In recent years, Wang and his group developed a software platform for Multi-physics Analysis of Plasma Sources (MAPS), which contains three models, i.e., the fluid model,^{[4–6,9–11]} the Particle-In-Cell/Monte Carlo (PIC/MC) model^[8,12–15] and the global model.^[16] The MAPS can be used to investigate the plasma behaviors in CCP and inductively coupled plasma (ICP) discharges. In this work, the fluid model is employed to study the N₂/Ar CCP discharge, in which 4 charged species (Ar⁺, , N⁺ and electrons) and 5 neutral species (Ar metastable, the ground-state N atom, and the excited-state N₂ (A³Σ_u), N₂ (B³Π_g), N₂ (a′Σ_u) molecules) are considered. Various chemical reactions between the electrons, ions and neutral species are listed in Table 1 and Table 2. Ar_m, N₂ (A), N₂ (B), and N₂ (a′) are used to indicate the Ar metastable and three excited N₂ molecules.

Table 1.

Table 1.

Table 1. Overview of the chemical reactions for electrons taken into account in the model. . No. Reaction Rate Coefficient (cm3·s−1) Threshold Ref. 1.1 Ar+e → Ar+ + 2e 2.34 × 10−8(Te)0.59exp(−7.44/Te) 15.76 [2] 1.2 Ar+e → Arm + e 2.48 × 10−8(Te)0.33exp(−12.78/Te) 11.55 [2] 1.3 Arm+e → Ar+ + 2e 6.8 × 10−9(Te)0.67exp(−4.2/Te) 4.43 [2] 1.4 7.76 × 10−9(Te)0.79exp(−16.75/Te) 15.60 [4] 1.5 N+e → N+ +2e 3.87 × 10−9(Te)0.86exp(−14.62/Te) 14.54 [4] 1.6 N2+e → N+ + N +2e 2.90 × 10−9(Te)0.72exp(−29.71/Te) 18.0 [4], [17] 1.7 N2+e → N2(A)+e 8.06 × 10−10(Te)−0.306exp(−8.87/Te) 6.17 [4] 1.8 N2+e → N2(B)+e 1.56 × 10−8(Te)−0.52exp(−9.16/Te) 7.35 [4] 1.9 N2+e → N2(a′)+e 6.6 × 10−9(Te)−0.66exp(−11.05/Te) 8.40 [4] 1.10 N2+e → N+N+e 2.15 × 10−8exp(−14.39/Te) 9.75 [4] 1.11 1.8 × 10−7(300/Te)−0.39 [4]

Table 1. Overview of the chemical reactions for electrons taken into account in the model. .

Table 2.

Table 2.

Table 2. Overview of the chemical reactions for ions and neutral species taken into account in the model. . No. Reaction Rate coefficient/cm3·s−1 Ref. 2.1 Arm+Arm → Ar++Ar+e 5.0 × 10−10 [9] 2.2 3.2 × 10−12 [4] 2.3 5.0 × 10−11 [4] 2.4 N2(A)+N → N2+N 2.0 × 10−12 [4] 2.5 N2(A)+N2 → N2+N2 3.0 × 10−18 [4] 2.6 N2(A)+N2(A) → N2(B)+N2 7.7 × 10−11 [4] 2.7 N2(B)+N2 → N2 +N2 1.5 × 10−12 [4] 2.8 N2(B)+N2 → N2(A)+ N2 2.85 × 10−11 [4] 2.9 N2(a’)+N2 → N2(B)+N2 1.9 × 10−13 [4] 2.10 N+N+N2 → N2(B)+N2 8.27 × 10−34exp(500.0/Tgas) (cm6·s−1) [4] 2.11 N+N+N → N2+N 1.0 × 10−32 (cm6·s−1) [4] 2.12 N2(B) → N2(A)+hν 2.0 × 105 (s−1) [4] 2.13 4.45 × 10−10 [18] 2.14 Ar++N → Ar+N+ 4.45 × 10−10 [18] 2.15 2.81 × 10−10 [18] 2.16 N2(B)+Ar → N2(A)+Ar 3.0 × 10−13 [18] 2.17 N2(a’)+Ar → N2(B)+Ar 1.0 × 10−14 [18]

Table 2. Overview of the chemical reactions for ions and neutral species taken into account in the model. .

In the fluid model, the plasma behaviors can be described by a series of fluid equations. For electrons, the density, flux and temperature can be obtained by solving Eqs. (1)–(3). Since the inertial term can be ignored due to the small electron mass, the electron flux is expressed as the drift-diffusion form.

Since the ion temperature is assumed at T_gas = 300 K based on the cold fluid approximation, only the continuity equation and momentum balance equation are needed for ions

The density of the neutral species is obtained by solving the continuity equation:

Under the conditions investigated in this paper, the induced electric field excited by the plasma is negligible. Thus only the electrostatic field is considered by solving the Poisson equation, and can be regarded as the total electric field E in Eq. (2).

