Prediction of multi-cracking in sub-micron films using the coupled criterion

Abstract

Sub-micron films deposited on a flexible substrate are now commonly used in electronic industry. The main damaging mode of these systems is a multi-cracking of the film under the action of thermal and mechanical stresses. This multi-cracking phenomenon is described using the coupled criterion based on the simultaneous fulfilment of an energy and a stress criteria. The coupled criterion is implemented in a representative volume element and it allows to decide whether the stress or the energy condition governs the cracking mechanism. It is found that the energy conditions predominates for very thin films whereas the stress condition can take place for thicker films. The initial density of cracks is determined and is in good agreement with the experimental measures. Further subdivisions, when increasing the load, are also predicted. Moreover, under some conditions, a master curve can rule the density of cracks function of the applied strain, showing a good agreement between predictions and experiments for a wide range of film thicknesses.

Appendices

Appendix 1

1.1 Special solutions

E is the Young modulus and $\nu$ the Poisson ratio, the index f holds for film s for substrate, l is the half length of the RVE.

• Tensile loading

  $$\begin{aligned} \left\{ {\begin{array}{l} V_1^\mathrm{t} =\frac{l-x_1 }{l}\hbox { (}e=1\hbox {); }\sigma _{12}^\mathrm{t} =\sigma _{22}^\mathrm{t} =0 \\ \sigma _{11}^\mathrm{t} =\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }\varepsilon _{11}^\mathrm{t} =-\frac{1}{l}\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }\hbox { in the film} \\ \sigma _{11}^\mathrm{t} =\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\varepsilon _{11}^\mathrm{t} =-\frac{1}{l}\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\hbox { in the substrate} \\ \end{array}} \right. \end{aligned}$$

  (28)

• Bending loading

  $$\begin{aligned} \begin{array}{l} \left\{ {\begin{array}{l} V_1^\mathrm{b} =x_2 \frac{l-x_1 }{l}\hbox { (}m=1\hbox {); }\sigma _{12}^\mathrm{b} =\sigma _{22}^\mathrm{b} =0 \\ \sigma _{11}^\mathrm{b} =\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }\varepsilon _{11}^\mathrm{b} =-\frac{x_2 }{l}\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }\hbox { in the film} \\ \sigma _{11}^\mathrm{b} =\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\varepsilon _{11}^\mathrm{b} =-\frac{x_2 }{l}\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\hbox { in the substrate} \\ \end{array}} \right. \\ \\ \end{array} \end{aligned}$$

  (29)

• Constrained thermoelastic loading

  $$\begin{aligned} \left\{ {\begin{array}{l} V_1^\mathrm{c} =0; \sigma _{12}^\mathrm{c} =\sigma _{22}^\mathrm{c} =0 \\ \sigma _{11}^\mathrm{c} =0\hbox { in the film} \\ \sigma _{11}^\mathrm{c} =-\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\varepsilon _{11}^{\mathrm{in}} =-\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\alpha \Delta \theta \hbox { in the substrate} \\ \end{array}} \right. \end{aligned}$$

  (30)

  $\Delta \theta$ is the temperature change and $\alpha$ is the coefficient of thermal expansion, it is taken 0 in the film in both examples.

Appendix 2

The $2\times 2$ system (16) when there is no crack in the film is

$$\begin{aligned} \left\{ {\begin{array}{l} R(\underline{V}^{\mathrm{t}})=\frac{1}{l}\left( {\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }h+\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }t} \right) \\ M(\underline{V}^{\mathrm{t}}) \,{=}\,R(\underline{V}^{\mathrm{b}}){=}\frac{1}{2l}\left( {\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }-\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }} \right) \left[ {\left( {\frac{h+t}{2}} \right) ^{2}-\left( {\frac{h-t}{2}} \right) ^{2}} \right] \\ M(\underline{V}^{\mathrm{b}})=\frac{1}{3l}\left[ \frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\left( {\left( {\frac{h+t}{2}} \right) ^{3}+\left( {\frac{h-t}{2}} \right) ^{3}} \right) \right. \\ \qquad \quad \qquad \left. +\,\frac{E_\mathrm{f} }{1-\nu _\mathrm{f}^2 }\left( {\left( {\frac{h+t}{2}} \right) ^{3}-\left( {\frac{h-t}{2}} \right) ^{3}} \right) \right] \\ R(\underline{V}^{\mathrm{c}})=h\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\alpha \Delta \theta \\ M(\underline{V}^{\mathrm{c}})=-\frac{1}{2}th\frac{E_\mathrm{s} }{1-\nu _\mathrm{s}^2 }\alpha \Delta \theta \\ \end{array}} \right. \end{aligned}$$

(31)

t is the film thickness and h the substrate height.