Machine learning assisted analysis of the micro-cantilever deflection
The investigated W1-x Tix thin film stacks consists of a silicon substrate Si (100), a 100 nm non-stoichiometric thick SiO layer, and the W1-xTix alloy thin film with a thickness of about 280 nm. Three different W1-xTix compositions are used with x= 15, 20 and 30 at%. The thin films are deposited on the Si (100) substrate by physical vapour deposition (PVD). The deposition is carried out using the same target varying the deposition conditions to achieve different concentrations of Ti in W1-xTix thin films.
For the local detection of the residual stress distribution within the thin films, we prepare a micro cantilever, as shown in Fig. 1a. We remove material within a defined area, the so-called ion layer removal- (ILR-) zone (Fig.1a), using an ion beam and monitor the resulting deflection of the cantilever, see further information in the method section. This micromechanical method is also referred to as the ion layer removal (ILR) approach26–30. The deflection of the micro cantilever as well as the removal process in the ILR-zone is monitored in operando with SEM. For the accurate and an efficient detection of the cantilever deflection on nm-scales, we apply a trained neural network-model (Fig.1a). Machine learning (ML) provides advantages compared to conventional image analysis approaches or manual interpretation with respect to objectiveness, yield and adaptive utilization. Once the model is sufficiently trained it is very valuable with respect to the gained accuracy in comparison to conventional image analysis (See Supplementary Note 3.). Up to 600 images are processed with the ML-based model. The deflection change, due to the abrasion in the ILR-zone is detected, by witnessing the position of a marker (line) imprinted at the end of the cantilever (moving part) and by comparing it´s relative change to an additional marker (line) located close-by on the non-moving part. The model is trained by a sufficient amount of labelled image data which has been segmented semi-automatically by using a conventional image analysis algorithm. Histogram of Gradient (HOG) features are extracted from a cell of 15*15 pixels of the labelled image. Furthermore, principal component analysis (PCA) is done as a part of pre-processing and to extract the relevant features. In total 400 features are extracted and each feature serves as an input to the neural network31. See further details in the method section.
We obtain for the thin film system with 15 and 20 at% Ti a positive deflection, respectively. For 30 at% Ti we observe a negative deflection. Notably, the deflection profile changes significantly within the different W1-xTix layer configurations and stays constant in the SiO and Si layers (Fig.1b).
FEM-Model to extract the residual stress from the ML-analysed deflection profiles
The depicted deflection data is used as an input for the finite element modelling (FEM) to extract the residual stress profile over the layer thickness. For this purpose, we developed a general model in ANSYS MAPDL to overcome past evaluation limitations26,27 e.g. in connection to orthotropic materials. The developed numerical approach is not limited to a certain type of material or material system. We use a general parameterized model32–35, which can be adapted with minimal effort to any material system. Therefore, an iterative bottom up approach for the FEM with an optimization routine was used (see Supplementary Note 2 and method section). Once the geometry is generated, mesh convergence33 is carried out to find a suitable element size for the model (Fig.1c). The used machine learning-based approach for the determination of the deflection provides the possibility to trace the stress distribution over the W1-xTix layer with 15 nm resolution. Figure 1d, highlights the numerically evaluated residual stress distribution over the thickness of the different thin film stacks with 15 at% Ti, 20 at% Ti and 30 at% Ti. A spline fit is used to obtain the stress gradient and the residual stresses along the thickness. The distribution significantly changes over the W1-xTix film thickness and differs for all three different sample configurations. We obtain a compressive stress distribution (positive deflection) for 15 and 20 at% Ti. A tensile stress distribution (negative deflection) is observed for 30 at% Ti. For 20 at% Ti, a maximum in the residual stress distribution is depicted at about 115 nm (measured from the surface). For 20 at% Ti the observed maximum is about 145 nm closer to the W1-xTix / SiO interface when compared to 15 or 30 at% Ti. For the 30 at% Ti sample we observe a shoulder at about 200 nm and a further increase of the residual stress at about 250 nm in the vicinity of the W1-xTix/SiO interface. The sample with 15 at% Ti provides the highest absolute mean stress value (Tab. 1). For 30 and 20 at% Ti, the mean stress shows a similar value, however the values differ in sign. The residual stress distribution for the 20 at% is broader compared to the 15 at% Ti film. By fitting the distribution with a Gaussian fit (see Supplementary Figure 4) an FWHM of about 163 nm and about 134 nm is observed for 15 and 20 at%, respectively. Note, that the Si substrate is stress-free, i.e. shows zero for all three sample configurations (Fig.1d).
