, in contrast to the others, to remain linear at small tilt angles. This fact additionally implies the absence of resolvable σ 33 contributions. In correlation with the increasing layer thickness, the compressive in-plane residual stress contributions are successively decreasing [4], while the magnitude of the tensile z-component becomes larger and finally tend to remain constant. Considering the interplay of all principal stress com ponents, the general trend of the overall residual stress state is very much towards tensile stress as the total film thickness increases. Similar results, but with much greater effort (synchrotron radiation), have been found by Malhotra et al. [17]. The mechanism of stress generation
Table 1 Summary of the residual stress results (in MPa) from the sin2Ψ method and those by pattern decomposition extracted ones including the associated uncertainties from the least-squares minimization (Levenberg–Marquardt algorithm). Mo-coating thickness
sin2Ψ σ11 =σ22
100 nm 200 nm 400 nm 600 nm 800 nm 1000 nm 1250 nm 1500 nm 1750 nm 2000 nm
− − − − − − − − − −
897 ± 60 947 ± 47 723 ± 53 675 ± 36 514 ± 36 457 ± 34 412 ± 31 350 ± 35 381 ± 31 200 ± 25
σ33
“combined analysis” σ11 =σ22
35 ± 28 105 ± 22 151 ± 27 102 ± 17 153 ± 17 140 ± 16 227 ± 14 209 ± 16 251 ± 14 202 ± 11
− − − − − − − − − −
871 ± 64 1015 ± 36 789 ± 27 690 ± 21 569 ± 29 485 ± 24 389 ± 22 361 ± 21 414 ± 20 191 ± 15
σ33
23 ± 24 117 ± 14 178 ± 11 113 ± 9 139 ± 12 174 ± 10 277 ± 10 311 ± 10 287 ± 9 288 ± 8
(notice the small corrugation in the difference plot of the 2000 nm coating in the high 2θ range, particularly pronounced on the {3 1 0}, {2 2 2} and {3 2 1} reflection (Fig. 4c)). The qualitative information of the involved texture is illustrated in Fig. 5 in terms of spherical harmonics and reflects the pronounced outof-plane [1 1 0] fiber texture within the sputtered Mo layers [4,40,41]. The resultant texture is similar for all investigated samples. This fact is also reflected in the relative intensity proportions within the diffraction patterns (Fig. 4a). A generally beneficial effect is that the more distinct the texture, the more accurate becomes the determination of the stresses [34]. The resulting residual stress magnitudes are listed in Table 1 and are additionally compared with those of the sin2Ψ method in Fig. 6. In accordance with theory, when comparing the measured peak positions with the corresponding peaks of the initial molybdenum powder, the tensile σ 33 component intensifies the peak shift of the {1 1 0} reflection towards larger d spacing (see Fig. 4a). For details concerning the impact of the stress state on the individual reflection shifts within the multiple hkl-method, it is referred to cited articles
Fig. 6. The residual stress magnitudes: (a) comparison of the results derived by the sin2Ψ method ( ) and those by “combined analysis” ( ) evaluated ones as function of the coating dimensions. (b) Illustration of the spatial organization of the stress components by the ellipsoid lame feature for a selection of coating dimensions. The directions of the acting stresses are denoted by the arrows: compressive stress (red), tensile stress (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
6
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
considering the interrelation of substrate and deposited coating, the role of entrapped Ar and the balancing of the principal stress components during the growth of thin polycrystalline Mo films have been briefly addressed by Debelle et al. [8]. Some other mechanistic aspects are discussed by Magnf¨ alt et al. [43].
