Journal of Materials Science & Technology, 2020, 54(0): 87-94 DOI: 10.1016/j.jmst.2020.03.043

Research Article

A unified model for determining fracture strain of metal films on flexible substrates

Xu-Ping Wua,b, Xue-Mei Luo,a,*, Hong-Lei Chena,b, Ji-Peng Zoua,b, Guang-Ping Zhang,a,*

a Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China

b School of Materials Science and Engineering, University of Science and Technology of China, Shenyang 110016, China

Corresponding authors: *E-mail addre:xmluo@imr.ac.cn(X.-M. Luo),gpzhang@imr.ac.cn(G.-P. Zhang).

Received: 2020-01-22   Accepted: 2020-03-2   Online: 2020-10-1

Abstract

Failure strain determination of polymer-supported thin films is a key for the design of the flexible devices. A theoretical model R/R0=(L/L0) 2 (R, L are the electrical resistance and the length of the stretched film, respectively. R0, L0 are the corresponding initial values.) has been widely used to determine the fracture strain of thin films on flexible substrates. However, this equation loses its function in some special cases. Here, a simple and universal theoretical model was proposed to determine the fracture strain of metal thin films on flexible substrates in more generally situations. With this model, we investigated the thickness-dependent failure strains of Cu-5 at.% Al films with thickness of 10 nm, 200 nm, 1000 nm, and Ti films with thickness of 50 nm, 100 nm, 300 nm. This model was also employed to study the published data available. The results showed that the new model provided a fairly good prediction of the failure strains of different films.

Keywords: Thin films ; Fracture ; Electrical resistance ; Microstructure ; Flexible substrate

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Cite this article

Xu-Ping Wu, Xue-Mei Luo, Hong-Lei Chen, Ji-Peng Zou, Guang-Ping Zhang. A unified model for determining fracture strain of metal films on flexible substrates. Journal of Materials Science & Technology[J], 2020, 54(0): 87-94 DOI:10.1016/j.jmst.2020.03.043

1. Introduction

Metal thin films are extensively used as interconnections of flexible devices [[1], [2], [3], [4], [5], [6]]. The broken of these interconnections leads to the fracture of electrical connections between electronic components. Therefore, it is of great importance to determine the fracture strain of metal thin films in order to secure the design of flexible devices. However, for polymer-supported metal films, the film fractures much earlier than the substrate, and it is difficult to observe the damage morphology of films at the micron scale and determine their failure strains with naked eyes. Resistance change of the polymer-supported films can effectively reflect the damage of its microstructure. Therefore, the damage evolution process of thin films on flexible substrates are typically evaluated by real-time resistance measurement during deformation. For example, Niu et al. [7] determined the fracture strain for crack nucleation as the strain where the electrical resistance-strain curve transformed from the linear first stage to the nonlinear second stage. However, there is no simple linear relationship between the resistance change and the strain before the fracture of the film. Researches have obtained the theoretical resistance-strain curves of polymer-supported metal films before failure based on the assumptions of volume invariance and electrical resistivity invariance [[8], [9], [10]]. The equation is presented as follows:

ΔR/R0=ε2+2ε

where ΔR=R-R0 is the resistance change of the stretched film, and ε is the engineering strain defined as ε=(L-L0)/L. R, L are the electrical resistance and the length of the stretched film, respectively. R0, L0 are the corresponding initial values.

They defined fracture strain as the strain where the experimental curve deviated from the theoretical curve by 0.05. The proposed model can give a well prediction of the initial resistance variation of the deformed film constrained by a substrate, and has been widely used to determine the fracture strain of metal films on flexible substrates [[11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]]. It should be noticed that in most cases, researchers have focused on the film with a uniform and homogeneous structure which contains few defects. The electrical resistivity of such a thin film is almost a constant during uniaxial tension. Besides, the volume of a high-quality film could generally be regarded as a constant before the film begins to deform plastically. Therefore, the assumptions of Eq. (1) are completely valid in general cases. That’s why Eq. (1) can be widely used to determine the fracture strain of thin films on flexible substrates. However, this theoretical formula is not applicable in some cases because the assumption in Eq. (1) is not valid for some films [7,13,[24], [25], [26], [27]].

