Computational Characterization of Microwave Planar Cuto ﬀ Probes for Non-Invasive Electron Density Measurement in Low-Temperature Plasma: Ring- and Bar-Type Cuto ﬀ Probes

: The microwave planar cuto ﬀ probe, recently proposed by Kim et al. is designed to measure the cuto ﬀ frequency in a transmission (S 21 ) spectrum. For real-time electron density measurement in plasma processing, three di ﬀ erent types have been demonstrated: point-type, ring-type (RCP), and bar-type (BCP) planar cuto ﬀ probes. While Yeom et al. has shown that the RCP and BCP are more suitable than the point-type probe for process monitoring, the basic characteristics of the ring- and bar-type probes have yet to be investigated. The current work includes a computational characterization of a RCP and BCP with various geometrical parameters, as well as a plasma parameter, through a commercial three-dimensional electromagnetic simulation. The parameters of interest include antenna size, antenna distance, dielectric thickness of the transmission line, and input electron density. Simulation results showed that the RCP has several resonance frequencies originating from standing-wave resonance in the S 21 spectrum that the BCP does not. Moreover, the S 21 signal level increased with antenna size and dielectric thickness but decreased with antenna distance. Among the investigated parameters, antenna distance was found to be the most important parameter to improve the accuracy of both RCP and BCP. This paper investigated the basic properties of two types of planar cutoff probes, the ring-type and the bar-type planar cutoff probes, with various geometrical parameters as well as a plasma parameter through a commercial 3D electromagnetic simulation. Simulation results showed that the RCP had several resonance frequencies that originated from standing-wave resonance on the detecting antenna, while the BCP did not. Moreover, the signal level of both the RCP and BCP increased with increasing antenna size and decreased with increasing antenna distance between the radiating and detecting antennae because of elevated and diminished capacitive coupling, respectively. Among the studied parameters, antenna distance was found to be the key parameter highly related to the accuracy of both probes. As a result, the BCP was more practical than the RCP for use in non-invasive electron density measurement systems since it allowed for much easier determination of cutoff frequency in S 21 spectra.


Introduction
Consisting of charged particles (electrons and ions) and neutral particles (atoms, molecules, radicals, excited and metastable species), plasma is controllable via electrostatic and electromagnetic fields [1]. Its application covers various fields such as material fabrication, nuclear fusion, and medical, environmental, and aerospace industries [1,2]. In particular, plasma is one of the key factors in semiconductor and display fabrication, since both its physical (energetic ions) and chemical (reactive radicals) properties can be exploited.
Advanced process control (APC) refers to the fine-tuning of plasma processing based on real-time feedback signals from various plasma process monitoring devices. In fabrication fields, recent demands and BCP considering various geometrical parameters, as well as a plasma parameter through the CST Microwave Studio. The main parameters include antenna size, antenna distance, dielectric thickness, and input electron density.
This paper consists of three parts. Section 2 gives the simulation details, such as simulation method, geometry, boundary conditions, and materials. Section 3 presents the simulation results and analysis of each case. Finally, Section 4 summarizes the paper.

