6.4.2 Lag Effect

The influence of the hole diameter on the final profile is further investigated. A Bosch process with $\SI{6}{\second}$ passivation followed by $\SI{12}{\second}$ etching cycles is applied on a $\SI {1}{\micro \metre }$ thick perforated mask with cylindrical holes of varying diameters ( $\SI {0.5}{\micro \metre }$ , $\SI {1}{\micro \metre }$ , $\SI {1.5}{\micro \metre }$ , $\SI {2}{\micro \metre }$ , and $\SI {2.5}{\micro \metre }$ ).

The simulation domain is resolved on a grid with extensions $500\times 140$ . 12.5 million particles are simulated for each time step. Using 8 cores of AMD Opteron 8222 SE processors ( $\SI {3}{\giga \hertz }$ ) the total computation time is approximately 2 days. Approximately 6500 time steps are necessary to simulate all 20 cycles of the Bosch process. The calculation time for one time step is $\SI{27}{\second}$ on average. The runtimes increase continuously during the entire simulation due to the increasing depths of the holes and the increasing surface area. Figure 6.10 shows the final profile after 20 cycles. The different etching depths due to the lag effect can be observed. With increasing aspect ratio, the effective etching rate decreases.

Figure 6.10: Deep reactive ion etching of holes with varying diameters ( $\SI {0.5}{\micro \metre }$ , $\SI {1}{\micro \metre }$ , $\SI {1.5}{\micro \metre }$ , $\SI {2}{\micro \metre }$ , and $\SI {2.5}{\micro \metre }$ ). The lag effect is the reason for the different depths. The structure is resolved on a grid with lateral extensions $500\times 140$ .

To analyze the reason behind the lag effect in more detail, the ion and neutral fluxes are calculated at the bottom center of idealized cylindrical holes for various aspect ratios $x$ , which is a ratio of the depth and the hole diameter. The ion fluxes obtained by ray tracing are in very good agreement with those calculated analytically (Figure 6.11). The analytical expression

$${F}_{\text{ion}}={F}^{\text{src}}_{\text{ion}}\cdot \left(1-\left(\frac{2x}{\sqrt{1+4x^2}}\right)^{{\nu}+1}\right)$$ (6.5)

can be derived from (5.10) by integrating over the open solid angle of the cylindrical hole. For the calculation of the neutral flux, the sticking coefficient is set to 0.1, which corresponds to the sticking probability of neutrals on the passivation layer, as used in this model. The results show that the aspect ratio affects the neutral flux much more than the directional ion flux. With increasing depth the hole surface area increases, leading to a smaller fraction of particles which remain sticking at the bottom and not at the sidewalls.

Figure 6.11: The characteristic dependence of the neutral and ion fluxes at the bottom center on the aspect ratio.

According to (6.4) and Table 6.1 the neutral flux is the main contributor to the deposition rate of the passivation layer. Hence, by increasing the aspect ratio, the thickness of the deposited passivation layer decreases due to the smaller neutral fluxes. However, unlike the neutral flux, the ion flux is not reduced significantly. As a consequence, the passivation layer is etched through much faster. The ion flux countervails the lag effect, because the substrate is attacked earlier in the etching cycle for larger aspect ratios. However, this head start is more than compensated by the larger substrate etch rate for smaller aspect ratios, due to the larger neutral fluxes.