3D grinding mark simulation and its applications for silicon wafer grinding

Abstract

A three-dimensional mathematical model based on homogenous coordinate transformation was developed and later experimentally validated to simulate the abrasive trajectories caused by the grit of grinding wheel onto the wafer. Those abrasive trajectories become the cross-hatch grinding marks on the ground wafer surface. The resulting convex or concave face profile of wafer after the grinding process as well as the abrasive trajectories which correlated to the wafer surface quality in terms of total thickness variation (TTV) can be predicted accurately. Simulations revealed that the relative orientation between the chuck table and the grinding wheel most affects TTV, followed by the offset distance if only the grinding geometry is considered. Moreover, the spindle speed should be coprime to chuck table in order to avoid overlapped trajectory, which may ensure a better grinding quality and grinding efficiency as well. Similarly, the spindle speeds for fine grinding should be coprime to rough grinding to effectively eliminate the grinding marks left by the rough grinding for better grinding quality. The model was implemented in software to generate grinding trajectories to predict wafer TTV given the process parameters such as rotational speeds and relative orientation between the grinding wheel and chuck table, which plays an essential tool for further process parameters optimization for wafer grinding.

Appendix

$${{}^{w}\mathbf{A}}_{g}=\mathbf{R}\mathbf{o}\mathbf{t}\left({{\text{Z}}}_{w},{\theta }_{w}\right)\mathbf{T}\left({{\text{S}}}_{x},{{\text{S}}}_{y},-{z}_{t}\right)\mathbf{R}\mathbf{o}\mathbf{t}\left({{\text{Y}}}_{w},-{\phi }_{ty}\right)\mathbf{R}\mathbf{o}\mathbf{t}\left({{\text{X}}}_{w},-{\phi }_{tx}\right)\mathbf{T}\left({R}_{g},0,0\right)\mathbf{R}\mathbf{o}\mathbf{t}\left({{\text{Z}}}_{w},-{\theta }_{g}\right)$$

$$=\left[\begin{array}{cccc}a& b& c& d\\ e& f& g& h\\ i& j& k& l\\ 0& 0& 0& 1\end{array}\right],$$

where

$$\begin{array}{c}a=cos\left({\omega }_{g}\Delta t\right)cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{ty}\right)-sin\left({\omega }_{g}\Delta t\right)(-cos\left({\phi }_{tx}\right)sin\left({\omega }_{w}\Delta t\right)\\ +cos\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)sin\left({\phi }_{ty}\right))\end{array},$$

(8)

$$\begin{array}{c}b=cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{ty}\right)sin\left({\omega }_{g}\Delta t\right)\\ +cos\left({\omega }_{g}\Delta t\right)\left(-cos\left({\phi }_{tx}\right)sin\left({\omega }_{w}\Delta t\right)+cos\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)sin\left({\phi }_{ty}\right)\right)\end{array},$$

(9)

$$c=-sin\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)-cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{tx}\right)sin\left({\phi }_{ty}\right),$$

(10)

$$d={S}_{x}cos\left({\omega }_{w}\Delta t\right)+{R}_{g}cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{ty}\right)-{S}_{y}sin\left({\omega }_{w}\Delta t\right),$$

(11)

$$\begin{array}{c}e=cos\left({\omega }_{g}\Delta t\right)cos\left({\phi }_{ty}\right)sin\left({\omega }_{w}\Delta t\right)\\ -sin\left({\omega }_{g}\Delta t\right)\left(cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{tx}\right)+sin\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)sin\left({\phi }_{ty}\right)\right)\end{array},$$

(12)

$$\begin{array}{c}f=cos\left({\phi }_{ty}\right)sin\left({\omega }_{g}\Delta t\right)sin\left({\omega }_{w}\Delta t\right)\\ +cos\left({\omega }_{g}\Delta t\right)\left(cos\left({\omega }_{w}\Delta t\right)cos\left({\phi }_{tx}\right)+sin\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)sin\left({\phi }_{ty}\right)\right)\end{array},$$

(13)

$$g=cos\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{tx}\right)-cos\left({\phi }_{tx}\right)sin\left({\omega }_{w}\Delta t\right)sin\left({\phi }_{ty}\right),$$

(14)

$$h={S}_{y}cos\left({\omega }_{w}\Delta t\right)+{S}_{x}sin\left({\omega }_{w}\Delta t\right)+{R}_{g}cos\left({\phi }_{ty}\right)sin\left({\omega }_{w}\Delta t\right),$$

(15)

$$i=cos\left({\phi }_{ty}\right)sin\left({\omega }_{g}\Delta t\right)sin\left({\phi }_{tx}\right)+cos\left({\omega }_{g}\Delta t\right)sin\left({\phi }_{ty}\right),$$

(16)

$$j=-cos\left({\omega }_{g}\Delta t\right)cos\left({\phi }_{ty}\right)sin\left({\phi }_{tx}\right)+sin\left({\omega }_{g}\Delta t\right)sin\left({\phi }_{ty}\right),$$

(17)

$$k=cos\left({\phi }_{tx}\right)cos\left({\phi }_{ty}\right),$$

(18)

$$l=-z_t+R_gsin{\left(\phi_{ty}\right)}.$$

(19)