Capability of X-ray diffraction for the study of microstructure of metastable thin films

The capability of X-ray diffraction for the microstructure investigations of metastable systems is illustrated on supersaturated and partially decomposed thin films of titanium aluminium nitrides with high aluminium content. The anisotropy of the elastic constants and their role in these investigations is discussed.


Introduction
Thin films play an increasingly important role in contemporary life. They are used in electronics, optics, chemistry, as well as in energy and engineering applications. In the field of electronics, thin film technology is employed for the production of integrated circuits, memories, conductors, isolators and diffusion barriers. In the field of optics, thin films are utilized as reflection, antireflection and decorative coatings, for manufacturing of semiconductor lasers and for optical recording. The majority of magnetic and gas sensors, solar cells and batteries are nowadays also produced in the form of thin films. The engineering applications are dominated by wear and corrosion-resistant coatings, which typically improve the hardness and/or friction coefficient of the substrate materials.
Many applications exploit the outstanding properties of thin-film nanocomposites, which can be tailored to be superior in comparison with the properties of the individual constituents. Such nanocomposites consist of at least two nanoscaled phases, which are formed either during the deposition process or after the thin-film deposition. Numerous methods for the production of thin-film nanocomposites are based on the deposition of metastable compounds, which decompose in the thermodynamically stable phases either during the deposition process (Rafaja et al., 2012) or during the subsequent annealing Haas et al., 2013). The deposition of metastable compounds utilizes a limited mobility of the deposited atoms (adatoms), which can easily be manipulated through ion bombardment during the deposition (Sundgren et al., 1983;Petrov et al., 1989). In general, the microstructure and thus the physical properties of thins films can be reliably modified through the deposition parameters.
Although thin-film research was always supported by transmission electron microscopy, some microstructure features can be much better or more reliably determined using X-ray diffraction (XRD). The main advantage of XRD is its excellent accuracy regarding the interplanar spacing measurement, which predestines XRD for a precise quantification of the micromechanical behaviour of thin films. Furthermore, XRD offers statistically relevant microstructure information, as the XRD measurements are performed on relatively large sample volumes. In this contribution, the capability of XRD for the microstructure analysis of thin films is illustrated on the example of supersaturated and partially decomposed (Ti, Al)N solid solutions. The main topics of this study are the assessment of the stability of the supersaturated (Ti,Al)N, the estimation of the degree of its decomposition and the description of the effect of the partial phase segregation on the micromechanical properties of the thin films.
(Ti, Al)N thin films were first produced for electronic applications in 1970s (Wasa & Hayakawa, 1972), but a breakthrough for (Ti, Al)N thin films came with their application as oxidation-resistant, hard protective coatings (Knotek et al., 1986;Mü nz, 1986). Since that time, (Ti, Al)N has been investigated as a model metastable system. The first attempt to describe the phase stability of (Ti,Al)N was made by Cremer et al. (1998), who reported that the metastable (Ti,Al)N with a cubic face-centered crystal structure of the NaCl-type exists up to approximately 65 mol% of AlN in TiN and remains metastable up to approximately 973 K. At higher temperatures, (Ti, Al)N decomposes into Ti-rich fcc-(Ti, Al)N and Alrich w-(Al, Ti)N having a wurtzitic crystal structure. Later on, the stability of the metastable supersaturated (Ti, Al)N solid solution was assessed by using ab initio methods assuming that fcc-(Ti, Al)N decomposes spinodally into fcc-TiN and fcc-AlN (Mayrhofer et al., 2006;Alling et al., 2007). The spinodal decomposition was suggested to be followed by the transformation of fcc-AlN into its thermodynamically stable wurtzitic form (see e.g. Rachbauer et al., 2011). Still, recent studies have shown that the spinodal decomposition followed by the w-AlN formation is not necessarily the sole mechanism of the fcc-(Ti,Al)N transformation. Alternatively, a direct formation of w-(Al, TiN) from fcc-(Ti, Al)N driven by the presence of stacking faults in fcc-(Ti, Al)N was considered as a competing process to the spinodal decomposition of fcc-(Ti,Al)N (Rafaja et al., 2014).