In Tab. 1, we compare the ILR mean stress values with X-ray diffraction (XRD) results. XRD, using the sin 2 Psi method, provides an integral method to determine global residual stresses in W1-xTix thin films. The mean residual stress determined from the ILR approach for 15 at% Ti and 20 at% Ti provides comparable values to the XRD result. However, for 30 at% Ti a deviation between the ILR mean residual stress and the XRD measurement with about 800 MPa is obtained.
Table 1: Comparison of residual stresses using the ILR and XRD method.
No
|
Sample
|
Residual Stress Results
|
XRD for W1-xTix stack
|
Mean value ILR for W1-xTix
|
1
|
30 at% Ti
|
2032 MPa ± 203 MPa
|
1279 MPa ± 289 MPa
|
2
|
20 at% Ti
|
-1328 MPa ± 103 MPa
|
-1211 MPa ± 146 MPa
|
3
|
15 at% Ti
|
-2110 MPa ± 153 MPa
|
-2047 MPa ± 389 MPa
|
Grain structure analysis of the nanocrystalline W1-x Tix thin films
We use field emission scanning electron microscopy (FESEM) in combination with electron back scattered diffraction (EBSD) as well as field emission scanning electron microscopy transmission Kikuchi diffraction (FESEM-TKD) to investigate the origin of the depicted different residual stress profiles. The secondary electron (SE)-FESEM images indicate on the W1-x Tix surface (x-y-plane) a lamella-like structure for all three samples, which is sensitive to the Ti concentration. A similar structure for W1-x Tix thin films was observed previousely19. The segmented lamella-like structure for the three different W1-x Tix thin film configurations is presented in Fig.2a (see Supplementary Note 4). The segmented lamellas are coloured according to their orientation from 0° to 90°. Clearly, for 30 at% Ti, a less dense lamella structure with thicker lamellas is depicted for the same region of interest (ROI). The evaluated lamella density is 1640 µm-2, 1449 µm-2 and 730 µm-2 for 15, 20 and 30 at% Ti, respectively.
Figure 2b shows the FESEM-EBSD IPF-Z data from the surface (x-y-plane), for a region of interest (ROI) with 5.8 X 5.2 µm2. The insets illustrate the corresponding secondary electron (SE)-FESEM images. The mean diameter and corresponding standard deviation (see Method section) for 15, 20 and 30 at-% Ti is 117 nm ± 58 nm, 125 ± 59 nm and 108 nm and ± 43 nm, respectively. In Fig. 2c we plot the intensity distribution extracted from the given IPF-Z to visualise the crystallographic orientation for the different W1-x Tix thin films (x-y plane). For 30 at% Ti, the orientation is distributed between [111] and [101]. For 15 and 20 at% Ti, the pole figures reveal a maximum in the intensity at the [101] grain orientation. The misorientation map shown in the supplementary (See Supplementary Figure 5) supports the results of the pole figures. Here, it is clearly demonstrated that the misorientation is highest for the Ti concentration with 30 at%. Figure 2d, shows the cross-section (x-z plane) using FESEM-TKD to resolve the nanocrystalline grain structure of the investigated thin films for an ROI of 500 x 300 nm2. We evaluate the grain shape for the surface and cross-section data by fitting the grains with ellipsoids (See supplementary Figure 6. and Supplementary Figure 10). The corresponding statistics with respect to the minor axis, aspect ratio and area of the fitted ellipses as well as the grain density for the surface (x-y plane) and FESEM-TKD cross-section (x-z plane) is summarized in Table 2. The mean aspect ratio for the cross-section and surface is increasing with increasing Ti content. A stronger increase is demonstrated for the cross-sectional data.
According to table 2, the 30 at% Ti-alloy shows the highest grain density with about 252 µm-2 (150 µm-2) for the cross-section (surface), respectively. Here, the mean area of the fitted ellipse shows the smallest value for the cross-section (surface) with 5.2 x 103 nm2 (3.9 x 103 nm2). This is illustrated by the thin (in x-direction) and densely packed columnar grain structure observed in Fig.2d.
