M Marciszko

A Baczma Ński

K Wierzbanowski

M Wróbel

C Braham

J.-P Chopart

A Lodini

J Bonarski

L Tarkowski

N Zazi

Application of multireflection grazing incidence method for stress measurements in polished Al-Mg alloy and CrN coating

Keywords:

Multi-reflection grazing incidence geometry, referred to as MGIXD, characterized by a small and constant incidence angle, was applied to measure low surface stresses in very thin layers of Al-Mg alloy and CrN coating. These two materials were selected in order to deal with the low and high levels of residual stress, respectively. The influence of different mechanical treatments on residual stresses was studied for Al-Mg samples. It was found that both rolling and mechanical polishing influence the distribution and amplitude of residual stress in surface layers. In the case of CrN coating, a very high compressive stress was generated during the deposition process. The stress distributions determined by the MGIXD method is in good agreement with the classic sin 2 technique results for all studied samples. In performing stress measurements for a powder sample, it was found that the application of the Göbel mirror in the incident beam strongly reduces statistical and misalignment errors. Additionally, the root mean square values of the third order lattice strain within diffracting grains were determined.

Introduction

Both the magnitude and the spatial distribution of residual stress play key roles in the behaviour of materials subjected either to heat treatment or plastic deformation. The residual stress influences the fatigue response of solids, stress corrosion, and is important for the manufacturing process of all products. Depending on its direction and magnitude, the residual stress superimposed with an external load can be destructive or beneficial to a component. For example, the mechanical strength of a surface layer is improved by machining, which creates a compressive residual stress. The same type of stress increases fracture resistance and minimises spalling of the coating. The opposite type of stress, tensile residual stress, can accelerate the growth of the crack and cause the coating's destruction, when external loads are superimposed.

Diffraction methods are commonly used for determining lattice elastic deformation and distortion (i.e., of macrostrains and microstrains) from the displacement and broadening of the diffraction peak [START_REF] Noyan | Residual Streess-Measurement by Diffraction and Interpretation[END_REF][START_REF] Welzel | [END_REF][3][4]. The stress present in the near surface volume can be measured using the standard X-ray sin 2 method. However, this method is not advisable for the analysis of the in-depth stress gradient, because the penetration depth of X-ray radiation varies significantly during measurement. Therefore, the grazing incidence X-ray diffraction geometry (GIXD) was applied to measure residual stresses in thin surface layers [5][6][7][8][9][10][11][START_REF] Marciszko | Thin Solid Films[END_REF]. In this work one of the version of the GIXD method referred to as the multi-reflection grazing incidence X-ray diffraction (MGIXD, [10][11][START_REF] Marciszko | Thin Solid Films[END_REF]) was used in the laboratory diffractometer equipped with the Göbel mirror in the incident beam optics. Using this method, it is possible to perform a non-destructive stress analysis for different (and well defined) volumes below the surface of the sample and the stress can be measured at very shallow depths of a few micrometers. The main advantage of this method is that apart from the residual stress the root mean square of the lattice strain, caused by defects within polycrystalline material, can also be determined using Williamson-Hall method [START_REF] Williamson | [END_REF]14].

The disadvantage of the MGIXD method was its low accuracy in stress determination, when a classic line focus and a parallel plate collimator in the reflected beam optics were used in the measurement. For example the systematic error of about 50 MPa for Fe powder was reported in [10] (this value corresponds to 20 MPa if the Al elastic constants are used in the analysis). This is why our previous measurements were performed only for samples with high surface stresses [10,11]. Recently, the accuracy was considerably improved using the Göbel mirror to collimate an incident beam. A perfectly collimated parallel beam radically decreases the uncertainty of the determined peak position and, consequently, one reaches the accuracy of a few MPa for an Al powder sample. Due to very good precision and high beam intensity, the new experimental setup can be applied to measure low surface stresses in thin layers.

The reproducibility of the experimental setup with the Göbel mirror was already tested repeating measurements for different powder specimens [START_REF] Marciszko | Thin Solid Films[END_REF]. It was found that the difference between the stresses measured using the MGIXD method was smaller than 15 MPa for the Al powder. This value can be considered as the uncertainty of measurements arising from misalignments of the diffractometer and the sample position. From the measurements on the powder samples and also from theoretical analysis it was found that the important source of systematic error is the shift of 2 zero value. The misalignment of the diffractometer equal to 2 = 0.01 • leads to the fictitious stress of about 10-15 MPa determined for the Al stress-free powder. This error can be minimised by the careful alignment of the diffractometer or the results obtained for the studied sample can be corrected by using the powder diffraction data.

The application of the MGIXD method is especially recommended for determining of the in-depth stress gradients close to the surface, which can significantly change the corrosion resistance of the material [15]. This methodology can be also used to determine the stress distribution in thin films. The latter stress influences the fracture and spalling resistance of the coating.

In the present work we demonstrate the possibilities of the MGIXD method on two examples of surface layers. The stress distribution is determined in a mechanically polished Al-Mg alloy (isotropic elastic properties) and a CrN coating (anisotropic single crystal elastic constants).

Experimental principles

Stress measurements using grazing incidence geometry

The standard method for stress determination (called sin 2 ) is based on the measurement of interplanar spacing for various directions of the scattering vector [START_REF] Noyan | Residual Streess-Measurement by Diffraction and Interpretation[END_REF][START_REF] Welzel | [END_REF][3]. These directions are defined by and angles (Fig. 1a). Using diffraction, the mean interplanar spacing d(, ) {h k l} , averaged over reflecting crystallites, is measured. As it has been mentioned, the orientation of the scattering vector varies in this method, however the reflecting plane index {h k l} is kept constant during strain measurements. Consequently, different penetration depths of X-ray radiation appear for different tilt angles in standard geometry.

The multi-reflection grazing incidence geometry (MGIXD) is a non destructive diffraction method, which can be used to determine lattice elastic distortion in the near surface layers of a material. The characteristic feature of this method is a small and constant incidence angle [10][11][START_REF] Marciszko | Thin Solid Films[END_REF]. Consequently, the penetration depth of X-ray radiation is well defined and does not change during the experiment. Measurements are performed for different sets of {h k l} planes using the appropriate values of 2 {h k l} scattering angles. The incidence angle ˛is fixed during measurement, while the orientation of the scattering vector is characterized by the angle {h k l} . The {h k l} angle depends on h k l reflection (2 {h k l} angle) and on a constant incidence angle ˛(Fig. 1b):

{h k l} = {h k l} -˛(1)
Consequently, possible values of {h k l} angles are limited to the number of h k l reflections used in the experiment.

