Capacitively Coupled Plasmas
Capacitively coupled plasmas (CCP) represent a monumental challenge when it comes to numerical modeling for the following reasons. The main challenge is the overwhelming number of RF cycles before the plasma evolves to a time periodic steady state solution. COMSOL Multiphysics includes a special physics interface for such applications, which removes the need to solve the problem in the time domain, while still capturing all the nonlinearities involved in the discharge. This is the Plasma, Time Periodic interface.
Consider the following test problem, which roughly describes the evolution of a metastable species in a capacitively coupled plasma:
where u is representative of the amount of the metastable species. The first term on the right hand side represents the periodic production of the metastable in the plasma sheath, oscillating with period ω. The second term represents losses due to collisions with the background gas, and the third term losses due to collisions between the metastables.
The evolution of the metastable density is shown in Figure 6-1. It takes close to 100 cycles before the value evolves to its periodic steady state solution. In many plasma reactors, it can be more like 50 or 100,000 RF cycles before this evolution is complete. Solving this in the time domain is computationally intractable.
Figure 6-1: Metastable density.
The above equation can be reformulated as a boundary value problem:
with periodic boundary conditions:
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Solving this boundary value problem will immediately produce the solution during the period when the periodic steady state solution is reached. Thus, a time-dependent problem with enormous computational cost is replaced by a simple boundary value problem. This is the basic technique used in the Plasma, Time Periodic interface.
In order to facilitate the modeling process, the conversion to a boundary problem is accomplished by attaching an extra dimension to the reactor geometry. This forms a product space where the boundary value problem can be solved while leaving the model setup practically the same as when using the Plasma interface. All of the dependent variables may be solved for in the product space, and periodic boundary conditions are applied across the extra dimension. The Time Periodic study is used to solve this problem, but it is basically the equivalent of a Stationary study.
As well as reducing the computational burden, the technique also has the following advantages:
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Integrals performed over both the extra dimension and the base geometry can be used in the equations themselves. This allows the contacts and terminals to be driven directly with a fixed power rather than a fixed voltage. This is important not only for numerical stability, but also for discharges where an alpha to gamma transition occurs. In these cases, a given voltage excitation could produce two different solutions, depending on the initial conditions, but with a fixed power, only one physical solution exists. Also, the discharge power is usually known, whereas the electrode potential is often not.
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The self DC bias can be computed easily with an additional equation, rather than the ad hoc methods used in traditional methods.
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Parametric sweeps over the operating conditions are fast and easy, since the problem is not solved in the time domain. So, for 1D models, a range of powers, pressures, frequencies, and so on can be swept in only a few minutes.
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It is relatively easily to include a matching network in the model, so the plasma can be driven from an L–type matching network. It is also easy to compute the plasma impedance at the fundamental frequency which is useful when designing matching networks.
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The harmonics generated by the plasma are still resolved by the method, there are no approximations in the model. It is possible to observe how these harmonics in the discharge current can lead to an impedance mismatch when driving with an external circuit.
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The method is well suited for a modern computer architecture, since re-assembling and paging in and out of memory at each time step is not necessary. Nearly all the time is spent performing the factorization of a sparse matrix with a direct solver, which is highly parallelized and runs at a very high FLOP rate.
There are, of course a couple of downsides worth mentioning:
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Since an extra dimension is attached to the base geometry, the memory requirements are high. For 2D models, 64+ GB of memory is recommended.
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The problem formulation is highly nonlinear, which is quite a challenge for the solver. Convergence might be very difficult for a given setup, depending on the exact geometry and operating conditions.