# Nanoscale stress and microstructure gradients across a buckled Mo-Cu bilayer: Cu self-annealing triggered by interface delamination

A. Lassnig, J. Todt, M. Tkadletz, S. Zak, C. Mitterer, A.A. Medjahed, M.

Burghammer, J. Keckes, M.J. Cordill, M. Meindlhumer

► To cite this version:

A. Lassnig, J. Todt, M. Tkadletz, S. Zak, C. Mitterer, et al., Nanoscale stress and microstructure gradients across a buckled Mo-Cu bilayer: Cu self-annealing triggered by interface delamination. *Acta Materialia*, 2025, 283, pp.120465. 10.1016/j.actamat.2024.120465. hal-05122135

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# Nanoscale stress and microstructure gradients across a buckled Mo-Cu bilayer: Cu self-annealing triggered by interface delamination

A. Lassnig<sup>a,1,*</sup>, J. Todt<sup>b</sup>, M. Tkadletz<sup>b</sup>, S. Zak<sup>a</sup>, C. Mitterer<sup>b</sup>, A.A. Medjahed<sup>c</sup>, M. Burghammer<sup>c</sup>, J. Keckes<sup>b</sup>, M.J. Cordill<sup>a</sup>, M. Meindlhumer<sup>b,*</sup>

## ARTICLE INFO

**Keywords:**

Thin films

buckle type delamination

high resolution stress gradients

transmission electron microscopy, cross-sectional X-ray nanodiffraction

## ABSTRACT

Residual stresses in thin film structures significantly impact their mechanical properties and affect interface delamination. Highly compressively stressed thin films buckling is the predominant interfacial failure mode due to strain energy release. In the present study the effect of cross-sectional stress and microstructural gradients of thin films on the buckling behavior are explored in a model material system consisting of a thin Cu film sputtered onto glass and a highly compressively stressed 500 nm thick Mo overlayer causing buckling delamination at the Cu-glass interface. Employing synchrotron cross-sectional X-ray nano-diffraction, multiaxial X-ray elastic strain and microstructure distributions were explored across the cross-section of the adhering and buckled bilayer, respectively. In the adhering state, a gradual thickness evolution of columnar microstructure and residual stress was found for Mo, while in Cu, no microstructure changes and only minimal stress variations were detected along the film thickness. After delamination, diffraction peak broadening and changes in unstrained lattice parameters in the Cu sublayer indicated structural defect annihilation and grain coarsening. These microstructural changes were further validated via cross-sectional transmission electron microscopy. The evaluated residual stress distributions across the two sublayers of the pristine and buckled bilayer were used to quantify the released strain energy per unit area due to buckling, amounting to 0.61 J/m². Further cross-validation of experimental stress results with finite element simulations strengthened the experimental findings, providing a comprehensive understanding of the stress distribution across the buckled bilayer.

## 1. Introduction

Metallic thin films are indispensable in nowadays technological applications and act as protective coatings [1], diffusion barriers [2] or serve as interconnection structures in microelectronic packaging [3] applications. The downscaling of such components leads to reliability challenges as a high density of different materials occur. Interfaces between thin films and their substrates are known to be mechanically weak and pose reliability challenges since thin film delamination has been recognized as one of the most problematic and prominent failure modes in such applications [4,5]. Especially when metal thin films are bonded to brittle substrates such as dielectric materials [3], their adhesion is known to be fragile [6]. Thus, mechanically stable interfaces between thin films and substrates in multilayer composites [7] are imperative to

ensure the reliability of microelectronic devices [8]. Extensive theoretical and experimental studies [9] have focused on understanding the mechanisms underlying thin film delamination and spallation, trying to identify factors that promote failure as well as to develop strategies to prevent interface delamination. Ultimately the study of interfaces allows to give guidelines to materials scientists to find suitable material combinations and subsequently to improve thin film deposition processes tuning the thin film parameters as to enhance the functionality of such devices.

The concept of interface adhesion strength elucidates the fracture behavior at interfacial regions [10,11], where intrinsic and extrinsic factors are acknowledged to exert influence. Such factors include film thickness, microstructure, interfacial interdiffusion, and operando conditions encompassing temperature gradients and elevated service

<footnote><sup>a</sup> Erich Schmid Institute of Materials Science, Austrian Academy of Science, Jahnstrasse 12, Leoben 8700, Austria</footnote>

<footnote><sup>b</sup> Department of Materials Science, Montanuniversität Leoben, Franz Josef-Straße 18, Leoben 8700, Austria</footnote>

<footnote><sup>c</sup> ESRF, The European Synchrotron, 71 Avenue des Martyrs, CS40220, Grenoble Cedex 9 38043, France</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: alice.lassnig@oeaw.ac.at (A. Lassnig), michael.meindlhumer@unileoben.ac.at (M. Meindlhumer).</footnote>

<footnote n="1">¹ current affiliation: Department of Materials Science and Engineering, University of California, Berkeley, Berkeley, 94720, CA, United States.</footnote>
temperatures, all of them playing pivotal roles in determining interface adhesion properties. Operating conditions can induce chemical reactions and interdiffusion, leading to the emergence of new interlayers with distinct physical characteristics. Another important and inevitable aspect in thin film structures and multilayers are residual stresses, which can also trigger delamination. Notably, these factors are often interconnected; for instance, microstructure and residual stresses are mutually influenced by the film synthesis process parameters and growth conditions and both in turn influence the mechanical properties [12–14].

For a film with uniform in-plane residual stress $\sigma$, film thickness $t$ and given mechanical properties (Poisson's ratio $\nu$, and Young's modulus $E$), steady-state propagation conditions [15] (that is, where the delamination morphology does not change across the buckle and the delamination is away from any edge) prevail during delamination and the energy release $G_{SS}$ is given by:

$$G_{SS} = \frac{1 - \nu^2}{2} \frac{\sigma^2 t}{E} \quad (1)$$

The condition to ensure that the film still adheres to the substrate is $G_{SS} < \Gamma_i$, where $\Gamma_i$ is the interface toughness at the corresponding mode I/mode II loading mode ratio. Eq. (1) shows that the interface fracture properties are influenced by the residual stresses within the film. In the presence of elevated compressive stresses, a characteristic failure morphology observed in thin film delamination is characterized by straight buckles [16].

During thin film buckling delamination, two simultaneous processes unfold: (i) the propagation of interface cracks, resulting in film delamination, and (ii) the buckling phenomenon similar to the deformation of a clamped cantilever beam. Here, the latter leads to the curvature of the thin film buckle [17]. The widely employed model proposed by Hutchinson and Suo (H&S) [18], which is rooted in the elastic beam theory, allows to determine experimentally the film stresses and interface fracture toughness as a function of the loading mode for experimentally obtained spontaneous delamination. Applying the isotropic mechanical properties of the thin film materials ($E$ and $\nu$) and incorporating the buckle morphology (width $2b$, and height $\delta_b$), the critical stress for interface separation and the residual stress within the film can be drawn from this model. These determinations, in turn, facilitate the calculation of loading mode-dependent interface fracture toughness.

Experimentally, thin film adhesion can be quantified by delaminating thin films from the desired interface by the addition of a highly compressively stressed overlayer, first proposed by Bagchi et al. [19]. There, a bilayer (the thin film and a stressed overlayer) delaminates from the substrate and stress (or energy) minimization triggers delamination in the shape of straight or telephone cord buckles [20]. To account for the mechanically different layers, the buckling stress definition was extended by adding a second moment of inertia term accounting for the different mechanical properties of the individual layers [21].

X-ray diffraction (XRD) experiments conducted in reflection geometry have demonstrated that stresses are reduced in the buckled thin film section, while the contiguous adhering section exhibits high compressive film stresses [22,23]. These findings were confirmed for WTi on borophosphosilicate glass, where a highly compressively stressed Mo overlayer was added to trigger delamination leading to a substantial decrease of the residual stresses in the delaminated portion [24]. A new FIB-based technique introduced in a previous study [25] could decouple experimentally the elastic and plastic contributions on Mo-Cu bilayers spontaneously buckling from a glass substrate. There, finite element (FE)-based simulations confirmed the experimental findings, namely that plasticity cannot be neglected in thin films undergoing delamination [26]. Another study on Au films delaminating from Si substrates as circular blisters explained the discrepancy between the measured blister height and the height predicted via pure elastic theory due to plastic deformation of the Au film, which was also confirmed via finite element modelling [27].

However, up to now the impact of stress gradients along the growth direction of thin films on their delamination behavior has not been investigated at high resolution. In particular, the evolution of these stresses in thin films, from the adhering state to the delaminated state, is still experimentally unexplored.

To fulfill this objective, microstructure and stress gradients will be assessed based on peak broadening analysis and X-ray elastic strains, respectively, measured through cross-sectional X-ray nanodiffraction (CSnanoXRD) using synchrotron radiation [28]. In previous studies, this approach has been used in scanning mode to reveal the localized multiaxial stresses and their changes in as-deposited and mechanically loaded thin films. The technique has been proven to be extremely beneficial in the study of multilayer materials [29,30] and how the stress field evolves during the propagation of a crack initiated by in situ indentation [31–35].