In the case of the MGIXD method the measurements of interplanar spacings d(, ) {h k l} are performed in the near surface volume, which is limited by radiation absorption. To define this volume, the path of the X-ray beam through the sample must be considered (Fig. 1b). The measured average interplanar spacings d(, ) {h k l} are equal to:

< d(, )> {h k l} = t 0 d(, , z) {h k l} exp[-l(z)]dz t 0 exp[-l(z)]dz and l(z) = z 1 sin ˛+ 1 sin(2 {h k l} -˛) (2)
where is the linear coefficient of absorption and the above formula can be used if ˛» ˛cr (˛c r is the critical angle for the total external reflection [START_REF] Welzel | [END_REF]), z is a depth below the surface and the average is calculated over the volume of all reflecting grains in the beam path l(z) = a(z) + b(z) (see Fig. 1b), i.e. from the surface (z = 0) to the thickness of the coating (z = t). If the stresses are measured in a monolithic material or in a thick coating (comparing to penetration of X-ray radiation) we assume that t→ ∞. Eq. ( 2) is usually expressed in the equivalent form:

< d(, )> {h k l} = t 0 d(, , z) {h k l} exp(-z/)dz t 0 exp(-z/)dz and = sin ˛+ sin(2 {h k l} -˛) -1 (3)
where "penetration depth" is defined as the distance from the sample surface, for which 1 -1/e = 0.63 part of total intensity of the incident beam is absorbed. The above average corresponds to the so called "information" or "effective" depth z, which can be understood as the mean value of z-depth weighted by an attenuation factor [8]:

z = t 0 z exp(-z/)dz t 0 exp(-z/)dz = ⎧ ⎨ ⎩ - t exp(-t/) 1 -exp(-t/) forlimited t for t → ∞ (4)
When small ˛angles are used (i.e., for long incident beam paths, i.e. a(z) » b(z) in Fig. 1b) the penetration depth can be approximated by the expression = sin ˛/ (see Eq. ( 3)). Consequently both the penetration depth (Eq. ( 3)) and the information depth z (Eq. ( 4)) do not depend on 2 and related (Eq. ( 1)) angles. The information depths z vs. sin 2 calculated for the MGIXD and the standard method ( mode [START_REF] Noyan | Residual Streess-Measurement by Diffraction and Interpretation[END_REF][START_REF] Welzel | [END_REF]) are shown in Fig. 2, where the materials studied in this work are considered. It is visible that only in the case of the MGIXD method, the information depth is almost constant for a fixed small ˛angle and for a large range of angle. Moreover, it is shown that stress can be measured for different layers under the sample surface by setting different ˛angle values (Fig. 2).

In the MGIXD method, the d(, ) {h k l} interplanar spacings are measured in the directions defined by and angles for different h k l reflections. Hence, the orientations of the scattering vector with respect to the sample are strictly determined by Eq. ( 1), and the Fig. 1. Orientation of the scattering vector with respect to the sample described by and angles (a) and the geometry of MGIXD method (b). The following symbols were used: is an angle between the scattering vector K and a normal to the sample surface x3, ˛is an incident angle, 2 is a diffraction angle, z is a depth below the surface, t is the coating thickness and l = a + b denotes length of the beam path in the sample.

measurements can be done only for a few angles. These experimental data can be easily analysed in a multi-reflection method, and the residual stress can be determined for every incidence angle ˛ [10][11][START_REF] Marciszko | Thin Solid Films[END_REF]. In this procedure, the equivalent lattice parameters are used (instead of d(,

) {h k l} ): < a(, )> {h k l} = h 2 + k 2 + l 2 < d(, )> {h k l} (5)
to determine the macrostress tensor. In the case of a quasiisotropic sample and when the forces perpendicular to the surface are equal to zero, the interplanar spacing measured in the direction of scattering vector ( K in Fig. 1a) is given by the well known relation [3,11,[START_REF] Marciszko | Thin Solid Films[END_REF]:

< a(, )> {h k l} = [F ij (h k l, , ) ij ]a 0 + a 0 , ( 6
)
where ij is the average macrostress for the information depth z (Eq. ( 4)) corresponding to a given incidence angle ˛(see Eq. ( 2)), a 0 is the strain free lattice parameter, while F ij (h k l,, ) are the stress factors calculated for different h k l reflections related to different and angles [3]. In this work, the free-surface self-consistent method [3] was used to calculate the F ij (h k l,, ) stress factors from single crystal elastic constants (given in Table 1) and experimentally determined crystallographic textures. It should be underlined that F ij (h k l,, ) factors depend on the h k l reflection, especially for crystals having high elastic anisotropy characterized by the Zener factor (also presented in Table 1):

A = 2C 44 C 11 -C 12 (7)
Using the least square method for Eq. ( 6), the fitting parameters (i.e., ij and a 0 ) can be determined by minimizing a merit function given by formula [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF]:

2 = 1 N -M N n=1 < a( n , n )> exp {h k l} -< a( n , n )> cal {h k l} ı n (< a(, )> {h k l} ) 2 (8)
where < a( n , n )> exp {h k l} and < a( n , n )> cal {h k l} are experimental and calculated equivalent lattice parameters determined using h k l reflections, ı n (< a(, ) > {h k l} ) is an experimental uncertainty of the spacing < a( n , n )> exp {h k l} for the n-th measurement, N and M Fig. 2. Information depth z vs. sin 2 calculated from Eq. ( 4) for Al (a) and CrN (b) materials using Cu radiation. Results for the MGIXD (different ␣ angles) and for the standard sin 2 methods ( mode) are shown. In the case of CrN the calculations correspond to the coating thickness equal to t = 6 m, while an infinite sample thickness was assumed for aluminium.

are the numbers of measured points and fitting parameters, respectively.

The value of 2 is a measure of goodness-of-fit, i.e. [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF]:

-2 = 1 means that the "good fit" was obtained (it corresponds to the fitting exactly within the limits of experimental uncertainty), -2 < 1 the uncertainties of experimental data ı n (< a(, ) > {h k l} ) are overestimated, -2 > 1 the uncertainties of experimental data are underestimated or calculated (theoretical) values < a( n , n )> cal {h k l} depending on stress factors F ij (h k l,, ) according to Eq.( 6) are not accurate.

Experimental setup

X-ray stress determination requires a precise measurement of diffracted peak positions. A perfectly collimated (parallel) beam radically decreases errors of the determined peak position, caused by a sample misalignment. Watkins et al. showed that parallel beam optics significantly improves data quality, allowing for accurate residual stress measurements [START_REF] Watkins | [END_REF]. In the present work, grazing incidence X-ray diffraction measurements were carried out with X-Pert Philips X-ray diffractometer (Cu K ␣ radiation) equipped with a Göbel mirror, in the incidence beam optics (Fig. 3). The experiment was repeated for various incidence angles ˛corresponding to different information depths z (see Eq. ( 2)). To verify the results of grazing incidence measurements, the standard sin 2 method in mode was also applied. Due to variations of information depth with the angle, only approximated average values of z were estimated in the standard method.

In order to test the experimental setup, a set of preliminary measurements was performed for the Al powder using two configurations, i.e., the Göbel mirror (divergence 0.02 • ) and a slit with a divergence of 1/2 • for the primary optics. In both cases a parallel plate collimator (Soller collimator with divergence of 0.18 • ) was used in the reflected beam optics. In this case the a(, ) {h k l} vs. sin 2 plots were measured for = 0 • and different incident angles ␣. The stress values determined using the least square method (based on Eqs. ( 6) and ( 8)) are given in Table 2, while the corresponding d(, ) {h k l} vs. sin 2 plots are shown in Fig. 4. In the analysis, the stress factors F ij (h k l,, ) were calculated from single crystal elastic constants (Table 1) assuming random orientations of powder grains.