First, when decreasing the film thickness to the micron scale, especially down to the nanoscale, the thickness-dependent effects of the surface and the grain boundary on electrical resistivity become prominent. A number of theories have been proposed to interpret the anomalous variation of the electrical resistivity with film thickness, including Fuchs-Sondheimer (F-S) model considering the interface scattering of electrons [28,29], Mayadas-Shatzkes (M-S) considering the grain boundary (GB) scattering [30,31], etc. Besides, a few researches have reported the combination of the F-S and M-S models to consider both interface dispersion and GB dispersion of electrons [[32], [33], [34]].

Furthermore, the microstructure of the small-scale films, especially the local imperfect regions, would influence the electrical resistance variation during deformation. The theoretical derivative results of Meiksin et al. [35] indicated that electrical resistance of ultrathin films increased two orders of magnitude faster with strain than the bulk metal, and it was closely related to the local imperfect regions. Besides, thin films composed of nanoparticle aggregates show much higher sensitivity than that of conventional metal film [24,36] due to a tunneling conduction mechanism [37]. Though Eq. (1) is not valid for some cases, it is still used to determine the fracture strain of thin films in some inappropriate situations [13]. For example, Sim et al. [13] used this formula to determine the failure strain of nanoparticle printed Ag films, even though the experimental electrical resistance-strain curves hardly coincided with the theoretical curve based on Eq. (1). Improper use of Eq. (1) would reduce the accuracy of fracture strain obtained from resistance-strain curve. Therefore, it is of great necessity to develop a more universal method to determine the fracture strain of various thin films. In this work, a more effective and universal model was proposed to precisely determine fracture strain based on the systematic results of polyimide-supported Cu-5 at.% Al alloy films with thickness of 10 nm, 200 nm, and 1000 nm, Ti films with thickness of 50 nm, 100 nm, and 300 nm, and the published data available.

2. Experimental

Ti films with thickness (h) of 50 nm, 100 nm, and 300 nm, as well as Cu-5 at.% Al films with h = 10 nm, 200 nm, and 1000 nm were fabricated on polyimide foils (Kapton HN, 125 μm thick) by magnetron sputtering deposition. Rectangular substrates with a width of 12 mm and a length of 10 cm were used for the deposition of Ti films, and dog-bone-shaped substrates with a width of 6 mm and a gauge length of 40 mm were used for the deposition of Cu-5 at.% Al films. The substrates were cleaned in acetone, isopropanol and DI water for 10 min under ultrasonic condition, respectively. The sputtering power of all the samples was 200 W. The working gas pressure was 5 × 10-3 mbar, with Argon gas used at a flow rate of 30 sccm. For Ti films, the substrates were heated to 200 °C and kept for 30 min before sputtering. Then the Ti films were deposited at 200 °C with a sputtering rate of 0.176 nm/s. The Cu-5 at.% Al films were deposited at room temperature with a sputtering rate of 0.713 nm/s. Then Cu-5 at.% Al films were annealed in the sputtering chamber for 30 min after sputtering, and the annealing temperature was 200 °C for h = 10 nm and h = 200 nm film, 300 °C for 1000 nm films, respectively. The samples were cooled down to room temperature after sputtering and then further kept in the chamber for 30 min to avoid oxidation.

The uniaxial tensile tests were performed on SHIMADZU microforce testing system (MMT-101NV-10) at a constant strain rate of 6.3 × 10-4 s-1. Rectangular tensile specimens with a width of 3 mm for Ti films and a width of 6 mm for Cu-5 at.% Al films were cut from the coated substrates. For the Cu-5 at.% Al films, the width of the tensile specimen was the same as that of the as-deposited film. All specimens were stretched with a gauge of 10 mm at room temperature. The electrical resistance of the films was measured in situ using an Agilent 34,410 multimeter with a resolution of ∼ 0.003 Ω in a four-point measurement method. Conductor wires were adhered to the grasp side of the specimen using copper conductive adhesive. At least 3 samples were tested for each film. Surface morphology of the as-deposited films and the stretched films was characterized by dual-beam focused ion beam microscopy (FEI Nova200 NanoLab) and scanning electron microscope (SEM, Zeiss Supra 35). Crack density of the stretched film was calculated as total length of microcracks per unit area according to the optical micrograph obtained by a laser scanning confocal microscope (LSCM, OLS4000).

