research papers
Analysis of stress gradients in physical vapour deposition multilayers by Xray diffraction at fixed depth intervals
^{a}RIF e.V. – Insitut für Forschung und Transfer, JosephvonFraunhoferStrasse 20, D44227 Dortmund, Germany, and ^{b}Institute of Materials Engineering, Faculty of Mechanical Engineering, Technische Universität Dortmund, D44221 Dortmund, Germany
^{*}Correspondence email: gottfried.fischer@rifev.de
The objective of this article is to develop and apply a model for the design and evaluation of Xray diffraction experiments to measure phasespecific residual stress profiles in multilayer systems. Using synchrotron radiation and angledispersive diffraction, the stress measurements are performed on the basis of the sin^{2}ψ method. Instead of the traditional Ω or χ mode, the experiments are carried out by a simultaneous variation of the goniometer angles χ, Ω and φ^{G} to ensure that the penetration and information depth and the measuring direction φ remain unchanged when the polar angle ψ is varied. The applicability of this measuring and evaluation strategy is demonstrated by the example of a multilayer system consisting of Ti and TiAlN layers, alternately deposited on a steel substrate by means of physical vapour deposition.
Keywords: residual stress measurement; depth dependency of residual stress; constant penetration depths; multilayer systems; Ti/TiAlN.
1. Introduction
Many cutting tools are coated by means of physical vapour deposition (PVD) to enhance their lifetime by an increase in wear resistance. An important trend is to manufacture multilayer coating systems in which layers of different phases, and hence of different material characteristics, alternate. This strategy enables one to combine the advantageous mechanical properties of the phases involved within one system. The aim of the development of new multilayer systems is to find optimum parameter settings for the substrate preparation, the multilayer design (material and thickness of single layers) and the deposition.
The behaviour of multilayer systems is mainly characterized by three features: resistance against wear, hardness and residual stress. Previous investigations have shown that a high hardness is accompanied by high compressive residual stresses in the brittle ceramic layers (phases), with
coefficients (CTE) that are low compared to the CTE of metallic substrates. Additionally, the residual stresses in these layers were found to be depth dependent.The standard nondestructive and phaseselective method to measure residual stresses in regions close to the surface is Xray ; Manns & Scholtes, 2013), the diffracting volume is limited by a system of partially absorbing slit masks deposited on the specimen surface by a photolithographic process. In an alternative approach, Denks (2008) used a system of two apertures above the specimen surface to realize a gauge volume with a rhombic cross section, which is shifted through the sample to scan depth profiles of stresses (stress scanning method). The author applied an experimental setup at the energy dispersive beamline EDDI in Berlin (BESSY) for the separation of stresses in buried sublayers of a multilayer system deposited on tungsten carbide. Another group of methods provides averaged stresses since the diffracting volume is only limited by the of radiation. This group includes amongst others the multiwavelength (Eigenmann, 1990; Eigenmann et al., 1990), the scattering vector (Genzel, 1994, 1999a,b), the combined ω–χ (especially at constant penetration depth) (Kumar et al., 2006a,b; Erbacher, 2006; Erbacher et al., 2008) and the LIBAD (low incidence beam angle diffraction) methods (Van Acker et al., 1994; Mohrbacher et al., 1996). Although it is not always the case that a true Laplacian transform is applied, the averaged stresses measured by these methods are referred to as Laplace space stresses. Residual stress analysis on multilayers with alternating layer stacks of different materials was carried out for the first time by using the LIBAD method (Celis et al., 1995; Saerens et al., 1997). Recently, Klaus (2009) developed the equivalent thickness model for the deconvolution of the measured quantities in multilayer systems. This model considers the thickness, the absorption properties and the contribution to the diffraction signal of each single layer to determine the intralayer residual stresses or gradients. Applying the equivalent thickness model, Klaus and coworkers (Klaus et al., 2008; Klaus, 2009) measured residual stress profiles in multilayer systems. The current article presents an alternative model to plan and evaluate diffraction experiments in order to analyse profiles of depthdependent residual stress in multilayer systems. First, key terms and equations from the literature are summarized as required for the derivation of new model equations.
