5.4.1 First approach of Local Oxidation

The first problem that has been successfully solved had a comparable simple geometry. Nevertheless it was a big progress in the development, since it was not sure if the developed model will lead to acceptable results because of the numerical conditions:

• Instead of decoupling the diffusion and displacement calculation a coupled system has to be solved:
  • generation of silicon dioxide
    

    $${\frac{\partial C_{SiO_2}}{\partial t}}$$

    
    

  • diffusion of oxygen including chemical reaction with silicon and the segregation effect at the interface
    

    $${\frac{\partial C_{O}}{\partial t}} \left[\vphantom{D\left(\eta(SiO_2)\right)\cdot \left( grad\left( C_{O}\right) -\frac{C_O}{m}\: grad\left( m\right) \right) }\right. \left(\vphantom{\eta(SiO_2)}\right. \eta \left.\vphantom{\eta(SiO_2)}\right) \left(\vphantom{ grad\left( C_{O}\right) -\frac{C_O}{m}\: grad\left( m\right) }\right. \left(\vphantom{ C_{O}}\right. \left.\vphantom{ C_{O}}\right) {\frac{C_O}{m}} \left(\vphantom{ m}\right. \left.\vphantom{ m}\right) \left.\vphantom{ grad\left( C_{O}\right) -\frac{C_O}{m}\: grad\left( m\right) }\right) \left.\vphantom{D\left(\eta(SiO_2)\right)\cdot \left( grad\left( C_{O}\right) -\frac{C_O}{m}\: grad\left( m\right) \right) }\right]$$

    
    

  • volume dilatation depending on the generated silicon dioxide
    

    $$\left(\vphantom{\int\limits_V{{\cal{L}^T}\cdot{\cal{D}}\left(\eta(SiO_2)\right)\cdot{\cal{L}}}\cdot\:dV}\right. \int\limits_{V}^{} \cal {L} \cal {D} \left(\vphantom{\eta(SiO_2)}\right. \eta \left.\vphantom{\eta(SiO_2)}\right) \cal {L} \left.\vphantom{\int\limits_V{{\cal{L}^T}\cdot{\cal{D}}\left(\eta(SiO_2)\right)\cdot{\cal{L}}}\cdot\:dV}\right) \int\limits_{V}^{} \cal {L} \cal {D} \left(\vphantom{\eta(SiO_2)}\right. \eta \left.\vphantom{\eta(SiO_2)}\right) \varepsilon_{o}^{}$$

    
    

• The characteristics of the quantities differ in magnitudes of order.
• The immediate jump of the material (silicon and silicon dioxide interface) within the transition zone may cause problems concerning convergence.
• High reactive terms (5.19) at the interface are reducing the convergence behavior of the total system.
• The implicitly solved boundary movement, that requires a Lagrange formulation, where even the shape functions itself have to be considered to calculate the Jacobian matrix.

As first example two typical three-dimensional effects occurring in the corners of the nitride mask have been calculated (Fig. 5.7 and Fig. 5.8). Starting from a pure silicon block the results show, that the introduced model can even handle structures without pad oxide below the nitride mask. This leads to the effect that the interface meets the nitride layer vertically that can hardly be solved by algorithms based on a sharp interface formulation. Nevertheless, the interface conditions are too smooth for a physical interpretation of the material interface and an improvement must be made using grid adaptation to accurately resolve the interface layer.

Figure 5.7: Oxide growth around a nitride mask covering one quarter of the geometry

Figure 5.8: Oxide growth around a nitride mask covering three quarters of the geometry