In the present work, the phase composition, size and preferred local orientation of nanocrystallites and the micromechanical properties of supersaturated solid solutions with the overall chemical compositions Ti 0.53 Al 0.47 N and Ti 0.44 Al 0.56 N are correlated with the aluminium content and with the adatom mobility. The surface mobility of the adatoms was modified by the kinetic energy of the impinging ions, which were accelerated by the bias voltage applied during the cathodic arc evaporation (CAE) process. Furthermore, the effect of the internal lattice strain on the stabilization of the metastable fcc-(Ti, Al)N is discussed. The internal lattice strain affects the deformation energy which is, according to Cahn (1961), a factor influencing the Helmholtz free energy and thus the phase stability of metastable fcc-(Ti,Al)N. Finally, this study illustrates how the microstructure parameters and the micromechanical characteristics determined using XRD can be employed to identify the particular decomposition mechanisms in metastable compounds.

Experimental details
The (Ti, Al)N thin films under study were deposited in an industrial scale CAE facility of the Balzers RCS-type (Durand-Drouhin et al., 2003). As substrates for the deposition, polished cemented carbide cutting inserts (SNUN-type, grade S40T) were used. The metallic atoms were sputtered from Ti-Al targets (PLANSEE Composite Materials GmbH), which were produced by a powder metallurgical route (Korb, 1988). Recently, it was shown that the CAE process activates the formation of intermetallic phases in powder metallurgically processed targets (Rafaja, Polzer et al., 2011); thus these targets behave like the conventional cast targets. The chemical compositions of the targets were 50 at.% Ti:50 at.% Al and 40 at.% Ti:60 at.% Al, respectively. According to the results reported by Mayrhofer et al. (2006) and Alling et al. (2007), these aluminium concentrations were chosen to be within the spinodal decomposition region at the deposition temperature of approximately 723 K and below the AlN solubility in fcc-TiN (65.3 mol% of AlN in fcc-TiN) reported by Makino et al. (2005). Nitrogen was supplied directly from the deposition atmosphere, which was pure nitrogen with a working pressure of 3.2 Pa. The thickness of the thin films was about 2 mm.
Glow discharge optical emission spectroscopy (GDOES) performed on the deposited samples confirmed that the (Ti, Al)N films contained 50 at.% nitrogen as expected. The [Ti]/[Al] ratio was slightly higher in the thin films than in the targets. The energy of the ion impact on the surface of the deposited films, which along with the chemical composition was the main factor influencing the microstructure of the thin films, was adjusted by the bias voltage. The bias voltage (U B ) was set to À40, À80 and À120 V in the respective deposition runs. In this range, the bias voltage has no effect on the chemical composition of the thin films measured using GDOES.
The X-ray diffraction experiments were performed in the glancing-angle diffraction mode (GAXRD) on the Rossendorf beamline (ROBL@BM20) at the European Synchrotron Radiation Facility (Grenoble). The measurements were carried out at a wavelength of 0.8857 Å . The data were collected using a scintillation detector, in front of which a Soller collimator was located. The angular range was 17-70 in 2, the step size 0.05 in 2. This angular range corresponds to the range of the diffraction vector 4 sin = = 2.09-8.14 Å À1 .
The main reason for using GAXRD geometry for the thinfilm experiments is to reduce the penetration depth of X-rays into the volume of the sample. When the penetration depth of the radiation is reduced, the diffraction signal from the thin film is more intense than the diffraction signal from the bulky substrate. In general, the penetration depth of X-rays is defined as the path of the radiation through the sample, which causes a decrease in the X-ray intensity to 1/e ' 37% of the primary intensity. Analogously in thin films measured using GAXRD geometry, the penetration depth of X-rays is defined as the distance of an infinitesimal slab of the material under study from the sample surface which delivers, after absorption, intensity that is equal to 1/e of the non-absorbed intensity. As the path of the X-rays in a thin film depends on the incident () and outgoing angle (2 À ), the penetration depth can be written as (see e.g. Rafaja et al., 1997): where 2 is the diffraction angle and is the mean linear absorption coefficient of the thin film. Other advantages of the GAXRD geometry are a weak dependence of the penetration depth on the diffraction angle and a concurrent strong decrease of the penetration depth with decreasing angle of incidence (Valvoda et al., 1990).
