Phenomenological Single-Particle
Modeling of Reactive Transport
in Semiconductor Processing

2.2 Langmuir Adsorption Kinetics

The process of etching or deposition involves physical and chemical transformations of the evolving surface. Thus, a crucial element in the overall simulation approach is the choice of an adequate model for the interaction of the particles with the surface. Additionally, such a reactive transport model must be connected with a velocity field, as discussed in Section 2.1. In the context of phenomenological modeling, said model should be as simple as possible to capture the relevant physical and chemical processes, therefore, coarse approximations are permitted as long as the simulation outcomes can be related to experiments.

The established approach for phenomenological modeling of surfaces builds upon the seminal work by Irving Langmuir in 1918 [68]. The key idea behind Langmuir adsorption kinetics is that impinging vapor species adsorb upon interacting with the force fields stemming from the surface atoms. Importantly, such force fields are not explicitly investigated, instead their properties are only phenomenologically described regarding general and approximate properties of the reactant-surface system. These forces are assumed to be very short in range, thus the impinging reactants adsorb forming a film with at most one monolayer. If such forces are relatively weak, the reactant can thermally evaporate spontaneously, leading to a reversible kinetic behavior.
This comparatively weak and reversible interaction is named physisorption (or physical adsorption), in contrast to chemisorption where the chemical bonds change more permanently [69]. Due to large unknowns present in phenomenological modeling, the limit between physisorption and chemisorption is often tenuous, however, insights from Langmuir modeling are usually associated with the physisorption regime.

Langmuir adsorption kinetics starts by assuming that the surface has a homogeneous distribution of surface sites with area $s_0$ which are available to interact with the impinging reactants. These surface sites are allowed to be partially saturated by the condensed vapor, leading to the fundamental concept of the surface coverage $\mathrm{\Theta }$: The local ratio of occupied to available surface sites, so that $\mathrm{\Theta }=0$ for an unoccupied, or clean, surface and $\mathrm{\Theta }=1$ for a fully-saturated site. From that, Langmuir postulates [68] a simple relationship for competitive adsorption and desorption

$$\begin{array}{}\text{(2.3)}& \frac{1}{s_0}\frac{d\mathrm{\Theta }}{dt}=\stackrel{\text{adsorption rate}}{\overbrace{{\mathrm{\Gamma }}_{\mathrm{imp}}{\beta }_0(1-\mathrm{\Theta })}}-\stackrel{\text{desorption rate}}{\overbrace{{\mathrm{\Gamma }}_{\mathrm{ev}}\mathrm{\Theta }}},\end{array}$$ where ${\mathrm{\Gamma }}_{\mathrm{imp}}$ is the local flux of impinging particles, whose calculation is discussed in Section 2.3; ${\mathrm{\Gamma }}_{\mathrm{ev}}$ is the evaporation flux for a fully covered surface; and ${\beta }_0$ is the clean surface sticking coefficient, i.e., the probability of an impinging species to adsorb on an unoccupied surface ($\mathrm{\Theta }=0$). From the steady-state limit of Eq. (2.3) the standard Langmuir adsorption isotherm is recovered [69]. Although such limit is very often assumed in applications for semiconductor processing [43], it fails in many applications where the transient times are relevant, such as in atomic layer deposition (ALD). More recent studies have suggested alternative forms for Langmuir adsorption, including a second-order term in $\mathrm{\Theta }$ [70, 71]. Thus, Eq. (2.3) can accurately be named first-order reversible Langmuir kinetics.

