Calculating Macropore Growth and Mass Transport

In contrast to the micro- and mesoporous regimes, for which only a few empirical laws for the growth rate and porosity are available, the detailed pore geometry for macropore arrays in n-type silicon can be pre-calculated by a set of equations. This is possible because every pore tip is in a steady-state condition characterized by = JPS [Le9]. This condition enables us to draw conclusions about the porosi- 

This equation holds true even for thick porous layers, if the decrease in the HF concentration in the pores, which leads to a decrease in JPS at the pore tip, is taken into account. 

Note that Eq. (9.1) applies to pore arrays as well as to randomly distributed pores. For simple orthogonal or hexagonal arrays of macropores with one pore per unit cell of the pattern, the porosity can be defined locally as the ratio between the cross-sectional area of the pore AP and the area of the unit cell AU as shown in Fig. 9.15 a  

The cross-section of a macropore may have all shapes between a circle and a four-pointed star, as shown in Fig. 9.12a-e. In addition the pore walls are covered with a microporous silicon layer, as shown in Fig. 9.12h, which makes the determination of AP difficult. In most cases, however, the approximation of the pore cross-section by a square of size d is found to be sufficient. Under this assumption and for a square pattern of pitch i, as shown in Fig. 9.15 a, d becomes simply  

For a square pattern and square pores the pore wall thickness w is given by  

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