Interface layer effect on the stress distribution of a wafer-bonded bilayer structure

Abstract

The interface layer plays an important role in stress transfer in composite structures. However, many interface layer properties such as the modulus, thickness, and uniformity are difficult to determine. The model developed in this article links the influence of the interface layer on the normal stress distribution along the layer thickness with the layer surface morphology before bonding. By doing so, a new method of determining the interfacial parameter(s) is suggested. The effects of the layer thickness and the surface roughness before bonding on the normal stress distribution and its depth profile are also discussed. For ideal interface case with no interfacial shear stress, the normal stress distribution pattern can only be monotonically decreased from the interface. Due to the presence of interfacial shear stress, the normal stress distribution is much more complex, and varies dramatically with changes in the properties of the interface layer, or the dimensions of the bonding layers. The consequence of this dramatic stress field change, such as the shift of the maximum stress from the interface is also addressed. The size-dependent stress distribution in the thickness direction due to the interface layer effect is presented. When the interfacial shear stress is reduced to zero, the model presented in this article is also demonstrated to have the same normal stress distribution as obtained by the previous model, which does not consider the interface layer effect.

Appendix

In the boundary conditions as given by Eqs. 7 and 8, the fourth and eighth are equivalent to the following two:

$$\sigma _{xz}^1 (z=0)-\sigma _{xz}^2 (z=0)=0$$

$$\sigma _{xz}^1 (z=0)+\sigma _{xz}^2 (z=0)=\frac{2G_i}{\eta}[u^{1}(z=\frac{t_1 }{2})-u^{2}(z=-\frac{t_2 }{2})].$$

We arrange them in the last two rows of the following matrix form:

$$\left[{{\begin{array}{llllllll}{m_{11}}& {m_{12}}& {m_{13}}& {m_{14}}& 0& 0& 0& 0 \\ {m_{21}}& {m_{22}}& {m_{23}}& {m_{24}}& 0& 0& 0& 0 \\ {m_{31}}& {m_{32}}& {m_{33}}& {m_{34}}& 0& 0& 0& 0 \\ 0& 0& 0& 0& {m_{45}}& {m_{46}}& {m_{47}}& {m_{48}} \\ 0& 0& 0& 0& {m_{55}}& {m_{56}}& {m_{57}}& {m_{58}} \\ 0& 0& 0& 0& {m_{65}}& {m_{66}}& {m_{67}}& {m_{68}} \\ {m_{71}}& {m_{72}}& {m_{73}}& {m_{74}}& {m_{75}}& {m_{76}}& {m_{77}}& {m_{78}} \\ {m_{81}}& {m_{82}}& {m_{83}}& {m_{84}}& {m_{85}}& {m_{86}}& {m_{87}}& {m_{88}} \\ \end{array}}} \right]\left({{\begin{array}{lllllll}{a_1} \\ {a_2} \\ {a_3} \\ {a_4} \\ {a_5} \\ {a_6} \\ {a_7} \\ {a_8} \\ \end{array} }} \right)=\left( {{\begin{array}{llllll} 0 \\ 0 \\ {-\frac{H_1}{2}} \\ 0 \\ 0 \\ {\frac{H_2}{2}} \\ 0 \\ 0 \\ \end{array}}} \right)$$

(13)