In the analysis of experimental data it is important to take the different sensitivity of the measured lattice strain on the value of scattering angle 2 {h k l} into account. In this work the fitting procedure was based on Eq. ( 8), in which the uncertainty of equivalent lattice parameters ı n (< a(, ) > {h k l} ) are treated as the weight in the calculation of the 2 value. The ı n (< a(, ) > {h k l} ) uncertainty is calculated directly from the uncertainty of peak position ı n (2 {h k l} ), i.e.:

ı n (< a(, )> {h k l} ) =< a( n , n )> exp {h k l} cot( {h k l} )ı n ( {h k l} ). ( 9
)
In the data analysis it can be assumed that that the ı n (2 {h k l} ) uncertainty is equal to the standard deviation of the peak position obtained from procedure of peak adjustment. However, these values are very small (smaller than 0.01 • ) and other experimental errors play a more significant role, for example those due to the misfit of the sample position, defocusing or misalignment errors. Errors having different reasons are in fact unknown, therefore we decided to assume a reasonable value of peak position uncertainty, the same for all reflections. As shown in Fig. 4 (see error bars) the values of ı n (< a(, ) > {h k l} ), calculated using Eq. ( 9) with ı n (2 {h k l} ) = 0.01 • , are different for different 2 {h k l} . This ensures different influences of measured equivalent parameters < a( n , n )> exp {hkl} on the fitting quality criterion (Eq. ( 8)) and consequently on the values of the determined stresses. As seen in Fig. 4 the uncertainties ı n (< a(, ) > {h k l} ) are larger for lower a value of 2 {h k l} scattering angle, i.e., the low 2 {h k l} angle reflections affect the fitting results less than those for which the scattering angle is higher (cf. Eq. ( 8)). It is also important to estimate the uncertainty of the determined stresses in the case of unknown the ı n (2 {h k l} ) values. Therefore, regardless of the reasons of the experimental errors or inaccuracy of the data treatment the stress uncertainties were calculated assuming a "good fit" for which 2 = 1 [START_REF] Press | Numerical Recipes in C: The Art of Scientific Computing[END_REF] (in this aim the assumed uncertainty ı n (2 {h k l} ) was varied in order to obtain condition 2 = 1).

The values of 11 stress component determined for the Al powder are compared for the two optics used. Moreover, the data treatment was repeated applying two different conditions, i.e., using all measured reflections presented in Fig. 4 or excluding two low 2 reflections {1 1 1} and {2 0 0}, for which < a( n , n )> exp {h k l} deviate significantly from the theoretical values. The results of the analysis are presented in Table 2. Small, but significant, values of fictitious stresses (between -8 and -33 MPa) were found, when the slit was used. As the real stress for the powder sample is equal to zero, the determined non-zero stresses can be treated as the values of systematic uncertainty caused by the diffractometer or sample misalignments. The latter uncertainties can be minimized using parallel optics of the incident beam. The near zero values of stresses measured in the Al powder (absolute values lower 5 MPa, see Table 2) show that the experimental errors were significantly reduced by use of the Göbel mirror.

Comparing the results obtained with and without two low 2 reflections (i.e. {1 1 1} and {2 0 0}) it can be stated that a small improvement of the results (lower fictitious stress and its uncertainty) was obtained when the latter reflections were excluded (Table 2). However, the difference is not significant because in our data treatment the influence of the < a( n , n )> exp {h k l} values on the fitting results decreases with the 2 angle. Therefore in the further analysis even the data obtained with the low 2 angle will be used, certainly with definitely smaller weight (Eqs. ( 8) and ( 9)).

For comparative purposes the results of standard sin 2 method ( mode, 422 reflection) are also shown in Table 2. In this case, a small value of uncertainty (less than 5 MPa) was obtained for both experimental setups.

Residual stress measurements

The diffractometer equipped with the Göbel mirror was used to measure residual stresses in two very different types of samples, i.e. in the mechanically treated surface of an Al-Mg alloy [15] and in a CrN coating deposited at high temperature on a 4H13 steel substrate [7]. As shown in Table 1, the first material exhibits a very low anisotropy of single crystal elastic constants, i.e. the Zener factor A is close to 1 for aluminium (due to the low contents of magnesium, shown in Table 3, the Al elastic constants were used for the Al-Mg alloy), while the elastic properties of the CrN crystals are strongly anisotropic (A = 0.34). Moreover, relatively low residual stresses were generated by the mechanical treatment in the Al-Mg samples and very high stress is expected in the CrN coating [START_REF] Buijnsters | [END_REF][21][22].

Residual stress in Al-Mg samples

The residual stress was measured in the Al-Mg alloy having the chemical composition given in Table 3. The samples were prepared from a sheet that was subjected to cold rolling followed by an industrial stabilisation (in such an industrial process the annealing temperature is maintained between 50 and 200 • C). Next, the material was annealed at 420 • C for 1.5 h in a salt bath and cooled in the air. Afterwards, a part of the material was cold rolled up to the Fig. 3. Geometry of the MGIXD method (a) and parabolic Göbel mirror reflecting the parallel beam (b). The source of X-rays is placed in the focus F and the surface of the mirror consists of nanolayers monochromating the reflected beam.

Table 2

Residual stress component 11 determined for Al powder using two optics of incidence beam: Göbel mirror and slit (stresses calculated excluding {1 1 1} and {2 0 0} reflections compared with results obtained from all reflections). reduction of 92%. Finally, the surfaces of the not rolled (specimen A) and rolled (specimen B) samples were mechanically polished by hand polishing device ESC 200 GTL -ESCIL and emery papers from 240 to 4000 grit, with 5 steps (240; 320; 600; 1200; 4000 grit). The surface preparation that was done is a standard laboratory treatment before corrosion analyses [15,23]. In industrial processes polishing is used before some surface treatments as anodization [23].

The average roughness parameters were similar for both samples and equal to approximately R a = 0.4 m. For both samples the orientation distribution function (ODF [START_REF] Bunge | Texture Analysis in Material Science: Mathematical Methods[END_REF]) characterizing crystallographic texture was determined from pole figures {1 1 1}, {2 0 0} and {2 2 0} measured using Co radiation (WIMV method for texture analysis was used [START_REF] Kallend | Operational Texture Analysis[END_REF]). The ODFs for initial sample (A) and cold rolled sample (B) are shown in Fig. 5.

The MGIXD and the standard sin 2 methods with parallel optics (Göbel mirror) were used to determine residual stresses in the Al-Mg alloy samples. The measurements were done for two directions ( = 0 • and = 90 • , see Fig. 1a) and for two incidence angles (˛= 5 • and ˛= 15 • ). Sample diffraction patterns obtained for the A and B samples (˛= 5 • ) are shown in Fig. 6. The positions of the peaks were determined by fitting pseudo-Voigt functions to the experimental diffraction peaks. Using the Bragg relation and Eq. ( 5) the equivalent lattice parameters a(, ) {h k l} were calculated. The a(, ) {h k l} vs. sin 2 plots, presented in Figs. 7 and8, show very good quality of fitting the theoretical relations to the experimental data.