In the present study, the primary objective is to obtain the spatial distribution of multiaxial stresses and microstructural changes in a buckling bilayer delaminating from a rigid glass substrate at nanometer resolution using CSnanoXRD. The investigated bilayer system featured a thin Cu film and a highly compressively stressed Mo overlayer on a glass substrate, which showed delamination in the form of straight buckles at the Cu-glass interface. Specialized sample preparation using femtosecond (fs) laser machining enabled the preservation of the delicate buckle structure as a cross-sectional lamella. Additionally, the experimental stress results were utilized to quantify both the stored elastic energy and the strain energy release rate arising from buckling. Correlative cross-sectional transmission electron microscopy and FE simulations provided a comprehensive understanding of the mechanisms involved in buckle delamination. The combined results illustrate that thin film buckling is not only an elastic process but irreversible processes accompanying interfacial delamination could be discovered, leading to a deeper understanding of the toughening mechanisms accompanying thin film delamination.

## 2. Experimental and modelling techniques

### 2.1. Material system

The investigated material system consists of a 100 nm thin Cu film deposited via magnetron sputtering onto a 0.5 mm thick borosaluminosilicate glass substrate (Corning EAGLE2000 AMLCD). Controlled delamination at the Cu-glass interface was triggered via the addition of a highly compressively stressed 500 nm thin Mo overlayer - further referred to as Mo stressed overlayer (Mo SOL) - on top of the Cu film leading to a delamination morphology of straight buckles, as described in our previous study [25].

For the deposition of the Cu film, a lab-scale magnetron sputtering chamber using a single high purity planar Cu target (purity of 99.99 %, diameter 50.8 mm × height 6 mm) was employed. During deposition, the Ar working gas pressure was set to 0.5 Pa (corresponding to an Ar flow of 30 sccm) and a DC current of 0.4 A was applied to the Cu target, while an asymmetric bipolar pulsed DC bias voltage of -100 V was applied to the substrate. The total deposition time of the Cu film lasted 1 min resulting in a film thickness of 100 nm. The 500 nm Mo SOL was then deposited on top of the Cu film using an industrial scale in-line magnetron sputtering system [36] with a cylindrical, rotatable Mo target (FHR Line.600 V, target size diameter 152 mm × height 600 mm). Prior deposition of the Mo SOL, an inverse sputter-etching cleaning step was introduced to remove possible oxides and contaminations on the Cu film surface to promote good adhesion between the Cu film and the Mo SOL. The Cu-glass samples were mounted on a sample carrier and transferred to the deposition chamber after reaching a base pressure of less than $1 \times 10^{-4}$ Pa. Sputter-etching of the Cu was conducted by oscillating the carrier with the Cu-glass samples for six cycles with a velocity of 20 mm/s in front of a radio frequency plasma with an Ar pressure of 0.28 Pa and a power of 0.8 kW. For the deposition of the Mo
film onto the Cu surface, the carrier was stationed in a central position opposite to the magnetron at a distance of 75 mm. The Mo SOL was deposited onto the Cu films with a discharge power of 10 kW, an Ar pressure of 0.37 Pa, and 10 rpm of the target.

After deposition of the Mo SOL, localized delamination at the Cu-glass interface manifested as straight buckles. This morphology is ideal when studying the cross-sectional stresses in the buckled Mo-Cu bilayer system as the buckle propagation direction is maintained (in contrast to undulating telephone cord buckles).

Fig. 1a shows an optical micrograph of the typical buckle morphology of the investigated bilayer in top-view. Straight buckles could be triggered via nanoindentation (highlighted by white circles) or formed spontaneously (highlighted by arrows) in this material system. Regardless of their origin, buckles exhibit the same geometries (same buckle width, 2b, and buckle height, $\delta_b$). The investigations of the buckles were conducted far from any indents to avoid any interactions with the indented portion of the film. A 3D height image (in Fig. 1b) of the buckle was measured using confocal laser scanning microscopy (CLSM) to determine the buckle height and width quantitatively, yielding a buckle width and a buckle height of approximately 30 and almost 1.4 µm, respectively, as shown by a representative height profile in Fig. 1c. The straight buckle morphology with sufficient spacing between the buckles allows site-specific synchrotron lamella preparation containing a single buckle as shown in the schematic in Fig. 1d. The site-specific sample preparation of the synchrotron lamella is detailed in Section 2.2.

## 2.2. Sample preparation for cross-sectional synchrotron X-ray nanodiffraction

For the CSnanoXRD experiments detailed in Section 2.3, a 50 µm thick cross-sectional specimen containing a single straight buckle in its center as shown in Fig. 1d was prepared from the bulk specimen. To avoid any mechanical impact of the sample preparation and to preserve the fragile buckle, the lamella was prepared using a femtosecond (fs)-laser at a wavelength of 532 nm utilizing a 3D-Micromac micro-PREP PRO FEMTO laser micromachining system. A pulse length of 238 fs was chosen to limit the laser-affected zone and the repetition rate was set at 60 kHz [37].

At first a thick, rectangular (3 mm × 1 mm) block containing the buckle in its center was cut out from the bulk sample using 500

repetitions at a laser power of 100 mW. In this step 5 focus levels of 250 µm were chosen. Final thinning was performed at a laser power of 50 mW to obtain an 800 µm wide and 50 µm thin portion of the lamella containing the buckle in its center. Here the repetitions were chosen until sufficient depth was achieved. Amorphous redeposition on top of the films occurred due to the large amount of ablated material.

## 2.3. Cross-sectional synchrotron X-ray nanodiffraction setup

The CSnanoXRD experiments were performed at the nanofocus extension of the beamline ID13 of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France [38]. A pair of multi-layer Laue lenses working with vertical and horizontal focus size of 75 nm and a focal depth of 50 µm were used. The setup was previously described in [39]. An X-ray photon energy of 13.0 keV resulting in a wavelength of 0.0953725 nm was employed. Prior to the experiment, the interface between the glass substrate and the Mo-Cu bilayer film was aligned parallel to the incident beam by performing a set of absorption line scans along the z-axis at various sample orientations $\varphi$ using a passivated, implanted, planar silicon diode to measure transmitted X-ray intensities. The optimal sample position and tilt were determined by maximizing the absorption contrast between the glass substrate and the Cu sublayer, as well as the interface between the Cu film and the Mo overlayer.

Fig. 2 shows a schematic of the setup with the orientation of the synchrotron lamella containing the buckle relative to the beam and the detector with the corresponding coordinate system. Diffraction patterns were collected across the buckle width (y-direction) and buckle height (z-direction). A total of 57,600 2D diffractograms were recorded on an Eiger X 4M CCD detector from a sample region of interest (ROI) with a size of $y \times z=72 \mu m \times 4.5 \mu m$, which corresponds to a step size of 75 nm. The exact detector geometry with respect to the sample was calibrated using a NIST corundum powder [30] yielding a sample-to-detector distance of 109.97 mm.

In the ROI, the cross-sectional variations of phase, microstructure and stress of the portion containing the buckled film were evaluated. For comparison, a line scan was measured in the pristine region, where the Mo-Cu bilayer still adheres to the glass substrate, at a distance of 100 µm away from the buckle, employing a step size of 50 nm. The 2D diffractograms were evaluated using the pyFAI software package [40].

To analyze X-ray elastic strains and quantify the full width at half maximum (FWHM) of X-ray diffraction peaks, cake-like azimuthal in-

<figure><img src="image_4.png" /><figcaption>Fig. 1. Investigated delaminating Mo-Cu bilayer on a glass substrate: (a) an optical image of nanoindentation-induced (indents triggering buckles are highlighted with white circles) and spontaneously formed straight buckles (highlighted with white arrows), (b) CLSM height image of a buckle, (c) corresponding buckle height profile extracted from the region highlighted by a white line in b) including an inset showing a schematic of the Mo-Cu-bilayer delaminating from the glass substrate (height scale is shared in b and c). (d) schematic of prepared cross-sectional lamella via fs laser used for the synchrotron measurements.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 2. A schematic of the CSnanoXRD experiment carried out on the buckled Mo-Cu bilayer thin film. A thin lamella with a thickness of 50 µm in beam direction was scanned along the y- and z-axes using a step size of 75 nm. BS denotes the beam stop.</figcaption></figure>

integrations of Debye-Scherrer rings were carried out in azimuthal angle (δ) steps of Δδ = 10 deg over the whole azimuthal range. Fitting a pseudo-Voigt function to all resulting radial intensity profiles yields for each of the 36 sectors a FWHM<sup>m,hkl</sup>(δ) value and a Bragg's angle positions θ<sup>m,hkl</sup>(δ), where hkl represent the index of the measured Debye-Scherrer ring and m is a particular phase. This analysis was carried out for the Cu 111 and 200 as well as Mo 110 and 200 Debye-Scherrer rings.