With m _ij s given as: $m_{11}=-2ke^{\sqrt{2}kt_1},$ $m_{12}=k(2\sqrt{2}-2kt_1-2\sqrt{2}\nu _1)e^{\sqrt{2}kt_1},$ $m_{13}=-2ke^{-\sqrt{2}kt_1},$ $m_{14}=k(-2\sqrt{2}-2kt_1+2\sqrt{2}\nu _1)e^{-\sqrt{2}kt_1}$ $m_{21}=2\sqrt{2}ke^{\sqrt{2}kt_1},$ $m_{22}=k(-2+4\nu _1+2\sqrt{2}kt_1)e^{\sqrt{2}kt_1},$ $m_{23}=-2\sqrt{2}ke^{-\sqrt{2}kt_1},$ $m_{24}=k(-2+4\nu _1-2\sqrt{2}kt_1)e^{-\sqrt{2}kt_1}$ $m_{31}=-\sqrt{2},$ m ₃₂ = 3−4ν₁, $m_{33}=\sqrt{2},$ m ₃₄ = 3−4ν₁ $m_{45}=-2ke^{-\sqrt{2}kt_2},$ $m_{46}=k(2\sqrt{2}+2kt_2-2 \sqrt{2}\nu _2)e^{-\sqrt{2}kt_2},$ $m_{47}=-2ke^{\sqrt{2}kt_2},$ $m_{48}=k(-2\sqrt{2}+2kt_2+2\sqrt{2}\nu _2)e^{\sqrt{2}kt_2}$ $m_{55}=2\sqrt{2}ke^{-\sqrt{2}kt_2},$ $m_{56}=k(-2+4\nu _2-2\sqrt{2}kt_2)e^{-\sqrt{2}kt_2},$ $m_{57}=-2\sqrt{2}ke^{\sqrt{2}kt_2},$ $m_{58}=k(-2+4\nu _2+2\sqrt{2}kt_2)e^{\sqrt{2}kt_2}$ $m_{65}=-\sqrt{2},$ m ₆₆ = 3−4ν₂, $m_{67}=\sqrt{2},$ m ₆₈ = 3−4ν₂, $m_{71}=\sqrt{2}\frac{E_1(1+\nu _2)}{E_2 (1+\nu _1)},$ $m_{72}=(-1+2\nu _1)\frac{E_1(1+\nu _2)}{E_2(1+\nu _1)},$ $m_{73}=-\sqrt{2}\frac{E_1(1+\nu _2)}{E_2(1+\nu _1)},$ $m_{74}=(-1+2\nu _1)\frac{E_1(1+\nu _2)}{E_2(1+\nu _1)},$ $m_{75}=-\sqrt{2},$ m ₇₆ = 1−2ν₂, $m_{77}=\sqrt{2},$ m ₇₈ = 1−2ν₂ $m_{81}=\frac{\sqrt{2}E_1k}{1+\nu _1}-\frac{2G_i}{\eta }e^{\frac{\sqrt{2}}{2}kt_1},$ $m_{82}=\frac{E_1k(-1+2\nu _1)}{1+\nu _1}-\frac{{G_i}kt_1}{\eta }e^{\frac{\sqrt{2}}{2}kt_1},$ $m_{83}=\frac{-\sqrt{2}E_1{k}}{1+\nu _1 }-\frac{2G_i } {\eta }e^{\frac{-\sqrt{2}}{2}kt_1 },$ $m_{84}=\frac{E_1k(-1+2\nu _1)}{1+\nu _1}-\frac{G_i kt_1}{\eta}e^{\frac{-\sqrt{2}}{2}kt_1},$ $m_{85}=\frac{\sqrt{2}E_2 k}{1+\nu _2}+\frac{2G_i}{\eta}e^{\frac{-\sqrt{2}}{2}kt_2},$ $m_{86}=\frac{{E_2}k(-1+2\nu _2)}{1+\nu _2}-\frac{G_i kt_2}{\eta}e^{\frac{-\sqrt{2}}{2}kt_2},$ $m_{87}=\frac{-\sqrt{2}E_2 k}{1+\nu _2}+\frac{2G_i}{\eta}e^{\frac{\sqrt{2}}{2}kt_2},$ $m_{88}=\frac{E_2 k(-1+2\nu _2)}{1+\nu _2}-\frac{G_i kt_2}{\eta}e^{\frac{\sqrt{2}}{2}kt_2}$

The strategy of solving Eq. 13 is to express a ₁, a ₂, a ₃ via a ₄ and a ₅, a ₆, a ₇ via a ₈ first, and then substitute them into the last two equations of Eq. 13 to solve a ₄ and a ₈. Therefore, we have

$$a_1=\frac{\left|{{\begin{array}{lll}{V_1}& {m_{12}}& {m_{13}} \\ {V_2}& {m_{22}}& {m_{23}} \\ {V_3}& {m_{32}}& {m_{33}} \\ \end{array}}} \right|}{D}=\frac{k_1}{D}a_4-\frac{H_1}{2D}(m_{12}m_{23}-m_{13}m_{22}),$$

(14)

$$a_2=\frac{\left|{{\begin{array}{lll}{m_{11}}& {V_1}& {m_{13}} \\ {m_{21}}& {V_2}& {m_{23}} \\ {m_{31}}& {V_3}& {m_{33}} \\ \end{array}}} \right|}{D}=\frac{k_2}{D}a_4+\frac{H_1}{2D}(m_{11}m_{23}-m_{13}m_{21}),$$

(15)

and

$$a_3=\frac{\left|{{\begin{array}{lll}{m_{11}}& {m_{12}}& {V_1} \\ {m_{21}}& {m_{22}}& {V_2} \\ {m_{31}}& {m_{23}}& {V_3} \\ \end{array}}} \right|}{D}=\frac{k_3}{D}a_4+\frac{H_1}{2D}(m_{12}m_{21}-m_{11}m_{22}),$$

(16)

here D, V ₁, V ₂, and V ₃ are defined as

$$D=\left|{{\begin{array}{lll}{m_{11}}& {m_{12}}& {m_{13}} \\ {m_{21}}& {m_{22}}& {m_{23}} \\ {m_{31}}& {m_{32}}& {m_{33}} \\ \end{array}}} \right|, V_1=-m_{14}a_4,\,V_2=-m_{24}a_4,\,V_3=-\frac{H_1}{2}-m_{34}a_4.$$