Most publications using this technique to obtain depthdependent stress profiles focus on homogeneous materials and single layers. To this end, suitable measuring and analysis strategies have already been developed. Basically, the residual stresses are determined indirectly from measured lattice strains, which are converted to stresses on the basis of Xray elastic constants. Two principles are used to measure the lattice strains: angleresolved Xray scattering using monochromatic Xrays and energyresolved scattering with white radiation. To obtain residual stresses and stress gradients in regions near the surface or in singlelayer coatings a group of methods is applied which directly provide stresses in real space. In this case, apertures are inserted into the beam to limit the diffracting volume and, thus, to avoid the averaging over gradients of the measured quantity. In the publications of Manns and Scholtes (Manns, 20101.1. Penetration depth
In Xray analysis, mainly two measures are used to characterize the degree of penetration of Xrays into a material: the τ and ξ, respectively. In the case of a homogeneous material the mathematical definition of the τ is based on Beer's law, which describes the dependency of the beam intensity I on the beam path s:
and the information depth, hereafter denominated asI_{0} denotes the beam intensity at the moment when the Xray enters the specimen and μ stands for the of the material. The intensity at the moment when the Xray leaves the specimen (hereinafter called final intensity I_{f}) can be obtained by replacing the path s in equation (1) with the sum of both the path to the site of diffraction at the depth z and the path of the diffracted partial beam to the specimen surface. Thus, one obtains the following relationship between I_{f} and the depth z (Genzel, 1999b):
Here the variable k denotes the which can be calculated from the incidence (α) and exit (β) angle, and from the gonimeter angles Ω, χ and the , according to the following equation:
This equation is valid for homogeneous specimens and specimens covered by a monolayer as well as for multilayer systems. In this work, the τ is always defined by the following condition:
The authors are aware of the fact that this socalled exp(1) criterion defines a τ that cannot be used as an appropriate measure to fix a depth to which the diffraction signal can be assigned.
Applying this generally valid condition for τ to equation (2), whose validity is limited to homogeneous specimens and monolayers, one obtains for τ
In this expression, τ specifies a depth interval , which contributes about 63% of the diffracted intensity received by the detector in an infinitely thick, homogeneous specimen. Hence, the validity of the definition of τ in equation (5) is also restricted to homogeneous specimens and monolayers with a thickness .
1.2. Information depth
Delhez et al. (1987) stressed the need for the information depth as a second measure for Xray penetration besides the τ. While τ specifies the depth interval that acts as a source of diffracted intensity, the information depth ξ is the depth to which the parameters derived in diffraction experiments (like composition, stress) can be assigned. Kumar et al. (2006b) defined the information depth as the average depth obtained upon weighting each depth z with an attenuation factor that accounts for the intensity reduction due to an attenuation of the signal originating from the depth z. According to Delhez et al. (1987) and Birkholz (2006), the information depth ξ is calculated from the weighted sum of all Xrays differing in the amount of damping that they are exposed to if diffracted in the depth interval :
The variable t denotes the thickness of a homogeneous specimen or the thickness of a monolayer. Based on the μ and the thickness t, the weight function w (k, z ) in equation (6) is defined by the following equation (see for instance Delhez et al., 1987; Birkholz, 2006):
This definition can be derived from the final intensity I_{f} (z ) given in equation (2) :
This means that the final intensities of the diffracted partial beams determine the weight function used for z averaging.
The integration from z = 0 to z = t in equation (6) and the denominator of equation (7) yields
By this definition, the information depth ξ is free of such arbitrary assumptions as are made in the case of the τ.
The limit of the information depth ξ as the thickness t of the specimen or monolayer tends to infinity is the τ:
At a low k (steep beam incidence, high penetration depth) and a low coefficient of linear attenuation μ the information depth tends to t/2 (Kumar et al., 2006b):
As already stated by Kumar et al. (2006a), equation (11) implies that the maximum achievable information depth is half the thickness of the specimen or monolayer.
1.3. Averaging depth profiles
Birkholz (2006) argues that the averaging formalism with the exponential term as the weight function, which was used to calculate the information depth ξ [see equations (6) and (7)], may be applied for any other depthdependent property g(z). He mentions, for instance, the concentration of a chemical phase, the amorphous fraction and the crystallite size. In much earlier publications this averaging formalism was applied to the gradients of lattice spacing (Delhez et al., 1987) as well as to the tensors of strain and stress (Dölle & Hauk, 1979; Noyan & Cohen, 1987).