Assuming the equi-biaxial stress state, the unstrained lattice plane spacing d<sup>m,hkl</sup> across the buckle can be calculated using Bragg's law as follows:

$$2 \cdot d_0^{m,hkl} \cdot \sin\theta_0^{m,hkl} (\delta_{m,hkl}^*) = n \cdot \lambda \quad (2)$$

where θ<sup>m,hkl</sup>(δ<sup>*</sup><sub>m,hkl</sub>) is the diffraction angle of the selected Debye-Scherrer ring at the azimuthal orientation δ<sup>*</sup><sub>m,hkl</sub> and n denotes the diffraction order. Given the already mentioned assumption of equi-biaxial stress state, the diffraction signal collected at the ring position δ<sup>*</sup><sub>m,hkl</sub> represents the diffraction on {hkl} crystallographic planes, which are not strained. This azimuthal orientation δ<sup>*</sup><sub>m,hkl</sub> can be determined as follows [41–43]:

$$\epsilon_{\delta 0}^{m,hkl}(y, z) = S_{1}^{m,hkl} \left[ \sigma_{xx}^{m}(y, z) + \sigma_{yy}^{m}(y, z) + \sigma_{zz}^{m}(y, z) \right] + \frac{1}{2} S_{2}^{m,hkl} \left[ \sin^2\theta \sigma_{xx}^{m}(y, z) + \cos^2\theta \sin^2\delta \sigma_{yy}^{m}(y, z) + \cos^2\theta \cos^2\delta \sigma_{zz}^{m}(y, z) - \sin^2\theta \cos\delta \sigma_{xx}^{m}(y, z) + \cos^2\theta \sin\delta \sigma_{yy}^{m}(y, z) - \sin^2\theta \sin\delta \sigma_{xy}^{m}(y, z) \right] \quad (6)$$

$$\sin\delta_{m,hkl}^* = \sqrt{\frac{2\nu_{hkl}^m}{1 + \nu_{hkl}^m}} \quad (3)$$

where ν<sup>m</sup><sub>hkl</sub> is the Poisson's ratio of the measured Debye-Scherrer ring hkl

of the material of interest m. Poisson's ratios ν<sup>cu</sup><sub>111</sub> and ν<sup>mo</sup><sub>110</sub> of 0.302 and 0.307, respectively, calculated from the X-ray elastic constants of the Cu [44] and Mo [45] phases using the Hill-type grain interaction model [46], were used to determine the strain-free lattice plane spacing d<sup>m,hkl</sup>.
The strain-free sample directions of δ<sup>*</sup><sub>Cu,111</sub> = ~42.9±0.2 and δ<sup>*</sup><sub>Mo,110</sub> = ~43.3±0.1 deg were found for Cu 111 and Mo 110 rings, respectively, considering the uncertainties of the X-ray elastic constants of Cu and Mo given in [44] and [45], respectively. Finally, the strain-free lattice parameters a<sup>m</sup><sub>0</sub> were calculated according to d<sup>m,hkl</sup> = a<sup>m</sup><sub>0</sub>/√(h<sup>2</sup> + k<sup>2</sup> + l<sup>2</sup>). Far off the buckle, strain-free lattice constants of a<sup>mo</sup><sub>0,f</sub> = 315.231±0.016 pm and a<sup>cu</sup><sub>0,f</sub> = 361.309±0.059 pm were evaluated for the Mo and Cu phases, respectively. The uncertainty of a<sup>m</sup><sub>0</sub> was evaluated by linear propagation of the fitting error of θ<sup>m,hkl</sup>, similar to [47] while other uncertainties (e.g. detector calibration and uncertainties of the used elastic constants [44,45] are comparably negligible (see above)). In order to evaluate buckling-induced changes of the strain-free lattice constants, the ratios between a<sup>m</sup><sub>0</sub>(y, z) and a<sup>m</sup><sub>0,f</sub> were calculated and displayed (cf. Fig. 5c), which may be caused by removal of interstitials or vacancies originating from the thin film deposition process [48] possibly due to Cu self-annealing across the buckle, a phenomenon widely recognized for nanostructured Cu regardless of their processing origin [49].

For quantification of the stress distributions through the thickness of the bilayer thin film system, the measured orientation-dependent lattice strain ε<sup>m,hkl</sup><sub>δ0</sub>(y, z) can be expressed for each thin film material m (Cu, Mo) and each Debye-Scherrer ring hkl as follows:

$$\epsilon_{\delta 0}^{m,hkl}(y, z) = \frac{d_{\delta 0}^{m,hkl}(y, z) - d_{0}^{m,hkl}}{d_{0}^{m,hkl}} \quad (4)$$

where θ is the diffraction angle, δ the azimuthal angle on the detector (see. Fig. 2), while d<sup>m,hkl</sup><sub>δ0</sub>(y, z) is the measured lattice plane spacing and d<sup>m,hkl</sup><sub>0</sub> = a<sup>m</sup><sub>0,f</sub>/√(h<sup>2</sup> + k<sup>2</sup> + l<sup>2</sup>) is the strain-free lattice plane spacing for the two materials (Cu and Mo) and hkl reflection determined above, respectively. The orientation-dependent lattice strain ε<sup>m,hkl</sup><sub>δ0</sub>(y, z) of the measured material is a function of unknown triaxial strain components ε<sup>m,hkl</sup><sub>ij</sub>(y, z) defined in the sample coordinate system (i,j ∈ x,y,z) as follows:

$$\epsilon_{\delta 0}^{m,hkl}(y, z) = \sin^2\theta \epsilon_{xx}^{m,hkl}(y, z) + \cos^2\theta \sin^2\delta \epsilon_{yy}^{m,hkl}(y, z) + \cos^2\theta \cos^2\delta \epsilon_{zz}^{m,hkl}(y, z) - \sin^2\theta \cos\delta \epsilon_{xx}^{m,hkl}(y, z) + \cos^2\theta \sin\delta \epsilon_{yy}^{m,hkl}(y, z) - \sin^2\theta \sin\delta \epsilon_{xy}^{m,hkl}(y, z) \quad (5)$$

Applying the X-ray elastic constants S<sup>m,hkl</sup><sub>1</sub> and 1/2 S<sup>m,hkl</sup><sub>2</sub> allows to transform the strain components ε<sup>m,hkl</sup><sub>ij</sub>(y, z) to the components of the stress tensor σ<sup>m</sup><sub>ij</sub>(y, z) defined in the sample coordinate system (i,j ∈ x,y,z) as follows:

X-ray elastic constants of S<sup>cu,111</sup><sub>1</sub> = -1.837·10<sup>-3</sup> GPa<sup>-1</sup>, 1/2 S<sup>cu,111</sup><sub>2</sub> = 7.912·10<sup>-3</sup> GPa<sup>-1</sup>, S<sup>cu,200</sup><sub>1</sub> = -4.279·10<sup>-3</sup> GPa<sup>-1</sup>, 1/2 S<sup>cu,200</sup><sub>2</sub> = 1.5237·10<sup>-2</sup> GPa<sup>-1</sup>, S<sup>mo,110</sup><sub>1</sub> = -9.83·10<sup>-4</sup> GPa<sup>-1</sup>, 1/2 S<sup>mo,110</sup><sub>2</sub> = 4.181·10<sup>-3</sup> GPa<sup>-1</sup>, S<sup>mo,200</sup><sub>1</sub> = -8.55·10<sup>-4</sup> GPa<sup>-1</sup> and 1/2 S<sup>mo,200</sup><sub>2</sub> = 3.796·10<sup>-3</sup> GPa<sup>-1</sup> were
adopted from literature for Cu [44] and Mo [45] single-crystal elastic constants and averaged using the Hill-type grain interaction model, similar as above.

At first residual stress depth-gradients from the line scan far off the buckle were evaluated supposing that the stress state was equi-biaxial with non-zero in-plane principal stress components $\sigma_{xx}^m(y, z) = \sigma_{yy}^m(y, z)$, further denoted as residual stress $\sigma_r^m(z)$. Thereby, all other stress components, in detail shear stress ($\sigma_{xx}^m(y,z)$, $\sigma_{yy}^m(y,z)$, $\sigma_{xy}^m(y,z)$) and out-of-

and therefore can be approximated with $\sigma_{LM}^m(y, z)$ evaluated using Eq. 7. Moreover, it was supposed that due to the absence deformation gradients across the lamella width (i.e. in direction of the incident beam), $\sigma_{xx}^m(y, z)$ and $\sigma_{xy}^m(y, z)$ will have negligible magnitudes and can be neglected, leaving essentially a 2D problem. The unknown stress components ($\sigma_{yy}^m(y, z)$, $\sigma_{yz}^m(y, z)$, $\sigma_{zz}^m(y, z)$) in Cu and Mo were evaluated as follows:

$$ \epsilon_{\delta\theta}^{m,hkl}(y, z) = S_1^{m,hkl} [\sigma_{xx}^m(y, z) + \sigma_{yy}^m(y, z) + \sigma_{zz}^m(y, z)] + \frac{1}{2} S_2^{m,hkl} [\sin^2\theta\sigma_{xx}^m(y, z) + \cos^2\theta\sin^2\delta\sigma_{yy}^m(y, z) + \cos^2\theta\cos^2\delta\sigma_{zz}^m(y, z) + \cos^2\theta\sin2\delta\sigma_{yz}^m(y, z)] \quad (9) $$

plane stress $\sigma_{xx}^m(y, z)$ components can be neglected to comply with the supposed stress state:

$$ \epsilon_{\delta\theta}^{m,hkl}(y, z) = 2 \cdot S_1^{m,hkl} \cdot \sigma_{yy}^m(y, z) + \frac{1}{2} S_2^{m,hkl} \cdot \sigma_{yy}^m(y, z) [\sin^2\theta + \cos^2\theta \sin^2\delta] \quad (7) $$

Using this simplified linear model, the in-plane lateral and cross-sectional residual stress gradients $\sigma_{yy}^m(y, z)$ were also determined for the mesh scan along the buckled film, which are further denoted as $\sigma_{LM}^m(y, z)$, respectively. The uncertainties for the residual stress were estimated by the uncertainties of the peak fit and the linear regression of Eq. 7, according to [50]. On average, uncertainties of $\Delta\sigma_{LM}^{Mo} = 53$ MPa and $\Delta\sigma_{LM}^{Cu} = 46$ MPa were evaluated for the Cu and Mo across the buckle, while the distributions of uncertainties are given in Suppl. Fig. A-1.

