As the feature size of nanoelectronic devices continues to shrink in response to the demand for smaller and faster low-power, high-density devices, traditional semiconductor fabrication methods that consume large numbers of wafers to determine optimal processing windows are no longer economically viable [1–4]. In response to this challenge, there is a need for new methods to determine the processing window based on plasma parameter measurements, moving away from the traditional wasteful approach. Among the crucial measurements for understanding the state of the processing plasma, the electron density is considered the most fundamental plasma parameter that determines the state of a plasma, including factors such as plasma potential, the number of radical species and ions contributing to etching or deposition, and spatial uniformity [5–7]. Achieving precise measurement of plasma density is imperative for plasma equipment and manufacturing industries.

Microwave diagnostic methods can be used to measure plasma parameters, even with polymer contamination, as electromagnetic (EM) waves penetrate dielectric films [8]. Cutoff [8–10], plasma absorption [11], impedance [12], multipole resonance [13], and hairpin probe [14] methods are all viable. Notably, the plasma-equivalent circuit model for cutoff probes was first proposed by Kim et al. [8], and their method provides the most precise electron plasma frequency measurement. Moreover, probes are simple to fabricate and do not require critical assumptions [15–17]. Na et al. [17] further examined plasma measurement characteristics based on probe geometries of chamber size, probe distance, probe length, and coaxial cable characteristics, confirming that there was little difference in the intensity of the transmission spectrum and cutoff frequency with respect to the length of the cable.

The dispersion relationship among EM waves propagating through plasma without a direct current magnetic field is given by ${\omega }^2={\omega }_{pe}^2+c^2{k}^2$, where ω denotes the EM frequency, ω_pe is the electron plasma frequency, k is the wavenumber, and c is the speed of light [18]. Based on the dispersion relation of the plasma, EM waves are reflected by the plasma when $\omega \le {\omega }_{pe}^2$; otherwise, they penetrate the plasma. Hence, by carefully selecting the frequency measurement range of the network analyzer, the transmission spectrum peaks at the EM cutoff location. Comparative investigations using plasma oscillation probes have confirmed that the minimum cutoff frequency corresponds precisely to the electron plasma frequency [9]. After obtaining this frequency, the electron density can be obtained using $n_e=\frac{{\omega }_{pe}^2{ϵ}_0{m}_e}{e^2}$, where n_e is the electron density, ϵ₀ is the permittivity of a vacuum, m_e is the electron mass, and e is the elementary charge. The theoretical error bar for the electron–density measurement originates from the choice of the minimum value point, noting that the cutoff point is small [19,20].

A cutoff probe for these purposes can be constructed using two 50 Ω coaxial cables. The probe tips, which are responsible for emitting and detecting EM waves, are crafted by removing the insulator and braided shielding around the core cable. For ease of fabricating and downsizing, a subminiature version-A cable that uses Teflon dielectric materials is widely used. However, because Teflon melts at 327 °C, it can be destroyed under high-density plasma conditions. Therefore, a change in the dielectric material of the coaxial cable may be considered for process plasma or high-density plasma measurements. However, altering the dielectric material of the coaxial cable can impact the impedance of the cable. However, despite the critical significance of understanding the material characteristics of these cables under plasma conditions, this aspect has received minimal attention from researchers.

To fill this gap in knowledge, this study examines the plasma measurement characteristics of cutoff probes based on the permittivity and thickness of the dielectric material using EM wave simulations and a circuit model of the cutoff probe. Although changes in the dielectric constant and thickness of the dielectric material are known to alter the impedance of the coaxial cable, the cutoff frequency and plasma density can still be accurately measured. Hence, this study focuses on the probe’s plasma measurement characteristics in response to changes in its dielectric behavior under high-density plasma conditions. Furthermore, an analysis of the characteristics based on the thickness of the dielectric material in the cable is provided. We find that these variations can influence the intensity of the transmission spectrum measured using the cutoff probe, and the interpreted results are expected to contribute to the optimized production and utility of cutoff probes used for plasma density measurements under harsh high-density or process plasma conditions.

