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Project supported by the National Natural Science Foundation of China (Grant No. 51172101).
A one-dimensional (1D) fluid model on capacitively coupled radio frequency (RF) argon glow discharge between parallel-plates electrodes at low pressure is established to test the effect of the driving frequency on electron heating. The model is solved numerically by a finite difference method. The numerical results show that the discharge process may be divided into three stages: the growing rapidly stage, the growing slowly stage, and the steady stage. In the steady stage, the maximal electron density increases as the driving frequency increases. The results show that the discharge region has three parts: the powered electrode sheath region, the bulk plasma region and the grounded electrode sheath region. In the growing rapidly stage (at 18 μs), the results of the cycle-averaged electric field, electron temperature, electron density, and electric potentials for the driving frequencies of 3.39, 6.78, 13.56, and 27.12 MHz are compared, respectively. Furthermore, the results of cycle-averaged electron pressure cooling, electron ohmic heating, electron heating, and electron energy loss for the driving frequencies of 3.39, 6.78, 13.56, and 27.12 MHz are discussed, respectively. It is also found that the effect of the cycle-averaged electron pressure cooling on the electrons is to “cool” the electrons; the effect of the electron ohmic heating on the electrons is always to “heat” the electrons; the effect of the cycle-averaged electron ohmic heating on the electrons is stronger than the effect of the cycle-averaged electron pressure cooling on the electrons in the discharge region except in the regions near the electrodes. Therefore, the effect of the cycle-averaged electron heating on the electrons is to “heat” the electrons in the discharge region except in the regions near the electrodes. However, in the regions near the electrodes, the effect of the cycle-averaged electron heating on the electron is to “cool” the electrons. Finally, the space distributions of the electron pressure cooling the electron ohmic heating and the electron heating at 1/4T, 2/4T, 3/4T, and 4/4T in one RF-cycle are presented and compared.
Capacitively coupled radio frequency (RF) glow discharge at low pressure can generate low temperature plasmas easily. The plasmas, which usually are called capacitively coupled plasmas or CCPs in short, have been widely used in many fields, such as microelectronics, materials processing, metallurgy, and biology. In order to improve efficiencies of all these applications, it is very important to understand the generating mechanism and physical characteristics of the plasmas.
In the past decades, many research studies of CCPs have been conducted both theoretically and experimentally.[1–10] Makabe et al. investigated the structure of RF glow discharge in argon at 13.56 MHz by modeling and diagnostics.[1] Richards et al. studied argon RF glow discharges by continuum modeling.[2] Kushner examined the electron properties in parallel plate capacitively coupled RF discharges by Monte–Carlo simulation.[6] Meyyappen and Govindan gave a fluid model for RF discharge simulation.[7] Lymberopoulos and Economou investigated the effect of the metastable atoms on the RF glow discharge in argon by fluid simulation.[8] They also developed a PIC/dynamic MC model to study the spatiotemporal electron dynamics in RF glow discharges.[9]
For CCP from RF glow discharge, the driving frequency is a very important operating parameter. Thus, many research efforts have been devoted to the driving frequency.[11–25] The modeling can be performed by using fluid, kinetic, or hybrid models. Nakano and Makabe investigated the effect of driving frequency on the structure of RF discharge by the relaxation continuum model in a narrow-gap reactive-ion etcher with parallel-plate geometry in SF6 for driving frequencies ranging from 100 kHz to 13.56 MHz.[15] They found that the electron and ion densities increased as the driving frequency increased, however, the sheath width and ion energy decreased with increasing the driving frequency because of a decrease in the magnitude of the dc bias. Segawa et al. analyzed the CCPs between parallel plates in CF4 as a function of the driving frequency between 13.56 and 200 MHz for 50 and 200 mTorr by using a fluid model.[16] They found that the sustaining mechanism and spatiotemporal structure depend on the driving frequency and the mean energy of ions. Also, they found that the degree of electro-negativity strongly depends on the pressure and the electro-negativity becomes weaker at low pressure. Colgan et al. used self-consistent fluid equations to study electrical characteristics of argon discharges at frequencies varying between 13.56 and 54.4 MHz.[17] They found that electron density scales with the square of driving frequency at constant pressure and applied voltage. Vahedi et al.[18] explained the influences of different frequencies by presenting the two-dimensional (2D) results based on direct implicit PIC/MC simulation for capacitive argon RF discharge, as being due to the fact that as the driving frequency increases, the sheath width decreases, and the bulk plasma becomes more uniform. The effects of single and dual driving frequency on glow discharges were also studied.[22]
In a gas discharge, the electrons obtain energy from the electric field and then they are ionized. Therefore, the electron heating mechanism plays an important role in the process of the gas discharge. The electron heating in low pressure discharge has been investigated by using different models.[21–35] Surendra and Graves used particle-in-cell simulations to study the structure of RF glow discharge in pure helium between parallel plate electrodes as well as the ohmic heating.[21] They found that in the absence of secondary electron emission, electron heating in both the sheath regions of the discharge is enhanced at higher voltages compared with ohmic heating in the bulk of the plasma. Kawamura et al. investigated the stochastic heating phenomena in single and dual frequency capacitive discharges.