With the semiconductor industry increasingly striving for greater integration and miniaturization, the difficulty of the semiconductor manufacturing process has increased significantly. As device dimensions decrease and process complexity increases, conventional methods of monitoring and controlling semiconductor processes are reaching their limits. One critical area where these limitations become apparent is process optimization. To improve process yield in semiconductor manufacturing, various parameters such as etch rate, uniformity, and selectivity need to be precisely controlled. However, as the processes approach the nanoscale, it is becoming increasingly difficult to manage the complexity and associated interactions using conventional techniques alone. Plasma diagnostics is expected to offer a promising solution for overcoming the limitations of conventional methods for process optimization in semiconductor manufacturing. By measuring plasma density, which is considered an important plasma parameter, these diagnostic methods help predict process results in terms of discharge characteristics, species concentration of radicals, and surface interactions. With this approach, plasma diagnostic tools can provide a more comprehensive understanding of these processes [1–5].

Among the diagnostic tools used in plasma manufacturing, flat cutoff sensors using microwaves are known for their ability to monitor processes non-invasively and non-perturbatively [6–8]. The measurement principle of these sensors is based on the wave cutoff method [9,10]. The dispersion relation for electromagnetic (EM) waves propagating through a plasma without an external magnetic field is represented by the equation ${\omega }^2={\omega }_{pe}^2+c^2{k}^2$, where ω is the frequency of the EM wave, ω_pe is the electron plasma frequency, k is the wave number, and c represents the speed of light. According to this relationship, EM waves are reflected by the plasma when their frequency ω is less than or equal to ω_pe, and they penetrate the plasma when the frequency is higher. Therefore, careful selection of the frequency measurement range of the network analyzer is crucial, as it helps to detect peaks in the transmission spectrum at the cutoff frequency [8]. Once this frequency is determined, the electron density n_e can be calculated using the formula $n_e=\frac{{\omega }_{pe}^2{\epsilon }_0{m}_e}{e^2}$, where ϵ₀ is the permittivity of the vacuum, m_e is the electron mass, and e is the elementary charge [4]. In addition, existing studies on flat cutoff sensors have addressed the structure, sheath thickness, and wafer type in relation to plasma measurement characteristics [7]. However, in previous studies, the antenna properties were modeled as a perfect electric conductor (PEC), which is an idealized metal. The electrical properties of metals need to be analyzed as they affect the radiation and reflection characteristics of the antenna [11,12]. In addition, because of the limited exposure of materials in the process chamber, an analysis of the measurement characteristics of the flat cutoff sensor based on the material properties is required.

In this study, the transmission spectrum and electron density measurement characteristics of a flat cutoff sensor were analyzed by changing the electrical conductivity of the antenna during a simulation. In addition, the attenuation of the transmission spectra was analyzed using an electric field profile for a comparative analysis.

To investigate the transmission spectra and characteristics of the plasma diagnostics, an EM wave simulation was conducted. CST Microwave Studio, a commercial EM wave simulation tool renowned for its robust time-domain solver for three-dimensional full Maxwell equations, was chosen. The accuracy of the simulation findings was confirmed by comparing the results with those of previous studies [13–15]. Figure 1(a) shows a top view of the flat cutoff sensor, which consists of two rectangular antennas surrounded by an insulator and a ground plane. The antennas transmit and receive signals via a 50 Ω coaxial cable. The specifications of the flat cutoff sensor are as follows: the length (l) of both the radiating and receiving antennas is 20 mm, the thickness (t) is 3 mm, the distance (d) between them is 5 mm, and the thickness (g) of the insulator is 1 mm. Figure 1(b) shows the crosssectional view of the flat cutoff sensor, which has a uniform sheath and plasma above it. The plasma height (h) is 150 mm, and the sheath thickness (s) is calculated using the formula $\text{s}=5{\lambda }_{De}$, where λ_De represents the electron Debye length [4]. In this simulation, the permittivity of the plasma was assumed to be the Drude model that can apply the dielectric constant according to the frequency of the EM wave [4], and the dielectric constant of the sheath was set to unity (ϵ_r=1). The insulating material of the cutoff sensor was made of Teflon (ϵ_r=2.1), the electrical conductivity (σ) of the antennas ranged from 0.1 to ∞ S/m, and the properties of the ground plate were adjusted to those of a perfect electrical conductor (σ = ∞). The electrical conductivities of the nichrome and 301 stainless steel [16] were approximately 1 × 10⁶ S/m, and the electrical conductivity of the doped wafer [17] was approximately 0.1 S/m. The calculations assumed an electron density of 1.24 × 10¹⁰ cm⁻³ and an electron temperature of 2 eV. Open boundary conditions were implemented to exclude effects, such as cavity resonance and reflections from the chamber walls.

Figure 1. Schematic representation of the flat cutoff sensor in the EM simulation: (a) top view and (b) cross view.

