(image) (image) [ Home ]

Phenomenological Single-Particle
Modeling of Reactive Transport
in Semiconductor Processing

4.3 Integration with the Level-Set Method

Ultimately, the reactive transport model presented in Section 4.2 must be integrated within a topography simulator to evaluate the experimental quantities of interest, especially the film thicknesses, as well as to enable simulations of device operation. To achieve this integration in a LS based topography simulator, a velocity field \(v(\vec {r},t)\) must be constructed, as discussed in Section 2.2. However, unlike the growth rate proposed for conventional CVD [30] in Eq. (2.5), a velocity field for ALP requires a careful consideration of the involved time scales.

As conventional CVD is a steady-state process, the actual reactor operation time can directly be used in the solution of the LS equation in Eq. (2.2). In ALP, the pulse time \(t_p\) does not play the same role due to the self-limiting nature of the reactions. Instead, the changes in profile are controlled by the \(N_\mathrm {cycles}\) and the involved GPC or EPC. Thus, to maintain unit consistency in Eq. (2.2), an artificial time \(t^*\) for the calculation of the evolution of the topography is defined as

\begin{align} \label {eq::artificial_time} t^* = \frac {N_\mathrm {cycles}}{C}\, , \end{align} where \(C\) is a numerical constant. For the reasons discussed below, it has been observed that choosing \(C\) such that \(t^* \approx 1\) is sufficient. Under the assumption that the reactive transport is limited by a single species, the factor limiting the conformality is the coverage distribution of said species. Therefore, the velocity field is constructed from \(\Theta (z) \) calculated using Eqs. (4.5) and (4.11) as

\begin{align} \label {eq::velocity_ALP} v(\vec {r},t) = v(z,t) = C \cdot \Theta (z) \cdot \begin{cases} \text {GPC, or} \\ -\text {EPC}\, . \end {cases} \end{align} This is in contrast to the modeling of aluminum oxidation, where the self-limiting nature is directly captured by the LS method [29]. Instead, Eq. (4.15) delegates the self-limiting aspects of the calculation to \(\Theta (z)\), while the surface advection proceeds normally for the duration of \(t^*\).

Effectively, the introduction of \(t^*\) in Eq. (4.14) means that multiple, but not all, ALP cycles are bundled in a single LS advection time step. LS based simulation treats \(v(\vec {r},t)\) as constant only during a single advection step which is limited to at most one grid spacing \(\Delta x\) according to the Courant-Friedrichs-Lewy (CFL) condition [59, 79]. That is, the actual bundling of cycles is given by \(\Delta x/{\max {|v(\vec {r},t)|}}\) and not \(t^*\), as the inputs \(d\) and \(L\) of the calculation of \(v(\vec {r},t)\) are updated after each advection step. Notwithstanding, \(\Delta x\) must be carefully chosen so that variations in the distribution of \(\Gamma _\mathrm {imp}(\vec {r})\) are minimal during the advection step.

The calculation of \(\Gamma _\mathrm {imp}(\vec {r})\) presented in Section 4.2 is performed on \(N_z\) slices which might not necessarily correspond to the required \(z\) coordinates for the solution of the LS equation. This is addressed by storing the values of \(\Theta \) at each \(k\) slice as a look-up table. Then, the values are linearly interpolated for the \(z\) coordinate of each evolving surface point \(\vec {r}\). Since the presented model does not require any modifications to the LS method, it has been implemented using the Python interface of ViennaLS [158] as well as the user-facing Open Model Library of Silvaco’s Victory Process [55].