In GAXRD experiments, it is reasonable to choose such an angle of incidence, for which the penetration depth is comparable with the film thickness, unless depth-resolved measurements are intended (Rafaja et al., 1997). If the selected penetration depth is comparable with the film thickness, the information obtained from XRD is averaged over the whole thin film, while the signal from the substrate is sufficiently suppressed. In our experiments, the angle of incidence of the primary beam on the sample surface was 0.5 . This angle of incidence reduced the penetration depth of the radiation with a wavelength of 0.8857 Å below 1.5 mm as calculated according to equation (1) with the linear absorption coefficient of 59.5 cm À1 . A smaller angle of incidence would further reduce the penetration depth, but the surface roughness of the samples would unnecessarily diminish the diffracted intensities (Suortti, 1972;Hermann & Ermrich, 1987).
The individual XRD lines from the measured diffraction patterns were fitted by the Pearson VII functions (Hall et al., 1977) in order to obtain the line positions and the line broadening needed for further analyses. For selected samples, the diffraction patterns were subjected to Le Bail analysis (Le Bail et al., 1988) and to Rietveld refinement (Rietveld, 1967(Rietveld, , 1969) by using the MAUD routine (Ferrari et al., 1996;Lutterotti et al., 1999).

Results and discussion
3.1. Effect of the chemical composition and mobility of the deposited atoms on the phase composition of metastable (Ti, Al)N thin films The XRD patterns of the samples with the overall chemical composition Ti 0.53 Al 0.47 N ( Fig. 1) show mainly the diffraction lines from fcc-(Ti, Al)N and hexagonal tungsten carbide (hex-WC) which is the main constituent of the substrates. In samples deposited at U B = À80 and À120 V, the presence of minor Al-rich fcc-(Al, Ti)N was indicated by weak diffraction maxima located at higher diffraction angles as compared with the positions of the XRD lines from the major phase, i.e. fcc- (   stay nearly symmetric, because the structure factors of the diffraction lines from fcc-(Al, Ti)N with odd diffraction indices are low (Rafaja et al., 2014). Distinct XRD lines from w-(Al, Ti)N were not observed. Instead, maxima of the diffuse scattering appeared near the diffraction angles which correspond to the crystalline w-(Al, Ti)N. These line positions are labelled by short bars located at the bottom of Fig. 1. Fig. 1 also illustrates a larger XRD line broadening in the samples deposited at U B = À80 and À120 V in comparison with the line broadening observed in the sample deposited at U B = À40 V. The maximum of the XRD line broadening is reached in the sample deposited at U B = À80 V. In general, a higher bias voltage makes the ion impact more intense. A higher energy for the ion impact diminishes the mean free path of the deposited atoms on the surface and thus the adatom mobility, but it simultaneously heats up the sample surface that increases the surface diffusivity of the adatoms (Oura et al., 2003) at the highest bias voltages. This interference of the involved phenomena has already been discussed by Wü stefeld et al. (2010), who showed that the minimum for the surface mobility occurs at U B = À80 V in this particular deposition system.
In samples with the overall chemical composition Ti 0.44 Al 0.56 N (Fig. 3), the XRD lines from fcc-(Ti, Al)N and hex-WC are complemented by broad diffraction maxima from w-(Al, Ti)N. The most pronounced diffraction maximum from w-(Al, Ti)N can be found at around 20 in 2. In samples deposited at U B = À80 and À120 V, a similar asymmetry of the fcc diffraction lines with even diffraction indices was observed as in the samples of Ti 0.53 Al 0.47 N. The simulation of the diffraction pattern for the wurtzitic structure and a subsequent Rietveld refinement using the Le Bail fit (Le Bail et al., 1988) in MAUD (Ferrari et al., 1996;Lutterotti et al., 1999) performed for sample Ti 0.44 Al 0.56 N deposited at U B = À40 V revealed that the strongest (and narrowest) diffraction lines from w-(Al, Ti)N are 100, 002, 110, 112, 210, 114 and 300 (cf. Fig. 4). These diffraction indices fulfil the relationship h = k or l = 0. According to Warren (1969), the diffraction lines with these diffraction indices are not broadened by the stacking faults, which are randomly distributed on the hexagonal lattice planes f001g. The apparently strong diffraction lines 103, 203 and 212 imitate the diffuse scattering from the fcc structure.
