| Citation: |
Yuqiang ZHANG, Xingang YU, Zongbiao YE. Particle-in-cell simulations of EUV-induced hydrogen plasma in the vicinity of a reflective mirror[J]. Plasma Science and Technology, 2024, 26(8): 085503. DOI: 10.1088/2058-6272/ad48d0
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Particle-In-Cell (PIC) simulations were performed in this work to study the dynamics of the EUV-induced hydrogen plasma. The Monte-Carlo Collision (MCC) model was employed to deal with the collisions between charged particles and background gas molecules. The dynamic evolution of the plasma sheath, as well as the flux and energy distribution of ions impacting on the mirror surface, was discussed. It was found that the emission of secondary electrons under the EUV irradiation on the ruthenium mirror coating creates a positively charged wall and then prevents the ions from impacting on the mirror and therefore changes the flux and energy distribution of ions reaching the mirror. Furthermore, gas pressure has a notable effect on the plasma sheath and the characteristics of the ions impinging on the mirrors. With greater gas pressure, the sheath potential decreases more rapidly. The flux of ions received by the mirror grows approximately linearly and at the same time the energy corresponding to the peak flux decreases slightly. However, the EUV source intensity barely changes the sheath potential, and its influence on the ion impact is mainly limited to the approximate linear increase in ion flux.
Radio-frequency capacitively coupled plasma sources (RF-CCPs) are widely utilized in the microelectronics industry [1, 2]. Two plate electrodes are usually utilized in typical RF-CCPs, and the electron density is often lower than 1015 m−3. Hence the etching and deposition rates are very low [3–5]. Raising the driving frequency [2, 6] or utilizing the magnetic field [7–10] can increase the electron density. However, the standing wave effect [11–14] and the skin effect [15, 16] may appear with a high driving frequency, and plasma non-uniformity occurs, which is not conducive to large-area plasma treatment. However, the standing wave effect and skin effect will not appear with a low driving frequency. Therefore, it is preferable for the microelectronics industry if plasma with high density can be obtained at a low driving frequency.
In addition to parallel-plate electrodes [17–19], hollow electrodes [20–25] are also frequently applied in RF and direct current (DC) discharges. With a hollow electrode, RF-CCPs can produce higher plasma density outside the cavity when a low frequency is utilized, which implies that plasma density enhancement effect exists in RF discharges with hollow electrodes [20–22, 26–28]. Moreover, utilizing hollow electrodes with different shapes, uniform plasma density can be obtained and the film deposition quality can be improved [29–31].
Due to the cavity structures, electrons are confined inside the hollow electrode [32] and oscillate between the side walls, forming the hollow cathode effect (HCE) [25, 33, 34]. Via the HCE, inelastic collisions increase [33], which cause higher ionization and plasma density inside the cavity [3, 4, 26]. Electron density up to (2–3)×1017 m−3 was obtained with a ring-shaped hollow electrode, which is ten times that of the density when parallel-plate electrodes are utilized [3]. In addition, RF-CCPs with multi-hole electrodes were also reported to enhance thin-film deposition rates [35] and increase the electron density [26, 36, 37] over large pressure ranges.
Although previous investigations have shown that RF-CCPs with hollow electrodes can enhance the electron density outside the cavity, the mechanism of density enhancement is still unclear. Previous studies have also found that discharge conditions affect the magnitude of electron density outside the cavity [21, 28, 38, 39]. For instance, electron density outside the cavity can be influenced by the pressure, magnetic field and electrode gap [28, 38, 40]. Furthermore, the shape, depth and diameter of the hollow electrode can also affect the magnitude of electron density outside the cavity [22, 26, 36]. However, the effect of these factors on the magnitude of electron density outside the cavity is not clear, which is important for the design of hollow electrodes and the modification of discharge conditions to improve plasma properties. In addition, in experimental studies, researchers often determine the strength of the HCE based on the magnitude of electron density outside the cavity [20, 21, 36], and they consider that the higher the electron density outside the cavity, the stronger the HCE inside the cavity. Hence, further investigation is needed to ascertain whether the above method to determine the HCE strength is suitable.