And k ₁, k ₂, and k ₃ are k ₁ = −m ₁₄(m ₂₂ m ₃₃−m ₂₃ m ₃₂) + m ₂₄(m ₁₂ m ₃₃−m ₁₃ m ₃₂)−m ₃₄(m ₁₂ m ₂₃−m ₁₃ m ₂₂), k ₂ = m ₁₄(m ₂₁ m ₃₃−m ₂₃ m ₃₁)−m ₂₄(m ₁₁ m ₃₃−m ₁₃ m ₃₁) + m ₃₄(m ₁₁ m ₂₃−m ₁₃ m ₂₁), k ₃ = m ₁₄(m ₂₂ m ₃₁−m ₂₁ m ₃₂)−m ₂₄(m ₁₂ m ₃₁−m ₁₁ m ₃₂) + m ₃₄(m ₁₂ m ₂₁−m ₁₁ m ₂₂).

a₅, a₆, and a₇ are:

$$a_5=\frac{\left|{{\begin{array}{lll}{V_5}& {m_{46}}& {m_{47}} \\ {V_6}& {m_{56}}& {m_{57}} \\ {V_7}& {m_{66}}& {m_{67}} \\ \end{array}}} \right|}{J}=\frac{l_1}{J}a_8+\frac{H_2}{2J}(m_{46}m_{57}-m_{47}m_{56}),$$

(17)

$$a_6=\frac{\left|{{\begin{array}{lll}{m_{45}}& {V_5}& {m_{47}} \\ {m_{55}}& {V_6}& {m_{57}} \\ {m_{65}}& {V_7}& {m_{67}} \\ \end{array}}}\right|}{J}=\frac{l_2}{J}a_8-\frac{H_2}{2J}(m_{45}m_{57}-m_{47}m_{55}),$$

(18)

and

$$a_7=\frac{\left|{{\begin{array}{lll}{m_{45}}& {m_{46}}& {V_5} \\ {m_{55}}& {m_{56}}& {V_6} \\ {m_{65}}& {m_{66}}& {V_7} \\ \end{array}}} \right|}{J}=\frac{l_3}{J}a_8-\frac{H_2}{2J}(m_{46}m_{55}-m_{45}m_{56}),$$

(19)

with J, V₅, V₆ and V₇ defined as

$$J=\left|{{\begin{array}{lll}{m_{45}}& {m_{46}}& {m_{47}} \\ {m_{55}}& {m_{56}}& {m_{57}} \\ {m_{65}}& {m_{66}}& {m_{67}} \\ \end{array}}} \right|,\,V_5=-m_{48}a_8,\,V_6=-m_{58}a_8,\,V_7=\frac{H_2}{2}-m_{68}a_8.$$

l₁, l₂, and l₃ are l₁ = −m₄₈(m₅₆m₆₇−m₅₇m₆₆) + m₅₈(m₄₆m₆₇−m₄₇m₆₆)−m₆₈(m₄₆m₅₇−m₄₇m₅₆), l₂ = m₄₈(m₅₅m₆₇−m₅₇m₆₅)−m₅₈(m₄₅m₆₇−m₄₇m₆₅) + m₆₈(m₄₅m₅₇−m₄₇m₅₅).

And l ₃ = m ₄₈(m ₅₆ m ₆₅−m ₅₅ m ₆₆)−m ₅₈(m ₄₆ m ₆₅−m ₄₅ m ₆₆) + m ₆₈(m ₄₆ m ₅₅−m ₄₅ m ₅₆).

Substitution of the expressions of a ₁, a ₂, a ₃in Eqs. 14, 15, 16 and a ₅, a ₆, a ₇ in Eqs.17, 18, 19 into the last two equations of Eq. 13 gives following equations of a ₄ and a ₈:

$$\left\{{{\begin{array}{l}{M_{11}a_4+M_{12}a_8=\frac{H_1}{2D}\alpha _1+\frac{H_2}{2J}\alpha _2} \\ {M_{21}a_4+M_{22}a_8=\frac{H_1}{2D}\gamma _1 +\frac{H_2}{2J}\gamma _2} \\ \end{array}}} \right.,$$

(20)

with M _ij s defined as $M_{11}=\frac{m_{71}k_1+m_{72}k_2+m_{73}k_3}{D}+m_{74},\,M_{12}=\frac{m_{75}l_1+m_{76}l_2+m_{77}l_3}{J}+m_{78},$ and $M_{21}=\frac{m_{81}k_1+m_{82}k_2+m_{83}k_3}{D}+\,m_{84},\,M_{22}=\frac{m_{85}l_1+m_{86}l_2+m_{87}l_3}{J}+m_{88}.$