3. Results and discussion

3.1. Failure of the existing method

Fig. 1 presents the surface morphology of h = 10 nm, 200 nm and 300 nm Cu-5 at.% Al films. The surface morphology of the film changes significantly with the increase of film thickness. For the h = 10 nm Cu-Al film, a relative continuous and uniform structure was observed in Fig. 1(a). However, as the film thickness increased to 200 nm, the surface of the Cu-Al film was composed of small clusters with inhomogeneous dimensions, as shown in Fig. 1(b). As the film thickness further increased to 1000 nm, the size of clusters increased a lot and several adjacent small clusters were united together to form larger clusters, as shown in Fig. 1(c). The gap between these large clusters was much wider than that between the small clusters. Fig. 2 presents the surface morphology of h = 50 nm, 100 nm and 300 nm Ti films. A compact structure composed of tiny particles was observed when h = 50 nm, as shown in Fig. 2(a). When h increased to 100 nm (see Fig. 2(b)), typical flake-like patterns with jagged circumferences were observed in the films. Similarly, these patterns become larger and the arrangement become looser as increasing the film thickness.

Fig. 1.

Fig. 1.   Surface morphologies of as-deposited Cu-5 at.% Al films with film thickness (h) of (a) 10 nm, (b) 200 nm and (c) 1000 nm.


Fig. 2.

Fig. 2.   Surface morphologies of as-deposited Ti films with (a) h = 50 nm, (b) h = 100 nm and (c) h = 300 nm.


Eq. (1) has been widely used to determine the fracture strain of polymer-supported metal films and has been proved to be valid in most cases. However, our experimental results suggest that Eq. (1) fail to fit the early stage of the resistance-strain curve for the Cu-5 at.% Al films and the Ti films. Fig. 3 shows the electrical resistance-strain curves of Cu-5 at.% Al films and Ti films with different thicknesses under uniaxial tensile loading. The electrical resistance of all the film exhibits a sudden increase after a period of gentle growth, and researches have demonstrated that the sudden rise of the change of electrical resistance reflects the fracture of polymer-supported metal film [7,8]. Now let us have a look at whether the widely cited theoretical curve based on Eq. (1) is in agreement with the experimental curve here. Black dotted line in Fig. 3(a) and (b) shows the theoretical curve based on Eq. (1), which deviates a lot from most of the experimental curves at the beginning of deformation. Only the experimental curve of h = 10 nm Cu-5 at.% Al film is basically consistent with Eq. (1). To further confirm that the theoretical model based on Eq. (1) is inapplicable in some special cases, we consult a lot of references and finally find that this anomaly also exists in several cases, two of which are discussed in detail latter. The electrical resistance-strain curves of Cu films of Niu et al. [7] and nanoparticle-based Ag films of Sim et al. [13] are shown in Fig. 4(a) and (b), respectively, and the theoretical curve based on Eq. (1) is still presented as black dotted line. For the Cu films in Fig. 4(a), the theoretical results of Eq. (1) is significantly higher than the experimental curves of most of the Cu films, except for that of the h = 60 nm Cu film. For the Ag films in Fig. 4(b), the theoretical results of Eq. (1) is in good agreement with the experimental results of evaporated Ag film, while it is a little lower than that of the h = 380 nm nanoparticle-based Ag films. Based on the above analysis, we can confirm that the previous widely used theoretical model (Eq. (1)) does fail to fit the experimental results in some special cases. Therefore, in order to determine the fracture strain of metal thin films on flexible substrates more accurately and more universally, a new and unified theoretical model needs to be developed to fit the electrical resistance-strain curve of polymer-supported thin films for some special cases.

Fig. 3.

Fig. 3.   Electrical resistance-strain curves of (a) Cu-5 at.% Al films with h = 10 nm, 200 nm and 1000 nm and (b) Ti films with h = 50 nm, 100 nm and 300 nm. The theoretical curves obtained by M model (Eq. (6)) and Eq. (1) are plotted as red dotted line and black dotted line, respectively.


Fig. 4.

Fig. 4.   Electrical resistance-strain curves of (a) Cu films of Niu et al. [7] and (b) Ag films of Sim et al. [13]. The theoretical curves obtained by M model (Eq. (6)) and Eq. (1) are plotted as red dotted line and black dotted line, respectively. Inset of (a) presents the fracture strain determined by M model and linear fitting.