The purpose of this study is to investigate the plasma density enhancement mechanism and influence factors of the plasma density enhancement magnitude in RF-CCPs with hollow electrodes. This article is elaborated in the following sequence. The 2D PIC/MCC simulation is described in section 2. The results are shown in section 3. Section 4 provides a summary.
The reactor geometry studied here can be found in figure 1 of our previous work [41]. An RF source and a DC bias voltage VDC = −30 V are both applied on the hollow electrode, and the frequency of the RF source is 13.56 MHz, while radii r of the powered hollow electrode and the grounded plate electrode are both 1.6 cm. The distance d between the two electrodes is set to be 1 cm. The pressure p of the working gas argon is 1 Torr. A cylindrical coordinate is adopted for the axis-symmetric discharge device, and the simulation area is that enclosed by the red dashed line. The origin of the coordinates (Z = 0 cm, R = 0 cm) is located at the center of the orifice, with Z > 0 cm and Z < 0 cm denoting the axial positions inside and outside the hollow electrode, respectively.
An electrostatic 2D PIC-MCC code (the XOOPIC code) is adopted in this study [42–45]. A detailed model description is included in our previous study [40]. Therefore, only a brief description of this model is included here. The initial temperatures of electrons and argon ions are 2 eV and 0.04 eV, with the same density. The temperature of the working gas argon is also set to be 0.04 eV, with an invariant density. The time step of electrons is set to be 3×10−13 s, and the time step of ions is set to be 3×10−12 s. The spatial grids are set at 256 (R)×192 (Z).
In RF-CCPs with hollow electrodes, the ionization degree is low and collisions occur mainly between neutral and charged particles. Collisions between electrons and neutral particles are elastic, excitation and ionization collisions, and the collision cross-sections are taken from [46]. Charge-exchange collisions and elastic collisions are considered between ions and neutral particles, and the collision cross-sections are cited from [47]. The same collision types were considered in other RF hollow electrode discharges [33, 48–50].
Figure 1 depicts the electron density profiles with different cavity depths h of 0–1.5 cm when the sheath inside the hole is fully collapsed or expanded, with the cavity diameter D = 1 cm, RF voltage amplitude V0 = 210 V and secondary electron emission coefficient γAr+ = 0.1. As can be seen from figure 1, with h = 0.5–1.5 cm, an electron density peak exists at the Z axis inside the cavity, manifesting the existence of the HCE in the cavity [49]. Due to the HCE, with h = 0.5–1.5 cm, the electron density inside the cavity is higher than that with the parallel-plate configuration (h = 0 cm). To do a comparison, two vertical dashed lines are presented at the hollow electrode sheath fully collapsed and expanded phases of each cavity depth. As shown in figure 1, with different cavity depths of h = 0.2–1.5 cm, when the sheath inside the hole expands, electrons inside the hole move towards the grounded electrode, causing a decrease in the volume of electrons inside the hole.
Time-averaged electron density profiles along the Z axis with different h are shown in figure 2(a). As can be seen from figure 2(a), from inside to outside the hole, the electron density increases with the depth of the cavity, and the electron density outside the cavity is higher than that with the parallel-plate electrodes (h = 0 cm). Figure 2(b) shows the time-averaged radial sheath thickness along the side wall of the hollow electrode at different cavity depths h of 0.5–1.5 cm. The radial sheath thickness is the width between the radial sheath edge and the side wall of the hollow electrode, while the position of the radial sheath edge is obtained by finding where ∂2φ/∂R2 becomes zero for the first time (here φ is the potential) [33]. As can be seen from figure 2(b), from the orifice to the cavity bottom, the radial sheath thickness at the side wall increases, which means that the sheath at the side wall of the hollow electrode is tilted. Figure 2(c) shows the variation of the time-averaged radial plasma bulk width at the orifice with the cavity depth h, while the radial plasma bulk width is equal to the value of the cavity diameter D subtract twice the radial sheath thickness. As shown in figure 2(c), with the cavity depth h increasing from 0.5 to 1.5 cm, the radial width of bulk plasma at the orifice increases.
