Capacitively coupled RF discharges were widely used in the microelectronics industry for the production of in- tegrated circuits specially processes such as etching and coating [1] -[5] . In these systems, the metallic electrodes were normally enclosed within the reactor and in direct contact with the plasma. External conditions, like pres- sure, power, geometry and driving frequency determine the properties of the plasma and thus its practical use in various applications [6] .

When RF voltage is applied between two parallel plate electrodes, two sheaths are formed in front of each electrode and plasma connecting these sheaths. These sheaths heat the electrons through a process in which electrons reflected from the moving sheath edge and gained energy on average. This process usually referred to Stochastic heating which is the dominant source of electron heating at low pressure besides the well-known oh- mic heating. Heating by RF radiation was very important in the fusion device [7] . Two kinds of electron heating mechanisms may be taken place. Firstly, ohmic heating due to the transfer of energy gained from the accelera- tion of electrons in electric field to thermal electron energy through collisional processes. Secondly, Stochastic electron heating has been found to be a powerful mechanism in capacitive coupling R-F discharge. Here elec- trons impinging on the oscillating sheath edge suffer a change of velocity upon reflection back into the bulk plasma. As the sheath moves into the bulk, the reflected ions gain energy, as the sheath moves away, the electron loss energy. However, averaging over an oscillation period, there is a net energy gain.

At sufficiently large RF voltage, the beams of fast electrons ensure the high rate of ionization in the quasi- neutral plasma. This ionization, as well as the diffusion of charged particles near the boundaries of the sheaths compensates the losses of electrons and ions in the central part of the discharge due to recombination and diffu- sion. The discharge becomes stable and continues to burn in the strong-current regime, and low-frequency oscil- lations disappear [8] .

The aim of the present work is using a self-consistent analysis of the collisional sheath of the capacitively coupled RF discharge to calculate the electron ohmic heating power, the average Stochastic power for a single sheath, the power dissipated in the sheath and the sheath thickness and compared the results with the available experimental data.

The capacitive discharge plasma was formed between two stainless-steel 20 cm diameter electrodes separated by 6 cm. The lower electrode was powered by RF (13.56 MHz), type ENI model OEM-6, generator through a matching box, while the upper electrode as well as the discharge chamber was grounded. The chamber was pumped down by a vacuum system to base pressure of 10⁻⁴ torr. During all measurements, continues flow of a gas through the discharge chamber was maintained. High purity Ar gas was used as the working gas and was fed to the chamber through a needle valve. The pressure of the working gas was varied between 1 - 10 torr and measured using a digital vacuum gauge (VAP 5). For displaying the waveform of the discharge current and vol- tage two channel digital oscilloscope (Tektronix TDS 320, 100 MHz) was used. The applied voltage “V” was measured by using HF voltmeter type B7-37, while the discharge current “I” was measured from its waveform with the help of an oscilloscope. Bird Model 4431-43 Series wattmeter was used to measure the input and/or dissipated powers.

Figure 1 shows a schematic drawing of the apparatus used in this work. Also, Figure 2 shows an example of the waveform of RF discharge at discharge voltage = 950 V and at Ar gas pressure = 1 torr.

At high pressure a self-consistent analysis of the collisional sheath is required, which has been given by Lieber- mann [9] and Godyak [10] . These authors assuming that, the ion mean free path λ_i is constant independent of the velocity [11] . Also, assuming the ionization occurs in the sheath, and the electrons are accelerated into the plasma.

The relation between the applied voltage “V” and the discharge current “I” of the RF discharge is measured. The measurements have been carried out in the pressure range (0.5 - 2 torr) for Ar gas. I-V characteristic curves for Ar gas at pressures of 0.5, 1, 1.5, and 2 torr are shown in Figure 3.

The relation between the applied voltage and the discharge current can be explained as the following:

When the applied voltage increased the amplitude of the RF electric field is increased, so, the oscillating electrons gain enough energy to make collisions with the neutral particles and increasing the ionization rate, consequently the discharge current would increase. At high pressure, the number of electrons-atom ionizing collision increases, thus, more electrons are produced and consequently, the discharge current is increased. Moreover, the voltage across the plasma sheath cannot be specified independently of the heating mechanism and the power absorbed by the plasma.