3.2. Establishment of a new model

Here, we start from the very basic definition of electrical resistance R:

R=ρL/S

where ρ is the electrical resistivity, L is the length of the sample, and S is the cross-sectional area. If we applied uniaxial tension to the film along the length direction, the relative change of electrical resistance can be expressed as:

dR/R=dρ/ρ-dS/S+dL/L

For a elastically deformed material, the Poisson’s ratio (ν), which is defined as the ratio of transverse strain to longitudinal strain in the elastic loading direction [38], can be employed to understand the shape and volume changes. Thus:

dR/R=dρ/ρ+(1 + 2ν)dL/L

Under the assumptions for Eq. (1) that electrical resistivity and volume of the film remain unchanged during deformation, we have dρ/ρ = 0 and ν = 0.5, respectively. Then we can obtain Eq. (1) by taking integrals of both sides of Eq. (4). Based on the facts that Eq. (1) is not applicable in some special cases, we realize that the assumptions used in Eq. (1) are not valid. Now we attempt to derive a more universal equation without taking account of the above assumptions. Considering that the Poisson’s ratio (ν) is regarded as a constant for a certain material, the latter term on the right side of Eq. (4) can be determined easily. Therefore, in order to figure out the dependence of electrical resistance change on the longitudinal strain, the term dρ/ρ is the most crucial term that we need to analyze in detail. Researches have suggested that the term dρ/ρ is influenced by many complicated factors during deformation, and the value of (dρ/ρ)/(dL/L) of most metals lies in the range of 0-3 [35,36,39]. Next, we will discuss the effects of the term dρ/ρ on the electrical resistance change during deformation from geometry and microstructure of the film.

First, we consider the effect of the size effect on the term dρ/ρ due to the thickness reduction during the uniaxial tensile deformation. Specifically, film thickness decreases slightly when the film is under uniaxial tensile stress, and researches have proved that the electrical resistivity of metal thin film is strongly related with film thickness [[28], [29], [30], [31], [32], [33], [34]]. Thus, we expect that thickness reduction would have an impact on the resistivity under tensile strain. Let us take two films with thickness in nanoscale (h = 10 nm) and submicron scale (h = 500 nm) for concrete examples. If we apply uniaxial tensile test on a rectangular metal film and Poisson’s ratio of 0.4, under a longitudinal strain of 20 %, there would be thickness reduction of 0.8 nm and 40 nm for h = 10 nm and h = 500 nm, respectively. According to the F-S model [29]:

$\frac{\rho }{{{\rho }_{0}}}=\{\begin{matrix} \frac{4}{3}\frac{1-p}{1+p}\frac{1}{\begin{matrix} (h/\lambda )log(\lambda /h) \\ h) \\\end{matrix}}(h\ll \lambda ) \\ 1+(3\lambda /(8h))\left( 1-p \right)(h\gg \lambda ) \\\end{matrix}$

where ρ0 is the electrical resistivity of bulk counterpart, λ is the mean free path of electrons (taken as 40 nm for Cu), h is the film thickness, and p is the fraction of elastically dispersed electrons by the film surface and the film/substrate interface (taken as 0.3). For ease of calculation, only film thickness is regarded as a variable, and all other parameters are treated as constant during deformation. Thus, the resistivity variation of the film as a function of strain can be presented as follows:

$\frac{\text{d}\rho }{\rho }=\left\{ \begin{matrix} \nu \left( 1-\frac{1}{\text{ln}10\text{log}(\lambda /h)} \right)\frac{\text{d}L}{L}(h\ll \lambda ) \\ \frac{(3\lambda /(8h))(1-p)\nu }{1+(3\lambda /(8h))(1-p)}\frac{\text{d}L}{L}(h\gg \lambda ) \\\end{matrix} \right.$

For the h = 10 nm film and the h = 500 nm film, we have dρ/ρ = 0.1115dL/L and dρ/ρ = 0.008227dL/L, respectively. Thus, it is found that the calculated value of dρ/ρ is much smaller than the latter term (1+2ν)dL/L in Eq. (4). Therefore, we can conclude that size effect (thickness reduction during deformation) on the resistivity of the thin film can be ignored.