In the calculations of the stress factors F ij (h k l,, ), the ODFs shown in Fig. 5 and the single crystal elastic constants of aluminium (Table 1) were used [3]. It should be noted that in the case of low anisotropy of the crystal elastic properties (A = 1.22, Table 1) the F ij (h k l,, ) factors do not depend significantly on the h k l reflections and on the crystallographic texture. Hence, the a(, ) {h k l} vs. sin 2 plots obtained using the MGIXD method are almost linear (similarly as for the standard sin 2 method, when only one reflection is used).

Two stress components, 11 (along the rolling direction) and 22 (along the transverse direction), were determined for the depths of 5.8 ± 0.2 m and 14.2 ± 2.0 m in the case of the MGIXD method and for the approximate depth between 12 and 34 m in the case of the standard sin 2 method (cf. ranges of information depth shown in Fig. 2a). The in-depth distributions of the stress below the sample surface are shown in Fig. 9. It should be stated, that in spite of its low spatial accuracy (large horizontal error bars for last points in Fig. 9), the results of the standard sin 2 method measurements confirm the in-depth profile of stress distributions determined by the MGIXD method.

Additionally, the diffraction peak profile was also analyzed for Al-Mg samples. It is well known that the peak position is determined by macrostresses, while its broadening depends on the density and structure of the defects inside polycrystalline grains [4,[START_REF] Williamson | [END_REF]14,[START_REF] Ungár | [END_REF]. The comparison of the diffraction peaks (Fig. 6) shows a low density of dislocation, i.e., narrow peak profiles, in the initial sample (A), while broad profiles were measured for cold rolled one (B; see Fig. 6). The advantage of the presented MGIXD method is that diffraction peaks were measured for different reflections at a given ˛angle corresponding to a constant information depth. Thus the quantitative analysis of the lattice strains caused by the lattice defects can be performed. In this work the Williamson-Hall method [ [START_REF] Williamson | [END_REF]14] was applied to determine the root mean square value of the third order strain ε 2 III characterizing the distortion of the lattice within grains [4,[START_REF] Williamson | [END_REF]14,[START_REF] Ungár | [END_REF]. As shown in Table 4, the value of ε 2 III for the cold rolled sample (B) does not depend on the depth. In the case of initial sample (A) the root mean square of the third order strain cannot be determined because its contribution to the peak broadening is too small in comparison with instrument peak broadening.

The following conclusions concerning stress fields in different Al-Mg samples can be drawn:

-nearly isotropic planar stresses (i.e., 11 = 22 ) were found in the polished initial sample (A), while 11 / = 22 occurred in the cold rolled and the polished sample (B), -the initial sample (A) exhibits a significant in-depth gradient of stress characteristic for polished surfaces, while the approximately constant stress was determined in the cold rolled and polished sample (B), -the cold rolling process introduced a significant change in the microstructure of the material causing significant distortion of the lattice within the grains (described by ε 2 III , which can be attributed to the lattice defects, mainly dislocations [START_REF] Ungár | [END_REF], -significant texture evolution caused by the cold rolling process was observed (Fig. 5).

Residual stress in CrN coating

This section presents the results of the diffraction measurements of stress in the CrN film (6 m thickness) deposited on the 4H13 steel substrate. The coating was obtained by means of the arc-vacuum method in a nitrogen atmosphere at the pressure of N 2 equal to 3.5 × 10 -2 mbar and the temperature of 450 • C. The average speed of deposition was 60 nm/min. As a result, the coating exhibiting the average surface roughness R a = 0.33 m and fibre crystallographic texture (ODF determined using Co radiation is shown in Fig. 10) was produced. The advantage of the CrN film is its high hardness, good oxidation resistance and a low friction coefficient. Therefore it is used to coat cutting tools, and due to a good biocompatibility, it can be applied to medical instruments or to surfaces of joint replacements. Residual stress created during the cooling of the deposited layer to room temperature significantly influences the strength of the coating. The favourable state of stress in the film is the compressive stress that causes the closing of the initiated cracks and prevents their propagation.

Stress measurements for the CrN layer were performed using configuration with the Göbel mirror and the parallel plate collimator in the primary and secondary optics, respectively. The diffraction pattern, presented in Fig. 11 for ˛= 5 • shows a large broadening of the diffraction peaks. The root mean square values of the third order strain ε 2 III characterizing the distortion of the lattice within the grains, calculated using the Williamson-Hall method [START_REF] Williamson | [END_REF]14], are shown in Table 5 (cf. [27]). The values of ε 2 III are much larger than in the case of the cold rolled Al-Mg alloy and they do not change vs. the information depth. The large value of third order stresses can be attributed to the high level of defects caused by high-energy ion deposition. It has been observed in [28] that the width of the diffraction peak significantly decreased due to the annealing of the similar samples.

The equivalent measured and theoretical lattice parameters lattice a(, ) {h k l} vs. sin 2 for ˛= 5 • , 10 • and 15 • are shown in Fig. 12. Due to the process symmetry, an isotropic planar stress state (i.e., 11 = 22 ) was assumed and the measurements were performed only in one direction. The stress factors F ij (h k l,, ) were calculated using the single crystal elastic constants of CrN (Table 1) and the ODF shown in Fig. 10. It can be noticed that a high anisotropy of crystal elastic constants (A = 0.34, Table 1) caused significant nonlinearities of the a(, ) {h k l} vs. sin 2 plots. Using our methodology of data analysis based on Eq. ( 6) (where F ij (h k l,, ) factors are calculated for different h k l reflections) the experimental points are matched by the fitted data (lines). It can be concluded that the elastic anisotropy of the material was properly taken into account. Additionally, the standard diffraction measurements were performed using 311 and 422 reflections. As shown in Fig. 13, the linear nature of the a(, ) {h k l} vs. sin 2 plot was obtained in both measurements. A high compressive stress, with a relatively small variation vs. depth, was found in the CrN coating (Fig. 14). The stresses obtained using the MGIDX and standard methods are very similar, however in the standard method it is not possible to precisely determine the depth for which the stress is measured (large horizontal error bars). Our measurements reveal a large compressive stress in the CrN layer. It results from different shrinking amplitudes of the CrN layer and the steel substrate during cooling (their thermal expansion coefficients are: ˛CTE = 6 × 10 -6 K [29] and ˛CTE = 11 -12 × 10 -6 K for steel substrate and for CrN coating, respectively [START_REF] Buijnsters | [END_REF]21]). It should be noted that the observed important compressive stress is caused not only by the temperature effect but also due to the peening of the growing coating by accelerated atoms, interdiffusion and the reactions with the substrate [21,28]. A similar level of the stresses in the CrN coating deposited on the steel base was previously observed in [7,21,28]. As already mentioned, this stress state enhances the fracture resistance and the strength of the coating.