The stored elastic energy was calculated from the residual stress using the $\sigma_r^m(z)$ according to the formula given by Hutchinson and Suo [18] (Eq. 1) adapted as follows:

$$ G_{SS}^{\text{pristine}} = \sum_{m} \sum_{i} \frac{1 - \nu_{m}^{2}}{2} \frac{\sigma_{r}^{m}(z_{i})^{2} \Delta z}{E_{m}} \quad (8) $$

In essence, this more involved model considers two additional stress components, compared to the linear model in Eq. 7. Subsequently, for every measured point in the ROI, a system of 72 linear equations (corresponding to 36 $\delta$ orientations on the detector and 2 reflections per material $m$) based on Eq. 6 was established according to [32] and the three unknown stress components ($\sigma_{yy}^m(y, z)$, $\sigma_{yz}^m(y, z)$, $\sigma_{zz}^m(y, z)$) were refined for Cu and Mo, respectively, while considering $\sigma_{xx}^m(y, z) = \sigma_{LM}^m(y, z)$ in Eq. 5 as a boundary condition. The uncertainties were estimated in this case using the squares of the diagonal elements of the covariance matrix. On average, the uncertainties derived by evaluating the covariance matrix [50] amounted to $(\Delta\sigma_{yy}^{Cu}(y, z), \Delta\sigma_{yz}^{Cu}(y, z), \Delta\sigma_{zz}^{Cu}(y, z)) = (52, 18, 33)$ MPa and $(\Delta\sigma_{yy}^{Mo}(y, z), \Delta\sigma_{yz}^{Mo}(y, z), \Delta\sigma_{zz}^{Mo}(y, z)) = (46, 27, 46)$ MPa. Detailed uncertainty distributions of the individual stress components across the buckle are presented in Suppl. Fig. A-2

Finally, in order to get information about the energy release as a consequence of the buckling process, the von Mises stress $\sigma_{von\ Mises}^m(y, z)$ was calculated from ($\sigma_{yy}^m(y, z)$, $\sigma_{yz}^m(y, z)$, $\sigma_{zz}^m(y, z)$) as follows:

$$ \sigma_{von\ Mises}^{m}(y, z) = \sqrt{(\sigma_{yy}^{m}(y, z))^{2} - \sigma_{yy}^{m}(y, z) \cdot \sigma_{zz}^{m}(y, z) + (\sigma_{zz}^{m}(y, z))^{2} + 3 \cdot (\sigma_{yz}^{m}(y, z))^{2}} \quad (10) $$

where $E_m$ and $\nu_m$ are the isotropic Young's modulus and the Poisson's ratio for Cu and Mo, $i$ denotes the individual residual stress data points gathered from the pristine film presented in Fig. 4e, and $\Delta z = 50$ nm is the step size used for scanning the pristine bilayer thin film.

In order to get further insight into the stress distributions along the buckled thin film it was supposed that the in-plane stress component $\sigma_{xx}^m(y, z)$ along the x-axis did not change significantly due to buckling

The total released elastic energy per unit thickness $E_r$ (in beam direction) was calculated from the von Mises stress and the stored elastic energy density $G_{SS}^{\text{pristine}}$ as follows

$$ E_r = 960 \cdot \Delta y \cdot G_{SS}^{\text{pristine}} - \sum_{m} \sum_{i} \frac{1 - \nu_m^2}{2} \frac{\sigma_{von\ Mises}^m(y, z)^2 \Delta z \cdot \Delta y}{E_m} \quad (11) $$

where $i$ denotes the individual stress data points gathered from the buckled portion of the film presented in Fig. 6d, and $\Delta y = \Delta z = 75$ nm is the step size used for scanning the buckle while spanning the range of 72 $\mu$m width with a step size of 75 nm results into 960 cross-sectional lines throughout the ROI. Using Eq. 8, the strain energy release rate (per buckle area) $G_r$ can then be calculated by dividing the total energy through the buckle width $2b$

$$ G_r = \frac{E_r}{2b} \quad (12) $$

## 2.4. Microstructural changes revealed by transmission electron microscopy

The Mo-Cu bilayer microstructure and changes induced due to buckling were analyzed by transmission electron microscopy (TEM)

<figure><img src="image_6.png" /><figcaption>Fig. 3. Schematic of the FE model highlighting the symmetry plane and artificially induced interface delamination between Cu and the glass substrate corresponding to the length of the half buckle width.</figcaption></figure>
using a Jeol 2200FS operating at 200 kV and compared with the stress evolution revealed by CSnanoXRD. Particular focus was set on revealing microstructural variations in the delaminated Mo-Cu bilayer along the buckle and comparing them to the microstructures of the Cu and Mo films, respectively, when still adhering to the glass substrate. Therefore, two cross-sectional TEM lamellae were extracted from a pristine and delaminated area of the same sample. The first lamella was taken from a region, where the film is still adhering to the glass substrate far away from any delamination and the second lamella was lifted from a buckle in the vicinity of the point of maximum deflection of the Mo-Cu bilayer close to the buckle top. Both lamellae were prepared in a Zeiss Auriga dual beam (SEM and FIB) workstation. The Mo overlayer surface was protected with a Pt layer; throughout the thinning process an acceleration voltage of 30 kV was employed with cutting currents ranging from 1 nA down to 50 pA for the final polishing steps.

## 2.5. Finite element model

The stress distribution of the delaminated, buckled Mo-Cu bilayer was additionally modelled by the finite element (FE) method to evaluate the stress and strain evolution due to buckling taking the present stress gradients of the pristine bilayer as detailed below in Fig. 4e into account. The comparison with experimental stress measurements and ideal, modelled situation can reveal internal mechanisms involved in the process. The model was created and results were processed in the Das-sault Systèmes Abaqus code in the 2018 version.

For a precise calculation combined with feasible computing time, a few simplifications were used when creating the FE model. A 2D plane stress assumption was used for the model, employing one symmetry plane in the geometry of the FE model, as shown in Fig. 3. The delamination process was not included in the simulation, instead, a half-crack with a final length corresponding to the half buckle width of $b = 15 \ \mu m$ was introduced. The geometry was meshed with CPS4R elements (plane stress 2D linear elements with reduced integration). The mesh size was set to be very fine to allow for a precise stress-strain field evaluation and proper implementation of the stress gradient as the initial condition in the simulation. The element size through the substrate thickness was set to be 1 nm on the glass-Cu interface and linearly increasing to 7 µm on the bottom of the model. The Cu layer was discretized in the thickness direction to have elements of length 1 nm on both interfaces (with glass and Mo) and 2.5 nm in the middle. The Mo layer was modelled with the element size in the thickness direction being 1 nm on the interface with the Cu, linearly increasing to 5 nm at the free surface. Along the buckle, the mesh size was set to be 15 nm and in the unbuckled area, the mesh size was set to increase from 15 nm at the end of artificial crack to 3 µm on the far side of the model.

For the purpose of this research, only purely elastic material properties were used for all materials involved. Isotropic material models were used with the assumption of elastic moduli and Poisson's ratios presented in Table 1.

Boundary conditions were set with respect to the symmetry plane (zero displacement in the direction perpendicular to the symmetry plane) and semi-infinite dimensions of the model (symmetry-like boundary conditions in the large distance from the region of interest). Additionally, the residual stresses measured in the unbuckled layers (see Fig. 4e in the Results section) were added as an initial condition for the

<caption>**Table 1**</caption>

Elastic constants (Young's moduli *E* and Poisson's ratios *ν*) used for the numerical modeling of the buckling in the Mo-Cu bilayer film and glass substrate. The anisotropic elastic constants for Cu and Mo were adopted from Refs [44] and [45], respectively, and transformed into averaged aggregates.

<table><thead><tr><th></th><th>Cu</th><th>Mo</th><th>glass [51]</th></tr></thead><tbody><tr><td><i>E</i> / GPa</td><td>129.63</td><td>320</td><td>64</td></tr><tr><td><i>ν</i></td><td>0.342</td><td>0.294</td><td>0.2</td></tr></tbody></table>

respective layers in the model. A tensile stress (with no through-thickness gradient) of 475 MPa was used for the Cu layer. However, a significant through-thickness stress gradient was measured in the Mo layer. Therefore, in the model, a linearly changing compressive stress between -0.83 GPa at the free surface and -1.74 GPa at the interface between Mo and Cu was introduced. This compressive residual stress was the driving force for the portion of the Mo-Cu bilayer to form a buckled geometry – the bilayer with compressive stress and no attachment to the glass substrate (via introduced crack) is only in a semi-stable state when lying parallel to the substrate surface, however, the buckled geometry is more favorable (from the energetic point of view) and the numerical procedures seeking the potential energy minimization switch to the buckled geometry naturally. Therefore, this modelling can be seen to reflect the real spontaneous buckling.

