2.2 Transport Kinetics

In dry processes, where the wafer is exposed to a gas phase, the flux distribution of particles on the surface is usually the crucial factor for the local deposition or etching rates. Common models for the particle transport within the process chamber to the wafer surface divide the gas phase region by a plane ${\mathcal {P}}$ just above the surface ${\mathcal {S}}$ as depicted in Figure 2.1 [32,80]. A first model describes the transport within the reactor-scale region to ${\mathcal {P}}$ and allows the determination of the flux distribution

$${\Gamma}_{\text{src}}={\Gamma}_{\text{src}}({\vec{x}};{q},{\vec{\omega}}, {E})\quad\text{with} \ {\vec{x}}\in{\mathcal{P}}$$ (2.2)

which describes the number of arriving particles of species ${q}$ with incident direction ${\vec{\omega}}$ and energy ${E}$ per unit area. Here ${\vec{\omega}}$ is a unit vector. The total flux ${F}^{\text{src}}_{q}$ of particles of type ${q}$ crossing ${\mathcal {P}}$ is the integral over all directions and energies

$${F}^{\text{src}}_{q}({\vec{x}})= \int\limits_0^\infty\int\limits_... ...Gamma}_{\text{src}}({\vec{x}}; {q},{\vec{\omega}}, {E}) \,{d}{\Omega} \,{d}{E}.$$ (2.3)

Here ${\vec{n}}_{\mathcal{P}}$ denotes the normal vector to ${\mathcal {P}}$ pointing towards ${\mathcal {S}}$ (see Figure 2.1).

A second model describes the continued particle transport from ${\mathcal {P}}$ to ${\mathcal {S}}$ within the feature-scale region using (2.2) as a boundary condition. The advantage of dividing the gas phase region is that the particle transport acts on different length scales, and hence, different transport models can be applied to each region.

Figure 2.1: The particle transport is broken up by describing the transport to ${\mathcal {P}}$ on reactor-scale and the transport from ${\mathcal {P}}$ to ${\mathcal {S}}$ on feature-scale.