To solve Eqs. (1)–(8), appropriate boundary conditions (BCs) are needed. The schematic diagram of the CCP reactor used in this study is shown in Fig. 1. An Si wafer (45 cm in diameter and h₁ = 0.08 cm in thickness), surrounded by an L-shaped dielectric ring, sits in contact with the grounded lower metal electrode (50 cm in diameter). The relative permittivities of the wafer and the dielectric ring are ε_rw = 12.5 and ε_rr, respectively.

Fig. 1.

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	Fig. 1. Schematic diagram of the reactor configuration.

Since charges are accumulated at the dielectric surface due to the “surface-charging effect”, as mentioned in Section 1, the potential at the plasma-wafer interface cannot be treated as that at the lower metal electrode (i.e., grounded). Moreover, it is observed from Fig. 1 that, a segment of the dielectric ring is sandwiched between the wafer and the lower metal electrode, which forms a two-layer dielectric structure. Therefore, the potential at the wafer edge is expected to be different from that at the central region. In order to obtain the plasma potential at the plasma-dielectric surface accurately, a general formula of the potential at the two-layer dielectric surface is given for the first time.

A typical two-layer dielectric structure is shown in Fig. 2. The first dielectric plate (with a thickness of d₁ and relative permittivity of ε₁) is placed above the second dielectric plate (with a thickness of d₂ − d₁ and relative permittivity of ε₂), and a metal plate is placed below them. The dielectric-plasma interface potential V_s is deduced as follows.

Fig. 2.

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	Fig. 2. Schematic diagram of the two-layer dielectric structure.

Since there are no free charges in the dielectrics, the potential in the first dielectric layer (V₁) and in the second layer (V₂) can be determined by the Laplace equation:

Since the upper electrode is powered by an rf source, and the lower electrode together with the side wall is grounded in this paper, the BCs of the potential at the dielectric-plasma interface can be expressed as:

Here, R_r is defined as the distance from the central axis to the left edge of the dielectric ring along the radial direction.

Besides, other BCs, which are also needed to complete the model, have been described previously in Ref. [4]. So a brief description is here. The ion densities are assumed to be continuous at the walls and the axial, i.e., ∇n_i = 0. Since the neutral species are considered to be lost at the walls with a surface-loss probability β_n in the simulation, the BCs for the neutral species continuity equation are given by

In this part, the plasma parameters are examined with different dielectric rings, i.e., silicon (Si), ceramic alumina (Al₂O₃) and silica (SiO₂), whose relative permittivities are 12.5, 8.5, and 3.9, respectively.

Radial profiles of the ion density along the mid-gap between electrodes (i.e., z = 0.75 cm, r = 0 − 30 cm) and the axial ion flux towards the wafer surface (i.e., z = 0 cm, r = 0 − 22.5 cm) with different ε_rr are presented in Figs. 3 and 4. It is obvious that the ion density and the flux at the wafer center (i.e., r < 15 cm) are quite uniform and almost constant for all ε_rr, which means the dielectric ring has no significant influence on the plasma properties at the central region. Whereas, the ion density and the flux at the wafer edge behave differently with different ε_rr. The radial profile of the ion density is characterized by a tiny bump near the electrode edge (i.e., r = 24 cm). The flux first increases along the radial direction at 15 cm< r < R_r (18.8 cm), then there is a significant drop at r = R_r, and finally it increases again at R_r < r < 22.5 cm. Besides, both the ion density and the flux at the wafer edge increase with ε_rr, due to the strong electric field at large ε_rr (as shown in Fig. 6(b)).

Fig. 3.

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	Fig. 3. Radial distributions of the density along the mid-gap between electrodes (z=0.75 cm) with different dielectric ring relative permittivities.

Fig. 4.

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	Fig. 4. Radial distributions of the axial flux of on the wafer (z = 0 cm) with different dielectric ring relative permittivities.

In practical processes, N atoms are also very important, as they are responsible for the film deposition. Fig. 5 illustrates spatial distributions of the N atom density at different ε_rr. The results indicate that the evolution of the N atom density with ε_rr is similar to the ion density. The peak of the N density increases from 2.8 × 10¹¹/cm³ to 3.6 × 10¹¹/cm³ when ε_rr increases from 3.9 to 12.5.

Fig. 5.

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	Fig. 5. Spatial distributions of the N density with different dielectric ring relative permittivities: (a) εrr = 3.9, (b) εrr = 8.5, (c) εrr = 12.5.