Figures 2(a) and 2(b) show the transmission spectra computed by EM simulations when varying the electrical conductivity of the antenna in the flat cutoff sensor. As shown in Fig. 2(a), the electrical conductivity of the antenna was set to a value from 2.0 to ∞ S/m, whereas in Fig. 2(b), it was set to a value from 0.1 to 1.0 S/m. The transmission spectra, calculated based on the electrical conductivity of the antenna in the flat cutoff sensor, showed that the cutoff frequency matches the input plasma frequency under all electrical conductivity conditions. However, as the electrical conductivity of the antenna decreased, the intensity of the minimum transmission spectrum also decreased. When the electrical conductivity of the antenna was 1 × 10⁶ S/m, the intensity of the minimum transmission spectrum was −109 dB, similar to results calculated with the antenna properties set as PEC. This indicates that the difference in the intensity of the minimum transmission spectrum between nichrome and materials with electrical conductivity greater than that of nichrome, that is, metals such as copper and silver, is small. In contrast, when the electrical conductivity of the antenna was 0.1 S/m, the intensity of the minimum transmission spectrum was −206 dB. This can be interpreted as an increased antenna resistance owing to lower electrical conductivity, resulting in a lower intensity of the minimum transmission spectrum [15]. The measurable range of the intensity of the minimum transmission spectrum was determined by the vector network analyzer. The dynamic range of vector network analyzers is approximately −118 dB (Pico VNA 106, Keysight P937xA, R&S ZNLE3), which makes it difficult to measure flat cutoff sensors with lower minimum transmission spectrum intensity. Therefore, the electrical conductivity of the antenna in the flat cutoff sensors that can measure the intensity of the minimum transmission spectrum above −118 dB is 4.0 S/m when the intensity of the minimum transmission spectrum is −116 dB. These results indicate a high potential for measuring plasma density, even in antenna materials such as doped semiconductors.

Figure 2. Transmission spectra of the flat cutoff sensor calculated by EM simulation with the electrical conductivity (σ) of antennas set to (a) σ=2.0−∞ S/m and (b) σ=0.1−1.0 S/m.

To analyze the effects of the reduced electrical conductivity on the reduction in the intensity of the minimum transmission spectrum, the profile and strength of the electric field were calculated using EM simulations. Figure 3(a) shows the electric field profile when the electrical conductivity of the antenna was 0.1 S/m, and Fig. 3(b) shows the electric-field profile when the electrical conductivity of the antenna was 1 × 10⁶ S/m. Figure 3(c) shows a plot of the electric-field strength calculated at different heights above the radiating antenna surface when the electrical conductivity varied in the EM simulation. In Figs. 3(a) and 3(b), the electric field profile is broader and more intense at higher electrical conductivity, whereas it is narrower and less intense at lower conductivity. Furthermore, Fig. 3(c) shows that the strength of the electric field increases with height as the electrical conductivity increases. These results can be interpreted in terms of Ohm’s law. Since R ∝ $\frac{1}{\sigma }$, R ∝ V, and V ∝ $\frac{1}{\sigma }$, when the electrical conductivity of the antenna is low, the voltage applied to the antenna increases, and so does the voltage drop. As the electric field is proportional to the voltage, it can be interpreted that the electric field applied to the plasma decreases.

Figure 3. Electric field profile of the flat cutoff sensor with (a) σ=0.1 and (b) σ=1 × 10⁶ S/m, (c) vertical E-field intensity from the surface of the radiating antenna according to electrical conductivity of antennas calculation by EM simulation.

In this study, the characteristics of the plasma measurement were analyzed using transmission spectra, electric field profiles, and the strength as a function of the electrical conductivity of the antenna in a flat cutoff sensor using EM simulations. Despite the changes in electrical conductivity, the cutoff frequency was constant. The effects of changes in the electrical conductivity of the antenna on the intensity of the minimum transmission spectrum were analyzed comparatively with an electric field. The same transmission spectrum as that of the PEC was calculated when a metal with low electrical conductivity, such as stainless steel, was used as the antenna. The feasibility of plasmadensity measurements in materials with properties similar to those of doped semiconductors was confirmed, and the importance of matching the properties of the antenna with the performance of the network analyzer when altering the properties of the antenna was established. Our research can support the fabrication of flat cutoff sensors.

This research was supported by the Material Innovation Program (Grant No. 2020M3H4A3106004) of the National Research Foundation (NRF) of Korea, and funded by: (i) Ministry of Science and ICT and 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) Korea Evaluation Institute of Industrial Technology (Grant No. 1415181740), and (iii) Korea Research Institute of Standards and Science (Grant No. KRISS GP2024-0012-04); (iv) Ministry of Trade, Industry & Energy (MOTIE) (Grant Nos. 1415187722, 1415188153), and KSRC (Korea Semiconductor Research Consortium) (Grant No. 00235950).

The authors declare no conflicts of interest.