Discussion and general conclusions

The multi-reflection method of the experimental data analysis was used for the determination of residual stresses using the multireflection grazing incidence geometry (the MGIDX method). In this technique the interplanar spacings d(, ) {h k l} are measured for different orientations of the scattering vector. The stress state can be determined for a chosen material layer, at a given distance below the sample surface, which can be changed easily. In contrast, in the standard sin 2 method the depth of measurement is not strictly determined, because it changes during measurement (large horizontal error bars in Figs. 9 and14). This method is not applicable for the analysis of in-depth stress heterogeneity if additional assumptions are not introduced.

Using parallel beam optics it was possible to increase the accuracy of the lattice strain measurements by X-ray radiation. The analysis of the experimental results performed in this work for the powder sample proved that the Göbel mirror installed in the primary optics of diffractometer can significantly reduce the uncertainty of the determined stresses.

It was shown that for different crystal anisotropies and various surface treatments, the multi-reflection method can be successfully applied to interpret the grazing incidence results. Both for elastically isotropic (Al-Mg alloy) and strongly anisotropic (CrN) crystals the a(, ) {h k l} vs. sin 2 plots were correctly predicted, when the diffraction elastic constants were calculated by the self-consistent method used for the sample surface (the free-surface method in which the crystallographic texture is taken into account [3]).

The residual stresses, measured using the grazing incidence geometry, were completed by the standard measurements and a very good accordance of the stress was found for both methods, especially when the stress value did not change significantly with the information depth (Figs. 9b and14). It should be stated that only MGIXD method provides a special resolution enough to observe the stress gradient (Fig. 9), while only the mean stress for large range of information depth can be obtained from the standard method.

The advantage of the multi-reflection grazing incidence method is that the diffraction peaks for many reflections are measured. Therefore not only the stresses but also the peak broadening can be analysed (using the Williamson-Hall method for example). In the present work, for the first time the macroscopic residual () as well as the third order lattice strain ε 2 III (within polycrystalline grains) were determined for different information depths using the MGIXD method.

The MGIXD and the standard sin 2 methods were applied to measure stresses in the mechanically processed Al-Mg alloy. The results show that the stress field in the surface layer as well as the microstructure (density of dislocation) depend strongly on the sample preparation. In the case of the as received material (specimen A), mechanical polishing generated stress gradient in the surface layer, while not significant changes in microstructure were observed after this treatment. The stress components measured in two perpendicular directions are very similar. On the contrary, the stress components in the rolling and transverse directions are different and the almost constant in-depth profile of stress was found for the cold rolled sample B. Moreover, the microstructure of the rolled sample was significantly changed and a high value of ε 2 III was found. A very high residual compressive stress, which does not change significantly with depth, was measured in the CrN coating. These results can be explained due to the difference in the shrinking of the coating and the base material during the sample cooling.

Additionally, the stress is generated by the peening of the film by accelerated atoms, diffusion and the reactions processes. Moreover, a large value of the measured third order strains ε 2 III in the CrN coating is caused by high-energy ion deposition.

  Method ˛[• ] and {h k l} information depth z [m] primary beam configuration 11[MPa] all reflections 11[MPa] {1 1 1}, {2 0 0} excluded Grazing incidence ˛= 5

Fig. 4 .Fig. 5 .

 45 Fig.4. Measured lattice parameters (points) and theoretical results of fitting (continuous lines) vs. sin 2 for the Al powder sample obtained using the grazing incidence method (a) for three angles ˛and standard sin 2 method (b). Experimental results determined for two configuration of the diffractometer, i.e. with the Göbel mirror or slit in the incident beam optics are shown.

Fig. 6 .

 6 Fig. 6. Diffractograms for the studied Al-Mg samples (MGIXD with the incident angle ˛= 5 • ). Significantly smaller peaks broadening occurs in the initial sample A (a splitting into K␣1 and K␣2 lines is visible) in comparison with the cold rolled sample B.

Fig. 7 .

 7 Fig. 7. Measured lattice parameters (points) and theoretical results of fitting (continuous lines) vs. sin 2 for two Al-Mg samples: only polished (a) and polished after cold rolling (b). Results of the grazing incidence method for two angles ˛and two directions are shown.

Fig. 8 .

 8 Fig. 8. Measured lattice parameters (points) and theoretical results of fitting (continuous lines) vs. sin 2 for Al-Mg samples ( = 0 • and 422 reflection). Results of the standard sin 2 method are shown for only polished (a) and polished after cold rolling (b) specimens.

Fig. 9 .

 9 Fig. 9. In-depth distribution of residual stress determined for Al-Mg samples determined by grazing incidence and standard methods. Results for only polished (a) and polished after cold rolling (b) specimens are shown.

Fig. 10 .

 10 Fig. 10. Orientation distribution function (ODF) for CrN coating determined by Xray diffraction. The sections through Euler space [20] with the step of 5 • along 2 axis are presented.

Fig. 11 .

 11 Fig. 11. Diffractogram for the studied CrN coating (MGIXD with incident angle 5 • ). Large diffraction peak broadening is shown for the 422 reflection.

Fig. 12 .

 12 Fig. 12. Measured lattice parameters (points) and theoretical results of fitting (continuous vs. sin 2 for the CrN coating. Results of the grazing incidence method are shown for three angles ˛.

Fig. 13 .

 13 Fig.[START_REF] Williamson | [END_REF]. Measured lattice parameters (points) and theoretical results of fitting (continuous lines) vs. sin 2 for the CrN coating. Results of standard sin 2 method are shown for 311 and 422 reflections.

Fig. 14 .

 14 Fig. 14. In-depth distribution of residual stress determined for CrN coating determined by grazing incidence and standard methods.

Table 1

 1 Single crystal elastic constants of aluminium (used also for Al-Mg alloy AA5083) and CrN coating[START_REF] Simoms | Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook[END_REF][START_REF] Birkholz | Thin Film Analysis by X-ray Scattering[END_REF].

	Material	Single crystal elastic constants (GPa)	Zener factor A
		C11	C12	C44
	Al	106.8	60.4	28.3	1.22
	CrN	542	27	88	0.34

Table 3

 3 Chemical composition of aluminium alloy AA5083.

	Element	Mg	Fe	Si	Mn	Cr	Zn	Ti	Cu	Other
	Concentration (%)	4.0-4.9	0.4	0.4	0.3-1.0	0.25-0.5	0.25	0.15	0.1	0.15

Table 4

 4 Root mean square of third order strain ε 2

	˛[• ]	Information depth z [m]	ε 2 III	[%]
	5	5.8 ± 0.2	0. 19 ± 0.01
	15	14.9 ± 2.1	0.17 ± 0.01

III

for Al-Mg cold rolled sample B.

Table 5

 5 Root mean square of third order strain ε 2

	III	for CrN coating.

Acknowledgements

The work was financed by the Polish National Centre for Science (NCN) basing on decision number: DEC-2011/01/B/ST8/07394 as well as by the Polish Ministry of Science and Higher Education (MNSiW) and its grant 3264/B/H03/2011/40.