Simulation Details
The high-frequency time domain solver in CST Microwave Suite was adopted in this study. This solver used the finite-difference time-domain method to solve Maxwell's equations in 3D space, typically in the microwave range. Although this software did not solve the basic plasma equations self-consistently, compared to fluid plasma simulation [21] or particle-in-cell simulation [22], it was a useful tool to study the characteristics of the microwave probes mentioned in Section 1. Since electron density was a controllable input parameter in this simulation, the software allowed us to easily establish the ideal measurement accuracy by defining it as the deviation of the output electron density from the input electron density.
Besides, it should be noted that this simulation did not need to define the plasma source, such as a capacitively or inductively coupled plasma source, since plasma is regarded as a dispersive dielectric material. Such a material can be represented by the Drude model, in which the plasma dielectric constant ( p ) is given by √ n e,input is the plasma oscillation frequency, ω is microwave frequency, ν m is electron-neutral collision frequency, and n e,input is the input electron density. Here, the ν m term only includes the momentum transfer collision between electrons and Argon atoms at an electron temperature of 2.0 eV, with Maxwellian distribution for simplicity. The p is related to the complex wavenumber (k) from the dispersion relation for electromagnetic waves (k = ω/c √ p ) [5]. The real and imaginary parts of k are related to the refractive index and attenuation constant, respectively. Figure 1a,b show schematic diagrams of the top and cross-sections of the RCP and BCP, respectively, which are embedded in a cylindrical holder. Both probes consisted of radiating and detecting antennae; the RCP had a point-type radiating antenna and ring-type detecting antenna, while the BCP had bar-type radiating and detecting antennae. All antennae had 1.0 mm height, were insulated by a dielectric of height 2.0 mm, and were connected with a 50 Ω coaxial line. The plasma was a cylinder of 330 mm diameter and 40 mm height, and was positioned with an interlayer (sheath) distance of 5.0 mm from the RCP and BCP, as shown in Figure 1c; here, the simulation considers the sheath as a vacuum dielectric material ( sheath = 0 ).  Figure 1a,b represent the geometrical simulation variables (a to h), and Table 1 lists the simulation conditions. For simple expression, except for antenna diameter (a) and antenna length (e), all parameters were normalized as (≡ / or / ), (≡ / or ℎ/ ), and (≡ / or / ). In each simulation case, α changed from 1 to 4 for the RCP, and β and γ changed from 4.5 to 7 and 0.5 to 1.5 for both probes, respectively. Additionally, the normalized factors (a and e) changed from Input and output ports were located at the ends of the 50 Ω coaxial lines. First, a Gaussian voltage pulse (V in (t)) with 1 V maximum, including frequencies from zero to 10 GHz, was applied to the input port and radiated by the radiating antenna. Then, a fraction of the pulses (V out (t)) entered the output port via the detecting antenna. The S 21 is defined as S 21 = 20 log 10 (V in ( f )/V out ( f )) (dB), where V in (f ) and V out (f ) are the processed data through a fast Fourier transform of V in (t) and V out (t).
All simulations were conducted in the open boundary condition except for the z = 0 plane where the input and output ports are located. On this plane, the ground boundary condition (zero tangential electric field) was applied. Figure 1a,b represent the geometrical simulation variables (a to h), and Table 1 lists the simulation conditions. For simple expression, except for antenna diameter (a) and antenna length (e), all parameters were normalized as α (≡ d/a or f /e), β (≡ b/a or h/e), and γ (≡ c/a or g/e). In each simulation case, α changed from 1 to 4 for the RCP, and β and γ changed from 4.5 to 7 and 0.5 to 1.5 for both probes, respectively. Additionally, the normalized factors (a and e) changed from 2.0 to 3.0 mm. Otherwise, the plasma parameter (input electron density) also changed from 1 × 10 9 to 1 × 10 11 cm −3 , a range common in plasma processing conditions. Table 1. Simulation conditions with various α, β, γ, and probe diameters. Parameters a-g are given in Figure 1.

Type
Ring-Type Probe Type Bar-Type Probe

Simulation Results and Discussion
To analyze the antenna characteristics of the two probes, Sections 3.1 and 3.2 include an investigation of the S 21 spectrum in a vacuum condition (without plasma) with various geometrical parameters, such as α, β, γ, a for the RCP, and e for the BCP. Furthermore, for simple understanding, visualizations of the antenna configurations in the RCP and BCP are provided for each condition. Afterward, in plasma conditions, the S 21 spectrum with various antenna distances is examined. Figure 2a shows the S 21 spectra with various α (normalized width of the detecting antenna) at fixed β and γ. An increase in α led to a slight elevation of the S 21 level. This was because of an increase in the capacitive coupling between the radiating and detecting antenna. Furthermore, the S 21 had several resonance frequencies at its extremes. Except for the lowest resonance frequency, with an increase in α, the resonance frequencies shifted toward lower values. When a increased, as shown in Figure 2d,g, the low-frequency shift became enlarged. The origin of the resonance frequencies will be analyzed later.