Figure 3 shows the electron density profiles with different RF voltages V0 of 150–210 V when the sheath inside the hole is fully collapsed or expanded, with the cavity diameter D = 1 cm, cavity depth h = 1.5 cm and γAr+ = 0.1. With different V0 of 150–210 V, an electron density peak exists at the Z axis inside the hole, manifesting the existence of the HCE in the hole. Due to the HCE, with V0 of 150–210 V, the electron density inside the cavity is higher than that with the parallel-plate configuration with V0 = 210 V (h = 0 cm). However, with V0 = 150 V, in the yellow dashed rectangular area, the electron density is lower than the initial electron density, which implies that discharges can only occur in the area between the hollow electrode hole and the grounded electrode when V0 is low and d is narrow. As also shown in figure 3, with different V0, when the sheath inside the hole expands, electrons inside the hole move towards the grounded electrode, causing a decrease in the volume of electrons inside the hole.
Time-averaged electron density profiles along the Z axis with different V0 are shown in figure 4(a). As can be seen from figure 4(a), from inside to outside the hole, the electron density increases with the RF voltage V0. Figure 4(b) shows the variation of the time-averaged radial plasma bulk width at the orifice with the RF voltage V0. The curve in figure 4(b) shows that the time-averaged radial plasma bulk width at the orifice increases with the RF voltage V0.
Figure 5 shows the electron density profiles with different D of 0–2 cm when the sheath inside the hole is fully collapsed or expanded, with the cavity depth h = 1.5 cm, RF voltage V0 = 210 V and γAr+ = 0.1. With the small cavity diameter D = 0.4 cm, even the sheath inside the hole is fully collapsed, there is no plasma inside the hole, and the electron density outside the hole is almost equal to that with the parallel-plate configuration (D = 0 cm). With D = 0.7 cm, an electron density peak with density up to 2.6×1016 m−3 exists at the Z axis inside the cavity, and the peak value is higher than that with D = 1–2 cm. The results indicate that the strongest HCE is obtained with D = 0.7 cm. With D increasing from 0.7 to 2 cm, the density of electrons inside the hole reduces, which means that the HCE intensity inside the cavity is gradually attenuated.
Time-averaged electron density profiles along the Z axis with different D are shown in figure 6(a). As can be seen from figure 6(a), at different axial positions, the electron density ne outside the cavity with D = 0.7–2 cm is higher than that with the parallel-plate configuration (D = 0 cm). However, with D = 0.7 cm, the electron density decreases dramatically along the Z axis, and after Z < −0.28 cm, the electron density with D = 0.7 cm is lower than that with D = 1–2 cm. Furthermore, it can be seen from figure 6(b) that the radial width of bulk plasma at the orifice with D = 0.7 cm is smaller than that with D = 1–2 cm.
Figure 7 shows the electron density profiles with different secondary electron emission coefficients γAr+ when the sheath inside the hole is fully collapsed or expanded, with the cavity diameter D = 1 cm, cavity depth h = 0.5 cm and RF voltage V0 = 210 V. As can be seen from figure 7, with different γAr+ of 0–0.2, an electron density peak exists at the Z axis inside the hole, manifesting the existence of the HCE in the hole, and the electron density inside the hole increases with γAr+. However, with γAr+ = 0.2, despite the sheath inside the hole is fully expanded, the electron density peak is located completely inside the cavity.
Time-averaged electron density profiles along the Z axis with different γAr+ are shown in figure 8(a), and the electron density with the parallel-plate configuration (h = 0 cm) with γAr+ = 0.1 is also shown for comparison. With γAr+ = 0.2, from inside to outside the cavity, the electron density decreases dramatically. Once Z < −0.15 cm, the electron density with γAr+ = 0.2 is lower than that with γAr+ = 0 and 0.1, and once Z < −0.3 cm, the electron density with γAr+ = 0.2 is almost equal to that with the parallel-plate configuration with γAr+ = 0.1. With γAr+ increasing from 0 to 0.2, the time-averaged radial width of bulk plasma at the orifice increases, as shown in figure 8(b).