Armel Menant
email: armel.menant@cnrs-orleans.fr

Pietro Sternai

Laurent Jolivet

Laurent Guillou-Frottier

Taras Gerya

Crustal deformation and magmatism at the transition between subduction and collisional domains: insight from 3D numerical modeling

Geodynamics evolution of subduction zones is often complex, notably due to the variable nature and rheology of interacting lithospheres, 3D slab geometry and related mantle flow. In order to understand the temporal evolution of this dynamics, the long-term geological record should be considered, including the present-day kinematics, the long-term history of deformation and magmatism within the upper plate. This leads to conceptual models that imply complex physical process that can be tested through numerical modeling.

In this study, we focus on the eastern Mediterranean region where a long-lived subduction zone has been active since the Mesozoic. In this region, the Tethyan active margin is notably characterized by successive subductions of oceanic and continental ribbons, slab retreat episodes and possible slab tearing processes, notably interpreted from tomographic models showing low-velocity anomaly below Anatolia [START_REF] Ricou | Geological constraints on the alpine evolution of the Mediterranean Tethys[END_REF][START_REF] Piromallo | P wave tomography of the mantle under the Alpine-Mediterranean area[END_REF]van Hinsbergen et al., 2005a;[START_REF] Jolivet | Aegean tectonics: Strain localisation, slab tearing and trench retreat[END_REF].

Using G-Plates software, we first modeled the kinematic evolution of the eastern Mediterranean in the last 100 Myrs, accounting for a large number of geological data, including the magmatic occurrences (figure 1).

Since 35-30 Ma, the geodynamics of this region has been dominated by the subduction and collision of Arabia with Eurasia and with the development of a large back-arc domain in the Aegean region, resulting from an increase of the rate of slab retreat. This slab retreat is associated with the southward migration of high-potassic magmatism in the back-arc domain with time, and with the southwest fast migration of the Aegean plutons during the middle Miocene from the Menderes to the Cyclades (figure 1).

In addition, a major change occurred in the dynamics of the back-arc opening in the middle Miocene (figure 1) with a new increase of the rate of trench retreat and large-scale block rotations on the edge of this extensional domain in the Hellenides and in western Anatolia [START_REF] Kissel | The Tertiary geodynamical evolution of the Aegean arc: a paleomagnetic reconstruction[END_REF]van Hinsbergen et al., 2005b]. This kinematic change is coeval with the southwest migration of the back-arc plutons and with the increase of their mantle component compared to their crustal component from east to west [START_REF] Altherr | I-type plutonism in a continental back-arc setting: Miocene granitoids and monzonites from the central Aegean Sea, Greece[END_REF]. Finally, sparse alkaline volcanism emplaced in the Aegean and western Anatolia since ~15 Ma with the same westward migration trend, indicating a hot asthenospheric upwelling associated with a low rate of partial melting [START_REF] Dilek | Geochemical and temporal evolution of Cenozoic magmatism in western Turkey: mantle response to collision, slab break-off, and lithospheric tearing in an orogenic belt[END_REF]. Using high-resolution 3D thermo-mechanical numerical modeling, we reproduce a relatively simple model of retreating subduction neighboring a collisional domain that we compare with these geological observations. These numerical experiments are performed with the code I3ELVIS, considering nonnewtonian visco-plastic rheologies and notably integrating partial melting and melt extraction processes as well as fluid and melt transport mechanism [START_REF] Gerya | Characteristics-based marker-in-cell method with conservative finite-differences schemes for modeling geological flows with strongly variable transport properties[END_REF][START_REF] Zhu | Four-dimensional numerical modeling of crustal growth at active continental margins[END_REF]. The resolved grid has a resolution of 3.4, 2.5 and 4.4 km in the x,y and z dimensions respectively and ~130 million additional randomly distributed moving markers.

International
The initial model setup shows an oceanic lithosphere and a continental plate (considered as Arabia)

converging toward a continental upper plate (considered as Eurasia) (figure 2a). The evolution of the model shows first the subduction of the oceanic crust then of the continental crust below the upper plate margin that is stretched as a result of the trench retreat (figure 2b). When the contient buoyancy and the slab pull forces

International Conference on Geoscientific Modeling -GEOMOD 2014

August 31 st -September 5 th overstep the strength of the lithosphere, the slab is torn first horizontally then vertically along the subducted continental margins (figure 2c). The slab tear results in the exhumation and the rebound of the subducted continental crust and the establishing of a collisional regime (figure 2c). Aside of the collisional domain, the oceanic subduction protracts for longer and slab roll-back and trench retreat occur as faster rates since the slab tearing reduced the along-strike dimension of the slab (figure 2c). Consequently, the upper plate extension rates increase and strike-slip deformation accommodates the rotation of crustal blocks. The rebound of the subducted continental crust following the slab tearing, combined with 3D mantle flow, carries partially molten mantle wedge toward the base of the stretched upper continental plate (figure 3a).

This material then migrates progressively toward the subducting oceanic plate replacing progressively the partially hydrated and molten crustal material previously developed in the back-arc region (figures 3b and 3c). Finally, asthenospheric material rises at the base of the crust, undergoing a possible adiabatic partial melting inducing the development of an alkaline volcanism at the surface (figure 3).

Results from these numerical experiments show a tectonic and magmatic evolution that can be compared to the geological observations across the eastern Mediterranean region. This exercise shows the potential of this double-sided approach involving an independent reconstruction of the long-term evolution of subduction zones based on kinematic reconstructions and 3D thermo-mechanical numerical modeling to investigate the complex interactions between slab behavior, magmatic history, mantle flow and crustal deformation.

International Conference on Geoscientific Modeling -GEOMOD 2014

August 31 st -September 5 th

Figure 1 :

 1 Figure 1: Kinematic reconstructions of eastern Mediterranean region at 30, 15 and 9 Ma, highlighting the main metamorphic, tectonic and magmatic features.

Figure 2 :

 2 Figure 2: 3D thermo-mechanical modeling of retreating subduction neighboring a collisional domain. (a) Initial setup of the 3D model domain with colors showing different rock types. (b, c) Evolution of the reference model at 17.6 and 20.8 Myr. The top layer ("sticky air"", y < ~12 km) and the asthenosphere are cut off for clarity.

Figure 3 :

 3 Figure 3: 35 km-depth horizontal cross-section of the reference model, corresponding to the base of the stretched continental crust, highlighting the different hydrated and partially molten phases.(a, b, c) Evolution of the model at18.2, 19.6 and 20.8 Myr.


Giorgio Dinatale

Sophie Dupuis

Marie-Lise Flottes

Bruno Rouzeyre

Identification of Hardware Trojans triggering signals

Keywords: Hardware Trojan, Hardware Trojan Detection, Hardware Trojan Activation, Logic testing. I

Hardware Trojans are malicious alterations to a circuit. These modifications can be inserted either during the design phase or during the fabrication process. Due to the diversity of Hardware Trojans (HTs), detecting and/or locating them are challenging tasks. Numerous approaches have been proposed to address this problem. Methods based on logic testing consist in trying to activate potential Hardware Trojans in order to detect erroneous outputs during simulation. However, traditional ATPG testing may not be sufficient to detect Hardware Trojans. Hardware Trojans are indeed stealthy in nature i.e. mostly inactive unless they are triggered by a rare value. The activation of a Hardware Trojan is therefore a major concern. In this paper, we propose a procedure to identify circuit sites where a possible HT may be easily inserted. The selection of the sites is based on the assumption that the HT is triggered (i) by signals that have potential rare values, (ii) in paths that are not critical, and (iii) combining multiple gates that are close one to the other in the circuit's layout, and close to available space. This identification is then used to automatically generate test patterns able to excite these sites.