Furthermore, we consider the microstructural difference between various thin films. The electrical resistivity of thin films is strongly influenced by their microstructure, including grain size, porosity, and so on [20,[40], [41], [42]]. For example, nanoparticle-based Ag films of Sim et al. [13] exhibited different microstructure with evaporated Ag films, and the resistance of the nanoparticle-based film grew slightly faster than that of the evaporated film during deformation. Researches have suggested that the tunneling model can be used to explain the faster growth of resistance of nanoparticle-based films [24,37]. According to the tunneling model, the tunnel resistance between two neighboring nanoparticle shows an exponential relationship with the separation gaps between the nanoparticles. Similarly, according to the surface morphologies of the h = 1000 nm Cu-5 at.% Al film (Fig. 1(c)) and the h = 300 nm Ti film (Fig. 2(c)), we can see that the gap between those clusters or particles would become wider with the increase of strain and thus the resistivity of the film would increase due to the tunneling model. Therefore, we can speculate that porous and discontinuous microstructure is a crucial factor that contributes to the term dρ/ρ during deformation. However, for metal thin films with inhomogeneous and loose structure, it is hard to quantify the electrical resistivity of the whole film.

In order to synthesize the influence of various factors on film resistance during deformation, such as microstructure, and geometry of the film, a parameter m which presents the electrical resistance-strain coefficient can be defined as [36]:

m≡(dR/R)/(dL/L)=(1 + 2ν)+dρ/ρ

where the former term refers to a geometrical effect and the latter term represents a physical effect. The value of m is usually regarded as a constant for a certain film [24,35,36,[43], [44], [45]]. It is remarkable that this coefficient connects the variation of electrical resistance with strain perfectly. Besides, based on its connection with the electrical resistance-strain curve, the correspondence between film failure and deviation of the curve from a corresponding theoretical model has not been proved yet. Based on the aforementioned analysis, the physical effect is predominantly decided by the microstructural features of the film, and specifically, it is closely related to the compactness of structure for our Cu-5 at.% Al films and Ti films and the nanoparticle films. It is remarkable that the parameter m synthesizes the influence of geometry effect and physical effect together. Nevertheless, this coefficient has not been used to interpret the deformation process of general thin films, but typically used to judge the sensitivity of resistive strain sensors [35,36]. Then we attempt to utilize m to establish a new theoretical model and demonstrate the correspondence between film fracture and deviation of the electrical resistance-strain curve from the theoretical model.

Taking the integral of both sides of Eq. (7), we can have:

ΔR/R0=(1+ε)m-1

where ε=ΔL/L0, and the exponent m is an undetermined parameter, which refers to the electrical resistance-strain coefficient before film fracture, i.e., the formation of microcracks. According to Eq. (7), we can get m = 2 when the assumptions of Eq. (1) stand, and in this situation, Eqs. (8) and (1) is identical with the widely cited Eq. (1). Actually, the Poisson’s ratio ν is in the range of 0.3-0.5 for most metals [38], therefore, the term m lies in the range of 1.6-2.0 if there is no resistivity change during the deformation. For this case, the value m quite close to that (m = 2.0) in the widely cited Eq. (1) where resistivity change is ignored. What’s more, the value of m will change a lot when the electrical resistivity change is considered, and thus, the case of m = 2 is only applicable if the term (dρ/ρ)/(dL/L) approximately equals (1-2ν).

Based on the above basic physic deduction, Eq. (8), describing the early stage of electrical resistance-strain curves of thin films, can also be used to determine the fracture strain. Similarly, the fracture strain (εf) is determined as the strain where the experimental curve deviates from the theoretical curve based on Eq. (8) by 0.05. We call this method as M model in the following for convenience.

