From the beginning of the development of the theory of Knudsen diffusion, there has been controversy about its proper application to tubes with non-cylindrical cross-sections [112]. Smoluchowski [113] provided a
correction to the calculation method originally proposed by Knudsen [72] which eventually led to the development of tables of transmission probabilities \(W\) for rectangular tubes of different widths \(w\) to heights
\(h\) [112]. These tubes are defined by the structural ratio
\(\seteqnumber{0}{3.}{31}\)
\begin{align}
\label {eq::ratio_rectangle} X = \frac {w}{h}
\end{align}
which is, in essence, the AR in the plane perpendicular to the preferential transport direction. However, the nomenclature \(X\) is preferred since AR is conventionally reserved for relating to the depth of the feature (c.f.
Fig. 2.7).
As the semiconductor processing community started applying Knudsen diffusion models to deposition processes in rectangular trenches, simplified expressions were sought [50]. In general, these models attempt to use the
results obtained for a long cylinder, discussed in Section 3.2, but assuming an equivalent diameter \(d_\mathrm {equiv}\). This is conventionally performed in analogy to
continuum fluid dynamics through the use of the hydraulic diameter approximation [109]:
\(\seteqnumber{0}{3.}{32}\)
\begin{align}
\label {eq::hydraulic_diameter} d_\mathrm {equiv} = \frac {2}{\frac {1}{w}+\frac {1}{h}} = h\, g_\mathrm {hd}(X)
\end{align}
That is, the equivalent diameter is found by identifying a characteristic length (\(h\)) and multiplying it by a geometric factor determined solely by the structural ratio \(X\):
\(\seteqnumber{0}{3.}{33}\)
\begin{align}
\label {eq::geometric_factor_hydraulic} g_\mathrm {hd}(X) = \frac {2}{1+\frac {1}{X}}
\end{align}
Equation (3.34) claims that a square hole (\(X{=}1\)) is equivalent to the inscribed cylinder (\(d{=}h\)), and that an infinitely wide
trench has an equivalent diameter twice as large as its opening, i.e., \(\lim _{X\to \infty } g_\mathrm {hd}(X) = 2\).
The approximation in Eq. (3.33) is further justified by Monte Carlo simulations for square holes [125]. This eventually led to Cremers
et al. proposing a mapping between several three-dimensional (3D) structures to equivalent cylinders through a purportedly structure-independent parameter: The equivalent aspect ratio (EAR) [110]. That is, all
studies of conformality in different structures, such as square holes or wide trenches, can be related to an equivalent cylinder with AR as conventionally defined in Eq. (2.19). They propose, among others
\(\seteqnumber{0}{3.}{34}\)
\begin{align}
\label {eq::EAR_square} \text {EAR}(\text {square hole}) =&\, \frac {L}{h} = \frac {L}{h\, g_\mathrm {hd}(1)}\, , \\ \label {eq::EAR_infinite} \text {EAR}(\text {infinitely wide trench})
=&\, \frac {L}{2h} = \frac {L}{h\, g_\mathrm {hd}(\infty )}\, .
\end{align}
These definitions, however, are still questionable. Already from the work of Smoluchowski [113], it is known that the transport properties in a square hole differ from a cylinder by a fixed factor [112].
A more rigorous definition of the geometric factor \(g_\mathrm {vf}(X)\) for long square holes can be obtained by using the expression for \(D_\mathrm {Kn}\) involving the view factors. After introducing \(A_\mathrm
{cross}{=}wh\) in Eq. (3.22) and identifying the characteristic length as \(h\):
\(\seteqnumber{0}{3.}{36}\)
\begin{align}
\label {eq::diffusivity_geometric} D_\mathrm {Kn} = \frac {1}{3}\overline {v}h \cdot \underbrace {\frac {3}{2}\frac {1}{wh^2}\int _0^\infty z' F_{dA'-A}(z',X)
dA'}_{g_\mathrm {vf}(X)}
\end{align}
For clarity, it is useful to calculate the contributions of each individual trench wall separately. This is shown in Fig. 3.5 for the exchange between a surface element
\(dA'\) on one of the walls with width \(w\) to the rectangle at the end of the trench. The view factor has been calculated in terms of \(Z=z'/w\) and \(X\) and takes the form [107]:
Figure 3.5: View factor calculation for one wall of a long rectangular trench with height \(h\) and width \(w\). The rectangular strip \(dA'\) on the trench wall is separated by \(z'\) from a rectangle of area \(A_\mathrm
{cross}\) at the end of the long trench.