INTRODUCTION

With ever-shrinking transistor technologies, the cost of new fabrication facilities is becoming prohibitive and outsourcing the fabrication process to low-cost locations has become a major trend in IC industry in the last decade. This raises the question about untrusted foundries in which circuit descriptions can be manipulated with the possible insertion of malicious circuitry or alterations, referred to as Hardware Trojans (HTs) [START_REF] Tehranipoor | A Survey of Hardware Trojan Taxonomy and Detection[END_REF]. Besides, recent issues arose from the possibility of getting HTs from untrusted IP vendors [START_REF] Jin | Proof Carrying-Based Information Flow Tracking for Data Secrecy Protection and Hardware Trust[END_REF].

Due to the diversity of HTs, different classifications have been proposed. The proposed classification in [START_REF] Wolf | Towards Trojan-Free Trusted ICs: Problem Analysis and Detection Scheme[END_REF] is based on the activation mechanism (referred as the triggering) and the introduced effect (referred as the payload). The triggering logic monitors a set of inputs to activate the payload at the proper event. A taxonomy is also presented in which HTs are classified based on their trigger and payload mechanisms (digital, analog, combinational, sequential…). The focus of our work is on digital, combinational HTs. The fundamental assumption in that case is that the HT activation should occur under very rare conditions i.e. the trigger is attached on nodes with low controllability. In addition, also for reasons of stealthiness, it is often assumed that the payload is attached on nodes with low observability. This is referred in [START_REF] Wolf | Towards Trojan-Free Trusted ICs: Problem Analysis and Detection Scheme[END_REF] as rare values based HTs. A model of this type of HT is presented in Figure 1. HTs detection methods are divided into two categories: methods based on side-channel analysis [START_REF] Agrawal | Trojan Detection using IC Fingerprinting[END_REF][START_REF] Jin | Hardware Trojan Detection Using Path Delay Fingerprint[END_REF], or logic testing [START_REF] Wolf | Towards Trojan-Free Trusted ICs: Problem Analysis and Detection Scheme[END_REF][START_REF] Chakraborty | MERO: A Statistical Approach for Hardware Trojan Detection[END_REF]. In the latter case, if an erroneous behavior of the IC is observed, it can be inferred that a HT has been inserted in the IC. The most important advantage of logic testing is that, as opposed to side channel analysis, it is robust with respect to environment and process variability. It seems therefore more suitable for the detection of small HTs (whose effects can be beyond the threshold of variability). Yet, traditional ATPG test vectors are not sufficient to detect HTs. The assumption is indeed that an attacker will try to hide the HT of ICs' functional behavior i.e. a HT is mostly inactive and is triggered under very rare conditions. The main concern is therefore to be able to activate potential HTs i.e. to find test vectors that can maximize the chances of triggering potential HTs. Design for hardware trust methods exist also. These methods consist in incorporating into the ICs some features that should improve the HT detectability [START_REF] Salmani | A Novel Technique for Improving Hardware Trojan Detection and Reducing Trojan Activation Time[END_REF][START_REF] Chakraborty | On-Demand Transparency For Improving Hardware Trojan Detectability[END_REF].

In this paper, we propose a procedure to identify circuit sites where a possible HT may be easily inserted. The selection of the sites is based on the assumption that the HT is triggered (i) by gates that have potential rare values as proposed in [START_REF] Wolf | Towards Trojan-Free Trusted ICs: Problem Analysis and Detection Scheme[END_REF][START_REF] Chakraborty | MERO: A Statistical Approach for Hardware Trojan Detection[END_REF], (ii) in paths that are not critical, and (iii) combining multiple gates that are close one to the other in the circuit's layout, and close to available space.

This paper is organized as follows. In Section II, we recall the different proposed logic testing HTs detection methods. In Section III, we present our technique. Finally, Section IV concludes the paper.

II. PRIOR WORK

In order to be able to detect a potential HT by logic testing, the main concern is to be able to activate the HT.

The assumption is that a HT has a stealthy nature and is activated under very rare conditions. Based on this assumption, the HT detection methods aim at optimizing pattern generation techniques in order to maximize the probability of inserted HTs getting activated and therefore detected by logic testing.

The first logic based detection approach is presented by Wolff et al. in [START_REF] Wolf | Towards Trojan-Free Trusted ICs: Problem Analysis and Detection Scheme[END_REF]. The goal is to find so-called HT test vectors i.e. vectors that can detect HTs triggered by rare values. It is assumed that HTs triggered by non-rare values should be detected by traditional manufacturing testing. Firstly, a logic simulator is used to find most likely target sites to attach a HT trigger, i.e. low controllability nodes. This results in a set of Q targets for q-input triggers HTs (note that an assumption is be done on the number of trigger inputs). Secondly, a fault simulator is used to identify low observability nodes, which results in a set of targets for payload. From these two sets, the trigger values and frequencies of each possible HT are computed, as well as the input trigger vectors associated with these trigger values. Then, an ATPG tool is used to check whether each vector from the set can be propagated to the circuit output. Simulations were performed on a small set of ISCAS'85 benchmarks. The results show that, assuming a 2-input HT, ATPG vectors are not sufficient for trigger coverage.

In [START_REF] Chakraborty | MERO: A Statistical Approach for Hardware Trojan Detection[END_REF], Chakraborty et al. propose also to generate a set of test patterns that is compact (in order to minimize test time) while maximizing the chance to detect a HT. The proposed methodology is Called MERO for Multiple Excitation of Rare Occurrence. The assumption is that the number of times a HT trigger condition is satisfied increases with the number of times the trigger nodes have been individually excited to their rare value. This results then in increasing the probability to trigger the HT. From the circuit netlist, a set of random patterns, a list of rare nodes and the number of times to activate each node to its rare value, the set of patterns is modified so that each node satisfies its rare value the desired number of times. To validate the method, a comparison with random and ATPG patterns has been done for a set of ISCAS'85 and ISCAS'89 benchmarks. The signal probabilities were estimated with a simulation tool (and 100 000 random vectors). 100 000 random instances of HTs have been considered, with 2 or 4 triggers and 1 payload. The conclusions are that the MERO set produces a better HT coverage with a reduced test set.

The identification of nodes with low controllability is also referred in [START_REF] Salmani | A Novel Technique for Improving Hardware Trojan Detection and Reducing Trojan Activation Time[END_REF]. To avoid simulations, Salmani et al. propose to compute the probabilities of each node to be '0' or '1'. This consists in propagating the probabilities of each node to be '0' or '1' from the inputs to the outputs, probabilities of ½ and ½ being put to the inputs of the circuits. Therefore, when the probabilities become unbalanced for a node, this node has a low controllability.