The formation of stacking faults in w-(Al,Ti)N accompanies a direct transition between fcc-(Ti, Al)N and w-(Al, Ti)N as reported by Rafaja et al. (2014). This transformation mechanism, which is facilitated by the stacking faults located on the wurtzitic lattice planes f001g and on the fcc lattice planes f111g, requires the orientation relationship 11 " 1 1 À Á fcc jj 002 ð Þ w and 1 " 1 10 Â Ã fcc jj 100 ½ w (or symmetrically equivalent ones) between the cubic and the wurtzitic counterparts. These orientation relationships were previously found in similar (Ti,Al)N thin films by using local fast Fourier transformation of high-resolution transmission electron micrographs . Analogue transformation pathways for the phase transitions in AlN were reported recently by Schmerler & Kortus (2014).
Furthermore, the Rietveld refinement of the diffraction pattern measured in sample Ti 0.44 Al 0.56 N deposited at U B = À40 V revealed the lattice parameters of w-(Al, Ti)N a w = 3.033 (5)  Diffraction patterns of samples with the overall chemical composition Ti 0.44 Al 0.56 N, which were deposited at U B = À40 (bottom), À80 (middle) and À120 V (top). The meaning of the vertical lines and the bars is the same as in Fig. 1. lattice parameters of w-AlN, a w = 3.11197 (2) and c w = 4.98089 (4) Å (Paszkowicz et al., 2004), the lattice parameter a w measured in this thin film is smaller, whereas c w is larger. A similar distortion of the elementary cell was reported by Christensen & Gorczyca (1994) for w-AlN under hydrostatic pressure, who gave a w = 3.081 and c w = 5.031 Å . In our case, the reduction of a w is caused by the intergrowth of the cubic and wurtzitic regions with the mutual orientation 11 " 1 1 À Á fcc jj 002 ð Þ w and 1 " 1 10 Â Ã fcc jj 100 ½ w and with the habitus planes 111 f g fcc and 001 f g w . This kind of heteroepitaxy implies a convergence of the interatomic distances within the habitus plane of the fcc and wurtzitic structures, which are equal to a fcc (2) À1/2 and a w for the respective crystal structure as calculated from the distances between the metallic atoms (Rafaja et al., 2014). As a fcc (2) À1/2 of (Ti,Al)N is always less than a w [even for Al-free fcc-TiN, where a fcc = 4.242 Å and a fcc (2) À1/2 = 3.000 Å ], the elementary cell of w-AlN is compressed in the hhk0i w directions and expands in the nonconstrained [001] w direction.
Another consequence of such intergrowth of the cubic and wurtzitic domains is the relatively high observed intensity of the diffraction line 002 from w-(Al, Ti)N, cf. Figs. 3 and 4. The high intensity of this diffraction line stems from the transient regions, in which the local crystal structure alternates between fcc-(Ti, Al)N and w-(Al, Ti)N and which consequently contain a high density of the stacking faults on the lattice planes f001g w and f111g fcc . In such regions, the translation periodicity is almost broken with the exception of the h001i w and h111i fcc directions, in which the heteroepitaxially grown cubic and wurtzitic domains are mutually coherent and produce an enhanced coherently diffracted intensity.
The XRD line broadening in the samples with the chemical composition Ti 0.44 Al 0.56 N behaves similarly to the XRD line broadening observed in samples with the chemical composition Ti 0.53 Al 0.47 N. The maximum line broadening was measured in the samples deposited at the bias voltage of À80 V. Even at U B = À40 V, the XRD lines are broader in sample Ti 0.44 Al 0.56 N than in sample Ti 0.53 Al 0.47 N.

Micromechanical properties of metastable (Ti, Al)N thin films
Additional information about the micromechanics of metastable (Ti, Al)N thin films were obtained from the lattice parameters of fcc-(Ti, Al)N. Within the Reuss model (Reuss, 1929), the cubic lattice parameters measured in a thin film being under equi-axial macroscopic stress () depend both on the inclination of the diffraction vector from the normal direction, , and on the crystallographic direction hkl (see e.g. Rafaja et al., 2010) where S 0 ¼ S 11 À S 12 À S 44 =2; S 11 , S 12 and S 44 are the three independent elements of the cubic compliance tensor and a 0 is the stress-free lattice parameter of a cubic material. The dependence of the lattice parameters on the crystallographic direction follows the cubic invariant: In GAXRD geometry, is related to the diffraction angle 2 and to the angle of incidence of the primary beam on the sample surface (): It should be noted that although equation (2) is based on the Reuss approximation, an analogous dependence of the measured lattice parameters on the crystallographic direction can be obtained for the Eshelby-Krö ner model (Eshelby, 1957;Krö ner, 1958) and for the Vook & Witt (1965) model, because all three models reveal the same functional dependence of the X-ray elastic constants on the cubic invariant À hkl [see e.g. Welzel et al. (2005)].