# 3. Results

## 3.1. CSnanoXRD analysis: cross-sectional stress and microstructure evolution

To quantify the stresses across the pristine Mo-Cu bilayer still adhering to the glass substrate, a line scan was measured 100 µm away from the buckle. For each probe position of the line scan, an azimuthal integration of the Debye-Scherrer rings (along $\delta$) was carried out. Collating the $I(2\theta)$ plots, obtained for each cross-sectional position, into a 2D contour map provided a phase plot in Fig. 4a, in which the glass substrate, 100 nm thick Cu film and 500 nm thick Mo overlayer can be clearly distinguished. For qualitative texture analysis, the azimuthal intensity distributions of the Cu 111 and Mo 110 Debye-Scherrer rings are shown in Fig. 4b and 4c, respectively. While the azimuthal intensity distribution of the Cu 111 Debye-Scherrer ring is stochastic and did not show pronounced azimuthal maxima and thus did not reveal any discernible texture (cf. Fig. 4b), the azimuthal intensity distribution of the Mo 110 Debye-Scherrer ring is rather continuous with unpronounced maxima at $\delta = 0, \pm 45, \pm 90$ and $\pm 135$ deg, coinciding with those expected for a weak (110) fiber texture for the Mo overlayer (Fig. 4c).

In a next step, cake-wise azimuthal integration was carried out for each Debye-Scherrer ring in steps of $\Delta\delta = 10$ deg over the whole azimuthal range in order to obtain $I(2\theta)$ dependencies. Fitting a Pseudo-Voigt function yields 36 instances each of FWHM($\delta$) value and a lattice plane spacing $d(\delta)$ for the Cu 111 and Mo 110 peaks. The FWHM was analyzed in the out-of-plane (parallel to the z axis) and in-plane (parallel to the substrate surface) diffraction vector orientations to gain qualitative insights about the grain morphology and defect densities. Generally, the FWHM evaluated from the Debye-Scherrer rings is sensitive to the size of coherently diffracting domains, the presence of high gradients of stresses of 1st order within the X-ray gauge volume, as well as the presence of structural defects such as dislocations, vacancies and other types of micro- and nanoscopic crystal lattice distortions (which are further denoted as microstrains of 2nd and 3rd order). FWHMs evaluated from the Cu 111 and Mo 110 Debye-Scherrer rings in-plane and out-of-plane diffraction vector orientations are presented in Fig. 4d. For Cu, only a slight difference between the in-plane and out-of-plane FWHMs was evaluated, indicating rather equiaxed grains (Fig. 4d). In case of Mo, the observed difference between the in-plane and out-of-plane FWHMs indicates columnar grain growth; similar to the behavior of columnar microstructures already reported in previous studies, see e.g. Fig. 10 in [52,53]. The decrease of the in-plane FWHM in Mo indicates (i) a widening of the Mo columns and (ii) decreasing defect density with the increasing overlayer thickness, where the former is in agreement with the TEM data presented in Fig. 7. In addition to the FWHM, the size of coherently diffracting domains (SCDD) was calculated using Scherrer's formula [54] giving a qualitative measure for the cross-sectional evolution of domain size in Fig. 4d and confirms the columnar growth. However, it has to be stressed that analysis of the grain size by Scherrer's
<figure><img src="image_7.png" /><figcaption>**Fig. 4.** CSnanoXRD data collected from the linescan on the pristine interface region of the thin film lamella (along the z-direction). In (a), the phase plot is presented showing (i) the glass substrate, (ii) the 100 nm thick Cu film and on top (iii) the 500 nm thick Mo overlayer. In (b) and (c) the azimuthal intensity distributions of the Cu 111 and Mo 110 Debye-Scherrer rings are presented, respectively, revealing no texture for Cu and a (110) fiber texture for Mo. The in-plane and out-of-plane FWHM distributions in (d) indicate columnar grain growth for both Cu and Mo, which is however more pronounced for Mo. In addition to the FWHM, the SCDD calculated from the FWHM of the Cu 111 and Mo 110 Debye-Scherrer rings is presented in (d). Finally, in (e) in-plane residual stress distributions evaluated from Cu 111 and Mo 110 Debye-Scherrer rings show tensile and compressive stress gradients for Cu and Mo through the film thicknesses, respectively. (b-e) share the same ordinate, being the film thickness in µm (thickness = 0 coincides with the Cu-glass interface).</figcaption></figure>

formula is only a qualitative measure, since peak broadening by lattice defects and grain size are inherently intertwined and the TEM data presented in Fig. 7 are considered more representative.

Stresses were determined from the orientation-dependent lattice plane spacing as detailed in Section 2.3. Assuming in-plane biaxial stress, the residual stresses were evaluated from the Cu 111 and Mo 110 Debye-Scherrer rings using Eq. 7 and are further denoted as $\sigma_r^m$. Fig. 4e shows the residual stress dependencies as a function of film thickness (along the film growth direction), revealing tensile and compressive stress gradients for Cu and Mo, respectively. In the vicinity of the Cu-glass interface, the Cu film shows tensile residual stresses of 530±50 MPa, which slightly decrease to 470±35 MPa close to the Mo-Cu interface. These tensile stresses are in agreement with previous literature data from 100 nm thin Cu films processed via magnetron sputtering [55,56] and our previous measurements in reflection geometry on the same film system [25].

In contrast, the Mo overlayer shows high compressive stresses of -1.76±0.09 GPa close to the Mo-Cu interface, which decrease to -0.92 ±0.08 GPa close to the Mo surface. The stress values obtained for the Mo films align with those acquired through reflection geometry for similarly

synthesized Mo films documented in the literature [57]. The decrease of the stress magnitude towards the Mo film surface agrees with the concomitant decrease in the in-plane FWHM (cf. Fig. 4d), indicating a decrease in the defect density with the increasing overlayer thickness.

Finally, the stress data obtained from the pristine Mo-Cu bilayer were subsequently used to calculate the stored elastic energy of the pristine Mo-Cu bilayer $G_{SS}^{pristine}$ according to Eq. 8, which summed up to 1.52 J/m², where the Cu and Mo layers contribute 0.13 and 1.39 J/m², respectively. The mode I adhesion energy of the same Cu-glass interface material system (the same Cu and Mo deposition parameters) was evaluated using the model of Hutchinson and Suo yielding a value of $\Gamma_i=(1.1\pm0.1)$ J/m² [25], meaning that the energy released during the delamination process is lower than the elastic strain energy stored in the pristine bilayer. This is expected as delamination can only occur for $\Gamma_i<\Gamma_{SS}$.

In the next step, the Mo-Cu bilayer film was mapped across the buckle using a step size of $\Delta y = \Delta z = 75$ nm. Fig. 5a shows the small angle X-ray scattering (SAXS) intensities obtained by integrating azimuthally and radially the signal around the beam stop. Generally, the diffuse X-ray scattering at (relatively) small diffraction angles can be
<figure><img src="image_8.png" /><figcaption>**Fig. 5.** SAXS signal distribution (a); averaged FWHM micrographs of the buckled Cu-Mo bilayer (b); relative lattice parameter variation of the thin film across the buckle (c). Please note the different color codes representing the SAXS intensity, FWHM, and relative lattice variations in (a), (b) and (c), respectively. Additionally, the SCDD calculated from the FWHM of the Cu 111 and Mo 110 Debye-Scherrer rings is given in the color codes in (b).</figcaption></figure>

attributed to electron density variations originating from the presence of
nanoscale structural heterogeneities within the X-ray gauge volume [58,
59]. In the present case the gauge volume was
(x × y × z) = (50 × 0.075 × 0.075) µm³. The density variations may
include alternation of phases, presence of grain boundaries, interfaces,
cracks, surface and interface roughness, precipitates and pores with sizes
in the magnitude of λ/θ, where λ represents the X-ray wavelength and θ
is half the Bragg angle [59]. Here the 2θ-range between 0.1 and 1.0 deg
corresponds to features with sizes of 55–5.5 nm, respectively. While in
Mo no significant variation of the SAXS intensity could be detected (cf.
Fig. 5a), the rise of the SAXS intensity in the buckled Cu portion (be-
tween y = -15 to 15 µm) may be attributed to an increase in surface
roughness and/or an increased density of cracks or pores in the Cu film.
Here, the SAXS intensity increase only stems from the increased surface
roughness of the Cu film, which was previously attached to the glass
substrate, as confirmed by cross-sectional TEM, which will be discussed
in. Section 3.2.

Fig. 5b shows the averaged FWHM distribution within the Mo-Cu
bilayer along the buckled thin film, where the image combines the
FWHM values obtained for both the Cu 111 and Mo 110 reflections. The

distribution was obtained by arithmetically averaging the orientation-
dependent 36 FWHM values. In Mo, in the still adhering part of the
film the averaged FWHM varies between 0.27 and 0.22 deg, in good
agreement with the data presented for the pristine film in Fig. 4d. A
slight increase in FWHM evaluated from the Mo overlayer at the center
and the edges of the buckle is visible (Fig. 5b), which can be explained
by pronounced gradients of stresses of 1st order within the gauge vol-
ume, as shown in Fig. 6. In Cu, in the still adhering part an average
FWHM slightly above 0.2 deg was evaluated. In contrast to Mo, in the
delaminated portion of the Cu film, the FWHM decreased to a value of
0.15 deg.

