# Material Removal Model for Lapping Process Based on Spiral Groove Density
## 2 Materials and Methods
### Material Removal Model according to Indentation Theory  
In the lapping process, material is removed with abrasives moving under the relative motion of the wafer and platen while a load is applied to the abrasives at the contact interface between the platen and wafer. The behavior of the slurry and abrasive at the contact interface is affected by the groove, viscosity, flow rate of the slurry, and abrasive size [18]. Among these, the platen groove is closely related to the lapping characteristics because it greatly influences the slurry flow and the distribution of abrasives [19]. Therefore, a theoretical approach to the material removal mechanism is required to explain the correlation between the groove and lapping characteristics; accordingly, a material removal model was developed using indentation theory to analyze the correlation between groove density and material removal rate. Diamond abrasives used for lapping are mainly synthesized using explosion synthesis to crush the diamond, after which abrasives of the desired size are selected and refined to high-purity diamond particles with nitric acid or hydrochloric acid [20]. Diamond abrasives synthesized in such a manner are irregular in shape with multiple cutting edges present on their surface, as shown in Figure 2, and the inconsistent abrasive size is measured through laser diffraction particle analysis. The scattering intensity and angle measured by the detector are used to calculate the projected area or the volume of the particle based on the Fraunhofer theory or Mie theory, which is then converted to an equivalent sphere to represent the abrasive size in terms of the diameter of the equivalent sphere. In addition, in many studies, the abrasive shape is assumed to be of an equivalent sphere shape to express the irregular shape of pad asperities or non-spherical abrasives [21; 22]. Likewise, in this paper, the abrasive size refers to the diameter of the equivalent sphere corresponding to the abrasive, and the particle shape is assumed to be spherical. Figure 1: Lapping platen and groove on platen surface. Figure 2: SEM images of the polycrystalline diamond abrasive. In a material removal model, the removal amount of the wafer can be expressed as the product of the amount of material removed by a single abrasive and the number of abrasives participating in lapping. The amount of material removed by a single particle is correlated to the indentation depth of the abrasive based on indentation theory. Figure 3 shows the relationship between indentation depth and cutting area when a single abrasive is under a load. The depth of indentation, \(\delta\) [mm], can be expressed as shown in Equation (2), and the contact radius, \(a\) [mm], and contact angle, \(\theta_{a}\) [radian], of the abrasive can be expressed as shown in Equations (3) and (4) [23]:  
\[\delta=r_{a}\cdot(1-cos\theta_{a}), \tag{2}\]  
\[a=\sqrt{\frac{2\cdot F}{\pi\cdot H}}, \tag{3}\]  
\[\theta_{a}=\sin^{-1}\left(\frac{a}{r_{a}}\right), \tag{4}\]  
where \(r_{a}\) [mm], \(F\) [N], and \(H\) [N/mm\({}^{2}\)] represent the radius of the abrasive, load applied to a single abrasive, and the Vickers hardness of the workpiece, respectively. The area of material removed by a single abrasive is expressed as shown in Equation (5):  
\[A_{a}=r_{a}^{2}\cdot\theta_{a}-a(r_{a}-\delta). \tag{5}\]  
The amount of material removed by a single abrasive is proportional to the distance of the abrasive's movement, which in turn is correlated to the relative velocity of the plate and wafer, assuming that the particle is fixed on the plate while moving. Thus, the volume of material removed by a single particle per minute, \(MRR_{V\_a}\) [mm\({}^{3}\)/min], is expressed as shown in Equation (6):  
\[MRR_{V\_a}=k_{m}\cdot A_{a}\cdot v, \tag{6}\]  
where \(v\) [mm/min] is the relative velocity of the plate and wafer, and \(k_{m}\) is the material removal coefficient, which indicates the contribution of other parameters toward material removal.