When Product space is used for the heavy species, the transport equations in the drift–diffusion approximation are solved in the product space of the base geometry and the extra dimension. With this option it is possible to define the Electric field applied to ions to be Instantaneous or Time averaged. These two options correspond to two physical limits where the ions respond instantaneously to the electric field or respond only to the time averaged electric field. The latter limit case, the high-frequency case, is valid when:

where ω is the electric excitation frequency, and ωpi is the ion plasma frequency. For typical conditions of an industrial CCP reactor using an excitation frequency of 13.56 MHz the ion density in the sheaths (where the electric fields are more intense) is found to be around 1015 m-3. For these conditions and for an argon plasma and consequently the ions only respond to the average potential. The frequency region between the two limit cases is only possible to be described by including the time derivative of the ion velocity in the momentum equation, which is not included in the Plasma Module at the present time.

At high excitation frequencies (in the MHz and above) it is also common to find that the ions and neutral species densities do not change during the excitation period. In these cases it is possible to obtain accurate solutions by solving the heavy species in the Base geometry. When using this option the heavy species transport equations are solved in the stationary limit (time derivatives of densities set to zero), the source terms are averaged along the period, and the ion migration is computed with the average field. Since the heavy species are not solved along the period in the product space there is an enormous reduction in degrees of freedom. Consequently, using the Base geometry for 2D models reduces considerable the computational time. However, for 1D models the reduction of degrees of freedom is not enough to compensate for the computational cost of numerical averages.

To exemplify some of the consequences of the options discussed above, a 1D model for helium was solved with the heavy species in the Product space and in the Base geometry. Figure 6-6 shows the results for the total current at the metal contact (only one curve is showed since they are practically the same in both cases), and the ion current. When using the Product space the ion current is period modulated through the electric field in the ion migration. When solving the heavy species in the Base geometry the ion current is effectively the period average of the one obtained in the Product space.