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6.1.1 Transport Kinetics

Figure 6.1: Transport characteristics for low- and high-pressure processes.
\begin{figure}\psfrag{D}[l][l]{$\frac{\partial c_i}{\partial t} = \nabla (D_i \n...
...[width=0.8\textwidth, height=5cm]{eps-pvd/pvd-cvd.eps}}

To clarify the term ``low-pressure'' which summarizes the processes described in this chapter, it is opposed to ``high-pressure'' which will characterize high-pressure chemical vapor deposition following in the next chapter. Fig. 6.1 shows the traces of two particles traveling towards a trench structure residing in a reactor evaporated to different pressures. One of the particles is assumed to move vertically towards the center of the trench-like feature, the second one travels parallel to the wafer surface and is directed towards the edge of the trench. On the left hand side the pressure in the chamber is assumed to be 1mTorr which is a typical value for low-pressure processes, on the right hand side it is assumed to be 10Torr. According to the theory on gas kinetics the mean free path $\bar{l}$ for a molecule with a diameter of approximately 10$^{-9}$m at a pressure of 1mTorr and a temperature of 300K

\bar{l}=\frac{kT}{\pi \sqrt{2} d_p^2 p}
\end{displaymath} (6.1)

is about 1cm, which is 10.000 times larger than the typical feature size of 1$\mu\mathrm m$ for semiconductor devices. Therefore only a few statistically negligible particle-particle collisions occur within the feature and the assumption of ballistic transport is correct. Deviations from the initial incidence direction of the particles can be neglected. This is illustrated on the left hand side of Fig. 6.1. The particle in the middle undergoes only a few collisions and moves almost directly to the bottom of the feature. The second particle with parallel incidence is trapped by the edge of the feature. Due to its straight trace the particle cannot pass the edge and reach the bottom or sidewalls of the feature.

A completely different situation is given on the right hand side of Fig. 6.1, where the pressure is assumed to be 10Torr. According to (6.1), the particle mean free path at this pressure is about 1$\mu\mathrm m$, which is comparable to the dimensions of the feature. Many particle-particle interactions occur. The particles are randomly scattered and the initial incidence direction loses its significance. The direction of motion will be stochastic with some directionality towards regions of lower concentration. Such a statistic scattering process can be described with diffusion. Diffusion allows the particle traveling parallel to the wafer surface to adapt its direction according to the concentration and hence to move around the corner.

Diffusion will be an important aspect in the next chapter about CVD processes. Within the scope of low-pressure processes it is important to capture, that the considerations about the mean free path confirm the assumption of ballistic transport. Therefore modeling of low-pressure processes concentrates on the investigation of the incident particles and the statistical evaluation of their distributions, which is the topic of the next section.

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W. Pyka: Feature Scale Modeling for Etching and Deposition Processes in Semiconductor Manufacturing