Lagrange-Kirchhoff Model for curvature of Circular Silicon Plate, Annular Silicon Plate and Taiko Silicon Plate

1. Stresses and the Curvatures in a Linear Elastic Plate

Plate theory is an approximate theory. It turns out to be an accurate theory provided the plate is relatively thin but also that the deflections are small relative to the thickness (deflection/t < 0.75) and when the ratio between the thickness t and the plate width L is t/L <1/20. The plate showing the same elastic behavior in all directions is called isotropic [1].

In case the plate has different elastic properties in the two orthogonal directions is called orthotropic. The elastic problem for an orthotopic plate, in the form developed by Lagrange, is

$${\sigma }_{xx}=\frac{E_x}{1-{\nu }_{xy}{\nu }_{yx}}\left({ϵ}_x+{\nu }_{yx}{\epsilon }_y\right)$$

(1.1)

$${\sigma }_{yy}=\frac{E_y}{1-{\nu }_{xy}{\nu }_{yx}}\left({ϵ}_y+{\nu }_{xy}{\epsilon }_x\right)$$

(1.2)

Ex, Ey Young's modules in x, y directions. νij (i, j = x, y) is Poisson's ratio of the substrate characterizing the compressive strain in the j direction (direction of the effect) produced by the tensile stress in the i direction (direction of the stress).

For the symmetry conditions expressed by Green's relations, we also have

$E_y{\nu }_{yx}=E_x{\nu }_{xy}$.

The Stresses and the Curvatures in a Linear Elastic Plate are $,$

$${\sigma }_{xx}=\frac{E_x}{1-{\nu }_{xy}{\nu }_{yx}}z\left(\frac{{\partial }^{2}w}{\partial x^2}+{\nu }_{yx}\frac{{\partial }^{2}w}{\partial y^2}\right)$$

(1.3)

$${\sigma }_{yy}=\frac{E_y}{1-{\nu }_{xy}{\nu }_{yx}}z\left(\frac{{\partial }^{2}w}{\partial y^2}+{\nu }_{xy}\frac{{\partial }^{2}w}{\partial x^2}\right)$$

(1.4)

for small deflection,

$$k_x\approx \frac{{\partial }^{2}w}{\partial x^2};k_x\approx \frac{{\partial }^{2}w}{\partial y^2}$$

(1.5)

Kx and Ky are the curvatures in the x and y direction.

This important assumption of small slope, $\raisebox{1ex}{$\partial w$}\!\left/ \!\raisebox{-1ex}{$\partial x$}\right., \raisebox{1ex}{$\partial w$}\!\left/ \!\raisebox{-1ex}{$\partial y$}\right.\ll 1$, means that the theory to be developed will be valid when the deflections are small compared to the overall dimensions of the plate.

In this work we will analytically calculate the deflection and curvature of a silicon plate and an annular silicon plate coated on one side with an aluminum layer with a thickness of 4.5 um

2. The Moment-Curvature Equations

Consider a plate subjected to bending moments Mx and My, with no other loading (Plates subjected to Pure Bending without twisting, $\partial w(x,y)/\partial x\partial y=0$). From equilibrium considerations, these moments act at all points within the plate they are constant throughout the plate. The moment-curvature equations are un the set of coupled partial differential equations,

$$M_x={\int }_{-t/2}^{t/2}{\sigma }_{xx}z{d}_z=\frac{E_x{t}^3}{12(1-{\nu }_{xy}{\nu }_{yx})}\left(\frac{{\partial }^{2}w}{\partial x^2}+{\nu }_{yx}\frac{{\partial }^{2}w}{\partial y^2}\right)=D_x{(k}_x+{\nu }_{yx}{k}_y)$$

(2.1)

$$M_y={\int }_{-t/2}^{t/2}{\sigma }_{yy}z{d}_z=\frac{E_y{t}^3}{12(1-{\nu }_{xy}{\nu }_{yx})}\left(\frac{{\partial }^{2}w}{\partial y^2}+{\nu }_{xy}\frac{{\partial }^{2}w}{\partial x^2}\right)=D_y{(k}_y+{\nu }_{xy}{k}_x)$$