[7] G. Abadias, Y.Y. Tse, Diffraction stress analysis in fiber-textured TiN thin films grown by ionbeam sputtering: Application to (001) and mixed (001)+(111) texture, J. Appl. Phys. 95 (2004) 2414–2428, https://doi.org/10.1063/1.1646444. [8] A. Debelle, G. Abadias, A. Michel, C. Jaouen, Stress field in sputtered thin films: Ion irradiation as a tool to induce relaxation and investigate the origin of growth stress, Appl. Phys. Lett. 84 (2004) 5034–5035, https://doi.org/10.1063/1.1763637. [9] M.E. Fitzpatrick, A.T. Fry, P. Holdway, F.A. Kandil, J. Shackleton, L. Suominen, Measurement Good practice guide No. 52: determination of residual stresses by Xray diffraction - Issue 2. National Physical Laboratory, Materials Science Centre, Manchester, 2005. [10] A. Baczmanski, C. Braham, W. Seiler, N. Shiraki, Multireflection method and grazing incident geometry used for stress measurement by Xray diffraction, Surf. Coat. Technol. 182 (2004) 43–54, https://doi.org/10.1016/j. surfcoat.2003.07.005. [11] V.M. Hauk, E. Macherauch, A useful guide for X-ray stress evaluation (XSE), Adv. X-ray Anal. 27 (1983) 81–99, https://doi.org/10.1154/S0376030800016992. [12] Ch. Genzel, X-ray residual stress analysis in thin films under grazing incidence – basic aspects and applications, Mat. Sci. Tech. 21 (1) (2005) 10–18, https://doi. org/10.1179/174328405X14100. [13] S. Wronski, K. Wierzbanowski, A. Baczma´ nski, A. Lodini, C. Braham, W. Seiler, Xray grazing incidence technique-corrections in residual stress measurement - a review, Powder Diffr. 24 (S1) (2009) 11–15, https://doi.org/10.1154/1.3139054. [14] S.J. Skrzypek, A. Baczma´ nski, Progress in X-ray diffraction of residual macro-stress determination related to surface layer gradients and anisotropy, Adv. X-Ray Anal. 44 (2001) 134–145. http://www.icdd.com/assets/resources/axasearch/VOL44 /v44_022.pdf. [15] L. Suominen, D. Carr, Selected methods of evaluating residual stress gradients measured by X-ray diffraction, traditional, full tensor, wavelet, Adv. X-ray Anal. 43 (2000) 21–30. http://www.icdd.com/resources/axa/vol43/v43_003.pdf. [16] I.C. Noyan, J.B. Cohen, Residual stress: measurement by diffraction and interpretation, in: B. Ilschner, N.J. Grant (Eds.), Springer Series on Materials Research and Engineering, Springer-Verlag, New York-Berlin-Heidelberg-LondonParis-Tokyo 1987, pp. 140–144. [17] S.G. Malhotra, Z.U. Rek, S.M. Yalisove, J.C. Bilello, Depth dependence of residual strains in polycrystalline Mo thin films using high-resolution X-ray diffraction, J. Appl. Phys. 79 (1996) 6872–6879, https://doi.org/10.1063/1.361509. [18] S.G. Malhotra, Z.U. Rek, S.M. Yalisove, J.C. Bilello, Analysis of thin film stress measurement techniques, Thin Solid Films 301 (1997) 45–54, https://doi.org/ 10.1016/S0040-6090(96)09569-7. [19] M. Marciszko, A. Baczma´ nski, K. Wierzbanowski, M. Wr´ obel, C. Braham, J. P. Chopart, A. Lodini, J. Bonarski, L. Tarkowski, N. Zazi, Application of multireflection grazing incidence method for stress measurements in polished AlMg alloy and CrN coating, Appl. Surf. Sci. 266 (2012) 256–267, https://doi.org/ 10.1016/j.apsusc.2012.12.005. [20] H. D¨ olle, V. Hauk, R¨ ontgenographische Spannungsermittlung für Eigenspannungssysteme allgemeiner Orientierung, H¨ art. Tech. Mitt. 31 (1976) 165–168. https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail &idt=PASCAL7780124177. [21] R.A. Winholtz, J.B. Cohen, Generalised least-squares determination of triaxial stress states by X-ray diffraction and the associated errors, Aust. J. Phys. 41 (1988) 189–199, https://doi.org/10.1071/PH880189. [22] A.A. Coelho, TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++, J. Appl. Cryst. 51 (2018) 210–218, https://doi.org/10.1107/S1600576718000183. [23] P. Schoderb¨ ock, Investigation of complex residual stress states in the near-surface region: Evaluation of the complete stress tensor by X-ray diffraction pattern decomposition, Appl. Surf. Sci. 466 (2019) 151–164, https://doi.org/10.1016/j. apsusc.2018.10.010. [24] P. Schoderb¨ ock, X-ray diffraction assessment of complex sample states on the basis of “combined analysis”, Appl. Surf. Sci. 509 (144870) (2020) 1–10, https://doi. org/10.1016/j.apsusc.2019.144870. [25] P. Schoderb¨ ock, Powder metallurgy and process induced residual stress states: a case distinction by X-ray diffraction, Materialia 12C (2020) 100714, https://doi. org/10.1016/j.mtla.2020.100714. [26] D.H. Chung, W.R. Buessem, The elastic anisotropy of crystals, J. Appl. Phys. 38 (1967) 2010–2012, https://doi.org/10.1063/1.1709819. [27] R.W. Cheary, A.A. Coelho, J.P. Cline, Fundamental parameters line profile fitting in laboratory diffractometers, J. Res. Natl. Inst. Stand. Technol. 