And α₁ and α₂ are defined as

$$\alpha _1=m_{71}(m_{12}m_{23}-m_{13}m_{22})-m_{72}(m_{11}m_{23}-m_{13}m_{21})-m_{73}(m_{12}m_{21}-m_{11}m_{22})$$

and

$$\alpha _2=-m_{75}(m_{46}m_{57}-m_{47}m_{56})+m_{76}(m_{45}m_{57}-m_{47}m_{55})+m_{77}(m_{46}m_{55}-m_{45}m_{56}).$$

γ₁ and γ₂ are

$$\gamma _1=m_{81}(m_{12}m_{23}-m_{13}m_{22})-m_{82}(m_{11}m_{23}-m_{13}m_{21})-m_{83}(m_{12}m_{21}-m_{11}m_{22}),$$

and

$$\gamma _2=-m_{85}(m_{46}m_{57}-m_{47}m_{56})+m_{86}(m_{45}m_{57}-m_{47}m_{55})+m_{87}(m_{46}m_{55}-m_{45}m_{56}).$$

a ₄ and a ₈ are solved as

$$a_4=A_1 H_1+A_2 H_2, a_8=B_1 H_1+B_2 H_2,$$

(21)

with A _i and B _i (i = 1, 2) given as

$$A_1=\frac{\alpha _1 M_{22}-\gamma _1 M_{12}}{2D(M_{11}M_{22}-M_{12}M_{21})}, A_2=\frac{\alpha_2 M_{22}-\gamma_2 M_{12}}{2J(M_{11}M_{22}-M_{12}M_{21})}.$$

and

$$B_1=\frac{\gamma_1 M_{11}-\alpha_1 M_{21}}{2D(M_{11}M_{22}-M_{12}M_{21})},B_2=\frac{\gamma_2 M_{11}-\alpha_2 M_{21}}{2J(M_{11}M_{22}-M_{12}M_{21})}.$$

Substitute the above solution of a ₈ and a ₄ into Eqs. 14, 15, and 16, and 17, 18, and 19, we have

$$\begin{aligned}a_1 &=C_1 H_1+C_2 H_2,\,a_2=F_1 H_1+F_2 H_2,\,a_3 =I_1 H_1+I_2 H_2\\ a_5 &=P_1 H_1+P_2 H_2,\,a_6=Q_1 H_1+Q_2 H_2,\,a_7=R_1 H_1+R_2 H_2 , \end{aligned}$$

(22)

with

$$C_1=\frac{k_1}{D}A_1-\frac{m_{12}m_{23}-m_{13}m_{22}}{2D},\,C_2 =\frac{k_1}{D}A_2,\,F_1=\frac{k_2}{D}A_1+\frac{m_{11}m_{23}-m_{13}m_{21}}{2D},\,F_2=\frac{k_2 }{D}A_2,$$

$$I_1=\frac{k_3}{D}A_1+\frac{m_{12}m_{21}-m_{11}m_{22}}{2D},I_2 =\frac{k_3 }{D}A_2,P_1=\frac{l_1}{J}B_1,P_2=\frac{l_1}{J}B_2+\frac{m_{46}m_{57}-m_{47}m_{56}}{2J},$$

and

$$Q_1=\frac{l_2}{J}B_1,Q_2=\frac{l_2}{J}B_2-\frac{m_{45}m_{57}-m_{47}m_{55}}{2J},R_1=\frac{l_3}{J}B_1,R_2=\frac{l_3}{J}B_2-\frac{m_{46}m_{55} -m_{45}m_{56}}{2J}$$

Equations 9 and 10 result in the following two equations:

$$\left\{{{\begin{array}{l} {H_1+H_2=2H} \\ {H_1+rH_2=0} \\ \end{array}}}\right..$$

(23)

Here r is defined as

$$r=\frac{-\beta C_2+\sqrt{2}\beta (1-\nu _1)F_2-\beta I_2-\sqrt{2}(1-\nu _1)A_2+P_2-\sqrt{2}(1-\nu _2)Q_2+R_2+\sqrt{2}(1-\nu _2)B_2}{-\beta C_1+\sqrt{2}\beta (1-\nu _1)F_1-\beta I_1-\sqrt{2}(1-\nu _1)A_1 +P_1-\sqrt{2}(1-\nu _2)Q_1+R_1+\sqrt{2}(1-\nu _2)B_1},$$

with

$$\beta=\frac{E_1(1+\nu _2)}{E_2(1+\nu _1)}$$

Therefore, H ₁ and H ₂ are solved as follows

$$H_1=\frac{-2rH}{1-r},H_2=\frac{2H}{1-r}.$$

(24)

Substitution of the above solution of H ₁ and H ₂ into Eqs. 21 and 22 gives the solutions of all a _i s.