5.1 Conclusion
A one dimensional sheath model is proposed for multi-frequency driven, inhomogeneous and capacitively coupled plasma considering the case that the sheath is collisional. In the modelling we assumed that the plasma chamber is driven by a current source having a .low frequency and N number of high frequencies. All the high frequencies are integer multiples of the low frequency. Ions crossing the sheath can respond to the instantaneous electric field. The ion sheath-plasma boundary is stationary, and ions enter the sheath with a Bohm pre-sheath velocity. The electrons are inertialess and can respond to the instantaneous electric field. The time-derivative terms in the ion fluid equations are neglected as it is showed that the time derivative terms do not play important role. As the ~lectric field considered here is time dependent, all the sheath parameters determined using this model should be time dependent. As collisional sheath is considered, this model is appropriate for intermediate and high pressure ranges where the plasma is assumed to be highly ionized. The proposed model is analyzed for two special cases of CCP- dual frequency and triple frequency capacitively coupled plasma.
For DF CCP the results obtained using this model are compared with the time dependent collisionless model proposed by Jiang et al. [12] and time independent collisional model proposed by Boyle et al. [6] using some standard plasma parameters.
The maximum values of the sheath motion and potential estimated using this model are found to be lower than those of the above two models. This is because the collisional effect is absent in the Jiang's model where the time-dependent effect is absent in the Boyle's model. Due to the lower sheath width and potential the ion bombarding energy estimated by this model will also be lower.
In DF CCP sheath the sheath property is significantly modulated by the two frequencies. The high and the low frequencies determine the fast and the slow oscillations, respectively. The effect of pressure and plasma density on the sheath parameters are investigated. The sheath parameter found decrease with the increase of pressure as well as plasma density. This is due to collisional effects within the sheath.
The phenomena is also evident in Boyle's model. However, time-impendent electric field present in the Boyle's model overestimates values of sheath parameter at all pressures and plasma densities. Jiang's model does not have any effect of pressure as it assumes no collisionality within the sheath. So, sheath parameters estimated by Jiang's model do not have any effect of pressure.
The effect of frequencies on the sheath parameters have been investigated. They found to have inverse relationship with the high frequency value. They become less sensitive to the variation of the frequency when high frequency becomes large. This is because at this stage the frequency approaches the ion critical frequency. Proper choice of phase difference can be used to control the peak position and value of the sheath parameters. It can playa crucial role in determining the ion bombarding energy.
Sheath parameters for TF CCP have also been determined. As in the case of DF CCP it is found that the sheath motion and potential is governed by LF frequency for slower oscillation and, MF and HF frequency for faster oscillation. Compared with Rahman et al. [33] the sheath parameters found to have lower values. This is because Rahman's model assumes a time-average collisional sheath.
The dependency of the sheath parameters on pressure and plasma density showed similar behaviour as of the DF CCP case. The frequency ratio showed interesting relationship with the sheath parameters. The parameters show maxima when
CtllIWt becomes an odd number and minima when the ratio becomes an even number. In an analysis it has been found that proper choice of phase differences can be useful to adjust the ion energies hitting electrode.
Compared with the DF CCP case more oscillating behaviour is found in both the sheath motion and potential in case of TF CCP. The higher oscillation will provide
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higher electron heating and higher ion-neutral collisions within the sheath. The oscillation pattern of these parameters depend on the frequency ratios.
As this model considers the instantaneous ion motion and the instantaneous electric field, so this model is more accurate compared with other models where time- average electric field is considered. It is not very far that three or more RF frequencies will be used in real plasma systems. So, multi-frequency modelling of CCP will help to estimate the sheath parameters using this model more accurately, that will facilitate the plasma users to have better control in etch properties.
5.2 Suggestions for Future Work
For future research in plasma modelling the following suggestions are made:
(a) In this we have analyzed two and three frequency system and compared the results. More frequencies can be used to extent the investigation.
(b) In the calculation of sheath parameters we used typical values for the input parameters due to the lack of experimental facility to acquire these parameters. These values could be measured in a real experiment and used in the calculations to have better results.
(c) Particle-in-Cell (PIC) simulation can be used to verify result obtained in this work.