109 (2004) 1–25. http s://www.ncbi.nlm.nih.gov/pmc/articles/PMC4849620/. [28] H. Neerfeld, Zur Spannungsberechnung aus r¨ ontgenographischen Dehnungsmessungen, Mitt. K.-Wilh.-Inst. Eisenforschg. 24 (1942) 61–70. https ://www.worldcat.org/title/zur-spannungsberechnung-aus-rontgenographische n-dehnungsmessungen/oclc/73010304&referer=brief_results. [29] R. Hill, The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc. London 65 (1952) 349–354, https://doi.org/10.1088/0370-1298/65/5/307. [30] H. Behnken, V. Hauk, Berechnung der r¨ ontgenographischen Elastizit¨ atskonstanten (REK) des Vielkristalls aus den Einkristalldaten für beliebige Kristallsymmetrie, Z. Metallkd. 77 (1986) 620–626. http://pascal-francis.inist.fr/vibad/index.php? action=getRecordDetail&idt=7756106. [31] P. Schoderb¨ ock, P. Leibenguth, M. Tkadletz, Pattern decomposition for residual stress analysis: a generalization taking into consideration elastic anisotropy and extension to higher-symmetry Laue classes, J. Appl. Cryst. 50 (2017) 1011–1020, https://doi.org/10.1107/S1600576717006616. [32] U. Welzel, J. Ligot, P. Lamparter, A.C. Vermeulen, E.J. Mittemeijer, Stress analysis of polycrystalline thin films and surface regions by X-ray diffraction, J. Appl. Cryst. 38 (2005) 1–29, https://doi.org/10.1107/S0021889804029516.
4. Conclusions For the sake of simplicity, the z-component σ33 is often neglected in the residual stress analysis of layers and coatings. But within this sys tematic study, the usual assumption of a biaxial stress nature turns out to be only applicable in individual cases and is justified only for thin layer thicknesses. Nevertheless, even in such cases, the question can be raised if a hydrostatic term is indeed not present? More sophisticated diffrac tion techniques show opposite results [17,18]. Does the detection of the σ 33 component in thin layers exceed the resolution capabilities of con ventional diffraction methods and the associated evaluation proced ures? Do the corresponding contributions get lost in the noise? However, within the investigation presented it was possible to demonstrate that the overall stress state in magnetron sputtered Mo films exceeding a certain thickness unequivocally consists of rotationally symmetric biaxial compressive in-plane stresses and a tensile hydrostatic compo nent, resulting in an overall triaxial stress situation. Author contributions P.S. carried out all X-ray diffraction experiments and the data anal ysis. H.K. was responsible for magnetron sputtering. Both authors summarized and discussed the results and wrote the manuscript. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Data availability statement The raw/processed data required to reproduce these findings cannot be shared at this time due to legal reasons (company policy). Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References [1] S.-F. Chen, S.-J. Wang, W.-D. Lee, M.-H. Chen, C.-N. Wei, H.-Y.Y. Bor, Preparation and characterization of molybdenum thin films by direct-current magnetron sputtering, Atlas J. Mat. Sci. 2 (1) (2015) 54–59, https://doi.org/10.5147/ajms. v2i1.123. [2] H. Cao, C. Zhang, J. Chu, The effect of working gas pressure and deposition power on the properties of molybdenum films deposited by DC magnetron sputtering, Sci. China Technol. Sci. 57 (2014) 947–952, https://doi.org/10.1007/s11431-0145537-x. [3] C. Paturaud, G. Farges, M.C. Sainte Catherine, J. Machet, The influence of particle energies on the properties of magnetron sputtered tungsten films, Surf. Coat. Technol. 98 (1998) 1257–1261, https://doi.org/10.1016/S0257-8972(97) 00404-0. [4] A.M. Hofer, J. Schlacher, J. Keckes, J. Winkler, C. Mitterer, Sputtered molybdenum films: Structure and property evolution with film thickness, Vacuum 99 (2014) 149–152, https://doi.org/10.1016/j.vacuum.2013.05.018. [5] G. Abadias, E. Chason, J. Keckes, M. Sebastiani, G.B. Thompson, E. Barthel, G. L. Doll, C.E. Murray, C.H. Stoessel, L. Martinu, Stress in thin films and coatings: Current status, challenges, and prospects, J. Vac. Sci. Technol. A 36 (2) (2018) 020801, https://doi.org/10.1116/1.5011790, 1-48. [6] J.-D. Kamminga, Th.H. de Keijser, R. Delhez, E.J. Mittemeijer, On the origin of stress in magnetron sputtered TiN layers, J. Appl. Phys. 88 (2000) 6332, https:// doi.org/10.1063/1.1319973.
