Ar/SF₆ plasma simulation for dual-frequency capacitively coupled plasma incorporating gas flow simulation and secondary electron emission

Figure 2 shows the simulation model. One-half of the cross section of the chamber was used as a simulation model. The central axis of the chamber corresponded to the center of the cylindrical coordinates, and the vertical and lateral directions were defined as the z- and r-axes, respectively. In this coordinate system, the position of the absorption probe at the center was (r, z) = (0, 80) and that at the edge was (r, z) = (35, 80).

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The boundary conditions for the simulation were set to be the same as those for experiment. The gas mixture Ar/SF₆ was set to be evenly supplied from the upper electrode and exhausted from the space between alumina and the chamber (lower right of Fig. 2). The gas flow rates were Ar/SF₆ = 50/2.5 and Ar/SF₆ = 50/5.0 sccm, and the total pressure was set to be 4 Pa for both gas mixtures. The relative permittivity of the insulating material Teflon of the upper electrode was 2.5, and that of alumina was 8.5. A frequency of 60 MHz and V_pp = 810 V were applied to the upper electrode, and a frequency of 2 MHz and V_pp = 905 V were applied to the lower electrode. The other metal surfaces were grounded.

Plasma hybrid module (PHM) of Pegasus Software Inc. was used as the fluid model for the plasma simulation. In PHM, the continuity equation of electrons is expressed as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{{\rm{e}}}}{\partial {\rm{t}}}+{\rm{\nabla }}\cdot {{\boldsymbol{\Gamma }}}_{{\rm{e}}}={R}_{{\rm{e}}},\end{eqnarray} \tag{ 1 }$$

where ${n}_{{\rm{e}}}$ is the electron density, ${{\boldsymbol{\Gamma }}}_{{\rm{e}}}$ is the electron flux, and ${R}_{{\rm{e}}}$ is the electron generation rate. Similarly, the ion density ${n}_{{\rm{i}}}$ is given as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{{\rm{i}}}}{\partial {\rm{t}}}+{\rm{\nabla }}\cdot {{\boldsymbol{\Gamma }}}_{{\rm{i}}}={R}_{{\rm{i}}},\end{eqnarray} \tag{ 2 }$$

where ${{\boldsymbol{\Gamma }}}_{{\rm{i}}}$ is the ion flux and ${R}_{{\rm{i}}}$ is the ion generation rate.

The gas reaction model in the plasma simulation was constructed referring to Refs. 23–26. Table I shows the species calculated in the simulation. Twenty-four types of ion and radical generated from SF_6, F₂, and Ar were calculated in the simulation. Table II summarizes the reactions of electrons with molecules, atoms, neutral radicals, and ions. Thirty-five types of reaction were incorporated into the plasma simulation referring to the reaction constants in Refs. 23–33. Table III shows the reactions between ions, molecules, and atoms. Sixty-four types of reaction were incorporated into the plasma simulation, mainly referring to the reaction rates in Refs. 27 and 34.

Secondary electrons are emitted by the collision between the positive ions and the electrode as shown in Fig. 2. In the simulation, the SEE coefficient was calculated as follows. In the ion flux ${{\boldsymbol{\Gamma }}}_{{\rm{i}}}$ of Eq. (2), the amount of ion flux flowing into the electrode Γ_iC was calculated. Assuming that the SEE coefficient is γ, the flux amount of emitted electrons Γ_eE was calculated as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{eE}=\gamma \,{{\rm{\Gamma }}}_{iC}.\end{eqnarray} \tag{ 3 }$$

Γ_eE was added to the electron flux Γ_e so as to increase the total Γ_e without considering the emission direction.

Here, we discuss the SEE coefficient γ. The coefficient has been reported in several papers. However, the coefficient changes depending on the electrode voltage, the electrode material, and surface condition, and it has not been clearly quantified. ^17–19) In our previous study, the average V_dc values of the upper and lower electrodes were −691 and −859 V. ²⁰⁾ The RF frequency of the lower electrode was as slow as 2 MHz, and the inflow of electrons changed in response to the RF frequency, and V_dc fluctuated between –816 and –902 V. On the other hand, the RF frequency of the upper electrode was fast, and it became almost constant –691. As a result, the difference between the upper and lower electrodes is 125–211 V, which is smaller than the average of each. Furthermore, the materials of the upper and lower electrodes are the same as Si. From these points, we assumed that the upper and lower electrodes had the same SEE coefficient.