3.3. Verification of the new model

To this extent, we have proposed a new model to determine the fracture strain of thin films on flexible substrates. However, the universality and validity of the M model remains to be further verified. Next, the M model is used to fit the early stage of electrical resistance-strain curves of the Cu-5 at.% Al films and the Ti films. The data for fitting are selected as follows. First, the curve based on Eq. (8) is plotted initially with an arbitrary m value (for example, m = 2). Then we adjust the value of m so that the plotted curve basically agrees with the early stage of the experimental curve. Then the data points that basically coincide with the drawn curve are selected for further fitting, and a more accurate m value is obtained. The theoretical curve based on the M model is plotted as red dotted line in Fig. 3, and the fracture strain obtained from the M model is presented in Fig. 5. The results indicate that the fracture strain of both Cu-5 at.% Al films and Ti films based on Eq. (8) decreases significantly with the increase of film thickness, at the same time, m value increases a lot. In order to verify the reliability of our model, the fracture strain determined by the M model is further proved to be accurate by comparing with the critical strain for the formation of microcracks. The microcrack densities (total length of microcracks per unit area, represented as D) of the h = 200 nm Cu-5 at.% Al film and the h = 300 nm Ti film under different strains are obtained statistically, as shown in Fig. 6. The critical strain for the formation of microcracks is obtained by extrapolating the D-ε curve to the point of D = 0. Therefore, we can get that the critical strains of the h = 200 nm Cu-5 at.% Al film and the h = 300 nm Ti film are 0.79 % and 1.43 %, respectively, which are exactly consistent with the values determined by the M model. Thus, we can conclude that the fracture strain determined by the M model effectively reflects the formation of microcracks in the film, indicating that the results based on the M model are reliable.

Fig. 5.

Fig. 5.   Fracture strain and m value of Cu-5 at.% Al films and Ti films based on M model as a function of film thickness.


Fig. 6.

Fig. 6.   Microcrack densities as a function of strain for h = 200 nm Cu-5 at.% Al film and h = 300 nm Ti film.


Furthermore, we use the M model to fit some published data available in order to prove its universality. On the one hand, for the experimental data that could not be fitted by Eq. (1), we take Cu films of Niu et al. [7] and Ag films of Sim et al. [13] for example, the corresponding theoretical curves of the M model is shown by red dotted line in Fig. 4. The M model is in excellent agreement with the experimental curves on the early stage of deformation. Critical strains of Cu films obtained from the M model are close to that of linear fitting, as shown in the inset of Fig. 4(a), demonstrating the validity of the M model in determining the fracture strain of thin films. On the other hand, our model remains valid for the experimental data that were well fitted by Eq. (1), such as the electrical resistance-strain curves of Cu films of Lu et al. [8]. Modeling results give m = 2 for these films, and fracture strains determined by our model are the same as that determined by Eq. (1). Therefore, it is demonstrated that we have developed Eq. (1) to a much more universal mode, which can be used to fit the early stage of electrical resistance-strain curve of more thin films and the deviation of the experimental curve from the theoretical curve based on the M model can reflect the fracture of thin films accurately.

3.4. Physical significance of the new model

Here, there is an open question, why the electrical resistances of these films show different dependence of the applied strain during the deformation? According to Eq. (7) which demonstrates the basic physical understanding of the electrical resistance of a conductor during the deformation, m is determined by the geometrical term and physical term. For the geometrical term, Poisson’s ratio is intimately connected with the way structural elements are packed [38]. For the physical term, the electrical resistivity of the thin film is dominated by the scattering of free electrons, which is also greatly influenced by the microstructure of films, such as concentration and distribution of imperfections, surface structure and grain size [20,[40], [41], [42]]. According to the fitting results of the M model on our Cu-5 at.% Al films and Ti films, the case of m∼2 can only fit with the experimental curve of the h = 10 nm Cu-5 at.% Al film, and larger m is obtained for thicker film, as shown in Fig. 5. Combined with the surface morphology of the as-deposited films, we can conclude that the case of m = 2 is applicable for continuous and homogeneous films. This can also be evidenced by the our former work [11]. Thus, it is reasonably that the Eq. (1) is widely applicable for the reported high-quality films [[8], [9], [10]]. However, m>2 corresponds to films composed of loosely arranged small clusters or particles, such as the Cu-5 at.% Al films (h = 200 nm and h = 1000 nm) and the Ti films (h = 50 nm, h = 100 nm and h = 300 nm), as shown in Fig. 1, Fig. 2, respectively. Besides, the electrical resistance of some nanoparticle-based metal films in published literature also increased faster with strain than the theoretical curve based on Eq. (1), i.e., m>2 for these films, such as Ag films of Sim et al. [13] and Au films of Herrmann et al. [24], and the nanoparticle-based metal films exhibited porous structure with voids and gaps between nanoparticles. It suggests that the microstructure difference between films plays a crucial role in the different dependence of the fracture strain on the applied strain. Some imperfections contained in the film, such as gaps or voids, make the electrical resistance changes faster with increasing strain due to the tunneling model.