7
P. Schoderb¨ ock and H. K¨ ostenbauer
Applied Surface Science xxx (xxxx) xxx
[33] H. D¨ olle, The influence of multiaxial stress states, stress gradients and elastic anisotropy on the evaluation of (residual) stresses by X-rays, J. Appl. Cryst. 12 (1979) 489–501, https://doi.org/10.1107/S0021889879013169. [34] M. Zaouali, J.L. Lebrun, P. Gergaud, X-ray diffraction determination of texture and internal stresses in magnetron PVD molybdenum thin films, Surf. Coat. Technol. 50 (1) (1991) 5–10, https://doi.org/10.1016/0257-8972(91)90185-Y. [35] K. Tanaka, Y. Akiniwa, T. Ito, K. Inoue, Elastic constants and X-ray stress measurement of cubic thin films with fiber texture, Jpn. Soc. Mech. Eng. Int. J. Ser. A 42 (2) (1999) 224–234, https://doi.org/10.1299/jsmea.42.224. [36] A. Benediktovich, I. Feranchuk, A. Ulyanenkov, Theoretical concepts of X-Ray nanoscale analysis, Springer Series in Materials Science vol. 183 (2014). [37] P. Schoderb¨ ock, J. Brechbühl, Application of the whole powder pattern decomposition procedure in the residual stress analysis of layers and coatings, Thin Solid Films 31 (2015) 419–426, https://doi.org/10.1016/j.tsf.2015.05.071. [38] Ch. Genzel, A self-consistent method for X-ray diffraction analysis of multiaxial residual-stress fields in the near-surface region of polycrystalline materials,
[39] [40] [41] [42] [43]
8
I. Theoretical concept, J. Appl. Cryst. 32 (1999) 770–778, https://doi.org/ 10.1107/S0021889899005506. B.H. Toby, R Factors in Rietveld Analysis: How good is good enough, Powder Diffr. 21 (2006) 67–70, https://doi.org/10.1154/1.2179804. O.P. Karpenko, J.C. Bilello, S.M. Yalisove, Combined transmission electron microscopy and X-ray study of the microstructure and texture in sputtered Mo films, J. Appl. Phys. 76 (1994) 4610–4617, https://doi.org/10.1063/1.357295. L. Chen, P. Shimpi, T.-M. Lu, G.-C. Wang, Fiber texture of sputter deposited molybdenum films and structural zone model, Mater. Chem. Phys. 145 (2014) 288–296, https://doi.org/10.1016/j.matchemphys.2014.02.010. A. Ulyanenkov, LEPTOS: a universal software for X-ray reflectivity and diffraction, in: Proc. SPIE 5536, Advances in Computational Methods for X-Ray and Neutron Optics, (Denver, Colorado, USA, 21 October 2004), https://doi.org/10.1117/12.563302. D. Magnf¨ alt, G. Abadias, K. Sarakinos, Atom insertion into grain boundaries and stress generation in physically vapor deposited films, Appl. Phys. Lett. 103 (5) (2013) 051910, https://doi.org/10.1063/1.4817669.