Plasma Modelling

Semiconductor manufacturing, microelectronic devices, lighting, flat panel displays, aerospace, steel, toxic waste treatment, food packaging are just some noteworthy examples of plasma-aided applications [1]. The continuous increase of the number of applications of RF Plasmas imposes the creation of analytic tools used for in depth understanding,design, forecast and optimization of these processes. In this direction, numerical simulation is indispensable for the interpretation of the quite complicated physical and chemical phenomena taking place. In addition,  plasma is one of the most challenging and interesting areas in the field of numerical modeling, requiring an intensive multi disciplinary approach.

The different phenomena taking place during plasma deposition, etching, cleaning, surface modification etc can be generally  grouped in the following subcategories:

- Physical Processes

- Chemical Gas Phase kinetics

- Chemical and Physical Gas-Surface Interaction

A complete simulation of a certain plasma process have to deal with all the above mentioned sub-processes.  One can find several examples of plasma simulations in recent literature, differing in the the theoretical approach, the method of solution etc.  A most general classification of plasma models is:

- Models relying on experimental measurements

- Self - consistent models

Models based on experimental measurements (i.e. voltage, current (total, ion, electron), impedance, electron density, electron temperature, ion flux, excited or ground-state species density, consumption of gas precursors, deposition rate etc) are used as model inputs. These models further analyze and interpret experimental measurements allowing to draw conclusions for quantities not directly measurable. They have the advantage of very short computational times while being able to achieve very good very good forecasts of measurable quantities. On the other hand, they have the disadvantage that they can only be directly implemented in plasma reactors for which the experimental results are available whereas the generalization of their results must be justified with care.

Self-consistent models require as input, only the applied voltage. They have the advantage that can be applied to any system if a specific geometry is given and that they handle all sub-processes. On the other hand, they have the disadvantage of much longer computational times that sometimes lead to practically not-applicable models. In addition, the assumptions often adopted in order to decrease the computational time lead to significant errors and results that lead to, many times even qualitative, discrepancies from experimental measurements.

"Experimental" or fully self-consistent glow discharge models can be further classified, according to the mathematical approach for the solution of the problem, to the following types:

- Analytical models

- Numerical models

Analytical models are usually based on a higher number of assumptions. They are used mostly for one dimensional simulations  and almost exclusively for the modeling of the plasma electrical properties. The most common approach in these models is to separate the discharges in three regions (two sheaths and the bulk plasma). It is the easiest and the fastest way to simulate electrical properties of discharges and in some cases have lead to excellent results [2-4].  However, the implementation of this kind of models to the rather complicated molecular gas discharges like the ones used for PECVD is impractical and their use is limited to the simulation of noble gas discharges.

Numerical models are generally more popular in self - consistent plasma modeling. They can be further distinguished, according to the methodology used for the management of electron and ion transport in RF discharges, in the following categories:

- Kinetic Models

- Fluid Models

- Hybrid Kinetic/Fluid Models

Kinetic models are time and spatially dependent solutions of the Boltzmann equation which produces electron and ion velocity distributions either by direct integration of the equation or by applying statistical techniques (Particle in Cell - Monte Carlo method). The kinetic approach although it is computationally intensive, is the least dependent on a-priori assumptions leading to more accurate results [5-7].

Fluid models solve moments of the Boltzmann equation in time and space, while the Electron Energy Distribution Function is calculated off-line and coupled to the fluid model providing the electron transport coefficients and the rate of electron molecule reactions. The fluid approach, although it is not so accurate compared to kinetic methods, due to the shorter computational times, allows for higher dimensionality (2D, 3D) and for the introduction of more detailed physics to the models. However, these models are limited to gas pressures above 200 mTorr, as they assume a local equilibrium between electrons and the electric field [8-10].

Hybrid models use the kinetic approach in order to handle the non-local transport of electrons and ions in the discharges and to derive transport coefficients of charged species. The fluid approach is simultaneously applied in order to provide the density of the charged species and the electric field distribution. Hybrid models have been developed in order to simulate rather complex chemistries of gas discharges. The transport coefficients and the rate of reactions of electrons with molecules are derived kinetically, while the density of species and the time and space variation of the electric field are calculated using the fluid flow approach. Nowadays, hybrid models are implemented very often especially in plasma-based applications that involve molecular gases since they combine the accuracy of kinetic models with the high dimensionality and short computational times of fluid models [11-13].

PTL_UP has been involved in plasma simulations for more than ten years. During this time, we have developed and expanded plasma models that deal mostly with amorphous and microcrystalline silicon deposition, the simulation of electrical properties of gases (H₂, noble gas) discharges as well as with other specific gas phase and plasma-surface interaction models.

Some of the PTL_UP  developed plasma models are described in more detail below:

- Electrical Model based on Experimental Measurements

- Mass Transfer Model based on Experimental Measurements

- Computational Fluid Dynamics Plasma Model

Bellow is a non-exhaustive and somewhat arbitrary selection of publications:

[01]  M. Meyyappan, Computational Modelling in Semiconductor Processing Artech House (1995).

Analytical Models:

[02]  M. A. Lieberman, J. Appl. Phys. 65, 4186 (1989).

[03] E. Kawamura, V. Vahedi, M.A. Lieberman and C. K. Birdsall, Plasma Sources Sci. Technol. 8, R45 (1999).

[04] F. A. Haas and N. St J Braithwaite, Plasma Sources Sci. Technol. 9, 77 (2000).

Kinetic Models:

[05] T. J. Sommerer, W. N. G. Hitchon, R. E. P. Harvey and J. E. Lawler  Phys. Rev. A 43, 4452 (1991).

[06] M. Surendra and D. Graves, IEEE Trans. Plasma Science 19 144 (1991).

[07] M. Yan and W. J. Goedheer, Plasma Sources Sci. Technol. 8, 349 (1999).

Fluid models:

[08] J. P. Boeuf and L. Pitchford, Phys. Rev. E 51, 1376 (1991).

[09] F.F. Young and C.H. Wu, IEEE Trans. Plasma Sci. 21, 312 (1993).

[10] E. Gogolides and H. Sawin, J. Appl. Phys. 72, 3971 (1992).

Hybrid Models:

[11] T. J. Sommerer and M. J. Kushner, J. Appl. Phys. 71, 1654 (1992).

[12] T. J. Sommerer and M. J. Kushner, JVST B 10, 2179 (1992).

[13] P. L. G. Ventzek, M. Grapperhaus and M. J. Kushner, JVST B 12, 3118 (1995).