The alloy with 20 at% Ti shows the lowest grain density with about 120 µm-2 (83 µm-2) for the cross- section (surface). Here, in comparison to the 30 at% Ti-alloy, a columnar grain structure with thicker (in x-direction) grains is observed (Fig. 2d). The extracted mean area (Tab.2) of the fitted ellipse provides the largest measure for the cross-section (surface), with about 11.1 x 103 nm2 (5.2 x 103 nm2).
The grain density for the cross-section (surface) of the 15 at% Ti sample (Fig.2d), is smaller than for the 30 at% Ti with about 178 µm-2 (105 µm-2). The mean area (Tab.2) of the fitted ellipses is for the cross- section (surface) 8.1 x 103 nm2 (4.6 x 103 nm2). That is, lies between the 30 and 20 at% Ti alloy. As shown in the EBSD cross-sectional data (Fig.2d) a rather less distinctive columnar structure is shown for 15 at% Ti compared to 20 and 30 at% Ti. Note, that for the cross-sectional data using a larger ROI with 3.26 µm X 300 nm and FESEM-EBSD in reflection, the statistical analysis indicates a similar trend (see Supplementary Table1).
Table 2: FESEM-EBSD surface (x-y plane) and FESEM-TKD cross-section (x-z) grain analysis illustrating the mean value and standard deviation for the minor axis, aspect ratio and area of the fitted ellipses as well as the grain density.
|
W-Ti
|
Minor axis of ellipse
|
Minor axis of ellipse
|
Aspect ratio of fitted ellipse
|
Aspect ratio of fitted ellipse
|
Grain Density
|
Area of ellipse
|
at% Ti
|
nm
|
nm
|
µm-2
|
µm-2
|
|
Mean
|
Standard deviation
|
Mean
|
Standard deviation
|
|
|
Surface
|
15
|
58
|
29
|
1.86
|
0.62
|
105
|
4.6
|
20
|
60
|
30
|
1.94
|
0.62
|
83
|
5.2
|
30
|
50
|
22
|
2.07
|
0.70
|
150
|
3.9
|
TKD Cross- section
|
15
|
31
|
56
|
2.68
|
1.16
|
178
|
8.1
|
20
|
34
|
46
|
2.92
|
1.98
|
120
|
11.1
|
30
|
27
|
25
|
3.37
|
1.17
|
252
|
5.2
|
Elemental analysis of the nanocrystalline W1-x Tix thin films
In addition to the microstructure analysis we perform elemental analysis for the cross-sectional (x-z plane) data to understand the W-Ti distribution within the nanocrystalline alloy. In Fig 3a, we show the scanning transmission electron microscope (STEM) image from the cross-section for the 30 at% Ti sample (Fig.3a). The observed columnar structure indicates a width of approx. 20 nm (in x-direction), similar to the mean value evaluated from the FESEM-TKD data shown in Table 2.
In Fig 3a (middle), we obtain the elemental distribution of W (red) and Ti (green) by using STEM-EDX for the 30 at% Ti sample. Figure 3b provides further information about the element variation along a defined line scan shown in Fig.3a. Significant fluctuations in the element distribution along the x-direction are observed. A detailed statistical analysis is performed in order to examine element distribution in the STEM-EDX cross-sectional data (x-z plane) in Fig.3b (see, Method section). The mean Ti atomic concentration is 28.2% ± 4.2 % for the nanocrystalline thin film. The maximum and minimum of Ti concentration is about 50.8% and 12.0%, respectively. The mean for all the maxima and the minima in the line scans of the Ti is 41.8% is 18.6%, respectively. The observed fluctuations of the local Ti content increase by almost 20 at% in the vicinity of the grain boundaries throughout the measured data. For the analysis of the maximum in Fig. 3b, a Gaussian fit is applied. The FWHM is 3.04 ± 0.29 (See Supplementary Figure 7).