(2.2)

$$D_x=\frac{E_x{t}^3}{12(1-{\nu }_{xy}{\nu }_{yx})}; D_y=\frac{E_y{t}^3}{12(1-{\nu }_{xy}{\nu }_{yx})}; k_x=\frac{{\partial }^{2}w}{\partial x^2};k_x=\frac{{\partial }^{2}w}{\partial y^2}$$

The moment curvature equations are analogous to the beam moment-deflection equation. The factor D_x, D_y, is called the plate stiffness or flexural rigidity and plays the same role in the plate theory as does the flexural rigidity term E*I in the beam theory.

Solving for the derivatives,

$$\frac{{\partial }^{2}w}{\partial x^2}=\frac{1}{(1-{\nu }_{xy}{\nu }_{yx})} \left(\frac{M_x}{D_x}-{\nu }_{yx}\frac{M_y}{D_y}\right)$$

(2.3)

$$\frac{{\partial }^{2}w}{\partial y^2}=\frac{1}{(1-{\nu }_{xy}{\nu }_{yx})}\left( \frac{M_y}{D_y}-{\nu }_{xy}\frac{M_x}{D_x}\right)$$

(2.4)

For a plate with axial symmetry (wafer), the relations 2.1, 2.2, 2.3 and 2.4 become

$$\frac{M_{rr}}{D_r}=\left(\frac{{\partial }^{2}w}{\partial r^2}+{\nu }_{\vartheta r}\frac{1}{r}\frac{\partial w}{\partial r}\right)={(k}_r+{\nu }_{\vartheta r}{k}_{\vartheta })$$

(2.4)

$$\frac{M_{\vartheta \vartheta }}{D_{\vartheta }}=\left(\frac{1}{r}\frac{\partial w}{\partial r}+{\nu }_{r\vartheta }\frac{{\partial }^{2}w}{\partial r^2}\right)={(k}_{\vartheta }+{\nu }_{r\vartheta }{k}_r)$$

(2.5)

and

$$\frac{{\partial }^{2}w}{\partial r^2}=\frac{1}{(1-{\nu }_{r\vartheta }{\nu }_{\vartheta r})} \left(\frac{M_r}{D_r}-{\nu }_{\vartheta r}\frac{M_{\vartheta }}{D_{\vartheta }}\right)$$

(2.5)

$$\frac{1}{r}\frac{\partial w}{\partial r}=\frac{1}{(1-{\nu }_{r\vartheta }{\nu }_{\vartheta r})}\left( \frac{M_{\vartheta }}{D_{\vartheta }}-{\nu }_{r\vartheta }\frac{M_r}{D_r}\right)$$

(2.6)

For an isotropic plate the equations are simplified,

$$\frac{{\partial }^{2}w}{\partial x^2}=\frac{M_x-\nu M_y}{D(1-{\nu }^2)}$$

(2.11)

$$\frac{{\partial }^{2}w}{\partial y^2}=\frac{M_y-\nu M_x}{D(1-{\nu }^2)}$$

(2.12)

$$M_x=D{(k}_x+\nu k_y)$$

(2.13)

$$M_y=D{(k}_y+\nu k_x)$$

(2.14)

$$D=\frac{E{t}_s^3}{12(1-{\nu }^2)}$$

(2.15)

The bending moment Mx and My induced by the layer are,

$$M_x={\sigma }_{xx}^f*t_f*\frac{t_s}{2}$$

(2.16)

$$M_y={\sigma }_{yy}^f*t_f*\frac{t_s}{2}$$

(2.17)

From equilibrium considerations, these moments act at all points within the plate – they are constant throughout the plate.

Equations 2.13 and 2.14 are combined to describe film stresses in terms of substrate curvatures:

$${\sigma }_{xx}^f=\frac{1}{6}{\frac{E}{1-{\nu }^2}\frac{t_s^2}{t_f}(k}_x+\nu k_y)$$

(2.18)

$${\sigma }_{yy}^f=\frac{1}{6}{\frac{E}{1-{\nu }^2}\frac{t_s^2}{t_f}(k}_y+\nu k_x)$$

(2.19)

where ${\sigma }_{xx}^f$, ${\sigma }_{yy}^f$ are the corresponding intrinsic film stresses in the respective directions, in the other hand, material properties E and v correspond to those of substrate.