Ring-Type Cutoff Probe (RCP)
resonance frequencies toward lower values.
As shown in Figure 2c, an increase in (normalized dielectric thickness) did not change the resonance frequencies. The signal levels changed with , while there was no trend compared with the α and effects. This might have resulted from a characteristic impedance mismatching between the radiation and detecting antennae, since they had no symmetry with each other. Impedance matching might deteriorate at = 1.0. To figure out the origin of the resonance frequencies as shown in Figure 2, the electric field on the detecting antenna surface was examined, the direction of which was normal to the antenna surface (z-axis as shown in Figure 1). Figure 3 exhibits the normalized electric field as a function of the normalized antenna length (θ -axis as shown in Figure 1) at resonance (2.61, 5.17, and 7.71 GHz) and non-resonance (3.90 and 6.50 GHz) frequencies of the #6 condition at a = 3.0 mm. The simulation result showed that the oscillation amplitude at the resonances was larger than that at the nonresonances. Furthermore, the wavelength at the three resonances was one, two, and three times the antenna length. Based on these two facts, the resonance frequencies resulted from the SW resonances of the electric field on the detecting antenna [20,24]. Due to this SW resonance, electric field energy  showing that the overall S 21 signal level reduced with increasing β due to a decrease in the capacitive coupling between the antennae [23]. Besides, an increase in β led to a low frequency shift of the resonance frequencies. Compared with the effect of α, increasing β more clearly shifted the resonance frequencies toward lower values.
As shown in Figure 2c, an increase in γ (normalized dielectric thickness) did not change the resonance frequencies. The signal levels changed with γ, while there was no trend compared with the α and β effects. This might have resulted from a characteristic impedance mismatching between the radiation and detecting antennae, since they had no symmetry with each other. Impedance matching might deteriorate at γ = 1.0.
To figure out the origin of the resonance frequencies as shown in Figure 2, the electric field on the detecting antenna surface was examined, the direction of which was normal to the antenna surface (z-axis as shown in Figure 1). Figure 3 exhibits the normalized electric field as a function of the normalized antenna length (θ-axis as shown in Figure 1) at resonance (2.61, 5.17, and 7.71 GHz) and non-resonance (3.90 and 6.50 GHz) frequencies of the #6 condition at a = 3.0 mm. The simulation result showed that the oscillation amplitude at the resonances was larger than that at the non-resonances. Furthermore, the wavelength at the three resonances was one, two, and three times the antenna length. Based on these two facts, the resonance frequencies resulted from the SW resonances of the electric field on the detecting antenna [20,24]. Due to this SW resonance, electric field energy was strongly localized on the detecting antenna surface such that it could not propagate toward the Signal Out ( Figure 1); the S 21 value at the resonances, therefore, dramatically decreased, as shown in Figure 2.
the cutoff frequency became equal to the plasma frequency (fpe in Figure 4). This might have resulted from a larger rate of reduction in the capacitive coupling in the sheath (Path 1 in Figure 1c) compared to that in the plasma (Path 2 in Figure 1c) with increasing antenna distance [25]. In other words, at a short antenna distance, capacitive coupling in the sheath was dominant, so the effective electron density became smaller than the input electron density due to the sheath, where the electron density was zero. This means that the cutoff frequency was lower than the plasma frequency. But at longer antenna distances, coupling in the plasma was dominant, and the effective electron density became the input electron density, and thereby, the cutoff frequency matched the plasma frequency. Deeper analysis of the different coupling reduction trends between Path 1 and Path 2 in terms of antenna distance is beyond the scope of this paper, but will be discussed in detail in a later paper with rigorous theory.
The trend of cutoff frequency saturating to the plasma frequency at large antenna distances was the same as for other high density cases, as shown in Figure 4b,c. If measurement accuracy is defined as the discrepancy between the cutoff frequency (fc) and the plasma frequency (fpe) as , the measurement accuracy of the RCP, therefore, strongly depended on the antenna distance; it was recommended that the distance be as large as possible. However, since the overall level of an S21 spectrum diminished as the antenna distance increased, there was a trade-off between signal-to-noise ratio and measurement accuracy. Additionally, the larger the antenna distance was, the more complex the spectrum shape became due to the SW resonance. Figure 5a shows the S21 spectra of the BCP with various α at fixed and in a vacuum condition. An increase in α led to a slight increase of the S21 level, which was similar to the RCP case and for the same reason. Here, there were no significant resonance peaks in the S21 spectra from the The RCP essentially featured resonance peaks induced by SW resonance in the simulated frequency ranges (<10 GHz), resulting from the length of the detecting antenna. This fact ultimately brought some negative aspects to the RCP, since the resonance peaks made finding the cutoff peak difficult in an S 21 spectrum, as discussed below. Figure 4 shows the S 21 spectra of the RCP with various electron density values ranging from 1 × 10 9 to 1 × 10 11 cm −3 with various β. In Figure 4a, an increase in β produced a high frequency shift of the cutoff frequency, which was the extreme in the S 21 spectrum and marked by an arrow. Eventually, the cutoff frequency became equal to the plasma frequency (f pe in Figure 4). This might have resulted from a larger rate of reduction in the capacitive coupling in the sheath (Path 1 in Figure 1c) compared to that in the plasma (Path 2 in Figure 1c) with increasing antenna distance [25]. In other words, at a short antenna distance, capacitive coupling in the sheath was dominant, so the effective electron density became smaller than the input electron density due to the sheath, where the electron density was zero.