To get further insights into the microstructural variations in Cu
across the buckled thin film, the ratio between the unstrained lattice
constant across the buckle and the unstressed lattice constant far off the
buckle $a_0^m(y, z)/a_{0f}^m$ was evaluated under the assumption of equi-biaxial
stress over the buckled part of the bilayer system. The $a_0^m(y, z)/a_{0f}^m$ ratio
is shown in Fig. 5c. In the case of Mo, it can be seen that no significant
variation of the lattice parameter could be detected, the maximum
variation of the lattice constant is below $a_0^{Mo}(y, z)/a_{0f}^{Mo}=1\pm0.001$.
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** The in-plane biaxial stress evaluated for Mo and Cu using the linear stress model is presented in (a), while von Mises stress calculated from the $\sigma_{yy}$, $\sigma_{yz}$ and $\sigma_{zz}$ components (see supplemental material for individual components) is presented in (b). Please note the different color codes representing the stress magnitudes of $\sigma_{LM}$, and $\sigma_{von Mises}$ in (a) and (b), respectively.</figcaption></figure>

Generally, the variation of $a_0^{Mo}(y, z)/a_0^{Mo}$ displays only gradients which can be associated with the rotation of the principal stress axis due to the buckling. This conclusion can be drawn by comparing $a_0^{Mo}(y, z)/a_0^{Mo}$ and the shear component $\sigma_{yz}^{Mo}$ in Fig. 5c and Suppl. Fig. A-3b, respectively, where positions with increased magnitudes of $\sigma_{yz}^{Mo}$ coincide very well with changes of the unstrained lattice constant.

In contrast, while in the adhering part of the bilayer no variation of the lattice constant can be seen in Cu, the $a_0^{Cu}(y, z)/a_0^{Cu}$ is reduced to 0.995 in the buckled portion of the film. This reduction of the unstrained lattice constant aligns well with the significant reduction of the FWHM in the buckled portion of the Cu layer. A possible reason is given by Mukherjee et al. [65] who compared grain size, lattice parameter, residual strain with total film thickness and found for films up to 100 nm, that the lattice parameter decreases with increasing grain size and film thickness. Another, although speculative, possibility is, that Ar interstitials commonly incorporated in thin films during magnetron-sputter deposition [66–68] were removed during delamination of Cu accompanied by plastic deformation which was followed by the subsequent self-annealing of Cu.

The color-coded in-plane biaxial stress denoted as $\sigma_{LM}^m$ is shown in Fig. 6a and for comparison the experimentally obtained von Mises stress $\sigma_{von Mises}^m$ distribution is shown in Fig. 6b. The $\sigma_{von Mises}^m$ was calculated according to Eq. 10 using the individual stress components $\sigma_{ij}^m$, which are shown in detail in Fig. A1 in the supplemental material section.

Figs. 6a and 6b show the in-plane equi-biaxial and von Mises stress variations along the entire scanned region of interest containing the buckled Mo-Cu bilayer. The results contain stress distributions from a still adhering portion on both sides of the buckle, respectively. The buckle shape, and hence the stresses are symmetrically arranged around $y = 0$ µm, which corresponds to the location of maximum buckle deflection, often referred to as the buckle top. While the film is only delaminated at distances of $-15$ µm $\le y \le 15$ µm, corresponding to the

buckle width 2b (cf. CLSM and SAXS data presented in Figs. 1 and 5a, respectively), the stress distributions in the adhering bilayer are deviating from the as-deposited film up to distances of $|y| \ge 25$ µm. Only in the outer region around $|y| \ge 25$ µm on both sides, i.e. more than another half buckle width b away from the buckle onset, the stress field of the adhering bilayer is not altered by the presence of the buckle and, thus, corresponds to the stress distributions of the as-deposited thin film (see linescan in Fig. 4e). In the adhering portion next to the delaminated buckle, a transitional zone could be identified which is located around the onset of the delamination and includes a delaminated and a still adhering part ($7$ µm $\le |y| \le 20$ µm). In this region, the buckle onset corresponds to the point of inflection. This transitional region has a concave curvature and is decisive for the final buckle shape.

In Fig. 6a, showing the in-plane stress variations across the buckling bilayer using the equi-biaxial stress model (Eq. 7), a clear change of the in-plane stress field is evidenced. Where the Cu film is still adhering, the film portions show moderate tensile stresses in the order of 500 MPa, which is in agreement with the line scan of the pristine region in Fig. 4e. In contrast, the Cu film in the buckled region is almost stress free. The change from +500 MPa in the adhering region to 0 MPa the delaminated region is abrupt and does not show any stress concentrations associated with an interface crack (as shown in the FE model discussed in Section 3.3). In the Mo overlayer, the residual stress starts to decrease already 25 µm away from the buckle top ($y = \pm 25$ µm) and decreases to between -50 MPa (surface) and -150 MPa (Mo-Cu interface) at the buckle onset ($y = \pm 15$ µm). Closer to the buckle top at $y = \pm 12$ µm (in the already delaminated part), a bending stress gradient between -600 MPa (surface) and +200 MPa (Mo-Cu interface) was evaluated, due to the concave buckle shape at the buckle onset. The evaluated bending stress gradient is dependent on the local curvature of the buckle and changes its sign towards the buckle top (i.e. at $y = 0$ µm), so that the stresses range from +800 to -1500 MPa close to the Mo film surface and the vicinity of the Mo-Cu interface film, respectively, due to the convex buckle shape. In detail, the residual stress distributions evaluated from
the Mo overlayer can be divided into (i) stresses originating from the deposition process superimposed by (ii) stress distributions apparently similar to the ones found in a clamped cantilever [34,69].

Additionally, Fig. 6b shows a different representation of the stress distribution showing the experimentally obtained von Mises stress distribution (Eq. 10). Here, it should be noted that the individual stress components $\sigma_{ij}^m$ retrieved from Eq. 9 are highly sensitive to the used unstrained lattice constant, which varies strongly in Cu along the buckle width (Fig. 5c): The $d_0$ variation observed in Cu cannot be accounted for in the model corresponding to Eq. 9, which leads to increased $\sigma_{von\ Mises}$ values along the buckle ($y = -15$ to $15$ µm). Therefore, an offset of $\sigma_{zz}^Cu$ magnitudes due to the lattice parameter change may shift the presented stress results obtained from the Cu layer. Following this, no significant stress variation is visible in the $\sigma_{von\ Mises}^Cu$ throughout the delaminated region of the sample in Fig. 6b. But as shown in Fig. 5c, the variation of the lattice parameter is nearly negligible in the Mo overlayer and the results presented fit very well to the data obtained using the equi-biaxial stress model. Far from the buckle onset, the $\sigma_{von\ Mises}^{Mo}$ stress distributions display a gradient ranging from 1600 to 1000 MPa (Fig. 6b), indicating that the evaluated stress gradient is slightly diminished. At the buckle onset ($y = \pm 15$ µm), a $\sigma_{von\ Mises}^{Mo}$ stress ranging between 450 and 280 MPa at the Cu-Mo interface and the surface was evaluated in the Mo overlayer. In the delaminated part ($y = \pm 12$ µm), approximately at the point of highest concave curvature, $\sigma_{von\ Mises}^{Mo}$ magnitudes between +250 and +700 MPa were calculated from the stress data at the Mo-Cu interface and the Mo surface, respectively. Furthermore, at the buckle top $\sigma_{von\ Mises}^{Mo}$ varied between 1500 and 350 MPa, where the maximum stress at 1500 MPa was evaluated at the Mo-Cu interface and a minimum stress of 350 MPa at the center of the Mo overlayer, while at the surface a stress magnitude of 700 MPa was found. The deviations of $\sigma_{von\ Mises}^{Mo}$ from the equi-biaxial stress model can be related to slight lattice parameter changes (Fig. 5c) in the Mo overlayer, which are not accounted for in the evaluation. However, most of the observed differences are rooted in the divergence of the actual stress state in the Mo film from the equi-biaxial model.

Finally, both the $\sigma_{LM}^m$ and $\sigma_{von\ Mises}^m$ stress distributions were used to evaluate the strain energy release rate using Eq. 11. When using the $\sigma_{LM}^m$ stress distributions, the strain energy release rate amounts to 0.61 J/m². As stated above, there is an offset in stress in the $\sigma_{ij}^Cu$ stress distributions due to a varying lattice parameter in the Cu layer, but the $\sigma_{ij}^Cu$ stress

magnitudes are nearly negligible compared to the $\sigma_{ij}^{Mo}$ stress magnitudes. Thus, the von Mises stress results can be used to calculate the strain energy released due to buckling according to Eq. 11 which amounted to 0.40 J/m². The differences to the strain energy release rate computed from the equi-biaxial stress distributions $\sigma_{LM}^m$ originates from the strain energy stored in Cu (0.13 J/m²) and can be related to the stress offset found in the $\sigma_{von\ Mises}^Cu$ distributions and a slightly reduced strain energy release rate in Mo when using $\sigma_{von\ Mises}^{Mo}$ from the calculation.