The non-uniform distribution of the axial ion flux is caused by the discontinuity of the dielectric materials. This phenomenon has also been observed in previous research though under different discharge conditions.^[21] Indeed, part of the dielectric ring is sandwiched between the wafer and the lower electrode (cf. Fig. 1). Hence, the surface charge density and the electric field above the wafer here (i.e., R_r < r < 22.5 cm) are different from those at the central region (i.e., r < 15 cm), and this gives rise to the different plasma properties. In order to highlight the different plasma parameters at the wafer edge, the surface charge density and the axial electric field in the range of R_r < r < 22.5 cm are shown in Fig. 6. From Fig. 6(a), it is clear that less negative charges are accumulated at the wafer edge at larger ε_rr. At the wafer center, the surface charge density is almost constant with low values (not shown here), because the wafer is very thin.

Fig. 6.

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	Fig. 6. Radial distributions of (a) the accumulated charge density and (b) the axial electric field at the wafer edge (i.e., z = 0 cm and Rr < r < 22.5 cm) with different dielectric ring relative permittivities. In panel (a), the minus sign (–) scale on the vertical axis indicates negative charge accumulation.

From Fig. 6(b), it is observed that the electric field at the wafer edge increases along the radial direction due to the edge effect, and therefore the plasma density and the ion flux increase with the radial distance at the wafer edge, as shown in Figs. 3–5. Since the so-called “surface-charging effect" is more remarkable at the wafer edge, the induced electric field in the dielectric materials (E_zr) weakens the axial electric field in the plasma more effectively. Therefore, the electric field at the center (not shown here) is stronger than at the wafer edge, and this gives rise to a rapid drop in the axial ion flux at R_r. However, the drop in the plasma density at R_r is less obvious, due to the diffusion. Since E_zr is proportional to the ratio of the surface charge density to the dielectric relative permittivity, E_zr becomes weaker at larger ε_rr, leading to the stronger electric field on the wafer. As a result, both the plasma density and the axial ion flux at the wafer edge increase with ε_rr, as shown in Figs. 3–5. When ε_rr is large, the stronger axial electric field at the wafer edge results in a higher plasma density, which indicates worse uniformity. However, the discontinuity in the ion flux becomes less obvious at larger ε_rr. Hence, the dielectric ring material needs to be chosen carefully to satisfy the specific requirements in actual production processes.

3.2. Effects of the dielectric ring thickness

In order to study the influence of the dielectric ring thickness on the plasma parameters, h₂ varies from 0.1 cm to 0.9 cm, with ε_rr fixed at 3.9. Since the dielectric ring thickness has little effect on the plasma behavior at the central region, only the parameters at the wafer edge are shown here. Fig. 7(a) shows the radial distribution of the potential on the wafer (V_s) in the range of R_r < r < 22.5 cm at different h₂. It is clear that V_s increases along the radial direction, and a higher V_s is obtained with a thicker dielectric ring. This is because the wafer and the dielectric ring can be treated as two parallel plate capacitors which are connected in series, and the capacitance of the dielectric ring (and the wafer) is in inverse proportion to its thickness h₂ (and h₁) and the potential difference between the upper and lower surfaces. Hence, for the “capacitor system”, there is a positive correlation between the thickness and the potential difference between the upper and lower surfaces. In other words, with a constant h₁, the potential difference between the upper surface of the wafer and the lower surface of the dielectric ring, which is equivalent to V_s, increases with h₂.

Fig. 7.

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	Fig. 7. Radial distributions of (a) the electric potential and (b) the axial electric field at the wafer edge (i.e., z = 0 cm and Rr < r < 22.5 cm) with different dielectric ring thicknesses.

The electric field on the wafer surface at the edge is plotted in Fig. 7(b). It is clear that the electric field increases gradually along the radial direction due to the edge effect, and it decreases with the increasing dielectric ring thickness due to the surface charging effect. By comparing Figs. 6(b) and 7(b), it is clear that the evolution of the electric field with the dielectric ring thickness is in contrast to that with the dielectric ring permittivity. This can be simply interpreted as the inverse relationship between the dielectric ring thickness and the permittivity in the capacitor system. This can also be understood by examining the potential distribution at and above the wafer. Although the potential above the wafer (not shown here) increase with h₂, the potential at the wafer surface increases faster. Therefore,we can conclude from Eq. (8) that the axial electric field at the wafer surface decreases with the ring thickness. The higher potential corresponding to a lower electric field was also observed in Ref. [3], by using the segmented dielectric electrodes.

Fig. 8.

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	Fig. 8. Radial distributions of the density along the mid-gap between electrodes (z = 0.75 cm) with different dielectric ring thicknesses.

Since the electric field in the plasma at the edge is stronger when the dielectric ring is thinner, higher ion density and ion flux are obtained under this condition, as is clear from Figs. 8 and 9. Similarly, the peak of the N atom density at the radial edge descends significantly when h₂ increases to 0.9 cm (not shown here), and the radial uniformity of the plasma density becomes better due to the weaker edge effect.

