Phenomenological Single-Particle
Modeling of Reactive Transport
in Semiconductor Processing

Chapter 3 Knudsen Diffusive Transport

Recently, increased research attention has been placed on the development of one-dimensional (1D) diffusion-based reactive transport models due to their applicability in atomic layer deposition (ALD) [109, 110, 111], following their introduction to aid the design of chemical vapor deposition (CVD) reactors in 1999 [50]. These models have as advantages their relatively straightforward implementation and the ability to provide physically meaningful insight. These diffusive transport approaches are based on Knudsen diffusion, that is, the diffusivity stems from molecule-geometry collisions instead of the conventional molecule-molecule interactions. Since its inception in the early 20th century by Martin Knudsen [72], the development of the theory of Knudsen diffusion has been littered with misconceptions [112], requiring corrections calculated by Smoluchowski [113] and Clausing [114, 115]. Nonetheless, many of these misconceptions persist to this day, possibly due to the well-established status of the theory which is often applied without critical examination.

This chapter presents a more rigorous discussion of Knudsen diffusive processes based on the excellent but often-overlooked work by Pollard and Present [116, 117]. First, some properties of ideal gases, stemming from the kinetic theory of gases, are reviewed. Then, the equations of Knudsen diffusion are reformulated in a more modern manner, including their analogy to radiative heat transfer and the role of the view factor. The classical result of the diffusivity coefficient in long cylinders is recovered, while problems originating from attempting to derive the commonly-used hydraulic diameter approximation for rectangular trenches [109, 110] are discussed. An approach to capture the transitional flow for intermediate Knudsen numbers, the Bosanquet approximation, is also reviewed. Finally, the original research contributions of this dissertation are presented.

3.1 Properties of Ideal Gases

In this section, the properties of ideal gases, that is, the fundamental results of the kinetic theory of gases, are shortly reviewed. They are the foundation of diffusive transport processes and must be carefully considered for a thorough discussion of reactive transport. All definitions follow Present [117], including the necessary adaptations in the labeling of variables for consistency.

Ideal gases, also named perfect gases, are a useful abstraction of real gas behavior under reasonable hypotheses. It is assumed that the gas is composed of molecules, defined as the smallest discrete quantity of a certain substance with defined chemical properties. This definition of molecule encompasses monoatomic gases (e.g., noble gases) and is used instead of "particle" throughout this chapter for consistency with the literature. The number of molecules must be large enough such that meaningful statistical information can be extracted from their average properties. These molecules are assumed to be separated from each other by a distance much greater than its molecular diameter, and they propagate on a straight line throughout a container with fixed volume. Upon colliding with each other or with the walls of the container, the molecules scatter perfectly elastically.

In essence, the assumptions of the kinetic theory of gases hold for systems with low density and without particle acceleration due to, e.g., plasma sheath acceleration. Therefore, in the context of semiconductor processing the ideal gas approximation is well-suited for systems involving gas-phase neutral particles in a vacuum chamber, such as low pressure CVD or ALD. In such situations, the ideal gas law can be derived:

$$\begin{array}{}\text{(3.1)}& p=n{k}_{B}T,\end{array}$$ where $p$ represents the pressure, $n$ the number density or concentration of molecules, $T$ the temperature, and $k_B$ Boltzmann constant.

Due to the large number of involved molecules, the properties of an ideal gas system can be more precisely studied by analyzing their statistical behavior. The most crucial statistical property of a gas is the distribution function of molecular velocities $f^{\ast }(\overrightarrow{v})$. If the container carrying the ideal gas is stationary, then trivially all components of the velocity are zero on average, i.e., ${\bar{v}}_x={\bar{v}}_y={\bar{v}}_z=0$. Thus, it is often more useful to focus on the distribution function of the molecular speed $v$ which can be obtained by integrating $f^{\ast }(\overrightarrow{v})$ over the solid angle $\mathrm{\Omega }$:

$$\begin{array}{}\text{(3.2)}& f(v)={\int }_0^{4\pi }{f}^{\ast }(\overrightarrow{v})d\mathrm{\Omega }\end{array}$$ It can be shown that, assuming that the velocities are isotropic and the velocity component in any direction is independent of any other direction, the speed distribution function of an ideal gas composed of molecules of mass $m$ takes the form of

$$\begin{array}{}\text{(3.3)}& f(v)={\left(\frac{m}{2\pi k_{B}T}\right)}^{\frac{3}{2}}{e}^{-\frac{m{v}^2}{2k_{B}T}}.\end{array}$$ Equation (3.3) is the Maxwell-Boltzmann distribution. The fundamental statistical property extracted from Eq. (3.2) is the average speed $\bar{v}$, also called thermal speed:

$$\begin{array}{}\text{(3.4)}& \bar{v}={\int }_0^{\mathrm{\infty }}vf(v)dv={\left(\frac{8k_{B}T}{\pi m}\right)}^{\frac{1}{2}}\end{array}$$ Two additional speeds can be computed from the distribution, notably the root-mean squared velocity $v_{\mathrm{rms}}$ and the most probable speed $v_m$. Their values differ from $\bar{v}$ only by fixed constants.