It is important to note that the average value of a depthdependent property g(z), calculated by equation (12), corresponds to the value that is measured at a k, since the same averaging takes place during the measuring process.
1.4. Xray diffraction at constant depth intervals
Xray diffraction measurements of stress based on the method are usually performed in the Ω or χ mode. In both cases, the tilting of the specimen by the angles Ω or χ is accompanied by a permanent change in the penetration and information depths. If significant stress gradients are present beneath the surface of the material, reliable stress values can be expected only when using an evaluation procedure that takes into account the occurrence of these stress gradients. This means that the inevitable variation of the must be utilized to get information on the stress profiles. An alternative approach is to leave the τ unchanged during the course of the lattice strain measurement (variation of polar angle ψ) by appropriate combinations of the χ and Ω angles. This approach was proposed by Bonarski et al. (1994) and later used in several other studies (for instance, Kumar et al., 2006a,b; Erbacher et al., 2008; Klaus et al., 2008; Klaus, 2009). In consequence, all data points of the plot for a certain hkl reflection correspond to a mixture of signals which originate from the depth interval . To implement this strategy, the authors of the abovementioned articles had to perform three main tasks when designing Xray diffraction experiments with a constant τ to obtain residual stress depth profiles:
(1) A set of desired penetration depths , , has to be defined on the basis of the thickness t of a homogeneous specimen or a monolayer, as well as the linear attenuation coefficients of the concerned phases.
(2) For each of these penetration depths appropriate incidence–exit angle pairs and, finally, settings of the instrumental angles Ω and χ have to be specified in such a way that a variation of the angle ψ is realized by keeping the predetermined value constant at the same time.
(3) If a nonequiaxed plane stress state is expected, the goniometer angle has to be varied concurrently so that the measuring direction (azimuth angle φ) remains constant.
The desired goniometer angles Ω and χ (item 2) can be calculated from τ, μ, ψ and θ, which are usually given while planning the experiments. The angle Ω is obtained by numerically solving the following implicit equation:
While for the right side of equation (13)
generally holds true, for the expression on the left side
is only valid in the case of homogeneous materials.
Subsequently, the angle χ can be calculated from the angles ψ, θ and Ω using the equation
To compensate for the change of the measuring direction caused by the variation of the goniometer angles Ω, and χ, which is described by the equation,
the azimuthal angle φ has to be corrected by a change of the gonimeter angle according to (item 3; see also Dümmer, 1999). Consequently, Erbacher et al. (2008) named this kind of measurement the Ω–χ–φ^{G} method.
Kumar et al. (2006a,b) used the concept of constant penetration/information depth to analyse the of equiaxed stresses in Ni monolayers. In the work of Erbacher et al. (2008), this concept was applied to a strongly graded nonequiaxed stress state in a homogeneous material (highdensity alumina, α—Al_{2}O_{3}); such a stress state is typical for surface layers produced by friction loading or grinding. These authors applied the method of Laplacian transformation to obtain profiles in regions close to the surface of this ceramic.
1.5. The analysis of stress profiles in multilayer systems
Klaus and coworkers (Klaus et al., 2008; Klaus, 2009) were the first to develop model equations for studying the depthprofile distribution of residual stress in a multilayer system. They applied these equations to a multilayer consisting of three Al_{2}O_{3} layers and three TiCN layers, alternately deposited on a WC/Co substrate using chemical vapour deposition. In their model, these authors have already outlined the path that was also chosen in our work to obtain a residual stress profile in multilayer systems. However, since they started with a different expression for the final beam intensity as a function of the depth of diffraction, they derived different equations to calculate the information depth and the averaged stress profile.