The results of least-squares fits for samples Ti 0.53 Al 0.47 N (Fig. 5) and Ti 0.44 Al 0.56 N (Fig. 6) show that the best agreement between the model from equation (2) and the experimental data can be achieved for samples deposited at U B = À40 V, although the sample Ti 0.44 Al 0.56 N produces a large amount of the diffuse scattering (cf. Figs. 1 and 3), which was explained above by the presence of segregated wurtzitic regions (Fig. 4). According to equation (2)  stresses in all samples under study are summarized in Fig. 7 as calculated from the parameters a 0 S 11 À S 12 ð Þ and 2a 0 S 12 þ a 0 , which were obtained from the least-squares fit of equation (3) to the measured lattice parameters, and from the elastic constants S 11 = 2.59 TPa À1 and S 12 = À0.637 TPa À1 calculated by Tasná di et al. (2010). It is worth noting that in samples with high residual stress (deposited at U B = À80 and À120 V), the Al-rich fcc-(Al, Ti)N was found (cf. Fig. 2). As fcc-AlN is a high-pressure phase (Vollstä dt et al., 1990), the Al-rich fcc-(Al, Ti)N is considered to be stabilized by the residual stress present in the samples deposited at high bias voltages.
From the refined factors a 0 S 11 À S 12 ð Þ and a 0 S 0 , the anisotropy of the cubic elastic constants (see e.g. Tasná di et al., 2010), can be calculated according to because S 44 ¼ 2 S 11 À S 12 À S 0 ð Þ . The anisotropy factors calculated by using equation (6)  The anisotropy factors determined for both samples deposited at U B = À40 V are much higher than the anisotropy factors predicted from the ab initio calculation. High-anisotropy factors obtained from the XRD experiment are a consequence of a large difference between the measured lattice parameters a 111 and a 200 (Figs. 5 and 6) in samples that are under relatively low residual compressive stress (Fig. 7). As the difference between a 111 and a 200 depends on and S 0 through the factor a 0 S 0 in equation (2), the elastic constant S 0 must be increased in order to describe the large difference between a 111 and a 200 at low residual stresses. Finally, the increase of S 0 leads to an apparent increase of the anisotropy factor calculated according to equation (6).
In the decomposed (Ti,Al)N thin films, the anisotropy of the elastic constants and especially the difference between a hhh Dependence of the lattice parameters on sin 2 and on the diffraction indices as measured in samples with chemical composition Ti 0.53 Al 0.47 N, which were deposited at U B = À40 (circles), À80 (boxes) and À120 V (triangles). The solid lines represent the least-squares fit of the measured lattice parameters according to equation (2).

Figure 6
Dependence of the lattice parameters on sin 2 and on the diffraction indices as measured for samples Ti 0.44 Al 0.56 N deposited at U B = À40 V (circles), À80 V (boxes) and À120 V (triangles). The solid lines represent the least-squares fit of the measured lattice parameters according to equation (2).  and a h00 are intensified by the interaction between fcc-(Ti,Al)N and w-(Al, Ti)N grown heteroepitaxially with the habitus planes f111g fcc jjf001g w as already discussed in x3.1. As the distances between the metallic atoms within the f001g w planes of w-(Al, Ti)N are larger than the interatomic distances within the f111g fcc planes of fcc-(Ti, Al)N, w-(Al, Ti)N is compressed mainly in the hhk0i w directions and expanded mainly in the perpendicular direction [001] w . The elementary cell of fcc-(Ti, Al)N is expanded mainly in the directions perpendicular to h111i fcc , e.g. h110i fcc and h211i fcc , and compressed in one of the equivalent crystallographic directions h111i fcc , which is perpendicular to the respective habitus plane. In analogy with equation (2), the deformation of the fcc lattice caused by the heteroepitaxy with w-(Al, Ti)N can be described as a linear function of sin 2 ', where ' is the angle between the cubic lattice planes ðhklÞ and ð111Þ: In samples with intergrown fcc-(Ti, Al)N and w-(Al, Ti)N, the interplanar distances of the lattice planes with sin 2 ' ¼ 0 such as (111) or (222) shrink, whereas the interplanar distances of the lattice planes with sin 2 ' ' 0:5, such as (200), (400), (420) or (511), are almost unaffected. Such a dependence of a hkl on sin 2 ' correlates with the À hkl dependence of a hkl from equation (2), which apparently enhances the elastic anisotropy of the nanocomposite, in particular if the anisotropy is calculated from equation (6) using the parameters a 0 S 0 and a 0 S 11 À S 12 ð Þobtained from fitting the experimental data by using equation (2).