Moreover, the energy received by a positive ion before collision with an electrode can be approximated by the product of V_dc and the amount of charge. Since the positive ions generated in the Ar/SF₆ plasma have a monovalent charge, it is considered that they collide with the electrodes with almost the same energy. Therefore, the SEE coefficient γ was set to be constant regardless of the type of positive ion. Furthermore, in our previous study on Ar plasma, the electron density calculated in the simulation of γ = 0.05 was close to the measured electron density. Therefore, in this study, we assumed that γ is in the range from 0 to 0.05. ²⁰⁾

The Neutral momentum equation module (NMEM) of PEGSUS Software Inc. was used to calculate the diffusion and flow of neutral particles. In NMEM, the continuity equation of Eq. (4) is solved to determine the distribution of gas flow velocity. ³⁵⁾

$$\begin{eqnarray}&&\displaystyle \frac{\partial \rho }{\partial t}+\displaystyle \frac{\partial \left(\rho {v}_{r}\right)}{\partial {\rm{r}}}+\displaystyle \frac{\partial \left(\rho {v}_{z}\right)}{\partial {\rm{z}}}=0,\end{eqnarray} \tag{ 4 }$$

where ρ is density and t is time. v_r , and v_z are the velocities in the r- and z-axes, respectively.

Figure 3 shows the flow diagram for the coupled calculation of plasma and gas flows in simulations. We calculated the collision reaction between electrons and Ar/SF₆ by plasma simulation, and we calculated the gas state by considering the particles generated in the flow simulation. The continuity equation in the coupled calculation is expressed as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\rho }_{a}}{\partial t}+\displaystyle \frac{\partial \left({\rho }_{a}{v}_{r}\right)}{\partial {\rm{r}}}+\displaystyle \frac{\partial \left({\rho }_{a}{v}_{z}\right)}{\partial {\rm{z}}}={S}_{a},\end{eqnarray} \tag{ 5 }$$

where ${\rho }_{a}$ is the mass density of a-type particles, and S_a is the amount of a-type particles generated and disappeared per unit time and unit volume. The densities calculated in the plasma simulation is reflected in ${S}_{a}.$ After the movement of various particles in the gas flow simulation was calculated, the density of particles generated by plasma was calculated again. The calculations were repeated until the changes in all particle densities converge.

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Firstly, assuming that there is no SEE, the coupled calculation of plasma and gas flows in simulations was performed at Ar/SF₆ = 50/5.0 sccm. As for the V_dc in the Ar/SF₆ mixed gas, similarly to the Ar gas described in Sect. 3.2, the V_dc of the lower electrode changed in the range of –772 to –868 V, and the V_dc of the upper electrode became almost constant –667 V. The voltage difference between the upper electrode and the lower electrode was 105–201 V, which was smaller than the average value of each. The calculation result of electron density is shown in Fig. 4. The region where the calculated electron density is low near the upper and lower electrodes corresponds to the sheath region of Ar/SF₆ plasma.

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From the results in Fig. 4, we extracted the electron density at the center at the probe position. The probe was set at a height of 23 mm from the lower electrode and the height position corresponded to z = 80  mm in the model shown in Fig. 2. Since r = 0 mm is the center of symmetry of the simulation model and is a singular point, the center position in the simulation was set to r = 2 mm.

The electron density calculated in the simulation was 4.93 × 10¹⁶ m⁻³, whereas that obtained by experiment was 10.2 × 10¹⁶ m⁻³. The result shows that the electron density in the simulation was 51.7% lower. We considered that the electron density was low because the effect of SEE was not included. To add the effect of SEE to the simulation, the value of the SEE coefficient γ was increased by 0.01 in the range from 0 to 0.05 and compared with the experimental results. From the results obtained in the simulation, the electron density at (r, z) = (2, 80) was extracted. Figure 5 shows the relationship between the SEE coefficient γ and the electron density. The electron density increased with the increase in γ, and when γ = 0.04, it became 10.26 × 10¹⁶ m⁻³, which was the closest to the measured electron density. The SEE γ in the Ar/SF₆ mixed gas was determined to be 0.04.