It must be mentioned firstly that, in all these techniques, all signals are considered individually, while the trigger can be driven by several signals. In other words, rare values signals are not synonymous of rare logic conditions. Thus, the goal of our approach is to identify sets of signals that conjunctively may trigger a HT. The number of such subsets being prohibitively large, our technique aims at reducing the search space. Furthermore, the position on the triggering signals on the layout is not taken into account, while it is unrealistic that a HT can be inserted anywhere in the circuit. This is an important parameter of our approach.

III. OUR HT DETECTION APPROACH

Our procedure is divided into two steps. First of all, we select a set of nodes that may be targets for the attacker to attach a HT trigger. Second of all, we generate test vectors that aim at activating a set of q triggers. We extend previous approaches by making the nodes' selection according to three criteria: -First, the nodes' controllability: nodes with a low controllability are difficult to set to a required logic value and are therefore good candidate for rare value triggering, -Second, the nodes' slack time i.e. nodes with a positive slack and for which the insertion of a HT does not generate a degradation of delay. The assumption is that the attacker will want to hide the HT from a delay point of view so that it is stealthy. HT detection methods based on delay analysis exist (e.g. [START_REF] Jin | Hardware Trojan Detection Using Path Delay Fingerprint[END_REF]), but are limited to HT impacting only critical paths. The idea behind this criterion is then that by inserting the HT into the available slack, these techniques would fail.

-Third, the position of these nodes in the layout i.e. the nodes for which there is enough free space around to insert an extra door without compromising the placement.

Once the sets of nodes are identified, the test vector generation can be done aiming at activating a subset of q nodes in this set. The subset generation is done according once again to the nodes placement in the layout: our assumption is that the different trigger inputs of a HT must be placed close to each other.

The criteria taken in our approach are intended to better reflect the choices that may be done by an attacker in an untrusted foundry i.e. an attacker who has access to the layout and wants to insert a HT as discreet as possible (both from the functional point of view and from the layout point view).

Our first criterion is the controllability of the nodes, as in previous approaches. The nodes that have a very low controllability i.e. that are difficult to set to a required logic value and are therefore good candidate for rare value triggering are selected. Our second criterion is based on the assumption that an attacker wants the HT to be not visible also from a delay point of view, otherwise it would have been detected by delay-testing. Therefore a selection is made of the nodes with a positive slack. Our third criterion is the "free space" around the nodes. In this case, the assumption is that, for an attacker to insert triggering gates connected to a node, there must be free space close to this node so that the gate(s) can be easily inserted. In this way, the placement of gates around is not compromised, which, if necessary, would be much more visible.

! !

A set of nodes that satisfy these three criteria is then obtained. From this set, an assumption has to be done on the number of triggers (e.g. t) of potential HTs, and test vectors have to be generated in order to activate each possible subset of t triggers. Depending on the number of selected nodes, the set of vectors to be generated can still be very large. Once again, so as to limit the number of vectors and hence, the simulation time, we consider the layout to generate the test vectors: the criterion is the proximity of the nodes to consider. Our assumption is that an attacker will not select nodes that are distant in the layout but nodes close to each other so that the connection of the HT is feasible.

A. Nodes with a low controllability

The first step is to be able to find rarely activated nodes in a circuit. To do that, simulation results can be used. However, an exhaustive simulation is a very time consuming task, and a simulation with only a small portion of input vectors can lead to imprecise results. Using probabilities seems like a good alternative such as presented in [START_REF] Salmani | A Novel Technique for Improving Hardware Trojan Detection and Reducing Trojan Activation Time[END_REF]. This consists in propagating the probabilities of each node to be '0' or '1' from the inputs to the outputs, probabilities of ½ and ½ being put to the inputs of the circuits (cf. Fig. 2). However, this probability-based method gives correct results as long as the input signals of a gate are independent. In order to manage realistic circuits, this method has to take into account reconvergence, i.e. correlations among input signals of a gate. To do that, the support-set of each node is needed i.e. the set of nodes representing a portion of the circuit that has only independent signals as inputs. An example of support-set is presented in Figure 3. In this figure, all gates except gate F have independent inputs. The support-sets if these gates consist in their two inputs. As for gate F, the support-set is {A, 3, 4, 5, 6} (A representing the output of gate A). Then, for each support-set showing reconvergence, the probabilities are computed given all inputs in the supportset. An exhaustive simulation can be performed up to a certain support-set size (e.g. 16 inputs). If the size of the support set is beyond this size, a simulation is performed with random input patterns (e.g. 2 16 patterns). For sequential circuits, flip-flops are considered as primary inputs / outputs i.e. with p 0 =p 1 =½. The p 0 and p 1 of the gates in the combinational part are computed accordingly.

Once the transition probability is known for each node, only nodes with a probability under a certain threshold are good candidates.

B. Slack time

The second step is to know the slack time of each node. HT detection techniques indeed exist that are based on the analysis of the delays of the circuits. The assumption is therefore that the attacker will hide the HT from a delay point of view so that it is stealthy.

Based on the gates delay model and the interconnect, a timing analysis is done to compute the circuit nodes' slack time such as presented in Figure 4: -From the set of arrival times asserted on starting points, the analysis propagates arrival times forward (As Soon As Possible), -From the set of required arrival times asserted on end points, the analysis propagates required arrival time backward (As Late As Possible), -Then, the slack time at any timing node is the difference of its required arrival time minus its arrival time. The more accurate the estimation of the delay is, the better. This computation is therefore done after place and route in order to take into account not only the gates' delay, but also the interconnect's delay. Furthermore, this is what reflects the best the information that an attacker can obtain (from the GDSII sent to the foundry).

Once this information is known for each node, only nodes with a slack time large enough to accommodate the insertion of a HT are good candidates.

C. Layout

The third criterion is the available free space around the gates for an attacker to insert triggering gates. Only nodes for which there is enough free space around to insert an

(1/2, 1/2) (1/2, 1/2) (1/2, 1/2) (1/2, 1/2) (1/2, 1/2) (1/2, 1/2) (1/2, 1/2) (1/2, 1/2)

D. Subsets creation

The last step is to create subsets among the nodes that correspond to the three previous criteria. The number of such subsets being still large, a last criterion is used to reduce it., Signals are partitioned into subsets according to the layout of the circuit: by combining multiple gates that are close one to the other in the circuit's layout Each subset is then a potential triggering support set. This step is also still under development.

IV. CONCLUSION

In this paper, we have presented a method for identifying potential support sets of multiple inputs triggering conditions. Three criteria are taken into account: low controllability, sufficient slack time and available space silicon area. Once these sites are identified, ATPG can be driven to produce input sequences intended to trigger the potential HTs.

Figure 1 .

 1 Figure 1. Rare value based HT circuit model [5].

Figure 2 .

 2 Figure 2. Nodes' probability.

Figure 3 .

 3 Figure 3. Support-set.

Figure 4 .

 4 Figure 4. Slack time computation.

(

  compromising the placement are good candidates. This step is still under development.