## 3.2. Microstructural evolution studied via cross-sectional TEM

Cross-sectional TEM micrographs of the Cu-Mo bilayer obtained in scanning TEM (STEM) mode were recorded for the pristine and delaminated areas to study microstructural and morphological changes in the Cu film after delamination. Micrographs of the pristine region and delaminated region are shown in Fig. 7a, b and 7c, d, respectively. Fig. 7a shows a bright-field (BF) STEM image of the pristine interface. The sample was taken far away from any delaminated regions where the Mo-Cu bilayer is still attached to the glass substrate. The corresponding dark field (DF) STEM image is shown in Fig. 7b highlighting the competitive microstructure in the Mo overlayer. A small-grained Cu microstructure is observed showing a bimodal morphology with multiple small equiaxed grains arranged within the Cu film thickness and narrow columnar grains. This agrees with the in-plane and out-of-plane FWHM plot obtained for the linescan through a pristine region of the Cu film presented in Fig. 4d. The bimodal nature of the as-deposited Cu synthesized with the same deposition parameters is already known from [25].

Figs. 7c and d show a BF- and DF-STEM image for the lamella extracted from the delaminated region at the buckle top. After delamination the Cu grains are larger, indicating that grain growth occurred during or after the delamination process. The average grain size in the Cu film was estimated in the TEM micrographs in Fig 7a (when still adhering) and Fig 7c (after delamination) by applying the average grain intercept method in the bottom part of the Cu layer (i.e. in the vicinity of the glass-Cu interface/free Cu surface) yielding an average grain size of approximately 66 nm in the adhering case and 113 nm in the delaminated case. This result shows the same trend as predicted by the SCDD evaluation using Scherrer's equation (24–31 nm in the adhering case vs. 31–61 nm in the delaminated case). Discrepancies of the results from both techniques (TEM and X-ray profile analysis) by a factor of two are a

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Cross-sectional micrographs of the pristine interface shown in (a) and (b) (BF-STEM and DF-STEM, respectively), compared to the cross-sectional micrographs of the delaminated bilayer BF-STEM in (c) white arrows highlighting twinned Cu grains and DF-STEM in (d) showing an overview of the delaminated bilayer: black arrows indicate an increased roughness/undulation at the free surface of the Cu film (this side was formerly attached to the glass substrate). The white layer beneath the Cu film is redeposited material due to the FIB preparation.</figcaption></figure>
well-known fact and are especially present in plastically deformed materials due to high presence of dislocation substructures [70].

A possible explanation for the observed grain growth is that after the delamination, which was possibly accompanied by plastic deformation in Cu, the Cu underwent self-annealing at room temperature. This phenomenon is known for nanocrystalline Cu processed by SPD, which serves as an analogy here, where after removal of the hydrostatic pressure, i.e., the mechanical constraint, grain growth is observed over time [64,71]. Additionally, twin boundaries are observed in the delaminated Cu in Fig. 7c (as indicated by white arrows), which is in agreement with Shinada et al., who observed an increasing fraction of twin boundaries in thin film Cu after room temperature self-annealing [60]. Furthermore, an undulating morphology of the newly-formed released Cu surface is observed in Fig. 7c, and hence increased surface roughness as highlighted by black arrows in the DF-STEM image (see Fig. 7d), which is indicative for a grain rearrangement due to the release of the constraint previously given by the glass substrate. The increased surface roughness supports the increase in SAXS signal for the delaminated Cu in Fig. 5a and was also reported in literature as sign for room temperature self-annealing of Cu [49].

In the pristine and in the delaminated state, the Mo overlayer shows a variation of the grain size along the film growth direction. In both cases an ultra-fine-grained lamellar seed layer is seen in the first 40 nm close to the Cu film, highlighted by a white dotted line in the Mo layer in Fig. 7a, b, and c, respectively, followed by a lamellar microstructure extending towards the Mo surface. This Mo film morphology is known from previous works employing similar sputter deposition parameters [57]. The observed microstructure evolution resembles competitive grain growth, which is commonly observed in thin films deposited by magnetron sputter deposition [12,28,72,73]. The competitive morphology is also in line with the stress gradient obtained from the CSnanoXRD line scan in Fig. 4e being more compressive near the interface to Cu (-1.76 GPa) and less compressive (-0.92 GPa) in the coarser grained lamellar structure close to the Mo film surface. Furthermore, stress gradients with decreasing compressive stress magnitudes are also regularly observed as a result of the competitive thin film morphology [12,28,34]. The columnar microstructure of the Mo overlayer is also confirmed by the in-plane and out-of-plane FWHM distributions shown in Fig. 4d.

### 3.3. Modelled stress distributions

To compare with the CSnanoXRD stress results, the stress distribution in the buckle was calculated using FE modelling, assuming the 2D plane stress conditions and initial state described in Section 2.5. For the comparison with the experimental measurements, the equivalent von Mises stress component was extracted and plotted over the buckled geometry.

The von Mises stresses from the FE simulation show a similar stress distribution as the experimentally obtained stress distribution presented in Fig. 6b, however with the absolute stress magnitudes being significantly higher due to the elastic assumptions for the model. The buckled region (Fig. 8 between details in Fig. 8a (buckle top) and Fig. 8b (buckle onset) show a strong bending stress field in the Mo layer with two different directions of bending, whereas the inflexion point is approximately in the middle of the half-buckle width. The bending stress shows larger values for the “compression” side (lower part in Fig. 8a and upper part in Fig. 8b) of the Mo than for the “tension” side (upper part in Fig. 8a and lower part in Fig. 8b) of the Mo layer. This corresponds well with the initial state of the model including pure compression of the Mo layer. In the region away from the delaminated portion of the bilayer in Fig. 8c, the Mo shows the initial stress value, since there was no geometrical change. It is worth noting that the change between the stress influenced by the buckling and the delamination crack tip stress field (see Fig. 8b) and the steady-state stress far away from the buckle (Fig. 8c) is not sudden or uniform. The transition between them is rapid for

approximately 1.1 µm from the end of the buckle (see the knee-like bend in the red contour in the right third of the Fig. 8b). However, after this rapid change, there is a slow, linear change between the right edge of the Fig. 8b and the steady state in Fig. 8c. This slow adjustment in the von Mises stress takes place over approximately 10 µm. The Cu thin film exhibits very small amounts of stresses in the buckled area (Fig. 8a) with an increase in proximity to the delamination crack tip (Fig. 8b) due to the stress concentration and stronger bending deformation. In the region away from the delamination, the stress is still at the level of the original Cu layer state. The detailed view of the delamination crack tip on Fig. 8b shows also a stress increase in the glass substrate to values almost at the 1 GPa level, which could not be measured during the CSnanoXRD measurements due to its amorphous structure. Finally, the energy mitigated by buckling of the bilayer could also be calculated from the elastic FE model and divided by the half buckle width *b* in order to receive the numeric strain energy release rate, which amounted to 0.85 J/m², somewhat higher than the experimentally obtained *G*<sub>IC</sub>.

## 4. Discussion

A high-resolution analysis of the stress distributions in a buckled Cu sublayer, which was triggered via a stressed Mo overlayer, was carried out using combined CSnanoXRD, TEM and FE methodologies. Summarizing briefly, the experimental stress distributions in Cu strongly deviated from the stresses calculated via finite element simulations and/or from geometrical considerations. More specifically, the FE calculations predicted a high von Mises stress in the top surface of Mo film around the buckle top, whereas experimentally it shows lower values. Additionally, FE calculations of the Mo film in the vicinity of the buckle predict too high von Mises stress values. Experimentally, microstructural changes in the delaminated Cu were found, which are most likely the reason for the unexpected stress relaxation in the delaminated Cu and will be further discussed in this section.

While it has been previously shown experimentally via XRD in reflection geometry that originally compressively stressed thin films showed an integral stress relief in buckled regions [22–24,74] we could show for the first time the multiaxial stress evolution in a buckled film at high resolution using CSnanoXRD. In the present study the Cu film undergoes a stress change from a moderately tensile (500 MPa) state in the adhering condition to a neutral stress state (approximately 0 MPa) in the buckled condition corresponding to a stress transition to a less tensile state, which contradicts elastic theory that buckle delamination is based on. Given the experimental (Fig. 6) and modelled stress (Fig. 8) distributions in the Mo overlayer, which both depict a deformation somewhat similar to a double-clamped cantilever, the Cu layer should be subjected to tensile and compressive stresses at the buckle onset and the buckle top, respectively. In the linear elastic FE model this is the case, but (re-iterating) the experimental stress distributions in the Cu layer deviate strongly from the modelled stress.

We can start our analysis with a highly simplified model, geometrically visualized in Fig. 9a. There, a comparatively more tensile in-plane stressed state of the buckled part film is rationalized by approximating one half of the buckled film by a straight line. Geometrically, using the hypotenuse, *l*, calculated from the legs, $\delta_b$ and *b*, the average elongation of the buckled portion of the film can thus be estimated as $\Delta\epsilon = l/b$, which amounts to 0.4 % of tensile strain.