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The spatial distributions of the experimental and simulation results were compared with those of the SEE coefficient γ = 0.04. Figure 6(a) shows the calculation results of electron density at γ = 0.04. Figure 6(b) shows the electron density distributions along the z-direction at the positions r = 2 (CNT) and r = 35 mm (EDG). The calculated electron density at the probe position (z = 80 mm) was extracted and compared with the experimental results.

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Table IV summarizes the calculated and experimental results. In the table, the experimental and calculation results at γ = 0 are also shown for comparison with those at γ = 0.04. At γ = 0, the electron densities in the simulation for both the center and the edge were lower than those in the experiment. The electron density at the edge to that at the center (EDG/CNT) in the experiment was 83.3% and that in the simulation was 92.70%. On the other hand, the calculation results at γ = 0.04 were close to the experimental results for both the center and the edge. In addition, the EDG/CNT in the experiment was 83.3% and that that in the simulation was 83.3%. By adding the effect of SEE, we found that the spatial distribution of the electron density in the simulation became close to that in the experiment.

Next, simulation was performed at γ = 0.04 and SF₆ flow rate of 2.5 sccm. The electron densities were extracted at the center and edge and compared with the results of the probe measurement. Figure 7 shows the simulation results and probe measurement results. For comparison, the results at SF₆ = 5 sccm are also shown. At SF₆ = 2.5 sccm, the simulation results at the center and edge were 15.20 × 10¹⁶ and 12.10 × 10¹⁶ m⁻³, respectively. The calculated electron density at the center was 4.11% higher than the experimental result, and that at the edge was 11.03% lower than the experimental result. It is considered that there is an error of about 10% in the probe measurement and the experimental results and the simulation results are almost in agreement. By setting the SEE coefficient γ at 0.04, we can realize a plasma simulation whose results reproduce the probe measurement result.

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In Sect. 4.2, the validity of setting the SEE coefficient γ at 0.04 was shown. Therefore, the simulation results at the SEE coefficient γ of 0.04 were investigated. Firstly, the positive and negative charges of the plasma were examined at the center position.

Figure 8(a) shows the densities of electrons and negative ions. The densities of negatively charged particles increased in the order of electrons, SF₆ ⁻, and SF₅ ⁻, and the total charge was 2.10 × 10¹⁶ m⁻³. Similarly, Fig. 8(b) shows the densities of positive ions. The densities of positive ions increased in the order of Ar⁺, SF₅ ⁺, and SF₂ ⁺. The total charge was 2.24 × 10¹⁶ m⁻³ and the sum of the densities of negatively charged particles and that of the positively charged ions are almost the same. These results show the validity of the plasma simulation from a viewpoint of charge neutrality in plasmas.

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Figures 9(a) and 9(b) respectively show the spatial distributions of the densities of electrons and SF₅ ⁻, which are higher than those of other densities in negatively charged particles. In addition, Figs. 9(c) and 9(d) respectively show the spatial distributions of the densities of Ar⁺ and SF₅ ⁺, which are higher than those of other positively charged ions. The electron density is high at the center of the chamber in the height direction, whereas the densities of Ar⁺, SF₅ ⁺, and SF₆ ⁻ are high near the lower electrode.

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Figure 10 shows the spatial distribution of species other than charged particles in the plasma. It shows (a) the Ar* density, (b) the SF density, (c) the F₂ density, and (d) the SF₅ density. Ar^* has been reported to have a long lifetime of more than 10 s. ³⁶⁾ Therefore, it is transported toward the exhaust port by the gas flow and has high density even near the port. In the coupled calculation of plasma and gas flows in simulations, this phenomenon was reproduced as reported in the previous paper. ²⁰⁾

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