2. Theoretical background of a novel stress analysis method for multilayer systems
2.1. Depth dependency of the final intensity I_{f}
The multilayer model assumed for the derivation of the equations for the final intensity consists of two materials (phases a and b) alternately deposited on a substrate (phase c; see Fig. 1a). The thicknesses of each single layer t_{ai} and t_{bi}, , and the number n of the double layers (a layer of phase b together with the next overlying layer of phase a forms a double layer) can be chosen freely. Just as in the cases of the homogeneous specimens and specimens with monolayers, the τ, the information depth ξ and the averaged stress are calculated in our multilayer model on the basis of the intensity with which the partial beam, diffracted at a depth z, leaves the specimen (final intensity). However, instead of only one final intensity–depth function I_{f} (z ) [see equation (2)], each diffracting layer i of the multilayer system and the substrate requires a function I_{i}(z ) of its own. Since we assume a multilayer system with two layer phases a and b, both of which can be the current diffracting phase, a total of two sets of intensity–depth functions I_{ai} (z ) and I_{bi} (z ) is needed. The index i denotes the number of the double layer to which the concerned diffracting layer belongs.
The index f formerly used to symbolize the final intensity in homogeneous materials is omitted to shorten the names of the variables. In the case of diffraction in the substrate (phase c), the index i is set to n. The index p is the phase index. Then
This formula was derived by taking into account the fact that the beam may pass step by step through both pure absorbing and further diffracting layers prior to entering and after leaving the diffracting layer (see Appendix A). The third factor in equation (18) describes the exponential z dependency of the final intensity of the partial beam, diffracted in the concerned layer and phase, assuming that the beam only passes through phase p. Therefore, the respective phasespecific , p = (a,b,c), is presented in this equation. For the second term in equation (18) the following three equations apply:
The variables t_{aj} and t_{bj} with stand for the layer thickness distribution of the phases a and b, respectively. The variables and in equations (19) specify a correction of the intensity loss in all pure absorbing layers above the concerned diffracting layer. Consequently, this correction depends on the difference of the attenuation coefficients and the thicknesses of the pure absorbing layers passed through by the Xray beam prior to entering and after leaving the layer of diffraction.
In the case of diffraction in the substrate, the expression for calculating the correction of the intensity loss [equation (20)] incorporates the linear attenuation coefficients of all phases and the thicknesses of all n double layers. The k in equations (18)–(20) is further calculated according to equation (3).
2.2. Penetration depths
The criterion in equation (4) which has to be met by the τ applies not only to homogeneous materials and monolayers but also to multilayer systems. However, instead of equation (5), equation (18) is now used to obtain the following formulae to calculate the phasespecific penetration depths , and :
It should be noticed that the penetration depths , and (like the depth z; see Fig. 1) are measured from the surface of the multilayer stack (cf. Fig. 5 in Appendix A).
Two cases of employing equations (21) and (22) can be distinguished:
(1) When planning diffraction experiments, the k has to be calculated for a given , specified by the phase p as well as the double layer number i of the diffracting layer. The corresponding equation must be solved for k.
(2) When evaluating diffraction experiments, the k is known and the phasespecific equation can be straightforwardly used to calculate the actual penetration depths.
2.3. Xray diffraction at constant penetration depth
The calculation of the goniometer angles Ω when planning a new experiment with a constant is now performed with modified implicit equations. Instead of equation (13) the relations with a phasespecific and layerspecific left side g_{pi} and an unchanged right side have to be used:
Depending on the diffracting phase p, for the left side of relation (23) the following equations hold true:
Equations (24)–(26) are derived from equations (21) and (22) by replacing the expression 1/k with
This general formula, which is dependent solely on the beam geometry, results from equations (3) and (16).
2.4. Weight functions w_{p}
The weight functions that are required to calculate the phasespecific information depths and the average of the stress profiles , , and , , , respectively, can be calculated in the case of multilayer systems by using the same basic equation as for homogeneous specimens and specimens with monolayers [equation (8)]. However, instead of the final intensity given in equation (2), now the phasespecific final intensities of equation (18) must be included in equation (8). Additionally, the lower limit 0 and the upper limit t of the integration have to be replaced by the phasespecific lower and upper limits u_{aj}, u_{bj}, u_{c} and v_{aj}, v_{bj}, v_{c}, respectively:
As pointed out above, the index j denotes the number of the double layer to which the concerned diffracting layer belongs. The upper limits v_{an} and v_{bn} specify the maximum depth of the concerned phase. In the case of the phases p = a and p = b, the integration in the denominator of equation (8), now denoted as C_{p}^{1}, yields the following equation:
with
In the case of phase p = c, the integration in the denominator of equation (8) with the limits u_{c} and v_{c} delivers
Using the variable C_{p}^{1} introduced in equations (31) and (34), the weight functions for the phases p = a and p = b are described by
In the case of the phase p = c a simpler equation for the weight function holds true:
2.5. Information depths ξ_{p}
In comparison to equation (6), the formulae to calculate the information depth in multilayer systems have to be modified regarding the weight functions and the limits of integration. Now for all three phases p the following equation holds true:
The lower and upper limits of integration u_{pj} and v_{pj}, and the weight functions w_{pj}, are given in equations (28)–(30), and (35) and (36), respectively. These limits and weight functions are the same as those already applied to calculate the denominators of the weight functions [cf. equations (31) and (34)]. The integration in equation (37) provides the following formula for the information depth of the phases p = a, b:
The corresponding equation in the case of the substrate phase p = c reads
In contrast to the phasespecific penetration depths given in equations (21) and (22), the information depths and are functions that are dependent on the order of the layers in the multilayer system.