Bar-Type Cutoff Probe (BCP)
This means that the cutoff frequency was lower than the plasma frequency. But at longer antenna distances, coupling in the plasma was dominant, and the effective electron density became the input electron density, and thereby, the cutoff frequency matched the plasma frequency [26]. Deeper analysis of the different coupling reduction trends between Path 1 and Path 2 in terms of antenna distance is beyond the scope of this paper, but will be discussed in detail in a later paper with rigorous theory.
The trend of cutoff frequency saturating to the plasma frequency at large antenna distances was the same as for other high density cases, as shown in Figure 4b,c. If measurement accuracy is defined as the discrepancy between the cutoff frequency (f c ) and the plasma frequency (f pe ) as f pe − f c f pe , the measurement accuracy of the RCP, therefore, strongly depended on the antenna distance; it was recommended that the distance be as large as possible. However, since the overall level of an S 21 spectrum diminished as the antenna distance increased, there was a trade-off between signal-to-noise ratio and measurement accuracy. Additionally, the larger the antenna distance was, the more complex the spectrum shape became due to the SW resonance.    Figure 5a shows the S 21 spectra of the BCP with various α at fixed β and γ in a vacuum condition. An increase in α led to a slight increase of the S 21 level, which was similar to the RCP case and for the same reason. Here, there were no significant resonance peaks in the S 21 spectra from the BCP, except for case #3, because the length of the radiating and detecting antennae was much smaller than the RCP detecting antenna; the SW resonance took place at high frequencies beyond the interested range (<10 GHz). When the antenna length e increased, as shown in Figure 5d,g, a low frequency shift of the resonance frequency with low Q-factor took place near 8 GHz (cases #2 and #3), but this could be negligible. The signal level of the BCP, however, was lower than that of the RCP, by as much as 20 dB due to the small size of the BCP antennae.  Figure 6 shows the S21 spectra of the BCP with various electron densities ranging from 1 × 10 09 cm −3 to 1 × 10 11 cm −3 and various β, with the same conditions as in Figure 4. In Figure 6a, an increase in produced the same results as the RCP case. With increasing antenna distance, the cutoff frequency saturated to the plasma frequency, which might have resulted from a reduction in capacitive coupling between the antennae for the same reason as in the RCP. Figure 6b,c show smooth spectral shapes, and it is therefore easy to determine the cutoff frequency in each spectrum. They also manifested the same trend as Figure 6a in terms of dependence.  Figure 6 shows the S 21 spectra of the BCP with various electron densities ranging from 1 × 10 9 cm −3 to 1 × 10 11 cm −3 and various β, with the same conditions as in Figure 4. In Figure 6a, an increase in β produced the same results as the RCP case. With increasing antenna distance, the cutoff frequency saturated to the plasma frequency, which might have resulted from a reduction in capacitive coupling between the antennae for the same reason as in the RCP. Figure 6b,c show smooth spectral shapes, and it is therefore easy to determine the cutoff frequency in each spectrum. They also manifested the same trend as Figure 6a in terms of β dependence. It should be noted for the BCP here that the critical antenna distance (dc), where the cutoff frequency matched the plasma frequency, was much lower than that in the RCP, at 14 mm compared to 30 mm as shown in Figure 7, which showed the ratio of the cutoff frequency to the plasma frequency by antenna distance. Additionally, the S21 spectrum of the BCP was much straighter, without any resonance peaks, than that of the RCP. This fact facilitated simple cutoff frequency monitoring by just measuring the minimum S21 value, which indicated the possibility for a simple plasma monitoring system based on the BCP. In summary, the BCP was more practical than the RCP in terms of miniaturization as well as electron density monitoring, since it was much easier to determine the cutoff frequency in the S21 spectrum of the BCP than that of the RCP. It should be noted for the BCP here that the critical antenna distance (d c ), where the cutoff frequency matched the plasma frequency, was much lower than that in the RCP, at 14 mm compared to 30 mm as shown in Figure 7, which showed the ratio of the cutoff frequency to the plasma frequency by antenna distance. Additionally, the S 21 spectrum of the BCP was much straighter, without any resonance peaks, than that of the RCP. This fact facilitated simple cutoff frequency monitoring by just measuring the minimum S 21 value, which indicated the possibility for a simple plasma monitoring system based on the BCP. In summary, the BCP was more practical than the RCP in terms of miniaturization as well as electron density monitoring, since it was much easier to determine the cutoff frequency in the S 21 spectrum of the BCP than that of the RCP.