The radial profiles of the time-averaged electron density ne with different h are shown in figure 9, and the parameters are the same as those used in section 3.1.1. As can be seen from figure 9, at different axial positions of Z = −0.3, −0.45 and −0.6 cm outside the cavity, the radial distributions of the electron density are not uniform, presenting a mountain-like distribution, with the electron density peak at the Z axis. In addition, the closer to the orifice, the higher the electron density outside the hole, and the worse the plasma uniformity. At different axial positions outside the cavity, with h = 0.2–1.5 cm, the electron density increases with the cavity depth h. However, with h increasing from 1.5 to 3 cm, the electron density decreases. Although the plasma uniformity needs to be improved, the maximum electron density can always be obtained outside the cavity with h = 1.5 cm. Therefore, with the parameter settings shown in section 3.1.1, the optimum cavity depth for the plasma density enhancement is h =1.5 cm.
Figure 10 shows the radial profiles of the time-averaged electron density ne with different D, and the other parameters are the same as those shown in section 3.1.3. It can be seen from figure 10 that the radial distributions of the electron density with different cavity diameters are not uniform, with electron density peaks at the Z axis. With different cavity diameters, the closer to the orifice, the higher the electron density outside the cavity. At axial positions Z = −0.3 and −0.45 cm, the electron density at the Z axis has the maximum value with D = 1 cm. However, at Z = −0.6 cm, the electron density at the Z axis has the maximum value with D = 1.5 cm. With the cavity diameter D increasing from 1 to 2 cm, the plasma uniformity is gradually improved.
As shown in figures 6(a) and 10, if Z > −0.5 cm, the maximum electron density outside the cavity is obtained with D = 1 cm, and high electron density plays a crucial role in the enhancement of the plasma etching rate. Therefore, if other parameters remain constant, as shown in sections 3.1.1 and 3.1.3, at Z > −0.5 cm, the optimum cavity depth and diameter for the plasma density enhancement are h = 1.5 cm and D = 1 cm, respectively, while the plasma uniformity can be improved by utilizing a multi-ring electrode [51].
According to the above simulation results, the enhancement mechanism of plasma density and influence factors of the plasma density enhancement magnitude outside hollow electrodes are analyzed.
The variations of the electron density with h and V0 are analyzed as follows. With different cavity depths h or RF voltages V0, an electron density peak exists at the orifice (see figures 1 and 3), which is generated via the HCE and contains numerous energetic electrons [4, 51]. When the sheath inside the hole expands, some energetic electrons in the electron density peak are pushed out of the cavity by the tilted sheath at the side wall and the expanding sheath at the cavity bottom [4, 51]. The electrons with high energy pushed out of the hole will cause high ionization and hence enhanced plasma density outside the hole [4, 51]. The electron energy distribution function (EEDF) at the orifice region and at the central region of two electrodes at ωt/2π = 0.41 is shown in figure 11, with V0 = 210 V, D = 1 cm, h = 1.5 cm and γAr+ = 0.1. Phase ωt/2π = 0.41 is close to phase ωt/2π = 0.5 when the sheath inside the hole is completely expanded. The EEDF is obtained by calculating the electron energy in a local area over 20 RF cycles after the discharge is stable. As shown in figure 11, at the orifice region, the probability of electrons with energy exceeding 15.76 eV (the first ionization energy of argon atom) is much higher than that at the central region of the two electrodes, which indicates that more energetic electrons are generated at the orifice region via the HCE.
In addition, with the cavity depth or the RF voltage increasing, both the radial plasma bulk width and the electron density at the orifice increase, and the electron density at the orifice with the cavity depth h = 1.5 cm or the RF voltage V0 = 210 V even reaches 1.3×1016 cm−3. This indicates that more energetic electrons can be pushed out of the hole through the expanding sheath of the hollow electrode and cause higher plasma density outside the hole. Therefore, with the depth of the hole or the RF voltage increasing, the electron density outside the hole also increases.
Since D varies from 0.4 to 2 cm, with D = 0.7 cm, the electron density has the maximum value at the orifice. However, outside the cavity, the electron density with D = 0.7 cm is lower than that with D = 1–2 cm after a certain axial distance from the orifice (see figure 6(a)). The reason is analyzed as follows. With D = 0.7 cm, the HCE in the cavity is the strongest and the electron density at the orifice is the highest (see figure 5). Hence, in the same orifice region, the probability of electrons with energy exceeding 15.76 eV has the maximum value with D = 0.7 cm, as shown in figure 12, and some electrons even reach an energy of 30 eV. High-energy electrons pushed out of the hole with the highest density cause the maximum electron density near the orifice. However, due to the minimum radial plasma bulk width with D = 0.7 cm (compared with D = 1–2 cm), the number of high-energy electrons pushed out of the hole is also the smallest, resulting in the sharp decrease in the plasma density along the axial distance. The results above also indicate that it is not suitable to determine the strength of the HCE through the magnitude of the electron density outside hollow electrodes.