There are two intrinsically empirical parameters in Eq. (2.3): ${\mathrm{\Gamma }}_{\mathrm{ev}}$ and ${\beta }_0$. From those, ${\beta }_0$ carries the most physical and chemical significance. Firstly, it implicitly considers the stoichiometry of the surface reaction. Although stoichiometric factors can be explicitly written in a Langmuir-type equation, they are not always observed in reality due to steric hindrance [68]. That is, the impinging species might effectively be larger than the underlying surface sites, preventing further molecules to attach even though stoichiometric considerations would otherwise permit. Additionally, ${\beta }_0$ connects the surface state with the transport of reactants given by ${\mathrm{\Gamma }}_{\mathrm{imp}}$, since a reactant which does not adsorb is free to propagate and perhaps impinge on a different surface element. For these first-order kinetics, the adsorption probability, i.e., the full sticking coefficient, is [30]

$$\begin{array}{}\text{(2.4)}& \beta ={\beta }_0(1-\mathrm{\Theta }),\end{array}$$ and the reflection probability is then simply $1-\beta$.

Finally, this surface kinetics relationship brings additional insight into the reactive transport calculation. Already from early experiments from Knudsen [72, 73] and Langmuir [74], it has been observed that the molecules which do not adsorb are reflected, however, they do not scatter away specularly. Instead, they follow a cosine distribution, that is, their reflection direction is independent of the incoming path. This observation is interpreted as evidence of a very fast mechanism such that during the reflection process the molecule thermalizes with the surface and loses information about its previous state. In semiconductor processing, this cosine dependence usually holds for uncharged reactants, while accelerated ions show specular reflections [75].

The Langmuir adsorption kinetics shown in Eq. (2.3) is valid for a single particle. Due to its phenomenological nature, such model can be employed regardless if the particle represents a well-defined chemical species or a phenomenological aggregate of multiple reactants. This dissertation focuses on semiconductor processing steps which can be effectively modeled using a single-particle. Nonetheless, it is notable that the semiconductor process modeling community has developed and applied multiple-particle variants of such models [43, 76, 77, 78]. La Magna et al. [43] propose, e.g., a steady-state, first-order reversible Langmuir kinetics for the competing adsorption of neutrals and polymers under ion bombardment for reactive ion etching (RIE).

Regardless of the number of involved chemical species, as discussed in Section 2.1 and Eq. (2.2), the final goal of describing the surface state is the determination of a velocity field $v(\overrightarrow{r},t)$. However, Langmuir adsorption kinetics only evaluate small changes in the surface structure of up to a monolayer. Thus, the construction of a coverage-dependent velocity field $v(\mathrm{\Theta }(\overrightarrow{r}),t)$ requires careful consideration as it involves substantial approximations. In the LS method, the velocity field is assumed to be constant as the surface advects while the Courant-Friedrichs-Lewy (CFL) condition is satisfied. The CFL condition is necessary in order to guarantee numerical stability [79], so after each CFL-limited advection step, the velocity field is recalculated. In summary, for fast marching LS methods, the surface advection step is limited to a single grid spacing $\mathrm{\Delta }x$, while for narrow band methods it is limited to $0.5\cdot \mathrm{\Delta }x$ [53]. Therefore, a constructed $v(\mathrm{\Theta }(\overrightarrow{r}),t)$ is only valid during an advection step if the following conditions are met: i) Transport equilibrates much more quickly than the surface advection speed, such that there are no changes in local impinging fluxes ${\mathrm{\Gamma }}_{\mathrm{imp}}(\overrightarrow{r})$ during the advection step; and ii) $\mathrm{\Delta }x$ is small enough that changes in the distribution $\mathrm{\Theta }(\overrightarrow{r})$ due to the evolving geometry are negligible. For example, Yanguas-Gil [30] proposes the following form for the growth rate of chemical vapor deposition (CVD) from a single impinging reactant

$$\begin{array}{}\text{(2.5)}& v(\mathrm{\Theta }(\overrightarrow{r}),t)=\mathit{\text{GR}}(\mathrm{\Theta }(\overrightarrow{r}),t)=\frac{M_m}{\rho }{\mathrm{\Gamma }}_{\mathrm{imp}}(\overrightarrow{r},t)\beta (\mathrm{\Theta }),\end{array}$$ where $M_m$ is the incorporated mass per impinging molecule and $\rho$ is the film density.