A powerful time-domain EM wave simulation tool (CST Microwave Studio) using full 3D Maxwell equations was adopted to investigate probes’ transmission spectra and cutoff frequency characteristics. The results were verified using data from previous cutoff probe studies. Open boundary conditions were selected to exclude effects such as cavity resonance and chamber wall reflections. Figure 1(a) shows a cross-sectional view of the cutoff probe. For this investigation, an argon discharge with a gas pressure of 10 mTorr was used, and the input plasma frequency was set to 1 GHz. The cutoff probe, as depicted in Fig. 1(a), was constructed using two 50 Ω coaxial cables and a metal tip. The tip distance, d, of the probe between radiating and detecting antennae was 7 mm, the length, h, was 5 mm, and the radius, r₁, was 0.1 mm. In our simulation, the dielectric constant of the coaxial cable was set in the range of 1–20, and the diameter of the dielectric material, 2r₂, in Fig. 1(b) ranged from 0.3 to 2.0 mm. It was assumed that the plasma was uniform around the probe structure and that its permittivity followed the Drude model [18]. In the simulation, all conductors were considered electrically perfect. The dielectric constant of the sheath between the cutoff probe and plasma was ϵ_r = 1, which is considered a vacuum layer. The sheath width was calculated as s = 5λ_De, where λ_De is the electron Debye length [18]. Under the conditions shown in Fig. 1, we assumed that the electron density of the calculation was 1.24 × 10¹⁰ cm⁻³ and the fixed electron temperature was 2 eV.

Figure 1. Schematic of the cutoff probe: (a) cross-sectional view, (b) view from above, and (c) circuit model including a lumped circuit element.

We developed a circuit model of the microwave cutoff probe to account for the coaxial cable and cutoff probe. Our circuit model, shown in Fig. 1(c), displays the cutoff probe immersed in plasma using an equivalent electrical circuit model. The plasma region is modeled as a collection of circuit elements with an admittance, $Y_p=i\omega C_0+1/\left(i\omega L_p+R_p\right)$, where $C_0=\text{π}{ϵ}_{0}h/{\cosh }^{-1}\left[d/2(r_1+s)\right]$ is the vacuum capacitance, $L_p={\omega }_{pe}^{-2}{C}_0^{-1}$ is the plasma inductance, $R_p={\text{ν}}_{\text{m}}{L}_p$ is the plasma resistance, and ν_m is the electron-neutral collision frequency. According to the circuit model of the cutoff probe, the cutoff frequency is the parallel resonance between L_p and C₂₁. In this model, the sheath surrounding the radiating and detecting antennae was conceptualized as a capacitor filled with vacuum space to account for the high frequency utilized in the cutoff probe: $C_{s1}=2\text{π}{ϵ}_0\left(h-\text{s}\right)/\ln \left[r_1+s/s\right]$. The total impedance of the plasma and sheath region was ${\text{Z}}_{\text{T}}=1/Y_p+2/i\omega C_s$, and the transmission spectrum, S₂₁, was calculated as $S_{21}=10\log Z_0/Z_0+Z_T\left(\text{dB}\right)$, where Z₂₁ is the characteristic impedance of the coaxial cable, given by $Z_0=138\log \left(r_2/r_1\right)/\sqrt{{ϵ}_r}$.