Electron density profiles at shallow h or small D are analyzed as follows. With the shallow cavity depth h = 0.2 cm, a small amount of plasma exists inside the hole when the sheath inside the hole is completely collapsed (see figure 1). Bulk plasma electrons inside the hole gain high energy via repeated interactions with the expanding side wall sheath [51] and are all pushed out of the hole by the expanding sheath of the hollow electrode (see figure 1), hence the plasma density outside the hole is enhanced slightly (see figure 2(a)). However, with the small cavity diameter D = 0.4 cm, there is no plasma inside the cavity even at the hollow electrode sheath fully collapsed phase (see figure 5), hence the electron density outside the cavity is not enhanced (see figure 6(a)). Therefore, an essential condition for the plasma density enhancement outside the hole is that plasma exists inside the hole when the sheath inside the hole is fully collapsed.
The variations of the electron density with γAr+ are analyzed as follows. With γAr+ = 0.2, even the sheath inside the hole is fully expanded, and the electron density peak is completely located inside the cavity (see figure 7), which means that energetic electrons in the electron density peak can hardly be pushed out of the hole when the sheath inside the hole expands. Therefore, with γAr+ = 0.2, although the highest electron density is inside the cavity and the largest radial width of bulk plasma at the orifice (see figures 8(a) and (b)), the electron density outside the hole still decreases sharply with the axial distance (see figure 8(a)). However, due to the largest secondary electron emission coefficient, more secondary electrons will be emitted from the plane part of the hollow electrode. Therefore, compared with that of γAr+ = 0 and 0.1, the electron density has the maximum value above the plane part of the hollow electrode with γAr+ = 0.2 (see figure 7). To further illustrate the importance of the existence of the electron density peak at the orifice for the plasma density enhancement outside hollow electrodes, the EEDF at the orifice region with γAr+ = 0.1 and 0.2 at ωt/2π = 0.41 is presented in figure 13. Since the electron density peak generated via the HCE is not located at the orifice with γAr+ = 0.2, there are few energetic electrons with energy exceeding 15.76 eV located at the orifice region at ωt/2π = 0.41. However, with γAr+ = 0.1, an electron density peak exists at the orifice. Hence more energetic electrons with energy exceeding 15.76 eV are located at the orifice region. When the sheath inside the hole expands, more electrons with high energy are pushed out of the hole and cause higher ionization and plasma density. Therefore, the electron density with γAr+ = 0.1 is higher than that with γAr+ = 0.2 outside the cavity (see figure 8(a)).
Through investigation into the electron density distributions inside and outside the hollow electrode hole with different hole depths, voltages, cavity diameters and γAr+, it is found that an electron density peak exists at the orifice generated via the HCE, which plays an important role in the density enhancement outside hollow electrodes.
The plasma density enhancement mechanism and influence factors of the density enhancement magnitude in RF-CCPs with hollow electrodes are investigated via a 2D PIC/MCC model. Results show that plasma exists inside the cavity when the sheath inside the hole is fully collapsed, which is an essential condition for the plasma density enhancement outside hollow electrodes. In addition, an electron density peak exists at the orifice generated via the HCE, which also plays a crucial role in the density enhancement. It is also found that the radial width of bulk plasma at the orifice significantly affects the plasma density enhancement magnitude outside hollow electrodes. If the radial width of bulk plasma at the orifice is the narrowest, even though the HCE is the strongest and the electron density at the orifice is the highest, the plasma density outside the cavity may also be low. Therefore, it is not suitable to determine the HCE strength through the magnitude of the electron density outside the hollow electrode hole. Higher electron density at the orifice, combined with larger radial plasma bulk width at the orifice, helps to obtain higher density enhancement magnitude outside hollow electrodes.
This work was supported by National Natural Science Foundation of China (Nos. 12172356 and U23B20110) and the Interdisciplinary and Collaborative Teams of CAS. The support is gratefully acknowledged.
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