Advances in Plasma-Grown Hydrogenated Films (eBook)

1.4 Plasma Modeling

All properties of a-Si:H are a result of operating conditions of the discharge (pressure, RF frequency, RF power, gas mixture, geometry). Optimization of properties mostly is done empirically, due to the complex chemistry of the silane–hydrogen discharge, including plasma–wall (= substrate) interaction. Nevertheless, serious attempts have been made to model the discharge, thus providing knowledge of the chemistry and the deposition process, with the ultimate goal to find the optimum parameter space of the specific reactor system used.

Various types of models have been used to describe and study glow discharges. These models differ in the approximations that are made [e.g., the dimension, self-consistency, approach (fluid or kinetic), and number of chemical processes], which will influence the physical relevance of the modeling results. However, a higher accuracy (fewer approximations) will require a larger computational effort. Much effort has been put into the development of self-consistent models for glow discharges. Examples are fluid models [220, 221], particle-in-cell/Monte Carlo (PIC/MC) models [222–224], and hybrid models [190, 225, 226]. More extensive overviews have been presented elsewhere [195, 196, 227, 228]. Most of these self-consistent models have been used for discharges with a relatively simple chemistry and no plasma–wall interaction of the neutrals. Other models include the silane–hydrogen chemistry and the surface interaction to a relatively high degree of completeness, but these models are not self-consistent with respect to the rates of the electron impact collisions (i.e., ionization, dissociation, excitation, and attachment); see for instance Perrin [211]. Models developed for plug-flow silane–hydrogen discharges [190, 195, 196] are self-consistent with respect to the silane–hydrogen chemistry, the electron impact collisions, the plasma–wall interaction, and the transport phenomena. However, in this type of models the composition of the neutral background gas is taken equal to the gas feedstock composition, that is, the depletion of silane will be low.

1.4.1 1D FLUID DISCHARGE MODEL

1.4.1.1 Model Description

Nienhuis et al. [189, 191] have developed a self-consistent fluid model that describes the electron kinetics, the silane–hydrogen chemistry, and the deposition process of a perfectly stirred reactor. Due to the discharge processes, the composition of the background neutrals will differ from the composition of the feedstock gases, because hydrogen and higher-order silanes are formed, while silane is consumed. Fluid models describe the discharge by a combination of balances for the particle, momentum, and energy densities of the ions, electrons, and neutrals obtained from moments of the Boltzmann transport equation, see e.g. [174, 229, 230]. A limitation of a fluid model is the requirement that the mean free path of all particles must be less than the characteristic dimensions of the discharge. The balance equations are coupled to the Poisson equation to calculate the electric field, where only electrostatic forces are taken into account. Solving these equations yields the behavior in space and time of the particle densities and the electric potential during one cycle of the periodic steady state RF discharge.

Rates for electron impact collisions as well as the electron transport coefficients depend on the EEDF. The collision rates, the electron transport coefficients, and the average electron energy are obtained by solving the Boltzmann equation for the EEDF. This EEDF has been expanded in two terms with respect to the velocity [231]. The EEDF is calculated as a function of the electric field for a given composition of the neutral background density. A lookup table has been constructed to obtain the collision rates and electron transport coefficients as functions of the average electron energy, which are used in the fluid model. The deposition process is modeled by the use of sticking coefficients, which actually specify the boundary conditions of the particle density balance equations.

The 1D fluid model describes a discharge created and sustained between two parallel plates, where the time-dependent discharge characteristics only vary along the z-axis, i.e., the direction normal to the plates. Plasma parameters such as the dc self-bias voltage, uniformity of the deposition, and radial transport cannot be studied. Self-consistent models in more than one dimension can model these transport phenomena more appropriately [220, 221, 232]. In reality radial dependences do exist in the discharge, due to radial transport processes. Silane is introduced and is transported from the inlet into the discharge, where it may be transformed into other species via chemical reactions. This new species may be transported outside the discharge toward the pump outlet. The model described here corrects for this by means of additional source terms. The reactor is divided into two volumes, the discharge volume (defined as the volume between the two electrodes), and the discharge-free volume; see also Figure 4a. The charged particles and the radicals are confined to the discharge volume. Radical reaction times or diffusion times are in the millisecond range, shorter than the diffusion time to the outside wall, and shorter than the residence time (~ 1 s) in the reactor. Hence, radial dependences are neglected, and the reactor is considered perfectly stirred. The radical densities are spatially resolved in the z-direction. while the nonradical neutral densities are assumed to be homogeneous throughout the whole reactor volume. In the remaining discharge-free volume only nonradical neutrals are considered.

The main reason for using a 1D model is the reduction of the computational effort compared to the higher-dimensional models. This reduction to one dimension is acceptable because it is possible to study the sustaining mechanisms and the chemistry in the discharge with a 1D self-consistent model.

To verify the model and to establish in which process parameter space it is valid, a comparison is made with experimental data. These experimental data are the partial pressures of silane, disilane, and hydrogen and the growth rate, obtained during deposition of amorphous silicon. Data are compared for various combinations of the total pressure in the reactor, the electrical power, and the frequency of the power source.

A sensitivity study of the influence of the elementary data (i.e., reaction coefficients, cross sections, and transport coefficients) has been performed to determine the importance of specific elementary data, so to guide further research in this area [189].

1.4.1.2 Particle Balances

For each nonradical neutral j in the discharge volume VD, the balance equation of their total number NjD can be written as

NjD∂t=∫DSreac,jdV+QjF→D

where the source term Sreac, j represents the creation or destruction of corresponding species j by electron impact collisions and/or by chemical reactions, e.g., A + B → C + D, where A, B, C, and D indicate the reaction species. The source term for the particles in such a reaction is

reac,C=Sreac,D=−Sreac,A=−Sreac,B=nAnBkr

where kr is the reaction rate. For electron collisions the reaction rate depends on the local (averaged) electron energy; see Figure 13 and Table II. Other chemical reaction rate coefficients are constant; see Table III. The term QjF → D represents the total number of neutrals j per second that are transported from the discharge-free (F) volume into the discharge (D) volume.

For each nonradical neutral j in the discharge-free volume VF, the balance equation of their total number NjF can be written as

NjF∂t=Qjin+Qjpump−QjF→D

where the term Qjin represents the number of particles that enter the reactor volume per second as feedstock gas. The term Qjpump represents the number of pumped particles per second, and is given by

jpump=−NjFτ

with τ the average residence time, which is computed so that the total pressure in the discharge ptot, given by the ideal gas law, equals the preset pressure.

Only the steady-state situation is considered; therefore the partial time derivatives are set equal to zero. Then, combining the above equations and solving for NFj yields (with nj = NjF/VF)

j=τVFQjin+∫DSreac,jdV

For each particle j in the discharge volume (electrons, ions, and radicals) the density balance can be written as

nj∂t+∂Γj∂z=Sreac,j

where nj is the density and Γj is the flux of particle j.

1.4.1.3 Particle Fluxes

In the fluid model the...