[22] They found that for a uniform fixed-ion discharge, in which the ions are assumed to have a uniform density profile, there is no stochastic heating as expected. Zhu et al. experimentally investigated the RF power distribution between the electron heating and the ion acceleration by measuring the electron densities and ion energies for different RF driving frequencies (13.56, 27.12, 60, and 156 MHz). They used an inhomogeneous model accounting for different heating mechanisms to compare with the experimental results.[25] Graves used fluid model simulations of a 13.56 MHz RF discharge to study the time and space dependence of rate of electron impact excitation.[26] He found that the electron heating by the RF field peaks at both the sheath regions boundary, resulting in a local rise in electron mean energy there. Nitschke and Graves studied period-averaged electron heating with PIC model and fluid model. They found a difference below 100 mTorr, i.e., in the period-averaged electron heating, the discharge properties predicted by different models are different. They estimated that the electron power input can be improved in the fluid simulation by including an analytic expression for stochastic heating in the electron energy balance equation.[31] Recently, Liu et al. used a fluid model to study the effect of the secondary electron emission on the discharge characteristics in a low pressure capacitive RF argon discharge.[34] They found that when the secondary electron emission coefficient varies from 0.01 to 0.3, the electron net power absorption and the electron heating rate have different degrees of enhancement and the electron heating mainly takes place in both sheath regions. Lafleur et al. revisited the problem of electron heating in CCPs, and they proposed a method of quantifying the levels of collisionless and collisional heating in plasma.[35]
The effect of the driving frequency on the electron heating in capacitive RF glow discharge at low pressure is a very important physical problem but it is still not clear. In this work, we focus on this effect by means of a numerical simulation. For this aim, a fluid model of the CCPs is established and numerically solved to obtain the numerical results for the driving frequencies of 3.39, 6.78, 13.56 and 27.12 MHz. Analyzing these results, we can draw some conclusions. This paper presents a foundational study on CCPs. The rest of this paper is organized as below. In Section
Considering an RF low pressure glow discharge between two parallel-plates. When the electric potential difference between the electrodes reaches a certain value, the gas discharge will take place between the electrodes, thus generating a plasma. The plasma can be described by using a fluid model. Although the sizes of the electrodes are much larger than the gap between the electrodes, a one-dimensional (1D) model can be used.
In the plasma generated in the capacitively coupled RF argon glow discharge at low pressure, the particle species are electron (e), argon atom (Ar), and argon ion (Ar+), and among them takes place the following reaction: e + Ar → Ar+ + 2e. Thus, the density of the ions satisfies[8,34]
In this work, the initial conditions are as follows:
The model mentioned in the above section is numerically solved by a finite difference method. In calculating, the parameter values are shown in Table
To analyze the effect of the driving frequency on the plasma characteristics in the argon glow discharge, the simulations are carried out for the cases of four different frequencies; that is, f = 3.39, 6.78, 13.56, and 27.12 MHz.
Figure
Figures
It is most important to study the electron energy transferring mechanism in the rapid growing stage. From Fig.
The electron ohmic heating can be calculated from Eq. (
Figure
Figure
The spatial distributions of the electron pressure cooling with the driving frequency 3.39 MHz in the 100th RF cycle, 6.78 MHz in the 200th RF cycle, 13.56 MHz in the 400th RF cycle, and 27.12 MHz in the 800th RF cycle at times t = 1/4T, 2/4T, 3/4T, and 4/4T are plotted in Fig.
The spatial distributions of the electron ohmic heating with the driving frequencies of 3.39 MHz in the 100th RF cycle, 6.78 MHz in the 200th RF cycle, 13.56 MHz in the 400th RF cycle, and 27.12 MHz in the 800th RF cycle at times t = 1/4T, 2/4T, 3/4T, and 4/4T are plotted in Fig.
The spatial distributions of the electron heating with the driving frequencies of 3.39 MHz in the 100th RF cycle, 6.78 MHz in the 200th RF cycle, 13.56 MHz in the 400th RF cycle, and 27.12 MHz in the 800th RF cycle at times t = 1/4T, 2/4T, 3/4T, and 4/4T are plotted in Fig.
A 1D fluid model for a capacitive RF argon discharge at low pressure is established to study the effect of the driving frequency on electron heating. The numerical results of the evolutions of the discharge process are obtained. The results show that the discharge process has three stages: the growing rapidly stage, the growing slowly stage and the steady stage. In the steady stage, the maximal electron density increases as the driving frequency increases. Also, the results of the cycle-averaged electric field, electron temperature, electron density and electric potential for the driving frequencies of 3.39, 6.78, 13.56, and 27.12 MHz are compared. The results show that the discharge region has three parts: the powered electrode sheath region, the bulk plasma region, and the grounded electrode sheath region. The width of the sheath regions decreases as driving frequency increases. At 18 μs, the cycle-averaged electron pressure cooling, electron ohmic heating, electron heating and electron energy loss for the driving frequencies of 3.39, 6.78, 13.56, and 27.12 MHz are compared. The results indicate that the electron pressure cooling cools the electrons and the electron ohmic heating heats the electrons, the electron heating heats the electrons in the discharge region except in the regions near the electrodes. The heating (cooling) effect increases as the driving frequency increases. Furthermore, the results of the electron pressure cooling, electron ohmic heating and electron heating, for the driving frequencies of 3.39 MHz in the 100th cycle, 6.78 MHz in the 200th cycle,13.56 MHz in the 400th cycle, and 27.12 MHz in the 800th cycle, at four times in one RF-cycle: 1/4T, 2/4T, 3/4T, and 4/4T are compared. It is found that the electron heating has the heating and cooling effect on the electrons alternatively in one cycle.
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