As discussed in Chapter 2, the reactive transport models are intrinsically tied to the local impinging flux ${\mathrm{\Gamma }}_{\mathrm{imp}}$ at the container wall. The ${\mathrm{\Gamma }}_{\mathrm{imp}}$ can be directly calculated for ideal gases by considering its definition: The number of striking molecules per unit area per time. This is shown in Fig. 3.1 for an area $dA$ of the container wall, aligned with the $xy$-plane without loss of generality. The number of molecules $dN$ which strike $dA$ during a time interval $dt$ with velocities in the neighborhood of $\overrightarrow{v}$ are contained in the oblique cylinder with base $dA$ and height $v_{z}dt$. Therefore, $dN$ can be calculated as the product of the number of molecules per unit volume $n{f}^{\ast }(\overrightarrow{v})dvd\mathrm{\Omega }$ with the cylinder volume $v_{z}dtdA$. The differential flux then reads

$$\begin{array}{}\text{(3.5)}& d{\mathrm{\Gamma }}_{\mathrm{imp}}=\frac{dN}{dAdt}=n{v}_z{f}^{\ast }(\overrightarrow{v})dvd\mathrm{\Omega }=nvf(v)\cos \theta dv\sin \theta \frac{d\theta d\varphi }{4\pi }.\end{array}$$ The impinging flux is then obtained after integrating over all possible values and directions of $\overrightarrow{v}$, i.e., over the hemisphere above $xy$ ($0<\theta <\pi /2$), obtaining

$$\begin{array}{}\text{(3.6)}& {\mathrm{\Gamma }}_{\mathrm{imp}}=\frac{1}{4\pi }{\int }_0^{\mathrm{\infty }}vf(v)dv{\int }_0^{2\pi }{\int }_0^{\frac{\pi }{2}}\cos \theta \sin \theta d\theta d\varphi =\frac{1}{4}n\bar{v}.\end{array}$$

By using Eqs. (3.1) and (3.4), the impinging flux can be summarized as

$$\begin{array}{}\text{(3.7)}& {\mathrm{\Gamma }}_{\mathrm{imp}}=\frac{1}{4}n\bar{v}=\frac{p}{\sqrt{2\pi m{k}_{B}T}}.\end{array}$$ The relationship in Eq. (3.7) between the gas properties and the local flux is a fundamental result of surface science and is named the Hertz-Knudsen equation.

Additionally, the kinetic theory of gases enables a direct calculation of the mean free path $\lambda$ of an ideal gas. Assuming that a gas is composed of spherical molecules with a fixed diameter $d$, the average distance between collisions for a single gas molecules can be calculated as

$$\begin{array}{}\text{(3.8)}& \lambda =\frac{1}{\sqrt{2}n\pi d^2}=\frac{k_{B}T}{\sqrt{2}\pi d^2}.\end{array}$$ The assumption of fixed $d$ is called the "billiard ball" or hard-sphere approximation. Although in reality the physical diameter of molecules is not clearly defined due to their inherent quantum chemical nature, the hard-sphere diameter is defined empirically from measurements of viscosity [120]. Notably, the calculation of ${\mathrm{\Gamma }}_{\mathrm{imp}}$ can be performed assuming a finite $\lambda$, that is, that the molecules might collide with each other before impacting the wall. This calculation yields the same result as in Eq. (3.6), further strengthening the Hertz-Knudsen equation.

Finally, the first indications of diffusive transport behavior can be directly encountered by assuming that the concentration $n$ is not constant, but instead it follows a gradient in the $z$ direction. In this situation, $n$ can be approximated via a first-order Taylor expansion at an arbitrary $z=0$ plane:

$$\begin{array}{}\text{(3.9)}& n(z)=n(0)+z{\frac{dn}{dz}|}_{z=0}\end{array}$$ A similar calculation to that shown in Fig. 3.1 can be performed at the $z=0$ plane which is not part of the container wall. However, unlike Eq. (3.6), the calculation cannot be restricted to the hemisphere aligned with $+z$. Instead, the net flux crossing the plane ${\mathrm{\Gamma }}_{\mathrm{cross}}(z=0)$ is the difference between the flux coming from the hemisphere below ${\mathrm{\Gamma }}_{z^{-}}$ and the hemisphere above ${\mathrm{\Gamma }}_{z^{+}}$. The sign convention is that a positive ${\mathrm{\Gamma }}_{\mathrm{cross}}$ is interpreted as a net flow in the $+z$ direction. After performing the involved integral, it can be shown that:

$$\begin{array}{}\text{(3.10)}& {\mathrm{\Gamma }}_{\mathrm{cross}}(z=0)={\mathrm{\Gamma }}_{z^{-}}-{\mathrm{\Gamma }}_{z^{+}}=-\frac{1}{3}\bar{v}\lambda \frac{dn}{dz}\end{array}$$ Equation (3.10) has the shape of a diffusion equation with diffusivity

$$\begin{array}{}\text{(3.11)}& D_{\mathrm{self}}=\frac{1}{3}\bar{v}\lambda .\end{array}$$ These equations represent the phenomenon of self-diffusion, that is, the diffusion of one gas through another with the same properties. For example, this is the expected behavior of the diffusion of isotopic tracer molecules through an inert gas composed of molecules with similar weights and diameters.