2.6. Averaging the stress profile
The same limits of integration and weight functions as applied to calculate the information depth are also used to average the stress profile in the multilayer systems. For the averaged stress of all phases p the following relationship holds true:
The residual stress profile in the integrand is approximated by a damped seconddegree polynomial:
This ansatz function was used earlier by Hauk & Krug (1984). In the case of the phases p = a,b, the integration yields
The variables newly introduced in equation (42) have the following definitions:
The variables d_{pj} (p = a,b) are defined in equations (32) and (33). Equations (43)–(48) reveal that the k is indeed the only independent variable in function (42) for the averaged stress profile as expressed by the denotation on the lefthand side of equation (42).
2.7. Calculating residual stress profiles by nonlinear fitting
Since a nonlinear ansatz function for the residual stress profile was selected [equation (41)], the averaged stress also depends nonlinearly on the set of the four unknown parameters a_{i}, . The bestfit parameter set is determined in our evaluation program by an iterative minimization using the Levenberg–Marquardt method. For this, the vector of derivatives , , was calculated analytically on the basis on equations (42)–(48) and inserted in the corresponding routine of the leastsquares algorithm.
3. Experimental procedure and data analysis
3.1. Multilayer deposition and design
The three multilayer systems studied in our work consist of five metallic Ti and five ceramic TiAlN layers alternately deposited by means of PVD on substrates made of the hot working steel 1.2343 (designs 1–3 in Table 1; see also Fig. 1b). In the case of design 4, only one Ti and one TiAlN layer were deposited. For each multilayer design, the thickness of the ceramic layers and the thickness of the metallic interlayers, t_{a} and t_{b}, respectively, were always equal. Since the uniform layer thicknesses varied from design to design, the total coating thickness of our specimens t_{total} ranges from 2510 to 3000 nm. The symbol m denotes the number of stress values measured in different depth intervals. The surface pretreatment of the substrate and the PVD process are described in detail by Selvadurai et al. (2013). The following values of attenuation length were used to plan and evaluate the experiments: TiAlN, = 55 µm; Ti, = 10.75 µm; Fe, = 4.016 µm.

3.2. Diffraction experiments
The diffraction experiments were carried out at beamline G3 of the synchrotron radiation facility HASYLAB at DESY, Hamburg, Germany. The reflection 111 was chosen to analyse the residual stress profile in the TiAlN layers of the multilayer systems of designs 1–4. This reflection lies in the forwardscattering range ( = 37.4°). The measurements using an energy of E = 8.0352 keV (wavelength λ = 1.54302 Å) were performed at positive and negative values of the angle ψ. Although an equiaxed plane stress state was to be expected and although no splitting of the – curves for positive and negative ψ angles was observed in the first measurements, the goniometer angle was varied to keep the measuring direction (expressed by the azimuthal angle ) constant. While planning the experiments, the same lower limit of the range of was defined for each of the four types of multilayer design (Table 1). The upper limits were set so that a significant portion of the beam is diffracted in the substrate. The ranges of the k, also specified in Table 1, are calculated from the τ range according to equations (21) and (22). The average stress values were estimated for each using the method. The estimation was based on the assumption , since the problem of the unknown unstressed lattice parameters as a function of the depth is not solved.
3.3. Evaluation
The residual stress profiles in the TiAlN layers of the multilayer designs 1–4 were approximated by a damped firstdegree polynomial. In view of the limited number m of measured stress values (see Table 1), the reduction of the number of unknown fit parameters from four to three was considered appropriate.