Figure 2(a) presents the transmission spectra calculated using an EM simulation based on the dielectric constant of the material. The r₂ was 0.46 mm, which corresponds to the condition in which the characteristic impedance of the coaxial cable becomes 50 Ω under ϵ_r = 1. The input plasma frequency was set to 1 GHz, and the calculated cutoff frequency was the same as that of the input plasma, with a negligible discrepancy. However, as the dielectric constant of the dielectric material increased, the intensity of the transmitted spectrum decreased. At a cutoff frequency of 1 GHz, the intensity of the transmission spectrum varied with the dielectric constant. Specifically, at ϵ_r = 1, the intensity was –68 dB, whereas at ϵ_r = 20, it decreased further to –81 dB. Changes in the dielectric constant of a coaxial cable can alter its characteristic impedance because Z₂₁ is inversely proportional to the square root of ϵ_r . Figure 2(b) illustrates the impedance of the coaxial cable with respect to variations in ϵ_r . Note that an increase in ϵ_r resulted in a decrease in Z₂₁, leading to a reduction in the S₂₁ intensity, as determined by its formula. Figure 2(c) presents the transmission spectrum calculated using circuit modeling. The results exhibit a consistent cutoff frequency despite changes in the cable’s ϵ_r , which is similar to the EM simulation results. However, it is evident that a decrease in ϵ_r led to a reduction in the intensity of the transmission spectrum. In summary, variations in the dielectric constant of the coaxial cable did not affect the cutoff frequency measurement of the cutoff excision probe. However, while measuring the transmission spectrum with the cutoff probe and a vector network analyzer, the measurement limit was reported to be around –100 dB [22]. The use of dielectric materials with a high dielectric constant implies potential difficulties in accurately measuring the cutoff frequency due to the limitations of the vector network analyzer. When utilizing high-dielectric-constant materials, such as YAG and Y₂O₃, compensating for the intensity of the transmission spectrum can be achieved by adjusting the tip distance and length of the cutoff probe [19]. Additionally, compensating for the characteristic impedance of the coaxial cable can be achieved by adjusting its dielectric thickness.

Figure 2. (a) Transmission spectra of the cutoff probe calculated by EM simulation, (b) characteristic impedance of the coaxial cable, and (c) transmission spectra of the cutoff probe calculated by the circuit model depending on the dielectric constant of the cable.

Figure 3(a) shows the transmission spectra calculated using an EM simulation based on the r₂ value of the dielectric material. The ϵ_r of the coaxial cable was 2.1 (Teflon), which corresponds to the condition in which the characteristic impedance of the able became 50 Ω under the condition of r₂ = 0.35 mm. The input plasma frequency was set to 1 GHz, and the calculated cutoff frequency was closely aligned with that of the input plasma. However, an increase in r₂ corresponded to a notable increase in the intensity of the transmitted spectrum. At a cutoff frequency of 1 GHz, the intensity of the transmission spectrum varied with the dielectric constant. Specifically, at r₂ = 0.15 mm, the intensity was –80 dB, whereas at r₂ = 1.00 mm, it increased further to –64 dB. Changes in the cable’s r₂ can alter its characteristic impedance because Z₂₁ is proportional to log(r₂/r₁). Figure 3(b) presents the impedance of the coaxial cable with respect to r₂, showing that an increase in r₂ corresponds to an increase in Z₂₁, resulting in heightened S₂₁ intensity. Figure 3(c) shows the transmission spectrum calculated using circuit modeling, revealing a consistent cutoff frequency analogous to the variations observed in the cable’s r₂, as indicated by the EM simulation results. Overall, it is evident that an increase in r₂ leads to an increase in transmission spectrum intensity.

Figure 3. (a) Transmission spectra of the cutoff probe depending on the coaxial cable insulator radius, r₂, calculated by EM simulation, (b) characteristic impedance of the cable, and (c) transmission spectra of the cutoff probe calculated by the circuit model depending on the cable’s r₂.

Our study analyzed the plasma measurement characteristics of cutoff probes, considering dielectric material permittivity and thickness, through EM wave simulations and the cutoff probe circuit model. Despite alterations in the dielectric constant and thickness affecting the line impedance of coaxial cables, consistent plasma density measurements remain achievable. Nevertheless, it is crucial to note that these variations may affect the intensity of the transmission spectrum recorded by the cutoff probe. Our research offers valuable insights into optimizing cutoff probes and enhancing the precision of plasma density measurements, particularly in challenging conditions of high-density or process plasmas.

This research was supported by the Material Innovation Program (Grant No. 2020M3H4A3106004) of the National Research Foundation (NRF) of Korea, and funded by (i) the Ministry of Science and ICT via the R&D Convergence Program (Grant No. CRC-20–01-NFRI) of the National Research Council of Science and Technology (NST) of the Republic of Korea, (ii) the Korea Evaluation Institute of Industrial Technology (Grant No. 1415181740), and (iii) the Korea Research Institute of Standards and Science (Grant No. KRISS GP2023-0012-08, GP2023-0012-09).

The authors declare no conflicts of interest.