In the present case, buckling of the Cu film was triggered by a compressively stressed Mo overlayer. The Mo overlayer undergoes a stress relief from an on average -1.28 GPa compressive stress state in the adhering case to an almost stress-free state (Fig. 6a). Assuming total stress relief and using Hooke's law the buckled Mo film with a Young's modulus of $E_{Mo} = 329$ GPa the tensile strain can thus be estimated as $\epsilon = 0.4 \%$, similar to assumptions made in [25] and the geometric considerations above. Consequently, the Cu film has to follow the same elongation leading to a tensile stress increase $\Delta\sigma_{Cu,buckle} = +440$ MPa when assuming a Young's modulus for sputtered Cu films of $E_{Cu} = 110$ GPa
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Von Mises stress distribution in the delaminated Mo-Cu-bilayer obtained via FE simulations (stress units: MPa); detail views: (a) central portion of the delamination buckle, (b) delamination crack tip area, (c) pristine region far away from delamination (Cu still adheres to glass substrate) (color scale applicable for a through c).</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Schematic showing the in-plane film expansion when undergoing buckling: film length in buckled condition will always exceed the film length in its adhering state (corresponding to 2b) is presented in (a). In (b), the experimental horizontal stress data $\sigma_{yy}^m$ are presented for one half of the buckle, while in (c) the horizontal stress component calculated by the FE model is shown. Full details of the horizontal stress components evaluated from the experiment and the FE model are shown in Fig A-3 and Fig. A-5 of the supplementary material, respectively.</figcaption></figure>

[75]. Following those assumptions it is expected that the initially tensile stressed Cu, on average, has a high in plane tensile stress after buckling at a magnitude of $\sigma_{Cu,buckled} \approx 0.9$ GPa, possibly surpassing the elastic limit of nanocrystalline copper [76]. Given this simplistic geometric assumption, a strain energy release rate of $2.2 \text{ J/m}^2$ was evaluated, considering the average stress and strain changes in Cu and Mo, significantly overestimating the $G_{IC}$ values of 0.4-0.61 J/m$^2$ and 0.85 J/m$^2$ from the experimental data and the FE model, respectively.

Similarly, the elastic stress data computed via FE modelling indicate maximum stresses (component in y-axis direction) of +1.2 and -0.2 GPa in Cu at the buckle onset and top, respectively, while a direct comparison between experimental and modelled stress distributions in y-direction is given in Fig. 9b and 9c, respectively. The comparison between CSnanoXRD experiment and FE model, yields obviously a good agreement of the stress distributions (Figs. 6, 7, 9b and 9c), while the magnitudes of the model are exaggerated due to the linear-elastic assumptions. However, the differences between the experiment and the FE model highlight the regions, where plastic deformation is indicated. Clearly, the tensile stress levels of up to 1.2 GPa evaluated in the FE model surpass the yield strength of nanocrystalline Cu ranging between 200-800 MPa [76–80], depending on sample and grain size. These tensile stress levels (Figs. 7 and 9c) would be then mediated by plastic deformation in the initially nanocrystalline Cu film, showcasing again the benefits of the linear-elastic FE model. Consequently, plastic deformation would lead to an introduction of lattice defects, such as dislocations and vacancies, which in turn are represented in the CSnanoXRD experiment by an increased FWHM [33] contrary to the observed experimental data in Fig. 5. This deviation from an ideal elastic(-plastic) case is reasoned by the microstructural changes observed by CSnanoXRD (Fig. 5) and TEM (Fig. 7). Cross-sectional and lateral FWHM results in the Cu film presented in Fig. 5b show a decrease from 0.2 to 0.15 deg in the adhering and the buckled (delaminated) portion of the Cu film, respectively, which is either indicative of dislocation or vacancy annihilation (reduction of stresses of 2^nd^ and 3^rd^ order) or an increase of the size of coherently diffracting domains [81]. The concomitant reduction of FWHM (Fig. 5b) and unstrained lattice constant $a_0^{Cu}(y, z)$ (Fig. 5c) show that both dislocation as well as interstitial and vacancy annihilation and an increase of the size of coherently diffraction domains took place during or after buckling. As shown by Mukherjee et al.

[65] the lattice parameter of nanocrystalline Cu films decreases with increasing grain size and film thickness. In turn, the observed concomitant decrease of FWHM (Fig. 5b) and unstrained lattice constant $a_0^{Cu}(y, z)$ (Fig. 5c) may be interpreted by an increased grain size. Additionally, during magnetron sputter deposition Ar interstitials are commonly incorporated in thin films [66–68], increasing the lattice parameter. It is reasonable, that Ar interstitials were removed during delamination of Cu, which is accompanied by plastic deformation as discussed above. Furthermore, the short diffusion pathway to the newly formed surface due to delamination may also enhance Ar removal. Moreover, the increase of grain size is further confirmed by the additional cross-sectional TEM investigations (Fig. 7). The grain growth at room temperature in the delaminated film can be caused by the removal of the constraint given by the attachment to the substrate. Room temperature grain growth is not uncommon in nanocrystalline Cu [82] and has also been previously observed for nanocrystalline Cu processed via severe plastic deformation [63,64] and Cu nanocrystalline thin films [60–62]. In SPD, the nanocrystalline material is synthesized by applying hydrostatic pressure and shear deformation simultaneously, while in case of Cu, grain growth is observed over time after removal of the hydrostatic pressure, i.e., the mechanical constraint [63,64]. Here, grain growth of Cu is observed after delamination accompanied by plastic deformation as discussed above, i.e. after the removal of the mechanical constraint induced from the bond to the glass substrate. Plastic deformation of ductile films delaminating from brittle substrates in the shape of buckles or blisters has previously been studied experimentally and numerically [25–27].

Sputter-deposited nanocrystalline Mo can release tensile stresses only via cracking at room temperature [83], which was neither observed by an increased SAXS intensity nor by TEM in Fig. 5a and Fig. 7, respectively. The absence of microstructural changes in the Mo overlayer highly suggests exclusively elastic deformation during buckling. In particular, the remaining stress distributions in the Mo overlayer fit well with the clamped and bearing-supported cantilever models of elastic theory and agree well with the FE model. Thus, the simplified model from above may overestimate the strain energy release rate, since the buckled portion of the Mo overlayer is not actually stress-free after the buckling process. The room temperature grain growth in the Cu film after delamination most probably resulted in a further relaxation of the
stress distributions in the Mo overlayer, thus in this case the estimated strain energy release rate evaluated from the experimental stress data is most probably an upper limit.

Finally, from the experimental and numerical observations of microstructure and stress in a buckled Mo-Cu bilayer film on glass it can be concluded that the observed state of the bilayer is divided into two separate consecutive parts: (i) buckle formation with elastic loading in the Mo film and plastic deformation in the Cu film, and (ii) room temperature grain growth of Cu accompanied by stress relief and elastic rebound in Mo. From the experimental data it was possible to quantify the stress contribution to energy release upon buckling, which was calculated between 0.61 J/m² using $a_{LM}^m$ and 0.40 J/m² using the von Mises stress, which is lower than the predicted value of 0.85 J/m² obtained via FE computation.

## Summary and Conclusion

In this study we explored the stress distribution of a sputter deposited Cu film undergoing delamination from a glass substrate by a stressed Mo overlayer, forming a straight buckle. Employing CSnanoXRD, we aimed to uncover the underlying mechanisms accompanying the buckle-type delamination of a thin and ductile film from brittle substrates by investigating high resolution microstructural changes and stress distributions across the buckled portion. Contrary to predictions based on geometric assumptions and theory, which suggest an increasingly tensile stress in a buckled film, the CSnanoXRD results revealed that post-delamination the Cu film is in a stress-free state in the buckled region. Elastic models are not sufficient to describe buckling due to plastic deformation of the Cu film during delamination. The zero-stress level in Cu could be attributed to grain coarsening at room temperature. In particular, the FWHM and $a_0^{Cu}(y, z)/a_0^{Cu,f}$ plots demonstrated a reduction in FWHM and unstrained lattice constant, respectively, indicating grain coarsening and defect annihilation in the delaminated Cu. Grain coarsening was further evidenced by TEM of the buckled Cu thin film. Due to the applied unique experimental conditions, which allowed to quantitatively evaluate the 2D stress tensor, the stress relaxation due to buckling could be uncovered for the first time, which was similar to the deformation of a clamped cantilever. As a final result of this investigation, the stress contribution to the energy release by buckling was quantified for the first time experimentally and amounted to 0.61 J/m² using the data from the equi-biaxial stress model. This new insight into delamination of thin films via buckling can help to further foster architectural designs with improved mechanical performance.

## CRediT authorship contribution statement

**A. Lassnig:** Writing – original draft, Investigation, Funding acquisition, Conceptualization. **J. Todt:** Writing – review & editing, Investigation, Data curation. **M. Tkadletz:** Investigation. **S. Zak:** Investigation. **C. Mitterer:** Writing – review & editing, Supervision. **A.A. Medjahed:** Investigation. **M. Burghammer:** Investigation. **J. Keckes:** Writing – review & editing, Supervision. **M.J. Cordill:** Writing – review & editing. **M. Meindlhumer:** Writing – original draft, Investigation, Data curation, Conceptualization.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

A. L. acknowledges funding of the Austrian Science Fund (FWF) under the grant numbers T891-N36 and Y1236-N37. M. M. and J. K.

gratefully acknowledge the financial support for this work provided by Österreichische Forschungsförderungsgesellschaft mbH (FFG), project "CuttingEdgeNanoMat" (No. 46568228/FO999899020). The provision of synchrotron facilities by the ESRF (European Synchrotron Radiation Facility) is gratefully acknowledged by A.L., J.K., J.T. and M.M. S. Z. received funding in the frame of the ESPRIT program under grant number ESP 41-N. Authors are grateful to Tanja Jörg, Velislava Terziyska and Karl-Heinz Pichler (all Montanuniversität Leoben) for the deposition of the thin film samples.

## Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.actamat.2024.120465.

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