Fig. 9.

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	Fig. 9. Radial distributions of the axial flux of on the wafer (z = 0 cm) with different dielectric ring thicknesses.

3.3. Effects of the dielectric ring length

From the results discussed above, it is clear to see the discontinuity in the ion flux at R_r. In this subsection, the effect of the dielectric ring length on the discharge is investigated to improve the ion flux distribution, by changing R_r in the range of 0 cm to 18.8 cm. Note that R_r = 0 cm indicates that the entire SiO₂ dielectric ring is sandwiched between the wafer and the metal electrode, as shown in Fig. 10.

Fig. 10.

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	Fig. 10. Schematic diagram of the entire-layer dielectric ring case (Rr = 0 cm).

Radial distributions of the ion density along the mid-gap between electrodes, and the axial flux of the ion on the wafer at different R_r are plotted in Figs. 11 and 12. It is obvious that the dielectric ring length exercises a great influence on the plasma parameters. The results in the case of R_r = 18.8 cm have been previously described in Subsections 3.1 and 3.2. As R_r is shortened to 10.5 cm, the ion density is lower than in the case of R_r = 18.8 cm. Moreover, the ion density first decreases to 5.0 × 10⁸/cm³ at about 15 cm, and then it increases along the radial direction. Apparently, the radial plasma uniformity is worse at this condition. This is because the stronger “surface-charging effect”, due to the existence of the dielectric ring at r > 10.5 cm, weakens the electric field there, and results in the lower ion density. Moreover, abundant ions at the wafer center and the edge diffuse to the low density region, and this gives rise to the lower ion density at the central and edge areas than that of the case R_r = 18.8 cm. The evolution of the N atom density with R_r is similar to the ion density (cf. Fig. 13(b)). The radial distribution of the ion flux at R_r = 10.5 cm is similar to that of the case of R_r = 18.8 cm, except for the lower value and the radial position where the ion flux exhibits a rapid decrease, as shown in Fig. 12. On one hand, the lower ion flux is caused by the lower ion density. On the other hand, the significant reduction in the ion flux at r = R_r demonstrates that the discontinuity of the dielectric materials affects the plasma properties remarkably. The results indicate that small R_r (R_r > 0 cm) leads to bad plasma radial uniformity. By enlarging R_r, a larger area of uniform plasma can be obtained, and this accordingly helps to improve the etching and deposition quality.

Fig. 11.

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	Fig. 11. Radial distributions of the density along the mid-gap between electrodes (z = 0.75 cm) with different Rr.

Fig. 12.

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	Fig. 12. Radial distributions of the axial flux of on the wafer (z = 0 cm) with different Rr.

Fig. 13.

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	Fig. 13. Spatial distributions of the N density with different Rr: (a) Rr = 0 cm, (b) Rr = 10.5 cm, and (c) Rr = 18.8 cm.

For the case of R_r = 0 cm, which is similar to a dielectric barrier discharges,^[2,22] the ion density and the N atom density at the central region are much lower than the other two cases (i.e., R_r = 18.8 cm and R_r = 10.5 cm), and they gradually increase along the radial direction. This is again because more negative charges are accumulated at the wafer center, and weaken the electric field there. Moreover, the ion flux on the wafer exhibits a strikingly different distribution, i.e., it shows a lower value at the center, and it increases monotonically along the radial direction. This indicates that the discontinuity of the dielectrics is effectively eliminated, although the plasma density and the flux decrease to some extent.

In this paper, a two-dimensional self-consistent fluid model is employed to study the effects of the dielectric ring on the plasma characteristics, especially on the plasma uniformity, for a large-area N₂/Ar discharge. The results indicate that the plasma parameters strongly depend on the properties of the dielectric ring.

First, the effects of the relative permittivity and thickness (or the ratio of the thickness to the relative permittivity, i.e., h₂/ε_rr) of the dielectric ring have been investigated, and an important influence on the plasma parameters, especially at the wafer edge, has been observed. This is because when h₂/ε_rr is large, the electric field at the wafer edge becomes much weaker. This gives rise to the lower ion density, ion flux and N atom density there, and a better uniformity of the plasma density is obtained. Moreover, it is also observed that the radial distribution of the ion flux exhibits a rapid drop near the radial position of R_r, which is caused by the discontinuity of the dielectric materials. Although the entire-layer dielectric ring (i.e., R_r = 0 cm) helps to eliminate the discontinuity, the better uniformity is obtained at the expense of a drop in the production efficiency. From the results discussed above, the relative high plasma density and ion flux can be obtained by decreasing h₂/ε_rr of the dielectric ring. Therefore, by employing a dielectric ring with larger R_r and lower h₂/ε_rr, the larger area uniform plasma with a higher deposition or etching rate can be achieved.