On the contrary, the anisotropy factors of samples deposited at U B = À80 and À120 V are lower than expected (Fig. 8).
In these samples, the low value of S 0 can be interpreted as an effective increase to the shear component of the compliance, S 44 ¼ 2 S 11 À S 12 À S 0 ð Þ , or as an effective reduction of the shear stiffness C 44 ¼ 1=S 44 , which facilitates both the formation of the stacking faults in fcc-(Ti,Al)N (Rafaja et al., 2014) and the shear-induced phase transitions in AlN (Schmerler & Kortus, 2014). However, strongly faulted w-(Al, Ti)N does not significantly contribute to the coherently scattered X-rays, thus the diffraction maxima from w-(Al, Ti)N become weaker, as it is clearly visible in Fig. 3.
To some extent, the low-anisotropy factors determined for samples deposited at U B = À80 and À120 V are caused by the segregation of AlN from fcc-(Ti, Al)N. According to Tasná di et al. (2010), the elastic anisotropy factor of Ti 1Àx Al x N decreases with decreasing x, until it reaches unity (as in an isotropic medium) for x ' 0.27. Using the XRD data, the Al content in fcc-(Ti, Al)N can be estimated from the Vegard-like dependence of the stress-free lattice parameter of Ti 1Àx Al x N on x , a = 4.2418 À 0.1432x. The stress-free lattice parameters of all samples under study and the corresponding Al concentrations are summarized in Fig. 9. The stress-free lattice parameters were determined from the refined factor a 0 S 11 À S 12 ð Þby using the residual stress and the elastic constants S 11 and S 12 discussed above. As already indicated by the shift of the measured lattice parameters in Figs. 5 and 6, the stress-free lattice parameter increases with increasing bias voltage for both chemical compositions, i.e. for Ti 0.53 Al 0.47 N (Fig. 5) and Ti 0.44 Al 0.56 N (Fig. 6). Still, the amount of Al in fcc-(Ti, Al)N ranges only between x = 0.50 and 0.45 in Ti 0.53 Al 0.47 N and between x = 0.60 and 0.55 in Ti 0.44 Al 0.56 N (Fig. 9). These composition ranges correspond to the variations of the anisotropy factors between 1.25 and 1.33 for Ti 0.53 Al 0.47 N and between 1.42 and 1.50 for Ti 0.44 Al 0.56 N (Tasná di et al., 2010), which cannot explain the observed anisotropy factors from Fig. 8. Therefore, the main reason for the observed changes in the elastic anisotropy is the interaction between the fcc-(Ti,Al)N crystallites and their Al-enriched vicinity.

Fragmentation of the microstructure
Complementarily, the segregation of AlN from fcc-(Ti, Al)N can be concluded from the XRD line broadening, which is shown in the form of the Williamson & Hall (1953) (8) is that the line broadening depends on the cubic invariant À hkl from equation (3), which represents the dependence of the contrast factors of dislocations and the dependence of the local variance of the interplanar spacings h" 2 hkl i ¼ h Ád=d ð Þ 2 i on the crystallographic direction hkl (Ungá r et al., 1999). The meaning of other symbols in equation (8) is as follows: D is the mean crystallite size, is a half of the diffraction angle, is the wavelength of the X-ray radiation and q is the amount of the crystallographic anisotropy.
The least-squares fitting of equation (8) (8). The same approximation was used for the line broadening measured at low diffraction angles in the samples deposited at U B = À80 (boxes) and À120 V (triangles). The line broadening measured in these samples at high diffraction angles was assumed to be constant.

