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Emulation and Simulation of
Microelectronic Fabrication Processes

B Geometric Modelling of Deep Reactive Ion Etching

The geometric advection models introduced in Section require some additional considerations due to their complexity. In the following, the necessary equations for the correct evaluation of the smooth profile, as well as the positioning of the lens distributions along the profile are derived and their effects on the advection are discussed.

B.1 Profile and Scallop Generation for the DEM Sequence

In this sequence, the depletion of etchant flux down the feature [261, 262] leads to a decrease in etch depth per cycle \(d_c\). For simplicity, it is assumed that \(d_c\) is constant until a certain depth \(L_t\), where tapering starts and \(d_c\) decreases linearly down the feature. Each distribution must be spaced by a distance \(d_c\) from the distribution above, which means that the scallops are more closely spaced down the feature. If the etch process is continued to infinity, the etch rate would, at some point, balance the deposition and \(d_c\) would go to zero. The depth at which \(d_c\) reaches zero is defined as \(L_0\), which is used to find the distance \(D\) along which \(d_c\) decreases from its initial value to zero:

\begin{equation} D = L_0 - L_t \label {eq::DEMEtchDepth} \end{equation}

Using \(L_t\) as the origin in the \(z\)-direction, the distance between the \(n\)th and \((n+1)\)th tapered scallop is

\begin{equation} z_{n+1} - z_n = \left (1 - \frac {z_{n+1}}{D}\right )\frac {d_c}{2} + \left (1 - \frac {z_n}{D}\right )\frac {d_c}{2} \quad . \label {eq::scallop_recursive_definition} \end{equation}

Rearranging Eq. (B.2) gives a recursive relation for the \(z\) coordinate of the centre of the \(n\)th scallop:

\begin{equation} z_{n+1} = \frac {d_c}{1+\frac {d_c}{2D}} + \frac {1-\frac {d_c}{2D}}{1+\frac {d_c}{2D}} z_n = a + b z_n \quad . \label {eq::geom_progression} \end{equation}

Since tapering only starts a distance \(L_t\) down the trench, where the origin in \(z\) is placed, the first scallop is centred at \(z_0 = 0\).

Eq. (B.3) is a geometric progression and therefore the origin of the \(n\)th scallop may also be expressed as

\begin{equation} z_{n} = a \frac {1 - b^n}{1 - b} \quad . \label {eq::circle_centers} \end{equation}

Therefore, if there are enough etch cycles to reach an etch rate of zero, the expression for \(D\) in Eq. (B.1) can be used to find the centres of all lens distributions.

However, experiments are not always conducted until the etch rate per cycle approaches zero, but are rather stopped after a certain number of cycles \(N_c\). In this case, \(D\) can be found from the number of cycles and the ratio \(r_e\) of the etch depths \(d_c\) and \(d_f\) of the first and last cycle, respectively. This ratio is also closely related to the tapering width \(w_t\) and is given by

\begin{equation} r_e = \frac {d_f}{d_c} = 1 - \frac {w_t}{w_{tot}} \quad , \end{equation}

where \(w_{tot}\) is the tapering width when the etch rate per cycle goes to zero. This final tapering width may also be geometry dependent in geometries with high aspect-ratios since it cannot exceed the radius of a via or half the width of a trench. Given either \(d_f\) and \(d_c\) or \(w_t\) and \(w_{tot}\), \(r_e\) can be calculated and used to find \(D\) with the relation:

\begin{equation} 1 - r_e = \left (1 - \left ( \frac {1 - \frac {d_c}{2D}}{1 + \frac {d_c}{2D}}\right )^{N_t} \right ) \left ( 1 + \frac {d_c}{2D} \right ) \quad . \label {eq::r_e2D} \end{equation}

\(D\) can therefore be found straight-forwardly using a root-finding algorithm, as it cannot be solved analytically. Since \(D\) is constant throughout the process, it only has to be calculated once, so the computational effort to find a numerical solution can be neglected.

This value of \(D\) is then used to find \(a\) and \(b\) defined in Eq. (B.3), which is used to find the values of all \(z_n\). Then, given the number of etch cycles \(N_c\) to be performed, the final feature depth is given by

\begin{equation} L_b = L_t + z_{N_t} = L_t + a \frac {1 - b^{N_t}}{1 - b} \quad , \end{equation}

which is used to generate the final smooth profile of the process.