In Fig 3c, we show the cross-sectional STEM image of 20 at% Ti. The elemental distribution obtained from STEM-EDX is shown for W (red) and Ti (green) in the middle. The sample with the nominal 20 at% Ti concentration exhibits a larger grain extension (x-direction) compared to the 30 at% Ti sample. The width of the grain is about 50 nm (x-direction), similar to the mean value evaluated from the FESEM-TKD data shown in Tab. 2. In Fig 3d, the corresponding element distribution along the defined scan in Fig.3a is shown. No significant fluctuations along the x-direction as shown for 30 at% Ti are observed. It provides for the atomic concentration of Ti a mean value of about 18% over the defined line scan width.
For both investigated Ti samples, the observed mean value of the concentration corresponds well with the targeted Ti-concentration. The grey value images (Fig3a and Fig3c) illustrate for the 30 at% Ti as well as for the 20 at% Ti alloy a V-shaped morphology i.e. grains are small on the bottom and larger at the top of the film (zone T)7,37. Commonly, this can be understood, that the grain boundaries become less mobile for larger grains as the film gets thicker during growth1,38.
Atomistic modelling of segregation behavior
In order to address the segregation state observed in Fig. 3, we carried out density functional theory (DFT) simulations of the Ti segregation towards a representative grain boundary in W. The results are illustrated in Fig. 4a. We considered the S3(110)( 11) grain boundary (GB) according to the coincident site lattice model, which has been shown in previous works to be representative for high angle and high energy GBs39–41. Although the real monocrystalline coating contains many different grain boundary orientations which may exhibit relative differences in GB segregation state, we expect that the general segregation trend reported below remains the same.
Segregation energies in the dilute limit are calculated as explained in more detail in the Supplementary Note 5. In the dilute limit, the concentration of Ti atoms in the bulk is low (below 1 at%) to ensure that no interactions between Ti atoms occur in the bulk. On the grain boundary concentration can be much higher if segregation is energetically favourable which according to our definition means that segregation energies are negative. Figure 4a shows the DFT results where different Ti configurations with increasing GB-Ti content are considered. Due to configurational differences (see Fig. 4 i-vi), a considerable spread in segregation energies up to about 1 eV can be observed at a given GB concentration. Additionally, a clear trend towards higher segregation energies can be recognized at higher GB-Ti contents. Importantly, all segregation energies are positive which implies that the GB is not energetically favourable for the solute and vice versa. Therefore, as long as segregation is considered in the dilute limit no enrichment of Ti at grain boundaries is expected from DFT simulations.
The situation changes once the Ti concentration in bulk increases. To understand the reason for this behaviour we focus on the formation energy of bcc W1-xTix in bulk and at the GB. As can be seen in Fig. 4b, the bulk formation energy taken from Ref.42 is negative up to concentrations of about 50 at% Ti with a clear minimum at 12.5 at%. This implies that Ti prefers to be coordinated by W atoms in this concentration range. The formation energy of the grain boundary shows quite a different trend. In general, the offset between the curves is equal to the grain boundary energy per GB atom. The curve shows a monotonic decrease of roughly parabolic shape. The segregation energy in the concentrated solution is given by the difference in the first derivative of the two formation energy curves (see Supplementary Note 5). The result is shown in Fig. 4c where the segregation energy for different bulk concentrations is shown as function of the Ti-GB concentration. For a low Ti-bulk concentration, the curve reproduces largely the segregation energies shown already in Fig. 4a. For higher Ti-bulk concentrations, the curve shifts towards lower values indicative of the attractive binding of Ti to the GB. With the use of a thermodynamic model (see Supplementary Note 5), the GB concentration is calculated for different temperatures. The resulting GB excess, i.e. the increases of Ti atoms at the GB compared to the bulk, is shown in Fig. 4d. Due to the particular dependency of the segregation energies with bulk concentration, a steep increase in GB excess is observed between 0 and 30 at% Ti where the excess shows a maximum. For higher Ti-bulk concentrations, the excess decreases almost linearly to zero. To highlight the importance of this results we also show in the Fig. 4d simplified calculation were the segregation energy is fixed to -0.5 eV irrespective of bulk concentration as was assumed in Ref7. In this case the GB excess would lead to a linear decay over the whole concentration range (see Fig. 4d). GB excess in the region between 0 and 30 at% Ti is, thus, strongly governed by concentration dependent interactions between Ti and W atoms in the bulk and at the GB.