Figure 11
Diffraction line broadening measured for fcc-(Ti, Al)N in Ti 0.44 Al 0.56 N and plotted as a function of sin . Circles denote the line broadening measured for U B = À40 V, boxes for U B = À80 V and triangles for U B = À120 V. The approximation of the line broadening was performed analogously to Fig. 10. Kužel (1988) to be the reason for the dependence of the contrast factors of dislocations on the crystallographic direction.
In the samples deposited at higher bias voltages (U B = À80 and À120 V) and thus at a lower adatom mobility, the line broadening (boxes and triangles in Figs. 10 and 11) is significantly higher than the line broadening in the samples deposited at higher mobility of the deposited species (U B = À40 V). Furthermore, the pronounced dependence of the line width on the crystallographic direction and even the linear dependence of the line broadening on sin disappear. At high diffraction angles, the line broadening increases abruptly and remains nearly constant.
Originally, this phenomenon was explained by a partial overlap of strongly broadened reciprocal lattice points belonging to slightly mutually misoriented neighbouring nanocrystallites (Rafaja et al., 2004). The extent of the overlap depends on the magnitude of the diffraction vector, i.e. on the distance of the respective reciprocal lattice point from the origin of reciprocal space, and on the mutual misorientation of the crystallites. Near the origin of the reciprocal space, i.e. at small diffraction vectors, the strongly broadened reciprocal lattice points overlap, which is recognized by XRD as a high partial coherence of the neighbouring nanocrystallites. Consequently, the resulting XRD lines become narrower in this region, which can be interpreted alternatively that XRD cannot distinguish the adjacent nanocrystallites with nearly identical orientations from each other. Instead, the Williamson-Hall analysis identifies agglomerates of such nanocrystallites (nanocrystalline clusters) as large but highly defective crystallites. The corresponding defects are mainly represented by 'small angle' boundaries between the mutually misoriented crystallites .
For partially coherent crystallites, the slope of the Williamson-Hall plot is basically determined by the loss of the partial coherence with increasing magnitude of the diffraction vector. Far from the origin of the reciprocal space, i.e. at large diffraction vectors, the partial coherence of neighbouring nanocrystallites disappears. In this region, XRD recognizes individual nanocrystallites as coherently diffracting domains and their intrinsic defects as the only source of the microstrain. Consequently, the XRD line broadening is much higher than in the vicinity of the origin of the reciprocal space and nearly constant, because the concentration of intrinsic defects (e.g. dislocations) in nanocrystallites is low.
In metastable thin films, the loss of the partial coherence of nanocrystallites can be used as an indicator of the fragmentation of the thin-film microstructure . For the Ti 0.53 Al 0.47 N sample, the analysis of the cluster size in terms of equation (8), i.e. from the intercept of the line broadening measured at low diffraction angles with the ordinate, revealed 10 (2) nm for both bias voltages (À80 and À120 V). The size of individual crystallites was 3.6 (4) nm for U B = À80 V and 4.2 (5) nm for U B = À120 V. In the Ti 0.44 Al 0.56 N sample, the size of the fcc-(Ti, Al)N clusters was 12 (1) and 16 (2) nm for U B = À80 and À120 V, respectively. The respective crystallite sizes were 3.2 (4) and 4.2 (4) nm.

Microstructure of supersaturated (Ti, Al)N solid solution
Based on the results above, the microstructure of the supersaturated (Ti,Al)N solid solutions and the underlying mechanisms of the microstructure formation can be summarized as follows. The supersaturated (Ti, Al)N solid solutions are metastable compounds, which tend to decompose into the thermodynamically stable fcc-TiN and w-AlN. The chemical driving force for the decomposition increases with increasing aluminium content. The stabilization of metastable phases like fcc-(Ti, Al)N and fcc-(Al,Ti)N is facilitated by intermixing of Ti and Al during the physical vapour deposition process and by a limited mobility of the deposited species, which already hinders the decomposition of the metastable compounds during the deposition process. The adatom mobility increases with increasing substrate temperature and decreases with increasing bias voltage. Thus, these parameters of the deposition process are crucial for tailoring the thin-film microstructure.