The relationships 2.18 and 2.19 are Stoney's formulas for isotropic coated rectangular substrate. Rearranging the relations (2.11) and (2.12),

$$\frac{{\partial }^{2}w}{\partial x^2}=k_x=\frac{{\sigma }_{xx}^f-\nu {\sigma }_{yy}^f}{N({\nu }^2-1)} \left(2.20\right); \frac{{\partial }^{2}w}{\partial y^2}=k_y=\frac{{\sigma }_{yy}^f-\nu {\sigma }_{xx}^f}{N\left({\nu }^2-1\right)}\left( 2.21\right); N=\frac{E{t}_s^2}{6t_f(1-{\nu }^2)}$$

Integrating the first two equations, the plate deflection can be written,

$$w\left(x,y\right)=\frac{1}{2}{\left(\frac{{\sigma }_{xx}^f-\nu {\sigma }_{yy}^f}{N\left({\nu }^2-1\right)}\right)x}^2+\frac{1}{2} \left(\frac{{\sigma }_{yy}^f-\nu {\sigma }_{xx}^f}{N\left({\nu }^2-1\right)}\right)y^2+Ax+By+C$$

A = B = C = 0, if $\frac{\partial w}{\partial x}=\frac{\partial w}{\partial y}=0inx=0ey=0$

$$w\left(x,y\right)=\frac{1}{2}\frac{{\sigma }_{yy}^f}{N({\nu }^2-1)}\left[{\left(\frac{{\sigma }_{xx}^f}{{\sigma }_{yy}^f}-\nu \right)x}^2+\left(1-\nu \frac{{\sigma }_{xx}^f}{{\sigma }_{yy}^f}\right)y^2\right]= \frac{1}{2}\left(k_x{x}^2+k_y{y}^2\right)$$

(2.20)

$$N=\frac{E{t}^2}{6t_f(1-{\nu }^2)}; \frac{{\sigma }_{xx}^f}{{\sigma }_{yy}^f}=\frac{k_x+\nu k_y}{k_y+\nu k_x}$$

$$-L_x\le x\le L_x;-L_y\le y\le L_y$$

Once the deflection w(x,y) is known, all other quantities in the plate can be evaluated.

For orthotropic plate, the deflection equations 2.17 is written,

$$w\left(x,y\right)=3\frac{{\sigma }_f*t_f}{t_s^2}\frac{1}{E_y} \left[\left(\frac{D_y}{D_x}-{\nu }_{yx}\right)x^2+\left(1-{\nu }_{xy}\frac{D_y}{D_x}\right)y^2\right]$$

(2.21)

and the curvatures are,

$$k_x=6\frac{{\sigma }_f*t_f}{t_s^2} \frac{1}{E_y}\left[\left(\frac{D_y}{D_x}-{\nu }_{yx}\right)\right]$$

(2.22)

$$k_y=6\frac{{\sigma }_f*t_f}{t_s^2} \frac{1}{E_y}\left[\left(1-{\nu }_{xy}\frac{D_y}{D_x}\right)\right]$$

(2.23)

The relationships 2.22 and 2.23 are Stoney's formulas for orthotropic coated rectangular substrate.

If we assume a homogeneously deposited film, the stress in the two axes is isotropic and homogeneous (Circular Plate),

$${\sigma }_{xx}^f={\sigma }_{yy}^f={\sigma }_{rr}^f$$

this for the general transformation rule ${\sigma }_{rr}^f={cos}^2\phi {\sigma }_{xx}^f+sin2\phi {\sigma }_{xy}^f+{sin}^2\phi {\sigma }_{yy}^f$ and ${\sigma }_{xy}^f=0$ and

$${ \sigma }_{rr}^f=\frac{1}{6}{\frac{E}{1-\nu }\frac{t^2}{t_f} k}_{rr }$$

(2.24)

$$k_{rr}=\frac{{\partial }^{2}w}{\partial x^2}=\frac{{\partial }^{2}w}{\partial y^2}$$