4. Results and discussion
4.1. and information depth
On the basis of equations (21) and (38) and the values of the attenuation length , p = a,b,c, of our multilayer system given in §3.1, the 1/k dependency of the and information depth were calculated for designs 3 and 4. As shown in Fig. 2(b), the τ versus 1/k plots are composed of linear graphs, the slopes of which correspond to the attenuation length . Thus, the penetration τ increases more in the TiAlN layers than in the Ti layers and in the substrate. The intercept of each line depends on both the phasespecific value and the double layer number [see also equations (21) and (22)]. In the period of planning diffraction experiments, it makes sense to take the τ into consideration, which provides information on whether a usable signal can be expected from a given depth interval or phase. τ–1/k relationships like those plotted in Fig. 2 can be used to estimate suitable values of the reciprocal k^{1} and, from these, to calculate sets of the goniometer angles Ω, χ and the azimuthal angle φ.
In the case of phase c, the information depth ξ obeys a linear relationship with the slope (see Fig. 3b). This linearity results directly from equation (39). The limit of as 1/k tends to zero (lower limit ) is
Since u_{a1} = 0 holds true, is always zero for phase a (see Fig. 3b). In the case of phase b, is only low for the multilayers of designs 1–3 ( = 500 nm; Fig. 3a). For design 4, there is only one layer of phase b and the lower integration limit is relatively high (u_{b1} = 2500 nm). Hence, the lower limit is close to the corresponding value of the substrate phase c (Fig. 3b).
4.2. Measured averaged stress and stress profile
Like the penetration and information depth, the measured average stress values are plotted against the reciprocal 1/k (Fig. 4a). These data are fitted by a versus 1/k function which was calculated using the multilayer model described in §2.6 [see equation (42)].
In Fig. 4(b), the residual stress profiles according to the ansatz function in equation (41) are plotted for the TiAlN layers of designs 1–4 using the parameters a_{i}, , of a damped firstdegree polynomial. The four curves are interrupted in each z range of the Ti interlayers. The total z range of the plotted curves approximately corresponds to the selected range of the The maximum information depth that was attained for each design (see also Table 1) is marked by a vertical line in the curves. In the z range the curves are drawn as dashed lines to point at possible deviations from the actual curve progression.
With the knowledge that the maximum information depth is the weighted average of depth z over all layers of phase p (p = a,b), the reliability of the extrapolated curve progression diminishes with increasing distance .
5. Discussion
In the present work, a model that is suitable for analysing depth profiles of residual stress in multilayer systems was developed. The model equations are derived for multilayers that consist of two materials (phases) alternately deposited on a substrate. The thicknesses of each single layer and the number n of double layers can be chosen freely.
The derivation started with three phasespecific equations for the intensity I_{pi} (z) of a partial beam that is diffracted at a depth z at the moment when leaving the specimen (final intensity) [equations (18)–(20)]. As a next step, equations for the penetration depths , the goniometer angle Ω, and the weight function w_{pj} (z) [equations (21)–(26), and (35) and (36), respectively] were derived directly from the final intensity functions. In order to obtain both the phasespecific information depth [equations (38) and (39)] and the average stress [equation (42)], the weight functions were subsequently used for a weighted averaging of the depth of the diffraction z and the stress profile function . was approximated by a damped seconddegree polynomial [equation (41)]. The authors are aware of the fact that using this single residual stress profile to describe several depth distributions in each individual sublayer of a phase is only the first step. Finally, the vector of , , was calculated to enable the fitting of the measured () data pairs by the model functions using the Levenberg–Marquardt method.
The sequence of the abovementioned equations shows the importance of the expressions for the final intensity functions I_{pi}(z ), which were derived by taking the design of the multilayer into account. They form the basis for all further equations. The parameter that is calculated by using these final intensity functions (like the information depth parameter) retains its physical meaning as in the case of homogeneous materials. These parameters, however, are of minor importance as they are not required for either the graphical representation of the measured stress or the calculated stress profiles . To check the quality of the fitting, the measured stress values should always be plotted together with the fit function against the reciprocal 1/k (cf. Fig. 4a). The parameter k or 1/k proved to be the only independent variable in function (42) for the averaged stress profile. To visualize the the calculated residual stress is plotted against the depth z (Fig. 4b).