On the other hand, the self-consistent model is used for high pressure capacitive RF discharge. Assuming the following: 1) The only loss mechanism is electron loss to the wall; 2) The ionization occurs in the sheath, and the electrons are accelerated into the plasma; 3) The plasma length d » l − 2S_m (where S_m is the maximum sheath thickness) l is the system length, with an initial estimate S_m » 1 cm for numerical computations which is a no-

minal value for high-pressure capacitive RF discharge.

The electron ohmic heating power per unit area and the average Stochastic power for a single sheath are cal- culated using Equations (7) and (8) respectively. The calculation results indicate that, at high pressure, the ohmic heating is usually the dominant heating mechanism. Ohmic heating due to the transfer of energy gained from the acceleration of electrons in electric field to thermal electron energy through collisional processes [14] .

The power dissipated in the sheath P_s is calculated using Equation (18), (where the constant k for Ar gas is equal to 103.75 [13] . Figure 4 shows the comparison between the measured dissipated power P_s, measured using the wattmeter, and the calculated one as a function of Ar gas pressure at constant discharge current I = 20 mA. When the gas pressure increased the dissipated power is decreased. At high gas pressure, the secondary electrons traversing the near-electrode sheath have an opportunity to experience one or several elastic and/or inelastic collisions that results in lesser energy acquired by them over the sheath width compared with the case without collisions.

Moreover, Figure 5 shows the relation between the input power of the discharge, which calculated from the measured waveform of the discharge currents, at two constant discharge voltage, as a function of the gas pres- sure. When the gas pressure increases the gas discharge is more confined in the electrode space due to the lower mean free path associated with a large number of collisions. It’s clear, however, that one can vary the power consumed by the discharge not only by varying the discharge voltage V but also by varying the chamber pres- sure at a constant V.

In contrast to the measurements presented, the total power measured by the wattmeter, before the matching network, shows the small variations of the order of ±10% when increasing the gas pressure, while the power re- ally consumed in the discharge changes more than that for higher applied voltage. The reason for that is, most of the power output of the power supply is consumed in the matching network and the stray impedance of the sys- tem. Thus a relation between the actual dissipated power P_s in the discharge to the total power P_T measured by the wattmeter which include, the power consumed in the discharge and the power losses in the matching net- work, is shown in Figure 6. The actual power is a small non constant fraction of the total power and the amount of power consumed in the discharge is higher at higher applied voltages and gas pressures. This is due to the fact that plasma impedance is changing and thus a large amount of power is transferred from the power supply to the discharge [15] .

Moreover, the sheath thickness d_s of a capacitively coupled RF discharge is calculated using Equation (19), d_s should be of the same order of the electron oscillation amplitude X_m in the RF field of the discharge. The higher sheath field will increase the amount of power consumed for the acceleration of ions and secondary electrons in the sheath. In the case of higher pressures, the smaller mean free path of the electrons leads to smaller sheath dimensions and thus to higher sheath capacitance [16] .

A self-consistent model of the collisional sheath of the capacitively coupled RF discharge at high gas pressure is used. The electron ohmic heating power, the average Stochastic power for a single sheath, the power dissipated in the sheath and the sheath thickness are calculated.

The plasma is produced, using 13.65 MHz RF source, using Ar gas. The relation between the applied voltage “V” and the discharge current “I” of the RF discharge is measured. The voltage across the plasma sheath cannot be specified independently of the heating mechanism and the power absorbed by the plasma. The calculation results indicate that, at high pressure, the ohmic heating is usually the dominant heating mechanism. The electrons are heated by ohmic heating resulting from the volume currents, rather than from sheath heating.

At constant discharge current I = 20 mA, the calculated and/or measured dissipated power in the sheath is de- creased when the gas pressure is increased.

The relation between the actual dissipated power P_s in the discharge to the total power P_T measured by the wattmeter which include, the power consumed in the discharge and the power losses in the matching network, is

given. The actual power is a small non-constant fraction of the total power and the amount of power consumed in the discharge is higher at higher applied voltages and gas pressures. The power coupling to the discharge is sensitive to the variation of both the discharge voltage and gas pressure.

Moreover, the sheath thickness d_s of a capacitively coupled RF discharge is calculated and found to be of the same order of the electron oscillation amplitude X_m in the RF field of the discharge.