The equation 2.23 is Stoney' s formula for coated Circular Plate and the deflection w(r) is,

$$w\left(r\right)=\frac{1}{2}\frac{{\sigma }^f\left(r\right)}{N(1+\nu )}{r}^2 , N=\frac{E{t}_s^2}{6t_f(1-{\nu }^2)}$$

(2.25)

In the linear region, an increase in stress leads to a proportional increase in the curvature and a preservation of the spherical shape. Outside the linear region a further increase in stress, the shape rapidly transforms from spherical to cylindrical with a dominant curvature in a preferential direction. This phenomenon is called bifurcation of curvature. Figure 1 show a 720 µm thick wafer coated with a 4.5 µm thick AlCu layer.

We can calculate the residual stress of the thin film from warp measurements before and after AlCu layer deposition and from equation 2.23. The ΔCBow measurement is about 100 um (parabolic shape) and the residual stress is about 85 MPa (Silicon Young Modulus E = 131 GPa, ν=0.27, ts = 720 um, tf = 4.5 um) and the curvature is k= 0.024 1/m. The warpage have been measured with an MX-204 equipment (E+H Metrology).

Another way to obtain the same result eq. 2.23 is to solve the Equation of the Elastic Surface w(x,y) for circular plate supported and loaded with radial moment Mo at the outer edge is,

$$\frac{d}{dr}\left[ \frac{1}{r}\frac{d}{dr}\left(r\frac{dw\left(r\right)}{dr}\right) \right]=0 (Load-free Plate)$$

(2.25)

By integrating this equation 2.25 we obtain

$$w\left(r\right)=C_1\frac{r^2}{4}+C_2\ln \left(\frac{r}{R}\right)+C_3$$

(2.26)

$C_1$,$C_2$ and $C_3$are determined by the boundary conditions,

$$\left(B.C.\right)\left\{\begin{array}{c}w \left(R\right)=0\\ M_{rr}\left(R\right)=M_0\\ {\left(\frac{dw\left(r\right)}{dr}\right)}_{r=0}=0\end{array}\right.\left\{\begin{array}{c}{C}_1\frac{R^4}{4}+C_3=0\\ -D\left(\frac{C_1}{2}-\frac{C_2}{R^2}+\nu \frac{C_1}{2}+\nu \frac{C_2}{R^2}\right)=M_0\\ C_2/0\to 0\end{array}\right. \to \left\{\begin{array}{c}{C}_1=-\frac{2M_0}{\left(1+\upsilon \right)D}\\ C_2=0 (must be)\\ C_3=-\frac{M_0{R}^2}{2\left(1+\upsilon \right)D}\end{array}\right.$$

$$w\left(r\right)=\frac{M_0}{2\left(1+\upsilon \right)D}\left(R^2-r^2\right)$$

(2.27)

$$f=w_{max}\left(0\right)=\frac{{\left(2R\right)}^2}{8\rho }=\frac{M_0{R}^2}{2\left(1+\upsilon \right)D}$$

(2.8)

and

$$k_{rr}=\frac{d^{2}w\left(r\right)}{d{r}^2}=\frac{C_1}{2}-\frac{C_2}{r^2}=-\frac{M_0}{\left(1+\nu \right)D}$$

(2.29)

$$k_{\vartheta \vartheta }=\frac{1}{r} \frac{dw}{dr}=\frac{C_1}{2}+\frac{C_2}{r^2}=-\frac{M_0}{\left(1+\nu \right)D}$$

(2.30)

$$M_{rr}=M_{\vartheta \vartheta }=M_0$$

$$k_{rr}=k_{\vartheta \vartheta }$$

Constant moments (and tensions) across the plate. The bending moment $M_0$ induced by the coated layer is,

$$M_0={\sigma }^f*t_f*\frac{t_s}{2}$$

(2.31)

replacing in the (2.24) and (2.25),

$$k_{rr}=k_{\vartheta \vartheta }=6\frac{1-\nu }{E}\frac{t_f}{t_s^2}{\sigma }^f$$

(2.32)

we have found the equation (2.23).