In Fig. 4(a), a strong effect of the design on the curve of the averaged stress profile is clearly recognizable. The comparison of Fig. 4(a) with Fig. 4(b) shows that for both designs the gradients of the stress profiles are higher than the gradients of the averaged stress profiles up to the maximum information depth (in first two TiAlN layers).
The advantage of our model is that it enables us to analyse the depth profiles of residual stresses without any restriction of the selection of the τ, whereas in the approach of Laplacian transformation, τ is subject to the condition . Thus, the application of the Laplacian approach is restricted to homogeneous specimens or monolayers with thicknesses much greater than the maximum attainable or desired . In our model, the data of the averaged stress that is measured at can be taken into account too. Here, t denotes the maximum depth of the actual phase of the multilayer. As a consequence, our measurements of the residual stress values in the TiAlN phase at penetration depths well into the substrate could be included in the analyses of the stress profiles presented in Fig. 4(b). In addition, it is not only possible but also useful to include measurements with penetration depths when varying the reciprocal 1/k in order to obtain a greater information depth ξ (as close as possible to the maximum value).
6. Conclusions
When planning future diffraction experiments, the distance between 1/k values should be better oriented according to the 1/k dependency of the information depth ξ. A low distance is recommended in the initial 1/k range in which the information depth increases rapidly as 1/k increases (Fig. 3a: TiAlN, Ti; Fig. 3b: TiAlN). Above this range, the distance of 1/k values can be raised. If the Xrays are diffracted far from the surface, i.e. in the substrate phase Fe (design 1–4) or in the Ti layer with a thick TiAlN layer above it (design 4, Fig. 3b), equidistant 1/k values should be chosen. The maximum value of the reciprocal should be set as high as possible. In future work, the authors will use data of nanodiffraction experiments with a submicrometre resolution like those carried out by Krywka et al. (Krywka et al., 2012, 2013; Bartosik et al., 2013) to check the depth profiles of the residual stresses presented in this article.
APPENDIX A
Derivation of the final intensity I_{f}
In the following, the final intensity I_{f} given in equation (18) will be derived using phase a in the double layer i as an example (cf. Fig. 5). To this end, the path of a beam that enters the specimen with an angle α (angle of incidence), penetrates the material to the depth z of diffraction and goes back to the surface with an angle β (exit angle) is considered. In Fig. 5 the beam trajectories are presented for the special case that the surface normal vector lies in the plane given by the incident beam and the diffracted beam. However, since the derivation is not based on this assumption, the derived expressions for the final intensity I_{f} apply without limitation to this special geometric case.
The paths of both the incident and the diffracted beam, s^{I} and s^{O}, respectively, are divided into two segments:
Segment is the beam path from the multilayer surface to the interface between the layer of phase b, belonging to the double layer i1, and the layer in the double layer i where the diffraction takes place (belonging to phase a). is the distance travelled by the diffracted beam from this interlayer back to the multilayer surface. For these paths the following equations hold true:
s^{I}_{aj}, s^{I}_{bj} and s^{O}_{aj}, s^{O}_{bj} are the segments that are passed through by the incident and diffracted beams in the layers of the phases a and b, respectively. Along the distance , the initial intensity I_{0} drops to
At the site of diffraction (the beam has additionally passed through the segment ), this intensity has further diminished:
Rearranging this expression leads to
To obtain the final intensity I_{f}, the intensity loss of the beam on its way to the surface must be taken into account:
Using the expressions
and
equation (55) for the final intensity can be further rearranged to
To introduce the depth z of the diffraction site and the layer thicknesses t_{bj}, , the segments s^{I}_{bj} and s^{I} of the beam path in equation (58) are replaced by the following expressions:
As a result, for the final intensity, the following equation holds true:
By using the definition of the geometry factors k in equation (3) one finally obtains
Acknowledgements
We thank Mr J. Donges for his support at beamline G3 at HASYLAB@DESY in Hamburg, Germany, and Mr A. Liehr for his help in planning the experiments. The funding of this research by the German Research Foundation (DFG) under contract Nos. FI 686/81 and TI 343/341 is gratefully acknowledged.
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