The expression for \(g_\mathrm {vf}(X)\) is then obtained by combining the two contributions from the walls aligned with \(w\) with those from the two walls aligned with \(h\). The latter have view factors equivalent to
Eq. (3.38) but with the role of \(w\) and \(h\) reversed, i.e., \(X \rightarrow 1/X\). After performing the necessary substitutions to
obtain integrals over a variable \(Z\):
The integral resulting from inserting the view factor from Eq. (3.38) into Eq. (3.39) does not have a closed-form solution. Nonetheless, it can be solved numerically with a program such as Mathematica [126]. For the case of a square
hole, a value \(g_\mathrm {vf}(1) \approx 1.11495\) is obtained which is numerically equivalent to the ratio of transmission probabilities \(W_\mathrm {square}/W_\mathrm {cylinder} \approx 1.11495\) calculated using
Smoluchowski’s expression [112]. Thus, it is clear that the EAR for a square proposed by Eq. (3.35) is incorrect, albeit only by a factor of
approximately \(10\,\%\).
A larger problem occurs when attempting to recover the wide trench limit. As it can be intuited from the first term on Eq. (3.39),
\(g_\mathrm {vf}(X)\) diverges on the infinite width limit \(X \to \infty \). This is represented in Fig. 3.6, where numerical calculations of both forms of the geometric
factor, \(g_\mathrm {vf}(X)\) and \(g_\mathrm {hd}(X)\), are compared. Figure 3.6 demonstrates that the hydraulic diameter approximation consistently underestimates
the more rigorous \(g_\mathrm {vf}(X)\). Additionally, it indicates that the divergence of \(g_\mathrm {vf}(X)\) occurs very slowly, as even for rather extreme values of \(X>100\), \(g_\mathrm {vf}\) remains in the same
order of magnitude as \(g_\mathrm {hd}\).
Figure 3.6: Comparison of the geometric factor from the hydraulic diameter approximation \(g_\mathrm {hd}(X)\) to that from the rigorous view factor \(g_\mathrm {vf}(X)\) calculation as a function of the rectangular structural
ratio \(X\) (c.f. Fig. 3.5).
Therefore, it must be concluded that Knudsen diffusion is not strictly defined for very wide trenches. However, in practice it remains a useful model in the limit \(w \to \infty \). In this situation, the molecule-molecule collisions,
and, therefore, the mean-free path \(\lambda \), can no longer be neglected. A rough analogy is that, as \(w\) increases, \(X\) does not increase asymptotically. Instead, it is bounded by \(\lambda \), that is,
\(\seteqnumber{0}{3.}{39}\)
\begin{align}
\label {eq::limit_X} \lim _{w \to \infty } X \approx \frac {\lambda }{h} = \text {Kn}
\end{align}
which can be identified as the definition of the Knudsen number in Eq. (2.6). Thus, assuming a pressure regime resulting in \(\text {Kn} \approx
100\), the hydraulic diameter approximation is only wrong by a factor of approximately \(2\). Since there are large uncertainties in the involved parameters, particularly the sticking coefficient, such a factor can be neglected in
many cases. However, unlike the claim in [110], results from different geometries cannot be directly compared without careful consideration.
Finally, it is noteworthy that \(g_\mathrm {vf}(X)\) only applies to the dependency on \(d\) for the calculation on the diffusivity \(D_\mathrm {Kn}\). The conservation of mass in Eq. (3.25) also involves \(d\) in the adsorption loss term in its right-hand side through the specific surface ratio \(\overline {s}=4/d\). For a rectangular trench
\(\seteqnumber{0}{3.}{40}\)
\begin{align}
\label {eq::specific_surface_rectangle} \overline {s} = \frac {2(h+w)}{hw} = \frac {4}{h\,g_\mathrm {hd}(X)}\, .
\end{align}
Therefore, the hydraulic diameter approximation must be applied to the right-hand side of Eq. (3.25). This is likely the source of the
lingering confusion, since Eq. (3.41) has been correctly applied since the first applications of Knudsen diffusion in semiconductor
processing [50].