Besides, it is also noticed that the change trend of fracture strain with film thickness is in contrast to that of magnetron sputtering Cu films [7] and evaporated Ag films [13], but similar to that of nanoparticle-based Ag films [13]. A qualitative explanation is given according to the surface morphology of our Cu-5 at.% Al films and Ti films. As shown in Fig. 1, Fig. 2, it is found that the surface morphology of the film changes from homogeneous to inhomogeneous and more voids and wider gaps appear with the increase of film thickness. As a result, the influence of tunneling model on the electrical resistance change during deformation become greater with increasing film thickness, and thus, the resistance grows faster with applied strain for thicker films. What’s more, the formation and propagation of microcracks is significantly affected by those gaps and voids. Fig. 7 shows the surface morphology of stretched Cu-5 at.% Al films. Isolated small cracks are observed for the h = 10 nm Cu-5 at.% Al films (Fig. 7(a)). It can be inferred that the strong adhesion at the substrate/film interface limits the propagation of cracks for the h = 10 nm Cu-5 at.% Al film. However, transverse channel cracks along the boundaries of the clusters are generated for the Cu-5 at.% Al films with thickness of 200 nm and 1000 nm, as shown in Fig. 7(b) and (c), respectively. The nucleation and propagation of cracks occur easily and quickly along the gaps between the clusters, which causes the formation of channel cracks vertical to loading direction under a very small strain. For Ti films, the fracture mode transforms from ductile mode (h = 50 nm and 100 nm) to brittle mode (h = 300 nm) with increasing film thickness, as presented in Fig. 8. Similarly, transformation of the fracture mode of Ti films can also be interpreted by the change of microstructure (from compact structure to loosely arranged particles), and consequently, the fracture strain of Ti films decreases with film thickness. That’s why the thicker the film, the smaller the fracture strain for the Cu-5 at.% Al films and the Ti films.

Fig. 7.

Fig. 7.   Surface morphologies of stretched Cu-5 at.% Al films with film thickness (h) of (a) 10 nm, (b) 200 nm and (c) 1000 nm.


Fig. 8.

Fig. 8.   Surface morphologies of stretched Ti films with (a) h = 50 nm, (b) h = 100 nm and (c) h = 300 nm.


Apart from determining the fracture strain of metal thin films on flexible substrates in an effective and universal way, the M model introduced a parameter m, which helps interpret the relationship between the electrical resistance-strain coefficient and the fracture strain of thin films on flexible substrates. For our Cu-5 at.% Al films and Ti films and the reported data [7,13], the difference of m is dominated by the discontinuities of the structure, and lager m reflects more voids in the film and looser arrangement of the clusters or particles and also smaller fracture strain. In addition, the m value might be quantified by investigating factors such as geometry, size, distribution and density of voids, width and depth of gaps, porosity, and surface morphology of the film. Therefore, the quantitative relationship between the parameter m and the microstructure of thin films would be a tough work and need further effort. Based on the above analysis, the fracture strain of magnetron sputtered metal thin films can be improved by avoiding the formation of a porous structure during deposition. However, films with a large m may hold great promise for sensor application. It is expected that the resistance-strain coefficient of the Cu-5 at.% Al films and Ti films are closely related to the diameter of particles and the separation gap between them. Further detailed studies are needed to figure out the specific relationship between them.

4. Conclusion

In summary, we have proposed a more general model to determine fracture strains of metal thin films on flexible substrates. Reliability of the model is verified by an extrapolation treatment to crack density-strain curves and the published data available. With this method, the thickness-dependent fracture strains of sputtering Cu-Al films and Ti films were investigated. It shows that the fracture strains of Cu-Al films and Ti films decrease with increasing the film thickness, which is tightly related with their loose and inhomogeneous microstructure.

Acknowledgments

This work was supported financially by the National Natural Science Foundation of China (Nos. 51601198 and 51571199), the Foundation for Outstanding Young Scholar, Institute of Metal Research (IMR), China, the Natural Science Foundation of Liaoning Province of China (No. 20180510025), and the Foundation for Outstanding Young Scholar, the Shenyang National Laboratory for Materials Science, China (No. L2019F23).

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