Conclusions
This paper investigated the basic properties of two types of planar cutoff probes, the ring-type and the bar-type planar cutoff probes, with various geometrical parameters as well as a plasma parameter through a commercial 3D electromagnetic simulation. Simulation results showed that the RCP had several resonance frequencies that originated from standing-wave resonance on the detecting antenna, while the BCP did not. Moreover, the signal level of both the RCP and BCP increased with increasing antenna size and decreased with increasing antenna distance between the radiating and detecting antennae because of elevated and diminished capacitive coupling, respectively. Among the studied parameters, antenna distance was found to be the key parameter highly related to the accuracy of both probes. As a result, the BCP was more practical than the RCP for use in non-invasive electron density measurement systems since it allowed for much easier determination of cutoff frequency in S21 spectra.

Conclusions
This paper investigated the basic properties of two types of planar cutoff probes, the ring-type and the bar-type planar cutoff probes, with various geometrical parameters as well as a plasma parameter through a commercial 3D electromagnetic simulation. Simulation results showed that the RCP had several resonance frequencies that originated from standing-wave resonance on the detecting antenna, while the BCP did not. Moreover, the signal level of both the RCP and BCP increased with increasing antenna size and decreased with increasing antenna distance between the radiating and detecting antennae because of elevated and diminished capacitive coupling, respectively. Among the studied parameters, antenna distance was found to be the key parameter highly related to the accuracy of both probes. As a result, the BCP was more practical than the RCP for use in non-invasive electron density measurement systems since it allowed for much easier determination of cutoff frequency in S 21 spectra.