A high adatom mobility, which was accomplished by the deposition temperature of 723 K and a low bias voltage (U B = À40 V) in our samples, already leads to a relaxation of the residual stress during the deposition process. Furthermore, it cannot prevent the segregation of TiN and AlN in supersaturated Al-rich (Ti, Al)N. The locally Al-enriched (Al, Ti)N forms highly faulted wurtzitic clusters, which are detectable in the XRD patterns as 'diffuse scattering' or as very broad diffraction lines. Preferentially, the w-(Al, Ti)N clusters grow on fcc-(Ti, Al)N with the habitus planes ð002Þ w and ð111Þ fcc . This mutual orientation relationship is facilitated by similar symmetry operations, which exist along the h001i fcc direction in the wurtzitic crystal structure and along the h111i fcc direction in the fcc crystal structure (Rafaja et al., 2014). Still, owing to the different size of the fcc and wurtzitic elementary cell within the respective habitus plane, this intergrowth increases the anisotropy of the elastic constants determined from the dependence of the measured cubic lattice parameters on the diffraction indices as discussed above. The high adatom mobility promotes the formation of fcc-(Ti, Al)N crystallites having a size between 10 and 12 nm. Owing to the heteroepitaxy between fcc-(Ti, Al)N and w-(Al, Ti)N, and to the small size of the w-(Al, Ti)N domains, the segregation of elements and even the formation of w-(Al, Ti)N domains do not lead to an additional fragmentation of the fcc-(Ti, Al)N crystallites. However, these phenomena are still recognized by the broadening of XRD lines from fcc-(Ti, Al)N as increasing microstrain.
At the lower adatom mobility that was achieved via higher bias voltages, TiN and AlN segregate as well, but Al-rich (Al, Ti)N does not form visible wurtzitic clusters in the composition range under study. An explanation of this effect is that the low adatom mobility does not support the relaxation of the residual stress. The high compressive residual stress (8-14 GPa) present in the thin films deposited at U B = À80 and À120 V stabilizes high-pressure fcc-(Al, Ti)N. Furthermore, the local accumulations of AlN result in a fragmentation of the fcc-(Ti, Al)N crystallites into nanocrystallites having a size between 3 and 4 nm. Adjacent fcc-(Ti,Al)N nanocrystallites research papers 454 David Rafaja et al. X-ray diffraction on metastable thin films are slightly misoriented, which was concluded from the loss of their partial coherence for X-rays.

Conclusions
On the example of (Ti, Al)N thin films with a high aluminium content, it was illustrated that X-ray diffraction is a very efficient experimental method for the microstructure assessment of thermodynamically metastable compounds, for the description of their micromechanical properties and for the identification of the decomposition pathways. Detailed analysis of X-ray diffraction patterns yields information about the phase composition, lattice parameters, residual stress, anisotropy of the elastic constants, size of coherently diffracting domains and amount of the microstructure defects. In the supersaturated solid solution of titanium nitride and aluminium nitride, (Ti, Al)N, these microstructure features were employed to describe the microstructure formation as a function of the surface mobility of the deposited atoms for different high degrees of the thermodynamic instability. It was confirmed that a higher degree of thermodynamic instability forces the segregation of TiN and AlN and that the surface mobility of deposited atoms affects the decomposition mechanism and the distribution of the segregated species already in the deposition process. A high atomic mobility (achieved by a low bias voltage during the cathodic arc evaporation process) was shown to contribute to the relaxation of the intrinsic residual stress in fcc-(Ti, Al)N and thus to the formation of wurtzitic AlN during segregation of the excessive AlN from (Ti, Al)N. The wurtzitic AlN was located predominantly outside of the fcc-(Ti, Al)N domains, which consequently persisted unfragmented. Some AlN was accommodated inside of the fcc-(Ti, Al)N domains, where it increased the shear stiffness. Low atomic mobility (accomplished by high bias voltage in the deposition process) was found to promote the formation of high compressive residual stresses, which stabilize the high-pressure fcc-AlN. Part of the aluminium-rich fcc-(Al,Ti)N starts to transform into w-(Al, Ti)N via shearing, which is manifested by a decrease of the shear stiffness of the cubic structure. The excessive AlN segregates from fcc-(Ti, Al)N, but it is located inside of the fcc-(Ti,Al)N domains, where it leads to their fragmentation and to the formation of nanosized fcc-(Ti, Al)N crystallites.